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7000 | https://static.bigideasmath.com/protected/content/pe/hsim/int_math2_pe_06_04.pdf | Section 6.4 Medians and Altitudes of Triangles 361 Medians and Altitudes of Triangles 6.4 Essential Question Essential Question What conjectures can you make about the medians and altitudes of a triangle?
Finding Properties of the Medians of a Triangle Work with a partner. Use dynamic geometry software. Draw any △ABC.
a. Plot the midpoint of — BC and label it D. Draw — AD , which is a median of △ABC. Construct the medians to the other two sides of △ABC.
0 1 2 3 4 5 6 0 1 2 B D G E A C 3 4 5 6 7 8 B medians b. What do you notice about the medians? Drag the vertices to change △ABC. Use your observations to write a conjecture about the medians of a triangle.
c. In the fi gure above, point G divides each median into a shorter segment and a longer segment. Find the ratio of the length of each longer segment to the length of the whole median. Is this ratio always the same? Justify your answer. Finding Properties of the Altitudes of a Triangle Work with a partner. Use dynamic geometry software. Draw any △ABC.
a. Construct the perpendicular segment from vertex A to — BC . Label the endpoint D. — AD is an altitude of △ABC. b. Construct the altitudes to the other two sides of △ABC. What do you notice?
c. Write a conjecture about the altitudes of a triangle. Test your conjecture by dragging the vertices to change △ABC.
Communicate Your Answer 3. What conjectures can you make about the medians and altitudes of a triangle? 4. The length of median — RU in △RST is 3 inches. The point of concurrency of the three medians of △RST divides — RU into two segments. What are the lengths of these two segments?
LOOKING FOR STRUCTURE To be profi cient in math, you need to look closely to discern a pattern or structure.
Sample Points A(1, 4) B(6, 5) C(8, 0) D(7 , 2.5) E(4.5, 2) G(5, 3) 0 1 2 3 4 5 6 0 1 2 B D A C 3 4 5 6 7 8 altitude 362 Chapter 6 Relationships Within Triangles Lesson 6.4 What You Will Learn What You Will Learn Use medians and fi nd the centroids of triangles. Use altitudes and fi nd the orthocenters of triangles.
Using the Median of a Triangle A median of a triangle is a segment from a vertex to the midpoint of the opposite side. The three medians of a triangle are concurrent. The point of concurrency, called the centroid, is inside the triangle. Using the Centroid of a Triangle In △RST, point Q is the centroid, and SQ = 8. Find QW and SW.
SOLUTION SQ = 2 — 3 SW Centroid Theorem 8 = 2 — 3 SW Substitute 8 for SQ. 12 = SW Multiply each side by the reciprocal, 3 — 2 .
Then QW = SW − SQ = 12 − 8 = 4. So, QW = 4 and SW = 12.
median of a triangle, p. 362 centroid, p. 362 altitude of a triangle, p. 363 orthocenter, p. 363 Previous midpoint concurrent point of concurrency Core Vocabulary Core Vocabulary Theorem Theorem Centroid Theorem The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side.
The medians of △ABC meet at point P, and AP = 2 — 3 AE, BP = 2 — 3 BF, and CP = 2 — 3 CD.
Proof BigIdeasMath.com Finding the Centroid of a Triangle Use a compass and straightedge to construct the medians of △ABC.
SOLUTION Step 1 Step 2 Step 3 A D E F B C A D E F B C A D E P F B C Find midpoints Draw △ABC. Find the midpoints of — AB , — BC , and — AC . Label the midpoints of the sides D, E, and F, respectively.
Draw medians Draw — AE , — BF , and — CD . These are the three medians of △ABC.
Label a point Label the point where — AE , — BF , and — CD intersect as P. This is the centroid.
A F C E B D P R W T V 8 Q S U Section 6.4 Medians and Altitudes of Triangles 363 Finding the Centroid of a Triangle Find the coordinates of the centroid of △RST with vertices R(2, 1), S(5, 8), and T(8, 3).
SOLUTION Step 1 Graph △RST.
Step 2 Use the Midpoint Formula to fi nd the midpoint V of — RT and sketch median — SV . V ( 2 + 8 — 2 , 1 + 3 — 2 ) = (5, 2) Step 3 Find the centroid. It is two-thirds of the distance from each vertex to the midpoint of the opposite side.
The distance from vertex S(5, 8) to V(5, 2) is 8 − 2 = 6 units. So, the centroid is 2 — 3 (6) = 4 units down from vertex S on — SV . So, the coordinates of the centroid P are (5, 8 − 4), or (5, 4).
Monitoring Progress Monitoring Progress Help in English and Spanish at BigIdeasMath.com There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P. 1. Find PS and PC when SC = 2100 feet. 2. Find TC and BC when BT = 1000 feet. 3. Find PA and TA when PT = 800 feet.
Find the coordinates of the centroid of the triangle with the given vertices. 4. F(2, 5), G(4, 9), H(6, 1) 5. X(−3, 3), Y(1, 5), Z(−1, −2) Using the Altitude of a Triangle An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.
JUSTIFYING CONCLUSIONS You can check your result by using a different median to fi nd the centroid. READING In the area formula for a triangle, A = 1 — 2 bh, you can use the length of any side for the base b. The height h is the length of the altitude to that side from the opposite vertex.
FINDING AN ENTRY POINT The median — SV is chosen in Example 2 because it is easier to fi nd a distance on a vertical segment. x y 4 2 8 6 4 2 10 8 6 T(8, 3) S(5, 8) P(5, 4) V(5, 2) R(2, 1) A B T S P R C Q P R Q P R altitude from Q to PR Orthocenter The lines containing the altitudes of a triangle are concurrent. This point of concurrency is the orthocenter of the triangle.
The lines containing — AF , — BD , and — CE meet at the orthocenter G of △ABC.
Core Core Concept Concept B F C A E G D 364 Chapter 6 Relationships Within Triangles As shown below, the location of the orthocenter P of a triangle depends on the type of triangle.
P P P Acute triangle P is inside triangle. Right triangle P is on triangle. Obtuse triangle P is outside triangle. Finding the Orthocenter of a Triangle Find the coordinates of the orthocenter of △XYZ with vertices X(−5, −1), Y(−2, 4), and Z(3, −1).
SOLUTION Step 1 Graph △XYZ.
Step 2 Find an equation of the line that contains the altitude from Y to — XZ . Because — XZ is horizontal, the altitude is vertical. The line that contains the altitude passes through Y(−2, 4). So, the equation of the line is x = −2.
Step 3 Find an equation of the line that contains the altitude from X to — YZ .
slope of ⃖⃗ YZ = −1 − 4 — 3 − (−2) = −1 Because the product of the slopes of two perpendicular lines is −1, the slope of a line perpendicular to ⃖⃗ YZ is 1. The line passes through X(−5, −1).
y = mx + b Use slope-intercept form. −1 = 1(−5) + b Substitute −1 for y, 1 for m, and −5 for x. 4 = b Solve for b.
So, the equation of the line is y = x + 4.
Step 4 Find the point of intersection of the graphs of the equations x = −2 and y = x + 4.
Substitute −2 for x in the equation y = x + 4. Then solve for y.
y = x + 4 Write equation. y = −2 + 4 Substitute −2 for x. y = 2 Solve for y. So, the coordinates of the orthocenter are (−2, 2).
Monitoring Progress Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. Then fi nd the coordinates of the orthocenter. 6. A(0, 3), B(0, −2), C(6, −3) 7. J(−3, −4), K(−3, 4), L(5, 4) READING The altitudes are shown in red. Notice that in the right triangle, the legs are also altitudes. The altitudes of the obtuse triangle are extended to fi nd the orthocenter.
x y 5 1 1 −1 −3 X Z Y (−2, 2) x = −2 y = x + 4 Section 6.4 Medians and Altitudes of Triangles 365 In an isosceles triangle, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base are all the same segment. In an equilateral triangle, this is true for any vertex. Proving a Property of Isosceles Triangles Prove that the median from the vertex angle to the base of an isosceles triangle is an altitude.
SOLUTION Given △ABC is isosceles, with base — AC . — BD is the median to base — AC .
Prove — BD is an altitude of △ABC.
Paragraph Proof Legs — AB and — BC of isosceles △ABC are congruent. — CD ≅ — AD because — BD is the median to — AC . Also, — BD ≅ — BD by the Refl exive Property of Congruence. So, △ABD ≅ △CBD by the SSS Congruence Theorem. ∠ADB ≅ ∠CDB because corresponding parts of congruent triangles are congruent. Also, ∠ADB and ∠CDB are a linear pair. — BD and — AC intersect to form a linear pair of congruent angles, so — BD ⊥ — AC and — BD is an altitude of △ABC.
Monitoring Progress Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. WHAT IF? In Example 4, you want to show that median — BD is also an angle bisector. How would your proof be different?
Segments, Lines, Rays, and Points in Triangles Example Point of Concurrency Property Example perpendicular bisector circumcenter The circumcenter P of a triangle is equidistant from the vertices of the triangle.
A C B P angle bisector incenter The incenter I of a triangle is equidistant from the sides of the triangle.
A C B I median centroid The centroid R of a triangle is two thirds of the distance from each vertex to the midpoint of the opposite side.
A C D R B altitude orthocenter The lines containing the altitudes of a triangle are concurrent at the orthocenter O.
A C O B A D C B Concept Summary Concept Summary 366 Chapter 6 Relationships Within Triangles Dynamic Solutions available at BigIdeasMath.com 1. VOCABULARY Name the four types of points of concurrency. Which lines intersect to form each of the points? 2. COMPLETE THE SENTENCE The length of a segment from a vertex to the centroid is __ the length of the median from that vertex.
Exercises 6.4 In Exercises 3–6, point P is the centroid of △LMN. Find PN and QP. (See Example 1.) 3. QN = 9 4. QN = 21 L N Q P M L N Q P M 5. QN = 30 6. QN = 42 L N Q P M L N Q P M In Exercises 7–10, point D is the centroid of △ABC. Find CD and CE. 7. DE = 5 8. DE = 11 A B C E D A B C E D 9. DE = 9 10. DE = 15 A B C E D C B E D A In Exercises 11–14, point G is the centroid of △ABC. BG = 6, AF = 12, and AE = 15. Find the length of the segment.
A F 12 6 C B D E G 11. — FC 12. — BF 13. — AG 14. — GE In Exercises 15–18, fi nd the coordinates of the centroid of the triangle with the given vertices. (See Example 2.) 15. A(2, 3), B(8, 1), C(5, 7) 16. F(1, 5), G(−2, 7), H(−6, 3) 17. S(5, 5), T(11, −3), U(−1, 1) 18. X(1, 4), Y(7, 2), Z(2, 3) In Exercises 19–22, tell whether the orthocenter is inside, on, or outside the triangle. Then fi nd the coordinates of the orthocenter. (See Example 3.) 19. L(0, 5), M(3, 1), N(8, 1) 20. X(−3, 2), Y(5, 2), Z(−3, 6) 21. A(−4, 0), B(1, 0), C(−1, 3) 22. T(−2, 1), U(2, 1), V(0, 4) CONSTRUCTION In Exercises 23–26, draw the indicated triangle and fi nd its centroid and orthocenter. 23. isosceles right triangle 24. obtuse scalene triangle 25. right scalene triangle 26. acute isosceles triangle Monitoring Progress and Modeling with Mathematics Monitoring Progress and Modeling with Mathematics Vocabulary and Core Concept Check Vocabulary and Core Concept Check Section 6.4 Medians and Altitudes of Triangles 367 ERROR ANALYSIS In Exercises 27 and 28, describe and correct the error in fi nding DE. Point D is the centroid of △ABC.
27. DE = 2 — 3 AE DE = 2 — 3 (18) DE = 12 ✗ 28. DE = 2 — 3 AD DE = 2 — 3 (24) DE = 16 ✗ PROOF In Exercises 29 and 30, write a proof of the statement. (See Example 4.) 29. The angle bisector from the vertex angle to the base of an isosceles triangle is also a median.
30. The altitude from the vertex angle to the base of an isosceles triangle is also a perpendicular bisector.
CRITICAL THINKING In Exercises 31–36, complete the statement with always, sometimes, or never. Explain your reasoning.
31. The centroid is __ on the triangle.
32. The orthocenter is _ outside the triangle.
33. A median is _ the same line segment as a perpendicular bisector.
34. An altitude is _ the same line segment as an angle bisector. 35. The centroid and orthocenter are _ the same point.
36. The centroid is _ formed by the intersection of the three medians. 37. WRITING Compare an altitude of a triangle with a perpendicular bisector of a triangle.
38. WRITING Compare a median, an altitude, and an angle bisector of a triangle.
39. MODELING WITH MATHEMATICS Find the area of the triangular part of the paper airplane wing that is outlined in red. Which special segment of the triangle did you use?
3 in.
3 in.
9 in.
40. ANALYZING RELATIONSHIPS Copy and complete the statement for △DEF with centroid K and medians — DH , — EJ , and — FG .
a. EJ = KJ b. DK = _ KH c. FG = KF d. KG = FG MATHEMATICAL CONNECTIONS In Exercises 41–44, point D is the centroid of △ABC. Use the given information to fi nd the value of x.
B E D C F G A 41. BD = 4x + 5 and BF = 9x 42. GD = 2x − 8 and GC = 3x + 3 43. AD = 5x and DE = 3x − 2 44. DF = 4x − 1 and BD = 6x + 4 45. MATHEMATICAL CONNECTIONS Graph the lines on the same coordinate plane. Find the centroid of the triangle formed by their intersections. y1 = 3x − 4 y2 = 3 — 4 x + 5 y3 = − 3 — 2 x − 4 46. CRITICAL THINKING In what type(s) of triangles can a vertex be one of the points of concurrency of the triangle? Explain your reasoning.
B E D C A B E D C A AE = 18 AD = 24 368 Chapter 6 Relationships Within Triangles 47. WRITING EQUATIONS Use the numbers and symbols to write three different equations for PE. A F C E B D P PE − + AP AE = 2 — 3 1 — 2 1 — 3 1 — 4 48. HOW DO YOU SEE IT? Use the fi gure.
J M L K h 17 9 9 10 N a. What type of segment is — KM ? Which point of concurrency lies on — KM ? b. What type of segment is — KN ? Which point of concurrency lies on — KN ? c. Compare the areas of △JKM and △KLM. Do you think the areas of the triangles formed by the median of any triangle will always compare this way? Explain your reasoning. 49. MAKING AN ARGUMENT Your friend claims that it is possible for the circumcenter, incenter, centroid, and orthocenter to all be the same point. Do you agree? Explain your reasoning.
50. DRAWING CONCLUSIONS The center of gravity of a triangle, the point where a triangle can balance on the tip of a pencil, is one of the four points of concurrency. Draw and cut out a large scalene triangle on a piece of cardboard. Which of the four points of concurrency is the center of gravity? Explain. 51. PROOF Prove that a median of an equilateral triangle is also an angle bisector, perpendicular bisector, and altitude. 52. THOUGHT PROVOKING Construct an acute scalene triangle. Find the orthocenter, centroid, and circumcenter. What can you conclude about the three points of concurrency?
53. CONSTRUCTION Follow the steps to construct a nine-point circle. Why is it called a nine-point circle?
Step 1 Construct a large acute scalene triangle.
Step 2 Find the orthocenter and circumcenter of the triangle. Step 3 Find the midpoint between the orthocenter and circumcenter.
Step 4 Find the midpoint between each vertex and the orthocenter. Step 5 Construct a circle. Use the midpoint in Step 3 as the center of the circle, and the distance from the center to the midpoint of a side of the triangle as the radius.
54. PROOF Prove the statements in parts (a)−(c).
Given — LP and — MQ are medians of scalene △LMN. Point R is on ⃗ LP such that — LP ≅ — PR . Point S is on ⃗ MQ such that — MQ ≅ — QS .
Prove a. — NS ≅ — NR b. — NS and — NR are both parallel to — LM .
c. R, N, and S are collinear.
Maintaining Mathematical Proficiency Maintaining Mathematical Proficiency Determine whether — AB is parallel to — CD . (Skills Review Handbook) 55. A(5, 6), B (−1, 3), C(−4, 9), D(−16, 3) 56. A(−3, 6), B(5, 4), C(−14, −10), D(−2, −7) 57. A(6, −3), B(5, 2), C(−4, −4), D(−5, 2) 58. A(−5, 6), B(−7, 2), C(7, 1), D(4, −5) Reviewing what you learned in previous grades and lessons |
7001 | https://openstax.org/books/biology-ap-courses/pages/4-6-connections-between-cells-and-cellular-activities | 4.6 Connections between Cells and Cellular Activities - Biology for AP® Courses | OpenStax
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Biology for AP® Courses
4.6 Connections between Cells and Cellular Activities
Biology for AP® Courses4.6 Connections between Cells and Cellular Activities
Contents Contents
Highlights
Table of contents
Preface
The Chemistry of Life
The Cell
4 Cell Structure
Introduction
4.1 Studying Cells
4.2 Prokaryotic Cells
4.3 Eukaryotic Cells
4.4 The Endomembrane System and Proteins
4.5 Cytoskeleton
4.6 Connections between Cells and Cellular Activities
Key Terms
Chapter Summary
Review Questions
Critical Thinking Questions
Test Prep for AP®Courses
Science Practice Challenge Questions
5 Structure and Function of Plasma Membranes
6 Metabolism
7 Cellular Respiration
8 Photosynthesis
9 Cell Communication
10 Cell Reproduction
Genetics
Evolutionary Processes
Biological Diversity
Plant Structure and Function
Animal Structure and Function
Ecology
A | The Periodic Table of Elements
B | Geological Time
C | Measurements and the Metric System
Index
Search for key terms or text.
Close
Learning Objectives
In this section, you will explore the following questions:
What are the components of the extracellular matrix?
What are the roles of tight junctions, gap junctions, and plasmodesmata in allowing cells to exchange materials with the environment and communicate with other cells?
Connection for AP® Courses
With the exception of gap junctions between animal cells and plasmodesmata between plant cells that facilitate the exchange of substances, the information presented in Section 4.6| Connections between Cells and Cellular Activities is not required for AP®. Concepts about cell communication and signaling processes that are required for AP®, including the features of cells that make communication possible, are covered in Chapter 9.
Teacher Support
Review the role of the extracellular matrix by comparing it to a scaffold outside of a building that buttresses and supports the main structure. It is an extension of the cell in animal tissues and provides essential functions in mechanical support, cell-cell communication, wound healing, and organism development.
Cell junctions can be compared to the piecing together of fabric. Pieces of fabric are tightly joined by a seam. Tight junctions similarly “glue” cells together. Rivets and snaps bind fabric tightly in specific spots as desmosomes form close associations in specific areas. Gap junctions are almost like pins and allow trafficking between the cytoplasm of adjacent cells.
Cells in tissues are usually inadequately visualized as puzzle pieces that interlock but that are functionally independent. Dispel this notion by stressing the importance of cell junctions. In multicellular organisms, cells interact with their neighboring cells.
You already know that a group of similar cells working together is called a tissue. As you might expect, if cells are to work together, they must communicate with each other, just as you need to communicate with others if you work on a group project. Let’s take a look at how cells communicate with each other.
The Science Practice Challenge Questions contain additional test questions for this section that will help you prepare for the AP exam. These questions address the following standards:
[APLO 4.5][APLO 3.32][APLO 1.16][APLO 3.33][APLO 1.14][APLO 2.7][APLO 4.4]
Extracellular Matrix of Animal Cells
Most animal cells release materials into the extracellular space. The primary components of these materials are proteins, and the most abundant protein is collagen. Collagen fibers are interwoven with carbohydrate-containing protein molecules called proteoglycans. Collectively, these materials are called the extracellular matrix (Figure 4.27). Not only does the extracellular matrix hold the cells together to form a tissue, but it also allows the cells within the tissue to communicate with each other. How can this happen?
Figure 4.27 The extracellular matrix consists of a network of proteins and carbohydrates.
Cells have protein receptors on the extracellular surfaces of their plasma membranes. When a molecule within the matrix binds to the receptor, it changes the molecular structure of the receptor. The receptor, in turn, changes the conformation of the microfilaments positioned just inside the plasma membrane. These conformational changes induce chemical signals inside the cell that reach the nucleus and turn “on” or “off” the transcription of specific sections of DNA, which affects the production of associated proteins, thus changing the activities within the cell.
Blood clotting provides an example of the role of the extracellular matrix in cell communication. When the cells lining a blood vessel are damaged, they display a protein receptor called tissue factor. When tissue factor binds with another factor in the extracellular matrix, it causes platelets to adhere to the wall of the damaged blood vessel, stimulates the adjacent smooth muscle cells in the blood vessel to contract (thus constricting the blood vessel), and initiates a series of steps that stimulate the platelets to produce clotting factors.
Intercellular Junctions
Cells can also communicate with each other via direct contact, referred to as intercellular junctions. There are some differences in the ways that plant and animal cells do this. Plasmodesmata are junctions between plant cells, whereas animal cell contacts include tight junctions, gap junctions, and desmosomes.
Plasmodesmata
In general, long stretches of the plasma membranes of neighboring plant cells cannot touch one another because they are separated by the cell wall that surrounds each cell (Figure 4.8b). How then, can a plant transfer water and other soil nutrients from its roots, through its stems, and to its leaves? Such transport uses the vascular tissues (xylem and phloem) primarily. There also exist structural modifications called plasmodesmata (singular = plasmodesma), numerous channels that pass between cell walls of adjacent plant cells, connect their cytoplasm, and enable materials to be transported from cell to cell, and thus throughout the plant (Figure 4.28).
Figure 4.28 A plasmodesma is a channel between the cell walls of two adjacent plant cells. Plasmodesmata allow materials to pass from the cytoplasm of one plant cell to the cytoplasm of an adjacent cell.
Tight Junctions
A tight junction is a watertight seal between two adjacent animal cells (Figure 4.29). The cells are held tightly against each other by proteins (predominantly two proteins called claudins and occludins).
Figure 4.29 Tight junctions form watertight connections between adjacent animal cells. Proteins create tight junction adherence. (credit: modification of work by Mariana Ruiz Villareal)
This tight adherence prevents materials from leaking between the cells; tight junctions are typically found in epithelial tissues that line internal organs and cavities, and comprise most of the skin. For example, the tight junctions of the epithelial cells lining your urinary bladder prevent urine from leaking out into the extracellular space.
Desmosomes
Also found only in animal cells are desmosomes, which act like spot welds between adjacent epithelial cells (Figure 4.30). Short proteins called cadherins in the plasma membrane connect to intermediate filaments to create desmosomes. The cadherins join two adjacent cells together and maintain the cells in a sheet-like formation in organs and tissues that stretch, like the skin, heart, and muscles.
Figure 4.30 A desmosome forms a very strong spot weld between cells. It is created by the linkage of cadherins and intermediate filaments. (credit: modification of work by Mariana Ruiz Villareal)
Gap Junctions
Gap junctions in animal cells are like plasmodesmata in plant cells in that they are channels between adjacent cells that allow for the transport of ions, nutrients, and other substances that enable cells to communicate (Figure 4.31). Structurally, however, gap junctions and plasmodesmata differ.
Figure 4.31 A gap junction is a protein-lined pore that allows water and small molecules to pass between adjacent animal cells. (credit: modification of work by Mariana Ruiz Villareal)
Gap junctions develop when a set of six proteins (called connexins) in the plasma membrane arrange themselves in an elongated donut-like configuration called a connexon. When the pores (“doughnut holes”) of connexons in adjacent animal cells align, a channel between the two cells forms. Gap junctions are particularly important in cardiac muscle: The electrical signal for the muscle to contract is passed efficiently through gap junctions, allowing the heart muscle cells to contract in tandem.
Link to Learning
To conduct a virtual microscopy lab and review the parts of a cell, work through the steps of this interactive assignment.
What are two similarities and two differences between plant and animal cells that can be seen under a microscope?
Plant cells have cell walls which provide structure to the plant and also chloroplasts which allow for photosynthesis. Animal cells do not have either of these structures. Both cells have nuclei, the command center of the cell, and cytoplasm, the gel-like solution that fills the cell.
Plant cells and animal cells have cell walls as well as nuclei. Plant cells have chloroplasts as well as plasmodesmata which are lacking in animal cells.
Plant cells have cell walls which provide structure to the plant and also chloroplasts which allow for photosynthesis. Animal cells do not have either of these structures. Animal cells and plant cells both have glyoxysomes as well cytoplasm.
Plant cells and animal cells both have a rigid plasma membrane as well as cytoplasm which is the gel-like solution that fills the cell. Plant cells have cell walls which provide structure to the plant and also chloroplasts which allow for photosynthesis. Animal cells do not have either of these structures.
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7002 | https://physics.stackexchange.com/questions/248756/electric-field-from-a-uniformly-charged-disk | Stack Exchange Network
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Electric Field from a Uniformly Charged Disk
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$\begingroup$
As I was reading the solution of this problem the author gave the electric field in the point P as follows:
$$ \vec{E} = \sigma /(2 \epsilon ) [1-x/(x^2+R^2)^{1/2}]\ \hat \imath$$
Where:
$\sigma$ is the surface charge density on the disk
$x$ is the distance from the center of the disk to the point P
$R$ is the radius of the disk
Here the question comes: for $R \rightarrow 0$ with keeping $Q$ constant (the total charge) why is it that $E$ should go to a point charge?
According to my knowledge $E$ goes to zero as we calculate the limit of $E$ while $R$ tends to zero. Even if the absolute value of $x$ gives $-x$ we still get another constant.
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edited Apr 11, 2016 at 13:55
Jeff
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asked Apr 10, 2016 at 21:21
EnthusiastiCEnthusiastiC
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1
$\begingroup$ Did you mean $R \rightarrow \infty$? $\endgroup$
Jeff
– Jeff
2016-04-10 22:00:25 +00:00
Commented Apr 10, 2016 at 22:00
1
$\begingroup$ Is this the source for the problem?: web.mit.edu/8.02t/www/materials/InClass/IC_Sol_W02D1_4.pdf $\endgroup$
Αδριανός
– Αδριανός
2016-04-11 01:19:51 +00:00
Commented Apr 11, 2016 at 1:19
$\begingroup$ @sagardipak has a complete solution, but his comments can be expanded. Any finite object will "look like" a point source if you get far enough away, assuming that you have real measuring equipment, with real limitations on precision. $\endgroup$
garyp
– garyp
2016-04-11 01:57:04 +00:00
Commented Apr 11, 2016 at 1:57
$\begingroup$ @Jeff i mean R->0 $\endgroup$
EnthusiastiC
– EnthusiastiC
2016-04-11 05:56:51 +00:00
Commented Apr 11, 2016 at 5:56
$\begingroup$ exactly that's the source @Adpiavoc $\endgroup$
EnthusiastiC
– EnthusiastiC
2016-04-11 05:59:02 +00:00
Commented Apr 11, 2016 at 5:59
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2 Answers 2
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$\begingroup$
First, I think a little intuition could help. If you imagine a disk with a charge $Q$ get smaller, but all the while keeping the charge Q intact, shouldn't it geometrically approach a point with charge Q? So, in theory, we should expect the field to approach that due to a point charge.
Now, intuition aside, let's go to the mathematics. While the factor $\left[ 1-x/(x^2 + R^2)^{1/2} \right]$ does go to zero as $R\to0$, the charge density $\sigma = Q/(\pi R^2)$ goes to infinity. This is the source of your problem, because you end up with an indeterminate form $\infty \times 0$ while calculating the limit, and not 0.
To evaluate the limit, you could use l'Hôpital's rule, after rewriting your formula as an appropriate fraction (and substituting $\sigma$ for its expression in terms of $R$).
As a bonus, a quick way to do this would be to use this handy approximation, which works for small $y$ values: $$ (1+y)^n \approx 1+ny. $$
You can use this if you factor $x^2$ from the square root: $$ \frac{x}{ \left(x^2 + R^2\right)^{1/2} } = \frac{x}{ \lvert x \rvert \left(1 + (R/x)^2\right)^{1/2} } = \frac{x}{\lvert x \rvert}\left(1+\left(R/x\right)^2 \right)^{-1/2}\approx\frac{x}{\lvert x \rvert} \left( 1 - \frac{R^2}{2x^2}\right). $$
Here I used $y=R/x$ and $n=-1/2$. If $R$ goes to 0, then $y$ goes to 0, and this approximation gets better. Replacing in the expression you have given, we end up with $$ \vec E = \frac{Q}{4 \pi \epsilon x^2} \frac{x \hat \imath}{\lvert x \rvert} = \frac{Q}{4 \pi \epsilon x^2} \frac{\vec x}{\lvert x \rvert}, $$ which is the expression for a field due to a point charge. (Notice that the term $\vec x / \lvert x \rvert$ only gives you the direction of the field, but doesn't change its magnitude.)
Edit: if you try to do the calculations for $x<0$ you'll end up in trouble. The actual formula for the electric field should be $$ \vec E = \frac{\sigma}{2\epsilon}\left[ \frac{x}{\lvert x \rvert} - \frac{x}{\left(x^2 + R^2 \right)^{1/2}} \right] \hat \imath, $$ which you can see if you follow the derivation of the equation.
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edited Apr 11, 2016 at 2:19
answered Apr 11, 2016 at 1:51
sagardipaksagardipak
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$\endgroup$
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$\begingroup$ till now it's good. but you said if i try to do calculations for x<0 i'll end up in trouble. that's right i did so. i found E =(4Qx^2-QR^2)/4 \pi \epsilon x^2R^2 so the limit here (R->0) doesn't exist right? what does this mean physically? is it for Q<0 we can't find E at the interior of disk?! $\endgroup$
EnthusiastiC
– EnthusiastiC
2016-04-14 21:51:57 +00:00
Commented Apr 14, 2016 at 21:51
$\begingroup$ Yes, in that case the limit will be infinity. There isn't any interpretation -- the formula that you were given just isn't right for those cases. You can check this if you consider the following. For x>0, the field is positive, so it "runs away" from the disk. But everything is symmetric about the plane containing the disk. So, for the other side of the plane, you should get the same value for the field but with a minus sign, so the field would be running away in the opposite direction. If you change x with -x in the formula that I gave in the end, that's exactly what you obtain. $\endgroup$
sagardipak
– sagardipak
2016-04-14 23:23:51 +00:00
Commented Apr 14, 2016 at 23:23
Add a comment |
1
$\begingroup$
I think the mystery comes from the fact that as $R \rightarrow 0$ you will find that $\sigma$, the area charge density, will become infinitely great. The two effects "cancel out" such that the resulting electric field is that of a single point charge.
Note that $dq=\sigma dA$, and because we are preserving $Q$, as $R$ shrinks, the charge on the disk will become more and more compactly assorted. Namely, $dq$ will grow larger for the given strip of area. Consequently $\sigma$ will grow.
In other words, (roughly) think of $\sigma=\frac{dq}{dA}$, where we are decreasing $dA$ while keeping the charge dq constant. Consequently $\sigma$ increases.
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edited Apr 11, 2016 at 1:23
answered Apr 11, 2016 at 1:17
ÎδÏιανÏÏÎδÏιανÏÏ
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$\endgroup$
4
$\begingroup$ When the surface becomes a point (on tending the radius to zero), the surface charge density reduces to a point charge. But you may misunderstand that as radius tends to zero, the surface charge density blows up. But that's misleading. We have no surface here. So there is only a point and there is no need to think about a surface charge. there is only a point charge now. It's similar to the case of observing the surface charge from a far off distance. In both cases, the electric field reduces to that of a point charge. $\endgroup$
UKH
– UKH
2016-04-11 15:23:01 +00:00
Commented Apr 11, 2016 at 15:23
$\begingroup$ @Unnikrishnan.K.H Right, but the surface disappears in a limiting manner, hence the blowing up of the density. We don't immediately become a point charge but it becomes one as the surface area disappears. I'll revisit the language in my answer as soon as I get to a computer... $\endgroup$
Αδριανός
– Αδριανός
2016-04-11 19:02:01 +00:00
Commented Apr 11, 2016 at 19:02
$\begingroup$ But is it necessary to speak a point charge as something with infinite charge density.....Point charge is a structure less entity $\endgroup$
UKH
– UKH
2016-04-12 11:21:19 +00:00
Commented Apr 12, 2016 at 11:21
$\begingroup$ Mathematically it would make sense for a point charge to have an "infinite" charge density-- just in the same way that division by zero may analytically seem to go to infinity. I agree though completely, that it is pointless to talk about the surface charge density in a physically meaningful way; as you say point charges are somewhat structure less. It is just a manifestation of the formula; the resulting electric field is that of a point charge. Perhaps I should clarify that the changing surface density is a purely mathematical object. $\endgroup$
Αδριανός
– Αδριανός
2016-04-12 11:30:53 +00:00
Commented Apr 12, 2016 at 11:30
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7003 | https://www.albert.io/blog/inverse-trig-functions-ap-precalculus-review/ | Skip to content
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AP® Precalculus
Inverse Trig Functions: AP® Precalculus Review
The Albert Team
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Inverse trig functions might sound intimidating, but they’re just another step in understanding the world of angles and lengths. Think of them as the “undo” for regular trigonometric functions. Mastering these functions is crucial for progressing in precalculus and understanding section 3.9 Inverse Trigonometric Functions in AP® Precalculus.
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What We Review
Understanding Inverse Trigonometric Functions
Inverse trig functions help find angles when the trigonometric ratios are given. They switch the roles of inputs and outputs in regular trig functions. The key functions include:
Arcsine (\sin^{-1} x or \arcsin x )
Arccosine (\cos^{-1} x or \arccos x )
Arctangent (\tan^{-1} x or \arctan x )
However, there’s a catch! These functions require restricted domains. This means you can only use certain input values to get a unique output (angle) as a result.
The Arcsine Function: Understanding \arcsin x
Definition and Notation:
\sin^{-1} x or \arcsin x
Outputs angles from the range [- \frac{\pi}{2}, \frac{\pi}{2}]
Example 1: Calculating \arcsin \left( \frac{1}{2} \right)
Determine which angle yields a sine of \frac{1}{2}.
Step 1: Recall the sine value \frac{1}{2} is associated with the well-known angle \frac{\pi}{6}.
Step 2: Verify the range. \frac{\pi}{6} is within [- \frac{\pi}{2}, \frac{\pi}{2}].
Solution: \text{arcsin} \left( \frac{1}{2} \right) = \frac{\pi}{6}
Arcsin Graph: It goes from (-1, -\frac{\pi}{2}) to (1, \frac{\pi}{2}) in a well-defined arc.
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The Arccosine Function: Understanding \arccos x
Definition and Notation:
\cos^{-1} x or \arccos x
Outputs angles from the range [0, \pi]
Example 2: Calculating \arccos(1)
Find the angle with a cosine value of 1.
Step 1: The angle 0 provides a cosine of 1.
Step 2: Verify the range. 0 is within [0, \pi].
Solution: \arccos(1) = 0
The arccos graph gently drops from (-1, \pi) to (1, 0).
The Arctangent Function: Understanding \arctan x
Definition and Notation:
\tan^{-1} x or \arctan x
Outputs angles from the range (- \frac{\pi}{2}, \frac{\pi}{2})
Example 3: Finding \arctan(-1)
Identify the angle with a tangent value of -1.
Step 1: Recognize that the tangent of -\frac{\pi}{4} is -1.
Step 2: Verify the range. -\frac{\pi}{4} fits in (- \frac{\pi}{2}, \frac{\pi}{2}).
Solution: \arctan(-1) = -\frac{\pi}{4}
The graph of arctan: Visualize it swooping from the bottom at (-\infty, -\frac{\pi}{2}) to the top at (\infty, \frac{\pi}{2}).
Characteristics and Properties of Inverse Trig Functions
Understanding these relationships aids problem-solving:
Arcsine links with sine over [- \frac{\pi}{2}, \frac{\pi}{2}].
Arccosine partners with cosine over [0, \pi].
Arctangent aligns with tangent over (- \frac{\pi}{2}, \frac{\pi}{2}).
Key properties include:
Inputs: Know the acceptable values for functions (domains).
Outputs: Derived angles (ranges) for each function.
Quick Reference Chart
| | | | |
--- --- |
| Function | Notation | Domain | Range |
| Arcsine | \sin^{-1} x | [-1, 1] | [- \frac{\pi}{2}, \frac{\pi}{2}] |
| Arccosine | \cos^{-1} x | [-1, 1] | [0, \pi] |
| Arctangent | \tan^{-1} x | (-\infty, \infty) | (- \frac{\pi}{2}, \frac{\pi}{2}) |
Conclusion
Inverse trigonometric functions might seem challenging, but with practice, they become easier and provide a broader understanding of relationships in math. Dive into more practice to master arcsin, arccos, and arctan for a solid grip on precalculus. Happy learning!
Sharpen Your Skills for AP® Precalculus
Are you preparing for the AP® Precalculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!
3.8.B The Tangent Function
3.8.C The Tangent Function
3.10 Trigonometric Equations and Inequalities
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7004 | https://zulfanaa.github.io/thempe/03_choropleth.html | Choropleth Map
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3.1
Choropleth Map
In this section, you will learn about the data for choropleth map, the color scheme, and the classification method.
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Figure 1. Choropleth Map - Population Density 2022
About Choropleth Map
'Choropleth' word comes from Greek
χῶρος (choros) = 'area/region' and
πλῆθος (plethos) = 'multitude'(from Wikipedia)
The choropleth map is one of common thematic maps. This map uses visual variable color value to visualize quantitative data.
Figure 2. Common Map Types from
Kraak MJ, RE Roth, B Ricker, A Kagawa, and G Le Sourd (2020)
Choropleth Map Data
From the previous section, we have learnt about data type just like it is shown in Figure 2. A choropleth map is used to represent Quantitative Data
As it says in the figure on the left, we can visualize 'Abrupt - Continuous' data in a choropleth map. Here is the examples of the data for A Choropleth Map:
→ Population density, forest coverage percentage per cities, poverty index of a specific region
Choropleth Map Color Scheme
As you can see in Figure 3 beside, color scheme is one of the most important cosiderations of making a Choropleth Map.
There are two main color scheme:
-Sequential (Single-hue or Multi-hue)
-Diverging
Figure 3. Color Brewer 2.0
Figure 4. Difference between equal interval and quantile histogram for choropleth data classification
Choropleth Map Classification Method
From Figure 3, we can conclude that data classification methods can affect the appearance of the Choropleth map. There are several methods that can be used to classify data before it is visualized into a map, namely:
-Equal Interval
-Quantile (Equal Count)
-Natural Breaks (Jenks)
-Standard Deviation
-Pretty Breaks
Create a Choropleth Map
create freely, change your data and color scheme to see the difference affected by both
Data Description⬇️
will be shown once you select the data
Quiz #3: Choropleth Map
Question 1:
What kind of quantitative data can be visualized in a Choropleth Map?
abrupt and discrete
smooth and continous
smooth and discrete
abrupt and continous
abrupt and smooth
NextPrevTry AgainGood Job! Learn other maps?
I understand, go to "Map Elements"
Hi! R U stuck here? Please contact me 😉⬇️
Which Map Type Do You Want to Learn Next?
| # | Thematic Map Type | Data Type | Symbol dimensionality | Most commonly applied visual variable |
--- ---
| 1 | Choropleth Map | Quantitative | Area | Color value |
| 2 | Proportional Symbol Map | Quantitative | Line (bars), Area, Volume | Size |
| 3 | Flow Map | Qualitative and Quantitative | Line | Size or Color Value (when representing magnitude) |
References
Kraak MJ, RE Roth, B Ricker, A Kagawa, and G Le Sourd. 2020. Mapping for a Sustainable World. The United Nations: New York, NY (USA).
Data Reference: United Nations Sustainable Development Goals Data Portal
Further Data Visualization Tutorial: Mapping for a Sustainable World - QGIS technical supplement
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Collisions are key events in physics, where objects interact and exchange energy and momentum. Understanding elastic and inelastic collisions helps us analyze everything from billiard ball impacts to car crashes, revealing how energy and momentum are conserved or transformed.
One-dimensional collisions are simpler to solve, using conservation laws and equations. Two-dimensional collisions require vector analysis, breaking motion into components. These principles apply to real-world scenarios, from particle physics experiments to astronomical events like comet impacts.
Types of Collisions
Elastic vs inelastic collisions
Elastic collisions
Total kinetic energy and momentum conserved during impact
Objects bounce off each other with no deformation (billiard balls, atomic collisions)
Inelastic collisions
Momentum conserved but kinetic energy partially lost
Energy converted to heat, sound, or deformation (car crashes, clay balls sticking together)
Perfectly inelastic collisions
Objects combine into single mass after impact
Maximum kinetic energy loss while conserving momentum (bullet embedding in wood)
One-dimensional collision problem-solving
Conservation of momentum: $p_1 + p_2 = p_1' + p_2'$ applies to all collisions
Elastic collisions
Kinetic energy conserved: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$
Relative velocity reversal: $v_1 - v_2 = -(v_1' - v_2')$ helps solve for final velocities
Inelastic collisions
Find final velocity using momentum conservation
Energy loss: $\Delta E = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 - \frac{1}{2}(m_1 + m_2)v_f^2$
Coefficient of restitution
Elasticity measure: $e = \frac{v_2' - v_1'}{v_1 - v_2}$ ranges 0 to 1
0 for perfectly inelastic, 1 for perfectly elastic collisions
Two-Dimensional Collisions
Vector analysis of two-dimensional collisions
Decompose vectors into x and y components
Apply momentum conservation separately in each direction
$p_{1x} + p_{2x} = p_{1x}' + p_{2x}'$ and $p_{1y} + p_{2y} = p_{1y}' + p_{2y}'$
Use vector addition to find final momenta
Calculate deflection angles with trigonometry (glancing collisions)
Conservation principles in collision problems
Vector momentum conservation: $\vec{p_1} + \vec{p_2} = \vec{p_1'} + \vec{p_2'}$ for all collisions
Apply kinetic energy conservation for elastic collisions
Solve simultaneous equations for unknown variables
Center of mass frame
Simplifies calculations, total momentum zero
Useful for analyzing particle collisions
2D collision types include glancing and head-on impacts
Real-world applications in particle physics experiments and astronomical events (comet impacts)
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7006 | https://www.commoncause.org/colorado/articles/first-past-the-post-voting-our-elections-explained/ | First Past the Post Voting: Our Elections Explained - Common Cause Colorado
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First Past the Post Voting: Our Elections Explained
Jun 22, 2020
First Past the Post voting often results in governments where the ratio of seats given to a certain party is not the same as the ratio of votes they got in the election.
Josh Franklin
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Despite the United State’s status as the world’s oldest standing democracy, many Americans believe that our democratic institutions aren’t satisfying them. While Common Cause strives to make democracy more inclusive for exactly this reason, representation in a democracy isn’t just about who votes or what we vote on, but also how we vote.
The current system is simple enough: public offices representing some group of people have vacancies. We could be referring to a seat on a city council, in the House of Representatives, or to the President of the United States. Each voter submits a ballot choosing one person they want to occupy the office in question, and conventionally, whoever gets the most votes wins. This system is called First-Past-the-Post (FPTP) or Winner-Takes-All. It is easy to understand and implement, and it intuitively seems fair. Unfortunately when examined critically and practically, it becomes clear that like any system, FPTP has its downsides.
Before pointing out the problems with FPTP, it is worthwhile to examine its benefits. The first is that it is very easy to understand: everyone gets one vote, and whoever has the most votes wins. The second major benefit is the ease of auditing; another result of its simplicity. If something happens that later calls the results of an election into question, the votes can simply be recounted. This simple recount should accurately determine the winner.
However, problems start appearing in any highly contested elections with many candidates competing for only a single seat. Imagine there is an election with ten candidates who are equally appealing to the voting population. The winner of this election receives only 12% of the vote, but the rest of the votes are spread equally among the other nine candidates so this is enough to be a victory. The 88% of the population that voted for someone else ends up being represented by a person they did not vote for, and who may not represent their views. This is called minority rule: the winner of the election is only appealing to a fraction of the voters instead of seeking a majority of the vote.
Flaws in FPTP voting combined with our two-party system also erode the variety of candidates that can get into office, until eventually there will only be two viable choices to vote for. This comes about due to the interaction of two problems with FPTP.
The first is how it molds voter behavior. Consider the same election scenario as before, with ten candidates where the winner only received 12% of the vote. In this scenario, imagine a voter whose candidate only got 7% of the vote. Unless there are major events that significantly shift the political landscape, voters should reasonably expect similar performance in future elections. Because of this, they may change their vote to someone that they don’t necessarily like, but who they think is more likely to win against other candidates that they deeply dislike. This is called strategic voting, and it’s a necessary decision for many voters to work within FPTP systems.
This is how things eventually filter down to a two party system. As voters abandon less popular candidates, these candidates typically drop out, lose in primaries, or run on third-party tickets with little chance of success, leading to a situation where there are only two candidates who have a realistic chance of winning. Voters who hold views centered between the two primary candidates become the focus of politicians’ persuasion efforts, and as this continues, it can also lead to disinterest in democracy from people who hold opinions that depart from the political center and who feel their opinions are not represented by either of the two viable options.
The second problem emerges after the political landscape has settled down to have two political parties. In the past, significant third party candidates have emerged. One example comes from the 2000 United States Presidential election, where Ralph Nader ran a campaign for president. As a left of center candidate, his policies were most similar to those of Democratic candidate Al Gore, and post-election surveys have indicated that Nader likely had a decisive impact on the election results:
The official Florida tally gave Bush the win by 537 votes (48.847 percent to 48.838 percent), while Nader racked up 97,488 votes. The national exit poll asked respondents how they would vote in a two-person race between Bush and Gore. Political scientist Gerald Pomper summed up the results in a 2001 Political Science Quarterly overview: “approximately half (47 percent) of the Nader voters said they would choose Gore in a two-man race, a fifth (21 percent) would choose Bush, and a third (32 percent) would not vote. Applying these figures to the actual vote, Gore would have achieved a net gain of 26,000 votes in Florida, far more than needed to carry the state easily.”
Essentially, because Nader appealed more to Democrats than Republicans, a significant number of Democrats who would have voted for Gore instead voted for Nader, causing Gore to lose the election. This is called the Spoiler Effect, and it makes it extremely difficult to escape a two party system. There are numerous examples of the Spoiler Effect, and it can affect both parties in the system negatively. For another example, in 1912, former Republican president Theodore Roosevelt challenged sitting president Republican president William Taft, splitting Republican votes and allowing for an easy victory for Democrat Woodrow Wilson.
In the end, FPTP voting often leads to a system with only two effective political parties. Strategic voting narrows the playing field down to two candidates, and the Spoiler Effect means that third parties can’t get a foothold to challenge the status quo. This leaves many people’s interests without representation and ensures many ideas will never be heard. In this system, the two political parties in power aren’t competing for all voters, but just a persuadable middle, which leaves many voters feeling unrepresented. Political parties can count on negative partisanship to encourage people to vote against the party they dislike more, or alienation and disinterest to cause them not to vote at all. Institutional effects can keep parties in power despite voter dissatisfaction with their views, or even in elections like 2016 where the majority of Americans did not vote.
In addition, FPTP voting often results in governments where the ratio of seats given to a certain party is not the same as the ratio of votes they got in the election. The difference between the ratio of seats earned and number of votes cast is called misrepresentation error, and it has been demonstrated in many recent elections. For example, in the United State’s 2012 elections for the House of Representatives, the Republican Party was awarded 54% of seats despite only winning 47% of the vote nationwide.
While the United States aspires to be a beacon of democracy, our voting systems give a small section of voters disproportionate power, force third parties into ‘spoiler’ roles, and can significantly misrepresent votes cast and the proportion of seats won in an election. To learn more about alternatives and improvements to first-past-the-post voting, stay tuned for updates right here on Democracy Wire, and check out our work in election access and representation at commoncause.org/colorado/our-work.
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7007 | https://www.quora.com/How-do-we-derive-the-formula-for-the-center-of-mass-of-a-solid-body-from-a-system-of-particles | How do we derive the formula for the center of mass of a solid body from a system of particles? - Quora
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How do we derive the formula for the center of mass of a solid body from a system of particles?
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Aaron Dunbrack
Physics PhD 2024 from Stony Brook University · Author has 3.1K answers and 4.1M answer views
·8y
Center of mass is defined as the weighted average (pun not intended) of the position, as weighted by the density.
If you have discrete points, then “weighted average by density” is simply “weighted average [in a sum] by mass of each particle.” For a continuum, “weighted average by density” means “weighted average [in an integral] by the local density.”
To go from “solid body” to “system of particles,” you plug in a bunch of delta functions. To go the other way, you have to recognize a solid body as a bunch of delta functions which increase in number but decrease in size, in the limit of infinite
Continue Reading
Center of mass is defined as the weighted average (pun not intended) of the position, as weighted by the density.
If you have discrete points, then “weighted average by density” is simply “weighted average [in a sum] by mass of each particle.” For a continuum, “weighted average by density” means “weighted average [in an integral] by the local density.”
To go from “solid body” to “system of particles,” you plug in a bunch of delta functions. To go the other way, you have to recognize a solid body as a bunch of delta functions which increase in number but decrease in size, in the limit of infinite number but zero size, in such a way that “number in a region times size [the density] remains constant. Once you have this, the rest is… perhaps mathematically informal, but trivial insofar as physics derivations go (sum of Dirac deltas turns into integral over space, in the limit).
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Andy Buckley
Prof in particle physics, visiting researcher at CERN · Upvoted by
Erik Kofoed
, M.Sc. Physics & Theoretical Physics, Lund University (2016) and
Stephen Selipsky
, ex-particle theorist; Stanford Ph.D., research at CERN, BU, Yale · Author has 397 answers and 891.5K answer views
·7y
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How can I prove that the centre of mass for a two-particle system always lies on the line joining the two masses?
Others have given an explicit answer, but I want to give an indication of how physicists and mathematicians approach problems like this when arguing from acquired (through practice and experience) intuition. One word: symmetry.
Let's say the centre of mass was not on the line joining the two masses. But in what direction would it be off-axis? There would need to be a reason for the choice of that direction, otherwise all directions of decision would be equally well motivated. (We're talking classical rather than quantum mechanics here, so the choice of a direction in which the symmetry is broke
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Others have given an explicit answer, but I want to give an indication of how physicists and mathematicians approach problems like this when arguing from acquired (through practice and experience) intuition. One word: symmetry.
Let's say the centre of mass was not on the line joining the two masses. But in what direction would it be off-axis? There would need to be a reason for the choice of that direction, otherwise all directions of decision would be equally well motivated. (We're talking classical rather than quantum mechanics here, so the choice of a direction in which the symmetry is broken has to be deterministic and repeatable.) Generating that asymmetric CoM would require some sort of off-axis mass in the system, but there are only two so that’s impossible. From this line of argument, we see that the whole system must be symmetric about the joining line: we can't allow ourselves to have any finite deviation of the CoM from the axis, hence it must be on the axis.
This applies even if you think about the rather complicated distribution of gravitational or other force fields due to the particles: by symmetry all the vectors perpendicular to the line joining the objects have to cancel, so the CoM must be on that line and to compute it you only need to think of the force component along the axis of symmetry.
This sort of argument from symmetry is very visual and powerful, and can help to shortcut a lot of explicit calculation — either completely, or by indicating profitable computation strategies like clever choices of coordinate system. We use it a lot; actually, I doubt detailed real-time conversations about physics ideas would be possible without this sort of tactic to turn calculations into intuitions.
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To derive the formula for the center of mass of a solid body from a system of particles, we start by understanding the concept of center of mass for a discrete system and then extend it to a continuous solid body.
Step 1: Center of Mass for a System of Particles
For a system of N N particles, each with mass m i m i located at position vectors r i r i, the center of mass R c m R c m is defined as:
R c m=1 M∑N i=1 m i r i R c m=1 M∑i=1 N m i r i
where M M is the total mass of the system, given by:
M=∑N i=1 m i M=∑i=1 N m i
Step 2: Transition to a Continuous Solid Body
For a continuous solid
Continue Reading
To derive the formula for the center of mass of a solid body from a system of particles, we start by understanding the concept of center of mass for a discrete system and then extend it to a continuous solid body.
Step 1: Center of Mass for a System of Particles
For a system of N N particles, each with mass m i m i located at position vectors r i r i, the center of mass R c m R c m is defined as:
R c m=1 M∑N i=1 m i r i R c m=1 M∑i=1 N m i r i
where M M is the total mass of the system, given by:
M=∑N i=1 m i M=∑i=1 N m i
Step 2: Transition to a Continuous Solid Body
For a continuous solid body, we can think of it as being made up of an infinite number of infinitesimally small particles. We replace the discrete sum with an integral.
Define Mass Density: Let ρ(r)ρ(r) be the mass density at point r r in the body. The mass of an infinitesimal volume element d V d V is given by:
d m=ρ(r)d V d m=ρ(r)d V
Total Mass: The total mass M M of the solid body can be calculated by integrating the mass density over the entire volume V V of the body:
M=∫V ρ(r)d V M=∫V ρ(r)d V
Center of Mass: The position of the center of mass R c m R c m for the continuous body is then given by integrating over the volume:
R c m=1 M∫V r d m=1 M∫V r ρ(r)d V R c m=1 M∫V r d m=1 M∫V r ρ(r)d V
Final Formula
Combining these expressions, we arrive at the final formula for the center of mass of a solid body:
R c m=1∫V ρ(r)d V∫V r ρ(r)d V R c m=1∫V ρ(r)d V∫V r ρ(r)d V
Summary
In summary, we start from the definition of the center of mass for a discrete system of particles and transition to a continuous solid body by using integrals to account for the mass distribution throughout the volume of the body. This results in a general formula that can be applied to various shapes and mass distributions.
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What is center of mass?
How is the center of mass outside of the body?
How do you calculate the center of mass for a uniform solid body?
Where does the center of mass of two particle system lie?
What is the center of mass of a system of particles?
Aman Raj
Software Engineer at Wells Fargo (company) (2025–present) · Author has 308 answers and 1.1M answer views
·6y
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Why do we divide by sum of masses in centre of mass formula?
Why do we divide by sum of masses in centre of mass formula?
It's a good question that invokes a lot of curiosity. Well, it seems that we need to first define the term “ Centre of Mass” and then only we can defy the confusion.
What is centre of mass?
We have so many ideas for centre of mass. A lot of them! But if we want to define the centre of mass, it will be good to say that
The centre of mass of a body is defined as the unique point, where we assume that the whole body could be replaced by that point.
Well, that's a bit practical definition. But that's good!
How can we able to get the centre of
Continue Reading
Why do we divide by sum of masses in centre of mass formula?
It's a good question that invokes a lot of curiosity. Well, it seems that we need to first define the term “ Centre of Mass” and then only we can defy the confusion.
What is centre of mass?
We have so many ideas for centre of mass. A lot of them! But if we want to define the centre of mass, it will be good to say that
The centre of mass of a body is defined as the unique point, where we assume that the whole body could be replaced by that point.
Well, that's a bit practical definition. But that's good!
How can we able to get the centre of mass of a body?
Well, what we can do is to get a body and then let it be balanced by our tips. There will be a point (2D bodies!) where the object will be balanced. This point will be the centre of mass.
From the above method, it is clear that in equilibrium, the sum of torque about the centre of mass is zero!
That is quite an important observation.
Let x i x i denote the n points in space and let their masses be m i m i respectively. What is the centre of mass of the system?
We begin with the above condition that sum of torque about the centre of mass is zero.
Then, let x m x m be the centre of mass.
Thus, τ n e t=0 τ n e t=0
So, consider x 1 x 1. The torque about the centre of mass is
τ 1=m 1 g(x−x 1)τ 1=m 1 g(x−x 1)
For the nth particle, we get
τ n=m n g(x−x n)τ n=m n g(x−x n)
So, summing the above from n = 1 to n, and since the resultant of all torque is zero, we get
0=(∑m)x−∑m i x i 0=(∑m)x−∑m i x i
So, we derived the formula
x=∑m i x i∑m x=∑m i x i∑m
Notice that the total mass has come below. So, we found an explanation to this question!
So, we are done.
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Ken Sandale
MIT Class of 1979 · Upvoted by
Jesse Raffield
, Master's degree in physics · Author has 685 answers and 350.7K answer views
·4y
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Why is the center of mass of a rigid body exactly equal to the mass weighted average of all its particles' positions?
“Why is the center of mass of a rigid body exactly equal to the mass weighted average of all its particles' positions?”
It’s how it is defined!!!
On a deeper level, the question should be “Why is it useful to define a thing that way. It comes up in places. For example, The some of the forces on a system of particles such as a rigid body (or a non-rigid system) turns out to be the total mass times a thing which numerically is the acceleration of this “center of mass” thing. Another example is that The total angular momentum of a system of particles (rigid or non-rigid) is the angular momentum aro
Continue Reading
“Why is the center of mass of a rigid body exactly equal to the mass weighted average of all its particles' positions?”
It’s how it is defined!!!
On a deeper level, the question should be “Why is it useful to define a thing that way. It comes up in places. For example, The some of the forces on a system of particles such as a rigid body (or a non-rigid system) turns out to be the total mass times a thing which numerically is the acceleration of this “center of mass” thing. Another example is that The total angular momentum of a system of particles (rigid or non-rigid) is the angular momentum around the center of mass plus the angular momentum of the center of mass. So for example the angular momentum of the Earth is the sum of angular momentum of the Earth around its axis plus the orbital angular momentum of the Earth orbiting the Sun. (It’s actually a little tricky—the spin number is a bit different from what it might seem)
The quantity keeps popping up in formulas, so it deserves a name, to simplify things.
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Dennis Lewis
Can not make Gravity work out · Author has 1.8K answers and 280.5K answer views
·2y
Related
What is the definition of centre of mass in physics? How do we calculate it for objects which consist of many particles?
It is a point in the mass where the sum of the moments of all particles is 0.
As if the mass is concentrated at one point.
The calculation for “regular” shapes is moment of inertia/area.
For irregular shapes it is a lot of work.
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Related questions
More answers below
Where does the centre of mass of a particle system lie?
What is the importance of the center of mass of a system of particles and why do we use this concept?
What is linear momentum of a system of particles about the center of mass, and how?
How can you derive the formula for the center of mass of a system of particles: R=1 M∑n i=1 m i r i R=1 M∑i=1 n m i r i where M M is the total mass, m i m i is the mass of the i-th particle, and r i r i is the position vector of the i-th particle?
Where does the centre of mass of a two particle system lie if one particle is more massive than the other?
Paul Dunkley
Associate · Author has 994 answers and 2M answer views
·1y
Related
How do you find the centre of mass and centre of gravity for a solid figure?
We have 27 examples of finding the centre of gravity of plane and solid figures, taken from the American Machinery's Handbook (1950)
Continue Reading
We have 27 examples of finding the centre of gravity of plane and solid figures, taken from the American Machinery's Handbook (1950)
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Vishakh Rajendran
M.S. in Aerospace and Aeronautical Engineering, Nanyang Technological University · Author has 682 answers and 3M answer views
·5y
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How will you obtain an expression for centre of mass of a 'n' particle system using the expression for centre of mass of a two particle system?
For a two particle system, you can refer this to obtain an expression for the centre of mass - Vishakh Rajendran's answer to How can I prove that the centre of mass for a two-particle system always lies on the line joining the two masses?
So, the centre of mass of a two particle system Rcm = ( m1 r1 + m2 r2 ) / ( m1 + m2 )
The same expression can be extended to include ’n’ number of particle. In that case,
Rcm = (m1r1 + m2r2 + m3r3 + … + mnrn) / (m1 + m2 + m3 + … + mn)
This can be written as Rcm = Σ (miri) / Σ (mi)
In terms of the cartesian co-ordinate system, the X and Y co-ordinates of the centre
Continue Reading
For a two particle system, you can refer this to obtain an expression for the centre of mass - Vishakh Rajendran's answer to How can I prove that the centre of mass for a two-particle system always lies on the line joining the two masses?
So, the centre of mass of a two particle system Rcm = ( m1 r1 + m2 r2 ) / ( m1 + m2 )
The same expression can be extended to include ’n’ number of particle. In that case,
Rcm = (m1r1 + m2r2 + m3r3 + … + mnrn) / (m1 + m2 + m3 + … + mn)
This can be written as Rcm = Σ (miri) / Σ (mi)
In terms of the cartesian co-ordinate system, the X and Y co-ordinates of the centre of mass can be given by -
X cm = (m1x1 + m2x2 + … + mnxn ) / (m1 + m2 + … + mn) where x1 , x2 , … , xn are the distance from the x axis (and)
Y cm = (m1y1 + m2y2 + … + mnyn ) / (m1 + m2 + … + mn) where y1 , y2 , … , yn are the distance from the y axis.
Upvote ·
9 1
Martin Stojanovikj
Math enthusiast, science lover, "Shut up and calculate" · Author has 91 answers and 182.9K answer views
·10y
Related
How are formulae first derived?
I cannot exactly understand the question, but I will try to elaborate a little bit on how I view your question.
Basically it is the same process that humans go through today... For example you walk through the forest picking leafs from the ground, when you notice they have an almost same structure... So you wonder why? You try and find a pattern, and you find a number series explaining that pattern... So you say maybe there is a way to simplify this series, so you find a formula for the series, in which if someone plugs a number it will come out exactly the next one, or the 1957th one. There y
Continue Reading
I cannot exactly understand the question, but I will try to elaborate a little bit on how I view your question.
Basically it is the same process that humans go through today... For example you walk through the forest picking leafs from the ground, when you notice they have an almost same structure... So you wonder why? You try and find a pattern, and you find a number series explaining that pattern... So you say maybe there is a way to simplify this series, so you find a formula for the series, in which if someone plugs a number it will come out exactly the next one, or the 1957th one. There you go on and check if that series holds a pattern in itself, and if there are other similar phenomenons, so the process goes on and on...
It is the same since the ancient Greeks, they were interested in a phenomenon, they found that math is the best way to describe it. Since the phenomena described were getting more complex, the math symbols were getting more complex, so fields like set theory were invented, so that you could simplify and group the symbols, in order to clarify your work.
If this wasn't the answer you were looking for, then hopefully at least I was able to get a hold of a small part of your question.
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Duvvuri Jathaveda
Former Aerospace Engineering Graduate at University of Sheffield · Author has 175 answers and 331.8K answer views
·7y
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Why is the center of mass defined? How can we assume all the mass at that point?
Classical Mechanics is a subject where we frequently come across oddly shaped bodies and a system of such oddly shaped bodies. In order to make the calculations easier for the motion produced or to be produced in the body, we define a point where the motion will be parallel to the applied force and the body will not rotate.
Coming to the question, centre of mass of an object is defined so as to have the knowledge of the application of force for any regular or irregular shaped body so that it will conserve the momentum given to it into linear motion rather than rotational motion.
We can assume th
Continue Reading
Classical Mechanics is a subject where we frequently come across oddly shaped bodies and a system of such oddly shaped bodies. In order to make the calculations easier for the motion produced or to be produced in the body, we define a point where the motion will be parallel to the applied force and the body will not rotate.
Coming to the question, centre of mass of an object is defined so as to have the knowledge of the application of force for any regular or irregular shaped body so that it will conserve the momentum given to it into linear motion rather than rotational motion.
We can assume the mass at that point so that the calculations will become simpler and easier. If not assumed at a single location, we will be calculating the outcome of the force applied on each and every particle of that body. Just because we don’t want to make any complications, we can assume the centre of mass at a single location.
Duvvuri!
Upvote ·
9 2
9 1
Florens de Wit
M.S. in Physics, Eindhoven University of Technology (Graduated 1999) · Author has 2.3K answers and 1.3M answer views
·6y
Related
What is the importance of the center of mass of a system of particles and why do we use this concept?
We use this concept in physics because it makes some calculations easier.
With A collection of particles with fixed relative positions we can do all calculations involving gravity on this centre of mass rather than on all particles separately. We can use the total moment of inertia under the same conditions when moment inducing forces play a role.
So in summary: we use concepts like centre of mass to make it easier to do calculations on many particles. This works as long as the particles' relevant relationships (i.e. position) do not change. It's an example of Mathematics as a tool to simplify c
Continue Reading
We use this concept in physics because it makes some calculations easier.
With A collection of particles with fixed relative positions we can do all calculations involving gravity on this centre of mass rather than on all particles separately. We can use the total moment of inertia under the same conditions when moment inducing forces play a role.
So in summary: we use concepts like centre of mass to make it easier to do calculations on many particles. This works as long as the particles' relevant relationships (i.e. position) do not change. It's an example of Mathematics as a tool to simplify calculations and mental models in natural science.
Upvote ·
9 3
Jan-Christoph Schlage-Puchta
Professor for Mathematics at University of Rostock · Author has 1.9K answers and 5.5M answer views
·1y
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How can you derive the formula for the center of mass of a system of particles: R=1 M∑n i=1 m i r i R=1 M∑i=1 n m i r i where M M is the total mass, m i m i is the mass of the i-th particle, and r i r i is the position vector of the i-th particle?
You can't. This is usually taken as the definition of the centre of mass. If you use a different definition, you have to show that these definitions are eqivalent to show that the above formula holds.
Upvote ·
Bob Collier
Former EE Designed Specialized Computers for 33 Years. · Author has 3.1K answers and 1.6M answer views
·2y
Related
How do you calculate the centre of gravity for a system having n particles?
I’m not sure you can.
There is something called the “3-body problem’” that we can’t solve exactly. Earth, sun & moon is a good enough example of that.
Evidently we can work out earth-moon with no sun to worry about or earth-sun with no moon, but not the 3 that exist. Your distributed body COG sounds like the same situation to me.
A different aspect to that problem is the fact that every piece of mass in the universe pulls on every other piece of mass in the universe. So no ‘exact’ solution is even theoretically possible. But the universe still happens, anyway.
So much for ‘theory’.
F = GmM/R^2 [tha
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I’m not sure you can.
There is something called the “3-body problem’” that we can’t solve exactly. Earth, sun & moon is a good enough example of that.
Evidently we can work out earth-moon with no sun to worry about or earth-sun with no moon, but not the 3 that exist. Your distributed body COG sounds like the same situation to me.
A different aspect to that problem is the fact that every piece of mass in the universe pulls on every other piece of mass in the universe. So no ‘exact’ solution is even theoretically possible. But the universe still happens, anyway.
So much for ‘theory’.
F = GmM/R^2 [that’s 2 masses: n & M] and R [distance between them] can be pretty ‘big’ but that F is never zero.
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Related questions
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How can you derive the formula for the center of mass of a system of particles: R=1 M∑n i=1 m i r i R=1 M∑i=1 n m i r i where M M is the total mass, m i m i is the mass of the i-th particle, and r i r i is the position vector of the i-th particle?
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Teaching sequences for AC V9
Year 3
Multiplication: reSolve Market
Year 3
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Task 7
Multiplication: reSolve Market
Task 7 • Making fact families
Students continue to build their understanding that related multiplication and division facts can be recorded as a fact family.
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Multiplication: reSolve Market
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Task 1 • Finding multiplication
Task 2 • Mangoes and apples
Task 3 • Making arrays
Task 4 • Hidden fruit
Task 5 • Rolling arrays
Task 6 • Lemon arrays
Task 7 • Making fact families
Task 8 • Finding fact families
Key learning goals
Related multiplication and division facts can be found in the array. These related facts are known as a fact family.
List of materials
Whole class
reSolve Market PowerPoint
Each group
Number Cards printed onto A4 card and cut out
Counters (at least 100)
Each student
Making fact families Student sheet
reSolve Market PowerPoint
PPTX • 14.9MB
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Number cards
DOCX • 23.9KB
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Making fact families Student sheet
DOCX • 0.2MB
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Build
Making fact families
This activity is a variation of the earlier activity, Making arrays.
Use slide 34 of the reSolve Market PowerPoint to explain to students the Making fact families activity.
Students work in pairs. Each pair needs a set of cards and a collection of counters. Each student needs their own copy of Making fact families Student sheet.
Completing the activity
The first student selects a number card. Both students collect that number of counters.
Each student works on their own to arrange their full collection of counters into an array. They record a picture of their array and the corresponding fact family on their student sheet.
The students compare their arrays and facts and discuss the similarities and differences that they notice.
The second student selects a card and the pair repeat the activity.
Estimated time
50 minutes
Task type
Group
Mathematics content
Making fact families
This activity is a variation of the earlier activity Making arrays. It has been designed to build students’ understanding of the array and different facts that can be derived using the array.
Making fact families Close modal
Student attention is focused on the structure of their array as they record its details in their student sheet. In this version of the game, students use the rows and columns in the array to determine the fact family for the array, that is, multiplication and division facts for that array.
The numbers selected for this activity each have multiple factors which will increase the likelihood that students will make different arrays to their partners. This allows students to compare arrays and see that some numbers can be represented using different multiplication facts. Building different rectangular arrays for the same number provides students with experiences that enable them to recognise and identify factors.
Making fact families
Student attention is focused on the structure of their array as they record its details in their student sheet. In this version of the game, students use the rows and columns in the array to determine the fact family for the array, that is, multiplication and division facts for that array.
The numbers selected for this activity each have multiple factors which will increase the likelihood that students will make different arrays to their partners. This allows students to compare arrays and see that some numbers can be represented using different multiplication facts. Building different rectangular arrays for the same number provides students with experiences that enable them to recognise and identify factors.
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7009 | https://perachi.weebly.com/ratios-rates-and-proportions.html | MRS. PERACHI 6TH GRADE MATH
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| Home Classes + 6th Grade Math > - Pre-Algebra - Whole Numbers and Decimals - Fractions - Percentages - Expressions - Equations - Integers and Inequalities - Ratios, Rates, and Proportions - Graphs and Data - Geometry - Integer Computation + B Period > - Telling Time - Money - Whole Numbers - Pre-Algebra - Decimal Numbers - Fraction Numbers - Percentages - Algebra - Ratios Rates and Proportions - Data - Graphing - Geometry - Integers Extra Help Multiplication | |
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Unit VocabularyVocabulary QuizletRatio vs Rate vs Proportion NotesReferencesRatios Rates and Proportions Reference SheetCompleted Ratios Rates and Proportions Reference SheetRatio VideoRatio Video NotesRatio Intro Video 1Ratio Video 2Ratio Wheel SimulatorRatiosWriting Ratios- PicturesRatios with Pictures and TablesRatio Scene with Cartoon CharactersRatio Scene AssignmentRatio Word ProblemsWhat is a Ratio WorksheetEE Equal Ratios with BlocksRatios/Tables/GraphsRatios and TablesRatios with BlocksRate of Pay ChallengeWritingRatesRates VideoRates Video NotesDistance=Rate x Timed=rt Video Tutoriald=rt Video Tutorial Notesd=rt Notesd=rt Practice 1Tape Diagrams and Double Number LinesRatio Tables and Double Number LinesReading Tape DiagramsTape Diagrams 1Tape Diagrams 2Tape Diagrams 3Unit RateUnit Rate VideoUnit Price VideoComparison ShoppingBetter Buy in the Classroom (Stations)What is a Rate WorksheetUnit Rate IntroUnit Rate PracticeUnit Price PracticeUnit Price Online PracticeSugary Beverages ActivityProportions (Equal Ratios)More Proportion Word ProblemsProportion Problem SolvingProportions- Equal FractionsKahootProportions (Cross Products)Solving One Step Multiplication EquationsProportion Word Problem VideoSolving Proportions IntroSolving Proportions PracticeSolving Proportions Online PracticeEasy Proportion Word ProblemsSetting up Proportions for Problem SolvingProportion Word ProblemsProportion Word Problems Online PracticePercent Word ProblemsPercent Word Sentences 1Percent Word Sentences 2Percent Word Sentences 3Percent Word Problems 1Percent Word Problems 2Percent Word Problems 3Percents at the Mall WebsiteSimilar FiguresSimilar Figures Practice 1Similar Figures Practice 2Similar Figures Practice 3Similar Figures Practice 4Converting MeasurementsConverting Measurements Using ProportionsUsing Proportion to ConvertProportion Problem Solving and Circle GraphsIntro to Circle GraphsPercent Proportions with Circle GraphsUnit ReviewQuiz ReviewRatios and Rates Review Survival ProjectUnit Study GuideReview PacketReview Packet AnswersMore review PracticeReview GamesRatio Game
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Question & Answer » Physics Questions » How do we convert 1 g/cm3 to kg/m3
How do we convert 1 g/cm3 to kg/m3
How do we convert 1 g/cm3 to kg/m3?- Find the answer to this question and access a vast question bank that is customized for students.
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Answer:
The density (D or ) of a substance is the measurement of how firmly it is packed together. It is expressed as mass per unit volume. The density mathematical formula is provided by,
ρ = m/V
Where
ρ is the density
m is the mass of the object
V is the volume of the object
Unit of density
According to the density equation ( ρ= m/V), mass density has any unit that is mass divided by volume. There are several units for density in use since there are many units of mass and volume encompassing many different magnitudes. The SI unit of kilogram per cubic meter (kg/m3) and the cgs unit of gram per cubic centimeter (g/cm3) are the most often used density units. 1000 kg/m3 is equivalent to 1 g/cm3. One milliliter is equal to one cubic centimeter (abbreviated cc). Other greater or smaller units of mass and volume are frequently more practical and utilized in industry.
Conversion of 1 g/cm3 to kg/m3
1 g/cm3 is equivalent to 1000 kg/cubic meter.
To convert 100 gram to kilogram, divide it by 1000, 100/1000 = 0.1 kg.
To convert any gm/cm3 amount to kg/m3, multiply it by 1000.
Example:
(1) 1 gm/cm3 = 1 x 1000 = 1000 kg/meter3
(2) 10 gm/cm3 = 10 x 1000 = 10,000 kg/meter3
This is how the conversion of 1g cm3 to kg m3 is done through the formula. You can convert this in between any units. The only thing needed is the unit of density, and all of them are mentioned above.
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7011 | https://pubmed.ncbi.nlm.nih.gov/11943958/ | Low serum vitamin B12 is common in coeliac disease and is not due to autoimmune gastritis - PubMed
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Eur J Gastroenterol Hepatol
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. 2002 Apr;14(4):425-7.
doi: 10.1097/00042737-200204000-00016.
Low serum vitamin B12 is common in coeliac disease and is not due to autoimmune gastritis
William Dickey1
Affiliations Expand
Affiliation
1 Department of Gastroenterology, Altnagelvin Hospital, Londonderry BT47 6SB, Northern Ireland. wildickey@aol.com
PMID: 11943958
DOI: 10.1097/00042737-200204000-00016
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Low serum vitamin B12 is common in coeliac disease and is not due to autoimmune gastritis
William Dickey. Eur J Gastroenterol Hepatol.2002 Apr.
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. 2002 Apr;14(4):425-7.
doi: 10.1097/00042737-200204000-00016.
Author
William Dickey1
Affiliation
1 Department of Gastroenterology, Altnagelvin Hospital, Londonderry BT47 6SB, Northern Ireland. wildickey@aol.com
PMID: 11943958
DOI: 10.1097/00042737-200204000-00016
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Abstract
Objective: Although coeliac disease is a disorder of the proximal small bowel, associated vitamin B12 deficiency has been reported. This study aimed to assess the prevalence of B12 deficiency in a large series of coeliac patients, and to exclude the possibility that it is due to associated autoimmune gastritis.
Design: Prospective routine measurement of serum B12 in coeliac patients, with investigations for pernicious anaemia/autoimmune gastritis in B12-deficient patients.
Setting: Gastroenterology department of a large district general hospital.
Interventions: If they were not taking vitamin B12 supplements already, patients had serum B12 measured before starting dietary gluten exclusion. Those with low levels also had gastric biopsies taken and plasma gastrin and serum gastric parietal cell and intrinsic factor antibodies measured.
Main outcome measures: Prevalence of low serum B12, and presence or absence of indicators of pernicious anaemia/autoimmune gastritis in patients with low serum B12.
Results: Of 159 patients, 13 had low serum B12 at diagnosis. A further six had been receiving B12 replacement therapy for 3-37 years before diagnosis, giving an overall prevalence of 12% (19 patients). Only 2/19 patients had gastric corpus atrophy, one with intrinsic factor antibodies and the other with hypergastrinaemia. There was no relationship between low B12 and clinical characteristics.
Conclusions: Low B12 is common in coeliac disease without concurrent pernicious anaemia, and may be a presenting manifestation. B12 status should be known before folic acid replacement is started.
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Morphometric analysis of small-bowel mucosa in Turkish children with celiac disease and relationship with the clinical presentation and laboratory findings.Arikan C, Zihni C, Cakir M, Alkanat M, Aydogdu S.Arikan C, et al.Dig Dis Sci. 2007 Sep;52(9):2133-9. doi: 10.1007/s10620-006-9606-2. Epub 2007 Apr 4.Dig Dis Sci. 2007.PMID: 17406838
Narrative Review: Nutrient Deficiencies in Adults and Children with Treated and Untreated Celiac Disease.Kreutz JM, Adriaanse MPM, van der Ploeg EMC, Vreugdenhil ACE.Kreutz JM, et al.Nutrients. 2020 Feb 15;12(2):500. doi: 10.3390/nu12020500.Nutrients. 2020.PMID: 32075276 Free PMC article.Review.
Anemia in celiac disease is multifactorial in etiology: A prospective study from India.Berry N, Basha J, Varma N, Varma S, Prasad KK, Vaiphei K, Dhaka N, Sinha SK, Kochhar R.Berry N, et al.JGH Open. 2018 Aug 2;2(5):196-200. doi: 10.1002/jgh3.12073. eCollection 2018 Oct.JGH Open. 2018.PMID: 30483589 Free PMC article.
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7012 | https://brainly.com/question/1496714 | [FREE] What is the molar volume of a gas at standard temperature and pressure (STP)? - brainly.com
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What is the molar volume of a gas at standard temperature and pressure (STP)?
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Standard Molar Volume is the volume occupied by one mole of any gas at STP. Remember that "STP" is Standard Temperature and Pressure. Standard temperature is 0 ° C or 273 K. Standard pressure is 1 atmosphere or 760 mm Hg (also called "torr"). 1 mole of any gas at STP occupies 22.4 liters of volume. T he volume occupied by any number of moles (or grams) can be determined.
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The Basics of General, Organic, and Biological Chemistry - David W Ball
Fundamentals of General Organic and Biological Chemistry - Mcmurry Et Al.
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At standard temperature and pressure (STP), which is 0 °C and 1 atm, the molar volume of any ideal gas is approximately 22.4 liters per mole. This allows for useful calculations in gas laws and stoichiometry. The ideal gas law can be applied to determine this volume easily using the formula P V = n R T.
Explanation
In chemistry, the concept of molar volume refers to the volume occupied by one mole of a gas. This is particularly important at standard temperature and pressure (STP), which is defined as a temperature of 0 °C (273 K) and a pressure of 1 atmosphere (atm). At these conditions, the molar volume of any ideal gas is approximately 22.4 liters (L).
Understanding STP:
The term 'standard temperature and pressure' provides a reference point that allows scientists to compare the properties of gases.
Ideal Gas Behavior:
At STP, gases behave ideally, meaning they follow the ideal gas law:
P V=n RT
Where:
P = pressure
V = volume
n = number of moles of the gas
R = ideal gas constant (0.0821 L·atm/(K·mol))
T = temperature in Kelvin
Molar Volume Calculation:
If we rearrange the ideal gas law to solve for volume at STP, we set P=1 atm, n=1 mol, and T=273 K.
Plugging in these values gives:
V=P n RT=1 atm(1 mol)(0.0821 L\cdotp atm/(K\cdotp mol))(273 K)≈22.4 L
Applications:
Knowing that one mole of any gas occupies 22.4 L at STP allows for practical calculations in various chemical processes and stoichiometry problems.
For example, if you know you have 2 moles of a gas at STP, you can quickly calculate the volume it occupies as 2×22.4 L=44.8 L.
Examples & Evidence
For example, if you have 1 mole of oxygen gas (O2) at STP, it will occupy 22.4 liters. If you have 3 moles of nitrogen gas (N2) at STP, it will occupy 67.2 liters (3 moles x 22.4 liters/mole).
The statement that the molar volume of a gas at STP is 22.4 liters per mole can be consistently validated through laboratory experiments and is grounded in the ideal gas law, which is widely accepted in scientific literature.
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### Free Chemistry solutions and answers
Community Answer 2 What is the molar volume of a gas at standard temperature and pressure? LOL 241 1121 2241
Community Answer 4.9 29 What is the molar volume of a gas at standard temperature and pressure? A . 1.0 L B. 2.4 L C. 11.2 L D. 22.4 L
Community Answer What is the molar volume of a gas at standard temperature and pressure? 1. 0 L 2. 4 L 11. 2 L 22. 4 L.
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7013 | https://courses.lumenlearning.com/wm-microeconomics/chapter/the-production-possibilities-frontier/ | Module 2: Choice in a World of Scarcity
The Production Possibilities Frontier
Learning Objectives
Explain the production possibilities frontier
Just as individuals cannot have everything they want and must instead make choices, society as a whole cannot have everything it might want, either. Economists use a model called the production possibilities frontier (PPF) to explain the constraints society faces in deciding what to produce.
As you read this section, you will see parallels between individual choice and societal choice. There are more similarities than differences, so for now focus on the similarities.
While individuals face budget and time constraints, societies face the constraint of limited resources (e.g. labor, land, capital, raw materials, etc.). Because at any given moment, society has limited resources, it follows that there’s a limit to the quantities of goods and services it can produce. In other words, the products are limited because the resources are limited.
Suppose a society desires two products: health care and education. This situation is illustrated by the production possibilities frontier in Figure 1.
Figure 1. A production possibilities frontier showing health care and education.
Health care is shown on the vertical (or y) axis, and education is shown on the horizontal (or x) axis. Where does the PPF come from? It comes from the production processes for producing the two goods, and the limited amounts of resources available to use for that purpose. For example, suppose one teacher can teach 25 students in school. If society has a total of 10 teachers, education can be provided to a maximum of 250 students. We would say one teacher could “produce” 25 students worth of education using the education processes available.
Suppose a society allocated all of its resources to producing health care. That is certainly one possible way of allocating a society’s resources, but it would mean there would be no resources left for education. This choice is shown in Figure 1 at point A. Similarly, the society could allocate all of its resources to producing education, and none to producing healthcare, as shown at point F. Alternatively, the society could choose to produce any combination of health care and education shown on the production possibilities frontier. In effect, the production possibilities frontier plays the same role for society as the budget constraint plays for an individual consumer. Society can choose any combination of the two goods on or inside the PPF. However, it does not have enough resources to produce outside the PPF.
Most importantly, the production possibilities frontier clearly shows the tradeoff between healthcare and education. Suppose society has chosen to operate at point B, and it’s considering producing more education. Because the PPF is downward sloping from left to right, the only way society can obtain more education is by giving up some health care. That’s the trade-off this society faces. Suppose it considers moving from point B to point C. What would be the opportunity cost for the additional education? The opportunity cost would be the health care that society has to give up.
Do you remember Charlie choosing combinations of burgers and bus tickets within his budget constraint? In effect, the production possibilities frontier plays the same role for society as the budget constraint plays for Charlie. Society can choose any combination of the two goods on or inside the PPF, but it doesn’t have enough resources to produce outside the PPF. Just as with Charlie’s budget constraint, the opportunity cost is shown by the slope of the production possibilities frontier.
Try It
Watch It
Take another look at the production possibilities frontier in this video about the imaginary “Econ Isle.”
Differences between a Budget Constraint and a PPF
We’re now ready to address the differences between society’s PPF and an individual’s budget constraint.
A budget constraint shows the different combinations of goods and services a consumer can purchase with their fixed budget. A production possibilities frontier shows the possible combinations of goods and services that a society can produce with its limited resources. The first difference between a budget constraint and a production possibilities frontier is that the PPF, because it’s looking at societal choice, is going to have much larger numbers on the axes than those on an individual’s budget constraint.
The most important difference between the two graphs, though, is that a budget constraint is a straight line, while a production possibilities curve is typically bowed outwards, i.e. concave towards the origin. The reason for this difference is pretty simple: the slope of a budget line is defined as the ratio of the prices of the two goods or services. No matter how many of each good or service a consumer buys, the prices stay the same. By contrast, the slope of a PPF is the cost to society of producing one good or service relative to the other good or service. When society reallocates resources from one product to another, the relative costs change, which means the slope of the PPF does also. Let’s dig into this.
To understand why the PPF is curved, start by considering point A at the top left-hand side of the PPF. Considering the situation in Figure 1 (shown again below), suppose we have only two types of resources: doctors and teachers. At point A, all available resources (i.e. all the doctors and all the teachers) are devoted to providing health care and none is left for education. Say the doctors are practicing medicine and the teachers are helping out as best they can. This situation would be extreme and even ridiculous. For example, children are seeing a doctor every day, whether they’re sick or not, but not attending school. People are having cosmetic surgery on every part of their bodies, but no high school or college education exists!
Now imagine that some of these resources are diverted from health care to education, so that the economy is at point B instead of point A. What type of resources are going to move to producing education? The doctors are good at medicine, but they’re not particularly good at teaching, so it doesn’t make sense for them to switch. The teachers, though, are good at education, and not very good at healthcare. After all, that’s not what they were trained for. So it makes sense for teachers to be reallocated from healthcare to education. And when they move, the society doesn’t lose much healthcare, because the teachers weren’t very good at that. But the amount of education gained is great, because that’s what teachers are trained for. What this means is that from point A to B, the decrease in healthcare is small, while the gain in education is large. Graphically, the rise is small and the run is large so the slope (which is the ratio of rise over run) is flat. In other words, the opportunity cost of education in terms of healthcare is low.
Figure 1 (shown again). A production possibilities frontier showing health care and education.
If we started at the other end of the PPF at point F and moved to point D, we would be moving doctors from teaching to healthcare with the result that the gain in healthcare would be large while the loss in education would be small (the same logic we used above). In short, the slope of the PPF from point F to D would be steep, and the opportunity cost of education in terms of healthcare would be high.
More generally, as society produces more and more of some good or service, the cost of production grows larger and larger relative to the cost of producing other goods or services. Thus, the slope of a PPF starts flat and becomes increasingly steeper. In the real world, of course, we have more than two goods and services, and we have more resources than just labor, but the general rule still holds.
There’s another way to think about this. For consumers, there is only one scarce resource: budget dollars. As we choose more of one good and less of another, we are simply spending dollars on different items, but every dollar is worth the same in purchasing any item. For society, there are many scarce resources. In our simple example above, there were two different resources: doctors and teachers, and each resource is better at one job than at the other. In other words, each resource is not worth the same at producing different products. The general rule is when one is allocating only a single scarce resource, the trade-off (e.g. budget line) will be constant, but when there is more than one scarce resources, the trade-off will be increasingly costly (e.g. the PPF).
Watch It
Watch this video to see another explanation as to why the PPF is curved.
Law of Diminishing Returns and the Curved Shape of the PPF
Figure 2. If you’ve ever pulled an all-nighter, you’re probably familiar with the law of diminishing returns: as the night wears on and you get tired, every additional hour you study is a little less productive than the one before.
The lesson is not that society is likely to make an extreme choice like devoting no resources to education at point A or no resources to health at point F. Instead, the lesson is that the gains from committing additional marginal resources to education depend on how much is already being spent. If, on the one hand, very few resources are currently committed to education, then an increase in resources used can bring relatively large gains. On the other hand, if a large number of resources are already committed to education, then committing additional resources will bring relatively smaller gains.
This pattern is so common that it has been given a name: the law of diminishing returns. This law asserts that as additional increments of resources are devoted to a certain purpose, the marginal benefit from those additional increments will decline. For example, after not spending much at all on crime reduction, when a government spends a certain amount more, the gains in crime reduction could be relatively large. But additional increases after that typically cause relatively smaller reductions in crime, and paying for enough police and security to reduce crime to zero would be tremendously expensive.
The curve of the production possibilities frontier shows that as additional resources are added to education, moving from left to right along the horizontal axis, the initial gains are fairly large, but those gains gradually diminish. Similarly, as additional resources are added to health care, moving from bottom to top on the vertical axis, the initial gains are fairly large but again gradually diminish. In this way, the law of diminishing returns produces the outward-bending shape of the production possibilities frontier.
Try It
Glossary
law of diminishing returns:
: as additional increments of resources are devoted to a certain purpose, the marginal benefit from those additional increments will decline
production possibilities frontier (or curve):
: a diagram that shows the productively efficient combinations of two products that an economy can produce given the resources it has available
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The Production Possibilities Frontier and Social Choices. Authored by: OpenStax College. Located at: License: CC BY: Attribution
Hard Studies. Authored by: Thomas Anderson. Located at: License: CC BY: Attribution
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The Production Possibilities Frontier, Part 3 The Economic Lowdown Video Series. Authored by: Federal Reserve Bank of St. Louis. Located at: License: Other. License Terms: Standard YouTube License
The Production Possibilities Frontier, Part 1 The Economic Lowdown Video Series. Authored by: Federal Reserve Bank of St. Louis. Located at: License: Other. License Terms: Standard YouTube License
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CC licensed content, Original
Revision and adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
CC licensed content, Shared previously
The Production Possibilities Frontier and Social Choices. Authored by: OpenStax College. Located at: License: CC BY: Attribution
Hard Studies. Authored by: Thomas Anderson. Located at: License: CC BY: Attribution
All rights reserved content
The Production Possibilities Frontier, Part 3 The Economic Lowdown Video Series. Authored by: Federal Reserve Bank of St. Louis. Located at: License: Other. License Terms: Standard YouTube License
The Production Possibilities Frontier, Part 1 The Economic Lowdown Video Series. Authored by: Federal Reserve Bank of St. Louis. Located at: License: Other. License Terms: Standard YouTube License |
7014 | https://www.quora.com/How-do-you-find-the-directivity-of-a-half-wave-dipole | Something went wrong. Wait a moment and try again.
Half Wave
Radio Technology
Dipole Antenna
Antenna Theory
Radio Wave Propagation
Half Wavelength Dipole An...
5
How do you find the directivity of a half-wave dipole?
Chuck McCown
I design antennas.
·
Author has 1.9K answers and 2.5M answer views
·
3y
I would say, to actually “find” the directivity of any real physical antenna, one of the better ways is to test it, normally on a test range. Nearfield ranges are fairly compact but a far field range can be constructed without too much effort with a minimum of test equipment. A half wave dipole has been pretty much exhaustively characterized both in theory and in actual practice over the years. Assuming the antenna is efficient, directivity and gain are equal. Half Wave Dipoles are commonly stated to have a gain of 2.14 or 2.15 dB.
If you want to “find” it in more of a theoretical sense, and do
I would say, to actually “find” the directivity of any real physical antenna, one of the better ways is to test it, normally on a test range. Nearfield ranges are fairly compact but a far field range can be constructed without too much effort with a minimum of test equipment. A half wave dipole has been pretty much exhaustively characterized both in theory and in actual practice over the years. Assuming the antenna is efficient, directivity and gain are equal. Half Wave Dipoles are commonly stated to have a gain of 2.14 or 2.15 dB.
If you want to “find” it in more of a theoretical sense, and don’t mind calculus or Maxwell you can find the formula for directivity or you can derive it from first principles if you are a glutton for punishment. Just Google Antenna Directivity Equation.
If you want to “find” it without putting pencil to paper or physically testing the antenna there are many computational methods, I would start with NEC (Numerical Electromagnetics Code ) programs that are in the public domain, or a commercial product like HFSS, Comsol etc. They all have a learning curve.
And you could just Google dipole calculator and probably find many online calculation programs that will aid in the design and characterization. I guess it all depends on your definition of “find”.
Perhaps look up the directivity on the spec sheet of the antenna in question.
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What is the difference between a half-wave dipole and a full-wave dipole?
Sean Hartman
Studied Electrical Engineering
·
2y
If you want to try and calculate it, the most general half wave radiation intensity, U, can be approximated by sin^3(theta) and directivity, D, is 4piU / P_rad, where P_rad is double integral [Usin(theta)d(theta)d(phi)].
phi evaluates to 2pi and the sin^4(theta)d(theta) evaluates to 3pi/8.
Plug it all into D and set it equal to 1/2.
Then with some algebra you’ll find, theta = sin^(-1)(cube root(1.5pi/16)) which comes out to about 41.7 degrees, and twice that is the HPBW. (Again, radiation intensity was approximate so there is a small error)
Finally, you can approximate with McDonald's or Pozar’
If you want to try and calculate it, the most general half wave radiation intensity, U, can be approximated by sin^3(theta) and directivity, D, is 4piU / P_rad, where P_rad is double integral [Usin(theta)d(theta)d(phi)].
phi evaluates to 2pi and the sin^4(theta)d(theta) evaluates to 3pi/8.
Plug it all into D and set it equal to 1/2.
Then with some algebra you’ll find, theta = sin^(-1)(cube root(1.5pi/16)) which comes out to about 41.7 degrees, and twice that is the HPBW. (Again, radiation intensity was approximate so there is a small error)
Finally, you can approximate with McDonald's or Pozar’s formula for directivity of omnidirectional antennas and get about 1.64 or 2.15dB
Clive V
Former 45 Years Electrical and Electronics
·
Author has 1.2K answers and 959K answer views
·
3y
If the dipole has reflector and director elements, the lobe is directed at the front, in the direction of the director elements.
This is the direction where reception signal and transmitted signal is greatest.
If there are no extra elements -just the dipole, the (horizontal) dipole elements are perpendicular to the signal, both front and back.
Iif the dipole is vertical, without extra elements, the pattern is unidirectional, along a terrestrial plane.
Ian Lang
Electronic Engineer BTEC L3
·
Upvoted by
Stanislovas Zacharovas
, PhD Physics, Vilnius, Lithuania (1990) · Author has 7.6K answers and 107.5M answer views
·
4y
Related
What is a half wave?
It’s what you do when you cant be bothered to lift your arm up to say goodbye.
Actually, no. I’m assuming you are talking about rectifiers, here. This is a half-wave:
You can see straight away what the advantage is. They are really easy and cheap to make. The problem is you’re wasting half the input. These days you’ll rarely say a half-wave. If you look at that output it’s rough. Most will go for a bridge:
You can buy bridges in a package ready made. I buy them for £3:00 a go and they last forever. You can get some at 75p a go but I don’t trust them at that price.
Once you’ve done the rectifying y
It’s what you do when you cant be bothered to lift your arm up to say goodbye.
Actually, no. I’m assuming you are talking about rectifiers, here. This is a half-wave:
You can see straight away what the advantage is. They are really easy and cheap to make. The problem is you’re wasting half the input. These days you’ll rarely say a half-wave. If you look at that output it’s rough. Most will go for a bridge:
You can buy bridges in a package ready made. I buy them for £3:00 a go and they last forever. You can get some at 75p a go but I don’t trust them at that price.
Once you’ve done the rectifying you can put in a smoothing capacitor. It goes like this:
You can see that the bridge uses both halves of the input waveform but it inverts the bottom half. This is much easier to smooth out and you get a tiny ripple.
Here’s a little film; from about fifteen minutes onward I’m banging on about half waves.
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Lisa Dawson
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Related questions
What is an expression for the far field component of a half wave dipole of an antenna?
Is the standing wave on a resonant half wave dipole a common mode or differential mode wave?
What is a half-wave dipole?
In which direction does a half-wave dipole antenna radiation have the strongest signal?
How do I derive an expression for the gain of a half wave dipole?
Assistant
Bot
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1y
The directivity of a half-wave dipole antenna can be calculated using the formula for directivity [math]D[/math], which is defined as the ratio of the maximum radiation intensity [math]U_{\text{max}}[/math] in a given direction to the average radiation intensity [math]U_{\text{avg}}[/math] over all directions.
For a half-wave dipole antenna, the directivity can be expressed as:
[math]D = \frac{U_{\text{max}}}{U_{\text{avg}}}[/math]
Steps to Calculate Directivity:
Maximum Radiation Intensity: For a half-wave dipole antenna, the maximum radiation intensity occurs in the broadside direction (perpendicular to the antenna) and is given by:
[math]U_{\text{[/math]
The directivity of a half-wave dipole antenna can be calculated using the formula for directivity [math]D[/math], which is defined as the ratio of the maximum radiation intensity [math]U_{\text{max}}[/math] in a given direction to the average radiation intensity [math]U_{\text{avg}}[/math] over all directions.
For a half-wave dipole antenna, the directivity can be expressed as:
[math]D = \frac{U_{\text{max}}}{U_{\text{avg}}}[/math]
Steps to Calculate Directivity:
Maximum Radiation Intensity: For a half-wave dipole antenna, the maximum radiation intensity occurs in the broadside direction (perpendicular to the antenna) and is given by:
[math]U_{\text{max}} = \frac{P_{\text{rad}}}{\Omega}[/math]
where [math]P_{\text{rad}}[/math] is the total power radiated by the antenna, and [math]\Omega[/math] is the solid angle over which the power is radiated.
Average Radiation Intensity: The average radiation intensity [math]U_{\text{avg}}[/math] can be calculated as:
[math]U_{\text{avg}} = \frac{P_{\text{rad}}}{4\pi}[/math]
This assumes that the radiation is isotropic, which is a simplification but useful for calculating directivity.
Solid Angle for Half-Wave Dipole: The solid angle [math]\Omega[/math] for a half-wave dipole is [math]2\pi[/math] steradians, since it radiates in a doughnut shape with no radiation along the axis of the dipole.
Calculating Directivity:
Substitute the values into the directivity formula:
[math]D = \frac{U_{\text{max}}}{U_{\text{avg}}} = \frac{U_{\text{max}}}{\frac{P_{\text{rad}}}{4\pi}} = \frac{P_{\text{rad}} \cdot 4\pi}{P_{\text{rad}} \cdot \Omega}[/math]
Since [math]\Omega = 2\pi[/math]:
[math]D = \frac{4\pi}{2\pi} = 2[/math]
Conclusion:
The directivity of a half-wave dipole antenna is approximately 1.64 (or 2.15 dBi), meaning it radiates more power in certain directions compared to an isotropic radiator. This is a crucial characteristic in antenna design and analysis, as it indicates how effectively the antenna can direct its radiated energy.
Grant Ryan Swan
live sound engineer, bicycle and classic car enthusiast, ex cylinder manager
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Author has 533 answers and 2.2M answer views
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Updated 8y
Related
How can I calculate the directivity of a finite length dipole?
The length should have nothing to do with the directivity of an antenna.
The polarity (vertical or horizontal) is the single largest factor.
In a perfect world, a single element vertical antenna would radiate equally 360 degrees around itself, with no radiation from either end.
If you turn it horizontal that same 360 degrees is now focused 180 degrees forward and 180 degrees backward along the length of the antenna, again with no radiation from the ends.
If you place a reflector behind the horizontal, you block the signals to and from that direction, but increase them from the opposite. The amount
The length should have nothing to do with the directivity of an antenna.
The polarity (vertical or horizontal) is the single largest factor.
In a perfect world, a single element vertical antenna would radiate equally 360 degrees around itself, with no radiation from either end.
If you turn it horizontal that same 360 degrees is now focused 180 degrees forward and 180 degrees backward along the length of the antenna, again with no radiation from the ends.
If you place a reflector behind the horizontal, you block the signals to and from that direction, but increase them from the opposite. The amount of change between the two is comparable.
A director(s) in front of the radiator will increase this effect.
This is but a rough guesstimate, but the original 360 degrees is still there, it has just changed shape.
Reflectors and radiators can also be used on verticals, but going horizontal is the best bang for the buck for getting that first 180 degree split.
Remember that gain in one direction is achieved by loss in another.
Christopher
Telecom technician. Been doing telecom since 1987.
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Author has 277 answers and 29.2M answer views
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Updated Aug 26
Related
Why are dipole antennas half-wave long?
There is no such thing as a “full-wave” antenna. A full wave antenna would not be easy to impedance match. Also, it would radiate in unpredictable ways, making it worse than useless.
The reason we use a half-wave antenna and not a full-wave, is because of the definition of a wavelength. In one wavelength (or one cycle), it goes positive and negative. The antenna has to resonate twice to make one wavelength.
Any antenna that is designed to be efficient will be 1/2 wavelength, which is resonant at the desired frequency. This is what is most important.
A quarter-wave antenna (very common) depends on
There is no such thing as a “full-wave” antenna. A full wave antenna would not be easy to impedance match. Also, it would radiate in unpredictable ways, making it worse than useless.
The reason we use a half-wave antenna and not a full-wave, is because of the definition of a wavelength. In one wavelength (or one cycle), it goes positive and negative. The antenna has to resonate twice to make one wavelength.
Any antenna that is designed to be efficient will be 1/2 wavelength, which is resonant at the desired frequency. This is what is most important.
A quarter-wave antenna (very common) depends on a ground plane to serve as the other half, because you still need 1/2 wave to be resonant. If no ground plain is available, then a special “No-Ground-Plane” base is used that contains a long enough coil to at least give the electric charge someplace to go while the exposed element does all the work. This element is usually half-wave. Five-eights wave is also used to provide a little more gain, but requires special tuning to get the impedance right.
Directional antennas start with a half-wave antenna, combined with a reflector and directors, which reflect and reinforce signal in a desired direction. This is a Yagi-Uda antenna.
Ready for something deep? We all know that antenna length is determined by the speed of electricity in a wire — about 90% the speed of light. The speed of light is determined by the maximum speed limit of the universe. Light and electro-magnetic fields, which have no mass, travel at this speed (in a vacuum). Therefore, antenna design is determined by the speed of causality, the speed limit of the universe.
So look at your car’s FM antenna and think, “This stick of metal is this length because of how long it takes for an electric charge to travel up and down its length at 100MHz.
-Chris
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Samuel S. Cummings
KA9PAZ (General class amateur radio license)
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Author has 66 answers and 49K answer views
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5y
Related
What is a quarter-wave dipole?
It is a dipole where each element is 1/4λ of the frequency its cut for.
For instance, let’s assume its a dipole cut for the 2 meter band. The center frequency for 2m is 146MHz.
The formula for calculating lambda (the wavelength) is
Let:
λ (Lambda) = Wavelength in meters
c = Speed of Light (Because we’re calculating in MHz and NOT in Hz, this is reduced from 300,000,000 to 300. This isn’t EXACTLY right, but for anything we're doing, this is close enough usually).
f = Frequency (146MHz, in this case)
Where the fomula to calculate is λ=c/f
So, the wavelength of 146MHz is 2.05m [let’s be as accurate as we
It is a dipole where each element is 1/4λ of the frequency its cut for.
For instance, let’s assume its a dipole cut for the 2 meter band. The center frequency for 2m is 146MHz.
The formula for calculating lambda (the wavelength) is
Let:
λ (Lambda) = Wavelength in meters
c = Speed of Light (Because we’re calculating in MHz and NOT in Hz, this is reduced from 300,000,000 to 300. This isn’t EXACTLY right, but for anything we're doing, this is close enough usually).
f = Frequency (146MHz, in this case)
Where the fomula to calculate is λ=c/f
So, the wavelength of 146MHz is 2.05m [let’s be as accurate as we can].
A dipole where the elements would be 2.05m would give us 1λ; and an isotropic radiation field.
Typically, at this frequency, the elements are lengthened or shortened to manipulate the radiation field, in essence, to point it one way, or another. To go from 1λ to 1/4λ, we simply divide the wavelength by 1/4.
Thus: λ/4 = 0.5125 meters, or 51.25 centimeters.
Hope this helps!
DE KA9PAZ AR
Charles Tom Rauch
RF and Analog System Designer (1983–present)
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Author has 919 answers and 3.1M answer views
·
2y
Related
Is a 1:1 balun required for a half-wave dipole of 10 meter antenna?
In some cases it is required. In some cases it is not.
The need for a balun is entirely dependent on the common mode impedances and voltages present at a balanced (antenna) to unbalanced (coaxial cable) transition in a particular system.
For example, if we fed a perfect balanced dipole antenna with an unbalanced coaxial feedline 1/4 wavelength long from a shield ground point, with the feedline hanging clear of other things by at least several inches, the system would be perfectly fine with no balun at all.
Here is an example of a real case, where coax feeds a dipole without a balun and the system
In some cases it is required. In some cases it is not.
The need for a balun is entirely dependent on the common mode impedances and voltages present at a balanced (antenna) to unbalanced (coaxial cable) transition in a particular system.
For example, if we fed a perfect balanced dipole antenna with an unbalanced coaxial feedline 1/4 wavelength long from a shield ground point, with the feedline hanging clear of other things by at least several inches, the system would be perfectly fine with no balun at all.
Here is an example of a real case, where coax feeds a dipole without a balun and the system has good balance and minimal common mode:
Here is an actual picture of that almost perfectly balanced antenna, 1/4 wave from the dipole center to the shield’s ground:
If the dipole’s coaxial feedline was 1/2 wavelength long between the dipole and the shield’s ground point, the system would have horrible pattern distortion, large antenna current skewing, and a great deal of common mode exciting the unbalanced feedline shield. A system like this would badly need a balun. Here is an example of that case in an actual real world antenna:
This is a picture of that antenna way up in the clear by itself, but with a 1/2 wave long coaxial feeder to earth where the shield first grounds:
This is the current percentages of this same high dipole antenna with the same 1/2 wave feeder but now with a modest (500 ohm) common mode impedance balun added:
Now with a simple modest common mode impedance balun, the system is for all purposes perfectly balanced.
The need for a balun has nothing to do with SWR and little to do with what is around the antenna. It is more about the feedline common mode impedance at the antenna attachment point.
Most people get this wrong. I would venture a guess that very few people actually understand antennas, feedlines, and common mode well enough to get this correct.
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Andrew Maicke
A few physics courses, and some other physics books
·
8y
Related
How do I calculate the half power beam width of a half wave dipole antenna?
You’re asking the wrong question. You don’t ‘calculate’ the half-power beam width of a half wave dipole antenna; you just know it’s 78 degrees, or 1.36 radians. It’s a constant, and doesn’t change no matter what the size or frequency of your antenna is (as long as the length stays much greater than the diameter) because you’ve established the relationship between the two as ‘half wave’. In all cases your antenna length is one half of the wavelength [math]\lambda[/math] for whatever frequency you’re using. This ratio in essence sets the half power beamwidth.
Instead, what you might be interested in is derivi
You’re asking the wrong question. You don’t ‘calculate’ the half-power beam width of a half wave dipole antenna; you just know it’s 78 degrees, or 1.36 radians. It’s a constant, and doesn’t change no matter what the size or frequency of your antenna is (as long as the length stays much greater than the diameter) because you’ve established the relationship between the two as ‘half wave’. In all cases your antenna length is one half of the wavelength [math]\lambda[/math] for whatever frequency you’re using. This ratio in essence sets the half power beamwidth.
Instead, what you might be interested in is deriving this value. If that’s what you intend, buckle up, because I’m starting from the beginning.
To begin, we take the magnetic vector potential:
[math]\displaystyle \mathbf A = \frac{\mu}{4 \pi} \int_C \mathbf I \frac{e^{-jkR}}{R} \, dl'[/math]
This might seem like an odd place to start, but the magnetic vector potential is actually fairly straightforward to derive from Maxwell’s equations and certainly not only used in antenna analysis, though it gets a lot of mileage there. The formula above is general, and not just true for antennas, so it’s not really necessary to show it’s derivation in this question, and instead serves as a good starting place for us.
The equation for the magnetic vector potential essentially tells you to integrate the current [math]\mathbf I[/math] on the dipole over the entire surface of the antenna [math]C[/math] by taking small surface elements [math]dl'[/math] (the [math]'[/math] denotes that it is on the antenna). I say surface, but really we will approximate the dipole antenna as a very thin wire in this case, and so our integration really is just a line going from the bottom of the dipole to the top. We do this for convenience, as it would be incredibly difficult to create nice closed-form solutions without this approximation. We evaluate the integral at a distance [math]R[/math] from the antenna, where [math]R[/math] is the distance from our observation point to each surface element [math]dl'[/math]. [math]k = \omega \sqrt{\mu \epsilon}[/math] is the wavenumber of the medium the antenna is in, and depends on frequency [math]\omega = 2 \pi f = 2 \pi \frac{c}{\lambda}[/math].
If we think of the total length of the dipole as [math]l[/math] and place its center at [math]z=0[/math], our bounds of integration become [math]-l/2[/math] and [math]+l/2[/math]. For now, let’s take the case when the dipole is very small in terms of wavelength, holding off on the half wavelength part for the moment. This allows us to approximate the current on the wire as constant, ie
[math]\mathbf I = I_0 \hat z[/math]
The wire is too small for current to be flowing in any direction except up and down it, and is too small to vary over that length. Thus we can pull the current outside the integral above. Further, the distance of our observation [math]R[/math] essentially does not change along the antenna either, since each [math]dl'[/math] is very close to each other in comparison to how far away our observation point is. So all terms with [math]R[/math] come outside as well. We can now write the magnetic vector potential as
[math]\displaystyle \mathbf A = \hat z \frac{\mu I_0}{4 \pi R} e^{-jkR} \int_{-l/2}^{+l/2} \, dl' = \hat z \frac{\mu I_0 l}{4 \pi R} e^{-jkR}[/math]
At this point, we have been working in rectangular coordinates, but it will be better if we switch to spherical. Doing a coordinate transformation, we wind up with the result
[math]\displaystyle \mathbf A = \hat \rho \frac{\mu I_0 l}{4 \pi R} e^{-jkR} cos(\theta) + \hat \theta \frac{\mu I_0 l}{4 \pi R} e^{-jkR} sin(\theta)[/math]
We can use this to calculate the electric field [math]\mathbf E[/math] radiated by the antenna, where
[math]\displaystyle \mathbf E = -j \omega \mathbf A - j \frac{1}{\omega \mu \epsilon} \nabla (\nabla \cdot \mathbf A)[/math]
This equation comes around when we derive [math]\mathbf A[/math] in the first place, which as I noted we skipped here. Just trust me on this one.
Carrying out the equation above, we find
[math]\displaystyle E_r = \eta \frac{I_0 l cos(\theta)}{2 \pi R^2} \left( 1 + \frac{1}{jkR} \right) e^{-jkR}[/math]
and
[math]\displaystyle E_{\theta} = j \eta \frac{I_0 k l sin(\theta)}{4 \pi R} \left( 1 + \frac{1}{jkR} - \frac{1}{(kR)^2} \right) e^{-jkR}[/math]
as the only two components of the [math]\mathbf E[/math] field in spherical coordinates ([math]E_{\phi} = 0[/math]).
Now, we make sure that we are very far away from this antenna in terms of wavelengths, so that all terms with a [math]\frac{1}{R^2}[/math] or a [math]\frac{1}{R^3}[/math] term go to zero (since [math]R[/math] is getting very big). This leaves us with the much simpler
[math]\displaystyle E_{\theta} = j \eta \frac{I_0 k l sin(\theta)}{4 \pi R} e^{-jkR}[/math]
Keep this in mind.
We switch our focus for a minute, and bring our antenna to the full half wavelength size (remember: before we had said it was very small in terms of wavelength). Now the current isn’t as constant as we’d like. Instead, it varies along the length of the wire, though we’ll still assume it moves only in the [math]\hat z[/math] direction as before since the diameter will still be much smaller than the length of the antenna (that’s what makes it a dipole, after all). Fortunately, we have a few rules we know about how this current should vary. First, it needs to go to zero at the edges of the antenna. This much should be obvious, since the current can’t fly off of the dipole! Second, we want the pattern of the current to be symmetrical around the center of the antenna. This allows it to be ‘center-fed’, meaning you plug your wires into the center of the antenna to use it, which is the most common type of antenna connection for dipoles. With these two pieces of information in mind, we can come up with the following approximation
[math]\displaystyle \mathbf I = \left{ \begin{align} \hat z I_0 sin(k \left( \frac{l}{2} - z' \right), 0 \le z' \le \frac{l}{2} \ \hat z I_0 sin(k \left( \frac{l}{2} + z' \right), \frac{-l}{2} \le z' \le 0 \end{align} \right.[/math]
You can see that when [math]z' = \pm \frac{l}{2}[/math], the equations go to zero, since [math]sin(0) = 0[/math]. The magnitude of sine also is symmetrical about 0, so both of our requirements are met. [math]I_0[/math] remains to allow us to scale the magnitude of this current up or down, though is just an arbitrary scalar constant.
We can’t approximate [math]R[/math] as nicely as we did before either, as the distances between our observation point and the different segments of the antenna are different enough from each other to matter. Though they don’t matter a ton, so we will still approximate, and our approximation is now
[math]R = r - z' cos( \theta )[/math]
when [math]R[/math] is a phase term, and
[math]R = r[/math]
when [math]R[/math] is an amplitude term. We make this approximation because tiny differences in the phase matter, while tiny differences in the amplitude don’t nearly as much. Now, [math]r[/math] is the distance from our observation to the origin, where [math]R[/math] had been the distance between observation and the antenna segments before.
We now think of our half wavelength antenna as if it were a series of very small antennas all stacked on top of each other. This gives us a bunch of small segments, each causing the [math]E_{\theta}[/math] I told you to remember earlier. We integrate this series of segments along the z axis, like so:
[math]\displaystyle {E_{\theta}}^{halfwave} = \int_{-l/2}^{+l/2} E_{\theta} \, dz' = j \eta \frac{k e^{-jkr}}{4 \pi r} sin( \theta ) \int_{-l/2}^{+l/2} \mathbf I e^{jkz' cos( \theta )} \, dz'[/math]
which leads to
[math]\displaystyle {E_{\theta}}^{halfwave} = j \frac{ \eta I_0 e^{-jkr}}{2 \pi r} \left( \frac{cos(\frac{kl}{2} cos( \theta ) ) - cos(\frac{kl}{2})}{sin( \theta )} \right)[/math]
We now have an equation for the [math]\mathbf E[/math] field which is solely a function of the angle [math]\theta[/math] away from the top of the antenna.
Our motivation for finding [math]\mathbf E[/math] is to use it to calculate the power radiated by the antenna [math]\mathbf S[/math] . We can do this with the equation
[math]\displaystyle \mathbf S(\theta) = \hat r \frac{1}{2 \eta} {|{E_{\theta}}^{halfwave}|}^2[/math]
The power will have a maximum value [math]S^{max}[/math] when [math]\theta = 90[/math] degrees, which you can see from [math]sin( \theta )[/math] being in the denominator of the [math]{E_{\theta}}^{halfwave}[/math] equation above.
Obviously, the ‘half power point’ of an antenna is the direction where the power radiated by an antenna’s radiation pattern goes down to half (thus it is also often called the ‘3-dB point’). The ‘half power beamwidth’ is then how wide a range you have where the power of the radiation pattern is at least half as strong as its maximum. In essence, we’re looking for the angle [math]\theta[/math] where
[math]\displaystyle S^{half power}( \theta) = .5 S^{max}[/math]
In our case, we’re looking at the case when the length [math]l = \frac{\lambda}{2}[/math]. If you carry out the calculations, you can find that the half power points should happen at [math]\theta = 51[/math] and [math]\theta = 129[/math] degrees. The difference between these two values?
78 degrees.
In this analysis, we didn’t set a size in meters, or inches, or anything like that for the antenna. We also didn’t pick a frequency. By using a half wavelength dipole, we require [math]l = \frac{\lambda}{2}[/math]. Thus size and frequency values end up canceling in the equations, and we know the half power beamwidth should be 78 degrees.
In practice of course, this will differ depending on antenna construction, as we had to make a few assumptions to arrive at a nice solution. But in all cases, if the half power dipole is made well, it should be very near this value.
You can also use the above to calculate half power beamwidths for other sizes of antennas; the analysis isn’t limited to the half wave dipole case.
Or, as I’d suggest, look it up somewhere else if you can.
Reference:
Balanis, Antenna Theory, 3rd Ed
David Newberry
Former At&T Technical Instructor at AT&T (company)
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Author has 4.3K answers and 7.2M answer views
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Updated 2y
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How do you handle more standing waves in a dipole antenna (antenna, dipole, electronics)?
This is an odd question, since you know you have a high SWR and don’t understand, how to correct the problem…
Anyway, SWR (Standing Wave Ratio) has to do with impedance mismatch…between your radio output and the receiving Tuned antenna. Meaning, if you have an HF radio and want to work in the HF range, and you’ve attached your radio to something designed around a CB radio…2 totally different animals.
Anyway, the length of the Dipole, is how the antenna should be tuned for the frequencies you want to use..
But they do make RF antenna tuners to help with impedance mismatches. There is a device
Hi..
This is an odd question, since you know you have a high SWR and don’t understand, how to correct the problem…
Anyway, SWR (Standing Wave Ratio) has to do with impedance mismatch…between your radio output and the receiving Tuned antenna. Meaning, if you have an HF radio and want to work in the HF range, and you’ve attached your radio to something designed around a CB radio…2 totally different animals.
Anyway, the length of the Dipole, is how the antenna should be tuned for the frequencies you want to use..
But they do make RF antenna tuners to help with impedance mismatches. There is a device called a Balun, more accurately, an RF Balun, to help with the impedance mismatches as well…Probably a tuner is a Balun device anyway…
Maybe look at the Amazon website. They seem to sell all kinds of antenna kits, tuner and some Balun units….
If it was me, just do some simple math to determine, the proper antenna length needed to match the frequencies you want to operate in.
I found something in Amazon, that's similar to the Y connection I used to use in the military…just use the correct wire length, and use it the correct direction…I easily contacted stations out of state all the time…
About 25 bucks…but there are all sorts of antenna kits you maybe able to use…to get your SWR issues under control…This is designed around the HF band…3 to 30 Megs…
Good Luck
This is AGAMH, out…
David Zimmerman
Intense interest in Physics due to interest in space flight
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Author has 8.2K answers and 3M answer views
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3y
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Why does an ideal dipole or half-wave antenna not radiate in all directions equally, but rather in one direction only?
Dipole antennae do not radiate and receive in one direction only. They radiate and receive in pattern that is perpendicular to the antenna which gives you a figure eight type pattern when viewed from above. . You may be thinking of a beam antenna with director and reflector elements. A non modified dipole antenna only has the active element with no reflectors or directors.
You can even get a quasi
Dipole antennae do not radiate and receive in one direction only. They radiate and receive in pattern that is perpendicular to the antenna which gives you a figure eight type pattern when viewed from above. . You may be thinking of a beam antenna with director and reflector elements. A non modified dipole antenna only has the active element with no reflectors or directors.
You can even get a quasi omnidirectional pattern by putting the legs of the dipole at 90 degrees to each other. I used that kind of antenna when I was a novice amateur radio operator on the 80 meter band. Luckily my parents had a bit of cleared land which made this possible.
In truth no one knows exactly why antenna work they way they do. We only know that they work i...
Kip Ingram
PhD in Electrical Engineering, The University of Texas at Austin Cockrell School of Engineering (Graduated 1992)
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Author has 20.1K answers and 21.6M answer views
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2y
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How do you plot the directivity of a dipole: D=1.67xsin^3 (theta)? I just started learning about antennas and dipoles.
Do you mean how do you produce the data you need to plot? That’s done by solving the electromagnetic field equations, considering your antenna elements as current sources. But if you mean what kind of tool do you use, gnuplot will do it, except it doesn’t know about antennas - you have to get the data to plot some other way. There are programs specifically for antenna analysis, such as xnec2c (an open source package for Linux). It will let you just describe your antenna (and any metal structures around it that might have an effect) and it will figure the fields out for you, and give you variou
Do you mean how do you produce the data you need to plot? That’s done by solving the electromagnetic field equations, considering your antenna elements as current sources. But if you mean what kind of tool do you use, gnuplot will do it, except it doesn’t know about antennas - you have to get the data to plot some other way. There are programs specifically for antenna analysis, such as xnec2c (an open source package for Linux). It will let you just describe your antenna (and any metal structures around it that might have an effect) and it will figure the fields out for you, and give you various types of plots.
Stay safe and well!
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Problems & Questions
9 Algebra Questions And Practice Problems To Do With Your Middle Schoolers
August 15, 2025 | 2 min read
Beki Christian
Algebra questions involve using letters or symbols to represent unknown values or values that can change. Here you will find \bf{9} algebra questions to test your knowledge and show you the different ways that algebra can be used to solve a problem to find an unknown value or to make generalizations.
Algebra in elementary school
Students as early as kindergarten begin to solve problems with algebraic thinking. Algebraic thinking starts with students understanding the properties of the operations, the relationships between them, and the composition of numbers.
They begin to relate concrete models to abstract expressions and equations, and continue this work throughout elementary school. They also look at patterns and the order of operations that will contribute to their understanding of algebra in later years.
This flexible understanding with the composition and representation of numbers, as well as the operations sets them up for more complex and abstract equations once they get to middle school.
Algebra Questions for Middle School
Here are 9 algebra questions for middle school, covering the key algebra skills and topics for middle school math, including writing expressions, simplifying expressions and using a formula.
Download Free Now!
Algebra in middle school
Algebra encompasses many skills and concepts to help us make sense of the world and solve problems. In middle school, we learn to write basic algebraic expressions, write and solve linear equations, write and solve a system of linear equations, and graph linear equations on the coordinate plane. Later, we further develop these skills which allow us to work with more complex equations such as quadratic equations, polynomial equations, and exponential equations.
How to solve algebraic equations
When you are presented with an algebraic problem, it is important to make sense of the problem. Here are some of the key terms along with what they mean:
Solve the equation – find out the value of the unknown.
Substitute – put the values you have been given into the algebraic expression.
Coefficients – the amount a term has been multiplied by. For example, in the expression 4c+2, \; 4 is the coefficient.
Constants – values which are not variable and stay the same. For example, for 4c+2, \; 2 is the constant.
Binomial – a binomial or binomial expression is one which has two parts, like our example 4c+2.
Simplify – collect together like terms to make the expression or equation look simpler.
Expand – multiply out the expressions inside parentheses.
Factorize – put into parentheses.
In terms of x – rewrite the equation in the form x = …
Remember, when working with algebra, we must still apply the order of operations, PEMDAS. i.e. Parenthesis, Exponents (powers, square roots), Multiplication, Division, Addition, Subtraction.
When working with algebraic expressions and equations we must consider carefully which operations to deal with first.
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Algebra practice questions for middle school
A chocolate bar costs c cents and a drink costs d cents. Write down an expression for the cost of 2 chocolate bars and 2 drinks.
c+d
2c+d
2c+2d
2c-2d
2 chocolate bars would cost 2 lots of c, or 2c, and 2 drinks would cost 2 lots of d, or 2d.
Simplify the expression 4m+5+2m-1.
10m
6m+4
6m+6
6m-4
We need to collect together like terms here so 4m + 2m = 6m and 5-1 = 4 (watch out for the negative).
In middle school, we learn a variety of different algebra techniques to answer algebra questions and to practice problem solving with algebra. These include:
Simplifying algebraic expressions
Expanding brackets and factoring
Forming algebraic equations from word problems
Solving algebraic equations and inequalities
Substituting into expressions
Changing the subject of an equation
Working with real life graphs and straight line graphs
Sequences
Algebra questions for middle school: basic algebra
In this pyramid, you add two adjacent blocks to find the value of the block above.
What expression will be in the top box?
8a-2b
12a-b
12a+5b
10a+b
4. Brian is a window cleaner. He uses the following formula to calculate the amount to charge his customers:
Charge = \$20 + 4n
Where n is the number of windows a house has.
If a house has 7 windows, how much would Brian charge?
\$24
\$67
\$48
\$27
In this question, n is 7 so we can substitute 7 into the formula.
Charge = \$20 + 4 × 7
Charge = \$48
The area of a rectangle is 4x-6.
Which of the following pairs could be the length and width of the rectangle?
4x and 6x
4 and x-6
2 and 2x-3
2x and 2x-3
There are two ways of attempting this question. We know that \text{area of a rectangle } = \text{ length} \times \text{width} so we could multiply each pair together to see which pair makes 4x-6.
\begin{aligned} &4x × 6x \quad \quad\quad\; \;24x^2 \ &4 × (x − 6) \quad \quad \quad4x − 24\ &2 × (2x − 3) \quad \quad \;\;4x − 6\ &2x × (2x − 3) \quad \quad 4x^2 − 6x\ \end{aligned}
Alternatively, if we factorise 4x-6 we get 2(2x-3) meaning the sides could be 2 and 2x-3.
6. The formula for changing degrees Celsius to degrees Fahrenheit is:
F=\cfrac{9C}{5}+32
Rearrange this formula to make C the subject.
C=\cfrac{5(F-32)}{9}
C=\cfrac{5F-32}{9}
C=\cfrac{5F}{9}-32
C=5F-\cfrac{32}{9}
\begin{aligned} F&=\frac{9C}{5}+32 \hspace{3cm} &\text{subtract 32}\\ F-32&=\frac{9C}{5} &\text{multiply by 5}\\ 5(F-32)&=9C &\text{divide by 9}\\ \frac{5(F-32)}{9}&=C \end{aligned}
Algebra questions for middle school: forming and solving equations
Work out the size of the smallest angle.
20^{\circ}
26^{\circ}
8^{\circ}
34^{\circ}
The angles in a triangle add up to 180^{\circ} therefore we can write
4x+2x-10+3x-8=180
Now we have an equation we can solve.
\begin{aligned} 9x-18&=180 \hspace{3cm} &\text{add } 18\ 9x&=198 &\text{divide by } 9\ x&=22^{\circ} \end{aligned}
The angles are :
\begin{aligned} 4\times22&=88^{\circ}\ 2\times22 -10&=34^{\circ}\ 3\times22 -8&=58^{\circ} \end{aligned}
The smallest angle is 34^{\circ} .
Jamie’s dad is 4 times older than Jamie. In 14 years time, Jamie’s dad will be twice the age of Jamie.
What is the sum of Jamie’s age now and Jamie’s dad’s age now?
70
42
22
35
To solve this we need to write an equation.
Let Jamie’s age now be x . Then Jamie’s dad’s age is 4x .
In 14 years time Jamie’s age will be x + 14 and Jamie’s dad’s age will be 4x + 14 .
Since we know Jamie’s dad’s age will be two times Jamie’s age, we can write
4x+14=2(x+14)
Now we have an equation we can solve.
\begin{aligned} 4x+14&=2(x+14) \hspace{3cm} &\text{expand the brackets}\ 4x+14&=2x+28 \hspace{3cm} &\text{subtract 2x}\ 2x+14&=28 \hspace{3cm} &\text{subtract 14}\ 2x&=14 \hspace{3cm} &\text{expand the brackets}\ x&=7 \hspace{3cm} \end{aligned}
Jamie is currently 7 years old meaning his dad is 28 years old. The sum of their ages is 35 .
Note: when algebraic equations contain denominators on either side, we can use the cross multiplication method to help us work out the answer. For example with the following expressions:
\cfrac{a}{b} = \cfrac{c}{d}
This can then become, ad = bc and so on.
Algebra questions for middle school: graphs
Which of the following lines passes through the point (2, 5)?
y=2x-1
y=2x+1
y=4x-2
y=2x+5
At the point (2, 5), \; x is 2 and y is 5. We can check which equation works when we substitute in these values:
\begin{aligned} y&=2x−1 \quad \quad \quad 5=2×2−1 \quad \quad \text{False}\ y&=2x+1 \quad \quad 5=2×2+1 \quad \quad \text{True}\ y&=4x−2 \quad \quad \quad 5=4×2−2 \quad \quad \text{False}\ y&=2x+5 \quad \quad 5=2×2+5 \quad \quad \text{False} \end{aligned}
Looking for more algebra math questions?
Try these:
Ratio questions
Probability questions
Trigonometry questions
Venn diagram questions
Long division questions
Pythagorean theorem questions
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Ratio estimator
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From Wikipedia, the free encyclopedia
Statistical estimator for ratio of means
The ratio estimator is a statistical estimator for the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical so symmetrical tests such as the t test should not be used to generate confidence intervals.
The bias is of the order O(1/n) (see big O notation) so as the sample size (n) increases, the bias will asymptotically approach 0. Therefore, the estimator is approximately unbiased for large sample sizes.
Definition
[edit]
Assume there are two characteristics – x and y – that can be observed for each sampled element in the data set. The ratio R is
The ratio estimate of a value of the y variate (θy) is
where θx is the corresponding value of the x variate. θy is known to be asymptotically normally distributed.
Statistical properties
[edit]
See also: ratio distribution
The sample ratio (r) is estimated from the sample
That the ratio is biased can be shown with Jensen's inequality as follows (assuming independence between and ):
where is the mean of the variate and is the mean of the variate .
Under simple random sampling the bias is of the order O( n−1 ). An upper bound on the relative bias of the estimate is provided by the coefficient of variation (the ratio of the standard deviation to the mean). Under simple random sampling the relative bias is O( n−1/2 ).
Correction of the mean's bias
[edit]
The correction methods, depending on the distributions of the x and y variates, differ in their efficiency making it difficult to recommend an overall best method. Because the estimates of r are biased a corrected version should be used in all subsequent calculations.
A correction of the bias accurate to the first order is[citation needed]
where mx is the mean of the variate x and sxy is the covariance between x and y.
To simplify the notation sxy will be used subsequently to denote the covariance between the variates x and y.
Another estimator based on the Taylor expansion is
where n is the sample size, N is the population size, mx is the mean of the x variate and sx2 and sy2 are the sample variances of the x and y variates respectively.
A computationally simpler but slightly less accurate version of this estimator is
where N is the population size, n is the sample size, mx is the mean of the x variate and sx2 and sy2 are the sample variances of the x and y variates respectively. These versions differ only in the factor in the denominator (N - 1). For a large N the difference is negligible.
If x and y are unitless counts with Poisson distribution a second-order correction is
Other methods of bias correction have also been proposed. To simplify the notation the following variables will be used
Pascual's estimator:
Beale's estimator:
Tin's estimator:
Sahoo's estimator:
Sahoo has also proposed a number of additional estimators:
If x and y are unitless counts with Poisson distribution and mx and my are both greater than 10, then the following approximation is correct to order O( n−3 ).
An asymptotically correct estimator is
Jackknife estimation
[edit]
A jackknife estimate of the ratio is less biased than the naive form. A jackknife estimator of the ratio is
where n is the size of the sample and the ri are estimated with the omission of one pair of variates at a time.
An alternative method is to divide the sample into g groups each of size p with n = pg. Let ri be the estimate of the ith group. Then the estimator
where is the mean of the ratios rg of the g groups, has a bias of at most O( n−2 ).
Other estimators based on the division of the sample into g groups are:
where is the mean of the ratios rg of the g groups and
where ri' is the value of the sample ratio with the ith group omitted.
Other methods of estimation
[edit]
Other methods of estimating a ratio estimator include maximum likelihood and bootstrapping.
Estimate of total
[edit]
The estimated total of the y variate ( τy ) is
where ( τx ) is the total of the x variate.
Variance estimates
[edit]
The variance of the sample ratio is approximately:
where sx2 and sy2 are the variances of the x and y variates respectively, mx and my are the means of the x and y variates respectively and sxy is the covariance of x and y.
Although the approximate variance estimator of the ratio given below is biased, if the sample size is large, the bias in this estimator is negligible.
where N is the population size, n is the sample size and mx is the mean of the x variate.
Another estimator of the variance based on the Taylor expansion is
where n is the sample size and N is the population size and sxy is the covariance of x and y.
An estimate accurate to O( n−2 ) is
If the probability distribution is Poissonian, an estimator accurate to O( n−3 ) is
A jackknife estimator of the variance is
where ri is the ratio with the ith pair of variates omitted and rJ is the jackknife estimate of the ratio.
Variance of total
[edit]
The variance of the estimated total is
Variance of mean
[edit]
The variance of the estimated mean of the y variate is
where mx is the mean of the x variate, sx2 and sy2 are the sample variances of the x and y variates respectively and sxy is the covariance of x and y.
Skewness
[edit]
The skewness and the kurtosis of the ratio depend on the distributions of the x and y variates. Estimates have been made of these parameters for normally distributed x and y variates but for other distributions no expressions have yet been derived. It has been found that in general ratio variables are skewed to the right, are leptokurtic and their nonnormality is increased when magnitude of the denominator's coefficient of variation is increased.
For normally distributed x and y variates the skewness of the ratio is approximately
where
Effect on confidence intervals
[edit]
Because the ratio estimate is generally skewed confidence intervals created with the variance and symmetrical tests such as the t test are incorrect. These confidence intervals tend to overestimate the size of the left confidence interval and underestimate the size of the right.
If the ratio estimator is unimodal (which is frequently the case) then a conservative estimate of the 95% confidence intervals can be made with the Vysochanskiï–Petunin inequality.
Alternative methods of bias reduction
[edit]
An alternative method of reducing or eliminating the bias in the ratio estimator is to alter the method of sampling. The variance of the ratio using these methods differs from the estimates given previously. Note that while many applications such as those discussion in Lohr are intended to be restricted to positive integers only, such as sizes of sample groups, the Midzuno-Sen method works for any sequence of positive numbers, integral or not. It's not clear what it means that Lahiri's method works since it returns a biased result.
Lahiri's method
[edit]
The first of these sampling schemes is a double use of a sampling method introduced by Lahiri in 1951. The algorithm here is based upon the description by Lohr.
Choose a number M = max( x1, ..., xN) where N is the population size.
Choose i at random from a uniform distribution on [1,N].
Choose k at random from a uniform distribution on [1,M].
If k ≤ xi, then xi is retained in the sample. If not then it is rejected.
Repeat this process from step 2 until the desired sample size is obtained.
The same procedure for the same desired sample size is carried out with the y variate.
Lahiri's scheme as described by Lohr is biased high and, so, is interesting only for historical reasons. The Midzuno-Sen technique described below is recommended instead.
Midzuno-Sen's method
[edit]
In 1952 Midzuno and Sen independently described a sampling scheme that provides an unbiased estimator of the ratio.
The first sample is chosen with probability proportional to the size of the x variate. The remaining n - 1 samples are chosen at random without replacement from the remaining N - 1 members in the population. The probability of selection under this scheme is
where X is the sum of the N x variates and the xi are the n members of the sample. Then the ratio of the sum of the y variates and the sum of the x variates chosen in this fashion is an unbiased estimate of the ratio estimator.
In symbols we have
where xi and yi are chosen according to the scheme described above.
The ratio estimator given by this scheme is unbiased.
Särndal, Swensson, and Wretman credit Lahiri, Midzuno and Sen for the insights leading to this method but Lahiri's technique is biased high.
Other ratio estimators
[edit]
Tin (1965) described and compared ratio estimators proposed by Beale (1962) and Quenouille (1956) and proposed a modified approach (now referred to as Tin's method). These ratio estimators are commonly used to calculate pollutant loads from sampling of waterways, particularly where flow is measured more frequently than water quality. For example see Quilbe et al., (2006)
Ordinary least squares regression
[edit]
If a linear relationship between the x and y variates exists and the regression equation passes through the origin then the estimated variance of the regression equation is always less than that of the ratio estimator[citation needed]. The precise relationship between the variances depends on the linearity of the relationship between the x and y variates: when the relationship is other than linear the ratio estimate may have a lower variance than that estimated by regression.
Uses
[edit]
Although the ratio estimator may be of use in a number of settings it is of particular use in two cases:
when the variates x and y are highly correlated through the origin.
In survey methodology when estimating a weighted average in which the denominator indicates the sum of weights that reflect the total population size, but the total population size is unknown.
History
[edit]
The first known use of the ratio estimator was by John Graunt in England who in 1662 was the first to estimate the ratio y/x where y represented the total population and x the known total number of registered births in the same areas during the preceding year.
Later Messance (~1765) and Moheau (1778) published very carefully prepared estimates for France based on enumeration of population in certain districts and on the count of births, deaths and marriages as reported for the whole country. The districts from which the ratio of inhabitants to birth was determined only constituted a sample.
In 1802, Laplace wished to estimate the population of France. No population census had been carried out and Laplace lacked the resources to count every individual. Instead he sampled 30 parishes whose total number of inhabitants was 2,037,615. The parish baptismal registrations were considered to be reliable estimates of the number of live births so he used the total number of births over a three-year period. The sample estimate was 71,866.333 baptisms per year over this period giving a ratio of one registered baptism for every 28.35 persons. The total number of baptismal registrations for France was also available to him and he assumed that the ratio of live births to population was constant. He then used the ratio from his sample to estimate the population of France.
Karl Pearson said in 1897 that the ratio estimates are biased and cautioned against their use.
See also
[edit]
Mark and recapture, another way of estimating population using a ratio.
Ratio distribution
References
[edit]
^ Scott AJ, Wu CFJ (1981) On the asymptotic distribution of ratio and regression
estimators. JASA 76: 98–102
^ Cochran WG (1977) Sampling techniques. New York: John Wiley & Sons
^ Jump up to: a b c van Kempen GMP, van Vliet LJ (2000) Mean and variance of ratio estimators used in fluorescence ratio imaging. Cytometry 39:300–305
^ Jump up to: a b c Ogliore RC, Huss GR, Nagashima K (2011) Ratio estimation in SIMS analysis. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 269 (17) 1910–1918
^ Pascual JN (1961) Unbiased ratio estimators in stratified sampling. JASA 56(293):70–87
^ Beale EML (1962) Some use of computers in operational research. Industrielle Organization 31: 27-28
^ Jump up to: a b Tin M (1965) Comparison of some ratio estimators. JASA 60: 294–307
^ Sahoo LN (1983). On a method of bias reduction in ratio estimation. J Statist Res 17:1—6
^ Sahoo LN (1987) On a class of almost unbiased estimators for population ratio. Statistics 18: 119-121
^ Jump up to: a b c d Choquet D, L'ecuyer P, Léger C (1999) Bootstrap confidence intervals for ratios of expectations. ACM Transactions on Modeling and Computer Simulation - TOMACS 9 (4) 326-348 doi:10.1145/352222.352224
^ Durbin J (1959) A note on the application of Quenouille's method of bias reduction to estimation of ratios. Biometrika 46: 477-480
^ Mickey MR (1959) Some finite population unbiased ratio and regression estimators. JASA 54: 596–612
^ Jump up to: a b Lohr S (2010) Sampling - Design and Analysis (2nd edition)
^ Lahiri DB (1951) A method of sample selection providing unbiased ratio estimates. Bull Int Stat Inst 33: 133–140
^ Midzuno H (1952) On the sampling system with probability proportional to the sum of the sizes. Ann Inst Stat Math 3: 99-107
^ Sen AR (1952) Present status of probability sampling and its use in the estimation of a characteristic. Econometrika 20-103
^ Särndal, C-E, B Swensson J Wretman (1992) Model assisted survey sampling. Springer, §7.3.1 (iii)
^ Tin M (1965). Comparison of Some Ratio Estimators. Journal of the American Statistical Association, 60(309), 294–307.
^ Beale EML (1965) Some use of computers in operational research. Industrielle organisation 31:27-8
^ Quenouille R Rousseau AN Duchemin M Poulin A Gangbazo G Villeneuve J-P (2006) Selecting a calculation method to estimate sediment and nutrient loads in streams: application to the Beaurivage River (Quebec, Canada). Journal of Hydrology 326:295-310
^ Quilbé, R., Rousseau, A. N., Duchemin, M., Poulin, A., Gangbazo, G., & Villeneuve, J. P. (2006). Selecting a calculation method to estimate sediment and nutrient loads in streams: Application to the Beaurivage River (Québec, Canada). Journal of Hydrology, 326(1–4), 295–310.
^ Pearson K (1897) On a form of spurious correlation that may arise when indices are used for the measurement of organs. Proc Roy Soc Lond 60: 498
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7017 | https://www.carislifesciences.com/the-difference-between-point-and-frameshift-mutations/ | Home / The Difference Between Point and Frameshift Mutations
The Difference Between Point and Frameshift Mutations
Video Transcript
The DNA in our cells is the blueprint for making RNA, which is then translated into making the proteins that drive many of the functions in our bodies.Proteins are present in all living organisms and do things such as, providing flexibility and rigidity to a cell, delivering oxygen to the muscles, transporting bodily fluids, and making antibodies that fight diseases.
WHAT ARE THE AMINO ACIDS IN DNA AND RNA?
As we look closer, we see that proteins are made up of smaller units called amino acids. For every protein, a unique set of amino acids are connected in a chain that is coded for by the DNA. As DNA is copied to make RNA, a very specific sequence of instructions, called bases, are written in sets of 3’s. Bases are the specific information used to dictate which amino acid is being coded for. There are 4 bases that make up RNA which are, uracil, cytosine, adenine and guanine, we’ll represent them with the letters U, C, A and G. The order of these bases directs how the amino acids line up, when they start, when they stop, and determines how long the chain will be. So, what happens when things go wrong? When errors occur in the DNA code, it is referred to as a mutation. The two main types of mutations are point mutations and frameshift mutations.
WHEN DO POINT MUTATIONS OCCUR?
A point mutation is where one of the three bases, is replaced with a different base. Sometimes, these mutations have minimal effect, but other times they can significantly alter the structure of the protein depending on how different the new amino acid is from the original. The new base may also alter the protein by creating a stop signal too early in the amino acid chain. Mutations like this often render the protein completely ineffective.
WHAT CAUSES FRAMESHIFT MUTATIONS
A frameshift mutation happens when one or more of the bases are inserted or deleted. Since bases are read in groups of 3, their addition or deletion in anything other than multiples of 3 disrupts the reading frame sequence. When this happens, the entire code following the addition or deletion will be read incorrectly. This could result in the wrong amino acid being coded, thereby changing the protein’s shape and function.
CARIS CAN HELP ONCOLOGISTS FIND MUTATIONS
Whether the errors occurred in the DNA blueprint, or in the RNA copy, Caris can help your doctor determine where the mutation happened by testing all of your bodies DNA, RNA and proteins. Knowing all of the mutations present in your tumor can shed light on how the cancer may behave, and what treatments will be most effective. Ask your doctor about Caris testing.
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What is Gene Fusion?
Genes are portions of our DNA that contain detailed instructions that determine the specific traits we exhibit.
Microsatellite Instability
DNA is a double-stranded molecule that encodes instructions for all living things’ growth, regulation, and adaptation. Sometimes errors occur as DNA is duplicated.
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7018 | https://www.themathdoctors.org/a-very-different-kind-of-sequence/ | Skip to content
A Very Different Kind of Sequence
November 21, 2018 January 15, 2024 / AQOTW, Arithmetic, Higher math / Proof / By Dave Peterson
(An archive problem of the week)
While gathering sequence/pattern questions for my last post, I ran across a very different problem. Here we are told what the pattern is (a good example of one that you would probably never discover on your own), and asked some questions about later terms. It can be understood either using tools of number theory, or with only knowledge of arithmetic and some good insights.
A sequence of digits
Here is the question, from 2004:
```
Interesting Number Sequence Pattern
The sequence of digits 1,2,3,4,0,9,6,9,4,8,7,... is constructed in the following way: every digit starting from the fifth is the last digit of the sum of the previous four digits.
a) Do the digits 2,0,0,4 appear in the sequence in that order?
b) Do the initial digits 1,2,3,4 appear again in the sequence in that order?
I haven't come across such a series before. I really don't know how to continue.
```
To make sure we understand the pattern, here is how the first few terms are obtained:
1+2+3+4 = 10 --> 0
2+3+4+0 = 9 --> 9
3+4+0+9 = 16 --> 6
4+0+9+6 = 19 --> 9
0+9+6+9 = 24 --> 4
9+6+9+4 = 28 --> 8
6+9+4+8 = 27 --> 7
Now, how could we tell whether a given number appears ever, or a second time, in this sequence?
Some background
Doctor Vogler answered; not knowing what background Kamsin (an adult) had, he started by mentioning the Number Theory tools he used to solve the problem:
```
Thanks for writing to Dr. Math. Do you know what modular arithmetic is? You can refer to
Mod, Modulus, Modular Arithmetic
Thinking of this in terms of modular arithmetic will probably be helpful. Then you have: Each term is the sum of the previous four, mod 10. By the Chinese Remainder Theorem, we can break this into two problems: Answer each question mod 2 and mod 5. That will make things easier.
In case none of that made sense, I will try to avoid talk of modular arithmetic. But if you wonder how I came up with these ideas, then you should start by learning modular arithmetic.
```
The rest of what he says doesn’t require any knowledge beyond ordinary arithmetic, but can be better understood from the higher perspective. The modular arithmetic comes in due to the fact that taking the ones digit of a number is the same as dividing the number by 10 and taking the remainder; we say that the number is congruent to the remainder, “modulo 10”. For information about the Chinese Remainder Theorem, see
Chinese Remainder Theorem and Modular Arithmetic
Can 2004 ever appear?
But we’ll move on without these.
```
Now arithmetic mod 2 is simply a matter of odds or evens. So look at the pattern of odds and evens in your sequence:
odd, even, odd, even, even, odd, even, odd, even, even, ...
You should notice a repeat after every five terms. Can you prove that this pattern will continue indefinitely? Now answer part (a).
```
To make things more compact, rather than write “even” and “odd” over and over, I will just replace each number in our sequence with the remainder after division by 2, so that “0” means “even”, and “1” means “odd”:
1, 2, 3, 4, 0, 9, 6, 9, 4, 8, 7, … becomes
1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, …
We can build this new “even/odd” or “0/1” sequence by doing exactly what we did before, but starting with 1, 0, 1, 0, and adding, writing the remainder after division by 2 at each step (so that 1 + 1 = 0, since 2 is even):
1+0+1+0 = 0
0+1+0+0 = 1
1+0+0+1 = 0
0+0+1+0 = 1
0+1+0+1 = 0
1+0+1+0 = 0
Did you notice that we are right back where we started? From here on, the pattern will repeat forever, so that we will get 0, 1, 0, 1, 0 for every five terms.
But 2, 0, 0, 4 has the pattern 0, 0, 0, 0 (that is, all even). We’ve shown that this can never happen — the most we’ll ever get are two evens in a row. That concludes part (a).
Will 1234 ever reappear?
```
Part (b) is a little harder, but it only requires you to establish that this pattern has to repeat. So I'll give you the general ideas and hope that you can fill in the details and understand what is going on. First of all, you should notice that if you know any four consecutive numbers in the sequence, then you can start listing off all of the numbers that follow it, and you can start listing off all of the numbers that preceded it.
Let's practice. Suppose that
5, 8, 1, 4
appears somewhere in the sequence (which it may or may not). If it does, then what do the next three or four numbers have to be? And what does the number before the 5 have to be? What about the two numbers before that?
```
This is the same idea we used to see that the pattern of odds and evens repeats, but this time applied to the original sequence, and going in both directions. The latter is true because we can find the previous number by subtracting and taking the remainder: if x + 5 + 8 + 1 = 4 (mod 10), then x = 4 – 1 – 8 – 5 (mod 10). (This is a little tricky: the subtraction gives -10, and if we add 10 to this we get 0, which is the answer. The number before that will be 1 – 8 – 5 – 0 = -12, and we have to add 20 to this, getting remainder 8. We can’t just divide -12 by 10 and say the remainder is 2, because that doesn’t take the sign into account.)
So each time a given group of four numbers appears, everything after it is just the same as the previous time it appeared.
```
Okay, so if the initial sequence
1, 2, 3, 4
ever appears later in the sequence (after the first time), then that means that the sequence repeats, right? Well, what if it doesn't repeat? What can happen?
Well, there are only 10^4 possible sequences of four numbers. So let's think about the first 10^4 + 4 numbers, and every string of four of them makes a total of 10^4 + 1 different strings of four numbers. Do you know the pigeonhole principle? It tells us that some string of four numbers has to occur twice (at least).
But that means that it repeats! Let's suppose this string (and we don't know what numbers it has) starts the first time at position n and the second time at position n+m. Then the m numbers (including the four from the first string) from the beginning of the first occurrence to the beginning of the second occurrence MUST be the next m numbers as well! Because the next m numbers are determined by the next four. So that means it has to repeat, every m numbers.
```
To put this very simply, suppose you made a list of the first 10,004 digits in this sequence, which constitute 10,001 groups of four digits. Since there are only 10,000 possible 4-digit numbers, the 10,001 numbers can’t all be different. There must be at least one number that repeats. But as we saw, if a number repeats once, then it repeats over and over — and not just that one number, but everything between the two occurrences we found.
We have used the pigeonhole principle in many answers, most of them cases where the student knows what it is, and didn’t need a full explanation. For one simple case including an explanation, see
Pieces on a Chess Board
Now, there is one possibility left. Maybe we start with 1234, and eventually reach a point where things start repeating, but the repeating part doesn’t include the 1234. Is that possible?
```
But now let's go backwards. The m numbers before the first occurrence must be the same as the m numbers after it, so that same pattern repeats. That means that every string of four numbers that appears in the sequence at all must repeat every m numbers!
Is that right? Why can't we start going along and then get stuck somewhere? For example, why can't we have lots of random numbers coming along until we suddenly find four zeros in a row? If that ever happens, then it will just keep giving us zeros. So why can't that happen? (Hint: What was the last nonzero digit?)
Does all of this discussion make sense to you? So what is the answer to the second question?
```
To answer the hint, if at some point we have 0000, then we have to continue with zeros both forward and backward — that is, we must have started with 0000. We can’t get there from any other start.
More questions
It’s easy to ask more questions about this sequence. For example when is the next occurrence of 1234? I made a spreadsheet to create the sequence, and found that it occurs after 1560 digits. Could we have worked that out without actually carrying this out? What if we started with a different set of digits? Under what conditions would we get a different number than 1560? What if we used more or less than four digits in defining the sequence?
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7019 | https://plato.stanford.edu/entries/critical-theory/ | Stanford Encyclopedia of Philosophy
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Critical Theory (Frankfurt School)
First published Tue Dec 12, 2023
[Editor’s Note: The following new entry by Robin Celikates and Jeffrey Flynn replaces the former entry on this topic by the previous author.]
“Critical theory” refers to a family of theories that aim at a critique and transformation of society by integrating normative perspectives with empirically informed analysis of society’s conflicts, contradictions, and tendencies. In a narrow sense, “Critical Theory” (often denoted with capital letters) refers to the work of several generations of philosophers and social theorists in the Western European Marxist tradition known as the Frankfurt School. Beginning in the 1930s at the Institute for Social Research in Frankfurt, it is best known for interdisciplinary research that combines philosophy and social science with the practical aim of furthering emancipation. There are separate entries on influential figures of the first generation of the Frankfurt School – Max Horkheimer (1895–1973), Theodor W. Adorno (1903–1969), Herbert Marcuse (1898–1979), and Walter Benjamin (1892–1940) – and the leading figure of the second generation, Jürgen Habermas (b. 1929).
In a broader sense, there are many different strands of critical theory that have emerged as forms of reflective engagement with the emancipatory goals of various social and political movements, such as feminist theory, critical race theory, queer theory, and postcolonial/decolonial theory. In another, third sense, “critical theory” or sometimes just “Theory” is used to refer to work by theorists associated with psychoanalysis and post-structuralism, such as Michel Foucault and Jacques Derrida (see these separate entries as well as the entry on postmodernism).
This entry is primarily focused on the critical theory of the Frankfurt School, but broadens outward at various points to discuss engagements by that tradition with a range of critical theories and social developments. The need for a broad approach to critical theory is prompted today by a range of contemporary social, political, economic, and ecological crises and struggles as well as the critique of Eurocentric forms of knowledge production.
1. The Frankfurt School: Origins, Influences, and Development
1.1 Origins and Generations
1.2 Influences
1.3 Critical Theory versus Traditional Theory
1.4 Studies on Authoritarianism and Mass Culture
1.5 The Dialectic of Enlightenment
1.6 The Communicative Turn
1.7 A Continuing and Contested Tradition
2. Critical Methods
2.1 Immanent Critique
2.2 Normative Foundations for Critique
2.3 Reconstructive Critique
2.4 Disclosive Critique, Genealogy, and the Critique of Normativity
2.5 Current Challenges
3. Critical Concepts
3.1 Alienation
3.2 Reification
3.3 Ideology
3.4 Emancipation
4. Critical Theories Today
4.1 Theorizing Struggles and Movements
4.1.1 Gender
4.1.2 Race
4.1.3 Colonialism and Post-colonialism
4.2 Diagnosing Crises
4.2.1 Economic Crises
4.2.2 Ecological Crises
4.2.3 Political Crises
4.3 Critical Practices
Bibliography
Academic Tools
Other Internet Resources
Related Entries
1. The Frankfurt School: Origins, Influences, and Development
The “Frankfurt School” of critical theory is not really a school at all. It is a loosely held together tradition constituted by ongoing debates among adherents about how best to define and develop that tradition. This includes disagreements about methods, about how to interpret earlier figures and texts in the tradition, about whether past shifts in focus were advances or dead ends, and about how to respond to new challenges arising from other schools of thought and current social developments. This section tells a largely chronological story, focusing on the origins, influences, and key texts of the Frankfurt School, and concludes with reference to ongoing debates on how to inherit and continue the tradition.
1.1 Origins and Generations
In their attempt to combine philosophy and social science in a critical theory with emancipatory intent, the wide-ranging work of the first generation of the Frankfurt School was methodologically innovative. They revised and updated Marxism by integrating it with the work of Sigmund Freud, Max Weber, and Friedrich Nietzsche while developing a model of radical critique that is immanently anchored in social reality. They used this model to analyze a wide range of phenomena – from authoritarianism as a political formation and as it manifests in both the nuclear family and deep-seated psychological dispositions, to the effects of capitalism on psychological, social, cultural, and political formations as well as on the production of knowledge itself (for excellent guides, see Thompson 2017 and Gordon et al. 2019).
Max Horkheimer outlined the original research agenda for the Frankfurt School in his 1931 inaugural lecture upon becoming director of the Institute for Social Research in Frankfurt (founded in 1923). He proposed an interdisciplinary research program combining philosophy and social theory with psychology, political economy, and cultural analysis (Horkheimer 1931). In that way, “social philosophy” aims at providing an encompassing interpretation of social reality as a whole – as “social totality,” to use a concept central to the Marxist tradition (Jay 1984).
Other key figures of the first generation include Theodor W. Adorno, Herbert Marcuse, and Walter Benjamin, along with Erich Fromm, Friedrich Pollock, Leo Löwenthal, Franz Neumann, Otto Kirchheimer, and figures like Siegfried Kracauer, who belonged to the broader circle for a few years (for rich historical accounts, see Jay 1973, Buck-Morss 1977, Dubiel 1978, Wiggershaus 1986, Wheatland 2009). The work of the largely Jewish members of the first generation was deeply marked by the rise of National Socialism, the experience of exile, and, for some of its inner circle, their return to Germany after 1945. After the Nazis closed the Institute, Horkheimer, who had already moved it to Geneva, re-established it at Columbia University in 1934, where he was soon joined by Pollock, Marcuse, and Löwenthal, while Adorno did not emigrate to the US until 1938. Horkheimer, Adorno, and Pollock moved the Institute from New York to Los Angeles in 1941. Those three reestablished the Institute in Germany after the War, with Horkeimer as director from 1951 to 1958 and Adorno from 1958 to 1969. Key figures who worked with first generation figures during this period emerged as the second generation: Jürgen Habermas, Alfred Schmidt, Albrecht Wellmer, Oskar Negt, and Claus Offe.
Habermas was the leading figure of this second generation, taking up Horkheimer’s chair in Frankfurt in 1964 before moving to a research post in Starnberg in 1971. Habermas returned to Frankfurt in 1981, retiring from this position in 1994. Axel Honneth worked closely with Habermas in the 1980s and took over the chair in social philosophy in Frankfurt in 1996; Honneth was also director of, and largely responsible for the revival of, the Institute for Social Research in Frankfurt from 2001 to 2018. He is considered a leading figure in the third generation, along with Seyla Benhabib, Nancy Fraser, and Christoph Menke (Anderson 2000, Allen 2010). Going beyond the second and third generations of the Frankfurt School, there are far too many figures to list; and the focal points for critical theory in this tradition have expanded, both geographically – with prominent figures in the United States and an active reception in Latin America – and thematically – for example, with a turn to feminism (see §4.1.1).
1.2 Influences
The first generation of the Frankfurt School took inspiration from an earlier generation of critical theorists: “Left Hegelians” in Germany who, after Georg Wilhelm Friedrich Hegel’s death in 1831, applied his philosophy critically to social and political phenomena like religion and the state, maintaining that the progressive realization of freedom in history that was central to Hegel’s thought was not yet complete and required a fundamental transformation of the status quo. Karl Marx became the most influential of this group. In a materialist transformation of Hegel’s thought, Marx analyzed the concrete conditions for realizing autonomy for all and viewed philosophy itself as conditioned by socioeconomic developments. By developing a critique of political economy in order to analyze the nature of capitalism and the possibilities for revolutionary social transformation, Marx set the standard for future generations of critical theory by combining radical philosophy with a critique of the best available social science of the day in the pursuit of emancipation.
Marx’s early writings, in the 1840s, were written when capitalist modernization was only just beginning in Germany, but he already saw contradictory social relations as the objective condition of capitalist society and exploited workers as a nascent revolutionary force. By the time the Frankfurt School began working out a critical theory of society in the 1930s much had changed as Germany had emerged as a leading economic power in an industrialized, capitalist Europe. Frankfurt School theorists were committed to social transformation, but the vehicle for change Marx identified – workers in advanced capitalist states like Germany – not only lacked revolutionary consciousness, but would soon embrace fascist politics when faced with economic crisis and mass unemployment. Radical social theorists would need revised analytical tools.
To study the psychology of individuals and groups along with social and cultural influences on that psychology, they could not rely on the then-dominant dogmatic versions of scientific Marxism (Pensky 2019). To understand how social conflicts get denied or repressed, and why individuals and groups turn to authoritarian politics that seem not to align with their class interests, they turned to Freudian psychoanalysis. In contrast to orthodox Marxism, they analyzed individual and group psychology, changes in the modern family, and the cultural “superstructure” of society, not just the material “base,” in order to understand how the rise of “mass culture” and the decline of authority figures in the family led to the decline of critical capacities both in the individual psyche and in society generally. This effort to combine Marx and Freud is one of the distinctive features of the Frankfurt School; exactly how to integrate psychoanalytic theory into critical theory has been a long-standing debate (Marcuse 1955, Whitebook 1995, Honneth 2010, Part IV; Allen and O’Connor 2019, Allen 2021).
In addition to incorporating insights from Freud’s psychoanalytic theory, early critical theorists drew on Max Weber’s social theory to analyze contemporary society. Crucial here was Weber’s theory of rationalization, which stressed the growing dominance of instrumental rationality, or means-end reasoning, through the expanding bureaucratization of society. Weber posited a loss of freedom, due to the “iron cage” of modern bureaucracy, and a loss of meaning generated by the “disenchantment of the world” associated with secularization. Weber’s work was crucial for Horkheimer and Adorno’s critique of instrumental reason (1947) as well as for Habermas’s later theory of communicative action (1981).
In synthesizing Marx and Weber, the first generation of critical theory was heavily influenced by Georg Lukács’s attempt to do the same in his ground-breaking 1923 essay “Reification and the Consciousness of the Proletariat” (see Brunkhorst 1983). Radically extending Marx’s analysis of commodities by analyzing how they transform the character of society as a whole, and drawing on Weber to describe a process of rationalization that extends to all aspects of life, Lukács used the term “reification” (see 3.2 below) to describe how the commodity form transforms the consciousness of those living in capitalist societies, who then see all social relations, even their relation to themselves, as taking on a “thing-like” character.
The classic philosophical influences on the Frankfurt School range widely, from Immanuel Kant and German Idealism to Nietzsche. In some form, Kant’s appeal to Mündigkeit (autonomy, maturity, responsibility) in his famous essay “What is Enlightenment?” – with its call for freely and publicly making use of reason – animates the ideal of emancipation throughout the work of the Frankfurt School, along with the Kantian conception of the critique of reason: the use of reason to reflect on the limits of reason. But its adherents follow Hegel and Marx in focusing on the social, cultural, and material conditions for achieving autonomy and insisting that reason is always socially and historically embedded. For first generation critical theorists, this entailed a critique of Kant’s own individualist and repressive understanding of autonomy as it arises within capitalist social conditions (Horkheimer 1933) and formalizes the domination of our own inner nature (Horkheimer and Adorno 1947, Excursus II, Adorno 1963a, Chs. 10–11). Some later critical theorists have engaged more positively with Kant, as in Habermas’s attempt to “detranscendentalize” core aspects of Kant’s transcendental philosophy (Habermas 2005, Ch. 2) and Rainer Forst’s Kantian constructivism in moral and political theory (Forst 2007, 2021a).
Hegel’s work has been a continual reference point for Frankfurt School philosophers, with key figures in the tradition – from Marcuse (1941) and Adorno (1963b) to Benhabib (1986, Part I) and Honneth (1992, 2001, 2011) – contributing both substantive studies and relying on Hegel’s methodology either for its holistic approach or as a paradigm of immanent critique while eschewing his metaphysical, teleological, and reconciliatory tendencies. Honneth first built on Hegel’s account of the struggle for recognition and the intersubjective conditions for living an autonomous life (Honneth 1992) before developing his own account of the practices and institutions of modern ethical life that realize freedom in a way that goes beyond its liberal and Kantian interpretations (Honneth 2011). Rahel Jaeggi builds on Hegel’s method of immanent critique in her account of progressive social change as learning processes in response to problems, contradictions, and crises that arise from within ethically thick forms of life (Jaeggi 2014).
In aiming to explain irrationality, the first generation extended the critique of reason, going beyond rationalist philosophers like Kant and Hegel to figures like Freud and Nietzsche. They turned to Nietzsche in particular as a critic of modern bourgeois culture and the violent formation of individual subjectivity. Engagement with Nietzsche’s thought extends from early essays by Horkheimer (1933, 1936a) through Horkheimer and Adorno’s shift toward doing critical theory in a more Nietzschean spirit with the genealogy of reason in Dialectic of Enlightenment (1947), and Habermas’s more critical take on Nietzsche’s supposed irrationalism (1985, Ch. 3–4), to contemporary authors such as Menke, who returns to Nietzsche as a positive reference point in the critique of the repressive dimensions of the modern ideal of equality (2000) and for a genealogical analysis of the modern subject who demands rights (2015).
One way of categorizing work by later generations of the Frankfurt School is to note how, even when drawing on a range of theoretical resources, they give pride of place to the legacy of a particular figure like Kant, Hegel, Marx, or Nietzsche (often via Foucault), or how they combine approaches. For instance, Honneth and Jaeggi are more Hegelian while Forst is more Kantian, and Benhabib is, like Habermas, a Hegelianized Kantian, and Fraser draws heavily on Marx in recent work while Amy Allen and Martin Saar are influenced by Foucauldian genealogy. The latter is part of a broader engagement between the Frankfurt School and post-structuralism, ranging from the more critical (Habermas 1985) through the more sympathetic (Honneth 1985, Menke 1988, 2000) to attempts to combine deconstructive and reconstructive approaches to critical theory (McCarthy 1991; see also Fraser 1989).
It is not easy to capture key features of an intellectual tradition shaped by such a variety of influences, including multiple figures whose own thinking changed over time, and a body of work addressing a vast range of topics spanning from the 1930s to the present. The rest of this section outlines some of the main arguments and focal points of key texts by key figures. It is not meant to be exhaustive, but to identify influential methodological approaches, arguments, and themes that are indicative of the work of the Frankfurt School and still provide important reference points for contemporary debates.
1.3 Critical Theory versus Traditional Theory
One largely undisputed reference for defining Frankfurt School critical theory is Horkheimer’s 1937 essay “Traditional and Critical Theory,” in which he defines critical theory by contrasting it with traditional theories that take the existing social order as given. Social sciences do this, for example, when they model themselves after the natural sciences in attempting to descriptively mirror a given set of facts or establish law-like generalizations. The point is not that empirical social research is invalid, but that traditional theories fail to analyze the broader social context in which they are embedded. This form of “positivism” views science as a purely theoretical undertaking divorced from practical interests even while it actually serves a particular social function in relying on established concepts and categories in a way that reinforces dominant ideologies and power structures. In that way, the forms of knowledge production that we rely on for insight into the social order become obstacles to social change.
Critical theory, by contrast, reflects on the context of its own origins and aims to be a transformative force within that context. It explicitly embraces an interdisciplinary methodology that aims to bridge the gap between empirical research and the kind of philosophical thinking needed to grasp the overall historical situation and mediate between specialized empirical disciplines. Critical theory aims not merely to describe social reality, but to generate insights into the forces of domination operating within society in a way that can inform practical action and stimulate change. It aims to unite theory and practice, so that the theorist forms “a dynamic unity with the oppressed class” (1937a [1972, 215]) that is guided by an emancipatory interest – defined negatively as an interest in the “abolition of social injustice” (ibid., 242) and positively as an interest in establishing “reasonable conditions of life” (ibid., 199). “The theory never aims simply at an increase of knowledge as such,” but at “emancipation from slavery” (1937b [1972, 246]) in the broadest sense of eliminating all forms of domination. The critique of traditional social science was further developed by Adorno and Habermas in the so-called positivism dispute in German sociology (Adorno et al. 1969, Wellmer 1969) and Horkheimer’s model of critical theory continues to inform discussions about how social critique might be carried out today in a variety of contexts (Outlaw 2005, Collins 2019, 57–65).
1.4 Studies on Authoritarianism and Mass Culture
Nothing epitomizes the Frankfurt School’s interdisciplinary approach to analyzing irrational elements of modern society better than their studies of authoritarianism, beginning with studies of German society in the 1930s and continuing with studies of the U.S. in the 1940s. This work combined philosophy, social theory, and psychoanalytic theory with empirical research.
The first substantial foray was Studies in Authority and the Family (Horkheimer 1936b), the product of five years of research carried out by members of the Institute as part of the research agenda outlined by Horkheimer when he became director in 1930. In an essay articulating the study’s theoretical framework, Erich Fromm argued that the “drives underlying the authoritarian character” are “the pleasure of obedience, submission, and the surrender of one’s personality” along with “aggression against the defenseless and sympathy with the powerful” (Fromm 1936 [2020, 39, 41]). A main concern of the Studies was that the nuclear family had lost the power it once had to counter other socializing forces, which could now more directly influence the individual, and that individuals who view the world as governed by irrational forces submit to powerful leaders who ease their feelings of powerlessness.
The focus on authoritarianism continued into exile, with Neumann and Kirchheimer focusing more on distinctly political phenomena such as law, the state structure, and competing political groups under the Nazi regime (see Neumann 1944, Scheuerman 1996). Neumann and Kirchheimer were the main legal and political analysts of the first generation, but were outside the inner circle and less influential on the trajectory the Frankfurt School took in the 1940s (see Scheuerman 1994 and Buchstein 2020 for attempts to revive interest in their legal and political analysis).
The work on authoritarianism that the Institute is most well-known for came with the publication of The Authoritarian Personality (1950), the result of research conducted by Adorno in collaboration with a team of psychologists at the University of California, Berkeley. The aim was to identify personality types that might be susceptible to authoritarianism, based not on explicit commitments to fascist political movements but on psychological characteristics and social attitudes (measured on an “F-scale”). The researchers posited that individuals with an authoritarian personality tend to exhibit traits such as rigid conformity to conventional norms, a tendency toward stereotypical thinking, a preference for strong authority figures and disdain for perceived weakness, a preoccupation with power and status, and a propensity for prejudice and hostility towards minority groups. The book explored the link between authoritarianism and antisemitism, highlighting the role of scapegoating and the projection of repressed aggression onto targeted minority groups.
The text was published in a series edited by Horkheimer, titled Studies in Prejudice, along with other innovative studies such as Prophets of Deceit: A Study of the Techniques of the American Agitator (1949), a psychoanalytic analysis of the rhetoric and tropes of American demagogues authored by Frankfurt School member Leo Löwenthal and Norbert Guterman. If The Authoritarian Personality studied the kinds of people potentially receptive to the messages of authoritarian leaders, Prophets of Deceit studied the content of the messaging itself. Adorno would later follow up on all these themes – both the form and content of fascist agitation and the social and psychological conditions under which it can succeed (1951b, 1967a).
The Authoritarian Personality had a major impact on the field of political sociology, inspiring a wave of similar studies and commentary. The recent resurgence of authoritarian populism has inspired renewed interest in Frankfurt School analysis of authoritarianism (see Section 4.2 below) in conjunction with publication of new editions of some of the classic texts along with previously untranslated work by Kracauer on totalitarian propaganda dating from the late 1930s (Kracauer 2013 ) and a 1967 lecture by Adorno on “Aspects of the New Right-Wing Extremism” (Adorno 1967a ).
One point of continuity between the studies of authoritarianism and Frankfurt School cultural analysis more broadly was the idea that “mass culture” was one of the powerful forces playing an increasing role in the direct socialization of individuals, a role that led to the “disappearance of the inner life” of the individual (Horkheimer 1941) and an increasing loss of the ability to imagine a world any different than the existing one. In its various forms, this general thesis was common to Horkheimer, Adorno, and Marcuse in their critiques of mass culture from the 1930s to the 1960s.
More generally, the Frankfurt School is known for its analysis of popular culture. By contrast to orthodox Marxist dismissal of cultural analysis for focusing on the less consequential “superstructure” of society, Frankfurt School theorists attentively analyzed the form and content of cultural objects along with the genres and modes of producing works of art and popular culture. In an early essay titled “Mass Ornament” (1927), Kracauer argued that analyzing the “inconspicuous surface-level expressions” of an epoch, by virtue of their “unconscious nature,” can disclose its “fundamental substance” and “unheeded impulses” (1927 [1975, 75]). Adorno would later maintain that “cultural criticism must become social physiognomy” (1951a [1967, 30]), a method he pursued in his interpretations of works of literature and music by interpreting the surface features and forms of various cultural artifacts in relation to underlying social conditions as a mode of disclosive critique.
The more pessimistic analysis of mass culture of Horkheimer, Adorno, and Marcuse can be distinguished from the more optimistic views developed by Kracauer and Walter Benjamin. Benjamin posited, in his famous essay, “The Work of Art in the Age of Technological Reproducibility” (1936), that the rise of technologies for mechanical reproduction, such as photography and film, led to the decline of the “aura” surrounding traditional works of art – the “authenticity” associated with the unique presence of the original in space and time – in part because it makes no sense to talk about the “original” version of a photograph. The resulting changes in perception and modes of collective experience and participation in cultural production could, Benjamin hoped, also bring about political forms of art and a more general democratization of culture. He contrasted this emancipatory potential of mass culture, through a politicization of aesthetics, with the aestheticization of politics under fascism (Buck-Morss 1992). Adorno expressed his disagreement in an earlier letter to Benjamin and in published work (Adorno 1936, 1938). As Wellmer puts it, “in technologized mass culture, Benjamin sees elements of an antidote to the psychic destruction of society, whereas Adorno regards it above all as a medium of conformism and psychic manipulation” (1985/86 [1991, 32–33]). While Benjamin placed hope in mass culture, Adorno saw it lying in the kind of autonomous art that resists reconciling subjects to their social world, instead offering a kind of “promise of happiness” in a transfigured future that lies beyond that social world (Adorno 1970, Finlayson 2015, Gordon 2023).
1.5 The Dialectic of Enlightenment
The critique of mass culture took its most dramatic form in the chapter on the “culture industry” in Horkheimer and Adorno’s Dialectic of Enlightenment (first circulated in 1944 and published in 1947). They introduced the term “culture industry” to underline the fact that “mass culture” is not something “the masses” spontaneously generate (Adorno 1967b [1991, 98]), but is manufactured using the same standardized and profit-oriented methods as any industrial production method. In this sense, culture is no longer a relatively autonomous realm of meaning (that might aim, at its best, at beauty, freedom, and truth) or source of critical awareness, but is thoroughly commodified by the “distraction factories” of the culture industry. “Cultural entities typical of the culture industry are no longer also commodities, they are commodities through and through” (ibid., 129). Entertainment replaces experience, numbing the audience’s capacity for critical thought and reconciling them to the status quo in a form of domination far more subtle than direct tyranny.
In this way, Dialectic of Enlightenment, which is perhaps the most influential text by Frankfurt School philosophers, analyzes two forms of mass society, fascist Germany and the United States, focusing primarily on the latter. Co-authored by Horkheimer and Adorno between 1939 and 1944 at the height of Nazi rule and World War II, the text opens with these lines:
Enlightenment, understood in the widest sense as the advance of thought, has always aimed at liberating human beings from fear and installing them as masters. Yet the wholly enlightened earth is radiant with triumphant calamity (1947 [2002, 1]).
The book is a genealogy of reason that traces its self-destruction from the dawn of human history to the present. Reason was supposed to liberate human beings. Instead, in the dominant form it takes as instrumental rationality, it has become the primary instrument of their domination. With reason taking this form, humans lose their capacity for critical reflection as their thinking is increasingly oriented solely toward self-preservation within a system in which they are powerless. “Thought is reified as an autonomous, automatic process, aping the machine it has itself produced, so that it can finally be replaced by the machine” (ibid., 19).
The root of the catastrophic dynamic lies not just with modernity or capitalism, but goes back to humanity’s earliest attempts to dominate nature. A core thesis of the book is that myth and enlightenment are entwined. The process of enlightenment began with the earliest attempts to overcome “mythic fear” as a way of explaining the unknown and mitigating threats from nature. This anthropological claim about enlightenment is combined with a historical claim about the Enlightenment and the rise of modern science and technology. This is when instrumental rationality truly comes to dominate, as means-end calculation is the kind of reasoning required for capitalist production and efficient bureaucracy. “Enlightenment is totalitarian” (ibid., 50), Adorno and Horkheimer argue; it subsumes everything under its dissolvent rationality. In this way, enlightenment reverts back to myth.
The book represents a shift away from the critique of political economy, indebted to Marx, to the critique of instrumental reason, indebted to Weber (Benhabib 1986, 149–163). Although this shift is sometimes attributed to the growing pessimism of its authors during National Socialism, it was also motivated by Pollock’s analysis of the shift from nineteenth-century liberal capitalism to “state capitalism”: increased intervention by the state into the economy meant that the primacy of the economy posited by Marx had been replaced by the primacy of politics (1941). This claim supported the focus in Dialectic of Enlightenment on the administered control of society by the state apparatus. The book paints a bleak picture of a society in which people live “totally administered lives” under the sway of efficient and calculating institutions. For the sake of self-preservation, they adapt themselves entirely to this apparatus. All the while the culture industry, as an “organ of mass deception,” keeps them entertained at the price of numbing their critical capacities, producing conformity, and undermining any sense of individuality or capacity for autonomy. The book also represents a shift away from the earlier idea of critical theory as interdisciplinary social theory, which could marshal the findings of empirical social science toward the practical aim of emancipation, and more toward speculative history. In the story they tell, the effects of domination are so ubiquitous that every form of scientific knowledge is corrupted.
If Horkheimer and Adorno’s Dialectic of Enlightenment was supposed to provide the grounds for a positive concept of enlightenment – as they maintained in its preface (1947 [2002, xviii]) – many critics have wondered what that is supposed to be (Wellmer 1983). Habermas would later argue that the authors needed to leave “at least one rational criterion intact for their explanation of the corruption of all rational criteria” in order to “set the normative foundations of critical social theory;” but they failed to do so (1985 [1987, 127–9]; see also Benhabib 1986).
Reappraisals of the text in recent decades range from defending its approach as a form of world-disclosive critique (Kompridis 2006) that reveals our familiar social world as pathological by using techniques like “rhetorical condensation” (Honneth 1998), to reading it as developing a dialectical conception of progress – not simply a history of decline – aimed at making us more aware of the inevitable entanglement of reason with power (Allen 2014, 2016), and attempts to build on the chapter on antisemitism, which analyzes its social function in providing a “release valve” that allows rage to be “vented on those who are both conspicuous and unprotected” (1947 [2002, 140]), thereby stabilizing domination by channeling potential resistance to social suffering into hatred of a group (Rabinbach 2000, Rensmann 2017).
Herbert Marcuse’s influential book One-Dimensional Man (1964) – best summarized by its subtitle, Studies in the Ideology of Advanced Industrial Society – can be read as an attempt to update Dialectic of Enlightenment in the form of a diagnosis of U.S. society and its perfected mechanisms of pacification and social control, ranging from art, sexuality, and politics to philosophy and the very act of thinking. Marcuse argues that all forms of critical thought and practice, having been wholly integrated into the wasteful, dehumanizing, profit-seeking, imperialist logic of advanced capitalism, are subsumed by one-dimensional ideology, a “flattening out of the antagonism between culture and social reality” (61).
Marcuse developed his influential concept of “repressive desublimation” to explain how the manipulated need for instant gratification has sanitized any transgressive forces within the domains of sexuality and art. Prior to the rise of the “affluent society,” art contained a transcendent capacity in the sense that it thought of, engaged with, and appropriated the idea of breaking out of the world in which one lived and embodied the hope for a better one to replace it. Within late capitalism, art has lost this critical aspect and dissolved into consumer culture and technological rationality, masking the “surplus repression” that shapes human instincts and needs in line with the functional requirements of social domination and the reproduction of the status quo.
Marcuse’s work has been criticized for its totalizing diagnosis of domination, his reliance on an objectivist account of human nature and needs, and the paternalistic or even authoritarian implications that possibly result from combining these two elements (Jaeggi 2014 [2018, 104–108]). Nonetheless, it has remained an important reference point for the critique of technology (Feenberg 2023a, 2023b, Fong 2016, Ch. 5) and of false needs, and of new right-wing forms of “repressive desublimation” that affirm the status quo in a transgressive mode (Brown 2019, 165–169). Regardless of how one today assesses Marcuse’s concrete analyses, his work exemplifies a tension that all critical theories have to address between the dominating forces of one-dimensionality and the possibility of breaking free of them.
First-generation critical theorists posited various responses to their own bleak diagnoses of society from the 1940s to the 1960s. Marcuse supported rebellious social movements in the 1960s and 1970s, in contrast to other leading representatives of critical theory who kept a conspicuous distance. In One-Dimensional Man, he placed hope for overcoming the repressive, one-dimensional society in a “Great Refusal” to abide by its norms, as carried out by the “substratum of the outcasts and outsiders, the exploited and persecuted of other races and other colors, the unemployed and the unemployable” (1964, 256). He later expressed solidarity with, and saw as examples of this refusal in, both the global student movement (1968, 119) and the feminist movement with its aim of overcoming dominant forms of aggressive masculinity (1974). He likewise praised counter-cultural movements for expressing sexual, moral, and political rebellion in a non-aggressive form of life that might generate a total change in values (1967). For Marcuse, emancipation involves a new morality that fulfills the vital needs for joy and happiness and encompasses an aesthetic-erotic dimension that is foreshadowed in alternative artistic tastes and new social and cultural practices. While Horkheimer and Adorno were less supportive of rebellious social movements, they did become important institutional figures and public intellectuals after their return to Germany (Müller-Doohm 2003, part IV; Demirović 2016). Adorno’s radio addresses in particular can be viewed as an attempt to educate the public for autonomy and so as a kind of response to their own bleak diagnoses of society.
But the core of Adorno’s response, from the early essay on the culture industry to his posthumously published Aesthetic Theory (1970), was to posit that “autonomous” or “authentic” art, by contrast to the products of the culture industry, maintains a utopian impulse insofar as it points beyond, and provides a moment of resistance to, the status quo. For example, atonal music by composers like Arnold Schoenberg generates dissonance in the listener by challenging the unity of the whole found in more harmonious music. Adorno maintained that such art, in challenging aesthetic norms and conventions, can provide aesthetic experiences that are resistant to the homogenizing forces of the culture industry. Critics of this turn to the aesthetic have wondered how this is supposed to provide a sound basis for a critical theory of society (Benhabib 1986, 222).
But one can argue that Adorno’s later work was an attempt to push against that kind of grounding for critical theory. The title of Adorno’s 1966 magnum opus, Negative Dialectics (1966a), refers to a methodology that takes from traditional Hegelian dialectics the emphasis on difference and mediation but abandons the attempt to overcome difference through a unifying synthesis. Instead, taking up an argument already developed in Dialectic of Enlightenment, Adorno argues that “identity thinking” and the “identity principle” have been at the basis of humanity’s destructive project of cognitive as well as practical domination of external as well as internal nature, thereby linking the philosophical to the social oppression of particularity. Adorno rejects “identity thinking” in favor of affirming the negative, namely “non-identity,” that is, the irreducible particularity of objects, experiences, and persons that cannot be subsumed under concepts.
This approach undermines the totalizing aspirations of theoretical systems in philosophy as traditionally understood. The struggle to recognize that which is nonidentical is not only an epistemological but also an ethical and political project that seeks to do justice to both the object and the subject of cognition in their irreducible individuality (Bernstein 2001). Linking epistemology and the philosophy of language to critical theory of society, this leads Adorno to reject not only Hegel’s affirmative synthesis but also Heideggerian ontology and Kantian dualism. Methodologically, Adorno explores alternative ways of thinking about how to use and develop philosophical concepts, taking up the Benjaminian notion of constellation and developing “critical models” in order to articulate the complexity of experience, and suffering, without reducing or constraining it. In Adorno’s view, negative dialectics is a form of immanent critique engaged in a dynamic and transformative process, as it “must transform the concepts which it brings, as it were, from outside into those which the object has of itself, into what the object, left to itself, seeks to be, and confront it with what it is” (Adorno 1957 [1976, 69]). In his cultural criticism and interventions in public debates, Adorno follows this paradigm by exploring how concrete experiences exemplify a form of social domination that is obscured by mass culture but also open up the possibility of transcending reified consciousness by articulating the internal contradictions within social reality.
1.6 The Communicative Turn
Jürgen Habermas, who worked closely with Horkheimer and Adorno in the 1950s until he fell out of favor with Horkheimer for seeming too radical, inherited one of the central claims of the Dialectic of Enlightenment, namely that Enlightenment is inseparable from the self-critique of Enlightenment, while also insisting on the context-transcending force of reason embedded in everyday practice.
Two works from the 1960s established his status as a leading figure in the second generation: The Structural Transformation of the Public Sphere (1962) and Knowledge and Human Interests (1968b). In the former, Habermas provides a historical and conceptual reconstruction of the idea of the public sphere in which subjects recognize each other as equals, submit to the “force of the better argument,” and subject legislation to the public use of reason. Against the backdrop of its emergence in eighteenth-century European societies, Habermas identifies the internal contradictions of the public sphere under the conditions of capitalism and traces its decline under the combined pressure of mass culture and mass media that has gradually transformed a reasoning public into passive consumers – a claim consistent with the “culture industry” thesis.
Critics argued that Habermas’s historical narrative of decline presupposes highly idealized versions of public debate and a “reasoning” public – a public that has always in truth been fragmented by class, gender, and race-based domination – and neglects the political significance of a multiplicity of subaltern and non-official public spheres and counter-publics (Negt and Kluge 1972, Fraser 1990, Warner 2002, Allen 2012). Nevertheless, his critical analysis of a contemporary public of consumers as the objects of processes of de-politicization, commercialization, political manipulation, and refeudalization seems to have lost nothing of its relevance (Seeliger and Sevignani 2022). The claim that a robust and independent public sphere is crucial to a healthy democracy is central to Habermas’s later, systematic contribution to democratic theory in Between Facts and Norms (1992), and he continues to analyze recent transformations in the structures and modes of communication within the public sphere (Habermas 2006, 2021).
Habermas’s Knowledge and Human Interests (1968b) was an ambitious attempt to ground critical social theory as a form of inquiry aimed at fostering a distinct type of knowledge tied to a deep-seated human interest in emancipation. This was a return to Horkheimer’s methodological aims in “Traditional and Critical Theory” (1937), but with a novel set of arguments, such as Habermas’s claim that the method of critical theory can be illuminated by way of an analogy with psychoanalysis – “the only tangible example of a science incorporating methodological self-reflection” (1968b [1971, 124]). Like Horkheimer, Habermas was critical of the positivist understanding of science for failing to see the connection between specific kinds of inquiry and fundamental human interests. Habermas posited that both the natural sciences and the “human sciences” (interpretive social sciences and humanities) are grounded in distinct practical interests. The natural sciences are a reflective extension of “labor” (instrumental action), which is tied to the practical interest in material reproduction. The human sciences are a reflective extension of “interaction” (linguistic communication), which is tied to the practical interest in symbolic reproduction. Habermas distinguished “critique” or “reflection” as a third practice organized around the interest in emancipation, understood in terms of overcoming various forms of heteronomy, domination, and dependency.
In the early 1970s, Habermas largely abandoned this framework, based in an anthropology of knowledge, though he did continue to pursue some of its themes, and epistemological questions have remained central to his work in at least two domains: first, in his “postmetaphysical” (non-foundationalist and fallibilistic) understanding of philosophy as a form of critical reflection at the intersection between science and society (Habermas 1983a, Ch. 1) and, second, in his critique of naturalism, especially neuroscience as a form of positivism or scientism that absolutizes the observer’s perspective, thereby negating the irreducibility of the participants’ perspective and occluding the normative structure of interpersonal communication (Habermas 2005, Ch. 6).
Habermas increasingly came to the view that critical theory needed more robust social-theoretical and normative foundations, since, in his eyes, the totalizing critique of the first generation had proven to be self-undermining (1985, Ch. 5) and his own approach in Knowledge and Human Interests had conflated the reconstruction of invariant structures of communication (formal pragmatics) with the critique of the false consciousness of particular persons and societies (1973a). Habermas’s alternative path, after abandoning that methodological framework, was to focus on communicative reason in a two-volume magnum opus titled The Theory of Communicative Action (1981). By contrast with an instrumentalist understanding of reason and action, Habermas’s “communicative turn” starts from a reconstruction of the rational and normative potential of everyday interactions.
This turn involves a multidimensional paradigm shift, illustrating the theoretical ambition of Habermas’ enterprise. He develops a theory of communicative action and rationality that is anchored in everyday practices of communication, in which we raise validity claims whose normative dynamic is context-transcendent and which allow for consensus-based coordination of action. He provides a historical reconstruction of modern rationalization processes, in which social integration via authority or shared tradition has been increasingly replaced by an expanded use of communicative reason in response to the pressure to cooperate. Finally, he constructs a two-level model of society based on the distinction between “system” and “lifeworld,” claiming that the regulation of coexistence in modern societies depends on both communication oriented towards mutual understanding (“lifeworld”) and on the anonymous systems of state bureaucracy and the capitalist market (“system”).
For the methodological renewal of critical theory, Habermas’s central claim is that within complex societies, social order always has a double form: It must simultaneously be viewed as lifeworld and as system. The lifeworld can only be understood from the hermeneutic perspective of its participants while the mechanisms of systemic integration only come into view from a system-theoretical or external perspective. Critical theory needs both perspectives in order to identify distorting effects of the system on the lifeworld. Habermas famously and controversially diagnoses a “colonization of the lifeworld” by the systemic media of money and power, which impose economic and administrative rationality – the main forms of “functionalist reason” – on areas of the lifeworld whose reproduction relies on communicative processes of cultural reproduction, social integration, and socialization that cannot be subsumed under the media of money and power without generating resistance. This provides a new foundation for critical theory by updating the critique of reification in the form of a critique of systematic distortions of communication. The “critique of functionalist reason” becomes a central task for critical theory, along with the aim of diagnosing the “selective pattern” of capitalist modernization that only partially realizes the actually available potential for rationality and learning within society.
In the ensuing discussion, Habermas was accused of reifying the “system” by conceptualizing the capitalist market and the bureaucratic state as functionally necessary and supposedly norm-free systems that lie beyond the theoretical reach of critical social theory and the political reach of emancipatory politics (Honneth and Joas 1991), of idealizing the lifeworld in ways that largely ignore the domination and exploitation of women and minorities (Fraser 1985), of subscribing to a progressivist theory of modernization and history that is Eurocentric and insensitive to the continuing effects of colonial domination (Allen 2016, Ch. 2), and of underestimating how deeply power penetrates into and distorts the very heart of communicative reason (Allen 2008, Chs. 5–6).
Habermas and his followers insist that while these phenomena are real, it is only the power of communicative reason – and the public discourses and deliberations in which it manifests itself and gets institutionalized – that allows us to detect, criticize, and ultimately overcome (if only partially and temporarily) those forms of domination. Whether one agrees or not that the communicative turn enables critical theory to analyze and bring to agents’ attention the distortions that block them from addressing and overcoming obstacles to emancipation, one important legacy of Habermas’s theory can be seen in opening up space for a methodologically pluralist critical theory in response to the fundamental need to capture the perspective of both participants and observers (Bohman 2003). Some Frankfurt School theorists have also built on Habermas’s system-lifeworld distinction in maintaining that social change must be viewed from the perspective of both “evolution” and “revolution” (Brunkhorst 2002, 2014).
1.7 A Continuing and Contested Tradition
One dominant story told about the Frankfurt School begins with Horkheimer’s original research program in the 1930s and views Horkheimer and Adorno’s radical departure from that vision in Dialectic of Enlightenment (1947) as an intellectual dead end from which Habermas rescued the tradition and returned it to its original methodology. From this perspective, the second generation, dominated by Habermas, superseded the first (see Kompridis 2006, 255–258, for a critique of this story). An alternative story would point out that Dialectic of Enlightenment was in many ways consistent with themes first articulated by Adorno in work from the 1930s – particularly his 1931 inaugural lecture, heavily inspired by Benjamin – that ultimately came to fruition in Negative Dialectics (1966a). To complicate matters in another way, while collaborating on Dialectic of Enlightenment in the 1940s Adorno also contributed to the interdisciplinary collaboration that culminated in The Authoritarian Personality (1950), a product of Horkheimer’s original vision for critical theory that combined social theory with empirical research.
Rather than viewing the second generation solely in terms of Habermas overcoming deficits in the first, this alternative story recognizes that there have always been multiple models and styles of critical theory operating simultaneously within the tradition and that Adorno was heavily influenced by Benjamin prior to collaborating with Horkheimer (Buck-Morss 1977, Wolin 1994, 166, 265–274). Moreover, Adorno’s influence is evident in work by figures in the second generation such as Albrecht Wellmer (1933–2018), who used Adorno’s work as a basis for challenging Habermas’s approach (Wellmer 1985/86, 1993) and was far more sympathetic with post-structuralism than Habermas – also true of Wellmer’s students in the third generation, Christoph Menke (1988, 2000) and Martin Seel. Adorno scholars have defended his work directly against Habermas’s criticisms (Cook 2004, O’Connor 2004: 165–170), and critical theorists continue to defend Adorno’s approach to critical theory (Allen 2016, 2021, 175–183, Marasco 2015, Ch. 3).
To complicate the story further, Benjamin’s work has had an enormous influence on work by a variety of critical theorists, though his wider influence had to wait until Adorno collected Benjamin’s essays for a German audience in 1955 and Hannah Arendt edited them for English readers in 1968. There have been significant studies of Benjamin’s work by scholars working within the Frankfurt School tradition (see Buck-Morss 1989 and Pensky 1993), while many critical theorists beyond the Frankfurt School have engaged Benjamin’s critique of linear notions of progress, and the ways in which they fail to break with the catastrophic continuity of the present (Benjamin 1940, see Löwy 2001), as well as his analysis of the constitutive relation between law and violence (Benjamin 1920/21; see the recently published critical edition, 2021), to mention only Jacques Derrida’s “Force of Law” (1990), Giorgio Agamben’s Homo Sacer (1995), and Judith Butler’s Parting Ways (2012) (see also Loick 2012).
Methodological debates within the Frankfurt School focus not only on the legacy of first-generation theorists but also on Habermas’s earlier work, with some arguing that Knowledge and Human Interests is worth revisiting because it was more attuned than his subsequent work to the dynamics of power and domination, making it more apt for addressing oppression based on gender (Allen 2008) or race (McCarthy 2004), or for developing a more comprehensive critical theory of domination (Klein 2020). Honneth (2017) has recently taken Habermas’s text as a jumping off point for refocusing critical theory on the task of elaborating the relation between emancipatory interests and emancipatory knowledge. Honneth nonetheless maintains that Habermas’s use of the methodology of psychoanalysis as a model for emancipatory critique is not apt, while others argue that it is still in many ways productive (Celikates 2009 [2018, 137–157]; see Allen 2021, Ch. 5 for a critique of Habermas, Honneth, and Celikates).
The latter debate is part of the resurging interest in psychoanalysis by some theorists working in the Frankfurt School tradition. Habermas’s own engagement with Freud and psychoanalysis in Knowledge and Human Interests was largely methodological in contrast to the substantive use of Freudian ideas by the first generation (in their analysis of the entanglement of reason and repression and the concrete forces of fascism and antisemitism), and Habermas (1983a) subsequently abandoned psychoanalytic theory entirely in favor of engagement with developmental psychologists like Jean Piaget and Lawrence Kohlberg. In developing his theory of recognition, Honneth (1992, Ch. 5) returned to psychoanalysis in the form of object relations theory, primarily in the work of Donald Winnicott, arguing that the experience of fusion and symbiosis that characterizes the early infant-mother relationship is foundational in two ways: It serves as the template for the type of recognition Honneth calls “love” and explains why individuals and groups continue to experience existing relations of recognition – that necessarily fall short of fusion and symbiosis – as unsatisfactory and continue to struggle for recognition. While Honneth’s use of Winnicott is controversial (McAfee 2019, Ch. 2; Whitebook 2021, Deranty 2021), recent debates have more generally focused on how to take up object relations within critical theory (Allen and O’Connor 2019). As a result, the divide now seems to be primarily between those who focus on the pro-social implications of psychoanalytic theory (Honneth 2010, Part IV) and those who also stress asocial or antisocial forces of Freud’s drive theory in general and the death drive in particular in order to avoid what they see as the risk of over-idealization and romanticization built into Honneth’s way of integrating psychoanalysis into his theory of recognition (Allen 2021, Ch. 5). Those critics advocate returning to the more negativistic approaches familiar from first-generation critical theorists (Fong 2016, McAfee 2019, Allen 2021).
Honneth’s return to the question of struggles oriented by emancipatory interests (2017) hearkens back to a shift that began in the 1980s, when a significant strand of Frankfurt School critical theory, including Honneth’s early work (1985), aimed at recovering the connection between theory and practice by linking the development of theory itself to social conflicts and movements. Oskar Negt and Alexander Kluge’s Public Sphere and Experience (1972) is an early example of a critique of the bourgeois (i.e. hegemonic) public sphere that invokes proletarian or plebeian non-state forms of the public and the divergent critical experiences they articulate as alternative sources of normativity, while also identifying blockages they face in the form of the “consciousness industry” and the pacification of social conflicts through “pseudo-publics.”
In a more explicit vein, Nancy Fraser contributed to the feminist turn in Frankfurt School critical theory – for which the work of Seyla Benhabib, Jean Cohen, and Amy Allen has also been decisive – in echoing Marx by arguing that critical theory should frame its “research program and its conceptual framework with an eye to the aims and activities of those oppositional social movements with which it has a partisan, though not uncritical, identification” (Fraser 1985, 97), and that the Frankfurt School in general and Habermas in particular had failed to theorize one of the most significant struggles against domination: the feminist movement (see §4.1.1).
Honneth has also sought to systematically reconstruct the link between theory development and struggles by taking experiences of misrecognition that lead to social struggles for recognition as a pre-theoretical reference point (1992). Drawing on a wide range of philosophical work, psychological and psychoanalytic accounts of identity-formation, and sociological and historical accounts of social movements struggling for recognition, Honneth has developed a theory of recognition that is the most prominent alternative paradigm, within Habermasian critical theory broadly construed, to Habermas’s theory of communicative action (Honneth 2000, Zurn 2015). Honneth maintains Habermas’s focus on intersubjectivity, but instead of linguistic practice and the ideal of “undistorted communication,” he focuses on relations of mutual recognition and the ideal of “undistorted recognition,” which then serve as the basis for the critique of “social pathologies” that he considers central to the project of critical theory (Honneth 2004).
In short, the Frankfurt School of critical theory is today constituted by lively debates, discussed more below, about how to deploy various critical methods (Section 2) and concepts (Section 3) while remaining attuned to social struggles and crises (Section 4) and positioning itself in relation to critical theories developed out of other traditions.
2. Critical Methods
Frankfurt School critical theory is best characterized by a set of methodological aspirations that set it apart from many other forms of social and political theorizing (both in philosophy and the social sciences): It aspires to be (1) self-reflexive, accounting for its own embeddedness in specific social and historical conditions, (2) interdisciplinary, integrating philosophical analysis with social theory and empirical social research, (3) materialist, grounding critical theorizing in social reality, and (4) emancipatory, orienting itself toward the goal of social emancipation. These commitments situate the Frankfurt School firmly in the Marxist tradition, and that tradition’s aim of overcoming the division between theory and practice without uncritically subsuming one under the other.
This has given rise to three interrelated methodological challenges: how to conceptualize (1) the relation of theory to social reality, (2) the role and standpoint of critical theorists, and (3) the normative foundations, content, or force of their critical theorizing. In light of historical developments in the first half of the twentieth century – the rise of fascism and Stalinism and the integration of the working class into the liberal welfare state – Frankfurt School theorists lost confidence in an identifiable direction of history or an identifiable collective subject like the proletariat to lead the way. It became increasingly unclear how to uphold a link between their theories and a pre-theoretical anchor within social reality – such as oppositional experiences, forms of consciousness, practices of resistance, or social struggles and movements – or even to see how the conditions for any of those things to emerge were present at all.
Against this backdrop, this section first sketches the common ground most Frankfurt School theorists find in the approach of immanent critique (§2.1) before tracing the various ways in which they have sought normative foundations (§2.2) in a more or less constructive or reconstructive (§2.3) register, then turns to methods such as disclosive and genealogical critique that are critical of those normative approaches (§2.4), and concludes by outlining a set of methodological challenges that shape contemporary debates (§2.5).
2.1 Immanent Critique
In responding to the three-pronged methodological challenge of relating theory to social reality, reflecting on the standpoint of critique, and spelling out its normativity, Frankfurt School critical theory moves beyond the usual juxtaposition between internal and external critique. Frankfurt School theorists rely on a third model of critique, which builds on Hegel and Marx and is often understood as immanent or reconstructive. Critique proceeds immanently or reconstructively when it seeks to secure its normative resources and epistemic standpoint from the (often implicit) normative structures and epistemic possibilities of the practices and self‐understandings that are constitutive of the (type of) society in question. Immanent critique avoids the dichotomy between an internal critique that refers to standards and standpoints that are already recognized by those criticized and an external critique that refers to standards and standpoints that are not (or not yet) recognized and therefore have to be derived independently from the agents’ perspective and their social context (see Jaeggi 2005, 2014, Celikates 2009, Stahl 2013a). Critical theory understood in this way is both grounded in social reality as it exists and emancipatory in seeking to radically transform this reality.
The critique of ideology can both serve as a paradigmatic example of immanent critique in this sense and illustrate some of the challenges this model faces (Ng 2015). Ideology critique is immanent insofar as it starts from the contradictions of a social and ideological constellation and the experience of those affected, which is shaped by these contradictions. It does not criticize an ideological form of consciousness because it is immoral or unethical, but because of its epistemic, functional, and genetic features, i.e. for being false or distorted, for contributing to the reproduction of relations of domination, and for arising from within such relations in ways that are relatively immune to self-reflection. Consequently, the critique of ideology does not focus primarily on the injustice or domination found in society, but on the forms of consciousness, culture, practice, habit, and affect that make this injustice or domination seem natural or unavoidable (Jaeggi 2008). On this view, any critical theory that aims at emancipation must first aim at diagnosing and overcoming those obstacles that keep agents from fully experiencing, critically reflecting on, and collectively acting against the unjust and dominating conditions under which they live. The question is how critical theorists can do so without falling back into epistemologically and politically problematic distinctions between false and true consciousness, between ideology and scientific insight, and between true (“objective”) and false (“purely subjective”) interests and needs (Celikates 2006; see Section 3.3 below).
These challenges are among the many challenges critical theorists face in developing an immanent critique that is linked to social reality and practice, a link that comes out in two ways. First, theory is anchored in social reality in terms of its genesis, as it is shaped by the social context from which it emerges. Second, theory aims at a practice that transforms social reality. This dual commitment to linking theory and practice is spelled out in two rather different ways, both in the history of the Frankfurt School and in contemporary discussions. One way of anchoring theory in social reality – call it the crisis approach – starts with social contradictions, antagonisms, and crises, along with the practical challenges and conflicts that result, and maintains that identifying those conflicts requires socio-theoretical analysis and sociological research (Jaeggi 2017a, Fraser and Jaeggi 2018). A second way of anchoring theory in social reality – call it the struggles approach – takes social struggles and movements and the practices of critique and resistance of oppressed groups as its starting point. This approach incorporates alternative standpoints and counter-hegemonic epistemologies into its theorizing with the aim of countering the potentially disempowering and anti-emancipatory effects that arise when critical theorists view crises mainly in terms of structural contradictions while ignoring or underestimating the ways that social and political movements themselves can produce and intensify crises (Collins 2019, Celikates 2022).
While this distinction between crisis and struggle is useful for heuristic purposes, it should not be overstated. Most critical theorists share a commitment to the emancipatory role of theory as well as an immanent anchoring of theory in social reality, whether qua crises or struggles. The distinction is a matter of degree and starting points, and it is usually agreed that crises and struggles stand in need of mutual articulation (see Benhabib 1986, 123–133, and Section 4.1 and Section 4.2 below).
Horkheimer maintained that a critical theory should not have an external relation to, but must enter into a “dynamic unity” with, practice, so that it is “not merely an expression of the concrete historical situation but also a force within it to stimulate change” (1937a [1972: 215], see also Marcuse 1937, Horkheimer 1937b). But under social conditions that neutralize social struggles or turn them into regressive backlash movements, the “dynamic unity” envisaged by Horkheimer can appear foreclosed. Even for Adorno, whose diagnosis of the “totally administered world” is the most radical example of this foreclosure, however, it would be a mistake to conceptualize existing society as a perfectly closed, monolithic, and functionally integrated self-reproducing totality. Rather, even when society is viewed as a totality, it has to be understood not in terms of homogeneity or frozen stability but in terms of structural antagonisms (Adorno 1957 [1976, 77]), conflict, and process (Adorno 1966b), i.e. as riddled with contradictions that, at least in principle, allow for forms of oppositional experience, consciousness, or practice that a critical theory can build on. In one of his last texts written shortly before his death, Adorno concludes that “critical theory is not aiming at totality, but criticizes it. This also means, however, that it is, in its substance, anti-totalitarian, with the utmost political determination” (Adorno 1969a; our translation). Even – or especially – in the face of the closure of political space, the political significance of a critical theory can consist in safeguarding the link between theory and the possibility of a radically different practice. At the same time, this defense of the relation to practice needs to be complemented by a defense of theory in the face of what Adorno identified as an “actionist” and anti-theoretical ideology of “pseudo-activity” in arguing that “praxis without theory, lagging behind the most advanced state of cognition, cannot but fail, and praxis, in keeping with its own concept, would like to succeed” (Adorno 1969b [1998, 265]).
2.2 Normative Foundations for Critique
Despite this more nuanced reading of Adorno on the relation between theory and practice, the broader diagnosis – put forth in different guises by Adorno, Horkheimer, and Marcuse – that social integration, the pacification of class conflict, and the internalization of conformist attitudes had robbed critical theory of any pre-theoretical anchor, provides an important background for Habermas’s break with the first generation. That break concerns not only their “pessimism,” but the basic methodological and substantial premises of their theories. In Habermas’s view, the first generation had navigated themselves into a dead end with their totalizing diagnosis of an all-encompassing state of delusion dominated by instrumental rationality. In response, and in order to provide firm normative foundations for critical theory, Habermas advocates a “communicative turn,” reformulating social critique in terms of a critique of the conditions of communication and grounding it in the normative content presupposed within the practice of linguistically mediated social interaction and argumentation.
This element of normative validity – as opposed to merely factual social validity that is forced, imposed, or presupposed – is elaborated in Habermas’s discourse theory, originally referred to as “discourse ethics” (Habermas 1983a, Ch. 4) but later evolving into a differentiated approach that distinguishes between ethical and moral norms (Habermas 1991) and a discourse theory of law and democracy (Habermas 1992). At the heart of discourse theory is a principle of discursive justification that Habermas refers to as the “discourse principle” or “D,” which states: “just those norms of action are valid if all persons affected could agree as participants in rational discourse” (Habermas 1992 [1996, 107]). He further specifies discourse theory with a universalization principle (“U”) that is operative when arguing about moral norms, and a democratic principle that is operative when attempting to justify legal norms within a democratic society. Habermas does not naively suggest that actually existing discourses correspond to these ideals, but maintains that in those discourses participants necessarily make idealizing presuppositions that can then be used to identify and criticize the shortcomings of actual discourse as distorted by interests, power relations, and ideologies.
As a response to the challenges of immanent critique outlined above, Habermas’s work can be understood in terms of a “dialectics of immanence and transcendence” (Cooke 2006, Ch. 3). Habermas maintains the need to situate reason historically and within social reality – the largely Hegelian, pragmatist, or reconstructive element of his thought. But the idealizations that are immanent in our linguistic practices point toward context-transcending validity claims that must be defended in a discursive procedure – the Kantian or constructivist element in his thought. Habermas now refers to his attempt to “de-transcendentalize Kant” as a form of “Kantian pragmatism” (Habermas 1999; see also Bernstein 2010, Ch. 8; Baynes 2016, Ch. 4; Flynn 2014b).
Some interpretations of Habermas stress that his theory of communicative action is still a form of immanent critique (Finlayson 2007, Stahl 2013b) while others object to his increasingly Kantian focus on moral norms (Heath 2014). To provide empirical confirmation of his rational reconstruction of the “moral point of view,” further situating it within social reality, Habermas drew on Kohlberg’s developmental moral psychology, itself decidedly Kantian in its defining the highest stage of moral development in terms of the ability to make universalizable moral judgements (Habermas 1983a, Ch. 4; for a critique, see Benhabib 1992, Chs. 5–6, which, drawing on Carol Gilligan’s critique of Kohlberg, distinguishes a “generalized other” from a “concrete other” whose experience cannot be accounted for by abstract conceptions of the moral standpoint).
Habermas’s shift toward a Kantian position is particularly evident in the Rawls-Habermas debate (Habermas 1995a, Rawls 1995, Habermas 1996), widely viewed as a “family quarrel” among two Kantian political philosophers. In his early work on discourse ethics, Habermas compared his own principle (U) to Rawls’s “original position,” arguing that his approach was the better way to “operationalize” the moral point of view as a form of moral constructivism that tests moral norms in a discursive procedure posited as a dialogical alternative to Kant’s categorical imperative (Habermas 1991). The debate shifted in the 1990s with their contributions to legal-political constructivism: Rawls’s Political Liberalism (1993) and Habermas’s Between Facts and Norms (1992), in which he provides a rational reconstruction of the institutions of constitutional democracy. In that context, Habermas argues that Rawls’s approach is not transcendent enough, since in Habermas’s view Rawls reduces normative validity to the notion of reasonableness immanent within liberal democratic societies (for the implications of their debate for multiple issues in moral and political philosophy, see Hedrick 2010, Baynes 2016, Chs. 6–7, and Finlayson 2019).
Habermas’s Kantian turn also came to the fore when extending his work in a cosmopolitan or “post-national” direction (beginning with Habermas 1995b), even if he has continued to combine a Kantian approach to justifying universal norms with a wide-ranging analysis of the empirical phenomena of globalization (1998). This combination of normative and empirical theorizing, a hallmark of the Frankfurt School, is present in a range of work by other critical theorists addressing global issues (Ingram 2019, Ibsen 2023), from the challenge of disaggregating citizenship from the nation-state (Benhabib 2004) to transnationalizing the public sphere (Fraser et al. 2014), and theorizing new forms of transnational democracy (Bohman 2007). Rather than simply defending abstract cosmopolitan norms, such approaches typically aim at some form of critical cosmopolitanism (Milstein 2015), with some stressing the crucial role of political contestation of allegedly universal norms “from below” (J. Ingram 2013) or of concrete struggles for rights as part of a broadly construed intercultural dialogue on human rights (Flynn 2014a).
In light of Habermas’s turn to Kant, a significant focus of debate among Habermasians and interpreters of Habermas has been the status of idealizing presuppositions and the ultimate status of the principles of justification within discourse theory. Defenders of discourse theory can be divided up into those who focus more on immanence – pointing in a Hegelian, pragmatist, contextualist, or reconstructive direction – and those who focus more on transcendence – pointing in a Kantian or constructivist direction. Among the former, some argue, echoing Hegel’s critique of Kant, that Habermas should situate reason more thoroughly within its social and historical context in order to avoid an overly rationalistic, abstract, or gendered approach (Benhabib 1986, 1992), while others have argued for Habermas to embrace a more pragmatist (McCarthy 1991, Bernstein 2010) or contextualist approach (Rorty 1985, Allen 2008, Ch. 6). Habermas’s most recent work (2019) attempts a kind of middle path, going in a decidedly historical direction by tracing the provincial, European origins of his “post-metaphysical” mode of theorizing as a preparatory stage to a fully inclusive, global intercultural dialogue as the way to establish its universal validity in a world characterized by “multiple modernities” (see Forst 2021b, Chambers 2022, and Flynn 2022 for critical assessments).
Those who have taken discourse theory in a more Kantian or transcendental direction include Habermas’s long-time interlocutor Karl Otto-Apel, who argued that the dynamic of universal validity claims in practices of argumentation transcendentally presupposes an ideal communication community from which universal normative foundations for the assessment of discourses can be derived (Apel 1985). Apel maintained that grounding reason, and thereby critique, requires a more transcendental justification (or “ultimate grounding”) than Habermas has provided (Apel 1989; see Habermas’s most recent reply to Apel in 2005, Ch. 3).
More recently, Rainer Forst has embraced Kantian constructivism in positing that every human being has a “right to justification,” a right to demand reciprocal and general reasons for the practices, institutions, and structures that affect them (Forst 2007). Forst views moral and political constructivism as distinct, but integrated stages. While the task of moral constructivism is to construct a list of basic moral rights that cannot be reasonably rejected, those abstract rights must be given concrete content by citizens in a process of political constructivism. He maintains that his approach is immanent insofar as the right to justification is “recursively grounded” by reconstructing the validity claims implicit in all morally justified claims, while maintaining a moment of transcendence since the right to justification can be justifiably claimed in any context. Forst views this as the normative core of a critical theory that understands society as an ensemble of practices of justification. In that sense, the concept of justification is both descriptive (referring to actual arguments given within a particular social order) and normative (referring to reasons that could or should be accepted), and Forst maintains both perspectives are needed for a critique of existing justification narratives and relations of justification (see the Introductions to Forst 2011 and 2021a).
Various critics of Habermas have argued that his normative turn and shift to Kant risks transforming critical theory into something that looks increasingly like a liberal theory of justice. They posit alternative approaches such as reconstructive, disclosive, and genealogical critique that also return to questions and arguments developed by the first generation.
2.3 Reconstructive Critique
Those who subscribe to the model of reconstructive critique emphasize the downsides of uncoupling normative argument from social analysis and social theory. In Axel Honneth’s work, this shift takes two forms. In his earlier work (1992), he argues that the relatively narrow rationalist focus on communicative reason occludes more fundamental and often prelinguistic experiences and intersubjective relations that give rise to struggles for recognition and that his Hegel-inspired theory is better able to articulate, thus reestablishing the link between theory and social reality in more substantial ways. Relatedly, Honneth insists that critical theory can be distinguished from other normative enterprises by its reference to “the pretheoretical resource in which its own critical viewpoint is anchored extra-theoretically as an empirical interest or moral experience” (Honneth 1994 [2007, 63–64]).
Expanding on this earlier commitment, in his later work Honneth argues against the division of theoretical labor in which (constructivist) philosophy engages in normative theorizing while empirical sociology investigates our social reality (2011). By contrast, he undertakes a “normative reconstruction” of how modern society – its legal, moral, political as well as social and economic practices and institutions – came to be centered around individual freedom as the highest value of this cultural formation. Honneth wants to show that we can only gain an adequate theoretical understanding of, and critical perspective on, modern society if we analyze its different social spheres as attempts to institutionalize the value of freedom. In contrast to both revolutionary and conservative approaches, he wants to show that the structure of this institutionalization allows for a progressive realization of the value of freedom as social actors appeal to the constitutive idea of freedom to challenge the concrete forms of unfreedom that remain characteristic of our social reality.
Similar to Honneth methodologically, Rahel Jaeggi argues, in her reconstructive approach to the critique of forms of life, that bracketing the question of how to rationally evaluate and criticize forms of life as a whole, as Rawlsians do in the name of liberal neutrality and Habermasians in the name of “ethical abstinence,” ends up hindering precisely the kind of experimental learning processes that are crucial for forms of life to remain dynamic and avoid stagnation and failure (2014 [2018, 9–24, 318–319]). But Jaeggi places a greater emphasis on contradictions, crises, and conflicts than the later Honneth (see also Schaub 2015).
The approaches of both Honneth and Jaeggi exemplify a conception of immanent critique that closely links analysis and critique, issuing in a critique that is neither a mere description of what exists nor a normative demand imposed on what exists from the outside. Accordingly, it does not proceed in a free-standing, normative way, but relies on a specific combination of philosophical reflection and social-theoretical as well as empirical research that is grounded in social developments and crises and actual social experiences and self-understandings. This methodological reorientation has also led to a more substantial engagement with questions of the economy and the sphere of work, both from a more Durkheim-inspired (Honneth 2022, 2023, Celikates, Honneth, and Jaeggi 2023) and a more Marx-inspired (Fraser 2022) position that has also resulted in a fundamental (non-reformist) critique of capitalism (Fraser and Jaeggi 2018).
2.4 Disclosive Critique, Genealogy, and the Critique of Normativity
While these approaches seek to develop a socially grounded form of normativity, critics argue that they are still too idealizing in their understanding of social reality and its historical genesis, as well as too normative in their methods from the point of view of yet another model of critique, which has been called disclosive or genealogical.
Disclosive critique typically takes its cue from Adorno (and sometimes other theoretical sources from Heidegger to contemporary aesthetics), moving beyond the dichotomy between literary world-disclosure and philosophical reason-giving or the quest for normative foundations. On this view, critique has the task of revealing the world in a new and different light, disclosing unrecognized suffering and intricate forms of domination that are not only occluded by dominant ideologies but also shape the norms that emanate from that order in ways that escape more strongly normative versions of immanent critique that build on them. Dialectic of Enlightenment can be read as an exercise in disclosive critique that seeks to defamiliarize the social world for its readers and thereby break open their unquestioned acceptance of how things appear to them (Honneth 1998).
This negative orientation of disclosive critique can be complemented by a more positive one, in which what is disclosed also involves potentialities and horizons that have no space or way to articulate themselves within the existing social and normative order. Walter Benjamin’s writings on the radical potential of mass culture or Judith Butler’s Gender Trouble (1990) can be seen as examples of disclosive critique that involve both the disruption of established and the experimental opening up of new experiences and schemas (Vogelmann 2016).
Some critical theorists attempt to integrate a more positive idea of disclosure into critical theory while maintaining that this is not at odds with expanded conceptions or normativity. Some draw on Heidegger to develop an account of world-disclosive critique that rethinks reason and agency, stressing receptivity and “self-decentering” as an alternative model to Habermas’s focus on procedural reason (Kompridis 2006). Others stress that while disclosure can and must be subject to intersubjective validation through argumentation, critical theory must have recourse to the disclosive power of imagination, which is revealed in the force of exemplarity (Ferrara 2008), in focusing attention on the aesthetic dimension of narratives that social movements use to imagine alternative possibilities (Lara 1998, 2021), or in the way that powerful representations of the good society function to disclose a transcendent object that cannot be fully known or represented but can nonetheless provide ethical orientation (Cooke 2006). In a variety of different ways, these approaches attempt to maintain the utopian dimension of critique (Marcuse 1937).
Genealogical critique, by contrast, can be seen as a form of disclosive critique that is more focused on problematizing, unmasking, and disrupting (Saar 2002, Koopman 2013). Given its association with Nietzsche and Foucault, it also has a distinct trajectory, set of methodological commitments, and theoretical implications. Taking aim at social practices, self-understandings, identities and normative commitments that are seen as natural or accepted as given, genealogical critique traces their historical emergence, highlighting their contingency and denaturalizing them with the aim of opening up the possibility of thinking and acting differently. From this perspective, the search for normative foundations is misguided as it both underestimates how normativity is shaped by unacknowledged histories and power relations and overestimates the transformative power of a normative critique that appeals to reason alone. Genealogical critique, by contrast, seeks to destabilize and decenter the subject and its fundamental commitments (Owen 2002, Hoy and McCarthy 1994; for a version of this claim that builds on psychoanalytic theory, see Allen 2021, Ch. 5).
While earlier engagements with genealogical critique, especially Foucault’s, were marked by criticisms of his supposed rejection of all normative and rational standards, lack of social theorizing, and relativism (Habermas 1985, Chs. IX–X, Fraser 1981, Dews 1987), more recently critical theorists have sought to emphasize the potential convergence and mutual illumination of genealogy and Frankfurt School critical theory in providing an analysis of the workings of contemporary forms of power and domination (Allen 2008, Koopman 2013, Ch. 7, Saar 2018). At the same time, a recent debate between Forst and Wendy Brown exemplifies how the earlier split between Habermas and Foucault is rearticulated today, with Forst taking a broadly Habermasian position in arguing that his “respect conception of tolerance” manages to safeguard the autonomy of individuals by grounding toleration in the right to justification, and Brown insisting, with Foucault, on the normalizing, disciplining, and depoliticizing effects of liberal discourses of toleration that ultimately obfuscate the complex operations of social power (Brown and Forst 2014, see also Vogelmann 2021).
A genealogical orientation also characterizes postcolonial critiques of Frankfurt School critical theory that point out the lack of explicit and sustained engagement with European colonialism and imperialism and its legacies, including contemporary forms of racism, and the ways in which these have enabled and shaped the processes of “modernization” and thus the formation of “modern”’ societies, subjects, and forms of knowledge and rationality, all of which critical theorists purport to investigate critically (see §4.1.3 below).
2.5 Current Challenges
Exponents of genealogical critique and struggle-centered approaches problematize forms of immanent or reconstructive critique that take institutional achievements as their starting point, challenging them by excavating the histories of domination and repression, as well as struggles, that have constituted these institutions and continue to shape their functioning. This gives rise to numerous challenges that continue to animate methodological debates in critical theory: (1) how (or even whether) to defend the putative normative achievements of liberal democracies, and if those are to be defended as achievements, then (2) how to theorize about the relation between struggles, crises, and institutional achievements, in contexts that may involve either (3) an absence of struggles, or (4), the opposite problem, a proliferation and fragmentation of struggles:
From the perspective of genealogical and post-colonial critique, the commitment to the institutions of the modern liberal nation-state (Habermas 1992, Honneth 2011) relies on an idealizing view of the history and present of this political formation that ignores, or treats as historically contingent and philosophically inconsequential, the forms of domination and exclusion that have accompanied it. At a time when the putative institutional achievements of liberal democracies – such as the separation of powers, the independence of the judiciary, the integrity of elections, or the protection of fundamental rights, especially for minorities – have come under attack from right-wing and neo-authoritarian movements and governments, the question is how critical theorists can defend the normative achievements of the existing order despite its systemic shortcomings. Offering a more radical challenge, some critical race theorists and post-colonial critics argue that those shortcomings reveal that what were thought to be normative achievements were historically premised on, and continue to functionally presuppose, domination and exclusion both at a societal and global level (see §4.1.2 and §4.1.3 below). On a methodological level, this involves the challenge of revising or going beyond the normative and sociological categories of critical theory that seem, at least in part, to be tied to a specifically Western experience.
Many critical theorists who accept the claim that these are normative achievements insist that a more complex view of the relation between institutions, struggles, and crises is necessary. As mentioned above, an alternative strand within critical theory that reaches from Negt and Kluge’s recovery of proletarian counterpublics through Fraser’s theorization of feminist movements to current attempts to reconnect critical theory with the struggles of our age, has insisted that abstracting from collective movements and struggles and relocating the emancipatory potential in the normative achievements of the existing institutional order risks underestimating how institutional dynamics, the inherent crisis tendencies of (more or less) liberal democracies, and social struggles are inextricably intertwined. Beyond a merely historical and social-theoretical point, how this question is answered will also affect how to conceptualize the role of emancipatory as opposed to regressive struggles in the face of the new authoritarianism (see §4.2.3 below).
More abstractly, critical theorists must account for situations in which there seem to be no struggles or forms of critical consciousness to latch onto, or only highly constrained forms of them. How can a critical theory respond to a situation in which domination is more or less total and has managed to suppress any critical consciousness and practice? Some of Marcuse’s descriptions of contemporary society come closest to this scenario. One might respond that “a society of happy slaves, genuinely content with their chains,” a society in which domination is experienced not as domination but as freedom, might be the critical theorists’ nightmare, but it “is a nightmare, not a realistic view of a state of society which is at present possible” (Geuss 1981, 83–84). Nevertheless, the challenge points to a dilemma critical theorists need to navigate. On the one hand, a critical theory requires a starting point in the forms of consciousness, experience, and practice of its addressees, but, on the other hand, critical theory should respond to and address distortions and blockages of precisely these forms of consciousness, experience, and practice. While these distortions and blockages will in most cases turn out to be partial rather than total and thus allow for some form of problematization to emerge (Celikates 2009, Part III), it seems equally important to not simply tie a critical theory to already existing social movements and thus to “goals that have already been publicly articulated” since this “neglects the everyday, still unthematized, but no less pressing embryonic form of social misery and moral injustice” (Honneth 2003, 114; see also Renault 2004, 2008).
The opposite problem can arise when critical theorists diagnose a proliferation of social struggles and lines of conflict beyond the classic antagonism of labor and capital. After the demise of the kind of philosophy of history that identified the proletariat as the revolutionary subject and the workers’ movement as the emancipatory force to which critical theory could and should attach itself, it has become unclear how critical theorists can determine with which of the different emancipatory movements of their day to enter into the kind of alliance envisaged by Marx and Horkheimer and which “forms of existing social critique” or “experiences of injustice” to pick up on. This difficulty is not only due to the plurality – or intersectionality – of movements, practices of critique, and experiences of injustice, but also due to the fact that struggles are often far from perfectly aligned and can operate at cross-purposes, with regard to both their aims and their methods. In answering this challenge, critical theorists can neither simply deduce the “correct” struggle from some overarching laws of historical development (the pole of determinism), nor claim that theorists simply have to decide which struggle or movement to link their theory to (the pole of voluntarism).
Insofar as critical theory is committed to immanent critique, focusing on the internal contradictions and crises of a specific social order and the struggles and movements that arise from within it, these challenges cannot be easily resolved. Rather than seeking to resolve them at an abstract level, they could instead be viewed as opening up a field of tensions that critical theorists need to navigate within the specific constellation they find themselves in. While critical theory needs to be anchored in actually existing forms of theoretical as well as practical critique, in the social struggles that people actually engage in, it also has the task of articulating the experiences of those who are blocked from engaging in struggles of their own and of contributing to the further theoretical articulation of existing struggles. At times, critical theory may need “to push beyond the ‘subjective’ elements of struggle and languages of claims-making to the more ‘objective’ dimensions of contradictions and crises, which turn more on the dynamics of systemic elements operating independently of whether or not people actually thematize them via struggle” (Fraser and Jaeggi 2018, 11), without losing sight of the epistemic and political risks this involves.
In addressing these risks, one way forward has been to embrace methodological pluralism and to understand critical theory less as a comprehensive social theory and more as a critical practice, as something critics do (Bohman 2003, Kompridis 2006, Celikates 2019a). This approach can more systematically incorporate alternative standpoints and epistemologies and the practices of epistemic resistance they are tied to, and more easily build on other traditions and paradigms of critical theory, such as feminist, anti-colonial, and anti-racist struggles and theorizing (Mills 1988, Collins 1990, 2019, Medina 2013, Loick 2021, Celikates 2022; see Section 4.1 below). Anchoring the perspective of critical theory within the social struggles and epistemic standpoints of the oppressed can serve as a counterweight – in the sense of “reflexive accountability” (Collins 2019) – to the tendency of actually existing critical theories to set in motion a disempowering spiral of epistemic asymmetries that denies the existence of theoretically sophisticated practices of critique and resistance on the ground and thereby reproduces existing obstacles to equal participation in knowledge production and to radical social transformation. On this view, critical theorizing is itself a social practice that recognizes its addressees as equal partners in a dialogical struggle for appropriate interpretations and realization of transformative potentials that is informed by social theory and sociological research. As such, it can make use of a variety of critical methods – reconstructive, constructive, disclosive, or genealogical (Freyenhagen 2018) – that are not easily subsumed under one unified metatheoretical framework, even if they can be seen as various attempts to spell out the idea of a critical theory as self-reflexive, interdisciplinary, materialist, and emancipatory.
3. Critical Concepts
The basic concepts of Frankfurt School critical theory – such as alienation, reification, ideology, but also emancipation – are expressive of the specific methodology, or set of methodologies, that critical theorists in this tradition employ. As explained in the previous section, critical theory in this tradition proceeds in an immanent way, and this implies that its concepts are both developed from within a certain social constellation and seek to go beyond the self-understanding characteristic of this constellation, they are both descriptive and evaluative, and they exemplify the unity of analysis and critique inherited from Marx. While some concepts are primarily “anticipatory-utopian” (like emancipation) and others primarily “explanatory-diagnostic” (like alienation, reification, and ideology, as obstacles to emancipation) (Benhabib 1986), they are all “thick concepts” whose descriptive content is irreducibly social-theoretically as well as evaluatively loaded.
In addition, some of the critical concepts developed by Frankfurt School authors – again alienation, reification and ideology are the clearest examples – point to second-order phenomena. In contrast to substantial first-order injustices, these concepts seek to critically diagnose what happens when unjust (or exploitative or oppressive) social relations are not experienced as unjust (or exploitative or oppressive) but are accepted as legitimate or natural, or if they are intuitively experienced but not explicitly recognized as such, or recognized but not adequately interpreted and articulated. These concepts pick out social phenomena that are often ignored by more mainstream approaches in moral and political philosophy that focus on the moral status of the individual and their actions, or the legitimacy of institutional arrangements, to the neglect of the domain of the social, with its distinct structure, dynamics, and challenges (see, e.g., Honneth 2000, Ch. 1, Zurn 2011, Neuhouser 2022, Ch. 1). The following subsections introduce four key concepts that exemplify both the critical methodologies discussed in the previous section and the substantial social-theoretical and diagnostic contributions to our understanding of contemporary society that Frankfurt School critical theory aspires to. There are of course other concepts used by critical theorists – from normativity, justice, and autonomy to power, domination, and oppression – but the focus here is on concepts less widely discussed in other traditions or to which Frankfurt School theorists have made distinctive contributions.
3.1 Alienation
The concept of alienation has a long history within critical theory. The basic concept refers to the idea of humans being separated, estranged, or distanced from something crucial to their freedom or capacity to flourish. One is alienated when one has a distorted or deficient relation to oneself or to the natural or social world. Critical theorists face a number of challenges in developing a critique of alienation. Classic critiques of alienation, Rousseau and the early Marx for example, relied on substantive conceptions of human nature or self-realization to ground their diagnoses and provide standards for critique. Thick accounts of human nature are less compelling today, which means contemporary critics of alienation have pursued alternative approaches. Since a critique of alienation attempts to diagnose a social pathology, not a problem with particular individuals, critical theorists must also provide a social theory that can convincingly diagnose the social causes of, and possible paths for overcoming, alienation.
Rousseau can be credited with inaugurating “social philosophy” as a domain of inquiry while developing a critique of alienation (Honneth 2000, Ch. 1). Although he does not refer to alienation in his “Second Discourse” (1755), the term captures his argument that living in society leaves human beings disconnected from their true desires and passions, which he explored by speculating about what humans would have been like in a state of nature. Within Hegelian and Marxist social criticism, the concept of alienation has been used to capture the idea that something produced by humans is wrongly taken by them as something given or outside their conscious control (Jaeggi 2005 [2014, 13–14]). In his Phenomenology of Spirit (1807), Hegel first develops a concept of alienation to describe the relation of the human mind to reality when the products of human reason are not recognized as our own creation but are instead experienced as alien forces. In his Economic and Philosophical Manuscripts (1844), Marx analyzed how wage labor within capitalist societies causes alienation. Workers produce a world of objects, but the products of their labor as well as their own productive activity are commodities over which they have no control; the world they create becomes an alien power with increasing control over them. They are alienated from the kind of spontaneous and creative productive activity that Marx, in his early work, posits as the essence of human nature.
The concept of alienation was influential among first-generation Frankfurt School theorists, particularly in the work of Marcuse and the later work of Erich Fromm (1961). In Dialectic of Enlightenment, Horkheimer and Adorno echo Rousseau in telling a story of alienation going back to the dawn of civilization. They maintain that human beings, in their quest to dominate the natural world (external nature) and to acquire mastery over themselves (inner nature), become estranged from both aspects of nature, failing to see what Enlightenment denies: that we are fundamentally natural beings (Vogel 1996, 69).
Contemporary critical theorists have attempted to rejuvenate the concept of alienation without relying on overly substantive accounts of human nature and without the totalizing diagnosis of Dialectic of Enlightenment. Rahel Jaeggi formalizes key elements of the Hegelian-Marxist approach in developing a philosophical account of alienation focused on how failure to adequately appropriate oneself or the world results in a “relation of relationlessnes” (2005). In this way, the non-alienated self is not defined by a substantive conception of human nature but by the quality of one’s relation to the world: whether this relation is sustained by successful processes of appropriation. Hartmut Rosa also defines alienation as a distorted relation to the world but with a more substantive approach to the quality of non-alienated relations to the world. For this, he has developed a multifaceted concept of “resonance” to capture a kind of vibrant or responsive relation to the world by contrast with the alienated experience of the world as ossified, mute, or hostile (2016). In contrast to these approaches, which are largely framed in terms of necessary conditions for living a good life, Rainer Forst has argued that deontological aspects of the critique of alienation have been neglected, and that there is a kind of “noumenal alienation” that results from not being recognized, or failing to recognize oneself, as an agent of justification (2017).
3.2 Reification
Reification is a concept with close ties to alienation. If alienation is viewed as diagnosing a distorted relation to the world, reification can be understood as one way of articulating the form that distortion can take. In the broadest sense, reification is a term used to critique cases in which some entity that should not be viewed as an object – oneself, other people, or some segment of the social or natural world – is treated as a thing-like object. It is instrumentalized, objectified, or quantified in a way that is inappropriate according to some critical standard. One challenge for critical theorists is articulating the standard or perspective – a non-reified relation or perspective – according to which the reified stance is not appropriate.
Georg Lukács’s classic 1923 essay on reification heavily influenced the Frankfurt School. Lukács combined Marx’s analysis of the “fetishism of commodities” – which causes social relations between human beings to appear as quantifiable and thing-like – with Weber’s analysis of bureaucracy – which extends this instrumentalizing attitude to all social domains. Reification becomes “the necessary, immediate reality of every person living in capitalist society” (1923 [1971, 197]), which can refer to an instrumentalizing attitude taken toward objects (whose qualitative feature are reduced to quantitative terms), other people, and features of one’s own personality when viewed solely from the perspective of their marketability.
Different critical theorists have appealed to the concept of reification to capture similar but not identical phenomena, with differing definitions corresponding to differences in the larger theoretical framework in which they deploy the concept. For Horkheimer and Adorno, the concept captures the dominance of instrumental reason and the totally administered world that results (1947). Habermas reinterpreted the concept to describe the ways in which systems such as the economy and the bureaucratic state, which function properly as spheres in which instrumental rationality dominates, extend too far into spheres of everyday life that he refers to as the lifeword (1981). This “colonization” of the lifeworld by the system results in the communicative structures of the lifeworld becoming reified. Honneth, by contrast, takes up the concept of reification in relation to his theory of recognition, arguing that reification involves a kind of forgetting of a primary relation of mutual recognition that he calls “empathetic engagement” (2005). Within Rosa’s theory of resonance, in which he attempts to capture one side of the history of modernity as a “catastrophe of resonance,” reification can be viewed as a “forgetfulness of resonance” (2016 [2019, 325]). The revival of this concept has been extended in other ways by using reification as a guiding concept for analyzing the relation between economics and subject formation within a “political economy of the senses” (Chari 2015) or pairing reification with a suitably modified notion of reconciliation to assess experiences of exclusion and integration within modern social orders (Hedrick 2019).
3.3 Ideology
Ideology is similar to alienation and reification in being both a concept critical theory inherits from the Marxist tradition and one that is used to identify a distorted relationship to the world and one’s own place in it (Eagleton 1991). In the Marxist tradition, it has played a prominent role in answering questions such as why people accept social and political conditions that seem to be contrary to their own interests, or how it is possible that subjects feel free although they are dominated. When people experience and describe relations of exploitation and domination as natural and without alternative or even as just, this seems to be an effect of ideology. Ideology, on this critical understanding, usually denotes a more or less coherent system of action-guiding beliefs, such as liberal individualism, that is said to obscure social reality – especially power relations, crisis tendencies, and social conflicts. As Marx (1844, 1846, 1867) and subsequent critical theorists argue, by obscuring these, ideology contributes to the reproduction of the prevailing order (Rosen 1996). Accordingly, any radically transformative and emancipatory practice presupposes that this ideological obfuscation must be recognized as such, criticized, and overcome. The challenge to such critical reflection is particularly acute when the possibility of even asking questions about how we might want to live, if we could transform society, is occluded by a technocratic ideology that reframes such practical questions as technical problems with narrow solutions (Habermas 1963, 1968a).
Ideology differs from mere deception, propaganda, or conspiracy theories. Because it is structurally anchored in social reality and plays a functional role for its reproduction, it cannot be explained with reference to the individual psyche or manipulation by others alone. Even if false consciousness is an element of ideology, critical theorists from Adorno to Jaeggi emphasize the practical nature of ideology as it shapes identities, is embedded in social practice, and functions via affects and habitus.
According to one influential interpretation, the critical notion of ideology developed in the Frankfurt School is characterized by three dimensions (Geuss 1981, Jaeggi 2008). In the first, epistemic dimension, ideologies always encompass epistemically deficient beliefs and attitudes that can range from substantially false beliefs to the confusion of particular and universal interests and inadequate concepts (such as “illegal alien” to refer to undocumented immigrants). In the second, functional dimension, ideologies are seen as playing a necessary, or at least supporting, role for the stabilization and legitimation of social relations of domination, i.e. for their more or less smooth reproduction. In the third, genetic dimension, ideologies are shaped, in ways that are not transparent to the agents themselves, by the social conditions under which they emerge, so that it is not an accident that people end up with the specific sets of beliefs they end up with in a specific type of society.
Radicalizing the Marxist notion of ideology as “necessarily false consciousness,” i.e. consciousness that is false (and not simply morally problematic) for structural reasons (and not just accidentally), Adorno and Marcuse often seem to argue that ideology reaches into the innermost core of subjects, who are shaped all the way down to their psychological and physical impulses, leading them to affirm the existing order and thereby preempting any resistance to domination. While this might help explain the resilience of ideology and its continued effectiveness, it also poses the challenge for critical theorists to find an anchor for their critique in the forms of consciousness, experience, and practice of its addressees (Celikates 2006, and Section 2.5 above).
Due to its emancipatory orientation, the critique of ideology must connect up with the self-understanding of those affected by trying to initiate learning processes, which in turn are supposed to lead to a transformation of those social conditions that are hidden behind ideologies. At the same time, without recourse to critical theories agents themselves will often continue to face obstacles to identifying, diagnosing, and explaining the effects of ideology on their critical capacities and practices. Arguably, showing that a contradiction is inscribed in the existing social order and can only be “dissolved” if this order itself is fundamentally transformed is also a task for a critical theory.
Although for most critical theorists ideology is not merely false consciousness but embedded in social practices and identities, ideology critique has been criticized for being overly cognitivist and underestimating the role of habitualized attitudes and cultural practices, for relying on an overly strong distinction between true and distorted consciousness, and for presupposing an idealized notion of the subject. Critics such as Foucault and Bourdieu speak instead of power-knowledge (Foucault 1973, 15) or of symbolic power and its embodiment (Bourdieu 1980, Ch. 8). The epistemological and political challenges the notion of ideology gives rise to continue to animate discussions (Celikates, Haslanger, and Stanley (eds.) forthcoming), including, more recently, on the relation between ideology and epistemic injustice (Fricker 2007, Mills 2017), cultural technē (Haslanger 2017a), and propaganda (Stanley 2015).
3.4 Emancipation
Frankfurt School critical theory inherits its emancipatory orientation from Marx, in the sense that it aims not only to understand, but also to contribute to a radical transformation of the social world that is already under way, and the commitment to real emancipation as requiring a radical, irreducibly social and political transformation that overcomes the fundamental contradictions of modern society instead of partial or local reforms aimed at surface-level symptoms. Emancipation is thus understood as liberation, including self-liberation, from domination by social, political, and economic powers, both personal and structural. Against this background, however, critical theorists have given different accounts of what emancipation is, what it requires, and how much can be said about it as a process and as an aim or state. While some (Horkheimer 1937a, Habermas 1968b) have thought of emancipation as a process of enlightenment and self-reflection that would allow for the realization of a rational organization of society, others thought of emancipation as sensual liberation (Marcuse 1969), or as emancipation from the (internalized) destructive imperatives of capitalism towards a state “of lying on water and looking peacefully at the sky” (Adorno 1951c [2005, 157]).
At the same time, and insofar as the working class has been integrated, fragmented, or at least reconstituted, it has become increasingly less clear who is to be emancipated (or self-emancipated) from which forms of domination and how. The challenges to the possibility of emancipation include reflections on the potentially overblown ideals of autonomy, sovereignty, and transparency that seem to underlie it (Laclau 1992), the limits of active self-transformation under conditions in which subjects have been shaped by power-ridden forms of subjectivation (Allen 2015), and the prospects of overcoming capitalism given the apparent lack of any clear and viable alternative. Today, critical theorists also face the challenge of reorienting the emancipatory project in the face of a catastrophic climate crisis that seems to privilege adaptation, mitigation, and sheer survival over utopian visions of emancipation that have also served historically as a pretext for an extractive and dominating relation to nature (Brown 2022).
In light of these challenges, a critical theory that wishes to hold on to its emancipatory orientation will need to articulate emancipation as an immanent possibility that is enabled and in some ways required by unprecedented historical developments. Whether in doing so it can build on the presumption of an emancipatory interest of the oppressed that theorists from Marx and Horkheimer to Habermas and Honneth (2017) have sought to identify remains contested. But thinking of emancipation as a second-order process that aims at enabling collective practices of self-determination over and against the obstacles picked out by concepts such as alienation, reification, and ideology, rather than as a substantial ideal or positive utopian vision of emancipation to be attained, might provide a starting point. Insofar as critical theory continues to see the existing social order as one of structurally entrenched domination, exploitation, and alienation, it will also continue to rely on some notion of an emancipatory process that points beyond those structures, even if this process is invariably plural, non-teleological, open-ended, and negative in orientation.
4. Critical Theories Today
Marx defined critical theory as the “self-clarification of the struggles and wishes of the age” (Marx 1843). The vitality of this approach to critical theory depends on continually taking up this task in new social contexts, as the first generation of the Frankfurt School did. Contemporary critical theorists continue this legacy by engaging with and theorizing in relation to contemporary struggles, crises, and practices. This has meant engaging a much wider range of emancipatory social movements than earlier generations of the Frankfurt School, who focused more on class struggle and capitalism (and the ways these were entangled with antisemitism and fascism) while largely neglecting issues like colonialism, racism, and the subordination of women. Contemporary critical theorists have expanded and enriched the Frankfurt School tradition by engaging with, and in some cases making contributions to, feminist theory, critical race theory, and postcolonial and decolonial theory (4.1), enlarging their analyses of crises beyond capitalism and its contradictions (4.2), and exploring a variety of critical practices ranging from civil disobedience to prefigurative, abolitionist, and revolutionary practices (4.3).
4.1 Theorizing Struggles and Movements
As emphasized above, Frankfurt School critical theory is methodologically interdisciplinary and defined by its aim of contributing to the emancipatory transformation of society by critically reflecting on the ways in which thinking itself can be distorted by structures of domination. This is also true of the various forms of critical theorizing that have emerged from and in relation to struggles against gendered oppression, racism, and colonialism and its legacies. Indeed, those critical theories bring to light structures of domination and modes of thinking (patriarchy, white supremacy, neocolonialism and Eurocentrism) that have until recently been neglected by the Frankfurt School and must be taken into account by any theory that aims to be critical and emancipatory.
4.1.1 Gender
More than one feminist theorist has argued that engaging feminism has been, and still is, crucial to renewing Frankfurt School critical theory both methodologically and in order to live up to its emancipatory aims (Fraser 1985, Ferrarese 2018). But analyzing the intersection between feminist theory and the Frankfurt School is complicated by the diverse array of theorists on both sides of that intersection. Some of the debates among feminist critical theorists mirror debates already discussed, for instance between those who draw on first generation versus Habermas or those who embrace Habermasian versus poststructuralist critical theory.
In most accounts, the first generation of the Frankfurt School is portrayed as not including any women and, with the exception of Marcuse in the 1970s (Marcuse 1974), its main protagonists largely failed to theorize about gender-based oppression or engage with feminist movements or the feminist theory of their time (there is, however, a new research project at the Institute for Social Research in Frankfurt that aims to challenge the dominant historiography by highlighting contributions of female researchers such as Käthe Weil and Else Frenkel-Brunswik and feminist work within the Frankfurt School). While fully acknowledging why feminists might find little of value in the first generation, some feminist theorists have highlighted important methodological affinities between, and potential for productive engagement with, that body of work (Brown 2006, Heberle 2006, Marasco 2006). In spite of the first generation’s nostalgia for the authority of the patriarchal family, their studies of authoritarianism were groundbreaking in analyzing the family as a political institution and breeding ground for fascism (Marasco 2018). Recent interest in Adorno’s work in particular builds on his theory of the nonidentical as support for the feminist critique of essentialist identities as well as affinities between feminist aims and his deconstruction of dualisms like nature and history or reason and desire, and his appeal to lived experience as crucial to philosophy and critique (Heberle 2006, 5–6). Attempts at synthesis include using his theory of the nonidentical, in dialogue with Lacan and Marx, to theorize a new approach to feminist political subjectivity (Leeb 2017), and combining Adorno’s insights into “bourgeois coldness” with the feminist ethics of care to rethink the fragility of our concern for others within a capitalist form of life that fosters “generalized indifference” while also producing a gendered form of attention to others (Ferrarese 2018).
Turning to the second generation, the critique of Habermas’s failure to adequately theorize gender in his Theory of Communicative Action (1981) was a turning point. In a now-classic essay, Nancy Fraser (1985) took a cue from the Marx quote about critical theory reflecting on the struggles of the age to criticize the Frankfurt School, and Habermas in particular, for failing to theorize one of the most significant struggles against domination. Seyla Benhabib raised similar concerns about whether the theory of communicative action could adequately theorize the feminist movement (1986, 252), and in Situating the Self (1992) aimed to make Habermasian discourse theory more cognizant of the self as gendered (see also the essays collected in Meehan 1995). In his later discourse theory of democracy, Habermas does engage the feminist theory and politics of equality to illuminate his core thesis about how private and public autonomy mutually presuppose each other (1992 [1996, 418–427]). But feminist critical theorists maintain that his rationalist approach fails to adequately capture the way power operates (Allen 2008, Ch. 5; McNay 2022, Ch. 1) or to incorporate forms of communication like narrative that have been crucial to feminism (Lara 1998, 2021; Young 2000).
The third generation of the Frankfurt School represents a crucial shift, with prominent feminist theorists like Fraser and Benhabib attempting to make critical theory more amenable to feminism from within the tradition, while also engaging in debates with leading figures in the poststructuralist strand of feminist critical theory like Judith Butler (Benhabib et. al. 1995). A core issue in these debates has been between Habermasian feminists who stress autonomy and poststructuralists who stress the idea of subjection – the ways in which power is central to the formation of subjects and their desires (Butler 1997). Amy Allen critically engages and synthesizes insights from both sides of this debate in viewing subjects as both constituted through relations of power and able to exercise autonomy in the form of critical reflection (Allen 2008). Axel Honneth, another key figure in the third generation, has engaged with feminist theory (Honneth 2000) and the feminist movement (2011 [2014, 154–176]), and in debates with feminist critical theorists including Fraser (Fraser and Honneth 2003) and Butler (Ikäheimo et al. 2021), but his work has also been the subject of sustained feminist critique of his conception of love, the family, and caring labor (Young 2007, Rössler 2007, Wimbauer 2023).
Fraser has, over several decades, developed a systematic defense of socialist feminism while charting various shifts in the feminist movement (see the essays collected in Fraser 2013), recently making the case that the contemporary crisis in care work must be understood as part of a larger general crisis in capitalist society (Fraser 2016, 2022, Ch. 3). Other feminist critical theorists also argue for a return to the critique of capitalism as crucial to feminist theorizing (Leeb 2017). From a different perspective, Lois McNay argues that recent Frankfurt School theorists, not only Honneth and Forst but also Fraser and Jaeggi, have failed to adequately incorporate the experience of gendered oppression into critical theory (McNay 2022). Another set of challenges arises from the need to develop an intersectional analysis of power and domination while engaging with a broader range of work in feminist and gender theory including queer and trans theory as well as transnational and postcolonial feminism (Allen 2019, 537–538).
4.1.2 Race
Apart from the influential studies on antisemitism and fascism by the first generation, Frankfurt School theorists have until recently shown little interest in issues of race and racism despite the prominence of anti-racist struggles and theorizing throughout the twentieth century and the present. The silence is of course not total. Early analyses point to prejudice toward Jews and other minority groups as an important part of the authoritarian personality and a key mechanism of providing “pseudo-orientation in an estranged world” (Adorno et al. 1950, 622), diagnose a culturalist transformation of the earlier biological racism at the center of fascism in post-war Europe that serves to maintain white supremacy (Adorno 1955, 148–9), and identify the phantasmatic dimension of racism and its fictions of homogeneity, purity, and essential difference (Adorno 1967a). Arguably, there are also broader methodological lessons from the relational and materialist theory of antisemitism developed by Adorno and Horkheimer that also hold for the study of racism (even if their relation remains contested, see Catlin 2023), namely the rejection of psychologizing and individualizing approaches, the insistence that the pathology always lies in the antisemitic or racist subjects and not in their victims, and the emphasis on structural factors that include the functional role of racism in the context of the crisis of capitalism and democracy (see Postone 1980 for an early attempt to explain modern antisemitism in relation to the nature of capitalism and the anti-capitalism of National Socialism).
Despite these openings, there has not been any sustained engagement with the phenomena of race and racism or with anti-racist struggles and theorizing, an eminently emancipatory form of knowledge production that, from W.E.B. Du Bois and Frantz Fanon to Black feminism (Collins 1990, 2019, Mills 2017), has been engaged in crossing the theory-practice divide and articulating dominated standpoints in ways that should have been of significant interest to Frankfurt School theorists (Outlaw 2005; for a relatively early exception see McCarthy 2009).
This missed opportunity is all the more astonishing as the intersection of class and race, of racism and capitalism has been at the center of theorists that share a Marxist orientation, and even some closeness to the Frankfurt School, most notably Angela Davis – who had studied with Marcuse in the US and with Adorno in Frankfurt, and, following Marcuse, insists on the need to bridge the gap between theory and practice and to combine the critique of racism as well as gender-based domination with a critique of capitalism (Davis 1983, 2004) – and Stuart Hall, who, building on Marxist and post-Marxist approaches, theorizes racism as a historically variable response to crisis and as a mechanism that allows capital to divide the working class (Hall 2021).
In contrast to the first generation’s focus on the “dark side” of modernity, later theorists, from Habermas to Honneth, developed a stronger commitment not only to Enlightenment values, but to the belief that these have been, more or less successfully, institutionalized in Western societies. As a result, their views clash with a core aim of Critical Race Theory (Crenshaw et al. 1995) – itself influenced by Marxist theories of the state and the law – namely, the aim of debunking the idea that the law and the state are neutral institutions that secure the common good and the rights of all as an ideology masking their character as instruments of racial (and class) oppression, as evidenced by massive and persistent inequalities that systematically disadvantage Blacks in the US in particular and racialized populations on a global scale, in various areas of life, from access to education, health, jobs, and housing to the risk of becoming a victim of police violence. According to this view, the forms of freedom and solidarity realized in liberal-democratic societies are not just contingently accompanied by exclusions of racialized groups, as if these values had only been insufficiently realized up to now and only need to be extended to those hitherto excluded. Rather, the thesis is that these exclusions have played a constitutive role in the history of these societies and their value systems and continue to shape them to this day, and that radical emancipation would therefore require developing entirely different visions of living together in freedom and solidarity (Kelley 2002).
More recently, Nancy Fraser (2022, Ch. 2) has picked up on Black Marxist discussions of racial capitalism (prominently Du Bois 1935) by arguing that capitalism provides a structural basis for racial oppression and thus exhibits an inherent (even if historically variable) tendency to racialize populations in order to more effectively expropriate and exploit them. Others have elaborated a relational and materialist understanding of racism that builds on how antisemitism was theorized in the early Frankfurt School, and how racism was rearticulated in a culturalist register in reaction to anticolonial and antiracist struggles (Balibar and Wallerstein 1991, Bojadžijev 2020). What these approaches share, and what might be a distinctive contribution of a critical theory of race and racism, is a commitment to understanding racism as a comprehensive social relation that needs to be understood in relation to broader (capitalist) social formations, “race” as an ideological effect rather than an unquestioned category for social analysis, and anti-racist struggles as a starting point for critical theorizing about race – commitments that are at least partially shared with important contributions in the critical philosophy of race (Mills 2003, Shelby 2003, Haslanger 2017b).
4.1.3 Colonialism and Post-colonialism
For all its focus on modes of domination in modern society, Frankfurt School critical theory has largely failed to address European colonialism and imperialism (Said 1993, 278) and their continuing effects in a world structured by massive inequalities and asymmetries between the Global North and the Global South. With a few recent exceptions to be discussed here, critical theorists in this tradition have not engaged much with the large body of postcolonial and decolonial theory, even if in recent years debates about the universal validity of human rights and cosmopolitanism, globalization and multiple modernities, religious pluralism and postsecularism, have provided ample occasion to go beyond still operative Eurocentric limitations and become more globally relevant (Mendieta 2007, Butler et al. 2011, Baum 2015, Ingram 2019, Kerner 2018; on some early Frankfurt School engagement with Chinese thought, specifically in Benjamin’s work, see Ng 2023).
The main target of postcolonial critique is the idea of a universal history in which the central engine of progress is located in modern Europe while non-Europeans are viewed as always lagging behind. The story has taken many forms, from narratives of progress in Enlightenment thinkers and their critics, such as Hegel (Buck-Morss 2009), to nineteenth-century theories of racial hierarchy and twentieth-century theories of development that have been shaped by, and in turn, rationalized, racism, slavery, and imperialism (McCarthy 2009, Bhambra and Holmwood 2021). Both anticolonial struggles and theorizing (in the work of Mahatma Gandhi, Aimé Césaire, Fanon and others) have insisted that the history and present of capitalism and of modern European and North American societies are constitutively entangled with colonialism, imperialism, and their afterlives, and that taking their trajectory as paradigmatically modern ends up representing a specific and heterogeneous trajectory and experience as universal and self-contained (Grüner 2010). While some aspects of postcolonial critique can be seen as overlapping with the critique of conceptions of the subject, reason, and universal history in the early Frankfurt School, the former also goes beyond the latter by understanding these as the effects of specifically colonial forms of domination and by tracing a different genealogy of fascism through its roots in the colonialism of the nineteenth century (Bardawil 2018).
Recent decades have seen attempts to bring postcolonial theory into dialogue with the Frankfurt School. From the side of decolonial theory, Enrique Dussel has been one of the most prominent decolonial philosophers to engage with Frankfurt School philosophers, developing a global ethics of liberation in critical dialogue with the discourse ethics of Apel and Habermas (Dussel 1998; see also Dusell 2011 and Allen and Mendieta 2021).
From the side of Frankfurt School critical theory, postcolonial critique has been taken up in a variety of ways (see also Vázquez-Arroyo 2018). In the same spirit of Horkheimer and Adorno’s attempt to critique enlightenment in the name of an alternative conception of enlightenment, both Susan Buck-Morss (2009) and Thomas McCarthy (2009) attempt to salvage something of the core idea that is the target of their critique: “universal history” for Buck-Morss, and “development” for McCarthy.
Amy Allen (2016), on the other hand, is more decidedly critical of the role of the discourse of “progress” and the role of such concepts in grounding normativity and shaping assumptions about historical development, modernization, and reason in the work of Habermas, Honneth, and Forst. She regards the latter approaches as deeply Eurocentric and contrasts them with a contextualist form of critique, inspired by Foucault and Adorno, that takes the form of a critical history of the present that uncovers the deep entanglement between reason and domination. Calling for an even more thorough revision of historical narratives, conceptual frameworks, and normative criteria, Gurminder Bhambra (2021) argues that the prevalent understanding of modernity as an endogenous European achievement obscures the fact that colonization and slavery were integral to and constitutive of the Enlightenment project of modernity in both its epistemic and institutional dimensions, a task for which historical and theoretical resources beyond Adorno and Foucault would be required. Fundamental questions about modernity, the human subject, and freedom also emerge from an encounter between critical theory in the Frankfurt School tradition and Caribbean thought (Sealey and Davis forthcoming). In a similar vein, contemporary critics of the persistence of colonial structures point to how a denial of the colonial past reaffirms a violent global color line (Mbembe 2016) that affects how societies treat Indigenous peoples (Coulthard 2014) and racialized and migrant populations (Celikates 2022).
4.2 Diagnosing Crises
Diagnosing crises, and the social contradictions that give rise to them, is a hallmark of Hegelian-Marxist critical theory. Marx famously diagnosed capitalism as a crisis-ridden social system, and the early work of the first generation of the Frankfurt School was a response to the economic, social, and political crises of their time. Dialectic of Enlightenment (Horkheimer and Adorno 1947) can be understood as addressing the crisis of reason that was experienced with the rise of National Socialism, but the critique of instrumental reason was disconnected from more concrete crises and struggles. Habermas aimed to restore the link between critique and crisis beginning with his 1973 book Legitimation Crisis (Benhabib 1986, 252–3, Cordero 2017, Ch. 3). Writing in the context of state-managed capitalism, Habermas diagnosed the distinctively political contradictions and potential for political crises within a social system that aims to steer the economy and manage economic crises (a point influentially elaborated by Offe 1984).
In subsequent decades, crisis critique, along with the critique of capitalism, was largely abandoned by Frankfurt School theorists (for a notable exception see Postone 1993). Renewed theoretical interest has coincided with rising public concern about social, political, and economic systems currently in, or always seemingly on the brink of, crisis, all against the backdrop of the unfolding effects of the ongoing climate catastrophe.
4.2.1 Economic Crises
Nancy Fraser was one of the first critical theorists to revive crisis critique and to do so as part of a comprehensive critique of capitalism that renews the link between analytical diagnosis and critique (Fraser 2011, 2014; see Wellmer 2014 for a critique of the Frankfurt School’s earlier neglect). What distinguishes Fraser’s approach is that it posits capitalism as the unifying causal link among seemingly distinct crises – in relation to care work, the environment, and political institutions – by viewing capitalism as an institutionalized social order in which the economic system “cannibalizes” the very conditions that make it possible within the spheres of social reproduction, the natural environment, and the political system (Fraser 2022). Fraser combines analysis of “objective” social conditions – contradictions and crises – with an orientation toward social movements by analyzing the “boundary struggles” that arise at the seams between the economic system and other domains, making the case for these struggles to unite around an anti-capitalist agenda.
In a similar way, Rahel Jaeggi has developed a crisis-oriented theory of immanent critique (2014, Ch. 6) that is not limited to diagnosing systemic dysfunction but includes the normative expectations and self-understandings of social agents (Jaeggi 2017a; see Fraser and Jaeggi 2018), but at a more abstract theoretical level than Fraser’s immanent analysis of capitalism as a social order. Like Fraser and Jaeggi, Albena Azmanova argues for renewed attention to the critique of capitalism but is skeptical about how helpful “crisis” talk is (2014) and maintains that radical social change is possible without crisis, revolution, or utopia through a united struggle against forms of precarity that are endemic to contemporary capitalism (2020). More generally, the turn to economic crisis dynamics has also led to a renewed interest in work – its general significance, pathologies, and emancipatory potential (Jaeggi 2017b, Dejours et al. 2018, Honneth 2023).
4.2.2 Ecological Crises
Turning specifically to the ecological crisis, Frankfurt School theorists have only recently begun taking seriously the task of rethinking their approach to critical theory in the current context of an ongoing ecological disaster on a global scale (for an early exception, see Vogel 1996). Some critical theorists argue that this situation calls for a new paradigm of “Critical Naturalism” (Gregoratto et al., 2022); others argue for a fundamental rethinking of Western conceptions of human freedom and a radical shift in conceptions of the ethically good life as a precondition for the kind of radical social change required by the current crisis (Cooke 2020, 2023). Fraser focuses on the role of capitalism in the climate catastrophe and the need for eco-politics to be anti-capitalist so that we can reassert control over, and begin to reinvent from the ground up, our relation to nature (Fraser 2021, 2022; see Bernstein 2022 for a recent approach to such rethinking).
In rethinking our conception of nature, given the lack of serious attention to theorizing about nature in the second and, until quite recently, third generation of the Frankfurt School, it is not surprising that many critical theorists have looked more to the first generation (see the collected essays in Biro 2011), with Adorno’s work viewed as a promising starting point for rethinking humans’ relation to nature (Cook 2011, Cassegård 2021, Ch. 3). Cook argues that the “project of showing that human history is always also natural history and that non-human nature is entwined with history… informs all Adorno’s work” and that there are important affinities between his work and proponents of radical ecology (Cook 2011, 1, 5–6). On the other hand, the view of nature as having a kind of otherness that is beyond and not fully graspable by humans – a view expressed at times by Adorno and Horkeimer as well as Marcuse – has been criticized in favor of a more Hegelian-Marxist approach that sees “nature” as a product of human activity (Vogel 1996, 2011). Others argue for reviving critical engagement with Marcuse’s work as a resource for addressing the ecological crisis, with its combination of a critique of science and technology (most radically, as a call for a “new science”) with the idea that social transformation must include a changed, aesthetic relation to nature (Feenberg 2023a, 2023b). At this point, it is clear that there must be more engagement between Frankfurt School theorists and the many “critical ecologies” being developed today, e.g., deep ecology, eco-feminism, eco-socialism, ecological Marxism, environmental justice, indigenous and decolonial ecologies, and new materialism (on the recent dialogue between Frankfurt School theorists and new materialism, see Rosa et al., 2021).
4.2.3 Political Crises
Finally, Frankfurt School theorists have turned their attention to political crises and the rise of right-wing populist, authoritarian, and neo-fascist movements, parties, and governments (Brown, Gordon, and Pensky 2018, Gordon 2017, Abromeit 2016). This crisis is particularly important because adequately addressing the economic and ecological crises of our time requires political solutions, which will be hindered by political systems that are themselves in crisis, thereby contributing to a regressive dynamic (Jaeggi 2022, Forst 2023).
From the perspective of critical theorists, there seem to be two aspects of the political crisis that are often missing from mainstream liberal accounts. The first pertains to the genesis and the causes of the crisis (Brown 2019, Gambetti 2020). Against accounts that see authoritarianism only in terms of a rupture with and as entirely foreign to liberal democracy, they argue that we need to examine the continuities and enabling conditions that allow authoritarian tendencies to arise from within liberal-democratic capitalist societies. Without analyzing the neoliberal restructuring of social relations and the ways in which populist and authoritarian movements exploit electoral strategies, fragmented public spheres, and liberal ideological frameworks such as “freedom of speech,” the critique of and resistance to them will necessarily remain truncated.
The second aspect pertains to the dynamic of authoritarianism and the political crisis it engenders. Beyond focusing on its political dimension (e.g. political aims and values), critical theorists have sought to analyze the socio-cultural, affective, and psycho-social dynamics of authoritarianism and its attractiveness to populations that seem to have little to gain from the election of populist leaders (Marasco 2018, Brown 2019, McAfee 2019, Redecker 2020, Zaretsky 2022). These approaches can draw on and are supported by Adorno’s analysis (1967a) of core features of authoritarian right-wing populism. First, it is not so much actual abandonment but a feared, anticipated, or imagined abandonment, along with a perceived loss of privileges that had come to seem natural, that are the driving force of the rise of a reactionary authoritarianism that then gets misdescribed as a revolt of the oppressed and exploited. Second, the proponents of authoritarianism, following an antisemitic and/or racist logic, personalize the blame for their fears and feelings of abandonment by projecting it onto groups they classify as alien, rather than attributing it to structural features of society.
In responding to all the crises discussed here – economic, ecological, and political – critical theorists must grapple with a number of challenges. Purely at the level of theory, there is the question of whether positing unity or convergence among crises is diagnostically accurate. At a practical level, it remains to be seen whether a unity thesis will be politically motivating and whether a convergence of social struggles is indeed on the horizon. The issue of practice also bears on the question of whether and to what extent the objective conditions of crisis and contradiction diagnosed by critical theorists actually affect people’s everyday lived experience and become motivating factors for political movements (see Section 2.5). Such questions about the relation between theory and practice have long been a focus of critical theorists and have recently gained attention in theorizing about a range of critical and political practices.
4.3 Critical Practices
While overcoming the gap between theory and practice has been a central methodological and political concern for critical theorists, critics have pointed out the prominent turn away from practice to theory in the first generation – accused by Lukács of taking up residence in the “Grand Hotel Abyss” (Lukács 1963, 22) – and the continued marginalization of critical praxis in later generations (Harcourt 2020). There are multiple grounds for challenging this assessment. Frankfurt School theorists had arguments all along about how to assess and relate to radical movements, such as the student movement of 1968 (Adorno and Marcuse 1969, Freyenhagen 2014, Pickford 2023), and there has always been a strand that continuously engaged with struggles and movements, from Marcuse to Negt and Kluge and Fraser. Some critical theorists have focused on deliberative democracy, the public sphere, and civil society (Habermas 1962, 1992, 2021, Cohen and Arato 1992, Benhabib 2004, Lafont 2019) as core fora for critical practice, while others have argued for critical theory itself to be democratized and understood as a critical practice (Bohman 2003, Celikates 2009).
Still, many of these approaches have been criticized for prioritizing institutional achievements over struggles and critical practices, and reform over revolution. Given the challenges outlined above (4.2), it is not surprising that some recent work tries to reverse this tendency by exploring more radical responses to such crises. These attempts notably push beyond the dichotomy between reform and revolution – for example, by promoting non-reformist reforms that could “alter the terrain on which future struggles will be waged, thus expanding the set of feasible options for future reforms” (Fraser, in Fraser and Honneth 2003, 79) – and mine the rich history and present of radical struggles outside traditional forms of political organizing such as the party or reimagine the party in radical ways (Dean 2016).
The range of critical practices engaged with by critical theorists past and present is extensive (for an inventory, see Harcourt 2020, Ch. 15). Frankfurt School theorists of earlier generations covered various forms of resistance, from the “Great Refusal” of the 1960s (Marcuse) and the potential for resistance in independent thinking and critical analysis in the face of universal reification (Adorno) to civil disobedience as a sign of a dynamic public sphere and civil society (Habermas 1983b, Cohen and Arato 1992, Ch. 11). More recently, critical theorists within various traditions have analyzed forms of disobedience (direct, digital, migrant etc.) as political practices of contestation and struggles for democratization “from below” (Young 2001, Smith 2013, Scheuerman 2018, Celikates 2019b). Other practices of resistance that do not directly engage with state institutions or appeal to the broader public include forms of sabotage (Malm 2020), fleeing, withdrawal, or defection. These turn away from what are seen as state-oriented struggles for visibility, recognition, or representation (Virno 2004, Roberts 2015) and towards subaltern forms of sociality (Moten and Harney 2013) and counter-communities that prefigure fundamental alternatives for living together (Loick 2021). Emphasizing the revolutionary dimension of critical practices, theorists have drawn on the abolitionist tradition of struggles against slavery and colonialism, and its revitalization in movements like Black Lives Matter, to call for a fundamental critique of racial capitalism and its entanglement with the punitive state, and a correspondingly radical transformation of all social relations and institutions (Davis 2005, Gilmore 2022). Critical theorists have also explored political practices of assembling (Butler 2015), occupying, striking, and reorganizing processes of social reproduction (Gago 2019), linking these to the need to rethink revolution beyond the model of a single break or event and more as an interstitial process (Redecker 2018, Saar 2020).
Whether the new revolutionary subjects and struggles that emerge in these critical practices will indeed converge to fundamentally challenge the existing order, open up new pathways to emancipation, and develop emancipated – more just, democratic, and sustainable – modes of living together remains to be seen. Horkheimer’s quip still holds: “if the proof of the pudding is in the eating, the eating here is still in the future” (1937a [1972, 220–1]). Against this background, theoretical explorations of critical practices – in the multiplicity of their forms, terrains, and actors – can be seen as part of the ongoing attempt to bring theory and practice together with an emancipatory orientation in light of the crises and struggles of the age. This approach has characterized the Frankfurt School from its very beginnings and has been a driving force in its continual (self-)transformation, making it into one of the most influential paradigms in social philosophy today.
Bibliography
Abromeit, John, 2016, “Critical Theory and the Persistence of Right-Wing Populism”, Logos: A Journal of Modern Society and Culture, 15(2) [Abromeit 2016 available online].
Adorno, Theodor W., 1931 [year this lecture was given], “Die Aktualität der Philosophie”, in Theodor W. Adorno, Gesammelte Schriften I: Philosophische Frühschriften, Rolf Tiedemann (ed.), Frankfurt am Main: Suhrkamp, 1973, pp. 325–344; translated as “The Actuality of Philosophy”, Benjamin Snow (trans.), Telos, 31 (1977): 120–133.
–––, 1936 “Wiesengrund-Adorno an Benjamin, 18. März 1936”, in Theodor W. Adorno and Walter Benjamin, Briefwechsel 1928–1940, Henri Lonitz (ed.), Frankfurt am Main, 1994, pp. 168–177; translated as March 18, 1936, “Wiesengrund-Adorno to Benjamin, March 18, 1936”, Theodor W. Adorno and Walter Benjamin, The Complete Correspondence 1928–1940,, Nicholas Walker (trans.), Henri Lonitz (ed.), Cambridge: Polity Press, 1999, pp. 127–133.
–––, 1938, “Über den Fetischcharakter in der Musik und die Regression des Hörens”, Zeitschrift für Sozialforschung, 7(3), 321–356; translated as “On the Fetish Character in Music and the Regression of Listening”, Susan L. Gillespie (trans.), in Theodor W. Adorno, Essays on Music, Richard Leppert (ed.), Berkeley: University of California Press, 2002, pp. 288–317.
–––, 1951a, “Kulturkritik und Gesellschaft”, in Karl Gustav Specht (ed.), Soziologische Forschung unserer Zeit, Köln and Opladen: Westdeutscher Verlag, 1951, pp. 228–240, republished in Theodor W. Adorno, Prismen: Kulturkritik und Gesellschaft, Frankfurt am Main: Suhrkamp, 1955, pp. 7–31; translated as “Cultural Criticism and Society”, in Theodor W. Adorno, Prisms, Samuel Weber and Shierry Weber (trans.), Cambridge, MA: MIT Press, 1981, pp. 17–34.
–––, 1951b, “Freudian Theory and the Pattern of Fascist Propaganda”, in Géza Roheim (ed.), Psychoanalysis and the Social Sciences Vol. 3, New York: International Universities Press, 279–300. Reprinted in The Culture Industry: Selected Essays on Mass Culture, Jay Bernstein (ed.), New York: Routledge, 1991, pp. 132–157.
–––, 1951c, Minima Moralia: Reflexionen aus dem beschädigten Leben, Frankfurt am Main: Suhrkamp; translated as Minima Moralia: Reflections from Damaged Life, Edmund Jephcott (trans.), London: Verso, 2005.
–––, 1955, “Schuld und Abwehr”, in Friedrich Pollock (ed.), Gruppenexperiment: Ein Studienbericht, Frankfurt am Main: Europäische Verlagsanstalt, 278–428; translated as Guilt and Defense: On the Legacies of National Socialism in Postwar Germany, Jeffrey K. Olick and Andrew J. Perrin (eds./trans.), Cambridge, MA: Harvard University Press, 2010.
–––, 1957, “Soziologie und empirische Forschung”, in K . Ziegler (ed.), Wesen und Wirklichkeit des Menschen: Festschrift für Helmuth Plessner, Göttingen: Vandenhoeck & Ruprecht, pp. 245–260; translated as “Sociology and Empirical Research”, Glyn Adey and David Frisby (trans.), in Theodor W. Adorno et al. (eds.), The Positivist Dispute in German Sociology, London: Heinemann, 1976, pp. 68–86.
–––, 1963a [year this transcribed lecture course was given], Probleme der Moralphilosophie, Frankfurt am Main: Suhrkamp, 1996; translated as Problems of Moral Philosophy, Rodney Livingstone (trans.), Stanford, CA: Stanford University Press, 2000.
–––, 1963b, Drei Studien zu Hegel, Frankfurt am Main: Suhrkamp; translated as Hegel: Three Studies, Shierry Weber Nicholsen (trans.), Cambridge, MA: MIT Press, 1993.
–––, 1966a, Negative Dialektik, Frankfurt am Main: Suhrkamp; translated as Negative Dialectics, E. B. Ashton (trans.), New York: Seabury Press, 1973.
–––, 1966b, “Gesellschaft”, in Hermann Kunst and Siegfried Grundmann (eds.), Evangelisches Staatslexikon, Stuttgart: Kreuz Verlag, pp. 636–643; translated as “Society”, Fredric R. Jameson (trans.), Salmagundi, 10–11 (1969): 144–153.
–––, 1969a, “Zur Spezifikation der kritischen Theorie”, in Theodor W. Adorno Archiv (ed.), Adorno. Eine Bildmonographie, Frankfurt am Main: Suhrkamp, p. 292.
–––, 1969b, “Marginalien zu Theorie und Praxis”, Die Zeit, No. 33 (1969); translated as “Marginalia to Theory and Praxis”, Henry W. Pickford (trans.), in Critical Models: Interventions and Catchwords, New York: Columbia University Press, 2005, pp. 259–278.
–––, 1967a [year this transcribed lecture was given], Aspekte des neuen Rechtsradikalismus, Berlin: Suhrkamp, 1999; translated as Aspects of the New Right-Wing Extremism, Wieland Hoban (trans.), Cambridge: Polity, 2020.
–––, 1967b, “Résumé über Kulturindustrie”, in Theodor W. Adorno, Ohne Leitbild: Parva Aesthetica, Frankfurt am Main: Suhrkamp, pp. 60–70; translated as “The Culture Industry Reconsidered”, Anson G. Rabinbach (trans.), in Jay Bernstein, (ed.), The Culture Industry: Selected Essays on Mass Culture, New York: Routledge, 1991, pp. 98–106.
–––, 1970, Ästhetische Theorie, Frankfurt am Main: Suhrkamp; translated as Aesthetic Theory, Robert Hullot-Kentor (trans.), Minneapolis: University of Minnesota Press, 1977.
Adorno, Theodor W., Else Frenkel-Brunswik, Daniel Levinson, and Nevitt Sanford, 1950, The Authoritarian Personality, with a new Introduction by Peter E. Gordon, New York: Verso, 2019.
Adorno, Theodor W., Hans Albert, Ralf Dahrendorf, Jürgen Habermas, Harald Pilot, and Karl R. Popper, 1969, Der Positivismusstreit in der deutschen Soziologie, Neuwied/Berlin: Luchterhand; translated as The Positivist Dispute in German Sociology, Glyn Adey and David Frisby (trans.), London: Heinemann, 1976.
Adorno, Theodor W. and Herbert Marcuse, 1969, “Correspondence on the German Student Movement”, New Left Review, 233 (1999), 123–136.
Agamben, Giorgio, 1995, Homo Sacer: Il potere sovrano e la nuda vita, Turin: Einaudi; translated as Homo Sacer: Sovereign Power and Bare Life, Daniel Heller-Roazen (trans.), Stanford: Stanford University Press, 1998.
Allen, Amy, 2008, The Politics of Our Selves: Power, Autonomy, and Gender in Contemporary Critical Theory, New York: Columbia University Press.
–––, 2010, “Third Generation Critical Theory: Benhabib, Fraser, and Honneth”, in Rosi Braidotti (ed.), After Poststructuralism: Transitions and Transformations, Durham: Acumen, pp. 129–48.
–––, 2012, “The Public Sphere: Ideology and/or Ideal?”, Political Theory, 40(6): 822–829. doi:10.1177/0090591712457664
–––, 2014, “Reason, Power and History: Re-reading the Dialectic of Enlightenment”, Thesis Eleven, 120(1): 10–25. doi:10.1177/0725513613519588
–––, 2015, “Emancipation without Utopia: Subjection, Modernity, and the Normative Claims of Feminist Critical Theory”, Hypatia, 30(3): 513–29. doi:10.1111/hypa.12160
–––, 2016, The End of Progress: Decolonizing the Normative Foundations of Critical Theory, New York: Columbia University Press.
–––, 2019, “Critical Theory and Feminism”, in Gordon, Peter, Espen Hammer, and Axel Honneth (eds.), The Routledge Companion to the Frankfurt School, London: Routledge, pp. 528–541.
–––, 2021, Critique on the Couch: Why Critical Theory Needs Psychoanalysis, New York: Columbia University Press.
Allen, Amy, and Eduardo Mendieta, (eds.), 2021, Decolonizing Ethics: The Critical Theory of Enrique Dussel, University Park, PA: Pennsylvania State University Press.
Allen, Amy, and Brian O’Connor (eds.), 2019, Transitional Subjects: Critical Theory and Object Relations, New York: Columbia University Press.
Anderson, Joel, 2000, “The ‘Third Generation’ of the Frankfurt School”, Intellectual History Newsletter 22, 49–61.
Apel, Karl-Otto, 1985, “Ist die Ethik der idealen Kommunikationsgemeinschaft eine Utopie? Zum Verhältnis von Ethik, Utopie und Utopiekritik”, in Wilhelm Vosskamp (ed.), Utopieforschung, Frankfurt: Suhrkamp; translated as “Is the Ideal Communication Community a Utopia? On the Relationship between Ethics, Utopia, and the Critique of Utopia”, David Frisby (trans.), in Seyla Benhabib and Fred Dallmayr (eds.), The Communicative Ethics Controversy, Cambridge, MA: MIT Press, 1990, pp. 23–59.
–––, 1989, “Normative Begründung der ‘Kritischen Theorie’” durch Rekurs auf lebensweltliche Sittlichkeit? Ein transzendentalpragmatisch orientierter Versuch, mit Habermas gegen Habermas zu denken”, in Axel Honneth, Thomas McCarthy, Claus Offe, and Albrecht Wellmer (eds.), Zwischenbetrachtungen im Prozess der Aufklärung: Jürgen Habermas zum 60. Geburtstag, Frankfurt am Main: Suhrkamp, 15–65; translated as “Normatively Grounding ‘Critical Theory’ through Recourse to Lifeworld? A Transcendental-Pragmatic Attempt to Think With Habermas Against Habermas”, William Rehg (trans.), in Axel Honneth, Thomas McCarthy, Claus Offe, and Albrecht Wellmer (eds.), Philosophical Interventions Into the Unfinished Project of Enlightenment, Cambridge, MA: MIT Press, 1992, pp. 125–70.
Azmanova, Albena, 2014, “Crisis? Capitalism is Doing Very Well. How is Critical Theory?”, Constellations, 21(3): 351–365. doi:10.1215/00382876-7165857
–––, 2020, Capitalism on Edge: How Fighting Precarity Can Achieve Radical Change Without Crisis or Utopia, New York: Columbia University Press.
Balibar, Etienne, and Immanuel Wallerstein, 1988, Race, nation, classe: les identités ambiguës, Paris: La Découverte; translated as Race, Nation, Class. Ambiguous Identities, Chris Turner (trans.), New York: Verso, 1991.
Bardawil, Fadi, 2018, “Césaire with Adorno: Critical Theory and the Colonial Problem”, South Atlantic Quarterly, 117(4): 773–789. doi:10.1215/00382876-7165857
Baum, Bruce, 2015, “Decolonizing Critical Theory”, Constellations, 22(3): 420–434. doi:10.1111/1467-8675.12169
Baynes, Kenneth, 2016, Habermas, New York: Routledge.
Benhabib, Seyla, 1986, Critique, Norm, and Utopia: A Study of the Foundations of Critical Theory, New York: Columbia University Press.
–––, 1992, Situating the Self: Gender, Community and Postmodernism in Contemporary Ethics, New York: Routledge.
–––, 2004, The Rights of Others: Aliens, Residents, and Citizens, Cambridge: Cambridge University Press.
Benhabib, Seyla, Judith Butler, Drucilla Cornell, and Nancy Fraser, 1995, Feminist Contentions: A Philosophical Exchange, New York: Routledge.
Benjamin, Walter, 1920/21, “Zur Kritik der Gewalt”, in Archiv für Sozialwissenschaft und Sozialpolitik, 47(3): 809–832; translated as “Toward the Critique of Violence”, Julia Ng (trans.), in Peter Fenves and Julia Ng (eds.), Toward the Critique of Violence: A Critical Edition, Stanford University Press, 2021, pp. 39–60.
–––, 1936, “Das Kunstwerk im Zeitalter seiner technischen Reproduzierbarkeit”, originally published as “L’œuvre d’art à l’époque de sa réproduction mécanisée”, Pierre Klossowski (trans.), Zeitschrift für Sozialforschung, 5(1), 40–68, all textual variants can be found in Walter Benjamin, Das Kunstwerk im Zeitalter seiner technischen Reproduzierbarkeit (Walter Benjamin Werke und Nachlaß: Kritische Gesamtausgabe vol. 16), Burkhardt Lindner (ed.), Berlin: Suhrkamp, 2012, the essay was translated as “The Work of Art in the Age of its Technological Reproducibility”, Michael W. Jennings (trans.), Grey Room, 39 (2010): 11–37. doi:10.1162/grey.2010.1.39.11
–––, 1940, “Über den Begriff der Geschichte”, in Walter Benjamin, Gesammelte Schriften I, Rolf Tiedemann and Hermann Schweppenhäuser (eds.), Frankfurt am Main, 1974, pp. 691–706; translated as “On the Concept of History”, Harry Zohn (trans.) in Howard Eiland and Michael W. Jennings (eds.) Walter Benjamin: Selected Writings (Volume 4), Cambridge, MA: Harvard University Press, 2006, pp. 389–400.
–––, 1955, Schriften, 2 vols., Theodor W. Adorno and Gretel Adorno (eds.), Frankfurt am Main: Suhrkamp; translated in parts as Illuminations, Harry Zohn (trans.), Hannah Arendt (ed.), New York: Harcourt, Brace & World, 1968.
Bernstein, Jay, 2001, Adorno: Disenchantment and Ethics, Cambridge: Cambridge University Press.
Bernstein, Richard, 2010, The Pragmatic Turn, Malden, MA: Polity.
–––, 2022, The Vicissitudes of Nature: From Spinoza to Freud, Malden, MA: Polity.
Bhambra, Gurminder K., 2021, “Decolonizing Critical Theory?: Epistemological Justice, Progress, Reparations”, Critical Times, 4(1): 73–89. doi:10.1215/26410478-8855227
Bhambra, Gurminder K., and John Holmwood, 2021, Colonialism and Modern Social Theory, Malden, MA: Polity.
Biro, Andrew, 2011, Critical Ecologies: The Frankfurt School and Contemporary Environmental Crises, Toronto: University of Toronto Press.
Bohman, James, 2003, “Critical Theory as Practical Knowledge: Participants, Observers, and Critics”, in S. P. Turner and P. A. Roth (eds.), The Blackwell Guide to the Philosophy of the Social Sciences, Oxford: Blackwell, pp. 91–109.
–––, 2007, Democracy Across Borders: From Dêmos to Dêmoi, Cambridge, MA: MIT Press.
Bojadžijev, Manuela, 2020, “Anti-Racism as Method”, in John Solomos (ed.), The Routledge International Handbook of Contemporary Racisms, London: Routledge 2020, pp. 193–204.
Bourdieu, Pierre, 1980, Le sens pratique, Paris: Minuit; translated as The Logic of Practice, Richard Nice (trans.), Stanford: Stanford University Press, 1992.
Brown, Wendy (ed.), 2006, “Feminist Theory and the Frankfurt School”: a Special Issue of differences: A Journal of Feminist Cultural Studies, 17(1).
–––, 2019, In the Ruins of Neoliberalism: The Rise of Antidemocratic Politics in the West, New York: Columbia University Press.
–––, 2022, “Rethinking Politics and Freedom in the Anthropocene”, Crisis and Critique, 9(2): 24–44 [Brown 2022 available online].
Brown, Wendy, and Rainer Forst, 2014, The Power of Tolerance: A Debate, New York: Columbia University Press.
Brown, Wendy, Peter Gordon, and Max Pensky, 2018, Authoritarianism. Three Inquiries in Critical Theory, Chicago: The University of Chicago Press.
Brunkhorst, Hauke, 1983, “Paradigmenkern und Theoriedynamik der Kritischen Theorie der Gesellschaft”, Soziale Welt, 34: 22–36; translated as “Paradigm-core and theory-dynamics in critical social theory: people and programs”, Peter Krockenberger (trans.), Philosophy & Social Criticism, 24(6) (1998): 67–110.
–––, 2002, Solidarität: Von der Bürgerfreundschaft zur globalen Rechtsgenossenschaft, Frankfurt am Main: Suhrkamp; translated as Solidarity: From Civic Friendship to a Global Legal Community, Jeffrey Flynn (trans.), Cambridge, MA: MIT Press, 2005.
–––, 2014, Critical Theory of Legal Revolutions: Evolutionary Perspectives, New York: Bloomsbury.
Buchstein, Hubertus, 2020, “Otto Kirchheimer and the Frankfurt School: Failed Collaborations in the Search for a Critical Theory of Politics”, New German Critique, 47(2): 81–106.
Buck-Morss, Susan, 1977, The Origin of Negative Dialectics: Theodor W. Adorno, Walter Benjamin and the Frankfurt Institute, New York: The Free Press.
–––, 1989, The Dialectics of Seeing: Walter Benjamin and the Arcades Project, Cambridge, MA: MIT Press.
–––, 1992, “Aesthetics and Anaesthetics: Walter Benjamin’s Artwork Essay Reconsidered”, October, 62: 3–41. doi:10.2307/778700
–––, 2009, Hegel, Haiti, and Universal History, Pittsburgh: University of Pittsburgh Press.
Butler, Judith, 1990, Gender Trouble, London: Routledge.
–––, 1997, The Psychic Life of Power: Theories in Subjection, Stanford, CA: Stanford University Press.
–––, 2012, Parting Ways: Jewishness and the Critique of Zionism, New York: Columbia University Press.
–––, 2015, Notes Toward a Performative Theory of Assembly, Cambridge, MA: Harvard University Press.
Butler, Judith, Jürgen Habermas, Charles Taylor, and Cornel West, 2011, The Power of Religion in the Public Sphere, New York: Columbia University Press.
Cassegård, Carl, 2021, Toward a Critical Theory of Nature: Capital, Ecology, and Dialectics, New York: Bloomsbury.
Catlin, Jonathon, 2023, “Antisemitism and Racism ‘After Auschwitz’: Adorno on the ‘Hellish Unity’ of ‘Permanent Catastrophe’ ”, in Marcel Stoetzler (ed.), Critical Theory and the Critique of Antisemitism, New York: Bloomsbury, pp. 203–230.
Celikates, Robin, 2006, “From Critical Social Theory to a Social Theory of Critique: On the Critique of Ideology after the Pragmatic Turn”, Constellations, 13(1): 21–40. doi:10.1111/j.1351-0487.2006.00438.x
–––, 2009, Kritik als soziale Praxis: Gesellschaftliche Selbstverständigung und kritische Theorie, Frankfurt am Main: Campus; translated as Critique as Social Practice: Critical Theory and Social Self-Understanding, Naomi van Steenbergen (trans.), London: Rowman & Littlefield, 2018.
–––, 2019a, “Critical Theory and the Unfinished Project of Mediating Theory and Practice”, in Peter Gordon, Espen Hammer, and Axel Honneth (eds.), The Routledge Companion to the Frankfurt School, London: Routledge, pp. 206–220.
–––, 2019b, “Constituent Power Beyond Exceptionalism: Irregular Migration, Disobedience, and (Re-)Constitution”, Journal of International Political Theory, 15(1): 67–81. doi:10.1177/1755088218808311
–––, 2022, “Remaking the Demos ‘from Below’? Critical Theory, Migrant Struggles, and Epistemic Resistance”, in Didier Fassin and Axel Honneth (eds.), Crisis Under Critique: How People Assess, Transform, and Respond to Critical Situations, New York: Columbia University Press, pp. 97–120.
Celikates, Robin, Sally Haslanger, and Jason Stanley (eds.), forthcoming, Analyzing Ideology, Oxford: Oxford University Press.
Celikates, Robin, Axel Honneth, and Rahel Jaeggi, 2023, “The Working Sovereign: A conversation with Axel Honneth”, Journal of Classical Sociology, 23(3): 318–338. doi:10.1177/1468795X231170980
Chambers, Simone, 2022, “Can Postmetaphysical Reason Escape its Provincial Roots?”, in Tom Bailey (ed.), Deprovincializing Habermas: Global Perspectives, Second Edition, New York: Routledge, pp. 229–248.
Chari, Anita, 2015, A Political Economy of the Senses: Neoliberalism, Reification, Critique, New York: Columbia University Press.
Cohen, Jean L., and Andrew Arato, 1992, Civil Society and Political Theory, Cambridge, MA: MIT Press.
Collins, Patricia Hill, 1990, Black Feminist Thought: Knowledge, Consciousness, and the Politics of Empowerment, London: Routledge. Second edition published in 2000.
–––, 2019, Intersectionality as Critical Social Theory, Durham and London: Duke University Press. doi:10.1515/9781478007098
Cook, Deborah, 2004, Adorno, Habermas, and the Search for a Rational Society, London and New York: Routledge.
–––, 2011, Adorno on Nature, Durham: Acumen.
Cooke, Maeve, 2006, Re-Presenting the Good Society, Cambridge, MA: MIT Press.
–––, 2020, “Ethics and Politics in the Anthropocene”, Philosophy & Social Criticism, 46(10): 1167–1181. doi:10.1177/0191453720903491
–––, 2023, “Reenvisioning Freedom: Human Agency in Times of Ecological Disaster”, Constellations, 30: 119–127. doi:10.1111/1467-8675.12681
Cordero, Rodrigo, 2017, Crisis and Critique: On the Fragile Foundations of Social Life, New York: Routledge.
Coulthard, Glen Sean, 2014, Red Skin, White Masks: Rejecting the Colonial Politics of Recognition, Minneapolis: University of Minnesota Press.
Crenshaw, Kimberlé, Neil Gotanda, Gary Peller, and Kendall Thomas, 1995, “Introduction”, Critical Race Theory: The Key Writings that Formed the Movement, New York: The New Press, pp. viii–xxxii.
Davis, Angela Y., 1983, Women, Race and Class, New York: Vintage.
–––, 2004, “Marcuse’s Legacies”, in John Abromeit and W. Mark Cobb (eds.), Herbert Marcuse: A Critical Reader. New York, Routledge, pp. 43–50.
–––, 2005, Abolition Democracy, New York: Seven Stories Press.
Dean, Jodi, 2016, Crowds and Party, London: Verso.
Dejours, Christophe, Jean-Philippe Deranty, Emmanuel Renault, and Nicholas H. Smith, 2018, The Return of Work in Critical Theory: Self, Society, Politics, New York: Columbia University Press.
Demirović, Alex, 2016, “The Frankfurt School, Critical Theory, and Sociology at the Institute for Social Research (1950 to 1960)”, in Gabriel R. Ricci (ed.), The Persistence of Critical Theory, London: Routledge, pp. 25–40.
Deranty, Jean-Philippe, 2021, “Negativity in Recognition: Post-Freudian Legacies in Contemporary Critical Theory”, in Ikäheimo, Heikki, Kristina Lepold and Titus Stahl (eds.), Recognition and Ambivalence, New York: Columbia University Press, pp. 223–255.
Derrida, Jacques, 1990 [year the lecture was given], Force de loi, Paris: Galilée, 1994; translated as “The Force of Law”, Mary Quaintance (trans.), in Drucilla Cornell, Michel Rosenfeld, and David G. Carlson (eds.), Deconstruction and the Possibility of Justice, New York: Routledge, 1992, pp. 3–67.
Dews, Peter, 1987, Logics of Disintegration: Poststructuralist Thought and the Claims of Critical Theory, London: Verso.
Dubiel, Helmut, 1978, Wissenschaftsorganisation und politische Erfahrung: Studien zur frühen Kritischen Theorie, Frankfurt am Main: Suhrkamp; translated as Theory and Politics: Studies in the Development of Critical Theory, Benjamin Gregg (trans.), Cambridge, MA: MIT Press, 1985.
Du Bois, W. E. B., 1935, Black Reconstruction in America, 1860–1880, New York: The Free Press, 1998.
Dussel, Enrique, 1998, Ética de la Liberación en la edad de globalización y de la exclusión; translated as Ethics of Liberation in the Age of Globalization and Exclusion, Eduardo Mendieta, Camilo Pérez Bustillo, Yolanda Angulo, and Nelson Maldonado-Torres (trans.), Durham, NC: Duke University Press, 2013.
–––, 2011, “From Critical Theory to the Philosophy of Liberation: Some Themes for Dialogue”, Transmodernity, 1(2): 16–43. doi.org:10.5070/T412011806
Eagleton, Terry, 1991, Ideology: An Introduction, London, Verso.
Feenberg, Andrew, 2023a, “Marcuse’s Critique of Technology Today”, Philosophy & Social Criticism 49(6): 672–685. doi:10.1177/01914537231164657
–––, 2023b, The Ruthless Critique of Everything Existing: Nature and Revolution in Marcuse’s Philosophy of Praxis, London: Verso.
Ferrara, Alessandro, 2008, The Force of the Example: Explorations in the Paradigm of Judgment, New York: Columbia University Press.
Ferrarese, Estelle, 2018, La fragilité du souci des autres: Adorno et le care, Lyon: ENS Éditions; translated as The Fragility of Concern for Others: Adorno and the Ethics of Care, Steven Corcoran (trans.), Edinburgh: Edinburgh University Press, 2020.
Finlayson, James Gordon, 2007, “Political, Moral and Critical Theory: On the Practical Philosophy of the Frankfurt School”, in Michael Rosen and B. Leiter (eds.), The Oxford Handbook of Continental Philosophy, Oxford: Oxford University Press, pp. 626–670.
–––, 2015, “The Artwork and the Promesse du Bonheur in Adorno”, European Journal of Philosophy, 23(3): 392–419. doi:10.1111/j.1468-0378.2012.00542.x
–––, 2019, The Habermas-Rawls Debate, New York: Columbia University Press.
Flynn, Jeffrey, 2014a, Reframing the Intercultural Dialogue on Human Rights: A Philosophical Approach, New York: Routledge.
–––, 2014b, “Truth, Objectivity, and Experience after the Pragmatic Turn: Bernstein on Habermas’s ‘Kantian Pragmatism’ ”, in Judith M. Green (ed.), Richard J Bernstein and the Pragmatist Turn in Contemporary Philosophy, Basingstoke and New York: Palgrave Macmillan, pp. 190–209.
–––, 2022, “Decentering Eurocentrism through Dialogue”, in Tom Bailey (ed.), Deprovincializing Habermas: Global Perspectives, Second Edition, New York: Routledge, pp. 249–270.
Fong, Benjamin Y., 2016, Death and Mastery: Psychoanalytic Drive Theory and the Subject of Late Capitalism, New York: Columbia University Press.
Forst, Rainer, 2007, Das Recht auf Rechtfertigung: Elemente einer konstruktivistischen Theorie der Gerechtigkeit, Berlin: Suhrkamp; translated as The Right to Justification: Elements of a Constructivist Theory of Justice, Jeffrey Flynn (trans.), New York: Columbia University Press, 2011.
–––, 2011, Kritik der Rechtfertigungsverhältnisse, Berlin: Suhrkamp; translated as Justification and Critique: Towards a Critical Theory of Politics, Ciaran Cronin (trans.), Cambridge: Polity, 2014.
–––, 2017, “Noumenal Alienation: Rousseau, Kant and Marx on the Dialectics of Self-Determination”, Kantian Review, 22(4): 523–551. doi:10.1017/S1369415417000267
–––, 2021a, Die noumenale Republik: Kritischer Konstruktivismus nach Kant, Berlin: Suhrkamp; translation, The Noumenal Republic: Critical Constructivism after Kant, Ciaran Cronin (trans.), forthcoming.
–––, (ed)., 2021b, “Symposium on Jürgen Habermas, Auch eine Geschichte der Philosophie”, Constellations, 28(1): 1–147.
–––, 2023, “The rule of unreason: Analyzing (Anti-)Democratic Regression”, Constellations, 30(3): 217–224. doi:10.1111/1467-8675.12671
Foucault, Michel, 1973 [year this lecture was given], “La vérités et les formes juridiques”, Chimères, 10 (1990): 8–28; translated as “Truth and Juridical Forms”, in Michel Foucault, Power, ed. James D. Faubion, New York: New Press, 2000, pp. 31–45.
Fraser, Nancy, 1981, “Foucault on Modern Power: Empirical Insights and Normative Confusions”, PRAXIS International, 3: 272–287.
–––, 1985, “What’s Critical about Critical Theory? The Case of Habermas and Gender”, New German Critique, 35: 97–131. doi:10.2307/488202
–––, 1989, Unruly Practices: Power, Discourse, and Gender in Contemporary Social Theory, Minneapolis: University of Minnesota Press.
–––, 1990, “Rethinking the Public Sphere: A Contribution to the Critique of Actually Existing Democracy.” Social Text, 25/26: 56–80.
–––, 2011, “Marketization, Social Protection, Emancipation: Toward a Neo-Polanyian Conception of Capitalist Crisis”, in Craig Calhoun and Georgi Derlugian (eds.), Business as Usual: The Roots of the Global Financial Meltdown, New York University Press, pp. 137–58.
–––, 2013, Fortunes of Feminism: From State-Managed Capitalism to Neoliberal Crisis, London: Verso.
–––, 2014, “Behind Marx’s Hidden Abode”, New Left Review, 86: 55–72.
–––, 2016, “Contradictions of Capital and Care”, New Left Review, 100: 99–117.
–––, 2021, “Climates of Capital”, New Left Review, 127: 94–127.
–––, 2022, Cannibal Capitalism: How Our System is Devouring Democracy, Care, and the Planet – And What We Can Do about It, New York: Verso.
Fraser, Nancy, et al., 2014, Transnationalizing the Public Sphere, Kate Nash (ed.), Cambridge: Polity.
Fraser, Nancy, and Axel Honneth, 2003, Redistribution or Recognition? A Philosophical-Political Exchange, New York: Verso.
Fraser, Nancy and Rahel Jaeggi, 2018, Capitalism: A Conversation in Critical Theory, Brian Milstein (ed.), Cambridge: Polity.
Freyenhagen, Fabian, 2014, “Adorno’s Politics: Theory and Praxis in Germany’s 1960s”, Philosophy & Social Criticism, 40: 867–893. doi:10.1177/0191453714545198
–––, 2018, “Critical Theory: Self-Reflexive Theorizing and Struggles for Emancipation”, in: Oxford Research Encyclopedia of Politics [Freyenhagen 2018 available online].
Fricker, Miranda, 2007, Epistemic Injustice, Oxford: Oxford University Press.
Fromm, Erich, 1936, “Studien über Autorität und Familie. Sozialpsychologischer Teil”, in Max Horkheimer (ed.), Schriften des Instituts für Sozialforschung, Vol. V: Studien über Autorität und Familie, Paris: Librairie Félix Alcan; translated as “Studies on Authority and the Family. Socio-psychological Dimensions”, Fromm Forum, 24 (2020): 8–58.
–––, 1961, Marx’s Concept of Man, New York: Continuum.
Gago, Verónica, 2019, La potencia feminista: O el deseo de cambiarlo todo, Buenos Aires: Tinta Limón; translated as Feminist International: How to Change Everything, Liz Mason-Deese (trans.), London: Verso, 2020.
Gambetti, Zeynep, 2020, “Exploratory Notes on the Origins of New Fascisms”, Critical Times, 3(1): 1–32. doi:10.1215/26410478-8189841
Geuss, Raymond, 1981, The Idea of a Critical Theory, Cambridge: Cambridge University Press.
Gilmore, Ruth Wilson, 2022, Abolition Geography: Essays Towards Liberation, London: Verso.
Gordon, Peter, 2017, “The Authoritarian Personality Revisited”, boundary 2, 44(2): 31–56. doi:10.1215/01903659-3826618
–––, 2023, A Precarious Happiness: Adorno and the Sources of Normativity, Chicago: University of Chicago Press. doi:10.4324/9780429443374
Gordon, Peter, Espen Hammer, and Axel Honneth (eds.), 2019, The Routledge Companion to the Frankfurt School, London: Routledge. doi:10.4324/9780429443374
Gregoratto, Federica, Heikki Ikäheimo, Emmanuel Renault, Arvi Särkelä, and Italo Testa, 2022, “Critical Naturalism: A Manifesto”, Krisis: Journal for Contemporary Philosophy, 42(1): 108–24. doi:10.21827/krisis.42.1.38637
Grüner, Eduardo, 2010, La oscuridad y las luces, Buenos Aires: Edhasa; translated as The Haitian Revolution: Capitalism, Slavery and Counter-Modernity, Ramsey McGlazer (trans.), Cambridge: Polity, 2019.
Habermas, Jürgen, 1962, Strukturwandel der Öffentlichkeit: Untersuchungen zu einer Kategorie der bürgerlichen Gesellschaft, Neuwied, Berlin: Luchterhand; translated as The Structural Transformation of the Public Sphere: An Inquiry Into a Category of Bourgeois Society, Thomas Burger and Frederick Lawrence (trans.), Cambridge, MA: MIT Press, 1989.
–––, 1963, Theorie und Praxis: Sozialphilosophische Studien, Neuwied am Rhein and Berlin: Luchterhand. New and extended edition Frankfurt am Main: Suhrkamp, 1971; translated as Theory and Practice, John Viertel (trans.), Boston: Beacon Press, 1973.
–––, 1968a, Technik und Wissenschaft als ‘Ideologie’, Frankfurt am Main: Suhrkamp; translated as Chapters 4–6 of Toward a Rational Society: Student Protest, Science, and Politics, Jeremy J. Shapiro (trans.), Boston: Beacon Press, 1970.
–––, 1968b, Erkenntnis und Interesse, Frankfurt am Main: Suhrkamp; translated as Knowledge and Human Interests, Jeremy J. Shapiro (trans.), Boston: Beacon Press, 1971.
–––, 1973a, “Nachwort”, in Jürgen Habermas, Erkenntnis und Interesse, Mit einem neuen Nachwort, Frankfurt am Main: Suhrkamp, pp. 367–417; translated as “A Postscript to Knowledge and Human Interests”, Christian Lenhardt (trans.), Philosophy of the Social Sciences, 3: 157–189.
–––, 1973b, Legitimationsprobleme im Spätkapitalismus, Frankfurt am Main: Suhrkamp; translated as Legitimation Crisis, Thomas McCarthy (trans.), Boston: Beacon Press, 1975.
–––, 1981, Theorie des kommunikativen Handelns, 2 vols., Frankfurt am Main: Suhrkamp; translated as The Theory of Communicative Action, 2 vols., Thomas A. McCarthy (trans.), Boston: Beacon Press, 1984.
–––, 1983a, Moralbewusstsein und kommunikatives Handeln, Frankfurt am Main: Suhrkamp, translated as Moral Consciousness and Communicative Action, Christian Lenhardt and Shierry Weber Nicholsen (trans.), Cambridge, MA: MIT Press, 1990.
–––, 1983b, “Ziviler Ungehorsam: Testfall für den demokratischen Rechtsstaat”, in Peter Glotz (ed.), Ziviler Ungehorsam im Rechtsstaat, Frankfurt am Main: Suhrkamp, pp. 29–53; translated as “Civil Disobedience: Litmus Test for the Democratic Constitutional State”, Berkeley Journal of Sociology, 30: 95–116.
–––, 1985, Der philosophische Diskurs der Moderne: Zwölf Vorlesungen, Frankfurt am Main: Suhrkamp; translated as The Philosophical Discourse of Modernity: Twelve Lectures, Frederick Lawrence (trans.), Cambridge, MA: MIT Press, 1987.
–––, 1991, Erläuterungen zur Diskursethik, Frankfurt am Main: Suhrkamp; translated as Justification and Application: Remarks on Discourse Ethics, Ciaran Cronin (trans.), Cambridge, MA: MIT Press, 1993.
–––, 1992, Faktizität und Geltung: Beiträge zur Diskurstheorie des Rechts und des demokratischen Rechtsstaats, Frankfurt am Main: Suhrkamp; translated as Between Facts and Norms: Contributions to a Discourse Theory of Law and Democracy, William Rehg (trans.), Cambridge, MA: MIT Press, 1996.
–––, 1995a, “Reconciliation Through the Public Use of Reason: Remarks on John Rawls’s Political Liberalism”, The Journal of Philosophy, 92(3): 109–131. doi:10.5840/jphil199592335
–––, 1995b, “Kants Idee des ewigen Friedens aus dem historischen Abstand von 200 Jahren”, Kritische Jusitiz, 3: 293–319; translated as “Kant’s Idea of Perpetual Peace, with the Benefit of Two Hundred Years’ Hindsight”, in James Bohman and Matthias Lutz-Bachmann (eds.), Perpetual Peace: Essays on Kant’s Cosmopolitan Ideal, Cambridge, MA: MIT Press, 1997, pp. 113–153.
–––, 1996, “Vernünftig versus Wahr oder die Moral der Weltbilder”, in Die Einbeziehung des Anderen: Studien zur politischen Theorie, Frankfurt am Main: Suhrkamp; translated as “‘Reasonable’ versus ‘True,’ or the Morality of Worldviews”, Ciaran Cronin (trans.), in The Inclusion of the Other: Studies in Political Theory, Ciaran Cronin and Pablo De Greiff (eds.), Cambridge, MA: MIT Press, 1998, pp. 75-105.
–––, 1998, Die postnationale Konstellation: politische Essays, Frankfurt am Main: Suhrkamp; translated as The Postnational Constellation: Political Essays, Max Pensky (ed./trans.), Cambridge, MA: MIT Press, 2001.
–––, 1999, Wahrheit und Rechtfertigung: philosophische Aufsätze, Frankfurt am Main: Suhrkamp; translated as Truth and Justification, Barbara Fultner (trans.), Cambridge, MA: MIT Press, 2003.
–––, 2005, Zwischen Naturalismus und Religion. Philosophische Aufsätze, Frankfurt am Main: Suhrkamp; translated as Between Naturalism and Religion, Ciaran Cronin (trans.), Malden, MA: Polity, 2008.
–––, 2006, “Political Communication in Media Society: Does Democracy Still Enjoy an Epistemic Dimension? The Impact of Normative Theory on Empirical Research”, Communication Theory, 16(4): 411–426. doi:10.1111/j.1468-2885.2006.00280.x
–––, 2019, Auch eine Geschichte der Philosophie, 2 vols., Berlin: Suhrkamp; part of volume 1 translated as Also a History of Philosophy, Volume 1: The Project of a Genealogy of Postmetaphysical Thinking, Ciaran Cronin (trans.), Cambridge: Polity, 2023.
–––, 2021, “Überlegungen und Hypothesen zu einem erneuten Strukturwandel der politischen Öffentlichkeit”, in: Martin Seeliger and Sebastian Sevignani (eds.), Ein neuer Strukturwandel der Öffentlichkeit?, Baden-Baden: Nomos Verlag, 470–500; translated as “Reflections and Hypotheses on a Further Structural Transformation of the Political Public Sphere”, Ciaran Cronin (trans.), Theory, Culture & Society, 39(4) (2022): 145–171. doi:10.1177/02632764221112341
Hall, Stuart, 2021, Selected Writings on Race and Difference, Durham, NC: Duke University Press.
Harcourt, Bernard E., 2020, Critique and Praxis, New York: Columbia University Press.
Haslanger, Sally, 2017a, “Culture and Critique”, Proceedings of the Aristotelian Society (Supplementary Volume), 91: 149–173. doi:10.1093/arisup/akx001
–––, 2017b, “Racism, Ideology, and Social Movements”, Res Philosophica, 94(1): 1–22. doi:10.11612/RESPHIL.1547
Heath, Joseph, 2014, “Rebooting Discourse Ethics”, Philosophy & Social Criticism, 40(9): 829–866. doi:10.1177/0191453714545340
Hedrick, Todd, 2010, Rawls and Habermas: Reason, Pluralism, and the Claims of Political Philosophy, Stanford, CA: Stanford University Press.
–––, 2019, Reconciliation and Reification: Freedom’s Semblance and Actuality from Hegel to Contemporary Critical Theory, Oxford: Oxford University Press.
Heberle, Renée (ed.), 2006, Feminist Interpretations of Theodor Adorno, University Park, PA: The Pennsylvania University Press.
Honneth, Axel, 1985, Kritik der Macht: Reflexionsstufen einer kritischen Gesellschaftstheorie, Frankfurt am Main: Suhrkamp; translated as Critique of Power: Reflective Stages in a Critical Social Theory, Kenneth Baynes (trans.), Cambridge, MA: MIT Press, 1991.
–––, 1992, Kampf um Anerkennung: Zur moralischen Grammatik sozialer Konflikte, Frankfurt am Main: Suhrkamp; translated as The Struggle for Recognition: The Moral Grammar of Social Conflicts, Joel Anderson (trans.), Cambridge: Polity, 1995.
–––, 1994, “Die soziale Dynamik von Mißachtung: Zur Ortsbestimmung einer kritischen Gesellschaftstheorie”, Leviathan, 22(1): 78–93; translated as “The Social Dynamics of Disrespect: On the Location of Critical Theory Today”, John Farrell (trans.), Constellations, 1(2): 255–69, reprinted in in Axel Honneth, Disrespect: The Normative Foundations of Critical Theory, Cambridge: Polity Press, 2007, pp. 63–79.
–––, 1998, “Über die Möglichkeit einer erschließenden Kritik. Die Dialektik der Aufklärung im Horizont gegenwärtiger Debatten über Sozialkritik”, Paradigmi. Rivista di critica filosofica, 16(48): 501–514; translated as “The Possibility of a Disclosing Critique of Society: The Dialectic of Enlightenment in Light of Current Debates in Social Criticism”, Constellations, 7(1) (2000): 116–127. doi:10.1111/1467-8675.00173
–––, 2000, Das Andere der Gerechtigkeit; translated as Disrespect: The Normative Foundations of Critical Theory, Joseph Ganahl (trans.), Cambridge: Polity, 2007.
–––, 2001, Leiden an Unbestimmtheit: Eine Reaktualisierung der Hegelschen Rechtsphilosophie, Stuttgart: Reclam; translated as The Pathologies of Individual Freedom: Hegel’s Social Theory, Ladislaus Löb (trans.), Princeton: Princeton University Press, 2010.
–––, 2003, “Redistribution as Recognition”, in N. Fraser and A. Honneth (eds.), Redistribution or Recognition? A Political-Philosophical Exchange, London: Verso, pp. 110–197.
–––, 2004, “Eine soziale Pathologie der Vernunft. Zur intellektuellen Erbschaft der Kritischen Theorie”, in Christoph Halbig and Michael Quante (eds.), Axel Honneth: Sozialphilosophie zwischen Kritik und Anerkennung, Münster: LIT-Verlag, pp. 9-32; translated as “A Social Pathology of Reason: On the Intellectual Legacy of Critical Theory”, in A. Honneth, Pathologies of Reason: On the Legacy of Critical Theory, New York: Columbia University Press, 2009, pp. 19-42.
–––, 2005, Verdinglichung: Eine anerkennungstheoretische Studie, Frankfurt am Main: Suhrkamp; translated as Reification, Martin Jay (ed.), New York: Oxford University Press, 2008.
–––, 2010, Das Ich im Wir: Studien zur Anerkennungstheorie, Berlin: Suhrkamp; translated as The I in We: Studies in the Theory of Recognition, Joseph Ganahl (trans.), Cambridge: Polity, 2012.
–––, 2011, Das Recht der Freiheit: Grundriß einer demokratischen Sittlichkeit, Berlin: Suhrkamp; translated as Freedom’s Right: The Social Foundations of Democratic Life, Joseph Ganahl (trans.), New York: Columbia University Press, 2014.
–––, 2017, “Is There an Emancipatory Interest? An Attempt to Answer Critical Theory’s Most Fundamental Question”, European Journal of Philosophy, 25: 908–920. doi:10.1111/ejop.12321
–––, 2022, “‘Labour’, A Brief History of a Modern Concept”, Philosophy, 97(2), 149–167. doi:10.1017/S003181912100036X
–––, 2023, Der arbeitende Souverän, Berlin: Suhrkamp; translation, The Working Sovereign, Cambridge: Polity, forthcoming.
Honneth, Axel, and Hans Joas (eds.), 1986, Kommunikatives Handeln: Beiträge zu Jürgen Habermas ‘Theorie des Kommunikativen Handelns, Frankfurt am Main: Suhrkamp; translated as Communicative Action: Essays on Jürgen Habermas’s The Theory of Communicative Action, Jeremy Gaines and Doris L. Jones (trans.), Cambridge, MA: MIT Press, 1991.
Horkheimer, Max, 1931, “Die gegenwärtige Lage der Sozialphilosophie und die Aufgaben eines Instituts für Sozialforschung”, Frankfurter Universitätsreden, XXXVII: 3–16; translated as “The Present Situation of Social Philosophy and the Tasks of an Institute for Social Research”, John Torpey (trans.), in Max Horkheimer, Between Philosophy and Social Science: Selected Early Writings, Cambridge, MA: MIT Press, 1993, pp. 1–14. Retranslated as “The State of Contemporary Social Philosophy and the Tasks of an Institute for Social Research”, Journal for Cultural Research, Peter Wagner (trans.), 22(2) (2018): 113–121. doi:10.1080/14797585.2018.1461354
–––, 1933, “Materialismus und Moral”, Zeitschrift für Sozialforschung, 2(2): 162–197; translated as “Materialism and Morality”, G. Frederick Hunter (trans.), in Max Horkheimer, Between Philosophy and Social Science: Selected Early Writings, Cambridge: MIT Press, 1993, pp. 15–47.
–––, 1936a, “Egoismus und Freiheitsbewegung: Zur Anthropologie des bürgerlichen Zeitalters”, Zeitschrift für Sozialforschung, 5(2): 161–234; translated as “Egoism and Freedom Movements: On the Anthropology of the Bourgeois Era”, G. Frederick Hunter (trans.), in Max Horkheimer, Between Philosophy and Social Science: Selected Early Writings, Cambridge: MIT Press, 1993, pp. 49–110.
––– (ed.), 1936b, Studien über Autorität und Familie: Forschungsberichte aus dem Institut für Sozialforschung, Paris: Librairie Félix Alcan.
–––, 1937a, “Traditionelle und kritische Theorie”, Zeitschrift für Sozialforschung, 6(2): 245–294; translated as “Traditional and Critical Theory”, Matthew J. O’Connell (trans.), in Max Horkheimer, Critical Theory: Selected Essays, New York: Continuum, 1972, pp. 188–243.
–––, 1937b, “Nachtrag”, Zeitschrift für Sozialforschung, 6(3): 625–631; translated as “Postscript”, Matthew J. O’Connell (trans.), in Max Horkheimer, Critical Theory: Selected Essays, New York: Continuum, 1972, pp. 244–252.
–––, 1941, “Art and Mass Culture”, Zeitschrift für Sozialforschung, 9(2), 290–304, republished in Max Horkheimer, Critical Theory: Selected Essays, New York: Continuum, 1972, pp. 273–290.
Horkheimer, Max, and Theodor W. Adorno, 1947, Dialektik der Aufklärung: Philosophische Fragmente, Amsterdam: Querido; translated as Dialectic of Enlightenment, Edmund Jephcott (trans.), Stanford, CA: Stanford University Press, 2002.
Hoy, David Couzens and Thomas McCarthy, 1994, Critical Theory, Oxford: Blackwell.
Ibsen, Malte Froslee, 2023, A Critical Theory of Global Justice: The Frankfurt School and World Society, Oxford: Oxford University Press.
Ikäheimo, Heikki, Kristina Lepold and Titus Stahl (eds.), 2021, Recognition and Ambivalence, New York: Columbia University Press.
Ingram, David, 2018, World Crisis and Underdevelopment: A Critical Theory of Poverty, Agency, and Coercion, Cambridge: Cambridge University Press.
Ingram, James, 2013, Radical Cosmopolitics: The Ethics and Politics of Democratic Universalism, New York: Columbia University Press.
–––, 2019, “Critical Theory and Postcolonialism”, in Gordon, Peter, Espen Hammer, and Axel Honneth (eds.), The Routledge Companion to the Frankfurt School, London: Routledge, pp. 500–513.
Jaeggi, Rahel, 2005, Entfremdung: Zur Aktualität eines sozialphilosophischen Problems, Campus; translated as Alienation, Frederick Neuhouser and Alan E. Smith (trans.), New York: Columbia University Press, 2014.
–––, 2008, “Re-Thinking Ideology”, in Christopher Zurn, Boujdewijn de Bruijn (eds.), New Waves in Political Philosophy, Basingstoke: Palgrave Macmillan, 2008.
–––, 2014, Kritik von Lebensformen, Berlin.: Suhrkamp; translated as Critique of Forms of Life, Ciaran Cronin (trans.), Cambridge, MA: Harvard University Press, 2018.
–––, 2017a, “Crisis, Contradiction, and the Task of a Critical Theory”, in Banu Bargu und Chiara Bottici (eds.), Feminism, Capitalism, and Critique. Essays in Honor of Nancy Fraser, Basingstoke: Palgrave Macmillan, pp. 209–224.
–––, 2017b, “Pathologies of Work”, Women’s Studies Quarterly, 45(3/4): 59–76. doi:10.1353/wsq.2017.0044
–––, 2022, “Modes of Regression: The Case of Ressentiment”, Critical Times, 5(3): 501–537. doi:10.1215/26410478-10030204
Jay, Martin, 1973, The Dialectical Imagination, Boston and Toronto: Little, Brown.
–––, 1984, Marxism and Totality, Cambridge: Polity.
Kelley, Robin D. G., 2002, Freedom Dreams: The Black Radical Imagination, Boston: Beacon Press. Revised and expanded edition published in 2022.
Kerner, Ina, 2018, “Postcolonial Theories as Global Critical Theories”, Constellations, 25(4): 614– 628. doi:10.1111/1467-8675.12346
Klein, Steven, 2020, The Work of Politics: Making a Democratic Welfare State, Cambridge: Cambridge University Press.
Kompridis, Nikolas, 2006, Critique and Disclosure: Critical Theory between Past and Future, Cambridge, MA: MIT Press.
Koopman, Colin, 2013, Genealogy as Critique, Bloomington: Indiana University Press.
Kracauer, Siegfried, 1927, “Das Ornament der Masse”, Frankfurter Zeitung, July 9–10, 1927; translated as “Mass Ornament”, Barbara Correll and Jack Zipes (trans.) New German Critique, 5 (1975): 67–76.
–––, 2013, Totalitäre Propaganda, Bernd Stiegler (ed.), Berlin: Suhrkamp. Selections are translated as part of “Studies of Totalitarianism, Propaganda, and the Masses (1936–1940)” in Siegfried Kracauer, Selected Writings on Media, Propaganda, and Political Communication, Jaeho Kang, Graeme Gilloch, and John Abromeit (eds.), New York: Columbia University Press, 2022.
Laclau, Ernesto, 1992, “Beyond Emancipation”, in Emancipation(s), London: Verso, 1996, pp. 1–19.
Lafont, Cristina, 2019, Democracy without Shortcuts: A Participatory Conception of Deliberative Democracy, Oxford: Oxford University Press.
Lara, Maria Pia, 1998, Moral Textures: Feminist Narratives in the Public Sphere, Cambridge: Polity.
–––, 2021, Beyond the Public Sphere: Film and the Feminist Imaginary, Evanston, IL: Northwestern University Press.
Leeb, Claudia, 2017, Power and Feminist Agency in Capitalism: Toward a New Theory of the Political Subject, Oxford: Oxford University Press.
Löwenthal, Leo, and Norbert Guterman, 1949, Prophets of Deceit: A Study of the Techniques of the American Agitator (Studies in Prejudice: Volume 5), with a new Introduction by Alberto Toscano, London: Verso, 2021.
Löwy, Michael, 2001, Walter Benjamin: Avertissement d’incendie, Paris: Presses Universitaires de France; translated as Fire Alarm: Reading Walter Benjamin’s ‘On the Concept of History’, Chris Turner (trans.), London: Verso, 2016.
Loick, Daniel, 2012, Kritik der Souveränität, Frankfurt: Campus; translated as A Critique of Sovereignty, Amanda DeMarco (trans.) London: Rowman & Littlefield, 2018.
–––, 2021, “The Ethical Life of Counter-Communities”, Critical Times, 4(1): 1–28. doi:10.1215/26410478-8855203
Lukács, Georg, 1923, “Verdinglichung und das Bewußtsein des Proletariats”, in Geschichte und Klassenbewußtsein, Berlin: Malik; translated as “Reification and the Consciousness of the Proletariat”, in History and Class Consciousness, Rodney Livingstone (trans.), Cambridge, MA: MIT Press, 1971, pp. 83–222.
–––, 1963, “Vorwort” to Die Theorie des Romans, Berlin: Luchterhand; translated as The Theory of the Novel, Anna Bostock (trans.), Cambridge, MA: MIT Press, 1974.
Malm, Andreas, 2020, How to Blow Up a Pipeline: Learning to Fight in a World on Fire, London: Verso.
Marasco, Robyn, 2006, “‘Already the Effect of the Whip’: Critical Theory and the Feminine Ideal”, differences, 17(1): 88–115. doi:10.1215/10407391-2005-005
–––, 2015, The Highway of Despair: Critical Theory After Hegel, New York: Columbia University Press.
–––, 2018, “There’s a Fascist in the Family: Critical Theory and Antiauthoritarianism”, South Atlantic Quarterly, 117(4): 791–813. doi:10.1215/00382876-7165871
Marcuse, Herbert, 1937, “Philosophie und kritische Theorie”, Zeitschrift für Sozialforschung, 6(3): 625–647; translated as “Philosophy and Critical Theory”, in H. Marcuse (ed.), Negations, Jeremy J. Shapiro (trans), London: MayFlyBooks, 1968, pp. 99–117.
–––, 1941, Reason and Revolution: Hegel and the Rise of Social Theory, Oxford: Oxford University Press.
–––, 1955, Eros and Civilization, Boston: Beacon Press.
–––, 1964, One-Dimensional Man: Studies in the Ideology of Advanced Industrial Society, Boston: Beacon Press.
–––, 1967, “Liberation from the Affluent Society” in Douglas Kellner (ed.), The New Left and the 1960s: Collected Papers of Herbert Marcuse, Volume 3, New York: Routledge, 2004.
–––, 1968, “Beyond One-Dimensional Man”, in Douglas Kellner (ed.), Towards a Critical Theory of Society, Collected Papers of Herbert Marcuse, Volume 2, New York: Routledge, 2001.
–––, 1969, An Essay on Liberation, Boston: Beacon Press.
–––, 1974, “Marxism and Feminism”, Women’s Studies, 2: 279–288.
Marx, Karl, 1843, “Brief an Ruge, September 1843”, in Karl Marx and Friedrich Engels, Briefwechsel bis April 1846 (Karl Marx/Friedrich Engels Gesamtausgabe, vol. III.1), Berlin: Dietz: 1975, pp. 54–57; translated as “Letter to Ruge, September 1843”, in Karl Marx: Early Writings (Karl Marx/Friedrich Engels Collected Works, vol. 1), Rodney Livingstone (trans.), London: Lawrence and Wishart, 1975, pp. 209–211.
–––, 1844, “Zur Kritik der Hegelschen Rechtsphilosophie: Einleitung”, in Karl Marx: Werke, Artikel, Entwürfe: März 1843 – August 1944 (Karl Marx/Friedrich Engels Gesamtausgabe, vol. I.2), Berlin: Dietz, 1975, pp. 170–184; translated as “Contribution to the Critique of Hegel’s Philosophy of Law”, Martin Milligan and Barbara Ruhemann (trans.), in Karl Marx March 1843 – August 1944 (Karl Marx/Friedrich Engels Collected Works, vol. 3), London: Lawrence and Wishart, 1975, pp. 3–127.
–––, 1846, Die deutsche Ideologie: Manuskripte und Drucke (Karl Marx/Friedrich Engels Gesamtausgabe, vol. I.5), Berlin: De Gruyter/Akademie, 2017; translated as The German Ideology (Karl Marx/Friedrich Engels Collected Works, vol. 5), Clemens Dutt, W. Lough, and C. P. Magill (trans.), London: Lawrence and Wishart, 1976.
–––, 1867, Das Kapital: Kritik der politischen Ökonomie. Erster Band (Hamburg 1867) (Karl Marx/Friedrich Engels Gesamtausgabe, vol. II.5), Berlin: Dietz, 1983; translated as Capital, Volume I (Karl Marx/Friedrich Engels Collected Works, vol. 35), London: Lawrence and Wishart, 1996.
Mbembe, Achille, 2016, Politiques de l’inimitié; translated as Necropolitics, Steven Corcoran (trans.), Durham, NC: Duke University Press, 2019.
McAfee, Noëlle, 2019, Fear of Breakdown: Politics and Psychoanalysis, New York: Columbia University Press.
McCarthy, Thomas, 1991, Ideals and Illusions: On Reconstruction and Deconstruction in Contemporary Critical Theory, Cambridge, MA: MIT Press.
–––, 2004, “Political Philosophy and Racial Injustice: From Normative to Critical Theory” in Seyla Benhabib and Nancy Fraser (eds.), Pragmatism, Critique, Judgment: Essays for Richard J. Bernstein, Cambridge, MA: MIT Press, pp. 147–168.
–––, 2009, Race, Empire, and the Idea of Human Development, Cambridge: Cambridge University Press.
McNay, Lois, 2022, The Gender of Critical Theory: On the Experiential Grounds of Critique, Oxford: Oxford University Press.
Medina, José, 2013, The Epistemology of Resistance: Gender and Racial Oppression, Epistemic Injustice, and Resistant Imaginations, Oxford: Oxford University Press.
Meehan, Johanna (ed.), 1995, Feminists Read Habermas: Gendering the Subject of Discourse, New York: Routledge.
Mendieta, Eduardo, 2007, Global Fragments: Globalizations, Latinamericanisms, and Critical Theory, New York: SUNY Press.
Menke, Christoph, 1988, Die Souveränität der Kunst: Ästhetische Erfahrung nach Adorno und Derrida; translated as The Sovereignty of Art: Aesthetic Negativity in Adorno and Derrida, Neil Solomon (trans.), Cambridge, MA: MIT Press, 1998.
–––, 2000, Spiegelungen der Gleichheit, Berlin: Akademie; translated as Reflections of Equality, Howard Rouse and Andrei Denejkine (trans.), Stanford, CA: Stanford University Press, 2006.
–––, 2015, Kritik der Rechte, Berlin: Suhrkamp; translated as Critique of Rights, Christopher Turner (trans.), Cambridge: Polity, 2020.
Mills, Charles, 1988, “Alternative Epistemologies”, Social Theory and Practice, 14(3): 237–63.
–––, 2003, From Class to Race: Essays in White Marxism and Black Radicalism, New York: Rowman & Littlefield.
–––, 2017, “Criticizing Critical Theory”, in Critical Theory in Critical Times: Transforming the Global Political and Economic Order, Penelope Deutscher and Cristina Lafont (eds.), New York: Columbia University Press, pp. 233–250.
Milstein, Brian, 2015, Commercium: Critical Theory from a Cosmopolitan Point of View, New York: Rowman & Littlefield.
Moten, Fred, and Stefano Harney, 2013, The Undercommons: Fugitive Planning and Black Study, London: Minor Compositions/Autonomedia.
Müller-Doohm, Stefan, 2003, Adorno: Eine Biographie, Frankfurt am Main: Suhrkamp; translated as Adorno: A Biography, Rodney Livingstone (trans.), Cambridge: Polity, 2005.
Negt, Oskar, and Alexander Kluge, 1972, Öffentlichkeit und Erfahrung: Zur Organisationsanalyse von bürgerlicher und proletarischer Öffentlichkeit, Frankfurt: Suhrkamp; translated as Public Sphere and Experience: Analysis of the Bourgeois and Proletarian Public Sphere, Peter Labanyi, Jamie Owen Daniel, and Assenka Oksiloff (trans.), London: Verso, 2016.
Neuhouser, Frederick, 2022, Diagnosing Social Pathology: Rousseau, Hegel, Marx, and Durkheim, Cambridge: Cambridge University Press.
Neumann, Franz L., 1944, Behemoth: The Structure and Practice of National Socialism, 1933–1944, New York: Oxford University Press.
Ng, Julia, 2023, “The Action of Non-Action: Walter Benjamin, Wu Wei and the Nature of Capitalism”, Theory, Culture & Society, 40(4–5): 219–238. doi:10.1177/02632764231169944
Ng, Karen, 2015, “Ideology Critique from Hegel and Marx to Critical Theory”, Constellations, 22(3): 393–404. doi:10.1111/1467-8675.12170
O’Connor, Brian, 2004, Adorno’s Negative Dialectic: Philosophy and the Possibility of a Critical Rationality, Cambridge, MA: MIT Press.
Offe, Claus, 1984, Contradictions of the Welfare State, John Keane (ed.), Cambridge, MA: The MIT Press.
Outlaw, Lucius T., Jr., 2005, Critical Social Theory in the Interests of Black Folks, London: Rowman & Littlefield.
Owen, David, 2002, “Criticism and Captivity: On Genealogy and Critical Theory”, European Journal of Philosophy, 10: 216–230. doi:10.1111/1468-0378.00158
Pensky, Max, 1993, Melancholy Dialectics: Walter Benjamin and the Play of Mourning, Amherst, MA: University of Massachusetts Press.
–––, 2019, “Western Marxism: Revolutions in Theory”, in Peter Gordon and Warren Breckman (eds.), The Cambridge History of Modern European Thought, Volume 2: The Twentieth Century, Cambridge: Cambridge University Press, pp. 259–288.
Pickford, Henry, 2023, “Adorno and the Categories of Resistance”, Constellations. doi:10.1111/1467-8675.12652
Pollock, Friedrich, 1941, “State Capitalism: Its Possibilities and Limitations”, Studies in Philosophy and Social Science, 9: 200–225.
Postone, Moishe, 1980, “Anti-Semitism and National Socialism: Notes on the German Reaction to ‘Holocaust’ ”, New German Critique, 19: 97–115. doi:10.2307/487974
–––, 1993, Time, Labor, and Domination: A Reinterpretation of Marx’s Critical Theory, Cambridge: Cambridge University Press.
Rabinbach, Anson, 2000, “Why Were the Jews Sacrificed? The Place of Anti-Semitism in Dialectic of Enlightenment”, New German Critique, 81: 49–64. doi:10.2307/488545
Rawls, John, 1993, Political Liberalism, New York: Columbia University Press.
–––, 1995, “Political Liberalism: Reply to Habermas”, The Journal of Philosophy, 92(3): 132–180. doi:10.2307/2940843
Redecker, Eva von, 2018, Praxis und Revolution, Frankfurt am Main: Campus; translated as Praxis and Revolution: A Theory of Social Transformation, Lucy Duggan (trans.), New York: Columbia University Press, 2022.
–––, 2020, “Ownership’s Shadow: Neoauthoritarianism as Defense of Phantom Possession”, Critical Times, 3(1): 33–67. doi:10.1215/26410478-8189849
Renault, Emmanuel, 2004, L’expérience de l’injustice: Reconnaissance et clinique de l’injustice, Paris: La Découverte; translated as The Experience of Injustice: A Theory of Recognition, Richard A. Lynch (trans.), New York: Columbia University Press, 2019.
–––, 2008, Souffrances sociales: Sociologie, psychologie et politique, Paris: La Découverte; translated as Social Suffering: Sociology, Psychology, Politics, Maude Dews (trans.), London: Rowman & Littlefield International, 2017.
Rensmann, Lars, 2017, The Politics of Unreason: The Frankfurt School and the Origins of Modern Antisemitism, Albany, NY: State University of New York Press.
Roberts, Neil, 2015, Freedom as Marronage, Chicago: University of Chicago Press.
Rorty, Richard, 1985, “Habermas and Lyotard on Postmodernism”, in Richard J. Bernstein (ed.), Habermas and Modernity, Cambridge, MA: MIT Press, pp. 161–75.
Rosa, Hartmut, 2016, Resonanz: Eine Soziologie der Weltbeziehung, Berlin: Suhrkamp; translated as Resonance: A Sociology of Our Relationship to the World, James C. Wagner (trans.), Cambridge: Polity, 2019.
Rosa, Hartmut, Christoph Henning, and Arthur Bueno (eds.), 2021, Critical Theory and New Materialisms, New York: Routledge.
Rosen, Michael, 1996, On Voluntary Servitude: False Consciousness and the Theory of Ideology, Cambridge, MA: Harvard University Press.
Rössler, Beate, 2007, “Work, Recognition, Emancipation”, in Bert van den Brink and David Owen (eds.), Recognition and Power: Axel Honneth and the Tradition of Critical Social Theory. Cambridge: Cambridge University Press, pp. 135–163.
Saar, M. 2002, “Genealogy and Subjectivity”, European Journal of Philosophy, 10(2): 231–245. doi:10.1111/1468-0378.00159
–––, 2018, “What Is Social Philosophy? Or: Order, Practice, Subject”, Proceedings of the Aristotelian Society, 118(2): 207–223. doi:10.1093/arisoc/aoy009
–––, 2020, “Rethinking Resistance: Critical Theory before and after Deleuze”, Coils of the Serpent: Journal for the Study of Contemporary Power, 5(6): 68–80. [Saar 2020 available online].
Said, Edward, 1993, Culture and Imperialism, New York: Vintage.
Schaub, Jörg, 2015, “Misdevelopments, Pathologies, and Normative Revolutions: Normative Reconstruction as Method of Critical Theory”, Critical Horizons, 16(2): 107–130. doi:10.1179/1440991715Z.00000000043
Scheuerman, William, 1994, Between the Norm and the Exception: The Frankfurt School and the Rule of Law, Cambridge, MA: MIT Press.
–––, (ed.), 1996, The Rule of Law under Siege: Selected Essays of Franz L. Neumann and Otto Kirchheimer, Berkeley: University of California Press.
–––, 2018, Civil Disobedience, Cambridge: Polity Press.
Sealey, Kris F. and Benjamin P. Davis (eds.), forthcoming, Creolizing Critical Theory: New Voices in Caribbean Philosophy, New York: Rowman & Littlefield.
Seeliger, Martin and Sebastian Sevignani (eds.), 2022, Special Issue: A New Structural Transformation of the Public Sphere?, Theory, Culture & Society, 39(4).
Shelby, Tommie, 2003, “Ideology, Racism, and Critical Social Theory”, Philosophical Forum, 34(2): 153–188. doi:10.1111/1467-9191.00132
Smith, William, 2013, Civil Disobedience and Deliberative Democracy, London: Routledge.
Stahl, Titus, 2013a, Immanente Kritik: Elemente einer Theorie sozialer Praktiken, Frankfurt: Campus; translated as Immanent Critique, John-Baptiste Oduor (trans.), London: Rowman & Littlefield, 2022.
–––, 2013b, “Habermas and the Project of Immanent Critique”, Constellations, 20(4): 533–552. doi:10.1111/1467-8675.12057
Stanley, Jason, 2015, How Propaganda Works, Princeton: Princeton University Press.
Thompson, Michael J. (ed.), 2017, The Palgrave Handbook of Critical Theory, New York: Palgrave Macmillan. doi:10.1057/978-1-137-55801-5
Vázquez-Arroyo, Antonio Y., 2018, “Critical Theory, Colonialism, and the Historicity of Thought”, Constellations, 25: 54–70. doi:10.1111/1467-8675.12348
Virno, Paolo, 2004, A Grammar of the Multitude: For an Analysis of Contemporary Forms of Life, New York: Semiotext(e).
Vogel, Steven, 1996, Against Nature: The Concept of Nature in Critical Theory, New York: SUNY Press.
–––, 2011, “On Nature and Alienation”, in Andrew Biro (ed.), Critical Ecologies: The Frankfurt School and Contemporary Environmental Crises, Toronto: University of Toronto Press.
Vogelmann, Frieder, 2016, “Measuring, Disrupting, Emancipating: Three Pictures of Critique”, Constellations, 24(1): 101–112. doi:10.1111/1467-8675.12254
–––, 2021, “One Step Forward, Two Steps Back: Idealism in Critical Theory”, Constellations, 28(3): 322–336. doi:10.1111/1467-8675.12548
Warner, Michael, 2002, Publics and Counterpublics, New York: Zone Books.
Wellmer, Albrecht, 1969, Kritische Gesellschaftstheorie und Positivismus, Frankfurt am Main: Suhrkamp; translated as Critical Theory of Society, John Gumming (trans.), New York: Herder and Herder, 1971.
–––, 1983, “Reason, Utopia, and the Dialectic of Enlightenment”, Praxis International, 3(2): 83–107.
–––, 1985/86, Zur Dialektik von Moderne und Postmoderne, Frankfurt am Main: Suhrkamp, 1985, and Ethik und Dialog, Frankfurt am Main: Suhrkamp, 1986. Portions of both books are translated in The Persistence of Modernity: Aesthetics, Ethics and Postmodernism, David Midgley (trans.), Cambridge, MA: MIT Press, 1991.
–––, 1993, Endspiele: Die unversöhnliche Moderne, Frankfurt am Main: Suhrkamp; translated as Endgames: The Irreconcilable Nature of Modernity, David Midgley (trans.), Cambridge, MA: MIT Press 1998.
–––, 2014, “On Critical Theory”, Social Research, 81(3): 705–733. doi:10.1353/sor.2014.0045
Wheatland, Thomas, 2009, The Frankfurt School in Exile, Minneapolis: University of Minnesota Press.
Whitebook, Joel, 1995, Perversion and Utopia: A Study in Psychoanalysis and Critical Theory, Cambridge, MA: MIT Press.
–––, 2021, “Misuse of Winnicott: On Axel Honneth’s Appropriation of Psychoanalysis”, Constellations, 28(3): 306–321. doi:10.1111/1467-8675.12582
Wiggershaus, Rolf, 1986, Die Frankfurter Schule: Geschichte, theoretische Entwicklung, politische Bedeutung, München: Hanser; translated as The Frankfurt School: Its History, Theories, and Political Significance, Michael Robertson (trans.), Cambridge, MA: MIT Press, 1994.
Wimbauer, Christine, 2023, “Taking Care Seriously: Gendering Honneth’s The Working Sovereign – A Normative Theory of Work”, Journal of Classical Sociology, 23(3): 389–400. doi:10.1177/1468795X231170827
Wolin, Richard, 1994, Walter Benjamin: An Aesthetic of Redemption, Berkeley: University of California Press.
Young, Iris Marion, 2000, Inclusion and Democracy, Oxford: Oxford University Press.
–––, 2001, “Activist Challenges to Deliberative Democracy”, Political Theory, 29(5): 670–90. doi:10.1177/0090591701029005004
–––, 2007, “Recognition of Love’s Labor: Considering Axel Honneth’s Feminism”, in Bert van den Brink and David Owen (eds.), Recognition and Power: Axel Honneth and the Tradition of Critical Social Theory. Cambridge: Cambridge University Press, pp. 189–212.
Zaretsky, Eli, 2022, “Donald Trump and the Paranoid Position”, Critical Historical Studies, 9(1): 133–157. doi:10.1086/719127
Zurn, Christopher, 2011, “Social pathologies as second-order disorders”, in Danielle Petherbridge (ed.), Axel Honneth: Critical Essays, Leiden: Brill, pp. 345–70.
–––, 2015, Axel Honneth: A Critical Theory of the Social, Cambridge: Polity.
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Adorno, Theodor W. | alienation | Benjamin, Walter | colonialism | critical philosophy of race | disability: critical disability theory | feminist philosophy, interventions: epistemology and philosophy of science | feminist philosophy, topics: perspectives on power | Foucault, Michel | Habermas, Jürgen | Horkheimer, Max | Lukács, Georg [György] | Marcuse, Herbert | postmodernism | recognition
Acknowledgments
The authors would like to thank Amy Allen, Axel Honneth, Noëlle McAfee, and Martin Saar for their very helpful comments on earlier drafts and Christian Meyer for judicious editorial assistance.
Copyright © 2023 by Robin Celikates <robin.celikates@fu-berlin.de> Jeffrey Flynn <jeflynn@fordham.edu>
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Review article
A review of design issues specific to hypersonic flight vehicles
2016, Progress in Aerospace SciencesD. Sziroczak, H. Smith
4.1 Aerodynamics
One of the greatest concerns with a hypersonic vehicle is the aerodynamic performance. Due to the extensive range of speeds the operational envelope covers, the designs have to fulfil several, often contradictory requirements. The main focus, however, is the reduction of drag during the ascent phase (and cruise for transports). Fig. 8 shows a comparison between various aerospace vehicles' maximum flight speed.
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Fig. 8. Typical maximum flight speeds for various vehicles, TAS [m/s].
4.1.1 Aerodynamic lift
Unlike conventional aircraft, where the lift has to balance the weight to sustain flight, a hypersonic vehicle can rely solely on thrust, provided the engines are powerful enough. Thus the importance of aerodynamic lift depends on the actual vehicle configuration. Non-lifting (ballistic) vehicles do not rely on aerodynamic lift, which result in sleeker, lower drag shapes, but their cross range capability and controllability are poor. High hypersonic lift, however results in higher deceleration at higher altitudes, which reduces the peak heat, and overall maximum deceleration loads on the vehicle. Also lift can be used to provide safe launch abort capabilities. In the case of a horizontal take-off vehicle, it is essential to provide large amounts of lift, as it is required during the take-off and climb phase. Utilising aerodynamic lift reduces the thrust required compared to lifting the vehicle vertically, but this mainly depends on the lift to drag ratio of the vehicle. Refer to Table 4 for typical L/D ratios.
Table 3. Compression and expansion methods used for hypersonic flows.
| Compression methods | Expansion methods |
--- |
| Modified Newtonian | Cp=0 |
| Newtonian–Prandtl-Meyer | Newtonian–Prandtl-Meyer |
| Tangent wedge | Prandtl-Meyer |
| Tangent wedge infinite Mach | Cone at angle of attack |
| Old tangent cone | VanDyke Unified |
| Cone at angle of attack | Vacuum |
| VanDyke unified | Shock expansion |
| Blunt body viscous | Input value |
| Shock expansion | Free molecular flow |
| Free molecular flow | Modified Dahlem–Buck |
| Input value of CpStag | ACM empirical |
| Hankey flat surface | Half Pradtl-Meyer from freestream |
| Smyth Delta Wing | |
| Modified Dahlem–Buck | |
| BlastWave | |
| OSUBluntBody | |
| Tangent cone (Edwards) | |
Table 4. Lift to drag ratio comparison [39–43].
| Vehicle | Lift to drag subsonic | Lift to drag supersonic | Lift to drag hypersonic |
--- --- |
| Space Shuttle | 4.5 | 2 | 1.5 |
| Buran | 5.0 | No data | 1.7 |
| X-15 | 4 | 2.5 | 2.5 |
| X-20 | 4.3 | No data | 1.5–1.9 |
| Common aero vehicle (X-41) | No Data | 2.5 | 3.5 |
| SpaceShipTwo | 7 | 0.5 | Not applicable |
The lift characteristics of hypersonic aircraft are fundamentally different from subsonic or linearised supersonic behaviour. The methods used to evaluate hypersonic aerodynamics depend on the vehicle shape.
For blunt shapes such as the flat underside of the Space Shuttle or similar vehicles, hypersonic aerodynamics can be approximated using the Newtonian flow theory. This theory models the flow as many small individual particles, impacting a surface, losing their normal, but retaining the tangential speed components. Investigating the behaviour of a flat plate shows remarkable similarity with the usual lifting surfaces, and thus understanding it gives insight into the fundamental behaviour of hypersonic lift.
Using the Newtonian sine-squared law, the lift and drag coefficient of a flat plate can be derived as:
(1)c l=2·Sin 2 α·Cos α
(2)c d=2·Sin 3 α
Neglecting friction, it can be seen in Fig. 9, that the theoretical lift curve at hypersonic speeds follows a non-linear behaviour, reaching its maximum at 54.7°. Unlike subsonic flows, where the peak is followed by a stall region, and the loss of lift is due to separation of the flow, here it is due to the behaviour of the trigonometric functions.
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Fig. 9. Lift and drag characteristics of flat plate in hypersonic flow (Source: ).
The Newtonian theory simplifies the flow from two important points of view. First, it ignores any cross-flows or pressure effects and approximates aerodynamic loads solely based on geometric angles. The other very important simplification, and limitation, is that it does not treat the flow over the “shadowed” side of the vehicle. This is appropriate for a flat plate, where there is essentially ambient pressure over most of the leeward side, but it introduces significant errors for example in the case of diamond shaped aerofoils used on Hypersonic Transports, where expansion effects need to be taken into account. One method such as this is the Shock-Expansion method, where only the principal characteristics of the flow through shockwaves derived from three dimensional characteristics theory are considered, and other secondary effects such as reflections from the shockwave and vortex lines are ignored. The method was introduced in 1931 by Epstein .
There were many other methods developed over the years to address flows around different geometries. There is no single method universally applicable to any vehicle shape, the designer needs to have good understanding of the fundamentals and underlying assumptions of the various methods. Table 3 shows a list of various compression and expansion methods used to estimate hypersonic aerodynamic performance. These methods form part of the SHABP software and for more details on the methods, their applicability and implementation as computer code, one should refer to the Program Formulation manual and Dirkx and Mooij for modelling methods and comparison with real vehicle flight data .
Those aircraft, which are utilising lift for ascent fall into either the winged or lifting body category. The winged configuration offers a higher lift to drag ratio at subsonic speeds, but increases structural mass and drag compared to the lifting body. At hypersonic speeds wings are disadvantageous, as long thin surfaces are not efficient structures, thus they have to be heavy to survive re-entry loads. The table below shows some winged and lifting body vehicles with their respective L/D ratios. According to NASA investigations , the theoretical maximum hypersonic lift-to-drag ratio (with a skin-friction coefficient of 10−3) for a conventional low-winged configuration, such as the Space Shuttle, is 5.29. However, a flat-top, high wing type configuration, X-43 for example, could reach up to 6.65 L/D at Mach 5 . Although, due to aerodynamic heating issues, the low wing vehicles, such as the X-20 Dyna-soar, were chosen as the first designs to develop (Fig. 10).
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Fig. 10. Low and high wing hypersonic shape comparison: Space Shuttle (left) X-43 (right) (Source: ).
There is a special class of lifting body aerospace vehicles called the waverider. This is a specially designed vehicle that utilises its own shockwave to generate extra lift to improve its lift to drag ratio. The concept was developed by Terence Nonweiler . Up to day, only the Boeing X-51 has actually demonstrated flight with this shape. The drawback of the waverider is that it can only achieve the lift increase at a specific Mach number and altitude combination, to which the geometry was optimised. There is no drawback to a waverider design compared to a conventional lifting body vehicle. The shock wave when positioned correctly could also provide ram compression for airbreathing powerplants.
4.1.2 Aerodynamic drag
It is important to understand that unlike airliners, where there is a significant component of lift induced drag, the drag of a spacecraft comprises mostly of base and wave drag, and so it depends more on the volume and cross-sectional area of the vehicle, than the mass. This is important, as it means that the payload carried by the vehicle must have strict volume limits in addition to mass limits. Furthermore, this means that some fuel types, especially liquid hydrogen and other low energy to mass ratio fuels could be disadvantageous despite their high specific impulse and good emission characteristics.
As one of the main parameters of a launcher is the total change in speed or ∆v it can achieve by burning all the fuel on board, the aerodynamic drag is often represented as a ∆v increment in addition to that required to reach a specific orbit. The atmospheric drag is often combined with the gravity drag to give a total ∆v increase. Gravity drag results from the fact that in addition to accelerating the spacecraft, we also have to resist the gravitational pull of Earth. As such gaining altitude and reaching the orbital speed as soon as possible is a preferred way to reduce these two drag components. From this respect a vertical launch vehicle seems more efficient than a horizontal take-off, because the flight path angle and acceleration is high, to leave the atmosphere in the shortest possible time, thus minimising gravity drag. This shows that the horizontal take-off configuration is only efficient if the lift to drag ratio is sufficiently high, to compensate for the additional time spent during the climb phase, and as such the increased gravity and aerodynamic drag losses compared to a vertical take-off vehicle. For a typical rocket launched to LEO, the atmospheric and gravity drag adds up to about 1.5–2 km/s ∆v increment, compared to the 7.8 km/s baseline as calculated from the Tsiolkovsky rocket equation. It has also to be noted, that the aerodynamic drag losses account for up to around 10% of the total ∆v increment, and as such the gravitational losses are the dominant component .
In the case of a hypersonic transport, the issues with aerodynamic and gravity drag are not significantly different from the launcher vehicle, as a transport would also have to reach high altitudes to enable the hypersonic cruise.
The key question here is whether it is better to follow a lifting trajectory or to ascend like a rocket. The benefits of lifting flight is that the vehicle can capitalise on the reduced thrust requirements due to the L/D ratio larger than unity, thus enabling the propulsion systems to be potentially significantly smaller. The propulsion system usually contributes to significant percentage of the vehicle's empty mass, thus the lower the empty mass to launch, the less fuel is required, which allows smaller structure, and so on, this can have a snowballing reduction effect on the total mass. Also, in the case of many concepts, the propulsion systems pose a minimum diameter constraint on the vehicle size. Reducing the required thrust reduces the propulsion system size, and enables the design of sleeker, lower drag configurations.
Lifting ascent on the other hand, comes with disadvantages as well. First of all, a lifting structure is required, which adds considerable mass to the structure. As the wings are not solely used during ascent, but at approach, landing and most importantly re-entry as well, where they are most beneficial to reduce heat fluxes, thus potentially TPS mass, this added mass can be justified, or even negated with other mass savings.
Where the lifting ascent falls behind a rocket type vehicle is the losses due to the chosen trajectory. The ultimate aim of these vehicles, both Space Launchers and Hypersonic Transports is to climb to a high altitude, and acquire high velocities. It can be seen, that the fastest way to achieve this is with a near vertical climb at maximum acceleration, which is exactly the rocket type ascent. When the vehicle starts to rely on aerodynamic lift to support its weight, it will reduce the required thrust, but must do so by also reducing flight path angles, no longer achieving the near vertical climb trajectories. When the flight path angle is reduced, it can be seen, that to climb to the same altitude would take longer than in the case of the vertical ascent. This additional time spent in the atmosphere is the source of their major disadvantage. The fuel used to reach the given altitudes is proportional to the product of the power used by the system and the time it takes to reach the given altitude. In many cases the reduction in the power required due to aerodynamic lift is significantly lower than the additional time required to climb, thus suggesting that the rocket type ascent is superior in many cases. However this has to be evaluated on a concept-by-concept basis, as the factors affecting these are many; mainly L/D ratio, propulsion system performance, fuel types, trajectories, all evaluated over the full speed and altitude range and with possible different control strategies, for example in the case where multiple propulsion systems are installed on the same vehicle. This is indeed one of the main challenges of hypersonic vehicle conceptual design.
4.1.3 Stability and control
Designing stability and control characteristics for a hypersonic vehicle is not an easy task, and the reason for it is twofold: first, the vehicle operates at a wide range of speeds and also distinctly different environments.
The effects of the wide speed range mainly concerns the change in lift distribution, and thus the position of the aerodynamic centre, the change of effectiveness in control authority of the control surfaces and the aeroelasticity effects. When the aerospace vehicle accelerates from subsonic to supersonic speeds, the aerodynamic centre travels rearwards, thus altering the static margin and the stability characteristics (Fig. 11).
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Fig. 11. Aerodynamic centre position with changing Mach number for various aspect ratio and sweep (Source: ).
To maintain the stability characteristics over the speed range, there are some options available. Aerodynamic surfaces can be employed, which can produce changes in the overall pitching moment slope, such as in the case of F111, variable sweep wings. Additionally, centre of gravity shift can be employed, by pumping fuel to trim tanks, such as on the Concorde, shown in Fig. 12.
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Fig. 12. Concorde CG limit variation with Mach number (Source: ).
Maintaining the stability margins is a delicate problem for hypersonic vehicles, as the large engines required to produce the large amount of thrust are usually mounted in the back, resulting in an aft CG position, which often results in trim and stability problems. Furthermore, the significant change in fuel quantity (over 90% of total mass) also shifts the CG (usually backward towards the heavy engines), further complicating the issue. The best example for this is the case of the HOTOL and Skylon vehicles, where the original stability problems with the HOTOL were mitigated by placing the engines at the wing tip, instead of the rear of the spacecraft .
As it can be seen in Fig. 13, the authority of the control surfaces generally tends to drop as the Mach number increases, which makes the control of the aircraft at high speeds increasingly difficult. Sometimes however, as it can be seen at negative deflection angles, the efficiency of the control can improve. While the actual effect of Mach number change to the controls has to be evaluated for quantification, the change in effectiveness must be expected at every stage of the design nevertheless. Also, the second of the two issues mentioned, with increasing altitude, density reduces thus aerodynamic surfaces become progressively less effective. To counter this problem, most of the vehicles are installed with a reaction control system, which not only augments the control power at high speeds, but also enables attitude corrections outside the atmosphere. The available RCS propellants can be cold gas, fuel and oxidiser ignited or hypergolic fuels. On small spacecraft reaction control systems are usually operated by releasing inert gases such as nitrogen or helium or other gases such as methane. Larger vehicles, such as the X-15, use hydrogen-peroxide while some proposed configurations, such as the HL-42, burn liquid methane with oxygen in the reaction control systems. The Space Shuttle uses Monomethyl Hydrazine and Nitrogen Tetroxide RCS system . The drawbacks of using an RCS is the added weight and complexity for the system as a whole. RCS systems comprise nozzles, control valves, fuel tanks (with fuel), insulation, structural supports, which all add to the overall mass, complexity. The Space Shuttle has 12,800 kg of propellant on board (shared between the OMS and RCS) and the RCS system total mass is 1276 kg .
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Fig. 13. X-33 pitching moment and elevon authority as a function of AOA and Mach number (Source: ).
Aeroelasticity and general structural deformations are also of great interest, due to the high performance requirements, and the many unknowns in hypersonic aerodynamics, the aircraft must operate close to the design point in order to maintain controlled, efficient and safe flight. This means, that the structural deformations allowed tend to be smaller than for a subsonic vehicle. The problem is further complicated by the addition of thermal effects due to friction. These aero-thermo-elasticity calculations still pose a great challenge to engineers and usually require considerable computational power to solve. Also, for some vehicles, specifically the Space Shuttle or the Buran, the heat shield for atmospheric re-entry is made of ceramic material, and the elastic behaviour of the vehicle is the reason for the 24,300 separate tiles used on the Shuttle, contributing towards the extremely high maintenance costs.
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Journal2016, Progress in Aerospace SciencesD. Sziroczak, H. Smith
Review article
A review of design issues specific to hypersonic flight vehicles
2016, Progress in Aerospace SciencesD. Sziroczak, H. Smith
4 Technological issues
4.1 Aerodynamics
One of the greatest concerns with a hypersonic vehicle is the aerodynamic performance. Due to the extensive range of speeds the operational envelope covers, the designs have to fulfil several, often contradictory requirements. The main focus, however, is the reduction of drag during the ascent phase (and cruise for transports). Fig. 8 shows a comparison between various aerospace vehicles' maximum flight speed.
Sign in to download full-size image
Fig. 8. Typical maximum flight speeds for various vehicles, TAS [m/s].
4.1.1 Aerodynamic lift
Unlike conventional aircraft, where the lift has to balance the weight to sustain flight, a hypersonic vehicle can rely solely on thrust, provided the engines are powerful enough. Thus the importance of aerodynamic lift depends on the actual vehicle configuration. Non-lifting (ballistic) vehicles do not rely on aerodynamic lift, which result in sleeker, lower drag shapes, but their cross range capability and controllability are poor. High hypersonic lift, however results in higher deceleration at higher altitudes, which reduces the peak heat, and overall maximum deceleration loads on the vehicle. Also lift can be used to provide safe launch abort capabilities. In the case of a horizontal take-off vehicle, it is essential to provide large amounts of lift, as it is required during the take-off and climb phase. Utilising aerodynamic lift reduces the thrust required compared to lifting the vehicle vertically, but this mainly depends on the lift to drag ratio of the vehicle. Refer to Table 4 for typical L/D ratios.
Table 3. Compression and expansion methods used for hypersonic flows.
| Compression methods | Expansion methods |
--- |
| Modified Newtonian | Cp=0 |
| Newtonian–Prandtl-Meyer | Newtonian–Prandtl-Meyer |
| Tangent wedge | Prandtl-Meyer |
| Tangent wedge infinite Mach | Cone at angle of attack |
| Old tangent cone | VanDyke Unified |
| Cone at angle of attack | Vacuum |
| VanDyke unified | Shock expansion |
| Blunt body viscous | Input value |
| Shock expansion | Free molecular flow |
| Free molecular flow | Modified Dahlem–Buck |
| Input value of CpStag | ACM empirical |
| Hankey flat surface | Half Pradtl-Meyer from freestream |
| Smyth Delta Wing | |
| Modified Dahlem–Buck | |
| BlastWave | |
| OSUBluntBody | |
| Tangent cone (Edwards) | |
Table 4. Lift to drag ratio comparison [39–43].
| Vehicle | Lift to drag subsonic | Lift to drag supersonic | Lift to drag hypersonic |
--- --- |
| Space Shuttle | 4.5 | 2 | 1.5 |
| Buran | 5.0 | No data | 1.7 |
| X-15 | 4 | 2.5 | 2.5 |
| X-20 | 4.3 | No data | 1.5–1.9 |
| Common aero vehicle (X-41) | No Data | 2.5 | 3.5 |
| SpaceShipTwo | 7 | 0.5 | Not applicable |
The lift characteristics of hypersonic aircraft are fundamentally different from subsonic or linearised supersonic behaviour. The methods used to evaluate hypersonic aerodynamics depend on the vehicle shape.
For blunt shapes such as the flat underside of the Space Shuttle or similar vehicles, hypersonic aerodynamics can be approximated using the Newtonian flow theory. This theory models the flow as many small individual particles, impacting a surface, losing their normal, but retaining the tangential speed components. Investigating the behaviour of a flat plate shows remarkable similarity with the usual lifting surfaces, and thus understanding it gives insight into the fundamental behaviour of hypersonic lift.
Using the Newtonian sine-squared law, the lift and drag coefficient of a flat plate can be derived as:
(1)c l=2·Sin 2 α·Cos α
(2)c d=2·Sin 3 α
Neglecting friction, it can be seen in Fig. 9, that the theoretical lift curve at hypersonic speeds follows a non-linear behaviour, reaching its maximum at 54.7°. Unlike subsonic flows, where the peak is followed by a stall region, and the loss of lift is due to separation of the flow, here it is due to the behaviour of the trigonometric functions.
Sign in to download full-size image
Fig. 9. Lift and drag characteristics of flat plate in hypersonic flow (Source: ).
The Newtonian theory simplifies the flow from two important points of view. First, it ignores any cross-flows or pressure effects and approximates aerodynamic loads solely based on geometric angles. The other very important simplification, and limitation, is that it does not treat the flow over the “shadowed” side of the vehicle. This is appropriate for a flat plate, where there is essentially ambient pressure over most of the leeward side, but it introduces significant errors for example in the case of diamond shaped aerofoils used on Hypersonic Transports, where expansion effects need to be taken into account. One method such as this is the Shock-Expansion method, where only the principal characteristics of the flow through shockwaves derived from three dimensional characteristics theory are considered, and other secondary effects such as reflections from the shockwave and vortex lines are ignored. The method was introduced in 1931 by Epstein .
There were many other methods developed over the years to address flows around different geometries. There is no single method universally applicable to any vehicle shape, the designer needs to have good understanding of the fundamentals and underlying assumptions of the various methods. Table 3 shows a list of various compression and expansion methods used to estimate hypersonic aerodynamic performance. These methods form part of the SHABP software and for more details on the methods, their applicability and implementation as computer code, one should refer to the Program Formulation manual and Dirkx and Mooij for modelling methods and comparison with real vehicle flight data .
Those aircraft, which are utilising lift for ascent fall into either the winged or lifting body category. The winged configuration offers a higher lift to drag ratio at subsonic speeds, but increases structural mass and drag compared to the lifting body. At hypersonic speeds wings are disadvantageous, as long thin surfaces are not efficient structures, thus they have to be heavy to survive re-entry loads. The table below shows some winged and lifting body vehicles with their respective L/D ratios. According to NASA investigations , the theoretical maximum hypersonic lift-to-drag ratio (with a skin-friction coefficient of 10−3) for a conventional low-winged configuration, such as the Space Shuttle, is 5.29. However, a flat-top, high wing type configuration, X-43 for example, could reach up to 6.65 L/D at Mach 5 . Although, due to aerodynamic heating issues, the low wing vehicles, such as the X-20 Dyna-soar, were chosen as the first designs to develop (Fig. 10).
Sign in to download full-size image
Fig. 10. Low and high wing hypersonic shape comparison: Space Shuttle (left) X-43 (right) (Source: ).
There is a special class of lifting body aerospace vehicles called the waverider. This is a specially designed vehicle that utilises its own shockwave to generate extra lift to improve its lift to drag ratio. The concept was developed by Terence Nonweiler . Up to day, only the Boeing X-51 has actually demonstrated flight with this shape. The drawback of the waverider is that it can only achieve the lift increase at a specific Mach number and altitude combination, to which the geometry was optimised. There is no drawback to a waverider design compared to a conventional lifting body vehicle. The shock wave when positioned correctly could also provide ram compression for airbreathing powerplants.
4.1.2 Aerodynamic drag
It is important to understand that unlike airliners, where there is a significant component of lift induced drag, the drag of a spacecraft comprises mostly of base and wave drag, and so it depends more on the volume and cross-sectional area of the vehicle, than the mass. This is important, as it means that the payload carried by the vehicle must have strict volume limits in addition to mass limits. Furthermore, this means that some fuel types, especially liquid hydrogen and other low energy to mass ratio fuels could be disadvantageous despite their high specific impulse and good emission characteristics.
As one of the main parameters of a launcher is the total change in speed or ∆v it can achieve by burning all the fuel on board, the aerodynamic drag is often represented as a ∆v increment in addition to that required to reach a specific orbit. The atmospheric drag is often combined with the gravity drag to give a total ∆v increase. Gravity drag results from the fact that in addition to accelerating the spacecraft, we also have to resist the gravitational pull of Earth. As such gaining altitude and reaching the orbital speed as soon as possible is a preferred way to reduce these two drag components. From this respect a vertical launch vehicle seems more efficient than a horizontal take-off, because the flight path angle and acceleration is high, to leave the atmosphere in the shortest possible time, thus minimising gravity drag. This shows that the horizontal take-off configuration is only efficient if the lift to drag ratio is sufficiently high, to compensate for the additional time spent during the climb phase, and as such the increased gravity and aerodynamic drag losses compared to a vertical take-off vehicle. For a typical rocket launched to LEO, the atmospheric and gravity drag adds up to about 1.5–2 km/s ∆v increment, compared to the 7.8 km/s baseline as calculated from the Tsiolkovsky rocket equation. It has also to be noted, that the aerodynamic drag losses account for up to around 10% of the total ∆v increment, and as such the gravitational losses are the dominant component .
In the case of a hypersonic transport, the issues with aerodynamic and gravity drag are not significantly different from the launcher vehicle, as a transport would also have to reach high altitudes to enable the hypersonic cruise.
The key question here is whether it is better to follow a lifting trajectory or to ascend like a rocket. The benefits of lifting flight is that the vehicle can capitalise on the reduced thrust requirements due to the L/D ratio larger than unity, thus enabling the propulsion systems to be potentially significantly smaller. The propulsion system usually contributes to significant percentage of the vehicle's empty mass, thus the lower the empty mass to launch, the less fuel is required, which allows smaller structure, and so on, this can have a snowballing reduction effect on the total mass. Also, in the case of many concepts, the propulsion systems pose a minimum diameter constraint on the vehicle size. Reducing the required thrust reduces the propulsion system size, and enables the design of sleeker, lower drag configurations.
Lifting ascent on the other hand, comes with disadvantages as well. First of all, a lifting structure is required, which adds considerable mass to the structure. As the wings are not solely used during ascent, but at approach, landing and most importantly re-entry as well, where they are most beneficial to reduce heat fluxes, thus potentially TPS mass, this added mass can be justified, or even negated with other mass savings.
Where the lifting ascent falls behind a rocket type vehicle is the losses due to the chosen trajectory. The ultimate aim of these vehicles, both Space Launchers and Hypersonic Transports is to climb to a high altitude, and acquire high velocities. It can be seen, that the fastest way to achieve this is with a near vertical climb at maximum acceleration, which is exactly the rocket type ascent. When the vehicle starts to rely on aerodynamic lift to support its weight, it will reduce the required thrust, but must do so by also reducing flight path angles, no longer achieving the near vertical climb trajectories. When the flight path angle is reduced, it can be seen, that to climb to the same altitude would take longer than in the case of the vertical ascent. This additional time spent in the atmosphere is the source of their major disadvantage. The fuel used to reach the given altitudes is proportional to the product of the power used by the system and the time it takes to reach the given altitude. In many cases the reduction in the power required due to aerodynamic lift is significantly lower than the additional time required to climb, thus suggesting that the rocket type ascent is superior in many cases. However this has to be evaluated on a concept-by-concept basis, as the factors affecting these are many; mainly L/D ratio, propulsion system performance, fuel types, trajectories, all evaluated over the full speed and altitude range and with possible different control strategies, for example in the case where multiple propulsion systems are installed on the same vehicle. This is indeed one of the main challenges of hypersonic vehicle conceptual design.
4.1.3 Stability and control
Designing stability and control characteristics for a hypersonic vehicle is not an easy task, and the reason for it is twofold: first, the vehicle operates at a wide range of speeds and also distinctly different environments.
The effects of the wide speed range mainly concerns the change in lift distribution, and thus the position of the aerodynamic centre, the change of effectiveness in control authority of the control surfaces and the aeroelasticity effects. When the aerospace vehicle accelerates from subsonic to supersonic speeds, the aerodynamic centre travels rearwards, thus altering the static margin and the stability characteristics (Fig. 11).
Sign in to download full-size image
Fig. 11. Aerodynamic centre position with changing Mach number for various aspect ratio and sweep (Source: ).
To maintain the stability characteristics over the speed range, there are some options available. Aerodynamic surfaces can be employed, which can produce changes in the overall pitching moment slope, such as in the case of F111, variable sweep wings. Additionally, centre of gravity shift can be employed, by pumping fuel to trim tanks, such as on the Concorde, shown in Fig. 12.
Sign in to download full-size image
Fig. 12. Concorde CG limit variation with Mach number (Source: ).
Maintaining the stability margins is a delicate problem for hypersonic vehicles, as the large engines required to produce the large amount of thrust are usually mounted in the back, resulting in an aft CG position, which often results in trim and stability problems. Furthermore, the significant change in fuel quantity (over 90% of total mass) also shifts the CG (usually backward towards the heavy engines), further complicating the issue. The best example for this is the case of the HOTOL and Skylon vehicles, where the original stability problems with the HOTOL were mitigated by placing the engines at the wing tip, instead of the rear of the spacecraft .
As it can be seen in Fig. 13, the authority of the control surfaces generally tends to drop as the Mach number increases, which makes the control of the aircraft at high speeds increasingly difficult. Sometimes however, as it can be seen at negative deflection angles, the efficiency of the control can improve. While the actual effect of Mach number change to the controls has to be evaluated for quantification, the change in effectiveness must be expected at every stage of the design nevertheless. Also, the second of the two issues mentioned, with increasing altitude, density reduces thus aerodynamic surfaces become progressively less effective. To counter this problem, most of the vehicles are installed with a reaction control system, which not only augments the control power at high speeds, but also enables attitude corrections outside the atmosphere. The available RCS propellants can be cold gas, fuel and oxidiser ignited or hypergolic fuels. On small spacecraft reaction control systems are usually operated by releasing inert gases such as nitrogen or helium or other gases such as methane. Larger vehicles, such as the X-15, use hydrogen-peroxide while some proposed configurations, such as the HL-42, burn liquid methane with oxygen in the reaction control systems. The Space Shuttle uses Monomethyl Hydrazine and Nitrogen Tetroxide RCS system . The drawbacks of using an RCS is the added weight and complexity for the system as a whole. RCS systems comprise nozzles, control valves, fuel tanks (with fuel), insulation, structural supports, which all add to the overall mass, complexity. The Space Shuttle has 12,800 kg of propellant on board (shared between the OMS and RCS) and the RCS system total mass is 1276 kg .
Sign in to download full-size image
Fig. 13. X-33 pitching moment and elevon authority as a function of AOA and Mach number (Source: ).
Aeroelasticity and general structural deformations are also of great interest, due to the high performance requirements, and the many unknowns in hypersonic aerodynamics, the aircraft must operate close to the design point in order to maintain controlled, efficient and safe flight. This means, that the structural deformations allowed tend to be smaller than for a subsonic vehicle. The problem is further complicated by the addition of thermal effects due to friction. These aero-thermo-elasticity calculations still pose a great challenge to engineers and usually require considerable computational power to solve. Also, for some vehicles, specifically the Space Shuttle or the Buran, the heat shield for atmospheric re-entry is made of ceramic material, and the elastic behaviour of the vehicle is the reason for the 24,300 separate tiles used on the Shuttle, contributing towards the extremely high maintenance costs.
4.2 Propulsion
A second very important aspect of hypersonic flight, closely connected to aerodynamics, is the propulsion. It is probably the most critical factor that limits efficient and routine space access. The choice of propulsion for an aerospace vehicle is either the jet or rocket engine, as these offer the required performance for high speed flight. The most important characteristic of a propulsion system is the specific impulse. It is a measure of efficiency for the propulsion, the ratio of thrust generated divided by the flow of fuel (fuel flow can be either in weight flow [N/s] or mass flow [kg/s]). If the weight/sec definition is considered so the dimension is [s], it could be best described as the amount of time, that 1 kg of fuel can provide 1 N of thrust. When dividing by mass flow, the dimension of specific impulse will be [N s/kg]. Whichever unit is chosen, as per the definition, the higher the specific impulse, the more efficient the engine is. Fig. 14 shows a comparison of the specific impulse of various propulsion systems as a function of Mach number.
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Fig. 14. Hydrogen and hydrocarbon fuel propulsion systems comparison .
It can be seen, that although rockets provide thrust throughout the whole Mach number range, their specific impulse is lower compared to the airbreathing engines. The explanation for this comes from the operational principles, as the airbreathers utilise the reaction force of large amounts of air propelled backwards, while only consuming a small amount of fuel. For a rocket engine all the reaction mass has to be carried on board and expelled, thus the small ratio. It can also be seen, that Hydrogen fuel provides a higher specific impulse for a rocket than hydrocarbon fuels.
In addition to the specific impulse, another very important aspect of rocket engines is the available thrust. Having high specific impulse does not mean that the propulsion system is also capable of providing high thrust. For example, a propulsion system with a very high specific impulse is the ion-propulsion utilised for spacecraft. Their specific impulse is in the magnitude of 6000–10,000 s, some claiming even 20,000 s but the thrust provided is in the 0.05–3 N range. This means, that they are very efficient means to accelerate a spacecraft over long period of time, but they are unable to produce the required thrust to lift-off from the ground or even manoeuvre. As a comparison, the Rolls-Royce/Snecma Olympus engines powering the Concorde had an I sp of 3000, and a (wet) maximum thrust of 169 kN.
4.2.1 A list of propulsion systems
4.2.1.1 Airbreathers
•
Turbojet: Turbojets offer the highest specific impulse due to the large amount of air moved. While pure turbojets offer the highest thrust, turbofans offer a higher efficiency. Military supersonic aircraft are usually powered by low bypass ratio turbofans with afterburners. These systems can be used to take-off and propel the vehicle to low supersonic speeds. Due to high dynamic pressures, unducted configurations, such as turboprop or propfan are not promising choices for a hypersonic vehicle.
•
Ramjet: A ramjet utilises the high speed of the aircraft to compress the air at the inlet of the engine, and as such do not require rotating compressor or turbine parts, greatly reducing the complexity compared to a conventional jet. However due to the way compression occurs, they are incapable of providing thrust at low Mach numbers, and they also have an upper limit around M 5.5. One of the definitions of hypersonic speeds is the limit, where ramjets stop generating thrust.
•
Scramjet: Scramjet is an abbreviation of supersonic combustion ramjet. The working principle is the same as a ramjet, but the combustion occurs at supersonic speeds, which enables the engine to operate above the Mach limit of normal Ramjets. Scramjet engines are still under development, some aircraft successfully flown with scramjets are the US X-43 Hyper-X , X-51 , the Russian AJAX , and the Australian HyShot test aircraft.
4.2.1.2 Rocket based
•
Rocket: The rocket carries both fuel and oxidiser on board, so it does not rely on atmospheric oxygen, which enables it to operate outside the atmosphere. They produce a constant amount of specific impulse, but their thrust depends on altitude because the exhaust nozzle is optimised to one expansion ratio and thus one altitude. Usually adaptable nozzles induce additional weight and complexity which does not justify the performance gained. Another subtype of rockets with altitude compensating nozzles is the aerospike engine . These use an inverted approach to bell nozzles, and such use a spike at the centre of flow and the atmospheric air as the outer boundary of the exhaust plume. They were proposed for the X-33 and Venture Star designs but as of today there are no aerospike engines in operation.
4.2.1.3 Hybrid/combined cycles
•
Air augmented/ducted rocket: Their working principle is somewhat akin to the turbofan concept. They utilise an additional duct around the main rocket, which collects ram air, and the exhaust gases from the rocket further compress the outer flow and utilise this additional reaction mass. It is a hybrid of a ramjet and conventional rocket .
•
Turborocket: Usually consist of a gas generator, which provides compressed air for a combustion chamber, where fuel is added and the hot exhaust leaves the nozzle as a rocket. At higher altitudes and speeds or in the absence of atmospheric oxygen, an on board supply can be used to supply the oxidiser for the rocket engine .
•
Turboramjet: a potential solution to solve the problem of the lack of low speed thrust from the ramjet engines. It is built up from an inner turbojet with an outer ring of ramjet duct. At low speeds an internal turbojet is utilised to accelerate while at high speeds, an outer ramjet is used to generate the high thrust required. The North Aviation Griffon II was the first aircraft in the world to use a turboramjet in 1953 . There are variations of the concept, for example the ATREX engine is an expander cycle turboramjet .
•
Detonation engines: There are various types of detonation engines: standing detonation , pulse detonation [64,65] and continuous (rotating) detonation engines. Detonation engines, similar to ramjets require a minimum flight speed to work, but it is claimed that continuous detonation wave engines are usable at low speed as well.
•
Rocket Based Combined Cycle: Utilises rocket propulsion at low speeds, while at higher speeds, the system switches to a scramjet mode for more efficient propulsion [67,68].
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Turbine Based Combined Cycle: A combined system, which relies on a turbojet or low bypass ratio turbofan at speeds up to about Mach 3, after which a ramjet takes over to provide thrust. One example is the Lockheed Martin SR-72 currently under development. Fig. 15 shows the concept of the combined cycle system of the SR-72.
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Fig. 15. SR-72 Turbine-based combined cycle propulsion (Source: ).
4.2.2 Additional methods
There are some additional methods to produce extra propulsive force or reduce drag on an aircraft. Some of these methods are still at an early stage of development, and as such might turn out to be non-feasible solutions. These methods presented cannot provide adequate propulsion on their own, rather they have to be combined with existing systems.
•
In flight air collection: by not carrying the oxidiser on board, but collecting it from the atmosphere, significant take-off weight reductions could be achieved. This atmospheric oxygen can be stored in some form on-board to be further used as the oxidiser. The novel SABRE engine being developed for the Skylon space launcher utilises an evolved version of their original Liquid Air Cycle Engine (LACE) system.
•
External supersonic burning: a potential drag reduction phenomenon, recently discovered, Based on Froning and Roach , it is claimed that inviscid drag could be reduced up to 55% compared to baseline configuration. Injecting fuel into the external flow increases both the shock wave intensity (additional drag), and vehicle base pressures (reduces drag). Whether the injection results in a net drag increase or loss depends on the position of the heat addition. According to the studies, aft positions are favourable.
•
Precooled jet: any airbreather propulsion system could benefit from the utilisation of precooling. According to Taguchi et al. . A pre-cooled turbojet could propel a hypersonic transport up to the speeds of Mach 5.
•
Magnetohydrodynamic assist: MHD bypass could be utilised to add additional performance for an airbreather engine. The working principle is to ionise air and use electromagnets to further accelerate the flow, resulting in increased thrust. The technology was originally used on the Russian AJAX scramjet demonstrator.
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Reformed fuel: Also featured in the AJAX concept , this technology involves the reformation of hydrocarbon fuels, resulting in a superior ignition quality compared to pure hydrogen. Also the energy required for the reforming process acts as a heat sink for thermal protection.
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Thrust augmented nozzle: Also known as afterburning nozzle, part of the propellant flow is combusted in the rocket's nozzle. It is utilised at sea level and turned off at higher altitudes.
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Altitude compensating nozzle: this type of nozzle alters the exhaust flow of the rocket as it changes altitude, ensuring that the propulsion system always works at its design expansion ratio, thus reducing losses. Although variable geometry is a solution, it usually imposes a severe mass penalty on the vehicle. A special version is an aerospike nozzle , using a central “spike” or “plug” to which the exhaust gas flow attaches itself, while their outer boundary is the atmosphere, resulting in an inverted bell nozzle. They should achieve altitude compensation with a much lower mass penalty compared to variable geometry nozzles.
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“Spikes”: A slender body protruding from the nose of the vehicle, according to investigations, could provide drag reduction or even thermal protection for the airframe. The various types of spikes are: “plain” , “flame”, “laser” and “counter flow” . Some designs have claimed to also reduce the sonic signature of the vehicle . Example spike geometries and their effects on the flow are shown in Fig. 16.
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Fig. 16. Schlieren flowfield photographs and surface temperature distributions for various spike geometries (Source: ).
For further reading, a summary of the various propulsion systems can be found in the works of Pratt and Heiser or Hasselrot and Montgomerie or Varvill and Bond .
4.3 Structures
Although the structure of a hypersonic vehicle is not necessarily subject to the extreme g loads, as in the case of a fighter aircraft, it is still a very complex design. There is intensive heat load generated during flight, both for Transports and Launchers. The main difference is in the distribution and peak heat flux. The heat generated is proportional to the atmospheric density and to the third power of velocity. However this heat is generated in the air molecules, and only a fraction of it reaches the vehicle: from a few percentage points up to about 50%.
Transports generate constant, but fairly low (compared to Launchers) heat flux during their relatively long hypersonic cruise phase, due to their sleek shapes and sharp geometries. The shockwave is ideally still attached to the vehicle to reduce drag, and for a waverider also to provide lift. On the contrary, during the atmospheric re-entry phase, launchers experience brief but very intense heat loads, thus the common designs incorporate relatively blunt features to detach the shockwave from the structure, effectively using it as a shield to, ideally, prevent the formation high enthalpy turbulent flow near the surface, which would pass an extreme amount of heat into the structure through convection.
The main heat transfer mechanism to the vehicles is convection. For launchers there is another primary mechanism due to catalytic reaction on the vehicle surface, which can account for up to 40% of stagnation heat loads. In addition to this, depending on the atmosphere, radiation from the superheated plasma surrounding the aircraft could also transfer significant amount of heat into the vehicles. This is typically an issue at high entry speeds (10 km/s and above) or in extra-terrestrial atmospheres, such as Mars or Venus (on the other hand, on Jupiter radiation heat loads are less significant). Transports experience different flow conditions. They normally operate at lower altitudes (30–40 km), thus at higher Reynolds numbers and turbulent flow conditions, which significantly increase the convective heat transfer coefficient. However, take note that their speed, and thus air temperature, is lower than for a launcher, thus overall heat flux is smaller.
Due to these high heat loads many commonly used aerospace materials might not be available for the designers, unless they are combined with Thermal Protection Systems (TPS). The TPS design requires the designers to conduct complicated aero-thermo-elastic simulations, also accounting for chemistry, depending on the type of TPS material.
4.3.1 Types of TPS
A hypersonic vehicle's structure from a thermal protection point of view can be the following:
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Hot structure: there is no separate TPS, rather the metallic airframe (usually manufactured from exotic alloys, usually Nickel based superalloys) is capable of withstanding loads at elevated temperatures. A typical example is the SR-71 Blackbird.
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Cold structure: An external TPS covers the internal load carrying airframe, which in return can be made from more conventional materials. Two main types exist:
○
Active cooled: a coolant flow under the thermal protective outer skin of the vehicle is responsible for absorbing heat, and thus maintaining temperature at acceptable levels. The most commonly used heat sink is cryogenic fuel.
○
Ablative: the outer skin of the vehicle thermally degrades under high heat flux loads. This decomposition frees gases which act as a thin insulation layer on the surface. Also, due to this ablation, the upper layers of the TPS material becomes porous, further improving insulation capability. Their drawback is limited (short) lifespan, and the need for meticulous and expensive inspection and maintenance procedures. Typical example is the Apollo heat shield.
Normally a hypersonic vehicle utilises a combination of different TPS materials. The Space Shuttle's cover for example used 6 different materials, which all had different thermal properties. The X-37 builds on the legacy of the Space Shuttle, employing a similar concept in TPS, but with more advanced material technology. A close up image shown in Fig. 17 shows the various TPS components the X-37 (Table 5).
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Fig. 17. X-37 Thermal protection systems (Source: online).
Table 5. Thermal Protection System material properties (Based on Space Shuttle).
| Category | Name | Maximum temperature [°C] | Mass |
--- --- |
| Ultra high tempreature ceramics | Hafnium/Zirconium Diboride (SHARP) | 1200–1600 [83,84,85] | 10,500/6085 kg/m 3 |
| HRSI/LRSI (High/Low Temperature Reusable Surface Insulation) | 1300/650 | 9.2/4 kg/m 2 |
| Composites | Reinforced carbon-carbon composites | 1500 | 44.7 kg/m 2 |
| TUFROC (Toughened Uni-piece Fibrous Reinforced Oxidation-resistant Composite) | 1700 [84,92] | 400 kg/m 3 |
| Metallic TPS: | γ-TiAl | 900 | 3800 kg/m 3 |
| Ni based ODS (oxide dispersion strengthened alloys) | 1200 | 7500–8300 kg/m 3 |
| Inconel, typical properties | 1400 | 8410 kg/m 3 |
| Flexible blankets | SPFI (Surface Protected Felt Insulation) | 1200 | ND |
| FRSI (Flexible Reusable Surface Insulation) | 400 [10,82] | 1.6 kg/m 2 |
4.3.2 TPS materials
The main types of materials used are metal alloys, composites and ceramics. There are also non-load carrying TPS used in the form of flexible blankets. The current and most promising future TPS materials are the following.
As an overview, most of the materials used for current aircraft primary structure cannot normally exceed 400 K with active cooling, 367 K being the usual limit for conventional aluminium alloys. Hot structures on high speed vehicles are usually limited to about 800 K, while insulated structures can have surfaces temperatures up to 1200 K.
Hafnium/Zirconium Diboride are ultra-high temperature ceramics under active development for heat shield applications. They have the highest melting point of known ceramics (above 3200°C) and they have very good oxidation resistance. Due to this, they can be used for more advanced shapes such as sharp leading edges or sharp nosecones. Their main drawback is the lack of economical processing capability.
High/Low Temperature Reusable Surface Insulation are the ceramic tiles installed on most of the windward surface area of the now retired Space Shuttle. HTRSI are the black and LTRSI are the white tiles. The low temperature white tiles are coloured to reflect most of the solar radiation when in orbit, while the black ones absorb, thus also emit more heat during re-entry. The material of the insulation tiles is known as Silicon Fibrous Refractory Composite Insulation 12 (FRCI 12). It is patented by NASA . The insulation is made from aluminoborosilicate and silica fibres. Silica has good long term high temperature life, and technically acts as a matrix for the more refractory aluminoborosilicate, which could not be easily shaped by itself, but is capable of sustaining higher temperatures. It is a low density insulation with improved strength and temperature capabilities to prior state of the art insulations. As their main disadvantage, the tragedy of the Space Shuttle Columbia shows that even limited damage to the TPS could lead disaster . Ceramic TPS materials tend to be fairly brittle, even small objects could cause significant damage, especially at the high speeds that the vehicles are travelling at, so careful inspection is a must before and after launch to evaluate their integrity. Furthermore, many TPS materials, such as the insulation blankets and tiles of the Space Shuttle, are hygroscopic and thus require continuous effort to waterproof them.
Reinforced Carbon–Carbon composites are built up using carbon fibres, embedded in an all-carbon matrix. This approach combines the strength of the fibres with the refractory capabilities of the matrix. Further advantages include dimensional stability and low outgassing; both essential for space applications. RCC can be used to construct complex geometries, such as the leading edges of lifting surfaces. Careful combination of matrix material and fibre types are required to avoid brittleness issues in the finished component .
Toughened Uni-piece Fibrous Reinforced Oxidation-resistant Composite is a NASA innovation, developed for space applications but now finds its use in various commercial applications such as racing cars, turbines or furnaces. The technology is based on a two-piece construction: an exposed surface cap with a specialist coating, covering an insulator base, also with specialist coating. The cap is built-up from ROCCI (Refractory Oxidation Resistant Ceramic Carbon Insulation) and its purpose provides dimensional stability for components such as leading edges or nosecones. The insulation is TUFI (Toughened Uni-piece Fibrous Insulation) treated AETB (Alumina Enhanced Thermal Barrier). It was originally developed for the X-37 wing leading edges .
Inconel is the name of a family of superalloys. It was originally developed by Wiggin Alloys in 1940s to support the development of the Whittle jet engine, and was later used on the X-15 test vehicle. They are nickel based, with chromium as the second alloying element, and depending on the actual alloy can have various other elements, such as Fe, Mo, Nb, Co and various others. They have very good oxidation and corrosion resistance even at high pressures and temperatures; heating Inconel forms a thick oxide layer on its surface passivating the alloy and preventing further deterioration.
γ-TiAl is a form of titanium superalloy, often used in metal matrix composites. It is designed to replace more conventional superalloys (such as Inconel) by providing similar structural performance, but with significantly reduced density. It also has good oxidation resistance, high modulus and thermal stability. The most common combination for MMC is γ-TiAl matrix and silicon carbide fibres. The drawbacks are poor room temperature ductility, low fracture toughness and fast fatigue crack growth rate. Metallurgical research is still ongoing to mitigate some of these drawbacks through alloying and heat correct heat treatment .
Ni based ODS (oxide dispersion strengthened) alloys are used for heat turbine blades and heat exchangers, and also for re-entry vehicles. They are formed by introducing metal oxide particles into the crystal structure, which reduce the movement along dislocations and thus the material's tendency to creep. They form a protective oxide layer similar to Inconel alloys. Their drawback is fairly high density and lower allowable temperatures than for example Inconel. Also some of the alloying metals are very expensive; refractory materials such as rhenium and ruthenium .
Surface Protected Felt Insulation was used on the Space Shuttle's leeward surfaces. It was originally developed to replace the white LRSI tiles on the upper surface . The insulation is made up of Nomex felt blankets, covered in a flexible waterproof coating and bonded to the airframe by a resin adhesive. They are light and flexible, but they can be used only at low temperatures, thus they only cover the regions which are shadowed from the flow during re-entry.
Flexible Reusable Surface Insulation (and the improved AFRSI Advanced Flexible Reusable Surface Insulation, also known as FIB Fibrous Insulation Blankets) are blankets of layered, pure silica felt sandwiched between silica and glass fabric layers. They are semi rigid and can be made large (about 30 by 30 in.) so the number of blankets can be kept low. Their application is similar to the Nomex blankets.
Heat not only has to be absorbed by the vehicle, but also transferred away. For a transport some of the heat is absorbed by the fuel and then removed through the propulsion system. The other mechanism, which is especially important for a Space Launcher, is radiation. All bodies radiate (and absorb) heat, and the amount of radiation is proportional to the fourth power of body temperature and the surface emissivity. Emissivity depends on the surface (material, finish, and colour) the Space Shuttle has an average ε of 0.8 . Thus based on the input heat flux and the emissivity, the surface temperature of the vehicle is converging towards an equilibrium steady state value. This equilibrium is likely to be reached for a Transport, for its long duration flight, but might not be reached during a re-entry, it depends mainly on the vehicle and the re-entry conditions. Some designs actually rely on large heat sink masses to absorb all the heat of re-entry, without reaching equilibrium. This approach was used on the initial Mercury spacecraft, however it is a heavy solution .
4.3.3 Special thermal protection methods
Thermal protection of vehicles can be enhanced by special design features. These can be relatively simple, such as the blunt nose cone discovered by von Kármán, which enables the front shockwave to detach and absorb a fraction of the flow's energy in the process and redirect the flow.
A recent development is the feathered entry used by the SpaceShipOne family of vehicles. By changing the tail configuration the vehicle greatly increases its drag, enabling it to slow down more at higher altitudes, where the air is less dense, thus generating lower heat loads in the lower atmosphere. The only issue of this method is the low entry speed of the SpaceShipOne, which is nowhere near the velocity of an object returning from orbit (Mach 3 as compared to the Mach 25 of the Space Shuttle).
A recent result of NASA developments is the Hypersonic Inflatable Aerodynamic Decelerator (HIAD). Being part of the NASA Game Changing Technology Development Program, it is aimed to develop a lightweight, inflatable structure capable of absorbing the heat loads present at atmospheric re-entry. It is not exclusively intended for Earth, but for any planet bearing an atmosphere. It is manufactured from Nextel, Pyrogel and Kapton, and, in theory, is usable up to 1260°C . NASA completed a successful test launch of the HIAD on the 23rd July 2012.
As it was mentioned, active cooling of a vehicle relies on some form of heat sink to absorb the generated heat loads. The usual solution is to use cryogenic fuel, which heats up in the process. The heat capacity of the fuel might not be high enough, and thus would require very high, maybe even unsustainable, flow rates especially for low density fuels such as hydrogen. To counter this problem, different concepts were developed to absorb more heat by changing the chemical structure of the coolant. Reformed fuels use part of the heat absorbed to drive an endothermic reaction inside the fuel, which would “reform” it, thus absorbing significantly more energy than just simply relying on its heat capacity. The concept was originally proposed for the Russian AJAX vehicle. According to a Joint stock company report , steam reforminghydrocarbon fuel can absorb 3.3 MJ/kg as “physical cooling resource”, while the “chemical cooling resource”, the heat of the endothermic reaction is on the order of 6.6 MJ/kg. In addition to reformed fuels, other endothermic reactions such as cracking, pyrolysis or depolymerisation are also a viable option; basically the process would fracture the longer carbohydrate chains present in the fuel, allowing the absorption of about 5 MJ/kg. As a reference, the heat sink capability of regular hydrocarbons (JP7) is in the region of 1 MJ/kg .
4.3.4 Structure mass fraction
In the case of a vehicle designed for space access it is vital to keep the total mass as low as possible. According to Tsiolkovsky's rocket equation, additional weight of the vehicle raises the all up mass exponentially. On the other hand, the maximum mass fraction of a vehicle is limited by the specific impulse of its propulsion system. This means, that in order for the size of the spacecraft not to escalate, the structural weight must be kept as low as possible. It is also worth noting, that the payload fraction is significantly lower than the vehicle mass fraction as shown in Table 6. The vehicle mass fraction includes the payload mass fraction, and the remainder of the all up mass is fuel. The X-37 mass is based on an estimation by Pienkowski et al. . 250 kg/crew was assumed for the people, gear and provisions, where payload was not available, based on Space Launcher mass estimation methods . Lynx data is from the payload user's guide .
Table 6. Payload and vehicle mass fractions.
| Vehicle name | Payload mass [t] | Vehicle mass (including payload) [t] | All up mass [t] | Payload mass fraction [%] | Vehicle mass fraction [%] |
--- --- --- |
| Existing vehicles |
| X-37B – Atlas V | No data | 4.4 | 334 | No data | 1.32 |
| Space Shuttle | 24.4 | 123 | 2041 | 1.20 | 6.03 |
| Buran | 30 | 105 | 2375 | 1.26 | 4.42 |
| Saturn V – LEO | 118 | 301.6 | 2970 | 3.97 | 10.15 |
| Saturn V – Apollo 11 | 45.7 | 229.3 | 2970 | 1.54 | 7.72 |
| Delta IV Heavy – LEO | 23 | 105.7 | 733 | 3.14 | 14.40 |
| Soyuz – LEO | 7.8 | 32 | 308 | 2.53 | 10.39 |
| Vehicle concepts |
| Skylon | 15 | 68 | 345 | 4.35 | 19.71 |
| Dream chaser – Atlas V | 1.75 (7 crew) | 11.3 | 334 | 0.50 | 3.38 |
| Lynx-II (Sub-orbital) | .28 | No data | 5.2 | 5.38 | No data |
4.4 Systems
The systems onboard a hypersonic vehicle are a combination of traditional aircraft-like systems, and those used on current launchers and spacecraft. The systems are of varying technology readiness levels: some, like avionics, fuel or landing system are readily available today. Others, especially those mentioned in the previous chapter, such as propulsion and thermal protection system, still require considerable research effort.
4.5 Reliability
If hypersonic flight and space access is to become an everyday activity like today's airliners, reliability has to undergo a significant improvement. The main barrier in front of raising reliability today is the expendable nature of the launchers: everything has to work perfectly for the first time, there is no possibility for incremental testing, and usually every malfunction or fault results in the loss of the vehicle and payload. In 1999 Parkinson investigated the cost of reliability on the launchers. According to him, at that time, launch failure was estimated to be between 5 and 12%. This rate would be unacceptably high for a transportation system, for many reasons. Even excluding the moral issues, there is a significant economic consequence: the very high insurance cost. According to Parkinson, insurance costs for a launch can contribute up to 20% of the total launch costs.
There are challenges in predicting reliability. It is very difficult to predict reliability for non-existing or low TRL systems, such as novel propulsion concepts. Even in the case of scramjet technology, which has been under development for decades, there are only estimations for system reliability based on the few successful flight tests. These reliability figures based on technology demonstrators then need to be extrapolated to in service systems, which inevitably introduces uncertainty.
The success of a hypersonic transport, however, depends greatly on its propulsion system. In addition to systems, there is also an issue associated with operations. There is rarely any information available for exotic manoeuvres such as aerial refuelling, as they have only been conducted with military personnel, thus are seldom published. Their suitability to civilian application is questionable. Also operations such as maintenance and inspection for new structural materials are challenging to estimate.
4.6 Maintainability
Maintainability was clearly the Achilles' heel of the Space Shuttle. The Shuttle required tens of thousands of man hours between each launch, most of which could be attributed to the intricate thermal protection system, which consisted of over 30,000 ceramic tiles, each requiring individual inspection. In the case of future hypersonic transportation systems, it is imperative that maintainability is designed into the vehicle, rather than just something evaluated at the end of the design process. Improving maintenance not only reduces the turnaround time for the vehicle, but provides additional benefits when the time and wages of the maintenance personnel is considered. Typical turnaround times are shown in Table 7.
Table 7. Typical processing times (Refs.[105–109]).
| Vehicle | Turnaround time [days] |
--- |
| Skylon (planned) | 1 |
| X-37 (source not verified) | 10–15 |
| Space Ship One (Based on X-Prize winning performance) | 5 |
| Space Shuttle (processing only) | 75 |
| HL-20 (planned average) | 46 |
| X-15 (average) | 44 |
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Journal2016, Progress in Aerospace SciencesD. Sziroczak, H. Smith
Review article
A review of design issues specific to hypersonic flight vehicles
2016, Progress in Aerospace SciencesD. Sziroczak, H. Smith
4.1.1 Aerodynamic lift
Unlike conventional aircraft, where the lift has to balance the weight to sustain flight, a hypersonic vehicle can rely solely on thrust, provided the engines are powerful enough. Thus the importance of aerodynamic lift depends on the actual vehicle configuration. Non-lifting (ballistic) vehicles do not rely on aerodynamic lift, which result in sleeker, lower drag shapes, but their cross range capability and controllability are poor. High hypersonic lift, however results in higher deceleration at higher altitudes, which reduces the peak heat, and overall maximum deceleration loads on the vehicle. Also lift can be used to provide safe launch abort capabilities. In the case of a horizontal take-off vehicle, it is essential to provide large amounts of lift, as it is required during the take-off and climb phase. Utilising aerodynamic lift reduces the thrust required compared to lifting the vehicle vertically, but this mainly depends on the lift to drag ratio of the vehicle. Refer to Table 4 for typical L/D ratios.
Table 3. Compression and expansion methods used for hypersonic flows.
| Compression methods | Expansion methods |
--- |
| Modified Newtonian | Cp=0 |
| Newtonian–Prandtl-Meyer | Newtonian–Prandtl-Meyer |
| Tangent wedge | Prandtl-Meyer |
| Tangent wedge infinite Mach | Cone at angle of attack |
| Old tangent cone | VanDyke Unified |
| Cone at angle of attack | Vacuum |
| VanDyke unified | Shock expansion |
| Blunt body viscous | Input value |
| Shock expansion | Free molecular flow |
| Free molecular flow | Modified Dahlem–Buck |
| Input value of CpStag | ACM empirical |
| Hankey flat surface | Half Pradtl-Meyer from freestream |
| Smyth Delta Wing | |
| Modified Dahlem–Buck | |
| BlastWave | |
| OSUBluntBody | |
| Tangent cone (Edwards) | |
Table 4. Lift to drag ratio comparison [39–43].
| Vehicle | Lift to drag subsonic | Lift to drag supersonic | Lift to drag hypersonic |
--- --- |
| Space Shuttle | 4.5 | 2 | 1.5 |
| Buran | 5.0 | No data | 1.7 |
| X-15 | 4 | 2.5 | 2.5 |
| X-20 | 4.3 | No data | 1.5–1.9 |
| Common aero vehicle (X-41) | No Data | 2.5 | 3.5 |
| SpaceShipTwo | 7 | 0.5 | Not applicable |
The lift characteristics of hypersonic aircraft are fundamentally different from subsonic or linearised supersonic behaviour. The methods used to evaluate hypersonic aerodynamics depend on the vehicle shape.
For blunt shapes such as the flat underside of the Space Shuttle or similar vehicles, hypersonic aerodynamics can be approximated using the Newtonian flow theory. This theory models the flow as many small individual particles, impacting a surface, losing their normal, but retaining the tangential speed components. Investigating the behaviour of a flat plate shows remarkable similarity with the usual lifting surfaces, and thus understanding it gives insight into the fundamental behaviour of hypersonic lift.
Using the Newtonian sine-squared law, the lift and drag coefficient of a flat plate can be derived as:
(1)c l=2·Sin 2 α·Cos α
(2)c d=2·Sin 3 α
Neglecting friction, it can be seen in Fig. 9, that the theoretical lift curve at hypersonic speeds follows a non-linear behaviour, reaching its maximum at 54.7°. Unlike subsonic flows, where the peak is followed by a stall region, and the loss of lift is due to separation of the flow, here it is due to the behaviour of the trigonometric functions.
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Fig. 9. Lift and drag characteristics of flat plate in hypersonic flow (Source: ).
The Newtonian theory simplifies the flow from two important points of view. First, it ignores any cross-flows or pressure effects and approximates aerodynamic loads solely based on geometric angles. The other very important simplification, and limitation, is that it does not treat the flow over the “shadowed” side of the vehicle. This is appropriate for a flat plate, where there is essentially ambient pressure over most of the leeward side, but it introduces significant errors for example in the case of diamond shaped aerofoils used on Hypersonic Transports, where expansion effects need to be taken into account. One method such as this is the Shock-Expansion method, where only the principal characteristics of the flow through shockwaves derived from three dimensional characteristics theory are considered, and other secondary effects such as reflections from the shockwave and vortex lines are ignored. The method was introduced in 1931 by Epstein .
There were many other methods developed over the years to address flows around different geometries. There is no single method universally applicable to any vehicle shape, the designer needs to have good understanding of the fundamentals and underlying assumptions of the various methods. Table 3 shows a list of various compression and expansion methods used to estimate hypersonic aerodynamic performance. These methods form part of the SHABP software and for more details on the methods, their applicability and implementation as computer code, one should refer to the Program Formulation manual and Dirkx and Mooij for modelling methods and comparison with real vehicle flight data .
Those aircraft, which are utilising lift for ascent fall into either the winged or lifting body category. The winged configuration offers a higher lift to drag ratio at subsonic speeds, but increases structural mass and drag compared to the lifting body. At hypersonic speeds wings are disadvantageous, as long thin surfaces are not efficient structures, thus they have to be heavy to survive re-entry loads. The table below shows some winged and lifting body vehicles with their respective L/D ratios. According to NASA investigations , the theoretical maximum hypersonic lift-to-drag ratio (with a skin-friction coefficient of 10−3) for a conventional low-winged configuration, such as the Space Shuttle, is 5.29. However, a flat-top, high wing type configuration, X-43 for example, could reach up to 6.65 L/D at Mach 5 . Although, due to aerodynamic heating issues, the low wing vehicles, such as the X-20 Dyna-soar, were chosen as the first designs to develop (Fig. 10).
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Fig. 10. Low and high wing hypersonic shape comparison: Space Shuttle (left) X-43 (right) (Source: ).
There is a special class of lifting body aerospace vehicles called the waverider. This is a specially designed vehicle that utilises its own shockwave to generate extra lift to improve its lift to drag ratio. The concept was developed by Terence Nonweiler . Up to day, only the Boeing X-51 has actually demonstrated flight with this shape. The drawback of the waverider is that it can only achieve the lift increase at a specific Mach number and altitude combination, to which the geometry was optimised. There is no drawback to a waverider design compared to a conventional lifting body vehicle. The shock wave when positioned correctly could also provide ram compression for airbreathing powerplants.
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Journal2016, Progress in Aerospace SciencesD. Sziroczak, H. Smith
Review article
An overview of research on wide-speed range waverider configuration
2020, Progress in Aerospace SciencesZhen-tao Zhao, ... Yan-guang Yang
7 Other wide-speed-range waverider designs
In addition to the above-mentioned design schemes of waveriders with a wide-speed range, there are the following additional design schemes.
In 2011, Takama proposed to improve the ideal waverider's low-speed aerodynamic characteristics. Its configuration is shown in Fig. 24. He applied numerical simulation to obtain the subsonic and hypersonic aerodynamic performance of this configuration. The results showed that the addition of outer wings can increase the ideal waverider's lift-to-drag ratio in the subsonic regime, while having little effect on its hypersonic performance. Thus, this design concept can be promising for the design of wide-speed-range hypersonic waveriders.
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Fig. 24. The practical waverider attached to outer wings .
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Journal2020, Progress in Aerospace SciencesZhen-tao Zhao, ... Yan-guang Yang
Review article
Assessment of Aerothermodynamic Flight Prediction Tools
2012, Progress in Aerospace SciencesLouis M.G. Walpot, ... Ferry Schrijer
1 Introduction
In recent years, both Europe and the US are developing hypersonic research and operational vehicles. These include (re)entry capsules (both ballistic and lifting) and lifting bodies such as ExoMars, EXPERT, ARV, CEV and IXV. The research programs are meant to enable technology and engineering capabilities to support during the next decade the development of affordable (possibly reusable) space transportation systems as well as hypersonic weapons systems for time critical targets. These programs have a broad range of goals, ranging from the qualification of thermal protection systems, the assessment of RCS performances, the development of GNC algorithms, to the full demonstration of the performance and operability of the integrated vehicles. Since the aerothermodynamic characteristics influence nearly all elements of the vehicle design, the accurate prediction of the aerothermal environment is a pre-requisite for the design of efficient hypersonic systems. Significant uncertainties in the prediction of the hypersonic aerodynamic and the aerothermal loads can lead to conservative margins in the design of the vehicle including its Outer Mould Line (OML), thermal protection system, structure, and required control system robustness. The current level of aerothermal prediction uncertainties results therefore in reduced vehicle performances (e.g. suboptimal payload to mass ratio, increased operational constraints).
On the other hand, present computational capabilities enable the simulation of three-dimensional flow fields with complex thermo-chemical models over complete trajectories and ease the validation of these tools by e.g. reconstruction of detailed wind tunnel tests performed under identified and controlled conditions (flow properties and vehicle attitude in particular). These controlled conditions are typically difficult to achieve when performing in flight measurements which in turn results in large associated measurement uncertainties. Similar problems arise when attempting to rebuild measurements performed in “hot” ground facilities, where the difficulty level is increased by the addition of the free-flow characterization itself. The implementation of ever more sophisticated thermo-chemical models is no obvious cure to the aforementioned problems since their effect is often overwhelmed by the large measurement uncertainties incurred in both flight and ground high enthalpy facilities.
Concurrent to the previous considerations, a major contributor to the overall vehicle mass of re-entry vehicles is the afterbody thermal protection system. This is due to the large acreage (equal or bigger than that of the forebody) to be protected. The present predictive capabilities for base flows are comparatively lower than those for windward flowfields and offer therefore a substantial potential for improving the design of future re-entry vehicles. To that end, it is essential to address the accuracy of high fidelity CFD tools exercised in the US and EU, which motivates a thorough investigation of the present status of hypersonic flight afterbody heating.
This paper addresses the predictive capabilities of afterbody flow fields of re-entry vehicles investigated in the frame of the NATO/RTO—RTG-043 task group and is structured as follows:
First, the verification of base flow topologies on the basis of available wind-tunnel results performed under controlled supersonic conditions (i.e. cold flows devoid of reactive effects) is performed. Such tests address the detailed characterization of the base flow with particular emphasis on separation/reattachment and their relation to Mach number effects. The tests have been performed on an Apollo-like re-entry capsule configuration.
Second, the tools validated in the frame of the previous effort are exercised and appraised against flight-test data collected during the Apollo AS-202 re-entry.
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Journal2012, Progress in Aerospace SciencesLouis M.G. Walpot, ... Ferry Schrijer
Chapter
Spacecraft Flight Mechanics
2016, Manned Spacecraft Design PrinciplesPasquale M. Sforza
8.3 Blunt Bodies in Hypersonic Flight
The preponderance of manned space vehicles are capsule shaped, such as the Gemini vehicle, as shown in Figure 8.6. This shadowgraph of a Gemini capsule model in hypersonic flight in a ballistic range illustrates the shock wave and wake flow field. Such vehicles may be characterized by the idealized form as shown in Figure 8.7 for the symmetric case of zero angle of attack. The important parameters for such vehicles are the heat shield radius R N, the maximum radius of the capsule R b, and the associated angle ϕ 1=sin−1(R b/R N).
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Figure 8.6. Shadowgraph of a model of the Gemini space capsule traveling at hypersonic speed in a ballistic range.
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Figure 8.7. Schematic diagram of a blunt Apollo-like space capsule at a zero angle of attack. The angle θ defines the local tangent to the spherical nose cap and ϕ c is the afterbody cone angle.
In the hypersonic regime of interest, we may determine the zero-lift drag coefficient, C D,0 using the Newtonian flow approximation. The local tangent to the spherical cap heat shield is shown in Figure 8.7 at a generic point on the heat shield (R N,ϕ). The outward normal to the surface at any point is defined by the nose cap radius R N. As will be seen in Chapter 9, a large nose radius is the foundation of the spacecraft’s passive thermal protection system because convection heating is inversely proportional to the local radius of curvature. At the same time, the large frontal area produces high drag due almost entirely to the adiabatically compressed high-pressure shock layer formed over it. Skin friction is a minor contributor to the drag unlike the case for the slender bodies treated subsequently in this chapter. Therefore, we will be primarily concerned with pressure forces in our consideration of blunt bodies.
Consider a segment of spherically blunted spacecraft forebody as shown in Figure 8.8. We see from the pressure field for the axisymmetric situation posed that the lift distribution is antisymmetric about ϕ=0 so the lift is zero. The drag distribution is symmetric about ϕ=0, so the differential drag coefficient is given by
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Figure 8.8. Axisymmetric segment of a spherically blunted nose of radius R N.
d C D=d D S=C p S 2 π r 2 cos ϕ sin ϕ d ϕ
We have shown previously that Newtonian theory for a blunt body gives the pressure coefficient on the surface as follows:
(8.24)C p=(2−ε)sin 2 θ=(2−ε)cos 2 ϕ
The quantity ε is the ratio of density upstream of the shock to that downstream of the shock. In Section A.2.2 of Appendix A, we show that for constant γ, the density ratio is given by
ρ 1 ρ 2=ε=γ−1 γ+1(1+2(γ−1)M 2)=ε lim M→∞(1+2(γ−1)M 2)
The density ratio ε is shown as a function of Mach number for different constant, but practical, values of γ in Figure 8.9. It is clear that ε becomes essentially constant once the hypersonic regime is entered. Because the Newtonian approximation is appropriate in the hypersonic limit where M≫1, the density ratio ε=ε lim=(γ−1)/(γ+1) is a good approximation. The error incurred by using ε lim rather than the actual value of ε are below 10% for 1.2<γ<1.4 when M>10. For γ~1.1, a 10% error threshold is not reached until M>15.
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Figure 8.9. The density ratio ε=ρ 1/ρ 2 as a function of Mach number for different constant values of γ.
Integrating around the segment of the nose, the zero-lift drag coefficient is found to be
(8.25)C D,0=∫0 ϕ 1(2−ε)2 π r 2 S(−d cos 4 ϕ 4)=(2−ε)π r 2 2 S[1−cos 4 ϕ 1]
Beyond ±ϕ 1, the Newtonian pressure coefficient is zero, so there is no contribution from the base region. In practice, there is actually a small contribution from the capsule afterbody, and this detail will be discussed subsequently. The reference area is chosen to be S=π R b 2, where R b is the maximum transverse radius of the segment, as illustrated in Figure 8.8. The quantity R b is related to R N by R b/R N=sin ϕ 1, and therefore the zero-lift drag coefficient may be written as
(8.26)C D,0=(1−ε 2)(1+cos 2 ϕ 1)=(2−ε)[1−1 2(R b R N)2]
When ϕ 1≪1, the drag coefficient C D,0~2−ε, the result for a flat plate normal to the flow, and when ϕ 1~π/2, the drag coefficient C D,0~1−ε/2, the result for a hemisphere.
Note that the shaded portion of Figure 8.8 resembles the shape of a typical capsule-type spacecraft. Several candidate capsule designs for the NASA Mercury program illustrating the different afterbody cone angles ϕ c considered as shown in Figure 8.10. The afterbody cone angle basically defines the maximum angle of attack that can be safely employed. Larger angles would require cone sidewalls to be protected from the high-temperature flow. Astronaut John Glenn’s Mercury spacecraft, Friendship 7, which is shown in Figure 8.11 following its return to Cape Canaveral after recovery in the Atlantic Ocean, has a configuration closest to capsule C in Figure 8.10.
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Figure 8.10. Early capsule designs showing the different afterbody cone angles considered. Bluntness ratio R b/R N=0.33, 0.33, 0.5, and 0.465 for capsules A, B, C, and D, respectively.
Courtesy NASA.
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Figure 8.11. Astronaut John Glenn’s Mercury spacecraft, Friendship 7, is shown here after recovery in the Atlantic Ocean.
Courtesy NASA.
The variation of C D,0 with capsule bluntness ratio R b/R N according to Eqn (8.26) is shown in Figure 8.12 for two different values of the ratio of specific heats: γ=1.4, the standard air value, and γ=1.2, a value more representative of the hot gas around a re-entering space capsule. The range of bluntness ratios for Apollo-like capsules is also shown on the figure. Note that as R b/R N increases, the forward face of the capsule, that is, the heat shield, becomes relatively rounder resulting in a lower drag coefficient. In order to produce some lift to aid in the reentry process, it is necessary to put the capsule at some angle of attack, as discussed in the next section.
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Figure 8.12. The zero-lift drag coefficient of Apollo-like space capsule as a function of the capsule bluntness ratio R b/R N; for Apollo R b /R N=0.426.
8.3.1 Blunt Body L/D
Past and current manned space capsules are illustrated in Figure 8.13, and the forces acting on a typical blunt capsule are illustrated in Figure 8.14. The lift and drag components of the resultant aerodynamic force may be obtained from the components of that resultant force resolved normal to and along the axis of the capsule. If we choose to consider the angle of attack α>0 as shown, the following relations apply:
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Figure 8.13. Scaled sketches of past and current space capsules with pertinent configuration data.
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Figure 8.14. Blunt Apollo-like space capsule at an angle of attack showing resultant force field, direction of positive pitching moment, and center of pressure.
(8.27)L=F a sin α−F n cos α
(8.28)D=F a cos α+F n sin α
Note that since only pressure forces on the spherical cap are considered important and because pressure acts normal to the surface, the resultant force must pass through the origin of the spherical cap. Because no moment can act at that point, it is also the center of pressure. Note that pressures are highest where the angle between the velocity vector and the surface normal is smallest. Figure 8.15 is a shadowgraph of a model of a Mars Lander capsule at high Mach number in a ballistic range in which the degree to which the shock follows the body shape is evident. A diagram of a capsule on an entry trajectory has a similar attitude, as shown in Figure 8.16. For completeness in considering the forces to be acting at the center of gravity of the capsule, we show a pitching moment that must be also acting there. The moment must be trimmed out, that is made zero, for equilibrium to be possible. This is typically achieved by offsetting the center of gravity as shown in Figure 8.14.
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Figure 8.15. Shadowgraph of a model of a Mars Lander capsule at high Mach number in a ballistic range showing the shock following the body shape.
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Figure 8.16. Orientation of capsule for positive lift coefficient at a practical angle of attack.
The remarkably general accuracy of the simple Newtonian theory was of great benefit during the early manned spaceflight program because it put calculation of hypersonic aerodynamic parameters within relatively easy reach of researchers and designers during a period when computational capabilities were quite limited. The clearest manifestation of the breadth of application of Newtonian theory may be found in Ried and Mayo (1963). They provide equations based on Newtonian theory for the important static and dynamic stability coefficients for bodies of revolution with an offset center of gravity location flying at angles of attack, yaw, and bank. Although the final equations are often quite lengthy, solutions are reduced to carrying out a single integration along the longitudinal axis of the vehicle for each of the aerodynamic coefficients sought. It is worthwhile to examine this work to appreciate the extent of effort on analysis that was necessary to formulate the equations in a manner which then-current computers could readily handle. Computational power is now within such easy reach that the Newtonian theory may be readily applied to an array of independent panels representing the actual vehicle, so that the primitive forces on each may be combined to generate overall forces and moments as described previously in this chapter.
The value of analytically based approaches is that they often provide general information about the behavior of a solution rather than just the solution itself. The equations for the axial and normal force coefficients given by Ried and Mayo (1963), when simplified for the conditions of zero sideslip and bank angles and zero pitch, yaw, and roll rates are as follows:
(8.29)C a=8(1−ε 2)(R N d)2∫0 ϕ 1(2 cos 2 α cos 3 ϕ sin ϕ+sin 2 α sin 3 ϕ cos ϕ)d ϕ
(8.30)C n=8(1−ε 2)(R N d)2 sin 2 α∫0 ϕ 1 sin 3 ϕ cos ϕ d ϕ
Noting that R N/d=(2 sin ϕ 1)−1, we may integrate these equations to yield
(8.31)C a=(1−ε 2)[cos 2 α(1+cos 2 ϕ 1)+1 2 sin 2 α sin 2 ϕ 1]
(8.32)C n=1 2(1−ε 2)sin 2 α sin 2 ϕ 1
We may also write Eqn (8.31) in terms of the zero-lift drag coefficient as follows:
(8.33)C a=C D,0 cos 2 α+1 2(1−ε 2)sin 2 α sin 2 ϕ 1
The moment coefficient, after applying the simplifying assumptions concerning the flight attitude of the vehicle, becomes
C m=8(R N d)2 sin 2 α∫0 ϕ 1(x−x 0)+(R N d)cos ϕd ϕ−(z 0 d)C a
Carrying out the integration yields
(8.34)C m=[(x 0 d)−(R N d)]C n−(z 0 d)C a
The angle of attack used in the general development presented by Ried and Mayo (1963) is defined in the opposite sense to that shown in Figure 8.14 because the latter is more convenient for presenting force and moment data for the specific case of space capsules. The only special care to be taken is that the sense of the axial and normal forces should be accounted for in the moment Eqn (8.34). That is, the contribution of the normal force coefficient to the moment about the center of gravity in Eqn (8.34) is positive because the term in square brackets is negative and, as depicted in Figure 8.14, C n acts in the negative z-direction. Of course, Eqn (8.34) might have been written down by inspection because, as mentioned previously, only pressure forces are considered and they pass through the origin of the circular nose making that point the center of pressure. The normal and axial forces may be considered to act there with no associated moment. Then, the moment about the center of gravity is simply the sum of the moments produced by the axial and normal forces acting at the center of pressure. In practical cases where boundary layer details, afterbody wake effects, and surface protuberances are considered, the center of pressure may be somewhat removed from the location of the center of the nose radius.
Using the above equations for the axial and normal force coefficients, along with our convention for α, in Eqns (8.27) and (8.28), the lift and drag coefficients become
(8.35)C L=[C D,0 cos 2 α+1 2(1−ε 2)sin 2 ϕ 1(3 sin 2 α−2)]sin α
(8.36)C D=[C D,0 cos 2 α+3 2(1−ε 2)sin 2 ϕ 1 sin 2 α]cos α
The variation of the lift and drag coefficients with angle of attack as given by Eqns (8.35) and (8.36) is shown in Figure 8.17 for γ=1.4 and two bluntness ratios R b/R N=0.426 and 0.5 which are characteristic of the Apollo and Soyuz capsules, respectively. The variation of the lift to drag ratioL/D is shown in Figure 8.18 for the same two bluntness ratios. Also shown on both figures are curves representative of experimental data for the Apollo command module (CM) presented by DeRose (1969). In general, the theoretical results for manned capsule shapes are limited to angles of attack about equal to the afterbody cone angle ϕ c. When α>ϕ c, the afterbody surfaces are exposed to the shock-heated flow environment rather than the relatively benign environment in the Newtonian shadow region. In addition, at such high angles, the lift begins to drop off very much like the stalling of the lift on a wing at high angle of attack. For relatively flat-backed axisymmetric bodies, like some unmanned planetary reentry vehicles, the results of Eqns (8.35) and (8.36) are applicable although the high angle of attack restriction still applies.
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Figure 8.17. Variation of lift and drag coefficients of Apollo-like capsules with angle of attack for two bluntness ratios, both with γ=1.4.
Experimental Apollo data taken from DeRose (1969).
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Figure 8.18. Variation of L/D for Apollo-like capsules with angle of attack for two bluntness ratios, both with γ=1.4.
Experimental Apollo data taken from DeRose (1969).
It is clear from Figure 8.17 that the Newtonian theory overestimates the experimentally observed drag coefficient by around 7.5%. Miller and Lawing (1966) present experimental data from their experiments and from those of several others on the pressure distribution over the blunt nose of an Apollo-like capsule. The results they present are for α=0 and α=33° (approximately the capsule cone angle ϕ c) over the Mach number range 6<M<24.5, and all data show quite good agreement with the predictions of Newtonian theory. Interestingly, the best agreement occurs for α=33°, whereas at α=0, the pressure is somewhat overestimated over the outermost portion of the heat shield. They also compare the experimental data for surface pressure with theoretical results based on the approach presented by Kaattari (1962), which show even better agreement than do the Newtonian predictions for the zero angle of attack case.
Kaattari (1962) also provides a method for quite accurately calculating the shock shape over a blunt body rather than relying on the Newtonian assumption that the shock and body shapes are coincident. His approach, although more accurate than the simple Newtonian theory, is more cumbersome to apply, and the relatively small improvement in the results for surface pressures, particularly in the practical case of non-zero angle of attack, has limited its use. The method does however give useful results for shock shape, shock standoff distance, and stagnation point locations for capsule-type bodies.
This overestimation of the front face pressure contributes to predictions for the zero-lift drag coefficient that are around 5–10% higher than those obtained by experiment. It must also be noted that part of the higher estimate for the drag coefficient arises from the fact that the Newtonian theory predicts the afterbody (shadow) region to have C p,a=0. However, Miller and Lawing (1966) also provide experimental data which show that actually C p,a>0, as described subsequently in Section 8.3.2, and this also contributes to a slight reduction in the drag coefficient from that predicted by Newtonian theory alone.
8.3.2 Capsule Afterbody Pressure
The assumptions of the Newtonian theory discussed in Appendix A require that the thin shock layer follow the blunt nose and not turn to follow the capsule afterbody. Instead, the shock layer departs from the body leaving the afterbody in the so-called shadow region where C p=0. However, experimental results from more than a half dozen different investigations were presented by Miller and Lawing (1966) for hypersonic flow over an Apollo-like capsule model which show that the afterbody pressure p a increases with Mach number from values close to free stream pressure (for which C p=0) at M~5 to as much as an order of magnitude greater than free stream pressure at M=20. The data presented could be approximated by the following simple relation:
(8.37)p a p∞≈1+0.025 M∞2
Then, the pressure coefficient on the afterbody would be essentially constant with Mach number and equal to
(8.38)C p,a=p a−p∞1 2 γ p∞M∞2≈0.05 γ
Miller and Lawing (1966) also show data for the circumferential variation of the afterbody pressure field at zero angle of attack and at the essentially maximum angle of attack α=33°. At α=0, the pressure is reasonably constant around the afterbody, and although at α=33° the pressure varies circumferentially, the average value is about the same as that at α=0. Under these conditions, the pressure force on the afterbody is C p,a q π R b 2, and the zero-lift drag coefficient predicted by Newtonian theory could probably be corrected as follows:
(8.39)C D,0=(2−ε)[1−1 2(R b R N)2]−0.05 γ
Such a correction amounts to about a 2–3% reduction in the drag coefficient. We mentioned previously that accounting for the Newtonian theory overestimation of the front face pressure is likely to contribute another 5% or so.
8.3.3 Forces on Spherically Blunted Cones
Some reentry vehicles, particularly ballistic missile warheads, are configured as spherically blunted cones. At high Mach numbers, Newtonian flow theory can be used to estimate the aerodynamic forces on these bodies as well. A schematic of the blunted cone showing all the variables appears in Figure 8.19. The behavior of the pressure coefficient for the spherical nose portion follows directly from the results of the previous sections, while on the conical portion of the body, the pressure coefficient remains constant. For the spherical nose cap, the differential drag coefficient is
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Figure 8.19. Schematic diagram of spherically blunted cone.
(8.40)d C D=(2−ε)2 π r 2 S[d(−cos 4 ϕ 4)]
On the conical portion of the body, the differential drag coefficient is
(8.41)d C D=(2−ε)sin 2 ϕ 2 tan ϕ(2 π r d x)
The complete zero-lift drag coefficient is given by
(8.42)C D,0=(2−ε)[1 2(R N R B)2(1−sin 4 ϕ 2)+sin 2 ϕ 2−(R N R B sin ϕ 2 cos ϕ 2)2]
Note that when R N=R B (and therefore ϕ 2=0), the body is a hemisphere-cylinder and C D=(1−ε/2), the same as that for a sphere. When R N=0, the body is a sharp cone and the drag coefficient C D=(2−ε)sin 2 ϕ 2, the same as for a cone of half-angle equal to ϕ 2. The variation of the zero-lift drag coefficient with the ratio of nose radius to base radius for various values of the cone half-angle is shown in Figure 8.20 for the case of γ=1.4. Other values of γ may be accommodated by using ε=(γ−1)/(γ+1).
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Figure 8.20. Zero-lift drag coefficient for spherically blunted cones as a function of the ratio of nose to base radius for various cone half angles and γ=1.4.
It is clear that small nose radii can be accommodated without substantial drag penalties. Nose bluntness mitigates reentry heating effects because the heat flux at the stagnation point is inversely proportional to the square root of the nose radius. Thus, thermal protection can be enhanced without much sacrifice in drag. The effect of the thermodynamic state of the gas is contained solely in the initial term multiplying the square brackets; recall that
(8.43)ε=γ−1 γ+1
Thus at higher temperatures where γ<1.4, the drag coefficient will be larger than the perfect gas value as shown in Figure 8.20. For example, when γ=1.2, multiply the results in Figure 8.20 by 126/111 which increases the drag coefficient by about 4%.
An example of a sphere-cone body used as a practical entry vehicle is the General Electric Mark-6 as shown in Figure 8.21. This entry vehicle is large, with a mass of 3360 kg, a length of 3.1 m, a base radius of 1.15 m, and a nose radius of 0.59 m. The ballistic coefficient B=mg/C D,0 S for this vehicle is therefore 7.86 kPa/C D. The cone half-angle is 12.5° and R B/R N=0.5 which, from Figure 8.20, suggests C D,0~0.3 and therefore a ballistic coefficient B~26 kPa, about five times higher than a Project Mercury capsule. This provides the capability for very steep high-velocity entry, as discussed in Chapter 6, making the Mark-6 difficult for ABM systems to intercept. Advances in ablative thermal protection system design led to smaller vehicles with smaller nose bluntness ratio compared to the Mk-6. A more modern sphere-cone entry vehicle is the 360 kg Mark-12 with a 10° cone half-angle and a very small nose bluntness resulting in C D,0~0.05 and B=325 kPa.
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Figure 8.21. The 3.1 m long Mark-6 reentry vehicle as shown in the Titan-2 ICBM exhibit at the National Atomic Museum Albuquerque, NM.
Photograph by Stephen Sutton.
8.3.4 Newtonian Flow with a P–M Expansion
Consider the hypersonic inviscid flow over a hemisphere-cylinder blunt body as shown in Figure 8.22. Under the Newtonian flow approximation, the ratio of the local surface value of C p to the stagnation point value C p,max is
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Figure 8.22. Schematic diagram of a hemisphere-cylinder in hypersonic flow.
(8.44)C p C p,max=sin 2 θ=cos 2 ϕ
The variation of C p/C p,max as a function of non-dimensional distance s/R b=ϕ along the surface starting from the stagnation point where ϕ=0 is shown in Figure 8.23. The predicted pressure distribution of Eqn (8.44) agrees well with the measurements on a hemisphere-cylinder presented by Crawford and McCauley (1957) for M 1=6.8. They also show data from other researchers for M 1=1.97, 3.8, and 5.8. Only for the case of M 1=1.97 is there a noticeable departure from the Newtonian prediction and that only for ϕ>45°, which is in the vicinity of the sonic point. This overexpansion compared to Newtonian theory is only in evidence for M 1<3. Otherwise, the pressure recovers to approximately the free stream value p~p 1 as the hemisphere-cylinder junction is approached, as predicted by the Newtonian theory. However, it can be seen in Figure 8.23 that the measured pressure coefficient beyond the junction is small, but not zero as predicted.
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Figure 8.23. C p/C p,max as a function of non-dimensional distance s/R b=ϕ along the body surface from the stagnation point to beyond the hemisphere-cylinder junction. Symbols denote M=6.8 data from Crawford and McCauley (1957).
According to the Newtonian pressure distribution, the sonic point on the surface of the body is approximately located at the angle ϕ c where
(8.45)cos ϕ c=(γ+1 2)−γ 2(γ−1)=pp t 2
Using the definition of the pressure coefficient, one may solve for the static-to-stagnation pressure ratio and therefore M on the surface of the body. This approach yields
(8.46)p p t 2=(1+γ−1 2 M 2)−γ γ−1=C p C p,max+(1−C p C p,max)(p t 2 p 1)−1
Solving Eqn (8.46) for the surface, Mach number results in
(8.47)M 2=2 γ−1{[C p C p,max+(1−C p C p,max)(p t 2 p 1)−1]−(γ−1)γ−1}
The ratio of the stagnation pressure behind a normal shock to the static pressure upstream of the shock p t 2/p 1 comes from the normal shock relations and is given by
(8.48)p t 2 p 1=[γ+1 2 M 1 2]γ γ−1[(γ+1)2 γ M 1 2−(γ−1)]1 γ−1
For M≫1, Eqn (8.48) becomes
(8.49)p t 2 p 1≈(γ+1 2)γ+1 γ−1 γ−1 γ−1 M 1 2≈γ 0.75 M 1 2
Equation (8.49) is accurate to within about 4% for M 1>3 and improves as M 1 increases. Rather than relying completely on the Newtonian pressure distribution as expressed in Eqn (8.44), one may apply the P–M expansion discussed in Appendix A for points on the body beyond the sonic point ϕ c, where M=1 on the body and, by definition, the P–M angle ν=0. The change in the P–M angle as one proceeds around the body is given by Δ ν=ϕ−ϕ c, and for each value of ν, there is associated a Mach number and static-to-stagnation pressure ratio as given in Tables A.1 and A.2 of Appendix A Table A.1 Table A.2.
In Figure 8.24, the Mach number calculated from Eqn (8.47) on the basis of the Newtonian pressure distribution of Eqn (8.44) is compared to that calculated using the P–M expansion beyond the sonic point as defined by Eqn (8.45). It is clear that up to s/R b=1.2 or ϕ=69°, the two results are approximately equal, but beyond that point, the results for the two methods are different with the coupled Newtonian and P–M approach showing better agreement with experiment.
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Figure 8.24. Mach number distributions around a hemisphere-cylinder according to two simple theories using γ=1.4 are compared with experimental results.
Using Eqn (8.47) with the Newtonian pressure distribution to calculate the surface, Mach number yields M=3.33 at the hemisphere-cylinder junction for a free stream Mach number of 6.8 which is about 19% high compared to experiment. Although the pressure difference between the two methods is small and the effect on pressure force is not significant, the difference in surface Mach number prediction has an effect on the boundary layer and therefore the skin friction and heat transfer.
If we consider the flow around a sphere alone, the Newtonian theory predicts C p=0 at ϕ=90° and a “dead water” region behind the body beyond that point where the theory does not apply. Instead, the P–M expansion, which takes over from the Newtonian theory when M>1, can be continued around the body continually dropping the pressure and increasing the inviscid Mach number on the body. This effect on the surface Mach number is illustrated in Figure 8.25 for flow of a gas with γ=1.4 over a sphere at M 1=6.8. The surface pressure is equal to the free stream pressure p 1 at M=3.33, which occurs at ϕ=90° according to Newtonian theory and about ϕ=97.4° for the continued P–M expansion. Also shown on Figure 8.25 is the point ϕ=120° and M=5.11 for which p=0.1 p 1, which is about as far as a real flow can proceed. In general, somewhere between these values of ϕ, viscous effects become important and a recirculation region is established behind the body causing the inviscid flow to separate from the body.
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Figure 8.25. Mach number distributions around a sphere according to two simple theories using γ=1.4. The dashed lines behind the sphere delineate the region within which Newtonian theory does not apply.
In the general panel method described in Section 8.2.3, we know θ as the angle between the free stream and the tangent to the surface on each panel, as shown in Figure 8.26. When θ<20°, the P–M expansion is more accurate than the Newtonian theory and should be applied. The flow deflection between adjacent panels 1 and 2 is simply Δ θ=θ 2−θ 1. Knowing M 1 on panel 1 at a given x-station, we also know ν 1 on that panel and we may find ν 2 on panel 2, the adjacent panel at the next x-station, because ν 2=ν 1+Δ θ. Using this value of ν 2, we can find the corresponding value of M 2 on panel 2 either by using Tables A.1 and A.2 in Appendix A Table A.1 Table A.2 or one the equations for ν given there. We also can determine p 2/p t and then proceed for the other variables as described previously in Section 8.2.3. This unit process may be carried out for any pair of panels and thereby determine the flow conditions on each.
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Figure 8.26. Schematic of a typical adjacent panels used to implement the P–M expansion for values of θ<20°.
An example of a blunt body like a hemisphere-cylinder is the entry vehicle for the KH-7 (Keyhole-7) high-resolution space reconnaissance system as shown in Figure 8.27. The entry vehicle was about 0.7 m in diameter and 0.8 m long and had a mass of 160 kg. Satellites with film-return capabilities were used for two basic functions: search or surveillance. CORONA satellites, first launched in 1960, were search systems that photographed wide swaths of land to identify airfields and missile sites, among other things. The KH-7 series of satellites were operational during 1963–1967 fulfilling the need for surveillance using stereo high-resolution cameras. These satellites were launched by Atlas-Agena rockets from Vandenberg AFB in California, and the film capsules were recovered in flight near Hawaii.
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Figure 8.27. KH-7 reconnaissance satellite the National Museum of the US Air Force. The blunt body protected film containers during entry from orbit.
Courtesy US Air Force.
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Book 2016, Manned Spacecraft Design PrinciplesPasquale M. Sforza
Chapter
Flight (Aerodynamics)
2003, Encyclopedia of Physical Science and Technology (Third Edition)John D. Anderson Jr.
IV How Aerodynamics is Subdivided
An understanding of aerodynamics, like that of any other physical science, is obtained through a “building-block” approach—we dissect the discipline, form the parts into nice polished blocks of knowledge, and then later attempt to reassemble the blocks to form an understanding of the whole. An example of this process is the way that different types of aerodynamic flows are categorized and visualized. Although nature has no trouble setting up the most detailed and complex flow with a whole spectrum of interacting physical phenomena, we must attempt to understand such flows by modeling them with less detail and neglecting some of the (hopefully) less significant phenomena. As a result, a study of aerodynamics has evolved into a study of numerous and distinct types of flow. The purpose of this section is to itemize and contrast these types of flow and to briefly describe their most important physical phenomena.
IV.A Continuum versus Free-Molecule Flow
Consider the flow over a body, say, for example, a circular cylinder of diameter d. Also, consider the fluid to consist of individual molecules, which are moving about in random motion. The mean distance that a molecule travels between collisions with neighboring molecules is defined as the mean free path λ. If λ is orders of magnitude smaller than the scale of the body measured by d, then the flow appears to the body as a continuous substance. The molecules impact the body surface so frequently that the body cannot distinguish the individual molecular collisions, and the surface feels the fluid as a continuous medium. Such flow is called continuum flow. The other extreme is where λ is on the same order as the body scale; here, the gas molecules are spaced so far apart (relative to d) that collisions with the body surface occur only infrequently, and the body surface can feel distinctly each molecular impact. Such flow is called free molecular flow. For manned flight, vehicles such as the space shuttle encounter free molecular flow at the extreme outer edge of the atmosphere, where the air density is so low that λ becomes on the order of the shuttle size. There are intermediate cases, where flows can exhibit some characteristics of both continuum and free-molecule flows; such flows are generally labeled “low-density flows,” in contrast to continuum flow. By far, the vast majority of practical aerodynamic applications involves continuum flows. Low-density and free-molecule flows are just a small part of the total spectrum of aerodynamics.
IV.B Inviscid versus Viscous Flow
A major facet of a gas or liquid is the ability of the molecules to move rather freely. When the molecules move, even in a very random fashion, they obviously transport their mass, momentum, and energy from one location to another in the fluid. This transport on a molecular scale gives rise to the phenomena of mass diffusion, viscosity (friction), and thermal conduction. All real flows exhibit the effects of these transport phenomena; such flows are called viscous flows. In contrast, a flow that is assumed to involve no friction, thermal conduction, or diffusion is called an inviscid flow. Inviscid flows do not truly exist in nature; however, there are many practical aerodynamic flows (more than you would think) where the influence of transport phenomena is small, and we can model the flow as being inviscid. For this reason, more than 70% of aerodynamic theory deals with inviscid flows.
Theoretically, inviscid flow is approached in the limit as the Reynolds number goes to infinity. However, for practical problems, many flows with high but finite Re can be assumed to be inviscid. For such flows, the influence of friction, thermal conduction, and diffusion is limited to a very thin region adjacent to the body surface (the boundary layer), and the remainder of the flow outside this thin region is essentially inviscid. This division of the flow into two regions is illustrated in Fig. 2. For flows over slender bodies, such as the airfoil sketched in Fig. 2, inviscid theory adequately predicts the pressure distribution and lift on the body and gives a valid representation of the streamlines and flow field away from the body. However, because friction (shear stress) is a major source of aerodynamic drag, inviscid theories by themselves cannot adequately predict total drag.
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FIGURE 2. Division of an aerodynamic flow into a viscous boundary layer adjacent to the surface and an inviscid outer flow. [From Anderson J. D., Jr. (2001). “Fundamentals of Aerodynamics” 3rd ed., McGraw-Hill, New York.]
In contrast, there are some flows that are dominated by viscous effects. For example, if the airfoil in Fig. 2 is inclined to a high incidence angle to the flow (high angle of attack), then the boundary layer will tend to separate from the top surface, and a large wake is formed downstream. The separated flow is sketched at the top of Fig. 3; it is characteristic of the flow field over a “stalled” airfoil. Separated flow also dominates the aerodynamics of blunt bodies, such as the cylinder at the bottom of Fig. 3. Here, the flow expands around the front face of the cylinder, but separates from the surface on the rear face, forming a rather fat wake downstream. The types of flow illustrated in Fig. 3 are dominated by viscous effects: no inviscid theory can independently predict the aerodynamics of such flows.
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FIGURE 3. Illustration of flow separation. [From Anderson, J. D., Jr. (2001). “Fundamentals of Aerodynamics” 3rd ed., McGraw-Hill, New York.]
IV.C Incompressible versus Compressible Flows
A flow in which the density ρ is constant is called incompressible. In contrast, a flow where the density is variable is called compressible. For our purposes here, we will simply note that all flows, to a greater or lesser extent, are compressible: truly incompressible flow, where the density is precisely constant, does not occur in nature. However, analogous to our discussion of inviscid flow, there are a number of aerodynamic problems that can be modeled as being incompressible without any detrimental loss of accuracy. For example, the flow of homogenous liquids is treated as incompressible, and hence, most problems involving hydrodynamics assume ρ is a constant. Also, the flow of gases at low Mach number is essentially incompressible; for M<0.3, it it always safe to assume ρ is a constant. This was the flight regime of all airplanes from the Wright Brothers' first flight in 1903 to just prior to World War II. It is still the flight regime of most small, general aviation aircraft of today. Hence, there exists a large bulk of aerodynamic experimental and theoretical data for incompressible flows. On the other hand, high-speed flow (near Mach 1 and above) must be treated as compressible; for such flows, ρ can vary over wide latitudes.
IV.D Mach-Number Regimes
Of all the ways of subdividing and describing different aerodynamic flows, the distinction based on Mach number is probably the most prevalent. If M is the local Mach number at an arbitrary point in a flow field, then by definition the flow is locally
Subsonic if M<1 Sonic if M=1 Supersonic if M>1
Looking at the whole flow field simultaneously, four different speed regimes can be identified using Mach number as the criterion.
1.
Subsonic flow (M<1 everywhere). A flow field is defined as subsonic if the Mach number is less than 1 at every point. Subsonic flows are characterized by smooth streamlines (no discontinuity in slope), as sketched in Fig. 4a. Moreover, since the flow velocity is everywhere less than the speed of sound, disturbances in the flow (say the sudden deflection of the trailing edge of the airfoil in Fig. 4a) propagate both upstream and downstream and are felt throughout the entire flow field. Note that a freestream Mach number M∞ less than 1 does not guarantee a totally subsonic flow over the body. In expanding over an aerodynamic shape, the flow velocity increases above the freestream value, and if M∞ is close enough to 1, the local Mach number may become supersonic in certain regions of the flow. This gives rise to a rule of thumb that M∞<0.8 for subsonic flow over slender bodies. For blunt bodies, M∞ must be even lower to insure totally subsonic flow. (Again, emphasis is made that the above is just a loose rule of thumb and should not be taken as a precise quantitative definition.) Also, note that incompressible flow is a special limiting case of subsonic flow where M→0.
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FIGURE 4. Characterization of flow fields based on the Mach number range. [From Anderson, J. D., Jr. (2001). “Fundamentals of Aerodynamics” 3rd ed., McGraw-Hill, New York.]
2.
Transonic flow (mixed regions where M<1 and M<1). As stated above, if M∞ is subsonic but near unity, the flow can become locally supersonic (M<1). This is sketched in Fig. 4b, which shows pockets of supersonic flow over both the top and the bottom surfaces of the airfoil, terminated by weak shock waves behind which the flow becomes subsonic again. Moreover, if M∞ is increased slightly above unity, a bow shock wave is formed in front of the body; behind this shock wave the flow is locally subsonic, as shown in Fig. 4c. This subsonic flow subsequently expands to a low supersonic value over the airfoil. Weak shock waves are usually generated at the trailing edge, sometimes in a “fishtail” pattern as shown in Fig. 4c. The flow fields shown in Figs. 4b and 4c are characterized by mixed subsonic–supersonic flows and are dominated by the physics of both types of flows. Hence, such flow fields are called transonic flows. Again, as a rule of thumb for slender bodies, transonic flows occur for freestream Mach numbers in the range 0.8<M∞<1.2.
3.
Supersonic flow (M<1 everywhere). A flow field is defined as supersonic if the Mach number is greater than 1 at every point. Supersonic flows are frequently characterized by the presence of shock waves across which the flow properties and streamlines change discontinuously (in contrast to the smooth, continuous variations in subsonic flows). This is illustrated in Fig. 4d for supersonic flow over a sharp-nosed wedge; the flow remains supersonic behind the oblique shock wave from the tip. Also shown are distinct expansion waves, which are common in supersonic flow. (Again, the listing of M∞>1.2 is strictly a rule of thumb. For example, in Fig. 4d, if θ is made large enough, the oblique shock wave will detach from the tip of the wedge and will form a strong, curved bow shock ahead of the wedge with a substantial region of subsonic flow behind the wave. Hence, the totally supersonic flow sketched in Fig. 4d is destroyed if θ is too large for a given M∞. This shock detachment phenomenon can occur at any value of M∞>1, but the value of θ at which it occurs increases as M∞ increases. In turn, if θ is made infinitesimally small, the flow field in Fig. 4d holds for M∞≥1.0. However, the above discussion clearly shows that the listing of M∞>1.2 in Fig. 4d is a very tenuous rule of thumb and should not be taken literally. In a supersonic flow, because the local flow velocity is greater than the speed of sound, disturbances created at some point in the flow cannot work their way upstream (in contrast to subsonic flow). This property is one of the most significant physical differences between subsonic and supersonic flows. It is the basic reason why shock waves occur in supersonic flows but do not occur in steady subsonic flow.
4.
Hypersonic flow (very high supersonic speeds). Refer again to the wedge in Fig. 4d. Assume θ is a given, fixed value. As M∞ increases above 1, the shock wave moves closer to the body surface. Also, the strength of the shock wave increases, leading to higher temperatures in the region between the shock and the body (the shock layer). If M∞ is sufficiently large, the shock layer becomes very thin, and interactions between the shock wave and the viscous boundary layer on the surface occur. Also, the shock layer temperature becomes high enough that chemical reactions occur in the air. The O 2 and N 2 molecules are torn apart; that is, the gas molecules dissociate. When M∞ becomes large enough such that viscous interaction and/or chemically reacting effects begin to dominate the flow (Fig. 4e), the flow field is called hypersonic. (Again, a somewhat arbitrary but frequently used rule of thumb for hypersonic flow is M∞<5.) Hypersonic aerodynamics received a great deal of attention during the period 1955–1970 because atmospheric entry vehicles encounter the atmosphere at Mach numbers between 25 (ICBMs) and 36 (the Apollo lunar return vehicle.) From 1985 to the present, this attention has shifted to air-breathing hypersonic cruise vehicles and single-stage-to-orbit vehicles. Today, hypersonic aerodynamics is just part of the whole spectrum of realistic flight speeds.
In summary, we attempt to organize our study of aerodynamic flows according to one or more of the various categories discussed in this section. The block diagram in Fig. 5 is presented to help emphasize these categories and to show how they are related.
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FIGURE 5. Block diagram for the various categories of aerodynamic flows. [From Anderson, J. D., Jr. (2001). “Fundamentals of Aerodynamics,” 3rd ed., McGraw-Hill, New York.]
There are several additional types of flows frequently assumed in aerodynamics.
1.
Irrotational flow. Flow where the individual elements of fluid moving along a streamline are in translational motion only and are not rotating. For irrotational flow, a scaler potential function ϕ=ϕ (x, y, z) can always be defined in such a fashion that
V=∇ϕ.
This is an important aspect of theoretical aerodynamics; ϕ is called the velocity potential . 2.
Adiabatic flow. Flow where no heat is added or taken away.
3.
Isentropic flow. Flow undergoing a thermodynamic process that is both adiabatic and reversible. For such flow, the thermodynamic entropy is constant along a streamline.
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Reference work 2003, Encyclopedia of Physical Science and Technology (Third Edition)John D. Anderson Jr.
Chapter
Assessment of Eddy Viscosity Models in 2D and 3D Shock/Boundary-Layer Interactions
1999, Engineering Turbulence Modelling and Experiments 4T. Coratekin, ... J. Ballmann
1 INTRODUCTION
Many crucial problems still remain unsolved in the field of supersonic and hypersonic aerodynamics. The interaction of turbulent boundary-layers with shock waves is one of them and therefore a subject of widespread interest in hypersonic aircraft design. If strong enough, the interacting shock can cause the boundary-layer to separate. The strong pressure gradients involved produce an increase in skin friction and heat flux. In the worst case, the heat flux can be so high that the local temperature reaches beyond the melting point of the surface material and hence, causes irreversible damage to the fuselage. To prevent this type of structural failure in future hypersonic aircrafts, it is necessary to gain detailed information on the phenomenon itself on one hand, and profound understanding of the physics involved on the other. Shock/boundary-layer interactions occur preferably at concave supersonic flow deflections like rudder actuations or inlet configurations.
The two-dimensional compression corner and the three-dimensional single or double fin configuration are well known examples of flows involving shock/boundary-layer interaction. For the 2D case, the numerical investigation of a 24° compression corner flow using a two-equation turbulence model is a common test case and hence, almost serves as validation. Settles conducted the corresponding experiment at Ma ∞ = 2.84 in full detail. A similar experiment was performed by Schulte-Rôdding et al. at Ma∞ = 7.39. In recent years however, attention has focused on more complex three-dimensional problems, like the symmetric crossing-shock interaction . Extensive experimental and computational research on this configuration has elucidated the wave and streamline structure. Good agreement has been observed between computed and experimental surface pressure and flowfield profiles.
Most of the published work on shock wave/turbulent boundary-layer interaction has incorporated the turbulent effects through either two-equation or full Reynolds stress models. Without any doubt, full Reynolds stress closures definitely contain more physics than two-equation models but they present an inherent difficulty with wall-bounded flows, where near-wall models that typically depend on the unit normal to the wall must be introduced - a feature that makes it virtually impossible to systematically integrate second order closures in complex geometries . Moreover, compressible Reynolds stress models are not as well developed as their incompressible counterparts. Therefore, until new methods are fully developed, it is preferable to use two-equation models - with an anisotropiceddy viscosity systematically obtained from an algebraic second order closure - in complex, compressible wall-bounded turbulent flows.
The purpose of this paper is to study the two-dimensional 24° compression corner on one hand, and the three-dimensional asymmetric 7°×ll° crossing-shock interaction on the other. Two different turbulence models are tested to account for Reynolds stress effects. One is Wilcox’s k—ω— model model including low-Reynolds and compressibility corrections. The other is the explicit algebraic Reynolds stress model of Johansson and Wallin based on the former k—ω— model. Results are compared with the experiments mentioned above for the 2D case and with the experimental studies of Zheltovodov et al. for the three-dimensional interaction.
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Book 1999, Engineering Turbulence Modelling and Experiments 4T. Coratekin, ... J. Ballmann
Chapter
Introduction
2023, Aerodynamic Heating in Supersonic and Hypersonic FlowsMostafa Barzegar Gerdroodbary
1.1 Introduction
Flight is the dream of humans to move in an inaccessible, unknown world. To achieve this purpose, humans have always tried to invent devices and vehicles, which enable them to move in the sky. Due to the high velocity of the fly, the airplane has been widely developed for transportation for a long-distance journey. The advantages of flight in the sky have motivated the government to use this to fight against their enemies. Hence, various fighters have been invented that move at a higher speed than civil applications. Then, scientists have believed that they could access space with high-speed spacecraft, which moves in the outer atmosphere. In the atmospheric domain, the highest speed craft is available since existing gas in outer space is low density and cold.
The range of the flight speed is mainly divided according to the unit Mach number known as sonic (M=1). When the flight speed is less than M=1, it is known as subsonic. For greater speed flight (M>1), the flow regime is called supersonic. When the flight speed is significantly higher than sonic (M>>1), it is recognized as hypersonic. The world of high-speed flight is directly linked with an entirely unknown environment, named “hypersonic regime” The term hypersonic is introduced to discriminate flow field phenomena and problems that occur at flight speeds, and they are greater enough than the typical supersonic velocities. The advent of new distinctive features in hypersonic flow feature justifies the use of a new term, different from the well-established one supersonic. Hydrodynamic and chemical and physical natures are the two main aspects of these hypersonic features. The former is due to high flight Mach, while the latter is related to the high energy of the flow (Kuchemann, 1978). Although there are several definitions for the hypersonic aerodynamic, flows with Mach number more than 5 are known as hypersonic regime as presented in Fig.1.1.
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Figure 1.1. Ellipse of motion regimes.
Indeed, the hypersonic regime is best defined as that regime where certain physical flow phenomena become increasingly more significant and imperative as the Mach number is increased to higher values. For example, the ratio of thermal to hydrodynamic boundary layer is one of the progressive phenomena for discrimination of hypersonic regime from a supersonic one. This ratio and other significant factors may vary according to the geometry and operating condition of the model in the high-speed domain. This book has mainly focused on these aspects of hypersonic flow in detail for the analysis of aerodynamics and aerothermodynamics of a high-speed vehicle.
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Book 2023, Aerodynamic Heating in Supersonic and Hypersonic FlowsMostafa Barzegar Gerdroodbary
Review article
Capabilities and limitations of existing hypersonic facilities
2020, Progress in Aerospace SciencesSangdi Gu, Herbert Olivier
2.1 Aerodynamic flow fields
2.1.1 Low enthalpy flows
Hornung in 1988 reported a good strategy for subscale simulation of hypersonic aerodynamic flow fields. The lowest speeds at which chemical reactions become an important part of the flow field are about 2 km/s in blunt bodyair flows at sea level conditions. This corresponds to a stagnation temperature of about 2000 K. According to Fig. 1, at speeds below 2 km/s, the perfect gas model applies for air as the isentropic exponent, γ, is either constant or varies only with temperature due to vibrational excitation. In this case any dimensionless quantity, Q, will depend on the dimensionless parameters of the flow. If the gas used is the same between flight and experiment, the model is scaled exactly with the flight vehicle and the model and flight vehicle are orientated in the same way relative to the flow ,
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Fig. 1. Variation of the isentropic exponent, γ, of air with temperature and pressure .
(1)Q=Q(M∞,Re,T w T 0,Fq)
where M∞ is the Mach number, Re is the Reynolds number, T w/T 0 is the wall temperature to total temperature ratio and Fq is a set of dimensionless parameters defining the flow quality in the wind tunnel. This is called the Mach-Reynolds-Simulation .
The importance of simulating M∞, Re and T w/T 0 is clear when observing, exemplarily, the rearranged flat plate laminar boundary layer thickness equation given by Ginoux .
(2)δ x=C∗R e∞,x(3.07+0.58(γ−1)2 M∞2+1.93 T w T∞)
where δ is the boundary layer thickness, x is the distance downstream of the leading edge, Re∞,x is the Reynolds number at the location x, C is the Chapman-Rubesin factor evaluated at the reference temperature, M∞ is the freestream Mach number and T∞ is the freestream temperature. From equation (2), M∞ and the T w/T∞ ratio, which is matched when M∞ and T w/T 0 in equation (1) are matched under perfect gas conditions, are particularly important parameters for simulating the boundary layer thickness, while the Reynolds number also needs to be simulated. For typical hypersonic flight vehicles (altitude=35 km, M∞=6, T∞=237 K, recovery temperature, Tr=1670 K), T w/T∞ has a value of around 4–5. In cold hypersonic facilities where T∞ can be around 60–80 K, using room temperature test models conveniently preserves the T w/T∞ ratio. However, in situations where wind tunnels are used to generate higher stagnation temperatures, e.g. in order to duplicate the total enthalpy and therewith the real flight velocity or real gas effects, which may influence viscous effects such as transition , heated test models must be used in order to preserve T w/T∞ . Consider a flat plate in a typical short duration wind tunnel flow with M∞,wt=6, T∞,wt=237 K and T w,wt=300 K i.e. (T w/T∞)wt=1.3, and a flat plate in flight with the same M∞ and T∞ i.e. same recovery temperature, but a wall temperature of T w,fl=1185 K i.e. (T w/T∞)fl=5. Further it is assumed that the Reynolds number for flight and wind tunnel are the same which is a requirement of equation (1). Then for the considered case from equation (2), it follows:
(3)(δ x)wt(δ x)fl=0.64
where subscripts wt and fl denote wind tunnel and flight respectively, resulting in large differences in the boundary layer thickness at any given point behind the leading edge if the wall to freestream temperature ratio is not preserved. Hence, for these cases heated models must be used for simulating wall temperature effects in the laboratory. It has been shown that the wall temperature ratio strongly influences the size of separation bubbles . It further influences the transition behaviour of boundary layers. Additionally, the heated model technique is also useful on blunt bodies in order to study ablation-radiation coupling as pioneered by Zander et al. .
On the other hand, the skin friction coefficient,
(4)C f=0.664 R e∞,x C∗
and the Stanton number,
(5)S t=0.41 R e∞,x C∗
have lower dependencies on M∞ and T w/T∞ within a Mach number range that is not too large, as shown by equations (4) and (5) for a laminar boundary layer and illustrated in Fig. 2. Hence more leniency on abiding by equation (1) may be given to the simulation of the skin friction coefficient and the Stanton number which depends strongly on the geometry instead. Furthermore, included in Fq in equation (1) are parameters related to the freestream noise level, freestream turbulence level and the surface roughness. These parameters are important for studies such as boundary layer transition. Current difficulties include defining the freestream noise and turbulence levels in high enthalpy facilities and, to a lesser extent, scaling the surface roughness in subscale test models so as to prevent artificial boundary layer transition or enhanced heating rates due to surface roughness . In practice, since the surface roughness needs to be scaled with the model size, a surface roughness of the order of 0.01 mm or even less is often required on the experimental test model, which is not always easy to obtain.
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Fig. 2. Flat plate skin friction coefficient, (a), and Stanton numbers, (b), for compressible laminar boundary layers .
Another phenomenon of interest in hypersonic flows is the entropy layer. Correct simulation of the entropy layer is important as it can influence the heat flux distribution, skin friction, separation behaviour and boundary layer transition. For calorically perfect air at hypersonic conditions (M∞»1), the entropy change, Δs, across a normal shock can be written as,
(6)Δ s=Rln(2.78∗10−3∗M∞5)
where R is the specific gas constant of air. The equation above shows that the entropy layer is strongly dependent on the Mach number. Hence, the duplication of the Mach number is important for the duplication of the entropy layer. The duplication of the Mach number is also important for the duplication of the laminar viscous interaction. The laminar viscous interaction parameter, χ‾, is given by,
(7)χ‾=M∞3 R e∞,x C∞
where C∞ is the Chapman-Rubesin factor. The above equation shows that the Mach number has a strong influence on the viscous interaction while the Reynolds number and the wall temperature, which influences the value of C∞, have weaker influence on the viscous interaction.
2.1.2 High enthalpy flows
For high enthalpy flows in principle the same requirements hold as those formulated in the previous subchapter for the low enthalpy flow regime. But at speeds greater than 2 km/s or about 2000 K total temperature in blunt body air flows, additional real-gas effects become important as shown in Fig. 1 where γ varies with both temperature and pressure due to chemical reactions. A portion of hypersonic flows involves situations where dissociation reactions are dominant. In these situations, preserving the binary scaling product, ρ∞L where ρ∞ is the freestream density and L is the characteristic length, preserves the normalized distribution of the chemical composition behind a shock wave. This law was first mentioned by Birkhoff in 1955 . From Anderson , the binary scaling law can be derived as follows: for simplicity consider a two-dimensional flow where the dominant chemical reaction is the oxygen dissociation reaction, O 2+M → 2O+M where M is the collision partner, the species continuity equation for atomic oxygen can be written as,
(8)u(d c O d x)+v(d c O d y)=M O ρ k f(ρ c O 2 M O 2)(ρ c M M M)
where c O and c O2 is the mass fraction of O and O 2 respectively, u and v is the flow velocity in the x and y directions respectively, M O, M O2 and M M is the molar mass of O, O 2 and M respectively, ρ is the flow density and k f is the forward reaction rate constant which is a function of temperature as described by the Arrhenius equation. Defining the nondimensional variables x'=x/L, y'=y/L, u'=u/U∞, v'=v/U∞ and ρ'=ρ/ρ∞ where U∞ and ρ∞ is the velocity and density of the freestream respectively and L is a characteristic length, equation (8) can be written as,
(9)u′(d c O d x′)+v′(d c O d y′)=K 1(ρ∞L)U∞ρ′c O 2 c M
where
(10)K 1=M O M O 2 M M k f
Equation (9) shows that for two steady flows with the same post-shock temperature, resulting in the same K 1, and the same freestream velocity, the mass fraction distribution along the normalized directions x' and y' will be identical between the two flows if the product ρ∞L is preserved. This is the statement of binary scaling. Also, from Hornung , for an ideal chemically reacting gas, the rate of dissociation R D is
(11)R D=ρ T n e−D k T(1−α)
and the rate of recombination R R is
(12)R R=ρ 2 T n α 2 ρ d
where ρ is the gas density, α is the mass fraction of the dissociated gas, T is the temperature, k is Boltzmann's constant, n is a dimensionless constant and ρ d is the characteristic density of dissociation. From the above equations, the dissociation rate, which is a two body reaction, can be rewritten as,
(13)R D=d α d t=d c O d t=u(d c O d x)+v(d c O d y)
for steady flows. Then,
(14)L U∞R D=u′(d c O d x′)+v′(d c O d y′)=(ρ∞L)U∞ρ′T n e−D k T(1−c O)
where 1 - c o=c o2. This is the same statement as given by equation (9). Accordingly, for the recombination reaction the scaling factor is ρ∞2 L. It is interesting to note that de Crombrugghe et al. made the discovery that the binary scaling law is not only applicable for preserving the chemical nonequilibrium behaviour in shock layers but also maintain the same diffusion processes in binary scaled chemically reacting boundary layers, resulting in further applications in subsonic high enthalpy wind tunnels such as plasma tunnels.
The simulation strategy given by Hornung for chemically reacting blunt body flows considering only dissociation reactions is therefore as follows,
(15)Q=Q(U∞,ρ∞L,α∞,T w T 0)
where α∞ is the mass fraction of dissociated gas in the freestream. For strong bow shocks (usually from blunt bodies), the Mach number is omitted from the above equation due to the Mach number independence principle. Reynolds number is omitted because the post-shock Reynolds number is automatically satisfied when U∞ and ρ∞L, is matched since the freestream static temperature has only a small influence on the post-shock temperature of a strong bow shock which is mainly influenced by U∞ . Consequently, besides the atomic mass fraction distributions, duplication of the parameters in equation (15) gives a simulation of the temperature field, viscosity, Prandtl number and the specific heat ratio in the flowfield after a bow shock, between experiment and flight. However, for more slender bodies where Mach number and real-gas effects are important the Mach number has also to be matched, which for static temperatures less than 2000 K subsequently results in a duplication of the freestream static temperature T∞ and freestream Reynolds number .
It is important to note that some important portions of hypersonic blunt body flows cannot be described by the binary scaling law which is derived based on the assumption of no radiation coupling and purely binary reactions. From de Crombrugghe et al. , when binary scaling is applied, the impact of non-binary chemistry causes the shock layer in the subscale test model to be hotter and less dissociated than in flight, while the strength of radiation coupling increases with the length-scale of the flow resulting in the subscale flow having less radiation coupling than in flight which subsequently leads to the laboratory shock layer containing a greater enthalpy than the shock layer in flight. Furthermore, it should be noted that the duplication of the freestream dissociation level α∞ in equation (15) is very difficult because the production of high enthalpy flows in the wind tunnels often requires heating the test gas to temperatures high enough to cause significant thermochemical excitation which normally does not relax to equilibrium at the test section. In flight, the freestream dissociation level is practically zero which is not the case for high enthalpy wind tunnels. A finite freestream dissociation level leads to a reduced density jump across a shock resulting in a larger shock stand-off distance . Also, Fq from equation (1) is omitted in equation (15) because the flow quality that can be achieved in current real gas simulation facilities is only very low . Consequently, these facilities are only used for investigations of effects that are less subtle than those that depend sensitively on Fq. Unfortunately, there are actually a number of important effects which do depend sensitively on Fq . So, Lawson and Austin are currently investigating new ways to generate low-disturbance, high-enthalpy test flows . It has to be mentioned that the flow quality in high enthalpy facilities strongly depends on the test condition. A good flow quality can be achieved, for example, in shock tunnels at low unit Reynolds numbers even at high enthalpies as it is proven by numerous heat flux measurements.
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Table of Contents
Central Limit Theorem (CLT)
Understanding the CLT
Key Components
CLT in Finance
FAQs
The Bottom Line
What Is the Central Limit Theorem (CLT)?
By Akhilesh Ganti
Updated June 04, 2025
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Definition
The Central Limit Theorem (CLT) surmises that the average of the sample means and standard deviations equals the population mean and standard deviation.
What Is the Central Limit Theorem (CLT)?
The Central Limit Theorem is useful when analyzing large data sets because it assumes that the sampling distribution of the mean will be normally distributed and typically form a bell curve. The CLT may be used in conjunction with the law of large numbers, which states that the average obtained from a large group of independent random samples converges to the true value.
Key Takeaways
CLT assumes that a large sample size can successfully predict the characteristics of a population.
Sample sizes equal to or greater than 30 are often considered sufficient under the Central Limit Theorem.
Investors may use CLT to study a random sample of stocks to estimate returns for a portfolio.
Understanding the Central Limit Theorem (CLT)
According to the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases, notwithstanding the actual distribution of the data. The concept can hold true regardless of whether the distribution of the population is normal or skewed.
As a general rule, sample sizes of 30 or more are typically deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. In addition, the more samples one takes, the more the graphed results should take the shape of a normal distribution.
The central limit theorem is often used in conjunction with the law of large numbers, which states that the average of the sample means will come closer to equaling the population mean as the sample size grows. This concept can be extremely useful in accurately predicting the characteristics of very large populations.
Although this concept was first developed by Abraham de Moivre in 1733, it was not formalized until 1920, when the Hungarian mathematician George Pólya dubbed it the central limit theorem.
Key Components of the Central Limit Theorem
The central limit theorem has several key components. They largely revolve around sampling technique.
Sampling is successive:This means some sample units are common with sample units selected on previous occasions.
Sampling is random:All samples must be selected at random so that they have the same statistical possibility of being selected.
Samples should be independent: The selections or results from one sample should have no bearing on future samples or other sample results.
Large sample size: As sample size increases, the sampling distribution should come ever closer to the normal distribution.
The Central Limit Theorem in Finance and Investing
The CLT can help examine the returns of an individual stock or broader stock indices because the analysis is simple, due to the relative ease of generating the necessary financial data. Consequently, investors often rely on the CLT to analyze stock returns, construct portfolios, and manage risk.
Suppose, for example, that an investor wishes to analyze the overall return for a stock index that consists of 1,000 different equities. In this scenario, the investor may simply study a random sample of stocks to arrive at an estimated return for the total index. To be safe in this instance, at least 30 to 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold.
Why Is the Central Limit Theorem Useful?
The central limit theorem is useful when analyzing large data sets because it allows one to assume that the sampling distribution of the mean will be normally distributed in most cases. This allows for easier statistical analysis and inference. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over some time.
What Is the Formula for Central Limit Theorem?
The central limit theorem doesn't have a formula used in its practical application. Its principle is simply applied. With a sufficiently large sample size, the sample distribution will approximate a normal distribution, and the sample mean will approach the population mean. It suggests that if we have a sample size of at least 30, we can begin to analyze the data as if it fit a normal distribution.
Why Is the Central Limit Theorem's Minimum Sample Size 30?
A sample size of 30 or more is fairly common across statistics as the minimum for applying the central limit theorem. The greater your sample size, the more likely the sample will be representative of your population set.
What Is the Law of Large Numbers?
In probability theory and statistics, the law of large numbers states that the larger the sample size, the more likely its mean is to reflect the mean of the entire population.
In business, the law of large numbers can have a different meaning, specifically that as a company grows in size, maintaining its rate of growth in percentage terms becomes more difficult.
The Bottom Line
The central limit theorem (CLT) holds that as a sample size gets larger, its mean will increasingly approximate the mean in a normal distribution. This concept can be useful in many applications, such as analyzing investment returns, because it requires only a sufficient sample size (generally interpreted as 30 or more data points) rather than the entire population.
Article Sources
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Boston University School of Public Health. "Central Limit Theorem."
Hans Fischer. "A History of the Central Limit Theorem," Page 1.
The Mathematics Enthusiast. "Studying Moments of the Central Limit Theorem, Benjamin A. Stark."
University of Massachusetts Amherst. "What Is Central Limit Theorem? Properties, Best Practices, Examples & Everything to Know."
Emory University. "Final Summary: The Central Limit Theorem."
Mascha, Edward J., and Thomas R. Vetter. "Significance, Errors, Power, and Sample Size: The Blocking and Tackling of Statistics." Anesthesia & Analgesia, vol 126, no. 2, February 2018, pp. 691-698.
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Nature - Journal of Human Genetics - Do monochorionic dizygotic twins increase after pregnancy by assisted reproductive technology?
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BMC Pregnancy and Childbirth - Rare spontaneous monochorionic dizygotic twins: a case report and a systematic review
National Center for Biotechnology Information - PubMed Central - Traces of embryogenesis are the same in monozygotic and dizygotic twins: not compatible with double ovulation
Also known as: DZ twin, dizygotic twin, nonidentical twin
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John M. Quinn
Contributor to Encyclopedia of Global Health. He contributed an article on “Dizygotic Twin” to SAGE Publications’ Encyclopedia of Governance (2007), and a version of this article was...
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dizygotic twin, two siblings who come from separate ova, or eggs, that are released at the same time from an ovary and are fertilized by separate sperm. The term originates from di, meaning “two,” and zygote, “egg.” The rate of dizygotic twinning varies considerably worldwide. For example, parts of central and western Africa have very high twinning rates; studies in Nigeria have reported rates of more than 45 sets of twins per 1,000 births. By comparison, the rate in South and Southeast Asia appears to be as low as 6 to 9 per 1,000 births. While these figures concern all twin births (monozygotic and dizygotic), dizygotic twinning accounts for the majority—at least two-thirds—of them.
Dizygotic twins develop in the uterus separately. Each zygote develops with its own chorion (or outer sac). The chorion is connected to the placenta, which is the protective membrane that surrounds the developing fetus. The placenta lines the uterine wall, partially envelops the fetus, and is attached to the umbilical cord. The placenta exchanges nutrients, wastes, and gases between maternal and fetal blood. Each zygote also has its own inner sac that contains amniotic fluid, and each develops its own placenta. However, if the two zygotes implant in the uterus close together, the two placentas may appear as one on an ultrasound.
Because of their being two separate zygotes, the genetic makeup of dyzygotic twins is different, and thus they are not identical. On average, these twins will share half of their genes, just like any other pair of siblings. Furthermore, they may be the same sex or be a male and female pair. Dizygotic twins may have different fathers, meaning that one egg is fertilized by a sperm from one father and the other egg is fertilized by a different sperm from a different father. These cases are rare and few have been documented.
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human genetics: Fraternal twins
Dizygotic twins are most common for older mothers, with the highest rates found among mothers over age 35. With the advent of technologies and techniques to assist women in becoming pregnant, the rate of dizygotic twins has increased markedly. In some cases, family history of dizygotic twinning is an important factor. Other possible predispositions to dizygotic twins include mothers who are taller and heavier, the recent discontinuation of oral birth control, emotional stress, and diet. There has been speculation that diet may play a pronounced role in African mothers, particularly among the Yoruba in Nigeria, where a high frequency of dizygotic twinning is coincident with the consumption of large quantities of phytoestrogen-rich yams. Environmental stress from chemical agents in highly polluted areas may have negative impacts on the frequency of dizygotic twinning. Thus, while certain background factors likely are important, the occurrence of dizygotic twins may also be influenced by a complex mix of events, many of which are yet to be fully understood.
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Proportional-Assist Ventilation for Minimizing the Duration of Mechanical Ventilation
Authors: Karen J. Bosma, M.D. Karen E.A. Burns, M.D. Claudio M. Martin, M.D., Yoanna Skrobik, M.D., Jordi Mancebo Cortés, M.D., Ph.D., Sorcha Mulligan, B.A., Myriam Lafreniere-Roula, Ph.D., +40 , Kevin E. Thorpe, M.Math., Juan Carlos Suárez Montero, M.D., Indalecio Morán Chorro, M.D., Ph.D., Núria Rodríguez-Farré, M.D., Ron Butler, M.D., Tracey Bentall, C.C.R.P., Gaëtan Beduneau, M.D., Pauline Enguerrand, R.N., Marlene Santos, M.Sc., Thomas Piraino, R.R.T., Savino Spadaro, M.D., Ph.D., Federica Montanaro, M.D., John Basmaji, M.D., Eileen Campbell, C.C.R.P., Alain Mercat, M.D., François M. Beloncle, M.D., Ph.D., Guillaume Carteaux, M.D., Ph.D., Tommaso Maraffi, M.D., Emmanuel Charbonney, M.D., Ph.D. Marie Lecronier, M.D., Martin Dres, M.D., Ph.D., Yaseen M. Arabi, M.D., Andre Carlos K.B. Amaral, M.D., Nicole Marinoff, R.N., Neill K.J. Adhikari, M.D.C.M. Anna Geagea, M.D., Phil Shin, M.D., Katerina Vaporidi, M.D., Eumorfia Kondili, M.D., Jason Shahin, M.D., Josie Campisi, M.Sc., Pablo O. Rodriguez, M.D., Ph.D. Mariano Setten, R.P.T., Ewan C. Goligher, M.D., Ph.D., Niall D. Ferguson, M.D., Vito Fanelli, M.D., Ph.D., Gabriela Ferreyra, C.R.P.T., Francois Lellouche, M.D., Ph.D., Stephanie Sibley, M.D., and Laurent Brochard, M.D., H.D.R., for the PROMIZING Study Investigators, the Canadian Critical Care Trials Group, and the REVA Network† -40Author Info & Affiliations
Published June 13, 2025
DOI: 10.1056/NEJMoa2505708
Copyright © 2025
Abstract
Background
In critically ill patients, acceleration of liberation from mechanical ventilation is important in order to reduce the risk of complications and to improve long-term outcomes. Whether the use of proportional-assist ventilation with load-adjustable gain factors (PAV+) results in a shorter time to successful liberation from mechanical ventilation than pressure-support ventilation (PSV) is unclear.
Methods
In this international clinical trial, we randomly assigned adult patients who had been receiving mechanical ventilation for at least 24 hours and were able to undergo partial ventilatory support with PSV but were not yet ready for liberation from ventilation to undergo PAV+ (which targeted normal work of breathing) or PSV (which targeted a normal respiratory rate and tidal volume). The primary outcome was the time from randomization to successful liberation from mechanical ventilation.
Results
Across 23 centers in seven countries, 722 patients were enrolled, and 573 underwent randomization and were included in the analysis. The median time to successful liberation from mechanical ventilation was 7.3 days (95% confidence interval [CI], 6.2 to 9.7) in the PAV+ group and 6.8 days (95% CI, 5.4 to 8.8) in the PSV group (P=0.58). The median number of ventilator-free days, the incidence of reintubation and tracheostomy, and the incidence of death by day 90 (29.6% in the PAV+ group and 26.6% in the PSV group), all of which were secondary outcomes, were similar in the two groups. With respect to sedative drugs, the mean (±SD) difference in the midazolam-equivalent dose at day 28 relative to the baseline dose was −1.51±3.28 mg per kilogram of body weight in the PAV+ group and 0.04±0.97 mg per kilogram in the PSV group. Serious adverse events occurred in 31 patients (10.8%) in the PAV+ group and in 28 patients (9.8%) in the PSV group (P=0.79).
Conclusions
The time to liberation from mechanical ventilation did not differ significantly between the group that underwent PAV+ and the group that underwent PSV. (Funded by the Canadian Institutes of Health Research and others; PROMIZING ClinicalTrials.gov number, NCT02447692.)
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Notes
This article was published on June 13, 2025, at NEJM.org.
A data sharing statement provided by the authors is available with the full text of this article at NEJM.org.
Supported by a project grant (PJT-162086) and an industry-partnered collaborative research grant (IPR-136755) from the Canadian Institutes of Health Research, with Covidien, a Medtronic company, as the industry partner (ISR 2014-10481), and by the University of Western Ontario Department of Medicine Program of Experimental Medicine. Laurent Brochard is supported by the Keenan Chair in Critical Care and Respiratory Failure.
Disclosure forms provided by the authors are available with the full text of this article at NEJM.org.
We thank the patients who participated in this trial and their families; the members of the data and safety monitoring committee for their contributions during the 7-year enrollment period; Dr. B. Taylor Thompson (Boston); Dr. Maureen Meade (Hamilton, ON, Canada); Dr. Jean-Daniel Chiche (Lausanne, Switzerland); Dr. Charles (Charlie) Keown-Stoneman (senior research biostatistician at the Applied Health Research Centre [AHRC], Toronto, ON, Canada) for insightful discussions on statistics throughout the analysis; all the research personnel at the AHRC who helped us with trial management during the 11-year trial; and all our collaborators.
This work is dedicated to the memory of our collaborator, mentor, and friend, Dr. Jordi Mancebo Cortés, as well as his collaborator, Dr. Juan Carlos Suárez Montero, whose help and dedication in successfully conducting the PROMIZING trial have been essential.
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History
Published online: June 13, 2025
Topics
Clinical Medicine General
Critical Care
Hospital-Based Clinical Medicine
Mechanical Ventilation
Pulmonary/Critical Care General
Authors
Affiliations
Karen J. Bosma, M.D.
Critical Care Western, Department of Medicine, Schulich School of Medicine and Dentistry, University of Western Ontario, London, Canada
Lawson Health Research Institute, London, ON, Canada
London Health Sciences Centre Research Institute, London, ON, Canada
Karen E.A. Burns, M.D.
Interdepartmental Division of Critical Care Medicine, University of Toronto, Toronto
Keenan Research Centre for Biomedical Science, Li Ka Shing Knowledge Institute, Unity Health Toronto–St. Michael’s Hospital, Toronto
Claudio M. Martin, M.D.
Critical Care Western, Department of Medicine, Schulich School of Medicine and Dentistry, University of Western Ontario, London, Canada
Lawson Health Research Institute, London, ON, Canada
London Health Sciences Centre Research Institute, London, ON, Canada
Yoanna Skrobik, M.D.
Department of Medicine, McGill University, Montreal
Jordi Mancebo Cortés, M.D., Ph.D.
Intensive Care Department, Hospital de la Santa Creu i Sant Pau, Universitat Autònoma de Barcelona, Barcelona
Sorcha Mulligan, B.A.
Applied Health Research Centre, Li Ka Shing Knowledge Institute, Unity Health Toronto–St. Michael’s Hospital, Toronto
Myriam Lafreniere-Roula, Ph.D.
Applied Health Research Centre, Li Ka Shing Knowledge Institute, Unity Health Toronto–St. Michael’s Hospital, Toronto
Kevin E. Thorpe, M.Math.
Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, Toronto
Juan Carlos Suárez Montero, M.D.
Intensive Care Department, Hospital de la Santa Creu i Sant Pau, Universitat Autònoma de Barcelona, Barcelona
Indalecio Morán Chorro, M.D., Ph.D.
Intensive Care Department, Hospital de la Santa Creu i Sant Pau, Universitat Autònoma de Barcelona, Barcelona
Núria Rodríguez-Farré, M.D.
Intensive Care Department, Hospital de la Santa Creu i Sant Pau, Universitat Autònoma de Barcelona, Barcelona
Ron Butler, M.D.
Critical Care Western, Department of Medicine, Schulich School of Medicine and Dentistry, University of Western Ontario, London, Canada
Department of Anesthesia and Perioperative Medicine, University of Western Ontario, London, Canada
Tracey Bentall, C.C.R.P.
London Health Sciences Centre Research Institute, London, ON, Canada
Gaëtan Beduneau, M.D.
Université de Rouen Normandie, GRHVN UR 3830, Rouen, France
Department of Medical Intensive Care, Centre Hospitalier Universitaire de Rouen, F-76000, Rouen, France
Pauline Enguerrand, R.N.
Department of Medical Intensive Care, Centre Hospitalier Universitaire de Rouen, F-76000, Rouen, France
Marlene Santos, M.Sc.
Keenan Research Centre for Biomedical Science, Li Ka Shing Knowledge Institute, Unity Health Toronto–St. Michael’s Hospital, Toronto
Thomas Piraino, R.R.T.
Department of Anesthesia, Faculty of Health Sciences, McMaster University, Hamilton, ON, Canada
Savino Spadaro, M.D., Ph.D.
Department of Translational Medicine, University of Ferrara, Ferrara, Italy
Azienda Ospedaliera–Universitaria di Ferrara, Ferrara, Italy
Federica Montanaro, M.D.
Azienda Ospedaliera–Universitaria di Ferrara, Ferrara, Italy
John Basmaji, M.D.
Critical Care Western, Department of Medicine, Schulich School of Medicine and Dentistry, University of Western Ontario, London, Canada
Lawson Health Research Institute, London, ON, Canada
London Health Sciences Centre Research Institute, London, ON, Canada
Eileen Campbell, C.C.R.P.
London Health Sciences Centre Research Institute, London, ON, Canada
Alain Mercat, M.D.
Département de Médecine Intensive–Réanimation, Vent’Lab, Centre Hospitalier Universitaire d’Angers, Université d’Angers, Angers, France
François M. Beloncle, M.D., Ph.D.
Département de Médecine Intensive–Réanimation, Vent’Lab, Centre Hospitalier Universitaire d’Angers, Université d’Angers, Angers, France
Guillaume Carteaux, M.D., Ph.D.
Service de Médecine Intensive–Réanimation, Assistance Publique–Hôpitaux de Paris (AP-HP), Hôpitaux Universitaires Henri-Mondor, Créteil, France
INSERM U955, Institut Mondor de Recherche Biomédicale, Créteil, France
Tommaso Maraffi, M.D.
Intensive Care Unit, Hôpital Intercommunal de Créteil, Créteil, France
Emmanuel Charbonney, M.D., Ph.D.
Critical Care Department, Centre Hospitalier de l’Université de Montréal, Université de Montréal Faculté de Médecine, Montreal
Hôpital du Sacré-Cœur-de-Montréal, Montreal
Marie Lecronier, M.D.
Service de Médecine Intensive–Réanimation, AP-HP, Sorbonne Université, Paris
Martin Dres, M.D., Ph.D.
Service de Médecine Intensive–Réanimation, AP-HP, Sorbonne Université, Paris
Yaseen M. Arabi, M.D.
King Saud bin Abdulaziz University for Health Sciences, Riyadh, Saudi Arabia
King Abdullah International Medical Research Center, Riyadh, Saudi Arabia
Andre Carlos K.B. Amaral, M.D.
Interdepartmental Division of Critical Care Medicine, University of Toronto, Toronto
Department of Critical Care Medicine, Sunnybrook Health Sciences Centre, Toronto
Nicole Marinoff, R.N.
Department of Critical Care Medicine, Sunnybrook Health Sciences Centre, Toronto
Neill K.J. Adhikari, M.D.C.M.
Interdepartmental Division of Critical Care Medicine, University of Toronto, Toronto
Department of Critical Care Medicine, Sunnybrook Health Sciences Centre, Toronto
Anna Geagea, M.D.
Critical Care Unit, Department of Medicine, North York General Hospital, Toronto
Phil Shin, M.D.
Critical Care Unit, Department of Medicine, North York General Hospital, Toronto
Katerina Vaporidi, M.D.
Department of Intensive Care Medicine, University Hospital of Heraklion, University of Crete School of Medicine, Heraklion, Greece
Eumorfia Kondili, M.D.
Department of Intensive Care Medicine, University Hospital of Heraklion, University of Crete School of Medicine, Heraklion, Greece
Jason Shahin, M.D.
Department of Critical Care, McGill University, Montreal
Josie Campisi, M.Sc.
Research Institute, of the McGill University Health Centre, Montreal
Pablo O. Rodriguez, M.D., Ph.D.
Intensive Care Unit, Instituto Universitario Centro de Educación Médica e Investigaciones Clínicas “Norberto Quirno,” Buenos Aires
Mariano Setten, R.P.T.
Intensive Care Unit, Instituto Universitario Centro de Educación Médica e Investigaciones Clínicas “Norberto Quirno,” Buenos Aires
Ewan C. Goligher, M.D., Ph.D.
Interdepartmental Division of Critical Care Medicine, University of Toronto, Toronto
Department of Medicine, Division of Respirology, University Health Network, Toronto
Toronto General Hospital Research Institute, Toronto
Department of Physiology, University of Toronto, Toronto
Niall D. Ferguson, M.D.
Interdepartmental Division of Critical Care Medicine, University of Toronto, Toronto
Department of Medicine, Division of Respirology, University Health Network, Toronto
Toronto General Hospital Research Institute, Toronto
Department of Physiology, University of Toronto, Toronto
Institute of Health Policy, Management, and Evaluation, University of Toronto, Toronto
Vito Fanelli, M.D., Ph.D.
Department of Surgical Sciences and Department of Anesthesia, Critical Care, and Emergency, University of Turin, Turin, Italy
Azienda Ospedaliero–Universitaria Città della Salute e della Scienza di Torino, University of Turin, Turin, Italy
Gabriela Ferreyra, C.R.P.T.
Department of Surgical Sciences and Department of Anesthesia, Critical Care, and Emergency, University of Turin, Turin, Italy
Azienda Ospedaliero–Universitaria Città della Salute e della Scienza di Torino, University of Turin, Turin, Italy
Francois Lellouche, M.D., Ph.D.
Department of Medicine and Research Center of Institut Universitaire de Cardiologie et de Pneumologie de Québec, Laval University, Quebec, QC, Canada
Stephanie Sibley, M.D.
Department of Emergency Medicine and Department of Critical Care Medicine, Queen’s University, Kingston, ON, Canada
Laurent Brochard, M.D., H.D.R.
Interdepartmental Division of Critical Care Medicine, University of Toronto, Toronto
Keenan Research Centre for Biomedical Science, Li Ka Shing Knowledge Institute, Unity Health Toronto–St. Michael’s Hospital, Toronto
the PROMIZING Study Investigators, the Canadian Critical Care Trials Group, and the REVA Network†
Notes
Dr. Bosma can be contacted at karenj.bosma@lhsc.on.ca or at University Hospital B2-194E, 339 Windermere Rd., London, ON, Canada N6A 5A5.
Jordi Mancebo Cortés, M.D., Ph.D., and Juan Carlos Suárez Montero, M.D., are deceased.
†
A list of the PROMIZING investigators and collaborators is provided in the Supplementary Appendix, available at NEJM.org.
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Section 1.5 : Solving Trig Equations with Calculators, Part I
In the previous section we started solving trig equations. The only problem with the equations we solved in there is that they pretty much all had solutions that came from a handful of “standard” angles and of course there are many equations out there that simply don’t. So, in this section we are going to take a look at some more trig equations, the majority of which will require the use of a calculator to solve (a couple won’t need a calculator).
The fact that we are using calculators in this section does not however mean that the problems in the previous section aren’t important. It is going to be assumed in this section that the basic ideas of solving trig equations are known and that we don’t need to go back over them here. In particular, it is assumed that you can use a unit circle to help you find all answers to the equation (although the process here is a little different as we’ll see) and it is assumed that you can find answers in a given interval. If you are unfamiliar with these ideas you should first go to the previous section and go over those problems.
Before proceeding with the problems we need to go over how our calculators work so that we can get the correct answers. Calculators are great tools but if you don’t know how they work and how to interpret their answers you can get in serious trouble.
First, as already pointed out in previous sections, everything we are going to be doing here will be in radians so make sure that your calculator is set to radians before attempting the problems in this section. Also, we are going to use 4 decimal places of accuracy in the work here. You can use more if you want, but in this class, we’ll always use at least 4 decimal places of accuracy.
Next, and somewhat more importantly, we need to understand how calculators give answers to inverse trig functions. We didn’t cover inverse trig functions in this review, but they are just inverse functions and we have talked a little bit about inverse functions in a review section. The only real difference is that we are now using trig functions. We’ll only be looking at three of them and they are:
Inverse Cosine:cos−1(x)=arccos(x)Inverse Sine:sin−1(x)=arcsin(x)Inverse Tangent:tan−1(x)=arctan(x)Inverse Cosine:cos−1(x)=arccos(x)Inverse Sine:sin−1(x)=arcsin(x)Inverse Tangent:tan−1(x)=arctan(x)
As shown there are two different notations that are commonly used. In these notes we’ll be using the first form since it is a little more compact. Most calculators these days will have buttons on them for these three so make sure that yours does as well.
We now need to deal with how calculators give answers to these. Let’s suppose, for example, that we wanted our calculator to compute cos−1(34)cos−1(34). First, remember that what the calculator is computing is the angle, let’s say xx, that we would plug into cosine to get a value of 3434, or
x=cos−1(34)⇒cos(x)=34x=cos−1(34)⇒cos(x)=34
So, in other words, when we are using our calculator to compute an inverse trig function we are really solving a simple trig equation.
Having our calculator compute cos−1(34)cos−1(34) and hence solve cos(x)=34cos(x)=34 gives,
x=cos−1(34)=0.7227x=cos−1(34)=0.7227
From the previous section we know that there should in fact be an infinite number of answers to this including a second angle that is in the interval [0,2π][0,2π]. However, our calculator only gave us a single answer. How to determine what the other angles are will be covered in the following examples so we won’t go into detail here about that. We did need to point out however, that the calculators will only give a single answer and that we’re going to have more work to do than just plugging a number into a calculator.
Since we know that there are supposed to be an infinite number of solutions to cos(x)=34cos(x)=34 the next question we should ask then is just how did the calculator decide to return the answer that it did? Why this one and not one of the others? Will it give the same answer every time?
There are rules that determine just what answer the calculator gives when computing inverse trig functions. All calculators will give answers in the following ranges.
0≤cos−1(x)≤π−π2≤sin−1(x)≤π2−π2<tan−1(x)<π20≤cos−1(x)≤π−π2≤sin−1(x)≤π2−π2<tan−1(x)<π2
If you think back to the unit circle and recall that we think of cosine as the horizontal axis then we can see that we’ll cover all possible values of cosine in the upper half of the circle and this is exactly the range given above for the inverse cosine. Likewise, since we think of sine as the vertical axis in the unit circle we can see that we’ll cover all possible values of sine in the right half of the unit circle and that is the range given above.
For the tangent range look back to the graph of the tangent function itself and we’ll see that one branch of the tangent is covered in the range given above and so that is the range we’ll use for inverse tangent. Note as well that we don’t include the endpoints in the range for inverse tangent since tangent does not exist there.
So, if we can remember these rules we will be able to determine the remaining angle in [0,2π][0,2π] that also works for each solution.
As a final quick topic let’s note that it will, on occasion, be useful to remember the decimal representations of some basic angles. So here they are,
π2=1.5708π=3.14163π2=4.71242π=6.2832π2=1.5708π=3.14163π2=4.71242π=6.2832
Using these we can quickly see that cos−1(34)cos−1(34) must be in the first quadrant since 0.7227 is between 0 and 1.5708. This will be of great help when we go to determine the remaining angles
So, once again, we can’t stress enough that calculators are great tools that can be of tremendous help to us, but if you don’t understand how they work you will often get the answers to problems wrong.
So, with all that out of the way let’s take a look at our first problem.
Example 1 Solve 4cos(t)=34cos(t)=3 on [−8,10][−8,10].
Show Solution
Okay, the first step here is identical to the problems in the previous section. We first need to isolate the cosine on one side by itself and then use our calculator to get the first answer.
cos(t)=34⇒t=cos−1(34)=0.7227cos(t)=34⇒t=cos−1(34)=0.7227
So, this is the one we were using above in the opening discussion of this section. At the time we mentioned that there were infinite number of answers and that we’d be seeing how to find them later. Well that time is now.
First, let’s take a quick look at a unit circle for this example.
The angle that we’ve found is shown on the circle as well as the other angle that we know should also be an answer. Finding this angle here is just as easy as in the previous section. Since the line segment in the first quadrant forms an angle of 0.7227 radians with the positive xx-axis then so does the line segment in the fourth quadrant. This means that we can use either -0.7227 as the second angle or 2π−0.7227=5.56052π−0.7227=5.5605. Which you use depends on which you prefer. We’ll pretty much always use the positive angle to avoid the possibility that we’ll lose the minus sign.
So, all possible solutions, ignoring the interval for a second, are then,
t=0.7227+2πnt=5.5605+2πnn=0,±1,±2,…t=0.7227+2πnt=5.5605+2πnn=0,±1,±2,…
Now, all we need to do is plug in values of nn to determine the angle that are actually in the interval. Here’s the work for that.
n=−2:t=−11.8437and−7.0059n=−1:t=−5.5605and−0.7227n=0:t=0.7227and5.5605n=1:t=7.0059and11.8437n=−2:t=−11.8437and−7.0059n=−1:t=−5.5605and−0.7227n=0:t=0.7227and5.5605n=1:t=7.0059and11.8437
So, the solutions to this equation, in the given interval, are,
t=−7.0059,−5.5605,−0.7227,0.7227,5.5605,7.0059t=−7.0059,−5.5605,−0.7227,0.7227,5.5605,7.0059
Note that we had a choice of angles to use for the second angle in the previous example. The choice of angles there will also affect the value(s) of nn that we’ll need to use to get all the solutions. In the end, regardless of the angle chosen, we’ll get the same list of solutions, but the value(s) of nn that give the solutions will be different depending on our choice.
Also, in the above example we put in a little more explanation than we’ll show in the remaining examples in this section to remind you how these work.
Example 2 Solve −10cos(3t)=7−10cos(3t)=7 on [−2,5][−2,5].
Show Solution
Okay, let’s first get the inverse cosine portion of this problem taken care of.
cos(3t)=−710⇒3t=cos−1(−710)=2.3462cos(3t)=−710⇒3t=cos−1(−710)=2.3462
Don’t forget that we still need the “3”!
Now, let’s look at a quick unit circle for this problem. As we can see the angle 2.3462 radians is in the second quadrant and the other angle that we need is in the third quadrant. We can find this second angle in exactly the same way we did in the previous example. We can use either -2.3462 or we can use 2π−2.3462=3.93702π−2.3462=3.9370. As with the previous example we’ll use the positive choice, but that is purely a matter of preference. You could use the negative if you wanted to.
So, let’s now finish out the problem. First, let’s acknowledge that the values of 3tt that we need are,
3t=2.3462+2πn3t=3.9370+2πnn=0,±1,±2,…3t=2.3462+2πn3t=3.9370+2πnn=0,±1,±2,…
Now, we need to properly deal with the 3, so divide that out to get all the solutions to the trig equation.
t=0.7821+2πn3t=1.3123+2πn3n=0,±1,±2,…t=0.7821+2πn3t=1.3123+2πn3n=0,±1,±2,…
Finally, we need to get the values in the given interval.
n=−2:t=−3.4067and−2.8765n=−1:t=−1.3123and−0.7821n=0:t=0.7821and1.3123n=1:t=2.8765and3.4067n=2:t=4.9709and5.5011n=−2:t=−3.4067and−2.8765n=−1:t=−1.3123and−0.7821n=0:t=0.7821and1.3123n=1:t=2.8765and3.4067n=2:t=4.9709and5.5011
The solutions to this equation, in the given interval are then,
t=−1.3123,−0.7821,0.7821,1.3123,2.8765,3.4067,4.9709t=−1.3123,−0.7821,0.7821,1.3123,2.8765,3.4067,4.9709
We’ve done a couple of basic problems with cosines, now let’s take a look at how solving equations with sines work.
Example 3 Solve 6sin(x2)=16sin(x2)=1 on [−20,30][−20,30]
Show Solution
Let’s first get the calculator work out of the way since that isn’t where the difference comes into play.
sin(x2)=16⇒x2=sin−1(16)=0.1674sin(x2)=16⇒x2=sin−1(16)=0.1674
Here’s a unit circle for this example.
To find the second angle in this case we can notice that the line in the first quadrant makes an angle of 0.1674 with the positive xx-axis and so the angle in the second quadrant will then make an angle of 0.1674 with the negative xx-axis. So, if we start at the positive xx-axis we rotate a half revolution and then back off 0.1674. Therefore, the angle that we’re after is then, π−0.1674=2.9742π−0.1674=2.9742.
Here’s the rest of the solution for this example. We’re going to assume from this point on that you can do this work without much explanation.
x2=0.1674+2πnx2=2.9742+2πn⇒x=0.3348+4πnx=5.9484+4πnn=0,±1,±2,…x2=0.1674+2πnx2=2.9742+2πn⇒x=0.3348+4πnx=5.9484+4πnn=0,±1,±2,…
n=−2:x=−24.7980and−19.1844n=−1:x=−12.2316and−6.6180n=0:x=0.3348and5.9484n=1:x=12.9012and18.5148n=2:x=25.4676and31.0812n=−2:x=−24.7980and−19.1844n=−1:x=−12.2316and−6.6180n=0:x=0.3348and5.9484n=1:x=12.9012and18.5148n=2:x=25.4676and31.0812
The solutions to this equation are then,
x=−19.1844,−12.2316,−6.6180,0.3348,5.9484,12.9012,18.5148,25.4676x=−19.1844,−12.2316,−6.6180,0.3348,5.9484,12.9012,18.5148,25.4676
Example 4 Solve 3sin(5z)=−23sin(5z)=−2 on [0,1][0,1].
Show Solution
You should be getting pretty good at these by now, so we won’t be putting much explanation in for this one. Here we go.
sin(5z)=−23⇒5z=sin−1(−23)=−0.7297sin(5z)=−23⇒5z=sin−1(−23)=−0.7297
Okay, with this one we’re going to do a little more work than with the others. For the first angle we could use the answer our calculator gave us. However, it’s easy to lose minus signs so we’ll instead use 2π−0.7297=5.55352π−0.7297=5.5535. Again, there is no reason to this other than a worry about losing the minus sign in the calculator answer. If you’d like to use the calculator answer you are more than welcome to. For the second angle we’ll note that the lines in the third and fourth quadrant make an angle of 0.7297 with the xx-axis. So, if we start at the positive xx-axis we rotate a half revolution and then add on 0.7297 for the second angle. Therefore, the second angle is π+0.7297=3.8713π+0.7297=3.8713.
Here’s the rest of the work for this example.
5z=5.5535+2πn5z=3.8713+2πn⇒z=1.1107+2πn5z=0.7743+2πn5n=0,±1,±2,…5z=5.5535+2πn5z=3.8713+2πn⇒z=1.1107+2πn5z=0.7743+2πn5n=0,±1,±2,…
n=−1:x=−0.1460and−0.4823n=0:x=1.1107and0.7743n=−1:x=−0.1460and−0.4823n=0:x=1.1107and0.7743
So, in this case we get a single solution of 0.7743.
Note that in the previous example we only got a single solution. This happens on occasion so don’t get worried about it. Also, note that it was the second angle that gave this solution and so if we’d just relied on our calculator without worrying about other angles we would not have gotten this solution. Again, it can’t be stressed enough that while calculators are a great tool if we don’t understand how to correctly interpret/use the result we can (and often will) get the solution wrong.
To this point we’ve only worked examples involving sine and cosine. Let’s now work a couple of examples that involve other trig functions to see how they work.
Example 5 Solve 9sin(2x)=−5cos(2x)9sin(2x)=−5cos(2x) on [−10,0][−10,0].
Show Solution
At first glance this problem seems to be at odds with the sentence preceding the example. However, it really isn’t.
First, when we have more than one trig function in an equation we need a way to get equations that only involve one trig function. There are many ways of doing this that depend on the type of equation we’re starting with. In this case we can simply divide both sides by a cosine and we’ll get a single tangent in the equation. We can now see that this really is an equation that doesn’t involve a sine or a cosine.
So, let’s get started on this example.
sin(2x)cos(2x)=tan(2x)=−59⇒2x=tan−1(−59)=−0.5071sin(2x)cos(2x)=tan(2x)=−59⇒2x=tan−1(−59)=−0.5071
Now, the unit circle doesn’t involve tangents, however we can use it to illustrate the second angle in the range [0,2π][0,2π].
The angles that we’re looking for here are those whose quotient of sinecosinesinecosine is the same. The second angle where we will get the same value of tangent will be exactly opposite of the given point. For this angle the values of sine and cosine are the same except they will have opposite signs. In the quotient however, the difference in signs will cancel out and we’ll get the same value of tangent. So, the second angle will always be the first angle plus ππ.
Before getting the second angle let’s also note that, like the previous example, we’ll use the 2π−0.5071=5.77612π−0.5071=5.7761 for the first angle. Again, this is only because of a concern about losing track of the minus sign in our calculator answer. We could just as easily do the work with the original angle our calculator gave us.
Now, this is where is seems like we’re just randomly making changes and doing things for no reason. The second angle that we’re going to use is,
π+(−0.5071)=π−0.5071=2.6345π+(−0.5071)=π−0.5071=2.6345
The fact that we used the calculator answer here seems to contradict the fact that we used a different angle for the first above. The reason for doing this here is to give a second angle that is in the range [0,2π][0,2π]. Had we used 5.7761 to find the second angle we’d get π+5.7761=8.9177π+5.7761=8.9177. This is a perfectly acceptable answer; however, it is larger than 2π2π (6.2832) and the general rule of thumb is to keep the initial angles as small as possible.
Here are all the solutions to the equation.
2x=5.7761+2πn2x=2.6345+2πn⇒x=2.8881+πnx=1.3173+πnn=0,±1,±2,…2x=5.7761+2πn2x=2.6345+2πn⇒x=2.8881+πnx=1.3173+πnn=0,±1,±2,…
n=−4:x=−9.6783and−11.2491n=−3:x=−6.5367and−8.1075n=−2:x=−3.3951and−4.9659n=−1:x=−0.2535and−1.8243n=0:x=2.8881and1.3173n=−4:x=−9.6783and−11.2491n=−3:x=−6.5367and−8.1075n=−2:x=−3.3951and−4.9659n=−1:x=−0.2535and−1.8243n=0:x=2.8881and1.3173
The seven solutions to this equation are then,
−0.2535,−1.8243,−3.3951,−4.9659,−6.5367,−8.1075,−9.6783−0.2535,−1.8243,−3.3951,−4.9659,−6.5367,−8.1075,−9.6783
Note as well that we didn’t need to do the n=0n=0 computation since we could see from the given interval that we only wanted negative answers and these would clearly give positive answers.
Before moving on we need to address one issue about the previous example. The solution method used there is not the “standard” solution method. Because the second angle is just ππ plus the first and if we added ππ onto the second angle we’d be back at the line representing the first angle the more standard solution method is to just add πnπn onto the first angle.
If using the calculator answer this would give,
2x=−0.5071+πn,n=0,±1,±2,…2x=−0.5071+πn,n=0,±1,±2,…
If using the positive angle corresponding to the calculator answer this would give,
2x=5.7761+πn,n=0,±1,±2,…2x=5.7761+πn,n=0,±1,±2,…
Then dividing by 2 either of the following sets of solutions,
x=−0.2535+πn2,n=0,±1,±2,…x=2.8881+πn2,n=0,±1,±2,…x=−0.2535+πn2,n=0,±1,±2,…x=2.8881+πn2,n=0,±1,±2,…
Either of these sets of solutions is identical to the set of solutions we got in the example (we’ll leave it to you to plug in some nn’s and verify that). So, why did we not use the method in the previous example? Simple. The method in the previous example more closely mirrors the solution method for cosine and sine (i.e. they both, generally, give two sets of angles) and so for students that aren’t comfortable with solving trig equations this gives a “consistent” solution method.
Many calculators today can only do inverse sine, inverse cosine, and inverse tangent. So, let’s see an example that uses one of the other trig functions.
Example 6 Solve 7sec(3t)=−107sec(3t)=−10.
Show Solution
We’ll start this one in exactly the same way we’ve done all the others.
sec(3t)=−107⇒3t=sec−1(−107)sec(3t)=−107⇒3t=sec−1(−107)
Now we reach the problem. As noted above, many calculators can’t handle inverse secant so we’re going to need a different solution method for this one. To finish the solution here we’ll simply recall the definition of secant in terms of cosine and convert this into an equation involving cosine instead and we already know how to solve those kinds of trig equations.
1cos(3t)=sec(3t)=−107⇒cos(3t)=−7101cos(3t)=sec(3t)=−107⇒cos(3t)=−710
Now, we solved this equation in the second example above so we won’t redo our work here. The solution is,
t=0.7821+2πn3t=1.3123+2πn3n=0,±1,±2,…t=0.7821+2πn3t=1.3123+2πn3n=0,±1,±2,…
We weren’t given an interval in this problem so here is nothing else to do here.
For the remainder of the examples in this section we’re not going to be finding solutions in an interval to save some space. If you followed the work from the first few examples in which we were given intervals you should be able to do any of the remaining examples if given an interval.
Also, we will no longer be including sketches of unit circles in the remaining solutions. We are going to assume that you can use the above sketches as guides for sketching unit circles to verify the claims in the following examples.
The next three examples don’t require a calculator but are important enough or cause enough problems for students to include in this section in case you run across them and haven’t seen them anywhere else.
Example 7 Solve cos(4θ)=−1cos(4θ)=−1.
Show Solution
There really isn’t too much to do with this problem. It is, however, different from all the others done to this point. All the others done to this point have had two angles in the interval [0,2π][0,2π] that were solutions to the equation. This only has one. If you aren’t sure you believe this sketch a quick unit circle and you’ll see that in fact there is only one angle for which cosine is -1.
Here is the solution to this equation.
4θ=π+2πn⇒θ=π4+πn2n=0,±1,±2,…4θ=π+2πn⇒θ=π4+πn2n=0,±1,±2,…
Example 8 Solve sin(α7)=0sin(α7)=0.
Show Solution
Again, not much to this problem. Using a unit circle it isn’t too hard to see that the solutions to this equation are,
α7=0+2πnα7=π+2πn⇒α=14πnα=7π+14πnn=0,±1,±2,…α7=0+2πnα7=π+2πn⇒α=14πnα=7π+14πnn=0,±1,±2,…
This next example has an important point that needs to be understood when solving some trig equations.
Example 9 Solve sin(3t)=2sin(3t)=2.
Show Solution
This example is designed to remind you of certain properties about sine and cosine. Recall that −1≤sin(θ)≤1 and −1≤cos(θ)≤1. Therefore, since sine will never be greater that 1 it definitely can’t be 2. So, THERE ARE NO SOLUTIONS to this equation!
It is important to remember that not all trig equations will have solutions.
Because this document is also being prepared for viewing on the web we’re going to split this section in two in order to keep the page size (and hence load time in a browser) to a minimum. In the next section we’re going to take a look at some slightly more “complicated” equations. Although, as you’ll see, they aren’t as complicated as they may at first seem.
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7027 | https://www.bjanaesthesia.org.uk/article/S0007-0912(17)34495-1/fulltext | Hypotension in obstetric spinal anaesthesia: a lesson from pre-eclampsia - British Journal of Anaesthesia
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EditorialsVolume 102, Issue 3p291-294 March 2009 Open Archive
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Hypotension in obstetric spinal anaesthesia: a lesson from pre-eclampsia
G.Sharwood-Smith
G.Sharwood-Smith
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Department of Anaesthesia, Critical Care, and Pain Medicine, Royal Infirmary, Edinburgh EH16 4HA, UK
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∙ G.B.Drummond
G.B.Drummond
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g.b.drummond@ed.ac.uk
Affiliations
Department of Anaesthesia, Critical Care, and Pain Medicine, Royal Infirmary, Edinburgh EH16 4HA, UK
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g.b.drummond@ed.ac.uk
Affiliations & Notes Article Info
Department of Anaesthesia, Critical Care, and Pain Medicine, Royal Infirmary, Edinburgh EH16 4HA, UK
DOI: 10.1093/bja/aep003 External LinkAlso available on ScienceDirect External Link
Copyright: © 2009 British Journal of Anaesthesia. Published by Elsevier Inc.
User License: Elsevier user license | Elsevier's open access license policy
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Vasopressor use to prevent hypotension occurs after 80% of spinal anaesthetics for Caesarean section.1
1.
Rout, CC ∙ Rocke, DA
Prevention of hypotension following spinal anesthesia for cesarean section
Int Anesthesiol Clin. 1994; 32:117-135
Crossref
PubMed
Google Scholar
The problem was first recognized 50 yr ago2
2.
Holmes, F
Collapse from spinal anaesthesia in pregnancy
Anaesthesia. 1959; 14:204
Crossref
Scopus (4)
Google Scholar
when it was attributed to caval compression. This theory became accepted as the basis for clinical management, and it remains current today.3
3.
Burns, SM ∙ Cowan, CM ∙ Wilkes, RG
Prevention and management of hypotension during spinal anaesthesia for elective Caesarean section: a survey of practice
Anaesthesia. 2001; 56:794-798
Crossref
Scopus (98)
PubMed
Google Scholar
4
4.
Birnbach, DJ ∙ Browne, IM
Anesthesia for obstetrics
Miller, RD (Editor)
Miller's Anesthesia
Elsevier Churchill Livingston, Philadelphia, 2006; 2308-2334
Google Scholar
However, using this theory as a basis for the management of hypotension has proved disappointingly ineffective.5
5.
Jackson, R ∙ Reid, JA ∙ Thorburn, J
Volume preloading is not essential to prevent spinal-induced hypotension at Caesarean section
Br J Anaesth. 1995; 75:262-265
Abstract
Full Text (PDF)
Scopus (211)
PubMed
Google Scholar
6
6.
Cyna, AM ∙ Andrew, M ∙ Emmett, RS ...
Techniques for preventing hypotension during spinal anaesthesia for caesarean section
Cochrane Database Syst Rev. 2006;
Crossref
Google Scholar
Important information from the ‘natural experiment’ of pre-eclampsia was overlooked, and fresh information from vascular biology now calls for a reconsideration of our management of hypotension in these circumstances.
Previously, the concept of compression of the vena cava and the aorta was linked to three features, which can coexist and are often considered together. However, each probably has a different mechanism. First, spinal anaesthesia almost always causes hypotension in normal pregnancy, and we will consider the reasons for this phenomenon later. Secondly, cardiac output can be reduced by aortocaval compression when some mothers lie in the supine position, although this is not necessarily clinically evident. Thirdly, a marked bradycardia with a reduction in cardiac output and severe hypotension can occur suddenly in a few subjects at some time after the mother moves to the supine position. This reflex effect is the relatively uncommon supine hypotensive syndrome of pregnancy (SHSP).7
7.
Kinsella, SM ∙ Tuckey, JP
Perioperative bradycardia and asystole: relationship to vasovagal syncope and the Bezold–Jarisch reflex
Br J Anaesth. 2001; 86:859-868
Full Text
Full Text (PDF)
Scopus (240)
PubMed
Google Scholar
Holmes8
8.
Holmes, F
Spinal analgesia and caesarean section; maternal mortality
J Obstet Gynaecol Br Emp. 1957; 64:229-232
Crossref
Scopus (26)
PubMed
Google Scholar
proposed that compression of the inferior vena cava by the gravid uterus caused hypotension after spinal anaesthesia because venous return was reduced and thus cardiac output decreased. Marx9
9.
Marx, GF
Supine hypotension syndrome during cesarean section
J Am Med Assoc. 1969; 7:1903-1905
Crossref
Scopus (9)
Google Scholar
developed the concept that blood was trapped in the legs, and introduced the treatment strategy of ‘acute hydration’ supported by a widely cited illustrative case history. Subsequently, fluid administration before spinal anaesthesia became the putative ‘prophylaxis’ and an almost universal therapy.10
10.
Burns, SM ∙ Cowan, CM
Spinal anaesthesia for caesarean section: current clinical practice
Hosp Med. 2000; 61:855-858
Crossref
Scopus (10)
PubMed
Google Scholar
The theory of caval compression and supine hypotension was based largely on studies by Scott and colleagues,11
11.
Lees, MM ∙ Scott, DB ∙ Kerr, MG ...
The circulatory effects of recumbent postural change in late pregnancy
Clin Sci. 1967; 32:453-465
PubMed
Google Scholar
who measured cardiac output by dye dilution in eight patients. Overall, cardiac output was 12% less in the supine compared with the lateral position. In three subjects, the mean reduction was only 6% and the investigators suggested that vena caval compression was relieved because the fetal head was engaged.11
11.
Lees, MM ∙ Scott, DB ∙ Kerr, MG ...
The circulatory effects of recumbent postural change in late pregnancy
Clin Sci. 1967; 32:453-465
PubMed
Google Scholar
However, in two patients, there was sudden bradycardia, hypotension, and a decrease in cardiac output by more than 50%, suggesting a reflex response. Clearly, this study reported a heterogeneous group of patients, and the patients with bradycardia developed the supine hypotensive syndrome, which is a different phenomenon from the hypotension seen in the other patients in that study. An extensive review of SHSP found a wide range of case selection, clinical features, definitions, and degrees of hypotension.12
12.
Kinsella, SM ∙ Lohmann, G
Supine hypotensive syndrome
Obstet Gynecol. 1994; 83:774-788
PubMed
Google Scholar
Severe hypotension was reported in 2.5–20% of these patients. In some patients, hypotension only occurred after 20 min in the supine position. The possible reasons given for hypotension in these patients were either vena caval obstruction or a vagal reflex bradycardia, which is a well-known phenomenon associated with poorly filled heart.13
13.
Dickinson, CJ
Fainting precipitated by collapse-firing of venous baroreceptors
Lancet. 1993; 342:970-972
Abstract
Scopus (53)
PubMed
Google Scholar
Later studies found much less difference between supine and lateral positions. Using transcutaneous Doppler, a maximum change of cardiac output of 6% occurred with moving from supine to a left 15° tilt, and fetal head engagement made no difference.14
14.
Newman, B ∙ Derrington, C ∙ Dore, C
Cardiac output and the recumbent position in late pregnancy
Anaesthesia. 1983; 38:332-335
Crossref
Scopus (24)
PubMed
Google Scholar
Undoubtedly, the vena cava is affected by the gravid uterus. Femoral venous and distal inferior caval pressures were greater in the supine position. In the lateral position, venous pressure was less, but still not as low as non-pregnant levels.15
15.
Scott, DB
Inferior vena caval occlusion in late pregnancy and its importance in anaesthesia
Br J Anaesth. 1968; 40:120-128
Abstract
Full Text (PDF)
Scopus (46)
PubMed
Google Scholar
Angiography showed occlusion of the inferior vena cava and distension of the collateral azygos circulation in 12 supine patients having Caesarean section under general anaesthesia. The abdominal vena cava remained partly occluded in the lateral position.16
16.
Kerr, MG ∙ Scott, DB ∙ Samuel, E
Studies of the inferior vena cava in late pregnancy
Br Med J. 1964; 1:532-533
Crossref
Scopus (230)
PubMed
Google Scholar
However, in these studies, the link between changes in venous behaviour and hypotension was inferred rather than directly proved. No early studies involved spinal anaesthesia because general anaesthesia and increasingly epidural anaesthesia had, by that time, largely replaced spinals for Caesarean section in the UK.
The proponents of the caval compression theory suggested three ways to prevent hypotension after spinal block, but none has withstood careful examination. First, infusion of crystalloid or colloid was proposed to compensate for the venous blood said to be trapped in the legs, but initial reports of success in preventing hypotension17
17.
Wollman, SB ∙ Marx, GF
Acute hydration for prevention of hypotension of spinal anesthesia in parturients
Anesthesiology. 1968; 29:374-380
Crossref
Scopus (108)
PubMed
Google Scholar
were not replicated in subsequent studies.1
1.
Rout, CC ∙ Rocke, DA
Prevention of hypotension following spinal anesthesia for cesarean section
Int Anesthesiol Clin. 1994; 32:117-135
Crossref
PubMed
Google Scholar
5
5.
Jackson, R ∙ Reid, JA ∙ Thorburn, J
Volume preloading is not essential to prevent spinal-induced hypotension at Caesarean section
Br J Anaesth. 1995; 75:262-265
Abstract
Full Text (PDF)
Scopus (211)
PubMed
Google Scholar
6
6.
Cyna, AM ∙ Andrew, M ∙ Emmett, RS ...
Techniques for preventing hypotension during spinal anaesthesia for caesarean section
Cochrane Database Syst Rev. 2006;
Crossref
Google Scholar
Colloid administration could increase cardiac output transiently, perhaps by haemodilution and reduced viscosity, but this effect was not sustained after sympathetic block with a spinal. Secondly, leg compression was attempted but was relatively ineffective, despite the success of the anti-G suit in preventing lower limb pooling and hypotension in aerospace medicine.18
18.
Peterson, DF ∙ Bishop, VS ∙ Erickson, HH
Anti-G suit effect of cardiovascular dynamic changes due to +GZ stress
J Appl Physiol. 1977; 43:765-769
PubMed
Google Scholar
Finally, the tilt manoeuvre was advocated to reduce caval occlusion. Although widely used, this procedure is variably applied,19
19.
Paech, MJ
Should we take a different angle in managing pregnant women at delivery? Attempting to avoid the ‘supine hypotensive syndrome
Anaesth Intensive Care. 2008; 36:775-777
PubMed
Google Scholar
and does not prevent hypotension after spinal anaesthesia.6
6.
Cyna, AM ∙ Andrew, M ∙ Emmett, RS ...
Techniques for preventing hypotension during spinal anaesthesia for caesarean section
Cochrane Database Syst Rev. 2006;
Crossref
Google Scholar
There is no escape from the fact that therapies based on the concept of caval compression do not reliably prevent hypotension after spinal anaesthesia in Caesarean section. Despite this, current books suggest routine use of strategies based on these putative explanations,4
4.
Birnbach, DJ ∙ Browne, IM
Anesthesia for obstetrics
Miller, RD (Editor)
Miller's Anesthesia
Elsevier Churchill Livingston, Philadelphia, 2006; 2308-2334
Google Scholar
and current teaching uses these concepts.
The original hypothesis underlying the mechanism of hypotension was that a reduction in central venous pressure would reduce cardiac output, and thus reduce arterial pressure. This concept should be reconsidered. The hypothesis was based on the view that central venous pressure controls cardiac output, as suggested by the experimental studies of Paterson and Starling20
20.
Paterson, SW ∙ Starling, EH
On the mechanical factors which determine the output of the ventricles
J Physiol (Lond). 1914; 48:357-379
Crossref
Scopus (342)
Google Scholar
and Guyton.21
21.
Guyton, AC
Determination of cardiac output by equating venous return curves with cardiac output curves
Physiol Rev. 1955; 35:123-129
PubMed
Google Scholar
A clear understanding of the limits of Starling's studies is vital. They were of an isolated heart, supplied with blood from a venous chamber which could be raised or lowered to adjust the atrial pressure. In this ‘open’ system, output was not related to supply. The supply to the venous reservoir was externally adjusted by the investigator to keep the atrial pressure constant. By raising the reservoir to increase inflow pressure, the stretch of the ventricular muscle was increased, and thus ejection volume increased. To maintain the atrial pressure, the atrial reservoir had to be replenished more rapidly. In these circumstances, atrial pressure regulated cardiac output. This did not mean that the increased flow from the venous reservoir had increased the cardiac output, only that the flow had to be increased to sustain the reservoir pressure. The entirely separate studies of Guyton in which he related atrial pressure and venous return were equally artificial. Venous return was controlled using an adjustable pump. When the pump rate and thus the experimentally controlled ‘venous return’ was increased, a limit was reached where a decrease in venous pressure occurred and venous return did not change, implying upstream flow limitation.21
21.
Guyton, AC
Determination of cardiac output by equating venous return curves with cardiac output curves
Physiol Rev. 1955; 35:123-129
PubMed
Google Scholar
In the whole body, the two factors of venous return and cardiac output are of course linked, in the long term, and neither is the ‘cause’ or ‘effect’ of changes in output or venous pressure, merely two sides of the same coin. The inextricable link between venous return and cardiac output, and the unrealistic question concerning which is the cause and which is the effect, was recognized by Guyton, despite his considering venous pressure to be an independent variable. Even at the time, this highly artificial experiment was recognized as unlikely to be applicable to the intact animal.22
22.
Rushmer, RF
Applicability of Starling's law of the heart to intact, unanesthetized animals
Physiol Rev. 1955; 35:138-142
PubMed
Google Scholar
In recent years, the relationship between venous pressure and cardiac output has been re-evaluated23
23.
Levy, MN
The cardiac and vascular factors that determine systemic blood flow
Circ Res. 1979; 44:739-747
Crossref
Scopus (116)
PubMed
Google Scholar
24
24.
Tyberg, JV
How changes in venous capacitance modulate cardiac output
Pflugers Arch. 2002; 445:10-17
Crossref
Scopus (77)
PubMed
Google Scholar
and this has led to robust controversy.25
25.
Brengelmann, GL
A critical analysis of the view that right atrial pressure determines venous return
J Appl Physiol. 2003; 94:849-859
Crossref
Scopus (85)
PubMed
Google Scholar
26
26.
Magder, S
Point: Counterpoint: The classical Guyton view that mean systemic pressure, right atrial pressure, and venous resistance govern venous return is/is not correct
J Appl Physiol. 2008; 101:1523-1525
Crossref
Scopus (49)
Google Scholar
Reddi and Carpenter27
27.
Reddi, BAJ ∙ Carpenter, RHS
Venous excess: a new approach to cardiovascular control and its teaching
J Appl Physiol. 2005; 98:356-364
Crossref
Scopus (33)
PubMed
Google Scholar
repeated previous suggestions23
23.
Levy, MN
The cardiac and vascular factors that determine systemic blood flow
Circ Res. 1979; 44:739-747
Crossref
Scopus (116)
PubMed
Google Scholar
25
25.
Brengelmann, GL
A critical analysis of the view that right atrial pressure determines venous return
J Appl Physiol. 2003; 94:849-859
Crossref
Scopus (85)
PubMed
Google Scholar
that it makes more sense to re-draw the Guyton plot with cardiac output on the abscissa (Fig. 1), to escape the common misconception that a decrease in right atrial pressure would act to increase blood flow through the veins. The important feature of the venous system is its compliance, not its resistance, and we can relate the central venous pressure to the volume held in the veins.28
28.
Gelman, S
Venous function and central venous pressure
Anesthesiology. 2008; 108:735-748
Crossref
Scopus (424)
PubMed
Google Scholar
A recent helpful view is that the volume in the venous system is more relevant than the pressure, and that ‘Venous Excess’ is the important regulating factor on the venous side of the circulation.27
27.
Reddi, BAJ ∙ Carpenter, RHS
Venous excess: a new approach to cardiovascular control and its teaching
J Appl Physiol. 2005; 98:356-364
Crossref
Scopus (33)
PubMed
Google Scholar
Venous capacitance and its regulation in pregnancy may be an important element in understanding the haemodynamic response to spinal anaesthesia. For example, the splanchnic component of this capacitance drains directly into the vena cava via the hepatic vein which is not directly compressed by the uterus. However, we lack basic information on these aspects of venous dynamics.
Figure viewer
Fig 1 Comparison of cardiac function and venous return curves. (a) Cardiac function curve, after Guyton.19
19.
Paech, MJ
Should we take a different angle in managing pregnant women at delivery? Attempting to avoid the ‘supine hypotensive syndrome
Anaesth Intensive Care. 2008; 36:775-777
PubMed
Google Scholar
This relationship is based on the function of the isolated heart. An increase in central venous pressure causes an increase in cardiac output. The dependent variable (cardiac output) is plotted on the y-axis. (b) Venous return curve (also termed systemic or vascular function) which is the relationship found when the venous return is modified as the independent variable: under these circumstances, an increase in venous return reduces right atrial pressure.21
21.
Guyton, AC
Determination of cardiac output by equating venous return curves with cardiac output curves
Physiol Rev. 1955; 35:123-129
PubMed
Google Scholar
Combining the two curves on one diagram condemns one of the relationships to have an independent variable expressed on the y-axis.
The sensors that normally control arterial pressure, in the carotid sinus and the aorta, lie on the arterial side of the circulation, and are the sensors of the baroreflex. Why does this reflex fail to maintain arterial pressure after spinal anaesthesia in pregnancy? Part of the answer to this question can be found in the pathophysiology of pre-eclampsia. Remarkably, studies done in the 1950s showed that pregnant women with toxaemia (severe pre-eclampsia) were far less likely to develop hypotension after spinal anaesthesia than normal pregnant or non-pregnant women. Similar differences were seen in response to autonomic ganglionic block, supporting the conclusion that withdrawal of sympathetic activity had less effect in the patient with pre-eclampsia.29
29.
Assali, NS ∙ Prystowsky, H
Studies on autonomic blockade. I. Comparison between the effects of tetraethylammonium chloride (TEAC) and high selective spinal anesthesia on blood pressure of normal and toxemic pregnancy
J Clin Invest. 1950; 29:1354-1366
Crossref
Scopus (47)
PubMed
Google Scholar
30
30.
Assali, NS ∙ Prystowsky, H
Studies on autonomic blockade. II. Observations on the nature of blood pressure fall with high selective spinal anesthesia in pregnant women
J Clin Invest. 1950; 29:1367-1375
Crossref
Scopus (9)
PubMed
Google Scholar
For some reason, these studies were downplayed, although the proponents of the caval compression theory knew of them.2
2.
Holmes, F
Collapse from spinal anaesthesia in pregnancy
Anaesthesia. 1959; 14:204
Crossref
Scopus (4)
Google Scholar
More recent studies corroborate the ability of pre-eclamptic patients to sustain arterial pressure after the spinal block.31
31.
Clark, VA ∙ Sharwood-Smith, GH ∙ Stewart, AV
Ephedrine requirements are reduced during spinal anaesthesia for caesarean section in preeclampsia
Int J Obstet Anesth. 2005; 14:9-13
Full Text
Full Text (PDF)
Scopus (60)
PubMed
Google Scholar
In pre-eclampsia, vascular epithelium is damaged by a process involving placental-derived proteins, leading to an imbalance between pro- and anti-angiogenic growth factors,32
32.
Romero, R ∙ Nien, JK ∙ Espinoza, J ...
A longitudinal study of angiogenic (placental growth factor) and anti-angiogenic (soluble endoglin and soluble vascular endothelial growth factor receptor-1) factors in normal pregnancy and patients destined to develop preeclampsia and deliver a small for gestational age neonate
J Matern Fetal Neonatal Med. 2008; 21:9-23
Crossref
Scopus (586)
PubMed
Google Scholar
33
33.
Chaiworapongsa, T ∙ Espinoza, J ∙ Gotsch, F ...
The maternal plasma soluble vascular endothelial growth factor receptor-1 concentration is elevated in SGA and the magnitude of the increase relates to Doppler abnormalities in the maternal and fetal circulation
J Matern Fetal Neonatal Med. 2008; 21:25-40
Crossref
Scopus (101)
PubMed
Google Scholar
which results in persistent vasoconstriction. In contrast, the normal pregnant patient is very sensitive to spinal anaesthesia, because of an altered balance of vascular tone. Responses to endogenous pressors, particularly angiotensin II, are reduced. This is caused by an endothelium-dependent alteration of vascular smooth muscle function. Additionally, there is increased synthesis of vasodilator prostaglandins and nitric oxide. These effects increase dependence on sympathetic vascular tone in normal pregnancy.34
34.
Whalley, PJ ∙ Everett, RB ∙ Gant, NF ...
Pressor responsiveness to angiotensin II in hospitalized primigravid women with pregnancy-induced hypertension
Am J Obstet Gynecol. 1983; 145:481-483
Abstract
Full Text (PDF)
Scopus (16)
PubMed
Google Scholar
35
35.
Sharwood-Smith, G ∙ Clark, V ∙ Watson, E
Regional anaesthesia for caesarean section in severe preeclampsia: spinal anaesthesia is the preferred choice
Int J Obstet Anesth. 1999; 8:85-89
Abstract
Full Text (PDF)
Scopus (43)
PubMed
Google Scholar
The use of sympathomimetic vasopressors to sustain arteriolar tone and thus arterial pressure has become the most important strategy for safe spinal anaesthesia in contemporary practice, despite the prevailing theory of caval occlusion being responsible for hypotension after a spinal in normal pregnancy. Indeed, those who suggested that caval compression caused circulatory disturbances had advised against pressor agents to treat hypotension, suggesting that they would cause vasoconstriction but would not improve venous return.16
16.
Kerr, MG ∙ Scott, DB ∙ Samuel, E
Studies of the inferior vena cava in late pregnancy
Br Med J. 1964; 1:532-533
Crossref
Scopus (230)
PubMed
Google Scholar
Nevertheless, aortocaval compression can reduce cardiac output and impair placental blood flow, so it remains rational to use tilt during anaesthesia, although the exact contribution of tilt to reducing hypotension in spinal anaesthesia is unclear.
After 40 yr, the relationship between spinal anaesthesia, pre-eclampsia, and hypotension can be properly acknowledged and put into clinical practice. These observations shed light on the circulatory effects of spinal anaesthesia in normal pregnancy. Research in obstetric anaesthesia can now move on from the legacy of an uncertain hypothesis by learning the lessons of pre-eclampsia and understanding how the features of this disorder illuminate our current concepts. Modern non-invasive methods such as ultrasound, MRI, and measures of skin blood flow36
36.
Holowatz, LA ∙ Thompson-Torgerson, CS ∙ Kenney, WL
The human cutaneous circulation as a model of generalized microvascular function
J Appl Physiol. 2008; 105:370-372
Crossref
Scopus (371)
PubMed
Google Scholar
should be used in pregnancy to explain the effects of spinal anaesthesia more exactly. Better management and training based on logical theories should follow.
Post Publication Comments
Archive (2.23 KB)
References
1.
Rout, CC ∙ Rocke, DA
Prevention of hypotension following spinal anesthesia for cesarean section
Int Anesthesiol Clin. 1994; 32:117-135
Crossref
PubMed
Google Scholar
2.
Holmes, F
Collapse from spinal anaesthesia in pregnancy
Anaesthesia. 1959; 14:204
Crossref
Scopus (4)
Google Scholar
3.
Burns, SM ∙ Cowan, CM ∙ Wilkes, RG
Prevention and management of hypotension during spinal anaesthesia for elective Caesarean section: a survey of practice
Anaesthesia. 2001; 56:794-798
Crossref
Scopus (98)
PubMed
Google Scholar
4.
Birnbach, DJ ∙ Browne, IM
Anesthesia for obstetrics
Miller, RD (Editor)
Miller's Anesthesia
Elsevier Churchill Livingston, Philadelphia, 2006; 2308-2334
Google Scholar
5.
Jackson, R ∙ Reid, JA ∙ Thorburn, J
Volume preloading is not essential to prevent spinal-induced hypotension at Caesarean section
Br J Anaesth. 1995; 75:262-265
Abstract
Full Text (PDF)
Scopus (211)
PubMed
Google Scholar
6.
Cyna, AM ∙ Andrew, M ∙ Emmett, RS ...
Techniques for preventing hypotension during spinal anaesthesia for caesarean section
Cochrane Database Syst Rev. 2006;
Crossref
Google Scholar
7.
Kinsella, SM ∙ Tuckey, JP
Perioperative bradycardia and asystole: relationship to vasovagal syncope and the Bezold–Jarisch reflex
Br J Anaesth. 2001; 86:859-868
Full Text
Full Text (PDF)
Scopus (240)
PubMed
Google Scholar
8.
Holmes, F
Spinal analgesia and caesarean section; maternal mortality
J Obstet Gynaecol Br Emp. 1957; 64:229-232
Crossref
Scopus (26)
PubMed
Google Scholar
9.
Marx, GF
Supine hypotension syndrome during cesarean section
J Am Med Assoc. 1969; 7:1903-1905
Crossref
Scopus (9)
Google Scholar
10.
Burns, SM ∙ Cowan, CM
Spinal anaesthesia for caesarean section: current clinical practice
Hosp Med. 2000; 61:855-858
Crossref
Scopus (10)
PubMed
Google Scholar
11.
Lees, MM ∙ Scott, DB ∙ Kerr, MG ...
The circulatory effects of recumbent postural change in late pregnancy
Clin Sci. 1967; 32:453-465
PubMed
Google Scholar
12.
Kinsella, SM ∙ Lohmann, G
Supine hypotensive syndrome
Obstet Gynecol. 1994; 83:774-788
PubMed
Google Scholar
13.
Dickinson, CJ
Fainting precipitated by collapse-firing of venous baroreceptors
Lancet. 1993; 342:970-972
Abstract
Scopus (53)
PubMed
Google Scholar
14.
Newman, B ∙ Derrington, C ∙ Dore, C
Cardiac output and the recumbent position in late pregnancy
Anaesthesia. 1983; 38:332-335
Crossref
Scopus (24)
PubMed
Google Scholar
15.
Scott, DB
Inferior vena caval occlusion in late pregnancy and its importance in anaesthesia
Br J Anaesth. 1968; 40:120-128
Abstract
Full Text (PDF)
Scopus (46)
PubMed
Google Scholar
16.
Kerr, MG ∙ Scott, DB ∙ Samuel, E
Studies of the inferior vena cava in late pregnancy
Br Med J. 1964; 1:532-533
Crossref
Scopus (230)
PubMed
Google Scholar
17.
Wollman, SB ∙ Marx, GF
Acute hydration for prevention of hypotension of spinal anesthesia in parturients
Anesthesiology. 1968; 29:374-380
Crossref
Scopus (108)
PubMed
Google Scholar
18.
Peterson, DF ∙ Bishop, VS ∙ Erickson, HH
Anti-G suit effect of cardiovascular dynamic changes due to +GZ stress
J Appl Physiol. 1977; 43:765-769
PubMed
Google Scholar
19.
Paech, MJ
Should we take a different angle in managing pregnant women at delivery? Attempting to avoid the ‘supine hypotensive syndrome
Anaesth Intensive Care. 2008; 36:775-777
PubMed
Google Scholar
20.
Paterson, SW ∙ Starling, EH
On the mechanical factors which determine the output of the ventricles
J Physiol (Lond). 1914; 48:357-379
Crossref
Scopus (342)
Google Scholar
21.
Guyton, AC
Determination of cardiac output by equating venous return curves with cardiac output curves
Physiol Rev. 1955; 35:123-129
PubMed
Google Scholar
22.
Rushmer, RF
Applicability of Starling's law of the heart to intact, unanesthetized animals
Physiol Rev. 1955; 35:138-142
PubMed
Google Scholar
23.
Levy, MN
The cardiac and vascular factors that determine systemic blood flow
Circ Res. 1979; 44:739-747
Crossref
Scopus (116)
PubMed
Google Scholar
24.
Tyberg, JV
How changes in venous capacitance modulate cardiac output
Pflugers Arch. 2002; 445:10-17
Crossref
Scopus (77)
PubMed
Google Scholar
25.
Brengelmann, GL
A critical analysis of the view that right atrial pressure determines venous return
J Appl Physiol. 2003; 94:849-859
Crossref
Scopus (85)
PubMed
Google Scholar
26.
Magder, S
Point: Counterpoint: The classical Guyton view that mean systemic pressure, right atrial pressure, and venous resistance govern venous return is/is not correct
J Appl Physiol. 2008; 101:1523-1525
Crossref
Scopus (49)
Google Scholar
27.
Reddi, BAJ ∙ Carpenter, RHS
Venous excess: a new approach to cardiovascular control and its teaching
J Appl Physiol. 2005; 98:356-364
Crossref
Scopus (33)
PubMed
Google Scholar
28.
Gelman, S
Venous function and central venous pressure
Anesthesiology. 2008; 108:735-748
Crossref
Scopus (424)
PubMed
Google Scholar
29.
Assali, NS ∙ Prystowsky, H
Studies on autonomic blockade. I. Comparison between the effects of tetraethylammonium chloride (TEAC) and high selective spinal anesthesia on blood pressure of normal and toxemic pregnancy
J Clin Invest. 1950; 29:1354-1366
Crossref
Scopus (47)
PubMed
Google Scholar
30.
Assali, NS ∙ Prystowsky, H
Studies on autonomic blockade. II. Observations on the nature of blood pressure fall with high selective spinal anesthesia in pregnant women
J Clin Invest. 1950; 29:1367-1375
Crossref
Scopus (9)
PubMed
Google Scholar
31.
Clark, VA ∙ Sharwood-Smith, GH ∙ Stewart, AV
Ephedrine requirements are reduced during spinal anaesthesia for caesarean section in preeclampsia
Int J Obstet Anesth. 2005; 14:9-13
Full Text
Full Text (PDF)
Scopus (60)
PubMed
Google Scholar
32.
Romero, R ∙ Nien, JK ∙ Espinoza, J ...
A longitudinal study of angiogenic (placental growth factor) and anti-angiogenic (soluble endoglin and soluble vascular endothelial growth factor receptor-1) factors in normal pregnancy and patients destined to develop preeclampsia and deliver a small for gestational age neonate
J Matern Fetal Neonatal Med. 2008; 21:9-23
Crossref
Scopus (586)
PubMed
Google Scholar
33.
Chaiworapongsa, T ∙ Espinoza, J ∙ Gotsch, F ...
The maternal plasma soluble vascular endothelial growth factor receptor-1 concentration is elevated in SGA and the magnitude of the increase relates to Doppler abnormalities in the maternal and fetal circulation
J Matern Fetal Neonatal Med. 2008; 21:25-40
Crossref
Scopus (101)
PubMed
Google Scholar
34.
Whalley, PJ ∙ Everett, RB ∙ Gant, NF ...
Pressor responsiveness to angiotensin II in hospitalized primigravid women with pregnancy-induced hypertension
Am J Obstet Gynecol. 1983; 145:481-483
Abstract
Full Text (PDF)
Scopus (16)
PubMed
Google Scholar
35.
Sharwood-Smith, G ∙ Clark, V ∙ Watson, E
Regional anaesthesia for caesarean section in severe preeclampsia: spinal anaesthesia is the preferred choice
Int J Obstet Anesth. 1999; 8:85-89
Abstract
Full Text (PDF)
Scopus (43)
PubMed
Google Scholar
36.
Holowatz, LA ∙ Thompson-Torgerson, CS ∙ Kenney, WL
The human cutaneous circulation as a model of generalized microvascular function
J Appl Physiol. 2008; 105:370-372
Crossref
Scopus (371)
PubMed
Google Scholar
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7028 | https://artofproblemsolving.com/wiki/index.php/Rational_number?srsltid=AfmBOoqAHPbiJG1q4QOlperY9nno25N2Zn-xBQoOhbdPEe1aaIfNcCX8 | Art of Problem Solving
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Rational number
A rational number is a number that can be expressed as the ratio of two integers.
Examples
All integers are rational because every integer can be represented as
Every number with a finite decimal expansion is rational (for example, )
Every number with a periodic decimal expansion (for example, 0.314314314...) is also rational.
Moreover, any rational number in any base satisfies exactly one of the last two conditions.
Properties
Rational numbers form a field. In plain English it means that you can add, subtract, multiply, and divide them (with the exception of division by ) and the result of each such operation is again a rational number.
Rational numbers are dense in the set of reals. This means that every non-emptyopen interval on the real line contains at least one (actually, infinitely many) rationals. Alternatively, it means that every real number can be represented as a limit of a sequence of rational numbers.
Despite this, the set of rational numbers is countable, i.e. the same size as the set of integers.
See also
Fraction
Rational approximation
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7029 | https://journals.lww.com/spinejournal/Fulltext/2013/03150/Lumbar_Scoliosis_in_Rheumatoid_Arthritis_.16.aspx?generateEpub=Article%7Cspinejournal:2013:03150:00016%7C10.1097/brs.0b013e3182843397%7C | Spine
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March 15, 2013 - Volume 38 - Issue 6
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Deformity
Lumbar Scoliosis in Rheumatoid Arthritis
Epidemiological Research With a DXA Cohort
Makino, Takahiro MD; Kaito, Takashi MD, PhD; Fujiwara, Hiroyasu MD; Yonenobu, Kazuo MD
Author Information
From the Department of Orthopaedic Surgery, National Hospital Organization Osaka Minami Medical Center, Osaka, Japan.
Address correspondence and reprint requests to Takahiro Makino, MD, Department of Orthopaedic Surgery, National Hospital Organization Osaka Minami Medical Center, 2-1 Kidohigashi, Kawachinagano, Osaka, 586-8521, Japan; E-mail: t-makino@za2.so-net.ne.jp
Acknowledgment date: August 3, 2012. First revision date: October 28, 2012. Acceptance date: December 19, 2012.
The manuscript submitted does not contain information about medical device(s)/drug(s).
No funds were received in support of this work.
No relevant financial activities outside the submitted work.
Spine 38(6):p E339-E343, March 15, 2013. | DOI: 10.1097/BRS.0b013e3182843397
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Abstract In Brief
Study Design.
A retrospective cross-sectional study.
Objective.
The aim of this study was to identify the prevalence of and risk factors for lumbar scoliosis in patients with rheumatoid arthritis (RA) using lumbar images obtained from dual-energy x-ray absorptiometry (DXA).
Summary of Background Data.
The prevalence of lumbar scoliosis in the normal adult population has been reported, but that in patients with RA remains unclear.
Methods.
Subjects comprised 241 patients with RA who underwent annual DXA. Cobb angles of the lumbar spine were measured by lumbar anteroposterior DXA and the prevalence of lumbar scoliosis (curvature ≥10°) was calculated. Correlations between lumbar scoliosis and potential risk factors (age, sex, duration of RA, T score of lumbar spine and hip, medications for RA [daily predonisolone dose, use of biological agents] and osteoporosis, disease activity score-C-reactive protein, progression stage and functional classification of RA, and severity of hand deformity were analyzed.
Results.
The prevalence of lumbar scoliosis in patients with RA was 32.0%. Mean Cobb angle was 7.1º ± 5.5º among all subjects, compared with 13.6º ± 4.4º (range, 10º–32º) among subjects with scoliosis. Subjects with scoliosis were significantly older (67.8 yr) than those without (61.6 yr, P< 0.0001). The daily prednisolone dose was significantly higher in subjects with scoliosis (4.14 mg) than in those without (3.46 mg, P = 0.0389). The T score of the hip was significantly smaller in subjects with scoliosis (−1.79) than in those without (−1.26, P = 0.0005). A multivariate logistic regression analysis revealed age as the sole risk factor for lumbar scoliotic changes in patients with RA (odds ratio, 1.068; 95% confidence interval, 1.031–1.107; P = 0.0003).
Conclusion.
The prevalence of lumbar scoliosis in patients with RA was 32.0%, about 3- or 4-times higher than its prevalence as obtained from previous reports of DXA cohorts irrespective of RA. Increased age represented an independent risk factor for lumbar scoliosis in patients with RA.
Level of Evidence: 4
The prevalence of lumbar scoliosis was 32% in a rheumatoid arthritis cohort, with age representing the sole greatest risk factor for deformity. This is the first report to clarify the prevalence of scoliotic changes and risk factors for deformity in rheumatoid arthritis.
© 2013 Lippincott Williams & Wilkins, Inc.
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Lumbar Scoliosis in Rheumatoid Arthritis: Epidemiological Research With a DXA Cohort
Spine38(6):E339-E343, March 15, 2013.
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7030 | https://openstax.org/books/biology-2e/pages/7-introduction | Skip to ContentGo to accessibility pageKeyboard shortcuts menu
Biology 2e
Introduction
Biology 2eIntroduction
Search for key terms or text.
Figure 7.1 This geothermal energy plant transforms thermal energy from deep in the ground into electrical energy, which can be easily used. (credit: modification of work by the U.S. Department of Defense)
Chapter Outline
7.1 Energy in Living Systems
7.2 Glycolysis
7.3 Oxidation of Pyruvate and the Citric Acid Cycle
7.4 Oxidative Phosphorylation
7.5 Metabolism without Oxygen
7.6 Connections of Carbohydrate, Protein, and Lipid Metabolic Pathways
7.7 Regulation of Cellular Respiration
The electrical energy plant in Figure 7.1 converts energy from one form to another form that can be more easily used. This type of generating plant starts with underground thermal energy (heat) and transforms it into electrical energy that will be transported to homes and factories. Like a generating plant, plants and animals also must take in energy from the environment and convert it into a form that their cells can use. Mass and its stored energy enter an organism’s body in one form and are converted into another form that can fuel the organism’s life functions. In the process of photosynthesis, plants and other photosynthetic producers take in energy in the form of light (solar energy) and convert it into chemical energy in the form of glucose, which stores this energy in its chemical bonds. Then, a series of metabolic pathways, collectively called cellular respiration, extracts the energy from the bonds in glucose and converts it into a form that all living things can use.
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7031 | https://www.quora.com/How-does-a-census-differ-from-a-sample-Why-is-the-census-method-better-than-sample | How does a census differ from a sample? Why is the census method better than sample? - Quora
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Population Counting Metho...
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How does a census differ from a sample? Why is the census method better than sample?
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Tim Newberry
Author has 5.3K answers and 2.7M answer views
·3y
A census of light bulbs in a University means looking at every single light bulb and recording how many there are, what type each is, what wattage each is and whether they're working or not (and maybe other things depending on what you want to know).
A sample will look at a (hopefully) representative building to determine these things and then multiply the answers by the number of buildings.
One gives you an accurate answer while the other gives you a good approximation.
If you're trying to figure how many bulbs you need to order, a sample is probably good enough as the cost of counting would exc
Continue Reading
A census of light bulbs in a University means looking at every single light bulb and recording how many there are, what type each is, what wattage each is and whether they're working or not (and maybe other things depending on what you want to know).
A sample will look at a (hopefully) representative building to determine these things and then multiply the answers by the number of buildings.
One gives you an accurate answer while the other gives you a good approximation.
If you're trying to figure how many bulbs you need to order, a sample is probably good enough as the cost of counting would exceed the cost of being out by a bit.
If you are paying tax on each bulb you don't want to overestimate and the government doesn't want you to underestimate, so a census might be necessary.
Some products, eg nuclear bombs I'd argue, need censuses every time coz it's kinda important you don't miss any.
Edit:- Ironically census comes from the Latin cencere meaning to estimate. Also, I wish the plural was censi - but it isn't.
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When should use a sample, and when should you use a census?
What is the difference between a census and a sample method? Which is better?
What is the difference between a census and a sample of a population?
Between census and sample, which method of data collection is always better?
In what type of situation is conducting a census more likely than sampling?
Braxton Anderson III
Studied Mathematics&Statistics (academic discipline) at Johannes Gutenberg University of Mainz (Graduated 1986) · Author has 8.2K answers and 13.9M answer views
·3y
Samples are actually done ANNUALLY by the Census People. The Census itself is done each decade and its MANDATED by the Constitution but Mathematically it validates all the samples to keep the error margins down to very acceptable amounts. And then figure this factor, WOULD YOU WANT the Electoral College to be set and established by a sample or hard data?
Samples are too easily manipulated to be used for that. But I am sure one Party would love it. And it’s the one always toying with the vote to STOP people from voting.
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9 1
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A census and a sample are both methods used to collect data, but they differ significantly in scope and application:
Census
Definition: A census involves collecting data from every individual or unit within a population.
Example: The U.S. Census, which aims to count every person living in the country every ten years.
Sample
Definition: A sample involves collecting data from a subset of the population, which is used to make inferences about the entire population.
Example: Polling a random selection of 1,000 voters to predict the outcome of an election.
Differences
Scope:
Census: Comprehensive; ai
Continue Reading
A census and a sample are both methods used to collect data, but they differ significantly in scope and application:
Census
Definition: A census involves collecting data from every individual or unit within a population.
Example: The U.S. Census, which aims to count every person living in the country every ten years.
Sample
Definition: A sample involves collecting data from a subset of the population, which is used to make inferences about the entire population.
Example: Polling a random selection of 1,000 voters to predict the outcome of an election.
Differences
Scope:
Census: Comprehensive; aims for complete data.
Sample: Limited; focuses on a representative subset.
Cost and Time:
Census: Generally more expensive and time-consuming due to the need to reach every individual.
Sample: Often cheaper and faster, as it requires less data collection.
Accuracy:
Census: Can provide more accurate and complete data, eliminating sampling error.
Sample: Subject to sampling error, which can affect the reliability of the results.
Why Census Method Can Be Better
Complete Representation: A census provides a complete picture of the population, capturing all variations and characteristics.
No Sampling Error: Since it includes every individual, there’s no risk of sampling error, leading to more precise data.
Detailed Data: A census allows for detailed analysis across various subgroups within the population.
Policy and Planning: Governments and organizations often rely on census data for critical decision-making, resource allocation, and policy formulation.
Limitations of Census
Despite its advantages, a census can be impractical due to high costs, logistical challenges, and the time required for data collection and processing. In many cases, a well-designed sample can provide sufficiently accurate insights without the need for a full census.
In conclusion, while a census is generally more comprehensive and accurate, the choice between a census and a sample depends on the specific goals, resources, and context of the data collection effort.
Upvote ·
Bengt Persson
Attempting to understand complex systems for 50 years · Author has 8.1K answers and 2.8M answer views
·3y
Census is accurate, but very costly and take time to execute. Sample could give good results at a fraction of the cost and lead time, but one have to assume an error on the level of ~1%.
The availability of a recent census allow subsequent samples to use the census as the base norm and thereby give more accurate result. A typical such method is to compare relative size of various groups (in multiple dimensions), in the sample to the census, and adjust the how each sample is counted based on if its groups were over or under represented.
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In what type of situation is the sampling method more appropriate than the Census method?
In which situation is it better to conduct a census than a sample survey, and vice versa? Why can you say this so confidently about both methods (census vs. sampling)?
What are two differences between a census and a sample survey?
What are the differences between sampling and census?
What is the definition of a sample in a census?
Eric Husher
BA in Anthropology (college major)&Paleontology, University of Wyoming (Graduated 1983) · Author has 17.2K answers and 5.2M answer views
·3y
A census counts ALL individuals, while a sample only counts SOME individuals and then extrapolates the population details from that sample. However, all samples are subject to ‘flukes’ and biases, which depend on the SIZE of the sample, and the selected elements of those within the sample, any or all of which can skew the sample in one way or another. That cannot happen with a census.
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Allen Ries
Math Major University of Alberta · Author has 25.1K answers and 9.7M answer views
·3y
Originally Answered: What is the difference between a census and a sample of a population? ·
The census attempts to find the total population by counting the total number of people living in houses apartments and trailers and - presumably - the people who do not have a current hom (living on the streets). Using a sample means that you take the population of a subset (sample) of the population that you believe represents the total population in a fair way. Then based on that sample you extrapolate the total population .
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9 1
Saravanamuttu Sri Ranjan
Retired Teacher of Maths/ Accountant (1972–present) · Author has 4.5K answers and 1.7M answer views
·3y
A census is an attempt to gather information about every individual in a population. A sample is a part of the population that is actually examined in order to represent the whole. As sample is included in the census, the Method of Census is more accurate than Sample in getting statistical information.
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·Sep 12
What are the best professional affordable and comfortable shoes for women?
I usually look at three things when I’m buying work shoes: comfort, cushioning and arch support; how sturdy the sole is; and whether I can actually afford to get more than one pair if I want them in different colours.
Ballerinas by Bata though, are what I wear the most. I didn’t know about them until recently, when a coworker recommended them to me, also spotted my favorite creator Siddhi Karwa styling them across Europe and I have been absolutely loving them.
They’re professional enough for work wear but don’t feel heavy and keep me comfortable throughout the day, even when I’m commuting to the
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I usually look at three things when I’m buying work shoes: comfort, cushioning and arch support; how sturdy the sole is; and whether I can actually afford to get more than one pair if I want them in different colours.
Ballerinas by Bata though, are what I wear the most. I didn’t know about them until recently, when a coworker recommended them to me, also spotted my favorite creator Siddhi Karwa styling them across Europe and I have been absolutely loving them.
They’re professional enough for work wear but don’t feel heavy and keep me comfortable throughout the day, even when I’m commuting to the office. I got mine for around ₹999 from Bata, which felt like a steal compared to some other brands I looked at. They’ve held up really well, and I can easily pair them with trousers, skirts for my work outfits. If you’re on a budget but still want something that is comfortable and follows fashion trends, Ballerinas by Bata are the perfect choice. I picked up mine from a Bata store near me, you can grab yours too.
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Bodavula Brahmanandam
Former Retd Dy. Director of M&HS (Demography) at Government of Andhra Pradesh (1998–2010) · Author has 807 answers and 100.9K answer views
·2y
Originally Answered: What is the difference between a census and a sample method? Which is better? ·
In a census method, all units in the selected universe will be covered where as in sample method only the selected units in the universe will be covered.
Census method is costly where as the sample method is cheap.
Census method requires lot of man power where as sample method requires less man power.
Census method requires lot of time where as the sample method requires less time.
Results of Census method is more accurate than sample method.
Upvote ·
Peter Falk
Former Engineer (1980–2017) · Author has 2.3K answers and 1.6M answer views
·3y
Originally Answered: What is the difference between a census and a sample of a population? ·
By definition, a census must count every item, while a sampling uses mathematical tools to draw conclusions of the WHOLE from a much smaller sample
It might seem like the census method is more accurate, but that is true only if you can guarantee that there is no miscount or over- or undercount. This assumption is questionable, and hard to quantify, while the error of the statistics is known
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Studied at York University (Canada) · Author has 35K answers and 568M answer views
·4y
Related
Why is conducting a sample survey preferable to conducting a census?
One of the issues with U.S. Census is that we know the results are wrong. The reason we know that is sample surveys.
In theory, the Census counts everyone. In practice it doesn’t. There will always be people who don’t fill out the forms. They tend to be concentrated in large urban areas, which tend to be undercounted.
One of the first population counts ever conducted was with a sample survey. It relied on reports of births and deaths in each French municipality. Those numbers were all reported annually to the central government. The survey took a random selection of municipalities, counted all t
Continue Reading
One of the issues with U.S. Census is that we know the results are wrong. The reason we know that is sample surveys.
In theory, the Census counts everyone. In practice it doesn’t. There will always be people who don’t fill out the forms. They tend to be concentrated in large urban areas, which tend to be undercounted.
One of the first population counts ever conducted was with a sample survey. It relied on reports of births and deaths in each French municipality. Those numbers were all reported annually to the central government. The survey took a random selection of municipalities, counted all the people along with the births and deaths, then extrapolated from the average to determine the population of France. That turns out to be more accurate than a large Census.
The current proposal is to select Census tracts and to do a complete count of each one, then extrapolate from there.
Upvote ·
9 4
Larry Storeling
Former Adjunct in Mathematics a While Back · Author has 758 answers and 478.3K answer views
·6y
Originally Answered: What are the differences between sampling and census? ·
Census is when you sample the full population. Of course, this is done mostly in/for smaller populations. Sample by contrasts are usually a strict subset of the full population, i.r. you do not consider or sample everyone, ad it may be too costly in many ways. As an example, it may be practical to find out the averageSDAT score of a highschool by averaging all the grades, since a highschool is not likely to have too many students to make this impractical, but you will likely take a sample if you wanted to estimate the a erage SAT in the US.
Upvote ·
9 1
Diogo Ferreira
Author has 952 answers and 544.8K answer views
·3y
Originally Answered: What is the difference between a census and a sample of a population? ·
Census ~ the whole population is included (almost in the real world)
Sample ~ a group of individuals that represent a subset of the population. The objective is to use this group as a fair aproximation of what one would expect from a population, or a specific target group within a population.
Upvote ·
Mario Luis Iovaldi
MD, Surgeon, not Statistician. Argentina · Author has 2.9K answers and 965.5K answer views
·3y
Originally Answered: What is the difference between a census and a sample of a population? ·
Census is a collection of data of the population.
A sample is a number of persons of that population. However, a sample may have many other purposes, opinion surveys, two groups of persons with diabetes to test 2 drugs to lower blood glucose levels, etc
Upvote ·
Related questions
When should use a sample, and when should you use a census?
What is the difference between a census and a sample method? Which is better?
What is the difference between a census and a sample of a population?
Between census and sample, which method of data collection is always better?
In what type of situation is conducting a census more likely than sampling?
In what type of situation is the sampling method more appropriate than the Census method?
In which situation is it better to conduct a census than a sample survey, and vice versa? Why can you say this so confidently about both methods (census vs. sampling)?
What are two differences between a census and a sample survey?
What are the differences between sampling and census?
What is the definition of a sample in a census?
Why is a sample often a better way to obtain information about a population than a census?
Which is more effective: Sample survey or Census survey?
What are the uses of census and sample method?
What is population and how does it differ from sample?
Why should we use a sample instead of population?
Related questions
When should use a sample, and when should you use a census?
What is the difference between a census and a sample method? Which is better?
What is the difference between a census and a sample of a population?
Between census and sample, which method of data collection is always better?
In what type of situation is conducting a census more likely than sampling?
In what type of situation is the sampling method more appropriate than the Census method?
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7032 | https://www.emathhelp.net/calculators/calculus-1/derivative-calculator/ | Home
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Find dxd(xsin(2x)).
Solution
Apply the product rule dxd(f(x)g(x))=dxd(f(x))g(x)+f(x)dxd(g(x)) with f(x)=x and g(x)=sin(2x):
(dxd(xsin(2x)))=(dxd(x)sin(2x)+xdxd(sin(2x)))
The function sin(2x) is the composition f(g(x)) of two functions f(u)=sin(u) and g(x)=2x.
Apply the chain rule dxd(f(g(x)))=dud(f(u))dxd(g(x)):
x(dxd(sin(2x)))+sin(2x)dxd(x)=x(dud(sin(u))dxd(2x))+sin(2x)dxd(x)
The derivative of the sine is dud(sin(u))=cos(u):
x(dud(sin(u)))dxd(2x)+sin(2x)dxd(x)=x(cos(u))dxd(2x)+sin(2x)dxd(x)
Return to the old variable:
xcos((u))dxd(2x)+sin(2x)dxd(x)=xcos((2x))dxd(2x)+sin(2x)dxd(x)
Apply the constant multiple rule dxd(cf(x))=cdxd(f(x)) with c=2 and f(x)=x:
xcos(2x)(dxd(2x))+sin(2x)dxd(x)=xcos(2x)(2dxd(x))+sin(2x)dxd(x)
Apply the power rule dxd(xn)=nxn−1 with n=1, in other words, dxd(x)=1:
2xcos(2x)(dxd(x))+sin(2x)(dxd(x))=2xcos(2x)(1)+sin(2x)(1)
Thus, dxd(xsin(2x))=2xcos(2x)+sin(2x).
dxd(xsin(2x))=2xcos(2x)+sin(2x)A |
7033 | https://math.stackexchange.com/questions/1524449/how-to-prove-that-this-kernel-has-codimension-1 | real analysis - How to prove that this kernel has codimension 1? - Mathematics Stack Exchange
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How to prove that this kernel has codimension 1?
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Let ϕ∈X′ϕ∈X′ , where X is a Hilbert space and X′X′ its dual. Then I want to check that ker ϕ kerϕ is a closed subspace of X X of codimension 1 (ϕ≠0 ϕ≠0).
So to see the closeness we pick a sequence (x n)∈ker ϕ(x n)∈kerϕ that converges to x x, then, since ϕ ϕ is a bounded functional it is continuous, thus ϕ((x n))→ϕ(x)=0 ϕ((x n))→ϕ(x)=0 therefore ker ϕ kerϕ is closed.
The thing is that I don't know how to attached the socond part, because I think I need to prove that the space X/ker ϕ={x+ker ϕ:x∈X}X/kerϕ={x+kerϕ:x∈X} has dimension 1.
Can someone help me to prove this assertion please?
Thanks a lot in advance.
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edited Nov 11, 2015 at 18:16
user162343user162343
asked Nov 11, 2015 at 17:52
user162343user162343
3,393 2 2 gold badges 32 32 silver badges 56 56 bronze badges
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1 Note that the image of ϕ ϕ is R R or C C (or whatever field your Hilbert space is over).SamM –SamM 2015-11-11 18:02:38 +00:00 Commented Nov 11, 2015 at 18:02
Can you elaborate more in this please? Thanks :)user162343 –user162343 2015-11-11 18:08:23 +00:00 Commented Nov 11, 2015 at 18:08
The dual space consists of all bounded linear functionals, which are linear maps from a normed (Banach, Hilbert, etc.) space X X into the field K K, which is usually R R or C C. Thus the image of ϕ ϕ is isomorphic (as a vector space) to K K.SamM –SamM 2015-11-11 18:11:54 +00:00 Commented Nov 11, 2015 at 18:11
Rigth :), but how does that shows that the kernel has codimension 1?user162343 –user162343 2015-11-11 18:14:01 +00:00 Commented Nov 11, 2015 at 18:14
Don't forget you need ϕ≠0 ϕ≠0...peter a g –peter a g 2015-11-11 18:14:42 +00:00 Commented Nov 11, 2015 at 18:14
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This is a purely algebraic result that you can prove without using the inner product or any other additional structure on your vector space:
Proposition. Let V V be a vector space over some field F F, and let ϕ:V→F ϕ:V→F be a nontrivial linear functional. Then ker(ϕ)ker(ϕ) has codimension 1.
The proof will consist of the following three steps:
Note that, by the first isomorphism theorem, V/ker(ϕ)≅im(ϕ)V/ker(ϕ)≅im(ϕ) (as F F-vector spaces).
Note that im(ϕ)im(ϕ) is a subspace of F F, and that the only subspaces of F F are {0}{0} and F F itself. Since ϕ ϕ is nontrivial, the only possibility is im(ϕ)=F im(ϕ)=F.
Use the fact that isomorphic spaces have the same dimension to conclude that codim F(ker(ϕ))=d e f dim F(V/ker(ϕ))=dim F(F)=1 codim F(ker(ϕ))=d e f dim F(V/ker(ϕ))=dim F(F)=1.
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edited Jan 8, 2021 at 22:16
answered Nov 26, 2017 at 19:33
Oskar HenrikssonOskar Henriksson
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Let z∉ker ϕ z∉kerϕ, the observe that for each x∈X x∈X,
x=ϕ(x)ϕ(z)z−ϕ(x)ϕ(z)z+x=ϕ(x)ϕ(z)z+[x−ϕ(x)ϕ(z)z]x=ϕ(x)ϕ(z)z−ϕ(x)ϕ(z)z+x=ϕ(x)ϕ(z)z+[x−ϕ(x)ϕ(z)z]
and [x−ϕ(x)ϕ(z)z]∈k e r ϕ[x−ϕ(x)ϕ(z)z]∈k e r ϕ. This means that X X can be written as the direct sum of its two subspaces that is
X=span{z}⊕ker ϕ.X=span{z}⊕kerϕ.
Therefore the ker ϕ kerϕ has codim 1 1.
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answered Nov 11, 2015 at 19:40
XiaoXiao
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1 Thank you , let me check it , if I have further questions , can I let you know ?user162343 –user162343 2015-11-11 19:43:43 +00:00 Commented Nov 11, 2015 at 19:43
1 @user162343 Sure! Glad to help.Xiao –Xiao 2015-11-11 19:44:43 +00:00 Commented Nov 11, 2015 at 19:44
@Xiao Correct me if I'm wrong, but this proof works for any linear functional ϕ ϕ on a vector space X X, right?zxmkn –zxmkn 2019-01-27 21:00:46 +00:00 Commented Jan 27, 2019 at 21:00
@zxmkn Yes, you are correct Xiao –Xiao 2019-01-28 20:58:35 +00:00 Commented Jan 28, 2019 at 20:58
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4Prove: every subspace of finite codimension is dense in L p L p with p∈(0,1)p∈(0,1)
2Prove that this space of functions is closed in C B(X,R)C B(X,R)
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7034 | https://www.bbc.co.uk/bitesize/articles/zyjtfdm | Introduction to trigonometry for right-angled triangles - KS3 Maths - BBC Bitesize
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Introduction to trigonometry for right-angled triangles
Part of MathsPythagoras and trigonometry
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Key points
Labelling sides of a right-angled triangle for trigonometry
Examples
Question
The three trigonometric ratios
Examples
Question
Practise trigonometric ratios in right-angled triangles
Quiz
Game - Divided Islands
Key points
Image caption,
The sides of right-angled triangles are labelled opposite, hypotenuse and adjacent.
Trigonometry explores the relationship between sides and angles in right-angled triangles.
The sides of a right-angled triangle are labelled with:
Hypotenuse (h) – the longest side, which is opposite the right angle.
Opposite (o) – the side opposite to the given angle.
Adjacent (a) – the side next to the given angle.
The Greek letter Ɵ (theta) is often used as a symbol for an unknown (given) angle.
Similarclosesimilar shapesOne shape is an enlargement of another. The angles in each shape are the same, and the side lengths are in the same proportion. triangles are enlargementscloseenlargementA transformation that changes the size of a shape, keeping the lengths in the same proportions. of each other. The angles in similar triangles are the same. The sides of similar triangles are in the same proportioncloseproportionA comparison between numbers so that when one number increases, the other does at the same rate..
Trigonometric ratios show how long one side of the triangle is compared to another. The 3 important ratios are known as the sine (sin), cosine (cos) and tangent (tan) of the angle:
sinƟ = oppositehypotenuse
cosƟ = adjacenthypotenuse
tanƟ = oppositeadjacent
Image caption,
The sides of right-angled triangles are labelled opposite, hypotenuse and adjacent.
Back to top
Labelling sides of a right-angled triangle for trigonometry
When a question about right-angled triangles involves all three sides and no angles, Pythagoras’ theoremclosePythagoras’ theoremPythagoras’ theorem states the relationship between sides in a right-angled triangle. It states that 𝒂² + 𝒃² = 𝒄², where 𝒄 is the hypotenuse (longest side), and 𝒂 and 𝒃 are the other two sides. is used to calculate the value of a missing side.
When the question involves two sides and an angle in a right-angled triangle, trigonometryclosetrigonometryThe study of sides and angles in triangles. is used.
The sides are labelled depending on where the angle is.
The sides are called the hypotenuseclosehypotenuseThe longest side of a right-angled triangle, which is always opposite the right angle. When labelling a length as the hypotenuse, it can be shortened to 𝒉. (h), the oppositecloseoppositeThe side of a right-angled triangle that is opposite the angle mentioned. When labelling a length as the opposite, it can be shortened to 𝒐. (o) side, and the adjacentcloseadjacentThe side of a right-angled triangle between the right-angle and the angle mentioned. When labelling a length as the adjacent, it can be shortened to 𝒂. (a) side.
The Greek letter theta, which has the symbol Ɵ, is often used to show an unknown angle.
Examples
Image gallerySkip image gallery
Image caption,
When using trigonometry in right-angled triangles, each side is labelled based on a given angle, Ɵ. The hypotenuse (𝒉) is the longest side of the triangle, which is always opposite the right angle.
Image caption,
The opposite side (𝒐) is the side opposite the given angle.
Image caption,
The adjacent side (𝒂) is the side next to the given angle. It is between the given angle and the right angle.
Image caption,
A fully labelled triangle.
Image caption,
Label the right-angled triangle with the hypotenuse, opposite and adjacent side.
Image caption,
The hypotenuse is the longest side and is always opposite the right angle. The opposite side is the side opposite the given angle. The adjacent side is the side next to the given angle. It is between the given angle and the right angle.
1 of 6
Previous imageNext image
Slide 1 of 6, Example one. An image of a right angled triangle. The right angle is the bottom left vertex of the triangle. The bottom right vertex has been labelled theta. A diagonal arrow points from the right angle to the diagonal side opposite. The side has been labelled, hypotenuse. The arrow and the word hypotenuse are coloured orange. The triangle is coloured pink., When using trigonometry in right-angled triangles, each side is labelled based on a given angle, Ɵ. The hypotenuse (𝒉) is the longest side of the triangle, which is always opposite the right angle.
End of image gallery
Question
Label the right-angled triangle with the hypotenuse, opposite and adjacent sides.
Show answerHide answer
The hypotenuse is the longest side and is always opposite the right angle.
The opposite side is the side opposite the given angle.
The adjacent side is the side next to the angle. It is between the given angle and the right angle.
Back to top
The three trigonometric ratios
Triangles that have the same three angles are enlargementscloseenlargementA transformation that changes the size of a shape, keeping the lengths in the same proportions. of each other and are known as similar triangles.
For two right-angled triangles, if an angle (other than the right angle) is the same, the third angle will also be the same. This is true because angles in a triangle add up to 180°.
Therefore, right-angled triangles that include the same angle are similarclosesimilar shapesOne shape is an enlargement of another. The angles in each shape are the same, and the side lengths are in the same proportion..
For similar right-angled triangles, the hypotenuseclosehypotenuseThe longest side of a right-angled triangle, which is always opposite the right angle. When labelling a length as the hypotenuse, it can be shortened to 𝒉., oppositecloseoppositeThe side of a right-angled triangle that is opposite the angle mentioned. When labelling a length as the opposite, it can be shortened to 𝒐. and adjacentcloseadjacentThe side of a right-angled triangle between the right-angle and the angle mentioned. When labelling a length as the adjacent, it can be shortened to 𝒂. sides stay in the same proportionclose. For example, the adjacent side divided by the hypotenuse will give the same answer no matter how much the triangle is enlarged.
The hypotenuse, opposite and adjacent side can be divided in three important ways.
The answer to these division calculations depends on a given angle, Ɵ.
The three divisions are written as fractions. They are called the three trigonometric ratiosclosetrigonometric ratioA ratio written as a fraction that calculates how long one side of a right-angled triangle is compared to another, based on a given angle, Ɵ. The three main ratios are sinƟ = opposite/hypotenuse; cosƟ = adjacent/hypotenuse; and tanƟ = opposite/adjacent. because they show how long one side is compared with another.
The trigonometric ratios are known as the sine (sin), cosine (cos) and tangent (tan) of the angle.
sinƟ = oppositehypotenuse
cosƟ = adjacenthypotenuse
tanƟ = oppositeadjacent
Examples
Image gallerySkip image gallery
Image caption,
The right-angled triangle on the left has sides 3, 4, and 5 units long. The right-angled triangle on the right has sides 6, 8, and 10 units long. The triangles are mathematically similar because one is an enlargement of the other. The lengths of the right triangle are double those on the left triangle
Image caption,
The sides have been labelled as the hypotenuse (𝒉), opposite (𝒐), and adjacent (𝒂) based on the angle Ɵ.
Image caption,
The opposite side divided by the hypotenuse for each triangle is 3/5 and 6/10. These two fractions are equivalent and both equal 0·6. No matter how much the triangle is enlarged, the opposite divided by the hypotenuse will always be 0·6. This is because the angle stays the same and the sides stay in the same proportion.
Image caption,
The answer to the opposite side divided by the hypotenuse depends on how big the angle is. The opposite side divided by the hypotenuse is called the sine of the angle, and is abbreviated to sin. For a given angle Ɵ, sinƟ = opposite ÷ hypotenuse. This is written as a fraction.
Image caption,
The adjacent side divided by the hypotenuse for each triangle is 4/5 or 8/10. These two fractions are equivalent and both equal 0·8. No matter how much the triangle is enlarged, the adjacent divided by the hypotenuse will always be 0·8. This is because the angle stays the same and the sides stay in the same proportion.
Image caption,
The answer to the adjacent side divided by the hypotenuse depends on how big the angle is. The adjacent side divided by the hypotenuse is called the cosine of the angle, and is abbreviated to cos. For a given angle Ɵ, cosƟ = adjacent ÷ hypotenuse. This is written as a fraction.
Image caption,
The opposite divided by the adjacent for each triangle is 3/4 or 6/8. These two fractions are equivalent and both equal 0·75. No matter how much the triangle is enlarged, the opposite divided by the adjacent will always be 0·75. This is because the angle stays the same and the sides stay in the same proportion.
Image caption,
The answer to the opposite side divided by the adjacent depends on how big the angle is. The opposite side divided by the adjacent side is called the tangent of the angle and is abbreviated to tan. For a given angle Ɵ, tanƟ = opposite ÷ adjacent. This is written as a fraction.
Image caption,
These three equations are true for all right-angled triangles. They are known as the three trigonometric ratios because they show how large one side is compared to the other based on an angle Ɵ.
1 of 9
Previous imageNext image
Slide 1 of 9, An image of a square grid. The grid has a length of eighteen squares and a width of twelve squares. Two right angled triangles have been drawn on the grid. The first triangle has a base of length three squares, a height of length four squares and a hypotenuse of length five squares. The right angle is the bottom left vertex of the triangle. The length of each side is labelled. The angle marked theta is between the sides measuring four and five squares. The second triangle has a base of length six squares, a height of length eight squares and a hypotenuse of length ten squares. The right angle is the bottom left vertex of the triangle. The length of each side is labelled. The angle marked theta is between the sides measuring eight and ten squares. The first triangle is coloured green. The second triangle is coloured orange., The right-angled triangle on the left has sides 3, 4, and 5 units long. The right-angled triangle on the right has sides 6, 8, and 10 units long. The triangles are mathematically similar because one is an enlargement of the other. The lengths of the right triangle are double those on the left triangle
End of image gallery
Question
Calculate the value of sinƟ, cosƟ and tanƟ for the right-angled triangle.
Give your answer as a fraction.
Show answerHide answer
Label the sides hypotenuse (h), opposite (o) and adjacent (a).
Write the trigonometric ratios and substitute the correct values into the equations.
sinƟ = 2853
cosƟ = 4553
tanƟ = 2845
Back to top
Practise trigonometric ratios in right-angled triangles
Practise trigonometric ratios in similar right angled triangles with this quiz. You may need a pen and paper to help you with your answers.
Quiz
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7035 | https://publications.ersnet.org/content/erj/45/6/1704 | Dead space: the physiology of wasted ventilation | European Respiratory Society
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Dead space: the physiology of wasted ventilation
H. Thomas Robertson
European Respiratory Journal 2015 45(6): 1704-1716; DOI:
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European Respiratory Journal
Vol 45 Issue 6
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Abstract
An elevated physiological dead space, calculated from measurements of arterial CO 2 and mixed expired CO 2, has proven to be a useful clinical marker of prognosis both for patients with acute respiratory distress syndrome and for patients with severe heart failure. Although a frequently cited explanation for an elevated dead space measurement has been the development of alveolar regions receiving no perfusion, evidence for this mechanism is lacking in both of these disease settings. For the range of physiological abnormalities associated with an increased physiological dead space measurement, increased alveolar ventilation/perfusion ratio (V′A/Q′) heterogeneity has been the most important pathophysiological mechanism. Depending on the disease condition, additional mechanisms that can contribute to an elevated physiological dead space measurement include shunt, a substantial increase in overall V′A/Q′ ratio, diffusion impairment, and ventilation delivered to unperfused alveolar spaces.
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A review of current understanding of factors accounting for abnormal physiological dead space measurements in disease
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Introduction
Recent recognition of the prognostic relevance of measurements of physiological dead space for patients with the acute respiratory distress syndrome (ARDS) has brought new attention to a simple gas exchange calculation described over 120 years ago. The current calculation of physiological dead space, utilising measurements of arterial CO 2 tension (P aCO 2) and mixed expired CO 2 tension (P ECO 2), was initially thought to include an anatomical dead space, representing the fraction of ventilation advancing no further than the conducting airways, and an alveolar dead space, representing the fraction of ventilation delivered to alveolar surfaces receiving no pulmonary artery perfusion. With the subsequent development of a simple measurement of anatomical dead space, it became apparent that the alveolar dead space component was substantially increased in a range of pulmonary diseases, and that its original interpretation as ventilation delivered to unperfused alveolar surfaces was not adequate to explain the pathophysiology of most disease conditions. This review will cover the findings in gas exchange pathophysiology that shape our current understanding of the factors that influence physiological dead space measurements, with a primary focus on dead space measurements acquired in patients with respiratory or cardiac diseases.
Bohr dead space
Because of the tidal nature of ventilation, every exhaled breath contains a fraction of the inspired gas that does not participate in gas exchange. In 1891, the Danish respiratory physiologist Christian Bohr introduced his calculation to represent the volume of gas within the conducting airways that constituted the respiratory dead space . The Bohr model conceptually divided the volume of the exhaled breath (V T) into two compartments, with the first (V A) representing the fraction of exhaled breath participating in gas exchange, and the second (V D) representing the fraction of exhaled breath penetrating no further than the conducting airways, the respiratory dead space:
The Bohr calculation utilised two measurements of exhaled CO 2: the fractional CO 2 concentration in the total mixed exhaled breath (F ECO 2), and an estimate of mean alveolar CO 2 concentration (F ACO 2) based on a sample of gas collected late in exhalation. The mass balance calculation using CO 2 assumes the dead space contributes no CO 2 to the exhaled breath (F DCO 2 = 0):
Substituting (V T−V D) for V A yields:
Rearranging the terms describes V D/V T, the fraction of ventilation not contributing to CO 2 exchange:
The Bohr estimates of the volume of respiratory dead space correlated reasonably well with anatomical measurements of the respiratory pathway, and represented a noninvasive means of measuring the extent of ventilation inefficiency attributable to the constraints of tidal ventilation.
In the decades following the introduction of the Bohr dead space estimate, two factors that interfered with the accuracy of the measurement as an index of conducting airways volume became apparent. The development of a method to sample multiple within-breath samples of CO 2 demonstrated modest variability in partial pressures within an exhaled breath in normal subjects, leading to the recognition that a gas sample captured within a breath would not necessarily reflect the true mean alveolar gas composition. While that effect was minor in normal subjects, it became a substantial source of variability in patients with significant underlying lung disease. Furthermore, Bohr calculations performed on such patients revealed increases in dead space fraction that could not be ascribed to conducting airway volume. While the latter finding was an insight that drove subsequent investigations of gas exchange pathophysiology, problems with identifying a simple method to identify mean alveolar CO 2 tension (P ACO 2) in patients with lung disease led to the later abandonment of the traditional Bohr measurement.
Current airway CO 2 monitoring systems for intubated patients provide accurate measurements of end-tidal CO 2 concentrations that are acceptable estimates of alveolar CO 2 in normal subjects, and reflect at least directional changes in alveolar CO 2 in patients with underlying lung disease, but cannot provide an accurate measurement of Bohr dead space in a diseased lung because the end-tidal CO 2 does not reflect the mean alveolar CO 2. One current application of the traditional Bohr measurement that substitutes end-tidal CO 2 measurements for mean alveolar CO 2 is seen with commercial exercise testing systems that calculate a Bohr dead space during exercise. At rest, CO 2 concentrations measured within a single exhalation increase modestly, a consequence of a constant mixed venous CO 2 delivery and the intermittent inspiration of fresh gas into the alveoli. However, during heavy exercise, when mixed venous CO 2 tension (P CO 2) may exceed 60 mmHg, exhaled CO 2 concentrations increase substantially within the course of each exhaled breath. Hence during heavy exercise, although mean alveolar CO 2 concentrations within an exhaled breath still approximate arterial values, the end-tidal values are substantially higher than the mean alveolar values. J ones et al. documented end-tidal P CO 2 measurements exceeding P aCO 2 measurements by 4–6 mmHg during heavy exercise. Hence during heavy exercise the Bohr estimate of dead space utilising measurements of end-tidal CO 2 yields an inappropriately high value because the end-tidal CO 2 consistently exceeds the mean alveolar (and arterial) P CO 2. A recent method to identify an accurate mean alveolar CO 2 from an expiratory capnogram has been validated in experimental animals, and could be incorporated in commercial ventilator systems to calculate a Bohr dead space . However for critically ill patients, that calculation substantially underestimates the more familiar Enghoff modification of the Bohr dead space, which substitutes the P aCO 2 measurement for the mean P ACO 2 measurement . (See the physiological dead space section for mechanisms accounting for increased arterial–alveolar P CO 2 differences in abnormal lungs.)
Fowler dead space
The final fraction of each inspired breath travels no further than the conducting airways, and that anatomical dead space volume includes the upper airways, the larynx and the volume of the tracheo-bronchial tree extending to the acini. In the late 1940’s, with the development of a rapidly responding nitrogen meter, Ward Fowler assembled apparatus to measure that anatomical dead space utilising measurements of exhaled nitrogen concentrations immediately following the inspiration of a breath of 100% oxygen . Fowler’s apparatus plotted the volume of the exhaled breath following a single inspiration of 100% oxygen against the exhaled nitrogen concentration, yielding the curve illustrated in figure 1. The initial fraction of the exhaled breath, termed phase I, contains no nitrogen. The following segment of the exhaled volume, termed phase II, is characterised by progressively increasing concentrations of nitrogen prior to reaching the plateau that represents the nitrogen concentration in alveolar gas, termed phase III. The rising slope of phase II reflects the length differences among the large airway pathways within the lung, with shorter length conducting airways contributing alveolar gas concentrations in phase II, and longer regions still contributing 100% oxygen from dead space gas. Fowler’s graphical method to account for the dead space contribution from phase II was to place a straight line along phase III and adjust a vertical line within phase II so that the two areas labelled A and B in figure 1 represent equal areas. The exhaled volume identified by the intersection of that vertical line with the volume axis on the abscissa identifies the dead space, a measurement he termed the physiological dead space, but is now described as the anatomical dead space or Fowler dead space .
FIGURE 1
Fowler’s illustration of the measurement of anatomical dead space, plotting exhaled concentration of nitrogen (C) following an inspired breath of 100% oxygen against exhaled volume (V exp), where C insp represents the inspired nitrogen concentration (0%) and C alv represents the alveolar concentration of nitrogen. The vertical dashed line is positioned so that a and b subsume equal areas, and the intersection of the dashed line with the exhaled volume axis defines V ds, the anatomical dead space volume. V alv: alveolar volume. The original notation used by Fowler has been retained. Reproduced from with permission from the publisher.
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The Fowler dead space measurement is dependent on the subject size. An approximation suggested by Fowler based on measurements in 45 subjects was that anatomical dead space in cubic centimetres roughly equalled a subject’s ideal body weight in pounds . While that anatomical dead space is usually considered a fixed quantity, conducting airway diameter is dependent on lung volume, and when Fowler compared measurements made at different end-inspiratory lung volumes he noted an average 100 cm 3 difference in dead space measurements between the largest and smallest starting volumes. In addition, Fowler demonstrated that the measured dead space would decrease if a 20-s breath-hold preceded the exhalation. He postulated this effect primarily arose because of gas diffusion from alveolar regions back into the conducting airways during the breath hold. While a subsequent study did not show significant differences among Fowler measurements carried out with both nitrogen and helium , an influence of gas diffusivity was later documented as part of a study utilising inspirations of helium (atomic weight 4) and SF 6 (molecular weight 146), demonstrating a smaller Fowler dead space with helium compared with SF 6 . Gas mixing between small airways and alveolar spaces also takes place as a result of cardiogenic movement, and this effect could also explain the progressive decrease in Fowler dead space measurements observed after a sustained breath hold. The importance of cardiogenic motion in Fowler dead space measurements was documented in an animal study utilising dead space measurements made before and after cardiac arrest, with a 20% larger dead space measurement obtained after cardiac arrest . A subsequent study established those cardiogenic oscillations as more important than molecular diffusion in reducing the dead space measured by the Fowler method . Hence cardiogenic lung motion and a more modest contribution from gas diffusion combine to decrease the measured Fowler dead space compared with the actual volume of the respiratory pathway between the lips and the respiratory bronchioles.
Physiological dead space
In 1938, in recognition of the problem with obtaining an appropriate estimate of mean P ACO 2 for calculations using the Bohr equation, Enghoff proposed the substitution of P aCO 2 for the mean P ACO 2 estimate in the Bohr calculation :
Where P ECO 2 is the P CO 2 in the total mixed exhaled breath. It is the Enghoff modification of the Bohr dead space calculation that is in general use today, and is described as the physiological dead space, where that term represents the sum of the anatomical dead space and the alveolar dead space, where in current usage the alveolar dead space component is defined by the difference between the physiological dead space and the anatomical dead space . The substitution of P aCO 2 for the P ACO 2 avoided the difficulty of identifying an appropriate mean P ACO 2. However this modification of the Bohr equation added a new component to the dead space estimate that is particularly relevant when the equation is utilised to describe gas exchange in disease. The mean P ACO 2 is always less than mean P aCO 2, although this difference is quite small in normal lungs. However, just as any gas exchange abnormality will increase the alveolar–arterial O 2 difference, the same statement holds for the arterial–alveolar CO 2 difference. Hence the physiological dead space, incorporating the P aCO 2, will always be greater than the classic Bohr dead space, and the physiological factors determining the arterial–alveolar CO 2 difference can be anything that influences the efficiency of gas exchange. Specifically, in contrast to the original Bohr dead space calculation, the physiological dead space calculation will be sensitive to intrapulmonary shunt and diffusion impairment, and will have a greater sensitivity to alveolar ventilation/perfusion ratio (V′A/Q′) heterogeneity. As the Enghoff modification of the Bohr dead space measurement is sensitive to a range of physiological gas exchange abnormalities, it provides a convenient index of overall gas exchange impairment, but it is important to understand that elevated measurements can arise from multiple mechanisms, dependent on the specific pathophysiology producing the gas exchange impairment.
Shunt contribution to physiological dead space
Shunt increases not only the alveolar–arterial O 2 difference but also the arterial–alveolar CO 2 difference and, therefore, increases calculated physiological dead space. Table 1 presents a simple two-compartment alveolar model without anatomical dead space, containing a shunt compartment and a normal alveolar unit. The alveolar space of the shunt compartment is gas that is in equilibrium with the mixed venous gases and does not contribute to the exhaled gas mixture. One third of the total perfusion of 4.5 L is delivered to the shunt unit and all of the ventilation is delivered to the normal lung unit, yielding a V′A/Q′ of 0 for the shunt unit, a V′A/Q′ of 1.5 for the normal unit, and an overall V′A/Q′ for the two units combined of 1.0. Assuming a mixed venous P CO 2 of 46 mmHg, and sufficient alveolar ventilation to reduce the mixed arterial CO 2 to 40 mmHg, the lower alveolar CO 2 in the single ventilated compartment produces a 7.5% contribution to that calculated alveolar dead space (V D/V T−alv), even though the model does not contain an unperfused alveolar unit. While a large shunt is required to achieve this result, note that this shunt-mediated contribution to the alveolar dead space calculation would be increased by either a decrease in cardiac output or increase in metabolic rate, where either change would increase the mixed venous P CO 2.
TABLE 1
A simple two-compartment alveolar model containing a shunt compartment and a normal alveolar unit
Compartment 1 (shunt)Compartment 2 (ventilated)Combined output
V′A0.0 L·min−1 4.5 L·min−1 4.5 L·min−1
Q′1.5 L·min−1 3.0 L·min−1 4.5 L·min−1
V′A/Q′0.0 L·L−1 1.5 L·L−1 1.0 L·L−1
P aCO 246 mmHg 37 mmHg 40 mmHg
P ACO 246 mmHg 37 mmHg 37 mmHg
Alveolar V D/V T7.5%
V′A: alveolar ventilation; Q′: perfusion; P aCO 2: arterial CO 2 tension; P ACO 2: alveolar CO 2 tension; V D/V T: alveolar dead space.
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V′A/Q′ heterogeneity contribution to physiological dead space
While regional ventilation/perfusion heterogeneity does contribute to the classic Bohr dead space, the physiological dead space is more heavily influenced by V′A/Q′ heterogeneity. Table 2 provides a simple three-compartment alveolar model of V′A/Q′ heterogeneity, again without anatomical dead space, with V′A/Q′ values of 0.1, 1.0 and 10 in the three compartments to illustrate the effect. As with the example in table 1, the mixed venous CO 2 is assumed to be 46 mmHg, and the overall alveolar ventilation is adjusted so that the mixed P aCO 2 will be 40 mmHg. For the three compartments, the overall summed ventilation is 4.2 L and overall summed perfusion is 4.2 L, with an overall V′A/Q′ of 1.0. Combining the CO 2 partial pressures in the ventilated and perfused components of the three units yields a calculated physiological dead space of 26% in this simple V′A/Q′ heterogeneity model that includes no unperfused alveolar spaces and no anatomical dead space. While the influence of V′A/Q′ heterogeneity on the alveolar–arterial O 2 difference is well known, its influence on CO 2 exchange receives little clinical attention, as most patients can simply increase minute ventilation (V′E) as CO 2 exchange becomes less efficient, thereby concealing the impaired exchange of CO 2. Unless a patient requires ventilator support, the requirement for relatively increased V′E is never noticed. John West employed a more sophisticated computer model of V′A/Q′ heterogeneity to illustrate how disease-relevant degrees of V′A/Q′ heterogeneity would increase physiological dead space, requiring compensatory increases in V′E to maintain normal P aCO 2 values .
TABLE 2
A simple three-compartment alveolar model of V′A/Q′ heterogeneity
Compartment 1 (low V′A/Q′)Compartment 2 (normal V′A/Q′)Compartment 3 (high V′A/Q′)Combined output
V′A0.2 L·min−1 2.0 L·min−1 2.0 L·min−1 4.2 L·min−1
Q′2.0 L·min−1 2.0 L·min−1 0.2 L·min−1 4.2 L·min−1
V′A/Q′0.1 L·L−1 1 L·L−1 10 L·L−1 1 L·L−1
P aCO 242 mmHg 40 mmHg 20 mmHg 40 mmHg
P ACO 242 mmHg 40 mmHg 20 mmHg 29.6 mmHg
Alveolar V D/V T26%
V′A: alveolar ventilation; Q′: perfusion; P aCO 2: arterial CO 2 tension; P ACO 2: alveolar CO 2 tension; V D/V T: alveolar dead space.
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Physiological dead space in an abnormal lung
In contrast to the simple compartmental models of tables 1 and 2, abnormal lungs ordinarily have combinations of factors that contribute to the physiological dead space measurement. Figure 2 illustrates a distribution of V′A/Q′ units typical for an abnormal lung that includes increased ventilation/perfusion heterogeneity, shunt and anatomical dead space. Note that a normal lung would have the same bell-shaped distribution of V′A/Q′ units centred on the overall mean V′A/Q′ of 1.0, but the lung units would be contained within a V′A/Q′ range between 0.5 and 2.0. To quantitatively characterise the relative roles of combinations of physiological abnormalities on the physiological dead space measurement requires some familiarity with features of the multiple inert gas elimination technique (MIGET).
FIGURE 2
Allocation of ventilation and blood flow in an abnormal lung that includes shunt, increased alveolar ventilation/perfusion ratio (V′A/Q′) heterogeneity and increased anatomical dead space. The lung has an overall V′A/Q′ of 1.0 and has the component lung units sorted according to their individual V′A/Q′ ratios. The broad base of the bell-shaped curve reflects substantial overall V′A/Q′ heterogeneity. The bar on the left represents the frequency of lung units compromising shunt, and the bar on the right represents lung units receiving ventilation but no pulmonary artery blood flow. Figure reproduced courtesy of R.W. Glenny (Division of Pulmonary and Critical Care Medicine, University of Washington, Seattle, WA, USA).
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To employ MIGET, partial pressures of six intravenously infused inert gases are measured in arterial and mixed venous blood and mixed expired gas to provide the data required for the mathematical model that describes the distribution of ventilation and perfusion in the lung [14, 15]. The infusion technique with its associated model provides a quantitative estimate of the allocation of pulmonary blood flow to shunt, to regions with V′A/Q′ ratios ranging between .001 and 100, and ventilation to inert gas dead space, a parameter that correlates well with anatomical dead space . While this technique provides a unique means of fully characterising gas exchange abnormalities in humans, useful insights into the physiological dead space calculation in disease can be obtained from examining the basic inert gas data itself [17, 18].
In a publication that provided the insights crucial to the subsequent development of MIGET, F arhi demonstrated that the alveolar or arterial partial pressure of an intravenously infused inert gas could be predicted based on the solubility of the gas in blood and the V′A/Q′ ratio of the lung unit. For any given V′A/Q′ value, the predicted arterial (or alveolar) partial pressures of infused inert gases covering a wide range of solubility in blood form a sigmoid curve when plotted against the log of gas solubilities. The single solid line in figure 3 illustrates this retention–solubility diagram for a homogenous lung. If lung units including a distribution of different V′A/Q′ ratios are combined, the arterial and alveolar lines on the diagram diverge (dashed lines in fig. 3). The vertical line on figure 3 labelled λ G represents the solubility appropriate for CO 2, ranging between 2 and 4 mL of gas per mL of blood, depending on the influence of the Haldane effect . The intersection of the λ G line with the arterial and alveolar curves identifies the two partial pressures needed to make an alveolar dead space calculation for the gas with the solubility λ G:
FIGURE 3
Arterial (P a) (retention) and alveolar (P A) (excretion) partial pressures for intravenously infused inert gases spanning a very large range of solubility in blood. The single solid curved line represents the arterial and alveolar curves of a perfectly homogenous lung (P aHOMO = P AHOMO), and the two dashed lines represent the influence of ventilation/perfusion heterogeneity that creates an arterial–alveolar partial pressure difference for both respiratory and inert gases. P v: mixed venous partial pressure; λ G: represents the solubility appropriate for CO 2. #: solubility is expressed as mL of gas per mL of blood at 1 Atm. Reproduced from with permission from the publisher.
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Applying this graphical inert gas analysis to lung models representing different combinations of abnormal lung physiology, H lastala and R obertson examined the influence of different degrees of V′A/Q′ heterogeneity, shunt and anatomical dead space to illustrate the influence of these abnormalities on a physiological dead space calculation, examined over a wide range of inert (and respiratory) gas solubility . Figure 4 illustrates the physiological dead space calculation made over the range of different gas solubilities in three different abnormal lungs, all containing 20% shunt and 20% anatomical dead space, in combination with no V′A/Q′ heterogeneity (fig. 4a), normal V′A/Q′ heterogeneity (fig. 4b) and increased V′A/Q′ heterogeneity (fig. 4c). Note that the physiological dead space in the CO 2 solubility range, ∼3 mL of gas per mL of blood, is most sensitive to the extent of V′A/Q′ heterogeneity, with a secondary sensitivity to shunt. In the recent context of investigating the mechanisms responsible for dead space abnormalities in ARDS, W agner compared the influence of different degrees of shunt and V′A/Q′ heterogeneity on physiological dead space, and emphasised the relatively greater influence of V′A/Q′ heterogeneity on the measurement. A final insight gained from the application of inert gas retention–solubility curves to the understanding of physiological dead space in disease is that any increase the overall V′A/Q′ ratio (a change that would shift the bell-shaped distribution in fig. 2 to the right on the V′A/Q′ axis) will also shift all of the curves on the retention–solubility diagram to the right [17, 18]. Hence a five-fold increase in overall V′A/Q′ ratio will shift all of the figure 4 curves five units to the right along the solubility axis. As gas solubility in blood is fixed, any increase in the mean V′A/Q′ value by increased ventilation and/or decreased perfusion will also increase the calculated physiological dead space. Of note, the influence of a substantial increase in the mean V′A/Q′ ratio on the physiological dead space measurement was first described by John West, based on calculations utilising his initial computer model of ventilation/perfusion interactions in the lung .
FIGURE 4
Plots based on inert gas retention and excretion values for three model lungs that have 20% shunt and 20% anatomical dead space, illustrating the influence of a) no alveolar ventilation/perfusion ratio (V′A/Q′) heterogeneity, b) normal V′A/Q′ heterogeneity, and c) a high extent of V′A/Q′ heterogeneity. The dashed lines identify the physiological dead space calculation for the entire range of inert gas solubility. The solid lines represent the inert gas arterial–alveolar differences and the dotted lines represent the venous admixture calculation for the inert gases. #: solubility is expressed as mL of gas per mL of blood at 1 Atm. Reproduced from with permission from the publisher.
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Physiological dead space during exercise
The initial studies of gas exchange during a progressive work exercise test by W asserman et al. demonstrated a progressive decrease in physiological dead space from rest to maximal exercise, with a normal response identified as a V D/V T of <20% at maximal effort. This decrease is readily explained by the progressive increase in tidal volume during the first half of a progressive work exercise test to ∼60% of the vital capacity, associated with an anatomical dead space that increases only trivially with the increase in end-inspiratory volume induced by exercise. The decrease in dead space during exercise was originally postulated to also include improved ventilation/perfusion matching, but measurements of ventilation/perfusion heterogeneity during exercise in normal humans assessed by MIGET have not demonstrated a decrease in overall ventilation/perfusion heterogeneity at low and moderate levels of exercise , the period in a progressive work exercise test when the V D/V T decreases most appreciably . At the highest tolerated levels of exercise, V′A/Q′ heterogeneity increases modestly , and is associated with the well-known increase in the alveolar–arterial O 2 difference. From these MIGET observations, it might be expected that physiological V D/V T should increase slightly with maximal effort, but increases at maximal effort are inconsistent and small in comparison with the overall decrease in V D/V T seen early during a progressive exercise work test.
Can “true” alveolar dead space also exchange CO 2?
The original interpretation of alveolar dead space was as a compartment in alveolar spaces that received no pulmonary artery perfusion, although the current definition of the term alveolar dead space refers to anything in the physiological dead space that does not include anatomical dead space . While ventilated alveolar regions without pulmonary artery perfusion (here defined as “true” alveolar dead space) develop following acute pulmonary emboli and should lead to abnormal dead space measurements, physiological dead space as a diagnostic test for pulmonary emboli was found to lack sensitivity for all but the most massive occlusions. Some of that lack of sensitivity may be secondary to gas exchange provided by the bronchial circulation. Alveolar regions of lung with acute loss of pulmonary artery perfusion can receive capillary perfusion in variable amounts from connections to the bronchial circulation and drain into the mixed pulmonary venous output to the left atrium . With sustained occlusion, this bronchial artery perfusion of alveolar capillaries can be appreciably augmented . Although the systemic bronchial artery blood perfusing those alveolar capillaries will not be further oxygenated during that second pass through the lung, it will exchange additional CO 2. Hence for segments of embolised lung that only receive blood from the bronchial arteries, those units function as high V′A/Q′ units from the perspective of CO 2 gas exchange, and physiological dead space measurements will underrepresent the true extent of pulmonary artery occlusion. In the setting of severe bronchiectasis, the elimination of CO 2 via the bronchial circulation can provide a clinically significant contribution to overall CO 2 exchange, a situation occasionally recognised following the embolisation of bronchial arteries for acute haemoptysis in patients with severe bronchiectasis .
Interaction between anatomical dead space and V′A/Q′ heterogeneity
The presence of both anatomical dead space and instrument dead space means that the first portion of each breath is re-inspired alveolar gas. In the presence of substantial V′A/Q′ heterogeneity, the re-inspired alveolar gas might be mixed and, therefore, the re-inspired dead space could alter the measured heterogeneity among the different units of an abnormal lung. The effect of gas re-inspired from a well-mixed “common” dead space in a lung with overall V′A/Q′ heterogeneity was first analysed utilising the perspective of the O 2–CO 2 diagram by R oss and F arhi . Based on the assumption that the re-inspired dead space gas was well mixed, they demonstrated that this influence would reduce the measured extent of overall V′A/Q′ heterogeneity and hence reduce the measured alveolar dead space contribution to the physiological dead space. A second possibility is that the exhaled dead space gas remains “personal”, that is, each V′A/Q′ unit would re-inspire its own exhaled gas, an effect that would have no influence on measured V′A/Q′ heterogeneity. The “common” and “personal” hypotheses for the distribution of re-inspired gas were modelled by F ortune and W agner in normal lungs. They demonstrated that for normal lungs, the assumption that re-inspired dead space gas was “personal” was an adequate description. However when successive increments of external instrument dead space were added in an animal study, progressively increasing the proportion of re-inspired gas, P etrini et al. demonstrated that there was an appreciable reduction in the extent of externally measured V′A/Q′ heterogeneity. Hence with the addition of larger amounts of instrument dead space, the calculated alveolar dead space component of the physiological dead space will decrease without changing the actual V′A/Q′ distribution within the lung.
Diffusion impairment and physiological dead space
Carbon dioxide, as a relatively soluble gas in tissue, is not ordinarily considered subject to diffusion impairment during pulmonary gas exchange. However, the delivery of CO 2 to the alveolar spaces from pulmonary capillary blood requires the catalysis of carbonic anhydrase within the erythrocyte, as the principal transport of metabolically produced CO 2 to the lung is in the form of bicarbonate rather than dissolved CO 2 . Since that chemical reaction requires time, a very fast capillary transit time could possibly lead to a transpulmonary partial pressure profile identical to that seen with a molecular diffusion impairment. Model studies have suggested that P CO 2 in alveolar gas might be slightly lower that the systemically sampled P aCO 2 if there was decreased alveolar capillary residence time for this reaction . In addition, any drug that impairs the action of erythrocyte carbonic anhydrase can increase the P aCO 2 to P ACO 2 difference, a reversible drug effect previously documented in burn patients treated with the old topical sulfonamides with substantial carbonic anhydrase inhibitory effects . Acetazolamide, a weak diuretic currently used to prevent symptoms in high altitude sojourners, acts by inhibition of renal carbonic anhydrase, and also influences erythrocyte carbonic anhydrase. While inhibition could produce a mild arterial–alveolar CO 2 gradient in the lung, at recommended clinical doses a measurable influence would probably only be apparent during heavy exercise .
Dead space measurements in ARDS
Clinical interest in the physiological dead space measurement was reawakened by the publication by N uckton et al. in 2002, linking dead space measurements to prognosis in the ARDS. Physiological dead space was measured in 179 mechanically ventilated ARDS patients on the day of the syndrome onset. The mean dead space fraction was 0.54 in eventual survivors and 0.63 in patients who succumbed to the syndrome, and the risk of death increased with every 0.05 increment in dead space. The physiological dead space measurement outperformed all of the previous prognostic measures including traditional measures of oxygenation impairment, lung compliance and illness severity. The authors postulated that the abnormal measurement was due to regions with blocked microcirculation that remained ventilated, in short, that the abnormality was secondary to the creation of “true” alveolar dead space. C epovka et al. performed a subsequent study of ARDS patients ventilated with 6 cm 3·kg−1 tidal volumes, demonstrating a nearly identical predictive power of physiological dead space measurements, but no significant correlation between pulmonary artery pressures and the dead space measurements. Further confirmation of the prognostic value of physiological dead space measurement in both acute lung injury (ALI) and ARDS was presented by S iddiki et al. utilising previously collected ARDS network data from 1896 patients. While arterial blood gas and ventilator measurements were available, the mixed expired CO 2 values were not measured, and hence mixed expired CO 2 had to be calculated based on predicted CO 2 production rates appropriate for acutely ill mechanically ventilated patients. Based on the arterial CO 2 measurements and predicted mixed expired CO 2 concentrations, S iddiki et al. found that at both day 1 and day 3 of ARDS diagnosis, patients with a dead space fraction in excess of 0.50 had a risk for death that increased with every additional 0.10 increment in dead space fraction, a risk prediction that almost exactly equalled the predictive power of the complete CO 2 measurements on ARDS patients described by N uckton et al. and C epovka et al. .
Factors contributing to the elevated V D/V T in ARDS
By contrast to the postulated development of unperfused regions of lung in ARDS to explain the elevated physiological dead space, both increased V′A/Q′ heterogeneity and shunt are the more likely contributors to that observation [21, 35]. However, because the gas exchange characteristics of ARDS are multifactorial, it is instructive to review previous studies of ALI performed using MIGET, which provides quantitative measurement of the influence of shunt, V′A/Q′ heterogeneity and dead space. C offey et al. utilised an oleic acid model of ALI to study the influence of positive end-expiratory pressure (PEEP) on dead space calculations, including MIGET measurements to fully characterise the gas exchange responses. After the injury stabilised in the anaesthetised animals, gas exchange measurements were acquired at randomly allocated levels of PEEP ranging between 0 and 20 cmH 2 O. Physiological dead space measurements were made at each level of PEEP, and the contributions of the components of shunt, V′A/Q′ heterogeneity, anatomical dead space, and alveolar dead space (defined as V′A/Q′ >100) were noted. In this animal model, progressive addition of PEEP improved both shunt and V′A/Q′ heterogeneity, with an initial reduction in the physiological dead space. However, higher levels of PEEP increased the Fowler dead space, accounting for increased ventilation to regions with V′A/Q′ >100, and modestly increased the physiological dead space. In this animal model with randomly allocated PEEP levels, the high V′A/Q′ regions resolved when PEEP levels were lowered, so the development of very high V′A/Q′ regions with high PEEP was the result of PEEP-created increases in Fowler dead space and zone 3 regions of lung rather than ventilated lung regions with microvascular blockade that had been postulated in the human ARDS study of N uckton et al. .
Three studies utilising MIGET to investigate gas exchange in ARDS patients, all performed in the era when application of high tidal volumes (10–15 cm 3·kg−1) was the clinical standard, described various combinations of shunt, very low V′A/Q′ regions and increased overall V′A/Q′ heterogeneity, but with an infrequent incidence of isolated high V′A/Q′ regions among all patients studied [37–39]. The study of R alph et al. included data obtained during progressive application of PEEP and did not observe the development of high V′A/Q′ units at the higher levels of PEEP that had been noted in the dog model of ALI. A current era MIGET study of ARDS patients by F eihl et al. comparing the gas exchange responses to ventilation with tidal volumes of 10 cm 3·kg−1 or 6 cm 3·kg−1 showed minimal evidence of high V′A/Q′ regions at 10 cm 3·kg−1 tidal volumes. With the 6 cm 3·kg−1 tidal volumes, the shunt fraction increased, but evidence for high V′A/Q′ regions had essentially vanished, as evidenced by the very small retention–excretion differences for the most soluble gases. The patient population in the F eihl et al. study had an average physiological V D/V T of 65% at both 10 cm 3·kg−1 and 6 cm 3·kg−1 tidal volumes, and while the lower tidal volume did have a proportionately larger anatomical dead space, the unchanged V D/V T observed at 6 cm 3·kg−1 was explained by the higher cardiac output and lower mixed venous P CO 2 at that lower tidal volume. This ARDS patient population, with no demonstrable high V′A/Q′ component to their gas exchange when receiving 6 cm 3·kg−1 tidal volumes, had a higher overall V D/V T than the mean values from patients in the N uckton et al. and C epovka et al. studies. Hence, MIGET studies of patients with ARDS have not revealed any consistent elevation in unperfused alveolar dead space, and indeed the MIGET study performed utilising the current 6 cm 3·kg−1 tidal volumes demonstrated a striking absence of any high V′A/Q′ component, despite severe elevations in the physiological V D/V T measurement.
A final potential influence on the physiological V D/V T in the most severely ill ARDS patients relates to the overall V′A/Q′ ratio. For patients with a very low cardiac output who are receiving very high V′E, the overall V′A/Q′ ratio may exceed 5 L·L−1. An increase in the mean V′A/Q′ ratio will shift all the curves on the retention–solubility diagram to the right, increasing the physiological dead space measurement, just as a decrease in the mean V′A/Q′ ratio would have the opposite effect. Hence while physiological dead space abnormalities in ARDS patients are a compelling indicator of prognosis, the interpretation that abnormal values primarily represent the creation of lung parenchyma that is ventilated but not perfused is not consistent with the human data obtained utilising MIGET. The physiological dead space abnormalities in ARDS patients supported with the currently utilised 6 cm 3·kg−1 tidal volumes arise primarily as a consequence of the presence of both shunt and increased low and mid-range V′A/Q′ heterogeneity .
Abnormal exercise dead space measurements in heart failure
Exercise studies done on patients with stable severe heart failure have demonstrated an abnormally elevated exercise ventilation response that is present at all levels of exertion. The ventilatory equivalent for CO 2 (V′E/carbon dioxide production (V′CO 2)), calculated as the slope of the two measurements acquired throughout a maximal progressive exercise test, has become a standard tool for evaluation of patients with severe heart failure . The finding of a V′E/V′CO 2>34 L·L−1 is associated with adverse cardiac outcomes, and the risk increases progressively for patients with additional increases in that measurement . The elevated V′E/V′CO 2 in a heart failure patient is primarily attributable to alveolar hyperventilation, as peripheral chemoreceptor hyperactivity, demonstrated by an enhanced ventilation response to hypoxia, is a consistent feature in heart failure patients with significant impairment . However, that augmented exercise ventilation response in the most impaired patients is also associated with an elevated physiological dead space during exercise [44–46]. The hypothesis that this elevated exercise dead space reflected regions of very high V′A/Q′ in the lung was not consistent with the single MIGET study of heart failure patients at rest that revealed no increases in high V′A/Q′ regions or increases in inert gas dead space . While there are currently no MIGET studies of severe heart failure patients during exercise, an alternative explanation for the increased dead space arises as a consequence of the exercise hyperventilation and impaired cardiac output. The consistent feature of the exercise response in severe heart failure patients is the exceptionally high ventilation relative to cardiac output during a progressive work exercise test. The most impaired patients can barely double their cardiac output from rest to maximal effort . That disproportionate increase in exercise ventilation associated with an impaired cardiac output can yield overall V′A/Q′ ratios that may approach 10 L·L−1 during exercise in the most impaired patients. That increase in overall V′A/Q′ ratio will shift the retention and excretion curves in figure 3 to the right, and therefore also shift all the derived curves in figure 4 to the right. As the solubility of CO 2 remains unchanged, the exercise associated V′A/Q′ shift leads to an increased physiological dead space measurement for CO 2. Hence a consistent explanation for the elevated dead space measurements seen during exercise in the most impaired heart failure patients is the abnormally elevated overall V′A/Q′ ratio and its interaction with V′A/Q′ heterogeneity [49, 50].
Abnormal exercise dead space measurements in pulmonary hypertension
While normal subjects ordinarily demonstrate at least a 50% reduction in physiological dead space during heavy exercise compared with their resting measurement, patients with pulmonary hypertension typically fail to show any reduction in dead space during exercise, despite demonstrating typical increases in tidal volume as the exercise intensity increases. The original explanation proposed for this observation was that exercise produced more high V′A/Q′ regions in the lungs of those patients. A MIGET study of resting patients with idiopathic pulmonary hypertension (IPH) or chronic thromboembolic pulmonary hypertension (CTEPH) by D antzker et al. revealed a modest increase in both V′A/Q′ heterogeneity and shunt, although resting physiological dead space still remained within normal limits. A follow-up study by D antzker et al. , which included MIGET measurements taken during exercise on seven pulmonary hypertension patients, found that the extent of overall V′A/Q′ heterogeneity during exercise remained unchanged. The exercise study patients had a mean resting physiological V D/V T of 42% that increased to 44% with maximal effort. The patients showed the characteristic haemodynamic and ventilatory responses to exercise seen in patients with severe pulmonary hypertension, with marked exercise hyperventilation and only modest increases in cardiac output leading to exercise mean V′A/Q′ ratios in the range of 3–12. As the extent of V′A/Q′ heterogeneity and shunt fraction remained unchanged in the transition from rest to exercise, this study represents the best documented instance of the influence of an exercise-elevated mean V′A/Q′ ratio on the physiological dead space measurement. While the increase in mean V′A/Q′ with exercise appears to be an important determinant of the abnormal physiological dead space measurement in pulmonary hypertension patients, their underlying V′A/Q′ heterogeneity and shunt are still important contributors to the exercise measurement abnormality, as normal subjects at maximal exercise will show at least two-fold increases in mean V′A/Q′, and still manifest the normal decrease in V D/V T with progressive exercise.
A recent study by Z hai et al. comparing exercise physiological dead space measurements in patients with IPH and CTEPH found higher V D/V T in the CTEPH patients, despite comparable resting haemodynamic measurements in the two groups. In addition, unlike the IPH patients, the functional status of the CTEPH patients did not correlate with the extent of physiological dead space abnormality, and the authors suggested that the balance of factors determining the physiological dead space in CTEPH during exercise might be different from that in IPH. D antzker et al. did not identify an increased inert gas dead space in the three CTEPH patients included in their study, but as the bronchial artery flow increases dramatically to chronically embolised regions of lung , those chronically embolised regions will function as high V′A/Q′ units, thereby increasing the V′A/Q′ heterogeneity contribution to the physiological dead space calculation. Another explanation for the higher exercise V D/V T observed in the CTEPH patients could be a relatively higher mean exercise V′A/Q′ in comparison with the IPH patients .
Summary and conclusion
The physiological dead space is defined as including anatomical dead space and alveolar dead space components. In normal subjects, the measurement is primarily determined by the contribution of the anatomical dead space, with a small addition from the alveolar dead space attributable to normal ventilation/perfusion heterogeneity. However, in a wide range of pulmonary disease conditions the alveolar dead space component becomes more important, and the original concept that it reflects the influence of regions of lung parenchyma receiving no pulmonary artery perfusion is never an adequate explanation. Any of the physiological mechanisms contributing to an increased arterial–alveolar CO 2 difference will increase the measured physiological dead space, but ordinarily the extent of overall V′A/Q′ heterogeneity is the most important contributor. In addition, in the presence of increased V′A/Q′ heterogeneity, any stimulus increasing the overall mean V′A/Q′, such as extreme hyperventilation with exercise, will further increase the physiological dead space measurement.
References
↵Bohr C. [Ueber die Lungenathmung]. Skand Arch Physiol 1891; 2: 236–268.Google ScholarCrossRef
↵Krogh A, Lindhard J. The volume of the dead space in breathing and the mixing of gases in the lungs of man. J Physiol 1917; 51: 59–90.Google ScholarCrossRefPubMed
↵Jones NL, Robertson DG, Kane JW. Difference between end-tidal and arterial PCO2 in exercise. J Appl Physiol Respir Environ Exerc Physiol 1979; 47: 954–960.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Suarez-Sipmann F, Santos A, Böhm SH, et al. Corrections of Enghoff’s dead space formula for shunt effects still overestimate Bohr’s dead space. Respir Physiol Neurobiol 2013; 189: 99–105.Google ScholarCrossRefPubMed
↵Fowler WS. Lung function studies; the respiratory dead space. Am J Physiol 1948; 154: 405–416.Google ScholarWeb of Science
↵Klocke RA. Dead space: simplicity to complexity. J Appl Physiol 2006; 100: 1–2.Google ScholarCrossRefAbstract/Full TextPubMedWeb of Science
↵Bartels J, Severinghaus JW, Forster RE, et al. The respiratory dead space measured by single breath analysis of oxygen, carbon dioxide, nitrogen or helium. J Clin Invest 1954; 33: 41–48.Google ScholarCrossRefPubMedWeb of Science
↵Crawford AB, Makowska M, Paiva M, et al. Convection- and diffusion-dependent ventilation maldistribution in normal subjects. J Appl Physiol 1985; 59: 838–846.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Engel LA, Menkes H, Wood LD, et al. Gas mixing during breath holding studied by intrapulmonary gas sampling. J Appl Physiol 1973; 35: 9–17.Google ScholarFull TextPubMedWeb of Science
↵Fukuchi Y, Roussos CS, Macklem PT, et al. Convection, diffusion and cardiogenic mixing of inspired gas in the lung; an experimental approach. Respir Physiol 1976; 26: 77–90.Google ScholarCrossRefPubMed
↵Enghoff H. [Volumen inefficax. Bermekungen zur Frage des shadlichen Raumes. Upsala Laekarefoeren]. Foerh 1938; 44: 191–218.Google Scholar
↵West JB. Respiratory Physiology – The Essentials. 6th Edn. Philadelphia, Lippincott, Williams & Wilkins, 2000Google Scholar
↵West JB. Causes of carbon dioxide retention in lung disease. N Engl J Med 1971; 284: 1232–1236.Google ScholarCrossRefPubMedWeb of Science
↵Wagner PD, Naumann PF, Laravuso RB. Simultaneous measurement of eight foreign gases in blood by gas chromatography. J Appl Physiol 1974; 36: 600–605.Google ScholarFull TextPubMedWeb of Science
↵Wagner PD, Saltzman HA, West JB. Measurement of continuous distributions of ventilation-perfusion ratios: theory. J Appl Physiol 1974; 36: 588–599.Google ScholarFull TextPubMedWeb of Science
↵Dueck R, Wagner PD, West JB. Effects of positive end-expiratory pressure on gas exchange in dogs with normal and edematous lungs. Anesthesiology 1977; 47: 359–366.Google ScholarCrossRefPubMedWeb of Science
↵Farhi LE. Elimination of inert gas by the lung. Respir Physiol 1967; 3: 1–11.Google ScholarCrossRefPubMedWeb of Science
↵West JB, Wagner PD, Derks CM. Gas exchange in distributions of VA-Q ratios: partial pressure-solubility diagram. J Appl Physiol 1974; 37: 533–540.Google ScholarFull TextPubMedWeb of Science
↵Hlastala MP, Robertson HT. Inert gas elimination characteristics of the normal and abnormal lung. J Appl Physiol Respir Environ Exerc Physiol 1978; 44: 258–266.Google ScholarFull TextPubMedWeb of Science
↵Robertson HT, Hlastala MP. Elevated alveolar PCO 2 relative to predicted values during normal gas exchange. J Appl Physiol Respir Environ Exerc Physiol 1977; 43: 357–364.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Wagner PD. Causes of a high physiological dead space in critically ill patients. Crit Care 2008; 12: 148.Google ScholarCrossRefPubMed
↵West JB. Ventilation-perfusion inequality and overall gas exchange in computer models of the lung. Respir Physiol 1969; 7: 88–110.Google ScholarCrossRefPubMedWeb of Science
↵Wasserman K, Van Kessel AL, Burton GG. Interaction of physiological mechanisms during exercise. J Appl Physiol 1967; 22: 71–85.Google ScholarFull TextPubMedWeb of Science
↵Wagner PD, Gale GE, Moon RE, et al. Pulmonary gas exchange in humans exercising at sea level and simulated altitude. J Appl Physiol 1986; 61: 260–270.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Deffebach ME, Charan NB, Lakshminarayan S, et al. The bronchial circulation. Small, but a vital attribute of the lung. Am Rev Respir Dis 1987; 135: 463–481.Google ScholarPubMedWeb of Science
↵Henig NR, Glenny RW, Aitken ML. A hypertrophied bronchial circulatory system may participate in gas exchange. Lancet 1998; 351: 113.Google ScholarPubMed
↵Ross BB, Farhi LE. Dead-space ventilation as a determinant in the ventilation-perfusion concept. J Appl Physiol 1960; 15: 363–371.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Fortune JB, Wagner PD. Effects of common dead space on inert gas exchange in mathematical models of the lung. J Appl Physiol 1979; 47: 896–906.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Petrini MF, Robertson HT, Hlastala MP. Interaction of series and parallel dead space in the lung. Respir Physiol 1983; 54: 121–136.Google ScholarCrossRefPubMedWeb of Science
↵Swenson ER, Maren TH. A quantitative analysis of CO 2 transport at rest and during maximal exercise. Respir Physiol 1978; 35: 129–159.Google ScholarCrossRefPubMedWeb of Science
↵White MG, Asch MJ. Acid-base effects of topical mafenide acetate in the burned patient. N Engl J Med 1971; 284: 1281–1286.Google ScholarCrossRefPubMed
↵Nuckton TJ, Alonso JA, Kallet RH, et al. Pulmonary dead-space fraction as a risk factor for death in the acute respiratory distress syndrome. N Engl J Med 2002; 346: 1281–1286.Google ScholarCrossRefPubMedWeb of Science
↵Cepkova M, Kapur V, Ren X, et al. Pulmonary dead space fraction and pulmonary artery systolic pressure as early predictors of clinical outcome in acute lung injury. Chest 2007; 132: 836–842.Google ScholarCrossRefPubMedWeb of Science
↵Siddiki H, Kojicic M, Li G, et al. Bedside quantification of dead-space fraction using routine clinical data in patients with acute lung injury: secondary analysis of two prospective trials. Crit Care 2010; 14: R141.Google ScholarCrossRefPubMed
↵Robertson HT, Swenson ER. What do dead-space measurements tell us about the lung with acute respiratory distress syndrome?. Respir Care 2004; 49: 1006–1007.Google ScholarFull TextPubMed
↵Coffey RL, Albert RK, Robertson HT. Mechanisms of physiological dead space response to PEEP after acute oleic acid lung injury. J Appl Physiol Respir Environ Exerc Physiol 1983; 55: 1550–1557.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Dantzker DR, Brook CJ, Dehart P, et al. Ventilation-perfusion distributions in the adult respiratory distress syndrome. Am Rev Respir Dis 1979; 120: 1039–1052.Google ScholarPubMedWeb of Science
↵Ralph DD, Robertson HT, Weaver LJ, et al. Distribution of ventilation and perfusion during positive end-expiratory pressure in the adult respiratory distress syndrome. Am Rev Respir Dis 1985; 131: 54–60.Google ScholarPubMedWeb of Science
↵Mélot C, Lejeune P, Leeman M, et al. Prostaglandin E1 in the adult respiratory distress syndrome. Benefit for pulmonary hypertension and cost for pulmonary gas exchange. Am Rev Respir Dis 1989; 139: 106–110.Google ScholarCrossRefPubMedWeb of Science
↵Feihl F, Eckert P, Brimioulle S, et al. Permissive hypercapnia impairs pulmonary gas exchange in the acute respiratory distress syndrome. Am J Respir Crit Care Med 2000; 162: 209–215.Google ScholarCrossRefPubMedWeb of Science
↵Coats AJ. Why ventilatory inefficiency matters in chronic heart failure. Eur Heart J 2005; 26: 426–427.Google ScholarCrossRefFull TextPubMedWeb of Science
↵Ponikowski P, Francis DP, Piepoli MF, et al. Enhanced ventilatory response to exercise in patients with chronic heart failure and preserved exercise tolerance: marker of abnormal cardiorespiratory reflex control and predictor of poor prognosis. Circulation 2001; 103: 967–972.Google ScholarCrossRefAbstract/Full TextPubMedWeb of Science
↵Ponikowski P, Chua TP, Anker SD, et al. Peripheral chemoreceptor hypersensitivity: an ominous sign in patients with chronic heart failure. Circulation 2001; 104: 544–549.Google ScholarCrossRefAbstract/Full TextPubMedWeb of Science
↵Myers J, Salleh A, Buchanan N, et al. Ventilatory mechanisms of exercise intolerance in chronic heart failure. Am Heart J 1992; 124: 710–719.Google ScholarCrossRefPubMedWeb of Science
↵Wasserman K, Zhang YY, Gitt A, et al. Lung function and exercise gas exchange in chronic heart failure. Circulation 1997; 96: 2221–2227.Google ScholarCrossRefAbstract/Full TextPubMedWeb of Science
↵Johnson RL Jr. Gas exchange efficiency in congestive heart failure II. Circulation 2001; 103: 916–918.Google ScholarCrossRefFull TextPubMedWeb of Science
↵Adnot S, Radermacher P, Andrivet P, et al. Effects of sodium-nitroprusside and urapidil on gas exchange and ventilation-perfusion relationships in patients with congestive heart failure. Eur Respir J 1991; 4: 69–75.Google ScholarAbstract/Full TextPubMedWeb of Science
↵Agostoni PG, Wasserman K, Perego GB, et al. Non-invasive measurement of stroke volume during exercise in heart failure patients. Clin Sci 2000; 98: 545–551.Google ScholarCrossRefAbstract/Full TextPubMed
↵Robertson HT. Gas exchange consequences of left heart failure. Compr Physiol 2011; 1: 621–634.Google ScholarPubMed
↵Robertson HT. Excess ventilation during exercise and prognosis in chronic heart failure. Am J Respir Crit Care Med 2012; 185: 227.Google ScholarCrossRefPubMed
↵Dantzker DR, Bower JS. Mechanisms of gas exchange abnormality in patients with chronic obliterative pulmonary vascular disease. J Clin Invest 1979; 64: 1050–1055.Google ScholarCrossRefPubMedWeb of Science
↵Dantzker DR, D’Alonzo GE, Bower JS, et al. Pulmonary gas exchange during exercise in patients with chronic obliterative pulmonary hypertension. Am Rev Respir Dis 1984; 130: 412–416.Google ScholarPubMedWeb of Science
↵Zhai Z, Murphy K, Tighe H, et al. Differences in ventilatory inefficiency between pulmonary arterial hypertension and chronic thromboembolic pulmonary hypertension. Chest 2011; 140: 1284–1291.Google ScholarCrossRefPubMedWeb of Science
Footnotes
The online version of this article has been corrected. The correction is outlined in the erratum published in the October 2015 issue of the European Respiratory Journal [DOI: 10.1183/09031936.50137614].
Previous articles in this series: No. 1: Naeije R, Vachiery J-L, Yerly P, et al. The transpulmonary pressure gradient for the diagnosis of pulmonary vascular diseases. Eur Respir J 2013; 41: 217–223. No. 2: Hughes JMB, van der Lee I. The T L,NO/T L,CO ratio in pulmonary function test interpretation. Eur Respir J 2013; 41: 453–461. No. 3: Vonk-Noordegraaf A, Westerhof N. Describing right ventricular function. Eur Respir J 2013; 41: 1419–1423. No. 4: Hamzaoui O, Monnet X, Teboul J-L. Pulsus paradoxus. Eur Respir J 2013; 42: 1696–1705. No. 5: Prisk GK. Microgravity and the respiratory system. Eur Respir J 2014; 43: 1459–1471. No. 6: Dempsey JA, Smith CA. Pathophysiology of human ventilatory control. Eur Respir J 2014; 44: 495–512. No. 7: Petersson J, Glenny RW. Gas exchange and ventilation–perfusion relationships in the lung. Eur Respir J 2014; 44: 1023–1041. No. 8: Wagner PD. The physiological basis of pulmonary gas exchange: implications for clinical interpretation of arterial blood gases. Eur Respir J 2015; 45: 227–243.
Conflict of interest: None declared.
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Article Sections
Top
Abstract
Introduction
Bohr dead space
Fowler dead space
Physiological dead space
Shunt contribution to physiological dead space
V′A/Q′ heterogeneity contribution to physiological dead space
Physiological dead space in an abnormal lung
Physiological dead space during exercise
Can “true” alveolar dead space also exchange CO 2?
Interaction between anatomical dead space and V′A/Q′ heterogeneity
Diffusion impairment and physiological dead space
Dead space measurements in ARDS
Factors contributing to the elevated V D/V T in ARDS
Abnormal exercise dead space measurements in heart failure
Abnormal exercise dead space measurements in pulmonary hypertension
Summary and conclusion
References
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7036 | https://www.merckmanuals.com/home/children-s-health-issues/infections-in-newborns/cytomegalovirus-cmv-infection-in-newborns | honeypot link
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MERCK MANUALConsumer Version
Cytomegalovirus (CMV) Infection in Newborns
(Congenital Cytomegalovirus Infection; Perinatal Cytomegalovirus Infection)
ByAnnabelle de St. Maurice, MD, MPH, UCLA, David Geffen School of Medicine
Reviewed/Revised Modified Apr 2025
v34436815
VIEW PROFESSIONAL VERSION
Cytomegalovirus is a common virus that usually causes few or no problems but can cause serious illness in infants who are infected before birth or around the time of birth.
Symptoms|
Diagnosis|
Treatment|
Prognosis|
Prevention|
Cytomegalovirus infection is caused by a virus.
Most newborns do not have symptoms, but some do depending on when they were infected.
Doctors diagnose the infection by identifying the virus in a sample of urine, saliva, blood, or tissue.
Cytomegalovirus infection cannot be cured, but some antivirals can limit the problems it causes.
Newborns may develop neurologic problems such as hearing loss.
Handwashing can help prevent spread of the virus.
(See also Overview of Infections in Newborns and Cytomegalovirus (CMV) Infection in adults.)
Infection with cytomegalovirus (CMV) is very common. Blood tests show that most adults have had a CMV infection at some time. The virus never goes completely away and remains dormant (inactive) in various tissues for life. Sometimes the virus reactivates (becomes active again).
Most people who have CMV infection, including newborns, do not have any symptoms.
When a baby contracts CMV infection in the uterus, it is called congenital CMV infection. When a baby develops the infection immediately before, during, or shortly after birth, it is called perinatal CMV infection. CMV infection is the most common congenital viral infection.
Pregnant people who have never had CMV infection may acquire it through contact with infected people, most commonly young children. In some pregnant people, a previous CMV infection reactivates. When pregnant people are infected, they can pass the virus to their fetus during pregnancy if the virus crosses the placenta (the organ that provides nourishment to the fetus) and infects the fetus.
Newborns may also become infected during passage through the birth canal, through human milk containing the virus, or through a contaminated blood transfusion.
Preterm infants are at higher risk of developing symptoms from CMV infection because they are less likely to have protective antibodies from their mother.
Symptoms of CMV Infection in Newborns
CMV infection may cause different problems in newborns depending on whether they were infected before or after birth.
Approximately 13% of newborns who are infected with CMV before birth have symptoms.
In newborns infected before birth, possible symptoms include
Preterm delivery
A small head
Low birth weight
Jaundice (yellowing of the skin or whites of the eyes)
Small bruises in the skin
An enlarged liver and spleen
Inflammation of the lungs or eyes
Hearing loss
In newborns infected during or after birth, possible symptoms include
An enlarged liver and spleen
Pneumonia
Hepatitis
A low platelet count
A high white blood cell count
Some newborns have all of these symptoms.
Newborns who are infected through human milk (breastfeeding) tend to have a milder infection.
Diagnosis of CMV Infection in Newborns
Polymerase chain reaction (PCR) test using urine, saliva, blood, or tissues
Testing of urine, saliva, or tissue samples
To diagnose CMV infection before birth, doctors take a sample of the amniotic fluid and test it.
To diagnose CMV infection after birth, doctors take samples of the newborn's urine, saliva, or tissues. The samples are sent to a laboratory so the organism causing the infection can be identified.
Doctors also do a PCR test on samples of the newborn's urine, saliva, blood, or tissues. This laboratory technique, which produces many copies of a gene to make the gene easier to detect, can be used to detect the CMV virus in the newborn.
Other tests, such as blood tests, imaging tests of the head, and an examination of the eyes, are done to look for infection and inflammation and to determine the seriousness of the symptoms. Doctors also may test the hearing of infected newborns. This testing is more extensive than the usual newborn hearing screening.
Doctors may evaluate the newborn for other infections present at birth that can cause symptoms similar to CMV, including toxoplasmosis, rubella, syphilis, and herpes simplex virus.
Treatment of CMV Infection in Newborns
Ganciclovir or valganciclovir for newborns who have symptomsGanciclovir or valganciclovir for newborns who have symptoms
There is no cure for CMV infection.
Ganciclovir and valganciclovir are medications that combat viral infections (Ganciclovir and valganciclovir are medications that combat viral infections (antivirals) and may help relieve some symptoms or make the infection less severe.
Infected newborns should have repeated hearing tests during the first year of life.
Prognosis for CMV Infection in Newborns
CMV infection is fatal in 5 to 10% of newborns who have symptoms. Most of the infants who have symptoms who survive develop hearing loss.
About 5 to 15% of newborns who do not have symptoms eventually develop neurologic problems (most commonly hearing loss). Some degree of hearing loss is the most common.
Prevention of CMV Infection in Newborns
Pregnant people should try to limit their exposure to the virus. For example, because CMV infection is common among children attending day care centers and easily spread, pregnant people should always wash their hands thoroughly after exposure to urine and saliva from children in day care.
Donated human milk can be pasteurized to reduce the risk of CMV transmission to newborns who are at increased risk of severe CMV infection after birth.
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7037 | https://www.youtube.com/watch?v=a8Ge0jApdGk | TUTORIAL: Equilibrium - ICE Tables and the Approximation Method
JFR Science
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Posted: 1 Apr 2020
Mr. Key models the solution for an ICE table problem involving a very small Kc, allowing for the use of the approximation method to solve.
Please note that the primary purpose of these videos is educational for students/educators. While all feedback, both positive and...hmmm...constructive, is appreciated I do not have the time to moderate or respond to all comments. As a result, comments on these videos have been disabled at least for the near future.
Transcript:
okay so welcome to this tutorial involving the use of the ice table and more specifically the use of the approximation method now if we take a look at this particular sample problem that we have here you will see that of course it is at equilibrium as denoted by the bi-directional arrow and we have a really low KC value and that really low KC value is important for us because if we recognize that it's a really low KC value then what that tells us is this reaction lies to the left and since it is so low it's very strongly reactant favored and what that in fact is gonna tell us is that when this reaction under these conditions comes to equilibrium that effectively the concentrations of the reactants aren't really going to change all that much and there's not really going to be all that much product as a result we can use something called the approximation method which is going to make our calculations a little bit easier for you what that means is even though it might look like we have to use the quadratic formula to solve this that's not going to be the case so you can see here that I've written down what our starting amounts are they are in moles per liter so we don't have to worry too much about converting them into concentrations because they're already there you'll notice that the product concentration is zero because initially we only have the reactants now as we move through this reaction we're going to see that we have to put in what changes we are going to have to this particular system now notice for our product here it is going to go up and the only way it can go is up because initially starts at zero because the coefficient there is two it's going to go up by 2x or whatever factor it is that it's going to increase by it's going to increase by a factor of 2x and if we take a look over at our reactant side we can see that these two the only place they can go is down by that same factor X now again we just have to pause for a second here and think about what those X's mean remember the X that we're looking at is whatever factor the nitrogen and oxygen are going to go down by and these are gonna be the same factor the same value because it's a one to one ratio keep in mind this is an equilibrium system it is closed so there's nothing coming in and going out so the only way that the nitrogen monoxide can be formed is if nitrogen and oxygen react and since it's a one-to-one ratio whatever nitrogen goes down by oxygen has to go down by that same amount on the other side if we take a look at nitrogen monoxide we have to remember that for every mole of nitrogen gas every mole of oxygen gas it is gonna go up by a factor of two because of the coefficient 2 so whatever nitrogen and oxygen go down by nitrogen monoxide is going to go up by that factor times 2 now just because the in this scenario are X's on the reactant side equal our X on the product side that doesn't always have to be the case it really just has to do with the coefficients so just because it works out that our negative X's and our positive X's so to speak equal or cancel each other out or at least it looks that way it does not mean it's going to happen all the time it's strictly based on the coefficients that we have so now given that information we're going to take a look at the EU remember here the equilibrium concentrations they can be determined by taking the sum of the I and the C so what we get there is zero decimal 0 8 5 minus X zero decimal 0 3 8 minus X and we get is 2x for our product side so now that we have our equilibrium concentrations we can take a look at what our equilibrium expression is going to be for this particular reaction notice that again its products over reactants raised to the coefficient or having an exponent representing the coefficient from the balanced chemical equation so we can see here that its products squared over the concentration of each of the reactants to the exponent 1 which of course we don't have to write so now that we have our equilibrium expression before we go through and plug these things in remember this is a really low kc value we can see here that it's four point two times 10 to the negative eight what that will allow us to do is undergo or perform a test now this test is going to tell us whether or not we can use the approximation method and so if we take a look the way that we do the test and you may recall this from the video that we did originally on the ice tables is that we take the initial concentration the smallest initial concentration and we divide it by the kay in this case it is k.c but we could be using any other equilibrium constant for any other type of equilibrium problem you so once we perform this calculation we can see that in fact our value is pretty large so the benchmark for our test is 500 now that is kind of an approximate value because sometimes if it's in between 200 and 500 it's okay to use but for the most part of its a door around or certainly greater than 500 we can assume that we can use the approximation method now what that means for us is that X is going to be so low once we calculate that it's not going to appreciably change the equilibrium concentrations of the nitrogen and oxygen meaning that it's so low that is not really going to differ that much from their initial concentrations once we take precision into account so what that's going to do is make our calculations a lot easier so now if we're going to look at our KC expression and we can exclude the X that's going to make our calculations a lot easier so now we're going to put in the values for our K see that is four point two times 10 to negative eight I'm going to put in the values from the e part of the ice table and I'm going to exclude the X because of the approximation method due to the validity of the tests that we perform so now I have all of this information into my equilibrium expression so at this point I can just start to expand and simplify and solve for X so right now I'm going to take these two product values I'm going to multiply it by my KC and what that's going to give me is a value it's going to be a relatively small value and I'm going to then expand the brackets here remember we have to multiply everything in the brackets so it's going to be 4x squared at this point I'm going to take the 4 I'm going to divide the 1.36 times 10 to the negative 10 by 4 in order to solve for x squared so now in order to solve for X because X was squared initially I'm going up to take the square root of both sides that's going to leave me with X being equal to the value that we obtained when we get one point three six times ten the negative 10 divided by four which is three point three nine times ten to the negative 11 and then then taking the square root of that value I'm gonna get a value for X of five point eight two times ten to the negative six now I'm not done here because ultimately what we're trying to find is not X but what the concentration of the product that is the nitrogen monoxide is at equilibrium so in order to do that in order to find the concentration of our nitrogen monoxide we're gonna have to recall that at equilibrium it is equal to 2x now since we have solved for X I can now take my value for X I can sub it in or multiply it by two so it's going to be two times five point eight two times ten to the negative six and once I perform that calculation we are going to obtain a final value for our nitrogen monoxide concentration of 1.2 times 10 to the negative five now keep in mind it is a concentration so we do have to include our units at this point now in terms of significant figures here take a look at this notice that my K values two significant figures that my initial concentrations were both of two significant figures and that I wasn't really performing any addition or subtraction here it was primarily just multiplication and division so as a result I can just take a look at the two significant figures and represent my final answer in two significant figures now if we take a look initially at Y we could subtract X notice how much smaller our x value is than our initial concentrations we're talking about initial concentrations that are times 10 to the negative 2 whereas if we take a look at our x value it's times 10 to the negative 6 so even if we were to take our x value and subtract it from our initial concentrations due to precision rules our initial concentrations would effectively remain as they at equilibrium so in this case it's fair to say that our test was valid hopefully this video gave you a better sense of generally use ice tables and how to ultimately use the approximation method to negate any changes in X thanks for watching |
7038 | https://math.libretexts.org/Bookshelves/Calculus/Map%3A_University_Calculus_(Hass_et_al)/13%3A_Partial_Derivatives/13.6%3A_Tangent_Planes_and_Differentials | 13.6.1
13.6.1
13.6.2
13.6.3
13.6.2
13.6.4
13.6.3
13.6.1
13.6.2
13.6.5
13.6.4
fx(x,y)
fy(x,y)
13.6.1
z=7x+8y−3
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13.6: Tangent Planes and Differentials
Last updated
: Nov 17, 2020
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13.5: Directional Derivatives and Gradient Vectors
13.7: Extreme Values and Saddle Points
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: 5538
( \newcommand{\kernel}{\mathrm{null}\,})
Learning Objectives
Determine the equation of a plane tangent to a given surface at a point.
Use the tangent plane to approximate a function of two variables at a point.
Explain when a function of two variables is differentiable.
Use the total differential to approximate the change in a function of two variables.
In this section, we consider the problem of finding the tangent plane to a surface, which is analogous to finding the equation of a tangent line to a curve when the curve is defined by the graph of a function of one variable, y=f(x)y=f(x). The slope of the tangent line at the point x=ax=a is given by m=f′(a)m=f'(a); what is the slope of a tangent plane? We learned about the equation of a plane in Equations of Lines and Planes in Space; in this section, we see how it can be applied to the problem at hand.
Tangent Planes
Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly. Therefore, in a small-enough neighborhood around the point, a tangent plane touches the surface at that point only.
Definition: Tangent Lines
Let P0=(x0,y0,z0)P0=(x0,y0,z0) be a point on a surface SS, and let CC be any curve passing through P0P0 and lying entirely in SS. If the tangent lines to all such curves CC at P0P0 lie in the same plane, then this plane is called the tangent planeto SS at P0P0 (Figure 13.6.113.6.1).
For a tangent plane to a surface to exist at a point on that surface, it is sufficient for the function that defines the surface to be differentiable at that point. We define the term tangent plane here and then explore the idea intuitively.
Definition: Tangent Planes
Let SS be a surface defined by a differentiable function z=f(x,y),z=f(x,y), and let P0=(x0,y0)P0=(x0,y0) be a point in the domain of ff. Then, the equation of the tangent plane to SS at P0P0 is given by
z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0).
z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0).(13.6.1)
To see why this formula is correct, let’s first find two tangent lines to the surface SS. The equation of the tangent line to the curve that is represented by the intersection of SS with the vertical trace given by x=x0x=x0 is z=f(x0,y0)+fy(x0,y0)(y−y0)z=f(x0,y0)+fy(x0,y0)(y−y0). Similarly, the equation of the tangent line to the curve that is represented by the intersection of SS with the vertical trace given by y=y0y=y0 is z=f(x0,y0)+fx(x0,y0)(x−x0)z=f(x0,y0)+fx(x0,y0)(x−x0). A parallel vector to the first tangent line is ⇀a=ˆj+fy(x0,y0)ˆka⇀=j^+fy(x0,y0)k^; a parallel vector to the second tangent line is ⇀b=ˆi+fx(x0,y0)ˆkb⇀=i^+fx(x0,y0)k^. We can take the cross product of these two vectors:
⇀a×⇀b=(ˆj+fy(x0,y0)ˆk)×(ˆi+fx(x0,y0)ˆk)=|ˆiˆjˆk01fy(x0,y0)10fx(x0,y0)|=fx(x0,y0)ˆi+fy(x0,y0)ˆj−ˆk.
a⇀×b⇀=(j^+fy(x0,y0)k^)×(i^+fx(x0,y0)k^)=∣∣∣∣∣∣i^01j^10k^fy(x0,y0)fx(x0,y0)∣∣∣∣∣∣=fx(x0,y0)i^+fy(x0,y0)j^−k^.
This vector is perpendicular to both lines and is therefore perpendicular to the tangent plane. We can use this vector as a normal vector to the tangent plane, along with the point P0=(x0,y0,f(x0,y0))P0=(x0,y0,f(x0,y0)) in the equation for a plane:
⇀n·((x−x0)ˆi+(y−y0)ˆj+(z−f(x0,y0))ˆk)=0(fx(x0,y0)ˆi+fy(x0,y0)ˆj−ˆk)·((x−x0)ˆi+(y−y0)ˆj+(z−f(x0,y0))ˆk)=0fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)−(z−f(x0,y0))=0.
n⇀⋅((x−x0)i^+(y−y0)j^+(z−f(x0,y0))k^)(fx(x0,y0)i^+fy(x0,y0)j^−k^)⋅((x−x0)i^+(y−y0)j^+(z−f(x0,y0))k^)fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)−(z−f(x0,y0))=0=0=0.
Solving this equation for zz gives Equation 13.6.113.6.1.
Example 13.6.113.6.1: Finding a Tangent Plane
Find the equation of the tangent plane to the surface defined by the function f(x,y)=2x2−3xy+8y2+2x−4y+4f(x,y)=2x2−3xy+8y2+2x−4y+4 at point (2,−1).(2,−1).
Solution
First, we must calculate fx(x,y)fx(x,y) and fy(x,y)fy(x,y), then use Equation with x0=2x0=2 and y0=−1y0=−1:
fx(x,y)=4x−3y+2fy(x,y)=−3x+16y−4f(2,−1)=2(2)2−3(2)(−1)+8(−1)2+2(2)−4(−1)+4=34fx(2,−1)=4(2)−3(−1)+2=13fy(2,−1)=−3(2)+16(−1)−4=−26.
fx(x,y)fy(x,y)f(2,−1)fx(2,−1)fy(2,−1)=4x−3y+2=−3x+16y−4=2(2)2−3(2)(−1)+8(−1)2+2(2)−4(−1)+4=34=4(2)−3(−1)+2=13=−3(2)+16(−1)−4=−26.
Then Equation 13.6.113.6.1 becomes
z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)z=34+13(x−2)−26(y−(−1))z=34+13x−26−26y−26z=13x−26y−18.
zzzz=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)=34+13(x−2)−26(y−(−1))=34+13x−26−26y−26=13x−26y−18.
(See the following figure).
Exercise 13.6.113.6.1
Find the equation of the tangent plane to the surface defined by the function f(x,y)=x3−x2y+y2−2x+3y−2f(x,y)=x3−x2y+y2−2x+3y−2 at point (−1,3)(−1,3).
Hint
: First, calculate fx(x,y) and fy(x,y), then use Equation 13.6.1.
Answer
: z=7x+8y−3
Example 13.6.213.6.2: Finding Another Tangent Plane
Find the equation of the tangent plane to the surface defined by the function f(x,y)=sin(2x)cos(3y)f(x,y)=sin(2x)cos(3y) at the point (π/3,π/4).(π/3,π/4).
Solution
First, calculate fx(x,y)fx(x,y) and fy(x,y)fy(x,y), then use Equation 13.6.113.6.1 with x0=π/3x0=π/3 and y0=π/4y0=π/4:
fx(x,y)=2cos(2x)cos(3y)fy(x,y)=−3sin(2x)sin(3y)f(π3,π4)=sin(2(π3))cos(3(π4))=(√32)(−√22)=−√64fx(π3,π4)=2cos(2(π3))cos(3(π4))=2(−12)(−√22)=√22fy(π3,π4)=−3sin(2(π3))sin(3(π4))=−3(√32)(√22)=−3√64.
Then Equation 13.6.1 becomes
z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)=−√64+√22(x−π3)−3√64(y−π4)=√22x−3√64y−√64−π√26+3π√616
A tangent plane to a surface does not always exist at every point on the surface. Consider the piecewise function
f(x,y)={xy√x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0).
The graph of this function follows.
Figure 13.6.3: Graph of a function that does not have a tangent plane at the origin. Dynamic figure powered by CalcPlot3D.
If either x=0 or y=0, then f(x,y)=0, so the value of the function does not change on either the x- or y-axis. Therefore, fx(x,0)=fy(0,y)=0, so as either x or y approach zero, these partial derivatives stay equal to zero. Substituting them into Equation gives z=0 as the equation of the tangent line. However, if we approach the origin from a different direction, we get a different story. For example, suppose we approach the origin along the line y=x. If we put y=x into the original function, it becomes
f(x,x)=x(x)√x2+(x)2=x2√2x2=|x|√2.
When x>0, the slope of this curve is equal to √2/2; when x<0, the slope of this curve is equal to −(√2/2). This presents a problem. In the definition of tangent plane, we presumed that all tangent lines through point P (in this case, the origin) lay in the same plane. This is clearly not the case here. When we study differentiable functions, we will see that this function is not differentiable at the origin.
Linear Approximations
Recall from Linear Approximations and Differentials that the formula for the linear approximation of a function f(x) at the point x=a is given by
y≈f(a)+f′(a)(x−a).
The diagram for the linear approximation of a function of one variable appears in the following graph.
The tangent line can be used as an approximation to the function f(x) for values of x reasonably close to x=a. When working with a function of two variables, the tangent line is replaced by a tangent plane, but the approximation idea is much the same.
Definition: Linear Approximation
Given a function z=f(x,y) with continuous partial derivatives that exist at the point (x0,y0), the linear approximation of f at the point (x0,y0) is given by the equation
L(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0).
Notice that this equation also represents the tangent plane to the surface defined by z=f(x,y) at the point (x0,y0). The idea behind using a linear approximation is that, if there is a point (x0,y0) at which the precise value of f(x,y) is known, then for values of (x,y) reasonably close to (x0,y0), the linear approximation (i.e., tangent plane) yields a value that is also reasonably close to the exact value of f(x,y) (Figure 13.6.5). Furthermore the plane that is used to find the linear approximation is also the tangent plane to the surface at the point (x0,y0).
Example 13.6.3: Using a Tangent Plane Approximation
Given the function f(x,y)=√41−4x2−y2, approximate f(2.1,2.9) using point (2,3) for (x0,y0). What is the approximate value of f(2.1,2.9) to four decimal places?
Solution
To apply Equation 13.6.3, we first must calculate f(x0,y0),fx(x0,y0), and fy(x0,y0) using x0=2 and y0=3:
f(x0,y0)=f(2,3)=√41−4(2)2−(3)2=√41−16−9=√16=4fx(x,y)=−4x√41−4x2−y2 sofx(x0,y0)=−4(2)√41−4(2)2−(3)2=−2fy(x,y)=−y√41−4x2−y2 sofy(x0,y0)=−3√41−4(2)2−(3)2=−34.
Now we substitute these values into Equation 13.6.3:
L(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)=4−2(x−2)−34(y−3)=414−2x−34y.
Last, we substitute x=2.1 and y=2.9 into L(x,y):
L(2.1,2.9)=414−2(2.1)−34(2.9)=10.25−4.2−2.175=3.875.
The approximate value of f(2.1,2.9) to four decimal places is
f(2.1,2.9)=√41−4(2.1)2−(2.9)2=√14.95≈3.8665,
which corresponds to a 0.2 error in approximation.
Exercise 13.6.2
Given the function f(x,y)=e5−2x+3y, approximate f(4.1,0.9) using point (4,1) for (x0,y0). What is the approximate value of f(4.1,0.9) to four decimal places?
Hint
: First calculate f(x0,y0),fx(x0,y0), and fy(x0,y0) using x0=4 and y0=1, then use Equation 13.6.3.
Answer
: L(x,y)=6−2x+3y, so L(4.1,0.9)=6−2(4.1)+3(0.9)=0.5 f(4.1,0.9)=e5−2(4.1)+3(0.9)=e−0.5≈0.6065.
Differentiability
When working with a function y=f(x) of one variable, the function is said to be differentiable at a point x=a if f′(a) exists. Furthermore, if a function of one variable is differentiable at a point, the graph is “smooth” at that point (i.e., no corners exist) and a tangent line is well-defined at that point.
The idea behind differentiability of a function of two variables is connected to the idea of smoothness at that point. In this case, a surface is considered to be smooth at point P if a tangent plane to the surface exists at that point. If a function is differentiable at a point, then a tangent plane to the surface exists at that point. Recall the formula (Equation 13.6.1) for a tangent plane at a point (x0,y0) is given by
z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)
For a tangent plane to exist at the point (x0,y0), the partial derivatives must therefore exist at that point. However, this is not a sufficient condition for smoothness, as was illustrated in Figure 13.6.3. In that case, the partial derivatives existed at the origin, but the function also had a corner on the graph at the origin.
Definition: Differentiable Functions
A function f(x,y) is differentiable at a point P(x0,y0) if, for all points (x,y) in a δ disk around P, we can write
f(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)+E(x,y),
where the error term E satisfies
lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2=0.
The last term in Equation 13.6.4 is referred to as the error term and it represents how closely the tangent plane comes to the surface in a small neighborhood (δ disk) of point P. For the function f to be differentiable at P, the function must be smooth—that is, the graph of f must be close to the tangent plane for points near P.
Example 13.6.4: Demonstrating Differentiability
Show that the function f(x,y)=2x2−4y is differentiable at point (2,−3).
Solution
First, we calculate f(x0,y0),fx(x0,y0), and fy(x0,y0) using x0=2 and y0=−3, then we use Equation 13.6.4:
f(2,−3)=2(2)2−4(−3)=8+12=20fx(2,−3)=4(2)=8fy(2,−3)=−4.
Therefore m1=8 and m2=−4, and Equation 13.6.4 becomes
f(x,y)=f(2,−3)+fx(2,−3)(x−2)+fy(2,−3)(y+3)+E(x,y)2x2−4y=20+8(x−2)−4(y+3)+E(x,y)2x2−4y=20+8x−16−4y−12+E(x,y)2x2−4y=8x−4y−8+E(x,y)E(x,y)=2x2−8x+8.
Next, we calculate the limit in Equation 13.6.5:
lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2=lim(x,y)→(2,−3)2x2−8x+8√(x−2)2+(y+3)2=lim(x,y)→(2,−3)2(x2−4x+4)√(x−2)2+(y+3)2=lim(x,y)→(2,−3)2(x−2)2√(x−2)2+(y+3)2≤lim(x,y)→(2,−3)2((x−2)2+(y+3)2)√(x−2)2+(y+3)2=lim(x,y)→(2,−3)2√(x−2)2+(y+3)2=0.
Since E(x,y)≥0 for any value of x or y, the original limit must be equal to zero. Therefore, f(x,y)=2x2−4y is differentiable at point (2,−3).
Exercise 13.6.3
Show that the function f(x,y)=3x−4y2 is differentiable at point (−1,2).
Hint
: First, calculate f(x0,y0),fx(x0,y0), and fy(x0,y0) using x0=−1 and y0=2, then use Equation 13.6.5 to find E(x,y).
Since you will find that E(x,y)≤0, show lim(x,y)→(x0,y0)|E(x,y)|√(x−x0)2+(y−y0)2=0
Answer
: f(−1,2)=−19,fx(−1,2)=3,fy(−1,2)=−16,E(x,y)=−4(y−2)2.lim(x,y)→(x0,y0)|E(x,y)|√(x−x0)2+(y−y0)2=lim(x,y)→(−1,2)|−4(y−2)2|√(x+1)2+(y−2)2=lim(x,y)→(−1,2)4(y−2)2√(x+1)2+(y−2)2≤lim(x,y)→(−1,2)4((x+1)2+(y−2)2)√(x+1)2+(y−2)2=lim(x,y)→(−1,2)4√(x+1)2+(y−2)2=0.
Since lim(x,y)→(x0,y0)|E(x,y)|√(x−x0)2+(y−y0)2=0, we know it's also true that
lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2=0
Therefore, f(x,y)=3x−4y2 is differentiable at point (−1,2).
This function from (Equation 13.6.2)
f(x,y)={xy√x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0)
is not differentiable at the origin (Figure 13.6.3). We can see this by calculating the partial derivatives. This function appeared earlier in the section, where we showed that fx(0,0)=fy(0,0)=0. Substituting this information into Equations 13.6.4 and 13.6.5 using x0=0 and y0=0, we get
f(x,y)=f(0,0)+fx(0,0)(x−0)+fy(0,0)(y−0)+E(x,y)E(x,y)=xy√x2+y2.
Calculating
lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2
gives
lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2=lim(x,y)→(0,0)xy√x2+y2√x2+y2=lim(x,y)→(0,0)xyx2+y2.
Depending on the path taken toward the origin, this limit takes different values. Therefore, the limit does not exist and the function f is not differentiable at the origin as shown in the following figure.
Differentiability and continuity for functions of two or more variables are connected, the same as for functions of one variable. In fact, with some adjustments of notation, the basic theorem is the same.
Theorem 13.6.1: Differentiability Implies Continuity
Let z=f(x,y) be a function of two variables with (x0,y0) in the domain of f. If f(x,y) is differentiable at (x0,y0), then f(x,y) is continuous at (x0,y0).
Theorem 13.6.1 shows that if a function is differentiable at a point, then it is continuous there. However, if a function is continuous at a point, then it is not necessarily differentiable at that point. For example, the function discussed above (Equation 13.6.2)
f(x,y)={xy√x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0)
is continuous at the origin, but it is not differentiable at the origin. This observation is also similar to the situation in single-variable calculus.
We can further explore the connection between continuity and differentiability at a point. This next theorem says that if the function and its partial derivatives are continuous at a point, the function is differentiable.
Theorem 13.6.2: Continuity of First Partials Implies Differentiability
Let z=f(x,y) be a function of two variables with (x0,y0) in the domain of f. If f(x,y), fx(x,y), and fy(x,y) all exist in a neighborhood of (x0,y0) and are continuous at (x0,y0), then f(x,y) is differentiable there.
Recall that earlier we showed that the function in Equation 13.6.2 was not differentiable at the origin. Let’s calculate the partial derivatives fx and fy:
∂f∂x=y3(x2+y2)3/2
and
∂f∂y=x3(x2+y2)3/2.
The contrapositive of the preceding theorem states that if a function is not differentiable, then at least one of the hypotheses must be false. Let’s explore the condition that fx(0,0) must be continuous. For this to be true, it must be true that
lim(x,y)→(0,0)fx(x,y)=fx(0,0)
therefor
lim(x,y)→(0,0)fx(x,y)=lim(x,y)→(0,0)y3(x2+y2)3/2.
Let x=ky. Then
lim(x,y)→(0,0)y3(x2+y2)3/2=limy→0y3((ky)2+y2)3/2=limy→0y3(k2y2+y2)3/2=limy→0y3|y|3(k2+1)3/2=1(k2+1)3/2limy→0|y|y.
If y>0, then this expression equals 1/(k2+1)3/2; if y<0, then it equals −(1/(k2+1)3/2). In either case, the value depends on k, so the limit fails to exist.
Differentials
In Linear Approximations and Differentials we first studied the concept of differentials. The differential of y, written dy, is defined as f′(x)dx. The differential is used to approximate Δy=f(x+Δx)−f(x), where Δx=dx. Extending this idea to the linear approximation of a function of two variables at the point (x0,y0) yields the formula for the total differential for a function of two variables.
Definition: Total Differential
Let z=f(x,y) be a function of two variables with (x0,y0) in the domain of f, and let Δx and Δy be chosen so that (x0+Δx,y0+Δy) is also in the domain of f. If f is differentiable at the point (x0,y0), then the differentials dx and dy are defined as
dx=Δx
and
dy=Δy.
The differential dz, also called the total differential of z=f(x,y) at (x0,y0), is defined as
dz=fx(x0,y0)dx+fy(x0,y0)dy.
Notice that the symbol ∂ is not used to denote the total differential; rather, d appears in front of z. Now, let’s define Δz=f(x+Δx,y+Δy)−f(x,y). We use dz to approximate Δz, so
Δz≈dz=fx(x0,y0)dx+fy(x0,y0)dy.
Therefore, the differential is used to approximate the change in the function z=f(x0,y0) at the point (x0,y0) for given values of Δx and Δy. Since Δz=f(x+Δx,y+Δy)−f(x,y), this can be used further to approximate f(x+Δx,y+Δy):
f(x+Δx,y+Δy)=f(x,y)+Δz≈f(x,y)+fx(x0,y0)Δx+fy(x0,y0)Δy.
See the following figure.
One such application of this idea is to determine error propagation. For example, if we are manufacturing a gadget and are off by a certain amount in measuring a given quantity, the differential can be used to estimate the error in the total volume of the gadget.
Example 13.6.5: Approximation by Differentials
Find the differential dz of the function f(x,y)=3x2−2xy+y2 and use it to approximate Δz at point (2,−3). Use Δx=0.1 and Δy=−0.05. What is the exact value of Δz?
Solution
First, we must calculate f(x0,y0),fx(x0,y0), and fy(x0,y0) using x0=2 and y0=−3:
f(x0,y0)=f(2,−3)=3(2)2−2(2)(−3)+(−3)2=12+12+9=33fx(x,y)=6x−2yfy(x,y)=−2x+2yfx(x0,y0)=fx(2,−3)=6(2)−2(−3)=12+6=18fy(x0,y0)=fy(2,−3)=−2(2)+2(−3)=−4−6=−10.
Then, we substitute these quantities into Equation 13.6.6:
dz=fx(x0,y0)dx+fy(x0,y0)dydz=18(0.1)−10(−0.05)=1.8+0.5=2.3.
This is the approximation to Δz=f(x0+Δx,y0+Δy)−f(x0,y0). The exact value of Δz is given by
Δz=f(x0+Δx,y0+Δy)−f(x0,y0)=f(2+0.1,−3−0.05)−f(2,−3)=f(2.1,−3.05)−f(2,−3)=2.3425.
Exercise 13.6.4
Find the differential dz of the function f(x,y)=4y2+x2y−2xy and use it to approximate Δz at point (1,−1). Use Δx=0.03 and Δy=−0.02. What is the exact value of Δz?
Hint
: First, calculate fx(x0,y0) and fy(x0,y0) using x0=1 and y0=−1, then use Equation 13.6.6.
Answer
: dz=0.18
Δz=f(1.03,−1.02)−f(1,−1)=0.180682
Differentiability of a Function of Three Variables
All of the preceding results for differentiability of functions of two variables can be generalized to functions of three variables. First, the definition:
Definition: Differentiability at a Point
A function f(x,y,z) is differentiable at a point P(x0,y0,z0) if for all points (x,y,z) in a δ disk around P we can write
f(x,y)=f(x0,y0,z0)+fx(x0,y0,z0)(x−x0)+fy(x0,y0,z0)(y−y0)+fz(x0,y0,z0)(z−z0)+E(x,y,z),
where the error term E satisfies
lim(x,y,z)→(x0,y0,z0)E(x,y,z)√(x−x0)2+(y−y0)2+(z−z0)2=0.
If a function of three variables is differentiable at a point (x0,y0,z0), then it is continuous there. Furthermore, continuity of first partial derivatives at that point guarantees differentiability.
Key Concepts
The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
Tangent planes can be used to approximate values of functions near known values.
A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
The total differential can be used to approximate the change in a function z=f(x0,y0) at the point (x0,y0) for given values of Δx and Δy.
Key Equations
Tangent plane
z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)
Linear approximation
L(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)
Total differential
dz=fx(x0,y0)dx+fy(x0,y0)dy.
Differentiability (two variables)
f(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)+E(x,y),
where the error term E satisfies
lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2=0.
Differentiability (three variables)
f(x,y)=f(x0,y0,z0)+fx(x0,y0,z0)(x−x0)+fy(x0,y0,z0)(y−y0)+fz(x0,y0,z0)(z−z0)+E(x,y,z),
where the error term E satisfies
lim(x,y,z)→(x0,y0,z0)E(x,y,z)√(x−x0)2+(y−y0)2+(z−z0)2=0.
Glossary
differentiable
: a function f(x,y) is differentiable at (x0,y0) if f(x,y) can be expressed in the form f(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)+E(x,y),
where the error term E(x,y) satisfies lim(x,y)→(x0,y0)E(x,y)√(x−x0)2+(y−y0)2=0
linear approximation
: given a function f(x,y) and a tangent plane to the function at a point (x0,y0), we can approximate f(x,y) for points near (x0,y0) using the tangent plane formula
tangent plane
: given a function f(x,y) that is differentiable at a point (x0,y0), the equation of the tangent plane to the surface z=f(x,y) is given by z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)
total differential
: the total differential of the function f(x,y) at (x0,y0) is given by the formula dz=fx(x0,y0)dx+fy(x0,y0)dy
13.5: Directional Derivatives and Gradient Vectors
13.7: Extreme Values and Saddle Points |
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Jan-Apr 2021 - Volume 11 - Issue 1
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INTRODUCTION
MATERIALS AND METHODS
RESULTS
Maxillary premolars
Mandibular premolars
DISCUSSION
CONCLUSION
Financial support and sponsorship
Conflicts of interest
Acknowledgment
REFERENCES
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Original Article
Cone-beam computed tomography evaluation of the root morphology of the maxillary and mandibular premolars in a Moroccan subpopulation
Number of roots and tooth length (part 1)
Dhaimy, Said; Diouri, Manal 1; Bedida, Lamiae 1; Dhoum, Sara; Elmerini, Hafsa; Benkiran, Imane
Author Information
Department of Conservative Dentistry and Endodontics, School of Dentistry, Hassan II University, Casablanca, Morocco
1 Private Practice, Casablanca, Morocco
Address for correspondence: Prof. Said Dhaimy, Department of Conservative Dentistry and Endodontics, School of Dentistry, Hassan II University, Casablanca, Morocco. E-mail: saiddhaimy@gmail.com
Received May 18, 2019
Received in revised form May 18, 2019
Accepted April 16, 2020
This is an open access journal, and articles are distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License, which allows others to remix, tweak, and build upon the work non-commercially, as long as appropriate credit is given and the new creations are licensed under the identical terms.
Saudi Endodontic Journal 11(1):p 54-58, Jan–Apr 2021. | DOI: 10.4103/sej.sej_76_19
Open
Abstract
Introduction:
The aim of the study was to evaluate the number of roots and tooth length of the maxillary and mandibular premolars of a Moroccan subpopulation using cone-beam computed tomography (CBCT).
Materials and Methods:
The number of roots and tooth length of 919 maxillary and mandibular premolars (358 maxillary and 561 mandibular) examined by CBCT was evaluated using “Planmeca viewer” software. Statistical analysis was carried out using the Epi info software.
Results:
The study showed a high prevalence of two roots for maxillary first premolars, and single root for maxillary second premolars and mandibular first and second premolars. The mean length of the maxillary first premolars (single rooted) was 21.74 mm, that of the buccal roots was 21.92 mm, and that of the lingual roots of the two-rooted teeth was 20.67 mm. The maxillary second premolars (21.4 mm) and the mandibular premolars (21.5 mm) exhibited the same mean length.
Conclusion:
All premolars had a single root in most cases, except for the maxillary first premolars where two-rooted teeth were more frequent with an average length of 21.4 mm. CBCT is an exciting tool for endodontic diagnosis and treatment planning.
INTRODUCTION
The root canal system is complex and can harbor a rich microbial flora if infected. A lack of knowledge of the root canal anatomy and/or technical skills may result in a failure to identify and treat all root canals. To achieve a successful endodontic treatment, the clinician should be aware of the multiple and complex variations that can occur during the root formation.
Numerous methods have been used for studying the root canal anatomy, including canal staining and tooth clearing, conventional radiographs, alternative radiographic techniques, radiographic assessment enhanced with contrast media and more recently, computed tomographic techniques and modified root canal-staining technique.
The canal staining and clearing technique and the cross-sectioning technique are invasive and result in irreversible damage to the samples. Radiographic examinations also provide only two-dimensional images, and anatomic structures can be superimposed in these images. Therefore, they do not reflect the actual morphology of the root canals. Microcomputed tomography (CT) is a nondestructive high-resolution laboratory method used to study the root morphology of the extracted teeth. The major problems of this technique are time-consuming and material costing. On the other hand, cone-beam CT (CBCT) imaging allows a three-dimensional (3D) evaluation of teeth and their adjacent structures. The images obtained by CBCT display axial, sagittal, and coronal sections and reduce the superimposition of anatomic structures. These advantages facilitate the clinician's understanding of the thorough morphology of the root canals.
Several in vivo and ex vivo research studies on the root and root canal anatomy of many subpopulations using CBCT have been published. No studies have been published to evaluate the tooth root anatomy of Moroccan subpopulation. Therefore, the aim of this study was to evaluate the number of roots and tooth length of the maxillary and mandibular premolars of a Moroccan subpopulation using CBCT.
MATERIALS AND METHODS
The study was approved by the Ethical Committee of School of Dentistry, Hassan II University, Casablanca, Morocco (#104/15).
In order to comply with the principles of radioprotection (principles of justification and optimization) of ionizing radiation examinations, no cone-beam examination has been prescribed specifically for this study.
The ability and validation of the examiner to correctly analyze the images, the accuracy of his/her observation, as well as the reproducibility of the results were tested in a preinvestigation involving the roots' number and tooth length before conducting the actual study. It was carried out starting with 278 CBCT images to reach an agreement between the examiners on what should be recorded.
A total of 919 premolar teeth randomly collected from a radiology center's database in the region of Casablanca between January and December 2017 including 180 first maxillary premolars, 178 secondary maxillary premolars, 304 first mandibular premolars, and 257 secondary mandibular premolars were evaluated. The samples were made of 53.6% of females (n = 493) and 46.6% of males (n = 426), who presented with fully formed premolars, with no root canal filling, post or crown restorations, and free from teeth anomalies. CBCT examinations presenting the kinetic of metallic artifacts which could lead to a difficult interpretation were excluded.
The CBCT images were obtained using a CBCT scanner called Planmeca Promax 3D plus (Planmeca Oy, Helsinki, Finland), with the following parameters: 90 kV, 10 mA, with a field of view of 601 × 601 × 601, and 150 μm voxels. Serial sagittal, coronal, and axial views of the CBCT images from the coronal portion to the root apex of each tooth were examined carefully by an experienced radiologist according to the following features: the roots' number [Figures 1 and 2] and the tooth length [Figure 3]. The data were collected in The Digital Imaging and Communications in Medicine (DICOM) format.
Figure 1:
Cone-beam computed tomography images showing the number of roots of maxillary premolars (a) single-rooted coronal section;(b) two.rooted sagittal section;(c) three-rooted axial section
Figure 2:
Cone.beam computed tomography images showing the number of roots of single-rooted mandibular premolars (a) sagittal section; (b) coronal section; (c) axial section
Figure 3:
Cone-beam computed tomography images showing the length measurement of maxillary premolars in the sagittal section
The CBCT images were analyzed with “Planmeca viewer” software (version 3.2.7, Planmeca company, Helsinki, Finland). Statistical analysis was performed using the software Epi info (version 6, Centers for Disease Control and Prevention (CDC), Atlanta, Georgia (US)). The roots' number data were described as percentages and the length data were described as mean and standard deviation.
RESULTS
The findings of the preexamination of the CBCT images demonstrated a high level of agreement between examiners and good reproducibility within examiners.
Maxillary premolars
Among the analyzed maxillary first premolars, 38.3% were single rooted, whereas 61.7% presented two roots. Premolars with two roots presented a mean length of 21.92 ± 1.860 mm and 20.67 ± 1.951 mm buccally and lingually, respectively; for the single-rooted premolars, the mean length was 21.74 ± 2.002 mm.
The maxillary second premolars were mostly single rooted (73.6%), whereas 26.4% presented two roots. The mean length of premolars with two roots was 21.56 ± 1.916 mm and 20.81 ± 2.053 mm buccally and lingually, respectively. The mean length of single-rooted premolars was 21.81 ± 1.804 mm. No three-rooted maxillary premolar was found.
Mandibular premolars
For the mandibular first premolars, 90.5% were single rooted and 9.5% had two roots. The mean length was of 22.11±1.355mm, 19.87±2.016mm buccally and lingually respectively for the two rooted premolars and 22.53 ± 1.970 mm for the single rooted.
Most of the mandibular second premolars were single rooted (97, 7%), and only 1 (9%) of them was two rooted. The second premolars presented a mean length of 24.76 ± 1.826 mm and 23.05 ± 2.287 mm buccally and lingually, respectively, although the single-rooted premolars presented a mean length of 22.50 ± 1.803 mm.
One three-rooted second premolar was found (0.4%), of which the mesiobuccal root measured 19.70 mm, the distobuccal root 20 mm, and the lingual root 19.30 mm.
DISCUSSION
In the current study, DICOM was used to collect the data. It is considered as the most widely adopted standard method for the exchange and management of medical images. By using DICOM, different types of images generated by any medical imaging devices such as CBCT can be integrated into picture archiving and communication systems that can be used by other applications in health-care delivery.
Each tooth in the present study was measured from the coronal portion to the root apex. The use of CBCT to measure the tooth length was evaluated by different researchers. It was proven to be more accurate and reliable than the conventional periapical radiographs for estimating preoperative working length and tooth length measurements. Several factors contribute to the variations found regarding the tooth length and number of roots, which include ethnicity and gender. Thus, clinicians must be fully aware of these variations for a better treatment of their patients.
In this study, most maxillary first premolars were found to have two roots (61.7%), which is in correspondence with many other studies. Higher incidences of single-rooted maxillary first premolars were noted in the Southern Chinese population (60%). No three-rooted form was detected, however, three-rooted maxillary first premolars were noted in the Spanish (2.6%), Indian (2.2%), German (1.2%), and Jordanian (0.8%) population. Our findings regarding the mean length of the maxillary first premolars were similar to those reported by Pécora et al. for both the single-rooted and the two-rooted teeth.
In the maxillary second premolars, the percentages of single-, two-, and three-rooted teeth ranged from 55.3% to 90.6%, 9.4% to 44.2%, and 0.4% to 1.6%, respectively. This complies with our findings where the majority of maxillary second premolars were single rooted (73.6%), followed by two rooted (26.4%). However, no three-rooted premolar was found. The mean length of this group of teeth according to our study was in compliance with the one found in the Brazilian population.
In the mandibular first premolars, the single-rooted form was the most prevalent (91.4%–100%) in many studies, whereas the two-rooted one was present in small quantities in only some of them. Our study showed the same results as the ones found in the literature, unlike the study conducted by Estrela et al., where most of the premolars had two roots (99%) and no three-rooted teeth were found. According to our study, the mean length of the mandibular first premolar was 21.5 ± 1.780 mm, which is in accordance with the findings of other researchers in different ethnic groups using different evaluation methods.
Lastly, for the mandibular second premolars, all the previous studies agreed that this group of teeth presented one root in most cases, ranging from 98.6% to 100%, which is in compliance with our results (97.7%) and contrary to the findings of Estrela et al., who conducted a study in a Brazilian subpopulation and found that all the examined mandibular second premolars had one root (100%). Three-rooted premolars were also found in our study (0.4%). The presence of three roots in the mandibular second premolar is a rare finding. This was reported in a German population (0.1%), and in the systematic review conducted by Kottoor et al. (0%–2%). The mandibular first and second premolars had almost identical lengths in the present study (21.5 ± 1.972 mm), as well as in other studies.
Identifying and accessing all root canals is particularly challenging in the endodontic treatment of teeth with atypical canal configuration; the maxillary first premolar has a highly variable canal and root morphology, frequently with two separate canals (88%) and two foramina (81%). With the aim to a hermetic obturation, Dadresanfar et al. suggested a tactile examination of all major buccal walls with a small, precurved K-file tip in order to find the additional buccal canal. The mandibular second premolars have earned the reputation for having aberrant anatomy. According to Vertucci classification, second premolars with type 2 configuration gained 1.8 % as stated in the research of Llena C et al. Al-Mahroos et al. claimed that the use of a dental operating microscope allowed early recognition of the C-shaped canal system. The use of high-quality preoperative radiographs at different horizontal angulations is needful to detect the presence of extra root canal, so predictable results could be possible.
The CBCT technology offers a high accuracy when the collected clinic and conventional radiography data are not sufficiently contributory to the diagnosis. The use of CBCT database in this study will make the number of roots and tooth length measurement more correct and reproducible quantitatively as well as qualitatively. However, subtle limitations made the current data more problematic for quantitative use. Kinetic or metallic scanning artifacts due to patient mobility during acquisition obscured details of interest or caused the CT value of a single material to change in different parts of an image. In addition, the presence of teeth with root canal treatment or even dental anomalies reduced the number of samples.
CONCLUSION
All premolars had a single root in most cases, except for the maxillary first premolars where two-rooted teeth were more frequent with an average length of 21.4 mm. Cone-beam technology offers precision in situations where the information provided by the clinical examination and conventional radiology is not sufficiently contributory to the diagnosis.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
Acknowledgment
The authors would like to thank Pr. Mohamed BAITE for his help and support.
REFERENCES
Abella F, Teixidó LM, Patel S, Sosa F, Duran-Sindreu F, Roig M. Cone-beam computed tomography analysis of the root canal morphology of maxillary first and second premolars in a Spanish population J Endod. 2015;41:1241–7
Cited Here
Baroudi K, Kazkaz M, Sakka S, Tarakji B. Morphology of root canals in lower human premolars Niger Med J. 2012;53:206–9
Cited Here
Neelakantan P, Subbarao C, Ahuja R, Subbarao CV. Root and canal morphology of Indian maxillary premolars by a modified root canal staining technique Odontology. 2011;99:18–21
Cited Here
Ok E, Altunsoy M, Nur BG, Aglarci OS, Çolak M, Güngör E. A cone-beam computed tomography study of root canal morphology of maxillary and mandibular premolars in a Turkish population Acta Odontol Scand. 2014;72:701–6
Cited Here
Al-Nazhan S, Al-Daafas A, Al-Maflehi N. Radiographic investigation of in vivo endodontically treated maxillary premolars in a Saudi Arabian sub-population Saudi Endod J. 2012;2:1–5
Cited Here
Chourasia HR, Boreak N, Tarrosh MY, Mashyakhy M. Root canal morphology of mandibular first premolars in Saudi Arabian southern region subpopulation Saudi Endod J. 2017;7:77–81
Cited Here
Alkaabi W, AlShwaimi E, Farooq I, Goodis HE, Chogle SM. A micro-computed tomography study of the root canal morphology of mandibular first premolars in an Emirati population Med Princ Pract. 2017;26:118–24
Cited Here
Bürklein S, Heck R, Schäfer E. Evaluation of the root canal anatomy of maxillary and mandibular premolars in a selected German population using cone-beam computed tomographic data J Endod. 2017;43:1448–52
Cited Here
Hosseini M, Dixon BE. Chapter 8-Syntactic Interoperability and the Role of Standards in Health Information Exchange 2016 USA Academic Press, Elsevier Inc.:123–36
Cited Here
Yilmaz F, Kamburoglu K, Senel B. Endodontic working length measurement using cone-beam computed tomographic images obtained at different voxel sizes and field of views, periapical radiography, and apex locator: A comparative ex vivo study J Endod. 2017;43:152–6
Cited Here
Adarsh K, Sharma P, Juneja A. Accuracy and reliability of tooth length measurements on conventional and CBCT images: An in vitro comparative study J Orthod Sci. 2018;7:17.
Cited Here
Awawdeh L, Abdullah H, Al-Qudah A. Root form and canal morphology of Jordanian maxillary first premolars J Endod. 2008;34:956–61
Cited Here
Walker RT. Root form and canal anatomy of maxillary first premolars in a southern Chinese population Endod Dent Traumatol. 1987;3:130–4
Cited Here
Pécora JD, Saquy PC, Sousa Neto MD, Woelfel JB. Root form and canal anatomy of maxillary first premolars Braz Dent J. 1992;2:87–94
Cited Here
Al-Ghananeem MM, Haddadin K, Al-Khreisat AS, Al-Weshah M, Al-Habahbeh N. The number of roots and canals in the maxillary second premolars in a group of Jordanian population Int J Dent. 2014;2014:797692.
Cited Here
Bulut DG, Kose E, Ozcan G, Sekerci AE, Canger EM, Sisman Y. Evaluation of root morphology and root canal configuration of premolars in the Turkish individuals using cone beam computed tomography Eur J Dent. 2015;9:551–7
Cited Here
Felsypremila G, Vinothkumar TS, Kandaswamy D. Anatomic symmetry of root and root canal morphology of posterior teeth in Indian subpopulation using cone beam computed tomography: A retrospective study Eur J Dent. 2015;9:500–7
Cited Here
Kartal N, Ozçelik B, Cimilli H. Root canal morphology of maxillary premolars J Endod. 1998;24:417–9
Cited Here
Pécora JD, Sousa Neto MD, Saquy PC, Woelfel JB. In vitro study of root canal anatomy of maxillary second premolars Braz Dent J. 1993;3:81–5
Cited Here
Abraham SB, Gopinath VK. Root canal anatomy of mandibular first premolars in an Emirati subpopulation: A laboratory study Eur J Dent. 2015;9:476–82
Cited Here
Alhadainy HA. Canal configuration of mandibular first premolars in an Egyptian population J Adv Res. 2013;4:123–8
Cited Here
Dou L, Li D, Xu T, Tang Y, Yang D. Root anatomy and canal morphology of mandibular first premolars in a Chinese population Sci Rep. 2017;7:750.
Cited Here
Llena C, Fernandez J, Ortolani PS, Forner L. Cone-beam computed tomography analysis of root and canal morphology of mandibular premolars in a Spanish population Imaging Sci Dent. 2014;44:221–7
Cited Here
Park JB, Kim N, Park S, Kim Y, Ko Y. Evaluation of root anatomy of permanent mandibular premolars and molars in a Korean population with cone-beam computed tomography Eur J Dent. 2013;7:94–101
Cited Here
Singh S, Pawar M. Root canal morphology of South Asian Indian mandibular premolar teeth J Endod. 2014;40:1338–41
Cited Here
Estrela C, Bueno MR, Couto GS, Rabelo LE, Alencar AH, Silva RG, et al Study of root canal anatomy in human permanent teeth in a subpopulation of Brazil's center region using cone-beam computed tomography-part 1 Braz Dent J. 2015;26:530–6
Cited Here
Awawdeh LA, Al-Qudah AA. Root form and canal morphology of mandibular premolars in a Jordanian population Int Endod J. 2008;41:240–8
Cited Here
Velmurugan N, Sandhya R. Root canal morphology of mandibular first premolars in an Indian population: a laboratory study Int Endod J. 2009;42:54–8
Cited Here
Cleghorn BM, Christie WH, Dong CC. The root and root canal morphology of the human mandibular second premolar: a literature review J Endod. 2007;33:1031–7
Cited Here
Kottoor J, Albuquerque D, Velmurugan N, Kuruvilla J. Root anatomy and root canal configuration of human permanent mandibular premolars: a systematic review Anat Res Int. 2013;2013:254250.
Cited Here
Yu X, Guo B, Li KZ, Zhang R, Tian YY, Wang H, et al Cone-beam computed tomography study of root and canal morphology of mandibular premolars in a Western Chinese population BMC Med Imaging. 2012;12:18.
Cited Here
Dadresanfar B, Khalilak Z, Shahmirzadi S. Endodontic treatment of a maxillary first premolar with type IV buccal root canal: a case report Iran Endod J. 2009;4:35–7
Cited Here
Vertucci FJ. Root canal anatomy of the human permanent teeth Oral Surg Oral Med Oral Pathol. 1984;58:589–99
Cited Here
Al-Mahroos SE, Al-Sharif A, Ahmad I. Mandibular premolars with unusual root canal configuration: A report of two cases Saudi Endod J. 2016;6:87–91
Cited Here
View full references list
Keywords:
Cone-beam computed tomography; mandibular premolars; maxillary premolars; root number; tooth length
© 2021 Saudi Endodontic Journal | Published by Wolters Kluwer – Medknow
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Cone-beam computed tomography evaluation of the root morphology of the maxillary and mandibular premolars in a Moroccan subpopulation: Number of roots and tooth length (part 1)
Saudi Endodontic Journal11(1):54-58, Jan-Apr 2021.
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7041 | https://www.quora.com/Is-there-any-number-which-has-2-representations-as-a-sum-of-2-squares | Something went wrong. Wait a moment and try again.
Sum of Squares
Arithmetic Number Theory
Algebraic Concepts
Squares (mathematics)
Math Theory
Mathematical Sciences
Math Geometry
5
Is there any number which has 2 representations as a sum of 2 squares?
Amitabha Tripathi
have more than a working knowledge of Z
·
Author has 4.7K answers and 13.8M answer views
·
3y
Assuming you are looking for a,b,c,d∈N such that a2+b2=c2+d2 with a≥b>0, c≥d>0, a>c, the smallest example is 72+12=52+52.
If you want a>b>0 and c>d>0, then the smallest example is 82+12=72+42.
Vadim Pelyushenko
Upvoted by
Michael Jørgensen
, PhD in mathematics
·
6y
Is there a number that can be expressed in two different sums of squares (i.e. x=a^2+b^2=c^2+d^2, where a, b, c and d are all distinct non-zero natural numbers)?
For a short answer, here’s is an example of something that would work:
7^2 + 24^2 = 15^2 + 20^2 = 625
Now as for finding such examples…
For this, I draw on some particularly nice math that I learned from this video:
Long story short though, the connection is that
finding a,b,c,d such that a^2 + b^2 = c^2 + d^2, is equivalent to finding lattice points that lie on the circle formed at the origin with some radius. (With the added restriction in this case of having them be non-zero and positive).
Now lets skip to the tools that the video provides that will help us find all fe
For a short answer, here’s is an example of something that would work:
7^2 + 24^2 = 15^2 + 20^2 = 625
Now as for finding such examples…
For this, I draw on some particularly nice math that I learned from this video:
Long story short though, the connection is that
finding a,b,c,d such that a^2 + b^2 = c^2 + d^2, is equivalent to finding lattice points that lie on the circle formed at the origin with some radius. (With the added restriction in this case of having them be non-zero and positive).
Now lets skip to the tools that the video provides that will help us find all feasible solutions.
There’s factoring with integers, and then there’s factoring with Gaussian integers(which are basically complex numbers). It turns out that a natural prime that is 3 above a multiple of 4 is also a Gaussian Prime(cannot be factored further). A prime number that is 1 above a multiple of 4 can be factored into 2 Gaussian integers, for instance 5 = (2 + i)(2 - i). And everything else with factoring a natural number is still the same.
To be specific, if a prime P = 4k + 1, then we can factor it into a complex conjugate pair. A complex conjugate pair will always be in the form (a + bi)(a - bi), and it evaluates to a^2 + b^2, so in this particular example 2^2 + 1^2 = 5^2. If we factor a number N into different complex conjugate pairs, we get different solutions for a^2 + b^2 = N. As for how to find the gaussian factorization of a prime 1 greater than a multiple of 4… the video doesn’t show how, and I couldn’t find anything that explains how to do so simply, but this site will give you it Gaussian integer factorization calculator
For the example I showed you at the start, 7^2 + 24^2 = 15^2 + 20^2, I derived that from starting with 5^4, which is 625, and decomposed it into its 8 factors, 4 factors of (2+i) and 4 factors of (2-i). In building the conjugate pairs, you must always “divy it up evenly”, meaning that if one factor of a prime goes to one part of the pair, the other factor must go to the other part of the pair.
For instance, one valid pair construction for 625 would be
(The bottom part is the complex conjugate pairs, above them are the divied factors)
another would be
and the last unique one for 625
All of these pairs corresponds to the fact that
7^2 + 24^2 = 15^2 + 20^2 = 25^2 + 0^2 = 625
(though of course, 25 and 0 don’t count for the original question because the question asks for non-zero naturals. That’s fine, we just have to avoid divying up the factors in such a way that each factor pairs with its conjugate).
It turns out that, if you have a number that has a Gaussian prime to an odd power, there are NO COMPLEX CONJUGATE PAIRS that can multiply to form that number. Take for instance, 151875 = 3^5 5^4. 3 is a Gaussian prime, and 151875 has it come up an odd number of times in its factorization, therefore there is no solution for a^2 + b^2 = 151875. But if you have a Gaussian prime with an odd power in the prime factorization, then you can just divy it up evenly.
Another note, if you have a number that has a power of 2 in its prime factorization, like 5^2 2^7. 2 can be factored into (1 + i)(1 - i), it doesn’t add nor subtract the number of pairs you can form, for reasons the video explains.
A couple more examples:
Example 1: 85 = 5 17
5 = (2 + i)(2 - i), 17 = (4 + i)(4 - i)
1)
2)
7^2 + 6^2 = 9^2 + 2^2 = 85
Example 2: 48841 = 13^2 17^2
13 = (3 + 2i)(3 - 2i), 17 = (4 + i)(4 - i)
1)
2)
3)
4)
21^2 + 220^2 = 195^2 + 104^2 = 85^2 + 204^2 = 171^2 + 140^2 = 48841
Mathematics Skills - Learn/Expertise
Answered by
Aravind Kumar
·
Author has 8.4K answers and 3.2M answer views
·
Feb 19, 2022
all the pythagorean triplets follow the pattern.
for example: 25=5^2=3^2+4^2
100=10^2=8^2+6^2
Related questions
Can every number be written as the sum of two squares?
How do I find if a number can be represented as a sum of two squares?
Can every number be written as a sum of two squares or their squares' sums?
Is there any number which can be expressed as the sum of two squares as well as two cubes also?
Can every natural number be written as a sum of two squares?
Assistant
Bot
·
1y
Yes, there are numbers that can be represented as a sum of two squares in more than one way. A classic example is the number 50, which can be represented as:
1
2
+
7
2
=
1
+
49
=
50
2. 5
2
+
5
2
=
25
+
25
=
50
Another example is the number 65:
1
2
+
8
2
=
1
+
64
=
65
2. 4
2
+
7
2
=
16
+
49
=
65
In general, certain numbers can have multiple representations as sums of two squares due to the properties of their prime factorization. Specifically, if a number can be expressed in the form
a
2
+
b
2
, it may have multiple distinct pairs
(
a
,
b
)
that satisfy this condition.
D Khandelwal
.
·
Author has 75 answers and 143.7K answer views
·
4y
Well it’s already that way …
Let’s take a number for example 26
then it’s 5^2 + 1^2 ..
So if we take ordered pairs then 26 can be represented as square of :-
(1,5)
(5,1)
(-1,-5)
(-5,-1)
(-1,5)
(5,-1)
(1,-5)
(-5,1)
But if we take unordered pairs :-
(1,5)
(-1,-5)
(-1,5)
(5,-1)
So there are at least 4 representation , if there is not given whether the numbers are distinct or non distinct ..
.
.
.
.
But if you want all numbers are distinct then also there are infinite ways if the numbers are any real number ..
ex. 5
(2,1)
(sqrt5,sqrt5) etc..
Also if you want the number to be integers then only 2 dist
Well it’s already that way …
Let’s take a number for example 26
then it’s 5^2 + 1^2 ..
So if we take ordered pairs then 26 can be represented as square of :-
(1,5)
(5,1)
(-1,-5)
(-5,-1)
(-1,5)
(5,-1)
(1,-5)
(-5,1)
But if we take unordered pairs :-
(1,5)
(-1,-5)
(-1,5)
(5,-1)
So there are at least 4 representation , if there is not given whether the numbers are distinct or non distinct ..
.
.
.
.
But if you want all numbers are distinct then also there are infinite ways if the numbers are any real number ..
ex. 5
(2,1)
(sqrt5,sqrt5) etc..
Also if you want the number to be integers then only 2 distinct no.(their negatives too) will only satisfy it !!!
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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?
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Kermit Rose
Studied Mathematics & Statistics (academic discipline) at Florida State University School
·
Author has 960 answers and 641.8K answer views
·
2y
Related
Can every number be written as a sum of two or three squares?
I presume you meant to ask,
can every positive integer be written as a sum of two or three square positive square integers.
The answer is no.
You can easily determine this by ovservation.
1 is square, but is not the sum of two or three square positive square integers.
Ok, let’s rephrase the question.
Can every positive non-square integer be written as a sum of two or three square square integers?
2 = 1^2 + 1^2 = 1 squared plus 1 squared.
3 = 1^2 + 1^2 + 1^2.
5 = 2^2 + 1^2.
6 = 2^2 + 1^2 + 1^2.
7 = 2^2 + 1^2 + 1^2 + 1^2 is the sum of 4 square integers,
but is not the sum of 3 square integers.
This question
I presume you meant to ask,
can every positive integer be written as a sum of two or three square positive square integers.
The answer is no.
You can easily determine this by ovservation.
1 is square, but is not the sum of two or three square positive square integers.
Ok, let’s rephrase the question.
Can every positive non-square integer be written as a sum of two or three square square integers?
2 = 1^2 + 1^2 = 1 squared plus 1 squared.
3 = 1^2 + 1^2 + 1^2.
5 = 2^2 + 1^2.
6 = 2^2 + 1^2 + 1^2.
7 = 2^2 + 1^2 + 1^2 + 1^2 is the sum of 4 square integers,
but is not the sum of 3 square integers.
This question is related to the fact that
Positive primes with remainder 1 when you divide by 4, are always the sum of two squares in exactly one way.
If two integers are each the sum of two squares, then so is their product.
Every odd integer is the difference of two squares.
For example, 7 = 4^2- 3^2.
Related questions
How can we express number as a sum of 2 squares?
What is the only natural number that cannot be written as the sum of two squares, besides 2 and 3?
How can I express a number as sum of four squares?
What is the largest square number that can be represented as the sum of four squares?
Can every number be written as a sum of two or three squares?
Girija Warrier
Author of the book ‘Madeira Math 500’ (2018–present)
·
Author has 5.9K answers and 13.8M answer views
·
3y
Related
Can every number be written as the sum of two squares?
No! Every number can not be written as the sum of two squares..
Here is how we find which number can be expressed as the sum of two squares…
1️⃣🔹: Prime number 'P' can be expressed as a^2 + b^2 with 'a' & 'b' as integers
if and only if 'P' is equivalent to 1 (mod 4)
ie, if 'P' & 1 are each divided by 4, it gives the same remainder 1
like prime number 97 is equivalent to 1(mod 4) , OR 97 = 4k + 1 so 97 can be expressed as the sum of 2 squares
but prime number 11 is not equivalent to 1 (mod 4) , OR 11 not equal to 4k + 1, so 11 can not be expressed as the sum of two squares.
97 = 9^2 + 4^2
2️⃣🔹:To find
No! Every number can not be written as the sum of two squares..
Here is how we find which number can be expressed as the sum of two squares…
1️⃣🔹: Prime number 'P' can be expressed as a^2 + b^2 with 'a' & 'b' as integers
if and only if 'P' is equivalent to 1 (mod 4)
ie, if 'P' & 1 are each divided by 4, it gives the same remainder 1
like prime number 97 is equivalent to 1(mod 4) , OR 97 = 4k + 1 so 97 can be expressed as the sum of 2 squares
but prime number 11 is not equivalent to 1 (mod 4) , OR 11 not equal to 4k + 1, so 11 can not be expressed as the sum of two squares.
97 = 9^2 + 4^2
2️⃣🔹:To find if any number 'n' ( not necessarily prime) is expressible as the sum of two squares or not….
First find its prime factors. If no prime factor is equivalent to 3 mod 4 ie no prime factor is of the form 4k + 3, then 'n' can be expressed as a^2 + b^2
244 = 2^2 x 61 can be expressed as 10^2 + 12^2
because here none of the prime factor is equivalent to 3 mod 4
34 = 2 x 17 , here none of the prime factors is equivalent to 3 mod 4 ,
=> 34 = 5^2 + 3^2
20 = 2^2 x 5 , here none of the prime factors are equivalent to 3 mod 4
=> 20 = 4^2 + 2^2
3️⃣🔹: 'n' can be expressed as the sum of two squares if its every prime factor , which is of the form 4k+3, or we can say every prime factor , which is equivalent to 3 mod 4 , has an even exponent.
If any such prime factor has odd exponent then 'n' can not be expressed as a^2 + b^2
Like 882 = 2 x 3^2 x 7^2 , here 3 & 7 both are equivalent to 3 mod 4 & both have even exponent.
=> 882 = 21^2 + 21^2
But, 294 = 2 x 3 x 7^2, here prime factor 7 is equivalent to 3 mod 4 & has even exponent. But prime factor 3 is equivalent to 3 mod 4 but it has odd exponent. So 294 can not be expressed as the sum of two squares
612 = 2^2 x 3^2 x 17
here, 3 is the prime factor in the form of 4k + 3 & it has even exponent
=> 612 = 24^2 + 6^2
but 204 = 2^2 x 3 x 17 can not be expressed as the sum of 2 squares , since 3 is in the form of 4k + 3 , having an odd exponent.
51 = 3 x 17 , here 3 is equivalent to 3 mod 4 but 3 has odd exponent , so 51 can not be expressed as the sum of two squares
Similarly 3^3 x 2 = 54 can not be expressed as a^2 + b^2 , as 3 is equivalent to 3 mod 4 , but 3 has odd exponent.
but 3^2 x 2 = 18 = 3^2 + 3^2, here 3 is equivalent to 3 mod 4 & has an even exponent..
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David Shaffer
Bachelors degree in Maths and Physics from a long time ago
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Upvoted by
Horst H. von Brand
, PhD Computer Science & Mathematics, Louisiana State University (1987) and
Alon Amit
, Lover of math. Also, Ph.D. · Author has 2K answers and 2.4M answer views
·
6y
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Is there a number that can be expressed in two different sums of squares (i.e. x=a^2+b^2=c^2+d^2, where a, b, c and d are all distinct non-zero natural numbers)?
Yes, there are infinitely many. For example 342+312=462+12=2117.
To see this we can rearrange the equation:
a2+b2=c2+d2
a2−c2=d2−b2
(a+c)(a−c)=(d+b)(d−b)
And now we can pick any number that factorizes in two different ways (provided the factors differ by an even number). For example I picked the number 3×5×7×11, noting that
(3×11)×(5×7)=(7×11)×(3×5)
33×35=77×15
[math]34^2+31^2=46^2+1^2=2117[/math]
Arbash Nazeer
Number theory aficionado
·
Upvoted by
David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 107 answers and 100K answer views
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Updated 5y
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Can every number be written as the sum of two squares?
Take the chosen number [math]n[/math] and write out its prime factorization [math]n=p_1^{e_1}p_2^{e_2} \ldots p_m^{e_m}[/math]. Figure out which are the primes that are of the form [math]p \equiv 3 \pmod 4[/math] i.e. the primes when divided by 4 gives a remainder of 3. If all such primes [math]p_i[/math] have an even exponent [math]e_i[/math], then [math]n[/math] can be written as the sum of two squares [math]a^2+b^2 \ a,b \in \Z[/math]. If at least one such prime [math]p_i[/math] has an odd exponent [math]e_i[/math], then it cannot be represented as the sum of two squares.
Sum of two squares theorem - Wikipedia
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David Smith
BSc (Hons) in Mathematics & Computer Science, University of Bristol (Graduated 1986)
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Author has 3.6K answers and 4M answer views
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3y
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Is it possible to write any natural number as the sum or difference of two perfect squares?
No.
Every odd natural number is the difference of two squares:
[math]\quad 2n+1\ =\ (n+1)^2-n^2[/math]
For even numbers: every multiple of four is the difference of two squares:
[math]\quad 4n\ =\ (n+1)^2-(n-1)^2[/math]
Even numbers which are not multiples of four cannot be the difference of two squares. But they may sometimes (but not always) be the sum of two squares. When they are the sum of two squares, those squares will both be odd.
In particular, all numbers which leave a remainder of [math]6[/math] when divided by [math]8[/math] cannot be written as either the sum or difference of two squares (e.g. [math]6,\ 14,\ 22,\ 30\dots[/math])
Those which leave a re
No.
Every odd natural number is the difference of two squares:
[math]\quad 2n+1\ =\ (n+1)^2-n^2[/math]
For even numbers: every multiple of four is the difference of two squares:
[math]\quad 4n\ =\ (n+1)^2-(n-1)^2[/math]
Even numbers which are not multiples of four cannot be the difference of two squares. But they may sometimes (but not always) be the sum of two squares. When they are the sum of two squares, those squares will both be odd.
In particular, all numbers which leave a remainder of [math]6[/math] when divided by [math]8[/math] cannot be written as either the sum or difference of two squares (e.g. [math]6,\ 14,\ 22,\ 30\dots[/math])
Those which leave a remainder of [math]2[/math] when divided by [math]8[/math] can sometimes be written as the sum of two squares (e.g. [math]2,\ 10,\ 18,\ 26,\ 34,\ 50[/math] but NOT [math]42[/math]).
Mark Gritter
recreational mathematician
·
Author has 5.7K answers and 11.7M answer views
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9y
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Is there any number which can be expressed as the sum of two squares as well as two cubes also?
A001481 - OEIS is a list of numbers that are sums of two squares.
A003325 - OEIS is a list of numbers that are sums of two cubes.
You can just eyeball them for common numbers, for example 65 and 128. The latter is unfortunately a trivial example, but
[math]65 = 1^2 + 8^2 = 4^2 + 7^2 = 1^3 + 4^3[/math]
Obviously if we have one solution then we can generate many more, since if [math]a^2 + b^2 = c^3 + d^3[/math] then
math^3+(4d)^3 = 64(c^3 + d^3) = 64(a^2 + b^2) = 64a^2 + 64b^2 = (8a)^2+(8b)^2 [/math]
Here's the smallest number I found that is representable as a sum of squares or sum of cubes in two different ways (excluding cases
A001481 - OEIS is a list of numbers that are sums of two squares.
A003325 - OEIS is a list of numbers that are sums of two cubes.
You can just eyeball them for common numbers, for example 65 and 128. The latter is unfortunately a trivial example, but
[math]65 = 1^2 + 8^2 = 4^2 + 7^2 = 1^3 + 4^3[/math]
Obviously if we have one solution then we can generate many more, since if [math]a^2 + b^2 = c^3 + d^3[/math] then
math^3+(4d)^3 = 64(c^3 + d^3) = 64(a^2 + b^2) = 64a^2 + 64b^2 = (8a)^2+(8b)^2 [/math]
Here's the smallest number I found that is representable as a sum of squares or sum of cubes in two different ways (excluding cases where a=b or c=d):
[math]4624776 = 1026^2 + 1890^2 = 1350^2 + 1674^2 = 51^3 + 165^3 = 72^3 + 162^3[/math]
Alain Merigot
Former Have taught Computer Architecture for 30 Years
·
Author has 54 answers and 34.5K answer views
·
2y
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Can every number be written as the sum of two squares (positive integers)?
Definitely NO. 3=1+2 is NOT the sum of two squares. Neither are 6, 7, 11, etc.
Probably you made a confusion with other related problems:
Lagrange theorem states that all numbers can be expressed as the sum of 4 squares Lagrange's four-square theorem - Wikipedia
Legendre 3 square theorem states that many numbers can be expressed as the sum of 3 squares (precisely all numbers, if and only if they are not of the form [math]4^a\times(8b+7)[/math] with [math]a,b \geq0[/math]) Legendre's three-square theorem - Wikipedia
Aleš Mihev
M.Sc. in Computer Science, University of Ljubljana (Graduated 1988)
·
Author has 791 answers and 235.9K answer views
·
1y
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Is it possible to determine if an unknown positive integer can be represented as the sum of two squares, excluding trivial cases such as being one less than a square?
Yes, this is indeed possible (but the integer must of course be known).
Theorem (sum of two squares):
An integer [math]n > 1[/math] can be written as a sum of two squares if and only if the prime decomposition of [math]n[/math] does not contains any odd power of a prime [math]p[/math], such that [math]p \bmod 4 = 3[/math]. In other words, let
[math]n = p_1^{k_1} \times p_2^{k_2} \times \cdots \times p_m^{k_m}[/math]
be the prime decomposition of the integer [math]n[/math]. Then the following holds:
[math]n = a^2 + b^2 \iff p_i \not \equiv 3 \pmod{4}, \text{ for any } n, a, b \in N, n > 1, \text{ and } k_i \text{ is odd. } \blacksquare[/math]
For example, let [math]n = 4165.[/math] Its prime decomposit
Yes, this is indeed possible (but the integer must of course be known).
Theorem (sum of two squares):
An integer [math]n > 1[/math] can be written as a sum of two squares if and only if the prime decomposition of [math]n[/math] does not contains any odd power of a prime [math]p[/math], such that [math]p \bmod 4 = 3[/math]. In other words, let
[math]n = p_1^{k_1} \times p_2^{k_2} \times \cdots \times p_m^{k_m}[/math]
be the prime decomposition of the integer [math]n[/math]. Then the following holds:
[math]n = a^2 + b^2 \iff p_i \not \equiv 3 \pmod{4}, \text{ for any } n, a, b \in N, n > 1, \text{ and } k_i \text{ is odd. } \blacksquare[/math]
For example, let [math]n = 4165.[/math] Its prime decomposition is:
[math]4165 = 5 \times 7^2 \times 17[/math]
Although it does contain a prime [math]7[/math], whose remainder after dividing by [math]4[/math] is [math]3[/math], the corresponding power is [math]2[/math], which is an even number. This means that the number [math]4165[/math] can be represented as a sum of two squares. Indeed, [math]4165 = 14^2 + 63^2[/math].
The proof of the sum of two squares theorem is somewhat lengthy, and I will therefore omit it. Please, consult any textbook on the theory of numbers to find this proof. By the way, it also turns out that any integer can be represented as a sum of 4 squares.
Related questions
Can every number be written as the sum of two squares?
How do I find if a number can be represented as a sum of two squares?
Can every number be written as a sum of two squares or their squares' sums?
Is there any number which can be expressed as the sum of two squares as well as two cubes also?
Can every natural number be written as a sum of two squares?
How can we express number as a sum of 2 squares?
What is the only natural number that cannot be written as the sum of two squares, besides 2 and 3?
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5
How can you determine the values of sine and cosine for special angles such as pi/3 or pi/6?
Brinton Butler
Math Tutor for 18 Yrs and smrt-a$$ for like forev (2006–present)
·
Author has 120 answers and 115.9K answer views
·
1y
There are a couple of special right triangles that help us understand sine, cosine and tangent. The first is a 30–60–90 triangle, with sides in the ratio 1:2:sqrt3. The other is 45–45–90 with sides in the ratio 1:1:sqrt2. Some other things are necessare to remember as well: Sine in a right triangle is defined as Opposite over Hypoteneus, Cosine is Adjacent over Hypoteneuse, and Tangent is Opposite over Adjacent (hence the mnemonic SOC-CAH-TOA). Finally, converting to degrees, pi/3 radians = 60 degrees, and pi/6 radians = 30 degrees.
Putting these all together, the Sine of pi/6 (also the Cosine
There are a couple of special right triangles that help us understand sine, cosine and tangent. The first is a 30–60–90 triangle, with sides in the ratio 1:2:sqrt3. The other is 45–45–90 with sides in the ratio 1:1:sqrt2. Some other things are necessare to remember as well: Sine in a right triangle is defined as Opposite over Hypoteneus, Cosine is Adjacent over Hypoteneuse, and Tangent is Opposite over Adjacent (hence the mnemonic SOC-CAH-TOA). Finally, converting to degrees, pi/3 radians = 60 degrees, and pi/6 radians = 30 degrees.
Putting these all together, the Sine of pi/6 (also the Cosine of pi/3) = 1/2, the Sine of pi/3 (also the Cosine of pi/6) = sqrt3/2, Tangent of pi/6 (also the Cotangent of pi/3) = sqrt3/1 and the Tangent of pi/3 (also the Cotangent of pi/6) = 1/sqrt3 or sqrt3/3.
You can use the 45–45–90 triangle to see that Sine and Cosine of pi/4 = 1/sqrt2 or sqrt2/2 and Tangent and Cotangent of pi/4 = 1.
Hope this helps - a picture would make it clearer but I can’t use drawing tools in this site.
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George Ivey
Former Math Professor at Gallaudet University
·
Author has 23.7K answers and 2.6M answer views
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1y
Imagine an equilateral triangle with each side of length s. Each angle is pi/3.
A line from one vertex to the midpoint of the opposite side divides the equilateral triangles into two two right triangles having hypotenuse of length a and one side of length a/2 so the third side √a2−a2/4=a√1−1/4=a√3/2.
The side opposite the pi/3 angle is a√3/2 and the hypotenuse is a so sin(pi/3) is √32. The near side to the pi/3 angle is a/2 and the hypotenuse is a so cos(pi/3) is 12.
Changing to pi/6 swaps “near” and “opposite” sides so swaps sine and cosine.
s
Imagine an equilateral triangle with each side of length s. Each angle is pi/3.
A line from one vertex to the midpoint of the opposite side divides the equilateral triangles into two two right triangles having hypotenuse of length a and one side of length a/2 so the third side √a2−a2/4=a√1−1/4=a√3/2.
The side opposite the pi/3 angle is a√3/2 and the hypotenuse is a so sin(pi/3) is √32. The near side to the pi/3 angle is a/2 and the hypotenuse is a so cos(pi/3) is 12.
Changing to pi/6 swaps “near” and “opposite” sides so swaps sine and cosine.
sin(pi/6) is 1/2 and cos(pi/6) is √32.
Audrey Liu
studied Algebra 2/Trig in school
·
6y
Related
What is sin of pi (90 degrees)?
First of all, pi (π) as a radian value is equal to 180°, not 90°.
sin(π) = 0, but sin(90°) = 1.
Refer to the unit circle for more info. Take the y value of the coordinates, and that’s the sin value (x value is the cosine value).
Hope I could help!
First of all, pi (π) as a radian value is equal to 180°, not 90°.
sin(π) = 0, but sin(90°) = 1.
Refer to the unit circle for more info. Take the y value of the coordinates, and that’s the sin value (x value is the cosine value).
Hope I could help!
Neil Chopra
Tutor
·
6y
Related
How do you calculate sin (pi/3)?
The sin of pi/3 can be found with the unit circle.
The sine of any angle is just it’s y-value on the unit circle. Pi/3 in radians, or 60 degrees in degrees has the ordered pair (1/2, √3/2), and we will look at the y-value of this ordered pair.
As a result, our answer is that sin(pi/3) = √3/2.
Alternatively, you could plug this into a calculator and get 0.86602540378, which is equal.
The sin of pi/3 can be found with the unit circle.
The sine of any angle is just it’s y-value on the unit circle. Pi/3 in radians, or 60 degrees in degrees has the ordered pair (1/2, √3/2), and we will look at the y-value of this ordered pair.
As a result, our answer is that sin(pi/3) = √3/2.
Alternatively, you could plug this into a calculator and get 0.86602540378, which is equal.
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Mark Regan
Former Inmate Teacher's Aide for G.E.D. at California Department of Corrections and Rehabilitation (1997–2002)
·
Jun 20
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How can one determine if the sine or cosine of an angle is positive or negative in trigonometry?
You draw the angle and build a right triangle within a unit circle (Or you can imagine it.) centered at the Origin, where the angle sits at the origin. Its hypotenuse is a radius drawn from the center of the circle to the perimeter of the circle. A line normal to the x-axis drawn from the intersection of the hypotenuse and the circle determines the sign and value of the cosine. A line normal to the y-axis drawn from the intersection of the hypotenuse and the circle determines the sign and value of the sine.
If the normal line from perimeter of the circle to the x-axis intersects the x-axis at a
You draw the angle and build a right triangle within a unit circle (Or you can imagine it.) centered at the Origin, where the angle sits at the origin. Its hypotenuse is a radius drawn from the center of the circle to the perimeter of the circle. A line normal to the x-axis drawn from the intersection of the hypotenuse and the circle determines the sign and value of the cosine. A line normal to the y-axis drawn from the intersection of the hypotenuse and the circle determines the sign and value of the sine.
If the normal line from perimeter of the circle to the x-axis intersects the x-axis at a point to the left of the origin, the cosine of the angle is negative. If it intersects the x-axis to the right of the origin the cosine is positive. If the angle terminates at a point above the x-axis the sine is positive. If the angle terminates at a point below the x-axis the sine is negative.
Lucas Curtis
lover of numbers
·
Author has 7.4K answers and 22.1M answer views
·
2y
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How do I find cos pi/3?
I assume π3 is measured in radians.
A full circle (360∘) is 2π radians, which means that a semi-circle (180∘) is π radians. So π3 radians is one-third of a semi-circle, or 60∘.
The basic trig functions can be defined in a few different ways. Here we’re interested in this definition:
The cosine of an angle θ is the x-coordinate of the intersection between the terminal ray of that angle and the unit circle, provided the angle’s vertex is at the origin.
In other words, we want to know the x-coordinate of this point:
Now for π3 radians and
I assume π3 is measured in radians.
A full circle (360∘) is 2π radians, which means that a semi-circle (180∘) is π radians. So π3 radians is one-third of a semi-circle, or 60∘.
The basic trig functions can be defined in a few different ways. Here we’re interested in this definition:
The cosine of an angle θ is the x-coordinate of the intersection between the terminal ray of that angle and the unit circle, provided the angle’s vertex is at the origin.
In other words, we want to know the x-coordinate of this point:
Now for π3 radians and a few other special angles, there is some helpful geometry that can help us figure out the values of sine and cosine. As I said before, π3 radians is the same as 60∘, which means we have two sides of an equilateral triangle here.
Since ¯¯¯¯¯¯¯¯AB=¯¯¯¯¯¯¯¯BC, we can say that the bisector of ∠B intersects ¯¯¯¯¯¯¯¯AC directly at its midpoint.
And since ¯¯¯¯¯¯¯¯AC lies along the x-axis, its midpoint has the same x-coordinate as B, which is 12.
Ergo, the x-coordinate of B is 12 and cos(π3)=12.
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Gary Ward
MaEd in Education & Mathematics, Austin Peay State University (Graduated 1997)
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Author has 4.9K answers and 7.6M answer views
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3y
Related
How can you find an angle from its sine, cosine or tangent values?
How can you find an angle from its sine, cosine or tangent values?
Take the inverse sine, cosine or tangent of the angle.
sin−1θ,cos−1θortan−1θ
sometimes written as arcsine, arccosine or arctangent.
On your calculator, it is usually Shift followed by the function key.
If you are old school, find the ratio in the body of the table closest to the ratio given for that particular function and read the degree and minutes that produced it to the left and above that value. Interpolating will give a better answer if you are using an abbreviated table.
Michael Oyekunle
Bachelor of Engineering from University of Ilorin (Graduated 1991)
·
Author has 831 answers and 74.9K answer views
·
9mo
Related
How can one determine if the sine or cosine of an angle is positive or negative in trigonometry?
I believe you have heard of the four cardinal points before. These are the North, East, South, and the West. These cardinal points are separated by an angle of 90° from the one adjacent or next to it. That means an angle of 90° separate the North from the East, and 90° separate North from the West. Likewise 90° each separate the South from the East and from the West. Each 90° between North and East, North and West, South and West, South and East are called quadrants. Angles are always measured from the East in the anticlockwise direction.
The 1st quadrant is the one between North and East. Here
I believe you have heard of the four cardinal points before. These are the North, East, South, and the West. These cardinal points are separated by an angle of 90° from the one adjacent or next to it. That means an angle of 90° separate the North from the East, and 90° separate North from the West. Likewise 90° each separate the South from the East and from the West. Each 90° between North and East, North and West, South and West, South and East are called quadrants. Angles are always measured from the East in the anticlockwise direction.
The 1st quadrant is the one between North and East. Here the angles measured from the East to the North are from 0° to 90° and all of sine, cosine and tangent are positive e.g given angle 89°, sine of the angle is sin(89°), likewise we have cos(89°) and tan(89°)
In the 2nd quadrant, the angles are from 90° to 180°. Here, only sine is positive while cosine and tangent values of angles are negative. You obtain the acute angle by subtracting the given angle from 180° e.g an angle of 119°=180–119=61°.
So sin(119°)=sin(61°) , cos(119°)= -cos(61°) and tan(119°)= -tan(61°)
In the 3rd quadrant, the angles are from 180° to 270°. Here only tangent is positive while sine and cosine are negative. To obtain the acute angle value of given angle, you subtract 180 from the given angle e.g angle 237° is equal to 237°–180°=57°.
So sin(237°)= -sin(57°), cos(237°)= -cos(57°) while tan(237°)=tan(57°)
In the 4th quadrant, the angles are from 270° to 360°. Here, only cosine is positive while sine and tangent are negative. To obtain the value of the acute angle of the given angle, subtract the given angle from 360° e.g angle of 296° is equal to 360°-296°=64°.
So sin(296°)= -sin(64°), cos(296°)=cos(64°) and tan(296°)= -tan(64°)
Note : angles are always measured from the East in the anticlockwise direction.
Angle 210° is in the 3rd quadrant. So acute angle value of 210°=210°-180°=30° as you can see in the diagram. The sine valve is -sin(30) and for cosine is -cos(30°) and for tangent is tan(30°)
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Making smart financial decisions doesn't have to be complicated. In 2025, there are several simple yet highly effective money hacks that can make a huge difference in your financial health. These aren't complicated investment strategies; they are practical, everyday habits that help you keep more of your hard-earned money.
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Karen Slinger
Former Former math (inc. AP Calc & Geometry) teacher at New Jersey (state) (1986–2019)
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Author has 507 answers and 363.9K answer views
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1y
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How can we determine the sign of the sine and cosine values of an angle in radians?
Given the unit circle - circle with radius 1 centered on the origin, the cosine of the angle of rotation (from the positive x axis), is the x coordinate of the point on the unit circle, and the sine of the angle is the y coordinate.
So, since x coordinates are positive in quadrants 1 and 4, the sign of the cosine of the angle is positive in quadrants 1 & 4, and negative in quadrants 2 & 3.
Since y coordinates are positive in quadrants 1 & 2, the sign of the sine of the angle is positive in quadrants 1 & 2, negative in quadrants 3 & 4.
As far as measuring the angle in radians, from 0 to π/2 ( + 2π
Given the unit circle - circle with radius 1 centered on the origin, the cosine of the angle of rotation (from the positive x axis), is the x coordinate of the point on the unit circle, and the sine of the angle is the y coordinate.
So, since x coordinates are positive in quadrants 1 and 4, the sign of the cosine of the angle is positive in quadrants 1 & 4, and negative in quadrants 2 & 3.
Since y coordinates are positive in quadrants 1 & 2, the sign of the sine of the angle is positive in quadrants 1 & 2, negative in quadrants 3 & 4.
As far as measuring the angle in radians, from 0 to π/2 ( + 2πn) is quadrant 1, from π/2 to π (+2πn) is quadrant 2, from π to 3π/2 (+2πn) is quadrant 3, and from 3π/2 to 2π (+2πn) is quadrant 4.
So - from 0 to π/2 (quadrant 1), both sine and cosine are positive
From π/2 to π (quadrant 2), cosine is negative and sine is positive
From π to 3π/2 (quadrant 3), both sine and cosine are negative
From 3π/2 to 2π (quadrant 4), cosine is positive and sine is negative :)
Kris Walker
BSc in Physics & Astrophysics, Monash University (Graduated 2022)
·
Author has 256 answers and 630.1K answer views
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8y
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In a unit circle, why is the sine of pi/2 equal to 1?
Pictured below is the relationship between angle, sine, and cosine in a unit circle:
As you can see,
sin
θ
is the length of a line running from the
x
-axis to where the unit line contacts the circle. As such, when
θ
=
π
2
(
) , the line of
[math]\sin{\theta}[/math]
is projected directly upwards from the origin, giving it a length of, in accordance with the definition of a unit circle,
[math]1[/math]
.
Pictured below is the relationship between angle, sine, and cosine in a unit circle:
As you can see,
[math]\sin{\theta}[/math]
is the length of a line running from the
x
-axis to where the unit line contacts the circle. As such, when
[math]\theta=\frac{\pi}{2}[/math]
(
[math]90°[/math]
) , the line of
[math]\sin{\theta}[/math]
is projected directly upwards from the origin, giving it a length of, in accordance with the definition of a unit circle,
[math]1[/math]
.
Dean Rubine
Been doing high school math since high school, circa 1975
·
Upvoted by
Siddhant Grover
, MSc in Statistics & BSc Mathematics, Hindu College, University of Delhi (2022) · Author has 10.6K answers and 23.7M answer views
·
7y
Related
How do I find the exact value of sine (11pi/12) using the angle sum identity for sine and the exact values of sine (3pi/4) and cosine (pi/6)?
The sum angle formula for sine is
[math]\sin(a+b) = \sin a \cos b + \cos a \sin b[/math]
so we need the sine and cosine of both [math]a[/math] and [math]b[/math] to run it.
[math]\sin(\dfrac{11 \pi}{12}) = \sin (\dfrac{3\pi}{4} + \dfrac{\pi}{6} )[/math]
[math]= \sin (\dfrac{3\pi}{4} ) \cos( \dfrac{\pi}{6} ) + \cos (\dfrac{3\pi}{4} ) \sin ( \dfrac{\pi}{6} )[/math]
Let’s use [math]\sin(\pi - a) = \sin a[/math] and [math]\cos(\pi - a)= -\cos a[/math] to rewrite [math]\sin(3\pi/4).[/math]
[math]= \sin (\dfrac{\pi}{4} ) \cos( \dfrac{\pi}{6} ) - \cos (\dfrac{\pi}{4} ) \sin ( \dfrac{\pi}{6} )[/math]
Now we reached the two examples of trig ([math]45^\circ[/math] and [math]30^\circ[/math]) and we can fill in the values,
[math]= \dfrac{ \sqrt{2}} {2} \cdot[/math]
The sum angle formula for sine is
[math]\sin(a+b) = \sin a \cos b + \cos a \sin b[/math]
so we need the sine and cosine of both [math]a[/math] and [math]b[/math] to run it.
[math]\sin(\dfrac{11 \pi}{12}) = \sin (\dfrac{3\pi}{4} + \dfrac{\pi}{6} )[/math]
[math]= \sin (\dfrac{3\pi}{4} ) \cos( \dfrac{\pi}{6} ) + \cos (\dfrac{3\pi}{4} ) \sin ( \dfrac{\pi}{6} )[/math]
Let’s use [math]\sin(\pi - a) = \sin a[/math] and [math]\cos(\pi - a)= -\cos a[/math] to rewrite [math]\sin(3\pi/4).[/math]
[math]= \sin (\dfrac{\pi}{4} ) \cos( \dfrac{\pi}{6} ) - \cos (\dfrac{\pi}{4} ) \sin ( \dfrac{\pi}{6} )[/math]
Now we reached the two examples of trig ([math]45^\circ[/math] and [math]30^\circ[/math]) and we can fill in the values,
[math]= \dfrac{ \sqrt{2}} {2} \cdot \dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2}[/math]
[math]\sin(\dfrac{11 \pi}{12}) = \frac 1 4 (\sqrt{6} - \sqrt{2})[/math]
[math]11\pi/12[/math] is [math]165^\circ, [/math] constructible as [math]135^\circ + 30 ^\circ[/math] as the problem asks, with [math]135^\circ = 45^\circ + 90^\circ[/math] also easily constructible. Any constructible length, such as the cosines and sines of these constructible angles, is gotten by combining integers with the operations of addition, subtraction, multiplication, division and square root taking. So they all start to look kinda like this one, though they can get very complicated with nested square roots and fractions.
Angle trisection generally requires cube root taking, so is not constructible.
Another approach uses the half angle formula. [math]165^o[/math] is half of [math]330^o,[/math] which is [math]-30^o.[/math]
[math]\sin \dfrac \theta 2 = \pm \sqrt{ \frac 1 2 (1 - \cos \theta) }[/math]
We can choose the positive sign for [math]\sin 165^o[/math] because we know we’re in the second quadrant.
[math]\sin 165^o = \sin(\frac 1 2 (330^o)) = \sqrt{ \frac 1 2 (1 - \cos 30^o) } = \sqrt{\frac 1 2 (1 - \frac{\sqrt{3}}{2}}) = \frac 1 2 \sqrt{2 - \sqrt{3}}[/math]
So we’re getting into Ramanujan territory where we discover denestings:
[math] \frac 1 4 (\sqrt{6} - \sqrt{2}) = \frac 1 2 \sqrt{2 - \sqrt{3}}[/math]
Mike Xavier
Former Student
·
Author has 64 answers and 230.2K answer views
·
8y
Related
Is there a mathematical representation of sine, cosine and tangent, like how pi is 22/7?
Good question!
To start off, trig functions can be derived a number of ways, but the most fundamental way they are derived is through right triangles and their ratios of sides relating to their angle. Here is the unit circle that defines sine and cosine. From the two, we can derive tangent as the ratio of sine to cosine.
Now, the relationship between the ratio, often through the units of radians, to the angle can be given by the trigonometric functions graphed with the angle on the x-axis and the value of the ratio on the y-axis.
It is important to note though, as Wikipedia states for sine “More
Good question!
To start off, trig functions can be derived a number of ways, but the most fundamental way they are derived is through right triangles and their ratios of sides relating to their angle. Here is the unit circle that defines sine and cosine. From the two, we can derive tangent as the ratio of sine to cosine.
Now, the relationship between the ratio, often through the units of radians, to the angle can be given by the trigonometric functions graphed with the angle on the x-axis and the value of the ratio on the y-axis.
It is important to note though, as Wikipedia states for sine “More modern definitions express the sine as an infinite series or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers”. In school, sine is taught from an elementary viewpoint with triangles, but as Wikipedia says, it came to be used as a practical function for solutions and modeling. Consider an oscillating pendulum:
The graph of cosine is useful in modeling this real world example.
If I wasn’t clear in parts of my explanation or I didn’t answer your question, please let me know!
Source: Sine - Wikipedia
Therion Tiberius Ware
Video On Demand Eng in Broadcast TV. Born 22,721 Days ago.
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6y
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What is sin of pi (90 degrees)?
Q: What is sin of pi (90 degrees)?
The are 2 pi radians in a circle, so 90 degrees = pi/2. that said the SIN of 90 degrees is 1, and the SIN of 180 degrees is 0.
Actually, the SIN of pi is gluttony, which is the first of the seven deadly sins! Which must mean something, to someone, somewhere!
Gula (gluttony)
Lu...
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7043 | https://europepmc.org/article/med/7431865 | Chromodacryorrhea in laboratory rats (Rattus norvegicus): etiologic considerations. - Abstract - Europe PMC
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Chromodacryorrhea in laboratory rats (Rattus norvegicus): etiologic considerations.
Harkness JE,
Ridgway MD
Laboratory Animal Science, 01 Oct 1980, 30(5):841-844
PMID: 7431865
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Abstract
Chromodacryorrhea was produced in 5, 7, and 16 week old intact and adrenalectomized rats following intravenous injection of acetylcholine or acute stress induced by limb restraint. Atropine blocked the lacrimal response to both stimuli, and epinephrine and corticosterone had no dacryogenic effects. The latent period between the onset of limb restraint and red tear release averaged 16 minutes.
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Pathology of tumours in laboratory animals. Tumours of the rat. Tumours of the salivary and lacrimal glands.
Elwell MR, Leininger JR
IARC Sci Publ, (99):89-107, 01 Jan 1990
Cited by: 3 articles | PMID: 2093670
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Pathology of tumours in laboratory animals. Tumours of the rat. Tumours of the Harderian gland.
Elwell MR, Boorman GA
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Chromodacryorrhea in rats: absence following soman poisoning.
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Gaertner DJ, Lindsey JR, Stevens JO
Lab Anim Sci, 38(1):79-82, 01 Feb 1988
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7044 | https://handwiki.org/wiki/Helly%27s_selection_theorem | Helly's selection theorem - HandWiki
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Short description: On convergent subsequences of functions that are locally of bounded total variation
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematicianEduard Helly. A more general version of the theorem asserts compactness of the space BV loc of functions locally of bounded total variation that are uniformly bounded at a point.
The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.
[x]
Contents
1 Statement of the theorem
2 Generalisation to BV loc
3 Further generalizations
4 See also
5 References
Statement of the theorem
Let (f n)n∈N be a sequence of increasing functions mapping the real line R into itself, and suppose that it is uniformly bounded: there are a,b∈R such that a≤f n≤b for every n∈N. Then the sequence (f n)n∈N admits a pointwise convergent subsequence.
Generalisation to BV loc
Let U be an open subset of the real line and let f n:U→R, n∈N, be a sequence of functions. Suppose that
(f n) has uniformly boundedtotal variation on any W that is compactly embedded in U. That is, for all sets W⊆U with compactclosureW̄⊆U,
sup n∈N(‖f n‖L 1(W)+‖d f n d t‖L 1(W))<+∞,where the derivative is taken in the sense of tempered distributions;
and (f n) is uniformly bounded at a point. That is, for some t∈U, {f n(t)|n∈N}⊆R is a bounded set.
Then there exists a subsequencef n k, k∈N, of f n and a function f:U→R, locally of bounded variation, such that
f n k converges to f pointwise;
and f n k converges to f locally in L 1 (see locally integrable function), i.e., for all W compactly embedded in U,
lim k→∞∫W|f n k(x)−f(x)|d x=0;
and, for W compactly embedded in U,
‖d f d t‖L 1(W)≤lim inf k→∞‖d f n k d t‖L 1(W).
Further generalizations
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, is due to Barbu and Precupanu:
Let X be a reflexive, separable Hilbert space and let E be a closed, convex subset of X. Let Δ:X→[0,+∞) be positive-definite and homogeneous of degree one. Suppose that z n is a uniformly bounded sequence in BV([0,T];X) with z n(t)∈E for all n∈N and t∈[0,T]. Then there exists a subsequence z n k and functions δ,z∈BV([0,T];X) such that
for all t∈[0,T],
∫[0,t)Δ(d z n k)→δ(t);
and, for all t∈[0,T],
z n k(t)⇀z(t)∈E;
and, for all 0≤s<t≤T,
∫[s,t)Δ(d z)≤δ(t)−δ(s).
See also
Bounded variation
Fraňková-Helly selection theorem
Total variation
References
Rudin, W. (1976). Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill. p.167. ISBN978-0070542358.
Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co.. p.xviii+397. ISBN90-277-1761-3.MR860772
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7045 | https://artofproblemsolving.com/wiki/index.php/Vieta%27s_Formulas?srsltid=AfmBOopVMtvQSGOtQXgBpylUtEsh6i0IhEW9fAbdXr5j-_C4hIBB-Q5z | Art of Problem Solving
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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
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How to calculate corner positions/marks of a rotated/tilted rectangle?
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I've got two elements, a 2D point and a rectangular area. The point represents the middle of that area. I also know the width and height of that area. And the area is tilted by 40° relative to the grid.
Now I'd like to calculate the absolute positions of each corner mark of that tilted area only using this data. Is that possible?
2d
rotation
grid
computational-geometry
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edited Aug 20, 2018 at 4:29
Pikalek
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asked Nov 2, 2014 at 0:14
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cpp
X = xcos(θ) - ysin(θ)
Y = xsin(θ) + ycos(θ)
This will give you the location of a point rotated θ degrees around the origin. Since the corners of the square are rotated around the center of the square and not the origin, a couple of steps need to be added to be able to use this formula. First you need to set the point relative to the origin. Then you can use the rotation formula. After the rotation you need to move it back relative to the center of the square.
```cpp
// cx, cy - center of square coordinates
// x, y - coordinates of a corner point of the square
// theta is the angle of rotation
// translate point to origin
float tempX = x - cx;
float tempY = y - cy;
// now apply rotation
float rotatedX = tempXcos(theta) - tempYsin(theta);
float rotatedY = tempXsin(theta) + tempYcos(theta);
// translate back
x = rotatedX + cx;
y = rotatedY + cy;
```
Apply this to all 4 corners and you are done!
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edited Feb 2, 2015 at 4:20
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answered Nov 2, 2014 at 18:05
AholioAholio
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A square with equal widths on all sides is fine, but what about rectangles?Rin –Rin 2023-02-09 00:49:34 +00:00 Commented Feb 9, 2023 at 0:49
@Rin This applies to any geometrical shape. As per the answer, the point is "rotated around the center". So it's about rotating one point around another. The actual shape of the "thing", these two points belong to, doesn't matter. You can apply the same logic, if instead of rotating everything around the shape's center, you can rotate it around one of its corners.Stacky –Stacky 2023-03-06 09:42:00 +00:00 Commented Mar 6, 2023 at 9:42
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It is a common technique to rotate a point about a pivot by translating to a coordinate system where the pivot is the origin, then rotating about this origin, then translating back to world coordinates. (A very good explanation of this approach is available at Khan Academy)
However you are not storing your rectangle corners in world coordinates so we can tailor an approach to suit the data you have available.
```cpp
Cx, Cy // the coordinates of your center point in world coordinates
W // the width of your rectangle
H // the height of your rectangle
θ // the angle you wish to rotate
//The offset of a corner in local coordinates (i.e. relative to the pivot point)
//(which corner will depend on the coordinate reference system used in your environment)
Ox = W / 2
Oy = H / 2
//The rotated position of this corner in world coordinates
Rx = Cx + (Ox cos(θ)) - (Oy sin(θ))
Ry = Cy + (Ox sin(θ)) + (Oy cos(θ))
```
This approach can then be easily applied to the other three corners.
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edited Nov 2, 2014 at 16:42
answered Nov 2, 2014 at 16:31
Kelly ThomasKelly Thomas
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Based on the other answers, and to complement them, I managed to create an example with P5 here.
Here is the code, in case you want to access it directly:
```javascript
function setup() {
createCanvas(400, 400);
}
var count = 0;
function draw() {
background(250);
rectMode(CENTER);
stroke(0,0,255);
fill(0,0,255);
count += 1;
var box1X = 100;
var box1Y = 100;
var box2X = 160;
var box2Y = 100;
var box1R = count;
var box2R = -60-count;
var box1W = 50;
var box1H = 50;
var box2W = 50;
var box2H = 50;
translate(box1X, box1Y);
rotate(radians(box1R));
rect(0, 0, box1W, box1H);
rotate(radians(-box1R));
translate(-box1X, -box1Y);
translate(box2X, box2Y);
rotate(radians(box2R));
rect(0, 0, box2W, box2H);
rotate(radians(-box2R));
translate(-box2X, -box2Y);
stroke(255,0,0);
fill(255,0,0);
var pointRotated = [];
pointRotated.push(GetPointRotated(box1X, box1Y, box1R, -box1W/2, box1H/2)); // Dot1
pointRotated.push(GetPointRotated(box1X, box1Y, box1R, box1W/2, box1H/2)); // Dot2
pointRotated.push(GetPointRotated(box1X, box1Y, box1R, -box1W/2, -box1H/2)); // Dot3
pointRotated.push(GetPointRotated(box1X, box1Y, box1R, box1W/2, -box1H/2)); // Dot4
pointRotated.push(createVector(box1X, box1Y)); // Dot5
for (var i=0;i<pointRotated.length;i++){
ellipse(pointRotated[i].x,pointRotated[i].y,3,3);
}
}
function GetPointRotated(X, Y, R, Xos, Yos){
// Xos, Yos // the coordinates of your center point of rect
// R // the angle you wish to rotate
//The rotated position of this corner in world coordinates
var rotatedX = X + (Xos cos(radians(R))) - (Yos sin(radians(R)))
var rotatedY = Y + (Xos sin(radians(R))) + (Yos cos(radians(R)))
return createVector(rotatedX, rotatedY)
}
```
```xml
```
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edited Feb 6, 2017 at 11:57
answered Feb 4, 2017 at 20:55
DanfusDanfus
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Possibly, there is some optimizations available by dividing the problem into two:
compute the center of the top and the bottom side i. e., center + the rotated height/2.
compute the corners relative to these center points using the rotated width/2
Compute the actual sine and cosine once and for all.
Code below, here the rectangle is called ruler. ruler.x, ruler,y is the rectangle center.
```php
/ Middle point on rulers's top side. /
function getRulerTopMiddle(cos, sin) {
return {
x : ruler.x + sin ruler.height/2,
y : ruler.y - cos ruler.height/2
}
}
/ Middle point on rulers's bottom side. /
function getRulerBottomMiddle(cos, sin) {
return {
x : ruler.x - sin ruler.height/2,
y : ruler.y + cos ruler.height/2
}
}
/ Update ruler's four corner coordinates. /
function getRulerCorners() {
const sin = Math.sin(ruler.angle);
const cos = Math.cos(ruler.angle);
const topMiddle = getRulerTopMiddle(cos, sin);
const bottomMiddle = getRulerBottomMiddle(cos, sin);
ruler.nw = {
x: topMiddle.x - (cos ruler.width/2),
y: topMiddle.y - (sin ruler.width/2)
}
ruler.ne = {
x: topMiddle.x + (cos ruler.width/2),
y: topMiddle.y + (sin ruler.width/2)
}
ruler.sw = {
x: bottomMiddle.x - (cos ruler.width/2),
y: bottomMiddle.y - (sin ruler.width/2)
}
ruler.se = {
x: bottomMiddle.x + (cos ruler.width/2),
y: bottomMiddle.y + (sin ruler.width/2)
}
}
```
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answered Feb 25, 2017 at 7:41
Alec LeamasAlec Leamas
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Refactoring the code above gives a cleaned up form which also highlights the simple fact that each corner is basically center + height/2 + width/2, with signs as appropriate for each corner. This also holds if you treat height/2 and width/2 as rotated vectors.
Trusting the interpreter to inline the helpers, this should be pretty effective, should we try to benchmark this.
```matlab
function addPoints(p1, p2) {
return { x: p1.x + p2.x, y: p1.y + p2.y }
}
function subPoints(p1, p2 ) {
return { x: p1.x - p2.x, y: p1.y - p2.y }
}
function multPoints(p1, p2 ) {
return { x: p1.x p2.x, y: p1.y p2.y }
}
function getRulerCorners() {
const sin = Math.sin(ruler.angle);
const cos = Math.cos(ruler.angle);
const height = { x: sin ruler.height/2, y: cos ruler.height/2 };
const heightUp = addPoints(ruler, multPoints({x: 1, y :-1}, height));
const heightDown = addPoints(ruler, multPoints({x: -1, y: 1}, height));
const width = { x: cos ruler.width/2, y: sin ruler.width/2 };
ruler.nw = subPoints(heightUp, width);
ruler.ne = addPoints(heightUp, width );
ruler.sw = subPoints(heightDown, width);
ruler.se = addPoints(heightDown, width);
}
```
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edited Feb 28, 2017 at 15:23
Gnemlock
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answered Feb 28, 2017 at 8:42
leamasleamas
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See the Wikipedia article on rotation. The essence is this:
(1) If c is the center point, then the corners are c + (L/2,W/2), +/- etc., where L and W are the length & width of the rectangle.
(2) Translate the rectangle so that center c is at the origin, by subtracting c from all four corners.
(3) Rotate the rectangle by 40 deg via the trig formulas cited.
(4) Translate back by adding c to each coordinate.
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edited Nov 2, 2014 at 15:33
answered Nov 2, 2014 at 1:33
Joseph O'RourkeJoseph O'Rourke
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2
Thanks for your answer but I'm afraid I don't get it. How am I supposed to substract the center (known) from the corners (unknown) if they are unknown? I mean, the coordinates of the corners are the very things I'm trying to find out.Stacky –Stacky 2014-11-02 10:40:03 +00:00 Commented Nov 2, 2014 at 10:40
I tried to clarify.Joseph O'Rourke –Joseph O'Rourke 2014-11-02 15:33:42 +00:00 Commented Nov 2, 2014 at 15:33
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A little late, but here's a compact function I've used. It calculates the top and left points, then just flips them for the opposite corners.
```java
rotatedRect(float x, float y, float halfWidth, float halfHeight, float angle)
{
float c = cos(angle);
float s = sin(angle);
float r1x = -halfWidth c - halfHeight s;
float r1y = -halfWidth s + halfHeight c;
float r2x = halfWidth c - halfHeight s;
float r2y = halfWidth s + halfHeight c;
// Returns four points in clockwise order starting from the top left.
return
(x + r1x, y + r1y),
(x + r2x, y + r2y),
(x - r1x, y - r1y),
(x - r2x, y - r2y);
}
```
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answered May 6, 2019 at 22:25
cmanncmann
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Old post, but here is another way to do it:
```java
public static Point[] GetRotatedCorners(Rectangle rectangleToRotate, float angle)
{
// Calculate the center of rectangle.
Point center = new Point(rectangleToRotate.Left + (rectangleToRotate.Left + rectangleToRotate.Right) / 2, rectangleToRotate.Top + (rectangleToRotate.Top + rectangleToRotate.Bottom) / 2);
Matrix m = new Matrix();
// Rotate the center.
m.RotateAt(360.0f - angle, center);
// Create an array with rectangle's corners, starting with top-left corner and going clock-wise.
Point[] corners = new Point[]
{
new Point(rectangleToRotate.Left, rectangleToRotate.Top), // Top-left corner.
new Point(rectangleToRotate.Right, rectangleToRotate.Top), // Top-right corner.
new Point(rectangleToRotate.Right, rectangleToRotate.Bottom), // Bottom-right corner.
new Point(rectangleToRotate.Left, rectangleToRotate.Bottom), // Botton-left corner
};
// Now apply the matrix to every corner of the rectangle.
m.TransformPoints(corners);
// Return the corners of rectangle rotated by the provided angle.
return corners;
}
```
Hope it helps!
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answered Aug 19, 2019 at 18:22
DiligentKarmaDiligentKarma
101
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7048 | https://www.reddit.com/r/mathematics/comments/m961u7/regarding_quadratic_functions_are_factoring_and/ | Regarding quadratic functions, are factoring and completing the square really necessary? : r/mathematics
Skip to main contentRegarding quadratic functions, are factoring and completing the square really necessary? : r/mathematics
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•5 yr. ago
Jim2718
Regarding quadratic functions, are factoring and completing the square really necessary?
I teach high school Algebra 1, and we are in the midst of two chapters that focus on quadratics. We have not yet gotten to the quadratic formula, but when we do I can foresee kids asking why they didn’t learn that to begin with.
The main arguments I have heard for factoring and completing the square is it allows one to go from standard form to factored form and from standard form to vertex form. Don’t get me wrong, I understand the importance of all three forms and to be able to connect them. To go from factored or vertex form to standard form, it is straightforward enough to simplify using double distribution. And certainly, if you are trying to solve for the roots of a quadratic that is already in factored form or vertex form, then it is simpler to solve using the zero product property or algebraically, respectively, rather than simplifying to standard form in order to use the quadratic formula.
However, it is not necessary to factor a quadratic using the classic method taught in schools. Any quadratic in standard form f(x) = ax 2 + bx + c can be converted to factored form f(x) = a(x - p)(x - q) by using the quadratic formula to find the values of p and q. Furthermore, I don’t think it is necessary to teach completing the square for the sake of going from standard form to vertex form f(x) = a(x - h)2 + k, since h and k can be found using h = -b/(2a) and k = f(h).
I just think that expecting students to master three different methods of working with and solving quadratics muddies the waters for them. Why not start with the quadratic formula and then build off of that.
Am I overlooking something where traditional factoring and completing the square methods are critical for future course?
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7049 | https://pn.bmj.com/content/4/1/50 | Cerebral Autosomal Dominant Arteriopathy With Subcortical Infarcts and Leukoencephalopathy (CADASIL) | Practical Neurology
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Cerebral Autosomal Dominant Arteriopathy With Subcortical Infarcts and Leukoencephalopathy (CADASIL)
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Neurological Rarities
Cerebral Autosomal Dominant Arteriopathy With Subcortical Infarcts and Leukoencephalopathy (CADASIL)
S. Saif M. Razvi,
Keith W. Muir
Department of Neurology, Institute of Neurological Sciences, Southern General Hospital, 1345 Govan Road Glasgow G51 4TF, United Kingdom; E-mail: k.muir@clinmed.gla.ac.uk
Abstract
INTRODUCTION
Cerebral autosomal dominant arteriopathy with subcortical infarcts and leukoencephalopathy (CADASIL) is an inherited, autosomal dominant condition with high penetrance and varying expression. It is an important cause of protracted disability in young adults, with recurrent strokes, psychiatric dysfunction, and dementia. Although some families were identified in the 1950s, the syndrome was characterized and named only in 1993. The prevalence remains unknown, but numbers are increasing in parallel with wider medical recognition.
GENETICS, PATHOLOGY AND PATHOPHYSIOLOGY
CADASIL is caused by mutations in the Notch3 gene (chromosome 19p13) (Joutel et al. 1996), which encodes a large single-pass transmembrane receptor. The receptor’s extracellular domain contains 34 epidermal growth factor-like tandem repeats (EGF repeats). This receptor is part of a highly conserved signalling pathway that is essential for normal maturation of blood vessels in both fetal and adult brain, and is maximally expressed in small to medium penetrating arteries in early postnatal development
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7050 | http://www.algebra.com/algebra/homework/divisibility/Divisibility_and_Prime_Numbers.faq.question.1205180.html | SOLUTION: What's the total of one, two, three-digit prime numbers that can be formed using the digits 2, 3, 5 and 7. No digit can be used more than once in a number.
SOLUTION: What's the total of one, two, three-digit prime numbers that can be formed using the digits 2, 3, 5 and 7. No digit can be used more than once in a number.
Algebra->Divisibility and Prime Numbers -> SOLUTION: What's the total of one, two, three-digit prime numbers that can be formed using the digits 2, 3, 5 and 7. No digit can be used more than once in a number.Log On
Algebra: Divisibility and Prime Numbers SectionSolvers SolversLessons LessonsAnswers archive Answers Click here to see ALL problems on Divisibility and Prime Numbers Question 1205180: What's the total of one, two, three-digit prime numbers that can be formed using the digits 2, 3, 5 and 7. No digit can be used more than once in a number. Found 3 solutions by ikleyn, math_tutor2020, greenestamps: Answer byikleyn(52786) (Show Source): You can put this solution on YOUR website! . Good problem to kill your time for nothing without any visible meaning/sense/benefit. Answer bymath_tutor2020(3817) (Show Source): You can put this solution on YOUR website! The numbers 2,3,5, and 7 are prime since the only factors are 1 and themselves. Make a 4x4 table listing the values 2,3,5,7 along the left and top like so 2 3 5 7 2 3 5 7 Cross out the northwest main diagonal. This is because we cannot re-use the same digit twice. 2 3 5 7 2 X 3 X 5 X 7 X The remaining cells are filled in by concatenating the headers. I'll have the left header go first and then the top next. 2 3 5 7 2 X 23 25 27 3 32 X 35 37 5 52 53 X 57 7 72 73 75 X The "2" column and "5" column can be crossed out because of the divisibility by 2 and divisibility by 5 rules. 27 is composite since 27 = 39 57 is composite since 57 = 319 Or you can use the divisibility by 3 rule to check 27 and 57 are multiples of 3. Of that table, the primes are: 23, 37, 53, 73 Refer to a list of primes. Or you can check each one by one. I'll let you handle the possible 3 digit primes that can be formed with 2,3,5,7. Answer bygreenestamps(13200) (Show Source): You can put this solution on YOUR website! 1-digit numbers.... 2, 3, 5, and 7 are all prime numbers. ANSWERS: 2, 3, 5, 7 2-digit numbers.... A 2-digit number is not prime if the last digit is 2 or 5, so the last digit has to be 3 or 7. The possibilities are 23, 27, 53, and 57. 27 and 57 are not prime because their digit sum is divisible by 3. 23 and 53 are both prime, by inspection ANSWERS: 23, 53 3-digit numbers.... Again the last digit can't be 2 or 5; it has to be 3 or 7. The 3-digit combinations 2,3,7 and 3,5,7 won't work because the digit sum is divisible by 3. The other 3-digit combinations are 2,3,5 and 2,5,7; in each of those combinations neither the 2 nor the 5 can be the last digit, so the possible prime numbers are 253, 523, 257, and 527. Assuming we aren't working the problem simply by checking a list of prime numbers, here we have some actual detective work to do to see which if any of these are prime numbers. I won't go into details (I leave that for the student); the prime numbers among these are 523 and 257. ANSWERS: 257, 523 Final list of 1-, 2-, or 3-digit prime numbers using digits 2, 3, 5, and 7 without repetition: 2, 3, 5, 7, 23, 53, 257, 523 Since "total" is not a formal mathematical term, I won't guess what the expected answer is to the given problem. The student can use this list to answer the question, based on his/her interpretation.
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Euler’s Formula & De Moivre’s Theorem: Step-by-Step Guide
Euler’s Formula & De Moivre’s Theorem: Step-by-Step Guide
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Applications of Euler’s Formula and De Moivre’s Theorem in Complex Numbers
We know about complex numbers(z). They are of the form z=a+ib, where a and b are real numbers and 'i' is the solution of equation x²=-1. No real number can satisfy this equation hence its solution that is 'i' is called an imaginary number. When a complex exponential is written, it is written as e^iθ.
Euler's formula explains the relationship between complex exponentials and trigonometric functions.
DeMoivers’ theorem is also a theorem used for complex numbers. This theorem is used to raise complex numbers to different powers.
State Euler's Theorem
Euler’s law states that ‘For any real number x, e^ix = cos x + i sin x.
where,e=base of natural logarithm
i=imaginary unit
x=angle in radians
This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number.
Let z be a non zero complex number; we can write z in the polar form as,
z = r(cos θ + i sin θ) = r e^iθ, where r is the modulus and θ is argument of z.
Multiplying a complex number z with e^iα gives, zei^α = re^iθ × ei^α = rei^(α + θ).The resulting complex number re^i(α+θ) will have the same modulus r and argument (α+θ).
Euler's Identity
When x=π Euler’s formula evaluates to e^iπ+1=0, which is known as Euler's Identity.
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Euler's Formula
Euler's Formula For Cube
Euler's formula is related to the Faces, Edges and vertices of any polyhedron.
Euler's formula for a cube says that in a cube, the number of vertices minus the number of edges plus the number of faces results in two.
It can be written as
V-E+F=2
Where, V=number of vertices
E=number of edges
F=number of faces
It can be proven as,
In a cube, the number of vertices = 8
number of edges= 12
number of faces= 6
Putting values into the formula, V-E+F=8-12+6
=2
Hence proved.
De Moiver's Theorem
State De Moiver's Theorem
It states that for any integer n,
(cos θ + i sin θ)^n = cos (nθ) + i sin (nθ)
We can prove this easily using Euler’s formula as given below,
We know that, (cos θ + i sin θ) = e^iθ
(cos θ + i sin θ)^n = e^i(nθ)
Therefore,
e^i(nθ) = cos (nθ) + i sin (nθ)
Image will be added soon
nth Roots of Unity
If any complex number satisfies the equation zn = 1, it is known as nth root of unity.
An equation of degree n will have n roots as said by the fundamental theory of algebra, there are n values of z which satisfies zn = 1.
To find the values of z, we can write,
1 = cos (2kπ) + i sin (2kπ), —(1) where k can be any integer.
We have,
z^n = 1
z = 1^(1/n)
From (1),
z = [cos (2kπ) + i sin (2kπ)]^(1/n)
By De Moivre’s theorem,
z = [cos (2kπ/n) + i sin (2kπ/n)], where k = 0,1,2,3,……..,n−1
For example; if n = 3, then k = 0,1,2
We know that, z = cos (2kπ/n) + i sin (2kπ/n) = e^i(2kπ/n)
Let ω = cos (2πn) +i sin (2πn) = e^i(2πn)
nth roots of unity are found by,
When k = 0; z = 1
k = 1; z = ω
k = 2; z = ω2
k = n; z = ωn − 1
Therefore, nth roots of unity are 1,ω,ω2,ω3,…….,ωn − 1
Sum of nth roots of unity is,1 + ω + ω2 + ω3 + ⋯ + ωn − 1It is geometric series having first term 1 and common ratio ω.By using sum of n terms of a G.P,1 + ω + ω2 + ω3 + ⋯ + ωn − 1 = 1 − ωn1 − ωSince ω is nth root of unity, ωn = 1Therefore, 1 + ω + ω2 + ω3 + ⋯ + ωn − 1 = 0
Cube Roots of Unity:
We know that nth roots of unity are 1,ω,ω2,ω3,…….,ωn − 1.
Therefore, cube roots of unity are 1,ω,ω2 where,
ω = cos (2π/3) + i sin (2π/3) = −1 + √3 i2
ω2 = cos(4π/3) + i sin (4π/3) = −1 − √3 i2
Sum of the cube roots of the unity,
1 + ω + ω2 = 0
Product of cube roots of the unity,
1 × ω × ω2 = ω3 = 1
De Moiver's Theorem Example
If z = (cosθ + i sinθ ) , show that z^n + 1/ z^n = 2 cos nθ and z^n – [1/ z^n] = 2i sin nθ .
Solution
Let z = (cosθ + i sinθ ) .
By de Moivre’s theorem ,
z^n = (cosθ + i sinθ )^n = cos nθ + i sin nθ
1/z^n=z^(-n)=cos nθ - i sin nθ
=> z^n+1/z^n = (cos nθ + i sin nθ)+(cos nθ - i sin nθ)
=> z^n+1/z^n = 2cosnθ
Also,=> z^n-1/z^n = (cos nθ + i sin nθ)-(cos nθ - i sin nθ)
=> z^n-1/z^n = 2i sin nθ
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FAQs on Euler’s Formula & De Moivre’s Theorem: Step-by-Step Guide
What is Euler's formula in the context of complex numbers?
Euler's formula establishes a fundamental relationship between trigonometric functions and the complex exponential function. For any real number θ (in radians), the formula is stated as: eiθ = cos(θ) + i sin(θ). Here, 'e' is the base of the natural logarithm, 'i' is the imaginary unit (√-1), and cos(θ) and sin(θ) are the trigonometric functions which represent the coordinates on a complex plane.
What is De Moivre's theorem, and what is it used for?
De Moivre's theorem provides a straightforward method for calculating powers of complex numbers. The theorem states that for any complex number in the polar form (cos θ + i sin θ) and any integer 'n', the following holds true: (cos θ + i sin θ)n = cos(nθ) + i sin(nθ). Its primary applications in Class 11 Maths are to simplify the process of finding powers and roots of complex numbers.
How do you represent a complex number in polar and rectangular forms?
A complex number can be represented in two main forms:
Rectangular Form: Expressed as z = a + ib, where 'a' is the real part and 'b' is the imaginary part. This corresponds to the coordinates (a, b) on the Argand plane.
Polar Form: Expressed as z = r(cos θ + i sin θ), where 'r' is the modulus (or distance from the origin) and 'θ' is the argument (or angle with the positive real axis). This form is particularly useful for multiplication, division, and finding roots or powers of complex numbers.
What are the main applications of De Moivre's theorem for a Class 11 student?
For a Class 11 student following the CBSE syllabus, De Moivre's theorem is primarily applied in two key areas:
Finding Powers of Complex Numbers: It simplifies raising a complex number to a high power without performing tedious repeated multiplications. For example, calculating (1 + i)10 becomes much easier.
Finding the nth Roots of a Complex Number: It forms the basis for finding all 'n' distinct roots of a complex number, including the nth roots of unity, which is a crucial concept.
How does Euler's formula help in understanding De Moivre's theorem?
Euler's formula provides one of the most elegant proofs for De Moivre's theorem. We start with Euler's formula: eiθ = cos(θ) + i sin(θ). If we raise both sides to the power of 'n', we get (eiθ)n = (cos θ + i sin θ)n. Using the laws of exponents, the left side becomes ei(nθ). Applying Euler's formula again to this result gives cos(nθ) + i sin(nθ). By equating the results, we directly arrive at (cos θ + i sin θ)n = cos(nθ) + i sin(nθ), thereby proving the theorem.
Why is the number 'e' so important in Euler's formula for complex numbers?
The number 'e' (Euler's number) is crucial because it connects exponential functions, which describe growth and decay, with trigonometric functions, which describe periodic motion or rotations. In the context of complex numbers, the expression eiθ represents a point on the unit circle in the Argand plane. This creates a powerful link, allowing rotational operations (trigonometry) to be handled with the simpler rules of exponents, which is the core reason it simplifies so many calculations in complex analysis.
What is the geometric interpretation of multiplying two complex numbers using Euler's formula?
Using Euler's formula, two complex numbers z1 = r1eiθ₁ and z2 = r2eiθ₂ can be multiplied easily: z1z2 = r1r2ei(θ₁+θ₂). The geometric interpretation of this is profound:
The magnitudes (lengths) of the vectors are multiplied (r1r2).
The arguments (angles) of the vectors are added (θ₁ + θ₂).
Essentially, multiplying by a complex number corresponds to a rotation and a scaling in the complex plane.
Can De Moivre's theorem be applied to non-integer exponents?
The standard statement of De Moivre's theorem is for integer values of 'n'. When 'n' is a rational number (like 1/2 or 3/4), the formula cos(nθ) + i sin(nθ) gives one of the possible values or roots. However, for a fractional exponent like 1/q, there are 'q' distinct roots. While the theorem is the foundation for finding these roots, a more generalised formula is required to find all of them, not just the principal value.
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Tautomerism: Unveiling Molecular Transformations
Tautomers are like shape-shifters in the molecular world—they don't stick to one form. In this section, we'll explore compounds that can change into various versions of themselves without any magic. It's all about the dynamic interaction of protons and electrons.
Think of a molecule that can take on different shapes, each with a slightly rearranged set of atoms. These shape-changing structures, called tautomers, constantly switch back and forth in a fascinating dance. This isn't just for show; it has important implications in fields like biochemistry, drug discovery, and material science. Let’s find out how this molecular magic is essential for biomolecules like DNA and proteins. And it doesn't stop there; it affects the taste of food, the smell of flowers, and even how strong medicines are.
What is Tautomerism?
Tautomerism is a phenomenon where a single chemical compound tends to exist in either two or more structures of interconvertible type, which are different considering the relative position of one atomic nucleus i.e. hydrogen, in general. The two structures are known as tautomers. These types of isomer compounds only differ usually in the number of electrons and protons. They also exist in the dynamic equilibrium.
If a reaction takes place between these compounds, there is only the transfer of protons. Tautomerism is otherwise termed as desmotropism. Other names that are widely used for tautomerism are kryptomerism, allotropism or merotropy. Tautomerism is the most widely accepted term in the field of chemistry. There are several types of tautomerism, keto-enol tautomerism being the most important. In this type, one tautomer exists as a ketone while the other tautomer exists as an enol. Acetone and phenol are the two most common examples.
Tautomerism usually happens in the presence of a catalyst.
Acid-Catalyst: Here, firstly, the protonation takes place, cation will be delocalized. Then, the deprotonation will take place in the adjacent position of the cation.
Whereas, for base catalysts, deprotonation is the primary step. Here, rather than cation delocalization, anion delocalization takes place and finally protonation to a different position of the anion.
Characteristics of Tautomerism-
There is a movement of atoms involved which are alpha hydrogen atoms.
The compounds can be separated and isolated as they are definite compounds.
Tautomeric forms have two different structures.
The compounds of tautomers are in equilibrium with each other.
Tautomerism does not have an effect on the bond length.
Tautomerism does not decrease the energy of the molecules and hence stabilises the molecule.
It occurs in planar as well as non- planar molecules.
Examples of Tautomerism
As a simple definition of Tautomerism, it is expressed as a type of isomerism, where the isomers interchange into/between one another easily to exist together in equilibrium. At the time of reaction, there occurs a proton transfer in an intramolecular fashion. Consider a tautomerism example given below.
Ketone-enol, enamine-imine,lactam-lactim are some of the examples of tautomers.
Meanwhile, some key features of Tautomerism are that this process provides more stability for the compound. In this phenomenon, there exists an exchange of a hydrogen atom between two other atoms while forming a covalent bond to either one. The reversible process is Tautomerism.
Tautomerism's Structure Needs:
Substances have molecules that are polar and include weakly acidic functional groups.
It includes moving an atom's position.
It doesn't impact things like bond length.
Usually happens in flat or non-flat molecules.
Tautomerism Types
In the 1880s, a scientist named Emil Erlenmeyer developed a rule for Tautomerism. He is one of the first people to have studied keto-enol Tautomerism. This rule states that the hydroxyl group in all alcohols is attached directly with a double-bonded carbon atom, and forms aldehydes or ketones. This happened due to the more stability of the keto form.
There are different types of Tautomerism. Among those, keto-enol Tautomerism is the most important one. Here, one structure is in enol, and the other is in ketone form. Both tautomeric forms are interconvertible to each other by the use of acid or base catalysts. This conversion process of the ketone into enol is called enolization.
The types of Tautomerism include:
Prototropy
This tautomerism type occurs due to the compound's acid-base behaviour. Here, the two forms differ only in the position of a proton. This structure will have the same number of charges and the empirical formula.
Annular Tautomerism
If a proton occupies either two or more positions of a heterocyclic system, then such a process is known as annular Tautomerism. In Tautomerism, because of the delocalization of protons, if an open structure is altered to a ring structure, then such a tautomer is known as ring-chain tautomers. An example of ring-chain tautomers is Glucose.
Valence Tautomerism
Valence tautomerism is a type of Tautomerism where there is a continuous formation and single and double bonds breaking in the compound without any migration of atoms or groups. It is different from the other tautomerism types and is a rapid process.
In this, there is a change in geometrical structure but no change available for mesomers or canonical resonance structure.
Tautomerism in Non-Carbonyl Compounds
Many non-carbonyl systems are available as mixtures of tautomers. A few examples are given below.
On the chemical properties of the compound, the Tautomerism can have a huge impact.
Tautomeric Form of Unsymmetrical Ketones
There is only one form of tautomer in symmetric form. But there can be two for an unsymmetrical form. Let us have a look at the example given.
Tautomerism Reaction Mechanism
Let us discuss the keto-enol tautomerization acid catalysis. It is a two-step method in an aqueous solution of acid. The carbon atom, which is closest to the functional group, is known as the alpha carbon atom. So, for this mechanism to take place, one hydrogen atom at least should be with the alpha carbon atom. It can also be known as an alpha hydrogen atom.
The resultant hydrogen atom is added to the antibonding pi-orbital of the carbonyl group parallelly. Then, this bond will undergo hyperconjugation with the C-H bond and decrease the electron density at the alpha carbon atom, where the alpha hydrogen atom will become more acidic compared to before. If the position of alpha hydrogen has not happened, the tautomerism process will be very slow. An example of this slow process is Adamatanone.
We should follow Markovnikov's rule in addition to this process. Firstly, in the mechanism, there exists a hydronium ion (H3O+), an electrophile, so the electrons exposed near the C=C bond will be donated. If the number of hydrogen atoms present in the compound is more, thereby, the addition of protons also increases.
The reaction mechanism can be given below.
Important Questions of Tautomerism
What is meant by the term tautomer?
Explain the structural requirements of a compound to possess tautomerization.
How is the ring-chain tautomerism obtained?
Give some examples of tautomerism, ring-chain tautomerism, and annular tautomerism.
What are the different types of tautomerism, and how are they classified? Explain.
How does tautomerism depend on the catalyzation process? Which is the type of such a process? Explain briefly.
Explain the mechanism of tautomerism in a step-by-step procedure with examples.
What is valence tautomerism?
Write the importance of the alpha carbon atom in tautomerism.
Explain the tautomerism related to the acid-base behavior of the molecule.
What is meant by annular tautomerism?
Protonation and deprotonation are the two essential processes in tautomerism. Justify.
List the applications of tautomerism in various fields.
Explain the stability of the keto-enol form of tautomer.
Conclusion
Tautomerism, a vital concept for JEE Main, involves isomers (same molecular formula) that differ in the position of protons and electrons. In simpler terms, molecules can exist in two forms that quickly switch between each other. This dynamic behavior impacts chemical reactions. For instance, in keto-enol tautomerism, a molecule can transform between a keto form (more common) and an enol form. Understanding tautomerism is crucial for predicting how molecules behave, influencing drug design and various chemical processes. In JEE Main, grasp the idea that molecules are dynamic, not fixed structures, to navigate questions related to tautomerism successfully.
FAQs on Tautomerism
1. How does acid show tautomerism?
Tautomerism is the special kind of functional isomerism in which migration OH H occurs in a dynamic equilibrium condition.
Acid Mechanism :
Step 1 - Acid will give H+ ion, and the Oxygen of ketone will donate its lone pair to H+ and carries a +ve charge on itself (this is a fast process)
CH3—C=0—CH3 + H+ → CH3—C=OH—CH3
Step 2 - Because Oxygen carries a +ve charge, it will attract an electron from Carbon, and a double bond between C=O will break, and it will become OH, and simultaneously Carbon will form a double bond with the neighbouring Carbon. (it's a slow and rate-determining process)
CH3—C=OH—CH3 - H+ → CH2=C-OH—CH3
Key Point - Tautomerism can be achieved in acidic and basic mediums (the major product is different for both cases) as well.
Basic Medium - A double bond is formed between Carbon and its functional group, which has a lower number of alkyl groups.
Acidic Medium - A double bond is formed between the Carbon and its functional groups, which has more alkyl groups.
For example, consider the below reactions given.
CH3—CO—CH2—CH3 + OH-(basic) → CH2=C(OH)—CH2—CH3
CH3—CO—CH2—CH3 + H+(acidic) → CH3-C(OH)=CH—CH3
2. What is the difference between tautomerization and resonance?
Tautomerism is a real phenomenon, whereas resonance happens hypothetically. There is a movement of atoms (alpha hydrogen) involved in Tautomerism. But in resonance, the electron movement of the p orbital happens. Thus, there's no movement of atoms in the latter. While Tautomerism is represented by the double arrows (same as equilibrium), resonance is represented by one double-headed arrow.
Tautomerism happens in solutions; it means the alpha hydrogen gets removed, and another hydrogen from the water gets attached to the oxygen atom. But resonance is not restricted like Tautomerism, but only happens in solutions. Tautomerism does not have any effect on the bond length but resonance has an effect on the bond length as during resonance the single bond gets shortened while the double bond becomes longer. Tautomerism does not decrease the energy of the molecule while resonance lowers the energy of the molecule.
3. How is tautomerism different from metamerism?
Tautomerism can be defined as a phenomenon where a single chemical compound tends to exist in either two or more structures of interconvertible type, which are different considering the relative position of one atomic nucleus i.e. hydrogen, in general. On the other hand, Metamerism occurs when alkyl groups on the sides of the functional groups differ from each other. This indicates that it is an unequal distribution of carbon atoms. The main difference between tautomerism and metamerism is that tautomerism means the dynamic equilibrium between two compounds having a same molecular formula whereas metamerism refers to structural isomerism in which alkyl groups which are attached to the same functional group are different. The isomerization process of tautomers is called tautomerization whereas the isomerization of metamers is called metamerization.
4. What are the requirements for tautomerism to happen?
The compounds which undergo Tautomerism need to have a weak acidic group and also the molecules should be polar. It also requires an atom to change its position. It occurs in both types of the compounds, that is the compounds which have their atoms lying in the same plane as well as in compounds which do not have their atoms in the same plane. Tautomerism is a reaction which also needs a catalyst. The catalyst can be of two types, acidic and basic.
5. From where can I study structural isomerism and its types?
Online notes are handy for students who do not like to make notes and want to have a quick recap before the exams. The notes are also useful for those students who could not attend the classes in their schools and wish to study the missed portions. Students can download free notes from the official website of Vedantu that is Vedantu.com. The notes are available in PDF format and are written in easy and simple language. Students can access these free notes and score good marks. Vedantu also offers NCERT solutions, previous year question papers and important questions with solutions. Students may also attend free live classes on Vedantu from the comfort of their homes.
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How to find effective partition of n n into k k?
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Here is a type of question that I find quite often on MO sites, that I couldn't quite solve:
How many ways can I put n n identical balls into k k identical boxes, with n>>k n>>k, such that each box has at least one ball?
For me, this question had n=600 n=600, and k=3 k=3. At first, I thought this was a combinations problem, as we could have 3 balls put in each box first, and then the other 600−3=597 600−3=597 balls can be placed in any boxes. The answer I first thought of was 3 597 3 597. I then realised this was wrong, because I was considering each as a separate ball, in which case it wasn't.
Afterwards, I did a little research, and I thought this question might have something to do with compositions of numbers. However, the equation provided, which is (n−1 k−1)(n−1 k−1) was not the correct answer, when I plugged in n n and k k respectively, which got me 179101 179101. I realised that in my question, the k k boxes were not distinguishable, however in the equation, the k k boxes are.
I believe the way I need to solve this problem is using partitions, instead of compositions, of the numbers, but I am not yet sure. I don't know how I can solve this problem with partitions of numbers.
P.S the answer is 30,000 30,000
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asked Sep 7, 2019 at 11:43
Aarony JamesysAarony Jamesys
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You want the number of partitions of 600 600 into exactly 3 3 parts. See the Wikipedia article. The recursive formula given there
p k(n)=p k(n−k)+p k−1(n−1)(1)(1)p k(n)=p k(n−k)+p k−1(n−1)
is best understood as counting the number of partitions of n n with largest part k k. The first term on the right-hand side of (1)(1) counts the partitions with more than one part equal to k k and the second term counts the partitions with exactly one part equal to k k. That the number of such partitions is equal to the number of partitions of n n with exactly k k parts is easily proved by transposing the Ferrers diagram, as the wiki article mentions in passing.
I checked the value with this python script:
```
def partitions(n,k):
if n==k==0: return 1
if n <=0 or k <=0: return 0
return partitions(n-k,k)+partitions(n-1,k-1)
print(partitions(600,3))
```
and I did indeed get 30,000 30,000.
This is a really inefficient way to program it, and I wouldn't use it except for a quick script that I'm going to use once and throw away.
A better way to do it, but a little harder to understand is
```
class Partitions(dict):
def init(self):
self[0,0] = 1
def missing(self,key):
n,k=key
if n <= 0 or k <= 0:
return 0
self[n,k] = self[n-k,k] + self[n-1,k-1]
return self[n,k]
partitions = Partitions()
print(partitions[600,3])
```
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edited Sep 7, 2019 at 16:03
answered Sep 7, 2019 at 14:14
saulspatzsaulspatz
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1 Just add @functools.lru_cache(None) to the first script and you get the caching of the second script without the ugliness.Peter Taylor –Peter Taylor 2019-09-07 22:26:14 +00:00 Commented Sep 7, 2019 at 22:26
@PeterTaylor Thanks, I wan't aware of that decorator.saulspatz –saulspatz 2019-09-08 14:48:27 +00:00 Commented Sep 8, 2019 at 14:48
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Note that the (599 2)(599 2) compositions are almost exactly 6 6 times the correct answer. This makes sense, because in general each partition into three parts corresponds to 3!=6 3!=6 compositions. However, the ones with repeated parts correspond to fewer compositions, so we can't just divide by 6 6.
There are three cases for a composition λ 1+λ 2+λ 3 λ 1+λ 2+λ 3:
All different (six compositions per partition)
Two the same (three compositions per partition)
All the same (one composition per partition)
The sum of all three cases comes to the aforementioned (599 2)(599 2).
The third case is a special case of the second. The total (600 600) minus the odd one out must be even, and can be any even number from 2 2 to 598 598, so we have 299 299 partitions, of which one is the third case.
This gives a final count of
1 6(179101−298×3−1)+299=30000 1 6(179101−298×3−1)+299=30000
as expected.
Obviously this is messier when k>3 k>3.
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answered Sep 7, 2019 at 22:24
Peter TaylorPeter Taylor
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Okay, so following up from the answer:
The final answer, 179101 179101 is right if the question regarded each of the 3 boxes as unique. However, the question did not consider each box as unique, and each box is the same. This means that different compositions, such as (1,3,596)(1,3,596) and (596,3,1)(596,3,1) are the same sets.
For any composition, we have:
If the combination has 3 3 distinct numbers as the set, then 6 6 compositions is equivalent to the corresponding partition
If the combination has 2 2 distinct numbers in the set, then there are 3 3 different combinations making up the corresponding partition
If the combination has no distinct numbers, then the amount of partitions is equal to the number of combinations for that specific number arrangement.
For the third case, we know only one is possible because if there are no distinct numbers in the combination, the value for each combination must be 600/3=200 600/3=200.
For the second case, there are 299 299 even numbers, excluding that of (200,200,200)(200,200,200), 298 298 different cases. These numbers must be removed when we divide the sum of the combinations by 6, and they are counted 3 times.
So, what we have is:
O r i g i n a l=179101−3∗298−1 6+298+1=30,000 O r i g i n a l=179101−3∗298−1 6+298+1=30,000
Just to make it slightly clearer.
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answered Sep 10, 2019 at 7:50
Aarony JamesysAarony Jamesys
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Question about ohmic conductors
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I'm having some trouble understanding Ohm's law. My trouble is with the different ways it is described when referring to ohmic and non-ohmic conductors. If someone can answer this question I think it will clear up my doubts. (I made up this question myself -- it's not homework!)
Which one of A and B is wrong, and why?:
A:
A non-ohmic conductor is one whose resistance changes with increasing temperature, while an ohmic conductor is one whose resistance doesn't change with increasing temperature.
B:
A non-ohmic conductor is one whose temperature changes with increasing voltage, while an ohmic conductor is one whose temperature doesn't change with increasing voltage.
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asked May 13, 2015 at 19:52
user45220user45220
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Ohm's law assumes the temperature remains constant.
An Ohmic conductor is one in which the current flowing through it is proportional to the voltage applied across it.
A non-ohmic conductor is one in which the voltage and current are not linear.
A) The resistance of most conductors increases as the temperature increases, however being ohmic and not ohmic is not the reason.
B)What causes heating in a conductor is the current flowing through a conductor. The Power = current^2 Resistance = voltage^2 / resistance
This power is converted into heat which increases the temperature as time goes on. The actual voltage does not matter as much as the how long it has been on the conductor.
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answered May 13, 2015 at 20:11
stevesteve
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"however, being ohmic and not ohmic is not the reason": Does that mean that the resistance of an ohmic conductors can change if the temperature increases? Thanks!user45220 –user45220 2015-05-13 20:18:42 +00:00 Commented May 13, 2015 at 20:18
Yes ohmic conductors resistances will change with in increase in temperature. But the resistance will not change if the temperature is kept constant, the actual temperature it is at is not important. Its important to remember that nothing is perfect, and most materials have a range in which they are ohmic, meaning voltage and current go up linearly together.steve –steve 2015-05-13 20:25:11 +00:00 Commented May 13, 2015 at 20:25
Thanks. So how do we explain metallic conductors? They are always ohmic according to my textbook. Does that mean they NEVER heat up no matter how much current we put through them?user45220 –user45220 2015-05-13 20:28:57 +00:00 Commented May 13, 2015 at 20:28
1 Metallic conductors heat up.Their resistance is higher when they are hotter. But they are ohmic because the Resistance measured with 1Volt is the same as the resistance measured with 10 volts. However if you measured with 1 volt when it was 20 Celsius, and measured at 1 Volt when it was 200 Celcius, you would have 2 different resistances steve –steve 2015-05-13 20:32:06 +00:00 Commented May 13, 2015 at 20:32
1 Good answer. Small point: because resistance of ohmic resistors is typically a function of temperature, they will exhibit a small non-linear effect (the resistance measured with 10 V will not be the same as the resistance measured with 1 V, because the device was heated by the additional current). But in practice we ignore this because implicitly we assume "at constant temperature". You more or less said that, I just wanted to emphasize it.Floris –Floris 2015-05-13 20:53:30 +00:00 Commented May 13, 2015 at 20:53
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Application of the term "non-Ohmic" can be a bit messy and neither A nor B is wholly correct.
Ohm's law strictly applies only for constant temperature, however in practice resistance always varies with temperature.
An ohmic device such as a resistor is designed to remain within tolerance over a wide range of temperatures, working voltages and currents. Any variation in resistance due to such factors will be either ignored or compensated for.
A thermistor is a non-ohmic device in which the resistance variation with temperature is the characteristic property of the device.
Other passive non-ohmic devices include the voltage-dependent resistor or VDR, the photocell whose resistance which varied with the amount of light falling on it, and I forget what all else.
Such passive non-ohmic devices still obey Ohm's law provided the characteristic parameter (temperature, voltage, light) remains constant, in that they still have a specific resistance for that value of the parameter. Moreover the variation with parameter value is usually linear over the operating range of the device.
Other non-ohmic devices can be highly non-linear. Diodes and transistors are classic examples.
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answered Aug 16, 2020 at 8:02
Guy InchbaldGuy Inchbald
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7056 | https://archive.lib.msu.edu/crcmath/math/math/c/c429.htm | Coin Tossing
Coin Tossing
An idealized coin consists of a circular disk of zero thickness which, when thrown in the air and allowed to fall, will rest with either side face up (heads'' H ortails'' T) with equal probability. A coin is therefore a two-sided Die. A coin toss corresponds to a Bernoulli Distribution with . Despite slight differences between the sides and Nonzero thickness of actual coins, the distribution of their tosses makes a good approximation to a Bernoulli Distribution.
There are, however, some rather counterintuitive properties of coin tossing. For example, it is twice as likely that the triple TTH will be encountered before THT than after it, and three times as likely that THH will precede HTT. Furthermore, it is six times as likely that HTT will be the first of HTT, TTH, and TTT to occur (Honsberger 1979). More amazingly still, spinning a penny instead of tossing it results in heads only about 30% of the time (Paulos 1995).
Let be the probability that no Run of three consecutive heads appears in independent tosses of a Coin. The following table gives the first few values of .
0 1
1 1
2 1
3
4
5
Feller (1968, pp.278-279) proved that
(1)
where
(2)
and
(3)
The corresponding constants for a Run of heads are , the smallest PositiveRoot of
(4)
and
(5)
These are modified for unfair coins with and to , the smallest PositiveRoot of
(6)
and
(7)
(Feller 1968, pp.322-325). See alsoBernoulli Distribution, Cards, Coin, Dice, Gambler's Ruin, Martingale, Run, Saint Petersburg Paradox
References
Feller, W. An Introduction to Probability Theory and Its Application, Vol.1, 3rd ed. New York: Wiley, 1968.
Finch, S. ``Favorite Mathematical Constants.''
Ford, J. ``How Random is a Coin Toss?'' Physics Today36, 40-47, 1983.
Honsberger, R. ``Some Surprises in Probability.'' Ch.5 in Mathematical Plums (Ed. R.Honsberger). Washington, DC: Math. Assoc. Amer., pp.100-103, 1979.
Keller, J.B. ``The Probability of Heads.'' Amer. Math. Monthly93, 191-197, 1986.
Paulos, J.A. A Mathematician Reads the Newspaper. New York: BasicBooks, p.75, 1995.
Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman, pp.238-239, 1990.
Spencer, J. ``Combinatorics by Coin Flipping.'' Coll. Math. J., 17, 407-412, 1986.
© 1996-9 _Eric W. Weisstein
1999-05-26_ |
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Clarifying why compactness in a topology, implies compactness in a coarser topology
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If (X,τ)(X,τ)is compact and τ′⊆τ τ′⊆τ, then (X,τ′)(X,τ′) is compact.
I have already read several posts on the subject, but it is still unclear to me. The usual argument is: "In a coarser space, more sets are compact, essentially because there are fewer open covers to need finite subcovers. That is, if a set is compact in the finer topology then it is compact in the coarser topology." (as found here What does compactness in one topology tell us about compactness in another (coarser or finer) topology?)
But still I am not very convinced, specifically because, if I go to a coarser topology, some open sets are missing with respect to the initial topology, and what if I needed those sets for the extract the fine subcover, what guarantees they aren't needed?
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asked Jul 7, 2020 at 18:45
some_math_guysome_math_guy
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1 Just a guess about intuition: the set is compact if every open cover has a finite subcover, so it's not like an "existence" result where you have a particular cover and are throwing away some of the sets, which is how it sounds like you are thinking about it ("what if I needed those sets").user762914 –user762914 2020-07-07 18:54:14 +00:00 Commented Jul 7, 2020 at 18:54
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The usual argument is the proof, which is very short and straightforward:
Let U⊆τ′U⊆τ′ be a τ′τ′-open cover of X X. Then U⊆τ U⊆τ, so U U is a τ τ-open cover of X X, and there is therefore a finite R⊆U R⊆U that covers X X. R⊆U⊆τ′R⊆U⊆τ′, so R R is a finite τ′τ′-open subcover of U U, and ⟨X,τ′⟩⟨X,τ′⟩ is therefore compact.
In words, if we start with a τ′τ′-open cover U U, it is also automatically a τ τ-open cover, so it has a finite subfamily that covers the X X. The members of that subfamily are members of U U, so we have the desired finite subcover; no extra sets can possibly be needed, because we’re using only sets that are in the original cover U U.
It would be different if we were asking for an open refinement with some particular property instead of for a subcover: then we might actually need some of the sets in τ∖τ′τ∖τ′. For example, let τ′τ′ be any non-paracompact topology on X X, and let τ τ be the discrete topology. Then τ′⊆τ τ′⊆τ, ⟨X,τ⟩⟨X,τ⟩ is paracompact, and ⟨X,τ′⟩⟨X,τ′⟩ is not paracompact.
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edited Jul 7, 2020 at 19:11
answered Jul 7, 2020 at 18:50
Brian M. ScottBrian M. Scott
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Are you considering τ′⊆τ τ′⊆τ or τ⊆τ′τ⊆τ′ ?some_math_guy –some_math_guy 2020-07-07 19:16:07 +00:00 Commented Jul 7, 2020 at 19:16
@J.C.VegaO: τ′⊆τ τ′⊆τ, as in the question.Brian M. Scott –Brian M. Scott 2020-07-07 19:16:47 +00:00 Commented Jul 7, 2020 at 19:16
Then why does it start with a τ′τ′-open cover U U, shouldn't it start with a with a τ τ-open cover U U since we have the hypothesis of τ τ being compact and then show it is a cover of the coarser one? (which is the source of my confusion, because, some of the sets may be missing in the coarser one that we needed to extract the fnite subcover)some_math_guy –some_math_guy 2020-07-07 19:23:07 +00:00 Commented Jul 7, 2020 at 19:23
1 @J.C.VegaO: No, it absolutely should not start with a τ τ-open cover: we’re trying to prove that ⟨X,τ′⟩⟨X,τ′⟩ is compact, so we must of course start with a τ′τ′-open cover.Brian M. Scott –Brian M. Scott 2020-07-07 19:23:56 +00:00 Commented Jul 7, 2020 at 19:23
1 @J.C.VegaO: Both are standard usage. R R covers X X, so it’s reasonable to call it a subcover of X X, but it’s a subset of the cover U U, so it’s equally reasonable to call it a subcover of U U. The question is whether one wants to emphasize the space being covered or the original cover of which we’re finding a subcover.Brian M. Scott –Brian M. Scott 2020-07-07 19:50:08 +00:00 Commented Jul 7, 2020 at 19:50
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It might be helpful to think about what happens if the set were somehow compact in the finer topology, but not in the coarser topology. If X X is compact in τ τ but not compact in τ′τ′, then there is an open cover U U of X X in τ′τ′ which admits no finite subcover. However, since τ′⊆τ τ′⊆τ we also have that U U is an open cover of X X in τ τ, which is a problem because this means X X is also not compact in τ.τ.
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answered Jul 7, 2020 at 18:59
DMcMorDMcMor
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1 Nice argument, thanks for your contribution some_math_guy –some_math_guy 2020-07-07 19:52:38 +00:00 Commented Jul 7, 2020 at 19:52
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An open cover with respect to the coarser topology is also an open cover with respect to the finer one, this should be clear. Due to compactness with respect to the finer topology, we can find a finite open subcover with respect to the finer topology. This subcover only contains elements which are open with respect to the finer topology, which are open with respect to the coarser topology as well. Meaning that the finite subcover we found is open with respect to the coarser topology, too.
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answered Jul 7, 2020 at 19:12
VercassivelaunosVercassivelaunos
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7059 | https://pmc.ncbi.nlm.nih.gov/articles/PMC307526/ | Importin α/β and Ran-GTP Regulate XCTK2 Microtubule Binding through a Bipartite Nuclear Localization Signal - PMC
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Importin α/β and Ran-GTP Regulate XCTK2 Microtubule Binding through a Bipartite Nuclear Localization Signal
Stephanie C Ems-McClung
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PMCID: PMC307526 PMID: 13679510
Abstract
The small GTPase Ran is essential for spindle assembly. Ran is proposed to act through its nuclear import receptors importin α and/or importin β to control the sequestration of proteins necessary for spindle assembly. To date, the molecular mechanisms by which the Ran pathway functions remain unclear. Using purified proteins, we have reconstituted Ran-regulated microtubule binding of the C-terminal kinesin XCTK2, a kinesin important for spindle assembly. We show that the tail of XCTK2 binds to microtubules and that this binding is inhibited in the presence of importin α and β (α/β) and restored by addition of Ran-GTP. The bipartite nuclear localization signal (NLS) in the tail of XCTK2 is essential to this process, because mutation of the NLS abolishes importin α/β-mediated regulation of XCTK2 microtubule binding. Our data show that importin α/β directly regulates the activity of XCTK2 and that one of the molecular mechanisms of Ran-regulated spindle assembly is identical to that used in classical NLS-driven nuclear transport.
INTRODUCTION
The process of chromosome congression and segregation is mediated by the mitotic spindle, which is comprised of microtubules (MTs) and their associated proteins. Proper spindle assembly requires the activities of both plus-end– and minus-end–directed MT motors, nonmotor MT-associated proteins, and other essential non-MT–associated proteins (Compton, 2000). Minus-end–directed cytoplasmic dynein and plus-end–directed Eg5 play key roles in the organization of spindle MTs in Xenopus egg extracts and in cells (Gaglio et al., 1996; Heald et al., 1996; Merdes et al., 1996; Walczak et al., 1998). In addition, the mitotic minus-end–directed C-terminal kinesins function as MT cross-linkers to promote proper spindle assembly in many organisms (Endow and Komma, 1996; Walczak et al., 1997, 1998; Matuliene et al., 1999; Mountain et al., 1999; Ovechkina and Wordeman, 2003) and KIFC1 and KIFC5 C-terminal kinesins may serve a similar cross-linking function during spermatogenesis (Navolanic and Sperry, 2000; Yang and Sperry, 2003). The mechanism that regulates this MT cross-linking is currently unknown.
Recently, the GTPase Ran was shown to be sufficient to induce MT aster formation as well as bipolar spindle assembly in egg extracts (Carazo-Salas et al., 1999; Kalab et al., 1999; Ohba et al., 1999; Wilde and Zheng, 1999; Nachury et al., 2001). Subsequently, it was reported to be involved in regulating numerous mitotic spindle assembly processes, including MT nucleation and dynamics (Wilde and Zheng, 1999; Carazo-Salas et al., 2001; Wilde et al., 2001), MT motor activity (Wilde et al., 2001), and nuclear envelope assembly (Bamba et al., 2002; Hetzer et al., 2002; Zhang et al., 2002). It is believed that Ran provides positional cues by the generation of a steep gradient of Ran-GTP between the condensed chromosomes and the surrounding cytoplasm (Kalab et al., 2002; Trieselmann and Wilde, 2002). After nuclear envelope breakdown, the localized concentration of Ran-GTP around chromatin is thought to contribute to proper spindle assembly and to be maintained by the chromatin-associated nucleotide exchange factor RCC1 (Bilbao-Cortes et al., 2002; Moore et al., 2002; Li et al., 2003). The molecular mechanisms by which Ran affects these mitotic processes are largely unknown, but they seem to involve the nuclear import proteins importin α and importin β (Nachury et al., 2001; Wiese et al., 2001; Askjaer et al., 2002; Dasso, 2002; Kalab et al., 2002; Zhang et al., 2002; Quimby and Dasso, 2003; Tsai et al., 2003).
Thus far, the MT binding proteins TPX2 and NuMA were identified as proteins that bind to importin α and induce spontaneous MT aster formation when added in excess to Xenopus extracts (Gruss et al., 2001; Nachury et al., 2001; Wiese et al., 2001; Schatz et al., 2003). This suggests that these proteins may be early mediators in the Ran pathway that lead to the formation of bipolar spindles. This has led to a working model in which importin β or a complex of importin α/β sequesters proteins necessary for spindle assembly, such as TPX2 and NuMA, so that upon nuclear envelope breakdown, high levels of Ran-GTP promote the release of spindle-promoting factors, allowing them to function (Kuersten et al., 2001; Walczak, 2001; Dasso, 2002; Hetzer et al., 2002). Consistent with this idea, importin α/β binding to TPX2 inhibits Aurora A MT-dependent activation in vivo and in vitro (Eyers et al., 2003; Tsai et al., 2003) and mutation of the importin α binding site in TPX2 allows mutant TPX2 to induce aster formation in extracts at concentrations below those required for wild-type TPX2 (Schatz et al., 2003). This provides evidence that the pathway used for nuclear protein import is conserved in spindle assembly. In addition, Ran increases the plus-end–directed MT motor activity in egg extract asters, in part due to Eg5 activity (Wilde et al., 2001), but it has not been demonstrated to directly regulate the MT binding or organizational activities of any microtubule motors.
We hypothesized that Ran and importin α/β do likely influence the activity of mitotic motor proteins. From our previous work, we know that the Xenopus C-terminal kinesin XCTK2 (Xenopus C-terminal kinesin 2) is required for spindle formation in egg extracts (Walczak et al., 1997). Specifically, loss of XCTK2 cross-linking activity by antibody addition to extracts results in a modest decrease in the efficiency of bipolar spindle assembly and results in spindle structures with splayed poles, whereas addition of excess XCTK2 stimulates the formation of bipolar spindles (Walczak et al., 1997; Walczak et al., 1998). XCTK2 function seems to be redundant with the activity of the minus-end–directed motor dynein, because coaddition of XCTK2 and dynein antibodies results in a large increase in the proportion of spindles with unfocused poles (Walczak et al., 1998). XCTK2 was also found to be the limiting component of a large protein complex and coimmunoprecipitates with 95- and 105-kDa proteins. We envisioned that XCTK2 might be a protein regulated by Ran because it is important for spindle assembly and is nuclear during interphase (Walczak et al., 1997). Because the molecular weight of the 95-kDa protein present in the XCTK2 complex is similar to the molecular weight of importin β, we speculated that XCTK2 might bind to importin α/β and be regulated by Ran. We provide biochemical data to support the model that Ran regulates the cross-linking activity of XCTK2 to promote spindle bipolarity by removing the inhibitory effect of importin α/β.
MATERIALS AND METHODS
Protein Expression and Purification
For expression of the bacterially purified recombinant proteins His 6-S-importin β (human importin β1), His 6-S-importin βΔ, importin α-His 6 (Xenopus importin α1a), GST-XCTK2-NM (G-NM; amino acids 2–289), His 6-XCTK2-NM (H-NM; amino acids 2–289), GST-XCTK2-Motor (G-M; amino acids 290–643), GST-Ran L43E, His 6-Ran L43E, and GST-Ran T24N DNA plasmids were induced in BL21(DE3) bacteria, and the protein A-importin α-ED-His 6 construct was induced in M15[pREP4]. Induction was carried out in the presence of 0.1 mM isopropyl β-d-thiogalactoside for 3–5 h. GST-XCTK2-NM, GST-Ran L43E, GST-Ran T24N, and His 6-XCTK2-NM were induced at 37°C, and the other constructs were induced at 20°C. Cells were pelleted, resuspended in phosphate-buffered saline (50 mM phosphate buffer pH 7.4), pelleted, frozen in liquid nitrogen, and stored as cell pellets until needed.
His 6-S-importin β, His 6-S-importin βΔ, importin α-His 6, protein A-importin α-ED-His 6, His 6-Ran L43E, and His 6-XCTK2-M were purified on Ni-nitrilotriacetic acid (NTA) agarose and GST-Ran L43E and GST-Ran T24N were purified on glutathione agarose by using published procedures (Chi et al., 1997; Wilde and Zheng, 1999). GST-XCTK2-NM and His 6-XCTK2-NM were purified on glutathione agarose and Ni-NTA, respectively, and XCTK2 was purified from baculovirus infected Sf-9 cells as described previously (Walczak et al., 1997). All bacterially purified proteins were dialyzed into XB (10 mM HEPES, pH 7.2, 100 mM KCl, 25 mM NaCl, 50 mM sucrose, 0.1 mM EDTA, 0.1 mM EGTA) before being aliquoted, flash frozen in liquid nitrogen and stored at –80°C.
Site-directed Mutagenesis
PSORT, a Web-based protein motif prediction program ( was used to predict putative NLS sequences within the amino acid sequence of XCTK2. The three predicted NLS sequences were mutated by sequential mutagenesis on the His 6-XCTK2-NM and GST-XCTK2-NM DNA plasmids using the QuikChange site-directed mutagenesis system (Stratagene, La Jolla, CA) to generate the NLS mutant plasmids. Based on the published nucleotide sequence of XCTK2 (accession number U82809), four primers were designed to modify the lysine and/or arginine codons to codons that encode for alanine. Primer NLS1a (5′G GAC TCC ACA GAC GCA GCG GTC CAA GTG GCT TCC CG) mutates K6 and K7; primer NLS1b2a (5′GCT TCC CGT TTG CCA GTG CCT CCG GCG GCA GCA TAT GTC TCT AAT GAT G) mutates K19, R20, and K21; primer NLS2b (5′GAA AAT CAA GAG CAG ATG CAG GCG GCG GCT CTC AGA TCC TCC CTA GAG TC) modifies R34, K35, and R36; and NLS3 modifies (5′GCA GCC ATT GGC GCT GAA GCG GCG GCG GCT GCT GCT TGG GAT CTT AAG G) K116, K117, K118, and R119. Modified nucleotides are in italics. The coding region of each mutant construct was sequenced to verify that no extraneous mutations were introduced. Protein was expressed from the induction of the resulting plasmids and purified as described above.
S-Protein Agarose and Ni-NTA Pull-Down Assays
Per reaction, 0.7 pmol of purified importin β and/or importin α and 0.35 pmol of XCTK2, His-/GST-NM domain, or motor domain protein were incubated with 30 μl of S-protein agarose (Novagen, Madison, WI) or Ni-NTA agarose (QIAGEN, Valencia, CA) in 10% FPLC buffer (20 mM PIPES, pH 7.2, 1 mM MgCl 2, 1 mM EGTA, 0.1 mM EDTA, 100 mM KCl) with 2 mM MgATP for 45 min at 4°C with rotation. The protein/bead complexes were pelleted and then washed. For the pull-downs, the beads were washed two times with 10% FPLC buffer, 2 mM MgATP and two times with Tris-buffered saline-Triton X (20 mM Tris, pH 7.5, 150 mM NaCl, 0.1% Triton X-100), and then the protein was eluted with 1× SDS-PAGE sample buffer. For the bind and release experiments, the protein/bead complexes were washed three times with 10% FPLC buffer, 2 mM MgATP, resuspended in the same buffer, and then aliquoted to separate tubes for mock buffer or 25 μM Ran addition. The Ran/bead solution was incubated at room temperature for 15 min, pelleted, and washed as for the pull-down experiment. Equivalent volumes of supernatant and pellet fractions were electrophoresed on 10% SDS-PAGE gels and either stained with Coomassie Brilliant Blue R250 or transferred to Protran (Schleicher & Schuell, Keene, NH) and probed with anti-CTP1 (2 μg ml–1) (Walczak et al., 1997), which recognizes the motor domain, followed by donkey anti-rabbit immunoglobulin (Ig) horseradish peroxidase (HRP)-linked whole antibody (1:20,000; Amersham Biosciences, Piscataway, NJ) and developed using SuperSignal West Pico Chemiluminescent Substrate (Pierce Chemical, Rockford, IL).
Cytostatic Factor-arrested Extracts, Immunofluorescence, and Immunoprecipitations
Cytostatic factor (mitotic) and cycled cytostatic factor arrested extracts were prepared as described previously (Desai et al., 1999a) from Xenopus laevis laid eggs. Spindles were induced with Xenopus sperm and exogenous nonmotor (NM) domain was added to 0.13 μM. Spindle and import reactions were allowed to procede for 30 min before 20 μl were fixed, spun onto coverslips, and processed for immunofluorescence as described previously (Desai et al., 1999a). Coverslips were probed with either anti-XCTK2 (1 μg ml–1) or anti-GST (2.6 μg ml–1) followed by donkey anti-rabbit fluorescein isothiocyanate. Images were taken on a Nikon E600 epifluorescence microscope with a 60× 1.4 numerical aperture objective. The microscope is equipped with a Roper Micromax 1300Y camera and is under control of MetaMorph Software (Universal Imaging, Downingtown, PA). All images were taken at equal exposures and processed equivalently in Adobe Photoshop before assembling figures in Adobe Illustrator.
Immunoprecipitations were performed as described previously (Walczak et al., 1997) except anti-XCTK2, nonimmune rabbit IgG, and anti-glutathione S-transferase (GST) were covalently coupled to the Affi-prep protein A beads (Bio-Rad, Hercules, CA) (Harlow and Lane, 1999). Where indicated, 25 μM purified GST-Ran L43E, His-Ran L43E, His-Ran T24N, or GST-Ran T24N was added to the immunoprecipitation reactions. Equivalent volumes of eluted protein were electrophoresed on 10% SDS-PAGE gels and stained with Coomassie or transferred to Protran. Western blots were probed with anti-importin β (1 μg μl–1; Sigma-Aldrich, St. Louis, MO) or anti-importin α (1:1000; from Mary Dasso, National Institutes of Health, Bethesda, MD) followed by sheep anti-mouse Ig HRP-linked whole antibody (1:20,000; Amersham Biosciences, Piscataway, NJ) or donkey anti-rabbit Ig HRP-linked whole antibody (1:20,000) and developed by chemiluminescence as described above.
Microtubule Pelleting and Microtubule Affinity
MTs were polymerized from purified tubulin with 0.5 mM guanosine-5′-[(α,β)-methyleno]triphosphate (Jena Bioscience) and 10 μM paclitaxel at a 5 μM final tubulin concentration for 30 min at 37°C. Polymerized MTs were pelleted at 90,000 rpm in a Beckman TLA100 rotor and resuspended in BRB80 (80 mM PIPES, pH 6.8, 1 mM MgCl 2, 1 mM EGTA), 1 mM dithiothreitol (DTT), 10 μM paclitaxel. MTs were then diluted to a working concentration in BRB80, 1 mM DTT. MT pelleting reactions were set up in 10% FPLC buffer with either 2 mM MgATP, 2 mM MgADP + 20 mM inorganic phosphate (P i), 10 mM MgATP, or 5 mM MgAMP-PNP. The reactions were started by the addition of MTs to the reaction in a 1:1 ratio. The XCTK2 to MT ratio was 1:3 unless otherwise indicated. For the initial pelleting assays, XCTK2 or NM domain was incubated with 1:32 M ratio of XCTK2 to importin α/β (0.0625 μM XCTK2, monomer, to 2 μM importin α/β). Casein was added to 1 mg ml–1 to stabilize proteins at these low protein concentrations. For all assays, protein and MTs were incubated in a 40- or 50-μl final volume for 15 min at room temperature before pelleting. After centrifugation, the supernatants were removed as the soluble fraction, and the pellets were resuspended in an equal volume of 2× sample buffer and then diluted to a final volume of 80 or 100 μl with 1/2× FPLC buffer, 1/2× BRB80, 0.5 mM DTT. Sample buffer was added to the supernatants for a final volume of 80 or 100 μl. For microtubule-pelleting assays in the presence of Ran, either 25 μM GST-Ran L43E, GST-Ran T24N, His 6-Ran L43E, or His 6-Ran T24N was added to the MT–protein complexes 15 min after MT addition and then incubated for an additional 15 min at room temperature before pelleting. Equal volumes of supernatant and pellet fractions were electrophoresed on 10% SDS-PAGE gels and stained with Coomassie Blue or transferred to Protran. Western blots were probed with either anti-XCTK2 (0.5 μg ml–1) or anti-CTP1 (2 μg ml–1) as described above. For MT pelleting onto coverslips, rhodamine-labeled MTs (6:1 unlabeled to labeled) were polymerized in 0.5 mM guanosine-5′-[(α,β)-methyleno]triphosphate, BRB80, 1 mM DTT, and reactions were fixed and sedimented through a glycerol cushion as described previously (Desai et al., 1999b; Desai and Walczak, 2001).
The apparent K d,MTs of XCTK2 or GST-XCTK2-NM domain constructs at 0.35 μM (monomer concentration) was determined with increasing concentrations of MTs (0.0875–2.8 μM) in 1/2× BRB80, 0.5 mM DTT, 5% FPLC buffer without added nucleotide for 15 min and pelleted as described above. Additional affinities for XCTK2 were determined identically except 2 mM MgADP, 20 mM P i was added to the reaction. Equal volumes of supernatants and pellets were electrophoresed and the gels stained with Coomassie as described above. Gels were scanned and the apparent K d of the constructs determined similarly to that described for Ncd (Foster et al., 1998). Briefly, the amount of pelleted protein was determined from the densitometry of scanned gels by using NIH Image. Nonspecific pelleting of the XCTK2 proteins was determined with 0 μM MTs for each experiment. Per experiment, the amount of protein that pelleted at each MT concentration was corrected for the nonspecific pelleting by subtraction. The amount bound in terms of monomeric protein (micromolar) was then plotted against the total MT concentration defined as the concentration of tubulin dimer (micromolar). Using the GraFit 5 software package, the data were fit to the equation below:
where bound is the monomeric micromolar amount of XCTK2, GST-XCTK2-NM, or NLS-2 that pelleted, K d is the dissociation constant, MT t is the total MT concentration in terms of tubulin dimer, and capacity is the maximum micromolar amount of XCTK2 or NM domain that will pellet under these conditions. The apparent K d in the presence of importin α and/or importin β was determined similarly. The values shown for each construct are from at least three separate experiments. The binding curves displayed are the fit from the averaged data points, but the enumerated K d and capacity values are the average and SE of the mean from the individual experiments. F tests for variance equivalence and Student's t tests based on the variance equivalence were performed using EXCEL.
RESULTS
Ran-GTP Regulates the Association of Importin α and Importin β with XCTK2 in Xenopus Egg Extracts
We sequenced the 95-kDa protein that coimmunoprecipitates with XCTK2 by using two separate antibodies (Walczak et al., 1997) and found that 12 of the 14 peptides sequenced were homologous to mouse importin β. In additional anti-XCTK2 immunoprecipitations from cytostatic factor-arrested (mitotic) egg extracts, antibodies to importin β recognized the 95-kDa band, confirming the identity of this protein (Figure 1A, lane 2, middle). Because importin β associates with importin α in the import of proteins into the nucleus, the immunoprecipitations were also probed for importin α. The importin α antibodies recognized a 55-kDa band in the anti-XCTK2 immunoprecipitates, indicating that importin α also associates with XCTK2 (Figure 1A, lane 2, bottom).
Figure 1.
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Importin α and importin β associate with XCTK2 in a Ran-dependent manner via a bipartite NLS in the tail of XCTK2. (A) Coomassie-stained gel (top) and importin β (middle) or importin α (bottom) Western blots of XCTK2 immunoprecipitations from Xenopus egg extracts. Lane 1, immunoprecipitation control with nonimmune rabbit immunoglobulin (IgG). Lane 2–4, immunoprecipitations with anti-XCTK2. The 95-kDa band was used for protein sequencing and identified as importin β (β). The immunoprecipitated XCTK2 protein (X) and coimmunoprecipitated importin α (α) bands are also indicated. Lane 3, XCTK2 immunoprecipitation in the presence of purified Ran L43E (L43E). Lane 4, immunoprecipitation in the presence of purified Ran T24N (T24N). Importin α is slightly larger than the heavy chain of the anti-XCTK2 antibody (IgG H) and migrates slower by SDS-PAGE as seen on the Coomassie-stained gel and the importin α Western blot. The positions of molecular weight standards (kilodaltons) are indicated to the left of the figure. (B) Diagram of the domains of XCTK2. XCTK2 is composed of an N-terminal NM and a C-terminal motor domain (Motor). The NM domain consists of the globular tail (Tail), which contains the NLS-(NLS) and MT-binding sites, and the central α-helical stalk that is important for dimerization. (C and D) Coomassie stained gels of S-importin β pull-downs (C) and importin α-His pull-downs (D). Lane 1 of parts C and D are input (I) amounts of purified protein. The remaining lanes contain the indicated combinations of proteins. The NM domain of XCTK2 associates with importin β through importin α and can bind directly to importin α. (E) Amino acid sequences of the three putative NLS sequences contained within the tail of XCTK2 and the three NLS mutants generated. The corresponding amino acid numbers are indicated to the left of the NLS and the individual putative NLS sequences are underlined or overlined. The consensus sequence for a bipartite NLS is 2 K/R 10–12 amino acids 3 K/R and for a simple NLS is KKxK. Data from importin α-His pull-downs, nuclear import, and spindle localization are indicated to the right of the mutant sequences and indicate binding, import or localization (+), partial binding (+/–), or no binding or no import (–). (F) S-Importin β pull-downs of the three putative NLS mutants. Lane 1 contains the input (I) amounts of purified protein. The remaining lanes contain the indicated combinations of importin α, importin β, and either wild-type (wt) or NLS-mutated versions of G-NM. The NLS-1 mutant (N1) partially binds to importin α, whereas the NLS-2 mutant (N2) binds importin α the least compared with wt G-NM. Mutation of the simple NLS does not affect importin α binding (N3). The input amount of only wt G-NM is shown, but the NLS mutant versions had similar amounts of input protein.
To test whether Ran regulated the association of importin α and importin β with XCTK2, we performed additional immunoprecipitations from extracts with the anti-XCTK2 antibody in the presence of Ran L43E (L43E), an effector domain mutant that binds tightly to GTP (Lounsbury et al., 1996), or in the presence of Ran T24N (T24N), a dominant negative allele that strongly associates with its nucleotide exchange factor RCC1 (Dasso et al., 1994; Klebe et al., 1995). Addition of Ran L43E to immunoprecipitation reactions abolished the ability of importin α and importin β to coimmunoprecipitate with XCTK2 (Figure 1A, lane 3). In contrast, addition of Ran T24N did not affect the association of importin α and β with XCTK2 (Figure 1A, lane 4). These results show that Ran regulates the association of XCTK2 with importin α and importin β in mitotic egg extracts.
Importin α/β Bind Directly to XCTK2 through a Bipartite NLS in the Nonmotor Domain of XCTK2
Because egg extracts are a complex mixture of proteins, it is possible that the Ran-regulated association of importin α and β with XCTK2 is mediated by other factors. To investigate whether XCTK2 could bind directly to importin α and/or importin β, we performed pull-down experiments by using purified proteins and a fragment of XCTK2 that contained the N-terminal globular tail and the central α-helical stalk that we refer to as the NM domain (Figure 1B). Pull-down experiments with importin β showed that full-length XCTK2 and a His-tagged version of the NM domain of XCTK2 (H-NM) did not bind well to importin β alone (Figure 1C, lanes 4 and 6). However, in the presence of importin α, both associated with importin β (Figure 1C, lanes 5 and 7). Because importin α and β are the nuclear import receptors for proteins that contain classical NLS sequences and because nuclear import by importin β is often mediated by importin α (Chook and Blobel, 2001), we performed additional pull-down experiments with importin α alone. Both XCTK2 and a GST-tagged version of the NM domain of XCTK2 (G-NM) bound directly to importin α (Figure 1D, lanes 3–4), whereas a fragment of XCTK2 containing only the motor domain was not able to bind to importin α (our unpublished data). As an additional control, we performed pull-down experiments with the ED mutant of importin α that does not bind to bipartite NLS sequences due to a mutation in each NLS binding pocket (E389R and D189K) (Conti et al., 1998; Gruss et al., 2001) and found that the NM domain of XCTK2 did not associate with mutant importin α (our unpublished data). These results suggest that the binding of XCTK2 to importin α is mediated by a bipartite NLS in the NM domain of XCTK2 and that the association of XCTK2 with importin β in extracts is through importin α.
We examined the sequence of XCTK2 and identified three putative classical NLS sequences in the N-terminal globular tail of XCTK2 (Figure 1E). Two were bipartite and one represented a simple NLS (Conti et al., 1998). To define the NLS responsible for the binding of XCTK2 to importin α, sequential site-directed mutagenesis was performed to mutate the lysine and/or arginine residues to alanine residues in the three putative NLS sequences (Figure 1E). We then assayed the mutant G-NM proteins for binding to importin α by using in vitro pull-down assays. Mutation of NLS-1 (amino acids 6–21) resulted in partial inhibition of importin α/β binding (Figure 1F, lane 3). Mutation of the second putative bipartite NLS (NLS-2, amino acids 19–36) resulted in essentially full inhibition of importin α binding (Figure 1F, lane 4), whereas mutation of NLS-3, the putative simple NLS (amino acids 116–119), had no detectable effect on the NM domain binding to importin α (Figure 1F, lane 5). These results suggest that NLS-2, residing in the tail of XCTK2, is the major NLS.
We next tested the competency of the wild-type and mutant G-NM proteins to undergo nuclear import in egg extracts. Mutation of either NLS-1 or NLS-2 caused a defect in nuclear import, whereas mutation of NLS-3 had no effect (Figure 1E). We suspect that the nuclear import defect of the NLS-1 mutant is a consequence of the amino acid sequence overlap with NLS-2. It has been shown that modification of lysine and/or arginine residues in the head or tail of a bipartite NLS can confer an import defect (Ishii et al., 1996; Efthymiadis et al., 1997; Taniguchi et al., 2002). None of the putative NLS mutations conferred a disruptive effect on MT localization in extracts, because all G-NM proteins localized to bipolar spindles (Figure 1E), identical to the C-terminal kinesin CHO2 stalk-tail construct transfected into Chinese hamster ovary cells (Matuliene et al., 1999). From these results, we conclude that NLS-2 is the bona fide NLS sequence in the XCTK2 tail and that this sequence is responsible for importin α binding to XCTK2.
Ran Can Directly Regulate the Binding of Importin α/β to XCTK2
The above-mentioned experiments established that XCTK2 associates with importin β through the direct binding of importin α to a bipartite NLS in the tail of XCTK2. We wanted to know whether Ran regulated this association in a manner similar to the nuclear import pathway. We devised a bind and release experiment where we first formed an importin β/importin α/XCTK2 ternary complex on agarose beads and then incubated the complex with buffer, Ran L43E, or Ran T24N to look for release of protein. We expected that if Ran regulated the binding of the importins to XCTK2, then incubation of the ternary complex with Ran L43E would result in the release of XCTK2 and importin α from importin β bound to the agarose beads. Incubation of this complex with concentrations of Ran L43E that are sufficient to induce asters in spindle assembly reactions (Wilde and Zheng, 1999; Wilde et al., 2001) resulted in the near complete release of XCTK2 from importin β (Figure 2A, lanes 3 and 4), whereas incubation with Ran T24N did not release XCTK2 or importin α from the beads (lanes 5 and 6). Similar results were obtained using either the H-NM or G-NM proteins. We determined that this Ran effect is directly dependent on the Ran binding domain of importin β by repeating the experiment with a deletion construct of importin β in which the Ran-binding domain is deleted (Nachury et al., 2001), called importin βΔ. In this experiment, Ran L43E did not dissociate XCTK2 from importin βΔ (Figure 2A, lanes 9–10). These results establish that Ran regulates the interaction between importin α/β and XCTK2 in a reconstituted system by using purified proteins.
Figure 2.
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XCTK2 binding to importin α/β is regulated by the nucleotide state of Ran and is dependent on the Ran-binding domain of importin β. Coomassie-stained gels and anti-CTP1 immunoblot of the supernatants and pellets of S-importin β and S-importin βΔ (βΔ) bind and release assays in the presence and absence of Ran (R). (A) S-importin β bind and release assay with importin α-His and XCTK2 in the absence (No Ran, lanes 1 and 2) and presence of GST-Ran L43E-GTP (Ran L43E, lanes 3 and 4) or GST-Ran T24N-GDP (Ran T24N, lanes 5 and 6). The asterisk indicates a contaminating, comigrating band from the Ran T24N purification. Bottom, an immunoblot of the same fractions probed with the anti-CTP1 antibody. Right, S-importin βΔ bind and release assay with importin α and XCTK2 in the absence (lanes 7 and 8) and presence of GST-Ran L43E-GTP (lanes 9 and 10). (B) Anti-GST immunoprecipitation of GST (lane 1), G-NM (wt, lane 2), NLS-1 (N1, lane 3), NLS-2 (N2, lane 4), and NLS-3 (N3, lane 5). G-NM (lane 6) and NLS-3 mutant (lane 7) anti-GST immunoprecipitations in the presence of His-Ran L43E (+Ran L43E).
To demonstrate that the NLS in the tail of XCTK2 is responsible for importin α/β binding in a more physiological context, we performed anti-GST immunoprecipitations from mitotic egg extracts to which we added either wild-type or NLS-mutant versions of the GST-tagged NM proteins. Immunoprecipitation of exogenously added wild-type G-NM or the NLS-3 mutant resulted in the coimmunoprecipitation of importin α and importin β (Figure 2B, lanes 2 and 5). In contrast, importin α and importin β failed to coimmunoprecipitate with the NLS-1 or NLS-2 mutant (lanes 3 and 4), consistent with their nuclear import defect. Similarly to full-length XCTK2, addition of Ran L43E abolished the ability of importin α and β to coimmunoprecipitate with wild-type G-NM and NLS-3 (lanes 6 and 7). Together, our results demonstrate that Ran regulates the direct binding of importin α/β to the tail of XCTK2 via the same mechanism used for the nuclear import of proteins containing classical NLS sequences (Chook and Blobel, 2001; Kuersten et al., 2001).
Importin α/β Sequester the Microtubule-binding Activity of the XCTK2 Tail
XCTK2 and other members of the C-terminal kinesin family are proposed to function by cross-linking and sliding MTs in the spindle via their motor domains and globular tails (Chandra et al., 1993; Walczak et al., 1997; Karabay and Walker, 1999; Matuliene et al., 1999; Mountain et al., 1999). The motor domain binds to MTs in an ATP-dependent manner, whereas the tail binds MTs in an ATP-independent manner through undefined and nonconserved MT binding sites (Chandra et al., 1993; Karabay and Walker, 1999; Matuliene et al., 1999; Karabay and Walker, 2003). A significant amount of research has been done on the motor domains of C-terminal kinesins, but little work has been done with regards to the tails, and the mechanism for regulating the cross-linking activity has not been elucidated. Because importin α/β bind to XCTK2 through the tail, we envisioned that importin α/β might modulate the MT cross-linking activity of XCTK2 by sequestering the MT binding of the tail and that this activity would be regulated by Ran.
We first wanted to test the effects of importin α/β on the ability of XCTK2 to bind to MTs. If importin α/β affects the MT binding of the tail, then the ability of full-length XCTK2 to bind to MTs would vary with nucleotide condition, whereas the H-NM and G-NM proteins would bind to MTs independent of nucleotide condition. In the presence of 2 mM MgATP, XCTK2 and the H-NM protein cosedimented with MTs in the absence of importin α/β. In contrast, in the presence of importin α/β, ∼50% of full-length XCTK2 and all of the H-NM protein no longer cosedimented with MTs (Figure 3A). The motor domain of XCTK2 probably still bound MTs under this physiological ATP condition and could account for the partial amount of XCTK2 that bound to MTs in the presence of importin α/β. These results imply that importin α/β sequestered only the NM domain away from MTs (Figure 3E). As a test for the independence between the tail binding to MTs and the motor domain binding to MTs, we next assayed the ability of XCTK2 to bind MTs in the presence of high concentrations of MgATP or MgADP plus P i. These conditions inhibit only the MT binding activity of the motor domain of C-terminal kinesins (Chandra et al., 1993; Foster et al., 1998). In the presence of either 2 mM MgADP + 20 mM P i or 10 mM MgATP, importin α/β efficiently inhibited the MT binding activity of full-length XCTK2 (Figure 3, B and C, F and G). In contrast, in the presence of the nonhydrolyzable ATP analog AMP-PNP, which prevents the motor head from releasing MTs (Chandra et al., 1993; Foster et al., 1998; Wendt et al., 2002), importin α/β no longer inhibited XCTK2 from binding to MTs (Figure 3, D and H). Under all conditions tested, the H-NM protein bound to MTs and its binding was inhibited by importin α/β. These results suggest that importin α/β likely functions to inhibit or moderate the MT cross-linking activity of XCTK2 by binding to the tail.
Figure 3.
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Importin α/β inhibits the nonmotor microtubule binding domain in full-length XCTK2 from binding microtubules. Immunoblots of the supernatants and pellets of XCTK2 and His-tagged NM domain microtubule pelleting assays under different nucleotide conditions, without (– α/β) and with (+ α/β) a 32:1 ratio of importin α/β to XCTK2. XCTK2 blots were probed with anti-CTP1 and H-NM blots were probed with the anti-XCTK2. (A) 2 mM MgATP. (B) 2 mM MgADP + 20 mM P i. (C) 10 mM MgATP. (D) 5 mM MgAMP-PNP. (E–H) Schematic representations of the results in a–d. (E) In 2 mM MgATP and the absence of importin α/β (green and purple), XCTK2 can bind to and pellet with MTs either through the tail (gray domain) only, by cross-linking MTs through the motor domain (red domain) and the tail or through the motor domain only (our unpublished data). The NM domain (gray and yellow domains) binds tightly to MTs. In the presence of importin α/β (E, bottom), the population of XCTK2 that only binds to MTs through the tail is sequestered away from MTs, whereas XCTK2 that cross-links MTs can still bind to and pellet with MTs through its motor domain. (F and G) In the presence of 2 mM MgADP + 20 mM P i or 10 mM MgATP, the motor domain does not bind tightly to MTs. Thus, in the absence of importin α/β, XCTK2 and H-NM bind to and pellet with MTs, whereas in the presence of importin α/β, both XCTK2 and H-NM are sequestered away from MTs. (H) In the presence of 5 mM MgAMP-PNP, the motor domain binds tightly to MTs with no effect by importin α/β.
Importin α/β Compete with Microtubules for G-NM
The working model of importin α/β sequestering spindle assembly factors in the absence of Ran-GTP dictates that importin α/β must compete effectively with the substrates of the proteins important in spindle assembly. To begin to understand the molecular mechanism by which importin α/β compete with MTs for binding to the XCTK2 tail, we determined the affinity of XCTK2 for MTs and the affinity of the NM domain for MTs in the absence or presence of various concentrations of importin α/β. We first measured the apparent affinity of XCTK2 for MTs in the absence of nucleotide and in the presence of 2 mM MgADP + 20 mM P i. XCTK2 had a threefold stronger apparent affinity for MTs in the absence of nucleotide (K d,MT = 34 ± 1 nM) compared with its affinity in the presence of ADP + P i (K d,MT = 110 ± 10 nM; p < 0.001), suggesting that the MT pelleting of the NM domain can be uncoupled from the ATP-dependent MT binding of the motor domain. Because we were interested in how importin α/β regulated the tail of XCTK2, we performed additional measurements for full-length XCTK2 in the presence of ADP + P i. Addition of importin α/β to 0.35 μM caused a threefold decrease in affinity (p < 0.05) and addition of 0.525 μM importin α/β caused a fivefold reduction in the affinity of XCTK2 for MTs (p < 0.05) (Figure 4, A and D). Under these same conditions, the capacity of XCTK2 for MTs was reduced by 20 and 35% (p < 0.05) (Figure 4, A and D).
Figure 4.
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Importin α/β compete with microtubules for the XCTK2 tail and alter the affinity and capacity of G-NM for microtubules in a dose-dependent manner that is dependent on the bipartite NLS. Averaged ligand binding curves for full-length XCTK2 (A), wild-type G-NM (B), and NLS-2 mutant (C) in the absence (open symbols) and presence of importin α/β at 0.35 μM (shaded symbols) or 0.525 μM (black symbols). Concentrations of XCTK2 and G-NM are in terms of protein monomer and MTs in terms of tubulin dimer. (D) Affinity and capacity values determined from the ligand binding curves in A–C. The resulting MT affinities (K d) and maximum binding (capacity) are presented in terms of nanomolar concentrations ± the SE of the mean of at least three separate experiments.
To characterize the ability of the NM tail to bind to MTs in the presence of importin α/β in more detail and without constraints potentially imposed by the motor domain, similar affinity assays were performed with the G-NM protein. The K d of G-NM for MTs was 22 ± 2 nM with complete MT binding (Figure 4, B and D). The affinity of the G-NM protein was fivefold higher than for full-length XCTK2 in the presence of ADP + P i (p < 0.01), suggesting the motor domain likely imposed additional constraints on the conformation of the stalk in full-length XCTK2 but not in our stalk-tail construct (Wendt et al., 2002). Although these are apparent K d values, the values in this range and resulting curves are highly suggestive of very tight binding affinity and suggest a binding stoichiometry of one tubulin dimer per one NM domain.
Similar to full-length XCTK2, the G-NM protein also had reduced affinity and capacity for MTs in the presence of importin α/β (Figure 4B). Addition of a twofold molar amount of importin α/β resulted in a sixfold decrease in affinity (p < 0.05) and a 20% decrease in capacity (p < 0.05). Threefold molar excess of importin α/β resulted in a 19-fold decrease in affinity (p < 0.01) and 28% decrease in capacity (p < 0.05) (Figure 4D). Note that this effect was strikingly similar to full-length XCTK2. Addition of an eightfold molar excess (1.4 μM) of importin α/β resulted in the inability of the NM domain to bind to MTs (our unpublished data). These results indicate that the XCTK2 tail has a greater affinity for importin α/β than for MTs and that importin α/β not only reduce the affinity of the tail for MTs but also prevent the tail from binding MTs.
To test the importance of the NLS in importin α/β-regulated inhibition of MT binding, similar experiments were performed with the NLS-2 mutant. Because importin α/β are unable to bind to NLS-2 (Figure 1E) we expected that importin α/β would not affect the affinity of NLS-2 for MTs. Indeed, importin α/β did not have a statistically significant effect on the affinity (p = 0.70) for MTs or the capacity (p = 0.50) (Figure 4, C and D). These results demonstrate that importin α/β directly influence the MT binding activity of XCTK2 through the NLS-2 sequence that interacts with importin α. Unexpectedly, the NLS-2 mutant protein displayed an eightfold reduced affinity for MTs compared with the wild-type G-NM (p < 0.001) but not a reduced capacity (p = 0.12). Although the MT binding sites of other C-terminal kinesin tails have not been defined, it is generally believed that the positively charged residues are important for MT binding (Woehlke et al., 1997; Karabay and Walker, 1999; Kikkawa et al., 2000). The reduced affinity for MTs suggests that mutation of the six lysine/arginine residues in the tail of our constructs either alters the MT binding site or results in conformational changes that are allosterically transferred to the MT binding site.
Ran-GTP Promotes the Binding of the Tail to Microtubules by Releasing the Sequestering Activity of Importin α/β
Because Ran can regulate the association of XCTK2 with importin α/β, we predicted that Ran could also regulate the ability of XCTK2 to bind to MTs. We developed a visual assay to examine the G-NM protein binding to MTs. The G-NM protein colocalized with MTs in the absence of importin α/β but had drastically reduced binding in the presence of a fourfold molar excess of importin α/β (Figure 5, A and B). In contrast, the MT binding of the NLS-2 mutant was not affected by the presence of importin α/β (Figure 5, C and D). Addition of Ran L43E to the G-NM/importin α/β ternary complex restored the ability of the G-NM protein to bind MTs (Figure 5E), whereas addition of Ran T24N had little effect (Figure 5F). Classical MT pelleting assays using either full-length XCTK2 or H-NM gave similar results. These data show that Ran-GTP regulates the binding of the tail of XCTK2 to MTs in the presence of importin α/β and is dependent upon the NLS.
Figure 5.
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Ran-GTP promotes XCTK2 microtubule binding in the presence of importin α/β. Immunofluorescence analysis of G-NM and NLS-2 MT binding in the absence (– α/β) and presence of importin α/β (+ α/β) and Ran (+ Ran L43E or + Ran T24N). (A and B, E and F) Micrographs of reconstituted Ran regulation of the MT binding of the G-NM domain to rhodamine-labeled MTs. (C and D) Micrographs of NLS-2 MT binding in the absence and presence of importin α/β. Bar, 20 μm for all panels.
DISCUSSION
The small GTPase Ran is essential for spindle assembly and regulates diverse spindle assembly processes (Kahana and Cleveland, 1999; Kuersten et al., 2001; Walczak, 2001; Dasso, 2002; Hetzer et al., 2002; Quimby and Dasso, 2003). Although several spindle processes regulated by Ran have been identified, it is unclear how Ran functions at the molecular level to organize the MTs within the spindle because most of these experiments were performed in egg extracts. One proposed model to account for Ran action in extracts is that importin α and importin β function as a complex to inhibit spindle formation by sequestering a set of proteins required for the process of spindle assembly (Gruss et al., 2001; Nachury et al., 2001; Walczak, 2001; Wiese et al., 2001). The model proposes that importin α/β binds to proteins with spindle-promoting activities in mitotic extracts, such that upon chromatin or exogenous Ran-GTP addition, the importins dissociate, which then activates the spindle-promoting factors (Figure 6A). Given the diversity of the spindle assembly processes regulated by Ran, the mechanism by which Ran acts must be robust and yet specific. We propose that for spindle assembly it is essential for importin α and β to be present in vast molar excess in the extract relative to proteins important in spindle formation and that importin α/β and/or importin β be able to effectively compete with the substrates of the spindle-promoting factors. The former seems to be true because importin α and β are estimated to be present at 10–20 μM in extracts (Jans et al., 2000) relative to an extract concentration of only ∼100 nM for TPX2 (Gruss et al., 2001) and ∼40 nM for NuMA (Merdes and Cleveland, 1997). We have estimated an extract concentration of XCTK2 to be only ∼10 nM, well below the endogenous concentration of importin α and importin β (Walczak et al., 1997). Furthermore, our previous work shows that all of the XCTK2 in egg extracts is present in a large complex and that upon addition of recombinant XCTK2 to 100 nM, the added XCTK2 is completely incorporated into the large complex, suggesting it is a limiting component of that complex (Walczak et al., 1997). The work presented here demonstrates that importin α/β binds to XCTK2 and is likely a component of the XCTK2 complex. Together, these results are supportive of the idea that importin α/β are in sufficient quantities to sequester the known spindle-promoting factors, as well as many others.
Figure 6.
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Models for Ran-regulated spindle assembly. (A) One proposed model of Ran-induced spindle assembly. The importin α/β complex sequesters factors that possess spindle-promoting activity (SPF) whereupon exogenous Ran-GTP or chromatin addition, Ran-GTP binds to importin β and causes the dissociation of importin α/β from the spindle promoting factor. The “activated” spindle-promoting activities then promote spindle assembly. (B) Our proposed model of Ran-induced XCTK2 MT cross-linking. The importin α/β complex sequesters the cross-linking activity of XCTK2 by binding tightly to the tail of XCTK2 and thus prevents it from binding to MTs. With the addition of Ran-GTP, the sequestering activity of importin α/β is inhibited, thereby promoting XCTK2 MT cross-linking.
More importantly, to effectively inhibit spindle assembly, importin α/β must be able to successfully compete with the substrates of the spindle-promoting factors. Using purified proteins, we show here that importin α/β and Ran oppose one another to regulate the ability of XCTK2, a protein with spindle promoting activity, to bind to MTs. Specifically, by tightly binding to the tail of XCTK2 through a bipartite NLS, importin α/β inhibit the activity of the tail. They do so by competing with MTs and efficiently sequestering the MT binding domain of XCTK2 that results in the inability of XCTK2 to cross-link MTs. This result is consistent with previous observations that the importin α/β complex has a very high affinity for NLS sequences (2–180 nM) (Fanara et al., 2000; Jans et al., 2000; Catimel et al., 2001; Harreman et al., 2003) and that high salt concentrations are required to cause the release XCTK2 from MTs by the addition of ATP (Walczak et al., 1997). In addition, neither importin α nor importin β alone has a significant effect on the affinity of XCTK2 for MTs even though XCTK2 can bind to importin α alone in pull-down experiments (our unpublished data). These results are consistent with previous studies that showed that the affinity of importin α for NLS sequences increases up to 300-fold in the presence of importin β (Efthymiadis et al., 1997; Hu and Jans, 1999; Fanara et al., 2000), which is likely due to importin β binding to and sequestering the autoinhibitory sequence in importin α (Kobe, 1999; Fanara et al., 2000; Hodel et al., 2001; Harreman et al., 2003). This suggests that importin α cannot compete with MTs in the absence of importin β because of a reduced affinity for the XCTK2 tail. Thus, the Ran-regulated mechanism used for the nuclear import of proteins is extremely well suited for regulating spindle assembly because it evolved for the efficient import of diverse proteins into the nucleus.
These results provide the first report of importin α/β directly affecting the MT-binding activity of a MT motor in an NLS-dependent manner, which could provide a very effective mechanism to regulate spindle formation. Our findings are distinct from the recent report showing that importin α mediates TPX2-induced MT nucleation because in that study importin α does not inhibit TPX2 MT association (Schatz et al., 2003). For XCTK2, we envision two models of how importin α/β binding might influence activity. If the importin α/β binding site on the XCTK2 tail were distinct from the MT binding site, then it might be possible for XCTK2 to simultaneously bind to both MTs and to importin α/β. This situation would result in incomplete sequestering by importin α/β. On the other hand, if the importin α/β binding site on the XCTK2 tail coincides with the MT binding site or if importin α/β binding sterically obstructs the MT binding site, complete sequestering would occur. We favor the latter scenario in which importin α/β sterically obstruct the MT binding site of the tail such that there is maximal sequestering of the MT binding activity. This is supported by several experimental findings. First, the addition of anti-XCTK2 NM antibodies to egg extracts causes mislocalization of XCTK2 toward the poles (Walczak et al., 1997). These antibodies must be sterically blocking the MT binding site in the tail of XCTK2. Second, importin β in addition to importin α is required to prevent the NM domain from binding to MTs, suggesting that importin β physically occludes the MT binding site. This steric exclusion mechanism is similar to the way in which the mammalian Partner of inscuteable, LGN, inhibits the MT binding and stabilization activity of NuMA (Du et al., 2002). We predict that where Ran-GTP levels are highest, Ran-GTP would have the greatest consequence on the activities of factors that are completely sequestered by importin α/β or importin β. Together, these data provide a mechanism by which Ran can discretely regulate many activities through importin α/β.
How can this molecular mechanism be incorporated into what we know about the physiological function of XCTK2 in spindle assembly? We know that cross-linking MTs by XCTK2 or its homologs is important for bipolar spindle assembly and is needed to help focus spindle poles. In agreement with this function, C-terminal kinesins are localized to both the MTs of the mitotic spindle and spindle poles (Walczak et al., 1997, 1998; Matuliene et al., 1999; Mountain et al., 1999). Specifically, we previously demonstrated that in addition to a moderate decrease in spindle formation, addition of an antibody raised against the NM domain of XCTK2 to spindle assembly reactions results in the absence of XCTK2 localization on spindle MTs and enrichment at spindle poles, suggesting that the antibody inhibits XCTK2 cross-linking activity. Consistent with this idea, addition of the same antibody to MT-binding experiments results in the prevention of XCTK2-induced MT bundling (Walczak et al., 1997). We have recently found that when the wild-type and NLS-2 mutant NM domains are added to aster assembly reactions in extracts, asters with differing morphology assemble. In addition, the localization of the NM domain proteins to MTs in these extracts is influenced by the presence of Ran (our unpublished data). Thus, the localization of Ran-GTP specifically around chromatin (Bamba et al., 2002; Kalab et al., 2002; Moore et al., 2002) and spindle poles (Keryer et al., 2003) puts Ran at precisely the correct place for the activation of C-terminal kinesins as well as additional essential players in spindle assembly. Here, we provide experimental data that supports a model for Ran regulation of XCTK2 MT cross-linking whereby importin α/β effectively compete with MTs to bind to and inhibit XCTK2 cross-linking activity where Ran-GTP levels are low (Figure 6B), e.g., outside the boundaries of the spindle.
Driven by the positional cues of the Ran-GTP gradient within the spindle, XCTK2 will only cross-link MTs in the vicinity of the chromosomes and the spindle where MT levels are high. The cross-linking activity at the chromosomes would help organize the MTs, and then as XCTK2 moved toward the minus-ends, the cross-linking activity would help focus the MTs into discrete poles. We expect XCTK2 homologs to be regulated similarly in cells. Our modeling shows that a steep Ran-GTP gradient can be established in cells (Li et al., 2003) and that perturbation of RCC1 localization or ectopic expression of Ran-GTP in the cytoplasm of cells causes spindle defects (Moore et al., 2002). The existence of a Ran-GTP gradient is well established in egg extracts; however, it is not clear whether a similar gradient is established in cells as mathematical modeling has led to different conclusions (Görlich et al., 2003; Li et al., 2003). To resolve this issue, experimental measurements of the Ran-GTP gradient is necessary. We speculate that in light of importin α/β and Ran precisely regulating the ability of the XCTK2 tail to bind to MTs that even a slight Ran-GTP gradient may be sufficient to provide positional cues. Additionally, due to the complexity of bipolar spindle assembly it will be essential in the future to elucidate the physiological contribution of the NLS sequence in XCTK2 and its homologs both in extracts and in cells. Finally, upon completion of mitosis XCTK2 and its homologs are likely transported into the nucleus at telophase to prevent unwanted MT cross-linking during interphase.
In summary, we show here that XCTK2 MT cross-linking activity is regulated by the importin α/β and Ran pathway through a classical bipartite NLS and that importin α/β can uniquely regulate XCTK2 by directly interfering with the MT binding site of the XCTK2 tail. This provides a biochemical mechanism for the functional observation that MT cross-linking induced by XCTK2 is important for bipolar spindle assembly. In the future, it will be of great interest to identify all of the spindle components that interact with the effectors of Ran and to determine how Ran temporally and spatially regulates their diverse functions.
Acknowledgments
We thank Kathleen Hertzer, Sarah Johnstone, Susan Kline-Smith, Jane Stout, and Chris Wiese for critical review of the manuscript; Steve Adam for the His-S-importin β/βΔ and importin α-His constructs; Iain Mattaj for the protein A-importin α ED-His construct; Mary Dasso for the importin α antibody; and Ona Martin for purification of His 6-Ran L43E and His 6-Ran T24N. We also thank Susan Gilbert and Martin Stone for their assistance with fitting the MT binding data. This work was supported by grants from the National Institutes of Health (to C.E.W. and Y.Z.) and by a fellowship from the Walther Cancer Institute (to S.E.M.). C.E.W. is a Scholar of the Leukemia and Lymphoma Society.
Article published online ahead of print. Mol. Biol. Cell 10.1091/mbc.E03–07–0454. Article and publication date are available at www.molbiolcell.org/cgi/doi/10.1091/mbc.E03-07-0454.
References
Askjaer, P., Galy, V., Hannak, E., and Mattaj, I.W. (2002). Ran GTPase cycle and importins α and β are essential for spindle formation and nuclear envelope assembly in living Caenorhabditis elegans embryos. Mol. Biol. Cell 13, 4355–4370 [DOI] [PMC free article] [PubMed] [Google Scholar]
Bamba, C., Bobinnec, Y., Fukuda, M., and Nishida, E. (2002). The GTPase Ran regulates chromosome positioning and nuclear envelope assembly in vivo. Curr. Biol. 12, 503–507 [DOI] [PubMed] [Google Scholar]
Bilbao-Cortes, D., Hetzer, M., Langst, G., Becker, P.B., and Mattaj, I.W. (2002). Ran binds to chromatin by two distinct mechanisms. Curr. Biol. 12, 1151–1156 [DOI] [PubMed] [Google Scholar]
Carazo-Salas, R.E., Gruss, O.J., Mattaj, I.W., and Karsenti, E. (2001). Ran-GTP coordinates regulation of microtubule nucleation and dynamics during mitotic-spindle assembly. Nat. Cell Biol. 3, 228–234 [DOI] [PubMed] [Google Scholar]
Carazo-Salas, R.E., Guarguaglini, G., Gruss, O.J., Segref, A., Karsenti, E., and Mattaj, I.W. (1999). Generation of GTP-bound Ran by RCC1 is required for chromatin-induced mitotic spindle formation. Nature 400, 178–181 [DOI] [PubMed] [Google Scholar]
Catimel, B., Teh, T., Fontes, M.R., Jennings, I.G., Jans, D.A., Howlett, G.J., Nice, E.C., and Kobe, B. (2001). Biophysical characterization of interactions involving importin-α during nuclear import. J. Biol. Chem. 276, 34189–34198 [DOI] [PubMed] [Google Scholar]
Chandra, R., Salmon, E.D., Erickson, H.P., Lockhart, A., and Endow, S.A. (1993). Structural and functional domains of the Drosophila ncd microtubule motor protein. J. Biol. Chem. 268, 9005–9013 [PubMed] [Google Scholar]
Chi, N.C., Adam, E.H., and Adam, S.A. (1997). Different binding domains for Ran-GTP and Ran-GDP/RanBP1 on nuclear import factor p97. J. Biol. Chem. 272, 6818–6822 [DOI] [PubMed] [Google Scholar]
Chook, Y.M., and Blobel, G. (2001). Karyopherins and nuclear import. Curr. Opin. Struct. Biol. 11, 703–715 [DOI] [PubMed] [Google Scholar]
Compton, D.A. (2000). Spindle assembly in animal cells. Annu. Rev. Biochem. 69, 95–114 [DOI] [PubMed] [Google Scholar]
Conti, E., Uy, M., Leighton, L., Blobel, G., and Kuriyan, J. (1998). Crystallographic analysis of the recognition of a nuclear localization signal by the nuclear import factor karyopherin α. Cell 94, 193–204 [DOI] [PubMed] [Google Scholar]
Dasso, M. (2002). The Ran GTPase: theme and variations. Curr. Biol. 12, R502–R508. [DOI] [PubMed] [Google Scholar]
Dasso, M., Seki, T., Azuma, Y., Ohba, T., and Nishimoto, T. (1994). A mutant form of the Ran/TC4 protein disrupts nuclear function in Xenopus laevis egg extracts by inhibiting the RCC1 protein, a regulator of chromosome condensation. EMBO J. 13, 5732–5744 [DOI] [PMC free article] [PubMed] [Google Scholar]
Desai, A., Murray, A., Mitchison, T.J., and Walczak, C.E. (1999a). The use of Xenopus egg extracts to study mitotic spindle assembly and function in vitro. Methods Cell Biol. 61, 385–412 [DOI] [PubMed] [Google Scholar]
Desai, A., Verma, S., Mitchison, T.J., and Walczak, C.E. (1999b). Kin I kinesins are microtubule-destabilizing enzymes. Cell 96, 69–78 [DOI] [PubMed] [Google Scholar]
Desai, A., and Walczak, C.E. (2001). Assays for microtubule destabilizing kinesins. In: Kinesin Protocols, ed. I. Vernos, Totowa, NJ: Humana Press, 109–121 [DOI] [PubMed]
Du, Q., Taylor, L., Compton, D.A., and Macara, I.G. (2002). LGN blocks the ability of NuMA to bind and stabilize microtubules: a mechanism for mitotic spindle assembly regulation. Curr. Biol. 12, 1928–1933 [DOI] [PubMed] [Google Scholar]
Efthymiadis, A., Shao, H., Hubner, S., and Jans, D.A. (1997). Kinetic characterization of the human retinoblastoma protein bipartite nuclear localization sequence (NLS) in vivo and in vitro. A comparison with the SV40 large T-antigen NLS. J. Biol. Chem. 272, 22134–22139 [DOI] [PubMed] [Google Scholar]
Endow, S.A., and Komma, D.J. (1996). Centrosome and spindle function of the Drosophila Ncd microtubule motor visualized in live embryos using Ncd-GFP fusion proteins. J. Cell Sci. 109, 2429–2442 [DOI] [PubMed] [Google Scholar]
Eyers, P.A., Erikson, E., Chen, L.G., and Maller, J.L. (2003). A novel mechanism for activation of the protein kinase Aurora A. Curr. Biol. 13, 691–697 [DOI] [PubMed] [Google Scholar]
Fanara, P., Hodel, M.R., Corbett, A.H., and Hodel, A.E. (2000). Quantitative analysis of nuclear localization signal (NLS)-importin α interaction through fluorescence depolarization. Evidence for auto-inhibitory regulation of NLS binding. J. Biol. Chem. 275, 21218–21223 [DOI] [PubMed] [Google Scholar]
Foster, K.A., Correia, J.J., and Gilbert, S.P. (1998). Equilibrium binding studies of non-claret disjunctional protein (Ncd) reveal cooperative interactions between the motor domains. J. Biol. Chem. 273, 35307–35318 [DOI] [PubMed] [Google Scholar]
Gaglio, T., Saredi, A., Bingham, J., Hasbani, J., Gill, S.R., Schroer, T.A., and Compton, D.A. (1996). Opposing motor activities are required for the organization of the mammalian mitotic spindle pole. J. Cell Biol. 135, 399–414 [DOI] [PMC free article] [PubMed] [Google Scholar]
Görlich, D., Seewald, M.J., and Ribbeck, K. (2003). Characterization of Randriven cargo transport and the RanGTPase system by kinetic measurements and computer simulation. EMBO J. 22, 1088–1100 [DOI] [PMC free article] [PubMed] [Google Scholar]
Gruss, O.J., Carazo-Salas, R.E., Schatz, C.A., Guarguaglini, G., Kast, J., Wilm, M., Le Bot, N., Vernos, I., Karsenti, E., and Mattaj, I.W. (2001). Ran induces spindle assembly by reversing the inhibitory effect of importin α on TPX2 activity. Cell 104, 83–93 [DOI] [PubMed] [Google Scholar]
Harlow, E., and Lane, D. (1999). Immunoaffinity purification. In: Using Antibodies: A Laboratory Manual, Cold Spring Harbor, NY: Cold Spring Harbor Laboratory Press, 321–325
Harreman, M.T., Hodel, M.R., Fanara, P., Hodel, A.E., and Corbett, A.H. (2003). The auto-inhibitory function of importin α is essential in vivo. J. Biol. Chem. 278, 5854–5863 [DOI] [PubMed] [Google Scholar]
Heald, R., Tournebize, R., Blank, T., Sandaltzopoulos, R., Becker, P., Hyman, A., and Karsenti, W. (1996). Self-organization of microtubules into bipolar spindles around artificial chromosomes in Xenopus egg extracts. Nature 382, 420–425 [DOI] [PubMed] [Google Scholar]
Hetzer, M., Gruss, O.J., and Mattaj, I.W. (2002). The Ran GTPase as a marker of chromosome position in spindle formation and nuclear envelope assembly. Nat. Cell Biol. 4, E177–E184. [DOI] [PubMed] [Google Scholar]
Hodel, M.R., Corbett, A.H., and Hodel, A.E. (2001). Dissection of a nuclear localization signal. J. Biol. Chem. 276, 1317–1325 [DOI] [PubMed] [Google Scholar]
Hu, W., and Jans, D.A. (1999). Efficiency of importin α/β-mediated nuclear localization sequence recognition and nuclear import. Differential role of NTF2. J. Biol. Chem. 274, 15820–15827 [DOI] [PubMed] [Google Scholar]
Ishii, N., Minami, N., Chen, E.Y., Medina, A.L., Chico, M.M., and Kasamatsu, H. (1996). Analysis of a nuclear localization signal of Simian virus 40 major capsid protein Vp1. J. Virol. 70, 1317–1322 [DOI] [PMC free article] [PubMed] [Google Scholar]
Jans, D.A., Xiao, C.-Y., and Lam, M.H.C. (2000). Nuclear targeting signal recognition: a key control point in nuclear transport. Bioessays 22, 532–544 [DOI] [PubMed] [Google Scholar]
Kahana, J.A., and Cleveland, D.W. (1999). Beyond nuclear transport: Ran-GTP as a determinant of spindle assembly. J. Cell Biol. 146, 1205–1209. [DOI] [PMC free article] [PubMed] [Google Scholar]
Kalab, P., Pu, R.T., and Dasso, M. (1999). The Ran GTPase regulates mitotic spindle assembly. Curr. Biol. 9, 481–484 [DOI] [PubMed] [Google Scholar]
Kalab, P., Weis, K., and Heald, R. (2002). Visualization of a Ran-GTP gradient in interphase and mitotic Xenopus egg extracts. Science 295, 2452–2456 [DOI] [PubMed] [Google Scholar]
Karabay, A., and Walker, R.A. (1999). Identification of microtubule binding sites in the Ncd tail domain. Biochemistry 38, 1838–1849 [DOI] [PubMed] [Google Scholar]
Karabay, A., and Walker, R.A. (2003). Identification of Ncd tail domain-binding sites on the tubulin dimer. Biochem. Biophys. Res. Commun. 305, 523–528 [DOI] [PubMed] [Google Scholar]
Keryer, G., Fiore, B.D., Celati, C., Lechtreck, K.F., Mogensen, M., Delouvée, A., Lavia, P., Bornens, M., and Tassin, A.-M. (2003). Part of Ran is associated with AKAP450 at the centrosome: involvement in microtubule organizing activity. Mol. Biol. Cell 14, 4260–4271 [DOI] [PMC free article] [PubMed] [Google Scholar]
Kikkawa, M., Okada, Y., and Hirokawa, N. (2000). 15 Å resolution model of the monomeric kinesin motor, KIF1A. Cell 100, 241–252 [DOI] [PubMed] [Google Scholar]
Klebe, C., Bischoff, F.R., Ponstingl, H., and Wittinghofer, A. (1995). Interaction of the nuclear GTP-binding protein Ran with its regulatory proteins RCC1 and RanGAP1. Biochemistry 34, 639–647 [DOI] [PubMed] [Google Scholar]
Kobe, B. (1999). Autoinhibition by an internal nuclear localization signal revealed by the crystal structure of mammalian importin α. Nat. Struct. Biol. 6, 388–397 [DOI] [PubMed] [Google Scholar]
Kuersten, S., Ohno, M., and Mattaj, I.W. (2001). Nucleocytoplasmic transport: Ran, beta and beyond. Trends Cell Biol. 11, 497–503 [DOI] [PubMed] [Google Scholar]
Li, H.Y., Wirtz, D., and Zheng, Y. (2003). A mechanism of coupling RCC1 mobility to RanGTP production on the chromatin in vivo. J. Cell Biol. 160, 635–644 [DOI] [PMC free article] [PubMed] [Google Scholar]
Lounsbury, K.M., Richards, S.A., Carey, K.L., and Macara, I.G. (1996). Mutations within the Ran/TC4 GTPase. J. Biol. Chem. 271, 32834–32841 [DOI] [PubMed] [Google Scholar]
Matuliene, J., Essner, R., Ryu, J.-H., Hamaguchi, Y., Baas, P.W., Haraguchi, T., Hiraoka, Y., and Kuriyama, R. (1999). Function of a minus-end-directed kinesin-like motor protein in mammalian cells. J. Cell Sci. 112, 4041–4050 [DOI] [PubMed] [Google Scholar]
Merdes, A., and Cleveland, D.W. (1997). Pathways of spindle pole formation: different mechanisms; conserved components. J. Cell Biol. 138, 953–956 [DOI] [PMC free article] [PubMed] [Google Scholar]
Merdes, A., Ramyar, K., Vechio, J.D., and Cleveland, D.W. (1996). A complex of NuMA and cytoplasmic dynein is essential for mitotic spindle assembly. Cell 87, 447–458 [DOI] [PubMed] [Google Scholar]
Moore, W.J., Zhang, C., and Clarke, P.R. (2002). Targeting of RCC1 to chromosomes is required for proper mitotic spindle assembly in human cells. Curr. Biol. 12, 1442–1447 [DOI] [PubMed] [Google Scholar]
Mountain, V., Simerly, C., Howard, L., Ando, A., Schatten, G., and Compton, D.A. (1999). The kinesin-related protein, HSET, opposes the activity of Eg5 and cross-links microtubules in the mammalian mitotic spindle. J. Cell Biol. 147, 351–365 [DOI] [PMC free article] [PubMed] [Google Scholar]
Nachury, M.V., Maresca, T.J., Salmon, W.C., Waterman-Storer, C.M., Heald, R., and Weis, K. (2001). Importin β is a mitotic target of the small GTPase Ran in spindle assembly. Cell 104, 95–106 [DOI] [PubMed] [Google Scholar]
Navolanic, P.M., and Sperry, A.O. (2000). Identification of isoforms of a mitotic motor in mammalian spermatogenesis. Biol. Reprod. 62, 1360–1369 [DOI] [PubMed] [Google Scholar]
Ohba, T., Nakamura, M., Nishitani, H., and Nishimoto, T. (1999). Self-organization of microtubule asters induced in Xenopus egg extracts by GTP-bound Ran. Science 284, 1356–1358 [DOI] [PubMed] [Google Scholar]
Ovechkina, Y., and Wordeman, L. (2003). Unconventional motoring: an overview of the Kin C and Kin I kinesins. Traffic 4, 367–375 [DOI] [PubMed] [Google Scholar]
Quimby, B.B., and Dasso, M. (2003). The small GTPase Ran: interpreting the signs. Curr. Opin. Cell Biol. 15, 338–344 [DOI] [PubMed] [Google Scholar]
Schatz, C.A., Santarella, R., Hoenger, A., Karsenti, E., Mattaj, I.W., Gruss, O.J., and Carazo-Salas, R.E. (2003). Importin α-regulated nucleation of microtubules by TPX2. EMBO J. 22, 2060–2070 [DOI] [PMC free article] [PubMed] [Google Scholar]
Taniguchi, E., Toyoshima-Morimoto, F., and Nishida, E. (2002). Nuclear translocation of plk1 mediated by its bipartite nuclear localization signal. J. Biol. Chem. 277, 48884–48888 [DOI] [PubMed] [Google Scholar]
Trieselmann, N., and Wilde, A. (2002). Ran localizes around the microtubule spindle in vivo during mitosis in Drosophila embryos. Curr. Biol. 12, 1124–1129 [DOI] [PubMed] [Google Scholar]
Tsai, M.-Y., Wiese, C., Cao, K., Martin, O., Donovan, P., Ruderman, J., Prigent, C., and Zheng, Y. (2003). A Ran signalling pathway mediated by the mitotic kinase Aurora A in spindle assembly. Nat. Cell Biol. 5, 242–248 [DOI] [PubMed] [Google Scholar]
Walczak, C.E. (2001). Ran hits the ground running. Nat. Cell Biol. 3, E69–E70. [DOI] [PubMed] [Google Scholar]
Walczak, C.E., Verma, S., and Mitchison, T.J. (1997). XCTK 2, A Kinesin-related protein that promotes mitotic spindle assembly in Xenopus laevis egg extracts. J. Cell Biol. 136, 859–870 [DOI] [PMC free article] [PubMed] [Google Scholar]
Walczak, C.E., Vernos, I., Mitchison, T.J., Karsenti, E., and Heald, R. (1998). A model for the proposed roles of different microtubule based motor proteins in establishing spindle bipolarity. Curr. Biol. 8, 903–913 [DOI] [PubMed] [Google Scholar]
Wendt, T.G., Volkmann, N., Skiniotis, G., Goldie, K.N., Muller, J., Mandelkow, E., and Hoenger, A. (2002). Microscopic evidence for a minus-end-directed power stroke in the kinesin motor ncd. EMBO J. 21, 5969–5978 [DOI] [PMC free article] [PubMed] [Google Scholar]
Wiese, C., Wilde, A., Moore, M.S., Adam, S.A., Merdes, A., and Zheng, Y. (2001). Role of importin-β in coupling Ran to downstream targets in microtubule assembly. Science 291, 653–656 [DOI] [PubMed] [Google Scholar]
Wilde, A., Lizarraga, S.B., Zhang, L., Wiese, C., Gliksman, N.R., Walczak, C.E., and Zheng, Y. (2001). Ran stimulates spindle assembly by altering microtubule dynamics and the balance of motor activities. Nat. Cell Biol. 3, 221–227 [DOI] [PubMed] [Google Scholar]
Wilde, A., and Zheng, Y. (1999). Stimulation of microtubule aster formation and spindle assembly by the small GTPase Ran. Science 284, 1359–1362 [DOI] [PubMed] [Google Scholar]
Woehlke, G., Ruby, A.K., Hart, C.L., Ly, B., Hom-Booher, N., and Vale, R.D. (1997). Microtubule interaction site of the kinesin motor. Cell 90, 207–216 [DOI] [PubMed] [Google Scholar]
Yang, W., and Sperry, A.O. (2003). C-terminal kinesin motor KIFC1 participates in acrosome biogenesis and vesicle transport. Biol. Reprod. 69, 1719–1729 [DOI] [PubMed] [Google Scholar]
Zhang, C., Hutchins, J.R.A., Mühlhäusser, P., Kutay, U., and Clarke, P.R. (2002). Role of importin-β in the control of nuclear envelope assembly by Ran. Curr. Biol. 12, 498–502. [DOI] [PubMed] [Google Scholar]
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7060 | https://www.stat.purdue.edu/~lfindsen/stat503/t-Dist.pdf | Critical Values for Student’s t-Distribution.
Upper Tail Probability: Pr(T > t) df 0.2 0.1 0.05 0.04 0.03 0.025 0.02 0.01 0.005 0.0005 1 1.376 3.078 6.314 7.916 10.579 12.706 15.895 31.821 63.657 636.619 2 1.061 1.886 2.920 3.320 3.896 4.303 4.849 6.965 9.925 31.599 3 0.978 1.638 2.353 2.605 2.951 3.182 3.482 4.541 5.841 12.924 4 0.941 1.533 2.132 2.333 2.601 2.776 2.999 3.747 4.604 8.610 5 0.920 1.476 2.015 2.191 2.422 2.571 2.757 3.365 4.032 6.869 6 0.906 1.440 1.943 2.104 2.313 2.447 2.612 3.143 3.707 5.959 7 0.896 1.415 1.895 2.046 2.241 2.365 2.517 2.998 3.499 5.408 8 0.889 1.397 1.860 2.004 2.189 2.306 2.449 2.896 3.355 5.041 9 0.883 1.383 1.833 1.973 2.150 2.262 2.398 2.821 3.250 4.781 10 0.879 1.372 1.812 1.948 2.120 2.228 2.359 2.764 3.169 4.587 11 0.876 1.363 1.796 1.928 2.096 2.201 2.328 2.718 3.106 4.437 12 0.873 1.356 1.782 1.912 2.076 2.179 2.303 2.681 3.055 4.318 13 0.870 1.350 1.771 1.899 2.060 2.160 2.282 2.650 3.012 4.221 14 0.868 1.345 1.761 1.887 2.046 2.145 2.264 2.624 2.977 4.140 15 0.866 1.341 1.753 1.878 2.034 2.131 2.249 2.602 2.947 4.073 16 0.865 1.337 1.746 1.869 2.024 2.120 2.235 2.583 2.921 4.015 17 0.863 1.333 1.740 1.862 2.015 2.110 2.224 2.567 2.898 3.965 18 0.862 1.330 1.734 1.855 2.007 2.101 2.214 2.552 2.878 3.922 19 0.861 1.328 1.729 1.850 2.000 2.093 2.205 2.539 2.861 3.883 20 0.860 1.325 1.725 1.844 1.994 2.086 2.197 2.528 2.845 3.850 21 0.859 1.323 1.721 1.840 1.988 2.080 2.189 2.518 2.831 3.819 22 0.858 1.321 1.717 1.835 1.983 2.074 2.183 2.508 2.819 3.792 23 0.858 1.319 1.714 1.832 1.978 2.069 2.177 2.500 2.807 3.768 24 0.857 1.318 1.711 1.828 1.974 2.064 2.172 2.492 2.797 3.745 25 0.856 1.316 1.708 1.825 1.970 2.060 2.167 2.485 2.787 3.725 26 0.856 1.315 1.706 1.822 1.967 2.056 2.162 2.479 2.779 3.707 27 0.855 1.314 1.703 1.819 1.963 2.052 2.158 2.473 2.771 3.690 28 0.855 1.313 1.701 1.817 1.960 2.048 2.154 2.467 2.763 3.674 29 0.854 1.311 1.699 1.814 1.957 2.045 2.150 2.462 2.756 3.659 30 0.854 1.310 1.697 1.812 1.955 2.042 2.147 2.457 2.750 3.646 31 0.853 1.309 1.696 1.810 1.952 2.040 2.144 2.453 2.744 3.633 32 0.853 1.309 1.694 1.808 1.950 2.037 2.141 2.449 2.738 3.622 33 0.853 1.308 1.692 1.806 1.948 2.035 2.138 2.445 2.733 3.611 34 0.852 1.307 1.691 1.805 1.946 2.032 2.136 2.441 2.728 3.601 35 0.852 1.306 1.690 1.803 1.944 2.030 2.133 2.438 2.724 3.591 36 0.852 1.306 1.688 1.802 1.942 2.028 2.131 2.434 2.719 3.582 37 0.851 1.305 1.687 1.800 1.940 2.026 2.129 2.431 2.715 3.574 38 0.851 1.304 1.686 1.799 1.939 2.024 2.127 2.429 2.712 3.566 39 0.851 1.304 1.685 1.798 1.937 2.023 2.125 2.426 2.708 3.558 40 0.851 1.303 1.684 1.796 1.936 2.021 2.123 2.423 2.704 3.551 41 0.850 1.303 1.683 1.795 1.934 2.020 2.121 2.421 2.701 3.544 42 0.850 1.302 1.682 1.794 1.933 2.018 2.120 2.418 2.698 3.538 43 0.850 1.302 1.681 1.793 1.932 2.017 2.118 2.416 2.695 3.532 44 0.850 1.301 1.680 1.792 1.931 2.015 2.116 2.414 2.692 3.526 45 0.850 1.301 1.679 1.791 1.929 2.014 2.115 2.412 2.690 3.520 46 0.850 1.300 1.679 1.790 1.928 2.013 2.114 2.410 2.687 3.515 47 0.849 1.300 1.678 1.789 1.927 2.012 2.112 2.408 2.685 3.510 48 0.849 1.299 1.677 1.789 1.926 2.011 2.111 2.407 2.682 3.505 49 0.849 1.299 1.677 1.788 1.925 2.010 2.110 2.405 2.680 3.500 50 0.849 1.299 1.676 1.787 1.924 2.009 2.109 2.403 2.678 3.496 60 0.848 1.296 1.671 1.781 1.917 2.000 2.099 2.390 2.660 3.460 70 0.847 1.294 1.667 1.776 1.912 1.994 2.093 2.381 2.648 3.435 80 0.846 1.292 1.664 1.773 1.908 1.990 2.088 2.374 2.639 3.416 90 0.846 1.291 1.662 1.771 1.905 1.987 2.084 2.368 2.632 3.402 100 0.845 1.290 1.660 1.769 1.902 1.984 2.081 2.364 2.626 3.390 120 0.845 1.289 1.658 1.766 1.899 1.980 2.076 2.358 2.617 3.373 140 0.844 1.288 1.656 1.763 1.896 1.977 2.073 2.353 2.611 3.361 180 0.844 1.286 1.653 1.761 1.893 1.973 2.069 2.347 2.603 3.345 200 0.843 1.286 1.653 1.760 1.892 1.972 2.067 2.345 2.601 3.340 500 0.842 1.283 1.648 1.754 1.885 1.965 2.059 2.334 2.586 3.310 1000 0.842 1.282 1.646 1.752 1.883 1.962 2.056 2.330 2.581 3.300 ∞ 0.842 1.282 1.645 1.751 1.881 1.960 2.054 2.326 2.576 3.291 60% 80% 90% 92% 94% 95% 96% 98% 99% 99.9% Confidence Level Note: t(∞)α/2 = Zα/2 in our notation. |
7061 | https://www.geeksforgeeks.org/maths/am-gm-inequality/ | AM GM Inequality - GeeksforGeeks
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AM GM Inequality
Last Updated : 23 Jul, 2025
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AM-GM Inequalityis one of the most famous inequalities in algebra. Before going through AM-GM Inequality, we first need to go through arithmetic and geometric mean concepts.
Arithmetic Mean: Arithmetic mean is the sum of all the quantities divided by the number of quantities.
Geometric Mean: Geometric mean is defined as the mean which is calculated by multiplying the n number together and then taking their (1/n)th root.
In this article, we will learn about AM-GM Inequality, the relationship between AM and GM, and solve examples and problems on it.
Table of Content
What is AM-GM Inequality?
AM–GM Inequality Relationship
AM-GM Inequality Formula
AM–GM Inequality Relationship Proof
Example on AM-GM Inequality
FAQs on AM-GM Inequality
What is AM-GM Inequality?
Arithmetic Mean-Geometric Mean (AM-GM) Inequality is a fundamental result in algebra that provides a relationship between the arithmetic mean and the geometric mean of a set of non-negative real numbers. This inequality states that for any list of non-negative real numbers, the arithmetic mean (average) is at least as great as the geometric mean.
For any, two numbers 'a', and 'b' their arithmetic mean is 1/2(a + b) and geometric mean is √(ab). Then AM-GM inequality is,
AM GM Inequality
AM–GM Inequality Relationship
AM–GM Inequality is discussed below in the article,
For two positive numbers a and b
Arithmetic Mean: A.M = (a+b)/2
Geometric Mean: G.M = √(ab)
A.M ≥ G.M
(a+b)/2 ≥ √(ab)
For 'n' positive numbers a 1, a 2, a 3, a 4, ... a n
Arithmetic Mean: A.M = (a 1+ a 2+ a 3+ a 4+.......... +a n) / n
Geometric Mean: G.M = (a 1 a 2 a 3 a 4 .......... a n) 1/n
A.M ≥ G.M
(a1+ a2+ a3+ a4+.......... +an) / n ≥ (a1a2a3a4.......... an)1/n
AM-GM Inequality Formula
For two positive numbers a and b,
(a+b)/2 ≥ √(ab)
For 'n' positive numbers a 1, a 2, a 3, a 4, ... a n
(a1+ a2+ a3+ a4+.......... +an) / n ≥ (a1 a2 a3 a4 .......... an) 1/n
AM–GM Inequality Relationship Proof
Statement: For any n positive numbers a 1, a 2, ... a n Arithmetic Mean is always greater than equal to Geometric Mean. A.M ≥ G.M
Proof:
For two numbers,
A.M - G.M = (a+b)/2 - √(ab)
A.M - G.M = ½ (a+b - 2 √(ab) )
A.M - G.M = ½ (√a - √b)2
We know that square of any number is positive,( i.e ≥0) Hence,
A.M - G.M ≥0
A.M ≥ G.M
Hence we conclude by above proof that for all positive Numbers, A.M ≥ G.M
Article Related to AM GM Inequality:
Arithmetic Progression
Geometric Progression
Example on AM-GM Inequality
Example 1: Find the arithmetic mean of 3 and 27
Solution:
Arithmetic Mean: A.M = (a+b)/2
A.M = (3+27)/2
A.M = 15
Example 2: Find the Geometric Mean of 3 and 27
Solution:
Geometric Mean: G.M = √(ab)
G.M = √(27 × 3) = √81
G.M = 9
Example 3: If x>0, then Prove That: x+ (1/x) ≥ 2
Solution:
Since x>0 we can apply A.M-G.M Inequality here,
A.M ≥ G.M
(x+1/x) / 2 ≥ (x . 1/x) ½
(x+1/x) /2 ≥ 1
x + (1/x) ≥ 2 (Proved)
Example 4: If x,y>0,then Prove That: x2+y2≥ 2xy
Solution:
Since x,y>0 we can apply A.M-G.M Inequality here,
A.M ≥ G.M
For two variables x and y
(x+y) /2 ≥ (xy)1/2
squaring both sides:
(x+y)2 /4 ≥ xy
x 2+ y 2≥ 2xy
Example 5: If x,y,z ≤ 0,then can we Prove That: (x+y)(y+z)(z+x) ≥ 8xyz through A.M-G.M Inequality
Solution:
Since x, y, z ≤ 0
So, we can't apply A.M-G.M Inequality here,
Example 6: If a,b,c ∈ R+, such that a+b+c = 3, find the maximum value of abc.
Solution:
Since a,b,c >0 we can apply A.M-G.M Inequality here.
We need to find the value of the product of a,b,c i.e abc.
Applying A.M ≥ G.M,
(a+b+c)/3 ≥ 3√(abc)
3/3 ≥ (abc)1/3
1 ≥ (abc)1/3
cubing both sides, 1 3 ≥ (abc)
so, abc ≤1
Hence the maximum value of abc = 1
Summary - AM-GM Inequality
AM-GM Inequality is a fundamental mathematical principle stating that for any set of non-negative real numbers, the arithmetic mean (AM) is always greater than or equal to the geometric mean (GM). Specifically, the inequality formula is expressed as
(a1+ a2+ a3+ a4+.......... +an) / n ≥ (a1a2a3a4.......... an)1/n
where,
a1, a2, .......an are Non-Negative Numbers
The equality holds if and only if all the numbers in the set are equal. This inequality is crucial in various mathematical contexts, especially in proving bounds and optimizing algebraic expressions.
It finds applications across diverse fields such as economics, engineering, and optimization problems, making it a versatile and powerful tool in theoretical and applied mathematics.
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7062 | https://www.youtube.com/watch?v=J0m28OW-tH8 | Use the Given Distance Between the Points to Solve for missing X Value
charlie Lindelof
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Posted: 31 May 2012
ALGEBRA MADE EASY A different way to see distance formula questions is to be given distance and then find an x or y value of a point.
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Transcript:
thought we'd do a distance problem from the other direction I thought we'd use the given distance between two points to solve for a missing in this case x value so we have the points X -10 and -4 I'm sorry 84 and the distance between them is seven square roots of 5 so it's distance is seven square roots of 5 so all I'm going to do is this going to set up the distance formula if you remember the distance formula is D the distance is x sub 2 - x sub 1^ 2ar + y sub 2 - y sub 1^ 2ar all under the radical okay and then we're going to fill in fill in what we have here and what we know is this we know that the distance is 7 square roots of 5 we also know that we have right we have x - 8 so I did that as x - -8 2ar Plus did -10 - 4 squar those are the Y values aren't they and then we're just going to take the square root of the of the lot right so this negative negative is a positive isn't it and this is4 I think yeah4 this is -14 I'm just cleaning up my math a little bit here and then I'm going to go ahead and do the binomial expansion of this piece right here here so what I'm doing here is I'm doing x + 8 x + 8 obviously because that's what x + 8 s means isn't it remember we cannot distribute the exponent over addition it's not distributable so this is not equivalent to x^2 + 82 it's equivalent to this and if you do this math you'll see it's different so I'm going to do foil here and the foil of this if you know it is x² + 16 x + 64 is that right right and then -14 SAR is well it's -14 -14 which I believe is positive 196 so plus 196 got all that is equal to this I'm just cleaning up my math as I go along is still equal to this distance and we're going to try to find these X this x value or X values that satisfy this going to clean this up add these two things together and 64 + 196 is 260 so 260 now we're finally getting somewhere but we have this on this side so check this out I want to get rid of this because I want to be able to get to this inside and do more with it so what I'm going to do here is square both sides because if I square a square root right it disappears but I have to square this thing also it thises where it gets a little bit dicey when I Square this we just get x^2 + 16 x + 260 now remember when I Square this exponents are I'm going to say it again exponents are distributable over multiplication so I have to square this and this so just to show you what that would look like over here 7 squ roots of 5 squared is the same as 7qu roots of 7 time 7 Square I'm sorry 7 roots of 5 7 roots of 5 7 7 is 49 I'm not Distributing I'm just using the commutative property and the sare < TK of 5 the squ < TK of 5 is the square < TK of 25 which is just five and I think if you multiply this out it comes out to be it 235 or 24 245 so 245 right so from here this should get really easy I hope right this should get really easy here matter of fact we're almost done have to have to recognize this is a quadratic so by recognizing this is a quadratic we it it gives us a signal that we should be putting it in this form equals zero form so what I'm going to do is I'm going to add - 245 or subtract 245 from each side 245 and that gives us if you don't mind I'm going to write it this way x^2 + 16 x + 15 is equal to is zero right so we are done cuz when we Factor this out we get x + 1 x + 15 so our answers are x = -1 or x = -15 somebody have the one that's not possible because distance can't be negative these are not distances this is these are possible X values all right hey I hope this video was really helpful for you this is the kind of a problem that you'll see on a specialty math exam on a uh sat or a math or a yeah math placement exam when you're in college so hope it helps |
7063 | https://www.cdc.gov/group-a-strep/about/impetigo.html | A .gov website belongs to an official government organization in the United States.
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About Impetigo
Key points
What it is
Impetigo is a skin infection caused by bacteria: either group A Streptococcus, Staphylococcus aureus, or both. This page focuses on impetigo caused by group A Streptococcus (group A strep bacteria).
Did you know?
Symptoms
It usually takes 10 days for sores to appear after someone is exposed to group A strep bacteria.
Symptoms include red, itchy sores that break open and leak a clear fluid or pus for a few days.
The sores can occur anywhere on the body but most often affect exposed skin.
Sores are commonly found on the following areas:
Next, a crusty yellow or "honey-colored" scab forms over the sore, which then heals without leaving a scar.
Complications
Very rarely, complications can include post-streptococcal glomerulonephritis and rheumatic fever.
Post-streptococcal glomerulonephritis is a kidney problem.
Rheumatic fever is a disease affecting the heart, joint, brain, and skin.
Risk factors
Anyone can get impetigo, but some factors increase the risk of getting this infection.
Age
Impetigo is most common in children 2 through 5 years old.
Infections or injuries that break the skin
People with scabies infection are at increased risk for impetigo. Participating in activities where cuts or scrapes are common can also increase someone's risk of impetigo.
Group settings
Close contact with another person with impetigo is the most common risk factor for illness. For example, if someone has impetigo, the bacteria often spread to other people in their household.
Crowded conditions can increase the risk of spreading impetigo. These settings include:
Climate
Impetigo can occur anywhere. It is more common in areas with hot, humid summers and mild winters (subtropics), or wet and dry seasons (tropics).
Poor personal hygiene
Lack of proper handwashing, body washing, and facial cleanliness can increase someone's risk of getting impetigo.
Causes
Group A Streptococcus (group A strep bacteria) cause impetigo and are contagious.
Prevention
There are things people can do to protect themselves and others from group A strep infections, including impetigo.
Testing and diagnosis
Healthcare providers typically diagnose impetigo by looking at the sores during a physical examination. Lab tests aren't needed.
Treatment and recovery
When to return to work or school
Healthcare providers treat impetigo with antibiotics. They can be topical antibiotics (medicine rubbed onto the sores) and oral antibiotics (medicine taken by mouth).
A healthcare provider might recommend a topical ointment for only a few sores. Oral antibiotics can be used when there are more sores.
Use the prescription exactly as the healthcare provider says to.
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7064 | https://www.geeksforgeeks.org/dsa/bitwise-algorithms/ | Bitwise Algorithms
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Bitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.
Basics
Introduction to Bitwise Algorithms
Bitwise Operators in C/C++
Bitwise Operators in Java
Python Bitwise Operators
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All about Bit Manipulation
Little and Big Endian Mystery
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Bitwise Hacks for Competitive Programming
Easy Problems on Bit Algorithms
Binary Representation
Turn off the rightmost set bit
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Set the K-th Bit
Modulus division by 2's Power
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Add Bit Strings
Check for Integer Overflow
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Swap Two Numbers
Russian Peasant
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Generate n-bit Gray Codes
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Cyclic Redundancy Check and Modulo-2 Division
Set Bits in a Range
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Gray to Binary and Vice Versa
Hard Problems on Bit Algorithms
Next higher with same set bits
Karatsuba Algorithm for fast Multiplication
Max Subarray XOR
Longest Sequence of 1’s in Binary with One Flip
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Count pairs with at least one digit common
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What are Bitwise Algorithms?
Bitwise algorithms are algorithms that operate on individual bits of data rather than on larger data types like integers or floating-point numbers. These algorithms manipulate bits directly, typically using bitwise operators such as AND, OR, XOR, shift left, shift right, and complement.
Common Bitwise Algorithms and Operations
Here are some common bitwise algorithms and operations:
Bitwise AND (&): Takes two numbers as input and performs a bitwise AND operation on their corresponding bits. It returns 1 only if both bits are 1; otherwise, it returns 0.
Bitwise OR (|): Performs a bitwise OR operation on the corresponding bits of two numbers. It returns 1 if at least one of the bits is 1.
Bitwise XOR (^): Performs a bitwise exclusive OR operation on the corresponding bits of two numbers. It returns 1 if the bits are different and 0 if they are the same.
Bitwise NOT (~): Performs a bitwise NOT operation, which flips each bit of the input (1 becomes 0 and 0 becomes 1).
Left Shift (<<) and Right Shift (>>): These operators shift the bits of a number to the left or right by a specified number of positions. Left shifting is equivalent to multiplying the number by 2, while right shifting is equivalent to dividing by 2.
Applications of Bitwise Algorithms
Bit manipulation (setting, clearing, toggling bits): Bitwise operators are often used to manipulate individual bits of numbers. This includes tasks such as setting bits (using OR), clearing bits (using AND with the complement), toggling bits (using XOR with 1), and checking the value of a specific bit.
Efficient storage of data: Bitwise algorithms play a crucial role in data compression techniques like Huffman coding. They can efficiently represent and process compressed data by manipulating bits directly.
Cryptography: Many cryptographic algorithms, such as AES (Advanced Encryption Standard), DES (Data Encryption Standard), and SHA (Secure Hash Algorithm), utilize bitwise operations for encryption, decryption, and hashing. Bitwise XOR, in particular, is commonly used in encryption algorithms for its simplicity and effectiveness.
Networking and Protocol Handling: Bitwise algorithms are used in networking protocols for tasks like IP address manipulation, subnet masking, and packet parsing. For example, bitwise AND is used in subnet masking to determine the network address from an IP address and subnet mask.
Low-Level System Programming: Bitwise operations are essential in low-level system programming for tasks such as device control, memory management, and bit-level I/O operations. They are used to manipulate hardware registers, set/clear flags, and optimize code for performance.
Error Detection and Correction: Bitwise algorithms are employed in error detection and correction techniques, such as CRC (Cyclic Redundancy Check) and Hamming codes. These algorithms use bitwise XOR and other operations to detect and correct errors in transmitted data.
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7065 | https://courses.lumenlearning.com/calculus3/chapter/lagrange-multipliers/ | Lagrange Multipliers | Calculus III
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Calculus III
Module 4: Differentiation of Functions of Several Variables
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Lagrange Multipliers
Learning Objectives
Use the method of Lagrange multipliers to solve optimization problems with one constraint.
Use the method of Lagrange multipliers to solve optimization problems with two constraints.
Solving optimization problems for functions of two or more variables can be similar to solving such problems in single-variable calculus. However, techniques for dealing with multiple variables allow us to solve more varied optimization problems for which we need to deal with additional conditions or constraints. In this section, we examine one of the more common and useful methods for solving optimization problems with constraints.
Lagrange Multipliers
Chapter Opener: Profitable Golf Balls was an applied situation involving maximizing a profit function, subject to certainconstraints. In that example, the constraints involved a maximum number of golf balls that could be produced and sold in 1 1 month (x)(x), and a maximum number of advertising hours that could be purchased per month (y)(y). Suppose these were combined into a budgetary constraint, such as 20 x+4 y≤216 20 x+4 y≤216, that took into account the cost of producing the golf balls and the number of advertising hours purchased per month. The goal is, still, to maximize profit, but now there is a different type of constraint on the values of x x and y y. This constraint, when combined with the profit function f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 f(x,y)=48 x+96 y−x 2−2 x y−9 y 2, is an example of anoptimization problem, and the function f(x,y)f(x,y) is called theobjective function. A graph of various level curves of the function f(x,y)f(x,y) follows.
Figure 1. Graph of level curves of the function f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 corresponding to c=150,250,350,and 400 c=150,250,350,and 400.
In Figure 1, the value c c represents different profit levels (i.e., values of the function f f). As the value of c c increases, the curve shifts to the right. Since our goal is to maximize profit, we want to choose a curve as far to the right as possible. If there was no restriction on the number of golf balls the company could produce, or the number of units of advertising available, then we could produce as many golf balls as we want, and advertise as much as we want, and there would not be a maximum profit for the company. Unfortunately, we have a budgetary constraint that is modeled by the inequality 20 x+4 y≤216 20 x+4 y≤216. To see how this constraint interacts with the profit function,Figure 2 shows the graph of the line 2 x+4 y=216 2 x+4 y=216 superimposed on the previous graph.
Figure 2.Graph of level curves of the function f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 corresponding to c=150,250,350,and 395 c=150,250,350,and 395.The red graph is the constraint function.
As mentioned previously, the maximum profit occurs when the level curve is as far to the right as possible. However, the level of production corresponding to this maximum profit must also satisfy the budgetary constraint, so the point at which this profit occurs must also lie on (or to the left of) the red line in Figure 2. Inspection of this graph reveals that this point exists where the line is tangent to the level curve of f f. Trial and error reveals that this profit level seems to be around 395 395, when x x and y y are both just less than 5 5. We return to the solution of this problem later in this section. From a theoretical standpoint, at the point where the profit curve is tangent to the constraint line, the gradient of both of the functions evaluated at that point must point in the same (or opposite) direction. Recall that the gradient of a function of more than one variable is a vector. If two vectors point in the same (or opposite) directions, then one must be a constant multiple of the other. This idea is the basis of themethod of Lagrange multipliers.
Theorem: method of lagrange multipliers: one constant
Let f f and g g be functions of two variables with continuous partial derivatives at every point of some open set containing the smooth curve g(x,y)=0 g(x,y)=0. Suppose that f f, when restricted to points on the curve g(x,y)=0 g(x,y)=0, has a local extremum at the point (x 0,y 0)(x 0,y 0) and that ∇g(x 0,y 0)≠0∇g(x 0,y 0)≠0. Then there is a number λ λ called aLagrange multiplier, for which
∇f(x 0,y 0)=λ∇g(x 0,y 0)∇f(x 0,y 0)=λ∇g(x 0,y 0)
Proof
Assume that a constrained extremum occurs at the point (x 0,y 0)(x 0,y 0). Furthermore, we assume that the equation g(x,y)=0 g(x,y)=0 can be smoothly parameterized as
x=x(s)x=x(s) and y=y(s)y=y(s)
where s is an arc length parameter with reference point (x 0,y 0)(x 0,y 0) at s=0 s=0. Therefore, the quantity z=f(x(s),y(s))z=f(x(s),y(s)) has a relative maximum or relative minimum at s=0 s=0, and this implies that d z d s=0 d z d s=0 at that point. From the chain rule,
d z d s=∂f∂x⋅∂x∂s+∂f∂y⋅∂y∂s=(∂f∂x^i+∂f∂y^j)⋅(∂x∂s^i+∂y∂s^j)=0,d z d s=∂f∂x⋅∂x∂s+∂f∂y⋅∂y∂s=(∂f∂x i^+∂f∂y j^)⋅(∂x∂s i^+∂y∂s j^)=0,
where the derivatives are all evaluated at s=0 s=0. However, the first factor in the dot product is the gradient of f f, and the second factor is the unit tangent vector T(0)T(0) to the constraint curve. Since the point (x 0,y 0)(x 0,y 0) corresponds to s=0 s=0, it follows from this equation that
∇f(x 0,y 0)⋅T(0)=0,∇f(x 0,y 0)⋅T(0)=0,
which implies that the gradient is either 0 0 or is normal to the constraint curve at a constrained relative extremum. However, the constraint curve g(x,y)=0 g(x,y)=0 is a level curve for the function g(x,y)g(x,y)so that if ∇g(x 0,y 0)≠0∇g(x 0,y 0)≠0 then ∇g(x 0,y 0)∇g(x 0,y 0)is normal to this curve at (x 0,y 0)(x 0,y 0)It follows, then, that there is some scalar λ λ such that
∇f(x 0,y 0)=λ∇g(x 0,y 0)∇f(x 0,y 0)=λ∇g(x 0,y 0)
■◼
To apply the Method of Lagrange Multipliers: One Constraint to an optimization problem similar to that for the golf ball manufacturer, we need a problem-solving strategy.
Problem solving strategy: steps for using Lagrange multipliers
Determine the objective function f(x,y)f(x,y) and the constraint function g(x,y)g(x,y). Does the optimization problem involve maximizing or minimizing the objective function?
Set up a system of equations using the following template:
[latex]\hspace{8cm}\begin{align}
\nabla{f}(x_0,y_0)\cdot{\text{T}}(0)&=0 \ g(x_0,y_0)&=0. \end{align}[/latex]
Solve for x 0 x 0 and y 0 y 0.
The largest of the values of f f at the solutions found in step 3 maximizes f f; the smallest of those values minimizes f f.
Example: using lagrange multipliers
Use the method of Lagrange multipliers to find the minimum value of f(x,y)=x 2+4 y 2−2 x+8 y f(x,y)=x 2+4 y 2−2 x+8 y subject to the constraint x+2 y=7 x+2 y=7.
Show Solution
Let’s follow the problem-solving strategy:
The optimization function is f(x,y)=x 2+4 y 2−2 x+8 y f(x,y)=x 2+4 y 2−2 x+8 y. To determine the constraint function, we must first subtract 7 7 from both sides of the constraint. This gives x+2 y−7=0 x+2 y−7=0. The constraint function is equal to the left-hand side, so g(x,y)=x+2 y−7 g(x,y)=x+2 y−7. The problem asks us to solve for the minimum value of f f, subject to the constraint (see the following graph).
Figure 3.Graph of level curves of the function f(x,y)=x 2+4 y 2−2 x+8 y f(x,y)=x 2+4 y 2−2 x+8 y corresponding to c=10 and 26 c=10 and 26.The red graph is the constraint function.
We then must calculate the gradients of both f f and g g:
∇f(x,y)=(2 x−2)i+(8 y+8)j∇g(x,y)=i+2 j.∇f(x,y)=(2 x−2)i+(8 y+8)j∇g(x,y)=i+2 j.
The equation ∇f(x 0,y 0)=λ∇g(x 0,y 0)∇f(x 0,y 0)=λ∇g(x 0,y 0) becomes
(2 x 0−2)i+(8 y 0+8)j=λ(i+2 j),(2 x 0−2)i+(8 y 0+8)j=λ(i+2 j),
which can be rewritten as
(2 x 0−2)i+(8 y 0+8)j=λ i+2 λ j.(2 x 0−2)i+(8 y 0+8)j=λ i+2 λ j.
Next, we set the coefficients of i i and j j equal to each other:
2 x 0−2=λ 8 y 0+8=2 λ.2 x 0−2=λ 8 y 0+8=2 λ.
The equation g(x 0,y 0)=0 g(x 0,y 0)=0 becomes x 0+2 y 0−7=0 x 0+2 y 0−7=0. Therefore, the system of equations that needs to be solved is
2 x 0−2=λ 8 y 0+8=2 λ x 0+2 y 0−7=0.2 x 0−2=λ 8 y 0+8=2 λ x 0+2 y 0−7=0.
This is a linear system of three equations in three variables. We start by solving the second equation for λ λ and substituting it into the first equation. This gives λ=4 y 0+4 λ=4 y 0+4, so substituting this into the first equation gives
2 x 0−2=4 y 0+4 2 x 0−2=4 y 0+4.
Solving this equation for x 0 x 0 gives x 0=2 y 0+3 x 0=2 y 0+3. We then substitute this into the third equation:
(2 y 0+3)+2 y 0−7=0 4 y 0−4=0 y 0=1.(2 y 0+3)+2 y 0−7=0 4 y 0−4=0 y 0=1.
Since x 0=2 y 0+3 x 0=2 y 0+3, this gives x 0=5 x 0=5.
Next, we substitute (5,1)(5,1) into f(x,y)=x 2+4 y 2−2 x+8 y f(x,y)=x 2+4 y 2−2 x+8 y, gives f(5,1)=5 2+4(1)2−2(5)+8(1)=27 f(5,1)=5 2+4(1)2−2(5)+8(1)=27. To ensure this corresponds to a minimum value on the constraint function, let’s try some other values, such as the intercepts of g(x,y)=0 g(x,y)=0, Which are (7,0)(7,0) and (0,3.5)(0,3.5). We get f(7,0)=35 f(7,0)=35 and f(0.3.5)=77 f(0.3.5)=77,so it appears f f has a minimum at (5,1)(5,1).
try it
Use the method of Lagrange multipliers to find the maximum value of f(x,y)=9 x 2+36 x y−4 y 2−18 x−8 y f(x,y)=9 x 2+36 x y−4 y 2−18 x−8 y subject to the constraint 3 x+4 y=32 3 x+4 y=32.
Show Solution
f f has a maximum value of 976 976 at the point(8,2)(8,2).
Let’s now return to the problem posed at the beginning of the section.
Example: golf balls and lagrange multipliers
The golf ball manufacturer, Pro-T, has developed a profit model that depends on the number x x of golf balls sold per month (measured in thousands), and the number of hours per month of advertising y y, according to the function
z=f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 z=f(x,y)=48 x+96 y−x 2−2 x y−9 y 2,
where z z is measured in thousands of dollars. The budgetary constraint function relating the cost of the production of thousands golf balls and advertising units is given by 20 x+4 y=216 20 x+4 y=216.Find the values of x x and y y that maximize profit, and find the maximum profit.
Show Solution
Again, we follow the problem-solving strategy:
The optimization function is f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 f(x,y)=48 x+96 y−x 2−2 x y−9 y 2. To determine the constraint function, we first subtract 216 from both sides of the constraint, then divide both sides by 4, which gives 5 x+y−54=0 5 x+y−54=0. The constraint function is equal to the left-hand side, so g(x,y)=5 x+y−54 g(x,y)=5 x+y−54. The problem asks us to solve for the maximum value of f f subject to this constraint.
So, we calculate the gradients of both f f and g g:
∇f(x,y)=(48−2 x−2 y)i+(96−2 x−18 y)j∇g(x,y)=5 i+j.∇f(x,y)=(48−2 x−2 y)i+(96−2 x−18 y)j∇g(x,y)=5 i+j.
The equation ∇f(x 0,y 0)=λ∇g(x 0,y 0)∇f(x 0,y 0)=λ∇g(x 0,y 0) becomes(48−2 x 0−2 y 0)i+(96−2 x 0−18 y 0)j=λ(5 i+j)(48−2 x 0−2 y 0)i+(96−2 x 0−18 y 0)j=λ(5 i+j),
which can be rewritten as
(48−2 x 0−2 y 0)i+(96−2 x 0−18 y 0)j=λ 5 i+λ j(48−2 x 0−2 y 0)i+(96−2 x 0−18 y 0)j=λ 5 i+λ j.
We then set the coefficients of i i and j j equal to each other:
48−2 x 0−2 y 0=5 λ 96−2 x 0−18 y 0=λ.48−2 x 0−2 y 0=5 λ 96−2 x 0−18 y 0=λ.
The equation g(x 0,y 0)=0 g(x 0,y 0)=0 becomes 5 x 0+y 0−54=0 5 x 0+y 0−54=0. Therefore, the system of equations that needs to be solved is
48−2 x 0−2 y 0=5 λ 96−2 x 0−18 y 0=λ 5 x 0+y 0−54=0.48−2 x 0−2 y 0=5 λ 96−2 x 0−18 y 0=λ 5 x 0+y 0−54=0.
3. We use the left-hand side of the second equation to replace λ λ in the first equation: 48−2 x 0−2 y 0=5(96−2 x 0−18 y 0)48−2 x 0−2 y 0=480−10 x 0−90 y 0 8 x 0=432−88 y 0 x 0=54−11 y 0.48−2 x 0−2 y 0=5(96−2 x 0−18 y 0)48−2 x 0−2 y 0=480−10 x 0−90 y 0 8 x 0=432−88 y 0 x 0=54−11 y 0.
Then we substitute this into the third equation:
5(54−11 y 0)+y+0=0 270−55 y 0+y 0=0 216−54 y 0=0 y 0=4.5(54−11 y 0)+y+0=0 270−55 y 0+y 0=0 216−54 y 0=0 y 0=4.
Since x 0=54−11 y 0 x 0=54−11 y 0, this gives x 0=10 x 0=10.
We then substitute (10,4)(10,4) into f(x,y)=48 x+96 y−x 2−2 x y−9 y 2 f(x,y)=48 x+96 y−x 2−2 x y−9 y 2, which gives f(10,4)=48(10)+96(4)−(10)2−2(10)(4)−9(4)2=480+384−100−80−144=540.f(10,4)=48(10)+96(4)−(10)2−2(10)(4)−9(4)2=480+384−100−80−144=540.
Therefore the maximum profit that can be attained, subject to budgetary constraints, is $540,000$540,000 with a production level of 10,000 10,000 golf balls and 4 4 hours of advertising bought per month. Let’s check to make sure this truly is a maximum. The endpoints of the line that defines the constraint are (10.8,0)(10.8,0) and (0,54)(0,54). Let’s evaluate f f at both of these points:
f(10.8,0)=48(10.8)+96(0)−10.8 2−2(10.8)(0)−9(0)2=401.76 f(0,54)=48(0)+96(54)−0 2−2(0)(54)−9(54)2=−21,060.f(10.8,0)=48(10.8)+96(0)−10.8 2−2(10.8)(0)−9(0)2=401.76 f(0,54)=48(0)+96(54)−0 2−2(0)(54)−9(54)2=−21,060.
The second value represents a loss, since no golf balls are produced. Neither of these values exceed 540 540, so it seems that our extremum is a maximum value of f f.
try it
A company has determined that its production level is given by the Cobb-Douglas function f(x,y)=2.5 x 0.45 y 0.55 f(x,y)=2.5 x 0.45 y 0.55 where x x represents the total number of labor hours in 1 1 year and y y represents the total capital input for the company. Suppose 1 1 unit of labor costs $40$40 and 1 1 unit of capital costs $50$50. Use the method of Lagrange multipliers to find the maximum value of f(x,y)=2.5 x 0.45 y 0.55 f(x,y)=2.5 x 0.45 y 0.55 subject to a budgetary constraint of $500,000$500,000 per year.
Show Solution
A maximum production level of 13,890 13,890 occurs with 5,625 5,625 labor hours and $5,500$5,500 of total capital input.
Watch the following video to see the worked solution to the above Try It
You can view the transcript for “4.38” here (opens in new window).
In the case of an optimization function with three variables and a single constraint function, it is possible to use the method of Lagrange multipliers to solve an optimization problem as well. An example of an optimization function with three variables could be the Cobb-Douglas function in the previous example: f(x,y,z)=x 0.2 y 0.4 z 0.4 f(x,y,z)=x 0.2 y 0.4 z 0.4, where x x represents the cost of labor, y y represents capital input, and z z represents the cost of advertising. The method is the same as for the method with a function of two variables; the equations to be solved are
∇f(x,y,z)=λ∇g(x,y,z)g(x,y,z)=0.∇f(x,y,z)=λ∇g(x,y,z)g(x,y,z)=0.
Example: lagrange multipliers with a three-variable optimization function
Find the minimum of the function f(x,y,z)=x 2+y 2+z 2 f(x,y,z)=x 2+y 2+z 2 subject to the constraint x+y+z=1 x+y+z=1.
Show Solution
The optimization function is f(x,y,z)=x 2+y 2+z 2 f(x,y,z)=x 2+y 2+z 2. To determine the constraint function, we subtract 1 from each side of the constraint: x+y+z−1=0 x+y+z−1=0 which gives the constraint function as g(x,y,z)=x+y+z−1 g(x,y,z)=x+y+z−1.
Next, we calculate∇f(x,y,z)∇f(x,y,z) and∇g(x,y,z)∇g(x,y,z):
∇f(x,y,z)=⟨2 x,2 y,2 z⟩∇g(x,y,z)=⟨1,1,1⟩∇f(x,y,z)=⟨2 x,2 y,2 z⟩∇g(x,y,z)=⟨1,1,1⟩
This leads to the equations
⟨2 x 0,2 y 0,2 z 0⟩=λ⟨1,1,1⟩x 0+y 0+z 0−1=0⟨2 x 0,2 y 0,2 z 0⟩=λ⟨1,1,1⟩x 0+y 0+z 0−1=0
which can be rewritten in the following form:
2 x 0=λ 2 y 0=λ 2 z 0=λ x 0+y 0+z 0−1=0.2 x 0=λ 2 y 0=λ 2 z 0=λ x 0+y 0+z 0−1=0.
3. Since each of the first three equations has λ λ on the right-hand side, we know that 2 x 0=2 y 0=2 z 0 2 x 0=2 y 0=2 z 0 and all three variables are equal to each other. Substituting y 0=x 0 y 0=x 0 and z 0=x 0 z 0=x 0 into the last equation yields 3 x 0−1=0 3 x 0−1=0, so x 0=1 3 x 0=1 3 and y 0=1 3 y 0=1 3 and z 0=1 3 z 0=1 3 which corresponds to a critical point on the constraint curve.
4. Then, we evaluate f f at the point (1 3,1 3,1 3)(1 3,1 3,1 3):
f(1 3,1 3,1 3)=(1 3)2+(1 3)2+(1 3)2=3 9=1 3.f(1 3,1 3,1 3)=(1 3)2+(1 3)2+(1 3)2=3 9=1 3.
Therefore, an extremum of the function is 1 3 1 3. To verify it is a minimum, choose other points that satisfy the constraint and calculate f f at that point. For example,
f(1,0,0)=1 2+0 2+0 2=1 f(0,−2,3)=0 2+(−2)2+3 2=13.f(1,0,0)=1 2+0 2+0 2=1 f(0,−2,3)=0 2+(−2)2+3 2=13.
Both of these values are greater than 1 3 1 3, leading us to believe the extremum is a minimum.
Try it
Use the method of Lagrange multipliers to find the minimum value of the function
f(x,y,z)=x+y+z f(x,y,z)=x+y+z
subject to the constraint x 2+y 2+z 2=1 x 2+y 2+z 2=1.
Show Solution
f(√3 3,√3 3,√3 3)=√3 3+√3 3+√3 3=√3 f(−√3 3,−√3 3,−√3 3)=−√3 3−√3 3−√3 3=−√3 f(3 3,3 3,3 3)=3 3+3 3+3 3=3 f(−3 3,−3 3,−3 3)=−3 3−3 3−3 3=−3
Problems with Two Constraints
The method of Lagrange multipliers can be applied to problems with more than one constraint. In this case the optimization function, w w is a function of three variables:
w=f(x,y,z)w=f(x,y,z)
and it is subject to two constraints:
g(x,y,z)=0 g(x,y,z)=0 and h(x,y,z)=0 h(x,y,z)=0
There are two Lagrange multipliers, λ 1 λ 1 and λ 2 λ 2, and the system of equations becomes
∇f(x 0,y 0,z 0)=λ 1∇g(x 0,y 0,z 0)+λ 2∇h(x 0,y 0,z 0)g(x 0,y 0,z 0)=0 h(x 0,y 0,z 0)=0.∇f(x 0,y 0,z 0)=λ 1∇g(x 0,y 0,z 0)+λ 2∇h(x 0,y 0,z 0)g(x 0,y 0,z 0)=0 h(x 0,y 0,z 0)=0.
Example: Lagrange Multipliers with Two constraints
Find the maximum and minimum values of the function
f(x,y,z)=x 2+y 2+z 2 f(x,y,z)=x 2+y 2+z 2
subject to the constraints z 2=x 2+y 2 z 2=x 2+y 2 and x+y−z+1=0 x+y−z+1=0.
Show Solution
Let’s follow the problem-solving strategy:
The optimization function is f(x,y,z)=x 2+y 2+z 2 f(x,y,z)=x 2+y 2+z 2. To determine the constraint functions, we first subtract z 2 z 2 from both sides of the first constraint, which gives x 2+y 2−z 2=0 x 2+y 2−z 2=0, so g(x,y,z)=x 2+y 2−z 2 g(x,y,z)=x 2+y 2−z 2. The second constraint function is h(x,y,z)=x+y−z+1 h(x,y,z)=x+y−z+1.
We then calculate the gradients of f f,g g, and h h:
∇f(x,y,z)=2 x i+2 y j+2 z k∇g(x,y,z)=2 x i+2 y j−2 z k∇h(x,y,z)=i+j−k.∇f(x,y,z)=2 x i+2 y j+2 z k∇g(x,y,z)=2 x i+2 y j−2 z k∇h(x,y,z)=i+j−k.
The equation ∇f(x 0,y 0,z 0)=λ 1∇g(x 0,y 0,z 0)+λ 2∇h(x 0,y 0,z 0)∇f(x 0,y 0,z 0)=λ 1∇g(x 0,y 0,z 0)+λ 2∇h(x 0,y 0,z 0) becomes
2 x 0 i+2 y 0 j+2 z 0 k=λ(2 x 0 i+2 y 0 j−2 z 0 k)+λ 2(i+j−k),2 x 0 i+2 y 0 j+2 z 0 k=λ(2 x 0 i+2 y 0 j−2 z 0 k)+λ 2(i+j−k),
which can be rewritten as
2 x 0 i+2 y 0 j+2 z 0 k=(2 λ 1 x 0+λ 2)i+(2 λ 1 y 0+λ 2)j−(2 λ 1 z 0+λ 2)k.2 x 0 i+2 y 0 j+2 z 0 k=(2 λ 1 x 0+λ 2)i+(2 λ 1 y 0+λ 2)j−(2 λ 1 z 0+λ 2)k.
Next, we set the coefficients of i i, j j, and k k equal to each other:
2 x 0=2 λ 1 x 0+λ 2 2 y 0=2 λ 1 y 0+λ 2 2 z 0=2 λ 1 z 0−λ 2.2 x 0=2 λ 1 x 0+λ 2 2 y 0=2 λ 1 y 0+λ 2 2 z 0=2 λ 1 z 0−λ 2.
The two equations that arise from the constraints are x 2 0=x 2 0+y 2 0 x 0 2=x 0 2+y 0 2 and x 0+y 0−z 0+1=0 x 0+y 0−z 0+1=0. Combining these equations with the previous three equations gives
2 x 0=2 λ 1 x 0+λ 2 2 x 0=2 λ 1 y 0+λ 2 2 x 0=2 λ 1 z 0−λ 2 z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.2 x 0=2 λ 1 x 0+λ 2 2 x 0=2 λ 1 y 0+λ 2 2 x 0=2 λ 1 z 0−λ 2 z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.
3. The first three equations contain the variable λ 2 λ 2. Solving the third equation for λ 2 λ 2 and replacing into the first and second equations reduces the number of equations to four:
2 x 0=2 λ 1 x 0−2 λ 1 z 0−2 z 0 2 y 0=2 λ 1 y 0−2 λ 1 z 0−2 z 0 z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.2 x 0=2 λ 1 x 0−2 λ 1 z 0−2 z 0 2 y 0=2 λ 1 y 0−2 λ 1 z 0−2 z 0 z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.
Next, we solve the first and second equation for λ 1 λ 1. The first equation gives λ 1=x 0+z 0 x 0−z 0 λ 1=x 0+z 0 x 0−z 0, the second equation gives λ 1=y 0+z 0 y 0−z 0 λ 1=y 0+z 0 y 0−z 0. We set the right-hand side of each equation equal to each other and cross-multiply:
x 0+z 0 x 0−z 0=y 0+z 0 y 0−z 0(x 0+z 0)(y 0−z 0)=(x 0−z 0)(y 0+z 0)x 0 y 0−x 0 z 0+y 0 z 0−z 0 2=x 0 y 0+x 0 z 0−y 0 z 0−z 0 2 2 y 0 z 0−2 x 0 z 0=0 2 x 0(y 0−x 0)=0.x 0+z 0 x 0−z 0=y 0+z 0 y 0−z 0(x 0+z 0)(y 0−z 0)=(x 0−z 0)(y 0+z 0)x 0 y 0−x 0 z 0+y 0 z 0−z 0 2=x 0 y 0+x 0 z 0−y 0 z 0−z 0 2 2 y 0 z 0−2 x 0 z 0=0 2 x 0(y 0−x 0)=0.
Therefore, either z 0=0 z 0=0 or y 0=x 0 y 0=x 0. If z 0=0 z 0=0, then the first constraint becomes 0=x 2 0+y 2 0 0=x 0 2+y 0 2 The only real solution to this equation is x 0=0 x 0=0 and y 0=0 y 0=0, which gives the ordered triple (0,0,0)(0,0,0). This point does not satisfy the second constraint, so it is not a solution.Next, we consider y 0=x 0 y 0=x 0, which reduces the number of equations to three:
y 0=x 0 z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.y 0=x 0 z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.
We substitute the first equation into the second and third equations:
z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.z 0 2=x 0 2+y 0 2 x 0+y 0−z 0+1=0.
Then, we solve the second equation for z 0 z 0, which gives z 0=2 x 0+1 z 0=2 x 0+1. We then substitute this into the first equation,
z 0 2=2 x 0 2(2 x 0+1)2=2 x 0 2 4 x 0 2+4 x 0+1=2 x 0 2 2 x 0 2+4 x 0+1=0 z 0 2=2 x 0 2(2 x 0+1)2=2 x 0 2 4 x 0 2+4 x 0+1=2 x 0 2 2 x 0 2+4 x 0+1=0
and use the quadratic formula to solve for x 0 x 0:
x+0=−4±√4 2−4(2)(1)2(2)=−4±√8 4=−4±2√2 4=−1±√2 2 x+0=−4±4 2−4(2)(1)2(2)=−4±8 4=−4±2 2 4=−1±2 2.
Recall y 0=x 0 y 0=x 0,so this solves for y 0 y 0 as well. Then, z 0=2 x 0+1 z 0=2 x 0+1, so
x 0=2 x 0+1=2(−1±√2 2)+1=−2+1±√2=−1±√2.x 0=2 x 0+1=2(−1±2 2)+1=−2+1±2=−1±2.
Therefore, there are two ordered triplet solutions:
(−1+√2 2,−1+√2 2,−1+√2)(−1+2 2,−1+2 2,−1+2) and (−1−√2 2,−1−√2 2,−1−√2).(−1−2 2,−1−2 2,−1−2).
We substitute (−1+√2 2,−1+√2 2,−1+√2)(−1+2 2,−1+2 2,−1+2) into f(x,y,z)=x 2+y 2+z 2 f(x,y,z)=x 2+y 2+z 2, which gives
f(−1+√2 2,−1+√2 2,−1+√2)=(−1+√2 2)2+(−1+√2 2)2+(−1+√2)2=(1−√2+1 2)+(1−√2+1 2)+(1−2√2+2)=6−4√2.f(−1+2 2,−1+2 2,−1+2)=(−1+2 2)2+(−1+2 2)2+(−1+2)2=(1−2+1 2)+(1−2+1 2)+(1−2 2+2)=6−4 2.
6+4√2 6+4 2 is the maximum value and 6−4√2 6−4 2 is the minimum value of f(x,y,z)f(x,y,z) subject to the given constraints.
Try it
Use the method of Lagrange multipliers to find the minimum value of the function
f(x,y,z)=x 2+y 2+z 2 f(x,y,z)=x 2+y 2+z 2
subject to the constraints 2 x+y+2 z=9 2 x+y+2 z=9 and 5 x+5 y+7 z=29 5 x+5 y+7 z=29.
Show Solution
f(2,1,2)=9 f(2,1,2)=9 is a minimum.
Watch the following video to see the worked solution to the above Try It
You can view the transcript for “CP 4.40” here (opens in new window).
Candela Citations
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CP 4.38. Authored by: Ryan Melton. License: CC BY: Attribution
CP 4.40. Authored by: Ryan Melton. License: CC BY: Attribution
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Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
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CC licensed content, Original
CP 4.38. Authored by: Ryan Melton. License: CC BY: Attribution
CP 4.40. Authored by: Ryan Melton. License: CC BY: Attribution
CC licensed content, Shared previously
Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
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Determine if a 2D point is within a quadrilateral
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I'm working on a JS program which I need to have determine if points are within four corners in a coordinate system.
Could somebody point me in the direction of an answer?
I'm looking at what I think is called a convex quadrilateral. That is, four pretty randomly chosen corner positions with all angles smaller than 180°.
Thanks.
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edited Mar 19, 2013 at 3:30
HenrikHenrik
asked Mar 19, 2013 at 3:04
HenrikHenrik
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I tried the inpolygon methon used here link, but it doesn't work. It gives me "Cannot call method 'inpolygon' of undefined". if (Math.inpolygon(5,6,[1,22,13,1],[1,1,21,31])){ return "yep"; }Henrik –Henrik 2013-03-19 03:06:11 +00:00 Commented Mar 19, 2013 at 3:06
and why you cannot just compare coordinates? can you describe the problem closer? what are points? corner?4pie0 –4pie0 2013-03-19 03:10:21 +00:00 Commented Mar 19, 2013 at 3:10
I have tons of quite randomly generated quadrilaterals. Then I need to check wether some (also quite randomly generated) points are "available" or already occupied by a quadrilateral.Henrik –Henrik 2013-03-19 03:13:34 +00:00 Commented Mar 19, 2013 at 3:13
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There are two relatively simple approaches. The first approach is to draw a ray from the point to "infinity" (actually, to any point outside the polygon) and count how many sides of the polygon the ray intersects. The point is inside the polygon if and only if the count is odd.
The second approach is to go around the polygon in order and for every pair of vertices v i and v i+1 (wrapping around to the first vertex if necessary), compute the quantity (x - x i) (y i+1 - y i) - (x i+1 - x i) (y - y i). If these quantities all have the same sign, the point is inside the polygon. (These quantities are the Z component of the cross product of the vectors (v i+1 - v i) and (p - v i). The condition that they all have the same sign is the same as the condition that p is on the same side (left or right) of every edge.)
Both approaches need to deal with the case that the point is exactly on an edge or on a vertex. You first need to decide whether you want to count such points as being inside the polygon or not. Then you need to adjust the tests accordingly. Be aware that slight numerical rounding errors can give a false answer either way. It's just something you'll have to live with.
Since you have a convex quadrilateral, there's another approach. Pick any three vertices and compute the barycentric coordinates of the point and of the fourth vertex with respect to the triangle formed by the three chosen vertices. If the barycentric coordinates of the point are all positive and all less than the barycentric coordinates of the fourth vertex, then the point is inside the quadrilateral.
P.S. Just found a nice page here that lists quite a number of strategies. Some of them are very interesting.
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edited Mar 20, 2013 at 3:57
answered Mar 19, 2013 at 3:30
Ted HoppTed Hopp
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Henrik
HenrikOver a year ago
I don't quite see how I am to draw a ray like in the first approach without involving a lot lot lot of (unnecessary) computation, but i reckon I'll get it to work with the second one. Thanks!
2013-03-19T03:37:54.717Z+00:00
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Ted HoppOver a year ago
@Henrik - Just pick a point on the X axis that is further out than the max X coordinate of the four vertices. The ray can then go from your test point to the point on the X axis. (You can use the Y axis equally well, of course.) Just remember that you need to test for the intersection of the line segments, not the lines.
2013-03-19T03:39:48.027Z+00:00
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Ted HoppOver a year ago
@Henrik - If you're going with the second approach, be aware that I had a typo in the formula. It's now fixed. (I had used 0 instead of i as a subscript in a couple of places.)
2013-03-19T03:42:06.883Z+00:00
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Andrew Mao
Andrew MaoOver a year ago
Your second approach is computing if the point is in the intersection of the four halfspaces formed by the sides of the quadrilateral. It's a bit of a convoluted formula :)
2013-03-19T03:51:05.483Z+00:00
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Ted HoppOver a year ago
@AndrewMao - Nah. I'm a programmer. For programmers, you plop out code, you don't explain things. :)
2013-03-19T03:57:19.47Z+00:00
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You need to use winding, or the ray trace method.
With winding, you can determine whether any point is inside any shape built with line segments.
Basically, you take the cross product of each line segment with the point, then add up all the results. That's the way I did it to decide if a star was in a constellation, given a set of constellation lines. I can see that there are other ways..
There must be some code for this in a few places.
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answered Mar 19, 2013 at 3:29
Tom AndersenTom Andersen
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It is MUCH easier to see if a point lies within a triangle.
Any quadrilateral can be divided into two triangles.
If the point is in any of the two triangles that comprise the quadrilateral, then the point is inside the quadrilateral.
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answered Jun 14, 2014 at 6:16
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7067 | https://ncse.ngo/vertebrate-limbs | Main navigation
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Vertebrate limbs
Summary of problems:
Common function can explain certain similarities of form, but cannot explain similar developmental pathways, or the particular components that make up certain structures in different species.
Full discussion:
The discussion of functional constraints in Explore Evolution is nearly impossible to state in a way which does not refute itself. They do not deny the remarkable similarity between the structures of species within various taxonomic groups. They do not deny that one can produce a hierarchal (branching) arrangement of the ways these structures vary within and among these groups, and that the branching pattern is consistent regardless of which particular structure you examine. In other words, their response to the evidence of the branching pattern predicted by the tree of life is to agree that it is all accurate. They simply argue that it is possible to invoke special explanations for each such structure, multiplying causes needlessly. This practice violates basic scientific and logical principles. By treating structures in isolation, they obscure the actual evidence examined by scientists.
For instance, EE cites biologists from 150 years ago, biologists whose arguments were tested and found lacking.
It is worth noting, to begin with, that vertebrate limbs do not have "a variety of bones in the segment farthest out." The number of fingers and toes is consistent. The numbers of bones in each finger and toe are consistent. The number of wrist bones is consistent. Even if functional constraints could predict the broad pattern, they do not explain why no living species has more than 5 fingers or toes, nor the consistency of the number and developmental histories of the wristbones.
Furthermore, functional constraints do not explain the broad pattern. Robotic arms do not typically have one element nearest the base, two further out, and a number in the "hand." They employ various sorts of joints and connections which do not exist in living species. The argument of functional constraints only make sense if you assume some evolutionary process acting on some common starting point. That explains vestigial fingers and toes in the legs of deer, it explains why our two legs, and all four legs in a deer, still have two bones in the middle segment, despite having no need to twist. It explains why no species has more than five fingers or toes, and why vestiges of all five can be found in vertebrates which seem to have fewer. These results would be surprising if there were not some common starting point, but are predicted and found because of evolutionary hypotheses.
The claim of consistency because of functional constraint also does not match the actual evidence. Because of the basic physics of flight, bird wings, bat wings and pterosaur wings must all be similarly shaped. If they were shaped much differently, flight would be (or have been) impossible. If the functional constraint hypothesis were the sole explanation for wing structure, we might predict that all three types of vertebrate wings would be similar in their anatomical structure, but this prediction fails. Pterosaurs have a wing consisting mostly of skin stretched between the 4th finger and the body, with the thumb and three fingers free of the wing. Bat wings consist of skin stretched across all 4 fingers and attaching to the leg, with the thumb free of the wing. Bird wings (like chicken wings you've eaten) do not have skin stretched across them, have fused the bones of the 2nd and 3rd fingers together for strength, and cover the structure with feathers.
Within each group, these traits are consistent, indicating that the wing can be treated as homologous within each group, but not across groups. All three wings share the same bones (the bones are homologous), but they are arranged very differently. Bats use all four fingers in the wing; birds use two, have lost one finger almost completely, and another is nearly functionless; pterosaurs used only one finger in flight, but adapted other fingers to different purposes. Since the anatomy of these wings is so different, this falsifies a prediction of the functional constraint hypothesis.
It confirms what we would expect from evolutionary explanations. Because vertebrates share a common ancestor, all three groups shared ancestors which possessed the basic vertebrate limb. Each group took steps toward flight at different times and from different ancestors with different initial traits. The differences in starting conditions meant that different sorts of genetic and structural changes were advantageous in each different group. Within each group, the structure of the wing is consistent, indicating common descent within pterosaurs, within bats and within birds. The differences in the structure of each type of wing indicates that wings evolved independently in each group. The similarity of the bones in the wings (and elsewhere in the body) indicate that all three groups share an ancestor farther back in their history. This nested pattern of shared characters is exactly what common descent predicts, and has not successfully been explained by other means.
Functional constraints cannot explain why vertebrate wings should consist of skin stretched over bone, since birds do not use skin for the flight surface. Indeed, insect wings do not have bones or skin, and are not derived from legs. Insect wings are structurally similar (homologous) to gills. Again, this pattern of similar structure is unexpected under the predictions of common functional constraint, but entirely predictable based on evolution and common descent.
An inquiry-based textbook could turn a discussion like that above into a fascinating exercise. Students could generate predictions based on various potential explanations, and then test them using data from various biological structures. Explore Evolution, despite its claim to be inquiry-based (and despite the nonsensical and meaningless exercise on pp. 46-47), does not invite students to examine any evidence at all, nor does it explain why students should ignore the research conducted in over a century since Agassiz defended his position. Inquiry-based instruction invites students to discuss the topic, but no useful discussion could possibly proceed from such a flawed foundation.
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While grizzly bears have long, flat, and somewhat blunt claws, black b [#permalink] 07 Jan 2023, 05:15
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While grizzly bears have long, flat, and somewhat blunt claws, black bears have short, curved and______ claws.
| Blank (i) |
| A. obtuse |
| B. abominable |
| C. barren |
| D. acute |
| E. fearful |
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Re: While grizzly bears have long, flat, and somewhat blunt claws, black b [#permalink] 21 Jan 2023, 06:43
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The blank is describing black bears…claws. The transition word [w]hile indicates that the claws of black bears are different from those of grizzly bears, which are described as long, flat, and somewhat blunt. Black bears’ claws are described in the sentence as short and curved, which are the
opposite of long and flat. Therefore, a good word for the blank is the opposite of somewhat blunt, so use “sharp” and evaluate the answer choices. Choice (A) obtuse is a synonym for blunt, so eliminate (A). Choice (B), abominable may describe a bear, but the word doesn’t mean “sharp” and so doesn’t match the clue, so eliminate (B). Choice (C), barren, is not a match for “sharp” so eliminate it. Choice (D), acute, is a good match for “sharp,” so keep (D). Choice (E), fearful does not match “sharp,” so eliminate it. The correct answer is (D)
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Re: While grizzly bears have long, flat, and somewhat blunt claws, black b [#permalink] 06 Jul 2024, 19:00
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7069 | https://www.facebook.com/nationalzoo/posts/happy-serpentday-gaboon-viper-fangs-are-the-longest-of-any-venomous-snake-reachi/10157914673057902/ | Happy #SerpentDay! 🐍 Gaboon viper fangs are the longest of any venomous snake, reaching up to two inches in length. They can control the injection of venom when they bite. The results range from no effect to rapid death. 💀 See it in the Reptile Discovery... - Smithsonian’s National Zoo and Conservation Biology Institute | Facebook
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Smithsonian’s National Zoo and Conservation Biology Institute's Post
Smithsonian’s National Zoo and Conservation Biology Institute
February 1, 2020 ·
Happy #SerpentDay! Gaboon viper fangs are the longest of any venomous snake, reaching up to two inches in length. They can control the injection of venom when they bite. The results range from no effect to rapid death. See it in the Reptile Discovery Center!
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Jared Lussier
See rapid death in the reptile center?
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Bronwyn Hogan
Are they from Gabon? I lived there as a young child. My father opened up Peace Corps Gabon.
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Joan Millett
Beautiful! However total respect wouldn't want to get bit
5y
Marion Schuller
Snakes and I have an agreement. I go thisaway, they go thataway.
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Vicki Bretzfelder
Oh my goodness
5y
Paul Farmer
Happy Serpent Day Mary Barnett!
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7070 | https://tutorial.math.lamar.edu/Solutions/Alg/Circles/Prob8.aspx | Algebra - Circles
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Section 3.3 : Circles
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Determine the radius and center of the following circle. If the equation is not the equation of a circle clearly explain why not.
x 2+y 2+8 x+20=0 x 2+y 2+8 x+20=0
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Start Solution
To do this problem we need to complete the square on the x x and y y terms. To help with this we’ll first get the number on the right side and group the x x and y y terms as follows.
x 2+8 x+y 2=−20 x 2+8 x+y 2=−20 Show Step 2
Here is the number we need to complete the square for both x x. Note that we don’t need to complete the square for the y y.
x:(8 2)2=(4)2=16 x:(8 2)2=(4)2=16 Show Step 3
Now, complete the square.
x 2+8 x+16+y 2=−20+16(x+4)2+y 2=−4 x 2+8 x+16+y 2=−20+16(x+4)2+y 2=−4
Don’t forget to add the number from Step 2 to both sides of the equation!
Show Step 4
Okay, at this point we can see that this equation is not the equation of a circle. The standard form of the circle is,
(x−h)2+(y−k)2=r 2(x−h)2+(y−k)2=r 2
The right side is r 2 r 2 and that must be a positive number (and the coefficients of the x x and y y must also be positive) which for our equation it is not. Therefore, this is not the equation of a circle.
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Proving $|e^{iθ}|=1$
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How do I show that $|e^{iθ}|=1$? So I got that the length will be $\sqrt{\cos^2(x)-\sin^2(x)}$ and it can be written as the square root of $\cos 2x$ but I don't see how that equals 1.
complex-analysis
complex-numbers
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edited Sep 6, 2013 at 21:41
user02138
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asked Sep 6, 2013 at 21:41
user93515user93515
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$\begingroup$ $|a+bi|=\sqrt{a^2+b^2}$ not $\sqrt{a^2+(ib)^2}$ $\endgroup$
anon
– anon
2013-09-06 21:47:24 +00:00
Commented Sep 6, 2013 at 21:47
$\begingroup$ When $x= \pi$ your formula yields $\sqrt{-1}$ as the length, that should be a sign that the formula you are using is not the right one ;) $\endgroup$
N. S.
– N. S.
2013-09-07 16:33:50 +00:00
Commented Sep 7, 2013 at 16:33
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For a complex number of the form $z=a+ib$, it's magnitude is given by $|z|=\sqrt{a^2+b^2}$.
I believe you have incorrectly used $|z|=\sqrt{a^2-b^2}$
See here: Magnitude of a Complex Number
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edited Sep 6, 2013 at 21:52
answered Sep 6, 2013 at 21:43
MufasaMufasa
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The length will be $\sqrt{\cos^2(\theta )+\sin^2(\theta)}=1$.
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answered Sep 6, 2013 at 21:43
TalexiusTalexius
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According to de Euler's formula you have that $$ e^{i\theta} = \cos(\theta) + i \sin (\theta) $$ Now the module $| \cdot | $ of a complex number is $x+i y $ is simply $(x^2+y^2)^{1/2}$. Note that we are ignoring the $i$ when taking squares. I think you derived your incorrect formula because you also squared the $i$. Use also that for any real number $\theta$ we have: $\sin^2(\theta)+\cos^2(\theta)=1$.
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edited Sep 7, 2013 at 16:12
answered Sep 6, 2013 at 21:44
Mauricio TecMauricio Tec
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From $e^z = \sum_{k = 0}^{\infty} \frac{z^k}{k!}$, we see that $\overline{e^z} = e^{\overline{z}}$, and from $e^a e^b = e^{a+b}$ we get
$$|e^z|^2 = e^z \times\overline{e^z} = e^z e^{\overline{z}} = e^{z + \overline{z}}$$
In particular, for $z = i\theta$, you get $|e^{i\theta}| = 1$.
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answered Sep 7, 2013 at 16:22
Joel CohenJoel Cohen
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Even if you wouldn't use the distance formula, when you graph z = cost + isint you will see in the triangle that all points Z are on the unit circle? That's really fundamental trigonometry. The hypotenuse is traditionally 1. This is the unit circle approach in the introduction of trigonometry and that comes back when trig finds its place as a representation of complex numbers in polar form
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answered Sep 7, 2013 at 1:13
imranfatimranfat
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20
Dr. Kevin Bracker; Dr. Fang Lin; and jpursley
Most of TVM analysis on your Financial Calculator can be done with the 5-key approach. The five keys are as follows
N ⇒ This key refers to the number of periods
I/Y ⇒ This key refers to the interest rate (do not enter as a decimal ⇒ 10% would be 10 not 0.10). Sometimes this interest rate is referred to as a discount rate or rate of return.
PV ⇒ This key refers to the Present Value
PMT ⇒ This key refers to the Annuity Payment
FV ⇒ This key refers to the Future Value
When entering values into your financial calculator you press the value you are entering first, then the key. For example, if we want to put in 10 periods, we would enter this as 10 N. Sometimes you will need to enter a negative value. To do this, you must use the “+/-“ key on your calculator instead of the “–“ key. The order that you enter the variables doesn’t matter as long as you enter the four that you know first, and then solve for the fifth. To solve, you just press the key representing what you are trying to find. Now let’s go through several examples.
EXAMPLE ONE – Future Value of a Single Cash Flow
You are investing $10,000 today and want to know how much you will have after 6 years if you earn a 7% rate of return over the 6-year time frame. Since you are starting with $10,000, that is your present value. You have 6 years, so the number of time periods is 6. The 7% rate of return means you have a 7% interest rate. In this example we are not using an annuity, so we are going to set the Annuity Payment to zero.
6 N7 I/Y10,000 PV0 PMTFV -15,007.30
Thus, you will have $15,007.30 at the end of the 6th year. Notice that the answer came out negative instead of positive. This is due to the way the calculator “thinks” when it is solving TVM problems. The calculator needs to keep track not only of the dollar amounts, but which way the money is flowing. Because you entered the Present Value (PV) as $10,000, the calculator assumed you were receiving $10,000. If you receive $10,000 today, the only way for the problem to “balance out” is for you to give back $15,007.30 at the end of the 6th year. In a problem like this you can just ignore the negative sign in front of the $15,007.30. However, there are certain problems where this is important. Specifically – IF YOU ENTER NON-ZERO VALUES FOR TWO OR MORE OF THE CASH FLOW KEYS (THE CASH FLOW KEYS IN THE 5-KEY APPROACH ARE THE PV, PMT, AND FV KEYS), YOU MUST BE CAREFUL OF CASH FLOW SIGNS. In our example, we only entered 1 non-zero value for a cash flow (the $10,000 PV), so the sign doesn’t matter. We will reintroduce this in a little bit.
PRACTICE PROBLEM ONE
You are investing $400 today and want to know how much you will have after 45 years if you earn a 9.5% rate of return over the 45-year time period. The solution to this and other practice problems can be found at the end of this tutorial.
EXAMPLE TWO – Present Value of a Single Cash Flow
You are going to receive $6000 in 5 years. Assuming a 9% discount rate, what is this worth to you today?
5 N9 I/Y0 PMT6000 FVPV -3899.59
Again, we can ignore the negative sign in the answer (since the only non-zero cash flow that we entered was the $6000 Future Value). Thus, $6000 received in 5 years is only worth $3899.59 today (assuming a 9% discount rate). In other words, we are indifferent between receiving $3899.59 today and receiving $6000 in 5 years – they both are worth the same to us. Alternatively, we would be willing to pay $3899.59 or less to receive $6000 in 5 years, but we would NOT be willing to pay any more than $3899.59. The reason for this is that if we invested $3899.59 today and let it compound for 5 years at 9%, it would grow to $6000 at the end of the 5th year. Present value will be an important concept in valuation because most investments are structured in a manner that we pay a set amount today to receive cash flows in the future. Once we know what those future cash flows are worth to us today, we can evaluate the investment.
PRACTICE PROBLEM TWO
You are offered an opportunity to make an investment today that will pay you $100,000 in 20-years. Assuming a 5% discount rate, what is the most you would be willing to pay for that investment today?
ANNUITIES
An annuity is a sequence of equal, periodic cash flows. Many financial situations can be modeled as an annuity. For instance, calculating a mortgage payment on a home is an annuity. Simple retirement analysis can be structured as an annuity. Also, bond valuation is partially modeled as an annuity since we receive a fixed coupon payment each year. With annuities, we assume cash flows come at the end of each period. Note that there is a variation referred to as an “Annuity Due” that assumes cash flows come at the BEGINNING of the period. We will not work with Annuity Due situations in this text, however it is relatively simple to do so by making a simple adjustment to your calculator. We will include a sample example to illustrate this, however, in this textbook we will assume cash flows come at the end of each period in all of our annuity problems.
EXAMPLE THREE – Lottery Jackpot
Assume you have just won a $10 Million Lottery Jackpot. However, instead of paying you the $10 Million up front, you have the choice of receiving $5 Million today or $400,000 per year at the end of each year for the next 25 years. Assuming a 6% discount rate, which would you prefer? In order to answer this, you need to find the PV of the $400,000 per year for 25 years. This is done as follows:
25 N6 I/Y400,000 PMT0 FVPV 5,113,342.46 (Note that we dropped the negative sign)
Since the annuity is worth more than $5 Million to us, we would prefer to take the $400,000 per year for the next 25 years.
EXAMPLE THREE A – Lottery Jackpot Annuity Due (OPTIONAL)
Since many lotteries actually give you your first installment TODAY if you take the installment plan, we could make the example more realistic by assuming the prize was paid as $400,000 per year at the BEGINNING of each year for the next 25 years and leave everything else the same. To adjust your calculator for this, you need to set it up to work with beginning of period payments. To do so, do the following:
SHIFT BEG/END
When you do this, you should see the word “BEGIN” show up on the bottom of your calculator screen to show you that your calculator is set for beginning of period payments. Now repeat the calculation from the Example Three
25 N6 I/Y400,000 PMT0 FVPV 5,420,143.01 (Note that we dropped the negative sign)
So, with beginning of period payments the jackpot is worth more to us (since we start receiving our money earlier). Now, set your calculator back to end of period payments so that you don’t end up with the wrong answer on all your other problems. To set it back, just toggle the BEG/END mode again as follows:
SHIFT BEG/END
PRACTICE PROBLEM THREE
You are offered an investment that pays you $1000 per year for the next 30 years. Assuming a 10% discount rate, what is this investment worth to you today?
EXAMPLE FOUR – I want to be a Millionaire
You want to become a millionaire and plan to do so through a savings/investment plan. Assuming you want to reach your goal in 20 years and anticipate earning a 10% rate of return, how much must you save at the end of each year in order to reach your goal?
20 N10 I/Y0 PV1,000,000 FVPMT $17,459.62
This means you will need to save $17,459.62 per year in order to achieve your goal.
PRACTICE PROBLEM FOUR
Since saving $17,459.62 per year is not realistic for most of us, let’s try some adjustments. Calculate how much you would need to save under the following conditions
30 years at 10%
40 years at 10%
30 years at 7.5%
30 years at 5%
Note the large difference that time and rate of return make on savings. Having a short savings horizon or earning a low rate of return mean you must save considerably more each year to reach the same goal. This is especially important for retirement planning.
EXAMPLE FIVE – Changing Periods per Year
Now assume that you want to accumulate $1 million in 30 years, but instead of saving each year, you are going to save every two weeks (we will earn a 10% annual rate of return). There are 26 2-week periods in each year, so now you have to adjust your calculator to work with 26 periods per year. You can do this as follows:
26 SHIFT P/YR
Now your calculator will recognize that you are not making annual contributions to your savings plan, but instead making a contribution every other week. Another issue when you change the number of periods per year is to recognize that the N key stands for periods and not necessarily years. Since all of our previous examples were done using 1 P/YR, the number of periods and years were the same. However, now 1 year will have 26 periods. Therefore, 30 years is equivalent to 780 periods (calculated by taking 26 times 30). Now that our adjustments have been made, we are ready to enter the problem into our calculator.
780 N10 I/Y0 PV1,000,000 FVPMT $202.75
If we save $202.75 every two weeks for the next 30 years and earn a 10% rate of return, we will have $1,000,000 at the end of the 30th year.
PRACTICE PROBLEM FIVE
Repeat the above example, but now assume weekly payments (52 weeks per year) instead of payments every two weeks. Once you are done, figure out how much you are saving per year under both the once per week and once every other week alternatives and compare this to the answer we got in Practice Problem Four-1. Why are the answers different? (NOTE – Remember to reset your calculator to 1 period per year after you finish this problem).
EXAMPLE SIX – Solving for Interest Rates
Let’s keep working with the goal of becoming a millionaire. However, instead of calculating how much you must save, we’ll assume you can save $3000 per year and want to find the rate of return you will need to earn to reach your goal. This time we will give ourselves 35 years of saving $3000 per year.
35 N0 PV-3000 PMT1,000,000 FVI/Y 10.89%
If we can save $3000 per year at the end of each year for the next 35 years, we will need to earn a 10.89% rate of return in order to become a millionaire. There is a very important step in this that must be done in order to get the right answer. Note that we made the annuity payment equal NEGATIVE 3000 instead of 3000. This is because we are now entering 2 non-zero values into our cash flow keys (PV, PMT, FV). When enter 2 or 3 non-zero values into our cash flow keys, we need to be careful with the signs of the cash flows. The signs indicate the direction of the cash flow. A negative sign indicates that the cash flow is flowing away from us. In this case, we are saving $3000 per year so we are giving up that amount and making it negative. At the end of the 35 years, we will receive back $1,000,000 so that is positive.
PRACTICE PROBLEM SIX
What rate of return would you need to earn if you were able to save $4500 per year each year for the next 35 years in order to become a millionaire?
EXAMPLE SEVEN – Combining PMT, FV, and PV
Here is one last variation on our millionaire example. This time, instead of starting with nothing, let’s assume that we already have $40,000 and plan to save an additional $3000 per year over the next 35 years. Now, what rate of return must we earn in order to accumulate $1,000,000 at the end of the 35th year?
35 N-40,000 PV-3000 PMT1,000,000 FVI/Y 7.63%
Note that here, we must make both the Present Value and the Annuity Payment negative as they both are flowing away from us into the savings plan. The Future Value will flow back to us at the end of the time period so it is positive.
PRACTICE PROBLEM SEVEN
You want to retire a millionaire and have accumulated $20,000 which you are putting into your retirement plan. In addition, you plan to earn a 9% rate of return. How much must you save PER MONTH over the next 35 years in order to reach your goal?
Video: Introduction and 5-Key Approach (HP10BII)
EXAMPLE EIGHT – Uneven Cash Flow Stream – Present Value
Assuming a 6.5% discount rate, solve for the present value of the following cash flow stream.
Here we can no longer use the 5-key approach (technically, we CAN…it would just be a lot more tedious). Instead we want to move to the cash flow worksheet on our financial calculator. The cash flow worksheet for the HP10BII follows a basic 4-step process:
Clear out any previous values in your cash flow worksheet
Enter the cash flows and frequencies in your cash flow worksheet starting with year 0
Enter the discount rate
Solve for present value
Let’s walk through this process with keystrokes using the example above:
SHIFT C ALL (this clears out the previous values from the worksheet)
0 CFj (this enters the CF0 which in this example is 0. Note that it will not always be zero and could have either a positive or negative value)
400 CFj (this enters the first cash flow sequence )
5 SHIFT Nj (this enters the first cash flow frequency. The default value for this variable is 1, so you only need to enter an Nj value when the frequency is something other than 1.)
600 CFj (this enters the second cash flow sequence)
4 SHIFT Nj (this enters the second cash flow frequency)
1000 CFj (this enters the third cash flow sequence. Note that there is no need to enter an Nj value here as the default frequency value is 1 – which is what we want – so we can skip the Nj and just go onto the next cash flow sequence.)
1300 CFj (this enters the fourth cash flow sequence)
6 SHIFT Nj (this enters the fourth – and final – cash flow frequency)
6.5 I/YR (this is where you put in the discount rate of 6.5%)
SHIFT NPV (this calculates the answer. Your final answer here should be $7,047.87.)
Note that the sum of your frequencies should add up to the length of the timeline (5+4+1+6 = 16). If not, you miscounted somewhere along the way.
EXAMPLE NINE – Uneven Cash Flow Stream – Future Value
We can use a similar process to solve for the future value of an uneven cash flow stream. However, we will start by doing the exact same steps we did to get the present value. The reason is that to get the future value of an uneven cash flow stream we first (A) solve for the present value of the cash flow stream and then (B) figure out what that value will grow to over the time horizon. So, if the problem would have given you the same cash flow stream as above, but instead asked what it would be worth as of year 16 (the end of the time horizon). As we found above, the present value of the cash flow stream (what is worth today) is $7047.87. So, if we want to know what the cash flow stream is worth in year 16, we just bring the present value ($7047.87) forward 16 years at the 6.5% rate of return using the 5 key approach as follows:
16 N6.5 I/Y7047.87 PV0 PMTFV -19,304.19
This tells us that the value of the cash flows will grow to $19,304.19 at the end of the 16 year time horizon if we can invest them to earn a 6.5% rate of return. Note that the PV of an uneven cash flow stream will always be less than the sum of all the individual cash flows ($13,200 in this example) and the FV of an uneven cash flow stream will always be more than the sum of all the individual cash flows.
EXAMPLE TEN – Uneven Cash Flow Stream – Rate of Return
Assume you could buy the cash flow stream in this example for $6000 today. Based on this, what would your rate of return be? To do this, we will use the IRR function on the calculator with the cash flow worksheet. This is similar to what we did above for NPV with two major changes. First, our CF0 is now the initial investment (-6000) and is negative because it is a cash flow. Second, instead of entering the discount rate and then solving for NPV, we solve for the discount rate by press the IRR/YR button. So, it looks like this:
SHIFT C ALL
0 CFj
400 CFj
5 SHIFT Nj
600 CFj
4 SHIFT Nj
1000 CFj
1300 CFj
6 SHIFT Nj
SHIFT IRR/YR (this calculates the answer. Your final answer here should be 8.39%.)
One of the most common mistakes we see on this type of problem is people putting the CF0 in as a positive value which will result in an error (no solution). The 8.39% represents the average annualized rate of return we earn over the 16 year time horizon on our $6000 investment.
PRACTICE PROBLEMS EIGHT, NINE AND TEN
Assuming a 12.5% discount rate, solve for the present value and future value of the following time line. Also, assuming you could buy the cash flow stream for $80,000, what would your rate of return be?
Video: Uneven Cash Flows (HP10BII)
EXAMPLE ELEVEN – Effective Annual Rate
You are offered the choice of 7.8% compounded quarterly or 7.6% compounded daily. Which is a better investment (assuming both have the same risk)? In order to address whether we are better with the higher interest rate compounded less frequently or the lower interest rate compounded more frequently, we need to make them stable comparison by converting both to their annual compounding equivalent. We do this with the effective annual rate. It can be done with a formula or your financial calculator. If we use the formula, it looks like this:
[latex]k_{eff}=\Big(1+\frac{k_{nom}}{m}\Big)^m-1[/latex]
where
keff represents the annual equivalent knom represents the nominal or stated interest rate m represents the number of compounding periods per year
Plugging in our values for the 7.8% compounded quarterly we would get:
[latex]k_{eff}=\Big(1+\frac{0.078}{4}\Big)^4-1[/latex][latex]k_{eff}=(1.0195)^4-1[/latex][latex]k_{eff}=1.0803-1[/latex][latex]k_{eff}=0.0803[/latex][latex]k_{eff}=8.03\%[/latex]
And for the 7.6% compounded daily we would get:
[latex]k_{eff}=\Big(1+\frac{0.076}{365}\Big)^{365}-1[/latex][latex]k_{eff}=(1.000208219)^{365}-1[/latex][latex]k_{eff}=1.0790-1[/latex][latex]k_{eff}=0.0790[/latex][latex]k_{eff}=7.90\%[/latex]
In this case, the 7.8% compounded quarterly is better. If using the formulas, be sure to (A) carry out your calculations to several decimal places (or better yet, don’t round at all until you are done), (B) plug in rates into the formula as decimals, and (C) round your final answer to 2 decimal places in percentage terms. You can also do this with the financial calculator as follows:
4 SHIFT P/YR (this sets your periods per year to 4 for quarterly compounding. Be careful here as this means all your time value of money calculations will use 4 periods per year until you change your P/YR again…just like if you changed the P/YR for a 5-key problem.)
7.8 SHIFT NOM% (this enters the 7.8% nominal rate)
SHIFT EFF% (this solves for the effective annual rate to generate your final answer of 8.03%)
For the 7.6% compounded daily it is:
365 SHIFT P/YR7.6 SHIFT NOM%SHIFT EFF%
This will give you your answer of 7.90% indicating that it is better to take 7.8% compounded quarterly than 7.6% compounded daily.
PRACTICE PROBLEM ELEVEN
You are offered investments of 12% compounded annually, 11.75% compounded quarterly, or 11.5% compounded weekly. Assuming the same risk, which would you prefer?
Video: Effective Annual Rate (HP10BII)
Practice Problem Solutions
Practice Problem 1
45 N9.5 I/Y400 PV0 PMTFV $23,751.74
Practice Problem 2
20 N5 I/Y0 PMT100,000 FVPV $37,688.95
Practice Problem 3
30 N10 I/Y1000 PMT0 FVPV $9,426.91
Practice Problem 4A
30 N10 I/Y0 PV1,000,000 FVPMT $6,079.25
Practice Problem 4B
40 N10 I/Y0 PV1,000,000 FVPMT $2,259.41
Practice Problem 4C
30 N7.5 I/Y0 PV1,000,000 FVPMT $9,671.24
Practice Problem 4D
30 N5 I/Y0 PV1,000,000 FVPMT $15,051.44
Practice Problem 5
Set Calculator to 52 Periods Per Year ⇒ 52 SHIFT P/YRCalculate Number of Periods ⇒ 52 x 30 = 15601560 N10 I/Y0 PV1,000,000 FVPMT $101.07
Remember to set your calculator back to 1 period per year when you finish the calculation.
Annual Savings Required to Accumulate $1,000,000 in 30 years at 10%A) Saving at the end of each year ⇒ $6,079.25B) Saving at end of every 2 weeks ⇒ $202.75 x 26 = $5,271.50C) Saving at end of each week ⇒ $101.07 x 52 = $5,255.64
The more frequently we make contributions, the less we have to save each year. This is because of the compounding effect. When we make annual contributions, we earn no return the first year. With weekly contributions we start earning a return during the second week.
Practice Problem 6
35 N0 PV-4500 PMT1,000,000 FVI/Y 9.13%
Practice Problem 7
420 N9 I/Y-20,000 PV1,000,000 FVPMT $183.13
Practice Problem 8
SHIFT C ALL0 CFj 10000 CFj 3 Nj 12000 CFj 5 Nj 5000 CFj 2 Nj 8000 CFj 10000 CFj 2 Nj 12.5 I/YR SHIFT NPV ⇒ $63,878.58
Practice Problem 9
Step 1: Solve for Present Value (See solution to 8) ⇒ $63,878.58Step 2: Bring forward to end of year 1313 N12.5 I/Y63,878.58 PV0 PMTFV $295,350.73
Practice Problem 10
SHIFT C ALL-80000 CFj 10000 CFj 3 Nj 12000 CFj 5 Nj 5000 CFj 2 Nj 8000 CFj 10000 CFj 2 Nj SHIFT IRR/YR ⇒ 7.93%
Practice Problem 11
Since the 12% compounded annual is already annual, there is no need for an effective annual rate.
The 11.75% compounded quarterly is4 SHIFT P/YR11.75 SHIFT NOM%SHIFT EFF% ⇒ 12.28%
The 11.5% compounded daily is365 SHIFT P/YR11.5 SHIFT NOM%SHIFT EFF ⇒ 12.17%
Making the 11.75% quarterly the best deal.
License
Business Finance Essentials Copyright © 2018 by Dr. Kevin Bracker is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.
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7073 | https://en.wikipedia.org/wiki/Plane%E2%80%93plane_intersection | Plane–plane intersection - Wikipedia
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From Wikipedia, the free encyclopedia
Two intersecting planes in three-dimensional space
In analytic geometry, the intersection of two planes in three-dimensional space is a line.
Formulation
[edit]
The line of intersection between two planes Π 1:n 1⋅r=h 1{\displaystyle \Pi {1}:{\boldsymbol {n}}{1}\cdot {\boldsymbol {r}}=h_{1}} and Π 2:n 2⋅r=h 2{\displaystyle \Pi {2}:{\boldsymbol {n}}{2}\cdot {\boldsymbol {r}}=h_{2}} where n i{\displaystyle {\boldsymbol {n}}_{i}} are normalized is given by
r=(c 1 n 1+c 2 n 2)+λ(n 1×n 2){\displaystyle {\boldsymbol {r}}=(c_{1}{\boldsymbol {n}}{1}+c{2}{\boldsymbol {n}}{2})+\lambda ({\boldsymbol {n}}{1}\times {\boldsymbol {n}}_{2})}
where
c 1=h 1−h 2(n 1⋅n 2)1−(n 1⋅n 2)2{\displaystyle c_{1}={\frac {h_{1}-h_{2}({\boldsymbol {n}}{1}\cdot {\boldsymbol {n}}{2})}{1-({\boldsymbol {n}}{1}\cdot {\boldsymbol {n}}{2})^{2}}}}c 2=h 2−h 1(n 1⋅n 2)1−(n 1⋅n 2)2.{\displaystyle c_{2}={\frac {h_{2}-h_{1}({\boldsymbol {n}}{1}\cdot {\boldsymbol {n}}{2})}{1-({\boldsymbol {n}}{1}\cdot {\boldsymbol {n}}{2})^{2}}}.}
Derivation
[edit]
This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product n 1×n 2{\displaystyle {\boldsymbol {n}}{1}\times {\boldsymbol {n}}{2}} (this cross product is zero if and only if the planes are parallel, and are therefore non-intersecting or entirely coincident).
The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in space may be written as r=c 1 n 1+c 2 n 2+λ(n 1×n 2){\displaystyle {\boldsymbol {r}}=c_{1}{\boldsymbol {n}}{1}+c{2}{\boldsymbol {n}}{2}+\lambda ({\boldsymbol {n}}{1}\times {\boldsymbol {n}}{2})}, since {n 1,n 2,(n 1×n 2)}{\displaystyle {{\boldsymbol {n}}{1},{\boldsymbol {n}}{2},({\boldsymbol {n}}{1}\times {\boldsymbol {n}}{2})}} is a basis. We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for c 1{\displaystyle c{1}} and c 2{\displaystyle c_{2}}.
If we further assume that n 1{\displaystyle {\boldsymbol {n}}{1}} and n 2{\displaystyle {\boldsymbol {n}}{2}} are orthonormal then the closest point on the line of intersection to the origin is r 0=h 1 n 1+h 2 n 2{\displaystyle {\boldsymbol {r}}{0}=h{1}{\boldsymbol {n}}{1}+h{2}{\boldsymbol {n}}_{2}}. If that is not the case, then a more complex procedure must be used.
Dihedral angle
[edit]
Main article: Dihedral angle
Given two intersecting planes described by Π 1:a 1 x+b 1 y+c 1 z+d 1=0{\displaystyle \Pi {1}:a{1}x+b_{1}y+c_{1}z+d_{1}=0} and Π 2:a 2 x+b 2 y+c 2 z+d 2=0{\displaystyle \Pi {2}:a{2}x+b_{2}y+c_{2}z+d_{2}=0}, the dihedral angle between them is defined to be the angle α{\displaystyle \alpha } between their normal directions:
cosα=n^1⋅n^2|n^1||n^2|=a 1 a 2+b 1 b 2+c 1 c 2 a 1 2+b 1 2+c 1 2 a 2 2+b 2 2+c 2 2.{\displaystyle \cos \alpha ={\frac {{\hat {n}}{1}\cdot {\hat {n}}{2}}{|{\hat {n}}{1}||{\hat {n}}{2}|}}={\frac {a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}.}
References
[edit]
^Plane-Plane Intersection - from Wolfram MathWorld. Mathworld.wolfram.com. Retrieved 2013-08-20.
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Find smallest set of natural numbers whose pairwise sums include 0..n
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$\begingroup$
Given a positive integer $n$, how do you find the smallest set of nonnegative integers $S$ such that for each integer $m$, where $0\leq m
For example, consider the case $n=50$. Suppose the length of $S$ is $L$. For a lower bound, if the elements of $S$ have pairwise distinct sums, then there are $\dbinom{L+1}{2}$ sums (the plus 1 is because numbers can be added to themselves). Thus, $$\binom{L+1}{2}\geq50\implies L\geq10$$.
I can acheive $L=12$ with the set {0, 1, 2, 3, 7, 10, 15, 18, 22, 23, 24, 25} (done with very inefficient program which searches randomly among all sets). For $L=10$, I feel like it should be impossible; we only have to show that more than 5 numbers can be expressed as a sum in more than 1 way, which should be able to be done through some casework. However, is $L=11$ possible? I think so.
Similarly, for $n=100$, I have $L=17$ from my program: {0, 1, 3, 4, 9, 11, 16, 20, 25, 30, 34, 39, 41, 46, 47, 49, 50}. But the lower bound only gives $L\geq 14$, so at least $L=15$ or $L=16$ should be possible.
In general, how do you do it efficiently for any given $n$?
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asked Jun 28, 2018 at 23:38
soktinpksoktinpk
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$\begingroup$ I just realized that both my answer and Fabio Lucchini's (perhaps following my mistake) shifted your $n$ by one; we assumed that $n$ itself also needs to be generated. This is also how $n$ is defined in OEIS A066063. Usually I'd correct the mistake in my answer, but since the OEIS sequence and two long answers now all use the same convention, might I perhaps ask you to change the question slightly to match it? Your examples would still be valid, since in both cases you can also form $n$ itself; all you'd have to change is $m\le n$ and $\binom{L+1}2\gt50$. $\endgroup$
joriki
– joriki
2018-07-01 08:10:44 +00:00
Commented Jul 1, 2018 at 8:10
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3 Answers 3
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$\begingroup$
Here are results for $n$ up to $80$, where $\min$ is the lower bound you derive from the binomial coefficient, $\max$ is the upper bound that Fabio Lucchini derived in his answer, and $L=|S|$ is the size of a minimal generating set. Actual subsets $S$ are only shown for the last entry for any given $L$, since this subset also works for all smaller $n$.
\begin{array}{c|c|c|c|l} n&\min&\max&L&S\\hline 2&2&2&2&{0,1}\ 3&3&3&3\ 4&3&3&3&{0,1,2}\ 5&3&4&4\ 6&4&4&4\ 7&4&4&4\ 8&4&4&4&{0,1,3,4}\ 9&4&5&5\ 10&5&5&5\ 11&5&5&5\ 12&5&5&5&{0,1,3,5,6}\ 13&5&6&6\ 14&5&6&6\ 15&6&6&6\ 16&6&6&6&{0,1,3,5,7,8}\ 17&6&7&7\ 18&6&7&7\ 19&6&7&7\ 20&6&7&7&{0,1,2,5,8,9,10}\ 21&7&8&8\ 22&7&8&8\ 23&7&8&8\ 24&7&8&8\ 25&7&8&8\ 26&7&8&8&{0,1,2,5,8,11,12,13}\ 27&7&9&9\ 28&8&9&9\ 29&8&9&9\ 30&8&9&9\ 31&8&9&9\ 32&8&9&9&{0,1,2,5,8,11,14,15,16}\ 33&8&10&10\ 34&8&10&10\ 35&8&10&10\ 36&9&10&10\ 37&9&10&10\ 38&9&10&10\ 39&9&11&10\ 40&9&11&10&{0,1,3,4,9,11,16,17,19,20}\ 41&9&11&11\ 42&9&11&11\ 43&9&11&11\ 44&9&11&11\ 45&10&12&11\ 46&10&12&11&{0,1,2,3,7,11,15,19,21,22,24}\ 47&10&12&12\ 48&10&12&12\ 49&10&12&12\ 50&10&12&12\ 51&10&12&12\ 52&10&12&12\ 53&10&13&12\ 54&10&13&12&{0,1,2,3,7,11,15,19,23,25,26,28}\ 55&11&13&13\ 56&11&13&13\ 57&11&13&13\ 58&11&13&13\ 59&11&13&13\ 60&11&13&13\ 61&11&14&13\ 62&11&14&13\ 63&11&14&13\ 64&11&14&13&{0,1,3,4,9,11,16,21,23,28,29,31,32}\ 65&11&14&14&\ 66&12&14&14&\ 67&12&14&14&\ 68&12&14&14&\ 69&12&15&14&\ 70&12&15&14&\ 71&12&15&14&\ 72&12&15&14&{0,1,3,4,9,11,16,20,25,27,32,33,35,36}\ 73&12&15&15&\ 74&12&15&15&\ 75&12&15&15&\ 76&12&15&15&\ 77&12&16&15&\ 78&13&16&15&\ 79&13&16&15&\ 80&13&16&15&{0,1,3,4,5,8,14,20,26,32,35,36,37,39,40}\ \end{array}
Here's the code I used to generate these results. It loops over $n$, making use of the solution for $n-1$ in each step. It first checks whether the set for $n-1$ also works for $n$. If not, it tries finding a new set also containig $L$ numbers, at first using only elements up to $\lfloor\frac n2\rfloor + 2$. Only if that doesn't work does it try all combinations with $L$ elements all the way up to $n$. If that also doesn't work, it increases $L$. This way, it spends almost all its time only on the values of $n$ where $L$ needs to be incremented; for all other values of $n$ it quickly finds a solution without having to search the entire space.
The least $n$ for which Fabio Lucchini's upper bound is not tight is $n=39$, for which the $10$-element set ${0,1,3,4,9,11,16,17,19,20}$ is sufficient whereas the upper bound is $11$.
The sequence $L(n)$ is OEIS A066063, and the only information in that entry is the lower bound you already found. Usually, OEIS is quite good at collecting information about sequences, so it's likely that nothing else was known.
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edited Jul 4, 2018 at 13:11
answered Jun 29, 2018 at 7:10
jorikijoriki
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$\begingroup$ Indeed it looks like when $n$ is a multiple of 4, the set has symmetry around $n/4$. This is true even for my possibly suboptimal set for $n=100$. 0 and 1 must be included since if 0 weren't there, there would be no way to make 0, and if the next smallest number after 0 wasn't 1, (say it was 0, 2 or 0, 3) there would be no way to make 1. Then $n/2$ and $n/2-1$ would follow from symmetry if that were proven. Also, there may be a generalization of the symmetry to non multiples of 4. $\endgroup$
soktinpk
– soktinpk
2018-06-29 12:44:20 +00:00
Commented Jun 29, 2018 at 12:44
$\begingroup$ @soktinpk: I've updated the answer with results up to $n=64$. $\endgroup$
joriki
– joriki
2018-07-01 12:43:24 +00:00
Commented Jul 1, 2018 at 12:43
$\begingroup$ @soktinpk: I've updated the answer with results up to $n=80$. $\endgroup$
joriki
– joriki
2018-07-04 13:11:53 +00:00
Commented Jul 4, 2018 at 13:11
Add a comment |
3
$\begingroup$
Let $n+4=s^2+r$ with $r,s\in\Bbb N$ and $0\leq r\leq 2s$. Then an upper bound is given by $$L\leq \begin{cases} \lceil{\frac rs}\rceil+2s-3&2\mid s\ \lceil{\frac{r+1}{s+1}}\rceil+2s-3&2\nmid s \end{cases}$$ which gives $L\leq 12$ for $n=50$ and $L\leq 18$ for $n=100$. In general, for large $n$ this gives $L=O(\sqrt n)$.
A set $S$ corresponding to this upper bound is given by \begin{align} S &={i\in\Bbb N:0\leq i
If $n+4=s^2+r$ and $s=2t+b$ with $r\leq 2s$ and $0\leq b\leq 1$, then $$t\leq\frac{\sqrt{n+4}}2
For $q=t$ we get \begin{align} |S_t| &=\left\lceil\frac{n+4}{2t}\right\rceil+2t-3\ &=\left\lceil\frac{s^2+r}{s-b}\right\rceil+s-b-3\ &=\left\lceil\frac{r+b}{s-b}\right\rceil+2s-3 \end{align} while for $q=t+1$ \begin{align} |S_{t+1}| &=\left\lceil\frac{n+4}{2t+2}\right\rceil+2t+2-3\ &=\left\lceil\frac{s^2+r}{s-b+2}\right\rceil+s-b-3\ &=\left\lceil\frac{r+4-3b}{s+2-b}\right\rceil+2s-3 \end{align} Since $|S_t|\leq|S_{t+1}|$ if and only if $b(2s+1)\leq 2s-r$, the formula on the top is proved.
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edited Jun 29, 2018 at 12:25
answered Jun 29, 2018 at 7:39
Fabio LucchiniFabio Lucchini
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$\begingroup$ You are right, I made a typo; the answer is now edited. $\endgroup$
Fabio Lucchini
– Fabio Lucchini
2018-06-29 09:27:33 +00:00
Commented Jun 29, 2018 at 9:27
1
$\begingroup$ I see. I think the expression $\lceil{\frac{r-1}{s+1}}\rceil+2s-3$ in the odd case is slightly off. For $n=32$, both my results and your analysis further down show that $9$ elements are required ($q=3$, $k=5$), but that expression yields an upper bound of $8$. $\endgroup$
joriki
– joriki
2018-06-29 09:55:20 +00:00
Commented Jun 29, 2018 at 9:55
1
$\begingroup$ Yes, edited another typo. Thank'you. Now for $n=32$ we get $s=6$ and $r=0$ which gives $L\leq 9$. $\endgroup$
Fabio Lucchini
– Fabio Lucchini
2018-06-29 10:01:29 +00:00
Commented Jun 29, 2018 at 10:01
1
$\begingroup$ Unfortunately, this bound is not minimal: for $n=100$ it gives $|S|=18$, while in the OP soktinpk find a solution with $|S|=17$. $\endgroup$
Fabio Lucchini
– Fabio Lucchini
2018-06-29 13:00:59 +00:00
Commented Jun 29, 2018 at 13:00
1
$\begingroup$ Ah, yes, I didn't check that case. I'll try to find the first $n$ at which it's not minimal. Anyway, it's a very good bound, we should perhaps enter it at OEIS. $\endgroup$
joriki
– joriki
2018-06-29 13:01:44 +00:00
Commented Jun 29, 2018 at 13:01
| Show 6 more comments
1
$\begingroup$
Here is the casework to show that for $n=50$, $L$ indeed has to be greater than $10$, assuming joriki's suggestion:
all minimal solutions contain $0$,$1$,$\lceil\frac n2\rceil-1$ and $\lceil\frac n2\rceil$
Since ${11 \choose 2} =55$, we just need to show that we can get at least $6$ 'matches' for any set of numbers.
OK, so by joriki's suggestion, we need $0,1,24,25$. So immediately we have one match: $24+1=25+0$
Now for the cases:
I) If we add $2$, we have two more: $1+1=2+0$ and $24+2=25+1$
To make 47, we need to either add $23$ or $22$:
I.A) Add $23$. Then we can add $23+1=24+0$, $24+2=25+1$, and $23+25=24+24$, so we have our $6$ matches
I.B) Add $22$. We can add $22+2=24+0$ (4 matches now)
To make $5$, we need to add $3$, $4$, or $5$:
I.B.i) Add $3$: $2+1=3+0$ and $24+3=25+2$ Got our $6$
I.B.ii) Add $4$: $2+2=4+0$ and $22+4=24+2$. Also $6$
I.B.iii) Add $5$: $22+5=25+2$. OK, need one more ... well, to create $45$ we need to add either $20$ (which gives $20+2=22+0$) or $21$ ($21+1=22+0$), so done here as well
OK, so adding $2$ definitely leads to $6$ matches. So, let's not add $2$ ... but now we need $3$ to create $3$:
II) Add $3$
Again, to create $47$, we have to add either $23$ or $22$:
II.A) Add $23$. We can add $24+24=23+25$, $23+1=24+0$, and $23+3=25+1$, so 4 matches now.
To create $5$, we need to add either $4$ or $5$:
II.A.i) Add $4$. Then we have $3+1=4+0$ and $23+4=25+3$. Done
II.A.ii) Add $5$. We have $5+1=3+3$ and $23+5=25+3$ Done
II.B) Add $22$. We can add $22+3=25+0$ and $22+3=24+1$ (3 matches now)
To make $5$, we need tpo add $4$ or $5$:
II.B.i) Add $4$. This gives $1+3=4+0$, $24+4=25+3$, and $22+4=25+1$ Done
II.B.ii) Add $5$. This gives $1+5=3+3$, $22+5=24+3$, so need just one more. But to create $45$ we need to add $21$ ($21+1=22+0$), or $20$ ($20+3=22+1$), so done here as well
.. and that concludes all cases. So, yes, as you suspected, for $n=25$ we have $L>10$
And frankly, seeing how quickly these possible matches increase, my guess is that for $n=50$, $L=12$
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edited Jun 29, 2018 at 14:36
answered Jun 29, 2018 at 14:28
Bram28Bram28
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UCK Word Family List
Using word families can help teach beginning spellers that words contain and share patterns. Use this word family list to introduce the sound of words ending with -UCK. Words: buck, puck, yuck, duck, luck, suck, muck, tuck, chuck, snuck, cluck, stuck, pluck, truck, struck, potluck.
Printing options: full color and black lines.
Download printableSKU: WS-625017 Category: Worksheets
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1st GradeKindergarten
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GrammarPhonicsShort UShort VowelsUCK Word FamilyWord Families
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OT Word Family List
Teach beginning spellers that words contain and share patterns. Use this word family list to introduce the sound of words ending with -OT.
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| Fact families and basic addition and subtraction facts This article explains how to use fact families to help children learn the basic addition and subtraction facts (with single-digit numbers), and also contains a complete example lesson with exercises and word problems about fact families where the sum is 13 or 14. What is a fact family? A fact family is a group of math facts using the same numbers. In the case of addition/subtraction, you use three numbers and get four facts. For example, you can form a fact family using the three numbers 10, 2, and 12: 10 + 2 = 12, 2 + 10 = 12, 12 − 10 = 2, and 12 − 2 = 10. Where do we use fact families? We can use fact families to reinforce or learn the connection between addition and subtraction, and to help children memorize the basic addition & subtraction facts. The two videos below explain several strategies for learning addition and subtraction facts, including number rainbows and fact families. For some extra practice, check out IXL's fact families lesson! A Lesson on Fact Families - sums with 13 and 14 1. Fill in. In each fact family, color the marbles so they match the numbers in it. | | | Fact families with 13 | | | | | | --- | 10, 3, and 13 | 10 + 3 = _____ 3 + 10 = _____ | 13 − 10 = _____ 13 − 3 = _____ | | | | | | | --- | 9, _____, and 13 | 9 + ____ = 13 ____ + ____ = _____ | _____ − ____ = ____ _____ − ____ = ____ | | | | | | | --- | 8, ____, and 13 | 8 + ____ = 13 ____ + ____ = _____ | _____ − ____ = ____ _____ − ____ = ____ | | | | | | | --- | 7, ____, and 13 | 7 + ____ = 13 ____ + ____ = _____ | _____ − ____ = ____ _____ − ____ = ____ | | 2. Connect with a line the problems that are from the same fact family. You don't need to write the answers. | | | | | | --- --- | 13 − 7 = 5 + = 12 11 − 3 = 8 + = 13 12 − 3 = 7 + = 11 | | 11 − 4 = 11 − 8 = 5 + = 13 12 − 5 = 6 + = 13 9 + = 12 | | 12 − 7 = 13 − 6 = 3 + = 12 13 − 5 = 3 + = 11 4 + = 11 | 3. Fill in. In each fact family, color the marbles so they match the numbers in it. | | | Fact families with 14 | | | | | | --- | 10, 4, and 14 | 10 + 4 = _____ 4 + 10 = _____ | 14 − 10 = _____ 14 − 4 = _____ | | | | | | | --- | 9, _____, and 14 | 9 + ____ = 14 ____ + ____ = _____ | _____ − ____ = ____ _____ − ____ = ____ | | | | | | | --- | 8, ____, and 14 | 8 + ____ = 14 ____ + ____ = _____ | _____ − ____ = ____ _____ − ____ = ____ | | | | | | | --- | 7, ____, and 14 | 7 + ____ = 14 ____ + ____ = _____ | _____ − ____ = ____ _____ − ____ = ____ | | 4. Subtract. | | | | | --- --- | | a. 13 − 8 = ____ 14 − 6 = ____ | b. 13 − 5 = ____ 13 − 4 = ____ | c. 12 − 7 = ____ 13 − 7 = ____ | d. 12 − 9 = ____ 14 − 9 = ____ | 5. Find the missing numbers. | | | | --- | a. 9 + = 14 | b. 6 + = 14 | c. 6 + = 12 | | d. − 9 = 4 | e. − 7 = 7 | f. − 9 = 3 | | g. 14 − = 8 | h. 12 − = 7 | i. 13 − = 8 | 6. Solve the word problems. | | | a. Ted arranged his toy cars in rows. The first row had seven cars, the second had seven, and the third row had four. How many cars does Ted have? | | b. If you have 14 strawberries and I have eight, how many more do you have? | | c. Dad has six cherries and Mom has five more than him. How many cherries does Mom have? | | d. At first Mom had 20 apples to make a pie, but she gave each of the four children one apple before she made the pie. How many apples did she have left for the pie? | 7. Figure out the patterns and continue them! | | | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- --- --- | | a. | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | 40 | | | 64 | 72 | _____ | _____ | _____ | _____ | | | | | | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- --- --- --- | | b. | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | | --- | | + | | | | | 17 | 21 | 25 | 29 | _____ | _____ | _____ | _____ | _____ | This lesson is taken from Maria Miller's book Math Mammoth Add & Subtract 2A, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller. Math Lessons menu | | | | --- | Place Value Grade 1+ Using a 100-bead abacus in elementary math + Teaching tens and ones + Practicing with two-digit numbers + Counting in groups of ten + Skip-counting practice (0-100) + Comparing 2-digit numbers + Cents and dimes Grade 2+ Three-digit numbers + Comparing 3-digit numbers Grade 3+ Place value with thousands + Comparing 4-digit numbers + Rounding & estimating + Rounding to the nearest 100 Grade 4+ Place value - big numbers | Add & subtract lessons Grade 1+ Missing addend concept (0-10) + Addition facts when the sum is 6 + Addition & subtraction connection Grade 2+ Fact families & basic addition/subtraction facts + Sums that go over over the next ten + Add/subtract whole tens (0-100) + Add a 2-digit number and a single-digit number mentally + Add 2-digit numbers mentally + Regrouping in addition + Regrouping twice in addition + Regrouping or borrowing in subtraction Grade 3+ Mental subtraction strategies + Rounding & estimating | Multiplication Grade 3+ Multiplication concept as repeated addition + Multiplication on number line + Commutative + Multiply by zero + Word problems + Order of operations + Structured drill for multiplication tables + Drilling tables of 2, 3, 5, or 10 + Drilling tables of 4, 11, 9 Grade 4+ Multiplying by whole tens & hundreds + Distributive property + Partial products - the easy way + Partial products - video lesson + Multiplication algorithm + Multiplication Algorithm — Two-Digit Multiplier + Scales problems - video lesson + Estimation when multiplying | | Division Grade 3+ Division as making groups + Division/multiplication connection + Division is repeated subtraction + Zero in division + Division that is not exact (remainder) + Divisibility Grade 4+ How to teach long division + Long division as repeated subtraction + Why long division works + Zero in dividend + Remainder & long division + Two-digit divisor + Review of division topics Divisibility+ Divisibility within 0-1000 + Divisibility rules + Prime factorization 1 + Prime factorization 2 + Sieve of Eratosthenes | Fraction Lessons + Understanding fractions + Finding fractional parts with division + Mixed numbers + Fractions to mixed numbers and vv. + Adding like fractions + Equivalent fractions + Adding unlike fractions 1 + Adding unlike fractions 2: Finding the common denominator + Adding mixed numbers + Subtracting mixed numbers + Subtracting mixed numbers 2 + Measuring in inches + Comparing fractions + Simplifying fractions + Multiply fractions by whole numbers + Multiply fractions by fractions + Multiplication and area + Simplify before multiplying + Dividing fractions by whole numbers + Dividing fractions: fitting the divisor + Dividing fractions: reciprocal numbers + Dividing fractions: using the shortcut | Geometry Lessons + Lines, rays, and angles + Measuring angles + Parallel & perpendicular + Acute, obtuse, and right triangles + Angle sum of a triangle + Equilateral & isosceles triangles + Circles + Symmetry + Altitude of a triangle + Polygons + Perimeter + Area of rectangles + Area of right triangles + Area of parallelograms + Area of triangles + Area versus Perimeter + Angles in Polygons (PDF) + Review: Area of Polygons (PDF) + Surface Area (PDF) | | Decimals Lessons + Decimals videos + Decimals (1 decimal digit) + Decimal place value (1 decimal digit) + Decimals (2 decimal digits) + Decimal place value (2 decimal digits) + Decimals (3 decimal digits) + Add & subtract (1 decimal digit) + Add & subtract (2 decimal digits) + Add and subtract decimals — practice + Comparing decimals + Multiply a decimal by a whole number + Multiply decimals by decimals + Divide decimals—mental math + Divide decimals by decimals + Multiply and divide decimals by 10, 100, and 1000 + Decimals review lesson | Percents Lessons + How to teach proportions + Percent - the basic concept + Percentage of a number—mental math + How to calculate a percentage of a number + How to calculate percentages + Basics of percent of change | General + Four habits of highly effective math teaching + Why are math word problems SO difficult for children? Hint: it has to do with a "recipe" that many math lessons follow. + The do's and don'ts of teaching problem solving in mathAdvice on how you can teach problem solving in elementary, middle, and high school math. + How to set up algebraic equations to match word problems Students often have problems setting up an equation for a word problem in algebra. To do that, they need to see the RELATIONSHIP between the different quantities in the problem. This article explains some of those relationships. + Seven reasons behind math anxiety and how to prevent it + Mental math "mathemagic" with Arthur Benjamin (video) + Keeping math skills sharp in the summer + Geometric vanish puzzles + Science resources Short reviews of the various science resources and curricula I have used with my own children. | | | |
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Transformation of general equation of straight line to normal form
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Consider the general form of equation of straight line.$$ax + by + c = 0$$ $$\Rightarrow ax + by = -c$$ $$\Rightarrow \frac{ax}{-c} + \frac{by}{-c} = 1$$ $$\Rightarrow \frac{x}{\frac {-c}{a}} + \frac{y}{\frac {-c}{b}} =1 $$this is equation of line in intercept form with x-intercept $\frac {-c}{a}$ and y-intercept $ \frac {-c}{b}$
In above fig. $\overline {OD}\bot\overline {AB}$Now consider $\Delta$OAD$$Cos\alpha =\frac {\overline{OD}}{\overline{OA}} = \frac{P}{\frac {-c}{a}} =\frac {-aP}{c} $$In $\Delta$OBD$$Cos(90-\alpha)=\frac {\overline{OD}}{\overline{OB}} = \frac{P}{\frac {-c}{b}} =\frac {-bP}{c} $$Since Cos($90-\alpha$)= Sin $\alpha$ Therefore,$$Sin\alpha = \frac {-bP}{c}$$Now$$tan\alpha =\frac {sin \alpha}{cos\alpha} = (\frac {-bP}{c})(\frac{-c}{aP})$$$$tan \alpha = \frac {b}{c}=> \frac {perpendicular}{base}$$$$\Rightarrow hyp = \pm\sqrt{a^2+b^2} (by Pythagorean theorem).........(a)$$ $$\Rightarrow cos\alpha = \frac {a}{\pm\sqrt {a^2+b^2}},sin\alpha = \frac {b}{\pm\sqrt {a^2+b^2}}...............(1)$$ Now for transforming general equation of straight line to normal form:Consider the general form of equation of straight line $$ax+by+c=0$$ $$\Rightarrow ax+by=-c$$ Now in order to to have Normal form of equation of straight line it is said that divide both sides by $\pm\sqrt{a^2+b^2}$, then we get $$\frac {a}{\pm\sqrt{a^2+b^2}}x+\frac {b}{\pm\sqrt{a^2+b^2}}y=\frac {-c}{\pm\sqrt{a^2+b^2}}$$ Using equations (1) in above we get $$xcos\alpha+ysin\alpha=\frac {-c}{\pm\sqrt{a^2+b^2}}$$ Now comparing with normal equation of straight line i.e.$xcos\alpha+ysin\alpha=P$ $$\implies P=\frac {-c}{\pm\sqrt{a^2+b^2}}$$ Now the question is:1: In equation (a) $hyp = \pm\sqrt{a^2+b^2}$ but according to figure in $\Delta OAD hyp=\overline{OA}= \frac{-c}{a}$ How $\pm\sqrt{a^2+b^2}=\frac{-c}{a}$?
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asked Feb 20, 2017 at 16:33
Ali JanAli Jan
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$\begingroup$ What makes you think that $\triangle{OAD}$ and whatever triangle has a hypotenuse of $\sqrt{a^2+b^2}$ are congruent? $\endgroup$
amd
– amd
2017-02-20 22:00:07 +00:00
Commented Feb 20, 2017 at 22:00
$\begingroup$ In general why it is said to divide ax+by+c=0 by $\pm\sqrt {a^2 + b^2}$ to convert to normal form? I mean where does this factor came from, I am really confused with this term. $\endgroup$
Ali Jan
– Ali Jan
2017-02-23 08:36:45 +00:00
Commented Feb 23, 2017 at 8:36
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The general equation $ax+by+c=0$ of a straight line in $\mathbb R^2$ can be rewrtten as $\mathbf n\cdot\mathbf x=-c$, with $\mathbf n\ne0$. This form of the equation tells us that a line can be characterized as the set of points that have the same dot product with some fixed vector $\mathbf n$. If we normalize $\mathbf n$ by dividing both sides of the equation by its length, we have $${\mathbf n\over\|\mathbf n\|}\cdot\mathbf x=-{c\over\|\mathbf n\|}.$$ This is the so-called normal form of the equation for the line, characterized by $\mathbf n$ being a unit vector. The left-hand side of this equation is the (signed) length of the orthogonal projection of $\mathbf x$ onto $\mathbf n$, so an equivalent way to charactere the line is as the set of vectors $\mathbf x$ that have the same projection $\mathbf x_\parallel$ onto $\mathbf n$. The remainder $\mathbf x-\mathbf x_\parallel$ is perpendicular to $\mathbf n$, from which we can see that $\mathbf n$ is perpendicular to the line, and so the (perpendicular) distance of the line from the origin is ${|c|\over\|\mathbf n\|}$. Substituting the coefficients from the original equation yields the normal equation $${a\over\sqrt{a^2+b^2}}x+{b\over\sqrt{a^2+b^2}}y={-c\over\sqrt{a^2+b^2}}.$$ Since we have a unit normal, we can replace the coefficients on the left-hand side: $$x\cos\alpha+y\sin\alpha={-c\over\sqrt{a^2+b^2}},$$ where $\alpha$ is the angle made by the normal vector with the $x$-axis.
Now, as far as your equation (a) is concerned, remember that in computing $\tan\alpha$, you canceled a constant of proportionality that appeared in both the numerator and denominator, so the most that you can conclude from it is that the length of the hypotenuse of $\triangle{OAD}$ is proportional to $\sqrt{a^2+b^2}$. Indeed, in your diagram you have two other triangles that are similar to $\triangle{OAD}$: $\triangle{BOD}$ and $\triangle{BOA}$, so we have $$\tan\alpha = {\overline{AD}\over\overline{OD}} = {\overline{OD}\over\overline{BD}}={\overline{OA}\over\overline{OB}}.$$ The legs of the largest of these are the hypotenuses of the smaller ones, so there are at least two different lengths of hypotenuse in play. Moreover, it might be that none of the lengths is equal to $\sqrt{a^2+b^2}$!
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answered Feb 24, 2017 at 1:12
amdamd
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2 x 2 invertible matrix
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2 x 2 invertible matrix
Jump to:NotesPrerequisites
Notes
In this section, we will learn about what an invertible matrix is. An invertible matrix is a square matrix that has an inverse. We say that a square matrix is invertible if and only if the determinant is not equal to zero. In other words, a 2 x 2 matrix is only invertible if the determinant of the matrix is not 0. If the determinant is 0, then the matrix is not invertible and has no inverse.
2x2 Invertible matrix
We are about to start a series of lessons dedicated to the inverses of matrices. The topic of today is to learn to identify those matrices which can be inverted and those which can't. On later lessons we will obtain the inverses of different size matrices and how to use them when solving systems of linear equations.
What is an invertible matrix
An invertible matrix, also called a nondegenerate matrix or a nonsingular matrix, is a type of square matrix containing real or complex numbers which is the most common in existence. Its main characteristic is that for an invertible matrix there is always another matrix which multiplied to the first, will produce the identity matrix of the same dimensions as them.
In other words, an invertible matrix is that which has an "inverse" matrix related to it, and if both of them are multiplied together (no matter in which order), the result will be an identity matrix of the same order. To explain this concept a little better let us define a 2x2 matrix (a square matrix of second order) called X. Then, X is said to be an invertible 2x2 matrix if and only if there is an inverse matrix X−1 which multiplied to X produces a 2x2 identity matrix as shown below:
For clarity purposes, let us repeat that in this case the resultant identity matrix I2 is of second order since the matrices producing it are of second order too. In general, we know we can invert a matrix of nxn dimensions which we define as A if the following condition is met:
Keep always in mind that there is a difference between an "invertible matrix" and an "inverted matrix". And invertible matrix is any matrix which has the capacity of being inverted due to the type of determinant it has, while an inverted matrix is one which has already passed through the inversion process. If we look at equation 2, A would be referred as the invertible matrix and A−1 would be the inverted matrix.
Before we pass to the next section where we will learn how to tell if a matrix is invertible and when is a matrix not invertible, let us say something about a non invertible matrix: Remember that a matrix is a rectangular array of ordered coefficients, in other words, it can be taken as an array of information values. We have mentioned before that an invertible matrix is the most common case in existence, in this case we are talking about a continuous uniform distribution of arrays with information values; therefore the denomination of "nonsingular" matrix for a matrix invertible comes from the fact that in such distribution, a study case (a selected array from the distribution) would almost always come to be an invertible array or, invertible matrix. Therefore, a non invertible matrix is called a singular matrix, because is rare to find on an ideal distribution of information.
This last bit of information is essential when studying statistics and probability theory, and although for now we will keep our focus in linear algebra (the topic of this course), it is always important to understand the extent of mathematical concepts throughout different areas of study.
How to determine if a matrix is invertible
So after the introduction above we arrive to the main question of this lesson: When is a matrix invertible? If we define a nxn matrix we say that:
The matrix is invertible if and only if its determinant is different to zero.
In later lessons we will talk about the invertible matrix theorem which gives a series of conditions equivalent to the statement above, that if met, define an invertible matrix.
Is the zero matrix invertible?
Since a matrix is invertible when there is another matrix (its inverse) which multiplied with the first one produces an identity matrix of the same order, a zero matrix cannot be an invertible matrix. If you think about it, no matter which matrix you multiply to a zero matrix, and no matter the order in which the multiplication occurs, the result of such matrix multiplication will always be a zero matrix because all of the element entries in the zero matrix are zeros.
Under the same logic, we can conclude a general rule: any square matrix which contains a complete row or a complete column filled with zeros, cannot be inverted since it cannot produce an identity matrix through matrix multiplication.
Is the identity matrix invertible?
Yes, the identity matrix is invertible. We know what makes a matrix invertible is the fact that there is another matrix out there, which we call the inverse matrix of the original, which multiplied by the original produces the identity matrix as a result. This definition may sound confusing if the matrix we are trying to invert is the identity itself, but simply said, the inverse of the identity matrix is itself, and it can be shown below:
Multiplying an identity matrix by itself produces the identity matrix once more, and so, the invertible matrix definition is met, as can be seen in equation 8.
Such characteristic places the identity matrix into a group of special matrices called involutory matrices. Involution is the name given to functions which are their own inverses, in the case of linear algebra, an involutory matrix is that which multiplied by itself (squaring the matrix) produces the identity matrix, and so, following the concept from general mathematics, an involutory matrix is that which is its own inverse. The identity matrix itself is the main involutory matrix since all of the involutory matrices existent are square roots of it.
Invertible matrix properties
Besides the fact that there is an inverse matrix out there for an invertible matrix to be multiplied with and obtain the same order identity matrix out, you may be wondering: what does it mean for a matrix to be invertible?
The answer to this question is not simple, but the idea can be summed up by saying that an invertible matrix would allow us to manipulate the information contained in the rectangular array of a matrix in ways that may be convenient while trying to solve systems of linear equations or performing other matrix operations.
For that matter, we have made a list of some of the most important properties to remember about an invertible matrix, which may be useful to you in future lessons. In order to start this list, we need to define A as a square matrix of any order (with any dimensions), then, for A to be an invertible matrix, the next conditions must hold true:
(A−1)−1=A
The inverse of a matrix is denoted as a the division of the unit by the matrix or the matrix with an exponent of -1. Thus, when inverting matrix A, the notation for its inverse is equal to A−1.
Having this in mind the expression above can be read as "the inverse of the inverse of A is equal to A" which makes sense and although it sounds redundant, it can be useful when a matrix needs to be inverted for a specific function, but then the original matrix is needed once more in an operation.
In simple words, this property says that if you invert matrix A, you will obtain A−1 (the inverse of A), and if you invert A−1 once more, you will obtain A back again.
2. (AT)−1=(A−1)T
The expression above affirms that it doesn't matter the order in which you invert the transpose of a matrix. In other words, you can obtain the transpose of the matrix and then invert the resulting matrix (as shown in the left hand side of the expression) or you can invert the given matrix and then obtain the transpose of the resulting inversion (as shown in the right hand side of the expression). Both processes will yield the exact same resulting matrix as long as you use the same matrix A to start with.
As a reminder, keep in mind the transpose of a matrix can be obtained by rearranging the columns of the original matrix as rows in the transpose. For the case of square matrices, the transpose matrix will remain to be the same order since it will continue to have the same amount of rows and columns than the original.
3. (kA)−1=k−1A−1 for non-zero scalar k
What we see on the expression above is the distributive property of an exponent, which works in the same way for a matrix notation. As mentioned before, the -1 exponent represents the inversion of the matrix; for this case, a matrix is multiplied to a constant coefficient and then the scalar multiplication result is inverted (elevated to the power of -1), this is equivalent to having done the matrix inversion first, and separately "invert" the scalar coefficient too (elevate it to the power of -1, same as dividing 1 by the coefficient) and then, perform the scalar multiplication between the inverted scalar and the inverted matrix.
Notice this has the clarification that k must be a non-zero scalar because a zero scalar would convert the whole expression into zero.
4. For any two square matrices A and B, (AB)−1=B−1A−1
Once again, this property describes the distributive nature of exponents when applied to a type of multiplication with matrices. If you take a look at the equation above, this is very similar to the expression shown on the third property, the only difference is that the third property shows a scalar multiplication inside the parenthesis on the left hand side of the equation, while the expression here (in property number four) contains a matrix multiplication inside the parenthesis on the left hand side of the equation.
In short, the expression for property number four says that the result of the multiplication of two matrices after they have been inverted is the same as multiplying the two original matrices first and then invert the resulting one.
5. det(A−1)=(detA)−1
In simple words, this property defines that the determinant of an inverted matrix is the same as obtaining the determinant of the original matrix and then "invert" this result by elevating it to the power of -1.
Proving a matrix is invertible
To finalize this lesson we will work on a few example exercises where we will be determining if a matrix is invertible. Notice we have not learned on this lesson how to invert a matrix, that will be explained in our next lesson named the inverse of a 2x2 matrix.
Example exercises
Given the matrix X as shown below:Is X an invertible matrix 2x2? Remember that the condition for a matrix to be invertible is that det(A)=0. And so, we obtain the determinant of matrix X:The determinant of matrix X is equal to zero, therefore, this matrix is NOT invertible.
Given the matrix A as shown below:Could we invert a 2x2 matrix such as A?The determinant of matrix A is equal to zero, therefore, this matrix is NOT invertible.
Given the 2x2 matrix E as shown belowIs E invertible?Since the determinant is not zero, then matrix E is invertible.
Given the 2x2 matrix F as shown belowIs inverting a 2x2 matrix such as F possible?The determinant of matrix F is equal to zero, therefore, this matrix is NOT invertible.
Given the 2x2 matrix Y as shown belowIs matrix Y invertible?Since the determinant is not zero, then matrix Y is invertible.
Given the 2x2 matrix Z as shown belowIs matrix Z invertible?Since the determinant is not zero, then matrix Z is invertible.
After learning what does it mean for a matrix to be invertible, and the process of proving a matrix is invertible, it is time for you to learn the calculation itself of inverting a matrix.
We finish this lesson by recommending you to visit the next handout on providing a summarized version of invertible matrix concepts and properties.
Now prepare for our next lesson, see you there!
An invertible matrix is a square matrix that has an inverse.
We say that a square matrix (or 2 x 2) is invertible if and only if the determinant is not equal to zero.
In other words, if X is a square matrix and det(X)=0, then X is invertible.
Notation of matrices
Notation of matrices
The determinant of a 2 x 2 matrix
The determinant of a 2 x 2 matrix |
7079 | https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:composing/e/evaluate-composite-functions-from-formulas | Evaluate composite functions (practice) | Khan Academy
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Precalculus
Course: Precalculus>Unit 1
Lesson 1: Composing functions
Intro to composing functions
Intro to composing functions
Composing functions
Evaluating composite functions
Evaluate composite functions
Evaluating composite functions: using tables
Evaluating composite functions: using graphs
Evaluate composite functions: graphs & tables
Finding composite functions
Find composite functions
Evaluating composite functions (advanced)
Math>
Precalculus>
Composite and inverse functions>
Composing functions
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Evaluate composite functions
AZ.Math: A2.F‑BF.A.1.b, P.F‑BF.A.1.c, PC.RFR.BF.2, PC.RFR.BF.3
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Problem
Given:
g(x)=3 x−2 h(x)=9 x
Danielle tried to evaluate (g∘h)(7), but she made a mistake. Here is her work.
Step 1 g(7)=3 7−2=3 5 Step 2 h(7)=9(7)=63 Step 3(g∘h)(7)=3 5⋅63=189 5
What is the mistake in Danielle's work?
Choose 1 answer:
Choose 1 answer:
(Choice A) (g∘h)(7)=g(h(7)), not g(7)⋅h(7). A (g∘h)(7)=g(h(7)), not g(7)⋅h(7).
(Choice B) (g∘h)(7)=h(g(7)), not g(7)⋅h(7). B (g∘h)(7)=h(g(7)), not g(7)⋅h(7).
(Choice C) g(7)=1 3, not 3 5. C g(7)=1 3, not 3 5.
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7080 | https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/using-scientific-calculator/content-section-1.2 | Using a scientific calculator: 1.2 Fractions or decimals? | OpenLearn - Open University
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Using a scientific calculator
1.2 Fractions or decimals?
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Introduction
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1 Getting to know your calculatorMenu toggle
1.1 Basic calculations
Current section: 1.2 Fractions or decimals?
1.3 Powers
1.4 Making corrections
2 Using your calculator for negative numbers
3 Using your calculator for fractions
4 Doing longer calculations using your calculatorMenu toggle
4.1 Reusing a previous result
4.2 Using the calculator memory
4.3 Other ‘M’ memory operations
4.4 Other memories
5 Scientific notation on your calculatorMenu toggle
5.1 Inputting numbers in scientific notation to your calculator
6 Powers and surds on your calculatorMenu toggle
6.1 Using roots on your calculator
6.2 Inserting a missing root
7 Trigonometric ratios on your calculator
8 Finding angles from trigonometric ratios
9 Radians on your calculator
10 Logarithms on your calculator
11 Natural logarithms and powers of e on your calculator
12 Calculator reference guideMenu toggle
12.1 Display indicators
12.2 Common operations
12.3 Entering mathematics
Conclusion
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Acknowledgements
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1.2 Fractions or decimals?
In Activity 1 you set up your calculator to use Math mode. In this mode, when the result of a calculation is not a whole number, it will be displayed as a fraction, such as , wherever possible.
To obtain the answer in decimal form, you need to press instead of , or you can toggle between the fractional and decimal outputs using the key.
Remember, your calculator is in Math mode if the word Math is shown at the top of the calculator display. If your calculator is not in Math mode, repeat the steps of Activity 1.
Activity 3 Fractions and decimals
Use your calculator to find in both fractional and decimal forms.
Reveal answer
Answer
In Math mode, calculating gives the result .
Converting this to a decimal using gives 0.1814671815.
Previous1.1 Basic calculations
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7081 | https://cdar.berkeley.edu/sites/default/files/Hot-or-Not.pdf | Hot or Not? A Nonparametric Formulation of the Hot Hand in Baseball Amanda K. Glazer Department of Statistics University of California Berkeley amandaglazer@berkeley.edu Lisa R. Goldberg Department of Statistics University of California Berkeley lrg@berkeley.edu March 18, 2020 Abstract A nonparametric analysis of player plate appearances (PA) in the 2018 Major League Baseball (MLB) season provides no evidence of a batter hot hand. Players with more than 100 PAs in the 2018 season are analyzed using one-sided permutation tests strat-ified by player.
Based on recent literature, we use the correlation between lagged on-base percentage (OBP)and a binary indicator of on-base performance. We discuss the strengths and weaknesses of this test statistic as well as others in the literature.
A common criticism of no-hot-hand findings for individual players is low power, and a frequently proposed remedy is pooling data across players. Through simulation, we show that pooling data conflates long-term ability and recent performance. Another common criticism of no-hot-hand findings is emphasis on recent performance. We show that long lags, which de-emphasize recent performance, can lead to counter intuitive results. In contrast to much of the recent literature, which uses parametric methods, we argue that our nonparametric method is the most appropriate way to analyze the existence of the hot hand in baseball as well as numerous other inference questions.
I never blame myself when I’m not hitting. I just blame the bat and if it keeps up, I change bats. After all, if I know it isn’t my fault that I’m not hitting, how can I get mad at myself?
- Yogi Berra 1 Introduction Streaks of hits and misses are familiar to everyone who plays or watches sports. Performance seems to be dictated by magical streaky periods in which players appear to be "hot" or "cold." Are we to believe in these streaky periods, or should we believe Yogi Berra that there is no one to blame for strings of hits and misses, and they are especially not the fault of the players themselves?
1 The hot hand in sports is the idea that a player who has recently been successful has an elevated probability of success in the near future. The hot hand has been a hot topic since Gilovitch, Vallone, and Tversky’s 1985 study found no statistical evidence of a hot hand in basketball. They argued that, perceptions notwithstanding, a basketball player’s chance of making a shot showed no dependence on whether it followed a hit or a miss .
This classical study examined the difference in player i’s shooting percentage conditional on having made the last k shots and missed the last k shots (with k = 1, 2, 3): ˆ P i(hit|k hits) −ˆ P i(hit|k misses) Paired t-tests indicate that there is no significant difference in the chance that a player makes the next shot based on whether they missed or hit the last k shots.
The findings in the original study were widely debated, with many players and sports enthusiasts refusing to believe that the hot hand did not exist. Thirty years after its pub-lication, this study was found to contain a small sample bias that some argue led to the incorrect conclusion . There is no consensus on whether the hot hand exists or not in basketball.
The original study and much of the literature since, including recent studies on baseball, have largely focused on parametric models to test for the hot hand [3, 2]. These parametric models rely on a significant p-value of the coefficient of interest (generally in a linear model) to conclude that a hot hand exists. They usually control for a number of factors, such as ballpark characteristics and ability of the pitcher.
In order to delve more deeply into the question of whether the hot hand exists in baseball, we must clearly formulate our question. In analyzing whether the hot hand exists, researchers tend to focus their attention on one of the following two questions: 1. Does a player perform better if they have performed well recently (e.g., in the last L plate appearances or shots), outside of the effects of all other factors?
2. Does fan’s perception of the hot hand, players that have performed well recently will continue to do so, exist?
The key difference in these questions is whether we think it is important to control for all other factors when searching for the existence of the hot hand. We argue that most fans and players perceiving a hot hand are not mentally adjusting for factors. Rather, they are reacting to a streak. Generally, someone does not think, "LeBron James has been making a higher number of shots then you would expect given the opposing team, level of defense, arena, day of the week, etc." A more realistic formulation centers around a fan’s heart rate increasing because LeBron James just made his tenth basket in a row. While question 1 is interesting and has its own merits and challenges (e.g., it is difficult to take into account all factors that could affect performance), for this paper we will consider question 2 because we believe it is a more realistic formulation of the hot hand phenomenon.
In trying to sort out the hot hand issue outlined in question 2, some previous research in basketball has considered a nonparametric approach stratified by player. For example, in Daks, Desai and Goldberg (2018), strings of hits (1) and misses (0) were considered (e.g., “100101110”) for Steph Curry, Klay Thompson, and Kevin Durant. The proportion of “11” 2 followed by “1” minus the proportion of “00” followed by “1” was used as a test statistic in a permutation test. This study found no evidence that the Golden State Warriors players exhibited hot hands .
Relative to a parametric analysis, a nonparametric approach is based on fewer assump-tions and is conceptually clearer. For example, the p-value of a permutation test is based on the proportion of random shufflings of a variable that result in a test statistic larger than or equal to the test statistic calculated on the actual data. The core assumption is that if we believe that the shuffled variable does not make a difference, then we should be able to scramble its realizations without significantly changing the test statistic.
While most researchers have not found evidence of the hot hand in baseball, those that have found evidence have relied on data pooled across players rather than considering them individually; see, for example, [2, 6]. While the motivation for pooling data in order to increase statistical power is understandable, pooling data leads to a large type I error rate and erroneous conclusions. We demonstrate this with a simple simulation.
In this paper, we adopt permutation tests, a nonparametric method, stratified by player, to examine whether or not the hot hand exists in baseball. This analysis is inspired by Daks, Desai and Goldberg’s 2018 study (although using a different test statistic) . We illustrate the benefits of a nonparametric approach over a parametric approach, demonstrate the perils of pooling data, and compare various test statistics. Finally, we illustrate the potential use of nonparametric analysis for a variety of inference questions in baseball.
2 Data and Methodology 2.1 Data We used play by play Major League Baseball (MLB) data from all 30 teams in the 2018 season. The data is publicly available from Retrosheet.org. We analyzed all 447 players with more than 100 plate appearances in the 2018 season. Table 1 gives summary statistics about the OBP and number of PAs for the 2018 players included in the analysis.
Max Mean Minimum OBP 0.460 0.315 0.162 PA 745 387 101 Table 1: Summary statistics about the 447 players from 2018 included in the analysis.
2.2 Defining the batter hot hand in baseball As discussed in the introduction, we consider the following question: Is a batter with a higher on base percentage (OBP) over the last L plate appear-ances (PAs) more likely to get on base at their next PA?
3 OBP is approximately equal to the number of times on base divided by PAs. If the batter hot hand exists, the answer to this question would be yes, higher OBPs over the last L PAs should be positively correlated with whether the batter gets on base in the next PA. In order to investigate this, we used permutation tests stratified by player and season.
The binary on-base vector OBk has jth entry equal to 1 if player k got on base at the jth PA and 0 if not. Following Green and Zwiebel (2018), we define player state to be OBP for the last L PAs, where the lag L can take on different values. Let statekj be the state of player k just prior to plate appearance j, statekj = 1 L j−1 X i=j−L OBki, and let statek denote the player k’s vector of states.
2.3 Test Statistic Our test statistic Tk is the correlation between OBk and statek: Tk = corr(OBk, statek).
If there is a batter hot hand, we would expect higher state values to correlate with a higher likelihood of reaching base. Thus, we would expect that a batter hot hand would lead to larger test statistic values relative to random shufflings of player PAs. This test statistic is similar to Green and Zwiebel (2018)’s use of the state coefficient, in the logistic regression regressing on base indicator on state as well as other controls, as a measure of the batter hot hand.
In Section 3 we consider other possible test statistics, autocorrelation and the test statistic in the Gillovich et al. study , but find that Tk has the highest power among the test statistics we consider.
2.4 Permutation Tests For each player in our data set, we ran a one-sided permutation test. The null hypothesis is that state has no effect on outcome of the next PA.
If the null hypothesis is true, then shuffling the PAs, recalculating state, and calculating the correlation between state and OB should lead to a correlation that is not too different from our original value for player k. The p-value is the proportion of shufflings that result in a correlation as large or larger than the correlation we observed.
Advantages of permutation tests over parametric methods are that permutation tests make fewer assumptions, and those assumptions they do make are conceptually very clear.
For example, permutation tests do not make assumptions about the distributions from which data are drawn. Instead, the p-value in a permutation test is derived from the assumption that under the null hypothesis, you should be able to shuffle, for example, player PAs and not see a large difference in the test statistic if the hot hand does not exist.
We must acknowledge and correct for the fact that we are running multiple hypothesis tests, and some will show significance purely by chance.
The same is true for running 4 numerous regression models. Under the null hypothesis, we expect, for a significance level α, that a fraction α of the tests will lead to a false rejection (i.e., an indication of a hot hand when it is not there). In other words, if there is no hot hand, we expect nevertheless to see significance at level α in a fraction α of the tests.
2.5 Choice of Lag There is no clear choice for the lag L. Green and Zwiebel (2018) used lags of length 10, 25 and 40. In streaky data, counter-intuitive results can occur when the lag length is longer than the streak length. We ran several simple simulations to illustrate potential issues with certain lag lengths.
Consider a string of plate appearances composed of 24 1s followed by 24 0s followed by 24 1s and so on for the entire season. We would hope that our test would pick up on the extreme streakiness in this sequence. However, a lag 25 results in a negative correlation of -0.136 between OB and state. On the other hand, lag 10 and lag 5 result in positive correlations. A rule of thumb is that our test statistic will miss a streak that is shorter than the lag used to compute state.
For this reason, lags of length 10 or 5 seem more appropriate as they seem more likely to pick up on a variety of streak lengths. In our hot hand analysis, we used values of 5, 10 and 25 for L to evaluate if results are sensitive to the length of history used to calculate state.
3 Type I Error and Power Previous studies have pooled data across players to increase power in parametric tests for the hot hand in baseball [2, 6]. But pooling data can lead to erroneous significant results. If we do not pool data, however, we might be concerned about the ability of our permutation tests to correctly identify a hot hand if it does exist. Simulations can give us a general sense of the power of permutation tests in this setting.
3.1 Pooling Data Pooling data can inflate the type I error rate. Consider, for example the simple simulation where we generate 400 baseball players, each of which has a different OBP p evenly spaced from .250 to .450. For each player, we generated on base indicators for 500 PAs from the binomial distribution: Binom(500, p). There is no hot hand because data is generated at random from this binomial distribution.
We follow the same set up as in Green and Zwiebel (2018) . Ability is defined as the OBP of a player outside of the 50 PAs before and after the current PA. If we regress the current plate appearance on state (as defined in the previous section with L = 25) and ability, we would hope that state would not be significant because there is no hot hand in this simulation. However, over 1000 simulations, we find that 99% of the time, state is significant (at the 0.05 level). In other words, the type I error rate (or false positive rate) is 99%.
The source of the significance is differences in player OBP and not a hot hand. How-ever, the linear regression conflates differences in OBP with differences in state, even when 5 we account for an ability variable. We must be cautious about drawing conclusions from regressions that pool data across players with varied abilities.
3.2 Power Since we are concerned with the hot hand at the player level, and since pooling data can lead to high type I error, it makes sense to consider tests for streakiness at the player level.
However, previous research has expressed concern about the potential loss of power due to the smaller sample size when considering players individually rather than in aggregate [2, 6].
Before we run permutation tests that are stratified by player, we investigate and address power concerns. There is no closed form formula for power in permutation tests. However we can use simulation to assess power is in various situations. There are lots of ways that binary sequences can deviate from random. We consider two different ways of creating correlated binary strings and evaluate the power of our stratified permutation tests in each.
3.2.1 Autocorrelated Binary Strings There is no precise level of autocorrelation in a binary string that corresponds to a hot hand, the two concepts are loosely connected. By generating binary strings with different levels of autocorrelation, we can get a better sense of what level of autocorrelation our method is able to detect.
We simulate correlated binary variables with specified marginal means and correlations.
Then, we calculate the probability that we can detect a hot hand (with lag equal to 5, 10 and 25), as characterized by correlation between state and on base performance, using a permutation test. We simulate binary variables Y1, ..., Yn) with correlation ρ using the conditional linear family method as outlined in : 1. Generate Y1 ∼Bern(p) 2. Generate Y2, ..., Yn each with mean E(Yj|Yj−1) = p + ρ(yj−1 −p) We set p = 0.318, the average OBP for the 2018 MLB season (for all players), and n = 500. We consider several different values of the correlation, ρ, which gives us an idea of what the power (for significance level 0.05) would be for our permutation tests with lag 5, 10 and 25. Results are reported in Table 2. Power is smallest for L = 25 and largest for L = 5.
Both L = 5, 10 have fairly reasonable power for autocorrelation greater than or equal to 0.4.
3.2.2 Markov Model Next, we consider a two-state Markov model with transition probability 0.05. The two states are hot and cold. We consider hot OBP/cold OBP of 0.6/0.2, 0.55/0.25 and 0.5/0.3. When a player is in a hot (cold) state whether he makes it on base will be a Bernoulli draw with probability equal to the hot (cold) OBP. According to this model we generate binary strings of length 500 and over 1000 simulations calculate the proportion of time that our permutation test methodology with various test statistics will detect a hot hand.
6 Lag ρ 0.2 0.4 0.6 5 0.635 0.991 1 10 0.343 0.841 0.994 25 0.139 0.478 0.761 Table 2: Power (significance level is 0.05) over 1000 simulations for permutation tests de-fined in section 2 for various lag lengths and correlated binary variables with correlation ρ.
Marginal mean is set to 0.318 and n is set to 500.
Test Statistic Hot/Cold OBP 0.6/0.2 0.55/0.25 0.5/0.3 Lag 5 Correlation 1 0.88 0.32 Lag 10 Correlation 0.99 0.85 0.34 Autocorrelation 0.9 0.51 0.18 Gilovich et al. Statistic 0.97 0.65 0.23 Table 3: Power (significance level is 0.05) over 1000 simulations for four different test statis-tics for a two-state markov chain with transition probability 0.05 and various hot/cold OBPs.
Binary strings of length 500 were generated.
The results in Table 3 show that in our two-state Markov model simulations, power dropped offsteeply as the difference between hot and cold states gets smaller. The Lag 5 and 10 correlation test statistics perform the best.
Results from both of our power simulations indicate that smaller lags have higher power.
Therefore, we believe that lag 5 and 10 are more reasonable lags to consider in our tests.
Not only can larger lags such as 25 and 40 produce counter-intuitive results, but larger lags also appear to have lower power.
4 Hot Hand Results We found no evidence of a hot hand in the 2018 MLB season. Table 4 shows the proportion of player p-values that were significant at level α for α equal to 0.05 and 0.1, with state calculated using lags 5, 10 and 25. Under the null hypothesis that there is no streakiness: we would expect the proportion of p-values that are significant at level α to be close to α.
Figure 1 shows CDFs for player p-values from the permutation tests with state calculated using lags 5, 10 and 25. Under the null hypothesis that there is no streakiness, we expect the CDF to look similar to the uniform distribution.
Our results are in line with what we would expect for the type I error rate, if there were no hot hand. Results from the 2012 season (see Appendix A) show similar results.
Our analysis yields no evidence of the batter hot hand in baseball. Other permutation test 7 α Lag 5 10 25 0.05 0.087 0.067 0.076 0.01 0.018 0.020 0.025 Table 4: Proportion of p-values for player permutation tests significant at level α for state calculated with various lag lengths.
Figure 1: CDFs for player p-values for permutation tests with L = 5, 10 and 25 statistics, such as autocorrelation and the state coefficient in a linear regression, also show similar results.
5 Other Nonparametric Formulations of the Hot Hand in Baseball In the previous section, we looked for streakiness in strings of player plate appearances with no attention to clustering by game. In this section, we consider other formulations of the hot hand in baseball, and we analyze them with non-parametric methods.
To analyze streakiness at the game level, we formulate the hot hand as whether a player exhibits cross-game streakiness. In other words, do games in which a batter has above average performance cluster? This clustering is a necessary (but perhaps not sufficient) condition for cross-game hot hand. There are challenges to this formulation. For example, the game OBPs will be very noisy due to the relatively small number of plate appearances in a game.
However, it will give us a sense of whether a player exhibits cross-game streakiness or not.
Consider, for example, Mike Trout’s 2019 season. His OBP for the 2019 was 0.438, and 8 we calculate his OBP for each game. For his plate appearances in a game, we calculate the proportions of hits (H), walks (BB) and hits by pitch (HBP, Trout’s game OBP). We then create an indicator variable, called above average indicator, that is 1 when the game OBP is above his season average and 0 when it is below his season average. If Trout were to exhibit streaky playing, we would expect that the autocorrelation of his above average indicator to be high relative to random shufflings of his game OBPs.
We run a permutation test in order to evaluate this claim. We randomly shuffle the game level OBP and calculate the autocorrelation of the above average indicator. The p-value is the proportion of random shuffles or permutations that have autocorrelation greater than or equal to the observed autocorrelation. The p-value is 0.743, providing no evidence that above and below average OBP games are clustered together.
Figure 2: Game OBP for Mike Trout’s 2019 season From Figure 2, it is clear that there are many game OBPs that are close to his season average. It is not necessarily true that a streak of 0.5 game OBPs would feel like a hot hand to a fan. However, since we observe an insignificant p-value, there is no reason to believe that even that level of streakiness exists.
6 Other Applications of Nonparametric Analysis in Base-ball Permutation tests are the ideal methodology for many inference questions in baseball. Since permutation tests are nonparametric, they make minimal assumptions about the data, as opposed to parametric methods such as linear or logistic regression, which make numerous assumptions. In this section, we sketch examples of other uses of permutation tests to answer inference questions in baseball.
Suppose we want to evaluate the dependence of a batter’s OBP on the number of outs prior to plate appearance. Consider Mike Trout’s 2019 season as an example.
If the number of outs do not make a difference in the likelihood that Trout will get on base or not, then we should be able to shuffle the columns that indicates the number of 9 Outs 0 1 2 OBP .430 .449 .431 PA 200 247 153 Table 5: Mike Trout’s OBP and PA based on number of outs for the 2019 season.
outs for a PA and not see much of a difference. Using a chi-squared test statistic in the permutation test, we get a p-value of 0.90. There is no evidence that Trout’s OBP depends on the number of outs at the time of his plate appearance.
We use similar analysis to evaluate whether Trout’s OBP is significantly different for home versus away games. In 2019 Trout had an OBP of .450 for home games (280 PA) and .428 for away games (320 PA). Running a permutation test with difference in OBP between home and away games as the test statistic yields a p-value of 0.326. There is no evidence of a significant difference between Trout’s 2019 OBP at home versus away games.
A similar method could be used to evaluate whether OBP (or any other statistic) varies depending on whether or not it is a clutch situation.
7 Conclusion Nonparametric tests of player on base performance showed no evidence of the hot hand phenomenon, despite its perception by sports enthusiasts. We ran permutation tests on all MLB players with more than 100 plate appearances in the 2018 season. We used lags of 5, 10 and 25 plate appearance to determine hot and cold states, and we asked whether these states were correlated with the subsequent plate appearance. The proportion of tests that were significant aligned with the type I error rate, providing no evidence that the hot hand as formulated exists.
Crucially, the permutation tests are stratified by player, so the results are not corrupted by the conflation of state and ability, which plagued the results in Green and Zwiebel 2018 . Through simulation, we show that if there is no hot hand but players with varying OBP are pooled together then linear regression will reveal a hot hand effect when there is not one.
Since permutation tests stratified by player adequately control for the type I error rate at the cost of statistical power. It is clear that this is the proper tradeoff.
In general, nonparametric tests are the ideal methodology for answering many inference questions in baseball because they rely on relatively few assumptions, and the assumptions they do make are conceptually very clear. If we believe that a factor should not make a difference in outcome, then we should be able to shuffle the realizations of that factor and not see much difference in our chosen test statistic. There are no assumptions about the distribution of our data which could lead to erroneous results. We advocate for the use of permutation tests for inference questions in baseball.
10 A Appendix A.1 2012 Results We also ran our permutation test analysis on players with more than 100 plate appearances in the 2012 season (459 players). Our results are in line with our 2018 results: we find no evidence of a batter hot hand.
Figure 3: CDFs for player p-values for permutation tests for lags L = 5, 10, 25 for the 2012 season α Lag 5 10 25 0.05 0.052 0.059 0.059 0.01 0.009 0.004 0.002 Table 6: Proportion of p-values for player permutation tests significant at level α for state calculated with various lag lengths for all players with more than 100 PAs in the 2012 MLB season.
11 References N. D. Alon Daks and L. Goldberg. Do the golden state warriors have hot hands? The Mathematical Intelligencer, 2018.
B. Green and J. Zwiebel. The hot-hand fallacy: Cognitive mistakes or equilibrium ad-justments? evidence from major league baseball. Management Science, 2018.
S. A. Michael Bar-Eli and M. Raab. Twenty years of “hot hand” research: Review and critique. Psychology of Sport and Exercise, 2006.
J. Miller and A. Sanjurjo. Surprised by the hot hand fallacy? a truth in the law of small numbers. Econometrica, 2018.
J. S. Preisser and B. F. Qaqish. A comparison of methods for simulating correlated binary variables with specified marginal means and correlations. Journal of Statistical Computation and Simulation, 2014.
H. S. Stern and C. N. Morris. A statistical analysis of hitting streaks in baseball: Com-ment. Journal of the American Statistical Association, 1993.
R. V. Thomas Gilovich and A. Tversky. The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 1985.
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How to solve polynomial functional equation P(x,y)=P(x−y2,y−x2)?
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Given P(x,y) which is a polynomial function, satisfying P(x,y)=P(x−y2,y−x2).
Then why should P(x,y) be ∑i=0nai(x−y)i? Is it unique?
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Define R(t)=P(t/2,−t/2). Then R is clearly a polynomial in t (because compositions of polynomials are polynomials), and your functional equation now says that P(x,y)=R(x−y). Substitute t=x−y into the canonical form of the polynomial R, and you have your conclusion.
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7083 | https://dec.alaska.gov/eh/fss/shellfish/paralytic-shellfish-poisoning/ | Paralytic Shellfish Poisoning | AK Dept. of Environmental Conservation
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What is Paralytic Shellfish Poisoning?
Paralytic shellfish poisoning (PSP) is a serious illness caused by eating shellfish contaminated with algae that contains Paralytic Shellfish Toxin (PST), a toxin harmful to humans. This toxin is extremely poisonous; as little as one milligram (0.000035 ounce) is enough to kill an adult.
How do shellfish become contaminated with PST?
Shellfish feed by filtering food particles, including algae, from ingested water. When they filter and eat biotoxin-producing algae, the PST accumulates in their tissue.
What types of shellfish might be contaminated?
All molluscan shellfish (those having a hinged shell), including clams, mussels, geoducks, oysters, snails and scallops may be contaminated. Crab meat is not known to contain the toxin, but they feed on shellfish, so it’s possible the guts may contain unsafe levels of PST. To be safe, clean crab thoroughly, removing all butter (the white-yellow fat inside the back of the shell), and discard the gut. Shrimp and finfish are not known to contain the toxin.
What about shellfish I buy in the store?
Restaurants and stores must purchase shellfish from certified growers. Certified growers are required to have their products regularly tested for PST.
What if I cook contaminated shellfish thoroughly?
The toxin is not destroyed by cooking or freezing.
Does shellfish containing PST taste different?
No. Shellfish containing PST may not look, smell or taste any different than uncontaminated shellfish.
What are the symptoms of PSP?
Early symptoms include tingling of the lips and tongue, which may begin within minutes of eating poisonous shellfish or may take an hour or two to develop. Depending upon the amount of toxin a person has ingested, symptoms may progress to tingling of fingers and toes and then loss of control of arms and legs, followed by difficulty in breathing. Some people have experienced a sense of floating or nausea. If a person consumes enough poison, muscles of the chest and abdomen become paralyzed. Death can result in as little as two hours, as muscles used for breathing become paralyzed.
What is the treatment for PSP?
The only treatment for severe cases is the use of a mechanical respirator and oxygen. If symptoms are exhibited, call 911, or get to a medical facility immediately.
PSP in Alaska – How to avoid Paralytic Shellfish Poisoning
Don’t eat shellfish harvested from untested Alaska beaches.
Don’t eat the viscera (guts) of crab. Because PSP has been found in crab viscera, it is recommended that crab be cleaned and eviscerated before being cooked.
Cooking will not destroy the toxin. Freezing will not destroy the toxin.
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For more information, contact Carol Brady, Alaska DEC shellfish expert.
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Paralytic Shellfish Toxin (PST)
A naturally occurring toxin, called a “biotoxin,” that is produced by some species of microscopic algae. PST can concentrate in shellfish and, when eaten, can cause severe illness and death.
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1989 AIME Problems/Problem 14
Problem
Given a positive integer, it can be shown that every complex number of the form , where and are integers, can be uniquely expressed in the base using the integers as digits. That is, the equation
is true for a unique choice of non-negative integer and digits chosen from the set , with . We write
to denote the base expansion of . There are only finitely many integers that have four-digit expansions
Find the sum of all such ,
Solution
First, we find the first three powers of :
So we solve the diophantine equation.
The minimum the left-hand side can go is -54, so since can't equal 0, so we try cases:
Case 1:
The only solution to that is .
Case 2:
The only solution to that is .
So we have four-digit integers and , and we need to find the sum of all integers that can be expressed by one of those.
:
We plug the first three digits into base 10 to get . The sum of the integers in that form is .
:
We plug the first three digits into base 10 to get . The sum of the integers in that form is . The answer is . ~minor edit by Yiyj1
See also
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7086 | https://chem.libretexts.org/Courses/Saint_Francis_University/CHEM_113%3A_Human_Chemistry_I_(Muino)/02%3A_Atoms_and_the_Periodic_Table/2.07%3A_Electron_Configurations | Last updated Aug 8, 2022
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Learning Objectives
Describe how electrons are arranged in an atom using electron configurations.
Previously we discussed the concept of electron _shells_, _subshells_, _orbitals_, and electron _spin_. It is the arrangement of electrons into shells and subshells that most concerns us here, so we will focus on that.
We use numbers to indicate which shell an electron is in. The first shell, closest to the nucleus and with the lowest-energy electrons, is shell 1. This first shell has only one subshell, which is labeled 1 _\s\_ and can hold a maximum of 2 electrons. We combine the shell and subshell labels when referring to the organization of electrons about a nucleus and use a superscript to indicate how many electrons are in a subshell. Thus, because a hydrogen atom has its single electron in the _\s\_ subshell of the first shell, we use 1 _\s\_ 1 (spoken as “one-ess-one”) to describe the electron arrangement or distribution of electrons in hydrogen. This structure is called an electron configuration and is unique to hydrogen.
Helium atoms have 2 electrons. Both electrons fit into the 1 _\s\_ subshell because _\s\_ subshells contain one _s_ orbital which can hold up to 2 electrons; therefore, the electron configuration for helium atoms is 1 _\s\_ 2(spoken as “one-ess-two”).
The 1 _\s\_ subshell can hold a maximum of 2 electrons, so the electron configuration for a lithium atom, which has three electrons, cannot be 1 _\s\_ 3. Two of the lithium electrons can fit into the 1 _\s\_ subshell, but the third electron must go into the second shell and the lower energy orbital, which is the 2 _s_ orbital. Therefore, we write the electron configuration of a lithium atom as 1 _\s\_ 2 2 _\s\_ 1 (spoken as “one-ess-two two-ess-one”).
The shell diagram for a lithium atom (Figure 2.7.1). The shell closest to the nucleus (first shell) has 2 dots representing the 2 electrons in 1_\s\_, while the outermost shell (2 _\s\_) has 1 electron.
Figure 2.7.1: Shell diagrams of hydrogen (H), helium (He), lithium (Li), and Berryellium (Be) atoms. (CC BY-SA 2.0 UK; Greg Robson modified by Pumbaa via Wikipedia)
There are a set of general rules that are used to figure out the electron configuration of an atomic species: Aufbau Principle, Hund's Rule and the Pauli-Exclusion Principle.
Rule 1 (Aufbau Principle): Electrons occupy the lowest-energy orbitals (closest to the nucleus) possible, starting with 1 _s_, then 2 _s_, 2 _p_, and continuing on to higher energy (further away from the nucleus). Shells increase in energy in order from 1 to 2 to 3, and so on. Within these shells, an _s_ subshell is the lowest energy followed by _p_, then _d_, then _f_.
Rule 2 (Hund's Rule): When electrons occupy degenerate orbitals (i.e. same shell and subshell), they must first singly occupy (half-fill) each empty orbital in a subshell before double occupying (completely filling) them. Furthermore, the most stable configuration results when the spins are parallel (i.e. all spin up or all spin down). For example, all three _p_ orbitals in a _p_ subshell will have one electron before a single _p_ orbital contains two electrons.
Rule 3 (Pauli-Exclusion Principle):Each electron is described with a unique set of four quantum numbers (a unique address). Therefore, if two electrons occupy the same orbital, they must have different spins. This is the reason all orbitals can hold a maximum of two electrons.
Continuing on the periodic table to the next largest atom, beryllium, with 4 electrons, the electron configuration is 1 _\s\_ 2 2 _\s\_ 2. Now that the 2 _\s\_ subshell is filled, electrons in larger atoms, starting with boron, begin filling the 2 _\p\_ subshell, which can hold a maximum of six electrons. The next six elements progressively fill up the 2p subshell:
B: 1s 2 2s 2 2p 1
C: 1s 2 2s 2 2p 2
N: 1s 2 2s 2 2p 3
O: 1s 2 2s 2 2p 4
F: 1s 2 2s 2 2p 5
Ne: 1s 2 2s 2 2p 6
At the end of the period the element neon, has filled the 2 _s_, and 2 _\p\_ subshells, which completes the second shell. Now atoms with more electrons now must begin the third shell starting with the 3 _s_ subshell. The first two subshells of the third shell are filled in order—for example, the electron configuration of aluminum, with 13 electrons, is 1 _\s\_ 2 2 _\s\_ 2 2 _\p\_ 6 3 _\s\_ 2 3 _\p\_ 1. However, a curious thing happens after the 3 _\p\_ subshell is filled: the 4 _\s\_ subshell begins to fill before the 3 _\d\_ subshell does. In fact, the exact ordering of subshells becomes more complicated at this point (after argon, with its 18 electrons), so we will not consider the electron configurations of larger atoms.
Table 2.7.1: Electron Configurations of the First 20 Elements| Atomic Number | Element Symbol | Outermost Shell | Electron Configuration | Noble Gas Configuration |
--- ---
| 1 | H | 1 | 1 _s_ 1 | 1 _s_ 1 |
| 2 | He | 1 | 1 _s_ 2 | 1 _s_ 2 |
| 3 | Li | 2 | 1 _s_ 2 2 _s_ 1 | [He] 2 _s_ 1 |
| 4 | Be | 2 | 1 _s_ 2 2 _s_ 2 | [He] 2 _s_ 2 |
| 5 | B | 2 | 1 _s_ 2 2 _s_ 2 2 _p_ 1 | [He] 2 _s_ 2 2 _p_ 1 |
| 6 | C | 2 | 1 _s_ 2 2 _s_ 2 2 _p_ 2 | [He] 2 _s_ 2 2 _p_ 2 |
| 7 | N | 2 | 1 _s_ 2 2 _s_ 2 2 _p_ 3 | [He] 2 _s_ 2 2 _p_ 3 |
| 8 | O | 2 | 1 _s_ 2 2 _s_ 2 2 _p_ 4 | [He] 2 _s_ 2 2 _p_ 4 |
| 9 | F | 2 | 1 _s_ 2 2 _s_ 2 2 _p_ 5 | [He] 2 _s_ 2 2 _p_ 5 |
| 10 | Ne | 2 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 | [He] 2 _s_ 2 2 _p_ 6 |
| 11 | Na | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 1 | [Ne] 3 _s_ 1 |
| 12 | Mg | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 | [Ne] 3 _s_ 2 |
| 13 | Al | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 1 | [Ne] 3 _s_ 2 3 _p_ 1 |
| 14 | Si | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 2 | [Ne]3 _s_ 2 3 _p_ 2 |
| 15 | P | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 3 | [Ne] 3 _s_ 2 3 _p_ 3 |
| 16 | S | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 4 | [Ne] 3 _s_ 2 3 _p_ 4 |
| 17 | Cl | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 5 | [Ne] 3 _s_ 2 3 _p_ 5 |
| 18 | Ar | 3 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 6 | [Ne] 3 _s_ 2 3 _p_ 6 |
| 19 | K | 4 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 6 4 _s_ 1 | [Ar] 4 _s_ 1 |
| 20 | Ca | 4 | 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 6 4 _s_ 2 | [Ar] 4 _s_ 2 |
Noble Gas Configuration
The electron configuration of sodium is 1 s 2 2 s 2 2 p 6 3 s 1 (Table 2.7.1). The first ten electrons of the sodium atom are the inner-shell electrons and the configuration of just those ten electrons is exactly the same as the configuration of the element neon (Z=10). This provides the basis for a shorthand notation for electron configurations called the noble gas configuration, which atom consists of the elemental symbol of the last noble gas prior to that atom, followed by the configuration of the remaining electrons. So for sodium, we make the substitution of [Ne] for the 1 s 2 2 s 2 2 p 6 part of the configuration. Sodium's noble gas configuration becomes [Ne]3 s 1.
The electron filling diagram shown below in Figure 2.7.2 is useful in remembering the order for electrons to occupy shells and subshells. Although it is much easier to use the periodic table as a guide for electron filling as you will see in the next section.
Figure 2.7.2: The order of electron filling in an atom. Follow each arrow in order from top to bottom. The subshells you reach along each arrow give the ordering of filling of subshells in larger atoms.
Example 2.7.1: Electronic Configuration of Phosphorus Atoms
Using Figure 2.7.2 as your guide, write the electron configuration of a neutral phosphorus atom. The atomic number of P is 15.
Solution
A neutral phosphorus atom has 15 electrons. Two electrons can go into the 1 _s_ subshell, 2 can go into the 2 _s_ subshell, and 6 can go into the 2 _p_ subshell. That leaves 5 electrons. Of those 5 electrons, 2 can go into the 3 _s_ subshell, and the remaining 3 electrons can go into the 3 _p_ subshell. Thus, the electron configuration of neutral phosphorus atoms is 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p_ 3.
Exercise 2.7.1: Electronic Configuration of Chlorine Atoms
Using Figure 2.7.2 as your guide, write the electron configuration of a neutral chlorine atom. The atomic number of Cl is 17.
Answer
A neutral chlorine atom has 17 electrons. Two electrons can go into the 1 _s_ subshell, 2 can go into the 2 _s_ subshell, and 6 can go into the 2 _p_ subshell. That leaves 7 electrons. Of those 7 electrons, 2 can go into the 3 _s_ subshell, and the remaining 5 electrons can go into the 3 _p_ subshell. Thus, the electron configuration of neutral chlorine atoms is 1 _s_ 2 2 _s_ 2 2 _p_ 6 3 _s_ 2 3 _p 5_.
Orbital Diagrams
An orbital diagram is the more visual way to represent the arrangement of all the electrons in a particular atom. In an orbital diagram, the individual orbitals are shown as squares and orbitals within a sublevel are drawn next to each other horizontally. Each sublevel is labeled by its shell and sublevel. Electrons are indicated by arrows inside of the squares. An arrow pointing upwards indicates one spin direction, while a downward pointing arrow indicates the other direction. The orbital filling diagrams for hydrogen, helium, and lithium are shown in the figure below.
Figure 2.7.3: Orbital diagrams for hydrogen, helium, and lithium.
According to the Aufbau Principle, sublevels and orbitals are filled with electrons in order of increasing energy. Since the s sublevel consists of just one orbital, the second electron simply pairs up with the first electron as in helium. The next element is lithium and necessitates the use of the next available sublevel, the 2 s.
The orbital diagram for carbon is shown in Figure 2.7.10. There are two 2 p electrons for carbon and each occupies its own 2 p orbital (Hund's Rule).
Figure 2.7.4: Orbital diagram for carbon.
Oxygen has four 2 p electrons. After each 2 p orbital has one electron in it, the fourth electron can be placed in the first 2 p orbital with a spin opposite that of the other electron in that orbital.
Figure 2.7.5: Orbital diagram for oxygen.
If you keep your papers in manila folders, you can pick up a folder and see how much it weighs. If you want to know how many different papers (articles, bank records, or whatever else you keep in a folder), you have to take everything out and count. A computer directory, on the other hand, tells you exactly how much you have in each file. We can get the same information on atoms. If we use an orbital filling diagram, we have to count arrows. When we look at electron configuration data, we simply add up the numbers.
Example 2.7.3: Carbon Atoms
Draw the orbital filling diagram for carbon and write its electron configuration.
Known
Atomic number of carbon, Z=6
Use the order of fill diagram to draw an orbital filling diagram with a total of six electrons. Follow Hund's rule. Write the electron configuration.
_Step 2: Construct the diagram._
_Orbital filling diagram for carbon._
Electron configuration 1s 2 2s 2 2p 2
_Step 3: Think about your result._
Following the 2s sublevel is the 2p, and p sublevels always consist of three orbitals. All three orbitals need to be drawn even if one or more is unoccupied. According to Hund's rule, the sixth electron enters the second of those p orbitals and has the same spin as the fifth electron. |
7087 | https://espanol.libretexts.org/Bookshelves/Quimica/Qu%C3%ADmica_F%C3%ADsica_y_Te%C3%B3rica/Qu%C3%ADmica_F%C3%ADsica_(LibreTexts)/27%3A_La_teor%C3%ADa_cin%C3%A9tica_de_los_gases/27.06%3A_Trayectoria_Libre_Media | Saltar al contenido principal
27.6: Trayectoria Libre Media
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: 30 oct 2022
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27.5: La distribución de Maxwell-Boltzmann ha sido verificada experimentalmente
27.7: Tasas de reacciones químicas en fase gaseosa
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Energía de colisión
Considera dos partículas y en un sistema. La energía cinética de estas dos partículas es
Podemos describir la energía cinética en términos de centro de masa y momento relativo, que vienen dados por
y
donde
es la masa total de las dos partículas, y
es la masa reducida de las dos partículas.
Sustituyendo estos términos en la ecuación, encontramos
Tenga en cuenta que la energía cinética se separa en una suma de un término de centro de masa y un término de impulso relativo.
Ahora la posición relativa de las dos partículas es tal que la velocidad relativa es o . Así, si las dos partículas se acercan entre sí de tal manera que , entonces . Sin embargo, al equiparticionar la energía cinética relativa, que es independiente de la masa, obtenemos
que se llama la energía de colisión
Sección transversal de colisión
Considera dos moléculas en un sistema. La probabilidad de que colisionen aumenta con el “tamaño” efectivo de cada partícula. Sin embargo, la medida de tamaño que es relevante es el área aparente de la sección transversal de cada partícula. Por simplicidad, supongamos que las partículas son esféricas, lo que no es una mala aproximación para moléculas pequeñas. Si estamos mirando una esfera, lo que percibimos como el tamaño de la esfera es el área de sección transversal de un gran círculo. Recordemos que cada partícula esférica tiene asociada una “esfera de colisión” que solo encierra dos partículas en el contacto más cercano, es decir, en el momento de una colisión, y que esta esfera es un radio, donde está el diámetro de cada partícula esférica. La sección transversal de esta esfera de colisión representa una sección transversal efectiva para cada partícula dentro de la cual es inminente una colisión. La sección transversal de la esfera de colisión es el área de un gran círculo, que es. Denotamos esta aparente área de sección transversal. Así, para partículas esféricas y con diámetros y, las secciones transversales individuales son
La sección transversal de colisión, está determinada por un diámetro efectivo característico de ambas partículas. Los incrementos de probabilidad de colisión de ambas partículas tienen diámetros grandes y disminuye si una de ellas tiene un diámetro menor que la otra. De ahí que una simple medida sensible a esto es el promedio aritmético
y la sección transversal resultante de la colisión se convierte
que, curiosamente, es un promedio de los dos tipos de promedios de las dos secciones transversales individuales, ¡las medias aritméticas y geométricas!
Frecuencia promedio de colisión
Considera un sistema de partículas con secciones transversales individuales. Una partícula de sección transversal que se mueve una distancia en un tiempo barrerá un volumen cilíndrico (ignorando las tapas esféricas) de volumen (Figura 27.6.1
). Si el sistema tiene una densidad numérica, entonces el número de colisiones que ocurrirán es
Definimos la frecuencia de colisión para una sola molécula, también conocida como la tasa promedio de colisión como, i.e.,
donde es la velocidad promedio de una partícula
La ecuación no es del todo correcta porque se basa en el supuesto de que sólo se mueve la molécula de interés. Si tomamos en cuenta el hecho de que todas las partículas se mueven una respecto a la otra, y suponemos que todas las partículas son del mismo tipo (digamos, tipo), entonces realizar el promedio sobre una distribución de velocidad Maxwell-Boltzmann da
donde esta la masa reducida.
Por lo tanto,
y
El recíproco de es una medida del tiempo promedio entre colisiones para una sola molécula.
Trayectoria Media Libre
El camino libre medio es la distancia que recorrerá una partícula, en promedio, antes de experimentar un evento de colisión. Esto se define como el producto de la velocidad promedio de una partícula y el tiempo entre colisiones. El primero lo es, mientras que el segundo lo es. Por lo tanto, tenemos
El camino libre medio también se puede describir usando términos de la ley de gas ideal, porque :
Paseos Aleatorios
En cualquier sistema, una partícula que sufre frecuentes colisiones tendrá la dirección de su movimiento cambiada con cada colisión y trazará un camino que parece ser aleatorio. De hecho, si tratamos el proceso como estadístico, entonces, de hecho, estamos tratando cada evento de colisión como un evento aleatorio, ¡y la partícula cambiará su dirección en momentos aleatorios de maneras aleatorias! Tal trayectoria podría aparecer como se muestra en la Figura. Tal camino a menudo se conoce como un camino de caminata aleatorio.
Para analizar tales caminos, consideremos un paseo aleatorio en una dimensión. Supondremos que la partícula mueve una longitud de trayectoria libre de medias entre colisiones y que cada colisión cambia la dirección del movimiento de las partículas, lo que en una dimensión, significa que la partícula se mueve ya sea hacia la derecha o hacia la izquierda después de cada evento. Esto se puede mapear en un “lanzamiento de moneda” metafórico que puede subir cabezas “H” o colas “T”, con “H” causando movimiento hacia la derecha, y “T” causando movimiento hacia la izquierda. Que haya tales tiradas de monedas, sea el número de veces que aparezca “H” y denote el número de veces que aparece “T”. Así, el avance de la partícula, que definimos como movimiento neto hacia la derecha, viene dado por . Dejando , esto es justo. Así, necesitamos saber cuál es la probabilidad de obtener un valor particular de en un número muy grande de tiradas de monedas. Denote esto.
En los tirados de monedas, el número total de secuencias posibles de “H” y “T” es. Sin embargo, el número de formas en las que podemos obtener cabezas y) j) colas, con es un coeficiente binomial. Ahora
para que . Del mismo modo,
para que . Por lo tanto, la probabilidad es
Ahora tomamos el logaritmo de ambos lados:
y usa la aproximación de Stirling:
y escribir como
Ahora, escribe
y
Ahora usamos las expansiones
Si nos detenemos en el término de segundo orden, entonces
para que
Ahora bien, si dejamos y , y si dejamos ser una variable aleatoria continua, entonces la distribución de probabilidad correspondiente se convierte en
que es una distribución gaussiana simple. Ahora bien, es el número de colisiones, que viene dado por, así podemos escribir la distribución de probabilidad para que la partícula difunda una distancia en el tiempo como
Definir como la constante de difusión, que tiene unidades de (longitud)/time. The distribution then becomes
Tenga en cuenta que esta distribución satisface la siguiente ecuación:
que se llama la ecuación de difusión. La ecuación de difusión es, de hecho, más general que la distribución gaussiana en la Ecuación. It is capable of predicting the distribution in any one-dimensional geometry subject to any initial distribution and any imposed boundary conditions.
En tres dimensiones, consideramos que las tres direcciones espaciales son independientes, por lo tanto, la distribución de probabilidad para que una partícula se difunda a una ubicación is just a product of the three one-dimensional distributions:
y si solo estamos interesados en la difusión a distancia, we can introduce spherical coordinates, integrate over the angles, and we find that
Frecuencia total de colisión por unidad de volumen
En la ecuación 27.6.18, representa la frecuencia de colisión para una molécula específica en una muestra de gas. Si se desea calcular la frecuencia de colisión total por unidad de volumen, se debe tener en cuenta la densidad numérica de las moléculas,,. La frecuencia de colisión total en una muestra que contiene solo moléculas A,, es
Se debe incluir el factor de para evitar colisiones de doble conteo entre moléculas similares. (Este es un razonamiento idéntico al hecho de que solo hay una manera de rodar el doble 3 con dos dados).
Si tiene una muestra de gas que contiene moléculas A y moléculas B, entonces
donde
y
Ejemplo
Calcular la frecuencia de colisiones hidrógeno-hidrógeno en un contenedor de 1.00 centímetros cúbicos a 1.00 bar y 298 K.
Solución
El valor de es 2.30 x 10 -19 m 2.
La densidad numérica es
La velocidad promedio es
27.5: La distribución de Maxwell-Boltzmann ha sido verificada experimentalmente
27.7: Tasas de reacciones químicas en fase gaseosa |
7088 | https://www.jrlinteriors.com/blog/2025/1/25/7-winter-color-palettes | JRL Interiors
7 Winter Color Palettes
Winter whites aren’t the only winter colors, though looking outside around here right now you might question that.
Since this is the 526th day of January, it seems a good idea to embrace the unique light and nature colors of the season and celebrate them.
Here are 7 color palettes inspired by the season
I specify Benjamin Moore and Sherwin Williams paint most often. All colors referenced are from Benjamin Moore.
Winter Berry Colors
Red and green are not just for Christmas anymore. As complementary colors they are high contrast and add a bit of dramatic flair to any room.
Try deep green cabinets with a touch of red in the decorative accessories, or a cozy green living room with the dramatic punch of throw pillows in a pattern containing red.
Or red library shelves with green in the accents…anything from plants to pillows!
custom built-ins painted red in this mezzanine library, JRL Interiors
Cocoa Inspired Colors
Winter is also hot chocolate season! And nothing is cozier than a rich chocolate brown room. Warmer colors are trending and they are especially appealing in the cold winter months.
The transformation painting this home office in chocolate brown was remarkable - you can see the before pictures when it was just white here.
Pair chocolate with touches of blue and marshmallow white for classic elegance and timeless appeal.
Winter Green Colors
Snow frosted evergreens and pinecones are the quintessential elements of winter. Use a variety of shades of green and add warm browns for a fresh, earthy mood.
green library under the eaves, JRL interiors
A dash of contrasting snow white highlights and sharpens the colors.
family room with cozy green sofas, soft green walls, brown stained wood tables, and crisp white built-ins and trim, JRL Interiors
Winter Sunrise Colors
The pastels of a winter sunrise sparkle against a snow coated landscape. These happy colors are inspired by the sunrise picture below taken by my friend Bernadette.
shades of pink from pale raspberry to deep fuchsia make this glam hoe office a standout, JRL Interiors
These shades from peachy pink to lavender would make for a peaceful, cheerful space. A touch of brown offers a beautiful counterpoint to keep a pastel room from being too syrupy sweet.
Winter Neutral Colors
Natures winter neutrals are anything but boring.
A neutral pale taupe and white kitchen, with stained wood island, JRL Interiors
Complex shades of warm pale oak to deep taupe are inspired by these deer in the winter woods.
With neutral colors, it is especially important to pay attention to texture layering a variety from wicker and sisal to velvet and chenille to create subtle sophistication. And of course, just like these deer, every room is improved by a touch of black!
Winter Cardinal Colors
Of all the winter birds, cardinals are perhaps the most striking against the winter landscape.
And while Mrs. Cardinal in this pair looks completely nonplussed about whatever Mr. Cardinal has been mansplaining, they nevertheless inspire a palette of warm reds and mustard yellows against a tree bark taupe backdrop.
Winter Sky Colors
And finally, I would be remiss to not include a palette with a brilliant blue winter sky. The crisp cold air and angle of the sun this time of year produces the most remarkable shade of blue on a clear day.
This new build project features a palette of crisp blue and white (more pictures from this project here).
Blue and white is THE most classic color combination.
Winter Whites
Winter whites come in literally hundreds of variations. Most whites are not actually really white. They contain a variety of pigments that tint them in the direction of another hue - yellow, green, lavender, gray or peachy-pink.
These 5 exhibit a range of colors and can act as a whisper of color in a room when paired with a whiter white, or as a white when paired with a deeper color.
What colors are YOU embracing this winter?
Dreaming of spring? Check out some of our favorite nature inspired spring colors in this post., and our fall favorites here!
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7089 | https://americanhistory.si.edu/collections/object-groups/geometric-models-plane-dissections | Skip to main content
Geometric Models – Plane Dissections
To illustrate the Pythagorean Theorem, makers of geometric models have long made models with pieces that could be arranged either as a square with a side equal to the length of the hypotenuse of a right triangle or as two squares, with sides equal to the length of the two other sides of the triangle. In 1928, R. H. Wood, a student of high school teacher A. Harry Wheeler in Worcester, Massachusetts, made such a model.
Any two polygons of equal area can be divided into a finite number of polygonal pieces that can be arranged to form either polygon. This result was well known from the mid-1800s. A few model makers, such as Wheeler, took great delight in developing specific models of dissected polygons and figuring out different ways to arrange the pieces. Surviving notes from the early 1930s indicate that Wheeler designed models of relatively complicated plane dissections for his own pleasure. Then, mindful of the popularity of jigsaw puzzles in the Depression years, he made and encouraged his students to make dissections of simpler forms. Some of these models were hinged at vertices.
Wheeler classified his dissections according to the number of pieces used. The arrangement of records below follows this scheme. In some instances, pieces that fit together were assigned separate numbers. The separate records have been kept, with text indicating what fits together. Records on documentation relating to these models are at the end of the group. Throughout, clicking on the title of an object brings up further images and description.
Showing 1 to 24 of 82 items — page 1 of 4
Geometric Model by A. Harry Wheeler, Hexagon Transformable into a Hexagon, Plane Dissection
date made
: ca 1932
: c. 1932
maker
: Wheeler, Albert Harry
Description
: The three pieces of this hinged dissection are all divided in half, with half red and half black. They may be arranged to form a regular hexagon with an inscribed black equilateral triangle or a regular hexagon with an inscribed red equilateral triangle.
: This is Wheeler's model 3HA. A drawing (indicating two colors) dated March 23, 1932 is in 1979.3002.55 (folder 1). The model name is not indicated there.
: On his list of dissections, under regular dissections, Wheeler calls this "Hexagon Inside Out," and denotes it by 3HA See 1979.3002.43 (folder 2).
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Triangle Transformable into Triangle Transformable into Quadrilateral Transformable into Quadrilateral, Plane Dissection
maker
: Wheeler, Albert Harry
Description
: The three pieces of this hinged dissection have white, black and red plastic tops attached to wooden bases. They may be arranged as a quadrilateral in two ways and as a triangle in two ways.
: This is model 3J in Wheeler's list of decompositions of polygons, listed with the triangles.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Isoceles Triangle Transformable into Rectangle, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The three pieces of this hinged dissection may be arranged as a rectangle or as an isoceles triangle.
: Sheets entitled Regular Hexagons in 1979.3002.43, folder 2 discuss this model, where it is shown in a list of dissected rectangles. Wheeler apparently denoted it by 3O or 3Alpha. There is a drawing in 1979.3002.43, folder 1, dated January 29, 1932 and labeled 3alpha.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Square Transformable into a Square and Two Rectangles, Plane Dissection
maker
: Wheeler, Albert Harry
Description
: The hinged dissection has three hinged pieces (two of the triangles and the small square are glued together). They may be arranged as a large square or as two rectangles and a smaller square. If the side of the large square is c and the sides of the triangles are a and b (a longer than b), then the area of the small square is (a-b)2. The area of each of the four triangles is 1/2ab. Hence c2 = (a-b)2 + 4(1/2)ab. Hence c2 = a2 + b2, as the Pythagorean theorem would indicate. The model resembles Wheeler's 6SI (square into gnommon), but has only five pieces and is quite different from MA.304723.763, which is labeled as an example of that model.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Triangle Transformable into Right Triangle, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged model are made of plastic glued to wood. The pieces can be arranged as a triangle, a right triangle, or a rectangle. The green plastic is cut, but it covers only one piece
: This is 4A on Wheeler's list of decompositions of polygons. It is a dissection that transforms a triangle into a right triangle.
: A pattern for this model found in 1979.3002.55 is dated 1932.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Triangle Transformable into a Triangle, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection can be arranged as a triangle and as an isosceles triangle. The pieces are made of yellow, purple, blue and red plastic, glued to a wooden base.
: This appears to be model 4B in the list of triangles in Wheeler's list of decompositions of polygons.
: Drawings for this dissection, dated February 2, 1932, are in 1979.3002.43 (folder 2). Other drawings, dated February 4, 1932, are in 1979.3002.55, folder 1
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle Transformable into Square, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four plastic pieces fit in metal container with cloth-lined lid. The pieces may be arranged as a square or an equilateral triangle. Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823. This small version of the model has no hinges.
: This is similar but not identical to Wheeler's Model 4ST. Two drawings for that puzzle, both dated January 17, 1932 and April 26, 1932, are in 1979.3002.43, folder 1. The drawings refer to the Mathematical Gazette - no volume or page is given. A drawing in 1979.3002.53 dated January 5, 1932 refers to this model as 4C.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle Transformable into Square, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this model may be arranged as a square or and equilateral triangle. The container is a reused aspirin tin. A drawing in 1979.3002.53 dated January 5, 193? refers to this model as 4C. This small version of the model does not have hinges. It is stored in a metal case marked: ASPIRIN.
: Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
Location
: Currently not on view
Square Transformable into an Equilateral Triangle, Plane Dissection (Incomplete)
Description
: This crude, unpainted, incomplete wooden model consists of three pieces that can be arranged to form a five-sided figure that, with an additional triangle, would form a square. There is a pencil line across part of the front of the model, and masking tape at various intersections of the pieces on the back. With the additional triangle, the pieces could be arranged to form am equilateral triangle.
: For a complete examples of this model, see MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Square Transformable into an Equilateral Triangle, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection may be arranged as a square or as an equilateral triangle.
: Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823. There are two drawings for a related puzzle, 4ST, both dated January 17,1932 and April 26, 1932, in 1979.3002.43, folder 1. The drawings refers to the Mathematical Gazette. A drawing in 1979.3002.53 dated January 5, 1932 refers to this model as 4C.
: This is Wheeler's Model 4ST.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Square Transformable into an Equilateral Triangle, Plane Dissection
date made
: ca 1932.
maker
: Wheeler, Albert Harry
Description
: These four hinged pieces may be arranged a an equilateral triangle or a square.
: Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
: This is similar but not identical to Wheeler's Model 4ST. Two drawings for that puzzle, both dated January 17, 1932 and April 26, 1932, are in 1979.3002.43, folder 1. The drawings refers to the Mathematical Gazette - no volume or page is given. A drawing in 1979.3002.53 dated January 5, 193? refers to this model as 4C.
: References:
: Fredrickson, Greg N., Dissections Plane and Fancy, pp. 136-137 attributes publication of this dissection to Dudeney (1902).
: H. E. Dudeney, The Canterbury Puzzles, pp. 49-50, 178-180. Dudeney specifically points out on p.180 that the puzzle may be made with the pieces hinged. The first edition of the book appeared in 1907, the second in 1919.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Triangle Transformable into Triangle, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection have plastic tops glued to wooden bases. They can be arranged to form a triangle two ways.
: This is model 4D in Wheeler's triangles in his list of decompositions of polygons.
: A drawing of this model, dated January 4, 1932, is in 1979.3002.55, folder 1. It refers to Macaulay, Math. Gaz, 1915-1916, p. 382. Another drawing, marked D and also dated Janury 4, 1932 also shows this dissection. It is in 1979.3002.55, folder 2. There is a drawing in 1979.3002.50, dated November 21, 1931, that refers to a hinged model using this dissection - this model is larger than that drawing.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Triangle Transformable into Triangle (Incomplete), Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: This partially hinged dissection has three pieces. It can be arranged as a near-parallelogram or as a triangle in two ways. A protrusion from one edge and a hole in another edge do not meet.
: These are three of the four pieces in Wheeler's model 4H. A drawing for the model, dated January 6, 1932, is in 1979.3002.55, folder 2. It indicates that the missing piece is a triangle which can be added, with the pieces arranged to form a triangle in three ways.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Hexagon Transformable into Hexagon, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection may be arranged as a regular hexagon in two ways. An inscribed rectangle in red goes to one in green. The wooden core of the pieces is covered with plastic on the top and on the bottom.
: In his list of dissections, under "Regular Hexagons," Wheler describes this model as "Hexagon with a Hexagon Inside out" and denotes it as 4HG.
: A drawing of model 4HG (without the inscribed rectangle) in 1979.3002.55, folder 1, is dated April 21, 1932.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Hexagon Transformable into a Triangle, Plane Dissection
date made
: ca 1933
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection have green and black sections and may be arranged as a regular hexagon with a black rectangle inscribed or as 30-60-90 degree triangle with a green rectangle inscribed. Compare to 304723.794. This is Wheeler's model 4HH. There is a drawing in 1979.3002.55 (folder1), although it is not marked 4HH and the inscribed rectangle is not shown. A second drawing in this folder is denoted 4HH and dated April 21, 1933.
: In Wheeler's list of "Regular Hexagons" this is denoted as 4HH. See 1979.3002.43, folder 2.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Hexagon Transformable into Triangle, Plane Dissection
date made
: ca 1933
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection can be arrnaged to form a hexagon or a triangle. Decorative plastic pieces make it possible to show a red rectangle inscribed in the hexagon or a green rectangle inscribed in the triangle.
: Compare to 304723.789. This is Wheeler's model 4HH.
: In Wheeler's list of "Regular Hexagons" this is denoted as 4HH. See 1979.3002.43, folder 2. A drawing is in 1979.3002.55 (folder1), although it is not marked 4HH and the inscribed rectangle is not shown. A second drawing in this folder is denoted 4HH and dated April 21, 1933.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle Transformable into a Hexagon with Hole, Plane Dissection
date made
: ca 1931-1933
maker
: Wheeler, Albert Harry
Description
: This model has three triangular pieces, with black plastic on top and wood on the bottom. They can be arranged to form an equilateral triangle or hexagon with a hole in the shape of an equilateral triangle.
: The object is shown (with hole) as model 4HI in the list of "regular hexagons" in 1979.3002.43, folder 2. A drawing in 1979.3002.55 (folder 1) shows the model but does not give its name. A second drawing, dated April 21, 1931, is in the same folder and has the name designated.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle Transformable into a Hexagon with Hole, Plane Dissection
date made
: ca 1931-33
: ca 1931 -1933
maker
: Wheeler, Albert Harry
Description
: The three pieces of this hinged dissection can be arranged to form an equilateral triangle or a into a hexagon with a hole in the shape of an equilateral triangle.
: With a second model of an equilateral triangle, this is Wheeler's model 4HI, a hexagon transformable into two equilateral triangles or a rhombus. This is in Wheeler's list of dissections, 1979.3002.43, folder 2. There is a drawing in 1979.3002.55 (folder 1), but the name is not designated on the drawing. There is a second drawing with the name designated in the same folder, dated April 21, 1931.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Hexagon Transformable into a Rhombus, Plane Dissection
date made
: ca 1932
: c. 1932
maker
: Wheeler, Albert Harry
Description
: Thr four pieces of this hinged dissection can be arranged as a regular hexagon or as a rhombus.
: A drawing dated April 16, 1932 in 1979.3002.55, folder 1 shows this model. In his list of dissections, under "Regular Hexagons," Wheeler describes this model as "Hexagon into a Rhombus." He denotes it by 4HO.
: Another related drawing is in 1979.3002.43, folder 1.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle with Inscribed Rectangle or Parallelogram with an Inscribed Square, Plane Dissection
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection have plastic attached to both sides and a wooden central core. The pieces form an equilateral triangle with an inscribed rectangle which transforms into a parallelogram with an inscribed square.
: This is a special case of Wheeler's model 4K (trapezoid into a triangle) in his list of decompositions of polygons.
: Compare MA.304723.814.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle with Inscribed Rectangle or Rhombus with Inscribed Rectangle, Plane Dissection
date made
: ca 1932
: ca 1932 03 10
: c. 1932
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection have plastic on both sides and a wooden core. They can be arranged as an equilateral triangle or a rhombus .The pieces on one side have smaller plastic parts on them so that the equilateral triangle and the rhombus both appear to have an inscribed rectangle
: A pattern for this model found in 1979.3002.55, folder 1, is dated March 10, 1932. See also 1979.3002.43, folder 2.
: Compare MA.304723.813 and MA.304723.814.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Equilateral Triangle with Inscribed Rectangle or Parallelogram with an Inscribed Square, Plane Dissection
maker
: Wheeler, Albert Harry
Description
: The four pieces of this hinged dissection have plastic on both sides and a wooden core. They can be arranged as an equilateral triangle or a square.The pieces on one side have smaller plastic parts on them so that the equilateral triangle appears to have an inscribed rectangle.
: This is a special case of Wheeler's model 4K (trapezoid into a triangle) in his list of decompositions of polygons.
: Compare MA.304723.813.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Isoceles Triangle Transformable into Rectangle, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: This hinged dissection has four pieces that may be arranged as a rectangle or a triangle.
: This is model 4P among the triangles in Wheeler's list of decompositions of polygons. A drawing dated January 9, 1932 , is in 1979.3002.55, folder 2.
Location
: Currently not on view
Geometric Model by A. Harry Wheeler, Inscribed Quadrilateral Transformable into Inscribed Quadrilaterals, Plane Dissection
date made
: ca 1932
maker
: Wheeler, Albert Harry
Description
: This is Wheeler's model 4Q. It shows an inscribed quadrilateral in four pieces. Each piece is a section of a circle. It is divided into segments of two colors of plastic, so that there is a yellow triangle with a vertex at the center of the circle and a lune in another color. The model is shown in drawing in 1979.3002.55, folder 1, and dated April 9, 1932 . The four parts can be permuted six ways to form six different inscribed quadrilaterals all inscriptible in the same circle.
Location
: Currently not on view
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How to determine the effectiveness of a code review process?
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We've introduced a code review process within our organisation and it seems to be working well. However, I would like to be able to measure the effectiveness of the process over time, i.e. are we not finding bugs because the code is clean or are people just not picking up on bugs?
Currently, we don't have an effective fully-automated test process. We primarily employ manual testing, so we can't rely on defects found at this stage to ensure that the code review process is working.
Has anyone come across this issue before or has any thoughts on what works well in measuring code reviews?
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edited Jan 12, 2012 at 18:35
Thomas Owens♦
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asked Jan 12, 2012 at 16:38
Johnv2020Johnv2020
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Finding bugs is not the only purpose of code reviews. They are also useful for reinforcing coding standards, cross-training, cross-pollinating of ideas and technologies, etc
Jason
–
Jason
01/12/2012 16:59:20
Commented
Jan 12, 2012 at 16:59
Thanks Jason & understood, however in this case I'm trying to figure out how to ensure that the process achieves its core aim of defect prevention as early in the dev process as possible
Johnv2020
–
Johnv2020
01/12/2012 17:05:08
Commented
Jan 12, 2012 at 17:05
@Johnv2020 That is not its core aim though... You completely miss the point of a code review. This would be like putting in a great new safety feature on a fleet of jet aircraft, then trying to judge its effectiveness based on the number of crashes. There are too many variables and other factors to consider to accurately make any claim that the safety feature improved the odds of a crash not occurring.
maple_shaft
–
maple_shaft
♦
01/12/2012 17:45:07
Commented
Jan 12, 2012 at 17:45
@maple_shaft: Weak analogy. Trying to gauge bug rates is more like trying to measure the number of coffins used for dead people from crashes.
S.Lott
–
S.Lott
01/12/2012 17:52:14
Commented
Jan 12, 2012 at 17:52
1
In all the code reviews I've attended, many bugs have been fixed already in unit and higher-level testing. That is, code is not ready for review just because it compiles.
Pete Wilson
–
Pete Wilson
01/12/2012 20:52:50
Commented
Jan 12, 2012 at 20:52
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There are a number of metrics that can be gathered from code reviews, some even extending throughout the lifecycle of the project.
The first metric that I would recommend gathering is defect removal effectiveness (DRE). For every defect, you identify what phase the defect was introduced in and what phase it was removed in. The various defect detection techniques that you use are all assessed simultaneously, so it applies equally to requirements reviews, design reviews, code reviews, unit tests, and so on. You would be particularly interested in the number of defects caught in the code phase, since this would probably encompass your unit tests as well as code reviews. If many defects from the code phase are making it through to the integration test phase or even the field, you know that you post-coding practices should be evaluated.
Various meeting metrics would also be relevant. These include the time to prepare, time in meeting, lines of code read, defects found in the review, and so on. Some observations can be made from this data. As an example would be if your reviewers are spending a large amount of time reading the code in preparation for the review, but finding very few problems. Coupled with the DRE data, you can draw the conclusion that if defects are being tested in integration testing or the field, then your team needs to focus on their review techniques to be able to find problems. Another interesting note would be the lines of code (or some other size measurement) read in a meeting compared to the time of the meeting. Research has found that the speed of a typical code review is 150 lines of code per hour.
With any of these metrics, it's then important to understand their impact on the process. Root cause analysis, using techniques such as why-because, Five Whys, or Ishikawa diagrams can be used to identify the reasons why code reviews (or any other quality improvement technique) are (in)effective.
You might also be interested in this article about inspections from The Ganssle Group and an article by Capers Jones in Crosstalk about Defect Potentials and DRE.
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answered Jan 12, 2012 at 17:51
Thomas Owens♦Thomas Owens
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While largely it is true that code review would pick up problems which are rather latent that testing may or may not catch. However, in my opinion you may have a really stable (practically bug free) code but still written in such a way that it is extremely non-readable or not quite maintainable. So it may be that code review may NOT find bugs if there are no real issues actually in the code.
Having said that, i would really ask, why would one want to do code review? The simple reason why it is important is that the code should be improved to be made more readable, maintainable and evolvable. Many people should be able to read cleaner code and make sense out of it. In that sense, simplest objective of the code review process is to produce clean code. So the measure of effectiveness is how much cleaner the code is now.
As you wanted to have a measurable effectiveness - here is what i would suggest:
Metric related to amount of rework -
Number of time the rework is applied in a same given module/object/work item is a measure of how poor that code is in terms of maintainability. When effective code-review is applied, how often are we able to reduce the re-work request on the same module?
Metric related to amount of change that every change request incurs.
When every time a change request occurs - a poorly factored code will always have larger number of modules get affected. A measure would probably indicate that after a code review - a was that any reduction of such spread of change for a similar change request in the past?
Metric related to average speed with which a change request can be responded.
When the code is cleaner - faster and better it is to respond to required changes. After the code was cleaned in the review process, die we find any speed up in responding the similar size request.
I am not putting exact units of measures - you can probably craft more accurate measure about this from this approach. There can be more extension formalism in the above approaches on this.
Basically, my point is that instead of looking at the number of bugs the code-review process identifies; we should measure the effectiveness in terms of whether code-review has been able to bring code to be more cleaner, leaner and easy-to-maintain state; hence, we can gauge that effectiveness if we see that similar change requests in future becomes more efficient to be responded.
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answered Jan 12, 2012 at 18:35
Dipan MehtaDipan Mehta
10.6k22 gold badges3535 silver badges6767 bronze badges
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Although not a "bug", a lack of readability, maintaibility, or evolvability is a defect in the code and should be treated as such. There's no reason why these can't be tracked in a defect tracker, right along with actual bugs in functionality. By doing this, you also open up the ability to track a number of other defect-related metrics in the coding phase.
Thomas Owens
–
Thomas Owens
♦
01/12/2012 18:41:26
Commented
Jan 12, 2012 at 18:41
1
As a developer I sure like to see clean code. However, code reviews are very costly. So as a manager funding a project, clean code is really not a compelling reason to add 5-10% in costs and time to my development budget. Especially when (as a manager) my bonus/review is tied to completing the current project on-time/in-budget. So your opinion that the main reason for code reviews is to get clean code would make any good manager say the ROI is not worth it. You can argue about long-term returns, but by then the manager that delivers on-time/on-budget will have been promoted away from that prob
Dunk
–
Dunk
01/12/2012 20:22:56
Commented
Jan 12, 2012 at 20:22
...problem. While the manager who promoted the code reviews will have successful maintenance projects but will have been reamed for not completing the original project on-time/in-budget like the manager who didn't. OTOH, if the code reviews helped find bugs that the lack of review didn't and that let the code review manager complete his project more on-time/in-budget then it is a different story. That is the story that needs to be sold. Which also means that clean code cannot be the reason for code reviews.
Dunk
–
Dunk
01/12/2012 20:25:49
Commented
Jan 12, 2012 at 20:25
@Dunk The cost of a code review depends on the type of code review. A formal inspection with 3-5 readers, a moderator, and the presence of the author (5-7 people in a room) is expensive. A desk check that consists of another developer glancing over the code for 10-15 minutes is also a code review, but much less formal and much cheaper. Even pair programming can be considered a "code review" technique of sorts. The appropriate technique is determined by factors including (but not limited to) the criticality of the code, the desired defect rate, and the amount of time/money to be invested.
Thomas Owens
–
Thomas Owens
♦
01/12/2012 20:43:36
Commented
Jan 12, 2012 at 20:43
@Dunk - I think you've made an argument for taking the "should we code review" decision out of the hands of the project manager, and placing it in the hands of the manager who has responsibility for the software platform long term. IMO, generally speaking, spending an extra 5-10% on development for decent code reviews is a worthwhile investment in terms of the longevity of the system being developed. But probably not in terms of the budget and timeline of the current project.
Dawood ibn Kareem
–
Dawood ibn Kareem
01/12/2012 21:42:40
Commented
Jan 12, 2012 at 21:42
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7091 | https://teachingcalculus.com/2017/01/10/density-functions/ | Teaching Calculus
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Density Functions
Density, as an application of integration, has snuck onto the exams. It is specifically not mentioned in the “Curriculum Framework” chapter of the 2016 Course and Exam Description, there is one example in the 2020 CED There is an example (#12 p. 58) in the AB sample exam question section of 2020 Course and Exam Description. The first time this topic appeared was in the 2008 AB Calculus exam. There was a hint in the few years before that with a question in the old Course Description book. Both questions will be discussed below. The idea is that students are supposed to understand integration well enough to apply their knowledge to a new situation (density).
The Mathematics
A density function gives the amount of something per unit of length, area, or volume, for example
The density can be used to find the amount. In each example, notice that the length, area, or volume of the region is multiplied by the density to find the amount.
Example 1: A 10 cm rod with a constant density of 3 g/cm has a mass of
In other situations, the density is not constant and is given by some function. Suppose our metal rod of length b has a density of grams/cm where x is measured from one end of the rod. To find the total mass we think of cutting the rod in the very small pieces. (Think partition: each piece has a length of in which the density is nearly constant say .) The sum of the mass of these pieces is the Riemann sum . The limit of this expression as gives the total mass in grams: Notice that is the length of the rod. This is multiplied by the density to find the mass.
Example 2: The next example is from the old Course Description book.
A city is built around a circular lake that has a radius of 1 mile. The population density of the city is people per square mile, where r is the distance from the center of the lake, in miles. Which of the following expressions gives the number of people who live within 1 mile of the lake?
(A) (B)
(C) (D)
(E)
We need to partition the region so that each piece has a close to a constant density. Thin rings around the lake will accomplish this. A ring, if straightened out is similar to a rectangle of length where is the distance from the center of the lake (this is the circumference of the ring), the width of this ring (rectangle) is . In this ring (rectangle) the population density is people per square mile, so the population in the ring is approximated by multiplying the area by the density: . Adding these gives a Riemann sum whose limit gives the total population:
Answer (D).
The limits of integration are from the edge of the lake, r = 1 to r = 2 (“one mile from the lake”). Another way to look at this is that is the area of the city; this is multiplied by the population density to find the population.
This type of density situation is called a radial density function.
Notice that the answer looks like the formula for volume by cylindrical shells; this is not quite an accident. The rings around the center are like the shells used when finding volume. It is the units that are different.
Example 3: From the 2008 AB Calculus exam #92.
A city located beside a river has a rectangular boundary as shown in the figure above. The population density of the city at any point along a strip x miles from the river’s edge is people per square mile. Which of the following expressions gives the population of the city?
(A) (B) (C)
(D) (E)
A thin vertical strip of the city miles to the right of the river has an area of . The population in each such strip is found by multiplying the area by the density function; this gives . These are then added forming a Riemann sum, etc.
Answer (B)
Alternative solution: When I first saw this question, not having thought about density for quite a while, I answered it by doing a unit analysis. Since unit analysis is a good thing for students to understand I’ll outline my thinking next.
We are looking for the population so the answer must be in units of “people.” The density function is in units of “people/square mile” (given). Both x and dx are in units of “miles” and the “7” also has units of “miles.” Therefore, the only choice that gives “people” is the one that multiplies the 7, the dx and the density function. This eliminates (A) and (D). The 28 in (C) must be square miles, making the overall units “people-miles” which is not what we’re going for. Finally, choice (E) is eliminated since the 7 and the dx are not in the same direction. This leaves (B).
Example 4: A volume problem adapted from Calculus by Hughes-Hallett, Gleason, et al.
The density of air h meters above the earth’s surface is . Find the mass of a column of air 25 km high with a square base 3 meters on a side sitting on the surface of the earth.
At any height, h meters above the earth the volume of a thin slice of the column of air is . The mass of this slice is . The sum of these slices gives a Riemann sum whose limit gives the total volume:
For other examples see 2018 BC 2, and 2021 AB 1, Good Question 15 and Good Question 16
Check the index of your textbook for density problems. Calculus by Hughes-Hallett Gleason, et al and Calculus by Rogawski (2nd edition) have good exercises and examples. My advice is not to make too big a deal of this, but if you have time, you can take a look. Should this kind of question appear on the free-response section I would guess that the question will be carefully worded so that students who never saw this kind of question would have a good chance of answering it.
The changing population density of Sydney, Australia in persons per hectare. Note the date changes in the key at the lower left.
Revised and updated 8-20-2019
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Section0.4Functions
¶
A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain. We would write f:X→Y to describe a function with name f, domain X and codomain Y. This does not tell us which function f is though. To define the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule).
For example, consider the function f:N→N defined by f(x)=x2+3. Here the domain and codomain are the same set (the natural numbers). The rule is: take your input, multiply it by itself and add 3. This works because we can apply this rule to every natural number (every element of the domain) and the result is always a natural number (an element of the codomain). Notice though that not every natural number actually is an output (there is no way to get 0, 1, 2, 5, etc.). The set of natural numbers that are actually outputs is called the range of the function (in this case, the range is {3,4,7,12,19,28,…}, all the natural numbers that are 3 more than a perfect square).
The key thing that makes a rule actually a function is that there is exactly one output for each input. That is, it is important that the rule be a good rule. What output do we assign to the input 7? There can only be one answer for any particular function.
The description of the rule can vary greatly. We might just give a list of the images of each input. You could also describe the function with a table or a graph or in words.
Example0.4.1
The following are all examples of functions:
f:Z→Z defined by f(n)=3n. The domain and codomain are both the set of integers. However, the range is only the set of integer multiples of 3.
g:{1,2,3}→{a,b,c} defined by g(1)=c, g(2)=a and g(3)=a. The domain is the set {1,2,3}, the codomain is the set {a,b,c} and the range is the set {a,c}. Note that g(2) and g(3) are the same element of the codomain. This is okay since each element in the domain still has only one output.
h:{1,2,3}→{1,2,3} defined as follows:
This means that the function f sends 1 to 2, 2 to 1 and 3 to 3: just follow the arrows.
The arrow diagram used to define the function above can be very helpful in visualizing functions. We will often be working with functions with finite domains, so this kind of picture is often more useful than a traditional graph of a function. A graph of the function in example 3 above would look like this:
It would be absolutely WRONG to connect the dots or try to fit them to some curve. There are only three elements in the domain. A curve suggests that the domain contains an entire interval of real numbers. Remember, we are not in calculus any more!
Since we will so often use functions with small domains and codomains, let's adopt some notation that is a little easier to work with than that of examples 2 and 3 above. All we need is some clear way of denoting the image of each element in the domain. In fact, writing a table of values would work perfectly:
| | | | | | |
--- --- --- |
| x | 0 | 1 | 2 | 3 | 4 |
| f(x) | 3 | 3 | 2 | 4 | 1 |
We simplify this further by writing this as a matrix with each input directly over its output:
f=(0123433241)
Note this is just notation and not the same sort of matrix you would find in a linear algebra class (it does not make sense to do operations with these matrices, or row reduce them, for example).
It is important to know how to determine if a rule is or is not a function. Drawing the arrow diagrams can help.
Example0.4.2
Which of the following diagrams represent a function? Let X={1,2,3,4} and Y={a,b,c,d}.
Solution
(f) is a function. So is (g\text{.}) There is no problem with an element of the codomain not being the image of any input, and there is no problem with (a) from the codomain being the image of both 2 and 3 from the domain. We could use our two-line notation to write these as
\begin{equation}
f= \begin{pmatrix} 1 \amp 2 \amp 3 \amp 4 \ d \amp a \amp c \amp b \end{pmatrix} \qquad g = \begin{pmatrix} 1 \amp 2 \amp 3 \amp 4 \ d \amp a \amp a \amp b \end{pmatrix}.
\end{equation}
However, (h) is NOT a function. In fact, it fails for two reasons. First, the element 1 from the domain has not been mapped to any element from the codomain. Second, the element 2 from the domain has been mapped to more than one element from the codomain ((a) and (c)). Note that either one of these problems is enough to make a rule not a function. In general, neither of the following mappings are functions:
It might also be helpful to think about how you would write the two-line notation for (h\text{.}) We would have something like:
\begin{equation}
h=\begin{pmatrix} 1 \amp 2 \amp 3 \amp 4 \ \amp a,c? \amp d \amp b\end{pmatrix}.
\end{equation}
There is nothing under 1 (bad) and we needed to put more than one thing under 2 (very bad). With a rule that is actually a function, the two-line notation will always “work”.
SubsectionSurjections, Injections, and Bijections
¶
We now turn to investigating special properties functions might or might not possess.
In the examples above, you may have noticed that sometimes there are elements of the codomain which are not in the range. When this sort of the thing does not happen, (that is, when everything in the codomain is in the range) we say the function is onto or that the function maps the domain onto the codomain. This terminology should make sense: the function puts the domain (entirely) on top of the codomain. The fancy math term for an onto function is a surjection, and we say that an onto function is a surjective function.
In pictures:
Example0.4.3
Which functions are surjective (i.e., onto)?
f:Z→Z defined by f(n)=3n.
g:{1,2,3}→{a,b,c} defined by g=(123caa).
h:{1,2,3}→{1,2,3} defined as follows:
Solution
(f) is not surjective. There are elements in the codomain which are not in the range. For example, no (n \in \Z) gets mapped to the number 1 (the rule would say that (\frac{1}{3}) would be sent to 1, but (\frac{1}{3}) is not in the domain). In fact, the range of the function is (3\Z) (the integer multiples of 3), which is not equal to (\Z\text{.})
(g) is not surjective. There is no (x \in {1,2,3}) (the domain) for which (g(x) = b\text{,}) so (b\text{,}) which is in the codomain, is not in the range. Notice that there is an element from the codomain “missing” from the bottom row of the matrix.
(h) is surjective. Every element of the codomain is also in the range. Nothing in the codomain is missed.
To be a function, a rule cannot assign a single element of the domain to two or more different elements of the codomain. However, we have seen that the reverse is permissible: a function might assign the same element of the codomain to two or more different elements of the domain. When this does not occur (that is, when each element of the codomain is the image of at most one element of the domain) then we say the function is one-to-one. Again, this terminology makes sense: we are sending at most one element from the domain to one element from the codomain. One input to one output. The fancy math term for a one-to-one function is an injection. We call one-to-one functions injective functions.
In pictures:
Example0.4.4
Which functions are injective (i.e., one-to-one)?
f:Z→Z defined by f(n)=3n.
g:{1,2,3}→{a,b,c} defined by g=(123caa).
h:{1,2,3}→{1,2,3} defined as follows:
Solution
(f) is injective. Each element in the codomain is assigned to at most one element from the domain. If (x) is a multiple of three, then only (x/3) is mapped to (x\text{.}) If (x) is not a multiple of 3, then there is no input corresponding to the output (x\text{.})
(g) is not injective. Both inputs (2) and (3) are assigned the output (a\text{.}) Notice that there is an element from the codomain that appears more than once on the bottom row of the matrix.
(h) is injective. Each output is only an output once.
From the examples above, it should be clear that there are functions which are surjective, injective, both, or neither. In the case when a function is both one-to-one and onto (an injection and surjection), we say the function is a bijection, or that the function is a bijective function.
SubsectionInverse Image
¶
When discussing functions, we have notation for talking about an element of the domain (say x) and its corresponding element in the codomain (we write f(x), which is the image of x). It would also be nice to start with some element of the codomain (say y) and talk about which element or elements (if any) from the domain it is the image of. We could write “those x in the domain such that f(x)=y,” but this is a lot of writing. Here is some notation to make our lives easier.
Suppose f:X→Y is a function. For y∈Y (an element of the codomain), we write f−1(y) to represent the set of all elements in the domain X which get sent to y. That is, f−1(y)={x∈X:f(x)=y}. We say that f−1(y) is the complete inverse image of y under f.
WARNING: f−1(y) is not an inverse function! Inverse functions only exist for bijections, but f−1(y) is defined for any function f. The point: f−1(y) is a set, not an element of the domain.
Example0.4.5
Consider the function f:{1,2,3,4,5,6}→{a,b,c,d} given by
f=(123456aabccc).
Find the complete inverse image of each element in the codomain.
Solution
Remember, we are looking for sets.
\begin{equation}
f\inv(a) = {1,2}
\end{equation}
\begin{equation}
f\inv(b) = {3}
\end{equation}
\begin{equation}
f\inv(c) = {4,5,6}
\end{equation}
\begin{equation}
f\inv(d) = \emptyset.
\end{equation}
Example0.4.6
Consider the function g:Z→Z defined by g(n)=n2+1. Find g−1(1), g−1(2), g−1(3) and g−1(10).
Solution
To find (g\inv(1)\text{,}) we need to find all integers (n) such that (n^2 + 1 = 1\text{.}) Clearly only 0 works, so (g\inv(1) = {0}) (note that even though there is only one element, we still write it as a set with one element in it).
To find (g\inv(2)\text{,}) we need to find all (n) such that (n^2 + 1 = 2\text{.}) We see (g\inv(2) = {-1,1}\text{.})
If (n^2 + 1 = 3\text{,}) then we are looking for an (n) such that (n^2 = 2\text{.}) There are no such integers so (g\inv(3) = \emptyset\text{.})
Finally, (g\inv(10) = {-3, 3}) because (g(-3) = 10) and (g(3) = 10\text{.})
Since f−1(y) is a set, it makes sense to ask for ∣∣f−1(y)∣∣, the number of elements in the domain which map to y.
Example0.4.7
Find a function f:{1,2,3,4,5}→N such that ∣∣f−1(7)∣∣=5.
Solution
There is only one such function. We need five elements of the domain to map to the number (7 \in \N\text{.}) Since there are only five elements in the domain, all of them must map to 7. So
\begin{equation}
f = \begin{pmatrix}1 \amp 2 \amp 3 \amp 4 \amp 5 \ 7 \amp 7 \amp 7 \amp 7 \amp 7\end{pmatrix}.
\end{equation}
Function Definitions
A function is a rule that assigns each element of a set, called the domain, to exactly one element of a second set, called the codomain.
Notation: f:X→Y is our way of saying that the function is called f, the domain is the set X, and the codomain is the set Y.
To specify the rule for a function with small domain, use two-line notation by writing a matrix with each output directly below its corresponding input, as in:
f=(12342131).
f(x)=y means the element x of the domain (input) is assigned to the element y of the codomain. We say y is an output. Alternatively, we call y the image of x under f.
The range is a subset of the codomain. It is the set of all elements which are assigned to at least one element of the domain by the function. That is, the range is the set of all outputs.
A function is injective (an injection or one-to-one) if every element of the codomain is the output for at most one element from the domain.
A function is surjective (a surjection or onto) if every element of the codomain is the output of at least one element of the domain.
A bijection is a function which is both an injection and surjection. In other words, if every element of the codomain is the output of exactly one element of the domain.
The image of an element x in the domain is the element y in the codomain that x is mapped to. That is, the image of x under f is f(x).
The complete inverse image of an element y in the codomain, written f−1(y), is the set of all elements in the domain which are assigned to y by the function.
SubsectionExercises
¶
1
Write out all functions f:{1,2,3}→{a,b} (using two-line notation). How many are there? How many are injective? How many are surjective? How many are both?
Solution
There are 8 different functions. In two-line notation these are:
\begin{equation}
f = \begin{pmatrix} 1 \amp 2 \amp 3 \ a \amp a\amp a \end{pmatrix} \quad f = \begin{pmatrix} 1 \amp 2 \amp 3 \ b \amp b \amp b \end{pmatrix}
\end{equation}
\begin{equation}
f = \begin{pmatrix} 1 \amp 2 \amp 3 \ a \amp a\amp b \end{pmatrix} \quad f = \begin{pmatrix} 1 \amp 2 \amp 3 \ a \amp b \amp a \end{pmatrix} \quad f = \begin{pmatrix} 1 \amp 2 \amp 3 \ b \amp a\amp a \end{pmatrix}
\end{equation}
\begin{equation}
\quad f = \begin{pmatrix} 1 \amp 2 \amp 3 \ b \amp b \amp a \end{pmatrix} \quad f = \begin{pmatrix} 1 \amp 2 \amp 3 \ b \amp a\amp b \end{pmatrix} \quad f = \begin{pmatrix} 1 \amp 2 \amp 3 \ a \amp b \amp b \end{pmatrix}
\end{equation}
None of the functions are injective. Exactly 6 of the functions are surjective. No functions are both (since no functions here are injective).
2
Write out all functions f:{1,2}→{a,b,c} (in two-line notation). How many are there? How many are injective? How many are surjective? How many are both?
Solution
There are 9 functions: you have a choice of three outputs for (f(1)\text{,}) and for each, you have three choices for the output (f(2)\text{.}) Of these functions, 6 are injective, 0 are surjective, and 0 are both:
\begin{equation}
f = \twoline{1 \amp 2}{a\amp a} \quad f = \twoline{1 \amp 2}{b \amp b} \quad f = \twoline{1 \amp 2}{c \amp c}
\end{equation}
\begin{equation}
f = \twoline{1 \amp 2}{a\amp b} \quad f = \twoline{1 \amp 2}{a \amp c} \quad f = \twoline{1 \amp 2}{b \amp c}
\end{equation}
\begin{equation}
f = \twoline{1 \amp 2}{b \amp a} \quad f = \twoline{1 \amp 2}{c \amp a} \quad f = \twoline{1 \amp 2}{c \amp b}
\end{equation}
3
Consider the function f:{1,2,3,4,5}→{1,2,3,4} given by the table below:
| | | | | | |
--- --- --- |
| x | 1 | 2 | 3 | 4 | 5 |
| f(x) | 3 | 2 | 4 | 1 | 2 |
Is f injective? Explain.
Is f surjective? Explain.
Write the function using two-line notation.
4
Consider the function f:{1,2,3,4}→{1,2,3,4} given by the graph below.
Is f injective? Explain.
Is f surjective? Explain.
Write the function using two-line notation.
5
For each function given below, determine whether or not the function is injective and whether or not the function is surjective.
f:N→N given by f(n)=n+4.
f:Z→Z given by f(n)=n+4.
f:Z→Z given by f(n)=5n−8.
f:Z→Z given by f(n)={n/2 if n is even(n+1)/2 if n is odd.
Solution
(f) is injective, but not surjective (since 0, for example, is never an output).
(f) is injective and surjective. Unlike in the previous question, every integers is an output (of the integer 4 less than it).
(f) is injective, but not surjective (10 is not 8 less than a multiple of 5, for example).
(f) is not injective, but is surjective. Every integer is an output (of twice itself, for example) but some integers are outputs of more than one input: (f(5) = 3 = f(6)\text{.})
6
Let A={1,2,3,…,10}. Consider the function f:P(A)→N given by f(B)=|B|. That is, f takes a subset of A as an input and outputs the cardinality of that set.
Is f injective? Prove your answer.
Is f surjective? Prove your answer.
Find f−1(1).
Find f−1(0).
Find f−1(12).
Solution
(f) is not injective. To prove this, we must simply find two different elements of the domain which map to the same element of the codomain. Since (f({1}) = 1) and (f({2}) = 1\text{,}) we see that (f) is not injective.
(f) is not surjective. The largest subset of (A) is (A) itself, and (|A| = 10\text{.}) So no natural number greater than 10 will ever be an output.
(f\inv(1) = {{1}, {2}, {3}, \ldots {10}}) (the set of all the singleton subsets of (A)).
(f\inv(0) = {\emptyset}\text{.}) Note, it would be wrong to write (f\inv(0) = \emptyset) - that would claim that there is no input which has 0 as an output.
(f\inv(12) = \emptyset\text{,}) since there are no subsets of (A) with cardinality 12.
7
Let A={n∈N:0≤n≤999} be the set of all numbers with three or fewer digits. Define the function f:A→N by f(abc)=a+b+c, where a, b, and c are the digits of the number in A. For example, f(253)=2+5+3=10.
Find f−1(3).
Find f−1(28).
Is f injective. Explain.
Is f surjective. Explain.
Solution
(f\inv(3) = {003, 030, 300, 012, 021, 102, 201, 120, 210, 111})
(f\inv(28) = \emptyset) (since the largest sum of three digits is (9+9+9 = 27))
Part (a) proves that (f) is not injective. The output 3 is assigned to 10 different inputs.
Part (b) proves that (f) is not surjective. There is an element of the codomain (28) which is not assigned to any inputs.
8
Let f:X→Y be some function. Suppose 3∈Y. What can you say about f−1(3) if you know,
f is injective? Explain.
f is surjective? Explain.
f is bijective? Explain.
Solution
(|f\inv(3)| \le 1\text{.}) In other words, either (f\inv(3)) is the emptyset or is a set containing exactly one element. Injective functions cannot have two elements from the domain both map to 3.
(|f\inv(3)| \ge 1\text{.}) In other words, (f\inv(3)) is a set containing at least one elements, possibly more. Surjective functions must have something map to 3.
(|f\inv(3)| = 1\text{.}) There is exactly one element from (X) which gets mapped to 3, so (f\inv(3)) is the set containing that one element.
9
Find a set X and a function f:X→N so that f−1(0)∪f−1(1)=X.
Solution
(X) can really be any set, as long as (f(x) = 0) or (f(x) = 1) for every (x \in X\text{.}) For example, (X = \N) and (f(n) = 0) works.
10
What can you deduce about the sets X and Y if you know …
there is an injective function f:X→Y? Explain.
there is a surjective function f:X→Y? Explain.
there is a bijectitve function f:X→Y? Explain.
11
Suppose f:X→Y is a function. Which of the following are possible? Explain.
f is injective but not surjective.
f is surjective but not injective.
|X|=|Y| and f is injective but not surjective.
|X|=|Y| and f is surjective but not injective.
|X|=|Y|, X and Y are finite, and f is injective but not surjective.
|X|=|Y|, X and Y are finite, and f is surjective but not injective.
12
Let f:X→Y and g:Y→Z be functions. We can define the composition of f and g to be the function g∘f:X→Z which the image of each x∈X is g(f(x)). That is, plug x into f, then plug the result into g (just like composition in algebra and calculus).
If f and g are both injective, must g∘f be injective? Explain.
If f and g are both surjective, must g∘f be surjective? Explain.
Suppose g∘f is injective. What, if anything, can you say about f and g? Explain.
Suppose g∘f is surjective. What, if anything, can you say about f and g? Explain.
Hint
Work with some examples. What if (f = \twoline{1\amp 2 \amp 3}{a \amp a \amp b}) and (g = \twoline{a\amp b \amp c}{5 \amp 6 \amp 7}\text{?})
13
Consider the function f:Z→Z given by f(n)={n+1 if n is evenn−3 if n is odd.
Is f injective? Prove your answer.
Is f surjective? Prove your answer.
Solution
(f) is injective.
###### Proof
Let (x) and (y) be elements of the domain (\Z\text{.}) Assume (f(x) = f(y)\text{.}) If (x) and (y) are both even, then (f(x) = x+1) and (f(y) = y+1\text{.}) Since (f(x) = f(y)\text{,}) we have (x + 1 = y + 1) which implies that (x = y\text{.}) Similarly, if (x) and (y) are both odd, then (x - 3 = y-3) so again (x = y\text{.}) The only other possibility is that (x) is even an (y) is odd (or visa-versa). But then (x + 1) would be odd and (y - 3) would be even, so it cannot be that (f(x) = f(y)\text{.}) Therefore if (f(x) = f(y)) we then have (x = y\text{,}) which proves that (f) is injective.
2. (f) is surjective.
###### Proof
Let (y) be an element of the codomain (\Z\text{.}) We will show there is an element (n) of the domain ((\Z)) such that (f(n) = y\text{.}) There are two cases: First, if (y) is even, then let (n = y+3\text{.}) Since (y) is even, (n) is odd, so (f(n) = n-3 = y+3-3 = y) as desired. Second, if (y) is odd, then let (n = y-1\text{.}) Since (y) is odd, (n) is even, so (f(n) = n+1 = y-1+1 = y) as needed. Therefore (f) is surjective.
14
At the end of the semester a teacher assigns letter grades to each of her students. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bijective, or neither?
Solution
Yes, this is a function, if you choose the domain and codomain correctly. The domain will be the set of students, and the codomain will be the set of possible grades. The function is almost certainly not injective, because it is likely that two students will get the same grade. The function might be surjective – it will be if there is at least one student who gets each grade.
15
In the game of Hearts, four players are each dealt 13 cards from a deck of 52. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bijective, or neither?
16
Suppose 7 players are playing 5-card stud. Each player initially receives 5 cards from a deck of 52. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bijective, or neither?
Solution
This cannot be a function. If the domain were the set of cards, then it is not a function because not every card gets dealt to a player. If the domain were the set of players, it would not be a function because a single player would get mapped to multiple cards. Since this is not a function, it doesn't make sense to say whether it is injective/surjective/bijective.
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7093 | https://dummit.cos.northeastern.edu/teaching_fa19_1341/calc1_4_introduction_to_integration_v1.50.pdf | Calculus I (part 4): Introduction to Integration (by Evan Dummit, 2019, v. 1.50) Contents 4 Introduction to Integration 1 4.1 De nite Integrals and Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 4.1.1 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 4.1.2 The De nite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 4.2 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 4.2.1 Statement and Proof of the Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . .
9 4.2.2 Evaluating De nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 4.2.3 Inde nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 4.2.4 Evaluating De nite and Inde nite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 4.2.5 Dierentiating Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 4 Introduction to Integration In this chapter, we discuss integration, which is motivated by the problem of calculating the area underneath the graph of a function. We motivate the de nition of the de nite integral using Riemann sums to calculate areas, and prove the Fundamental Theorem of Calculus, which describes the close relationship between derivatives and integrals. We then introduce inde nite integrals of basic functions and discuss substitution techniques for evaluating de nite and inde nite integrals, and close with a discussion of some basic applications of integration to computing areas, arclengths, volumes, and moments and masses.
4.1 De nite Integrals and Riemann Sums • We originally motivated our development of the derivative by asking how to determine the instantaneous rate of change of a function.
• We now pose a new question with a similar avor: given a continuous, positive function f(x) on an interval [a, b], what is the area of the region that lies under the graph of y = f(x) and above the x-axis, between x = a and x = b?
4.1.1 Riemann Sums • In some cases we can nd the area under a curve using basic geometry.
• Example: Find the area under the graph of f(x) = x between x = 0 and x = 2.
◦Since f(x) = x is linear and passes through the origin, the area forms a right triangle, with base and height both equal to 2. The area is therefore 1 2 · 2 · 2 = 2 .
◦Below on the left is a graph of the area in question: 1 • Example: Find the area under the graph of g(x) = √ 9 −x2 between x = 0 and x = 3.
◦If we write y = g(x) = √ 9 −x2 we can see that x2 + y2 = 9, and so the graph of y = g(x) is the upper half of a circle of radius 3 centered at the origin, as can be easily seen by the graph above on the right.
◦Aided by the picture, we can see that the region is the interior of a quarter-circle of radius 3, so since the area of the circle is 9π, the desired region has area 9π 4 .
• However, to evaluate areas more complicated than those which have formulas from basic geometry, we will need a more general method.
◦Here is one possible approach (which was, in fact, essentially rst used by Archimedes): divide the interval [a, b] into pieces, and then in each interval draw a rectangle with base on the x-axis with one vertex on the graph of y = f(x). Then add up the areas of all of the small rectangles: this will give an approximation to the area under the graph.
◦As we use more and smaller rectangles, the collective area of the rectangles will approximate the total area under the graph more and more closely.
◦Here are some illustrations of this idea for the function f(x) = x on [0, 2], dividing the interval into 4 and 20 equally-sized subintervals respectively: ◦The same procedure will work for any continuous function, such as f(x) = x2: 2 • To formalize these ideas we rst need to de ne some terminology: • De nition: For an interval [a, b], a partition of [a, b] into n subintervals is a list of x-coordinates x0, x1, x2, · · · , xn−1, xn with x0 = a, xn = b, and x0 < x1 < x2 < · · · < xn−1 < xn. This list of x-coordinates divides [a, b] into the subintervals [x0, x1], [x1, x2], . . . , [xn−1, xn]. A tagged partition is a partition of [a, b] together with a point x∗ i is a point in the ith interval [xi−1, xi] for each 1 ≤i ≤n.
◦The only partitions we will be interested in are partitions of [a, b] into n equally-sized subintervals. In that case, for 0 ≤i ≤n we have xi = a + i · b −a n .
◦However, in some applications (and also when working in a more formal context) it is useful to use partitions where the intervals have dierent sizes.
◦Example: The partition of [0, 8] having 4 equal subintervals is [0, 2], [2, 4], [4, 6], [6, 8]. If we wish to give a tagged partition, we simply select a point to go along with each interval.
◦In general, we say the norm of the partition P is the width of the largest subinterval.
• Now we can give the formal de nition of a Riemann sum, which represents the sum of the areas of the rectangles we described above: • De nition: Suppose that f(x) is a continuous function and P ∗is a tagged partition of the interval [a, b] into n subintervals. If x∗ i is the tagged point in the ith interval [xi−1, xi], we de ne the associated Riemann sum of f(x) on [a, b] corresponding to P ∗to be RSP ∗(f) = n X i=1 f(x∗ i ) · [xi −xi−1].
◦Recall that if g(x) is a function, then the notation n X k=1 g(k) means g(1) + g(2) + g(3) + · · · + g(n). Thus n X i=1 f(x∗ i ) · [xi −xi−1] is the sum f(x∗ 0) · [x1 −x0] + f(x∗ 1) · [x2 −x1] + · · · + f(x∗ n−1) · [xn −xn−1].
◦Although this de nition is somewhat complicated, it is simply a formalization of what we discussed above: on each interval [xi−1, xi] in the partition, we draw a rectangle above the interval [xi−1, xi] whose height is f(x∗ i ), so that it lies on the graph of y = f(x). The area of this rectangle is the length of its base xi −xi−1 times its height f(x∗ i ). We then add up the areas of all of these rectangles, which is the sum given above.
◦We will primarily be interested in three case: the rst case is where x∗ i = xi−1 is the left endpoint of its interval which we call the left-endpoint Riemann sum, the second case is where x∗ i = xi is the right endpoint of its interval which we call the right-endpoint Riemann sum, and the third case is where x∗ i = (xi−1 + xi)/2 is the midpoint of its interval which we call the midpoint Riemann sum.
• In the case where P is the partition with n equally-sized subintervals, we can write the Riemann sum as n X i=1 f(x∗ i ) ∆x, where ∆x = b −a n is the common width of the subintervals.
3 ◦We can give explicit formulas for the left-endpoint, midpoint, and right-endpoint Riemann sums as well.
◦Speci cally, we have RSleft(f) = n X i=1 f[a + (i −1)∆x] ∆x, RSmid(f) = n X i=1 f[a + (i −1 2)∆x] ∆x, and RSright(f) = n X i=1 f[a + i∆x] ∆x.
• Here are some examples of Riemann sum computations: • Example: Find the left-endpoint, midpoint, and right-endpoint Riemann sums for f(x) = x2 on the interval [0, 4] with (i) 4 equal subintervals, and (ii) 10 equal subintervals.
◦First, we have a = 0 and b = 4. If there are 4 subintervals, then ∆x = b −a 4 = 1, and the subintervals themselves are [0, 1], [1, 2], [2, 3], and [3, 4].
◦The left-endpoint Riemann sum is then f(0)·1+f(1)·1+f(2)·1+f(3)·1 = 02·1+12·1+22·1+32·1 = 14 .
◦The midpoint Riemann sum is then f(0.5) · 1 + f(1.5) · 1 + f(2.5) · 1 + f(3.5) · 1 = 0.52 · 1 + 1.52 · 1 + 2.52 · 1 + 3.52 · 1 = 21 .
◦The right-endpoint Riemann sum is then f(1)·1+f(2)·1+f(3)·1+f(4)·1 = 12·1+22·1+32·1+42·1 = 30 .
◦Here are plots of the rectangles for these Riemann sums against the graph of y = f(x): ◦In a similar way we can compute the Riemann sums with 10 subintervals: in this case ∆x = b −a 10 = 0.4, and the subintervals are [0, 0.4], [0.4, 0.8], [0.8, 1.2], ... , [3.6, 4].
◦The left-endpoint Riemann sum is 10 X i=1 f[0.4 · (i −1)] · 0.4 = 02 ·0.4+0.42 ·0.4+0.82 ·0.4+· · ·+3.62 ·0.4 = 18.24 .
◦The midpoint Riemann sum is 10 X i=1 f[0.4 · (i −1/2)] · 0.4 = 0.22 ·0.4+0.62 ·0.4+12 ·0.4+· · ·+3.82 ·0.4 = 21.28 .
◦The right-endpoint Riemann sum is 10 X i=1 f[0.4 · i] · 0.4 = 0.42 · 0.4 + 0.82 · 0.4 + 1.22 · 0.4 + · · · + 4.02 · 0.4 = 24.64 .
◦Here are plots of the rectangles for these Riemann sums against the graph of y = f(x): 4 ◦We can see from the values that the Riemann sums with 10 rectangles give much better approximations of the actual area under the curve than the Riemann sums with 4 rectangles do.
◦We also observe that because the function f(x) = x2 is increasing, all of the left-endpoint rectangles lie below the graph, and thus the left-endpoint Riemann sum is less than the total area under the curve.
Likewise, all of the right-endpoint rectangles lie above the graph, and thus the right-endpoint Riemann sum is greater than the total area under the curve.
• Example: Find the left-endpoint, midpoint, and right-endpoint Riemann sums for f(x) = sin(x) on the interval [ π 2 , π] with 10 equal subintervals.
◦First, we have a = π 2 and b = π. If there are 10 subintervals, then ∆x = b −a 10 = π 20, and the subintervals themselves are [ π 2 , 11π 20 ], [ 11π 20 , 12π 20 ], [ 12π 20 , 13π 20 ], ... , and [ 19π 20 , π].
◦The left-endpoint Riemann sum is 10 X i=1 f[ π 2 + π 2 (i −1)] · π 20 = sin( π 2 ) · π 20 + sin( 11π 20 ) · π 20 + sin( 12π 20 ) · π 20 + · · · + sin( 19π 20 ) · π 20 ≈1.076 .
◦The midpoint Riemann sum is 10 X i=1 f[ π 2 + π 2 (i −1 2)] · π 20 = sin( 21π 40 ) · π 20 + sin( 23π 40 ) · π 20 + sin( 25π 40 ) · π 20 + · · · + sin( 39π 40 ) · π 20 ≈1.001 .
◦The right-endpoint Riemann sum is 10 X i=1 f[ π 2 + π 2 (i)] · π 20 = sin( 11π 20 ) · π 20 + sin( 12π 20 ) · π 20 + sin( 13π 20 ) · π 20 + · · · + sin(π) · π 20 ≈0.919 .
◦Here are plots of the rectangles for these Riemann sums against the graph of y = f(x): ◦Based on the values we have computed, it seems like the area under the graph of y = sin(x) above the x-axis on the interval for π/2 ≤x ≤π is equal to 1. In fact, this is true, but in order to establish this fact formally, we must rst develop some more results about Riemann sums.
5 • For su ently simple functions, we can evaluate certain Riemann sums exactly, for an arbitrary number of equal subintervals.
• Example: Compute the left-endpoint and right-endpoint Riemann sums for f(x) = x2 on the interval [0, 1] with n equal subintervals. By using the behavior as n →∞and the fact that f is increasing, show that the region under y = x2 above the x-axis on [0, 1] has area 1/3.
◦For this interval we have a = 0 and b = 1, and also ∆x = b −a n = 1 n. The intervals are [0, 1 n], [ 1 n, 2 n], [ 2 n, 3 n], ... , [ n−1 n , 1].
◦Then RSleft(f) = 02 · 1 n + ( 1 n)2 · 1 n + ( 2 n)2 · 1 n + · · · + (n −1 n )2 · 1 n = 02 + 12 + 22 + · · · + (n −1)2 n3 .
◦Also, RSright(f) = ( 1 n)2 · 1 n + ( 2 n)2 · 1 n + ( 3 n)2 · 1 n + · · · + 12 · 1 n = 12 + 22 + · · · + n2 n3 .
◦By using the summation formula 12 + 22 + · · · + k2 = k(k + 1)(2k + 1) 6 , we can evaluate both sums.
◦Using the summation formula, we see that RSleft(f) = (n −1)n(2n −1)/6 n3 = 1 3 −1 2n + 1 6n2 , while RSright(f) = n(n + 1)(2n + 1)/6 n3 = 1 3 + 1 2n + 1 6n2 .
◦Now since f is increasing, the left-endpoint Riemann sum is less than the total area (since all of its rectangles lie under the graph) while the right-endpoint Riemann sum is greater than the total area (since all of its rectangles lie above the graph).
◦Hence, if A is the desired area, we see that 1 3 −1 2n + 1 6n2 < A < 1 3 + 1 2n + 1 6n2 .
◦If we let n →∞, then since lim n→∞ 1 3 −1 2n + 1 6n2 = 1 3 = lim n→∞ 1 3 + 1 2n + 1 6n2 , we must have A = 1 3.
• In principle, we could employ a similar procedure to compute the area under the graph of other continuous functions (at least on intervals where the function is increasing, or where it is decreasing).
◦However, even for a comparatively simple function like f(x) = x2, these explicit calculations are already quite lengthy.
◦Instead, we will take a slightly dierent approach: we will instead de ne the integral of a continuous function to be the limit of its Riemann sums over partitions with smaller and smaller rectangles.
◦By de nition, the integral will give the value of the area under the graph of y = f(x), at least when f(x) is positive.
◦We will then relate integrals to derivatives, and thereby obtain methods for calculating areas.
4.1.2 The De nite Integral • We now give a precise de nition for the integral of a continuous function f on an interval [a, b], which is a formalization of the area under the graph: • De nition: The function f(x) is Riemann-integrable on the interval [a, b] if there exists a value L such that, for every ϵ > 0, there exists a δ > 0 (depending on ϵ) such that for every tagged partition P ∗all of whose subintervals have width less than δ, it is true that |RSP ∗(f) −L| < ϵ.
◦Essentially, what this de nition means is: the function f(x) is integrable if L is the limiting value of the Riemann sums of f as the size of the subintervals in the partition becomes small.
◦Like the formal de nition of the limit of a function, it takes a great deal of time and eort to become comfortable with this de nition1.
1In fact, most modern analytic treatments of integration typically use a slightly dierent formulation of integrability: instead of using Riemann sums, it is more technically convenient to use what are called upper and lower sums, which leads to what is called the Darboux integral, rather than the Riemann integral. However, the Darboux integral can be shown to be the same as the Riemann integral (in that the class of functions that can be integrated is the same, and the resulting integrals always have the same value). In treatments of elementary calculus, most authors nevertheless use Riemann sums, since they have an older history.
6 • De nition: If f(x) is Riemann-integrable on [a, b], we de ne the de nite integral of f on [a, b], denoted ´ b a f(x) dx, to be the limiting value L of the Riemann sums for f.
◦Example: Our analysis of f(x) = x2 on [0, 1] shows that x2 is integrable on this interval, and that ´ 1 0 x2 dx = 1 3.
◦Notation: All of the parts of the de nite integral notation are needed when writing an integral. The dx part labels the variable of integration (and behaves exactly as a dierential), and f(x) indicates the function being integrated. The values a and b are called the limits of integration, and specify the range [a, b] on which the function is to be integrated.
◦Observe the similarity between the notation n X i=1 f(x∗ i ) ∆x for a Riemann sum, and the notation ´ b a f(x) dx for the de nite integral. This similarity is deliberate: the idea of Leibniz (who developed the notation) is that in the limit of ∆x →0, the term ∆x becomes the dierential dx, and the sum becomes an integral.
• A fundamental result is that every continuous function is Riemann-integrable: • Theorem (Continuous Functions are Integrable): If f(x) is continuous on [a, b], then f(x) is Riemann-integrable on [a, b].
◦The proof is quite technical (so we will omit the lengthy details), but we can outline the basic idea: rst, one shows piecewise-constant functions are integrable. Next, one shows that on su ciently small intervals, a continuous function can be approximated closely by a piecewise-constant function. By taking a su ciently ne partition, it then follows that the corresponding Riemann sums for the two functions must also be close together. Finally, by taking an appropriate limit, one may establish that continuous functions are integrable.
• We will mention that there exist discontinuous functions that are also integrable, and also discontinuous functions that are not integrable.
◦An example of a discontinuous function that is integrable on the interval [0, 1] is the step function f(x) = ( 0 for 0 ≤x ≤1/2 1 for 1/2 < x ≤1. It is not hard to show that for this function f(x), its Riemann integral on [0, 1] is 1/2.
◦An example of a discontinuous function that is not integrable on the interval [0, 1] is the function g(x) = ( 1 if x is a rational number 0 if x is irrational .
◦For any partition of [0, 1], no matter how small the intervals, if we choose all of the tagging points x∗ i to be rational numbers then the corresponding Riemann sum for g is 1, while if we choose all of the tagging points x∗ i to be irrational numbers then the corresponding Riemann sum for g is 0. This means that the Riemann sums do not converge to a limit, and so g is not integrable.
◦Because there exist non-integrable discontinuous functions, we will focus from this point only on contin-uous functions.
• Note that our geometric motivation for integration involved nding the area under the graph of a function y = f(x), where we implicitly assumed that f(x) ≥0. However, the de nition via Riemann sums does not require that f(x) be nonnegative: it makes perfectly good sense for negative-valued functions as well.
◦If we follow the de nition through and evaluate Riemann sums for −f(x) where f(x) is positive, we obtain −1 times the result for +f(x).
◦So we can interpret the de nite integral of a negative function as giving a negative area: that is, if we interpret the area as being negative if f(x) < 0, the de nite integral makes sense for all functions.
7 • As with limits and derivatives, it is much easier to work with integrals after we have proven some basic results on manipulating them. Here are some properties of de nite integrals which are more or less immediate consequences of the Riemann sum de nition: • Proposition (Properties of De nite Integrals): Let a < b < c be arbitrary constants, let C be an arbitrary constant, and let f(x) and g(x) be continuous functions. Then the following properties hold: 1. Integral of constant: ´ b a C dx = C · (b −a).
2. Integral of constant multiple: ´ b a C · f(x) dx = C · ´ b a f(x) dx.
3. Integral of sum: ´ b a f(x) dx + ´ b a g(x) dx = ´ b a [f(x) + g(x)] dx.
4. Integral of dierence: ´ b a f(x) dx − ´ b a g(x) dx = ´ b a [f(x) −g(x)] dx.
5. Nonnegativity: If f(x) ≥0, then ´ b a f(x) dx ≥0.
6. Integral of inequality: If f(x) ≤g(x) for all x in [a, b], then ´ b a f(x) dx ≤ ´ b a g(x) dx.
7. Union of intervals: ´ b a f(x) dx + ´ c b f(x) dx = ´ c a f(x) dx.
8. Backwards interval: ´ a b f(x) dx = − ´ b a f(x) dx. In particular, ´ a a f(x) dx = 0.
◦Proof: Properties (1)-(4) and (7) follow from algebraic manipulations of Riemann sums: for example, (3) follows by observing that the Riemann sum for f + g is the sum of a Riemann sum for f with a Riemann sum for g.
◦Property (5) follows by observing that any Riemann sum for a nonnegative function is also nonneg-ative.
◦Property (6) follows by applying property (5) to the nonnegative function g(x) −f(x) ≥0, and then using property (4).
◦The statements in property (8) are actually notational conventions (rather than actual facts to be proven). They are chosen so that property (7) is true for any choice of a, b, c, regardless of order.
• By using these properties in tandem with some of the results we have already found using Riemann sums or geometry, we can evaluate a small number of integrals.
◦Using geometry, we can see that for a > 0, we have ´ a 0 x dx = 1 2a2, since the corresponding area is a right triangle with base and height both equal to a.
◦We also showed that ´ 1 0 x2 dx = 1 3.
◦So, for example, we can nd ´ 1 0 (x2 + 2x + 3) dx = ´ 1 0 x2 dx + 2 ´ 1 0 x dx + ´ 1 0 3 dx = 1 3 + 2 · 1 2 + 3 = 13 3 using the various properties listed above.
• For integrals that we cannot evaluate, we can in some cases give upper and lower bounds using the properties of inequalities.
◦Example: Because 0 ≤sin(x) ≤1 for 0 ≤x ≤π, the integral ´ π 0 p sin(x) dx is between ´ π 0 0 dx = 0 and ´ π 0 1 dx = π.
4.2 The Fundamental Theorem of Calculus • We would now like to extend our ability to evaluate integrals directly, without resorting to cumbersome Riemann sum calculations.
◦To do this, we will establish a fundamental relation between dierentiation and integration, namely that they are essentially inverse to one another.
◦More explicitly, we will show that integrating the derivative of a continuous function, or dierentiating the integral of a continuous function, will (essentially) give back the original function.
8 4.2.1 Statement and Proof of the Fundamental Theorem of Calculus • Our starting point is a simple inequality that follows from the observation that the integral of a nonnegative function is always nonnegative: • Theorem (Min-Max Inequality): If f(x) is a continuous function on [a, b], then (b−a)·mina,b ≤ ´ b a f(t)dt ≤ (b −a) · maxa,b.
◦The notation mina,b refers to the absolute minimum of f on the interval [a, b], while maxa,b refers to the absolute maximum. These values are guaranteed to exist by the Extreme Value Theorem, since f is continuous on a closed interval.
◦Proof: By de nition of the minimum and maximum values, for any x in the interval [a, b], it is true that min(f)[a,b] ≤f(x) ≤max(f)[a,b].
◦Now we apply the integration of an inequality integral property (6) twice to see that ´ b a mina,bdx ≤ ´ b a f(x)dx ≤ ´ b a maxa,bdx.
◦Then since the rst and last integrals are integrals of constants, evaluating them yields (b−a)·mina,b ≤ ´ b a f(t)dt ≤(b −a) · maxa,b, as claimed.
• Using this inequality, we can establish a version of the Mean Value Theorem for integrals: • Theorem (Mean Value Theorem for Integrals): If f(x) is a continuous function on [a, b], then there exists some c in (a, b) for which f(c) = 1 b −a ´ b a f(t)dt.
◦The value 1 b −a ´ b a f(t)dt is called the average value (or mean value) of f on the interval [a, b]. Intuitively, the Mean Value Theorem says that there is a point in the interval where the function is equal to its average value.
◦Proof: Dividing through by (b−a) everywhere in the Min-Max inequality gives mina,b ≤ 1 b −a ´ b a f(t)dt ≤ maxa,b.
◦Then since f is continuous, it attains its minimum and maximum values, and then by the Intermediate Value Theorem it takes every value in between.
◦But the inequalities above say that the average value 1 b −a ´ b a f(t)dt is between the minimum and maximum, and therefore is one of the values attained.
• We can now establish both parts of the Fundamental Theorem of Calculus: • Theorem (Fundamental Theorem of Calculus, Part 1): For any continuous function f on [a, b], the function F(x) = ´ x a f(t)dt is continuous, dierentiable, and has the property that F ′(x) = f(x) on [a, b].
◦In other words, this result says that the function F(x) = ´ x a f(t)dt is an antiderivative of f on the interval [a, b].
◦Note that the integration variable is t and not x: this is necessary because the limits of integration cannot contain the variable of integration (an expression like ´ x a f(x) dx does not make sense: when interpreted literally, it would say to integrate the function f(x) from x = a to x = x). Since we cannot use x for the variable of integration, we replace it with a dierent variable t instead.
◦Proof: To show that F ′(x) = f(x), we look at the dierence quotient lim h→0 F(x + h) −F(x) h = lim h→0 1 h "ˆ x+h a f(t)dt − ˆ x a f(t)dt # = lim h→0 1 h "ˆ x+h x f(t)dt # .
◦The quantity inside the limit is the mean value of f on the interval [x, x + h].
◦Applying the Mean Value Theorem for integrals shows that there exists a value ch in (x, x+h) for which 1 h h´ x+h x f(t)dt i = f(ch).
9 ◦Then lim h→0 F(x + h) −F(x) h = lim h→0 f(ch), and this last limit is just f(x) because f is continuous and the points ch approach x as h →0, since ch is in the interval (x, x + h).
◦Therefore lim h→0 F(x + h) −F(x) h exists and is equal to f(x), so F is dierentiable and F ′(x) = f(x).
Finally, since F is dierentiable, it is continuous.
• From our results on antiderivatives, we know that any two antiderivatives of a function de ned on an interval must dier by a constant.
◦Therefore, if we are able to nd an antiderivative of the function f(x) somehow, it must dier by a constant from the function F(x) = ´ x a f(t)dt.
◦This key insight allows us to evaluate de nite integrals, and is the second part of the Fundamental Theorem of Calculus: • Theorem (Fundamental Theorem of Calculus, Part 2): If F is any antiderivative of the continuous function f on the interval [a, b], then ´ b a f(t)dt = F(b) −F(a).
◦Proof: By the rst part of the Fundamental Theorem of Calculus, we know that G(x) = ´ x a f(t)dt is an antiderivative of f, since G′(x) = f(x).
◦But we have also shown that any two antiderivatives of a function on an interval dier by a constant: therefore, G(x) = F(x) + C for some constant C.
◦We also can see easily that G(a) = ´ a a f(t) dt = 0 and that G(b) = ´ b a f(t)dt.
◦Hence ´ b a f(t)dt = G(b) −G(a) = [F(b) + C] −[F(a) + C] = F(b) −F(a), as desired.
4.2.2 Evaluating De nite Integrals • The content of the second part of the Fundamental Theorem of Calculus is that we can evaluate de nite integrals using antiderivatives, instead of using the complicated process of computing limits of Riemann sums.
◦Speci cally, if F(x) is an antiderivative of f(x), then ´ b a f(x)dx = F(x) b x=a = F(b) −F(a).
◦The notation F(x) b x=a means to evaluate the function f from x = a to b, and is simply shorthand for F(b) −F(a).
◦Thus, in order to compute a de nite integral ´ b a f(t)dt, we need only nd an antiderivative of f, and then evaluate it at the endpoints a and b and subtract the results.
• Example: Evaluate ´ 1 0 x2 dx using the Fundamental Theorem of Calculus, and interpret the result as an area.
◦As we can easily see, the function F(x) = 1 3x3 is an antiderivative of f(x) = x2.
◦Therefore, by the Fundamental Theorem of Calculus, we have ´ 1 0 x2 dx = ´ 1 0 f(x) dx = F(x) 1 x=0 = F(1) −F(0) = 1 3 .
◦This evaluation ´ 1 0 x2 dx = 1 3 corresponds to the area of the region underneath the graph of y = x2 above the x-axis for 0 ≤x ≤1.
◦Remark: Note how much simpler this calculation was, in comparison to the very lengthy arguments using Riemann sums we needed earlier to compute the area of this region!
• Example: Evaluate ´ 16 1 √x dx using the Fundamental Theorem of Calculus, and interpret the result as an area.
◦We wish to nd an antiderivative of f(x) = √x = x1/2. Since the derivative of x3/2 is 3 2x1/2, we see the derivative of 2 3x3/2 is x1/2, and so we may take F(x) = 2 3x3/2.
10 ◦Then by the Fundamental Theorem of Calculus, we have ´ 16 1 √x dx = 2 3x3/2 16 x=1 = 2 3163/2 −2 313/2 = 42 .
◦This evaluation ´ 16 1 √x dx = 42 corresponds to the area of the region underneath the graph of y = √x above the x-axis for 1 ≤x ≤16.
• Example: Evaluate ´ e 1 1 x dx using the Fundamental Theorem of Calculus.
◦Observe that an antiderivative of f(x) = 1 x is F(x) = ln(x).
◦Then by the Fundamental Theorem of Calculus, we have ´ e 1 1 x dx = ln(x) |e x=1 = ln(e) −ln(1) = 1 .
• Example: Evaluate ´ 2 1 2x dx.
◦Since the derivative of 2x is 2x ln(2), we see that an antiderivative of 2x ln(2) is 2x.
◦Since we want the antiderivative of f(x) = 2x itself, we can just divide by ln(2) to see that an antideriva-tive is F(x) = 2x ln(2).
◦Then by the Fundamental Theorem of Calculus, we have ´ 2 1 2x dx = 2x ln(2) 2 x=1 = 4 ln(2) − 2 ln(2) = 2 ln(2) .
• Example: Evaluate ´ π/4 0 [4 sin(x) −2 cos(x)] dx.
◦Since an antiderivative of sin(x) is −cos(x), and an antiderivative of cos(x) is sin(x), we can see that an antiderivative of f(x) = 4 sin(x) −2 cos(x) is F(x) = −4 cos(x) −2 sin(x).
◦Then by the Fundamental Theorem of Calculus, we have ˆ π/4 0 [4 sin(x) −2 cos(x)] dx = [−4 cos(x) −2 sin(x)] π/4 x=0 = [−4 · √ 2 2 −2 · √ 2 2 ] −[−4 · 1 −2 · 0] = 4 −3 √ 2 .
4.2.3 Inde nite Integrals • In evaluating integrals via the Fundamental Theorem of Calculus, we need to compute general antiderivatives.
We refer to such antiderivatives as inde nite integrals, since they essentially tell us the value of the integral of a function on an unspeci ed interval: • De nition: The inde nite integral of f(x) with respect to x, denoted ´ f(x) dx, is the set of all antiderivatives of f(x).
◦By our results on antiderivatives, if f is de ned on the interval I and F(x) is one antiderivative of f on I, then any other antiderivative of f is of the form F(x) + C for some arbitrary constant C.
◦We traditionally write +C at the end of an inde nite integral to ensure that the arbitrary constant is not lost.
◦Some examples are ´ x dx = 1 2x2 + C, ´ x2 dx = 1 3x3 + C, and ´ ex dx = ex + C.
◦Extremely Important Note: When writing an inde nite integral, the +C must always be included!
• Computing inde nite integrals is in general very di cult: unlike with derivatives, there is no straightforward procedure for computing antiderivatives of arbitrary functions.
11 ◦In fact, it is known that there exist elementary functions which have no elementary antiderivative (a function is elementary if it can be written in terms of polynomials, radicals, exponentials, logarithms, and trigonometric and inverse trigonometric functions), meaning that there is no nice formula for the antiderivative in terms of familiar functions.
◦Some examples of simple functions with no elementary antiderivative are ex2, √ 1 −x4, sin(x2), 1 ln(x), ln(ln x), and sin(x) x .
• From the derivatives we have calculated, we can write a list of simple inde nite integrals: ˆ xn dx = xn+1 n + 1 + C, n ̸= −1 ˆ 1 x dx = ln(x) + C ˆ ax dx = ax ln(a) + C, a ̸= 1 ˆ sin(x) dx = −cos(x) + C ˆ cos(x) dx = sin(x) + C ˆ sec2(x) dx = tan(x) + C ˆ sec(x) tan(x) dx = sec(x) + C ˆ 1 √ 1 −x2 dx = sin−1(x) + C ˆ 1 1 + x2 dx = tan−1(x) + C ˆ 1 x √ x2 −1 dx = sec−1(x) + C ◦Remark: For the inde nite integral of 1 x, there are many sources which write ´ 1 x dx = ln |x| + C, with absolute values. This has the advantage of being de ned for negative values of x (which ln(x) is not), but the downside is that this formula may appear to give nite values for de nite integrals that are actually unde ned. If one takes the viewpoint of de ning logarithms of negative numbers as having non-real values, then the two formulas ´ 1 x dx = ln(x) + C and ´ 1 x dx = ln |x| + C actually are equivalent: the minus sign gets absorbed into the constant of integration when x is negative. (Many computer algebra systems declare that ´ 1 x dx = ln(x) + C for this reason.) We will take the convention of avoiding the absolute values, but by using ln(−x) + C for de nite integrals where x is negative.
• Here are some other antiderivatives of basic functions that may be veri ed by dierentiation: ˆ ln(x) dx = x ln(x) −x + C ˆ tan(x) dx = −ln(cos(x)) + C ˆ sec(x) dx = ln(sec(x) + tan(x)) + C ˆ csc(x) dx = −ln(csc(x) + cot(x)) + C ˆ cot(x) dx = ln(sin(x)) + C 12 4.2.4 Evaluating De nite and Inde nite Integrals • By using these basic antiderivatives along with our rules for combining them, we can evaluate a moderately wide array of de nite and inde nite integrals: • Example: Find ´ sin(x) + x2 dx and ´ π 0 sin(x) + x2 dx.
◦From the basic integrals we see ´ sin(x) + x2 dx = −cos(x) + 1 3x3 + C .
◦Then ´ π 0 sin(x) + x2 dx = −cos(x) + 1 3x3 π x=0 = 2 + 1 3π3 .
• Example: Find ˆ x2 + 2x√x + 3 √x x3 dx.
◦We can distribute the fraction in the integrand, and then integrate each term separately.
◦This yields ˆ x2 + 2x√x + 3 √x x3 dx = ˆ 1 x + x−1/2 + x−8/3 dx = ln(x) + 2x1/2 −3 5x−5/3 + C .
• Example: Find ˆ 4 cos(x) + 2 sec2(x) dx and ˆ π/4 0 4 cos(x) + 2 sec2(x) dx.
◦From the basic integrals we see ˆ 4 cos(x) + 2 sec2(x) dx = 4 sin(x) + 2 tan(x) + C .
◦Then ˆ π/4 0 4 cos(x) + 2 sec2(x) dx = [4 sin(x) + 2 tan(x)] π/4 x=0 = 2 √ 2 + 2 .
• Example: Find ˆ 2 0 (2x + 3x + 4x) dx.
◦From the basic integrals we see ˆ (2x + 3x + 4x) dx = 2x ln(2) + 3x ln(3) + 4x ln(4) + C.
◦Then ˆ 2 0 (2x + 3x + 4x) dx = 2x ln(2) + 3x ln(3) + 4x ln(4) 2 x=0 = 3 ln(2) + 8 ln(3) + 15 ln(4) .
• Example: Find ˆ 1 −cos2(x) + sin(2x) sin(x) dx.
◦We can use trigonometric identities to simplify the integrand, and then integrate each term separately.
◦This yields ˆ 1 −cos2(x) + sin(2x) sin(x) dx = ˆ sin2(x) + 2 sin(x) cos(x) sin(x) dx = ˆ [sin(x) + 2 cos(x)] dx = −cos(x) + 2 sin(x) + C .
• Example: Find ´ π/3 π/4 tan(x) dx.
◦From the basic integrals we have ´ tan(x) dx = −ln(cos(x)) + C.
◦Then ´ π/3 π/4 tan(x) dx = (−ln(cos(x))) π/3 x=π/4 = (−ln(cos(π/3))) −(−ln(cos(π/4))) = ln( √ 2) = 1 2 ln(2) .
• Example: Find ´ 8 2 dx.
◦Note here that dx = 1 · dx, so ´ 8 2 dx is really just shorthand for ´ 8 2 1 dx = x 8 x=2 = 6 .
13 • Example: Find ˆ √ 3/2 1/2 1 √ 1 −x2 dx.
◦Since ´ 1 √ 1 −x2 dx = sin−1(x) + C, we see ˆ √ 3/2 1/2 1 √ 1 −x2 dx = sin−1( √ 3 2 ) −sin−1(1 2) = π 6 .
4.2.5 Dierentiating Integrals • By the rst part of the Fundamental Theorem of Calculus, we can compute derivatives of integrals.
In particular, by using integration, we can construct new functions.
• Example: The function erf(x) is de ned via erf(x) = 2 √π ´ x 0 e−t2 dt. Find the derivative of erf(x).
◦To nd the derivative, we can just use the Fundamental Theorem of Calculus: we have erf′(x) = d dx 2 √π ´ x 0 e−t2 dt = 2 √π e−x2 directly from the rst part of the Fundamental Theorem.
◦Remark: This function is called the error function and shows up very often in statistics due to its close connection to the normal distribution. It can be proven (though it is hard!) that there is no way to write the error function erf(x) in terms of the elementary functions (i.e., as a sum, product, or composition of any number of polynomials, exponentials, logarithms, and trigonometric or inverse trigonometric functions of x). Thus, the description above as an integral is, in some sense, the simplest way of describing the error function. So we have actually constructed a new function that we could not have described without using integration.
◦From our calculation of the derivative, we can see that this function is always increasing (since erf′(x) > 0 for all x). From the second derivative erf′′(x) = −4x √π e−x2 we can see that erf is concave up for negative x and concave down for positive x.
• Example: If g(x) = ´ x2 x sin(t) t dt, nd g′(x).
◦The idea here is to rearrange g(x) into simpler pieces to which we can apply the Fundamental Theorem of Calculus.
◦Speci cally, the Fundamental Theorem tells us how to nd the derivative of the function h(x) de ned by h(x) = ´ x 0 sin(t) t dt: we have h′(x) = sin(x) x .
◦Now, by integration properties, we can write g(x) = ´ x2 0 sin(t) t dt − ´ x 0 sin(t) t dt = h(x2) −h(x).
◦Now we can compute g′(x) using these observations along with the Chain Rule. We obtain g′(x) = d dx h(x2) −h(x) = 2x h′(x2) −h′(x) = 2x · sin(x2) x2 −sin(x) x = 2 sin(x2) −sin(x) x .
• Example: If J(x) = ´ x2 −x ln(1 + et) dt, nd J′(x).
◦By the Fundamental Theorem of Calculus, for F(x) = ´ x 0 ln(1 + et) dt, we have F ′(x) = ln(1 + ex).
◦By integration properties, we have J(x) = ´ x2 0 ln(1 + et) dt − ´ −x 0 ln(1 + et) dt = F(x2) −F(−x).
◦Then, by the Chain Rule, we get J′(x) = F ′(x2) · 2x −F(−x) · (−1) = 2x · ln(1 + ex2) + ln(1 + e−x) .
Well, you're at the end of my handout. Hope it was helpful.
Copyright notice: This material is copyright Evan Dummit, 2012-2019. You may not reproduce or distribute this material without my express permission.
14 |
7094 | https://pure.tue.nl/ws/portalfiles/portal/65364242/Keijsers2017_Regulation.pdf | Modeling regulation of vascular tone following muscle contraction: model development, validation and global sensitivity analysis Citation for published version (APA): Keijsers, J. M. T., Leguy, C. A. D., Narracott, A. J., Rittweger, J., van de Vosse, F. N., & Huberts, W. (2018).
Modeling regulation of vascular tone following muscle contraction: model development, validation and global sensitivity analysis. Journal of Computational Science, 24, 143-159. DOI: 10.1016/j.jocs.2017.04.007 Document status and date: Published: 01/01/2018 Document Version: Accepted manuscript including changes made at the peer-review stage Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
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Download date: 29. Sep. 2025 Authors version; accepted for publication in Journal of Computational Science on April 23, 2017 Modeling regulation of vascular tone following muscle contraction: model development, validation and global sensitivity analysis J.M.T. Keijsers1,2,∗, C.A.D. Leguy2, A.J. Narracott3,4, J. Rittweger2, F.N. van de Vosse1 and W. Huberts5 1 Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands 2 Institute of Aerospace Medicine, German Aerospace Center, Cologne, Germany 3 Medical Physics Group, Department of Cardiovascular Science, University of Sheffield, Sheffield, United Kingdom 4 INSIGNEO In-stitute for in silico Medicine, University of Sheffield, Sheffield, United Kingdom 5 Department of Biomedical Engineering, Maastricht University, Maastricht, The Netherlands ABSTRACT In this study the regulation of vascular tone inducing the blood flow increase at the onset of exercise is examined. Therefore, our calf circulation model was extended with a reg-ulation model to simulate changes in vascular tone depending on myogenic, metabolic and baroreflex regulation. The simulated blood flow corresponded to the in vivo re-sponse and it was concluded that metabolic activation caused the flow increase shortly after muscle contraction. Secondly, the change in baseline flow upon tilt was a result of myogenic and baroreflex activation. Based on a sensitivity analysis the myogenic gain was identified as most important parameter.
Keywords: regulation of vascular tone, metabolic regulation, myogenic regulation, barore-flex, 1D pulse wave propagation.
Corresponding address: J.M.T. Keijsers, Department of Biomedical Engineering. Eind-hoven University of Technology, PO Box 513, 5600 MB, Eindhoven. The Netherlands.
Phone: +3140 247 5675. Email: j.m.t.keijsers@tue.nl 1 A B artery deep vein superficial vein proximal valve perforator micro-circulation distal valve pex perforator artery deep vein superficial vein proximal valve perforator micro-circulation distal valve perforator Figure 1: Schematic representation of the muscle pump effect during the A contrac-tion and B relaxation phase. During contraction, the deep vein collapses due to the ex-travascular pressure pex exceeds the intravascular pressure. Venous return is increased, whereas back-flow and flow to the superficial system is blocked by the distal and per-forator valves. During the relaxation, the deep vein is refilled from both the artery and the superficial vein, while the perforator valves open and the proximal valve is closed to prevent back-flow. Figure was adapted from Keijsers et al. 1 Introduction During exercise several complex haemodynamic regulation mechanisms are activated to ensure sufficient supply of oxygen and nutrients, and removal of waste products. In-creased understanding of these individual mechanisms and their interaction is needed to fully characterize the dynamics of blood flow during exercise. At the onset of exercise, the blood flow within muscle in the lower limb increases significantly depending on the arterio-venous pressure drop and the peripheral resistance. These are influenced by both mechanical effects during muscle contraction and relaxation (the muscle pump effect) and the vasodilatory state of the arterioles. However, the exact contribution of these two mechanisms to the blood flow increment at the onset of exercise is still a matter of debate [Tschakovsky and Sheriff, 2004]. Three hypotheses are currently described in literature: the first states that flow augmentation is primarily caused by the muscle pump effect, the second claims that regulation of peripheral resistance is the major determinant. The third hypothesis considers both mechanisms to be important.
As a result of calf muscle activation, the muscle pump effect increases venous return by collapsing the deep veins embedded within the muscle. Furthermore, backflow towards the arterial system and into the superficial venous system is prevented by closure of the distal and perforating venous valves respectively (Figure 1A) [Rowell, 1993]. Arterial inflow rises during subsequent muscle relaxation as the perfusion pressure is increased due to the pressure shielding of the closed proximal valve (Figure 1B) [Rowell, 1993].
Furthermore, the opening of the distal and perforating valves [Meissner, 2005] allows venous refilling from both the arterial and superficial venous system. In summary, the muscle pump effect increases blood flow through an increase in arterio-venous pressure drop.
In a previous study [Keijsers et al., 2015], we examined the role of venous valves, hydro-static pressure and the superficial veins during the muscle pump using a mathematical 2 model. Although the model was able to simulate the increased venous return during muscle contraction and the elevated arterial flow during muscle relaxation, the predicted flow augmentation was low compared to the increase in arterial flow increase measured during in vivo calf muscle contractions [N˚ adland et al., 2009]. Based on the debate in literature [N˚ adland et al., 2009, Tschakovsky et al., 1996], it was proposed that vasodila-tion could be the missing component in the model. Furthermore, the simulated arterial baseline flow in the tilted position was equal to the baseline flow in the supine posi-tion, which was in strong contrast with the 50% decay observed in vivo [N˚ adland et al., 2009]. These postural changes can be attributed to changes in peripheral resistance due to myogenic vasoconstriction and a global increase in peripheral resistance [N˚ adland et al., 2009]. Therefore, in this study our previous model is extended to include regulation of vascular tone.
Regulation of vascular tone in skeletal muscle tissue is not based on a single mech-anism, but involves the interaction between the local myogenic, local metabolic and global baroreflex regulation [Joyner and Casey, 2015]. Myogenic regulation protects the capillaries against high pressures by vasoconstriction as transmural pressure increases [Boron and Boulpaep, 2003]. The metabolic mechanism induces vasodilation when the amount of metabolites accumulates, thereby regulating the oxygen delivery and removal of waste products [Boron and Boulpaep, 2003, Joyner and Casey, 2015]. Finally, the barore-flex initiates vasoconstriction when central pressure detected by the baroreceptors in the aortic arch and the carotid artery decreases. In addition, the baroreflex affects heart rate, cardiac contractility and venous unstressed volume [Boron and Boulpaep, 2003, Rowell, 1993]. A combined regulation model including all three components is thus required to describe the resulting vascular tone. As all three mechanisms respond to different param-eters and with different time delays, each should be modelled as a separate component.
The definition of specific parameters for each mechanism, allows us to examine the rela-tive activation of the three mechanisms during muscle contraction and relaxation.
Previous numerical studies of regulation of vascular tone have focussed on cerebral auto-regulation or the baroreflex. Ursino developed a model for cerebral auto-regulation including a neurogenic and endothelial response in addition to the myogenic and metabolic mechanism. This model was used to investigate the relation between cerebral blood volume and intracranial pressure changes [Ursino and Giammarco, 1991] and applied to examine cerebral regulation under squat exercise and visual stimulation [Spronck et al., 2012]. Models of the baroreflex have been applied to study various physiological re-sponses, e.g. the interaction between the baroreflex and a pulsating heart model [Ursino, 1998], heart rate variability [Ursino and Magosso, 2003], fetal welfare during labor [van der Hout-van der Ja 2013] and heart rate regulation under orthostatic stress [Olufsen et al., 2006]. However, to our knowledge, no model exists that combines myogenic, metabolic and baroreflex regulation to simulate the vascular tone response to a skeletal muscle contraction.
The aim of this study was to determine the importance of the myogenic, metabolic and baroreflex regulation during the different phases of muscle contraction. Therefore, the 1D arterio-venous model as described in Keijsers et al. was extended with a regulation model for the vascular tone, which includes both the myogenic and metabolic effects de-scribed by Spronck et al. , combined with the baroreflex model of Ursino to include all three mechanisms. In an initial explorative analysis, the intuitively most important model parameters representing the gain of the myogenic and metabolic mech-anism were fitted to match the measured in vivo flow response to a muscle contraction in the supine position. Secondly, the same parameters were used to predict the response to a muscle contraction in the tilted position. Finally, a sensitivity analysis was performed to quantify which parameters are most important for the variance in the flow response.
For the sensitivity analysis the two step approach as described by Donders et al. 3 model physiological data muscle contraction sensitivity analysis post-SA analysis circulation model regulatory model - 1D pulse wave propagation large vessels - 0D venous valves - 0D micro-circulation - myogenic mechanism - metabolic mechanism - baroreflex - in supine and tilted position - collapse deep veins in circulation model - increase metabolism in regulation model Figure 4.2B Figure 4.2A Figure 4.3 Figure 4.8 Figure 4.7 Figure 4.6 Figure 4.5 - assess flow response muscle contraction using ultrasound - derive supine and tilted fit for validation of the simulations - Morris screening - polynomial chaos expansion - analyse importance parameters - fit supine response varying Gmyo + Gmeta - predict response tilted position - analyse activation regulation importance regulation mechanisms importance systemic pressure variation - fit the either 8 or 4 most important parameters found by SA - analyse parameter distribution - include pressure variation in: - pressure BC - baroreflex Figure 2: Schematic overview of the methods as described in this study including: circu-latory and regulation model, simulation of a muscle contraction, physiological data, and the four analyses performed. For each part the main points are given together with the corresponding figures.
was used. This approach consists of an initial Morris screening and a subsequent general-ized polynomial chaos expansion (gPCE). We conclude with an analysis varying the most important parameters, identified by the sensitivity analysis, to fit the in vivo response.
2 Methods In this section the methods are described and a schematic overview can be observed in Figure 2.
2.1 Model To study the hemodynamic and local regulatory response to muscle contraction in the lower limb a simplified model of the calf circulation was constructed (Figure 3). The model includes an artery, supplying the muscle tissue with oxygen, a deep vein, embed-ded in the muscle tissue, and a superficial vein, between the skin and the muscle. To model regulation of vascular tone the mean response of the arterioles is included into a single model variable which represents the regulatory state. This state is based on the myogenic, metabolic and baroreflex regulation (Figure 3B). Changes in peripheral resis-tance are induced by relating the resistance to the regulation state. The following section describes the physiological background and governing equations of the various model components.
4 p0 p0 p0 poutlet pinlet 0D inlet BC 0D outlet BC 0D micro-circulation Ra/2 Ra/2 Rv/2 Rv/2 Cv Ca AR1 DV1 DV2 DV3 PV4 PV3 PV2 PV1 VV2 VV1 A p0 SV1 SV2 SV3 SV4 SV5 VV3 VV4 1D artery 1D deep vein 1D superficial vein 0D venous valve B Baroreflex [Ursino1998] Myogenic [Spronck2012] Metabolic [Spronck2012] time delay Laplace’ law CO2 production q pcarotid T CO2 level Ameta Amyo xmeta xmyo xbaro time delay time delay Gmeta Gmyo xtot T ra Ra Ca n2 n3 Figure 3: Model. A Model configuration of the calf circulation including: 1D artery (AR), a 1D deep (DV) and superficial (SV) vein, 0D venous valves (VV), a 0D micro-circulation, a 0D inlet and outlet boundary condition (BC). The length and radius of the 1D elements are not true to scale (geometrical parameters of all 1D segments can be found in Table 1) B Schematic overview of the regulation model including baroreflex, myogenic and metabolic regulation 5 2.1.1 1D Pulse wave propagation: arteries and veins The hemodynamics in the large arteries and veins is captured using the 1D equations for mass and momentum balance, with blood assumed to be an incompressible Newtonian fluid. The resulting equations read: C ∂ptr ∂t + ∂q ∂z = 0, (1) ∂q ∂t + ∂Av2 z ∂z + A ρ ∂p ∂z = 2πa ρ τw + Agz, (2) where C is the compliance per unit length, ptr = p −pex is the transmural pressure, p and pex are the intra- and extravascular pressure respectively, q is the flow, t is the time and z is the axial coordinate. Furthermore, A is the cross-sectional area, vz is the velocity in axial direction averaged over the cross-sectional area, ρ is blood fluid density, a = p A/π is the radius and τw is the wall shear stress. Additionally, gz = g eg · ez is the contribution of the gravitational acceleration in the axial direction, g is the magnitude of the gravitational acceleration on earth, eg is the unit vector in the direction of gravity and ez is the unit vector in axial direction.
To obtain an estimation of the wall shear stress τw and the advection term ∂Av2 z ∂z the ap-proximate velocity profile is used (see Bessems et al. for more details). Here, the pressure gradient and the gravitational forces are assumed to be in balance with viscous forces in the boundary layer close to the vessel wall. In the central core inertia forces are assumed to be in balance with the pressure gradient and the gravitational forces. Finally, a constitutive law relating cross-sectional area and pressure, is defined for both arteries and veins.
As the arterial cross-sectional area variations during the cardiac cycle are small under normal conditions, the mechanical characteristics of the arterial wall are modeled with the following linear A, p relation A = Aref,A + C(ptr −pref,A), (3) where Aref,A is the reference cross-sectional area at reference pressure pref,A and C the linearized compliance per unit length at reference pressure pref,A. The compliance is de-termined using thin-walled-cylinder theory for a linear isotropic elastic material: C = ∂A ∂ptr ptr=pref,A = 2π(1 −ν2)r3 ref,A hE , (4) where ν is the Poisson’s ratio, rref,A = p Aref,A/π is the reference radius, h ≈rref,A/10 is the vessel wall-thickness [Westerhof et al., 1969] and E is the Young’s modulus.
Because veins are prone to collapse under low transmural pressures due to e.g. increas-ing extravascular pressure during muscle contraction or gravitational stress, a nonlinear pressure area relationship needs to be considered. Therefore, Shapiro derived a tube law capturing the venous collapse with an p, A-relation. In order to solve the full system of equations for pressure a fit of the tube law is used as derived in Keijsers et al.
.
A = Aref,V h(p∗)f +(p∗) + (1 −h(p∗))f −(p∗) }, (5) where Aref,V is the reference cross-sectional area at zero transmural pressure, p∗= ptr/Kp is the dimensionless pressure and Kp is the bending stiffness. The functions f + and f − are fits of the positive and negative pressure part of the original tube law of Shapiro and 6 Table 1: Geometrical parameters of the various 1D vessels [M¨ uller and Toro, 2014] as de-picted in Figure 3. The four perforating veins consist of a deep (PV#-D) and a superficial (PV#-S) vein of which the parameters are noted separately.
Vessel Numbering (Figure 3) Radius [mm] Length [cm] artery AR1 2.5 34 deep vein DV1 1.5 2 DV2 1.5 26 DV3 1.5 2 superficial vein SV1 3.5 2 SV2 1.5 2 SV3 1.5 26 SV4 1.5 2 SV5 3.5 2 perforating vein PV#-S 0.5 1.5 PV#-D 0.5 1.5 h(p∗) is a scaling function.
f +(p∗) = A+ 0 π tan−1 p∗−p+ a p+ b + π 2 , (6) f −(p∗) = B + A− 0 π tan−1 p∗−p− a p− b + π 2 , (7) and h(p∗) = 1 π tan−1 γp∗ π + π 2 , (8) where B, A− 0 , p− a , p− b , A+ 0 , p+ a , p+ b and γ are fitting constants determining the shape of the A,p-relation. Venous compliance is calculated as the derivative of cross-sectional area with respect to the transmural pressure.
2.1.2 0D Venous valves The pressure-flow relation of a venous valve is included using the versatile valve model of Mynard et al. . As the flow through venous valves is much lower compared to heart valves, the linear viscous forces are included, as in Keijsers et al. .
∆p = Rq + Bq|q| + L∂q ∂t , (9) where the Poiseuille resistance R, Bernouilli resistance B and the inertance L are defined by R = 8πηleff A2 eff , B = ρ 2A2 eff and L = ρleff Aeff , (10) where Aeff is the effective cross-sectional area, η is the dynamic blood viscosity, and leff = βl · rref,V is the effective valve length defined as a multiple βl of the venous reference radius rref,V = p Aref,V/π [Keijsers et al., 2016]. To include valve opening and closing, the effective cross-sectional area is defined to be a function of valve state ζ via the following relation Aeff = (Aeff,max −Aeff,min) ζ + Aeff,min, (11) where Aeff,min and Aeff,max are the minimal and maximal effective cross-sectional area respectively. Here, maximal effective cross-sectional area Aeff,max = βA · Aref,V is defined as a multiple βA of the reference cross-sectional area Aref,V of the connecting vein. The valve state is defined to vary between zero and one (fully closed: ζ = 0, fully open: 7 ζ = 1). Its value is related via two differential equations for valve opening and closing respectively: dζ dt = ( (1 −ζ) Kvo (∆p −dpvalve,0) , if ∆p > dpvalve,0 ζKvc (∆p −dpvalve,0) , if ∆p < dpvalve,0 , (12) where Kvo and Kvc are coefficients determining the opening and closing speed of the valve. Furthermore, dpvalve,0 is the pressure drop above and below which opening and closing is initiated.
2.1.3 0D micro-circulation To account for the pressure drop over the micro-circulation (in the current study defined to include the arterioles, capillaries and venules) and its storage capacity, the micro-circulation model consists of both resistances and compliances. The micro-circulation is split into an arteriolar and venular part, both consisting of two resistances Ri (i = a, v) in series and a compliance Ci (i = a, v) connected to the extravascular pressure, for which the following relations hold (Figure 3A).
∆p = Riq and ∂ptr ∂t = 1 Ci q.
(13) Under baseline conditions the total resistance of the two parts is determined by the pres-sure drop over the micro-circulation ∆pbl and the time-avaraged baseline flow qbl accord-ing to Rtot = ∆pbl qbl = Ra + Rv, (14) where Rv is chosen such that the pressure drop over the venules is 400 Pa [Boron and Boulpaep, 2003]. Furthermore the baseline total compliance Ctot is derived from a typical time-constant τRC as in a classical single windkessel micro-circulation [Keijsers et al., 2015].
The compliance of the venules is assumed to be much larger than arteriolar compliance [Boron and Boulpaep, 2003]. Therefore, the compliances are distributed as follows Ca = 0.3 · Ctot and Cv = 0.7 · Ctot.
(15) The above equations for resistance and compliance relate to the baseline conditions.
However, for the arteriolar part of the micro-circulation the resistance and compliance are regulated by vascular tone as described in the following subsection.
2.1.4 Regulation of vascular tone Regulation of the vascular tone in muscular tissue is based on the following mechanisms (Figure 3B): • Myogenic regulation: protecting the capillaries against excessive pressures • Metabolic regulation: matching the blood flow to the oxygen demand • Baroreflex regulation: aiming to maintain the level of systemic pressure The regulation model is based on the implementation of cerebral auto-regulation as de-scribed by Spronck et al. . In this study, each regulation mechanisms is included in-dividually and represented by a regulatory state xi. The myogenic regulatory state xmyo is derived from the arteriolar wall tension T and has a time constant τmyo. The metabolic regulatory state xmeta with time constant τmeta is modeled to depend on tissue CO2-level, 8 Spatial pressure m(z) [−] Z−coordinate [m] A 0 0.1 0.2 0.3 0 0.5 1 Temporal pressure k(t) [−] Time [s] B 0 2 4 0 0.5 1 Figure 4: Extravascular pressure of the deep veins is increased to simulate a muscle con-traction. Plot A and B show the spatial m(z) and temporal k(t) course of extravascular pressure as applied to the deep venous elements respectively (see Appendix A for the full equations of m(z) and k(t)). The grey areas in the spatial plot indicate the location of the venous valves. [Keijsers et al., 2015] which is derived from the CO2-production and the blood flow. The latter two regulatory mechanism are included as described by Spronck et al. , but the metabolic mecha-nism is adjusted to induce metabolic activation upon muscle contraction instead of cere-bral activity, as included by Spronck et al. . Furthermore, tissue specific parameters are updated to match muscle tissue. Finally, the baroreflex regulatory state xbaro is based on the carotid pressure, which is derived from the pressure at the heart level based on the hydrostatic column. The baroreflex implementation is based on the model based on the study of Ursino . The total regulatory state is calculated as the weighted sum of the three mechanisms, each having a specific gain: Gmyo, Gmeta and Gbaro. The total regula-tory state is translated to arteriolar wall tension, which is subsequently converted to the arteriolar radius using Laplace’s law. Finally, the arteriolar radius is used to determine the change in peripheral resistance and compliance. For completeness, the equations de-scribing the activation of the three regulation mechanisms and how they affect a change in resistance and compliance are given in Appendix B.
2.1.5 Boundary conditions Both the inlet of the 1D arterial and the outlet of the 1D venous part are connected to a three element windkessel model representing the proximal vasculature. Each wind-kessel element consists of two resistances in series and a compliance connected to the extravascular pressure p0 (Equation (13)). The total windkessel resistance is the sum of the Poiseuille resistances of the proximal vasculature, based on the geometrical param-eters of the arterial and venous tree published by M¨ uller and Toro . Similarly, the inlet and outlet compliance is the sum of the compliances of the proximal vasculature based on Equation (4) and the derivative of Equation (5) times the length respectively.
At the inlet and outlet the pressure is set to a time-averaged pinlet and poutlet respectively.
When a head up tilt position is simulated the hydrostatic column up to the heart is added to both the inlet and outlet pressure.
The model formulation described above is completed for the current application by defin-ing the form of the muscle contraction.
2.1.6 Simulating muscle contraction The effect of a muscle contraction is included in the current model both in a mechanical and a metabolic manner. The mechanical effect on the deep veins is expected to be large due to their location, embedded in the muscle tissue, and the low intravascular pres-9 sure. Similar to Keijsers et al. , a muscle contraction is simulated by an increase in extravascular pressure included in the equation for mass balance and the venous consti-tutive law (Equation (1) and (5) respectively). The extravascular pressure is defined by the following relation pex = pex,max · k(t) · m(z), (16) where pex,max is the maximal extravascular pressure, and k(t) and m(z) are the tempo-ral and spatial course of extravascular pressure, respectively. The latter can be observed in Figure 4, and the full equations are give in Appendix A. The influence of the mus-cle contraction on the superficial veins is assumed to be negligible due to their location outside the muscle tissue. Furthermore, due to the high arterial pressure the influence of the muscle contraction on the arterial cross-sectional area is also assumed to be neg-ligible. Finally, the mechanical influence on the micro-circulation is also assumed to be negligible due to its viscous character (in Equation (13) and (24)) [Gray et al., 1967]. Al-though contradicted in some studies [Tschakovsky et al., 1996], few experimental studies hypothesize the decrease in transmural pressure could induce myogenic vasodilation [Tschakovsky and Sheriff, 2004], but implementation of this theory requires more accu-rate knowledge of the magnitude of extravascular pressure and is therefore neglected.
The increase in metabolism due to a muscle contraction is included in the metabolic mechanism of the regulation of vascular tone via muscle activation Amc (Equation (37)).
Because the flow increase due to a muscle contraction increases linearly with increas-ing contraction intensity [Tschakovsky et al., 2004], muscle activation is defined to follow the contraction pattern defined by the extravascular pressure and reaches a maximum of Amc,max corresponding to the percentage of maximum electromyogram (EMG) activity: Amc = Amc,max · k(t).
(17) 2.1.7 Numerical implementation The model equations were implemented in the finite element package SEPRAN (Inge-nieursbureau SEPRA, Leidschendam, the Netherlands) using the computational method described by Kroon et al. . Time discretization was included based on an implicit Euler scheme with a time step of ∆t = 1.0 ms and spatial discretization based on the trapezium rule with element size ∆z = 1.0 cm for arterial and superficial venous seg-ments, and ∆z = 0.5 cm for the deep venous segments, which is necessary to capture the collapse accurately. The model parameters that are not included in the sensitivity analy-sis are summarized in Table 2. Pre- and post-processing was performed using MATLAB R2012b (MathWorks, Natick, MA, USA).
2.2 Physiological data Pressure and flow measurements were performed on twelve healthy subjects (29 ± 3 years, six male, six female, BMI: 23.4 ± 2.3 kg m−2) during muscle contraction in both the supine and 70◦head up tilt positions. These experiments were approved by the ethi-cal committee of the Northern Rhine Medical Association, Germany (Ethikkommission der ¨ Arztekammer Nordrhein). Subjects were asked to perform a contraction of the left calf mus-cle corresponding to 30% of maximal electromyography (EMG) activity (Ambu Blue Sen-sor N, Ballerup, Denmark). Visual feedback of the relative muscle activity was provided to enable the subjects to maintain muscle activity at the prescribed level. During the experiment blood pressure waveforms were measured at the finger using photoplethys-mography (Finometer Midi, AD instruments), while maintaining the wrist at heart level.
Furthermore, femoral artery blood flow was assessed using a Mylab 25 ultrasound scan-ner (Esaote, the Netherlands) equipped with a linear array probe and having a center 10 Table 2: Constant model parameters Symbol Value Unit Description ρ 1050 kg m−3 Blood mass density [Kenner, 1989] g 9.81 m s−1 Gravitational acceleration pref,A 13 kPa Arterial reference pressure [Bessems et al., 2007] ν 0.5 -Poisson’s ratio [Westerhof et al., 1969] E 1.6 MPa Arterial Young’s modulus [Westerhof et al., 1969] Kp 425 Pa Bending stiffness [M¨ uller and Toro, 2014] A+ 0 1.37 -Fitting constant [Keijsers et al., 2015] p+ a -2.53 -Fitting constant [Keijsers et al., 2015] p+ b 3.02 -Fitting constant [Keijsers et al., 2015] B 0.108 -Fitting constant [Keijsers et al., 2015] A− 0 1.28 -Fitting constant [Keijsers et al., 2015] p− a -1.49 -Fitting constant [Keijsers et al., 2015] p− b 2.03 -Fitting constant [Keijsers et al., 2015] γ 4 -Fitting constant [Keijsers et al., 2015] η 4.5 mPa s Dynamic blood viscosity [Letcher et al., 1981] βl 1.0 -Effective valve length ratio [Keijsers et al., 2016] Aeff,min 1.0 10−20 m2 Minimal effective valve cross-sectional area [Mynard et al., 2012] βA 0.65 -Effective valve cross-sectional area ratio [Keijsers et al., 2016] Kvo 0.3 Pa−1s−1 Valve opening constant [Mynard et al., 2012] Kvc 0.3 Pa−1s−1 Valve closing constant [Mynard et al., 2012] dpvalve,0 0 Pa Valve opening and closing pressure drop [Keijsers et al., 2016] τRC 2.0 s Typical time constant for windkessel element [Keijsers et al., 2016] pex,max 20 kPa Maximal extravascular pressure [Keijsers et al., 2016] frequency of 10 MHz. The blood flow measurement were performed in pulsed-Doppler mode. Blood flow was estimated from mean blood flow velocity and vessel diameter using the Poiseuille formulation [Leguy et al., 2009].
To use the experimental data for validation of the simulated muscle flow, the in vivo flow decay after muscle contraction (10 s < t < 50 s) was captured using the following exponential decay relation and a non-linear least squares fit.
qfit(t) = q0 + (qmax −q0)e−(t−tmax)/τ, (18) where tmax = 10 s, q0 is the baseline flow, qmax is the flow at t = tmax and τ is the time constant of the flow decay. Measurements are excluded from postprocessing when (1) average arterial pressure is below 50 mmHg for a whole experiment, (2) femoral artery flow was only measured successful during part of the experiment or (3) the quality of the flow fit was too low (R2 adj < 0.6). An average of the pressure and femoral artery flow was derived in the supine and head up tilt positions using the following relation x(t) = 1 Nsubj Nsubj X isubj 1 NMC,isubj NMC,isubj X iMC xisubj,iMC(t), (19) where Nsubj is the number of subjects, NMC,isubj is the number of muscle contractions per-formed by subject isubj and xisubj,iMC(t) is the waveform obtained during muscle contrac-tion iMC of subject isubj. The corresponding intersubject standard deviation was derived using the following relation: σ2(t) = 1 Nsubj −1 Nsubj X isubj xisubj −x(t) 2 , (20) 11 Art pres [mmHg] A MC A MC −10 0 10 20 30 40 50 0 50 100 Time [s] Fem art flow [m3/s] B MC B MC −10 0 10 20 30 40 50 0 5 10 15 x 10 −6 Supine Tilted Supine fit Tilted fit Figure 5: Heartbeat average of A the finger pressure and B femoral artery flow response to a muscle contraction in supine (red) and head up tilt (blue) position. The gray area indicates the 4-s muscle contraction (MC). Furthermore, the fit (−−) to the flow response is included, where its uncertainty is indicated with the shaded area around (light gray indicates the overlap). The dotted line (··; 0 < t < 10 s) indicates the uncertain part of the flow curve due to measurement difficulties. Fitting parameters and their standard deviation can be found in Table 3.
Table 3: Fitting parameters of the flow decay after muscle contraction using the following equation: qfit(t) = q0 + (qmax −q0)e−(t−tmax)/τ q0 [mL/s] qmax [mL/s] τ [s] Supine 2.8 ± 1.5 12.1 ± 5.6 8.4 ± 1.6 Head up tilt 1.2 ± 1.1 9.2 ± 5.0 6.4 ± 2.2 where xisubj(t) is the mean response of subject isubj. The resulting time averaged pressure and femoral artery flow are shown in Figure 5A and B respectively, with the supine mea-surement in red and the head up tilt in blue. The area around the fitted curve represents one standard deviation from the average fit.
2.3 Simulations and analysis The first aim is to match the simulated flow response during a supine muscle contrac-tion to the fit of the measured data (Figure 5B), to gain insight into the importance of the various regulation mechanisms. An explorative local analysis including variation of xinit, Tmax,0, Gmeta and Gmyo, identified Gmeta and Gmyo as the major determinants. Therefore, Gmeta and Gmyo were varied (−25 < Gmeta < −15; dGmeta = 1 and 0.5 < Gmyo < 1.5; dGmyo = 0.25) during the fitting procedure, while keeping all other regulation param-eters at their baseline values (55 model evaluations). The best three sets of gains are derived based on the least square of the difference between the simulated flow response and the in vivo fit.
ǫ = Z 50 s t=10 s p (qsim −qfit)2dt.
(21) These parameters are then used to repeat the simulation in the head up tilt position.
Apart from obtaining a nice fit, the second aim of the first analysis is to determine the relative importance of the regulation mechanisms. Because the regulation mechanisms are included individually their relative importance can be analyzed directly.
12 Time [s] B A Sx1 Sx1,x2 Sx1,x3 Sx3 Sx1,x2,x3 MC Arterial flow 50 in vivo fit qhut,bl εsup εhut simulation 10 0 qsup,10 qhut,10 30 supine tilted Figure 6: Sensitivity analysis A Schematic visualisation of the distribution of the output variance over the various input parameter and their interactions. Si = main sensitivity index, Sij = second order effect, Sijk = third order effect. B Output of interest visualized in a plot of the flow response to a muscle contraction in supine and tilted position.
In the above mentioned analysis a constant pressure is used as an input for the baroreflex and the inlet boundary condition. However, in vivo the systemic pressure shows a small increase during and a decrease after muscle contraction (Figure 5). In a second analysis the influence of this pressure fluctuation via the systemic pressure and the baroreflex reg-ulation on the flow response was investigated. For this the best parameter set, derived in the first analysis, was used to repeat the supine and tilted simulations with the following adaptations: (1) in vivo pressure is used as an input for the baroreflex and pinlet remains unchanged compared to the previous simulations; (2) in vivo pressure is used as an input for the baroreflex as well as for pinlet.
2.4 Sensitivity analysis To investigate the importance of all regulation parameters on the flow response to a mus-cle contraction and to validate the choice to derive the fit based on only Gmeta and Gmyo as described in the previous section, a global sensitivity analysis was performed. Si-multaneous variation of the input parameters within their uncertainty range enables the derivation of the variance in the simulated flow response. Each fraction of this output variance can be allocated to individual parameters or interaction between two or more input parameters (Figure 6A). The influence of an individual input parameter is captured by the main sensitivity index Si, which can be interpreted as the expected reduction in output variance if the true value would have been known. The contribution of interac-tion between two or more parameters is captured by the higher order effects (Sij, Sijk, ...) [Eck et al., 2016].
2.4.1 Output of interest The following parameters, describing the flow response to a muscle contraction in both supine and tilted position, are used as outputs of interest: • qsup,max: Flow in supine position 10 s after the onset of muscle contraction.
• ǫsup = R 50 s t=10 s q (qsim,sup −qfit,sup)2: Root mean square of the difference between the simulation and the fit of the flow response to a muscle contraction in the supine position.
• qhut,bl: Baseline flow in the tilted position.
• qhut,max: Flow in the tilted position 10 s after the onset of muscle contraction.
13 • ǫhut = R 50 s t=10 s p (qsim,hut −qfit,hut)2: Root mean square of the difference between the simulation and the fit of the flow response to a muscle contraction in the tilted position.
2.4.2 Input parameters The sensitivity analysis was performed while varying all input parameters of the regula-tion model. A description of these parameters can be found in Table 4, along with their baseline value and the range used for the sensitivity analysis. The uncertainty ranges are based on literature values or values resulting in a physiological flow response determined by a local sensitivity analysis (results not shown). From this local sensitivity analysis, it was concluded thatrm (the radius at which maximal tension can be reached) and rt (the constant defining the shape of the maximal tension curve) should be fixed, as even small variation resulted in non-physiological responses or decreased model stability.
Table 4: Model input parameters included in the sensitivity analysis. Uncertainty range is given in percentages, unless indicated with superscript ABS when the absolute range is given. The uncertainty range is based on literature values and is adapted when the local sensitivity analysis indicated unphysiological outputs or decreasing in model stability.
Symbol Value Unit Description Range σe,0 1.49 kPa Parameter for elastic tension model (Laplace) [Ursino and Giammarco, 1991] -10,7.5 Kσ 4.5 -Parameter in tension model (Laplace) [Ursino and Giammarco, 1991] -10,10 ra,0 75.0 µ m Arteriolar radius in un-stressed condition (Laplace) [Ursino and Giammarco, 1991] -10,10 σc 5.51 kPa Stress contribution of col-lagen fibers (Laplace) [Ursino and Giammarco, 1991] -10,10 rha,0 0.33 -Unstressed arteriolar wall thick-ness relative to radius (Laplace) [Nordborg et al., 1985] -10,10 ηa 6.37 kPa s Arteriolar wall viscosity (Laplace) [Ursino and Giammarco, 1991] -10,10 nm 1.75 -Parameter for smooth mus-cle tension model (Laplace) [Ursino and Giammarco, 1991, Ursino and Lodi, 1998] -10,7.5 Tmax,0 5.0 Pa Smooth muscle tension in basal condition (Laplace) [Ursino and Giammarco, 1991, Ursino and Lodi, 1998] 4.0,5.5ABS xinit −0.5 -Offset regulation state (Laplace) -0.6,-0.45ABS pn 13.3 kPa Reference pressure barore-flex model (Baroreflex) [Boron and Boulpaep, 2003] -10,10 fab,min 2.52 s−1 Minimal afferent firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,30 fab,max 47.78 s−1 Maximal afferent firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,20 14 Table 4 – continued from previous page Symbol Value Unit Description Range kdp 1.5676 kPa Parameter defining slope of affer-ent firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000] -20,30 fsp,∞ 2.1 s−1 Sympathetic firing rate at infinite afferent firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,30 fsp,0 16.11 s−1 Sympathetic firing rate at zero affer-ent firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,30 kes 0.0675 s Parameter defining the shape of the sympathetic firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,20 fsp,max 60 s−1 Maximal sympathetic firing rate (Baroreflex) [Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,30 GR 0.33 MPa s m−3 Gain baroreflex (Baroreflex) [Ursino and Magosso, 2000] -30,15 DR 2.0 s Pure time delay sympathetic firing rate (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,30 fes,min 2.66 s−1 Minimal sympathetic firing rate affecting resistance (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000] -30,20 τR 6.0 s Time constant low pass filter baroreflex (Baroreflex) [Ursino, 1998, Ursino and Magosso, 2000, van der Hout-van der Jagt et al., 2013] -30,30 V 300 mL Volume estimation of per-fused muscle tissue (Metabolic) [Elliott et al., 1997] -10,10 Ca,CO2 20.65 mol m−3 Arterial CO2 concentration (Metabolic) 20.0,20.9ABS fm 75 -Ratio of metabolism in rest and under maximal activity (Metabolic) [Boron and Boulpaep, 2003] 75,85ABS Amc,max 0.3 − Percentage of maximum EMG reached during muscle contraction (Metabolic) [Tschakovsky et al., 2004] -30,15 ρm 1055 kg m−3 Muscle density (Metabolic) [Segal et al., 1986] 1040,1070ABS MCO2,0,m 12.9 µmol s−1 kg−1 Basal metabolic CO2 production per kg muscle tissue (Metabolic) [Boron and Boulpaep, 2003] 9.0,13.5ABS αt,v 0.49 -Fitting constant venous CO2 con-centration (Metabolic) [Irving et al., 1932] 0.43,0.55ABS 15 Table 4 – continued from previous page Symbol Value Unit Description Range βt,v 11.5 mol m−3 Fitting constant venous CO2 con-centration (Metabolic) [Irving et al., 1932] 9.7,13.3ABS Cv,CO2,0 22.34 mol m−3 Venous CO2 concentration at rest (Metabolic) [Geers and Gros, 2000] 22.1,23.0ABS Gmeta −15 -Gain for metabolic mechanism (Metabolic) -25,-10ABS τmeta 15.0 s Time-constant metabolic regulation (Metabolic) [Ursino and Lodi, 1998] 12,18ABS Gmyo 0.75 -Gain for myogenic mechanism (Myogenic) 0.1,5ABS τmeta 7.0 s Time-constant myogenic regulation (Myogenic) [Ursino and Lodi, 1998] 4,9ABS 2.4.3 Morris screening and general polynomial chaos expansion To derive the output variance and the sensitivity indices in a computationally efficient manner, the two-step approach described by Donders et al. was used. In the first step non-important model parameters are identified by using a Morris screening. In the second step the generalized polynomial chaos expansion method is applied to the re-duced input space, resulting in a metamodel from which the sensitivity indices can be calculated straightforwardly [Huberts et al., 2014]. The metamodel consists of orthogonal polynomials dependent on the model parameters and with output-specific coefficients, which are derived by a least-square regression of the metamodel and N simulations. The accuracy of the metamodel is determined by the quality of the regression, for which a sufficient number of model evaluations is needed. In the current study a metamodel con-taining orthogonal polynomials up to the third order is derived based on 13485 model evaluations (CPU ≈63 h, using 25 cores). The number of model evaluations is based on: N = z+k z · q, where z = 3 is the order of the metamodel, k = 28 is the number of input parameters of the reduced input space and q is set to 3 to have sufficient simulations to obtain a good regression.
2.4.4 Post sensitivity analysis To investigate how well the important parameters identified in the sensitivity analysis can fit the in vivo response two additional sets of simulations were performed. First, all parameters with ST > 0.05 for at least one output of interest were varied randomly over k ∗500 simulations, with k the number of parameters. Second, the same process was carried out for all parameters with ST > 0.10. For both sets of simulations it was investigated which simulations were in good agreement with the in vivo fit, i.e. within the standard deviation of the in vivo fit. A second subset is defined to include all simulation within half the standard deviation. Finally, it is analysed how the input parameters of these subsets of simulations were distributed over the input space.
3 Results This section first reports how the activation of the regulation mechanisms influences the agreement between the flow response and the in vivo data. Secondly, the influence of systemic pressure on the regulation is reported. Finally, the results of the sensitivity analysis are presented.
16 Regulation state [−] Supine position A vaso−constriction vaso−dilation MC −10 0 4 10 20 30 40 50 −3 −2 −1 0 1 2 3 Arterial flow [mL/s] Time [s] C MC −10 0 4 10 20 30 40 50 0 1 2 3 4 5 Tilted position B MC −10 0 4 10 20 30 40 50 −3 −2 −1 0 1 2 3 Time [s] D MC −10 0 4 10 20 30 40 50 0 1 2 3 4 5 xbaro xmeta xmyo xtot In vivo Simulations Figure 7: Regulatory response to muscle contraction in supine and tilted position, de-picted in the left and right column respectively. In plot A and B the regulation state of the various mechanisms is shown over time: baroreflex (−−), metabolic (−·), myogenic (··) regulation and the sum of the three (−). Here, the negative state corresponds to va-sodilation and the positive state to vasoconstriction. The resulting arterial flow is shown in plot C and D together with the fit to the in vivo response (−·). The three simulations best matching the in vivo response are depicted in color. The remaining simulations (as described in Section 2.3) are visualized together in the gray area. To show the general patterns some individual responses are depicted in dark gray.
3.1 Baseline simulations The regulatory response to a muscle contraction in the supine position was simulated while varying the gain of the myogenic and metabolic mechanism. The variation in reg-ulatory responses and arterial flow are indicated by the gray region either side of the curve in Figure 7A and C respectively. The period of muscle contraction (MC) is indi-cated by the shaded region (0 < t < 4 s). The arterial flow responses which best match the in vivo measurement (−· plus the standard deviation indicated by the blue area) are visualised in color. These parameters values are used to repeat the simulation in the tilted position, for which the results are shown in Figure 7B and D. For the best flow results the regulatory state of the baroreflex (−−), metabolic (−·) and myogenic (··) mechanism are also shown in color.
Before the onset of muscle contraction in the supine position all the regulation states are equal to zero. After muscle contraction (t > 4 s), the metabolic mechanism induces a strong vasodilation, whereas the myogenic mechanism and baroreflex show a mild and small vasoconstriction respectively (Figure 7A). Arterial flow shows an increase due to 17 the muscle contraction and a gradual decay starting at t ≈10 s, which closely matches the in vivo response (Figure 7C). Most of the remaining flow responses show waveforms that are parallel to each other, although some simulations cross due to a difference in decay (Figure 7C).
In the tilted position, the baroreflex and to a lesser extent the myogenic mechanisms induce a vasoconstriction at baseline (−10 < t < 0 s). After muscle contraction (t > 4 s), the metabolic response induces a vasodilation, slightly inhibited by the vasoconstrictive response of the myogenic mechanism and baroreflex (Figure 7B). Finally, arterial flow increases after muscle contraction and decreases back to baseline, matching the in vivo response (Figure 7D).
3.2 Influence of systemic pressure The influence of the fluctuation in systemic pressure on the flow response to a muscle contraction is investigated. For this, the best fit found in the previous section is compared to a simulation with the pressure fluctuations included only in terms of the baroreflex regulation and a simulation with the pressure fluctuation applied at the inlet as well as the baroreflex. The regulatory (top) and flow (bottom) response to a muscle contraction in the supine (left) and tilted (right) position are shown in Figure 8.
In the supine position the three simulations all start at the same baseline and show a similar decay for t > 10 s (Figure 8C). Peak flow (5 < t < 10 s) is lower once the pressure fluctuation is applied via baroreflex regulation (green line). In the case where the pressure fluctuation is also applied as an inlet boundary condition (orange line) a fast decrease is observed shortly after muscle contraction followed by a plateau. In the tilted position all three simulations start at the same baseline (Figure 8D). The flow decay for t > 10 s is faster for both simulations with the in vivo pressure applied compared to the original simulation, but remain close to the fit of the in vivo response (dashed dark blue line).
Furthermore, peak flow is higher and is reached sooner following muscle contraction if the in vivo pressure is used.
3.3 Sensitivity analysis 3.3.1 Morris screening From the Morris screening the following parameters were found to be unimportant: pa-rameter for smooth muscle tension model nm, minimal afferent firing rate (baroreflex) fab,min, parameter defining shape of sympathetic firing rate (baroreflex) kes, pure time de-lay of sympathetic firing rate (baroreflex) DR, percentage of maximum EMG Amuscle,max and basal metabolic CO2-production MCO2,0,m. Excluding these six parameters from the polynomial chaos expansion reduces the required number of simulations from 23310 to 13485.
3.3.2 Polynomial chaos expansion The quality of the derived metamodels, captured by the descriptive error, is shown in Table 5. This gives the part of the variance that could not be captured by the metamodel.
For the ǫsup and ǫhut the descriptive error is relatively large; 0.14 and 0.10 respectively.
The total sensitivity indices for all outputs of interest are shown in Table 6. The input parameters are arranged in order of importance and only contributions greater than 1% are shown. The myogenic gain Gmyo is the most important parameter for all outputs of interest. Furthermore, the metabolic gain Gmeta, the initial regulation state xinit and the 18 Regulation state [−] Supine position A vaso−constriction vaso−dilation MC −10 0 4 10 20 30 40 50 −3 −2 −1 0 1 2 3 Arterial flow [mL/s] Time [s] C MC −10 0 4 10 20 30 40 50 0 1 2 3 4 5 Tilted position B MC −10 0 4 10 20 30 40 50 −3 −2 −1 0 1 2 3 Time [s] D MC −10 0 4 10 20 30 40 50 0 1 2 3 4 5 xbaro xmeta xmyo xtot In vivo fit Original In vivo pbaro In vivo pbaro and pinlet Figure 8: Influence of variation in systemic pressure on the regulatory response to mus-cle contraction in supine and tilted position (left and right column respectively). In plot A and B the regulation state of the various mechanisms is shown over time: baroreflex (−−), metabolic (··), myogenic (−·) regulation and the sum of the three (−). Here, the negative state corresponds to vasodilation and the positive state to vasoconstriction. The resulting arterial flow is shown in plot C and D together with the fit to the in vivo response (−·). The various colors represent the original simulation (red line), in vivo pressure ap-plied at the baroreflex (green line) and in vivo pressure applied to both the baroreflex and the inlet boundary condition (orange line).
Table 5: Quality of the metamodel for each output of interest: qmax,sup, ǫsup, qbl,hut, qmax,hut and ǫhut. The error measure 1 −R2 can be interpreted as the residual variance that could not be captured by the metamodel.
qmax,sup ǫsup qbl,hut qmax,hut ǫhut 1 −R2 0.04 0.14 0.01 0.06 0.10 19 Table 6: Total sensitivity indices of all the outputs of interest: qmax,sup, ǫsup, qbl,hut, qmax,hut and ǫhut. The input parameters are arranged in order of importance and only contribu-tions starting at 1% are shown.
qmax,sup ǫsup qbl,hut qmax,hut ǫhut Gmyo 0.79 0.72 0.55 0.64 0.47 Gmeta 0.08 0.17 0.18 0.30 xinit 0.02 0.38 0.04 0.11 τmeta 0.16 0.02 0.09 Cv,CO2,0 0.02 0.06 0.05 0.09 Ca,CO2 0.02 0.06 0.05 0.09 r0 0.06 0.06 0.02 0.01 0.02 Tmax,0 0.03 0.03 0.01 0.03 0.05 τmyo 0.08 0.03 fab,max 0.01 0.02 0.02 0.05 fm 0.02 0.05 0.01 0.02 GR 0.02 0.02 0.02 0.04 pn 0.01 0.02 0.04 0.01 0.02 kdp 0.01 0.01 0.02 0.03 Kσ 0.03 0.03 fes,min 0.02 0.01 0.03 fsp,∞ 0.01 0.01 0.03 fsp,0 0.02 0.01 σe0 0.02 rh0 0.02 V 0.02 αtv 0.02 σc 0.01 ηa 0.01 fsp,max 0.01 τR 0.01 ρm 0.01 βtv 0.01 metabolic time constant τmeta all contribute more than 10% to the variance for at least one output of interest. This is in line with the first local analysis where a fit was derived based on Gmyo and Gmeta. Four other parameters have a contribution larger than 5%: Cv,CO2,0, Ca,CO2, r0 and τmyo. All other parameters have a smaller contribution, but they do all contribute to the variance of the output.
The main sensitivity indices and higher order interactions are shown in Figure 9, where the main sensitivity indices Si are shown as ellipsoids, the second order interactions are indicated by an arrow and the third order interactions by a shaded area. The myogenic gain contributes most to the output variance; it has the highest main sensitivity index for all outputs of interest and is present in all of the main interactions. Furthermore, the metabolic gain Gmeta, the initial regulation state xinit and the metabolic time constant τmeta all have a main sensitivity index and/or interaction larger than 0.05 for at least one output of interest. The sums of the sensitivity indices (bottom of each subfigure) show that most of the variance is captured by individual contributions (Si). However, for ǫsup and ǫhut a significant contribution to the variance comes from interactions between parameters.
The contribution of the parameters varies for each regulation mechanism. The influence of the metabolic parameters is mainly observed in the maximum flow and ǫ outputs.
The baroreflex parameters, on the other hand, are of more importance for the variance in baseline flow in the tilted position and the ǫ in tilted position. The parameters describing the myogenic mechanism and Laplace law are present in all outputs of interest.
20 Σ Si = 0.87 Σ Sijk = 0.07 Σ Sij = 0.06 E qmax,hut Ca,CO2 Si= 0.03 Cv,CO2,0 Si= 0.03 Tmax,0 Si= 0.02 xinit Si= 0.01 Gmyo Si= 0.58 Gmeta Si= 0.14 Σ Si = 0.87 Σ Sijk = 0.04 Σ Sij = 0.09 Sij= 0.03 Sij= 0.01 Ks r0 B qmax,sup Ca,CO2 Si= 0.02 Cv,CO2,0 Si= 0.02 Tmax,0 Si= 0.02 fm Si= 0.02 Gmyo Si= 0.71 Gmeta Si= 0.07 Cv,CO2,0 Si= 0.03 Ca,CO2 Si= 0.02 Σ Si = 0.59 Σ Sijk = 0.15 Σ Sij = 0.26 F εhut Tmax,0 Si= 0.02 fab,max Si= 0.01 GR Si= 0.01 xinit Si= 0.04 Sij= 0.02 Sij= 0.02 Sijk= 0.01 Sijk= 0.01 τmeta Si= 0.06 Sij= 0.09 Gmeta Si= 0.13 Gmyo Si= 0.23 Σ Si = 0.92 Σ Sijk = 0.01 Σ Sij = 0.07 D qbl,hut r0 Si= 0.02 pn Si= 0.04 GR fab,max Sij= 0.01 Sij= 0.01 Sij= 0.01 Gmyo Si= 0.48 xinit Si= 0.36 Gmeta Cv,CO2,0 Ca,CO2 fm Σ Si = 0.46 Σ Sijk = 0.16 Σ Sij = 0.38 C εsup r0 Si= 0.02 τmyo Si= 0.04 τmeta Si= 0.04 Sij= 0.02 Sij= 0.03 Sij= 0.03 Sij= 0.04 Sij= 0.01 Sij= 0.12 Sij= 0.08 Gmyo Si= 0.34 xi Si = ...
Second order interaction Sij = ... if Sij > 0.05 xj xi Sij= ...
xi xk xj Sijk= ...
Third order interaction Main sensitivity index if Si > 0.05 A Legend Figure 9: Results of the sensitivity analysis for all the outputs of interest: B qmax,sup, C rmsqsup, D qbl,hut, E qmax,hut and F rmsqhut. The main sensitivity index is visualised in a circle, the second order interaction with an arrow and the third order interaction with an area. For clarity only the contributions larger than 1% are shown. The most important parameters and interactions (with a contribution more than 5%) are highlighted with a gray background or bold font respectively. The sum of the main sensitivity indices, the sum of the second order interactions and the sum of the third order interactions are shown at the bottom of each subfigure.
21 Flow supine [mL/s] Variation parameters ST > 0.05 A −10 0 10 20 30 40 50 0 2 4 6 Flow tilted [mL/s] Time [s] C −10 0 10 20 30 40 50 0 2 4 6 Gmyo Gmeta xinit τmeta Cv,CO 2,0 Ca,CO 2 τmyo r0 Scaled input [−] E 0 0.5 1 Variation parameters ST > 0.10 B −10 0 10 20 30 40 50 0 2 4 6 Time [s] D −10 0 10 20 30 40 50 0 2 4 6 All sims Good sims (std) Good sims (std/2) Best sims In vivo Gmyo Gmeta xinit τmeta F 0 0.5 1 Figure 10: Post sensitivity analysis. The flow response of additional simulations varying the input parameters with ST > 0.05 (left column) and ST > 0.10 (right column) in both supine (AB) and tilted position (CD). For both sets the good simulations (present within one standard deviation and half the standard deviation) are presented in dark gray and the best 10 simulations in color. In the bottom plots (EF) the distribution of the corresponding input parameter is shown.
3.3.3 Post sensitivity analysis The important parameters identified though the sensitivity analysis were used to perform two sets of simulations: (1) varying all parameters with ST > 0.05 (k = 8) and (2) varying all parameters with ST > 0.10 (k = 4). The flow response in both the supine and tilted positions together with the distribution of the input parameters is shown in Figure 10.
Both sets of simulations are divided into four subsets: (1) the simulations that converged (light gray) (2) the simulations that had a flow response within the standard deviation of the in vivo fit (middle gray) (3) the simulation that had a flow response within half a standard deviation (dark gray) and (4) the 10 simulations which best matched the mean in vivo response (colors).
For the first set of 4000 simulations (ST > 0.05) 1610 of the 3880 (41%) converged simu-lations had a flow response within one standard deviation of the in vivo response (mid gray in Figure 10AC). Taking only half the standard deviation into account only 385 (10%) simulations remained (dark gray). In Figure 10E it can be observed that all values of the input parameters can result in a flow response within the in vivo uncertainty, because the 22 light gray area covers the whole input space. However, some combinations never occur; e.g. Gmyo and Gmeta never have their maximum value simultaneously. When considering the simulations within half a standard deviation, a decrease in the input range of Gmyo is observed (Figure 10E). The ten best simulations closely match the mean in vivo response in both positions (Figure 10AC). The distribution of the input parameters is more spread over the input domain once the importance of the parameter decreases (parameter im-portance decreases from left to right). Whereas the most important parameter Gmyo has relative values between 0.10 and 0.27, values of the less important parameters, Cv,CO2,0, Ca,CO2,0 and τmyo, cover the full input domain.
In the second set of 2000 simulations (ST > 0.10) 1104 out of 1961 (56%) converged simu-lations showed a flow response within the in vivo uncertainty (mid gray in Figure 10BD).
Considering only half a standard deviation results in only 277 (14%) simulations. Similar to the larger set of simulations, the input parameters of the good simulations (within one standard deviation) had values within their whole uncertainty range (mid gray Figure 10F). Again, not all combinations were present, especially at the lower and upper lim-its of the domains. For the subset within half a standard deviation a decrease in input range of Gmyo is observed as for the first set of simulations. Although the value of ǫsup and ǫhut slightly increased (same order of magnitude), the 10 best simulations still closely matched the in vivo fit. However, now the values of Gmeta show a stronger correlation with the values of Gmyo, which is in line with the high values found in the sensitivity analysis for the interaction between Gmyo and Gmeta. Furthermore, the range of Gmeta has shifted to the upper part of the domain, which indicates Gmeta could be fixed within this range to obtain a good fit.
4 Discussion The flow augmentation observed at the onset of exercise is hypothesized to be a result of the muscle pump effect, the regulation of vascular tone or a combination of both. In a previous study [Keijsers et al., 2015], we showed that the muscle pump effect alone cannot induce the flow increase observed in vivo. Therefore, in the current study the im-portance of the major mechanisms regulating blood flow during the different phases of a muscle contraction has been investigated in both the supine and tilted position. To inves-tigate these effects our arterio-venous 1D pulse wave propagation model [Keijsers et al., 2015] has been extended with a regulation model accounting for baroreflex, metabolic and myogenic regulation. Model parameters were either taken from literature or deter-mined by fitting the simulated arterial flow response to the measured in vivo response to a muscle contraction in the supine position. The model was then validated by comparing simulated results with the in vivo measurements in the tilted position without changing the parameter values obtained from the fit in the supine position. Furthermore, a sensi-tivity analysis has been performed to quantify the importance of the input parameters in the regulation model.
The model was able to capture the in vivo response in the supine position when only optimizing the values of the myogenic and metabolic gain (Figure 7C). When the same parameters were used to simulate a muscle contraction in the tilted position, again good agreement was found (Figure 7D). The model response replicates two of the main fea-tures of flow variation. Firstly, it matches the flow decay back to baseline after the va-sodilation is initiated following muscle contraction. Secondly, the model captures the decreased baseline flow in the tilted position observed in vivo. Examining the activation of the various regulation mechanisms, the metabolic mechanism is the main vasodilator after muscle contraction in both the supine and tilted position, which is in line with in vivo studies [N˚ adland et al., 2009, Tschakovsky and Sheriff, 2004]. Furthermore, these simu-23 lations support the hypothesis of N˚ adland et al. that the decrease in baseline flow in the tilted position is a result of the global baroreflex and local myogenic activation.
The latter is a result of the decreased carotid pressure and increased arteriolar pressure respectively.
The influence of the variation in systemic pressure via the baroreflex mechanism and the boundary conditions of the model is assessed and is most clearly observed within the first 10 s after the onset of muscle contraction (Figure 8). The lack of reliable in vivo data shortly after muscle contraction, does not allow any conclusions to be drawn on which implementation is closest to physiology. The relatively small effect during the remaining part of the response (t > 10s) can be explained by the fact that most of the variation in systemic pressure is present shortly after muscle contraction. In the in vivo study of N˚ adland et al. it was stated that the systemic pressure reduction was too small to have an effect on femoral artery flow. However, based on the combination of the current model results and in vivo measurements this statement can neither be confirmed nor re-jected. Because the current study focusses on the flow decay after muscle contraction and the baseline flow, which are both hardly affected, the systemic pressure variation is not expected to have a large influence on the results.
Based on the sensitivity analysis, the spread in myogenic gain Gmyo is clearly the most im-portant parameter (both individually and through interactions) of the regulation model for variance in the simulated flow response to muscle contraction (Figure 9). The uncer-tainty in metabolic gain Gmeta also has a significant contribution to the output variance.
The importance of both gains was expected, because they determine the magnitude of vasodilation. Furthermore, it confirms the choice to vary Gmyo and Gmeta in the first anal-ysis. The fact that Gmyo is more important than Gmeta may be a result of the sigmoid function (Equation (30)) that is applied to the total regulation state. Even a small myo-genic activation (i.e. vasoconstriction) will shift the total regulation towards the more sensitive part of the regulation curve. A third important parameter is xinit, which is the offset of the regulation state. A change in xinit can shift the regulatory response to a less or more sensitive region of the regulation curve. This explains the large importance of xinit for qbl,hut. The fourth important parameter is the metabolic time-constant τmeta, which is expectedly important for both ǫ outputs. The fact that the metabolic parameters dom-inate the ǫ output is logical, because the metabolic activation was concluded to be the main vasodilator after muscle contraction. The baroreflex is almost inactive in the supine position, which is confirmed by the fact that the baroreflex parameters are not present for the supine outputs. Whilst the current model may seem complex, the large contribution of the higher order terms (Sij and Sijk in Figure 9) indicates the need of all parameter in-teractions in capturing the complex physiology of the system and thereby that the model is not too complex.
In the post sensitivity analysis it was concluded that even when only varying the 4 most important parameters (each contributing more than 10%), it was still possible to find sim-ulations that strongly resemble the in vivo response. The small range found for Gmyo for the 10 best fits confirms the importance of Gmyo. Furthermore, the interaction between Gmyo and Gmeta was also confirmed, because high values of the two parameters never oc-cur simultaneously. Examining the relation between Gmyo and Gmeta for the 10 best fits, even suggests defining a relation between the two. The large spread of input parameters observed for the subset of simulations within the measurement uncertainty, could indi-cate that the whole input space is not covered. However, analysing the simulations with a flow response within half a standard deviation indicates that if one could reduce the measurement uncertainty, the input space of the most important parameter Gmyo could be decreased.
To model the regulation of vascular tone a general approach is taken using the mean ar-24 teriolar radius as a measure for the regulatory state, because the explicit representation of individual arterioles was not of interest in the current study. Metabolic regulation was included based on a single metabolite, whereas many metabolites are known to act as vasodilator and no single metabolite has been shown to account for the full vasodila-tory response [Joyner and Casey, 2015]. However, the current implementation is in good agreement with the in vivo response, which indicates that the tissue CO2-concentration is a good surrogate for the general metabolic response. For a correct myogenic activa-tion an accurate pressure level is necessary. As only the calf circulation is included in the 1D part, the hydrostatic column applied to the pressure boundary condition might be overestimated, especially on the venous side. This could be overcome if the proximal vasculature would also be included in the 1D part of the model. However, as the current model is able to accurately match the in vivo response, it is concluded that the current model contains sufficient detail to capture the flow response after muscle contraction.
For validation of the developed model, in vivo ultrasound measurements were performed capturing the flow response to a calf muscle contraction in both the supine and tilted po-sitions (Figure 5B). Measured baseline flow in the supine position was observed to be 2.3 times higher than in the 70◦head up tilt position (Table 3), which is in line with the flow decrease observed by N˚ adland et al. in the 30◦head up tilt position. Flow changes observed following muscle contraction reach peak flow within 10 s followed by a decay back to baseline within a further minute. This is in accordance with the changes observed by Tschakovsky et al. following a single forearm contraction and those observed by Wesche following quadriceps contraction. Although the general flow response is in accordance with previous in vivo studies, the first 10 s after the onset of muscle contraction are excluded from the validation, because this part of the measure-ment is less accurate due to measurement difficulties during and shortly after muscle contraction. Improved measurements are necessary for validation of the simulated flow response in the first 10 s after muscle contraction, which could possible be obtained by fixing the ultrasound probe to the subject.
The quality of the metamodel, captured in the coefficient of determination (1 −R2), was observed to be lower for the outputs ǫsup and ǫhut. Because both outputs cover a time range of 40 s, they include more information, which is more likely to be hard to capture in a metamodel. Furthermore, these effects could be due to the fact that the importance of the parameters excluded by the Morris screening was underestimated. However, the post sensitivity analysis shows that even when varying only the four most important param-eters the model is capable to capture the flow response to a muscle contraction. Another more likely reason is that the variance that could not be captured by the metamodel is a result of the high frequency vibrations present in some simulations, because the ǫ out-puts are affected most by these instabilities. Further research is needed to improve model stability. However, the values of the coefficient of determination are still acceptable and are not expected to influence the results.
This study has described how the developed model can be used to study the regulation of vascular tone in healthy individuals under muscle contraction. However, this model has potential application in the study of chronic venous disease. Extending the current model with regurgitating valves [Mynard et al., 2012] or valve prolapse [Pant et al., 2015], would allow examination of valve dynamics and hemodynamics in the presence of dis-ease. Furthermore, the model could be used to simulate the effect of multiple contrac-tions, as studied by Simakov et al. , or even exercise. For the latter application, an extension of the model to the full circulation [M¨ uller and Toro, 2014, Mynard and Smolich, 2015] is needed to account for venous return and baroreflex regulation of heart rate and heart contractility. This would also improve the model with a better representation of the full hydrostatic column.
25 5 Conclusion A 1D pulse wave propagation model was developed including the baroreflex, meta-bolic and myogenic regulation, which enables the simulation of the flow response to a muscle contraction. In addition to our previously presented model [Keijsers et al., 2015], which considered only the mechanical effect of a muscle contraction (muscle pump), we added a regulation model and now the simulated flow response accurately mimicks the in vivo measurements in both the supine and tilted positions. This confirms the hypothe-sis that regulation of peripheral resistance is an important mechanism inducing the flow increase at the onset of exercise. From the activation of the regulatory mechanisms it is concluded that (1) metabolic activation is the main vasodilator after muscle contraction and (2) baroreflex and myogenic activation are responsible for the decrease in baseline flow in the tilted position. The sensitivity analysis confirmed Gmyo as the most important parameter.
Acknowledgements J.M.T. Keijsers received a scholarship of the Helmholtz SpaceLife Sciences Research School (SpaceLife) which was funded by the Helmholtz Association and the German Aerospace Center (Deutsches Zentrum f¨ ur Luft- und Raumfahrt e.V., DLR). The contribution of Dr.
A.J. Narracott to this research was supported by funding from the Research Mobility Pro-gramme of the Worldwide Universities Network. The contribution of Dr. C.A.D Leguy was performed with the support of the Marie Curie International Outgoing fellowship of the European’s 7th Framework Programme for Research under contract number MC-IOF-297967 References References D. Bessems, M. Rutten, and F. Van De Vosse. A wave propagation model of blood flow in large vessels using an approximate velocity profile function. J Fluid Mech, 580:145–168, 2007.
W. F. Boron and E. L. Boulpaep. Medical Physiology: Updated Edition. Elsevier Health Sciences, 2003.
W. P. Donders, W. Huberts, F. N. van de Vosse, and T. Delhaas. Personalization of models with many model parameters: an efficient sensitivity analysis approach. Int J Numer Method Biomed Eng, 31(10):e02727, 2015.
V. G. Eck, W. P. Donders, J. Sturdy, J. Feinberg, T. Delhaas, L. R. Hellevik, and W. Hu-berts. A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications. Int J Numer Method Biomed Eng, 32(8):e02755, 2016.
M. A. Elliott, G. A. Walter, H. Gulish, A. S. Sadi, D. D. Lawson, W. Jaffe, E. K. Insko, J. S. Leigh, and K. Vandenborne. Volumetric measurement of human calf muscle from magnetic resonance imaging. MAGMA, 5(2):93–98, 1997.
Y. Fung. Biomechanics: mechanical properties of living tissues. Springer New York, 1993.
C. Geers and G. Gros. Carbon dioxide transport and carbonic anhydrase in blood and muscle. Physiol Rev, 80(2):681–715, 2000.
26 S. D. Gray, C. Staub, D. Sarah, E. Carlsson, and N. C. Staub. Site of increased vascular resistance during isometric muscle contraction. Am J Physiol, 213(3):683–689, 1967.
W. Huberts, W. P. Donders, T. Delhaas, and F. N. van de Vosse. Applicability of the poly-nomial chaos expansion method for personalization of a cardiovascular pulse wave propagation model. Int J Numer Method Biomed Eng, 30:1679–1704, 2014.
L. Irving, H. C. Foster, and J. K. W. Ferguson. The carbon dioxide dissociation curve of living mammalian muscle. J Biol Chem, 95(1):95–113, 1932.
M. J. Joyner and D. P. Casey. Regulation of increased blood flow (hyperemia) to muscles during exercise: a hierarchy of competing physiological needs. Physiol Rev, 95:549–601, 2015.
J. M. T. Keijsers, C. A. D. Leguy, W. Huberts, A. J. Narracott, J. Rittweger, and F. N. van de Vosse. A 1D pulse wave propagation model of the hemodynamics of calf muscle pump function. Int J Numer Method Biomed Eng, 31(7):e02714, 2015.
J. M. T. Keijsers, C. A. D. Leguy, W. Huberts, A. J. Narracott, J. Rittweger, and F. N. van de Vosse. Global sensitivity analysis of a model for venous valve dynamics. J Biomech, 49 (13):2845–2853, 2016.
T. Kenner. The measurement of blood density and its meaning. Basic Res Cardiol, 84(2): 111–124, 1989.
W. Kroon, W. Huberts, M. Bosboom, and F. van de Vosse. A numerical method of reduced complexity for simulating vascular hemodynamics using coupled 0D lumped and 1D wave propagation models. Comput Math Methods Med, 2012:156094, 2012.
C. A. D. Leguy, E. M. H. Bosboom, A. P. G. Hoeks, and F. N. van de Vosse. Model-based assessment of dynamic arterial blood volume flow from ultrasound measurements.
Med Biol Eng Comput, 47:641–648, 2009.
R. L. Letcher, S. Chien, T. G. Pickering, J. E. Sealey, and J. H. Laragh. Direct relationship between blood pressure and blood viscosity in normal and hypertensive subjects: role of fibrinogen and concentration. Am J Med, 70(6):1195–1202, 1981.
E. Lim, G. S. H. Chan, So. Dokos, S. C. Ng, L. A. Latif, S. Vandenberghe, M. Karunanithi, and N. H. Lovell. A Cardiovascular Mathematical Model of Graded Head- Up Tilt.
PLoS One, 8(10):e77357, 2013.
M. H. Meissner. Lower extremity venous anatomy. Semin Intervent Radiol, 22(3):147–156, 2005.
L. O. M¨ uller and E. F. Toro. A global multiscale mathematical model for the human circulation with emphasis on the venous system. Int J Numer Method Biomed Eng, 30(7): 681–725, 2014.
J. P. Mynard and J. J. Smolich. One-Dimensional Haemodynamic Modeling and Wave Dynamics in the Entire Adult Circulation. Ann Biomed Eng, 43(6):1443–1460, 2015.
J. P. Mynard, M. R. Davidson, D. J. Penny, and J. J. Smolich. A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models. Int J Numer Method Biomed Eng, 28:626–641, 2012.
I. H. N˚ adland, L. Walløe, and K. Toska. Effect of the leg muscle pump on the rise in muscle perfusion during muscle work in humans. Eur J Appl Physiol, 105(6):829–841, 2009.
27 C. Nordborg, K. Fredriksson, and B. B. Johansson. The Morphometry of Consecutive Segments in Cerebral Arteries of Normotensive and Spontaneously Hypertensive Rats.
Stroke, 16(2):313–320, 1985.
M. S. Olufsen, H. T. Tran, J. T. Ottesen, Research Experiences for Undergraduates Pro-gram, L. A. Lipsitz, and V. Novak. Modeling baroreflex regulation of heart rate during orthostatic stress. Am J Physiol Regul Integr Comp Physiol, 291:R1355–R1368, 2006.
S. Pant, C. Corsini, C. Baker, T. Y. Hsia, G. Pennati, and I. E. Vignon-Clementel. Data as-similation and modelling of patient-specific single-ventricle physiology with and with-out valve regurgitation. J Biomech, pages 1–12, 2015.
L. B. Rowell. Human Cardiovascular Control. Oxford University Press, 1993.
S. S. Segal, T. P. White, and J. A. Faulkner. Architecture, composition, and contractile properties of rat soleus muscle grafts. Am J Physiol, 250(19):C474–C479, 1986.
A. H. Shapiro. Steady flow in collapsible tubes. J Biomech Eng, 99(3):126–147, 1977.
S. Simakov, T. Gamilov, and Y. N. Soe.
Computational study of blood flow in lower extremities under intense physical load. Russ J Numer Anal M, 28(5):485–503, 2013.
B. Spronck, E. G. H. J. Martens, E. D. Gommer, and F. N. van de Vosse. A lumped pa-rameter model of cerebral blood flow control combining cerebral autoregulation and neurovascular coupling. Am J Physiol Heart Circ Physiol, 303:H1143–H1153, 2012.
M. E. Tschakovsky and D. D. Sheriff. Immediate exercise hyperemia: contributions of the muscle pump vs. rapid vasodilation. J Appl Physiol, 97(2):739–747, 2004.
M. E. Tschakovsky, J. K. Shoemaker, and R. L. Hughson. Vasodilation and muscle pump contribution to immediate exercise hyperemia.
Am J Physiol, 271(4):H1697–H1701, 1996.
M. E. Tschakovsky, A. M. Rogers, K. E. Pyke, N. R. Saunders, N. Glenn, S. J. Lee, T. Weiss-gerber, and E. M. Dwyer.
Immediate exercise hyperemia in humans is contraction intensity dependent: evidence for rapid vasodilation. J Appl Physiol, 96(2):639–644, 2004.
M. Ursino. A mathematical study of human intracranial hydrodynamics part 1 - the cerebrospinal fluid pulse pressure. Ann Biomed Eng, 16(4):379–401, 1988.
M. Ursino. Interaction between carotid baroregulation and the pulsating heart: a mathe-matical model. Am J Physiol, 275(44):H1733–H1747, 1998.
M. Ursino and P. D. Giammarco. A Mathematical Model of the Relationship Between Cerebral Blood Volume and Intracraniai Pressure Changes: The Generation of Plateau Waves. Ann Biomed Eng, 19(1):15–42, 1991.
M. Ursino and C. A. Lodi. Interaction among auto-regulation, CO2 reactivity, and in-tracranial pressure: a mathematical model. Am J Physiol, 274(43):H1715–H1728, 1998.
M. Ursino and E. Magosso. Acute cardiovascular response to isocapnic hypoxia. I. A mathematical model. Am J Physiol Heart Circ Physiol, 279:H149–H165, 2000.
M. Ursino and E. Magosso. Role of short-term cardiovascular regulation in heart period variability: a modeling study. Am J Physiol Heart Circ Physiol, 284:H1479–H1493, 2003.
28 M. B. van der Hout-van der Jagt, S. G. Oei, and P. H. M. Bovendeerd. Simulation of reflex late decelerations in labor with a mathematical model. Early Hum Dev, 89:7–19, 2013.
J. Wesche. The time course and magnitude of blood flow changes in the human quadri-ceps muscles following isometric contraction. J Physiol, 377:445–462, 1986.
N. Westerhof, F. Bosman, C. J. de Vries, and A. Noordergraaf. Analog studies of the human systemic arterial tree. J Biomech, 2:121–143, 1969.
A Extravascular pressure The temporal course of the extra vascular pressure is defined by k(t) = 0 if t < T0 1 2 1 + sin(π(t−T0) Tr −π 2 ) if T0 < t < T0 + Tr 1 if T0 + Tr < t < T0 + Tr + Tc 1 2 1 + sin(π(t−T0−Tr−Tc) Tf + π 2 ) if T0 + Tr + Tc < t < T0 + Tr + Tc + Tf 0 if t > T0 + Tr + Tc + Tf (22) where T0 = 0 s is the starting time of the compression, Tr = 1 s and Tf = 1 s are the times for the extravascular pressure to rise and fall respectively and Tc = 2 s is the time for the extravascular pressure to remain constant. The spatial course of the extravascular pressure is defined by m(z) = 0 if z < l0 1 2 1 + sin(π(z−l0) lr −π 2) if l0 < z < l0 + lr 1 if l0 + lr < z < lend −lf 1 2 1 + sin(π(z−lend) lf + π 2 ) if lend −lf < z < lend 0 if z > lend (23) where l0 = 0.07 m and lend = 0.27 m are the coordinates below and above which no extravascular pressure is applied and lr = 0.10 m and lf = 0.10 m are the lengths over which the extravascular pressure rises and falls. The z-coordinate is defined to be zero at the distal side of the vein and increases to z = 0.34 m at the proximal side.
B Regulation model The regulation of vascular tone is based on three regulation mechanisms: myogenic reg-ulation, metabolic regulation and the baroreflex. The activation of the different mecha-nisms and how they result in a change in resistance and compliance is explained in detail in the following sections.
B.1 Laplace’s law The arteriolar wall tension Ttot is the parameter determining vascular tone and is related to the arteriolar radius ra via the Laplace law [Fung, 1993].
Ttot = para −pex(ra + ha), (24) where pa is the arteriolar pressure and ha is the arteriolar wall thickness. The total ten-sion is divided into three components: Ttot = Te + Tv + Tm, where Te, Tv and Tm are 29 the elastic, viscous and active smooth muscle tension respectively. The passive elas-tic tension is based on experimental studies and is defined by the exponential function [Ursino and Giammarco, 1991] Te = haσe = ha σe,0 e Kσ· ra−ra,0 ra,0 −1 −σc , (25) where σe,0 and Kσ are parameters defining the shape of the function, ra,0 is the ves-sel radius in the unstressed condition and σc is the stress contribution of the collagen fibers. Furthermore, the arteriolar wall thickness ha is defined by assuming no longitu-dinal stretch and conservation of mass: ha = q r2 a + 2ra,0ha,0 + h2 a,0 −ra, (26) where ha,0 is the unstressed arteriolar wall thickness.
The second component of the passive tension is the viscous tension Tv, which is in accor-dance with the viscous component of the Voigt model [Ursino and Giammarco, 1991] Tv = σvha = ηa ra,0 dra dt ha, (27) where ηa is the arteriolar wall viscosity.
The active smooth muscle tension Tm is known to decrease for very small and very large arteriolar radius and is therefore based on the following bell-type curve [Ursino and Giammarco, 1991] Tm = Tmax · e− ra−rm rt−rm nm , (28) where rm is the radius at which the smooth muscle cell exerts maximal tension, and rt and nm are constants. Furthermore, the maximal active tension Tmax is defined by Tmax = Tmax,0(1 + Ms), (29) where Tmax,0 is the maximal tension at baseline, i.e. when the regulatory state Ms is equal to zero. The latter is defined by the following relation Ms = e2Ms,1 −1 e2Ms,1 + 1, (30) where Ms,1 = Gmyoxmyo + Gmetaxmeta + xbaro + xinit, (31) where xi and Gi are the state and gain of the regulation mechanism i and xinit is the reg-ulatory state at baseline. The state equations for xi are described later on. Summarizing, the Laplace law is used to translate a change in regulatory state to a change in arteriolar radius. Actual changes in resistance and compliance of the micro-circulation are derived by coupling the arteriolar radius ra to the arteriolar resistance Ra and volume Ca via Ra = Ka,R r4 a (32) and Ca = Ka,Cr2 a pa , (33) where • Ka,R is chosen such that the baseline arteriolar radius ra and pressure pa result in baseline flow qa. Where baseline means the supine position and Ms,1 = xinit.
30 • Ka,C is chosen such that it corresponds with a total RC-time of 2.0 [s] under baseline conditions.
Summarizing, the arteriolar resistance and compliance are regulated based on the arteri-olar radius ra. The latter is derived from the arteriolar wall tension based on the Laplace law. The muscular tension Tm is the part of the tension affected by the state of the regula-tion mechanisms xi. In the following sections the state equations for the three regulation mechanisms are explained.
B.2 Myogenic regulation mechanism Myogenic regulation protects the micro-vasculature against high pressures by increasing vascular tone upon increasing circumferential stresses and strains. Myogenic activation Amyo is therefore based on the current arteriolar tension Ttot [Spronck et al., 2012] Amyo = Ttot −Tmyo,0 Tmyo,s , (34) where Tmyo,0 is the tension at baseline pressure for Ms,1 = xinit and Tmyo,s = 0.2Tmax,0 is a normalization tension. The myogenic regulation state xmyo is defined by dxmyo dt = Amyo −xmyo τmyo , (35) where τmyo is the myogenic time constant.
B.3 Metabolic regulation mechanism Metabolic regulation can be initiated via different metabolites, such as potassium ions, adenosine, lactate and CO2. These metabolites are generated during a muscle contraction and are washed out by the blood flow. In the current model CO2 is chosen to be the determining metabolite for the regulation of blood flow during muscle contraction. First, the tissue CO2-concentration Ct,CO2 is defined as the balance between metabolic rate and muscle perfusion qd dCt,CO2 dt = 1 V (MCO2 −qd (Cv,CO2 −Ca,CO2)) , (36) where V is an estimate of the perfused muscle tissue volume, Cv,CO2 and Ca,CO2 are the venous and arterial CO2-concentration respectively, of which the latter is fixed [Spronck et al., 2012]. The metabolic rate MCO2 is related to muscle activity as follows: MCO2 = MCO2,0 (1 + Amc (fm −1)) , (37) where fm is the ratio of metabolic rate at rest and maximal activity and Amc is muscle activity, which is defined to follow the contraction pattern (see Section 2.1.6 for the full definition). Furthermore, MCO2,0 is the basal metabolic production of CO2 by a tissue of volume V MCO2,0 = MCO2,0,mρmV, (38) where ρm is the muscle density and MCO2,0,m is the basal metabolic CO2-production per kg tissue. Muscle perfusion qd in Equation (36) is the flow leaving the tissue, which is calculated using qd = p2 −p3 Ra/2 , (39) 31 where p2 and p3 are the pressure at node n2 and n3 respectively (Figure 3). Further-more, Cv,CO2 is the venous CO2-concentration and is determined by the following rela-tion [Irving et al., 1932] Cv,CO2 = αt,vCt,CO2 + βt,v, (40) where αt,v and βt,v are fitting constants. The metabolic activation Ameta is determined by the CO2-concentration in the tissue using Ameta = Ct,CO2 −Ct,CO2,0 Ct,CO2,s , (41) where Ct,CO2,0 is the steady state solution of Equation (36) and Ct,CO2,s = Cv,CO2,0 −Ca,CO2 is a scaling term. Finally, the metabolic regulation state xmeta is defined by dxmeta dt = Ameta −xmeta τmeta , (42) where τmeta is the time constant governing metabolic regulation.
B.4 Baroreflex regulation The baroreflex is a global regulation mechanism which aims to maintain systemic pres-sure by affecting the heart rate, heart contractility, venous unstressed volume and pe-ripheral resistance. In this study, only the effect of the baroreflex on the peripheral re-sistance is included, which is based on the model of Ursino (also implemented in other studies [Lim et al., 2013, van der Hout-van der Jagt et al., 2013]). The carotid pres-sure pcarotid is used as an input parameter and is defined as mean systemic pressure (see Section 2.3 for details) plus a hydrostatic column of 20 cm in tilted position. First, carotid pressure is compared to a reference pressure pn, which is defined as the baseline pressure in the supine position ∆pbaro = pcarotid −pn.
(43) This pressure difference ∆pbaro is converted to an afferent baroreflex firing fab rate via a sigmoidal transfer function fab = fab,min + fab,max · e ∆pbaro kdp 1 + e ∆pbaro kdp , (44) where fab,min and fab,max are the firing rates reached for minimal and maximal ∆pbaro and kdp is a constant determining the slope of the afferent firing rate. The firing rate for the sympathetic innervation of the peripheral micro-circulation fsp is calculated via the following relation: fsp = fsp,∞+ fsp,0 −fsp,∞ · e- kes fab for fsp < fsp,max fsp,max for fsp ≥fsp,max , (45) where kes is a parameter defining the shape of the sympathetic firing rate. The parameters fsp,0 and fsp,∞are the firing rates at zero and infinite afferent firing rate, and fsp,max is the maximal sympathetic firing rate. The sympathetic innervation is converted to an unfiltered change in resistance ∆R∗ ∆R∗= GR · ln fsp(t −DR) −fsp,min + 1 for fsp ≥fsp,min 0 for fsp < fsp,min , (46) 32 where GR is a constant gain, DR is a pure time delay and fsp,min is the minimal sympa-thetic firing rate affecting the resistance. The actual change in resistance ∆R is calculated based on ∆R∗using a low pass filter d∆R dt = 1 τR · (∆R∗−∆R) , (47) where τR is the time constant of the low pass filter. Finally, the relative change in resis-tance cbaro is calculated using the following relation cbaro = ∆R −Rref Rref,Ursino + 1 = Rbaro Rref , (48) where Rref is the resistance in baseline conditions in supine position and Rref,Ursino is the baseline resistance in the model of Ursino . To combine the baroreflex with the local auto-regulation mechanisms, the resistance is converted into a regulation state xbaro similar to the regulation state of the other mechanisms. Firstly, the resistance is converted to arteriolar radius using Equation (32) abaro = KR Rbaro 1/4 .
(49) Using Laplace’s law (Equation (24)) the arteriolar radius is converted into a change in muscular tension due to the baroreflex Tm,baro = paabaro −Te,baro −Tv,baro, (50) where Te,baro and Tv,baro are calculated using ra = abaro (Equation (25) and (27) respec-tively). From the muscular tension the total regulation state Ms,baro is derived using Equation (28) and (29) Ms,baro = Tm,baro Tmax,0 · e- abaro−am at−am n −1.
(51) Finally, the total regulation state is converted to the regulation state via Equation (30) and (31) xbaro = Ms,1,baro −xinit = tanh−1 (Ms,baro) −xinit = 1 2 ln 1 + Ms,baro 1 −Ms,baro −xinit.
(52) 33 |
7095 | https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/21 | Mathematical methods for economic theory
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3.2.3 Quadratic forms: conditions for semidefiniteness
Two variables
First consider the case of a two-variable quadratic form
Q(x, y) = ax2 + 2bxy + cy2.
If a = 0 then Q(x, 1) = 2bx + c. This expression is nonnegative for all values of x if and only if b = 0 and c ≥ 0, in which case ac − b2 = 0.
Now assume a ≠ 0. As before, we have
| | | |
---
| Q(x, y) | = | a[(x + (b/a)y)2 + (c/a − (b/a)2)y2]. |
Both squares are nonnegative, so if a > 0 and ac − b2 ≥ 0 then this expression is nonnegative for all (x, y). If these two conditions are satisfied then c ≥ 0.
We conclude that if a ≥ 0, c ≥ 0, and ac − b2 ≥ 0, then the quadratic form is positive semidefinite.
Conversely, if the quadratic form is positive semidefinite then Q(1, 0) = a ≥ 0, Q(0, 1) = c ≥ 0, and Q(−b, a) = a(ac − b2) ≥ 0. If a = 0 then by the previous argument we need b = 0 and c ≥ 0 in order for the quadratic form to be positive semidefinite, so that ac − b2 = 0; if a > 0 then we need ac − b2 ≥ 0 in order for a(ac − b2) ≥ 0.
We conclude that the quadratic form is positive semidefinite if and only if a ≥ 0, c ≥ 0, and ac − b2 ≥ 0.
A similar argument implies that the quadratic form is negative semidefinite if and only if a ≤ 0, c ≤ 0, and ac − b2 ≥ 0.
Note that in this case, unlike the case of positive and negative definiteness, we need to check all three conditions, not just two of them. If a ≥ 0 and ac − b2 ≥ 0, it is not necessarily the case that c ≥ 0 (try a = b = 0 and c < 0), so that the quadratic form is not necessarily positive semidefinite. (Similarly, the conditions a ≤ 0 and ac − b2 ≥ 0 are not sufficient for the quadratic form to be negative semidefinite: we need, in addition, c ≤ 0.)
Thus we can rewrite the results as follows: the two variable quadratic form Q(x, y) = ax2 + 2bxy + cy2 is
positive semidefinite if and only if a ≥ 0, c ≥ 0, and |A| ≥ 0
negative semidefinite if and only if a ≤ 0, c ≤ 0, and |A| ≥ 0
where
| | | | | | | |
--- --- ---
| A = | | | | --- | | a | b | | b | c | | . |
It follows that the quadratic form is indefinite if and only if |A| < 0. (Note that if |A| ≥ 0 then ac ≥ 0, so we cannot have a < 0 and c > 0, or a > 0 and c < 0.)
Many variables
As in the case of two variables, to determine whether a quadratic form is positive or negative semidefinite we need to check more conditions than we do in order to check whether it is positive or negative definite. In particular, it is not true that a quadratic form is positive or negative semidefinite if the inequalities in the conditions for positive or negative definiteness are satisfied weakly. In order to determine whether a quadratic form is positive or negative semidefinite we need to look at more than simply the leading principal minors. The matrices we need to examine are described in the following definition.
Definition
: Let A be an n × n symmetric matrix. For each k = 1, ..., n, the kth order principal minors of A are the determinants of the k × k matrices obtained by deleting n − k rows and the corresponding n − k columns of A.
Note that the kth order leading principal minor of a matrix is one of its kth order principal minors.
Example 3.2.3.1
: Let
| | | | | | | |
--- --- ---
| A = | | | | --- | | a | b | | b | c | | . |
The first-order principal minors of A are a and c, and the second-order principal minor is the determinant of A, namely ac − b2.
Example 3.2.3.2
: Let
| | | | | | | | | | | | |
--- --- --- --- --- --- |
| A = | | | | | --- | 3 | 1 | 2 | | 1 | −1 | 3 | | 2 | 3 | 2 | | . |
This matrix has 3 first-order principal minors, obtained by deleting
the last two rows and last two columns
the first and third rows and the first and third columns
the first two rows and first two columns
which gives us simply the elements on the main diagonal of the matrix: 3, −1, and 2. The matrix also has 3 second-order principal minors, obtained by deleting
the last row and last column
the second row and second column
the first row and first column
which gives us −4, 2, and −11. Finally, the matrix has one third-order principal minor, namely its determinant, −19.
The following result gives criteria for semidefiniteness.
Proposition 3.2.3.1
: Let A be an n × n symmetric matrix. Then
A is positive semidefinite if and only if all its principal minors are nonnegative.
A is negative semidefinite if and only if for k = 1, ..., n all of its kth order principal minors are nonpositive for k odd and nonnegative for k even.
Source
: For a proof, see Gantmacher (1959), Theorem 4 on p. 307.
Example 3.2.3.3
: Let
| | | | | | | |
--- --- ---
| A = | | | | --- | | 0 | 0 | | 0 | −1 | | . |
The two first-order principal minors and 0 and −1, and the second-order principal minor is 0. Thus the matrix is negative semidefinite. (It is not negative definite, because the first leading principal minor is zero.)
Procedure for checking the definiteness of a matrix
Procedure for checking the definiteness of a matrix
: 1. Find the leading principal minors and check if the conditions for positive or negative definiteness are satisfied. If they are, you are done. (If a matrix is positive definite, it is certainly positive semidefinite, and if it is negative definite, it is certainly negative semidefinite.)
2. If the conditions are not satisfied, check if they are strictly violated. If they are, then the matrix is indefinite.
3. If the conditions are not strictly violated, find all its principal minors and check if the conditions for positive or negative semidefiniteness are satisfied.
Example 3.2.3.4
: Suppose that the leading principal minors of the 3 × 3 matrix A are D1 = 1, D2 = 0, and D3 = −1. Neither the conditions for A to be positive definite nor those for A to be negative definite are satisfied. In fact, both conditions are strictly violated (D1 is positive while D3 is negative), so the matrix is indefinite.
Example 3.2.3.5
: Suppose that the leading principal minors of the 3 × 3 matrix A are D1 = 1, D2 = 0, and D3 = 0. Neither the conditions for A to be positive definite nor those for A to be negative definite are satisfied. But the condition for positive definiteness is not strictly violated. To check semidefiniteness, we need to examine all the principal minors. |
7096 | https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Map%3A_Inorganic_Chemistry_(Housecroft)/03%3A_Introduction_to_Molecular_Symmetry/3.08%3A_Chiral_Molecules | Skip to main content
3.8: Chiral Molecules
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3.7C: Vibrational Spectroscopy of Linear and Bent triatomic Molecules
4: Experimental Techniques
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( \newcommand{\kernel}{\mathrm{null}\,})
Introduction
Around the year 1847, the French scientist Louis Pasteur provided an explanation for the optical activity of tartaric acid salts. when he carried out a particular reaction, Pasteur observed that two types of crystals precipitated. Patiently and carefully using tweezers, Pasteur was able to separate the two types of crystals. Pasteur noticed that the types rotated the plane polarized by the same amount but in different directions. These two compounds are called enantiomers.
What are Enantiomers?
Two compounds are enantiomers if they are non-superimposable mirror images of each other. As was mentioned, enantiomers are characterized by their ability to rotate plane-polarized light. They also have the same physical properties (e.g., melting point, etc.) relative to each other. As a result, they are also referred to as being optically active. When it comes to symmetry, there are some general rules of thumb that help determine whether a molecule is chiral or achiral. This can be very useful because sometimes molecules can have relatively complicated structures and geometries that knowing whether or not they are chiral becomes a daunting task. The goal, as a result, is to determine the point group of the molecule and the symmetry elements associated with it, then inferring the chirality of the molecule.
Using Symmetry to Determine Chirality
For a molecule to be chiral, it must lack:
Center of inversion i and a plane of symmetry σ.
An improper rotation axis (rotation-reflection axis) Sn.
However, since, by definition, an improper rotation axis is a rotation about an certain axis followed by reflection about a plane perpendicular to that axis, and an inversion center is simply S2, the absence of an improper axis requires, in most cases, that absence of both a plane of symmetry and an inversion center. As a result, it suffices, in most cases, to check for improper rotation axes to determine whether a molecule is chiral or not.
As a result of the previous discussion, there are a few classes of point groups that lack an improper axis. Those classes are C1, Cn, and Dn. Cis-dichlorobis(ethylenediamine)cobalt (III) has two enantiomers that are chiral (figure 1), but the trans compound is achiral.
3.7C: Vibrational Spectroscopy of Linear and Bent triatomic Molecules
4: Experimental Techniques |
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Treasure Island (FL): StatPearls Publishing; 2025 Jan-.
Eosinophilic Esophagitis
Jordan M. Roussel; Sudha Pandit.
Author Information and Affiliations
Authors
Jordan M. Roussel1; Sudha Pandit2.
Affiliations
1 Louisiana State University HSC
2 Louisiana State University HSC
Last Update: August 8, 2023.
Continuing Education Activity
The esophagus physiologically lacks eosinophils, and when present, the condition is considered to be pathologic. Eosinophilic esophagitis (EoE) was once thought to be a component of gastroesophageal reflux disease (GERD). However, it is now known to be a separate entity as the knowledge of the esophagus as an active immunogenic organ has evolved. Eosinophils can be found in the esophagus in response to various stimuli or antigen. This activity explains how to properly evaluate for eosinophilic esophagitis, and highlights the role of the interprofessional team in caring for patients with this condition.
Objectives:
Describe the pathophysiology of eosinophilic esophagitis.
Review the evaluation of a patient suspected of eosinophilic esophagitis.
Summarize the treatment of eosinophilic esophagitis
Outline the importance of improving care coordination among interprofessional team members to improve outcomes for patients affected by eosinophilic esophagitis.
Access free multiple choice questions on this topic.
Introduction
The esophagus physiologically lacks eosinophils, and when present, the condition is considered to be pathologic. Eosinophilic esophagitis was once thought to be a component of gastroesophageal reflux disease (GERD). However, it is now known to be a separate entity as we understand more about the esophagus being an active immunogenic organ. Eosinophils can be found in the esophagus in response to various stimuli or antigen. Eosinophilic esophagitis (EoE) is a chronic immune or antigen-mediated process. Clinically, it presents with various esophageal dysfunction, and pathologically, there is mucosal inflammation predominantly with eosinophils, which is confined to the esophagus only. Diseases which can cause eosinophilia should be ruled out before diagnosing a patient with EoE.
Etiology
The exact etiology of EoE is unknown; however, it is thought to be a result of the interactions of environmental, genetic, and host immune factors. A food allergy may trigger EoE, but food anaphylaxis is a rare phenomenon among these patients. There is a strong correlation between atopy and EoE, with patients commonly reporting a history of chronic seasonal allergy, asthma, atopic dermatitis, or other allergic/immunologic conditions.
Epidemiology
EoE is common in both pediatric and adult populations. Epidemiologic studies have reported EoE cases in many countries on all continents except Africa. Based on many population studies, the reported incidence of EoE is between 0.1/10,000 to 1.2/10,000 worldwide. In the pediatric population, EoE is more common among boys. In the adult population, Caucasian and non-Hispanic white men are more likely to have EoE than women of respective races, 76% compared to 48%. EoE can occur in all age groups; however, it is most common in men during their 20s and 30s, and the mean age of diagnosis is 34.
Pathophysiology
EoE occurs as a result of an immunogenic reaction to various antigens which are commonly found in food and air. There is a strong genetic component involved in the pathogenesis of EoE and a high concordance reported for EoE among family members. The pioneer study that described the genetic basis for EoE was a study of genome-wide microarray expression profile analysis. This study reported that the gene responsible for EoE was TSLP (thymic stromal lymphopoietin) which is located in the 5q22 region of male X chromosome. TSLP stimulates Th2 cells and induces eotaxin-3. The stimulated Th2 cells activate various proinflammatory cytokines such as IL5, IL13, and IL15, which recruit eosinophils. Eotaxin-3 is overexpressed in the esophageal mucosa in EoE patients. Overall, this immunogenic process starts as an allergic response to various environmental antigens, food, or aeroallergens and leads to the inflammation of esophageal mucosa.
The other important cytokine involved in the pathogenesis is TGF-B (transforming growth factor-beta), which is released by eosinophils and mast cells recruited after immune activation. TGF-B is responsible for remodeling of esophageal mucosa and smooth muscle dysfunction. The remodeling of inflamed mucosa can occur with repeated exposure to the antigens, leading to remodeling and fibrosis which clinically manifests as various esophageal dysfunction that includes dysphagia, epigastric pain, dyspepsia, chest pain, and food impaction. It has been reported that a single exposure to airway antigen challenge and cutaneous antigen exposure may lead to recruitment of eosinophils in the esophagus leading to EoE.
Histopathology
For patients suspect for EoE, esophageal biopsies usually should be taken from the proximal, mid, and distal esophagus. During the endoscopy, biopsies also should be taken from the antrum and duodenum to rule out other possible causes of eosinophilia.
Histopathology is an important aspect of making a diagnosis of EoE. The histopathology reveals extensive eosinophils infiltrated esophageal mucosa, in addition to mast cells, basophils, basal cell hyperplasia, elongated papillae, superficial layering of eosinophils, extracellular eosinophilic granules, and fibrosis of sub-epithelium.
History and Physical
History is very important when considering a diagnosis of EoE as there are many overlapping symptoms of EoE that coincide with gastroesophageal reflux (GERD). The most common manifestation in adults is dysphagia to solid food. An emergency department visit due to food impaction has been the most common presenting symptom in patients with EoE. Other symptoms such as chest pain or heartburn are common as well. Pediatric patients can present with nausea, vomiting, food intolerance, abdominal pain, and weight loss. A history of various atopic conditions such as asthma, atopic dermatitis, seasonal allergy, food allergy, allergic rhinitis, and eczema may be present as well.
A physical exam is less useful than the history in making the diagnosis of EoE. The most common finding is tenderness to palpation of the abdomen without signs of peritonitis.
Evaluation
Clinicians should arrive at the diagnosis of EoE only after positive findings on clinical, endoscopic, and histopathologic examinations. Patients who present with food impaction, dysphagia, and history of atopy should undergo an upper endoscopy evaluation with esophageal biopsy to diagnose EoE.
Upper endoscopy with esophageal biopsy also should be done on patients with a presumed diagnosis of GERD who are resistant to optimal proton pump inhibitor (PPI) dose (20 to 40 mg orally twice daily) and duration (8 to 12 weeks). Esophageal biopsies normally should be taken from the proximal, mid, and distal esophagus. During the endoscopy, biopsies also should be taken from the antrum and duodenum to rule out other possible causes of eosinophilia.
Endoscopic findings of EoE include corrugated mucosa, longitudinal mucosal furrows, fixed esophageal rings or trachealization, whitish mucosal plaque or exudate, stricture, superficial mucosa tear upon passing endoscope, diffusely narrow lumen, and mucosal friability giving the appearance of crepe paper. Clinicians also should note that some patients may have normal esophagus in upper endoscopy.
The pathological diagnosis of EoE is made when eosinophils are present greater than or equal to 15 per high power field (HPF). Other histological findings suggestive of EoE include basal cell hyperplasia, elongation of papillae, superficial layering of eosinophils, extracellular eosinophilic granules, and fibrosis of sub-epithelium.
There is no diagnostic laboratory test available for EoE, but a mildly elevated serum IgE level is present in patients with EoE. Another common nonspecific finding would be a barium swallow study. Findings can show different types of strictures or a ringed esophagus that could be caused by EoE.
An allergist and immunologist should evaluate patients with a history of atopy or food allergy and a diagnosis of EoE.
Treatment / Management
The goal of EoE treatment is to control the symptoms by decreasing the number of eosinophils in the esophagus and, subsequently, reducing the esophageal inflammation. Management consists of dietary, pharmacological, and endoscopic treatment.
Dietary Treatment
Patients with a history of atopy to food generally respond well to dietary therapy. The approach to dietary therapy is to avoid the specific food if present. If no specific allergenic food or agent is present, a trial of the six food elimination diet (SFED) can be pursued. The six most common allergenic food that should be avoided in EoE patients are cow's milk, wheat, peanut/tree nut, egg, soy, and seafood/shellfish. Alternative options to SFED is elemental diet, which is an amino acid based diet. Patient on elemental diet sometimes require gastrostomy tube placement for adequate caloric intake. Research has shown that elemental diet is superior to SFED or modified SFED (avoidance of food detected by allergic skin test plus SFED). It is also recommended, although the evidence is low, that the clinical response should be measured based on esophageal symptom control and endoscopically with esophageal biopsy to ascertain that the numbers of eosinophils have decreased or not. Upper endoscopy with esophageal biopsy should be done whenever food is reintroduced or removed from the dietary regimen to assess the success of therapy.
Pharmacological Treatment
In patients diagnosed with EoE, trial of PPI 20 mg to 40 mg oral daily or twice daily as a first line therapy is a reasonable option. Those who respond to PPI therapy with symptomatic improvement, endoscopy with esophageal biopsy should be repeated. If no eosinophils present in repeat biopsy, the diagnosis is either acid mediated GERD with eosinophilia or non GERD PPI responsive EoE with unknown mechanism. If both symptoms and eosinophils persists after treatment with PPI, the diagnosis is immune mediated EoE. In case of immune mediated EoE, the American College of Gastroenterology (ACG) highly recommend to use topical (swallowed not inhaled) steroids for total 8 weeks. Oral suspension of fluticasone 880-1760 mcg per day or budesonide 1 mg to 2 mg per day is available options in the United States. Patients who do not respond to topical steroid, systemic steroid, Prednisone 2 mg per Kg per day (maximum 60 mg per day), may be used. Patients who initially respond but symptom recur, longer duration of topical steroid or systemic steroid may be used in addition to elemental diet or SFED.
Endoscopic Management
Patients who present with food impaction, flexible upper endoscopy is recommended to remove impacted food. Dilation is deferred in EoE until patients are adequately treated with pharmacological or dietary therapy, and the result of a response to therapy is available.The goals of therapy for treating EoE is to improve the patient’s symptoms as well as a reduction in the eosinophils on biopsy. The initial treatment is started after failure to improve after 2 months of PPI therapy to make the diagnosis of EoE.
Patient with persistent symptoms of dysphagia even after treatment with dietary elimination and medical therapy, endoscopic dilation is performed. Esophageal strictures and rings can be safely dilated in EoE. It is recommended to use a graduated balloon catheter for gradual dilation. The patient should be informed that after dilation they might experience chest pain and in addition risk of esophageal perforation and bleeding.
Due to the strong association of EoE with allergies, it is also suggested that all patients with diagnosed EoE undergo evaluation by an allergist or immunologist.
Differential Diagnosis
Achalasia
Celiac disease
Crohn’s disease
Connective tissue disease
Drug hypersensitivity
GERD
Graft-versus-host disease
Hypereosinophilic disease
Pemphigus vegetans
Vasculitis
Pearls and Other Issues
The challenge in diagnosing EoE is the differential of GERD as there is much overlap between the two diseases. GERD also can have eosinophils in the esophagus on pathology. The major difference between the diseases is the response to a PPI. Due to this difference, endoscopy with biopsy should be done at least two months after a trial of PPI therapy.
Other disease conditions that can have esophageal eosinophilia should be excluded before diagnosing a patient with EoE. Some of the diseases that have esophageal eosinophilia are gastroesophageal (GI) reflux disease, eosinophilic GI disease, PPI-responsive esophageal eosinophilia, Celiac disease, Crohn disease, infection, Achalasia, drug hypersensitivity, vasculitis, connective tissues disorders, and the use of a PPI.
Enhancing Healthcare Team Outcomes
The diagnosis of EoE is not always easy and the condition is best managed by an interprofessional team that includes a gastroenterologist, primary care provider, pathologist, nurse practitioner and an internist. A biopsy is needed to confirm the diagnosis.
The goal of EoE treatment is to control the symptoms by decreasing the number of eosinophils in the esophagus and, subsequently, reducing the esophageal inflammation. Management consists of dietary, pharmacological, and endoscopic treatment.
Patients who remain compliant with dietary and medication instructions do have a good outcome but for those who fail to make any adjustments in their lifestyle or diet continue to have symptoms.
Review Questions
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References
1.
: Wen T, Rothenberg ME. Clinical Applications of the Eosinophilic Esophagitis Diagnostic Panel. Front Med (Lausanne). 2017;4:108. [PMC free article: PMC5509802] [PubMed: 28770203]
2.
: Dellon ES, Hirano I. Epidemiology and Natural History of Eosinophilic Esophagitis. Gastroenterology. 2018 Jan;154(2):319-332.e3. [PMC free article: PMC5794619] [PubMed: 28774845]
3.
: O'Shea KM, Aceves SS, Dellon ES, Gupta SK, Spergel JM, Furuta GT, Rothenberg ME. Pathophysiology of Eosinophilic Esophagitis. Gastroenterology. 2018 Jan;154(2):333-345. [PMC free article: PMC5787048] [PubMed: 28757265]
4.
: Lucendo AJ, Molina-Infante J, Arias Á, von Arnim U, Bredenoord AJ, Bussmann C, Amil Dias J, Bove M, González-Cervera J, Larsson H, Miehlke S, Papadopoulou A, Rodríguez-Sánchez J, Ravelli A, Ronkainen J, Santander C, Schoepfer AM, Storr MA, Terreehorst I, Straumann A, Attwood SE. Guidelines on eosinophilic esophagitis: evidence-based statements and recommendations for diagnosis and management in children and adults. United European Gastroenterol J. 2017 Apr;5(3):335-358. [PMC free article: PMC5415218] [PubMed: 28507746]
5.
: Akhondi H. Diagnostic approaches and treatment of eosinophilic esophagitis. A review article. Ann Med Surg (Lond). 2017 Aug;20:69-73. [PMC free article: PMC5498267] [PubMed: 28721213]
6.
: Peiris CD, Tarbox JA. Eosinophilic Esophagitis. JAMA. 2019 Apr 09;321(14):1418. [PubMed: 30964530]
7.
: Mehr S, Brown-Whitehorn T. What do allergists in practice need to know about non-IgE-mediated food allergies. Ann Allergy Asthma Immunol. 2019 Jun;122(6):589-597. [PubMed: 30935977]
8.
: McGowan EC, Platts-Mills TAE, Wilson JM. Food allergy, eosinophilic esophagitis, and the enigma of IgG4. Ann Allergy Asthma Immunol. 2019 Jun;122(6):563-564. [PMC free article: PMC6555674] [PubMed: 30928417]
9.
: Sawada A, Hashimoto A, Uemura R, Otani K, Tanaka F, Nagami Y, Yamagami H, Tanigawa T, Watanabe T, Fujiwara Y. Association between endoscopic findings of eosinophilic esophagitis and responsiveness to proton pump inhibitors. Endosc Int Open. 2019 Apr;7(4):E433-E439. [PMC free article: PMC6428677] [PubMed: 30931374]
10.
: Safroneeva E, Schoepfer AM. Symptom-based patient-reported outcomes in adults with eosinophilic esophagitis: value for treatment monitoring and randomized controlled trial design. Curr Opin Allergy Clin Immunol. 2019 Apr;19(2):169-174. [PubMed: 30649010]
: Disclosure: Jordan Roussel declares no relevant financial relationships with ineligible companies.
: Disclosure: Sudha Pandit declares no relevant financial relationships with ineligible companies.
Copyright © 2025, StatPearls Publishing LLC.
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Roussel JM, Pandit S. Eosinophilic Esophagitis. [Updated 2023 Aug 8]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan-.
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7098 | https://static.hlt.bme.hu/semantics/external/pages/Mesters%C3%A9ges_Neuron/en.wikipedia.org/wiki/Piecewise.html | Piecewise - Wikipedia
Piecewise
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In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain. Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial function is a function that is a polynomial on each of its sub-domains, but possibly a different one on each.
The word piecewise is also used to describe any property of a piecewise-defined function that holds for each piece but not necessarily hold for the whole domain of the function. A function is piecewise differentiable or piecewise continuously differentiable if each piece is differentiable throughout its subdomain, even though the whole function may not be differentiable at the points between the pieces. In convex analysis, the notion of a derivative may be replaced by that of the subderivative for piecewise functions. Although the "pieces" in a piecewise definition need not be intervals, a function is not called "piecewise linear" or "piecewise continuous" or "piecewise differentiable" unless the pieces are intervals.
[x]
Contents
1 Notation and interpretation
2 Continuity
3 Applications
4 Common examples
5 See also
6 References
Notation and interpretation[edit]
Graph of the absolute value function, y=|x|.
Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. Crucially, in most settings, there must only be a finite number of subdomains, each of which must be an interval, in order for the overall function to be called "piecewise". For example, consider the piecewise definition of the absolute value function:
|x|={−x,if x<0 x,if x≥0{\displaystyle |x|={\begin{cases}-x,&{\mbox{if }}x<0\x,&{\mbox{if }}x\geq 0\end{cases}}}
For all values of x less than zero, the first function (−x) is used, which negates the sign of the input value, making negative numbers positive. For all values of x greater than or equal to zero, the second function (x) is used, which evaluates trivially to the input value itself.
Consider the piecewise function f(x) evaluated at certain values of x:
| x | f(x) | Function used |
---
| −3 | 3 | −x |
| −0.1 | 0.1 | −x |
| 0 | 0 | x |
| 1/2 | 1/2 | x |
| 5 | 5 | x |
Thus, in order to evaluate a piecewise function at a given input value, the appropriate subdomain needs to be chosen in order to select the correct function and produce the correct output value.
Continuity[edit]
A piecewise function comprising different quadratic functions on either side of x 0{\displaystyle x_{0}}.
A piecewise function is continuous on a given interval if the following conditions are met:
it is defined throughout that interval,
its constituent functions are continuous on the corresponding intervals (subdomains),
there is no discontinuity at each endpoint of the subdomains within that interval.
The pictured function, for example, is piecewise continuous throughout its subdomains, but is not continuous on the entire domain, as it contains a jump discontinuity at x 0{\displaystyle x_{0}}. The filled circle indicates that the value of the right function piece is used in this position.
Applications[edit]
In applied mathematical analysis, piecewise functions have been found to be consistent with many models of the human visual system, where images are perceived at a first stage as consisting of smooth regions separated by edges. In particular, shearlets have been used as a representation system to provide sparse approximations of this model class in 2D and 3D.
Common examples[edit]
Specific instances of piecewise functions include:
Step function, a piecewise function composed of constant functions
Boxcar function,
Heaviside step function
Sign function
Piecewise linear function, a piecewise function composed of line segments
Absolute value
Broken power law, a piecewise function composed of power laws
Spline, a piecewise function composed of polynomial functions, possessing a high degree of smoothness at the places where the polynomial pieces connect
B-spline
PDIFF
See also[edit]
Wikibooks has a book on the topic of: Gnuplot#Piecewise-defined functions
References[edit]
^Kutyniok, Gitta; Labate, Demetrio (2012). "Introduction to shearlets"(PDF). Shearlets. Birkhäuser: 1–38.
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7099 | http://mathcentral.uregina.ca/QQ/database/QQ.09.12/h/pardeep1.html | Is a line lying in a plane an open or a closed region? - Math Central
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Question from pardeep, a student: please help with this question on the open and closed intervals :: Is a line lying in a plane an open or a closed region? my teacher says it is a closed region reasoning out that it contains all it boundary points . please help how is it so?
Hi,
Your teacher is correct, a line in the plane is a closed subset of the plane. I would have said that it is closed since it contains all its limit points.
Harley
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