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Foci of a Hyperbola
In geometry, a hyperbola is a type of curve that looks like two symmetrical bowls placed back-to-back. It is defined by two points, called foci (plural of focus), which are connected by a line segment called the major axis. In this blog post, we will explore the concept of foci in detail and see how they relate to the overall shape of a hyperbola.
What are Foci?
In geometry, the term "focus" refers to a special point on a curve. A hyperbola has two foci, which are located on opposite sides of the major axis. The major axis is the line segment that connects the two foci.
The positions of the foci can be used to define the shape of the hyperbola. For example, if the distance between the two foci is equal to the length of the major axis, then the hyperbola is said to be "symmetrical." On the other hand, if the distance between the two foci is less than or greater than the length of the major axis, then the hyperbola is said to be "asymmetrical."
How do Foci Relate to the Overall Shape of a Hyperbola?
The placement of the foci relative to each other has a direct impact on the shape of the hyperbola. If the two foci are close together, then the resulting hyperbola will be narrow and elongated. If the two foci are far apart from each other, then the resulting hyperbola will be wide and flat.
In conclusion, we have seen that a hyperbola is defined by two points called foci, which are connected by a line segment called the major axis. The distances between the foci relative to each other and to the length of major axis directly impactsHyperbolasthe shape of resulting hyperbola. Understanding how foci work can help you better visualize and manipulate complex geometric shapes.
FAQ
How do you find foci of a hyperbola?
There are a few different ways to find the foci of a hyperbola. One way is to use the equation of the hyperbola. The equation of a hyperbola is typically written in the form:
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
Where a and b are the lengths of the semi-major and semi-minor axes, respectively.
The foci of the hyperbola are located at:
$$\left(\pm a\sqrt{1+\frac{b^2}{a^2}},0\right)$$
Another way to find the foci of a hyperbola is to use the standard form of the equation of a hyperbola. The standard form of the equation of a hyperbola is:
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
Where (h,k) is the center of the hyperbola and a and b are the lengths of the semi-major and semi-minor axes, respectively.
The foci of the hyperbola are located at:$$\left(h\pm a\sqrt{1+\frac{b^2}{a^2}},k\right)$$
Another way to find the foci of a hyperbola is to use the properties of a hyperbola. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. The foci of a hyperbola are located at:
$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci. Another way to find the foci of a hyperbola is to use the distance formula.
The distance formula is:$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Where (x1,y1) and (x2,y2) are the coordinates of two points in a plane.
The foci of a hyperbola are located at:$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
Another way to find the foci of a hyperbola is to use the definition of a hyperbola. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. The foci of a hyperbola are located at:$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
One final way to find the foci of a hyperbola is to use the properties of a hyperbola. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant.
The foci of a hyperbola are located at:$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
What is the formula to find foci?
There are a few different ways to find the foci of a hyperbola. One way is to use the equation of the hyperbola. The equation of a hyperbola is typically written in the form:
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
Where a and b are the lengths of the semi-major and semi-minor axes, respectively.
The foci of the hyperbola are located at:
$$\left(\pm a\sqrt{1+\frac{b^2}{a^2}},0\right)$$
Another way to find the foci of a hyperbola is to use the standard form of the equation of a hyperbola. The standard form of the equation of a hyperbola is:
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
Where (h,k) is the center of the hyperbola and a and b are the lengths of the semi-major and semi-minor axes, respectively.
The foci of the hyperbola are located at:$$\left(h\pm a\sqrt{1+\frac{b^2}{a^2}},k\right)$$
Another way to find the foci of a hyperbola is to use the properties of a hyperbola. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. The foci of a hyperbola are located at:
$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci. Another way to find the foci of a hyperbola is to use the distance formula.
The distance formula is:$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Where (x1,y1) and (x2,y2) are the coordinates of two points in a plane.
The foci of a hyperbola are located at:$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
Another way to find the foci of a hyperbola is to use the definition of a hyperbola. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. The foci of a hyperbola are located at:$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
One final way to find the foci of a hyperbola is to use the properties of a hyperbola. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant.
The foci of a hyperbola are located at:$$\left(\frac{c}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
Do hyperbolas have foci?
Yes, hyperbolas have two foci. The foci of a hyperbola are located at:$$\left(\frac{c
}{2},0\right) \text { and } \left(-\frac{c}{2},0\right)$$
Where c is the distance between the foci.
How do you find the vertices and foci of a hyperbola?
The vertices and foci of a hyperbola can be found using the equation of the hyperbola. The equation of a hyperbola is typically written in the form:
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
Where a and b are the lengths of the semi-major and semi-minor axes, respectively.
The vertices of the hyperbola are located at:
$$\left(\pm a,0\right) \text { and } \left(0,\pm b\right)$$
The foci of the hyperbola are located at:
$$\left(\pm a\sqrt{1+\frac{b^2}{a^2}},0\right)$$
Functions and Graphs
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189401 | https://en.wikipedia.org/wiki/Butterfly_theorem | Published Time: 2004-12-28T04:46:14Z
Butterfly theorem - Wikipedia
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1 Proof
2 History
3 References
4 External links
Butterfly theorem
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From Wikipedia, the free encyclopedia
About the midpoint of a chord of a circle, through which two other chords are drawn
For the "butterfly lemma" of group theory, see Zassenhaus lemma.
Butterfly theorem
The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:: p. 78
Let M be the midpoint of a chordPQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.
Proof
[edit]
Proof of Butterfly theorem
A formal proof of the theorem is as follows: Let the perpendicularsXX′ and XX″ be dropped from the point X on the straight lines AM and DM respectively. Similarly, let YY′ and YY″ be dropped from the point Y perpendicular to the straight lines BM and CM respectively.
Since
△M X X′∼△M Y Y′,{\displaystyle \triangle MXX'\sim \triangle MYY',}M X M Y=X X′Y Y′,{\displaystyle {MX \over MY}={XX' \over YY'},}△M X X″∼△M Y Y″,{\displaystyle \triangle MXX''\sim \triangle MYY'',}M X M Y=X X″Y Y″,{\displaystyle {MX \over MY}={XX'' \over YY''},}△A X X′∼△C Y Y″,{\displaystyle \triangle AXX'\sim \triangle CYY'',}X X′Y Y″=A X C Y,{\displaystyle {XX' \over YY''}={AX \over CY},}△D X X″∼△B Y Y′,{\displaystyle \triangle DXX''\sim \triangle BYY',}X X″Y Y′=D X B Y.{\displaystyle {XX'' \over YY'}={DX \over BY}.}
From the preceding equations and the intersecting chords theorem, it can be seen that
(M X M Y)2=X X′Y Y′X X″Y Y″,{\displaystyle \left({MX \over MY}\right)^{2}={XX' \over YY'}{XX'' \over YY''},}=A X⋅D X C Y⋅B Y,{\displaystyle {}={AX\cdot DX \over CY\cdot BY},}=P X⋅Q X P Y⋅Q Y,{\displaystyle {}={PX\cdot QX \over PY\cdot QY},}=(P M−X M)⋅(M Q+X M)(P M+M Y)⋅(Q M−M Y),{\displaystyle {}={(PM-XM)\cdot (MQ+XM) \over (PM+MY)\cdot (QM-MY)},}=(P M)2−(M X)2(P M)2−(M Y)2,{\displaystyle {}={(PM)^{2}-(MX)^{2} \over (PM)^{2}-(MY)^{2}},}
since PM = MQ.
So,
(M X)2(M Y)2=(P M)2−(M X)2(P M)2−(M Y)2.{\displaystyle {(MX)^{2} \over (MY)^{2}}={(PM)^{2}-(MX)^{2} \over (PM)^{2}-(MY)^{2}}.}
Cross-multiplying in the latter equation,
(M X)2⋅(P M)2−(M X)2⋅(M Y)2=(M Y)2⋅(P M)2−(M X)2⋅(M Y)2.{\displaystyle {(MX)^{2}\cdot (PM)^{2}-(MX)^{2}\cdot (MY)^{2}}={(MY)^{2}\cdot (PM)^{2}-(MX)^{2}\cdot (MY)^{2}}.}
Cancelling the common term
−(M X)2⋅(M Y)2{\displaystyle {-(MX)^{2}\cdot (MY)^{2}}}
from both sides of the equation yields
(M X)2⋅(P M)2=(M Y)2⋅(P M)2,{\displaystyle {(MX)^{2}\cdot (PM)^{2}}={(MY)^{2}\cdot (PM)^{2}},}
hence MX = MY, since MX, MY, and PM are all positive, real numbers.
Thus, M is the midpoint of XY.
Other proofs exist, including one using projective geometry.
History
[edit]
Proving the butterfly theorem was posed as a problem by William Wallace in The Gentleman's Mathematical Companion (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel posed the question again in a letter to Wallace. Reverend Thomas Scurr asked the same question again in 1814 in the Gentleman's Diary or Mathematical Repository.
References
[edit]
^Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
^Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", Forum Geometricorum 16, 2016, 337–338.
^, problem 8.
^William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.
External links
[edit]
Wikimedia Commons has media related to Butterfly theorem.
The Butterfly Theorem at cut-the-knot
A Better Butterfly Theorem at cut-the-knot
Proof of Butterfly Theorem at PlanetMath
The Butterfly Theorem by Jay Warendorff, the Wolfram Demonstrations Project.
Weisstein, Eric W."Butterfly Theorem". MathWorld.
Retrieved from "
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189402 | https://www.pas.rochester.edu/~stte/phy415F13/hw2.html | Problem Set 2
Home Contact Info Course Info Calendar Homework Lecture NotesPHY 415: Electromagnetic Theory I Prof. S. Teitel stte@pas.rochester.edu ---- Fall 2013 Problem Set 2 ------------- Due Wednesday, October 2, in lecture Problem 1 [10 points] Prove this mean value theorem: For charge-free space in the electrostatic limit, the value of the electrostatic potential φ at any point in space is equal to the average of the potential over the surface of any sphere centered on that point. Hint: Use the fact that where there are no charges ∇2 φ=0. Then relate the averge φ over the surface of a shpere to the flux of electric field through the surface. Functions which satisfy Laplace's equation are called harmonic functions; harmonic functions obey the above mean value theorem. Problem 2 [10 points] Prove Green's reciprocation theorm: If φ is the potential due to a volume charge density ρ within a volume V and a surface charge density σ on the conducting surface S bounding the volume V, while φ' is the potential for the same geometry but for a different ρ' and σ', then ∫V d 3 r ρ φ' + ∫S da σ φ' = ∫V d 3 r ρ'φ + ∫S da σ'φ Hint: Consider Green's 2nd identity. Problem 3 [10 points] Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use Green's reciprocation theorem to prove that the total induced charge on one of the planes is equal to -q times the fractional perpendicular distance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one whose charge densities and potential are known and simple.) Problem 4 [10 points] a) In lecture we solved the problem of the electric field from a spherical shell of radius R with uniform surface charge density σ=q/(4πR 2). Consider now the problem where this shell is of finite thickness d. That is, there is a uniform charge density ρ in a spherical shell of finite thickness from radius R to radius R+d, such that the total charge on this shell is q. Find the potential φ(r) by solving Poisson's equation (there may be easier ways to do it, but do it this way!), then take the gradient to get E(r). Sketch φ(r) and E(r) vs r. Now take the limit d→0 keeping ρd=σ constant. Compare your result with the case of the infinitesmally thin shell done in lecture. b) Consider an infinitesmally thin spherical shell of radius R with a total charge q uniformly distributed over its surface, and a concentric infinitesmally thin spherical shell of radius R+d with total charge -q uniformly distributed over its surface. Find the potential φ(r) by solving Poisson's equation for this geometry, then take the gradient to get E(r). Sketch φ(r) and E(r) vs r. Now take the limit d→0 keeping qd constant. What do you find? This is the limit of an infinitesmally thin dipole layer. |
189403 | https://www.merriam-webster.com/word-of-the-day/immutable-2017-01-09 | Word of the Day
immutable
What It Means
: not capable of or susceptible to change
immutable in Context
"There's an immutable attraction between fingers and potato chips, making resistance, as the saying goes, futile." — Michele Henry, The Toronto Star, 30 Nov. 2016
"Like much of the American heartland, the summertime landscape in Iowa's Webster County is dominated by several immutable features: hot sun and lots of it; a ruler-straight grid of byways …; shining grain silos towering above the plains; and farmhouses…." — Michelle Donahue, PCMag.com, 8 Nov. 2016
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Immutable comes to us through Middle English from Latin immutabilis, meaning "unable to change." Immutabilis was formed by combining the negative prefix in- with mutabilis, which comes from the Latin verb mutare and means "to change." Some other English words that can be traced back to mutare are commute (the earliest sense of which is simply "to change or alter"), mutate ("to undergo significant and basic alteration"), permute ("to change the order or arrangement of"), and transmute ("to change or alter in form, appearance, or nature"). There's also the antonym of immutable—mutable—which of course can mean "prone to change" and "capable of change or of being changed."
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189404 | https://www.aao.org/eyenet/article/ocular-toxoplasmosis-a-refresher | This website uses tracking technologies to enable our website functionalities, to enhance user experience or to analyze performance and traffic. We may also share or sell information about your use of our site with our social media, advertising, and analytics partners. This allows us to perform targeted advertising and to select ads and content that will be more relevant to you. Below you can Accept Default Settings, Reject All trackers, or exercise your right to opt -in or -out of the sale of personal data, targeted advertising, profiling, and the processing of sensitive data by clicking on “Manage Your Privacy Choices.” For more details on the data we process and how to exercise your rights, see our Privacy Policy Learn More
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189405 | https://stackoverflow.com/questions/66365776/calculate-weighted-average-of-percentages | python - Calculate weighted average of percentages - Stack Overflow
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Calculate weighted average of percentages
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Asked 4 years, 7 months ago
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I have a problem to solve in order to make the correct calculations. I have a list of percentages for the performance of a specific marketing channel before and after specific actions and I want to make the percentage increment. I can't just take the average of the following values I have to take the weighted average of the values.
Values below are percentages ( 0.10 is 10% )
python
percentage_list_after_actions = [ 0.10, 0.3, 0.16, 0.22 ]
percentage_list_before_actions = [ 0.14, 0.26, 0.12, 0.25 ]
Which is the best way of doing that?
Is the weighted average the correct metric as we have percentages or I have to take the moving_average/rolling_average?
python
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statistics
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asked Feb 25, 2021 at 9:28
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Create the function that calculates this.
```python
def weighted_average_m1(distribution, weights):
numerator = sum([distribution[i]weights[i] for i in range(len(distribution))])
denominator = sum(weights)
return round(numerator/denominator,2)
weighted_average_m1(distribution, weights)
```
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answered Feb 25, 2021 at 9:32
Murilo BarbosaMurilo Barbosa
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dapo
dapoOver a year ago
Thanks for your answer but I dont have weights its just the percentages.
2021-02-25T09:45:27.367Z+00:00
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189406 | https://www.turito.com/learn/english/context-clues-2 | Context Clues Types and Examples | Turito
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EnglishContext Clues Types and Examples
Context Clues Types and Examples
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Word Wizard – Context Clues Words
Context clues are hints that are present in a text to help the reader understand the meaning of a difficult or unusual word within the text. There are five ways in which you can identify the meaning of a word based on the context:
1. Synonym
Another word meaning the same or similar to the unfamiliar word is present.
Example:
I was astonished and amazedwhen I won the lottery.
2. Antonym
A word meaning the opposite of what the unfamiliar word means is present.
Example:
While Sarah is usually very compassionate, she can be really mean when she gets angry.
3. Explanation
The definition or an explanation of the unfamiliar word is present.
Example:
George finished his homework hastily. He was quick and careless with it.
4.Examples:
Examples of unfamiliar words are present.
Example:
The children were able to observe several crustaceans, including crabs, lobsters, and shrimp.
5. Inference
Clues present before and after the unfamiliar word from which the meaning can be inferred.
Example:
The haberdashery was Lou’s favorite place. He loved shopping for nice suits. The people who worked there were so kind and helpful.
Here are a few things to remember when trying to identify the meanings of words using context clues:
The context clues are not always present in the same sentence as the unfamiliar word.
Sometimes, the context clues are not very obvious, and you will have to infer the meaning by reading a few sentences before and after the unfamiliar word.
Context clues do not always lead you to the exact meaning of the word. Sometimes, you will get an approximate meaning of the word.
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189407 | https://resources.nu.edu/statsresources/samplingmethods | Skip to Main Content
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Introduction
Sampling, for the purposes of this guide, refers to any process by which members of a population are selected to participate in research. There are many methods for sampling, each with a slightly different purpose. In the box below you can learn more about these common sampling techniques:
simple random sampling
stratified sampling
cluster sampling
systematic sampling
non-probability sampling
Before you can obtain a sample, you must first identify a target population. The target population refers to all of the people who are the focus of a study. For example, a study about elementary school teacher burnout would include all elementary school teachers in the population. In some cases, you may need to consider an accessible population. This is a subset of the target population that can reasonably be accessed by the researcher for sampling. Oftentimes, researchers will use a sampling frame to facilitate their sampling methods. A sampling frame is a list of all of the members of the population.
Sampling Techniques
Simple Random Sampling
Stratified Sampling
Cluster Sampling
Systematic Sampling
Through simple random sampling (SRS), all members of the population have an equal chance of being selected. Therefore, this is a type of probability sampling. A rudimentary method of SRS is drawing names out of a hat. Each slip of paper has the same chance of being chosen on every draw. You could also use a random number generator to facilitate random selection from the population.
Simple random sampling assumes that all members of the population are accessible. If your population is "people in the United States" and you are attempting to sample via the Internet, members of the population without Internet access do not have a chance to be selected. This would not be an appropriate use of simple random sampling.
Researchers use SRS when the intention is to obtain a representative sample that can provide data for generalizing to the population. If members are chosen randomly, the sample is less subject to bias that may exist by non-random sampling methods.
Stratified sampling is a two-step sampling procedure. First, the population is divided into groups or strata. How this is done will depend on your specific population. Using the example of elementary school teachers, we could divide the teachers up based on state or school district, with each state (or school district) representing one strata. Next, members of eachstrata are selected for participation. When they are selected randomly from within each strata, it is called stratified random sampling.
In the above figure, the population was divided into four strata. Members of each strata were selected to participate using simple random sampling. Thus all four strata are represented in the final sample of participants. This method is effective in ensuring all strata are included in the sample. For example, making sure teachers from all 50 states are included in the sample.
You can also adjust the proportion of the sample that comes from each strata to maintain proportional alignment with the population. For example, if 50% of the population is in Group Three, 50% of the sample can be randomly selected from that strata, thus ensuring the final makeup of the sample aligns with the makeup of the population.
Like stratified sampling, cluster sampling is a two-step sampling procedure that also starts with dividing the population into groups called clusters. As with stratified sampling, how groups are divided will depend on your population. For example, teachers could be divided into clusters based on school district or grade level taught. The primary difference between stratified sampling and cluster sampling is that whole clusters are randomly selected and everyone in that cluster is included in the sample.
In the above figure, the population was divided into four clusters. Two of these clusters were randomly chosen. All members of Group One and Group Four will participate in the study, making up the sample.
Cluster sampling is ideal when there are not major differences between the clusters. Consider dividing teachers based on school district versus grade level taught. Each school district will include the same grade levels, though may have some variability in factors like school size and geographic location. Selecting school districts at random can help create a representative sample that covers an array of factors. If the clusters are grade level, randomly selecting grade levels mean entire grades are not included in the sample. These grades may have meaningful differences from the included grades. So, cluster sampling may not be as effective in this situation.
Systematic sampling occurs when participants are selected at set intervals. For example, choosing every third person from a list. To ensure this method aligns with probability sampling conditions, the starting point is randomly selected. Consider the following visual that shows systematic selection, beginning with the second person in the line (the randomly selected starting point).
Systematic sampling offers benefits similar to simple random sampling but is often perceived as being simpler to carry out. It also combats the potential problem of clusters that can occur with random sampling. While random sampling aims to select a variety from the population, there is also no way to regulate who it selects. So, clusters of individuals could be selected at random, thus potentially biasing the research. Systematic sampling ensures and even distribution across the population.
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Intro all right so one of the things that you must be able to do in both geometry and trigonometry is solve triangle problems now when you study geometry the triangle problems that you solve are pretty easy you use things like the Pythagorean theorem and some other basic concepts of triangles but when you get into things like trigonometry well triangle problems get much more challenging and you're going to need to know things like the law of signs and cosine and what I have for you here is a uh an example problem of using the law of s so it would basically go something like this matter of fact I'm going to show you the solution to this problem here in just one second but let me go ahead and set this up so we want to solve this triangle and we're given that angle a is 72° angle C is 50° and uh a side a is 40 units long so what you need to do is figure out uh this triangle where is a where is B where is C and to solve a triangle means that you need to actually figure out um all the degrees of each angle and all the sides so this can get pretty complicated and again you're going to be using things like the law of signs and cosiness to figure this out this is a pretty involved process matter of fact let me show you the law of ss here so this is the law of s sin a over a is equal to sin B over B is equal to sin C over C now the solution to this video actually comes from my um precalculus course so this is one problem of many uh to include full uh lesson instruction so um this particular problem again is uh only one segment of a bigger section that I teach in my precalculus course so if you need additional help with trigonometry or the law of signs or cosiness things like that then and in the description of this video you'll see a link to my precalculus course all right so uh enjoy this video and hopefully you're going to uh learn a thing or two about the law of Problem signs we're given an angle uh 72 we're given another angle 50° so 72° 50° and here we got a length of 40 for a now here are the answers if we want to go ahead and uh check your solution here on on this particular problem but this is an angle angle side scenario now the first thing that you want to do is draw a sketch and here um I just sketched freehand a triangle and you should do the same thing I'm not using a ruler it doesn't have to be perfect but what you really want to do is kind of play around with your your sketches until you get something that you think is fairly representative of um the given information so let's see see here okay so I have an angle 72 so if I draw angle a I'm not going to make angle a look like this that's not 72 right 72 is much more open like so okay so you want to have your angle a kind of looking like this and then we have another angle 50 so 50 might be somewhere in between like so not quite like a 45 so play around with your sketch until you get something that that pretty much you know um looks or you know as representative as accurate as possible it doesn't have to be perfect now but you don't want to just have any old triangle with angles in there because the results can confuse you and if you do a pretty good job with your sketch you can kind of you know reality check your answers as well really comes in handy okay so let's go ahead and start off so I have um an angle at 72 degrees that's angle a and then I have another angle uh 50° and I'm giving uh the information the length for a so let's call this here A so a is going to be 40 so that's what I'm given right now but if you look here I have two angles of the triangle so it's not going to be a problem for me to find that third angle okay so let's go ahead and find that third Solution angle all right so very easy right so just uh take these two angles subtract them uh take 180 subtract away these two angles and you get angle B being 58 now at this point I'm looking say okay what can I do now remember the next thing you want to do is uh just find a pair of you're looking for an angle and a its uh respective side okay you need that that pair in order to really start solving the triangle using a law of um signs so here I have 72 and I have 40 that's all I need I can create this uh nice ratio right here and now that I have all the other angles I can go get the remaining sides which are going to be B and C so let's go ahead and find uh length C first we have to use 50° okay so s of 50° with its respective side C right here okay once I set that up I'm going to go ahead and solve for C all right so hopefully at this point all of you are professionals at uh because you watch the lesson right and reference the lesson we're going to use the cross product so C s of 72 de will be this and then 40 s of 50° will be so and let's uh solve for seed by dividing both sides of the equation by S of 72 then I go into my calculator very carefully I plug all this in and you'll get approximately 32.2 okay so what is that okay well we just solved for this length here in the triangle length C okay so now we need to get uh that uh length Right Here length B and I think that'll uh be it for this triangle all right so let's keep our same proportion we can keep using this right here this s of 72 over 40 or I could have used now s of 50 over 32.2 so if you used a different um setup that's perfectly uh fine it's not a big deal um because remember the law of sign says you can use any um ratio because all three um angles and their respective sides are proportional okay that's what the law of sence says but we'll stick with this information because I already have it now I'm going to uh look here and I have to um see okay angle B is 58° so I'm solving for side B so I have to use its respective angle B okay so carefully you know make sure you have the problem set up correctly so that's going to be S of 58 over length B again I'm going to go ahead and use the cross product here so B s of 72° 40 s of 58° going to divide everything by S of 72 I get this plug it into my calculator and I get approximately 35.679204 |
189409 | https://journals.publishing.umich.edu/ptpbio/article/id/2102/print/ | Research Article
Consensus and a Unified Species Paradigm: Reality or Idle Hope
Author: Stijn Conix (Katholieke Universiteit, Leuven)
Abstract
Many systematic biologists claim that a new paradigm about species classification has been established in their discipline. This paradigm, which I call the "unified species paradigm", consists in a set of theoretical claims and methodological practices centered around the view that species are independently evolving lineages. This paper sets out the basic theoretical and methodological principles of this new paradigm, and looks at biological textbooks, publication patterns and citation patterns to evaluate the claim that there is growing consensus about it.
Keywords: Evolutionary Species Concept, paradigm shift, species delimitation, species classification, taxonomy, textbook analysis, bibliometric analysis
How to Cite: Conix, S. (2022) Consensus and a Unified Species Paradigm: Reality or Idle Hope, Philosophy, Theory, and Practice in Biology. 14(0). doi:
1 Introduction
It has long been common for papers about the species concept to start by pointing out how much has been written about the species problem, how vexing a problem it is, and how unlikely it is that biologists will ever agree about the definition of species. This was confirmed by a recent survey among biologists by PuÅ¡iÄ et al. (2017), which revealed great diversity in the preferred species concept of biologists. At the same time, many recent systematic studies suggestoften in passingthat biologists have reached agreement about what species are, namely, independently evolving lineages (Carstens et al. 2013). For example, Weisrock et al. (2010, 1) write that through decades of diverging opinions, at least one component of the species problemthe disagreement over what exactly species arehas found resolution in the consensus view that species are solely defined as separately evolving metapopulation lineages. Similarly, Leliaert et al. (2009, 122) write that important conceptual progress has been made in thinking about species concepts and that a vast majority of evolutionary biologists now accepts that species are lineages. This view is so widespread that in a recent review of the field Padial and De La Riva (2020; see also Leliaert and Clerck 2017; FiÅ¡er, Robinson, and Malard 2018) even speak of a paradigm shift in our view of species.
This paper will not evaluate whether this new paradigm is consistent, useful and an effective solution to the species problem. Similarly, I do not commit to a particular view on scientific concepts and the extent to which they are determined by pragmatic concerns or by what the world is like. Rather, the objective of this paper is purely descriptive: I aim to investigate whether, contrary to claims about the irresolvable nature of disagreements about species concepts, the paradigm of species as independently evolving lineages has indeed gained broad acceptance among taxonomists and other systematic biologists. To do this, Section 2 summarizes the main components of the paradigm associated with the view that species are independently evolving lineages. Sections 3, 4 and 5 then investigate biological textbooks, recent publications on species concepts, and the citations of these publications to investigate the extent to which claims about the resolution of the species problem by this paradigm reflect the opinion of most systematic biologists. Section 6, finally, concludes the paper and lists some limitations of this study.
2 The Unified Species Paradigm
Because of its continued use of the Linnaean system, taxonomy is one of the few fields of research that is still recognizably similar to what it was 300 years ago. However, behind the seemingly stable cover of Linnaean ranks, taxonomy has gone through fundamental and radical changes over the past centuries. I will not discuss or evaluate the various historical accounts of this change, which differ in the pivotal thinkers they emphasize and the revolutions they identify (e.g., Mishler 2009; Zachos 2016; Wilkins 2018). Rather, I will investigate where taxonomy stands now, and how it has changed over the past two decades. More precisely, I will investigate whether there is indeed growing consensus about the view that species are independently evolving lineages. For convenience of expression, I will refer to this view, and the set of other theoretical claims and methodological practices associated with it, as the unified species paradigm.
The theoretical basis of the unified species paradigm has long been implicitly assumed by many biologists. It has been expressed explicitly and famously by de Queiroz (1998, 2007) and Mayden (1997) among others. It can be summarized by three closely related points:
The Evolutionary Species Concept: Biological species are defined as independently evolving lineages. This is the conception of species described by Simpson (1951), Wiley (1978) and Mayden (1997) as the Evolutionary Species Concept (ESC), and, more recently and very popularly, by de Queiroz (1998; 2007) as the General Lineage Concept or Unified Species Concept.1 According to this view, species are ancestor-descendant population-level lineages that are on a unique evolutionary trajectory and evolve independently from other such lineages. In this paper, I will refer to this concept as the ESC.
Conception vs. Delimitation: The ESC conception of species is distinguished from the practical delimitation of species. Because evolutionary independence is realized in many different ways in different groups, and because practical constraints on research also vary between taxa, delimiting species requires the use of a wide range of different methods and criteria. In some groups, evolutionary independence is signaled by differences in mating calls or other barriers to reproduction, while in other groups morphological criteria, genotypic clusters, or reciprocal monophyly provide a more appropriate gauge. One major consequence of this is that the unified species paradigm considers traditional species concepts such as the Biological Species Concept (BSC) or the various forms of the Phylogenetic Species Concept (PSC) particular operationalizations of the more general ESC.
Species as fallible hypotheses: It follows from the first two points that species delimitation consists in raising hypotheses about the existence and independent evolution of population-level lineages.2 These hypotheses can and should be tested by a wide range of methods that target different aspects of evolutionary independence. Together, such different tests allow taxonomists to understand the unique and often complex way in which different lineages are (or are not) evolving independently. Thus, while a positive signal from multiple such tests often makes a species-hypothesis better supported, it is also possible that a lineage is evolving independently even if only one of many available methods suggests that this is the case (Padial and De La Riva 2010).
These three principles are closely related: Because the theoretical conception of species is very abstract (first principle), it can encompass the various traditional species concepts as different but compatible methods of species delimitation (second principle); and it is because species delimitation has been severed from the species concept (second principle), that we can think of species as hypotheses to be tested by a wide range of methods (third principle).
Various authors (Mallet 2001, 8; Ereshefsky 2011, 7475) have pointed out that this theoretical framework does not solve the species problem, as the ESC is so abstract and broad that the problem of defining species is simply moved to the choice of operationalizations in taxonomic practice. In light of this criticism, it is perhaps better to say that this framework resolves rather than solves the species problem: the distinction between the concept of species and species delimitation severs the practice of identifying species from theoretical debates about what species are. As long as taxonomists agree that species are independently evolving lineages, this framework allows them to focus on the practice of testing the independence of lineages. And indeed, the past two decades have witnessed a true renaissance in work on methods of species delimitation, often connected explicitly to the ESC and de Queirozs exposition of it (e.g., Sites and Marshall 2003; Camargo and Sites 2013).
Thus, methods of species delimitation are a core component of this paradigm of species: the species problem ought not to be solved through conceptual advancements or further philosophical debate, but by improving our methods for identifying concrete lineages and by collecting more data. In accordance with the second and third principles, these methods are diverse and span a wide range of biological disciplines. However, four more general trends are worth emphasizing:
Lineage-level and population-level processes: Because species delimitation consists in identifying independently evolving lineages, one major aim of methods of species delimitation is distinguishing patterns caused by population-level processes from patterns caused by lineage-level processes. While both population-level and lineage-level processes can cause various kinds of similarity between organisms, only the latter indicate independent evolution. By focusing on the relation between these two, the unified species paradigm has led to methods that bring together population genetics and phylogenetics. Most popular here are methods employing the multispecies coalescent model to estimate species trees while taking into account population-level processes that cause discordance between species trees and gene-trees (Degnan and Rosenberg 2009; Fujita et al. 2012).3
Contrast with traditional taxonomy: Traditionally, species delimitation relied heavily on the use of morphological traits. This way, cryptic diversity in morphologically similar groups often remained unnoticed. Hence, even though morphological evidence can be useful for identifying independently evolving lineages too, species delimitation under the unified species paradigm rarely relies solely on morphology and often has the explicit aim of checking for independently evolving lineages that are not morphologically distinct (Lumbsch and Leavitt 2011; Carstens et al. 2013).
Integrative taxonomy: Proponents of the unified species paradigm often urge to use multiple lines of evidence for species delimitation, and integrate these in a comprehensive view on the investigated groups. This is important because there is no single operational criterion that always indicates evolutionary independence and that all independently evolving lineages consistently instantiate. By combining different methods, an integrative taxonomy minimizes the chance of mistakenly recognizing or rejecting a species because the single line of evidence used was not appropriate for that group (Padial et al. 2010; Schlick-Steiner et al. 2010).
Methodological sophistication: Over the past two decades, biologists have revealed enormous complexity in the evolutionary processes that shape groups of organisms. The ubiquity of lateral gene transfer (in prokaryotes) and introgression (in eukaryotes) means that genes can be exchanged between different lineages, and the ubiquity of incomplete lineage sorting means that it is probably never the case that all gene-trees of a population reflect the correct history of speciation. In response to this complexity, a wide range of new, often model-based methods of species delimitation has been developed over the past two decades. These methods are increasingly sophisticated and able to take into account increasingly more variables and confounding factors (Sites and Marshall 2003; Camargo and Sites 2013).
Note that I do not claim that these four trends are due to a theoretical paradigm shift. They are clearly also driven by technological progress and new techniques, models and kinds of data that have become available. Moreover, it is theoretically possible to do research that fits in these trends without subscribing to the unified species paradigm. However, there is a clear and tight fit between the unified species paradigm and these methodological trends, and those who support the former are typically actively engaged in the latter. Rather than causing the unified species paradigm, then, it is most accurate to say that these methodological practices are a part of it.
These three theoretical principles and four methodological trends capture the main tenets of the unified species paradigm. In the remainder of this paper, I look at recent biology textbooks, recent publications, and citations to investigate whether claims about broad and increasing acceptance of this paradigm are justified.
3 Textbooks and Species Concepts
Thomas Kuhn (1962) famously emphasized the role textbooks play in consolidating and perpetuating scientific paradigms. Textbooks summarize the ruling paradigm as well as the main problems, aims and values that shape what Kuhn calls normal science in that paradigm. As new generations of scientists learn their trade through these textbooks, they ensure that these scientists work within this paradigm. Whenever a scientific revolution occurs, according to Kuhn, textbooks have to be rewritten so as to reconstruct scientific history in a way that represents it as linear progression towards the ruling paradigm.
Regardless of whether Kuhns view on scientific change is accurate, it is plausible that scientific textbooks typically contain brief summaries of the paradigm that is in place around the time of publication. It is also plausible that they have substantial influence on the paradigms that biologists come to accept and work in. Hence, surveying textbooks is one way of testing the claim that the unified species paradigm in taxonomy is firmly established, and surveying change in textbooks over time is one way of tracking changes in this paradigm over the past decades.
3.1 Methods
I collected various editions of authoritative, widely used and long-running textbooks in biology and evolutionary biology. For convenience, I refer to these textbooks by one of their authors. Table 1 provides an overview of these textbooks and the editions that were surveyed, and more information on each edition as well as a list of the consulted chapters is included in the Appendix (A1). I had nearly complete coverage of Campbells Biology and Solomons Biology from the 1990s until now. The editions that I consulted of Futuymas textbooks on evolutionary biology provide similarly consistent coverage, but were published with more time in between two editions and include two distinct but similar series of books (Evolutionary Biology until 1998 and Evolution after that). The coverage of the other textbooks was more patchy. I also checked textbooks in ecology (Townsend, Begon, and Harper 2008; Krebs 2009; Smith and Smith 2015) and conservation biology (Hunter and Gibbs 2009). As these generally contained no discussion of species concepts, classification or systematics, they were not included in this study.
Table 1:
Overview of the textbooks surveyed for this study.
| Textbook | Time period | Editions included | n |
--- --- |
| General Biology |
| Campbell | 19902020 | 211 | 10 |
| Solomon | 19852019 | 1, 311 | 10 |
| Raven | 19992017 | 59, 11 | 6 |
| Kimball | 19741982 | 35 | 3 |
| Evolutionary Biology |
| Futuyma | 19862017 | 2, 3; 1, 3, 4 | 5 |
| Freeman | 20012014 | 2, 5 | 2 |
| Ridley | 19932004 | 1, 3 | 2 |
For each available edition of each textbook, I read all chapters or sections devoted to the species concept, classification or systematics more generally. For each edition, I checked the number of species concepts that are discussed, whether the ESC is mentioned as the overarching concept, and whether an emerging consensus about the species concept in taxonomy is mentioned. Because the unified species paradigm involves a shift from theoretical to practical issues, I also tracked how much space in the chapters on systematics was devoted to classification and theoretical taxonomic debates as compared to practical, methodological and operational discussions.
3.2 Results
In the three textbooks for which there was coverage of the full period (Campbell, Solomon, Futuyma), there was a clear increase in the discussed number of species concepts until the early 2000s. The three textbooks initially only discuss the BSC, and then gradually add additional concepts from the mid-1990s. The number of species concepts discussed in Futuyma and Campbell then decreased strongly (from 8 and 6 to 2 and 3 respectively) over the past decade (see figure 1), but remained constant in Solomon (3). One explanation for this difference is that the chapter on species concepts in Solomon has not changed since 2005, while in Campbell and Futuyma was updated multiple times.
Figure 1:
Number of species concepts discussed in the three textbooks with good coverage.
Of the remaining textbooks, Ravens shows a similar pattern with 2 concepts up to 2005 and after 2017, and 3 in between. As only two editions of Ridleys (7 and 3 concepts) and Herrons (3 concepts) were available, no trend could be derived from those textbooks. In the final editions that I surveyed, all textbooks mention the BSC. Other species concepts still commonly discussed are the PSC, the Morphological Species Concept and the Ecological Species Concept.
All the general textbooks (Solomon, Campbell, Raven) emphasize that there are many species concepts, and that there probably is no single concept that is always correct. In line with this, none of these textbooks claims that there is an emerging consensus that species are independently evolving lineages. If the ESC is discussed (Campbell, 5; Raven, 6; Solomon, 711), it is as one among many possible species concepts. In contrast with this, two of the textbooks on evolutionary biology do mention consensus concerning the ESC. Futuymas 2013 (460) cites de Queiroz 2007 and claims that all definitions of species share the notion that species are independently evolving lineages, and that they differ as to the properties that define those lineages. However, this passage is removed again in the dramatically shortened discussion of species concepts in the next edition. In both surveyed editions of Freeman, the authors explicitly state that biologists agree that species are independently evolving lineages, and that different traditional species concepts provide different operational criteria for species delimitation.
In all textbooks with long-term coverage, the space devoted to taxonomy, classification, and theoretical discussion of classification decreases over time. This is most obvious in the way that so-called schools of taxonomy are discussed. The early editions of all textbooks contain a substantial discussion of classical evolutionary taxonomy, phenetics, and cladistics, often with an explicit preference for classical evolutionary taxonomy and explained by means of the example of the legitimacy of Aves and Reptilia as valid taxa. Over time, these discussions then get shortened and limited to cladistics. In Campbell, the discussion of schools of taxonomy disappears in the 1999 edition, in Raven and Futuyma in 2005, and in Solomon this is moved to a historical section in 2015. Parallel to this, discussion of topics such as naming, ranks, taxonomic criteria, the Phylocode, classification and lumping/splitting were gradually shortened or completely removed in all these textbooks over the past two decades. Over the same period, all long-running textbooks gradually expanded their discussion of phylogenetics, with increasingly more attention for cladistics and various molecular methods used for it.
These trends of decreasing coverage of taxonomy/classification and increasing coverage of phylogenetics and methodological issues can be illustrated by comparing the early and late editions of the textbooks. For example, the second edition of Campbells Biology (published in 1990) contained about 5 pages on taxonomy and classification, and about 2 pages on other, methodological aspects of systematics. The chapter that deals with systematics in the eleventh edition of the same textbook (published in 2017), contains about 2 pages on classification and between 10 and 15 on other aspects of systematics.4 Similarly, the chapter on systematics in the third edition of Solomons Biology (1993) is titled the classification of organisms, and contains about 9 pages on classification and about 2 on other aspects of systematics. In the eleventh edition (2019), this title has changed to Systematics: Understanding diversity, and classification and other aspects of systematics get respectively 6 and 11 pages. Finally, the sixth edition (1999) of Ravens Biology has 6 pages on classification, and no focused discussion of phylogenetics or cladistics. The eleventh edition (2017) has 4 pages on classificationmostly devoted to species conceptsand 13 pages on other aspects of systematics.
3.3 Discussion
The textbooks analyzed here do not reflect an emerging consensus on the ESC. Instead, they all devote most attention to the BSC and often even explicitly endorse a pluralist position concerning species concepts. Thus, claims about the unified species paradigm are not confirmed in this way. However, while general textbooks do not mention the consensus concerning species as evolutionary lineages, textbooks on evolutionary biology do. This suggests that it might be that textbooks on evolutionary biology are a better gauge of trends in systematics, and that it takes more time for changes in systematics to be taken up in general textbooks. This explanation is supported by the fact that Solomon and Campbell still explicitly endorsed classical evolutionary taxonomy when it had already been largely abandoned (in 1996 and 2002 respectively), while neither of the earliest consulted editions of textbooks in evolutionary biology (Futuyma in 1986 and Ridley in 1993) endorse this approach to classification.
Other trends in the textbooks are more in line with the claims about the unified species paradigm. First, the number of species concepts discussed in various textbooks was at a peak in the early 2000s, and steadily decreased after that. Together with shortening sections on taxonomy and the preferred theoretical framework for it, this suggests that interest in theoretical questions concerning species classification decreased. At the same time, discussion of cladistics and phylogenetic methods strongly increased, suggesting that interest in methods of systematics increased. Both trends are in line with the way the unified species paradigm resolves the species problem by focusing on methodological matters. However, it should be noted here that the time-span of these textbooks coincides with the introduction and popularization of molecular data in systematics and evolutionary biology. This led to the development of a wide range of new methods and revolutionized these fields in a way that would be taken up in textbooks regardless of conceptual changes or changing evaluation of the relative importance of theoretical and practical issues. Hence, the extent to which the increasing attention devoted to phylogenetic methods reflects a paradigm change in taxonomy is unclear.
4 Documents and citations on species concepts
While textbooks provide insight into paradigms that have been fully accepted for a substantial time, they may not be up to date with more recent changes in a field. Particularly when the textbooks are not devoted solely to the field in questionas was the case herethey are unlikely to be the best tool to track recent changes. Scientific publications are in that sense complementary to textbooks, as they appear more regularly and typically deal with the state of the art of the field. Such publications are also useful to test how broadly certain ideas are used in a field, namely, through citation counts. This does not work for textbooks, as their influence often extends beyond their cited use. This section looks at both publications and citations for further tests of the paradigm-change in taxonomy.
At least two testable predictions can be derived from the claim that consensus concerning the unified species paradigm is gradually emerging. If there is growing agreement to define species as independently evolving lineages instead of using other traditional species concepts, it should be expected that documents describing the ESC tend to get cited more often over time, and documents describing the traditional species concepts tend to get cited less often over time (call this prediction 1). Because the unified species paradigm focuses on operationalization rather than the species concept, one would also expect that the number of documents dealing with conceptual issues tends to decrease while the number of documents dealing with methods of species delimitation should increase (call this prediction 2).
4.1 Methods
To test prediction 1, I compiled a list of the main publications connected to the various species concepts. For the ESC, I selected the two most cited papers by de Queiroz (namely, his 1998 and 2007) on his solution to the species problem as well as Maydens (1997) influential paper on the same topic. Even though they were not the first to discuss the ESC (see Simpson 1951; Wiley 1978), their names and work have become closely associated with the unified species paradigm. For the other species concepts, I relied on the comprehensive list of Zachos (2016, chapter 4) and selected the main references that were given for each of the 32 species concepts discussed there, as well as the main references for two new species concepts discussed in Zachos (2018). Of these 34 concepts, the Nothospecies Concept, Taxonomic Concept, Morphological Concept, Phenetic Concept, Successional Concept, and Agamospecies Concept could not be included as their main references (or citations of these references) were not indexed in the online version of Clarivate Analytics Web of Science (WoS).5 In addition, the notions of Evolutionary Significant Unit, Least Inclusive Taxonomic Unit and Non-dimensional Species were not included because they are not strictly speaking species concepts. The Reproductive Competition Concept was considered a version of the BSC and therefore not included separately. Finally, the ESC and de Queirozs Unified Species Concept and General Lineage Concept were considered the same and not included in the list of non-ESC concepts. For each of the remaining concepts I selected multiple references (BSC, diagnosability PSC, monophyly PSC) or the most prominent reference (all other concepts).6
I then pooled the ESC documents in one group, and the other documents in another group, and retrieved citation numbers per year for both groups from WoS. Because this test aims to track the change over time in the proportion of systematic biologists that uses these documents (rather than change in absolute numbers of use), citation numbers for each year were made relative to the total number of relevant documents published that year. More precisely, citation numbers for each year where divided by the total number of documents for that year in the 15 WoS categories that cite documents about species most often.7 This way, the results do not merely reflect the increasing coverage of WoS or general increase in scientific publications. Note that because the two groups of documents differ in size and consist of records published at widely different times, these relativized citation numbers cannot be used to compare the influence of these groups. Rather, they show how interest for each of these groups changed over time, and allow us to compare these trends. To do this, a Spearman correlation between year (19982019) and relativized citation number was calculated using the cor.test() function from the stats package in R.8 The correlation estimates (Ï) were tested using a two-sided test from the same function.
To test prediction 2, I searched WoS in all databases for documents published between 1990 and 2019 with species delimitation in their title, and for documents with either species concept or species definition in their title. In addition, I searched for documents on at least one of 9 main species concepts (excluding the ESC).9 Because almost no documents mention these in their title, I also included documents with one of these species concepts in their abstract or keywords. For each of the three search results, I also retrieved how often they were cited each year between 1990 and 2019. Again, the results were divided by the same number of relevant WoS documents for the year of publication or citation to control for the increase in WoS coverage and general publication and citation numbers over time. The Spearman correlation was then calculated between year and the relativized citation and publication numbers for 19982019. Note that, as there are more citable documents as we move closer to the current year (e.g., documents published in 1998 can still be cited in 2000, but documents published in 2000 could not yet be cited in 1998), the number of citations for each year can be expected to increase regardless of the popularity of these documents. Still, it is relevant to compare this measure between the three groups.
4.2 Results
The correlation estimates for prediction 1 are summarized in table 2, and show a strong correlation between year (19982019) and relativized citation numbers of the ESC group, and a strong negative correlation between year (19982019) and relativized citations numbers for all other concepts. A similarly strong negative correlation was found between year and the relativized citations of a selection of non-ESC concepts consisting of the BSC, the monophyletic PSC, the diagnosability PSC, the Recognition Species Concept, the Cohesion Species Concept, and the Genotypic Cluster Species Concept.10 Figure 2 visualizes the changes in citations of the ESC (top, 19982019) and other concepts (bottom, 19902019) over time. Citations of the ESC documents have increased more or less steadily since their publication, and ESC documents have never been cited more often than over the past two years. In contrast with this, citations of all other species concepts and selected other concepts increased up to the early 2000s, and decreased after that.
Table 2:
Spearmans rank correlation coefficient for the change over time of relativized ESC-citations, citations of all other concepts, and citations of the 6 selected concepts.
| Trend | Spearmans Ï | p-value |
---
| ESC documents | 0.894 974 6 | 3.301 52Ã106 |
| All concepts documents | 0.883 681 5 | 2.729 49Ã106 |
| Selected concepts documents | 0.625 070 6 | 0.002 329 987 |
Figure 2:
Number of citations of the ESC documents (top) and the documents about other species concepts (bottom) relative to the total number of relevant publications in WoS. The dots indicate the relative number for each year. The nonlinear trend line has been generated using a LOESS smoother in Microsoft Excel (PTS Loess add-in; α = 0.33).
The correlation estimates for prediction 2 are summarized in table 3. There was a strong positive correlation between year and documents with species delimitation in their title, but no significant correlation between year and documents with species concept in their title or one of the various species concepts in abstract or keywords. Figure 3 (top) visualizes this, and shows that while the numbers of documents on species concepts (both in title and in abstract/keywords) has been stable since 1990, there was a strong increase in documents on species delimitation in the early 2000s. As expected (as more documents are added to the pool throughout the years), there was also a strong correlation between year and relativized citations for all three groups. However, as shown by figure 3 (bottom), this increase is much weaker for documents on species concept and the various species concepts than for documents on species delimitation. To compare the number of documents with species concept and species delimitation in their title, figure 4 represents the proportion of these two groups in all documents with either in their title. This figure shows clearly that until 2005, documents on the species concept were far more common, whereas after that point documents on species delimitation became dominant.
Table 3:
Spearmans rank correlation coefficient for the change in relativized documents and citations of Species delimitation, Species concept, and the various species concepts.
| Measure | Spearmans Ï | p-value |
---
| Species delimitation documents | 0.945 | 4.03 Ã 106 |
| Species concept documents | 0.333 | 1.31 Ã 101 |
| Various species concepts documents | 0.34 | 1.21 Ã 101 |
| Species delimitation citations | 0.98 | 3.15 Ã 106 |
| Species concept citations | 0.976 | 3.26 Ã 106 |
| Various species concepts citations | 0.96 | 3.72 Ã 106 |
Figure 3:
Number of documents (top) and citations (bottom) on species delimitation, species concept and the various species concepts relative to the number of WoS documents. The dots indicate the relative number for each year. The nonlinear trend line has been generated using a LOESS smoother in Microsoft Excel (PTS Loess add-in; α = 0.33).
Figure 4:
Proportion of documents with species delimitation or species concept in their title, relative to the total number of WoS documents.
4.3 Discussion
Prediction 1that citations of the ESC would increase while those of other species concepts would decrease over timewas confirmed by the test in this section. Citations of documents on other species concepts increased until around 2005, and decreased gradually after that. It was also around this time that citations of the ESC started increasing strongly. Interestingly, this matches the time of the peak of species concepts in the textbooks discussed in the previous section. This suggests that the debate about species concepts was influential and particularly active over the 1990s and with a peak in the early 2000s, after which its influence gradually decreased. Conversely, the ESC as a resolution to the species problem clearly won in importance over the past two decades, and seemingly continues to do so.
Prediction 2that the number of documents on species delimitation would increase while those on species concepts would decrease over timewas also confirmed by the test in this section. The number of documents on the species concept and documents on particular species concepts remained rather stable over the past decades, while documents on species delimitation increased strongly from around 2005 on. This confirms the pattern of the absolute number of documents on species delimitation and related terms (cryptic species and species tree) reported in Camargo and Sites (2013). The decreasing influence of debates on species concepts as compared to species delimitation is also reflected in the citation numbers, which over the past decade increased far more strongly for documents on species delimitation than for documents on species concepts. While this is of course also the direct effect of the different rate at which the total number of documents on each of these topics increases, it shows clearly which of these topics currently attracts most attention.
5 Citation context analysis
The tests in the previous section use citation numbers to estimate how use and discussion of different species concepts has evolved over time. One important limitation of this is that citations are not always used to express support for or use of a particular concept. Indeed, concepts are also likely to be cited whenever they are criticized. Hence, high citation numbers need not always indicate that the concepts discussed in those documents are generally accepted. The study of citations that looks at the context of the citation to determine the kind of link between documents it constitutes, is known as citation context analysis (Small 1982; Bornmann and Daniel 2008; Petrovich 2018). This section uses such citation context analysis to get a better understanding of the citation numbers used in the previous section. More particularly, it will look into citations of the ESC and of the traditional species concepts to distinguish between citations that indicate acceptance, support or use, and citations that do not indicate this and perhaps even indicate rejection or criticism. This way, it can be determined whether the different trends truly reflect an emerging consensus on the former, and a loss of interest in the latter. If claims about the unified species paradigm hold, it should be expected that over the past two decades documents on the ESC were more often cited to indicate use or support, while documents on the other species concepts were less often cited in this way.
5.1 Methods
Like in the previous section, two groups of documents were selected to represent the ESC on the one hand, and the traditional species concepts on the other. De Queirozs 2007 and 1998 were used for the reasons already described in the previous section: they are, more than any other papers, associated with the unified species paradigm. Because citation numbers were too high to code all 30 documents on the traditional species concepts, a selection was made to include a diversity of influential species concepts with highly cited papers published at the same time or before the two de Queiroz papers. This way, the monophyletic PSC (Donoghue 1985), the diagnosability PSC (Cracraft 1983), the BSC (Mayr 1942), the Ecological Species Concept (Van Valen 1976), the Recognition Species Concept (Paterson 1985), the Cohesion Species Concept (Templeton 1989) and the Genotypic Cluster Species Concept (Mallet 1995) were included.
For both groups of documents, the program Publish or Perish was used to download citation details from Google Scholar (which are more complete than WoS citation numbers).11 Because for both groups the total number of citations was in the order of several thousandsand thus, too much to code manuallya selection was made. For the two de Queirozs papers, the 100 most cited documents of each paper were included, as well as the 5 most cited documents for each year between 1998 and 2018. Citation numbers are a useful method of selection here because the test aims to gauge the influence of de Queirozs papers on scientific research. By coding the most cited documents, the results of the test will reflect how the most influential documents use the ESC. The top 5 most cited documents of each year since 1998 were included to make sure that the sample also includes recent influential documents which, because of their publication date, had much less opportunity to be cited. Finally, any document citing both the 1998 and 2007 was included only once and replaced by the next most highly cited document. Documents authored by de Queiroz or documents that did not have the citation in text were also replaced. Using these criteria, 217 documents citing either de Queirozs 1998 or 2007 were selected for coding.
As the group of papers on the traditional species concepts contained more different papers and papers published sometimes long before the study period, a slightly different method of selecting citing documents had to be used here. First, I selected only documents published between 1999 (the year of the earliest included document citing de Queiroz) and 2018. This makes sure that both groups cover the same time period and that the results of the study reflect the recent influence of the ESC and the traditional species concepts. Second, to ensure that citations of all 6 papers were equally represented, I selected the 35 most cited documents for each of the 6 included papers. By using this number, the sample could be expected to be the same size as the one selected for the ESC. Third, for the reasons discussed in the previous paragraph I included the 5 most cited documents for each year and the most cited document for each author for each year. Finally, documents authored by the cited author and documents that did not contain the citations in text were excluded and replaced. Using these criteria, 298 documents citing at least one of 6 selected species concepts were selected for coding.
As the main aim of this test is to uncover whether the ESC and other species concepts were cited to indicate support, the coding scheme focused on this criterion. Table 4 provides an overview of the categories of the coding scheme. Note that documents were only included in these categories if the citation was clearly included in the document with the purpose described by the category. For example, if a document cites de Queirozs 1998 and 2007 to refer to the species problem in general and then uses de Queirozs framework without explicitly saying that the framework will be used, this was not coded as use or support. As one document can cite another in more than one way, some documents were placed in multiple categories.
Table 4
Coding scheme for citations of the ESC and of the traditional concepts.
| | |
--- |
| Use or support | Any document in which the concept is explicitly endorsed or explicitly used as the correct or best species concept. |
| Criticism | Any document in which explicit arguments against the concept are given. |
| Neutral reference | Any document which cites a paper to refer to a concept or the species problem more generally without endorsing or criticizing the concept. |
| Not relevant to species concepts | Any document which cites a paper to refer to its contents not related to the species concept. |
|
In addition to coding citations, I tracked how often documents refer to a paradigm-change or substantial conceptual shift in taxonomy, and indicate that there is a consensus or emerging consensus about one of the concepts. This provides a way of testing the extent to which claims about the unified species paradigm are shared by biologists citing documents about species concepts.
5.2 Results
Figure 5 shows how often papers from the two groups were cited in the ways described by the four categories of the coding scheme. While both groups of papers are often cited in a neutral sense, there are large differences in the categories of use and criticism: while just more than half of the documents citing the ESC did this to indicate their use of or support for his concept, less than 10% of the documents citing other concepts did this for that reason; conversely, those papers were cited substantially more often to criticize the species concepts discussed in them. For both use and criticism, a Fishers Exact Test shows that the difference between the ESC papers and the other papers is highly significant.12
Figure 5:
The proportion of documents citing the ESC and other concepts to indicate each of the four coding categories.
In total, 30 documents (14%) citing the de Queirozs papers explicitly indicated that there is (emerging) consensus among biologists about this concept and/or that this concept constitutes a major conceptual shift in taxonomy. In addition, 41 documents (19%) indicated that the ESC unifies the traditional species concepts. In comparison, only three documents indicate a near consensus about one of the other species concepts.
5.3 Discussion
The classification of citation types in this section confirms the predictions about an emerging consensus concerning the unified species paradigm. De Queirozs papers were cited more frequently to indicate use, while papers about the other species concepts where cited more often to criticize them. This is compatible with the claim that there is a growing consensus about the ESC as the best conceptual framework to use in scientific research on species. Moreover, the low number of criticism-citations (compared to the other species concepts) suggests that influential documents rarely criticize this conceptual framework. All this means that the results from the tests in the previous section can safely be interpreted as supporting prediction 1 tested there: increasing citations of the ESC indicate growing support for this concept. Indeed, the results from the test in this section suggest that a stronger interpretation is even justified. It is likely that more of the citations of other species concepts are for criticizing these concepts, and that more of the citations of the ESC are for supporting or using this concept. Hence, the gap between support for the ESC and for the other concepts is likely to be even bigger than suggested by the difference in the trends of the citation numbers of both groups.
The community-support for the unified species paradigm suggested by these tests is also reflected in the way many of the authors of these documents evaluate the conceptual position of the community: 14% of the documents citing the ESC explicitly state that there is growing consensus about this view among biologists.
The seemingly growing consensus about the ESC is in contradiction with the findings of PuÅ¡iÄ et al.s (2017) survey about the species concept in biology. They asked 193 biologists which species concept they prefer, and only 14 of those indicated preferring the ESC. About one third of all respondents stated they use the BSC, which was by far the most popular option. One plausible explanation for the divergence between the results of that survey and the tests in this paper is that none of the respondents identified as systematists or taxonomists. The most common disciplines in the survey population were ecology and molecular biology. Hence, these biologists are unlikely to closely follow conceptual changes in taxonomy or be aware of the most popular views in that field. Indeed, the popularity of the BSC among participants suggests that non-systematists might suffer from the same delay on conceptual change that could also explain the gap between biology textbooks and the systematic literature. The textbooks influence on biologists might also be a partial explanation for the popularity of the BSC among non-systematists, as many of them might have had little contact with debates about species concepts since their education.
6 Conclusions
The tests in this paper largely confirm the claims about a growing consensus among taxonomists about the unified species paradigm as the most appropriate conceptual framework for thinking about species. If indeed widely accepted, this paradigm would resolve the species problem by shifting attention from theoretical questions to operational ones.
The evidence from textbooks forms one exception to this this conclusion, as for the largest part these textbooks did not endorse or even discuss this paradigm. However, these textbooks did bring to light another trend that was confirmed in subsequent tests, namely, that the theoretical debate on species concepts became increasingly influential throughout the 1990s, reached its most influential point in early in the twenty-first century, and then gradually decreased in influence. This is in line with the unified species paradigms resolution of the species concept by focusing on operational rather than conceptual issues. In line with this, publication and citation numbers on species delimitation increased strongly over the past two decades, while the papers on the species concept clearly score worse on both parameters. One exception to this are papers on the ESC, which have been cited frequently over the past two decades by papers that use or support it.
To end this paper, I wish to make three remarks. First, it is interesting to note that the operational focus of the unified species paradigm fits nicely with Kuhns (1962) view on scientific revolutions. Kuhn argues that a scientific field in crisis tends to focus on fundamental theoretical questions. Once a new paradigm arises from this period of crisis, Kuhn argues, the focus shifts from fundamental theoretical questions to solving practical puzzles that are suggested by whichever theoretical framework came out triumphant. Relevant is that the theoretical issues that dominated science during the period of crisis need not be solved but typically just shift out of focus. This is because scientists engaged in normal science typically do not question the fundamental theoretical framework of the paradigm. Rather, they focus on solving the more fine-grained puzzles that emerge within the theoretical framework of the paradigm (see also Lakatos 1976).
This Kuhnian picture mirrors the ESCs resolution of the species problem, which consists in a broad, abstract definition of species as evolutionary lineages and a strong focus on developing sophisticated methods for identifying such lineages. Thus, rather than working out in detail what evolutionary independence means, systematic biologists now focus on developing methods that can reconstruct species trees despite complexities like extensive introgression and incomplete lineage sorting. The focus thus lies on difficult operational problems rather than fundamental theoretical issues, and the latter are mostly dismissed as irrelevant to biological practice.
This parallel with Kuhns crisis science and normal science should not be taken to mean that change in taxonomy over the past three centuries perfectly fits the Kuhnian view: the field has gone through many minor and major changes, with competing paradigms often operating side by side, and with both theoretical and practical problems continuously present in taxonomic debates (Mishler 2009; Wilkins 2018). In addition, some of the theoretical principles associated to the unified species paradigm may have been implicitly assumed or used by many taxonomists for a long time before they were spelled out in print. However, mapping paradigm change in taxonomy to Kuhns view and investigating the implications for this view are subjects for a different paper.
A second point I want to draw attention to is that there are at least three important limitations to the conclusions we can draw from the tests in this paper. First, this paper only surveyed a small number of papers from taxonomy, and systematics more generally. More precisely, the papers that were taken into account here were mostly papers that are highly cited. This means that the results only reflect the conceptual views of a small part of the systematic community, and that they are biased towards the well-cited areas of systematics and taxonomy. This is significant because there are many small journals, often dedicated to particular taxa, that rarely accumulate citation numbers such as the ones of the papers used for this investigation. Moreover, amateurs form a substantial proportion of the taxonomic workforce (Fontaine et al. 2012). These amateurs typically do not have access to the same genomic or genetic methods as professionals. Hence, it is plausible that the views of amateurs on species concepts and methods of species delimitation are substantially different from the views reflected in the documents sampled for this investigation.
While an exhaustive investigation of the ruling paradigm in taxonomy should thus also test the views that are prevalent in areas of taxonomy that are cited less frequently, I do not think that this invalidates the results of this study completely. After all, it is likely that highly-cited papers and professional taxonomists have a stronger influence on what isand will bethe ruling taxonomic paradigm: citation scores and influential publications weigh heavily in the distribution of research funds, and professional taxonomists are typically also responsible for educating new generations of taxonomists. Hence, even if there is a divide currently, it is likely that the view of highly cited, professional taxonomists becomes more influential over time.
A second limitation of this study is that it only considered what taxonomists say, either in words or through citation. This means that the results do not reflect what taxonomists, and systematists more generally, actually do (Luckow 1995). Thus, it is possible that taxonomists claim to support the ESC and cite de Queiroz to support this, but use the same methods as taxonomists that a few decades ago claimed to rely on the BSC or PSC. A more exhaustive study should thus also look at taxonomic methods. This, of course, is tricky as there is no one-on-one link between species concepts and methods of species delimitation.
Finally, it is worth pointing out that the emerging consensus highlighted by the tests in this paper is more complex than might appear at first sight. Even among biologists that explicitly adopt de Queirozs resolution of the species concepts, there are subtle disagreements about how this resolution should be interpreted. For example, both Padial and De La Riva (2020) and Zachos et al. (2019) explicitly endorse the ESC, but disagree on the question of whether there are subjective decisions in species delimitation. According to Zachos et al., grouping organisms into lineages is an objective matter while ranking lineages as species is a subjective matter. According to Padial and De La Riva, taxonomy does not require the subjective decision of ranking, as all and only phylogenetic lineages should be recognised as species. Despite the consensus on the ESC, then, there are still unresolved questions that continue to fuel debates between taxonomists. The tests in this paper are limited in that they gloss over these different interpretations of the paradigm, and only record explicit endorsement.
A third remark I wish to make is that, regardless of whether the unified species paradigm is conceptually sound or correct, its popularity should motivate philosophers of biology interested in species to (also) devote their attention to practical questions concerning species delimitation. While a substantial number of highly cited papers on species concepts were authored by philosophers, none of the highly cited papers on species delimitation were authored by philosophers. Indeed, a search for species delimitation in PhilPapers yields almost no results. Recent philosophical work (Haber 2019; Sterner 2019) suggests that there are plenty of tricky conceptual issues in species delimitation that philosophers could explore and help clarifying. This paper has shown that in doing this, they would help developing a view that is increasingly important in systematic biology.
Acknowledgments
I am grateful to Andreas De Block, Lin Gen, and Lin Li for their comments on previous drafts of this paper. I also want to thank the two reviewers, Frank Zachos and John Wilkins, for their helpful suggestions. Research for this paper was funded by the Research CouncilFlanders (FWO; Grant 3H200026).
Notes
© 2022 Author(s)
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International license, which permits anyone to download, copy, distribute, display, or adapt the text without asking for permission, provided that the creator(s) are given full credit.
de Queiroz (1998) does not consider his two species concepts fully identical to the ESC. However, for the purposes of this paper, the differences between them are not relevant. ®
The view that species delimitation consists in testing hypotheses about the independence of lineages should not be confused with Kirk Fitzhughs (2009) view of species as explanatory hypotheses. ®
It is debatable, however, whether these methods are always successful at distinguishing population-level processes and lineage-level processes. Notably, Sukumaran and Knowles (2017) show that methods of species delimitation using the multispecies coalescent track genetic structure, which does not always reflect species boundaries. ®
These comparisons of pages dedicated to classification and pages dedicated to parts of systematics not involved in classification are inevitably approximations, as often sections deal with a mixture of both. However, the differences between early and late editions are large enough for the result to be clear despite this difficulty. ®
www.webofknowledge.com ®
For the full list of concepts (and corresponding papers) included in the search, see Appendix (table A4. ®
See Appendix (table A5) for the list of Web of Science categories included. ®
®
See Appendix (A) for the precise search terms. ®
The results for these 6 species concepts, based on only 1 rather than multiple papers, are included here because they figure in the citation context analysis discussed in the next section. For the list of papers of these 6 concepts, see Appendix (table A3). ®
®
For use, p < 0.00001 (odds ratio: 11.65); for criticism, p < 0.00001 (odds ratio: 0.17) ®
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Appendix: Supplementary Materials
A Search terms
Species delimitation: I performed a search in Clarivates Web of Science, Core Collection for all document types in English on April 1, 2020 for the following: TI=species delimitation.
Species concept: I performed a search in Clarivates Web of Science, Core Collection for all document types in English on April 1, 2020 for the following: TI=species concept OR species definition OR concept of species OR definition of species.
Various species concepts: I performed a search in Clarivates Web of Science, Core Collection for all document types in English on April 1, 2020 for the following: TS= biological species concept OR phylogenetic species concept OR ecological species concept OR recognition species concept OR cohesion species concept OR genotypic cluster species concept OR genealogical concordance species concept OR genetic species concept OR taxonomic species concept.
B Textbooks consulted
Table A1: Textbooks about biology in general. For each textbook, I consulted the chapters indicated in the final column, and searched in the index for species, taxonomy and classification.
Table A1:
All general biology textbooks included in the textbook survey.
| Name | Title | Authors | Ed. | Date | Publisher | Chapters |
--- --- ---
| Campbell | Biology | Neil Campbell | 2 | 1990 | Benjamin/Cummings Publishing Co. | 22, 23 |
| Biology | Neil Campbell | 3 | 1993 | Benjamin/Cummings Publishing Co. | 22, 23 |
| Biology | Neil Campbell | 4 | 1996 | Benjamin/Cummings Publishing Co. | 22, 23 |
| Biology | Neil Campbell, Jane Reece, Lawrence Mitchell | 5 | 1999 | Addison Wesley Longman | 24, 25 |
| Biology | Neil Campbell, Jane Reece | 6 | 2002 | Addison Wesley Longman | 24, 25 |
| Biology | Neil Campbell, Jane Reece | 7 | 2005 | Pearson Benjamin Cummings | 24, 25 |
| Biology | N. Campbell, J. Reece, L. Urry, M. Cain, S. Wasserman, P. Minorsky, R. Jackson | 8 | 2008 | Pearson Benjamin Cummings | 24, 26 |
| Biology | N. Campbell, J. Reece, L. Urry, M. Cain, S. Wasserman, P. Minorsky, R. Jackson | 9 | 2011 | Benjamin Cummings | 24, 26 |
| Biology | N. Campbell, J. Reece, L. Urry, M. Cain, S. Wasserman, P. Minorsky, R. Jackson | 10 | 2013 | Pearson | 24, 26 |
| Biology | Lisa Urry, Michael Cain, Steven Wasserman, Peter Minorsky, Robert Jackson | 11 | 2017 | Pearson | 24, 26 |
| Solomon | Biology | Claude Villee, Eldra Solomon, William Davis | 1 | 1985 | Saunders College Publishing | 17, 45 |
| Biology | Eldra Solomon, Linda Berg, Diana Martin, Claude Villee | 3 | 1993 | Saunders College Publishing | 19, 22 |
| Biology | Eldra Solomon, Linda Berg, Diana Martin, Claude Villee | 4 | 1996 | Saunders College Publishing | 19, 22 |
| Biology | Eldra Solomon, Linda Berg, Diana Martin | 5 | 1999 | Saunders College Publishing | 19, 22 |
| Biology | Eldra Solomon, Linda Berg, Diana Martin | 7 | 2005 | Brooks/Cole | 19, 22 |
| Biology | Eldra Solomon, Linda Berg, Diana Martin | 8 | 2008 | Brooks/Cole | 20, 23 |
| Biology | Eldra Solomon, Linda Berg, Diana Martin | 9 | 2011 | Brooks/Cole | 20, 23 |
| Biology | Eldra Solomon, Charles Martin, Diana Martin, Linda Berg | 10 | 2015 | Cengage Learning | 20, 23 |
| Biology | Eldra Solomon, Charles Martin, Diana Martin, Linda Berg | 11 | 2019 | Cengage Learning | 20, 23 |
| Raven | Biology | Peter Raven, George Johnson | 5 | 1999 | McGraw-Hill | 21, 28 |
| Biology | Peter Raven, George Johnson | 6 | 2002 | McGraw-Hill | 22, 32 |
| Biology | Peter Raven, George Johnson, Jonathan Losos, Susan Singer | 7 | 2005 | McGraw-Hill | 23, 25 |
| Biology | Jonathan Losos, Kenneth Mason, Susan Singer | 8 | 2008 | McGraw-Hill | 22, 23 |
| Biology | Kenneth Mason, Jonathan Losos, Susan Singer | 9 | 2011 | McGraw-Hill | 22, 23 |
| Biology | Kenneth Mason, Jonathan Losos, Susan Singer | 11 | 2017 | McGraw-Hill | 22, 23 |
| Kimball | Biology | John Kimball | 3 | 1975 | Addison Wesley Publishing Co. | 32 |
| Biology | John Kimball | 4 | 1978 | Addison Wesley Publishing Co. | 32, 33 |
| Biology | John Kimball | 5 | 1982 | Addison Wesley Publishing Co. | 33, 35 |
Table A2: Textbooks about evolutionary biology. For each textbook, I consulted the chapters indicated in the final column, and searched in the index for species, taxonomy and classification.
Table A2:
All textbooks on evolutionary biology included in the survey.
| Name | Title | Authors | Edition | Publication year | Publisher | Chapters consulted |
--- --- ---
| Futuyma | Evolutionary Biology | Douglas Futuyma | 2 | 1986 | Sinnauer Associates Inc. | 8, 10 |
| Evolutionary biology | Douglas Futuyma | 3 | 1998 | Sinnauer Associates Inc. | 2, 17 |
| Evolution | Douglas Futuyma | 1 | 2005 | Sinnauer Associates Inc. | 2, 15 |
| Evolution | Douglas Futuyma | 3 | 2013 | Sinnauer Associates Inc. | 5, 15 |
| Evolution | Douglas Futuyma, Mark Kirkpatrick | 4 | 2017 | Sinnauer Associates Inc. | 2, 9 |
| Freeman | Evolutionary analysis | Scott Freeman, John Herron | 2 | 2001 | Prentice Hall | 12, 13 |
| Evolutionary analysis | John herron, Scott Freeman | 5 | 2014 | Pearson | 4, 16 |
| Ridley | Evolution | Mark Ridley | 3 | 2004 | Blackwell | 13, 16 |
| Evolution | Mark Ridley | 1 | 1993 | Blackwell | 14, 15 |
C List of all species concepts
Table A3 lists all papers used for the test of prediction 1 in section 3.
Table A3:
The non-ESC species concepts selected for the citation context analysis, each with the reference used to retrieve citing documents from WoS, and how often these documents were cited in google scholar on 1/4/2020 (used as a selection criterion).
| Species Concept | Reference | Citations Google Scholar |
---
| Phylogenetic Species Concept (monophyly) | Donoghue (1985) | 612 |
| Phylogenetic Species Concept (Diagnosability) | Cracraft (1983) | 1748 |
| Biological Species Concept | Mayr (1942) | 8551 |
| Ecological Species Concept | Valen (1976) | 927 |
| Recognition Species Concept | Paterson (1985) | 1252 |
| Cohesion Species Concept | Templeton (1989) | 1078 |
| Genotypic Cluster Concept | Mallet (1995) | 939 |
D List of selected species concepts
Table A4 lists the selection of non-ESC species concepts used for the citation context analysis, and the reference used for each of these concepts.
Table A4:
List of all species concepts with their main references, the group they were included in for prediction 1, and reasons for exclusion of excluded concepts.
| Species Concept | Selected References | Group Prediction 1 |
---
| Agamospecies Concept | Cain (1954) | Excluded (no citations) |
| Biological Species Concept | Mayr (2000, 1996, 1940) | Other concepts |
| Biosimilarity Species Concept | Stamos (2003) | Other concepts |
| Cladistic Species Concept | Ridley (1989) | Other concepts |
| Cohesion Species Concept | Templeton (1989) | Other concepts |
| Compilospecies Concept | Harlan & De Wet (1963) | Other concepts |
| Differential Fitness Species Concept | Hausdorf (2011) | Other concepts |
| Ecological Species Concept | Valen (1976) | Other concepts |
| Evolutionary Significant Unit | Moritz (1994) | Excluded (not a species concept) |
| Evolutionary Species Concept | Wiley (1978), Simpson (1951) | ESC |
| Genealogical Concordance Species Concept | Avise & Ball (1990) | Other concepts |
| Genealogical Species Concept | Baum & Shaw (1995) | Other concepts |
| General Lineage Species Concept | de Queiroz (1998, 2007) | ESC |
| Genetic Species Concept | Baker & Bradley (2006) | Other concepts |
| Genic Species Concept | Wu (2001) | Other concepts |
| Genotypic Cluster Species Concept | Mallet (1995) | Other concepts |
| Hennigian Species Concept | Meier & Willmann (2000) | Other concepts |
| inclusive species concept | Shanker et al. (2017) | Other concepts |
| Internodal Species Concept | Kornet (1993) | Other concepts |
| Least Inclusive Taxonomic Unit | Pleijel (1999) | Excluded (not a species concept) |
| mitonuclear compatibility species concept | Hill (2017) | Other concepts |
| Morphological Species Concept | Cronquist (1978) | Excluded (not indexed) |
| Non-dimensional species concept | Mayr (2000, 1996, 1940) | Excluded (not a species concept) |
| Nothospecies Concept | Wagner (1983) | Excluded (not indexed) |
| Phenetic Species Concept | Sokal & Sneath (1963) | Excluded (no citations) |
| Phylogenetic Species Concept (Diagnosability Version) | Cracraft (1983), Nixon & Wheeler (1990), Wheeler & Platnick (2000) | Other concepts |
| Phylogenetic Species Concept (monophyly version) | Mishler & Brandon (1987), Donoghue (1985), Mishler (1985), Mishler & Donoghue (1982) | Other concepts |
| Phylo-Phenetic Species Concept | Rossello-Mora R & Amann R (2001) | Other concepts |
| Pragmatic Species Concept | Seifert (2014) | Other concepts |
| Recognition Species Concept | Paterson (1985) | Other concepts |
| Reproductive Competition Species Concept | Ghiselin (1974) | Excluded (identical to BSC) |
| Successional Species Concept | Simpson (1961) | Excluded (no citations) |
| Taxonomic Species Concept | Blackwelder (1967) | Excluded (no citations) |
| Unified Species Concept | de Queiroz (1998, 2007) | ESC |
E List of relevant Web of Science Research Areas
Table A5 lists the top 15 research areas in Web of Science that cite the references listed in table A4 most often.
Table A5:
Top 15 research areas in WoS citing papers about species concepts.
| Research Areas | Total citations of species concept papers | Proportion of total |
---
| Evolutionary biology | 6540 | 71% |
| Genetics heredity | 6379 | 70% |
| Life sciences biomedicine other topics | 5852 | 64% |
| Zoology | 5654 | 62% |
| Environmental sciences ecology | 4156 | 45%1 |
| Biochemistry molecular biology | 3791 | 41% |
| Biodiversity conservation | 2237 | 24% |
| Plant sciences | 2049 | 22% |
| Anatomy morphology | 1741 | 19% |
| Microbiology | 1308 | 14% |
| Science technology other topics | 1185 | 13% |
| Reproductive biology | 977 | 10% |
| Marine freshwater biology | 862 | 9% |
| Immunology | 857 | 9% | |
189410 | https://rileymath.com/index.php/2022/09/08/desmos-reviews-for-ratios-proportions-rates-percent/ | Skip to content
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Desmos Reviews for Ratios, Proportions, Rates, Percent
John Riley
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Here are some Desmos activities that focus on Ratios, Proportions, Rates and Percent and my comments and reviews.
DESMOS Activities
Ratios
Paint
Students use up to four colors to mix colors and complete the wall. If the paint matches, they are successful.
In the example above, the mix is 12 cups of blue, 9 cups of red and 14 cups of white to create the color of the wall behind the couch. The challenge is to start with 4 cups of blue and create the same color.
There are five warm up slides and then they proceed to class challenges where they use four colors to create their own.
I like this problem and would ask them to explain their answers to get to an understanding of ratios.
Click Battle
Click Battle does not have any actual battles. As you can see in the slide, students select which bot will click more times in 10 seconds. There are 14 total slides with six battles and then a final battle.
This is rather simple. After they select their answer, they are asked to explain their reasoning. This may be where the best thinking can be accomplished. I had one student claim that Bot #5 would click more because his arms are closer to his body.
In the early slides, some may have used unit rate without realizing. I would have a discussion around this in groups to guide them towards the concept of unit rate before introducing it. In slide 8, unit rate is introduced.
Des-Farm
This is essentially a 100 square with plants. Students use the 100 square to play with Ratios, Fractions, Decimals and Percents.
This is convenient, but I would spend some time doing Concrete work with manipulatives prior to this activity since it is Representational and then Abstract.
If they have trouble or get stuck, it may be time to get out a concrete manipulative.
Balloon Float
This one is closer to concrete. You choose the number of balloons and then test to see if it floats. Students will use ratios to determine the number of balloons.
There is one slide that asks students to model with linear equations. Here it is:
Nana's Chocolate Milk
Nana's Chocolate Milk: Dan Meyer Three-Act Task
This DESMOS activity is based on a Dan Meyer three-Act task, which you can see by clicking the button above. It begins with a video of Dan putting too many scoops of chocolate in the Milk. It is a lesson on using a double number line to get the correct ratio.
I actually prefer the original three-act task because it is more open and allows for better group work. If you use the DESMOS activity, you may want to use the pacing tool to stop them after the first slide to give them time to think before going beyond the video. You could even have them standing at vertical white boards in small groups.
Exchange Rates
The student exchanges between Aloha, Beta and Gamma coins. This is a lesson in ratios and also linear equations. The ratios are represented as a table, graph and equations.
Water Slide
Students adjust the ratio of height to width to create slides with the same slope. This lesson could be use to give a concrete example of slope. It is a very simple short example of making equal slopes. It could be used as an early activity in Algebra or later in Pre-Algebra.
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189411 | https://www.sciencedirect.com/science/article/pii/S1110116423000352 | Skip to article
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The Egyptian Rheumatologist
Volume 45, Issue 3, June 2023, Pages 229-234
Management of knee osteoarthritis using percutaneous high tibial osteotomy for correction of genu varum deformity in adolescents and young adults
Author links open overlay panel, , ,
rights and content
Under a Creative Commons license
Open access
Abstract
Background
Genu varum is not uncommon among young adults and adolescents and may increase the risk of knee osteoarthritis (OA).
Aim of the work
To analyse the ability to correct and achieve osseous union of genu varum with high tibial osteotomy (HTO) using a dynamic axial fixator (DAF) and to evaluate knee function and pain.
Patients and methods
Fifteen knees in 13 patients with genu varus and medial compartment knee pain underwent open wedge HTO using DAF. Clinical evaluation was done using Modified Lysholm score and new Knee Society score (KSS). Long-standing X-rays were performed.
Results
The mean age of the patients was 21.9 ± 2.57 years (1828 years) and they were 8 males and 5 females. The mean medial proximal tibial angle (MPTA) improved from 78.9 ± 4.35 (6984) pre-operative to 91.7 ± 2.37 (8896) post-operative. The mean mechanical tibiofemoral angle (mTFA) improved from 168.5 ± 4.06 (160176) pre-operative to 180.4 ± 3.11 (176186) post-operative. The target correction was achieved in 12 knees (80%) and under-correction occurred in three (20%). Significant post-operative improvements in pain and function of daily life were seen. All osteotomies healed before 4 months. The mean Modified Lyshlom and KSS scores improved from 66.4 ± 24.02 and 135.1 ± 43.95 preoperatively to 95.5 ± 9.51 and 236.9 ± 11.38 respectively (p < 0.0001 both). 6/15 (40%) knees showed complications in the form of superficial pin tract infection that all resolved without any sequela.
Conclusion
For the treatment of proximal tibia vara, HTO with DAF was safe and effective, allowing for correct post-operative alignment modification with high union rates and trivial sequelae.
Keywords
Osteoarthritis
High Tibial Osteotomy
Adolescents
Adults
Genuvarum
Cited by (0)
: Peer review under responsibility of Egyptian Society of Rheumatic Diseases.
© 2023 Egyptian Society of Rheumatic Diseases. Publishing services provided by Elsevier B.V. |
189412 | https://pmc.ncbi.nlm.nih.gov/articles/PMC9218397/ | Imaging in headache disorders - PMC
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. 2022 Jun 1;45(3):88–92. doi: 10.18773/austprescr.2022.023
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Imaging in headache disorders
Jason C Ray
Jason C Ray
1 Department of Neurology, Alfred Hospital
2 Department of Neurology, Austin Health
3 Department of Neuroscience, Monash University
4 Melbourne
Neurologist, Headache fellow
Find articles by Jason C Ray
1,2,3,4, Elspeth J Hutton
Elspeth J Hutton
1 Department of Neurology, Alfred Hospital
2 Department of Neurology, Austin Health
3 Department of Neuroscience, Monash University
4 Melbourne
Neurologist, Head, Peripheral Neuropathy, Pain and Headache Group
Find articles by Elspeth J Hutton
1,2,3,4
Author information
Article notes
Copyright and License information
1 Department of Neurology, Alfred Hospital
2 Department of Neurology, Austin Health
3 Department of Neuroscience, Monash University
4 Melbourne
Roles
Jason C Ray: Neurologist, Headache fellow
Elspeth J Hutton: Neurologist, Head, Peripheral Neuropathy, Pain and Headache Group
Issue date 2022 Jun.
(c) NPS MedicineWise
This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial No Derivatives (CC BY-NC-ND) 4.0 License.
PMC Copyright notice
PMCID: PMC9218397 PMID: 35755987
SUMMARY
Patients with a suspected change in intracranial pressure or a trigeminal autonomic cephalgia require MRI.
The need for investigation for other headache disorders is guided by the clinical evaluation of the patient. Particular care should be taken to identify any ‘red flags’.
Incidental findings on MRI occur in approximately 2% of patients. Patients with migraine have an increased rate of white matter lesions, but these are of uncertain clinical significance.
Keywords: headache, migraine, MRI, trigeminal autonomic cephalgias
Introduction
Headache disorders are a leading cause of disability. Worldwide, migraine is the second leading cause of years lived with disability and, in Australia, it is among the top 20 reasons for consulting a GP.1,2 While migraine is the most common disabling headache disorder, patients and clinicians are often concerned a headache could be a symptom of secondary pathology.
In a Norwegian population study, the one-year prevalence of secondary headaches was 2.14%. In 80% of these, the cause of the secondary headache could be diagnosed on the patient’s history.3
In a UK tertiary referral headache service, 3655 consecutive patients were screened using ‘red flags’ to identify the need for imaging. Over a five-year period, 14.5% underwent imaging, with 11 patients having a significant finding. This represents 2.1% of patients scanned or 0.3% of the clinic population.4
There are several evidence-based guidelines that recommend that imaging of patients with headaches should not be routine. The need for imaging should be guided by clinical evaluation.5-7
Clinical evaluation
A thorough clinical assessment is essential for evaluating a patient who presents with headache and for determining the need for imaging. The key features of a headache history can be summarised by ‘the 5Ps’:
patient
pattern
phenotype
precipitants
pharmacology.8
A thorough headache history, considering potential red flags (Table 1)8 and ‘green flags’ (Table 2),9 coupled with a detailed neurological examination is required. This not only determines the need for imaging, but also guides which imaging modality to use.
Table 1. The SNOOP4 list of ‘red flags’ for secondary headaches8.
| | Mnemonic | Examples of red flags | Possible secondary headache |
:--- :--- |
| S | Systemic symptoms | Fever, weight loss | Meningitis, encephalitis, giant cell arteritis |
| Secondary risk factor | Malignancy, immunosuppression | Metastasis, leptomeningeal carcinomatosis |
| N | Neurological deficit | Focal neurological sign, altered conscious state | Stroke, space-occupying lesion, hydrocephalus |
| O | Onset | Thunderclap, abrupt onset | Includes subarachnoid haemorrhage, pituitary apoplexy, cerebral venous sinus thrombosis |
| O | Older age | New or progressive headache (>50 years) | Mass lesion, giant cell arteritis |
| P4 | Positional | Changes with change in posture | Intracranial hypotension or hypertension |
| Pattern change | Change in character from baseline | Mass lesion |
| Precipitated by | Valsalva, coughing, sneezing | Posterior fossa lesion |
| Papilloedema | Visual obscuration | Idiopathic intracranial hypertension |
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Table 2. Potential ‘green flags’ for primary headaches9.
| Green flag | Rationale |
:--- |
| The current headache was present during childhood | Secondary headaches are uncommon in childhood and common secondary causes in childhood (viral, post-trauma) do not usually persist. |
| The headache is temporally related to the menstrual cycle | Menstrually related migraine is common, and the probability of a migraine during the first three days of the menstrual cycle is elevated. |
| The patient has headache-free days | Most primary headache disorders are intermittent, whereas secondary causes (excepting brain tumours) are less commonly so, and secondary causes are less commonly associated with an identifiable trigger. |
| Close family members have the same headache type | Migraine and cluster headache can be inherited, and so the presence of a family history is supportive of the diagnosis. |
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The recognition of red flags is useful for identifying which patients need further evaluation, however the specific differential diagnosis should be considered. This guides the choice of investigation and its urgency. For example, a patient with suspected stroke or meningitis requires urgent evaluation, while a patient with a recent change in the pattern of their headache is likely to be suitable for outpatient evaluation.
Green flags are reassuring features in a headache history (Table 2). They suggest a secondary cause of headache is unlikely. The green flags were determined by an expert group of the International Headache Society,9 but have not been validated in a prospective study.
Patient
When deciding on the need for investigation, patient factors such as age and general health are the most critical consideration. A patient with new headaches late in life, or in the setting of malignancy or immunosuppression, always requires further evaluation, regardless of other factors. The presence of neurological or systemic signs in relation to the headache also requires further evaluation. Conversely, the presence of a strong family history of similar headaches is a reassuring factor.
Pattern
The temporal pattern of a patient’s headache can help distinguish primary and secondary causes. A headache that has been present and unchanged from childhood, or is consistently related to menstruation, is less likely to have a secondary cause.9 Conversely, a recent onset or new pattern is suspicious for a secondary cause of headache. The timing of the change in pattern can give a clue as to the cause, such as in the case of medication-overuse headache.
Phenotype
The characteristics of a headache in an individual are called the phenotype. Accurate evaluation of the phenotype is key to determining the headache disorder. In the setting of an established, recurrent phenotype, the presence of a new phenotype requires increased clinical vigilance. However, the presence of a phenotype with features of a primary headache disorder, such as tension-type headache or migraine, should not provide false reassurance if there are red flags. For example, in one study of patients who were found to have primary or metastatic brain tumours, 77% presented with headaches phenotypically in keeping with tension-type headache.10 Some phenotypes always require further evaluation. These include the ‘thunderclap’ headache and trigeminal autonomic cephalgias, such as cluster headache.
Precipitating factors
The relationship of the headache to precipitating or provoking factors can provide a further clue to the underlying aetiology. A trigger, for example alcohol, may suggest a primary headache disorder such as migraine or cluster headache, whereas eating tyramine-containing food while taking a monoamine oxidase inhibitor suggests a secondary cause. Precipitating factors such as the valsalva manoeuvre or a change with posture are concerning because they may be due to posterior fossa pathology or raised intracranial pressure. Headaches can occur solely in ‘task-specific’ settings, such as exertion, intercourse or sleep, and the clinician should be alert to these factors in the patient’s history. Finally, new headaches that are ‘precipitated’ in the setting of pregnancy, postpartum, or ischaemic heart disease (cardiac cephalgia) may be suspicious for a secondary cause, and require specific consideration.
Pharmacology
Prescription and non-prescription medicines may precipitate or perpetuate headaches. As such, a detailed history noting the timing of new drugs and the pattern of headaches is required. The overuse of acute analgesia is a critical issue to be addressed in patients with a primary headache disorder. Medication-overuse headache may occur in over 70% of patients with a chronic daily headache.11 Patients who regularly use opioid or triptan analgesia for more than 10 days/month or simple analgesia for more than 15 days/month are at risk of increased neuronal hyperexcitability, peripheral and central sensitisation, and further potentiation of their headaches.11
Headache may also be an adverse reaction to a prescribed drug. The product information of many medicines lists headache as a possible adverse effect. Careful attention should therefore be paid to the temporal relationship when evaluating the relationship between a new drug and headaches. There are several classes of drugs that are well known to precipitate headaches. These include tacrolimus, interferon-beta, nitric oxide donors, phosphodiesterase inhibitors, some antidepressants and ciclosporin.12 Other drugs such as tetracyclines and vitamin A analogues may raise intracranial pressure, increasing the risk of idiopathic intracranial hypertension.13
Imaging of primary headaches
If imaging is indicated, MRI provides the most useful information. However, incidental findings are common and often result in patient anxiety, referral, and more imaging. Incidental findings on MRI occur in 2% of the general population.14 These findings include neoplasia in 0.7%, aneurysm in 0.35%, arachnoid cysts in 0.5%, Chiari I malformations in 0.24% and demyelination in 0.06%.14
In patients without red flags, there is not an absolute need for imaging in every patient. The decision to proceed to imaging should be made with consideration of the possibility of incidental findings and the overall clinical picture.
In selected scenarios, CT may be considered, depending on the question to be addressed by the imaging. It may be adequate at identifying subdural or epidural haematoma, skull fracture, sinus infection or subarachnoid haemorrhage (depending on the timing of the scan following the index event).
Migraine
Migraine, as the most common disabling primary headache disorder, is frequently investigated to exclude secondary pathology. In population studies, women with migraine are at an increased risk of white matter lesions (odds ratio 2.1, 95% CI 1.0–4.1) and hyperintense lesions in the brainstem (4.4% vs 0.7%).15 These findings are more common in patients with migraine aura, longer disease duration and higher attack frequency.15 The clinical significance of these lesions is still a matter of ongoing research, however they are not believed to be associated with cognitive changes.16,17 They can generally be differentiated from demyelination by an experienced reviewer, however serial imaging may be required.
Trigeminal autonomic cephalgias
The trigeminal autonomic cephalgias are a group of primary headache disorders characterised by unilateral (side-locked) headaches and ipsilateral cranial autonomic symptoms. All patients with a trigeminal autonomic cephalgia are required to have MRI primarily to exclude pathology in the pituitary region.
Ideally, the MRI would be of the brain and pituitary region, however it is not uncommon that just the brain is imaged. A review has now recommended that further dedicated pituitary imaging is only required if there are atypical features (older age, prolonged duration, higher frequency of attacks, bilateral attacks (rare, and should precipitate specialist review) or the absence of autonomic symptoms), pituitary-related symptoms, an abnormal examination or a poor response to treatment.18,19 Among 376 patients with cluster headache, the rate of pituitary adenomas was similar to the rate in the general population. Only patients with suggestive symptoms therefore require an additional MRI of the pituitary.19
Other primary headache disorders
Several primary headache disorders specifically require imaging to evaluate the patient for a possible secondary cause. Primary headache associated with sexual activity should be considered as attributed to reversible cerebral vasoconstriction syndrome until proven otherwise by angiographic study.20 Similarly, the diagnosis of primary exertional headache first requires evaluation for other causes of thunderclap headache.20 Primary cough headache should be evaluated with MRI particularly to check for posterior fossa pathology or structural malformations such as the Chiari I malformation.20,21
Imaging of secondary headaches
When investigating for a secondary headache, the clinical situation needs to be considered.
Thunderclap headache
Thunderclap headaches are sudden and severe. They are often due to cerebrovascular disorders, such as subarachnoid haemorrhage (see Box)22-24 and (Fig. 1)25. A non-contrast CT is frequently ordered for a patient presenting with a thunderclap headache. If performed within six hours of onset, CT has a sensitivity of 98.7% (CI 97.1–99.4%),26 however this drops considerably after six hours.27 A negative CT scan therefore may be falsely reassuring for ruling out subarachnoid haemorrhage, depending on the timing. CT is also likely to miss differential diagnoses that may be clinically relevant, including cerebral venous sinus thrombosis (see Fig. 2),28 reversible cerebral vasoconstriction syndrome, pituitary apoplexy or arterial dissection.
Box. Selected possible causes of thunderclap headache in order of frequency22-24.
Subarachnoid haemorrhage
Reversible cerebral vasoconstriction syndrome
Cerebral venous thrombosis
Other primary headache: primary thunderclap, cough, sexual and exertional headaches
Cervical artery dissection
Infection (e.g. sinusitis, meningitis, encephalitis)
Spontaneous intracranial hypotension
Stroke (haemorrhagic or ischaemic)
Posterior reversible encephalopathy syndrome
Pituitary apoplexy
Third ventricular colloid cyst
Sentinel headache (preceding a subarachnoid haemorrhage)
Retroclival haematoma
Fig. 1.
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Diffuse subarachnoid haemorrhage on a CT scan25
Fig. 2.
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CT venogram showing extensive venous thrombosis in the superior sagittal sinus28
Full evaluation for a patient with a thunderclap headache therefore includes non-contrast CT, with a lumbar puncture if the onset was more than six hours before, or the image is technically inadequate.27 When subarachnoid haemorrhage is excluded, there are many alternative diagnoses to consider (see Box).22 MRI with venography and angiography is recommended for investigating these causes.23
Disorders of intracranial pressure
Patients with a history or clinical examination suggestive of raised intracranial pressure always require further investigation. This is to exclude hydrocephalus, a space-occupying lesion and cerebral venous sinus thrombosis. Ideally, MRI of the brain and orbits and venography are performed.13 MRI features in keeping with raised intracranial pressure include flattening of the globe, optic nerve distension or tortuosity, empty sella, posterior displacement of the pituitary stalk, slit-like ventricles and an inferior position of the cerebellar tonsils (see Fig. 3).29,30 However, MRI findings are not pathognomonic, nor does their absence completely exclude idiopathic intracranial hypertension, so all patients with papilloedema should be referred for expert opinion. Conversely in patients with spontaneous intracranial hypotension, MRI may reveal diffuse pachymeningeal enhancement, descent of the tonsils (mimicking the Chiari I malformation), hygromas, or engorgement of the pituitary and the cerebral venous sinuses.31 These patients generally require expert evaluation and management.
Fig. 3.
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A case of idiopathic intracranial hypertension29
Conclusion
The overall rate of significant pathology found on MRI is relatively low, with incidental findings in approximately 2% of people. Investigation should therefore be guided by a thorough clinical assessment, to ensure the appropriate type and speed of investigation.
Footnotes
Conflicts of interest: Jason Ray has received compensation from the Pharmaceutical Society of Australia, sponsored by Viatris for educational material.
Elspeth Hutton has served on advisory boards for Sanofi-Genzyme, Novartis, Teva, Eli Lilly, Allergan, Lundbeck, been involved in clinical trials sponsored by Novartis, Teva, Xalud, Daewong and Novotech, and received payment for educational presentations from Allergan, Teva, Eli Lilly and Novartis.
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2018 AIME II Problems/Problem 6
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1 Problem
2 Solution
3 Video Solution
4 See Also:
Problem
A real number is chosen randomly and uniformly from the interval . The probability that the roots of the polynomial
are all real can be written in the form , where and are relatively prime positive integers. Find .
Solution
The polynomial we are given is rather complicated, so let's use Rational Root Theorem to see if it has any retinal roots. By Rational Root Theorem, are all possible rational roots. Upon plugging these roots into the polynomial, and make the polynomial equal 0 and thus, they are roots that we can factor out.
The polynomial becomes:
Since we know and are real numbers, we only need to focus on the quadratic.
Set the discriminant of the quadratic greater than or equal to 0, to ensure the remaining roots are real.
.
This simplifies to:
or
This means that the interval is the "bad" interval. The length of the interval where can be chosen from is 38 units long, while the bad interval is 2 units long. Therefore, the "good" interval is 36 units long.
~First
Video Solution
~Shreyas S
See Also:
2018 AIME II (Problems • Answer Key • Resources)
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Problem 5Followed by
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189414 | https://www.youtube.com/watch?v=qu-jyrwW6hw | Lecture 8: Quantum Harmonic Oscillator
MIT OpenCourseWare
5880000 subscribers
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Description
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Posted: 18 Jun 2014
MIT 8.04 Quantum Physics I, Spring 2013
View the complete course:
Instructor: Barton Zwiebach
In this lecture, Prof. Zwiebach covers the quantum mechanics of harmonic oscillators. He begins with qualitative discussion on bound state solutions and then moves on to the quantitative treatment of harmonic oscillators.
License: Creative Commons BY-NC-SA
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Transcript:
PRESENTER: The
following content is provided under a
Creative Commons license. Your support will help MIT
OpenCourseWare continue to offer high quality
educational resources for free. To make a donation or
view additional materials from hundreds of MIT courses,
visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we begin
with the harmonic oscillator. And before we get into
the harmonic oscillator, I want to touch on
a few concepts that have been mentioned in class
and just elaborate on them. It is the issue of nodes,
and how solutions look at, and why solutions have
more and more nodes, why the ground state has no nodes. This kind of stuff. So these are just a collection
of remarks and an argument for you to understand a
little more intuitively why these properties hold. So one first thing
I want to mention is, if you have a Schrodinger
equation for an energy eigenstate. Schrodinger equation for
an energy eigenstate. 1 minute, 28 seconds You have an equation of the
from minus h squared over 2m. d second dx squared psi of
x plus v of x psi of x equal e times psi of x. Now the issue of
this equation is that you're trying to solve for
two things at the same time. If you're looking at what
we call bound states, now what is a bound state? A bound state is something that
is not extended all that much. So a bound state will
be a wave function that goes to 0 as the absolute
value of x goes to infinity. So it's a probability function
that certainly doesn't extend all the way to infinity. It just collapses. It's normalizable. So these are bound
states, and you're looking for bound
states of this equation. And your difficulty is
that you don't know psi, and you don't know E either. So you have to solve a problem
in which, if you were thinking oh this is just a plain
differential equation, give me the value of E.
We know the potential, just calculate it. That's not the way it
works in quantum mechanics, because you need to have
normalizable solutions. So at the end of the day,
as will be very clear today, this E gets fixed. You cannot get
arbitrary values of E's. So I want to make a couple of
remarks about this equation. Is that there's this
thing that can't happen. Certainly, if v of x is a smooth
potential, then if you observer that the wave function
vanishes at some point, and the derivative
of the wave function vanishes at that same
point, these two things imply that psi of
x is identically 0. And therefore it
means that you really are not interested in that. That's not a solution of
the Schrodinger equation. Psi equals 0 is obviously solves
this, but it's not interesting. It doesn't represent
the particle. So what I claim here
is that, if it happens to be that you're
solving the Schrodinger problem with some
potential that is smooth, you can take derivatives of it. And then you encounter that
the wave function vanishes at some point, and its slope
vanishes at that same point. Then the wave function
vanishes completely. So you cannot have
a wave function, a psi of x that
does the following. Comes down here. It becomes an inflection
point and goes down. This is not allowed. If the wave function
vanishes at some point, then the wave function
is going to do this. It's going to hit at an angle,
because you cannot have that the wave function is 0 and needs
the derivative 0 at the same point. And the reason is simple. I'm not going to
prove it now here. It is that you have a second
order differential equation, and a second order
differential equation is completely determined
by knowing the function at the point and the the
derivative at the point. And if both are 0s, like
the most trivial kind of initial condition, the only
solution consistent with this is psi equals 0 everywhere. So this can't happen. And it's something good
for you to remember. If you have to do a
plot of a wave function, you should never have this. So this is what we call the
node in a wave function. 6 minutes, 4 seconds It's a place where the
wave function vanishes, and the derivative
of the wave function better not vanish at that point. So this is one claim that
it's not hard to prove, but we just don't try to do it. And there's another claim that
I want you to be aware of. That for bound states
in one dimension, in the kind of thing that we're
doing now in one dimension, no degeneracy is possible. What do I mean by that? You will never find
two bound states of a potential that
are different that have the same energy. It's just absolutely impossible. It's a very wonderful
result, but also I'm not going to prove it here. Maybe it will be given as an
exercise later in the course. And it's discussed
in 805 as well. But that's another statement
that is very important. There's no degeneracy. Now you've looked at
this simple potential, the square well infinite one. And how does it look? You have x going from 0 to a. And the potential is 0. From 0 to a is
infinite otherwise. The particle is bound
to stay inside the two walls of this potential. So in here we've plotted the
potential as a function of x. Here is x. And the wave functions are
things that you know already. Yes? AUDIENCE: Is it true if you have
two wells next to each other that there's still no degeneracy
if it's an infinite barrier? PROFESSOR: If there's
two wells and there's an infinite barrier
between them, it's like having two universes. So it's not really a
one dimensional problem. If you have an infinite
barrier like two worlds that can talk to each other. So yes, then you
would have degeneracy. It's like saying you can have
here one atom of hydrogen or something in one energy
level, here another one. They don't talk to each other. They're degenerate states. But in general, we're talking
about normal potentials that are preferably smooth. Most of these things are true
even if they're not smooth. But it's a little more delicate. But certainly two
potentials that are separated by
an infinite barrier is not part of what we
really want to consider. OK so these wave functions
start with m equals 0, 1, 2, 3, and psi ends of x are
square root of 2 over a sin n plus 1 pi x over a. Things that you've seen already. And En, the energies
are ever growing as a function of the integer
n that characterizes them. 2ma squared. And the thing that you
notice is that psi 0 has technically no nodes. That is to say these
wave functions have to varnish at the end, because
the potential becomes infinite, which means a particle
really can go through. The wave function
has to be continuous. There cannot be any wave
function to the left. So it has to vanish here. These things don't
count as nodes. It is like a bound state
has to vanish at infinity. And that's not what
we count the node. A node is somewhere
in the middle of the range of x where
the wave function vanishes. So this is the ground state. This is psi zero has no nodes. Psi one would be something
like that, has one node. And try the next ones. They have more and more nodes. So psi n has n nodes. And the interesting thing
is that this result actually is true for extremely
general potentials. You don't have to just
do the square well to see that the ground
state has no nodes. That first excited state was one
node, and so on and so forth. It's true in general. This is actually a
very nice result, but its difficult to prove. In fact, it's pretty
hard to prove. So there is a nice argument. Not 100% rigorous,
but thoroughly nice and really physical
that I'm going to present to you why
this result is true. So let's try to do that. So here is the general case. So I'm going to take
a smooth v of x. That will be of this kind. The potential, here is
x, and this potential is going to be like that. Smooth all over. And I will actually
have it that it actually goes to infinity as
x goes to infinity. Many of these things are
really not necessary, but it simplifies our life. OK, so here is a result
that it's known to be true. If this thing grows to
infinity, the potential never stops growing, you get
infinite number of bound states at fixed energies. One energy, two
energy, three energy, infinite number of bound states. Number of bound states. That's a fact. I will not try to prove it. We'll do it for the
harmonic oscillator. We'll see those infinite
number of states, but here we can't
prove it easily. Nevertheless, what
I want to argue for you is that these
states, as the first one, will have no nodes. The second state, the
first excited state, will have one node. Next will have two nodes,
three nodes, four nodes. We want to understand what
is this issue of the nodes. OK. You're not trying
to prove everything. But we're trying to
prove or understand something that is
very important. So how do we prove this? Or how do we understand that
the nodes-- so there will be an infinite number
of bound states, psi 0, psi 1, psi 2, up
to psi n, and it goes on. And psi n has n nodes. All right. So what I'm going to do, in
order to understand this, is I'm going to produce what we
will call screened potentials. Screened potentials. I'm going to select the
lowest point of the potential here for convenience. And I'm going to call it x
equals 0 is the lowest point. 14 minutes, 35 seconds And the screen potential
will have a parameter a. It's a potential
which is equal to v of x if the absolute
value of x is less than a. And its infinity if the absolute
value of x is greater than a. So I come here,
and I want to see this is this
potential, v of x, what is the screened
potential for sum a? Well, I wanted colored
chalk, but I don't have it. I go mark a here minus a. Here are the points
between x and a. Absolute value of x less than a. Throughout this region
the screened potential is the potential that you have. Nevertheless, for the
rest, its infinite. So the screened
potential is this thing. Is infinite there, and
it's here this thing. So it's just some potential. You take it a screen
and you just see one part of the potential,
and let it go to infinity. So that's a screen potential. So now what I'm going
to do is that I'm going to try to
argue that you could try to find the bound state
of the screen potential. Unless you remove the
screen, you will find, as you let a go to
infinity, you will find the bound states of
the original potential. It's reasonable
that that's true, because as you
remove the screen, you're letting more of
the potential be exposed, and more of the
potential be exposed. And the wave functions
eventually die, so as the time that you're very far away,
you affect the wave functions less and less. So that's the argument. We're going to try
to argue that we're going to look at
the bound states of the screened potentials and
see what happened, whether they tell us about the bound
states the original potential. So for this, I'm going
to begin with a screen potential in which
a goes to 0 and say that a is equal to
epsilon, very small. So what potential so do I have? A very tiny potential here
from epsilon to minus epsilon. Now I chose the original point
down here to be the minimum. So actually, the bottom part of
the potential is really flat. And if you take
epsilon going to 0, well, the potential
might do this, but really at the bottom for
sufficiently small epsilon, this is an infinite square
well with psis to epsilon. I chose the minimum so that you
don't get something like this. If it would be a
point with a slope, this would be an ugly thing. So let's choose the minimum. And we have the screen
potential here, and that's it. Now look what we do. We say all right, here
there is a ground state. Very tiny. Goes like that. Vanishes here. Vanishes there. And has no nodes. Very tiny. You know the two 0s are
very close to each other. And now I'm going
to try to increase the value of the screen a. So suppose we've
increased the screen, and now the potential is here. And now we have a finite screen. Here is the potential. And I look at the wave function. How it looks. Here is psi 0. This ground state psi 0. Well, since this thing in here,
the potential becomes infinite, the wave function
still must vanish here and still must vanish here. Now just for your
imagination, think of this. At this stage, it still more
or less looks like this. Maybe. Now I'm going to ask, as I
increase, can I produce a node? And look what's going to happen. So suppose it might happen
that, as you increase, suddenly you produce a node. So here's what I'm saying here. I'm going to show it here. Suppose up to this
point, there is no node. But then when I double
it, when I increase it to twice the size, when I go
to screen potential like that, suddenly there is a
node in the middle. So if there is a
node in the middle, one thing that could have
happened is that you have this. And now look what must
have happened then. As I stretch this, this slope
must have been going down, and down, and down, until
it flips to the other side to produce a node here. It could have
happened on this side, but it's the same, so
the argument is just done with this side. To produce a node you
could have done somehow the slope here must
have changed sine. But for that to happen
continuously, at some point the this slope must have been 0. But you cannot have
a 0 and 0 slope. So this thing can't
flip, can't do this. Another thing that
could have happened is that when we
are here already, maybe the wave function
looks like that. It doesn't flip at the
edges, but produces something like that. But the only way this
can happen continuously, and this potential is
changing continuously, is for this thing at
some intermediate stage, as you keep stretching the
screen, this sort of starts to produce a depression here. And at some point, to get
here it has to do this. But it can't do this either. It cannot vanish and have
derivative like that. So actually, as you
stretch the screen, there's no way to
produce a node. That property forbids it. So by the time you go and
take the screen to infinity, this wave function has no nodes. So that proves it that the
ground state has no nodes. You could call this
a physicist proof, which means-- not in
the pejorative way. It means that it's
reasonable, it's intuitive, and a mathematician working
hard could make it rigorous. A bad physicist proof is
one that is a little sloppy and no mathematician could
fix it and make it work. So I think this is a good
physics proof in that sense. Probably you can
construct a real proof, or based on this, a
very precise proof. Now look at excited states. Suppose you take now here
this screen very little, and now consider the third
excited state, psi three. I'm sorry, we'll call this psi
2 because it has two nodes. Well, maybe I should do psi 1. Psi 1. One node. Same thing. As you increase
it, there's no way to create another
node continuously. Because again, you have
to flip at the edges, or you have to
depress in the middle. So this one will evolve
to a wave function that will have one node in
the whole big potential. Now stayed does that
state have more energy than the ground state? Well, it certainly begins with
a small screen with more energy, because in the square well
psi 1 has more energy. And that energy should
be clear that it's not going to go below the
energy of the ground state. Why? Because if it went below the
energy of the ground state slowly, at some point for
some value of the screen, it would have the same
energy as the ground state. But no degeneracy is possible
in one dimensional problems. So that can't happen. Cannot have that. So it will always
stay a little higher. And therefore with one node you
will be a little higher energy. With two nodes will
be higher and higher. And that's it. That's the argument. Now, we've argued by this
continuous deformation process that this potential not
only has these bound states, but this is n nodes and En is
greater than En prime for n greater than n prime. So the more nodes,
the more energy. Pretty nice result,
and that's really all I wanted to say
about this problem. Are there any questions? Any? OK. So what we do now is
the harmonic oscillator. That's going to keep us busy
for the rest of today's lecture. It's a very interesting problem. And it's a most famous quantum
mechanics problem in a sense, because it happens to be useful
in many, many applications. If you have any
potential-- so what is the characteristic of
the harmonic oscillator? Harmonic oscillator. Oscillator. Well, the energy operator is p
squared over 2m plus, we write, one half m omega
squared x squared where omega is this
omega that you always think of angular velocity,
or angular frequency. It's more like
angular frequency. Omega has units of 1 over time. It's actually put 2pi over
the period of an oscillation. And this you know from
classical mechanics. If you have a harmonic
oscillator of this form, yeah, it actually oscillates
with this frequency. And E is the energy
operator, and this is the energy of the oscillator. So what defines an oscillator? It's something in which the
potential energy, this term is v of x. v of x is quadratic in x. That is a harmonic oscillator. Then you arrange the
constants to make sense. This has units of
energy, because this has units of length squared. 1 over time squared. Length over time
is velocity squared times mass is kinetic energy. So this term has
the units of energy. And you good with that. And why is this useful? Because actually in any sort
of arbitrary potential, rather general potential
at least, whenever you have a minimum where
the derivative vanishes, then the second derivative
need not vanish. Then it's a good
approximation to think of the potential at the minimum
as a quadratic potential. It fits the potential
nicely over a good region. And therefore when you have
two molecules with a bound or something oscillating,
there is a potential. It has a minimum at the
equilibrium position. And the oscillations
are governed by some harmonic oscillator. When you have photons
in space time traveling, there is a set of
harmonic oscillators that correspond to photons. Many, many applications. Endless amount of applications
for the harmonic oscillator. So we really want to
understand this system quantum mechanically. And what does that mean? Is that we really want
to calculate and solve the Schrodinger equation. This is our first step in
understanding the system. There's going to
be a lot of work to be done even once we have
the solutions of the Schrodinger equation. But the first thing
is to figure out what are the energy
eigenstates or the solutions of the Schrodinger
equation for this problem. So notice that here
in this problem there's an energy quantity. Remember, when you have
a harmontonian like that, and people say so what is
the ground state energy? Well, have to find the
ground state wave function. Have to do things. Give me an hour, I'll find it. And all that. But if you want an
approximate value, dimensional analysis will do it,
roughly what is it going to be. Well, with this constant how
do you produce an energy? Well, you remember
what Einstein did, and you know that h bar
omega has units of energy. So that's an energy associated
with Lagrangian energy like quantity. And we expect that
that energy is going to be the relevant energy. And in fact, we'll find
that the ground state energy is just one half of that. There's another quantity
that may be interesting. How about the length? How do you construct a
length from these quantities? Well, you can start
doing m omega h bar and put powers and struggle. I hate doing that. I always try to find some
way of doing it and avoiding that thing. So I know that
energies go like h over h squared over
m length squared. So I'm going to call
the length a quantity a. So ma squared. That has units of energy. And you should remember
that because energy is b squared over 2m,
and b by De Broglie is h bar over sub lamda. So h bar squared, lambda
squared, and m here, that's units of energy. So that's a length. On the other hand,
we have another way to construct an energy
is with this thing, m omega squared length squared. So that's also m omega
squared a squared. That's another energy. So from this equation I
find that a to the fourth is h squared over m
squared omega squared. And it's a little
complicated, so a squared is h bar over m omega. So that's a length. Length squared. I don't want to take
the square root. We can leave it
for a moment there. But that's important because
of that's a length scale. And if somebody would ask
you in the ground state, how far is this
particle oscillating, you would say probably
about a square root of this. Would be a natural answer
and probably about right. So OK, energy and
units is very important to begin your analysis. So what is the
Schrodinger equation? The Schrodinger
equation for this thing is going to be minus h
squared over 2m, d second psi, dx squared plus the potential,
one half m omega squared x squared psi is equal E psi. And the big problem is I don't
know psi and I don't know E. Now there's so many elegant
ways of solving the harmonic oscillator. You will see those next lecture. Allan Adams will be back here. But we all have to go
through once in your life through the direct, uninspired
method of solving it. Because most of the times
when you have a new problem, you will not come up with
a beautiful, elegant method to avoid solving the
differential equation. You will have to struggle with
the differential equation. So today we struggle with
the differential equation. We're going to just do it. And I'm going to do it slow
enough and in detail enough that I hope you
follow everything. I'll just keep a
couple of things, but it will be one line
computations that I will skip. So this equation is some sort
of fairly difficult thing. And it's complicated and
made fairly unpleasant by the presence of
all these constants. What kind of equation is that
with all these constants? They shouldn't be there,
all this constants, in fact. So this is the first step,
cleaning up the equation. We have to clean it up. Why? Because the nice
functions in life like y double prime is equal to
minus y have no units. The derivatives create no units. y has the same units of that,
and the solution is sine of x, where x must have no units,
because you cannot find the sine of one centimeter. So this thing, we should
have the same thing here. No units anywhere. So how can we do that? This is an absolutely
necessary first step. If you're going to be
carrying all these constants, you'll get nowhere. So we have to clean it up. So what I'm going
to try to see is that look, here is
psi, psi, and psi. So suppose I do the
following thing, that I will clean
up the right hand side by dividing by something
with units of energy. So I'm going to do
the following way. I'm going to divide all
by 1 over h bar omega. And this 2 I'm going
to multiply by 2. So multiply by 2
over h bar omega. So what do I achieve
with that first step? I achieve that
these 2s disappear. Well, that's not too bad. Not that great either, I think. But in the right hand side,
this has units of energy. And the right hand side will
not have units of energy. So what do we get here? So we get minus. The h becomes an h alone over--
the m disappears-- so m omega. The second psi the x squared. The 1/2 disappeared,
so m omega over h bar x squared psi equals
2 E over h bar omega psi. It looks actually
quite better already. Should agree with that. It looks a lot nicer Now
I can use a name for this. I want to call this
the dimensionless value of the energy. So a calligraphic e. It has no units. It's telling me if I find some
energy, that that energy really is this number, this pure
number is how many times bigger is e with respect
to h omega over 2. So I'll write this now as e psi. And look what I have. I have no units here. And I have a psi. And I have a psi. But things have
worked out already. Look, the same factor
here, h over m omega is upside down here. And this factor has
units of length squared. Length squared times d d
length squared has no units. And here's 1 over
length squared. 1 over length squared
times length squared. So things have worked out. And we can now
simply say x is going to be equal to au,
a new variable. This is going to be your new
variable for your differential equation in which is this thing. And then this
differential equation really has cleaned
up perfectly well. So how does it look now? Well, it's all gone
actually, because if you have x equals au, d dx by
chain rule is 1 over a d du. And to derivatives
this with respect to x it's 1 over a squared
times the d second du squared. And this thing is a squared. So actually you
cancel this factor. And when I write x
equals to au, you get an a squared times this. And a squared times this is 1. So your differential
equations has become minus the
second psi du squared, where u is a dimensionless
quantity, because this has units of length, this
has units of length. No units here. You have no units. So minus d second
du squared plus u squared psi is equal to e psi. 40 minutes, 12 seconds Much nicer. This is an equation
we can think about without being distracted
by this endless amount of little trivialities. But still we haven't
solved it, and how are we going to solve this equation? So let's again think
of what should happen. Somehow it should happen
that these e's get fixed. And there is some solution
just for some values of e's. It's not obvious at this stage
how that is going to happen. Yes? AUDIENCE: [INAUDIBLE]. PROFESSOR: Here for example,
let me do this term. h bar over m omega is minus,
from that equation, a squared. But dx squared is 1 over
a squared d du squared. So a squared cancels. And here the x is equal
a squared times u, so again cancels. OK so what is the problem here? The problem is that most likely
what is going to go wrong is that this solution for
arbitrary values of e's is going to diverge at
infinity, and you're never going to be able
to normalize it. So let's try to understand
how the solution looks as we go to infinity. So this is the first
thing you should do with an equation like that. How does this solution
look as u goes to infinity? Now we may not be able to solve
it exactly in that case either, but we're going to gain
insight into what's happening. So here it is. When u goes to infinity,
this term, whatever psi is, this term is much
bigger than that, because we're presumably
working with some fixed energy that we still don't
know what it is, but it's a fixed number and,
for you, sufficiently large. This is going to dominate. So the equation
that we're trying to solve as u goes to infinity,
the equation sort of becomes psi double prime-- prime
is for two derivatives-- is equal to u squared psi. 43 minutes, 2 seconds OK, so how do we get
an idea what solves this is not all that obvious. It's certainly not a power of u,
because when you differentiate the power of u, you
lower the degree rather than increase the degree. So what function increases
degree as you differentiate? It's not the trivial function. Cannot be a polynomial. If it could be
even a polynomial, if you take two derivatives,
it kind cannot be equal to x squared times a polynomial. It's sort of upside down. So if you think about
it for a little while, you don't have an
exact solution, but you would imagine
that something like this would do it, an e
to the u squared. Because an e to
the u squared, when you differentiate with respect
to us, you produce a u down. When you one derivative. When you take
another derivative, well, it's more complicated,
but one term you will produce another u down. So that probably is quite good. So let's try that. Let's try to see if we
have something like that. So I will try something. I'll try psi equals 2. I'm going to try
the following thing. e to the alpha u squared over
2 where alpha is a number. I don't know how much it is. Alpha is some number. Now could try this
alone, but I actually want to emphasize
to you that if this is the behavior
near infinity, it won't make any difference
if you put here, for example, something
like u to the power k. It will also be
roughly a solution. So let's see that. So for that I have
to differentiate. And let's see what we get. So we're trying to see how the
function behaves far, far away. You might say well
look, probably that alpha should be negative. But let's see what
the equation tells us before we put anything in there. So if I do psi prime,
you would get what? You would get one
term that would be alpha u times this u to
the k into the alpha u squared over 2. I differentiated
the exponential. I differentiated
the exponential. And then you would
get a term where you differentiate the power. So you get ku to the k
minus 1 into the alpha u squared over 2. If you take a second
derivative, well, I can differentiate the
exponential again, so I will get alpha
u now squared, because each derivative
of this exponent produces a factor of alpha u. u to the k into the
alpha u squared over 2. And a couple more terms that
they all have less powers of u, because this term
has u to the k plus-- already has u to the k plus 1. And this has u to the k minus 1. They differ by two powers of u. So for illustration,
please, if you want, do it. Three lines, you should skip
three lines in your notebook if you're taking notes
and get the following. No point in me doing
this algebra here. Alpha u squared over 2. Because actually it's
not all that important. Over alpha 1 over u squared
plus k minus 1 over alpha squared 1 over u to the fourth. That's all you get. Look, this is alpha
squared u squared times psi times these things. 1 plus 2 k plus 1 over
alpha 1 over u squared. So when u goes to
infinity, your solution works, because these
thing's are negligible. So you get a number
times u squared. That is the equation you are
trying to solve up there. And therefore, you get
that the equation if alpha squared is equal to 1. And that means and really that
alpha can be plus minus 1. And roughly this
solution near infinity, probably there's two solutions. This is a second order
differential equation, so even near infinity there
should be two solutions. So we expect as u goes
to infinity psi of u will be some constant A
times u to the k times e to the minus u squared over 2. That's where alpha
equal minus 1. Plus Bu to the k into the
plus u squared over 2. And what is k? Well, we don't know what is k. It seems to work for all k. That may seem a little
confusing now, but don't worry. We'll see other things
happening here very soon. So look at what has happened. We've identified that most
likely your wave function is going to look like
this at infinity. So we're going to want to
this part not to be present. So presumably we're going
to want a solution that just has this, because
this is normalizable. The integral of any power times
a Gaussian is convergence. So this can be normalized. The Gaussian falls so
fast that any power can be integrated
against a Gaussian. Any power however big
doesn't grow big enough to compensate a Gaussian. It's impossible to
compensate a Gaussian. So we hope for this. But we want to
translate what we've learned into some
technical advantage in solving the
differential equation, because, after all, we wanted
be insight how it looks far way, but we wanted to solve
the differential equation. So how can we use
this insight we now have to simplify the solution
of the differential equation? The idea is to change
variables a little bit. So write psi of u to be
equal to h of u times e to the minus u squared over 2. Now you're going to say
wait, what are you doing? Are you making an
approximation now that this is what is
going to look far away? Or what are you putting there? I'm not making
any approximation. I'm just saying whatever
pis is, it can always be written in this way. Why? Because if you have
a psi of u, you can write it as psi of u
over e to the minus u squared over 2 times e minus
u squared over 2. Very trivially this
can always be done. As long as we say
that h is arbitrary, there's nothing,
no constraint here. I have not assume
anything, nothing. I'm just hoping that I have a
differential equation for psi. That because this is a very
clever factor, the differential equation for h will be simpler. Because part of the dependence
has been taken over. So maybe h, for example, could
be now a polynomial solution, because this product
has been taken care. So the hope is that by
writing this equation it will become an
equation for h of u, and that equation
will be simpler. So will it be simpler? Well, here again this
is not difficult. You're supposed to plug
into equation one-- this is the equation
one-- plug into one. 53 minutes, 3 seconds I won't do it. It's three lines of
algebra to plug into one and calculate the
equation for h of u. You should do it. It's the kind of thing that
one should do at least once. So please do it. It's three, four lines. It's not long. But I'll just write the answer. So by the time you
substitute, of course, the e to the minus
u squared over 2 is going to cancel
from everywhere. It's very here. You just need to
take two derivatives, so it becomes a second
order differential equation. And indeed, it becomes
a tractable differential equation. The second h, du squared minus
2u dh du plus e minus 1 h equals 0. OK, that is our equation now. So now we face the problem
finally solving this equation. So before we start,
maybe there's some questions of what
we've done so far. Let's see. Any questions? Yes? AUDIENCE: Do you have
right there in the middle would be-- this equation
is linear, so can we just [INAUDIBLE] minus u squared over
2 and you stuck it to that u to the k. PROFESSOR: It's here? This thing? AUDIENCE: Yeah. Could you then just power series
what's going on at 0 with those u to the k terms [INAUDIBLE]? PROFESSOR: No. This is the behavior
as u goes to infinity. So I actually don't know
that the function near 0 is going to behave
like u to the k. We really don't know. It suggest to you that
maybe the solution is going to be near 0 u to
the k times some polynomial or something like that. But it's not that, because
this analysis was just done at infinity. So we really have no information
still what's going on near 0. Other questions? Yes? AUDIENCE: So is k some arbitrary
number or is it an integer? PROFESSOR: At this moment,
actually, it doesn't matter. Is that right? Doesn't matter. The analysis that we did here
suggests it could be anything. That's why I just didn't
put it into h or u. I didn't put it because
would be strange to put here a u to the k. I wouldn't know
what to make of it. So at this moment,
the best thing to say is we don't know what it is,
and maybe we'll understand it. And we will. In a few seconds, we'll
sort of see what's going on. OK, so how does one
solve this equation? Well, it's not a
trivial equation, again. But it can be solved
by polynomials, and we'll see that. But the way we
solve this equation is by a power series expansion. Now you could do
it by hand first, and I did it when
I was preparing the lecture yesterday. I said I'm going to
just write h of u equal a constant a0 plus
a1u plus a2u squared plus a3u cubed. And I plugged it in here. And I just did the
first few terms and start to see what happened. And I found after
a little thinking that a2 is determined
by a0, and a3 is determined by a1
once you substitute. It's not the obvious when you
look at this, but that happens. So when you face a
problem like that, don't go high power
to begin with. Just try a simple series
and see what happens. And you see a little pattern. And then you can do a more
sophisticated analysis. So what would be a more
sophisticated analysis? To write h of u
equal the sum from j equals 0 to infinity
aju to the j. Then if you take a
derivative, because we're going to need the
derivative, dh du would be the sum from
j equals 0 to infinity. j times aju to the j minus 1. You would say that
doesn't look very good because for j equals
0 you have 1 over u. That's crazy. But indeed for j equals 0,
the j here multiplies it and makes it 0. So this is OK. Now the term that we actually
need is minus 2u dh du. So here minus 2u dh du would
be equal to the sum from j equals 0 to infinity, and I
will have minus 2jaju to the j. The u makes this j minus 1 j,
and the constant went there. So here is so far h. Here is this other
term that we're going to need for the
differential equation. And then there's the
last term that we're going to need for the
differential equation, so I'm going to go here. So what do we get
for this last term. We'll have to take
a second derivative. So we'll take-- h
prime was there, so d second h du squared
will be the sum from j equals 0 of j times j minus
1 aju to the j minus 2. 59 minutes, 58 seconds Now you have to rewrite this
in order to make it tractable. You want everything
to have u to the j's. You don't want actually to
have u to the j minus 2. So the first thing
that you notice is that this sum actually begins
with 2, because for 0 and 1 it vanishes. So I can write j times j
minus 1 aj u to j minus 2. Like that. And then I can say let j
be equal to j prime plus 2. Look, j begins
with 2 in this sum. So if j is j prime plus 2,
j prime will begin with 0. So we've shifted the sum so it's
j prime equals 0 to infinity. And whenever I have a j I
must put j prime plus 2. So j prime plus 2. j prime plus 1 aj prime
plus 2 u to the j prime. Wherever I had j, I
put j prime plus 2. And finally you say j or
j prime is the same name, so let's call it j. j equals 0. j plus 2. j plus 1. aj plus 2 uj. 1 hour, 1 minute, 43 seconds So we got the series
expansion of everything, so we just plug into the
differential equation. So where is the
differential equation? It's here. So I'll plug it in. Let's see what we get. We'll get some from j
equals 0 to infinity. Let's see the second
derivative is here. j plus 2 times j plus 1 aj
plus 2 uj, so I'll put it here. So that's this second
derivative term. Now this one. It's easy. Minus 2j aj and the uj is there. So minus 2jaj. Last term is just e
minus 1, because it's the function this
times aj as well. That's h. And look, this whole
thing must be 0. So what you learn is that
this coefficient must be 0 for every value of j. Now it's possible to--
here is aj and aj, so it's actually
one single thing. Let me write it here. j plus 2 times j plus
1 aj plus 2 minus 2j plus 1 minus e aj uj. I think I got it right. Yes. And this is the same sum. And now, OK, it's a lot of
work, but we're getting there. This must be 0. So actually that solves for
aj plus 2 in terms of aj. What I had told you that you
can notice in two minutes if you try it a little. That a2 seems to be
determined by a0. And a3 seems to be
determined by a2. So this is saying
that aj plus 2 is given by 2j plus 1 minus e
over j plus 2 j plus 1 aj. 1 hour, 4 minutes, 36 seconds A very nice recursive relation. So indeed, if you
put the value of a0, it will determine for you a2,
a4, a6, a8, all the even ones. If you put the value of a1, it
will determine for you a3, a5. So a solution is determined by
you telling me how much is a0, and telling me how much is a1. Two constants, two numbers. That's what you expect from
a second order differential equation. The value of the
function at the point, the derivative at a point. In fact, you are
looking at a0 and a1 as the two constants that
will determine a solution. And this is the value of h at 0. This is the
derivative of h at 0. So we can now write
the following facts about the solution
that we have found. So what do we know? That solutions fixed
by giving a0 and a1. That correspond to the
value of the function at 0 and the derivative
of the function at 0. And this gives one solution. Once you fix a0, you get a2, a4. And this is an even
solution, because it has only even powers. And then from a1, you fixed
a3, a5, all the other ones with an odd solution. OK. Well, we solve the
differential equation, which is really,
in a sense, bad, because we were expecting
that we can only solve it for some values of the energy. Moreover, you have a0,
you get a2, a4, a6, a8. This will go on forever
and not terminate. And then it will be an
infinite polynomial that grows up and doesn't
ever decline, which is sort of
contradictory with the idea that we had before
that near infinity the function was going to be
some power, some fixed power, times this exponential. So this is what we're looking
for, this h function now. It doesn't look
like a fixed power. It looks like it goes forever. So let's see what
happens eventually when the coefficient, the
value of the j index is large. For large j. aj plus 2 is roughly equal
to, for large a, whatever the energy is, sufficiently
large, the most important here is the 2j here, the j and the j. So you get 2 over j aj. So roughly for large j,
it behaves like that. And now you have to ask
yourself the question, if you have a power series
expansion whose coefficients behave like that, how
badly is it at infinity? How about is it? You know it's the
power series expansion because your h was all
these coefficients. And suppose they
behave like that. They grow in that way
or decay in this way, because they're decaying. Is this a solution
that's going to blow up? Or is it not going to blow up? And here comes an
important thing. This is pretty bad
behavior, actually. It's pretty awful behavior. So let's see that. That's pretty bad. How do we see that? Well you could do it
in different ways, depending on whether
you want to derive that this is a bad
behavior or guess it. I'm going to guess something. I'm going to look at how
does e to the u squared behave as a power series. Well, you know as a
power series exponential is 1 over n u squared to the n. Here's n factorial. n equals 0 to infinity. Now these two n's, u to the
2n, these are all even powers. So I'm going to
change letters here, and I'm going to work with j
from 0, 2, 4, over the evens. So I will write u to the j here. And that this correct, because
you produce u to the 0, u to the 2, u to the
fourth, these things. And j is really 2n,
so here you will have one over j
over 2 factorial. Now you might say, j over
2, isn't that a fraction? No, it's not a fraction,
because j is even. So this is a nice factorial. Now this is the
coefficient, cj u to the j. And let's see how this
coefficients vary. So this cj is 1 over
j over 2 factorial. What is cj plus 2 over cj? Which is the analogue
of this thing. Well, this would be 1 over
j plus 2 over 2 factorial. And here is up there,
so j over 2 factorial. Well, this has one more
factor in the denominator than the numerator. So this is roughly
one over j over 2 plus 1, the last value of this. This integer is just
one bigger than that. Now if j is large,
this is roughly 1 over j over 2,
which is 2 over j. Oh, exactly that stuff. So it's pretty bad. If this series
goes on forever, it will diverge like
e to the u squared. And your h will be like e to the
u squared with e to the minus u squared over 2 is going
to be like e to the plus. u squared over 2 is going
to go and behave this one. So it's going to do
exactly the wrong thing. If this series
doesn't terminate, we have not succeeded. But happily, the
series may terminate, because the j's are integers. So maybe for some energies
that are integers, it terminates, and
that's a solution. The only way to get a solution
is if the series terminates. The only way it can
terminate is that the e is some odd number over here. And that will solve the thing. So we actually need to do this. This shows the energy. You found why it's quantized. So let's do it then. We're really done
with this in a sense. This is the most important
point of the lecture, is that the series
must terminate, otherwise it will
blow up horrendously. If it terminates
as a polynomial, then everything is good. So to terminate you can choose
2j plus 1 minus e to be 0. This will make aj
plus 2 equal to 0. And your solution, your
h of u, will begin. aj will be the last
one that is non-zero, so it will be aj
times u to the j, and it will go down like aj
minus 2 u to the j minus 2. It will go down in steps of
2, because this recursion is always by steps of two. So that's it. That's going to be the solution
where these coefficients are going to be fixed by
the recursive relation, and we have this. Now most people
here call j equal n. So let's call it n. And then we have 2n
plus 1 minus e equals 0. And h of u would be an u to
the n plus all these things. That's the h. The full solution is h
times e to the minus u squared over 2 as we will see. But recall what e was. e here is 2n plus 1. But he was the true energy
divided by h omega over two. That was long ago. It's gone. Long gone. So what have you
found therefore? That the energy,
that' we'll call en, the energy of the
nth solution is going to be h omega
over 2 2n plus 1. So it's actually h omega, and
people write it n plus 1/2. Very famous result. The nth level of the harmonic
oscillator has this energy. And moreover, these
objects, people choose these-- you
see the constants are related by steps of two. So just like you could start
with a0, or a1 and go up, you can go down. People call these functions
Hermite functions. And they fix the notation so
that this an is 2 to the n. They like it. It's a nice normalization. So actually h of
n is what we call the Hermite function of u sub n. And it goes like 2 to the
n u to the n plus order u to the n minus 2 plus
n minus 4, and it goes on and on like that. OK, a couple things
and we're done. Just for reference,
the Hermite polynomial, if you're interested
in it, is the one that solves this equation. And the Hermite
sub n corresponds to e sub n, which is 2n plus 1. So the Hermite solution
from that the equation is that the Hermite polynomial
satisfies this minus 2u d Hn du plus 2n. Because en is 2n plus 1. So it's 2n Hn equals 0. That's the equation for
the Hermite polynomial, and interesting thing to know. Actually, if you want to
generate the efficiently the Hermite polynomials, there's
something called the generating function. e to the minus z
squared plus 2zu. If you expand it in
a power series of z, it actually gives you
n equals 0 to infinity. If it's a power series of z,
it will be some z to the n's. You can put a factor here
n, and here is Hn of u. So you can use your
mathematic program and expand this in powers of z. Collect the various powers of
u that appear with z to the n, and that's Hn It's the most
efficient way of generating Hn And moreover, if you want
to play in mathematics, you can show that
such definition of Hn satisfies this equation. So it produces the solution. So what have we found? Our end result is the following. Let me finish with that here. We had this potential, and the
first energy level is called E0 and has energy h omega over 2. The next energy is E1. It has 3/2 h omega. Next one is E2 5/2 h omega. This polynomial is
nth degree polynomial. So it has n zeros,
therefore n nodes. So these wave functions will
have the right number of nodes. E0, the psi 0,
will have no nodes. When you have psi 0, the
Hn becomes a number for n equals zero. And the whole solution
is the exponential of u squared over 2. The whole solution, in
fact, is, as we wrote, psi n Hn of u e to the
minus u squared over 2. In plain English,
if you use an x, it will be Hn u with x over
that constant a we had. And you have minus x
squared over 2a squared. Those are your eigenfunctions. These are the solutions. Discrete spectrum, evenly
spaced, the nicest spectrum possible. All the nodes are there. You will solve this in a
more clever way next time. [APPLAUSE] |
189415 | https://www.mathopenref.com/lawofcosinesproof.html | Math Open Reference
Home Contact About Subject Index
Proof of the Law of Cosines
The Law of Cosines states that for any triangle ABC, with sides a,b,c For more see Law of Cosines.
In the right triangle BCD, from the definition of cosine: or,
Subtracting this from the side b, we see that
In the triangle BCD, from the definition of sine: or
In the triangle ADB, applying the Pythagorean Theorem
Substituting for BD and DA from (2) and (3)
Multiplying out the parentheses:
Rearranging the terms:
Factoring out a2
Looking at the terms in the parentheses above, recall that this is one of the trig identities, which states that (See Pythagorean trig identites.) So the terms in the parentheses can be removed since mutiplying a2 by one leaves it unchanged.
- Q.E.D
(C) 2011 Copyright Math Open Reference. All rights reserved |
189416 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3711056/ | IL-1 and T Helper Immune Responses - PMC
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Front Immunol
. 2013 Jul 15;4:182. doi: 10.3389/fimmu.2013.00182
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IL-1 and T Helper Immune Responses
Veronica Santarlasci
Veronica Santarlasci
1 Department of Experimental and Clinical Medicine, University of Florence, Florence, Italy
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1, Lorenzo Cosmi
Lorenzo Cosmi
1 Department of Experimental and Clinical Medicine, University of Florence, Florence, Italy
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1, Laura Maggi
Laura Maggi
1 Department of Experimental and Clinical Medicine, University of Florence, Florence, Italy
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1, Francesco Liotta
Francesco Liotta
1 Department of Experimental and Clinical Medicine, University of Florence, Florence, Italy
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1, Francesco Annunziato
Francesco Annunziato
1 Department of Experimental and Clinical Medicine, University of Florence, Florence, Italy
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1 Department of Experimental and Clinical Medicine, University of Florence, Florence, Italy
Edited by: Alberto Mantovani, University of Milan, Italy
Reviewed by: Massimo Gadina, National Institutes of Health, USA; Hideki Ogura, Osaka University, Japan
✉
Correspondence: Francesco Annunziato, Department of Experimental and Clinical Medicine, University of Florence, V.le Pieraccini 6, Florence 50134, Italy e-mail: f.annunziato@dmi.unifi.it, francesco.annunziato@unifi.it
This article was submitted to Frontiers in Inflammation, a specialty of Frontiers in Immunology.
Received 2013 Apr 9; Accepted 2013 Jun 24; Prepublished 2013 May 12; Collection date 2013.
Copyright © 2013 Santarlasci, Cosmi, Maggi, Liotta and Annunziato.
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PMCID: PMC3711056 PMID: 23874332
Abstract
CD4 T cells play a critical role in mediating adaptive immunity to a variety of pathogens as well as in tumor immunity. If not adequately regulated, CD4 T cells can be also involved in autoimmunity, asthma, and allergic responses. During TCR activation in a particular cytokine milieu, naïve CD4 T cells may differentiate into one of several lineages of T helper (Th) cells, including Th1, Th2, and Th17, as defined by their pattern of cytokine production and function. IL-1, the prototypic proinflammatory cytokine, has been shown to influence growth and differentiation of immunocompetent lymphocytes. The differential expression of IL-1RI on human CD4 T cell subsets confers distinct capacities to acquire specific effector functions. In this review, we summarize the role of IL-1 on CD4 T cells, in terms of differentiation, activation, and maintenance or survival.
Keywords: Th17 cells, Th2 cells, Th1 non-classic, IL-1RI and T cells, IL-1 and T cells
T Helper Cell Subsets
CD4+ T helper (Th) lymphocytes represent a heterogeneous population of cells that play an essential role in adaptive immunity. These cells include effector cells, which are devoted to protection against pathogens, and regulatory T cells (T regs), which protect against effector responses to autoantigens, and also to exogenous antigens when they become dangerous for the host. The term Th derived from the observation that these cells were critical for helping B cells to produce antibodies in the primary response (humoral immunity). CD4+ T cells were also found to be responsible for the so called cell-mediated immunity, or delayed-type hypersensitivity, which is characterized by reaction involving activation of macrophages. The distinct protective function of different effector CD4+ T lymphocytes, enables the best type of response according to the nature of the invading microorganism. Th1 cells produce high levels of IFN-γ and are responsible for both phagocyte activation and the production of opsonizing and complement-fixing antibodies, thus playing an important role in protection against intracellular pathogens. Th2 cells produce IL-4, IL-5, IL-9, and IL-13. Th2 cells, because of their ability to produce IL-4 and IL-13, can induce IgE class switching by B lymphocytes, enabling in this way mast cells and basophils sensitization and possible subsequent activation. In addition, IL-5 produced by Th2 cells has specific activity on differentiation, activation, and recruitment of eosinophils. Finally, IL-9 has an important role in the proliferation and accumulation of mast cells and the induction of mucus production by cells of the respiratory tract and the gut (1–4). Because of all the mentioned characteristics, Th2 cells are effective in the protection against helminthes (5). The more recently discovered Th17 subset is characterized by the production of IL-17A, IL-17F, IL-8, IL-21, and IL-22. Th17 cells play a critical role in the recruitment, activation, and migration of neutrophil granulocytes, both directly, through IL-8 production (6) and indirectly, by inducing, via IL-17, the production of colony stimulatory factors (CSF) and CXCL8 (7) in tissue resident cells. Because of their unique ability to recruit neutrophils, the main protective function of Th17 cells appears to be the clearance of extracellular pathogens, including fungi (8, 9). The distinctive features of the various CD4 effector/regulatory subpopulations are determined largely by the set of transcription factors they express and the genes they transcribe. The induction of the distinctive patterns of gene expression is dependent on the milieu of microenvironmental cytokines during the antigen-mediated activation of a naïve T cell.
In addition to their protective functions against invading pathogens, Th1, Th2, and Th17 cells contribute to the development of human disorders: Th1 and Th17 cells have been thought to be involved in the pathogenesis of organ-specific autoimmune diseases, as well as other chronic inflammatory disorders, such as Crohn’s disease (CD), psoriasis, and rheumatoid arthritis (RA); Th2 cells certainly play a central role in the development of allergic disorders (10–12).
IL-1 Family of Cytokines
Although the original IL-1 family comprised only IL-1α and IL-1β, the IL-1 family has expanded considerably in the last few years and nowadays comprises 11 members (IL-1α, IL-1β, IL-1RA, IL-18, IL-33, IL-36α, IL-36β, IL-36γ, IL-36RA, IL-37, and IL-38) which have similar gene structure. All these cytokines use heterodimeric receptors for signaling. IL-1 (α and β) binds to IL-1RI, IL-33 to T1/ST2 and IL-36 (α, β, and γ) to IL-1Rrp2, and subsequently, they recruit the same coreceptor IL-1R accessory protein (IL-1RAcP). IL-18 signals through the IL-18Rα associated to the coreceptor IL-18Rβ. On receptor binding, all IL-1 family cytokines activate similar intracellular signals. The signal is initiated with recruitment of the adaptor protein MyD88 to the Toll-IL-1 receptor (TIR) domain. Several kinases are then phosphorylated, NF-κB translocates to the nucleus, and the expression of a large portfolio of inflammatory genes takes place (13) IL-1 receptor antagonist (IL-1RA) and IL-36RA, act as natural inhibitors for the biologic activities of IL-1 (α and β) and of IL-36 (α, β, and γ), respectively.
In addition to signaling receptors, also decoy receptors and inhibitory receptors for IL-1 cytokine family members had been described. One of these is the IL-1RII that does not signal because it lacks the TIR cytoplasmic domain. IL-1RII binds IL-1β with higher affinity than IL-1RI but does not transduce a signal, acting therefore as a decoy receptor (14). The IL-1RII–IL-1β complex is able to bind to IL-1RAcP, in this way the decoy receptor also serves to sequester the accessory receptor from participating in IL-1 signaling from the IL-1RI. Other receptors characterized by the ability to deliver inhibitory signals in response to IL-1 family members are SIGIRR and IL-1RAcPb (15, 16).
Each member of the IL-1 family, IL-1Ra is the unique exception, is first synthesized as a precursor without a clear signal peptide for processing and secretion, and none are found in the Golgi. IL-1α and IL-33 are similar in that their precursor forms can bind to their respective receptor and trigger signal transduction. The precursor forms of IL-18 and IL-1β do not bind their respective receptors, are not active, and require cleavage by either intracellular caspase-1 or extracellular neutrophilic proteases (17).
Since the discovery of this family of cytokines their “immunostimulant activity” was evident, but wasn’t that clear on which and how this cytokine could interact on different T lymphocytes. The present review will focalize exclusively on two members of the IL-1 family of cytokines, IL-1α and IL-1β, and it will discuss, on the basis of the last 30 years literature, their involvement in the differentiation, activation, and maintenance or survival of the different Th cell subsets.
The IL-1α precursor is produced constitutively in all epithelial cells and fibroblasts and can be also found on the surface of several cells, particularly on monocytes and B lymphocytes. The primary sources of IL-1β are the blood monocytes, tissue macrophages, and dendritic cells; B lymphocytes and NK cells also produce IL-1β (18).
IL-1 and the Th2 Immune Response
In the 80s the first studies taking in account the direct effects of the prototypic proinflammatory cytokine, IL-1, on T lymphocytes were published (19–21). These studies indicated IL-1 as a cytokines possibly influencing growth and differentiation of immunocompetent lymphocytes. IL-1 costimulatory role for T cells was at that time attributed to two complimentary effects: (1) IL-1 can enhance transcription and secretion of the T cell growth factor IL-2; (2) IL-1 stimulates the expression of the membrane receptors for IL-2. The combination of these complimentary effects of IL-1 on T cells could explain its T-cell stimulating function. In 1988, Lichtman and colleagues (20) were the first to evaluate IL-1α costimulatory function on, at that time recently discovered, murine Th1 and Th2 cells. In this study, the authors demonstrated that only Th2 cells express high affinity receptors for IL-1 and that, accordingly, only this cell subset proliferate in response to IL-1α, whereas Th1 cells do not.
Few years later, Taylor-Robinson and colleagues (22), examining the expression of selected interleukin receptors by cloned CD4+ T cells specific for the murine malaria parasite Plasmodium chabaudi representative of the Th1 and Th2 subsets, found that while IL-1RI was constantly expressed by Th2 clones, its expression by the Th1 clones was either negligible or undetectable. Since then, the scientific community assumed that just the Th2 cell subset expresses IL-1RI, but lacking to confirm this data on human cells. Considering the pathogenic role of Th2 cells in allergic diseases, IL-1 activity was therefore investigated in several murine models of allergy. Nakae and colleagues (23) demonstrated that the ovalbumin-induced airway hypersensitivity response (AHR) in IL-1α/β-double deficient mice was significantly reduced when compared to wild type mice, whereas the response seen in IL-1RA-deficient mice was profoundly exacerbated, suggesting that IL-1 is required for Th2 cell activation during AHR. Accordingly, the authors showed that ovalbumin-specific IL-4 and IL-5 production by T cells, and IgG1 and IgE production by B cells in IL-1α/β-double deficient mice were markedly reduced compared with these responses in wild type mice. Similar results were obtained by Schmitz and colleagues (24) that investigated the role of IL-1 in models of allergic asthma using IL-1R1-deficient mice. The authors showed that in a model of mild asthma, based on repeated sensitization of mice with low doses of ovalbumin in the absence of any adjuvant, the pulmonary eosinophilic inflammation, the goblet cell hyperplasia, as well as antibody responses including IgG, IgE, and IgA were strongly reduced in IL-1R1-deficient as compared to wild type mice. In contrast, sensitization of mice in the presence of alum adjuvant, a more severe asthma model, rendered the IL-1 pathway dispensable for the development of pulmonary allergic Th2 responses. The role of IL-1 in sustaining the Th2 immune responses comes also from animal models of parasites infestation. Helmby and Grencis (25) showed that Th2 response-associated resistance to gastrointestinal nematode Trichuris muris is mediated was dependent on the presence of IL-1α and IL-1β. Indeed, they demonstrated that both IL-1α- and IL-1β-deficient mice were susceptible to chronic Trichuris muris infection and that the inability to eliminate the worms was associated with a defect in the development of a Th2 response in the mesenteric lymph nodes. Opposite data were obtained by Satoskar and colleagues (26) that found significantly increased IL-4 and IL-10 production by lymph node cells from Leishmania major infected IL-1RI-deficient mice when compared to wild type mice.
These findings are contradictory to the one showed by Helmby and Grencis, possibly because of differences in the type of cytokine/receptor KO utilized, the choice of experimental model, as well as the genetic background of the host.
The first description of IL-1RI expression and modulation on human T cells, however without distinguishing on which particular subsets, was made by Shirakawa and colleagues (21). Few years later Manetti and colleagues (27) analyzed the effects exerted by IL-1α on the growth and differentiation of human Th1 and Th2 cells. In this study, the authors showed that neither IL-1α nor the IL-1RA had detectable activity toward the antigen- or anti-CD3 antibody-induced proliferative response of already established Th1 or Th2 clones. However, allergen-specific T-cell lines, derived in the presence of anti-IL-1α Ab or IL-1RA, exhibited reduced and increased ability to produce IL-4 and IFN-γ, respectively. These data suggested that IL-1α was not required for the growth of already established human Th1 or Th2 clones, but it played a critical role in the development of Th2 cells, whereas Th1 development was unaffected. In light of the above mentioned data, the lack of an effect, described by Manetti, in terms of proliferative response to IL-1α on already established human Th2 cells and the reduction in the Th2 polarization in IL-1α neutralizing conditions, could be interpreted today as an indirect effect on non-Th2 subsets that are expanded in the presence of IL-1α (27).
The first data relative to the expression of IL-1RI on a particular Th cell subsets came out only in 2010. Indeed, Cosmi and colleagues (28) demonstrated the lack of IL-1-RI mRNA on human established Th2 clones, while Wang and colleagues (29) observed a slight membrane expression of the receptor on freshly enriched human CRTH2 positive cells, being CRTH2 a surface molecule selectively expressed by human Th2 cells (30, 31).
In any case neither Cosmi nor Wang analyzed the ability of the human Th2 cells to respond to IL-1, i.e., monitoring the activation signal transduction molecules downstream the IL-1RI, therefore it is not known if the receptor has functional activity.
Since the activity of IL-1 on human Th2 cells is not unequivocally established, caution is needed in considering this cytokine as potential new therapeutic target for human bronchial asthma as some studies suggest (32).
Both Mice and Humans Th17 Express IL-1RI and are Modulated by Its Signaling
For the in vitro differentiation of naïve T cell into Th17 cells in the mouse, the scientific community was fairly unanimous in defining TGF-β and IL-6, as the key cytokines. Yet in 2006 Veldhoen and colleagues (33) described a synergistic role of IL-1β and TNF-α in the Th17 differentiation initiated by TGFβ and IL-6 and in the same year Sutton and colleagues (34) described a lower induction of Th17 cells in IL-1RI-deficient mice, than in wild type mice and also a resistance to experimental autoimmune encephalomyelitis (EAE). Interestingly, in models of autoimmune diseases, such as EAE and collagen-induced arthritis (CIA), the induction of the Th17 cells require the presence of a mixture of killed Mycobacterium tuberculosis, that has been recently discovered to induce, via dectin-1 and TLR4, the release of IL-1β (35). Therefore it’s possible to speculate that IL-1β plays a pivotal role in Th17 induction. This hypothesis is confirmed observing mice deficient in caspase1-enzyme that cleaves IL-1β precursor into a mature form-, where EAE is markedly attenuated. On the other hand when IL-1β activity is unopposed, like in IL-1RA knot-out mice (C57BL/6J), causes autoimmunity and arthritis that closely resembled RA in humans (36). These data have been confirmed, observing that mice specifically deficient in endogenous IL-1RA developed an increased Th17 response, and CIA appears to be because of unrestrained IL-1 activity (37, 38), which may in turn, contribute to a more severe form of CIA. In keeping with these observations Coccia and colleagues (39) showed that IL-1β promotes intestinal inflammation by augmenting the recruitment of granulocytes and the accumulation and activation of innate lymphoid cells (ILCs) in a model of in Helicobacter hepaticus-triggered intestinal inflammation. In particular, the observation that synergistic interactions between IL-1β and IL-23 sustain innate and adaptive inflammatory responses in the gut, promoting intestinal pathology, suggests that targeting IL-1β may represent a useful therapeutic approach in IBD. To further support the possibility of IL-1β play significant role not only in the induction of Th17 phenotype, but also in their expansion and homeostatic maintenance, Sutton and colleagues reported that IL-1β can promote Th17 expansion and cytokine production in vitro even in the absence of TCR stimulation. The mechanisms underlying these in vivo phenomena became more clear since it has been described IL-1RI expression first on IL-17+CD4+ T cells of SKG mice (that spontaneously develop arthritis) (40), and later on, by the demonstration that IL-1 signaling is required for the upregulation of IRF4 and RORC (two fundamental Th17 transcription factors) during the early Th17 lineage programing and to sustain its differentiation (41).
When the differentiation process, from naïve to effector cells, was analyzed in humans, a number of evidence showed soon that a predominant role was led by IL-1β, alone or in combination with other cytokines. Annunziato and colleagues (42) described the expression of IL-1RI on Th17 cell clones derived from peripheral blood (PB) and gut specimens of Crohn’s affected patients, and, accordingly, Acosta-Rodriguez and colleagues (43) was able to induce a Th17 phenotype by culturing naïve T cells in presence of IL-1β and IL-6. In particular, IL-1β was sufficient to induce the expression of RORC and production of both IL-17 and IFN-γ. Cosmi and colleagues reported that all humans IL-17-producing cells originate from CD161+ naïve CD4+ T cells of umbilical cord blood, as well as of the postnatal thymus, in response to the combined activity of IL-1β and IL-23. Confirmation that IL-1β is important in the differentiation of Th17 cells comes from studies conducted on the CD161 positive fraction of naïve CD3+CD4+ cell from the thymus as well from the cord blood of newborns where the combination of IL-1β and IL-23 allows the Th17 polarization (44, 45).
These data were recently confirmed by Lee and colleagues (46) who, demonstrated the upregulation of IL-1RI on naïve cord blood CD4+ T cell after exposure to common γ-chain cytokines (IL-7, IL-15) plus TGF-β and establishing that such condition promote the differentiation into Th17 cells upon TCR triggering and IL-1β stimulation, which is enhanced by IL-23 and blocked by IL-1RA. The same upregulation of IL-1RI was described by Raffin and colleagues (47) on PB naïve CD4+ T cell in the presence of the combination of IL-2, IL-1β, IL-23, and TGF-β.
In human disease, several clinical studies support a role for IL-1β secreted by colon lamina propria monocytes in disease activity during IBD. IL-1β levels in the colon during active phase of IBD correlated with disease activity and high levels of IL-1β were associated with active lesions (48), suggesting an important role of this cytokine in promoting localized inflammation.
A human example of IL-1β dysregulation is the heterozygous mutation of NLRP3 gene (encoding for the inflammasome component, cryopyrin) that leads to an abnormal secretion of IL-1β by monocytes, leading to different clinical inflammatory manifestations, but all hampered by inhibition of IL-1β. Indeed Lasiglie (49) analyzing 11 patients carrying this mutation (Cryopyrin-associated periodic syndromes, CAPS) observed a skewed Th17 phenotype in PB lymphocytes, as well as an increased production of IL-1β and IL-23 by monocyte-derived dendritic cells. The anti-IL-1β treatment in vivo reduce the secretion of IL-1β by monocytes and both IL-1β and IL-23 by monocyte-derived dendritic cells in vitro. The observation that IL-1RA treatment leads to a down modulation of IL-23 in PBMC of celiac patients may support the hypothesis that the over expression of IL-23 in CAPS patients is actually related to an IL-1β dependent mechanism, likely associated to the activation of the inflammasome (50). The second arm of “IL-1 system” has been enlighted in 2009 with the identification (51) of the cause of a human autoinflammatory syndrome of skin and bone in a homozygous truncating mutations in the IL-1RN gene that leads to the lack of secretion of this receptor antagonist (IL-1RA), and as a consequence in an unopposed IL-1 signaling. Increased number of IL-17 secreting cells was found in biopsy samples of inflamed skin from patients with deficiency of the IL-1RA (DIRA patients); as expected treatment with Anakinra, a recombinant IL-1RA, leaded to symptoms remission.
We can conclude that many evidence in mice and humans support the concept that IL-1β, acting concurrently with other cytokines, is a key cytokine in the early phases of Th17 development, acting through its specific receptor expressed already by the naïve CD4+ Th17’s precursor.
Moreover, even if IL-1β plays an important role in combating the invading pathogen as part of the innate immune response, its dysregulation is responsible for a number of autoinflammatory disorders in which Th17 cells are involved. As a consequence, its inhibition has proved therapeutically beneficial in the treatment of a spectrum of serious, yet relatively rare, heritable pathologies. This raises the possibility that anti-IL-1 therapeutics may have broader applications than previously believed, and may be utilized across diverse disease states that are linked insidiously through heightened inflammasome activity.
Th1 “Non-Classic” Human T Cell Express IL-1RI: A New Point of View
Because of the scientific community had assumed the absence of IL-1RI expression by Th1 cells, very few works subsequently investigated the possible expression and function of this receptor on Th cell subset. Ben-Sasson and colleagues (52) were the first that described an activation effect of IL-1β on Th1 cells in a mouse model. The first data on human Th1 cells came from the study of Cosmi and colleagues mentioned before (44). In this study, the authors demonstrated that the combination of IL-1β and IL-23 was able to induce the development of Th17 cells in CD4+CD161+ cells, and also the Th1 phenotype, in both CD4+CD161+ and CD4+CD161− cell fractions. This observation leads the authors to hypothesize that also Th1 cells able to respond to IL-1β could exist. In keeping with this hypothesis, it has been recently found (53) that the CD4+CD161+ clones and inflamed tissue derived cells able to produce IFN-γ expressed IL-1RI mRNA. Interestingly, in the synovial fluid of JIA patients, the CD4+CD161+ IFN-γ-producing cells showed higher IL-1RI mRNA expression when compared to the CD4+CD161− counterpart (54). Of note, in this paper Cosmi and colleagues has highlighted the plasticity of Th17 cells showing that Th1 clones, expressing CD161 (named as “non-classic Th1 cells”), derive from an in vivo shifting of Th17 cells toward a Th1 phenotype. Interestingly, the authors found significantly increased levels of IL-12 in the SF of JIA patients and that Th17 cells from the PB of healthy children could be induced to shift to Th1 cells when cultured in vitro in the presence of JIA SF. More importantly, this effect was completely reversed by a neutralizing anti-IL-12 mAb, strongly suggesting that the shifting of Th17 cells toward the Th1 phenotype was related mainly to the activity of IL-12 present in the SF.
The late plasticity of Th17 cells to Th1 cells has been recently confirmed also in mice, where it has been found that IL-12, or the prolonged exposure to IL-23, is able to polarize Th17 cells toward the Th1 phenotype (55). Furthermore, similar results were recently reported by Nistala and colleagues (56), that showed Th17 plasticity to Th1 to be driven by the inflammatory environment in human autoimmune arthritis. Finally, very recently, the instability of the Th17 phenotype has been definitively demonstrated, at a genetic level, in mice (57).
These new data leads us to argue that up to now the scientific community overlooked this population of “non-classic Th1” cells expressing CD161 and IL-1RI (53). As mentioned before in this review many animal models of autoimmune disorders demonstrate the pivotal role of IL-1 in the pathogenesis of the disease and its relationship to Th17, but failed to look at the Th1 cells that could be affected by a lack of IL-1 signaling. In this context, the presence, and sometimes the prevalence, of Th1 cells in the inflammatory tissues have been interpreted as a protective, rather than proinflammatory, function. In humans many autoinflammatory disorders are treated blocking IL-1β [i.e., Familial Mediterranean fever (FMF), Pyogenic arthritis, pyoderma gangrenosum, acne (PAPA), CAPS, Hyper IgD syndrome (HIDS), Adult and juvenile Still disease Schnitzler syndrome, TNF receptor-associated periodic syndrome (TRAPS), Blau syndrome; Sweet syndrome, Deficiency in IL-1 receptor antagonist (DIRA)]. Initially, Anakinra (IL-1RA) was used to treat several chronic inflammatory diseases, today, these diseases are also successfully treated with neutralization by human anti-IL-1β monoclonal Abs. It’s time to speculate that the improvements observed during these treatments are not only due to the general anti-inflammatory effects and to the reduction in terms of production, survival and differentiation of Th17 cells, but also on the activity on Th1 effector cells in particular Th1 expressing IL-1RI probably derived from a Th17 phenotype (42, 53).
Concluding Remarks
Since the discovery of the lack of IL-1RI on murine Th1 cells, most of the studies focused their attention to in vivo animal models of Th2 related diseases. Different models of mice either deficient for IL-1RI or for IL-1α/IL-1β were analyzed to verify the effects in the Th2 response; most of the studies agree with the idea that both cytokines promote proliferation and differentiation of Th2 cells in vitro and in vivo, but some other found no effects or even the opposite. The contradictory findings could be related to the different animal models used, the different protocol of disease induction, the different genetic background; furthermore the analysis conducted on ex vivo bulk cultures may induce to overestimate some observations that are actually side or indirect effects. As most of the conclusions made on the basis of animal models could have an impact in clinical practice, we would have expected to find many papers confirming or disproving these data on human cells. Surprisingly very few studies focused on the effects of IL-1α either IL-1β in human Th2 cells, and the findings do not enlighten if human Th2 cells express a functional IL-1RI and therefore can be modulated by these cytokines.
Conversely, as we look to the relationship between IL-1β and Th17, human’s studies and animal models supported both the concept that IL-1β has a fundamental role in Th17 modulation. Two genetic human diseases carrying an impairment in IL-1β either in the expression or in its regulation and showing a skewed Th17 phenotype, is for sure of great confirmation of in vitro/ex vivo data. The in vitro assays clarify that IL-1β is able to induce those transcription factors necessary for Th17 development, as soon as its own receptor is upregulated in naïve T cells upon TCR triggering in the presence of γ-chain cytokines. The cooperation with other cytokines, i.e., IL-23, IL-6, IL-21 leads to the differentiation and the stabilization of the phenotype (39, 43, 58, 59); IL-1RI expression is long lasting, maintained on effector Th17 cells and its signaling is probably responsible for their survival during inflammation. The recent discovery of Th17 plasticity toward a Th1 phenotype in the presence of an inflammatory environment is driving the scientific community to focus attention also to those Th1 highly present in many autoimmune diseases that have been so far considered protective rather than pathogenic; it is intriguing, that also a sub population of human Th1 cells expressing CD161 and deriving from Th17 (named “Th1 non-classic”) do express IL-1RI and most likely respond to IL-1β (Figure 1). It is therefore likely that the therapeutically approaches were the IL-1β activity is blocked, like in JIA patients, are effective because acting on these Th1 CD161+ IL-1RI cells whose number correlate with some parameter disease. Moreover other autoimmune diseases, where Th1 CD161+ IL-1RI cells were increased, could benefit of an anti-IL-1 treatment.
Figure 1.
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IL-1β and IL-23 induce the differentiation of human CD161+ precursor toward both the Th17 and the non-classic Th1 phenotype. IL-1 β together with IL-23 acts on human CD4+CD161+ precursor to induce Th17 and non-classic Th1 effector cells. Th17 can shift toward a non-classic Th1 phenotype in the presence of IL-12.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Articles from Frontiers in Immunology are provided here courtesy of Frontiers Media SA
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Abstract
T Helper Cell Subsets
IL-1 Family of Cytokines
IL-1 and the Th2 Immune Response
Both Mice and Humans Th17 Express IL-1RI and are Modulated by Its Signaling
Th1 “Non-Classic” Human T Cell Express IL-1RI: A New Point of View
Concluding Remarks
Conflict of Interest Statement
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189417 | https://www.biologyonline.com/dictionary/net-primary-productivity | Dictionary > Net primary productivity
Net primary productivity
Net primary productivity
n., plural: net primary productivities
[nɛt ˈpɹaɪməɹi ˌprɒdʌkˈtɪvɪtɪ]
Definition: difference between gross primary productivity and the respiratory rate
Table of Contents
In order to keep the biosphere running, different organisms play different roles and functions. Some help in oxygen production while others consume it for survival. Some die and decay, so that some others can absorb the nutrition and feed on them. There are innumerable interlinks and interconnections that keep the biosphere running in a loop year after year, era after era, and from times immemorial to the times to come.
The primary fuel for running the biosphere is sourced from the autotrophs (primary producers) of a system. In this context, two topics become of significant importance – gross primary productivity (GPP) and net primary productivity (NPP). Let’s learn what each of these terms stands for and how one should define NPP and GPP.
Net Primary Productivity Definition
Let’s first understand what’s productivity and the different types of productivity. Then, we will move ahead and try to decipher the meaning and differences between NPP and GPP.
Productivity (ecology) stands for the rate of biomass generation/energy addition/carbon fixation/organic matter addition.
Primary productivity (ecology) stands for the production of biomass from inorganic molecules. Here the main conversion is of inorganic minerals to organic biomass. This duty is performed by the “autotrophs” of the ecosystem.
Secondary productivity (ecology) stands for the production of biomass from organic molecules. Here the main conversion is of organic molecules to some other form of organic molecules. This duty is performed by the “heterotrophs” of the ecosystem.
Gross primary productivity (ecology) stands for all the carbon that’s fixed by the primary producers of the ecosystem i.e., the main product of photosynthesis.
Net primary productivity (ecology) stands for the difference between all the carbon that’s fixed by the primary producers of the ecosystem and the part of that fixed carbon that the primary producer utilizes for its own cellular respiration needs.
In biology, net primary productivity is defined as the difference of the energy that’s fixed by the autotrophs and their own respiration losses. It is the net primary productivity that provides the channel for secondary productivity. For all the activities of consumers and heterotrophs, it’s the NPP that provides the energy and biomass source. And that is how NPP is converted to secondary productivity in the form of heterotrophic tissue. Another point to note here is that secondary productivity is “always” limited by the NPP. So, when asked what is net primary productivity, one can explain it and the significance associated with it.
Biology definition:
Net primary productivity is s the difference of the energy that’s fixed by the autotrophs and their own respiration losses. It is the rate at which an ecosystem accumulates energy or biomass, excluding the energy it uses for the process of respiration. This typically corresponds to the rate of photosynthesis, minus respiration by the photosynthesizers.
How Do You Calculate NPP?
Now, as you have become familiar with the NPP meaning, what is NPP, and the importance of net primary production, let’s move ahead and learn the net primary productivity formula.
NPP = GPP – RR
Where,
NPP= Net primary productivity
GPP= Gross primary productivity
RR= Respiratory Rate
This formula is important as it diligently associates GPP to NPP and also makes the metabolic losses and respiratory losses of autotrophs/producers count.
Generally, NPP is calculated and measured at the level/scale of “ecosystem” and over relatively longer frames/intervals of time(such as a year/s).
Some Important Pointers Related to NPP
NPP is very commonly associated with the increase in biomass per unit time and surface area (land).
In a way, NPP can be calculated by relating it to photosynthesis. The rate of anabolic photosynthesis reactions can be measured and used to quantify NPP.
Common query: how does oxygen production relate to the rate of photosynthesis? And the answer to it is this: oxygen is the product of anabolic photosynthesis reactions, hence it’s a product of NPP.
Not all the light energy captured by plants during photosynthesis contributes to NPP as parts of it get converted to organic matter, some other parts are lost as metabolic costs; others are dissipated as heat. Only the fixed organic matter that goes in the food chains, food webs, and in the subsequent trophic levels contribute to NPP.
Human colonies and societies are a part of the global NPP cycle but influence it in an inordinate way.
There are many factors on which NPP depends like:
Sunlight
Humidity
Temperature
Nutrient availability
Precipitation
Climate
Plant species and management
Initial biomass of the ecosystem (carbon-fixing plants/photosynthetic tissues)
The rate at which new biomass is assimilated
Net Primary Productivity Example
Some examples of net primary productivity are:
As we already know that the NPP is dependent on the “Initial biomass of the ecosystem (carbon-fixing plants/photosynthetic tissues)”, hence the “stands” or “ecosystems” with extensive-standing biomass will always have higher NPP. So, when asked what is a high NPP, one can answer That- “stands” with high initial biomass and higher rates of biomass production. Example: Production forests are having higher NPP than stable forests.
Legume-rich agricultural fields and pastures tend to have higher NPP than legume-deficient pastures as legume biomass accumulation is never hindered due to nutrient deficiency (especially nitrogen).
Temperate forests have higher NPP than temperate grasslands.
Try to answer the quiz below to check what you have learned so far about net primary productivity.
Time's up
References
Diaz S. (2013). Ecosystem Function Measurement, Terrestrial Communities. Encyclopedia of Biodiversity (Second Edition), 72-89.
DellaSala D.A. (2021). Imperiled Terrestrial Ecosystems: Nature in Retreat. Reference Module in Earth Systems and Environmental Sciences.
lark, D.A.; Brown, S.; Kicklighter, D.W.; Chambers, J.Q.; Thomlinson, J.R.; Ni, J. (2001). “Measuring net primary production in forests: Concepts and field methods”. Ecological Applications. 11 (2): 356–370. doi:10.1890/1051-0761(2001)011[0356:MNPPIF]2.0.CO;2. ISSN 1051-0761
Sigman, D.M.; Hain, M.P. (2012). “The Biological Productivity of the Ocean” (PDF). Nature Education Knowledge. 3 (6): 1–16.
Wright R.T., Boorse D.F. Environmental Science: Toward a Sustainable Future (11th Edition) 11th Edition. ISBN-13: 978-0321598707
©BiologyOnline.com. Content provided and moderated by Biology Online Editors.
Last updated on August 25th, 2023
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189418 | https://arec.alabama.gov/pages/laws/ViewLaw.aspx?LawSectionID=103 | AREC View Law
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| Category:CHAPTER 790-X-3 Rule 790-X-3-.13.Agency/Brokerage Services Disclosure Agency/Brokerage Services Disclosure 1. The Real Estate Consumers Agency and Disclosure Act (RECAD) requires the Alabama Real Estate Commission to write a Real Estate Brokerage Services Disclosure form which describes the alternative types of brokerage services available to consumers in Alabama transactions. The use of this form is mandatory as required by RECAD and this rule. Additionally, the Commission has written a Consumer Information Booklet which is optional and may be used by any licensees who choose to use it. 2. Licensees, except those engaged in rental or property management services, and those in transactions set out in Section 34-27-82(d), are required to provide the Real Estate Brokerage Services Disclosure form to the consumer as soon as reasonably possible for his or her signature. Consumers are not required by law to sign the form, although the licensee should encourage that it be signed. If the consumer declines to sign, the licensee shall make a note to this effect on the form. The texts of the optional Consumer Information Booklet and the mandatory form follow: A Consumer Information Booklet ============================== ### What Consumers Need to Know When Working with a Real Estate Broker As real estate transactions have become more complex and varied, the types of real estate brokerage arrangements available to the public have evolved to meet the changing needs of consumers entering this market.This booklet is intended to provide buyers and sellers with a description of the different types of brokerage arrangements so that consumers can choose the type of brokerage services best suited to their needs. #### Know Your Rights At the initial contact between a licensee (both sales persons and brokers) and the public, the licensee shall be considered to be a transaction broker. As soon as reasonably possible and before the exchange of confidential information, Alabama law requires the licensee to provide you with a written disclosure form that describes different types of brokerage arrangements. You are encouraged to read and sign this disclosure form.After disclosure you may then choose the type of brokerage agreement best suited to your needs. This brokerage agreement will contain a statement of the specific brokerage services the broker will provide. In the absence of a signed brokerage agreement, the transaction brokerage relationship will remain in effect. Make sure you talk to the real estate licensee with whom you are working to determine the type of services you need or will receive under alternative brokerage arrangements. ### Customer or Client? The most important thing you need to know when working with a real estate licensee is whether you are a client or a customer. A licensee owes certain duties to a client that are different from the services the licensee performs for a customer. ### Agent and Client An agent is a person who acts for or represents you in negotiations with other parties. The client or principal is the person the agent represents. The licensee when acting as an agent must loyally represent the best interest of the client by placing the interests of the client ahead of the interests of any other party. In a real estate transaction, when a real estate salesperson is employed as an agent, the salesperson is obligated to negotiate the best price and terms for his or her client. ### What is a Customer? A customer is a person who is provided services by a real estate broker,but who is not a client of the broker. In this case, the real estate licensee is not acting as an agent. The actual services you receive from a real estate broker depend on the arrangement that is established between you and the licensee. The different types of real estate agreements are described below. There are basically three types of real estate brokerage relationships that can be established between the consumer and a real estate licensee: Single agency, limited consensual dual agency, and a transaction brokerage arrangement. ### Transaction Brokerage Transaction brokerage describes a brokerage arrangement whereby the real estate licensee assists one or more parties, who are customers, in a contemplated real estate transaction, without being the agent, fiduciary, or advocate of that party to the transaction. This means that real estate brokers and salespeople can act as intermediaries between buyers and sellers. With this type of brokerage arrangement, home buyers and sellers are customers and not clients of the licensees with whom they are working. The basic function of the licensee is to bring buyers and sellers together so that a real estate sale can be completed. Sellers will employ the licensee to help market their real estate by identifying qualified buyers and showing their properties to prospective purchasers. This will usually also involve advertising properties for sale in newspapers and other media. Sellers will commonly also rely on the expertise, experience, and advice of the real estate licensee to help make their property ready for sale and determine an appropriate asking price. Buyers, in turn, rely on the services of brokers to find and show them suitable real estate that they can afford and have the desired characteristics. Real estate professionals may also help consumers obtain mortgage financing as well as assist them with finalizing the real estate sale and recording the deed and other documents associated with the sale. Transaction brokerage arrangements are usually best suited for consumers who are primarily interested in the marketing services and expertise that can be provided by real estate professionals, but who do not need an agent to represent them in the negotiations for the sale or purchase of real estate. Under transaction brokerage, the licensee must provide brokerage services to all parties honestly and in good faith and avoid showing favoritism to either buyer or seller. Alabama law also requires all licensees exercise reasonable care and skill when providing brokerage services, answer all questions completely and accurately, and present all written purchase offers to sellers promptly and in a truthful manner. Licensees must also keep confidential any information given to them in confidence, unless disclosure of this information is required by law. For sellers, this means that licensees must answer a buyer's questions about the condition of the property completely and honestly. In addition, the buyer must be told about any hidden defects known to the licensee that could affect the health or safety of occupants. ### Single Agency A single agency arrangement describes a relationship whereby the real estate licensee represents only one party in a real estate sales transaction. In the case of a single agency brokerage arrangement, the real estate licensee represents either the buyer or the seller, but not both parties to the real estate transaction. This type of brokerage arrangement is most appropriate for consumers who need the advice and negotiating skills of real estate professionals in addition to their marketing services. If a seller enters into a single agency agreement with a real estate broker, the broker is referred to as a seller's agent. Under this arrangement the broker must represent only the seller in the negotiations with buyers. Here the broker will seek the highest possible price and best possible sale terms for the seller. This type of brokerage arrangement can involve the use of subagents, especially in situations where properties are marketed through a multiple listing service. Subagents are empowered to act for another broker in performing real estate services for that broker. The subagent owes the same duties to the broker's client as the broker. If a broker is an agent of the seller, then the subagent is also the seller's agent. When examining properties advertised through a multiple listing service it is important for buyers to determine whether the licensee that is showing them properties is acting in the capacity of a transaction broker, seller's agent or as a subagent of the seller. Buyers should exercise care with respect to the information they reveal to licensees working as seller agents. For example, if you are the customer it would not be wise to tell a licensee the maximum price you would be willing to pay for a particular property when considering making a formal purchase offer. If you are the customer, the broker's primary responsibility is to the seller. In this case, the licensee, as the seller's agent, must convey such information to the seller. A buyer's agent describes a real estate licensee who is employed by and represents only the buyer in a real estate transaction. This relationship is created by a written contract. This contract should clearly state the service the agent will perform for the buyer as well as specify how the licensee is paid for services rendered in connection with the real estate sale. In this case, the buyer is the client or principal and the real estate broker is the agent of and represents the buyer in dealings with sellers. This type of real estate brokerage agreement should be used when the buyer needs guidance and representation when negotiating with sellers to purchase real estate. Buyers moving to a new location and who are unfamiliar with local market conditions would be those consumers most likely to benefit from this type of agency arrangement. It is becoming increasingly common in multiple listing situations for the selling broker (a licensee working with and showing properties to the buyer) to be an agent of the buyer and the listing broker to represent the owner-seller. Here, both the buyer and seller, working through their respective agents, could negotiate at arm's length with the benefit of professional help. ### Limited Consensual Dual Agent Limited consensual dual agency is an agency relationship where the real estate brokerage company represents both the buyer and the seller in the same real estate transaction. Consensual dual agency requires the licensee to obtain the written consent of both the buyer and the seller to act as their agent. The two most common circumstances where dual agency is encountered are (1) when two or more salespersons licensed under the same broker each represent a different party to the transaction, and (2) when one licensee represents both the buyer and seller in the same sales transaction. One major advantage of limited consensual dual agency is that it allows broader marketing opportunities than single agency arrangements. With this type of contract, salespeople can show houses of owners that they represent as agents to their buyer clients. Consensual dual agency is common in the larger real estate markets where real estate companies often have a large number of properties listed for sale. Many of these properties may be desirable to their buyers. With a dual agency agreement, these properties can be shown to their buyer clients. In the case of dual agency, the principle function of the licensee is to help both parties reach mutually satisfactory outcome to their negotiations. The dual agent must avoid showing favoritism to either party and refrain from revealing confidential information that could prove detrimental to one side or the other. Although buyers and sellers may not benefit from the full range of services or agent loyalty that could otherwise be provided with a single agency arrangement, consensual dual agency does offer consumers more assistance and guidance than would be possible under a transaction brokerage arrangement. When considering signing a dual agency agreement, it is very important that you talk with the broker to determine the types of services that will be provided, and what types of information you will share with the broker and broker's other clients. ### Conclusion After reading this consumer information booklet, you should sign a brokerage agreement that contains a statement of the services to be provided by the real estate professional. Remember, if you do not sign a brokerage agreement, by law, the licensee working with you will be considered a transaction broker. Ask your real estate licensee to clarify and explain anything in this booklet that you do not fully understand before signing a contract for real estate brokerage services. You are encouraged to sign the disclosure form and retain a copy for your records. THIS IS FOR INFORMATION PURPOSES THIS IS NOT A CONTRACT REAL ESTATE BROKERAGE SERVICES DISCLOSURE Alabama law requires you, the consumer, to be informed about the types of services which real estate licensees may perform. The purpose of this disclosure is to give you a summary of these services. A SINGLE AGENT is a licensee who represents only one party in a sale. That is, a single agent represents his or her client. The client may be either the seller or the buyer. A single agent must be completely loyal and faithful to the client. A SUBAGENT is another agent/licensee who also represents only one party in a sale. A subagent helps the agent represent the same client. The client may be either the seller or the buyer. A subagent must also be completely loyal and faithful to the client. A LIMITED CONSENSUAL DUAL AGENT is a licensee for both the buyer and the seller. This may only be done with the written, informed consent of all parties. This type of agent must also be loyal and faithful to the client, except where the duties owed to the clients conflict with one another. A TRANSACTION BROKER assists one or more parties, who are customers, in a sale. A transaction broker is not an agent and does not perform the same services as an agent. Alabama law imposes the following obligations on all real estate licensees to all parties, no matter their relationship: 1. To provide services honestly and in good faith; 2. To exercise reasonable care and skill; 3. To keep confidential any information gained in confidence, unless disclosure is required by law or duty to a client, the information becomes public knowledge, or disclosure is authorized in writing; 4. Present all written offers promptly to the seller; 5. Answer your questions completely and accurately. Further, even if you are working with a licensee who is not your agent, there are many things the licensee may do to assist you. Some examples are: 1. Provide information about properties; 2. Show properties; 3. Assist in making a written offer; 4. Provide information on financing. You should choose which type of service you want from a licensee, and sign a brokerage service agreement. If you do not sign an agreement, by law the licensee working with you is a transaction broker. The licensee's broker is required by law to have on file an office policy describing the company's brokerage services. You should feel free to ask any questions you have. The Alabama Real Estate Commission requires the real estate licensee to sign, date, and provide you a copy of this form. Your signature is not required by law or rule, but would be appreciated. Name of licensee ________. Signature _______. Date _________. Consumer name _______. Signature ________. (Acknowledgment for Receipt Purposes, Only) Date ________. Author:Alabama Real Estate Commission Statutory Authority:Code of Ala.1975, §§ 34-27-8,34-27-82, 34-27-87. History:New Rule: Filed January 25,1996;effective February 29,1996.Amended: Filed August 3,1998;effective September 7,1998.Amended: Filed February 27,2002;effective April 3,2002.Amended: Filed November 26,2002;effective December 31,2002. |
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189419 | https://www.themathdoctors.org/the-many-meanings-of-quartile/ | Skip to content
The Many Meanings of “Quartile”
October 15, 2018 December 1, 2023 / Statistics / Definitions, Inconsistency / By Dave Peterson
Some time ago I discussed various issues pertaining to the concept of median in statistics. The same issues, and more, affect the concept of quartile (the median being the second quartile), so much so that different statistical software packages produce many different answers for quartiles. I have seen this affect students, who are taught one method for doing it by hand, but then find that their software uses a different definition. Let’s look at some questions in this area.
Dueling software
First, from 2002:
Defining Quartiles We have a project for statistics class where we have to collect a set of data, then find the mean, median, mode, range, upper quartile, lower quartile, interquartile range, and standard deviation. We also have to plot the data in a stem-and-leaf plot, dot plot, histogram, and box-and-whisker plot. I decided to collect data on the heights of the players on our soccer team, and got the following data: {70", 71", 71", 71", 72", 73", 74", 74", 74", 74", 75", 75", 77", 77", 77", 82"} I didn't have any problems until I was checking my work using my calculator and a computer. All the values agreed with my hand calculations except the upper quartile, lower quartile, and interquartile range. When I calculated the quartiles and IQR following the textbook, I got 77" (UQ), 71" (LQ) and 6" (IQR). But when I plugged the values into my calculator (a TI-83), it gave the upper quartile as 76", the lower quartile as 71.5", and the IQR as 4.5". I then tried making an Excel spreadsheet and it gave the upper quartile as 75.5", the lower quartile as 71.75", and the IQR is 3.75". Then I went to the computer lab at school and tried using Minitab. That program gave the upper quartile as 76.5", the lower quartile as 71.25", and the IQR as 5.25". If they just disagreed with my calculations, I'd figure that I made a mistake, or there's some sort of rounding going on, since we're told to take the nearest data point and these programs obviously don't. But they don't even agree with each other. They can't all be right! What's going on? Please help clear up this mystery.
Doctor Twe gave a long, detailed answer that has become a classic; in fact, while searching for further information, I later discovered an excellent article (which will be referred to below) that referred to this answer. I will quote selected parts of what he wrote.
Quartiles are simple in concept but can be complicated in execution. The concept of quartiles is that you arrange the data in ascending order and divide it into four roughly equal parts. The upper quartile is the part containing the highest data values, the upper middle quartile is the part containing the next-highest data values, the lower quartile is the part containing the lowest data values, while the lower middle quartile is the part containing the next-lowest data values. Here's where it starts to get confusing. The terms 'quartile', 'upper quartile' and 'lower quartile' each have two meanings. One definition refers to the subset of all data values in each of those parts. For example, if I say "my score was in the upper quartile on that math test", I mean that my score was one of the values in the upper quartile subset (i.e. the top 25% of all scores on that test). But the terms can also refer to cut-off values between the subsets. The 'upper quartile' (sometimes labeled Q3 or UQ) can refer to a cut-off value between the upper quartile subset and the upper middle quartile subset. Similarly, the 'lower quartile' (sometimes labeled Q1 or LQ) can refer to a cut-off value between the lower quartile subset and the lower middle quartile subset. The term 'quartiles' is sometimes used to collectively refer to these values plus the median (which is the cut-off value between the upper middle quartile subset and the lower middle quartile subset). John Tukey, the statistician who invented the box-and-whisker plot, referred to these cut-off values as 'hinges' to avoid confusion. Unfortunately, not everyone followed his lead on that.
I’ll be saying some similar things soon in discussing percentiles, which likewise have multiple meanings. But this is not what Tom was concerned about.
Five methods
It gets worse. Statisticians don't agree on whether the quartile values ('hinges') should be points from the data set itself, or whether they can fall between the points (as the median can when there are an even number of data points). Furthermore, if the quartile value is not required to be a point in the data set itself, most data sets don't have a unique set of values {Q1, Q2, Q3} that divides the data into four "roughly equal" portions. The SAS statistical software package, for example, allows you to choose from among five different methods for calculating the quartile values. How then do we choose the "best" value for the quartiles? The answer to that question depends in part on the statisticians' objective in finding quartile values. Tukey wanted a method that was simple to use, "without the aid of calculating machinery." Others seek to minimize the bias in selecting the quartile values. Still others want methods that can be extended to other quantiles (for example, quintiles or percentiles). Thus, different methods have been developed for calculating the quartile values.
This is the underlying reason for the differences: Different contexts or goals lead to different choices. Unfortunately, for students, the context is just whatever they are taught, and the goal is only to satisfy their teacher! Doctor Twe first described two specific methods, from Tukey and the TI-83 (based on a textbook called Moore and McCabe), both of which are based on dividing the data into four parts and including or excluding boundaries. He summarizes:
Those methods involve only simple arithmetic and are easily extendable to octiles (eighths), hexadeciles (sixteenths), etc. They are not, however, extendable to quintiles (fifths) or percentiles (hundredths), etc. Furthermore, they tend to have a high bias. (That is, the quartile values calculated on subsets of the data set tend to vary more, and are not good predictors of the quartile values of the entire data set.)
Then he discussed three methods based on formulas for the index at which to find a “hinge”, used by Mendenhall and Sincich, Minitab, and Excel.
As we can see, these methods sometimes (but not always) produce the same results. To further illustrate, consider the following data sets: A = {1, 2, 3, 4, 5, 6, 7, 8} B = {1, 2, 3, 4, 5, 6, 7, 8, 9} C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Here are the upper and lower quartile values, as calculated by each method described above: Tukey M&M M&S Mini Excel ----- --- --- ---- ----- Set A LQ: 2.5 2.5 2 2.25 2.75 UQ: 6.5 6.5 7 6.75 6.25 Set B LQ: 3.0 2.5 3 2.50 3.00 UQ: 7.0 7.5 7 7.50 7.00 Set C LQ: 3.0 3.0 3 2.75 3.25 UQ: 8.0 8.0 8 8.25 7.75 Set D LQ: 3.5 3.0 3 3.00 3.50 UQ: 8.5 9.0 9 9.00 8.50
He closes with three links, which are all now broken (16 years later). I’ll be applying each of these methods, and another, to Tom’s data below.
A better method
In 2012, we got an inquiry from a user of a different program, Origin, who had found that it gave results differing from any of those we had shown:
Origin of Origin's Outputs I have read this question and answer: Defining Quartiles But another question has arisen. When I draw this box-plot using Origin (version 8.5 of the software), it resulted in LQ = 3 and UQ = 7 for data set B = {1, 2, 3, 4, 5, 6, 7, 8, 9} LQ = 3 and UQ = 9 for data set D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Apparently, the Origin program included the median when calculating the lower and upper quartiles for data set B, but excluded the median when calculating those values for D. So Origin seems to follow neither Tukey's method nor others. But what method could it have used for these calculations?
I responded, and after gathering some more information, I did some further research:
You're right, it appears they do not consistently use the M&S method; for even numbers, they might use Tukey or M&M. But since those have a very different style from M&S, I can't imagine they would mix the two. More likely, they follow some other method entirely. For a larger study of different methods of calculation of quartiles (actually percentiles, the quartiles being the 25th and 75th percentiles), see Quartiles in Elementary Statistics (Eric Langford) Table 2 is similar to Dr. TWE's table, but with slightly smaller sample data sets. Perhaps if you try those, you will recognize one that agrees with your software. Let me know what values you get for the quartiles.
I had discovered the Langford paper in 2007, noting that it mentioned Doctor TWE’s answer:
What are students to do when they check a MINITAB or SAS or Microsoft Excel calculation on their TI-83 Plus calculator and get a different answer, all of which differ from the answer in the back of the book? This is not an idle concern; a very confused student wrote to the “Ask Dr. Math” section of The Math Forum@Drexel inquiring why his TI-83, Excel, MINITAB, and his own paper-and-pencil calculations all gave different answers for the quartiles of his data set.
Langford lists 12 primary methods, one of which provided a likely answer to our question:
I suspect they may be using the "CDF method" (4): METHOD 4 ("CDF"): The Pth percentile value is found as follows. Calculate np. If np is an integer, then the Pth percentile valueis the average of #(np) and #(np + 1). If np is not an integer,the Pth percentile value is #ceil(np); that is, we round up.Alternatively, one can look at #(np + 0.5) and round off unlessit is half an odd integer, in which case it is left unrounded. As an example, if S5 = {1, 2, 3, 4, 5} and p = 0.25, then#(np) = 1.25, which is not an integer. So we take the nextlargest integer, and hence, Q1 = 2. Using the alternativecalculation, we would look at #(np + 0.5) = #(1.75), whichwould again round off to 2. Note that this method can beconsidered as "Method 10 with rounding." Translated into Dr. TWE's terms, L = n/4 if it is not an integer, round up and use Lth data point; if it is an integer, average the Lth and (L + 1)th data points U = 3n/4 if it is not an integer, round up and use Lth data point; if it is an integer, average the Lth and (L + 1)th data points For {1, 2, 3, 4, 5, 6, 7, 8}, this gives LQ: L = 8/4 = 2; average 2nd and 3rd data points, giving 2.5 UQ: U = 38/4 = 6; average 6th and 7th data points, giving 6.5 For {1, 2, 3, 4, 5, 6, 7, 8, 9}, we get LQ: L = 9/4 = 2.25; round up and use 3rd data point, 3 UQ: U = 39/4 = 6.75; round up and use 7th data point, 7 These cases agree with what you've told me.
He wrote back with conformation that his software gave the expected results for this method. I replied with additional details:
Yes, it does look very likely that they are using this method, or at least something equivalent. Note that at the end of his article, Langford says that this is the best method, and offers an equivalent method that is easy to teach: Thus, the following method is equivalent to the CDF Method 4, yet has the flavor of the Inclusive and Exclusive Methods 1 and 2, and thus should be more accessible to students. SUGGESTED METHOD: Divide the data set into two halves, a bottom half and a top half. If n is odd, include or exclude the median in the halves so that each half has an odd number of elements. The lower and upper quartiles are then the medians of the bottom and top halves, respectively. So for these four examples, the work looks like these: --+-- <-- bottom half 1 | 2 | 3 | 4 | | --+-- <-- top half 1.5 2.5 3.5 <-- Q1, Q2, Q3 ----+---- 1 (2) (3) (4) 5 | ----+---- 2 3 4 ----+---- 1 (2) 3 | 4 (5) 6 | | ----+---- 2 3.5 5 ----+---- 1 (2) 3 (4) 5 (6) 7 | | ----+---- 2 4 6 So it looks like your software has made a good choice.
Example
In my excerpts above, I skipped the detailed explanations of the first five methods; let’s demonstrate them with Tom’s data from the first question. The data were
70, 71, 71, 71, 72, 73, 74, 74, 74, 74, 75, 75, 77, 77, 77, 82 (n = 16)
Tukey and M&M/TI-83 (same when n is even): Split into halves at the median, use median of each half as quartiles.
[70, 71, 71, 71, 72, 73, 74, 74], [74, 74, 75, 75, 77, 77, 77, 82]Q1 = 71.5Q3 = 7670 71 71 71|72 73 74 74|74 74 75 75|77 77 77 82
M&S: For Q1, use index L = (1/4)(n+1), round to nearest integer (up if halfway). For Q3, use index U = (3/4)(n+1), round to nearest integer (down if halfway).
L = (1/4)(16+1) = 4.25, rounded to 4; Q1 = 71U = (3/4)(16+1) = 12.75, rounded to 13; Q3 = 7770 71 71 71 72 73 74 74|74 74 75 75 77 77 77 82
Minitab: Same as M&S but with linear interpolation:
L = (1/4)(16+1) = 4.25; Q1 = 71 + 0.25(72 – 71) = 71.25U = (3/4)(16+1) = 12.75; Q3 = 75 + 0.75(77 – 75) = 76.570 71 71 71|72 73 74 74|74 74 75 75|77 77 77 82
Excel: For Q1, use index L = (1/4)(n+3) with linear interpolation. For Q3, use index U = (1/4)(3n+1) with linear interpolation.
L = (1/4)(16+3) = 4.75; Q1 = 71 + 0.75(72 – 71) = 71.75U = (1/4)(316+1) = 12.25; Q3 = 75 + 0.25(77 – 75) = 75.570 71 71 71|72 73 74 74|74 74 75 75|77 77 77 82
Langford’s method, for n even, is the same as Tukey and M&M. Note that, in Doctor Twe’s explanation, this method agrees with Tukey in case B, with M&M in case D, and with both in cases A and C.
Tom, using his book’s unspecified method, got 71 and 77, which is the same as M&S, so that may be what he was taught.
Note that almost all methods yield quartiles that are in the gaps where we would expect them to be (where I put bars); only M&S, which always uses numbers in the data set, gave quartiles that don’t lie in the gaps. None of them can really be called wrong. And for large data sets, the differences are insignificant.
But, then, what is right?
Definitions vs. methods
In 2015, another question about varying definitions led to an important distinction. (I referred to this previously in my discussion of medians.) Quoting only part of this long and detailed question,
Quartile Conflict Consider the following sample (n = 10): 17, 21, 22, 22, 26, 30, 38, 59, 67, 85 Find the median and the lower and upper quartiles (LQ and UQ). After considering the position of the quartiles, is linear interpolation required for a sample with an even number of data points? This does not seem to be typical, but is it more accurate?
I responded,
The problem is that the definition of quartiles varies. Not all sources will give the same method. I would follow the method that gives 22 and 59: 17, 21, 22, 22, 26, 30, 38, 59, 67, 85 == ^ == This preferred method takes the median of the two halves; and, in the odd case, either includes or excludes the median as needed to make each "half" odd. This is discussed at length here: Defining Quartiles I discussed similar issues here, and gave a link to a paper by Langford that I consider definitive: Origin of Origin's Outputs Evidently, your text (or other source) gives a definition based on the location; that approach can accomplish the same results as my method (Langford's recommended method) if you handle rounding appropriately. This is specifically discussed in the second link above. In your example, you would use n rather than n + 1, giving 2.5 and 7.5, then round 2.5 up to 3, and 7.5 up to 8. Perhaps you can tell me more about the method you are using, and its source.
Then I had second thoughts about how to best answer him:
Perhaps I should add something that explicitly addresses your underlying question of whether or not linear interpolation is more accurate. The problem here is: what should your quartiles "more accurately" reflect? What is the actual DEFINITION of a quartile that the result of the METHOD must agree with? In what I wrote before, I confused these two different concepts, because texts often present the method as the definition. This question is addressed somewhere in the links I gave, but can more easily be seen in this answer relating to the same problem in the definition of the median: A Closer Look at the Definition of Median Adapting the definition I gave there to the first quartile: The first quartile of a set of data is a number such that NO MORE THAN 1/4 are LESS than the first quartile, and NO MORE THAN 3/4 are GREATER than the first quartile. The number 22 fits this definition: 17, 21, 22, 22, 26, 30, 38, 59, 67, 85 less ====== greater No more than 1/4 of the ten numbers (2 <= 2.5) are less than 22. No more than 3/4 of the ten numbers (6 <= 7.5) are greater than 22. If we used your linear interpolation to get 21.75, would that fit? 17, 21, 22, 22, 26, 30, 38, 59, 67, 85 less ^ greater No more than 1/4 of the ten numbers (2 <= 2.5) are less than 21.75. MORE than 3/4 of the ten numbers (8 > 7.5) are greater than 21.75. So in fact, linear interpolation does not fit the definition! Lesson: what matters is having an adequate definition, and then fitting that definition, rather than applying a method that sounds appropriate -- and I have to agree that your idea does sound good -- without reference to an actual definition!
This discussion continued with an examination of a couple other methods; my ultimate conclusion was that it was appropriate to use the assigned textbook’s method, and just make students aware that (a) there are other options, and (b) it doesn’t make any significant difference in real-life:
It certainly makes sense to follow the authority you are under. It's certainly good that the authors of HIGHER GCSE MATHEMATICS FOR WJEC recognize the variability of the concept; and also emphasize that it is only with small data sets that it matters. But it's disappointing to see them claim their method as the "most accurate," without, apparently, saying what the "exact" result is that they are comparing it to, or how.
So what is right?
In school, as just mentioned, what is “right” is what you are taught; and what is right to teach is what your curriculum says. If you are a curriculum writer, you might want to follow Langford’s advice.
In America, many schools take their direction from the Common Core standards, which apparently has sided with the “M&M” method:
First quartile. For a data set with median M, the first quartile is the median of the data values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6.2
2Many different methods for computing quartiles are in use. The method defined here is sometimes called the Moore and McCabe method. See Langford, E., “Quartiles in Elementary Statistics,” Journal of Statistics Education Volume 14, Number 3 (2006).
I wouldn’t be surprised if this choice is partly motivated by the widespread use of the TI-83 calculator family, as well as by ease of use; they are almost, but not quite, following the recommendation of their source.
But beyond education, it is important to be aware that there are many definitions in use (though they make little difference in large data sets). Wikipedia lists three such definitions (and refers to Doctor Twe’s answer); MathWorld compares five.
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Node (N):if the cancer has spread to the lymph nodes
Metastasis (M):whether the cancer has metastasized (spread) to other areas of the body
The results of the TNM system are combined to determine the stage of cancer for each person, with possible stages being 0, 1, 2, 3, or 4. The lower the number, the less the cancer has spread. A higher number, like stage 4, means a more serious cancer that has spread from where it started to other parts of the body.
T Categories for Staging Oral Cancer
"T" plus a letter or number (0 to 4) is used to describe the size and location of the tumor. Tumor size is measured in millimeters (mm) and centimeters (cm). To provide more detail about the tumor, the stage may also be divided into smaller groups. The specific tumor stage information is as follows:
TX:The primary tumor cannot be evaluated.
Tis:A very early cancer, called carcinoma in situ, where cancer cells are found only in 1 layer of tissue.
T1:The tumor is 2 cm or less. It has invaded nearby tissues to a depth of 5 mm or less (called depth of invasion).
T2:The tumor is 2 cm or smaller, and the depth of invasion is between 5 and 10 mm. Or, the tumor is larger than 2 cm but not larger than 4 cm, and the depth of invasion is 10 mm or less.
T3:The tumor is greater than 4 cm, or it is any tumor with a depth of invasion greater than 10 mm.
T4:Tumor is considered a moderately advanced or very advanced local disease.
T4a:The tumor has invaded nearby structures in the mouth, such as the jaw, sinuses, or skin of the face.
T4b:The tumor has invaded the muscles and bones that form the mouth or the base of the skull, and/or it encases the internal arteries.
N Categories for Staging Oral Cancer
There are 2 methods of categorizing N in the TNM system: clinical and pathological. Clinical N is used for patients who have not received a neck dissection. Pathological N is used for those who have had a lymph node dissection. When cancer has spread through a lymph node and into the tissues directly surrounding it, it is called extranodal extension (ENE).
Clinical
NX:The regional lymph nodes cannot be evaluated.
N0:No evidence of cancer in the regional lymph nodes.
N1:The cancer has spread to a single lymph node on the same side as the primary tumor, and the cancer found in the node is 3 cm or smaller. There is no ENE.
N2a:The cancer has spread to a single lymph node on the same side as the primary tumor and is greater than 3 cm but less than 6 cm. No ENE is present.
N2b:The cancer has spread to more than 1 lymph node on the same side as the primary tumor, and none measures larger than 6 cm. No ENE is present.
N2c:Cancer has spread to more than 1 lymph node on either side of the body, and none measure greater than 6 cm. No ENE is present.
N3a:Cancer is found in a lymph node and measures greater than 6 cm. No ENE is present.
N3b:ENE is present in any lymph node.
Pathological
NX:Regional lymph nodes cannot be evaluated.
N0:There is no evidence of cancer in the regional lymph nodes.
N1:Cancer has spread to a single lymph node on the same side as the primary tumor, and the cancer found in the node measures 3 cm or less. No ENE is present.
N2a:Cancer has spread to 1 lymph node and is 3 cm or smaller, but no ENE is present. Or, cancer has spread to a single lymph node on the same side as the primary tumor and is greater than 3 cm but no larger than 6 cm, and no ENE is present.
N2b:Cancer has spread to more than 1 lymph node on the same side as the primary tumor, and none measures greater than 6 cm. No ENE is present.
N2c:Cancer has spread to more than 1 lymph node on either side of the body, and none measure greater than 6 cm. No ENE is present.
N3a:Cancer is found in a lymph node and is greater than 6 cm. No ENE is present.
N3b:ENE is present in a single lymph node on the same side as the primary tumor, and it is greater than 3 cm. Or, cancer has spread to many lymph nodes, and at least 1 has ENE. Or, ENE is present in a single lymph node on the opposite side of the primary tumor that is 3 cm or less.
M Categories for Staging Oral Cancer
The "M" in the TNM system describes whether the cancer has metastasized (spread) to other parts of the body.
M0:The cancer has not spread to other parts of the body.
M1:The cancer has spread to other parts of the body.
Oral Cancer Stage Grouping
The T, N, and M information is combined to determine what stage the cancer is.
Stage 0:Abnormal cells are found in the lining of the lips and oral cavity. These abnormal cells may become cancerous and spread into nearby normal tissue (Tis, N0, M0). Stage 0 is also called carcinoma in situ.
Stage I:Cancer has formed. Tumor is 2 cm or smaller, and the depth of invasion is 5 mm or less. It has not spread to lymph nodes or other parts of the body (T1, N0, M0).
Stage II:Tumor is 2 cm or smaller, and the depth of invasion is between 5 and 10 mm. Or, the tumor is greater than 2 cm but not greater than 4 cm, and the depth of invasion is 10 mm or less. It has not spread to lymph nodes or other parts of the body (T2, N0, M0).
Stage III:One of the following applies:
Tumor is greater than 4 cm, or it is any tumor with a depth of invasion greater than 10 mm. It has not spread to lymph nodes or other parts of the body (T3, N0, M0).
Tumor is any size, but it has not invaded nearby structures of the oral cavity. There is cancer in a single lymph node on the same side as the primary tumor, and the cancer is 3 cm or less. No ENE is present. Cancer has not spread to other parts of the body (T1 to T3, N1, M0).
Stage IVA:One of the following applies:
Tumor has invaded nearby structures in the mouth, such as the jaw, sinuses, or skin of the face. If cancer has spread to a lymph node, it is to only 1 node on the same side as the primary tumor, and the cancer is 3 cm or less, with no ENE. Cancer has not spread to other parts of the body (T4a, N0 or N1, M0).
Tumor is small or it may have invaded nearby structures. Cancer has spread to 1 or more lymph nodes, but none is greater than 6 cm. There is no ENE. Cancer has not spread to other parts of the body (T1 to T4a, N2, M0).
Stage IVB:One of the following applies:
Tumor is any size. The cancer is found in a lymph node and is greater than 6 cm, but there no ENE is present; or ENE is present in any lymph node. Cancer has not spread to other parts of the body (any T, N3, M0).
Tumor has invaded the muscles and bones that form the mouth or the base of the skull, and/or it encases the internal arteries. The cancer may involve the lymph nodes, but it has not spread to other parts of the body (T4b, any N, M0).
Stage IVC:Cancer has spread to other parts of the body (any T, any N, M1).
Grading Oral Cancer
The grade of cancer gives the oncologist a better understanding of how quickly the cancer is likely to spread. The closer the cancer cells look to healthy cells, the lower the grade and the less chance that it will spread quickly.
G1:The cells look more like normal tissue and are well differentiated.
G2:The cells are only moderately differentiated.
G3 and G4:The cells do not resemble normal tissue and are poorly differentiated.
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189422 | https://brilliant.org/wiki/arithmetic-mean-geometric-mean/ | Arithmetic Mean - Geometric Mean | Brilliant Math & Science Wiki
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Arithmetic Mean - Geometric Mean
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Jubayer Nirjhor, Sandeep Bhardwaj, Mikael Marcondes, and
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The arithmetic mean-geometric mean (AM-GM) inequality states that the arithmetic mean of non-negative real numbers is greater than or equal to the geometric mean of the same list. Further, equality holds if and only if every number in the list is the same. Mathematically, for a collection of n n n non-negative real numbersa 1,a 2,...,a n a_1,a_2,...,a_n a 1,a 2,...,a n, we have
a 1+a 2+⋯+a n n≥a 1 a 2…a n n,\frac{ a_1 + a_2 + \cdots + a_n } { n}\ge\sqrt[n] { a_1 a_2 \ldots a_n} , n a 1+a 2+⋯+a n≥n a 1a 2…a n,
with equality if and only if a 1=a 2=⋯=a n a_1=a_2=\cdots =a_n a 1=a 2=⋯=a n.
This wiki page will familiarize you with the AM-GM inequality and its applications in several scenarios. We will also prove this inequality through several methods and further generalize it for deeper insights.
Contents
Statement of AM-GM inequality
Proof of AM-GM Inequality
Examples with AM-GM inequality
Generalization: Weighted AM-GM Inequality
See Also
Statement of AM-GM inequality
The Arithmetic Mean - Geometric Mean inequality, or AM-GM inequality, states the following:
The geometric mean cannot exceed the arithmetic mean, and they will be equal if and only if all the chosen numbers are equal. That is,
a 1+a 2+⋯+a n n≥a 1 a 2…a n n\frac{ a_1 + a_2 + \cdots + a_n } { n}\ge\sqrt[n] { a_1 a_2 \ldots a_n}n a 1+a 2+⋯+a n≥n a 1a 2…a n
with equality if and only if a 1=a 2=⋯=a n a_1=a_2=\cdots =a_n a 1=a 2=⋯=a n.
More precisely,
∑i=1 n a i n≥∏i=1 n a i n.{\frac{\displaystyle \sum { i=1 }^{ n }{ { a }{ i } }}{n} \ge \sqrt [ n ]{ \prod { i=1 }^{ n }{ { a }{ i } } } }.n i=1∑na i≥n i=1∏na i.
To get comfortable, let's consider the case when n=2,n=2,n=2, i.e. when there are only two variables, say x x x and y y y. Then the AM-GM inequality says
x+y 2≥x y.\dfrac{x+y}{2}\ge \sqrt{xy}.2 x+y≥x y.
Cross multiplying and rearranging terms, we get x−2 x y+y≥0 x-2\sqrt{xy}+y\ge 0 x−2 x y+y≥0 or (x−y)2≥0\left(\sqrt x-\sqrt y\right)^2\ge 0(x−y)2≥0. This is true because squares are always non-negative. Also equality occurs when x=y\sqrt x=\sqrt y x=y or x=y x=y x=y.
The general case is slightly harder to show, as we cannot just cross multiply and manipulate. A common approach would be inducting on the number of variables. We state the proof by Cauchy below.
Here is a simple example based on the AM-GM inequality.
If the product of two positive numbers is 100 100 100, what is the minimum value of their sum?
Let the two numbers be a a a and b b b. We are given that a b=100 ab = 100 ab=100, and want to find the minimum value of a+b a+ b a+b.
The AM-GM inequality states that
a+b 2≥a b, \dfrac{ a + b } { 2 } \geq \sqrt{ab } ,2 a+b≥ab,
implying for this problem that a+b≥2 100=20.□ a + b \geq 2 \sqrt{100} = 20 . \ _\square a+b≥2 100=20.□
Note: This minimum is achieved when a=b=10 a = b = 10 a=b=10.
You can try a problem similar to the above example:
Reveal the answer
The sum of two positive real numbers is 100. Find their maximum possible product.
The correct answer is: 2500
The three red jellies The three geen jellies They are both equal. No definite relation can be made.
Reveal the answer
A jelly shop sells its products in two different sets: 3 red jelly cubes and 3 green jelly cuboids.
The 3 red cubes are of side lengths a<b<c,a<b<c,a<b<c, while the 3 green cuboids are identical with dimensions a×b×c,a\times b\times c,a×b×c, as shown above.
Which option would give you more jelly?
The correct answer is: The three red jellies
Proof of AM-GM Inequality
AM-GM inequality can be proved by several methods. Some of them are listed here.
The first one in the list is to prove by some sort of induction. Here we go:
At first, we let the inequality for n n n variables be asserted by P(n)P(n)P(n). Traditional inductions check a base case and then show that P(n)⟹P(n+1)P(n)\implies P(n+1)P(n)⟹P(n+1). But we'll show these:
P(2)P(2)P(2) holds.
P(n)⟹P(2 n).P(n)\implies P(2n).P(n)⟹P(2 n).
P(n)⟹P(n−1).P(n)\implies P(n-1).P(n)⟹P(n−1).
Why should this work? Note that we jump from n n n to 2 n 2n 2 n by showing P(n)⟹P(2 n)P(n)\implies P(2n)P(n)⟹P(2 n). Then using P(n)⟹P(n−1)P(n)\implies P(n-1)P(n)⟹P(n−1), we can induct backwards from 2 n 2n 2 n to n+1,n+1,n+1, to verify that all numbers between n n n and 2 n 2n 2 n (inclusive) satisfy the assertion. This is known as forward-backward induction.
Now we move on to prove those points. It's already shown above that P(2)P(2)P(2) holds, and now we prove P(n)⟹P(2 n)P(n)\implies P(2n)P(n)⟹P(2 n). Consider positive reals a 1,a 2,...,a 2 n a_1,a_2,...,a_{2n}a 1,a 2,...,a 2 n. Since we assume that P(n)P(n)P(n) is true, for any n n n positive reals ∑i=1 n a i n≥∏i=1 n a i n\frac{ \sum { i=1 }^{ n }{ { a }{ i } }}{n} \ge \sqrt [ n ]{ \prod { i=1 }^{ n }{ { a }{ i } } } n∑i=1 na i≥n∏i=1 na i holds. Then
1 2 n∑i=1 2 n a i=1 2 n(∑i=1 n a i+∑i=n+1 2 n a i)≥n AM-GM 1 2(∏i=1 n a i n+∏i=n+1 2 n a i n)≥2 AM-GM∏i=1 n a i n×∏i=n+1 2 n a i n=∏i=1 2 n a i 2 n,\begin{array}{rcl} \dfrac{1}{2n}\displaystyle\sum_{i=1}^{2n} a_i &=& \dfrac{1}{2n}\left(\displaystyle\sum_{i=1}^n a_i +\displaystyle \sum_{i=n+1}^{2n} a_i\right) \ &\stackrel{n\text{ AM-GM}}{\ge} &\dfrac{1}{2}\left(\sqrt[n]{\prod_{i=1}^n a_i} + \sqrt[n]{\prod_{i=n+1}^{2n} a_i}\right) \ &\stackrel{2\text{ AM-GM}}{\ge} &\sqrt{\sqrt[n]{\prod_{i=1}^n a_i} \times \sqrt[n]{\prod_{i=n+1}^{2n} a_i}} \ &=& \sqrt[2n]{\prod_{i=1}^{2n} a_i}, \end{array}2 n 1i=1∑2 na i=≥n AM-GM≥2 AM-GM=2 n 1(i=1∑na i+i=n+1∑2 na i)2 1(n∏i=1 na i+n∏i=n+1 2 na i)n∏i=1 na i×n∏i=n+1 2 na i2 n∏i=1 2 na i,
where n AM-GM n\text{ AM-GM}n AM-GM means AM-GM inequality applied on n n n variables. We've also used the base case, with 2 2 2 variables. So the second part of our proof is also complete. It remains to show that P(n)⟹P(n−1)P(n)\implies P(n-1)P(n)⟹P(n−1). To show this, we take n n n positive reals a 1,a 2,...,a n−1 a_1,a_2,...,a_{n-1}a 1,a 2,...,a n−1 and a n=1 n−1∑i=1 n−1 a i a_n=\frac{1}{n-1} \sum_{i=1}^{n-1} a_i a n=n−1 1∑i=1 n−1a i. Then notice that 1 n∑i=1 n a i=1 n(∑i=1 n−1 a i+1 n−1∑i=1 n−1 a i)=1 n−1∑i=1 n−1 a i.\dfrac{1}{n}\sum_{i=1}^n a_i =\dfrac{1}{n}\left(\sum_{i=1}^{n-1} a_i + \dfrac{1}{n-1}\sum_{i=1}^{n-1} a_i\right)=\dfrac{1}{n-1}\sum_{i=1}^{n-1} a_i.n 1i=1∑na i=n 1(i=1∑n−1a i+n−1 1i=1∑n−1a i)=n−1 1i=1∑n−1a i.
So we have
1 n−1∑i=1 n−1 a i=1 n∑i=1 n a i≥∏i=1 n a i n=1 n−1(∑i=1 n−1 a i)(∏i=1 n−1 a i)n⇒(1 n−1∑i=1 n−1 a i)n≥1 n−1(∑i=1 n−1 a i)(∏i=1 n−1 a i)⇒(1 n−1∑i=1 n−1 a i)n−1≥∏i=1 n−1 a i⇒1 n−1∑i=1 n−1 a i≥∏i=1 n−1 a i n−1.\begin{array}{rcl} \dfrac{1}{n-1}\displaystyle\sum_{i=1}^{n-1} a_i &=& \dfrac{1}{n}\sum_{i=1}^n a_i \ &\ge &\sqrt[n]{\prod_{i=1}^n a_i} \ &=& \sqrt[n]{\dfrac{1}{n-1}\left(\sum_{i=1}^{n-1} a_i\right) \left(\prod_{i=1}^{n-1} a_i\right)} \ \Rightarrow \left(\dfrac{1}{n-1}\sum_{i=1}^{n-1}a_i \right)^n &\ge & \dfrac{1}{n-1}\left(\sum_{i=1}^{n-1} a_i\right) \left(\prod_{i=1}^{n-1} a_i\right) \ \Rightarrow \left(\dfrac{1}{n-1}\sum_{i=1}^{n-1} a_i \right)^{n-1} &\ge & \prod_{i=1}^{n-1} a_i \ \Rightarrow \dfrac{1}{n-1}\sum_{i=1}^{n-1} a_i &\ge & \sqrt[n-1]{\prod_{i=1}^{n-1} a_i}. \end{array}n−1 1i=1∑n−1a i⇒(n−1 1∑i=1 n−1a i)n⇒(n−1 1∑i=1 n−1a i)n−1⇒n−1 1∑i=1 n−1a i=≥=≥≥≥n 1∑i=1 na in∏i=1 na in n−1 1(∑i=1 n−1a i)(∏i=1 n−1a i)n−1 1(∑i=1 n−1a i)(∏i=1 n−1a i)∏i=1 n−1a in−1∏i=1 n−1a i.
As you must've noticed, the first inequality here is n n n variable AM-GM. So the third part of the proof is complete, and the induction as well. Clearly the equalities, inductively, hold if and only if a 1=a 2=⋯=a n a_1=a_2=\cdots =a_n a 1=a 2=⋯=a n. □_\square□
We see that the AM-GM inequality is one of the special cases of Jensen's inequality. So let's approach it that way.
Consider the function f(x)=logx∀x>0.f(x) = \log x \ \forall x > 0.f(x)=lo g x∀x>0.
We observe that f′′(x)=−1 x 2<0 f^{\prime\prime}(x) = \frac{-1}{x^2} < 0 f′′(x)=x 2−1<0. Therefore, f(x)f(x)f(x) turns out to be a concave function. (\big((Also notice that we can conclude f(x)f(x)f(x) is concave using the graph of logx.)\log x.\big)lo g x.) Thus, by Jensen's inequality, we have
f(∑i=1 n a i n)≥∑i=1 n f(a i)n⇒log(∑i=1 n a i n)≥∑i=1 n log(a i)n.\begin{aligned} f\left(\dfrac{\sum_{i=1}^n a_i}{n}\right) & \geq \dfrac{\sum_{i=1}^n f\left(a_i\right)}{n} \ \Rightarrow \log \left(\dfrac{\sum_{i=1}^n a_i}{n}\right) & \geq \dfrac{\sum_{i=1}^n \log \left(a_i\right)}{n}. \end{aligned}f(n∑i=1 na i)⇒lo g(n∑i=1 na i)≥n∑i=1 nf(a i)≥n∑i=1 nlo g(a i).
By the property of logarithm, we have logx+logy=logx y\log x + \log y = \log xy lo g x+lo g y=lo g x y and b loga=loga b b\log a = \log a^b b lo g a=lo g a b. Therefore, we can simplify the terms on the RHS as
log(∑i=1 n a i n)≥∑i=1 n loga i n=loga 1+loga 2+⋯+loga n n=log(a 1 a 2 a 3…a n)n=log(∏i=1 n a i)1/n.\begin{aligned} \log \left(\dfrac{\sum_{i=1}^n a_i}{n}\right) & \geq \dfrac{\sum_{i=1}^n \log a_i}{n} \ & = \dfrac{\log a_1+\log a_2+ \cdots + \log a_n}{n} \ & = \dfrac{\log \left(a_1 a_2 a_3 \ldots a_n\right)}{n} \ & = \log \left(\prod_{i=1}^n a_i\right)^{1/n}. \end{aligned}lo g(n∑i=1 na i)≥n∑i=1 nlo g a i=n lo g a 1+lo g a 2+⋯+lo g a n=n lo g(a 1a 2a 3…a n)=lo g(i=1∏na i)1/n.
Thus,
∑i=1 n a i n≥∏i=1 n a i n.□\dfrac{ \sum_{i=1}^n a_i}{n} \geq \sqrt[n]{\prod_{i=1}^n a_i}.\ _\square n∑i=1 na i≥n i=1∏na i.□
Another proof, made famous by a mathematician George Pólya, does not rely on induction. This shows us a more general version of the AM-GM inequality.
Suppose that for a sequence of positive reals a k a_k a k with 1≤k≤n 1\leq k\leq n 1≤k≤n and a sequence of positive reals p k p_k p k with 1≤k≤n 1\leq k \leq n 1≤k≤n such that ∑k=1 n p k=1 \sum_{k=1}^{n} p_k = 1∑k=1 np k=1, a 1 p 1 a 2 p 2 a 3 p 3⋯a n p n≤a 1 p 1+a 2 p 2+a 3 p 3+⋯+a n p n.a_1^{p_1}a_2^{p_2}a_3^{p_3}\cdots a_n^{p_n}\leq a_1p_1+a_2p_2+a_3p_3+\cdots+a_np_n.a 1 p 1a 2 p 2a 3 p 3⋯a n p n≤a 1p 1+a 2p 2+a 3p 3+⋯+a np n. Pólya's proof begins with the observation that 1+x≤e x,1+x\leq e^x,1+x≤e x, which can easily be verified graphically with equality occurring only at x=0 x=0 x=0. This intuition allegedy came to Pólya in a dream, where he claimed it was "the best mathematics he had ever dreamt." If we make the change of variables x↦x−1 x\mapsto x-1 x↦x−1, our initial observation becomes x≤e x−1 x\leq e^{x-1}x≤e x−1. Applying this to our sequence a k,a_k,a k, we get a k≤e a k−1 a k p k≤e a k p k−p k.\begin{aligned} a_k &\leq e^{a_k -1}\ a_k^{p_k} &\leq e^{a_kp_k - p_k}. \end{aligned}a ka k p k≤e a k−1≤e a kp k−p k. Now, we can see that G=∏k=1 n a k p k≤exp(∑k=1 n a k p k−∑k=1 n p k)≤exp(∑k=1 n a k p k−1).\begin{aligned} G=\prod_{k=1}^{n} a_k^{p_k} &\leq \exp \left(\sum_{k=1}^{n} a_kp_k - \sum_{k=1}^{n} p_k\right)\ &\leq \exp \left(\sum_{k=1}^{n} a_kp_k - 1\right). \end{aligned}G=k=1∏na k p k≤exp(k=1∑na kp k−k=1∑np k)≤exp(k=1∑na kp k−1). However, we can also see from the latter of our initial observations that A=∑k=1 n a k p k≤exp(∑k=1 n a k p k−1),A=\sum_{k=1}^{n} a_kp_k \leq \exp \left(\sum_{k=1}^{n} a_kp_k -1\right),A=k=1∑na kp k≤exp(k=1∑na kp k−1), which is the same bound we just found for G G G. So we could say A,G≤exp(∑k=1 n a k p k−1).A, G \leq \exp \left (\sum_{k=1}^{n} a_kp_k -1 \right ).A,G≤exp(k=1∑na kp k−1). So we have related A A A and G G G by inequality, but we have not separated them. Now we look closer at the case where A A A and G G G are equal to the expression on the right. The idea of "normalization" then comes to mind. In other words, we can somewhat manipulate the sequence a k a_k a k to our advantage. Define a new sequence α k\alpha_k α k with 1≤k≤n,1\leq k \leq n,1≤k≤n, where we have α k=a k A A=a 1 p 1+a 2 p 2+a 3 p 3+⋯+a n p n.\begin{aligned} \alpha_k &= \frac{a_k}{A}\ A&=a_1p_1+a_2p_2+a_3p_3+\cdots+a_np_n. \end{aligned}α kA=A a k=a 1p 1+a 2p 2+a 3p 3+⋯+a np n. Applying our earlier bound for G G G for our new variables α k\alpha_k α k, we get ∏k=1 n α k p k≤exp(∑k=1 n α k p k−1)∏k=1 n(a k A)p k≤exp(∑k=1 n a k A p k−1).\begin{aligned} \prod_{k=1}^{n} \alpha_k^{p_k} &\leq \exp \left(\sum_{k=1}^{n} \alpha_kp_k-1\right)\ \prod_{k=1}^{n} \left(\frac{a_k}{A}\right)^{p_k} &\leq \exp \left(\sum_{k=1}^{n} \frac{a_k}{A}p_k-1\right). \end{aligned}k=1∏nα k p kk=1∏n(A a k)p k≤exp(k=1∑nα kp k−1)≤exp(k=1∑nA a kp k−1). Then we have ∑k=1 n a k A p k=a 1 p 1+a 2 p 2+a 3 p 3+⋯+a n p n A=A A=1,\sum_{k=1}^{n} \frac{a_k}{A}p_k = \frac{a_1p_1+a_2p_2+a_3p_3+\cdots+a_np_n}{A} = \frac{A}{A} = 1,k=1∑nA a kp k=A a 1p 1+a 2p 2+a 3p 3+⋯+a np n=A A=1, and it follows that exp(∑k=1 n a k A p k−1)=1.\exp \left(\sum_{k=1}^{n} \frac{a_k}{A}p_k-1\right) = 1.exp(k=1∑nA a kp k−1)=1. So now, we are led to the fact that ∏k=1 n(a k A)p k≤1∏k=1 n a k p k∏k=1 n A p k≤1.\begin{aligned} \prod_{k=1}^{n} \left(\frac{a_k}{A}\right)^{p_k} &\leq 1\ \frac{\prod_{k=1}^{n} a_k^{p_k}}{\prod_{k=1}^{n} A^{p_k}} &\leq 1. \end{aligned}k=1∏n(A a k)p k∏k=1 nA p k∏k=1 na k p k≤1≤1. Since we have ∑k=1 n p k=1,\displaystyle \sum_{k=1}^{n} p_k = 1,k=1∑np k=1, we can finally say G A≤1 G≤A a 1 p 1 a 2 p 2 a 3 p 3...a n p n≤a 1 p 1+a 2 p 2+a 3 p 3+⋯+a n p n.□\begin{aligned} \frac{G}{A} &\leq 1\ G &\leq A\ a_1^{p_1}a_2^{p_2}a_3^{p_3}...a_n^{p_n} &\leq a_1p_1+a_2p_2+a_3p_3+\cdots+a_np_n.\ _\square \end{aligned}A GG a 1 p 1a 2 p 2a 3 p 3...a n p n≤1≤A≤a 1p 1+a 2p 2+a 3p 3+⋯+a np n.□
The following proof is way more intuitive and requires a bit of combinatorics.
Let S S S be a set of positive real numbers. This set contains n n n terms, and the n th n^\text{th}n th power (which is a positive integer because n n n represents the number of elements on a set) of this summation can be expressed as follows: (a 1+a 2+⋯+a n)n=(a 1+a 2+⋯+a n)(a 1+a 2+⋯+a n)⋯(a 1+a 2+⋯+a n)⏟n times.(a_1+a_2+\cdots+a_n)^{n}=\underbrace{(a_1+a_2+\cdots+a_n) (a_1+a_2+\cdots+a_n) \cdots (a_1+a_2+\cdots+a_n)}_{n \text{ times}}.(a 1+a 2+⋯+a n)n=n times(a 1+a 2+⋯+a n)(a 1+a 2+⋯+a n)⋯(a 1+a 2+⋯+a n). We know that there exists a term which is the product of all the terms of S S S. As we can take one term per bracket to combine with the other brackets, the combined term a 1 a 2 a 3…a n a_1 a_2 a_3 \ldots a_n a 1a 2a 3…a n appears n!n!n! times, because there are n n n ways to choose a term on the first bracket, n−1 n-1 n−1 on the second one, and so on.
As every term is positive, any n th n^\text{th}n th power expansion is greater than its summand, and thus we get (∑i=1 n a i)n>n!(∏i=1 n a i).\left(\sum_{i=1}^n a_i \right)^{n} >n! \left(\prod_{i=1}^n a_i\right).(i=1∑na i)n>n!(i=1∏na i). Now, we only need to remember the definition of geometric mean, which is the n th n^\text{th}n th root of the product between all terms on a set followed by some algebraic manipulations. Then we achieve ∑i=1 n a i(n!)1 n=k>m g,\frac{ \sum_{i=1}^n a_i}{(n!)^{\frac{1}{n}}}=k> m_g,(n!)n 1∑i=1 na i=k>m g, where m g m_g m g is the geometric mean. Now we can prove by induction that for every positive integer n n n, n n≥n!n^{n} \geq n!n n≥n!, and this proof will help us out to eliminate the factorial on the denominator of the last inequality. For n=1 n=1 n=1 this becomes 1 1≥1!1^{1} \geq 1!1 1≥1!, which is true. Then we assume the inequality is valid for n=k n=k n=k and apply the inequality for n=k+1 n=k+1 n=k+1, and prove that must be true too. Then, there we go: (k+1)k+1=k k+1+(k+1)⋅k k⋅1 1+k⋅(k+1)2⋅k k−1⋅1 2+⋯+1 k+1>k k+1+(k+1)⋅k k>k k⋅(2 k+1).\begin{aligned} (k+1)^{k+1} &=k^{k+1}+(k+1)\cdot k^{k}\cdot 1^{1}+\frac{k\cdot (k+1)}{2}\cdot k^{k-1}\cdot 1^{2}+\cdots +1^{k+1}\ &>k^{k+1}+(k+1)\cdot k^{k}\ &>k^{k}\cdot (2k+1). \end{aligned}(k+1)k+1=k k+1+(k+1)⋅k k⋅1 1+2 k⋅(k+1)⋅k k−1⋅1 2+⋯+1 k+1>k k+1+(k+1)⋅k k>k k⋅(2 k+1). But from induction hypothesis, we have k k≥k!k k.(2 k+1)≥k!.(2 k+1)k k.(2 k+1)≥k!.[k+(k+1)]k k⋅(2 k+1)≥k⋅k!+(k+1)!>(k+1)!⇒k k⋅(2 k+1)>(k+1)!.\begin{aligned} k^{k} &\geq k! \ k^{k}.(2k+1) &\geq k!.(2k+1) \ k^{k}.(2k+1) &\geq k!.[k+(k+1)]\ k^{k}\cdot (2k+1) &\geq k\cdot k!+(k+1)! \ &>(k+1)! \ \Rightarrow k^{k}\cdot(2k+1)&>(k+1)!. \end{aligned}k k k k.(2 k+1)k k.(2 k+1)k k⋅(2 k+1)⇒k k⋅(2 k+1)≥k!≥k!.(2 k+1)≥k!.[k+(k+1)]≥k⋅k!+(k+1)!>(k+1)!>(k+1)!.
With (k+1)k+1>k k⋅(2 k+1)(k+1)^{k+1}>k^{k}\cdot (2k+1)(k+1)k+1>k k⋅(2 k+1) and k k⋅(2 k+1)>(k+1)!k^{k}\cdot (2k+1)>(k+1)!k k⋅(2 k+1)>(k+1)!, we get (k+1)k+1>(k+1)!(k+1)^{k+1} >(k+1)!(k+1)k+1>(k+1)!. But for k=0 k=0 k=0 we already saw that there's equality, and it becomes (k+1)k+1≥(k+1)!(k+1)^{k+1} \geq (k+1)!(k+1)k+1≥(k+1)! as we wished to prove. Now, we can use the inequality n n≥n!n^{n} \geq n!n n≥n!, which is valid for every positive integer. Some algebraic manipulations can be made as follows: n n≥n!1 n n≤1 n!1 n≤1(n!)1 n∑i=1 n a i n≤∑i=1 n a i(n!)1 n m a≤∑i=1 n a i(n!)1 n=k,\begin{aligned} n^{n} &\geq n! \ \frac{1}{n^{n}} &\leq \frac{1}{n!} \ \frac{1}{n} &\leq \frac{1}{(n!)^{\frac{1}{n}}} \ \frac{ \sum_{i=1}^n a_i}{n} &\leq \frac{ \sum_{i=1}^n a_i}{(n!)^{\frac{1}{n}}} \ m_a &\leq \frac{ \sum_{i=1}^n a_i}{(n!)^{\frac{1}{n}}}=k, \end{aligned}n n n n 1n 1n∑i=1 na im a≥n!≤n!1≤(n!)n 11≤(n!)n 1∑i=1 na i≤(n!)n 1∑i=1 na i=k, where m a m_a m a is the arithmetic mean. Now, a geometric interpretation on the real line makes the results easy to understand: k>m g>0⟹m g∈(0,k)k≥m a>0⟹m a∈(0,k].\begin{aligned} k>m_g>0 &\implies m_g \in (0, k) \ k\geq m_a>0 &\implies m_a \in (0, k]. \end{aligned}k>m g>0 k≥m a>0⟹m g∈(0,k)⟹m a∈(0,k]. When m a m_a m a equals its limit, m g m_g m g can only be smaller than m g m_g m g. Otherwise, both equal each other. Hence, the inequality we were searching for: m g≤m a.□m_g \leq m_a.\ _\square m g≤m a.□
Examples with AM-GM inequality
Main Article: Applying AM-GM
You can refer to the article linked above for further problem solving on the applications of AM-GM inequality. In this section, we will work through some examples and problems based on the usage of AM-GM inequality.
If x,y∈R+x,y\in{\Bbb R^{+}}x,y∈R+ and x+y=8 x+y=8 x+y=8, then find the minimum value of (1+1 x)(1+1 y)\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)(1+x 1)(1+y 1).
The given expression can be rewritten as
(1+1 x)(1+1 y)=1+x+y+x y x y=9+x y x y=9 x y+1.\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)=\frac{1+x+y+xy}{xy}=\frac{9+xy}{xy}=\frac{9}{xy}+1.(1+x 1)(1+y 1)=x y 1+x+y+x y=x y 9+x y=x y 9+1.
From AM-GM
x+y 2≥x y 4≥x y 16≥x y 1 16≤1 x y⇒9 x y+1≥25 16.\begin{aligned} \frac{x+y}{2}&\ge\sqrt{xy}\ 4&\ge\sqrt{xy}\ 16&\ge xy\ \frac{1}{16}&\le\frac{1}{xy}\ \Rightarrow \frac{9}{xy}+1&\geq\frac{25}{16}. \end{aligned}2 x+y4 16 16 1⇒x y 9+1≥x y≥x y≥x y≤x y 1≥16 25.
Therefore, the minimum value of (1+1 x)(1+1 y)\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)(1+x 1)(1+y 1) is 25 16\frac{25}{16}16 25. □_\square□
Note: The equality holds true when x=y=4.x=y=4.x=y=4. These values can be derived by noting that the minimum value is achieved when x y=16,xy=16,x y=16, along with the given constraint that x+y=8.x + y = 8.x+y=8.
Show that 2 a 3+b 3≥3 a 2 b 2a^3 + b^3 \geq 3a^2 b 2 a 3+b 3≥3 a 2 b for a,b>0 a, b >0 a,b>0.
We apply the 3-variable version of AM-GM with x 1=a 3,x 2=a 3 x_1= a^3, x_2 = a^3 x 1=a 3,x 2=a 3 and x 3=b 3 x_3 = b^3 x 3=b 3 to obtain
a 3+a 3+b 3 3≥a 3⋅a 3⋅b 3 3=a 2 b. \frac { a^3 + a^3 + b^3 } {3} \geq \sqrt{a^3\cdot a^3 \cdot b^3} = a^2 b.3 a 3+a 3+b 3≥3 a 3⋅a 3⋅b 3=a 2 b.
Then we multiply both sides by 3 to obtain a 3+a 3+b 3≥3 a 2 b a^3 + a^3 + b^3 \geq 3a^2 b a 3+a 3+b 3≥3 a 2 b. □_\square□
Find all real solutions to 2 x+x 2=2−1 2 x 2^x + x^2 = 2 - \frac {1}{2^x}2 x+x 2=2−2 x 1.
We have 2≤2 x+1 2 x=2−x 2 2 \leq 2^x + \frac {1}{2^x} = 2 - x^2 2≤2 x+2 x 1=2−x 2, so 0≥x 2 0 \geq x^2 0≥x 2. This implies x=0 x=0 x=0 is the only possible value. Since 2 0+0 2=1 2^0 + 0^2 = 1 2 0+0 2=1 and 2−1 2 0=1 2 - \frac {1}{2^0} = 1 2−2 0 1=1, we have verified x=0 x=0 x=0 is the only solution. □_\square□
Find all positive real solutions to
4 x+18 y=14,2 y+9 z=15,9 z+16 x=17. \begin{array}{c}&4x + \frac {18}{y} = 14, &2y + \frac {9}{z} = 15, &9z + \frac {16}{x} = 17. \end{array} 4 x+y 18=14,2 y+z 9=15,9 z+x 16=17.
By AM-GM, we have 4 x+16 x≥16,2 y+18 y≥12,9 z+9 z≥18 4x + \frac {16}{x} \geq 16, 2y + \frac {18}{y} \geq 12, 9z + \frac {9}{z} \geq 18 4 x+x 16≥16,2 y+y 18≥12,9 z+z 9≥18. Summing these three inequalities, we obtain
4 x+16 x+2 y+18 y+9 z+9 z≥16+12+18=46. 4x + \frac {16}{x} + 2y + \frac {18}{y} + 9z + \frac {9}{z} \geq 16 + 12 + 18 = 46 .4 x+x 16+2 y+y 18+9 z+z 9≥16+12+18=46.
Furthermore, summing the three given equations, we obtain
4 x+16 x+2 y+18 y+9 z+9 z=46. 4x + \frac {16}{x} + 2y + \frac {18}{y} + 9z + \frac {9}{z} = 46.4 x+x 16+2 y+y 18+9 z+z 9=46.
Hence, equality must hold throughout, implying x=2,y=3 x=2, y=3 x=2,y=3 and z=1 z=1 z=1. By substituting these values into the original equations, we see that (x,y,z)=(2,3,1) (x, y, z) = (2, 3, 1)(x,y,z)=(2,3,1) is indeed a solution. □_\square□
[2-variable Cauchy Schwarz Inequality]
Show (a 2+b 2)(c 2+d 2)≥(a c+b d)2.\left(a^2 + b^2\right)\left(c^2 + d^2\right) \geq (ac+bd)^2 .(a 2+b 2)(c 2+d 2)≥(a c+b d)2.
Solution 1: Expanding both sides, we can cancel terms a 2 c 2 a^2c^2 a 2 c 2 and b 2 d 2 b^2d^2 b 2 d 2, so we need to show that a 2 d 2+b 2 c 2≥2 a c b d a^2 d^2 + b^2c^2 \geq 2acbd a 2 d 2+b 2 c 2≥2 a c b d. This follows from the 2-variable AM-GM by setting x 1=a 2 d 2 x_1 = a^2 d^2 x 1=a 2 d 2 and x 2=b 2 c 2 x_2 = b^2 c^2 x 2=b 2 c 2, to obtain
a 2 d 2+b 2 c 2≥2∣a b c d∣≥2 a b c d. a^2d^2 + b^2c^2 \geq 2 \lvert abcd \rvert \geq 2 abcd.a 2 d 2+b 2 c 2≥2∣ab c d∣≥2 ab c d.
Solution 2: From completing the square's Fermat's two square theorem, we have
(a 2+b 2)(c 2+d 2)=(a c+b d)2+(a d−b c)2. \big(a^2+b^2\big)\big(c^2+d^2\big) = (ac+bd)^2 + (ad-bc)^2 .(a 2+b 2)(c 2+d 2)=(a c+b d)2+(a d−b c)2.
Since squares are non-negative, the right-hand side is greater than or equal to (a c+b d)2 (ac+bd)^2(a c+b d)2. □_\square□
Show that if a,b a, b a,b, and c c c are positive real numbers, then
a 4+b 4+c 4≥a b c(a+b+c). a^4 + b^4 + c^4 \geq abc(a+b+c). a 4+b 4+c 4≥ab c(a+b+c).
A direct application of AM-GM doesn’t seem to work. Let's consider how we can get terms on the right hand side through AM-GM. To get a 2 b c a^2bc a 2 b c, we will need more of a a a than of b b b or c c c (as in the first example in this section). This gives a hint to try a 4+a 4+b 4+c 4≥4 a 2 b c. a^4 + a^4 + b^4 + c^4 \geq 4 a^2 b c.a 4+a 4+b 4+c 4≥4 a 2 b c. Similarily, we have a 4+b 4+b 4+c 4≥4 a b 2 c a^4 +b^4 +b^4 +c^4 \geq 4ab^2 c a 4+b 4+b 4+c 4≥4 a b 2 c and a 4+b 4+c 4+c 4≥4 a b c 2. a^4 + b^4 + c^4 + c^4 \geq 4abc^2.a 4+b 4+c 4+c 4≥4 ab c 2. Adding these 3 inequalities and dividing by 4 yields a 4+b 4+c 4≥a 2 b c+a b 2 c+a b c 2=a b c(a+b+c).□ a^4 + b^4 + c^4 \geq a^2bc + ab^2c + abc^2 = abc(a+b+c).\ _\square a 4+b 4+c 4≥a 2 b c+a b 2 c+ab c 2=ab c(a+b+c).□
Reveal the answer
For positive real numbers a,a,a,b,b,b, and c,c,c, what is the minimum value of a b+c+b c+a+c a+b?\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}?b+c a+c+a b+a+b c?
The correct answer is: 1.5
Reveal the answer
For all rectangles with perimeter P P P and area A A A, what is the minimum value of P 2 A?\frac{P^2}{A}?A P 2?
The correct answer is: 16
Let's work out the following example problems and try-it-yourself problem:
Find the minimum value of
4+9 x 2 sin2 x x sinx for 0<x<π. \frac{ 4+ 9x^2\sin^2 x}{x\sin x} ~\text{ for }~ 0 < x <\pi.x sin x 4+9 x 2 sin 2 xfor 0<x<π.
We can rewrite the expression 4+9 x 2 sinx 2 x sinx\frac{4+9x^{2}\sin{x}^{2}}{x \sin{x}}x s i n x 4+9 x 2 s i n x 2 as
4 x sinx+9 x sinx.\frac{4}{x \sin{x}}+9x\sin{x}.x sin x 4+9 x sin x.
Since x>0 x>0 x>0, we can find the minimum by using AM-GM:
4 x sinx+9 x sinx 2≥36 4 x sinx+9 x sinx≥12.\begin{aligned} \frac{\frac{4}{x \sin{x}}+9x\sin{x}}{2}&\geq\sqrt{36}\ \frac{4}{x \sin{x}}+9x\sin{x}&\geq12. \end{aligned}2 x s i n x 4+9 x sin xx sin x 4+9 x sin x≥36≥12.
So the minimum value is 12 12 12. □_\square□
Reveal the answer
For x>0 x> 0 x>0, maximize f(x)=(1+x)(1+x)(1−x)f(x)=(1+x)(1+x)(1-x)f(x)=(1+x)(1+x)(1−x).
Give your answer up to 3 decimal places.
The correct answer is: 1.185
Generalization: Weighted AM-GM Inequality
The weighted AM-GM inequality states that for non-negative numbers a 1,...,a n a_1,...,a_n a 1,...,a n and non-negative weights ω 1,...,ω n \omega_1,...,\omega_n ω 1,...,ω n
∑ω i a i∑ω i≥∏a i ω i∑ω i.□ \large\dfrac {\displaystyle\sum \omega_i a_i } { \displaystyle\sum \omega_i } \geq \sqrt[\sum\omega_i]{\displaystyle\prod a_i ^ {\omega_i }}.\ _\square∑ω i∑ω ia i≥∑ω i∏a i ω i.□
To prove the weighted AM-GM inequality, we will use the same approach of Jensen's inequality as we used above to prove AM-GM inequality. Here we will use the finite form of Jensen's inequality for the natural logarithm.
All values a k a_k a k with weight ω k=0\omega_k = 0 ω k=0 have no influence on the inequality, so let's assume that all weights are positive. If all a k a_k a k are equal, then equality holds. Therefore, it remains to prove strict inequality if they are not all equal. If at least one a k a_k a k is zero (but not all), then the weighted geometric mean is zero, while the weighted arithmetic mean is positive, and hence strict inequality holds. Thus, assuming all a k a_k a k are not equal, since the natural logarithm is strictly concave, the finite form of Jensen's inequality and the functional equations of the natural logarithm say
ln(ω 1 a 1+⋯+ω n a n ω)>w 1 w lna 1+⋯+w n w lna n=ln(a 1 ω 1 x 2 ω 2⋯a n ω n∑ω i). \begin{aligned} \ln\Bigl(\frac{\omega_1a_1+\cdots+\omega_na_n} \omega \Bigr) &>\frac{w_1}w\ln a_1+\cdots+\frac{w_n}w\ln a_n \ & =\ln \left( \sqrt[\sum\omega_i]{a_1^{\omega_1} x_2^{\omega_2} \cdots a_n^{\omega_n}} \right) . \end{aligned} ln(ω ω 1a 1+⋯+ω na n)>w w 1ln a 1+⋯+w w nln a n=ln(∑ω ia 1 ω 1x 2 ω 2⋯a n ω n).
We know that the natural logarithm is strictly increasing, and therefore we have
ω 1 a 1+⋯+ω n a n ω>a 1 ω 1 a 2 ω 2⋯a n ω n∑ω i.□\frac{\omega_1a_1+\cdots+\omega_na_n}{\omega} > \sqrt[\sum\omega_i]{a_1^{\omega_1} a_2^{\omega_2} \cdots a_n^{\omega_n}}.\ _\square ω ω 1a 1+⋯+ω na n>∑ω ia 1 ω 1a 2 ω 2⋯a n ω n.□
See Also
Applying the Arithmetic Mean - Geometric Mean Inequality
Power Mean (QAGH)
Cauchy-Schwarz Inequality
Inequalities with Strange Equality Conditions
Cite as: Arithmetic Mean - Geometric Mean. Brilliant.org. Retrieved 21:57, September 28, 2025, from
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Application CenterApplicationsLesson 5: Application: Mixing ProblemsPreview
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Lesson 5: Application: Mixing Problems
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Lesson05.mw
Outline of Lesson 5
Initialization
5.A One-Tank Mixing Problem
Problem Statement
Solution
(a)
(b)
(c)
5.B Variable Volume Mixing Problem
Problem Statement
Solution
(a)
(b)
(c)
(d)
(e)
5.C Two-Tank Mixing Problem
Problem Statement
Solution
ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL
Lesson 5 -- Application: Mixing Problems
Prof. Douglas B. Meade
Industrial Mathematics Institute
Department of Mathematics
University of South Carolina
Columbia, SC 29208
URL:
E-mail: meade@math.sc.edu
Copyright 2001 by Douglas B. Meade
All rights reserved
>
Outline of Lesson 5
5.A One-Tank Mixing Problem
5.B Variable Volume Mixing Problem
5.C Two-Tank Mixing Problem
>
Initialization
>restart;
>with( DEtools ):
>with( plots ):
>with( PDEtools ):
Warning, the name changecoords has been redefined
>
5.A One-Tank Mixing Problem
Problem Statement
A tank initially contains 40 gal of sugar water having a concentration of 3 lb of sugar for each gallon of water. At time zero, sugar water with a concentration of 4 lb of sugar per gal begins pouring into the tank at a rate of 2 gal per minute. Simultaneously, a drain is opened at the bottom of the tank so that the volume of the sugar-water solution in the tank remains constant.
(a) How much sugar is in the tank after 15 minutes?
(b) How long will it take the sugar content in the tank to reach 150 lb? 170 lb?
(c) What will be the eventual sugar content in the tank?
>
Solution
The mathematical formulation of this problem must express the physical requirement that
( amount of sugar in tank) = ( rate sugar is added to tank ) - ( rate sugar is removed from tank )
Let denote the amount of sugar (pounds) in the tank at time (minutes). Then, the rates in and out are respectively
>rate_in := 4 2;
>rate_out := (x(t)/40) 2;
>
so that the governing ODE is
>ode := diff( x(t), t ) = rate_in - rate_out;
>
The amount of sugar in the tank initially, that is, when , gives the initial condition
>ic := x(0)=40 3;
>
The ODE in this IVP is first-order and linear. The integrating factor is
>mu(t) = intfactor( ode );
>
The solution to the IVP is
>sol := dsolve( { ode, ic }, x(t), [linear] );
>
(a)
The amount of sugar (in pounds) in the tank after 15 minutes is
>eval( sol, t=15. );
>
(b)
The tank will contain 150 pounds of sugar at a time (in minutes) satisfying
>eq150 := eval( sol, x(t)=150 );
>
Thus, the desired time is found by the calculations
>t150 := solve( eq150, t ):
>t[150 lbs] = t150;
>`` = evalf(t150);
>
Repeating the same steps for the time when 170 pounds of sugar are in the tank leads to the equation
>eq170 := eval( sol, x(t)=170 );
>
whose solution is
>t170 := solve( eq170, t ):
>t[170 lbs] = t170;
`` = evalf(t170);
>
This complex-valued solution is clearly not physically realistic. A quick inspection of the solution, graphed in Figure 5.1,
>plot( rhs(sol), t=0..120, title="Figure 5.1" );
>
shows that the amount of sugar in the tank reaches a steady-state limit that is well below 170 pounds. Therefore, at no time is there ever 170 pounds of sugar in the tank.
>
(c)
In (b) it was noted that the amount of sugar in the tank levels off below 170 pounds. The exact limit can be determined from the solution by looking at the limit as , that is, at
>steady_state := map( Limit, sol, t=infinity );
>
whose value is
>value( steady_state );
>
Note that is an equilibrium solution for this ODE. However, be careful to avoid the common error of concluding that the limit is 160 pounds because is an equilibrium solution. (Recall the logistic growth model, Lesson 3, Section B , which has two equilibria.)
>
>
5.B Variable Volume Mixing Problem
Problem Statement
Consider the previous problem, except that the outflow from the tank is at a rate of 3 gallons per minute.
(a) Find the formula for the volume of sugar water in the tank at any time. When is the tank empty?
(b) Find the IVP for the amount of sugar in the tank.
(c) Find the IVP for the concentration of sugar in the water.
(d) When is the tank empty? What is the concentration of sugar immediately before the tank is empty? How much sugar is in the tank at this time?
(e) Plot the amount of sugar and concentration of sugar in the tank up to the time the tank becomes empty. What happens to these solutions at later times?
>
Solution
(a)
The volume starts at 40 gallons. Every minute 2 gallons of sugar water are added to the tank and 3 gallons are removed; the net change is a loss of 1 gallon per minute. The rates "in" and "out" are respectively
>Vrate_in := 2;
>Vrate_out := 3;
>
so the time-varying volume in the tank is
>V := 40 + (Vrate_in-Vrate_out)t;
>
Note that is the solution of the IVP
>odeV := diff( v(t), t ) = 2 - 3;
>icV := v(0) = 40;
>
as confirmed via
>dsolve( { odeV, icV }, v(t), [separable] );
>
(b)
The IVP for the amount of sugar in the tank is similar to the one in the previous example. There is no difference in the rate at which sugar enters the tank. The concentration of sugar exiting the tank is and this is different because is no longer constant. The rates "in" and "out" are now respectively
>Srate_in := rate_in;
>Srate_out := (x(t)/V) 3;
>
Thus, the governing ODE is
>ode2 := diff( x(t), t ) = Srate_in - Srate_out;
>
The initial condition is unchanged:
>ic2 := ic;
>
(c)
The IVP for the concentration of sugar in the tank is obtained from the ODE in (b) and the definition of the concentration, , which is
>conc_eq := c(t) = x(t)/'V';
>
Rather than deriving the differential equation for manually, the dchange command from the PDEtools package will be used to automate the process. It gives
>odeC := dchange( x(t) = c(t)V, ode2, [c] );
>
The initial condition for the concentration is
>icC := c(0) = 3;
>
(d)
The tank is empty when the volume of sugar water is zero. This occurs after 40 minutes, as obtained by
>t_empty := solve( V=0, {t} );
>
The concentration is found by solving the (linear) IVP found in (c), which is done via
>solC := dsolve( { odeC, icC }, c(t), [linear] );
>
Thus, the concentration at the instant the tank empties is
>subs( t_empty, solC );
>
The amount of sugar in the tank as the tank empties is obtained by evaluating the solution
>sol2 := dsolve( {ode2,ic2}, x(t), [linear] );
>
at the time the tank empties. This gives
>subs( t_empty, sol2 );
>
which is exactly what one would expect. (If not, think about it!)
>
(e)
The requested plots are
>plot( rhs(sol2), t=0..40, title=Amount of sugar );
>
and
>plot( rhs(solC), t=0..40, title=Concentration of sugar );
>
The concentration remains positive until , but after , the volume and amount of sugar become negative. Even though the IVPs have solutions for all time, for these results are not physically meaningful.
>
5.C Two-Tank Mixing Problem
Problem Statement
Two tanks, tank I and tank II, are filled with gal of pure water. A solution containing lb of salt per gallon is poured into tank I at a rate of gal per minute. The solution leaves tank I at a rate of gal/min and enters tank II at the same rate ( gal/min). A drain is adjusted on tank II and the solution leaves tank II at a rate of gal/min. This keeps the volume of solution constant in both tanks ( gal). Show that the amount of salt solution in tank II, as a function of time t, is given by
.
NOTE: This is Exercise 14 (p. 73) from Differential Equations,
by Polking, Boggess, and Arnold (Prentice-Hall, 2001).
>
Solution
Let and denote the amount of salt, in pounds, in tanks I and II, respectively, at time . The initial conditions for the two tanks are respectively
>icI := x(0) = 0;
>icII := y(0) = 0;
>
The "rate in" for tank I is lb/gal gal/min and the outflow is lb/gal at a rate of gal/min. Therefore, the ODE governing the amount of salt in tank I is
>unassign('V');
>odeI := diff( x(t), t ) = ab - b(x(t)/V);
>
For tank II, the inflow is the same as the outflow from tank I and the outflow exactly matches the inflow. Thus, the ODE governing the amount of salt in tank II is
>odeII := diff( y(t), t ) = b(x(t)/V) - b(y(t)/V);
>
The first equation in this pair is independent of the variable . Hence, it can be solved first for , and that result substituted into the second equation, making the second equation depend only on . In general, both equations of a system will contain both variables, and the equations will then be coupled. Solution techniques for such systems will be developed in succeeding lessons.
In this example, the first equation is a first-order linear ODE with integrating factor
>mu1 = intfactor( odeI );
>
The solution to the IVP for tank I is therefore
>solI := dsolve( {odeI,icI}, x(t), [linear] );
>
To find the amount of salt in tank II, substitute the solution for tank I into the ODE for tank II, obtaining
>odeIIa := subs( solI, odeII );
>
Despite the somewhat complicated appearance of this ODE, note that it is linear, as is verified by
>odeadvisor( odeIIa );
>
The integrating factor is then found to be
>mu2 = intfactor( odeIIa );
>
The solution to the IVP for tank II is
>solII := dsolve( {odeIIa,icII}, y(t), [linear] );
>
Thus, the amount of salt in tank II at any time , is
>expand( solII );
>
[Back to ODE Powertool Table of Contents]
>
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189424 | https://www.irishstatutebook.ie/eli/2014/act/38/section/972/enacted/en/html | Companies Act 2014, Section 972
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189425 | http://www.nhn.ou.edu/~gut/notes/cm/lect_03.pdf | September 3, 2003 16:35:04 P. Gutierrez Physics 5153 Classical Mechanics Principle of Virtual Work 1 Introduction The first variational principle we encounter in mechanics is the principle of virtual work. It estab-lishes the equilibrium condition of a mechanical system, and is fundamental for the later develop-ment of analytical mechanics (Lagrangian and Hamiltonian methods).
The concept of virtual work is centered on the idea of calculating the amount of work done on a system of particles through a virtual displacement. We will start by defining what we mean by a virtual displacement, then discuss virtual work, state the principle of virtual work, and consider an example of its use.
1.1 Virtual Displacement To define what we mean by a virtual displacement, let’s consider a system composed of N particles, possibly subject to some set of constraints, defined by 3N Cartesian coordinates (xi) relative to an inertial frame. Let’s assume that at some instant of time the system undergoes infinitesimal displacements that are virtual in the sense that they occur without the passage of time (instanta-neous), do not necessarily conform to the constraints. This change, δ⃗ xi, in the configuration of the system is known as a virtual displacement.
In the usual case, a virtual displacement conforms to the instantaneous constraints, that is, moving constraints are assumed stopped during the displacement. For example, consider a system subject to n holonomic constraints fi(x1, . . . , x3N, t) = 0 (1) A total derivative corresponds to a infinitesimal displacement of the system and is give by d fi = X j ∂fi ∂xi dxj + ∂fi ∂t dt = 0 (2) notice that this gives both a spacial and a temporal displacement. In the case of a virtual dis-placement, we assume that the temporal displacement is zero, therefore the constraint changes by δfi = X j ∂fi ∂xi δxj = 0 (3) It is important to note the difference, the displacement occurs in zero time.
One question that may be asked, are there any conditions under which a real and virtual displacement are the same? The answer can be seen by comparing Eqs. 2 and 3. If the constraint equation is scleronomic, a virtual displacement is the same as a real displacement. Therefore, in the general case virtual and real displacements are not the same, but in the scleronomic case they are.
Principle of Virtual Work-1 lect 03.tex September 3, 2003 16:35:04 P. Gutierrez Before concluding our discussion of virtual displacements, let’s consider the nonholonomic case where the constraint is given in terms of derivatives. Assume n constraint equations on a system of 3N degrees of freedom X i ajidxi + ajtdt = 0 (4) where j corresponds to the jth constraint. Based on our definition of a virtual displacement con-sistent with the constraints, a virtual displacement for nonholonomic constraints is given by X i ajiδxi = 0 (5) This equation will become important later when we discuss calculating forces of constraint through the Lagrange multiplier method.
So far we have considered virtual displacements in terms of Cartesian coordinates.
Virtual displacements in terms of generalized coordinates are also possible. Simply transform the Cartesian constraint equations to the generalized coordinates. The form of the constraint equation is given by X i ajidqi + ajtdt = 0 (6) where replacing the a with ∂fj ∂qi and ∂fj ∂t (7) gives the holonomic constraint. For a virtual displacement, the constraint equation becomes X i ajiδqi = 0 (8) Therefore the form is the same using any set of coordinates.
1.2 Virtual Work Let’s again consider a system of N particles with 3N degrees of freedom whose configuration is given by the Cartesian coordinates x1 . . . x3N. In addition, suppose that the forces F1 . . . F3N are acting on the particles at the corresponding coordinates in a positive sense. The virtual work is given by δW = X i Fiδxi = X i ⃗ Fi · δ⃗ ri (9) The second equality implies that the virtual work is independent of coordinates used. The equation can be transformed as follows to any set of generalized coordinates δW = X j ÃX i Fi ∂xi ∂qj !
δqj (10) From this equation, we define the generalized force as Qj = ÃX i Fi ∂xi ∂qj !
⇒ δW = X j Qjδqj (11) Principle of Virtual Work-2 lect 03.tex September 3, 2003 16:35:04 P. Gutierrez where we note that the generalized force does not have to have units of a force, just like the generalized coordinates do not have to have units of a length. But, the product of generalized force and coordinates has the units of work (energy).
In the expression for virtual work, the forces are assumed to remain constant throughout the vir-tual displacement. This is true even if the forces vary drastically over a infinitesimal displacement.
A sudden change of force with position can occur in certain nonlinear systems.
Now assume that the system is subject to constraints. The force can be separated into applied forces ⃗ Fa and constraint forces ⃗ Fc. The virtual work of the constraint forces in terms of generalized coordinates is given by δWc = X i Qc iδqi (12) If the displacement is consistent with the constraint, the virtual work is zero since the force does not act in the direction of the force δWc = X i Qc iδqi = 0 (13) which is referred to as a workless constraint. These will be the type of constraint that we will deal with most often. If the constraints are workless, then the total virtual work on the system is given by the applied forces δW = X i Qa i δqi (14) 1.3 Principle of Virtual Work One of the important applications of the idea of virtual work arises in the study of static equilibrium of mechanical system.
Assume a scleronomic system of N particles.
If the system is in static equilibrium, then Newton’s laws for each of the N particles give ⃗ Fa i + ⃗ Fc i = 0 (15) The virtual work for this system is given by δW = X i ⃗ Fa i · δ⃗ ri + ⃗ Fc i · δ⃗ ri = 0 (16) If we now assume that the constraints are workless, and the virtual displacements reversible (one can replace δ⃗ r with −δ⃗ r), then the condition for static equilibrium is δW = X i ⃗ Fa i · δ⃗ ri = 0 ⇒ δW = X i Qa i δqi = 0 (17) where the second equation is given using generalized coordinates. A very important point to note here is that unlike the Newtonian approach, we do not need to know what the constraint forces are. We only need to know the applied forces.
Now assume that the system is initially motionless, but not in equilibrium. Then one or more of the particles has a net applied force on it, and in accord with Newton’s laws, it will start to move in the direction of the force. Since any motion must be compatible with the constraints, the Principle of Virtual Work-3 lect 03.tex September 3, 2003 16:35:04 P. Gutierrez virtual displacements can be chosen to be in the direction of the actual motion at each point. In this case the virtual work is positive δW = X i ⃗ Fa i · δ⃗ ri + ⃗ Fc i · δ⃗ ri > 0 (18) Since the constraints are workless, the condition becomes δW = X i ⃗ Fa i · δ⃗ ri > 0 (19) If the virtual displacements are reversed, then the virtual work is negative. None-the-less, if the system is not in equilibrium, one can find a set of virtual displacements that will result in the virtual work being nonzero.
These results can be summarized in the principle of virtual work: The necessary and sufficient condition for the static equilibrium of an initially motionless scleronomic system that is subject to workless constraints is that zero virtual work be done by the applied forces in moving through an arbitrary virtual displacement satisfying the constraints.
1.4 Example As a simple consider the system described in Fig. 1,where we want to determine the force F that will keep the system in equilibrium. If we use the Newtonian approach, we require 3 equations to solve the problem X Fx = 0 N1 −F = 0 (20) X Fy = 0 N2 −2mg = 0 X τ = 0 mgℓcos θ −N1ℓsin θ = 0 From this point it is fairly straight forward to solve the problem. One finds F = mg cot θ.
Using the principle of virtual work, we set up the equation as follows mgδy −Fδx = 0 (21) PSfrag replacements m m mg N1 N2 ℓ θ F Figure 1: Two blocks on frictionless surfaces constrained by by a rod to move together.
Principle of Virtual Work-4 lect 03.tex September 3, 2003 16:35:04 P. Gutierrez with the constraint between x and y x = ℓcos θ y = ℓsin θ ) ⇒ ( δx = −δθℓsin θ δy = δθℓcos θ ) ⇒ δx cos θ −δy sin θ = 0 (22) since I have already assumed directions for δx and δy in Eq. 21, the sign here is dropped between the middle and final equations. Combining the two equations (mg cot θ −F)δx = 0 (23) Since the displacement is arbitrary, and this equation must hold for all possible virtual displace-ments, the quantity inside the parenthesis must be zero F = mg cot θ (24) the same as the Newtonian method.
The point of this example is not to show that one method is superior to the other, but that differences in the two methods. In the Newtonian method, we required the constraint force and a set of 3 equations to specify the problem. Using the method of virtual work we need only two equations, one describing the work done and the second describing the constraints. We don’t need the constraint forces.
Principle of Virtual Work-5 lect 03.tex |
189426 | https://www.reliance-foundry.com/blog/acero-de-carbono-vs-acero-inoxidable-es?srsltid=AfmBOophvWk--W70fVLt6nsoqzIxrmGCuHGpj96b2_NWHDT0yKIATm9z | Acero de Carbono vs Acero Inoxidable | Blog Reliance
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Home>Blog>Uncategorized>Acero de Carbono versus Acero Inoxidable
Uncategorized
Acero de Carbono versus Acero Inoxidable
Posted byDavid Hilscher
On August 2, 2022
Comments Off on Acero de Carbono versus Acero Inoxidable
ENGLISH | FRANÇAIS
Las diferencias importan más de lo que cree
El acero de alto carbono es frecuentemente usado para cuchillos de cocina de alta calidad debido a que es resistente al desgaste, lo que significa que permanece afilado por más tiempo.
El acero se puede encontrar en todo tipo de productos, desde la industria aeroespacial hasta utensilios de cocina. Estas aplicaciones tan diversas requieren de un material versátil – y el acero parece ser el adecuado.
“Acero” actualmente describe una familia entera de aleaciones metálicas, con cientos de grados específicos para cada aplicación, sin embargo, la mayoría entiende al acero en dos amplias categorías: acero de carbono y acero inoxidable.
El acero de carbono y el acero inoxidable tienen los mismos ingredientes básicos de hierro y carbono. Su principal diferencia es el contenido de aleación: el acero al carbono tiene un contenido de aleación inferior al 10,5%, mientras que el acero inoxidable debe contener un 10,5% de cromo o más. Esta diferencia esencial es la que le da al acero al carbono y al acero inoxidable sus distintas características físicas.
Composición del Acero
Los elementos esenciales del acero son hierro y carbono. En general, los aceros con alto contenidos de carbono son duros y frágiles, mientras que los aceros con bajo contenido de carbono son dúctiles y resistentes.
Por supuesto, raramente es así de simple. Se pueden añadir elementos de aleación como cromo, molibdeno, níquel, manganeso o silicio para aumentar la resistencia a la corrosión o conseguir un mejor equilibrio entre fuerza y dureza.
Acero de Carbono
El acero de carbono está compuesto de hierro y 0.12 – 2.00 por ciento de carbono. La definición más amplia incluye aceros de aleación, que también pueden contener hasta 10.5 por ciento de contenidos de aleación. Aún dentro de confines de menos de dos puntos porcentuales de carbono, existe una enorme variación en las características físicas, especialmente en la dureza.
Cuando la gente habla acerca del acero de carbono, usualmente se refieren al acero de alto contenido de carbono usado en los cuchillos y herramientas. Los aceros de alto carbono son muy duros, lo que los hace ideales para resistir la abrasión y retener la forma. Estos pueden soportar una fuerza significativa antes de deformarse. Desafortunadamente, los metales duros también son frágiles: cuando se encuentran bajo tensión extrema, los aceros con alto contenido en carbono son más propensos a agrietarse que a doblarse.
Los aceros de bajo carbono son más comunes que los de alto carbono debido a (1) bajos costos de producción, (2) mayor ductilidad, y (3) uso fácil en manufactura. Los aceros de bajo carbono tienden a deformarse bajo estrés en vez de quebrarse y esa ductilidad hace que los aceros de bajo carbono sean fáciles de mecanizar y soldar. Son frecuentemente usados en paneles carrocería de automóviles, pernos, accesorios, tubos sin soldadura y placas de acero.
Acero Inoxidable
El acero inoxidable es usado en plantas químicas y alimenticias debido a su capacidad de funcionar en ambientes húmedos sin oxidarse.
El acero inoxidable contiene hierro, carbono y por lo menos 10.5% de contenido de cromo. El cromo es clave – reacciona con el oxígeno para crear una capa pasiva que protege al acero de la corrosión. Esta protección reduce la probabilidad de oxidación del acero inoxidable – esto es importante para el mobiliario de exteriores, como los bolardos, que se instalan en entornos húmedos. Mientras más alto sea el contenido de cromo, mayor será la resistencia a la corrosión.
Es importante prestar atención al grado de acero inoxidable al comprar utensilios y otros artículos de gran valor. No todos los aceros son creados iguales. Un acero inoxidable con un mínimo de 10.5% de contenido de cromo es mucho más barato y menos duradero que uno de 16% y la diferencia se mostrará en los costos de mantenimiento y vida de servicio.
Cuchillos de Acero Inoxidable vs. Acero de Carbono
Los cocineros en casa frecuentemente eligen cuchillos de acero inoxidable. Los profesionales frecuentemente eligen acero de alto carbono.
¿Por qué? Los cuchillos de acero inoxidable son más comunes y son fáciles de almacenar. Los aceros inoxidables martensíticos son usados generalmente para los cortes más precisos. Estos aceros inoxidables son más duros que otros grados. Sin embargo, no son tan frágiles como los aceros con alto carbono: se pueden meter en el lavavajillas o golpear contra el mármol.
Para el cocinero en casa que no realiza trabajo de precisión esto suele ser suficiente. La prevención de la oxidación ayuda a mantener la hoja afilada en este contexto. Los óxidos de hierro en el filo desafilan un cuchillo tanto como el desgaste.
Los chefs profesionales frecuentemente, (¡pero no siempre!) tienen una preferencia para cuchillos de acero de alto carbono. Estos cuchillos pueden afilarse más y son más resistentes al desgaste. Los grados de alto carbono mantienen un filo más fino incluso frente al duro uso habitual. El metal muy duro es también es muy frágil: cuando las fuerzas de impacto son demasiado elevadas, el metal no se dobla, se rompe. Esto significa que es menos probable que los cuchillos de alto contenido en carbono se deformen al impactarse, perdiendo el filo por una serie de deformaciones. En cambio, mantendrán una línea limpia y agradable, y cuando se rompan, se astillarán.
El acero duro como este necesita de más cuidados. Aplicar aceite antes de guardar para evitar la oxidación, enjuagar después de usar en condiciones muy saladas o ácidas y afilar contra una piedra (en lugar del acero). Estos pasos de mantenimiento hacen que los aceros de alto carbono sean más recomendables para los cocineros de alto nivel o profesionales que cortan rodajas finas como el papel, que para el cocinero doméstico promedio para el que la precisión es menos importante que el mantenimiento fácil, y este perderá el filo por una serie de deformaciones.
¿Debo elegir Acero de Carbono o Acero Inoxidable?
El acero de carbono y el acero inoxidable son diferentes, pero uno no es necesariamente superior al otro. Se trata del contexto. Cada material tiene ventajas y desventajas – la clave es adecuar el acero a los requisitos del trabajo.
ACERO DE ALTO CARBONO
ACERO INOXIDABLE
Vulnerable al oxidación
Frágil
Resistente al uso
Resistente a la oxidación
Menos frágil
Menos resistente al uso
Para más información en acero o acero inoxidable, o para solicitar un cotización para un proyecto personalizado, por favor contáctenos.
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189427 | https://www.wyzant.com/resources/answers/728889/compare-the-graph-of-f-x-x-and-g-x-1-2x | WYZANT TUTORING
Angelia H.
compare the graph of f(X)=X AND G(X)=1.2x
1 Expert Answer
Anderson D.
answered • 11/19/19
M.Ed. in Math Education with a Decade of Tutoring Experience
Hi Angelia!
I hope you've had a good day! Here's your answer:
f(x) = x is a "line with a slope of 1"
Why? Well f(x) is in the form f(x) = mx + b (recognize this?)
That's the equation of a line! "Slope-Intercept Form", to be exact. The only difference here is that f(x) = x is a line with a slope of 1 (m=1), and a y-intercept (b) of 0.
If we compare that to G(x) = 1.2x, it's in the same "y=mx+b" form, and I hope we can see that the slope (the m, or the 'coefficient on x') is 1.2!
So a good answer would include that both f(x) and g(x) are lines, where f(x)'s slope is 1 and g(x)'s slope is 1.2.
Hope this helps!
-Anderson
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189428 | https://brightchamps.com/en-us/math/calculators/dividing-decimals-calculator | Table Of Contents
Summarize this article:
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Last updated on August 5, 2025
Dividing Decimals Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you're cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about dividing decimals calculators.
What is Dividing Decimals Calculator?
A dividing decimals calculator is a tool used to perform division operations involving decimal numbers. Decimal division can be complex and prone to errors without a calculator, especially when dealing with long decimals or repeating patterns. This calculator simplifies the process, providing accurate results quickly and efficiently.
How to Use the Dividing Decimals Calculator?
Given below is a step-by-step process on how to use the calculator: Step 1: Enter the dividend and divisor: Input the decimal numbers you wish to divide into the respective fields. Step 2: Click on calculate: Click the calculate button to perform the division and obtain the result. Step 3: View the result: The calculator will display the result instantly.
How to Divide Decimals Manually?
To divide decimals manually, adjust the divisor to be a whole number by multiplying both the dividend and divisor by the same power of ten. Then, divide as you would with whole numbers. Example: Divide 6.4 by 0.8. Multiply both by 10 to get 64 ÷ 8 = 8. Therefore, 6.4 ÷ 0.8 = 8.
Tips and Tricks for Using the Dividing Decimals Calculator
When using a dividing decimals calculator, there are a few tips and tricks to consider for better accuracy and efficiency: - Ensure both numbers are correctly entered as decimals. - Understand that the calculator provides precise results, but consider the context of your problem. - Use rounding appropriately when interpreting your results, especially in practical applications. - Double-check results for very small or large numbers as human error can occur in input.
Common Mistakes and How to Avoid Them When Using the Dividing Decimals Calculator
While using a calculator, errors can happen if inputs are incorrect or misunderstood.
While using a calculator, errors can happen if inputs are incorrect or misunderstood.
Mistake 1
Misplacing the decimal point in the input
Misplacing the decimal point in the input
Ensure the decimal point is correctly placed in both the dividend and divisor before calculating, as an error can lead to incorrect results.
Ensure the decimal point is correctly placed in both the dividend and divisor before calculating, as an error can lead to incorrect results.
Mistake 2
Rounding too early in the process
Rounding too early in the process
Wait until the final result is obtained before rounding. For example, rounding during intermediate steps can lead to inaccuracies.
Wait until the final result is obtained before rounding. For example, rounding during intermediate steps can lead to inaccuracies.
Mistake 3
Ignoring trailing decimals
Ignoring trailing decimals
Sometimes, calculators provide results with long decimal places. Consider whether the context requires truncating or rounding those decimals.
Sometimes, calculators provide results with long decimal places. Consider whether the context requires truncating or rounding those decimals.
Mistake 4
Forgetting to adjust divisor
Forgetting to adjust divisor
When calculating manually, always adjust the divisor to a whole number by multiplying both numbers by the same factor of ten.
When calculating manually, always adjust the divisor to a whole number by multiplying both numbers by the same factor of ten.
Mistake 5
Relying solely on the calculator for understanding
Relying solely on the calculator for understanding
While calculators provide quick results, having a good grasp of decimal division concepts helps in verifying results and applying them correctly in various scenarios.
While calculators provide quick results, having a good grasp of decimal division concepts helps in verifying results and applying them correctly in various scenarios.
Dividing Decimals Calculator Examples
Problem 1
Divide 15.5 by 1.25.
Use the calculator or follow the steps: Make the divisor a whole number by multiplying both by 100: 1550 ÷ 125 = 12.4 Therefore, 15.5 ÷ 1.25 = 12.4
Use the calculator or follow the steps: Make the divisor a whole number by multiplying both by 100: 1550 ÷ 125 = 12.4 Therefore, 15.5 ÷ 1.25 = 12.4
Explanation
By adjusting the divisor and dividend by the same power of ten, the division becomes straightforward.
By adjusting the divisor and dividend by the same power of ten, the division becomes straightforward.
Problem 2
What is 7.2 divided by 0.3?
Use the calculator or follow the steps: Convert the divisor to a whole number: 72 ÷ 3 = 24 Therefore, 7.2 ÷ 0.3 = 24
Use the calculator or follow the steps: Convert the divisor to a whole number: 72 ÷ 3 = 24 Therefore, 7.2 ÷ 0.3 = 24
Explanation
Multiplying both numbers by 10 simplifies the division process.
Multiplying both numbers by 10 simplifies the division process.
Problem 3
Calculate 9.81 divided by 0.09.
Use the calculator or follow the steps: Convert the divisor to a whole number: 981 ÷ 9 = 109 Therefore, 9.81 ÷ 0.09 = 109
Use the calculator or follow the steps: Convert the divisor to a whole number: 981 ÷ 9 = 109 Therefore, 9.81 ÷ 0.09 = 109
Explanation
Adjusting the divisor and dividend by multiplying by 100 simplifies the division.
Adjusting the divisor and dividend by multiplying by 100 simplifies the division.
Problem 4
How to divide 0.48 by 0.12?
Use the calculator or follow the steps: Make the divisor a whole number: 48 ÷ 12 = 4 Therefore, 0.48 ÷ 0.12 = 4
Use the calculator or follow the steps: Make the divisor a whole number: 48 ÷ 12 = 4 Therefore, 0.48 ÷ 0.12 = 4
Explanation
By converting both numbers to whole numbers by multiplying by 100, the division becomes simple.
By converting both numbers to whole numbers by multiplying by 100, the division becomes simple.
Problem 5
Find the result of 5.6 divided by 0.7.
Use the calculator or follow the steps: Convert the divisor to a whole number: 56 ÷ 7 = 8 Therefore, 5.6 ÷ 0.7 = 8
Use the calculator or follow the steps: Convert the divisor to a whole number: 56 ÷ 7 = 8 Therefore, 5.6 ÷ 0.7 = 8
Explanation
Multiplying both the dividend and divisor by 10 allows for easy division.
Multiplying both the dividend and divisor by 10 allows for easy division.
FAQs on Using the Dividing Decimals Calculator
1.How do you divide decimals by whole numbers?
Divide as you would with whole numbers, maintaining the decimal point in the correct position in the quotient.
Divide as you would with whole numbers, maintaining the decimal point in the correct position in the quotient.
2.Can dividing decimals result in a whole number?
Yes, dividing decimals can result in a whole number depending on the values involved.
Yes, dividing decimals can result in a whole number depending on the values involved.
3.Why adjust the divisor to a whole number?
Adjusting the divisor simplifies the division process by eliminating decimal places, making the division straightforward.
Adjusting the divisor simplifies the division process by eliminating decimal places, making the division straightforward.
4.How do I use a dividing decimals calculator?
Input the dividend and divisor into the respective fields, then click calculate to see the result.
Input the dividend and divisor into the respective fields, then click calculate to see the result.
5.Is the dividing decimals calculator accurate?
Yes, the calculator provides accurate results based on the entered inputs, ensuring precision in decimal division.
Yes, the calculator provides accurate results based on the entered inputs, ensuring precision in decimal division.
Glossary of Terms for the Dividing Decimals Calculator
Dividing Decimals Calculator: A tool to perform division operations involving decimal numbers. Dividend: The number to be divided in a division operation. Divisor: The number by which the dividend is divided. Rounding: Approximating a decimal number to a specified degree of precision. Decimal Point: A dot used to separate the whole number from the fractional part in decimal notation.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
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: She has songs for each table which helps her to remember the tables |
189429 | https://www.youtube.com/watch?v=kviwvLpnZSY | Hidden Dice Faces - Numberphile
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Posted: 14 Jun 2021
Featuring Ben Sparks. More links & stuff in full description below ↓↓↓
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Transcript:
I've been reading some Martin Gardner books, all sorts of gems in there, here's one I haven't practiced very much. It just needs a die. I want you to pick a number on the die, don't
tell me what it is. So I can't actually see anything at all which means that you can feel free to sort of,
I don't know, show the the video watchers- (Brady: Let's go for that number.)
- A whole number right? You haven't got a picked pi or something? (No it's a- it's an integer.)
- Okay (And it's on a die.)
- Good. You should be able to see three numbers on the die.
(Yes I can) Can you see your number?
- (No) Interesting. Can you see your number now?
- (Yes) That's interesting. Can you see your number now?
- (Yes) (Is it the one on top?)
I think so. (Yep, you got it.) [Laughter] Well I'm excited. [Laughter] (How did you do that? Can you can) (you teach it so people can do it to) (their parents?)
- I can do. Do you have any idea- because I've never tried this trick on anyone before and, first of all, with all magic tricks there's a trick right? But
when you know the trick the thing's lame.
I've just done a lame thing. Is it at all interesting that I could know with- blindfolded what your number was? (Yeah!)
- Okay, can we try it one more time? And I think you might start to spot what's going on, like all magic tricks you repeat it and you're like wait a second. So so you you decide a number again.
- (Let's go) (for - uh - let's go for that number.)
I'm trying to point a corner towards you so- it's interesting on a D6, you can only ever see three numbers. Can you see your number now?
- (No) Interesting. So let's try the
question again there, can you see your number now? (Yes)
- That's interesting, what about, what about now?
- (No) That's very intriguing, why would you choose that number? (Yes!) Why would you choose a 1? I mean that's what lost you in that pig game When I first saw this I was like, ah it's like what they call a binary search. A binary search is where you're hunting for something that
you don't know and you try and eliminate half the possibilities in one go. And that's kind of the key to this; I'm asking three questions. And that should be enough information, it's just that doing it with a blindfold somehow makes it feel surprising. I'm still surprised that I get it right- Audrey I know you're impressed too- (Audrey!) (Audrey - out!)
- Do you want to roll a die? - No? Okay.
- (Out. How do we-) (how do we do how do you do it?)
- So a binary search - get rid of half the options with each question. And a moment's thought reveals that I've only got six options. So first of all I could ask six questions; the trick would be a bit rubbish - is it 1? No, is it 2? I guarantee you- I think I did those in three questions and if you halve something three times actually you've reduced from eight to one; so three halvings should be plenty of questions to get this. So that's the that's the base principle; like I'm just asking enough questions. So I think I can talk you through this; the first thing is is one of the six numbers I ask a question which is, can you see your number? And if you say yes what do I know?
- (Narrowed it to three.) Yeah and I'm blindfolded but I can still feel there's three faces there there and there that I know you can see. If you say yes I know it's those three, if you say no I know it's one of those three. So I've halved it. And then the next question is, if I ask you then- let's say you say no on that and I spin it round, if I then ask you can you see your number? It's a stupid question right because I know you can, so I need to hide one of these. And all I did was spin that forward and now I know that two
of them are ones that might be your number. And if I say can you see your number and you say no, I knew that these two were possibles and the one on the bottom is also possible. And you can't see it so the one on the bottom is your number. But if you said yes then it's one of these two so I just need to hide one of those two and now that one's the only possible one that you can see, or it's the one that you can't see. So if you say no it's that one, if you say yes there's that one.
- (Let's) (use the number 1 again.)
- Okay so you've chosen 1 but I don't know that; I'm blindfolded, I can't see anything. But I know that if I spin the die so that you can see three then I ask a question, yes or no,
can you see your number? And in this case you would say yes, so I know it's one of these three. If you'd said no I would have known it's one of those three. In this case it's not so I've already got half the solutions out of the way. I now want to ask again; if I didn't move it I wouldn't be getting any new information so I'm going to hide one of these three, gonna hide that one. So now I know that these two faces are possibles and the one on the bottom. So can you see your number?
- (Yes..) So I know, even though I'm blindfolded,
that it's one of these two, so let's hide one of them. Can you see your number?
- (No) It's the one I just hid
which is on the bottom and that's it. And because of
this halving business I'm guaranteed to get down to only one left in at most three turns. In fact it might go even quicker. (Let's do it now if the
one wasn't visible) (in the first go.)
- Okay, and I'm also going to rig it slightly to show how it could go even quicker but- can you see your number?
- (No.) So under the cover of my hands, if I'm blindfolded I'd spin this all the way around and that's my mental check, right it's one of those three numbers; and I would hide one of them. Now I don't know which one to hide but I've just got a sort of consistent way of rolling it so I'm convinced I know where it is. And now I say, can you see your number?
- (No) And I know it must be the one I've just hidden then so which
point I just clean up, walk away, in two turns and that's nicer. Although as a performer for me I'm like it just feels really lame, I've just like just asked a couple of questions. But I still think the effect is quite nice and it's a nice demonstration of searching efficiently. And it also means that if eight is what doubling three times gives you - so halving three times gets you back to one - I've got more than enough information, I should be able to do this on a D8. (Can you do it with the D8?)
- Mathematics says I can; practice means I'm a bit nervous but we can have a try. This is an octahedron and it does have eight sides-
- (But I can see four at a time now!) Yeah that's true, so we've got- and I think if I aim a corner towards you you can only see those four - can you see those four? Okay, the most difficult thing with me in my blindfold is like orienting the dice so you can actually see the four, because if I put it like that I don't know what you can see. (Well do you want to do this one) (unblindfolded?)
- No we'll do it blindfolded (All right, should I choose a number?) Choose a number; don't tell me whatever you do.
- (All right,) (let's go for that number.)
- So you and the camera can see four numbers? Can you see your number?
- (No) You can't see your number? So I'm kind of hiding what I'm doing, you might want to speculate about what I'm doing. Let's go with- can you see your number?
- (Uh no I can't.) You can't see your number? This is the bit that I'm just super feeling my way about. Can you see- Oh dear - can you see your number now? (No!)
- Interesting Your number is on top.
- (On top there?) (Do you want the good
news or the bad news?) I want both. (You got it!)
- Yes! But I knew I was going to get it all along because I'm supremely confident in mathematics and my tactile blind handling of a D8. (Let's do the five again.)
- So you go for five so can you see your number, yes? But all I know is it's one of these four. My next job is to halve the amount of information so I need to hide two of these faces; so I did that in this case by just rolling over like that. I know that two of the ones you could see are now missing but two of them are visible. So can you see your number? (Yes)
- So I know it's one of these two. I'm kind of feeling with my first two fingers each time thinking it's one of those two. If you'd said no I knew it'd be one of those two that I first showed you, but I've got to keep track of where they are
with my blindfold. But since he said yes I know it's one of these two, so I just got to hide one of those two. So let's just roll it
onto there, so I know that one of them is on the bottom, that helps me know where it is, and one of them is visible and I know it's the front one. I say can you see your number and you say no, in which case I know it's the one I just hid so I flip it over and flourish or something. And sort of start sweating about whether I've actually got it.
- (Nice!) But it's it's a nice example of like searching efficiently if you've got a lot of things, trying to halve it each time is one of the most efficient; and that's partly why binary as a number system is like used in computers because it's kind of an efficient way of thinking about breaking up numbers. And that's just a nice demonstration of a binary search. The fact that you can do it on a D6 - Martin Gardner talked about doing on a D6 and I was like, that means it must be possible on a D8. (Can you do a higher D now?)
- There's some options right? There's a D10 - that could be done in four questions. D12, the dodecahedron icosahedron, that could be done in four; and that could be done in five. But it's hard, I mean I'm finding the octahedron hard enough. So I'd need a lot more practice but it could be done and the thing is I start to doubt whether the impressiveness- like four questions feels like quite a lot. The fact that the D6 sometimes works out in two is what gives that little kicker like boom boom, got it. Brush up on your dice mathematics as part of this Casino Probability course on Brilliant, today's episode sponsor. Now it doesn't matter if you get things right, or occasionally wrong, you're always learning with Brilliant's fantastic quizzes and puzzles and their deep dive courses. I do really enjoy doing these, interactive, gets my brain turning, yeah. Premium subscribers get access to everything on Brilliant but there's also plenty of stuff to see for free, so why not go check them out see what you make of it? If you do want to become a premium subscriber, Numberphile viewers are going to get 20% off by going to brilliant.org/Numberphile, that's brilliant.org/Numberphile there on the screen. And if you're already a member, why not consider gifting one of those premium subscriptions to the student or the lifelong learner in your life? ...Well of course we know what it means in practice, we mean if we throw it, you know, it should land on each of the 30 faces a 30th of the time. But let me talk about that for a second. When I was a graduate student we had a guy who was a retired executive who wanted to test the laws of chance- And well this means more often than that that's the end of the story. So the totem pole continues like this, this one beats this one and this one beat this one. So I think- this is almost for the simple game you're not playing
against someone else, aim for 20 is a good piece of advice. |
189430 | https://www.facebook.com/groups/kohatianchemist/posts/6564189973596171/ | General Chemistry | Why Sulphuric acid is called King of Compounds | Facebook
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Attaullah Soomro
· January 7, 2023 ·
Why Sulphuric acid is called King of Compounds?
Why Sulphuric acid is called King of Compounds?
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Devarapalli RPdarsini
because it is used in the preparation of a very large number of other useful chemicals such as hydrochloric acid, nitric acid, dyes, drugs etc....
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It releases a lot of H+ when it undergoes electrolysis with comparison to other compounds....
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189431 | https://math.stackexchange.com/questions/743324/maximizing-the-product-of-projections-of-a-vector-on-another-vectors | complex analysis - Maximizing the product of projections of a vector on another vectors - Mathematics Stack Exchange
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Maximizing the product of projections of a vector on another vectors
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I want to get the N×1 N×1 complex vector x x which maximizes this real valued function
f=x H(a 1 a H 1 x x H a 2 a H 2)x f=x H(a 1 a 1 H x x H a 2 a 2 H)x, subject to ||x||=1||x||=1
where a 1 a 1 and a 2 a 2 are N×1 N×1 known complex vectors, and (.)H(.)H is the complex conjugate operator
Can any one help by closed form solution or an iterative algorithm?
complex-analysis
eigenvalues-eigenvectors
nonlinear-optimization
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edited Apr 7, 2014 at 13:04
UnknownUnknown
asked Apr 7, 2014 at 9:23
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Could we just write this as (x⋅a 1)2(x⋅a 2)2(x⋅a 1)2(x⋅a 2)2?orion –orion 2014-04-07 10:14:30 +00:00 Commented Apr 7, 2014 at 10:14
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You may write this as
|x⋅a 1|2|x⋅a 2|2=|(x⋅a 1)(x⋅a 2)|2|x⋅a 1|2|x⋅a 2|2=|(x⋅a 1)(x⋅a 2)|2
Minima and maxima of the parenthesized thing at the right side will be maxima of the squared version. We are thus looking to minimize
(x⋅a 1)(x⋅a 2)=x⃗A x⃗(x⋅a 1)(x⋅a 2)=x→A x→
where A i j=a 1 i a∗2 j A i j=a 1 i a 2 j∗ is a generalized projection matrix.
Findng a minimum of
x⃗A x⃗−λ|x|2 x→A x→−λ|x|2
where λ λ is a Lagrange multiplier to ensure unit length of x⃗x→, you arrive at the eigenvalue problem
A x⃗=λ x⃗A x→=λ x→
Because you are maximizing the product of projections to two axes, the solution will naturally be a superposition of these two axes (any rotation out of this plane minimizes both projections).
x=X 1 a 1+X 2 a 2 x=X 1 a 1+X 2 a 2
This reduces this problem to a 2×2 2×2 eigenvalue problem.
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edited Apr 7, 2014 at 12:55
answered Apr 7, 2014 at 10:22
orionorion
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Only, we are considering the complex case, and this complicates things because Lagrange multipliers does not lead to the eigenvalue problem, in this case.Omran Kouba –Omran Kouba 2014-04-07 11:20:20 +00:00 Commented Apr 7, 2014 at 11:20
Thanks Orion and Omran Kouba. The superposition of the vectors is the best solution for real valued vectors but for the complex case it is not the best. I used the superposition solution as an initial solution and performed MATLAB optimization function "fmincon" and "interior point algorithm" and it gives a solution with larger value for the objective function f f. I want to know if I can use an iterative algorithm to reach the global solution from the superposition solution as an initial one?Unknown –Unknown 2014-04-07 12:57:37 +00:00 Commented Apr 7, 2014 at 12:57
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189432 | https://www.unitsconverters.com/en/Cubicmillimeter-To-Cubiccentimeter/Unittounit-8010-8014 | Cubic Millimeter to Cubic Centimeter
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mm³ stands for cubic millimeters and cm³ stands for cubic centimeters. The formula used in cubic millimeters to cubic centimeters conversion is 1 Cubic Millimeter = 0.001 Cubic Centimeter. In other words, 1 cubic millimeter is 1000 times smaller than a cubic centimeter. To convert all types of measurement units, you can used this tool which is able to provide you conversions on a scale.
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1 Cubic Millimeter is equal to 0.001 Cubic Meter. 1 Cubic Millimeter is 1000 times Smaller than 1 Cubic Meter.
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1 Cubic Millimeter is equal to 0.001 Cubic Inch. 1 Cubic Millimeter is 1000 times Smaller than 1 Cubic Inch.
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189433 | https://www.deepchecks.com/glossary/true-positive-rate/ | Deepchecks Agentic Workflow Evaluation is live now! Learn More
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True Positive Rate
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You simply need to allocate a business value to commercial applications to four sorts of results: true negatives, true positives, false negatives, and false positives. You can be sure you’re using the best model by multiplying each bucket’s number of results by the related business values.
The confidence values offered by the model further complicate the matter. Almost all machine learning models may be programmed to provide a level of confidence in their output. Multiplying this value with the findings is a high-level technique to apply it in accuracy measurement, effectively rewarding the model for delivering high confidence levels for its right assessments.
The true positive rate, also known as sensitivity or recall in machine learning, is a metric that measures the percentage of actual positives that are accurately identified
More advanced techniques, on the other hand, are feasible. If all low confidence predictions are going to be manually checked, allocating a manual labor cost to them and excluding their outcomes from the model accuracy measurement is a more accurate approximation of the business value created by the model.
Results in the machine learning model
Individual predictions in a model can be true or false, indicating whether the model is correct or incorrect. The data point’s actual value is also significant. You might be wondering why we need a model that predicts values when we already know what they are. We’re talking about the model’s performance on the training data, which we already know the answers to.
The actual value of the data points can be either the values we’re looking for in the dataset (positives) or something else entirely (negatives). As a result, the four possible outcomes of a model’s individual predictions are as follows:
True positive: The prediction is true, and the value is positive.
False positive: The forecast is incorrect, but the value is positive.
True negative: The prediction is true, but the value is negative.
Negative false: The prediction is incorrect, and the value is negative.
When you actually want to correctly forecast the cases in the true class, you need to know about the positive rate. If you have a test for a serious form of cancer, for example, you want it to be able to detect all of the situations where someone actually has cancer. So you’re really concerned about the positive rate.
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Divide the TP (true positives) with the sum of TP (true positives) and FN (false negatives) to get the true positive in machine learning:
Recall aka True Positive Rate = TP / (TP+FN)
The best possible TPR is one, while the lowest possible recall is zero. In circumstances where recall is critical, there is another thing we can do to accurately anticipate more true cases: adjust our decision threshold.
A scikit-learn classification model’s decision threshold is set to.5 by default. This means that if the model believes an observation has a 50% or greater chance of being a member of the positive class, it is projected to be a member of the positive class.
If TPR is important to us, we can reduce the decision threshold to catch more of the true affirmative cases. For example, suppose you want the model to predict true with a probability of 20% or greater for every observation.
After all, you might predict that every observation would be favorable and have a perfect 100 percent TPR. But that isn’t always a good idea.
When the cost of false positives is large, it’s important to be aware of them. You’ll need a statistic to track how successfully your model distinguishes between genuine and false positives. You must pay close attention to precision.
Summary
It’s important to remember that a positive rate is also known as sensitivity or recall. The true negative rate is another name for specificity.
These two criteria normally trade-off against one another inside any statistical evaluation tool, although dominating approaches are theoretically feasible across multiple statistical inferential procedures, where one way delivers larger positive rates without necessarily delivering a greater false-positive rate.
The false-negative rate is 1 minus the false positive rate, while the false-negative rate is 1 minus the positive rate
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189434 | https://www.sanbi.org/animal-of-the-week/black-mamba/ | Published Time: 2020-10-06T08:24:42+00:00
Black mamba - SANBI
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Home » Animal of the week » Black mamba
Black mamba
06/10/2020 11/05/2021
Common names: Black mamba, black-mouthed mamba, southern brown mamba (Eng.); swartmamba (Afr.), imamba emnyama (isiZulu & Ndebele); mokwepa (Setswana and Sepedi), khangala (tshiVenda), mamba ya ntima (Xitsonga), hangara (Shona).
The deadliest and fastest snake in Africa! The death kisser.
How to recognize a Black mamba
Black mambas are known for their large size, quickness and extremely potent venom. They have coffin-shaped heads and are graceful, athletic snakes. Black mambas live in sub-Saharan Africa and are one of the continent’s most dangerous snakes. The average adult black mamba is 2.0–2.5 metres long, with a maximum length of 4.3 metres (14 feet). Despite its name, the snake is not black. Instead, it ranges in colour from grey to dark brown, with a lighter underside. The black actually refers to the colour of the inside of its mouth.
Black mambas are shy but aggressive, and if cornered, the snake may rear up and threaten with an open mouth and a slightly expanded or flattened neck (or hood) before striking. Black mambas can strike repeatedly, injecting its deadly neurotoxic venom which causes difficulty in breathing within half an hour. They have a specialised organ known as the vomeronasal organ, or the Jacobson’s organ, which is involved in the snake’s chemical communication and in hunting prey. The organ processes stimuli and helps the snake to detect what is in its environment and where potential prey is. They often use their eyesight, mainly for detection of motion, and sudden movements will cause them to strike.
Black mamba (Dendroaspis polylepis) – © Johan Marais
Getting around
A black mamba moves using lateral undulation, which essentially involves moving in an ‘S’ shape and pushing off from objects in its path, such as trees, rocks or even piles of soil or dirt, to gain momentum. These sideways movements propel the animal throughout their environment successfully.
Communicating
They have no external ears, but are quite adept at detecting vibrations from the ground. Like many snakes, when threatened, they will display aggression with a set of signals warning of the possibility of attack.
Distribution
The black mamba inhabits a wide range in sub-Saharan Africa; its range includes Burkina Faso, Cameroon, Central African Republic, Democratic Republic of the Congo, South Sudan, Ethiopia, Eritrea, Somalia, Kenya, Uganda, Tanzania, Burundi, Rwanda, Mozambique, Eswatini, Malawi, Zambia, Zimbabwe, Botswana, South Africa, Namibia and Angola. The black mambas distribution in parts of West Africa has been disputed. In 1954, the black mamba was recorded in the Dakar region of Senegal. This observation, and a subsequent observation that identified a second specimen in the region in 1956, has not been confirmed and thus the snake’s distribution in this area is inconclusive.
The species is regarded as common in sub-Saharan Africa, occurrence in Senegal and Guinea has also been reported (Spawls 2010)
The black mamba is found in the northern regions of southern Africa. In South Africa, they live along the coastal regions from Kwa-Zulu Natal to Port St Johns and elsewhere, but are absent from the desert (Håkansson & Madsen 1983).
Habitat
Black mambas reside in South and East African savannas, rocky hills and open woodlands. They like low, open spaces and enjoy sleeping in hollow trees, rock crevices, burrows or empty termite mounds. They are found in both tropical and temperate regions. In the temperate region they are found in the savanna, grassland and forest biomes. They also inhabit riparian zones. They are comfortable in trees but occur mostly on the ground.
Black mamba (Dendroaspis polylepis) –© M. Burger
Food
The Black Mamba is a heterotroph, as are all organisms in the kingdom Animalia, which means that it derives its nutritional requirements from complex organic substances. It is also a carnivore, meaning that it eats other animals as well as a predator, which means that it hunts and kills its food. Black mambas are diurnal snakes, which means that they will hunt during the day in order to capture its prey. They are cold-blooded and do require a significant amount of basking time in the sun to heat up their bodies. The prey of the black mamba includes a variety of organisms that include smaller mammals such as mice, squirrels, rats, and birds. They have also been found to feed on other snakes such as the forest cobra.
Sex and life cycles:
Black Mambas mate in the spring and/or summer. They need warm temperatures so the time frame can vary based on location. The males go to the females. They may have to fight other males along the way so their level of aggression and tension will increase during this period of time. The female’s preferences regarding males is not understood.
Once mating has taken place, the female will need the male to leave and she will become extremely aggressive if he doesn’t.
Black mambas are oviparous (producing eggs that develop and hatch outside the maternal body). Females will release the eggs into an area covered with vegetation that is warm as the eggs need heat to complete the process of incubation. She may have 10–25 eggs in her body and lay them in underground chambers. After 2–3 months the eggs hatch and the young are able to fend for themselves straight away, as the mother would have long since disappeared – straight away after laying eggs.
Only a handful of the young will reach the age of maturity. The first few days and weeks of life end quickly for most of the young because they aren’t fast enough or skilled enough to stay away from predators.
Family life:
The black mamba is void of any symbioses, and so doesn’t live in close interaction with any other organisms. Its main interactions involve its prey and predators and the black mamba is near the top of the food chain in its habitat. It dominates smaller mammals and some birds. There is not much information about the lifespan of snakes in the wild. The longest recorded lifespan of a captive mamba was 11 years, but actual lifespans could be much greater (FitzSimons 1970).
The big picture
Very little information is available on the ecosystem roles of black mambas. They are important in controlling rodent populations (FitzSimons 1970).
Friends and foes
The black mamba does not live in close interaction with other animals except when the mating season begins when it will travel in search of a mate.
Smart strategies
The presence of the Jacobson’s organ provides the snake with an advantage over its prey by enhancing its taste/smell senses. The fangs of the black mamba are front fixed, and they can be folded up into the roof of the mouth while they aren’t in use. These snakes have also evolved a hood-like structure around its neck that can flare out when threatened, this behavior makes the snake appear much larger to its enemies. Its venom is extremely potent.
Black mamba (Dendroaspis polylepis) – © Johan Marais
One of the more amazing adaptations that the back mamba, and all snakes for that matter, have, is the ability to move without legs! . Another recently discovered evolutionary trait that black mambas possess, which helps them move, are their scales. A study done at Georgia Institute of Technology, Atlanta in Georgia (United States) has shown that the scales on snakes have evolved to act like hooks to create more friction to help drive the snake forward.
Poorer world without me
Black mambas control the population of small rodents.
People & I
The fear of this snake by humans has given rise to many myths among Africans about the deadly power of the snake. One such myth says that the black mamba has the ability to grab its tail, forming a loop with its body, roll down a hill and lash out and attack someone. Other myths show the mamba having exceptional intelligence, having the ability to plan attacks on humans and ambush them in cars when they reach their destinations. The other common myth is that it grows a feather on top of its head and will block the way for humans biting them on the encounter. The mythical feathered mamba is called iNdlondlo in isiZulu.
Conservation status and what the future holds
Dendroaspis polylepis has been assessed as Least Concern owing to its very large distribution throughout sub-Saharan Africa. No specific threats have been reported and this species is not undergoing significant population declines (Spawls 2010).
While black mambas are not endangered in any way, they do face a future threat due to human expansion. They are nervous animals and prefer to stay far away from humans. Human population expansion into their habitat could cause considerable habitat destruction and conflicts with human interests (FitzSimons 1970).
People who encounter the snake are encourage to move away slowly and call a professional snake handler for removal if the snake is inside any building.
Relatives
In South Africa there are two species of Dendroaspis namely D. angusticeps (green mamba) and D. polylepis (black mamba). Other genera within the family Elapidae include Aspidelaps, Boulengerina, Bungarus, Calliophis, Elapsoidea, Hemachatus, Hemibungarus, Micruroides, Micrurus, Naja, Ophiophagus, Paranaja, Pseudohaje and Walterinnesia.
Official common name: Black mamba
Scientific name and classification:Dendroaspis polylepis
Kingdom: Animalia
Phylum: Chordata
Class:Reptilia
Order:Squamata
Family: Elapidae
Genus: Dendroaspis
Species: D. polylepis Günther, 1864
Derivation of scientific name: The genus name was derived from the Latin words dendro-, meaning tree; because many of the snakes in this genus are tree-living snakes; and aspis, which comes from a word meaning ‘shield’, which is used to describe a snake coiling itself. The species name, polylepis is from the Greek words poly meaning ‘many’ and lepis meaning ‘scales’. So in English, Dendroaspis polylepis literally translates to ‘tree snake with many scales’.
References and further reading
African Snakebite Institute. (2020). Retrieved from Black mamba: (2020-09-11)
FitzSimons, V. (1970). A Field Guide to the Snakes of Southern Africa. London: Collins.
Håkansson, Thomas, and Thomas Madsen. (1983). ‘On the distribution of the Black Mamba (Dendroaspis polylepis) in West Africa.’ Journal of Herpetology 17.2: 186-189. (n.d.).
Marais, J. (1985). Snake versus man. Johannesburg: Macmillan South Africa.
Marais, J. (1992). A Complete Guide to the Snakes of Southern Africa. Halfway House: Southern Book Publishers.
Spawls , S. (2020). Dendroaspis polylepis. Retrieved from IUCN Redlist: (2020-09-08).
Author:Simphiwe Dlamini
E-mail: S.Dlamini@sanbi.org.za
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189436 | https://www.cedengineering.com/userfiles/M06-042%20-%20Fundamentals%20of%20Aerodynamics-Part%201%20-%20US.pdf | Fundamentals of Aerodynamics - Part 1 Course No: M06-042 Credit: 6 PDH Elie Tawil, P.E., LEED AP Continuing Education and Development, Inc. P: (877) 322-5800 info@cedengineering.com www.cedengineering.com This course was adapted from the Naval Aviation Schools Command, Publication No. CIN Q-9B-0020L CH1, “Fundamentals of Aerodynamics” – Training Guide, which is in the public domain. Note: This course is based on sections Basic Theory, Lift Production and Drag, and Stalls (pages 3 to 79) of this PDF document. Fundamentals of Aerodynamics: Part 1 – M06-042 OUTLINE SHEET 2-1-1 BASIC THEORY A.
INTRODUCTION This lesson is a basic introduction to the theory of aerodynamics. It provides a knowledge base in aerodynamic mathematics, air properties, airspeed and altitude definitions and measurements, airfoil and wing design, and the importance of center of gravity (CG).
B.
ENABLING OBJECTIVES 2.1 DEFINE scalar, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.2 DEFINE vector, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.3 DEFINE mass, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.4 DEFINE volume, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.5 DEFINE density, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.6 DEFINE force, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.7 DEFINE weight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.8 DEFINE moment, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.9 DEFINE work, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.10 DEFINE power, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 13 Fundamentals of Aerodynamics: Part 1 – M06-042 2.11 DEFINE energy, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.12 DEFINE potential energy, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.13 DEFINE kinetic energy, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.14 EXPLAIN Newton's Law of Equilibrium, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.15 STATE the requirements for an airplane to be in equilibrium flight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.16 STATE the requirements for an airplane to be in trimmed flight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.17 EXPLAIN Newton's Law of Acceleration, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.18 EXPLAIN Newton's Law of Interaction, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.19 DEFINE static pressure, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.20 DEFINE air density, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.21 DEFINE temperature, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.22 DEFINE lapse rate, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.23 DEFINE humidity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.24 DESCRIBE the relationship between humidity and air density, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.25 DEFINE viscosity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 14 Fundamentals of Aerodynamics: Part 1 – M06-042 2.26 DESCRIBE the relationship between temperature and viscosity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.27 DEFINE local speed of sound, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.28 DESCRIBE the relationship between temperature and local speed of sound, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.29 STATE the values for standard atmosphere, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.30 DESCRIBE the General Gas Law, given static pressure, air density, temperature, and altitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.31 EXPLAIN Bernoulli's Equation, given dynamic pressure, static pressure, and total pressure, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.32 DEFINE steady airflow, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.33 DEFINE streamline, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.34 DEFINE streamtube, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.35 EXPLAIN the continuity equation given density, cross-sectional area, and velocity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.36 DEFINE indicated altitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.37 DEFINE Above Ground Level (AGL) altitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.38 DEFINE Mean Sea Level (MSL) altitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 15 Fundamentals of Aerodynamics: Part 1 – M06-042 2.39 DEFINE pressure altitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.40 DEFINE density altitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.41 DESCRIBE the pitot-static system given the system components and Bernoulli's equation, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM- SG-200 2.42 DEFINE indicated airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.43 DEFINE calibrated airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.44 DEFINE equivalent airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.45 DEFINE true airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.46 DEFINE ground speed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.47 DESCRIBE the factors affecting the different types of airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.48 DEFINE an aircraft, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.49 DEFINE an airplane, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.50 DESCRIBE the five components of an airplane, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.51 STATE the advantages of the semi-monocoque fuselage construction, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.52 DEFINE full cantilever wing construction, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 16 Fundamentals of Aerodynamics: Part 1 – M06-042 2.53 DESCRIBE the airplane three-axis reference system, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.54 DEFINE chord line, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.55 DEFINE chord, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.56 DEFINE root chord, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.57 DEFINE tip chord, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.58 DEFINE average chord, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.59 DEFINE mean camber line, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.60 DEFINE symmetric airfoil, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.61 DEFINE positive camber, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.62 DEFINE negative camber, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.63 DEFINE spanwise flow, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.64 DEFINE chordwise flow, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.65 DEFINE pitch attitude, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.66 DEFINE flight path, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.67 DEFINE relative wind, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 17 Fundamentals of Aerodynamics: Part 1 – M06-042 2.68 DEFINE angle of attack, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.69 DEFINE angle of incidence, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.70 DEFINE dihedral angle, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.71 DEFINE wingspan, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.72 DEFINE wing area, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.73 DEFINE wing loading, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.74 DEFINE taper ratio, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.75 DEFINE sweep angle, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.76 DEFINE aspect ratio, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.77 DEFINE the center of gravity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.78 DEFINE the aerodynamic center, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.79 DESCRIBE the motions that occur around the airplane center of gravity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 18 Fundamentals of Aerodynamics: Part 1 – M06-042 C.
TOPIC OUTLINE 1.
Introduction 2.
This Lesson Topic 3.
Basics 4.
Atmospheric Properties 5.
Properties of Airflow 6.
Altitude Measurement 7.
Airspeed Measurement 8.
Major Components of an Airplane 9.
Airfoils and Wing Properties 10.
Center of Gravity (CG) 11.
Summary and Review 12.
Application 13.
Assignment 19 Fundamentals of Aerodynamics: Part 1 – M06-042 INFORMATION SHEET 2-1-2 BASIC THEORY A.
INTRODUCTION This lesson is a basic introduction to the theory of aerodynamics. It provides a knowledge base in aerodynamic mathematics, air properties, airspeed and altitude definitions and measurements, airfoil and wing design, and the importance of center of gravity (CG).
B.
REFERENCES 1.
Aerodynamics for Naval Aviators, NAVAIR 00-80T-80 2.
Introduction to the Aerodynamics of Flight, NASA SP-367 3.
Publication, U.S. Standard Atmosphere, 1976, NOAA-S/T 76-1562 4.
T-6 Joint Primary Pilot Training, AETC / CNATRA Syllabus P-V4A-J NATOPS Flight Manual C.
INFORMATION MATHEMATICAL SYSTEMS A scalar is a quantity that represents only magnitude, e.g., time, temperature, or volume. It is expressed using a single number, including any units. A vector is a quantity that represents magnitude and direction. It is commonly used to represent displacement, velocity, acceleration, or force. Displacement (s) is the distance and direction of a body’s movement (an airplane flies east 100 nm). Velocity (V) is the speed and direction of a body’s motion, the rate of change of position (an airplane flies south at 400 knots). Speed is a scalar equal to the magnitude of the velocity vector. Acceleration (a) is the rate and direction of a body’s change of velocity (gravity accelerates bodies toward the center of the earth at 32.174 ft/s2). A force (F) is a push or pull exerted on a body (1,000 lbs of thrust pushes a jet through the sky). A vector may be represented graphically by an arrow. The length of the arrow represents the magnitude and the heading of the arrow represents the direction. Vectors may be added by placing the head of the first vector on the tail of the second and drawing a third vector from the tail of the first to the head of the second. This new vector (Figure 1-1) is the resulting magnitude and direction of the original two vectors working together. DEFINITIONS Mass (m) is the quantity of molecular material that comprises an object. Volume (v) is the amount of space occupied by an object. Density (ρ) is mass per unit volume. It is expressed: Figure 1-1 Vector Addition 20 Fundamentals of Aerodynamics: Part 1 – M06-042 ρ = mass / volume Weight (W) is the force with which a mass is attracted toward the center of the earth by gravity. Force (F) is mass times acceleration: F = m x a A moment (M) is created when a force is applied at some distance from an axis or fulcrum, and tends to produce rotation about that point. A moment is a vector quantity equal to a force (F) times the distance (d) from the point of rotation that is perpendicular to the force (Figure 1-2). This perpendicular distance is called the moment arm. Work (W) is done when a force acts on a body and moves it. It is a scalar quantity equal to the force (F) times the distance of displacement (s). W = F x s Power (P) is the rate of doing work or work done per unit of time. P = W / t Energy is a scalar measure of a body’s capacity to do work. There Figure 1-2 Moment are two types of energy: potential energy and kinetic energy. Energy cannot be created or destroyed, but may be transformed from one form to another. This principle is called conservation of energy. The equation for total energy is: TE = PE + KE Potential energy (PE) is the ability of a body to do work because of its position or state of being. It is a function of mass (m), gravity (g), and height (h): PE = weight x height = mgh Kinetic energy (KE) is the ability of a body to do work because of its motion. It is a function of mass (m) and velocity (V): KE = ½ mV2 Work may be performed on a body to change its position and give it potential energy or work may give the body motion so that it has kinetic energy. Under ideal conditions, potential energy may be completely converted to kinetic energy, and vice versa. The kinetic energy of a glider in forward flight is converted into potential energy in a climb. As the glider’s velocity (KE) diminishes, its altitude (PE) increases. NEWTON’S LAWS OF MOTION NEWTON’S FIRST LAW - THE LAW OF EQUILIBRIUM “A body at rest tends to remain at rest and a body in motion tends to remain in motion in a straight line at a constant velocity unless acted upon by some unbalanced force.” 21 Fundamentals of Aerodynamics: Part 1 – M06-042 The tendency of a body to remain in its condition of rest or motion is called inertia. Equilibrium is the absence of acceleration, either linear or angular. Equilibrium flight exists when the sum of all forces and the sum of all moments around the center of gravity are equal to zero. An airplane in straight and level flight at a constant velocity is acted upon by four forces: thrust, drag, lift and weight. When these forces exactly cancel each other out, the airplane is in equilibrium (Figure 1-3). Trimmed flight exists when the sum of all moments around the center of gravity is equal to zero. In trimmed flight, the sum of the forces may not be equal to zero. For example, an airplane in a constant rate, constant angle of bank turn is in trimmed, but not equilibrium, flight. An airplane in equilibrium flight, however, is always in trimmed flight. Figure 1-3 Equilibrium Level Flight Figure 1-4 Equilibrium Climbing Flight An airplane does not have to be in straight and level flight to be in equilibrium. Figure1-4 shows an airplane that is climbing, but not accelerating or decelerating, i.e., there are no unbalanced forces. It is another example of equilibrium flight. Thrust must overcome drag plus the parallel component of weight. Lift must overcome the perpendicular component of weight. An airplane with sufficient thrust to climb vertically at a constant true airspeed can achieve an equilibrium vertical flight condition. Thrust must equal weight plus total drag, and lift must be zero (Figure 1-5). NEWTON’S SECOND LAW - THE LAW OF ACCELERATION “An unbalanced force (F) acting on a body produces an acceleration (a) in the direction of the force that is directly proportional to the force and inversely proportional to the mass (m) of the body.” In equation form: When an airplane’s thrust is greater than its drag (in level flight), the excess thrust will accelerate the airplane until drag increases to equal thrust. Figure 1-5 Equilibrium Vertical Flight 22 Fundamentals of Aerodynamics: Part 1 – M06-042 NEWTON’S THIRD LAW - THE LAW OF INTERACTION “For every action, there is an equal and opposite reaction; the forces of two bodies on each other are always equal and are directed in opposite directions.” This law is demonstrated when the rearward force from an aircraft propeller’s propwash causes an aircraft to thrust forward at an equal amount of force (Figure 1-6). Figure 1-6 Action and Reaction PROPERTIES OF THE ATMOSPHERE The atmosphere is composed of approximately 78% nitrogen, 21% oxygen, and 1% other gases, including argon and carbon dioxide. Air is considered to be a uniform mixture of these gases, so we will examine its characteristics as a whole rather than as separate gases. Static pressure (PS) is the pressure particles of air exert on adjacent bodies. Ambient static pressure is equal to the weight of a column of air over a given area. The force of static pressure always acts perpendicular to any surface that the air particles collide with, regardless of whether the air is moving with respect to that surface. As altitude increases, there is less air in the column above, so it weighs less. Thus atmospheric static pressure decreases with an increase in altitude. At low altitudes, it decreases at a rate of approximately 1.0 in. Hg per 1000 ft. Air density (ρ) is the total mass of air particles per unit of volume. The distance between individual air particles increases with altitude resulting in fewer particles per unit volume. Therefore, air density decreases with an increase in altitude. Air consists of very many individual particles, each moving randomly with respect to the others. Temperature (T) is a measure of the average random kinetic energy of air particles. Air temperature decreases linearly with an increase in altitude at a rate of 2 °C (3.57 °F) per 1000 ft until approximately 36,000 feet. This rate of temperature change is called the average lapse rate. From 36,000 feet through approximately 66,000 feet, the air remains at a constant −56.5 °C (−69.7 °F). This layer of constant temperature is called the isothermal layer.
Humidity is the amount of water vapor in the air. As humidity increases, water molecules displace an equal number of air molecules. Since water molecules have less mass and do not change the number of particles per unit volume of air, density decreases. Therefore, as humidity increases, air density decreases. Viscosity (μ) is a measure of the air’s resistance to flow and shearing. Air viscosity can be demonstrated by its tendency to stick to a surface. For liquids, as temperature increases, viscosity decreases. Recall that the oil in a 23 Fundamentals of Aerodynamics: Part 1 – M06-042 car gets thinner when the engine gets hot. Just the opposite happens with air: Air viscosity increases with an increase in temperature. Sound is caused by disturbances of the air that causes a sudden compression or vibration. This creates a series of alternating compressions and rarefactions which is transmitted to our ears as sound. The compressions and rarefactions are transmitted from one particle to another, but particles do not flow from one point to another. Sound is wave motion, not particle motion. The local speed of sound is the rate at which sound waves travel through a particular air mass. The speed of sound, in air, is dependent only on the temperature of the air. The warmer the air, the more excited the particles are in that air mass. The more excited the molecules are, the more easily adjacent molecules can propagate a sound wave. As the temperature of air increases, the speed of sound increases. THE STANDARD ATMOSPHERE The atmospheric layer in which most flying is done is an ever-changing environment. Temperature and pressure vary with altitude, season, location, time, and even sunspot activity. It is impractical to take all of these into consideration when discussing airplane performance. In order to disregard these atmospheric changes, an engineering baseline has been developed called the standard atmosphere. It is a set of reference conditions giving representative values of air properties as a function of altitude. A summary may be found in Appendix C. Although it is rare to encounter weather conditions that match the standard atmosphere, it is nonetheless representative of average zero humidity conditions at middle latitudes. Unless otherwise stated, any discussion of atmospheric properties in this course will assume standard atmospheric conditions. English Metric (SI) Static Pressure PS0 29.92 in. Hg 1013.25 mbar Temperature T0 59 °F 15 °C Average Lapse Rate 3.57 °F / 1000 ft 2 °C / 1000 ft ρ0 .0024 slugs / ft3 1.225 g / l Local Speed of Sound 661.7 knots 340.4 m / s Table 1-1 Sea Level Standard Atmospheric Conditions THE GENERAL GAS LAW The General Gas Law sets the relationship between three properties of air: pressure (P), density (ρ), and temperature (T). It is expressed as an equation where R is a constant for any given gas (such as dry air): P = ρ RT One method to increase pressure is to keep density constant and increase temperature (as in a pressure cooker). If pressure remains constant, there is an inverse relationship between density and temperature. An increase in temperature must result in a decrease in density, and vice versa. ALTITUDE MEASUREMENT Altitude is defined as the geometric height above a given plane of reference. True altitude is the actual height above mean sea level. Pressure altitude (PA) is the height above the standard datum plane. The standard datum plane is the actual elevation at which the barometric pressure is 29.92 in. Hg. Since the standard datum plane is at sea level in the standard atmosphere, true altitude will be equal to pressure altitude. 24 Fundamentals of Aerodynamics: Part 1 – M06-042 Density altitude (DA) is the altitude in the standard atmosphere where the air density is equal to local air density. It is found by correcting pressure altitude for temperature and humidity deviations from the standard atmosphere. In the standard atmosphere, density altitude is equal to pressure altitude. But as temperature or humidity increase, the air becomes less dense, with the effect that the actual air density at one altitude is equal to that of a higher altitude on a standard day. A high DA indicates a low air density. Density altitude is not used as a height reference, but as a predictor of aircraft performance. A high DA will decrease the power produced by an engine because less oxygen is available for combustion. It will also reduce the thrust produced by a propeller or jet engine because fewer air molecules are available to be accelerated. The reduced power and thrust will reduce an airplane’s acceleration and climb performance. A high DA also requires a higher true airspeed for takeoff and landing and will therefore increase takeoff and landing distances. Figure 1-7 Density Altitude as a Function of Temperature and Pressure Altitude Over a typical day, static pressure and pressure altitude remain virtually constant. However, as the sun heats the air, the reduced density causes a dramatic increase in density altitude. This will have a noticeable impact on aircraft performance. Figure 1-7 can be used to determine density altitude from pressure altitude and temperature (but does not take into account the effects of humidity). 25 Fundamentals of Aerodynamics: Part 1 – M06-042 Properties of Airflow The atmosphere is a uniform mixture of gases with the properties of a fluid and subject to the laws of fluid motion. Fluids can flow and may be of a liquid or gaseous state. They yield easily to changes in static pressure, density, temperature and velocity. Steady airflow exists if at every point in the airflow static pressure, density, temperature and velocity remain constant over time. The speed and/or direction of the individual air particles may vary from one point to Figure 1-8 Streamline in Steady Airflow another in the flow, but the velocity of every particle that passes any given point is always the same. In steady airflow, a particle of air follows the same path as the preceding particle. A streamline is the path that air particles follow in steady airflow (Figure 1-8). In steady airflow, particles do not cross streamlines. A collection of many adjacent streamlines forms a streamtube, which contains a flow just as effectively as a tube with solid walls (Figure 1-9). In steady airflow, a streamtube is a closed system, in which mass and total energy must remain constant. If mass is added to the streamtube, an equal amount of mass will be removed. An analogy is a garden hose in which each unit of water that flows in displaces another that flows out. Energy cannot be added to or removed from the system, although it can be transformed from one form to another. The Continuity Equation Figure 1-9 Streamtube Let us intersect the streamtube with two planes perpendicular to the airflow at points a-b and c-d, with cross- sectional areas of A1 and A2, respectively (Figure 1-10). The amount of mass passing any point in the streamtube may be found by multiplying area by velocity to give volume/unit time and then multiplying by density to give mass/unit time. This is called mass flow rate ( M ) and is expressed as: M AV Figure 1-10 Continuity of Flow The amount of mass flowing through A1 must equal that flowing through A2, since no mass can flow through the walls of the streamtube. Thus, an equation expressing the continuity of flow through a streamtube is: 1 A 1V1 2 A2V2 Our discussion is limited to subsonic airflow, so we can ignore changes in density due to compressibility. If we assume that both ends of the streamtube are at the same altitude, then 1 is equal to 2 and we can cancel them from our equation. The simplified, subsonic, continuity equation that we will use is: A 1V1 A2V2 If the cross sectional area decreases on one side of the equation, the velocity must increase on the same side so both sides remain equal. Thus velocity and area in a streamtube are inversely related. 26 Fundamentals of Aerodynamics: Part 1 – M06-042 Bernoulli’s Equation Daniel Bernoulli, a Swiss mathematician, described the variation of pressure exerted by a moving mass of fluid. His equation shows that the total energy of a fluid can be separated into potential energy (static pressure) and kinetic energy (dynamic pressure). Bernoulli’s equation applies in a frictionless, incompressible airflow. Static pressure (pS) is the pressure particles of air exert on adjacent bodies. Ambient static pressure is equal to the weight of a column of air over a given area. The force of static pressure always acts perpendicular to any surface that the air particles collide with, regardless of whether the air is moving with respect to that surface. Dynamic Pressure (q) is a measure of impact pressure of a large group of air molecules moving together. If a gas is static and not flowing, the measured pressure is the same in all directions. However, if the gas is flowing, the measured pressure depends on the direction of motion. It is expressed as: q 1 V 2 2 Total Pressure (H) is the sum of static and dynamic pressure. For a closed system, total pressure is constant; therefore, an increase in static pressure results in a decrease in dynamic pressure, and vice versa. It is expressed as: H pS 1 V 2 2 Where: H = total pressure p = static pressure ρ = density V = velocity Substituting for dynamic pressure gives us Bernoulli’s equation: Airspeed Measurement H pS q Airspeed is a necessary calculation for aircraft. Airspeed is critical for ensuring you have sufficient airflow over the wings for takeoff but not enough to cause structural damage, for navigation, for weapon employment and many other scenarios. However, unlike automobiles, aircraft cannot measure velocity, or dynamic pressure, directly. However, using Bernoulli’s equation, dynamic pressure can be calculated by measuring the total and static pressure acting on the aircraft. The system that accomplishes this measurement is the pitot static system. The pitot static system consists of a pitot tube that senses total pressure (H), a static port that senses static pressure (PS), and a differential pressure gauge. q H P S 27 Fundamentals of Aerodynamics: Part 1 – M06-042 Pitot tubes are used to measure total pressure. A pitot tube is a hollow tube open at one end and closed at the back. The open end is exposed into the airflow and begins to fill with air (Figure 1-1-13). Soon, the pitot tube is completely filled with air and since there is nowhere for the air to go, all motion ceases. The motion ceases at the inlet to the tube as well. According to Bernoulli’s equation, if velocity is zero, static pressure is equivalent to total pressure. The total pressure is measured at the closed end of the pitot tube. Now that we have total pressure, we need to measure static pressure. A Static port measures static pressure. The static pressure port is a hole or series of small holes on the surface of the airplane’s fuselage that are flush with the surface, and parallel to the airflow (Figure 1-11). Since the static port is parallel to the airflow, only ambient static pressure (PS) affects the static port; no dynamic pressure is sensed. Figure 1-11 Pitot Static System 28 Fundamentals of Aerodynamics: Part 1 – M06-042 Now that we have both total pressure (H) from the pitot tube, and static pressure (Ps) from the static port, the differential pressure gauge measures the difference and has an output of dynamic pressure (q). Recalling Bernoulli’s equation, and substituting for q: H P s 1 V 2 2 By rearranging Bernoulli’s equation, we are able to solve velocity for total and static pressure: 𝑉= √2(𝐻−𝑃 𝑆) 𝜌 Where: V = velocity H = total pressure Ps = static pressure Ρ = density Using the equation above, we can now turn the difference measured by the pitot static system into useable velocity. The velocity is read on the airspeed indicator in the cockpit. Indicated airspeed (IAS) is the actual instrument indication of the dynamic pressure the airplane is exposed to during flight. Factors such as altitude other than standard sea level, errors of the instrument and errors due to the installation, etc., may create great variances between instrument indication and the actual flight speed. The airspeed indicator is calibrated in knots of indicated airspeed (KIAS). Instrument error is caused by the indicator errors and errors due to the physical location of the static port on the aircraft. Indicator error is small by design and is often negligible in properly maintained equipment. When the aircraft is operated through a large range of angles of attack, the static pressure distribution varies greatly and it becomes quite difficult to minimize the position error. In most instances a compensating group of static sources may be combined to reduce the position error. When indicated airspeed is corrected for instrument error, it is called calibrated airspeed (CAS). Compressibility error is caused by the ram effect of air in the pitot tube resulting in higher than normal airspeed indications at airspeeds approaching the speed of sound. The aerodynamics and mathematics for compressible flow are complex and outside the scope of this chapter; however, the compressibility corrections can usually be found in charts. Equivalent airspeed (EAS) is the true airspeed at sea level on a standard day that produces the same dynamic pressure as the actual flight condition. It is found by correcting calibrated airspeed for compressibility error. 29 Fundamentals of Aerodynamics: Part 1 – M06-042 True airspeed (TAS) is the actual velocity at which an airplane moves through an air mass. It is found by correcting equivalent airspeed for the difference between the local air density (ρ) and the density of the air at sea level on a standard day (ρ0): TAS 2 0 EAS 2 𝑇𝐴𝑆= 𝐸𝐴𝑆√ 𝜌0 𝜌 As instrument error is typically small, and compressibility error is minor at subsonic velocities, we will ignore them and derive TAS directly from IAS: 𝑇𝐴𝑆= 𝐼𝐴𝑆√ 𝜌0 𝜌 The pitot static system is calibrated for standard sea level density, so TAS will equal IAS only under standard day conditions at sea level. Since air density decreases with an increase in temperature or altitude, if IAS remains constant while climbing from sea level to some higher altitude, TAS must increase. A rule of thumb is that for a constant IAS, TAS will increase approximately three knots for every thousand feet increase in altitude. Ground speed is the airplane’s actual speed over the ground. Since TAS is the actual speed of the airplane through the air mass, if we correct TAS for the movement of the air mass (wind), we will have ground speed. It is calculated using the following formulas: GS TAS HEADWIND GS TAS TAILWIND “ICE-TG” is a helpful mnemonic device for the order of the airspeeds. Major components of an airplane An aircraft is any device used or intended to be used for flight in the air. It is normally supported either by the buoyancy of the structure (e.g. a balloon or dirigible) or by the dynamic reaction of the air against its surfaces (e.g. an airplane, glider or helicopter). An airplane is a mechanically driven fixed-wing aircraft, heavier than air, which is supported by the dynamic reaction of the air against its wings. The T-6B is single-turboprop-engine, two-place, pressurized, low-wing training aircraft. It will be the primary example of a conventional airplane used throughout this course. The components of a conventional airplane are the fuselage, wings, empennage, landing gear, and engine(s) (Figure 1-12). The fuselage is the basic structure of the airplane to which all other components are attached. It is designed to hold crewmembers, passengers, cargo, etc. Three basic fuselage types are possible: Truss, full monocoque, and semi-monocoque. The truss type consists of a metal or wooden frame over which a light skin is stretched. The truss supports the entire stress load of the airplane. It is very strong and easily repaired, but quite heavy. Full 30 Fundamentals of Aerodynamics: Part 1 – M06-042 monocoque is extremely light and strong because it consists of only a skin shell which is highly stressed, but is almost impossible to repair if damaged. The skin supports the entire stress load of the aircraft. Semi- monocoque is a modified version of monocoque, having skin, transverse frame members, and stringers, which all share in stress loads and may be readily repaired if damaged. The T-6B uses a semi-monocoque fuselage. The wing is an airfoil attached to the fuselage and is designed to produce lift. It may contain control surfaces, fuel cells, engine nacelles, and landing gear. Ailerons are one of three major control surfaces of an airplane and are attached to the wing to control roll. Flaps and slots are high lift devices attached to the wing to increase lift at low airspeeds. The T-6B has a single low-mounted wing with split flaps integrated into the trailing edge inboard of the ailerons. Since all bracing is internal, the wings are considered to be full cantilever. The empennage is the assembly of stabilizing and control surfaces on the tail of an airplane. It provides the greatest stabilizing influence of all the components of the conventional airplane. The empennage consists of the aft part of the fuselage, the vertical stabilizer, and the horizontal stabilizer. The rudder is one of three major control surfaces of an airplane and is the upright control surface attached to the vertical stabilizer to control yaw. Elevators are one of the three major control surfaces of an airplane and are the horizontal control surfaces attached to the horizontal stabilizer to control pitch. The landing gear permits ground taxi operation and absorbs the shock encountered during takeoff and landing. The T-6B has retractable tricycle landing gear that includes a steerable nosewheel and two main wheels. During taxi operations steering is controlled by either nosewheel steering, which is accomplished using rudder pedals, rudder, and/or differential braking. The engine provides the thrust necessary for powered flight. Military and commercial airplanes may be fitted with multiple turboprop, turbojet, or turbofan engines. The type of engine depends on the mission requirements of the aircraft. The T-6B has a PT6A-68 turboprop engine. ENGINE FUSELAGE LANDING GEAR EMPENNAGE WING Figure 1-12 Airplane Components 31 Fundamentals of Aerodynamics: Part 1 – M06-042 AIRFOILS AND WING PROPERTIES Aircraft wings are designed to take advantage of atmospheric and airflow properties. The cross-sectional shape obtained by the intersection of the wing with a perpendicular plane is called an airfoil (Figure 1-13). An airfoil is a streamlined shape designed to produce lift as it moves through the air. The major design characteristic of an airfoil is the mean camber line. The mean camber line (MCL) is a line halfway between the upper and lower surface of an airfoil. The most forward and rearward points of the mean camber line are the leading edges and trailing edges, respectively. The chord line of an airfoil is an infinitely long, straight line which passes through its leading and trailing edges. Chord is the precise measurement between the leading and trailing edges measured along the chord line. Chord will typically vary from the wingtip to the wing root. The root chord (cR) is the chord at the wing centerline and the tip chord (cT) is measured at the wingtip. The average chord (c) is the average of every chord from the wing root to the wingtip. The camber of an airfoil is the maximum distance between the mean camber line and the chord line, measured perpendicular to the chord line. Mean Camber Line Figure 1-13 Wing Cross-Section View There are three types of airfoils: positively cambered, symmetric and negatively cambered. Positively cambered airfoils have the MCL above the chord line. A positively cambered airfoil produces lift at zero angle of attack. A symmetric airfoil has zero camber, which indicates that the MCL and the chord line are the same. A symmetric airfoil produces no lift at zero angle of attack. A negatively cambered airfoil has the MCL below the chord line and will produce negative lift at zero angle of attack. The aerodynamic center is the point along the chord line around which all changes in the aerodynamic force take place. On a subsonic airfoil, the aerodynamic center is located approximately one-quarter (between 23% and 27%) of the length of the chord from the leading edge. The aerodynamic center will remain essentially stationary unless the airflow over the wings approaches the speed of sound. Transonic and supersonic flight are not discussed in course. Wingspan (b) is the length of a wing, measured from wingtip to wingtip. It always refers to the entire wing, not just the wing on one side of the fuselage. The wingspan of the T-6B is 33’5”. Wing area (S) is the apparent surface area of a wing from wingtip to wingtip (Figure 1-14). More precisely, it is the area within the outline of a wing in the plane of its chord, including that area within the fuselage, hull or nacelles. The formula for S is: S bc 32 Fundamentals of Aerodynamics: Part 1 – M06-042 Taper is the reduction in the chord of an airfoil from root to tip (Figure 1-14). The wings of the T-6B are tapered to reduce weight, improve structural stiffness, and reduce wingtip vortices. Assuming the wing to have straight leading and trailing edges, taper ratio (λ) is the ratio of the tip chord to the root chord. cT cR Sweep angle (Λ) is the angle between the lateral axis and a line drawn 25% aft of the leading edge (Figure 1- 14). Sweep angle is not parallel to the leading edge on a tapered wing. Wing sweep affects maximum lift and stall characteristics. The T-6B wing is swept. Figure 1-14 Wing Planform Views Aspect ratio (AR) is the ratio of the wingspan to the average chord. An aircraft with a high aspect ratio (35:1), such as a glider, would have a long, slender wing. A low aspect ratio (3:1) indicates a short, stubby wing, such as on a high performance jet. AR b c Wing loading (WL) is the ratio of an airplane’s weight to the surface area of its wings. There tends to be an inverse relationship between aspect ratio and wing loading. Gliders have high aspect ratios and low wing loading. Fighters with low aspect ratios maneuver at high g-loads and are designed with high wing loading. The wing loading formula is: WL W S 33 Fundamentals of Aerodynamics: Part 1 – M06-042 The angle of incidence of a wing is the angle between the airplane’s longitudinal axis and the chord line of the wing (Figure 1-15). Figure 1-15 Angle of Incidence Dihedral angle is the angle between the spanwise inclination of the wing and the lateral axis (Figure 1-16). More simply, it is the upward slope of the wing when viewed from the front. A negative dihedral angle is called an anhedral angle (sometimes cathedral). The T-6B has dihedral wings to improve lateral stability. Figure 1-16 Dihedral Angle AIRPLANE REFERENCE SYSTEM An airplane’s reference system consists of three mutually perpendicular axes referenced to the center of gravity (Figure 1-17). As an airplane moves through the air, the axis system also moves. Therefore, the movement of the airplane can be described by the movement of its center of gravity. The center of gravity (CG) is the point at which all weight is considered to be concentrated and about which all forces and moments are measured. Theoretically, the airplane will balance if suspended at the center of gravity. As fuel burns, ordnance is expended or cargo loaded/unloaded, the CG can shift. 34 Fundamentals of Aerodynamics: Part 1 – M06-042 The longitudinal axis passes from the nose to the tail of the airplane. Movement around the longitudinal axis is called roll. The lateral axis passes from wingtip to wingtip. Movement around the lateral axis is called pitch. The vertical axis passes vertically through the center of gravity. Movement of the longitudinal axis around the vertical axis is called yaw. Figure 1-17 Airplane Reference System Pitch attitude (θ) is the angle between an airplane’s longitudinal axis and the horizon. An airplane’s flight path is the path described by its center of gravity as it moves through an air mass. Relative wind is the airflow the airplane experiences as it moves through the air. It is equal in magnitude and opposite in direction to the flight path. Figure 1-18 Flight Path, Relative Wind and Angle of Attack Angle of attack (α) is the angle between the relative wind and the chord line of an airfoil. Angle of attack is often abbreviated AOA. Flight path, relative wind, and angle of attack should never be inferred from pitch attitude. 35 Fundamentals of Aerodynamics: Part 1 – M06-042 Spanwise flow is airflow that travels along the span of the wing, parallel to the leading edge. Spanwise flow is normally from the root to the tip. This airflow is not accelerated over the wing and therefore produces no lift. Chordwise flow is air flowing at right angles to the leading edge of an airfoil. Since chordwise flow is the only flow that accelerates over a wing, it is the only airflow that produces lift. 36 Fundamentals of Aerodynamics: Part 1 – M06-042 ASSIGNMENT SHEET 2-1-3 BASIC THEORY REVIEW A.
INTRODUCTION This lesson is a basic introduction to the theory of aerodynamics. It provides a knowledge base in aerodynamic mathematics, air properties, airspeed and altitude definitions and measurements, airfoil and wing design, and the importance of center of gravity (CG).
B.
ENABLING OBJECTIVES C.
STUDY ASSIGNMENT 1.
Review Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 1 2.
Read Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 2 D.
STUDY QUESTIONS 1. A moment is . a. the force with which a mass is attracted toward the center of the earth b. the result of mass x acceleration c. a body’s capacity to do work d. created when a force is applied at some distance from an axis or fulcrum, producing rotation about that point 2. Mass per unit of volume defines which of the following?
a. Work b. Play c. Density d. Power 3. Potential energy is . a. the ability of a body to do work due to its motion b. dependent on mass and velocity c. a scalar measure of a body’s capacity to do work d. the ability of a body to do work due to its position or state of being 37 Fundamentals of Aerodynamics: Part 1 – M06-042 4. The Law of Interaction states . a. “For every action there is an equal and opposite reaction.” b. “A body at rest tends to stay at rest.” c. “Two opposite forces cannot interact.” d. “An unbalanced force acting upon a body produces an acceleration in the direction of the force that is directly proportional to the force and inversely proportional to the mass of the body.” 5. True airspeed will be _ a. less than b. equal to c. greater than indicated airspeed at 10,000 feet MSL. d. not enough information to determine 6. The chord of an airfoil is defined as . a. the angle between the longitudinal axis and the horizon b. a measure of the wing along the chord line, from the leading edge to the trailing edge c. an infinitely long line drawn through the leading and trailing edges of an airfoil d. the length of a wing 7. Air density is a. lower/5000 b. equal/5000 c. equal/20,000 d. lower/20,000 at 10,000 feet compared to feet. 8. The altimeter setting for standard day where pressure altitude equals true altitude is in-Hg.
a.
29.92 b.
92.29 c.
92.92 d. 1013.2 9. The center of gravity is the point at which . a. lift is generated b. all aerodynamic forces are concentrated c. elevator authority is lost d. all weight is concentrated 10. The a. center b. yaw c. pitch d. roll axis is which the aircraft nose moves left or right about the center of gravity. 38 Fundamentals of Aerodynamics: Part 1 – M06-042 Answers: 1.D 6. B 2.C 7. A 3.D 8. A 4.A 9. D 5.C 10. B 39 Fundamentals of Aerodynamics: Part 1 – M06-042 OUTLINE SHEET 2-2-1 LIFT PRODUCTION AND DRAG A.
INTRODUCTION This lesson is a continuation of aerodynamic principles from Lesson 2.1. Definitions of lift, weight, thrust and drag and how each relates to one another on an aircraft will be presented. Basic configuration principles and effects on lift production will also be discussed.
B.
ENABLING OBJECTIVES 2.80 EXPLAIN the aerodynamic relationship of the four primary forces of equilibrium flight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.81 DESCRIBE how the primary aerodynamic forces affect each other, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.82 STATE the pressure distribution around an airfoil, given changes in angle of attack and camber, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.83 DEFINE the lift component of aerodynamic force, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.84 DESCRIBE how factors in the lift equation affect lift production, given density, velocity, surface area, and coefficient of lift, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.85 LIST the factors affecting coefficient of lift that the pilot can directly control, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.86 DEFINE the drag component of aerodynamic force, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.87 DEFINE parasite drag and its components: form, friction, and interference drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 40 Fundamentals of Aerodynamics: Part 1 – M06-042 2.88 DESCRIBE the measures that can be taken to reduce each of the components of parasite drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.89 STATE the effects of upwash and downwash on an infinite wing, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.90 STATE the effects of upwash and downwash on a finite wing, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.91 DEFINE induced drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.92 STATE the cause of induced drag on a finite wing, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.93 DESCRIBE factors affecting induced drag, given the induced drag equation, and changes in lift, weight, density velocity, and wingspan, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.94 STATE when a plane will enter ground effect, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.95 STATE the effects of ground effect on lift, effective lift, and induced drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.96 DESCRIBE effects of angle of attack changes on coefficient of lift and coefficient of drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM- SG-200 2.97 EXPLAIN the lift to drag ratio, using the lift and drag equations, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.98 EXPLAIN the importance of L/D MAX, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.99 DEFINE total drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.100 DESCRIBE the effects of changes in velocity on total drag, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.101 DEFINE thrust components: thrust required and thrust available, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 41 Fundamentals of Aerodynamics: Part 1 – M06-042 2.102 DEFINE power components: power required and power available, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.103 DESCRIBE the effects of throttle setting, velocity, and density, on thrust available and power available, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.104 DEFINE thrust horsepower and components: shaft horsepower and propeller efficiency, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.105 STATE the maximum rated shaft horsepower in the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.106 EXPLAIN how propeller efficiency affects thrust horsepower, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.107 DESCRIBE power required in terms of thrust required, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.108 STATE the location of L/D MAX on the thrust required and power required curves, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.109 DESCRIBE how thrust required and power required vary with velocity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.110 DEFINE excess thrust and excess power, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.111 DESCRIBE the effects of excess thrust and excess power, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.112 DESCRIBE the effects of changes in weight on thrust and power components: thrust required, power required, excess thrust, and excess power, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.113 DESCRIBE the effects of changes in altitude on thrust and power components: thrust required, power required, thrust available, power available, excess thrust, and excess power, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM- SG-200 42 Fundamentals of Aerodynamics: Part 1 – M06-042 2.114 DESCRIBE the effects of changes in configuration on thrust and power components: thrust required, power required, excess thrust, and excess power, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.115 DESCRIBE the aerodynamic effects of raising or lowering the flaps, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.116 DESCRIBE the aerodynamic effects of raising and lowering the landing gear, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.117 EXPLAIN the aerodynamic effects of each primary flight control on the aircraft, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.118 DESCRIBE how the trim tab system holds an airplane in trimmed flight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.119 DEFINE aerodynamic balancing and mass balancing, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.120 STATE the methods for aerodynamic and mass balancing employed on the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.121 STATE the characteristics of the three basic types of control systems, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.122 STATE how trim tabs can be used to generate artificial feel on a control surface, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.123 DESCRIBE the purpose of bobweights and downsprings, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 43 Fundamentals of Aerodynamics: Part 1 – M06-042 C.
TOPIC OUTLINE 1.
Introduction 2.
This Lesson Topic 3.
Equilibrium 4.
Aerodynamic Force 5.
Pressure Distribution 6.
Factors Affecting Lift 7.
Parasite Drag 8.
Induced Drag 9.
Lift to Drag Ratio 10.
Thrust and Power Available 11.
Thrust and Power Required 12.
Horsepower and Propeller Efficiency 13.
Excess Thrust and Power 14.
Factors Affecting Excess Thrust and Power 15.
Flight Controls 16.
Trim 17.
Aerodynamic Balancing and Mass Balancing 18.
Control Systems 19.
Artificial Feel 20.
Summary and Review 21.
Application 22.
Assignment 44 Fundamentals of Aerodynamics: Part 1 – M06-042 INFORMATION SHEET 2-2-2 LIFT PRODUCTION AND DRAG A.
INTRODUCTION This lesson is a continuation of aerodynamic principles from Lesson 2.1. Definitions of lift, weight, thrust and drag and how each relates to one another on an aircraft will be presented. Basic configuration principles and effects on lift production will also be discussed.
B.
REFERENCES 1.
Aerodynamics for Naval Aviators, NAVAIR 00-80T-80 2.
Introduction to the Aerodynamics of Flight, NASA SP-367 3.
T-6 Joint Primary Pilot Training, AETC / CNATRA Syllabus P-V4A-J NATOPS Flight Manual C.
INFORMATION PRIMARY FORCES There are four primary forces that act on an aircraft: Lift, Weight, Thrust, and Drag. Weight is the force of the Earth’s gravity acting on the mass of the aircraft. It is always pointed towards the center of the Earth. Thrust is the force produced by a jet engine or engine/propeller combination.
Lift is the force that primarily acts against weight. Drag is the force that primarily acts against thrust and retards aircraft motion. Lift and drag are components of the aerodynamic force which will be discussed in greater detail next.
45 Fundamentals of Aerodynamics: Part 1 – M06-042 2 2 2 2 2 2 AERODYNAMIC FORCE Figure 2-1 Aerodynamic Forces The only mechanisms nature has for communicating a force to a body moving through a fluid are pressure and shear stress distributions on the body surface. Pressure acts normal (perpendicular) to the body while sheer stress acts tangential to the surface. When integrated over the entire surface, the resultant vector is the aerodynamic force. The aerodynamic force (AF) is the net force that results from pressure and shear stress distribution over an airfoil, and can be broken down into two components, lift and drag (Figure 2-1). Lift (L) is the component of the aerodynamic force acting perpendicular to the relative wind. Drag (D) is the component of the aerodynamic force acting parallel to and in the same direction as the relative wind. Aerodynamic force can be expressed as the product of dynamic pressure (q), surface area (S) of the airfoil, and a coefficient of aerodynamic force (CF). This coefficient represents the shape and orientation of the surface area with respect to the relative wind. The equation is: AF qSCF 1 V SCF Because lift and drag are components of aerodynamic force, they are also functions of dynamic pressure, surface area and a coefficient that represents the shape and orientation of the surface area. They are expressed as: L qSCL 1 V SCL Where: CL = Coefficient of Lift CD = Coefficient of Drag LIFT D qSCD 1 V SCD PRODUCTION OF LIFT As stated above, pressure and sheer stress are the only two aerodynamic forces that can act on a body. Because lift is the component of the resultant force in the perpendicular direction and because the pressure on the surface, at reasonable angles of attack, acts mainly in the lift direction 46 Fundamentals of Aerodynamics: Part 1 – M06-042 we can say that lift is mainly due to an imbalance of pressure distributions over the top and bottom surface. b a A A B Figure 2-2 Airflow Around a Positive Cambered Airfoil Figure 2-2 shows two streamtubes flowing around a positively cambered airfoil. The shaded streamtube A flows over the airfoil and streamtube B flows underneath the airfoil. Both streamtubes originate in the free stream ahead of the airfoil. As the streamtube A flows toward the airfoil, it senses the upper portion of the airfoil as an obstruction and must flow around it. In doing so, streamtube A is reduced to a smaller cross-sectional area as it flows over the leading edge of the airfoil. Because of the mass continuity equation (ρAV = constant), the velocity in the streamtube must increase. The higher velocity at point a is shown by the long arrow. As the flow continues past point a, the cross-sectional area increases and flow velocity decreases, as shown by the smaller velocity arrow at point b. Because the airfoil is positively cambered, streamtube B sees less of an obstruction when flowing over the bottom of the airfoil. Therefore, streamtube B does not have the same reduction in cross-sectional area as streamtube A, resulting in a lower velocity. Figure 2-3 Pressure Distribution Around Positively Cambered Airfoil at Zero and Positive AOA Bernoulli’s equation stated that for incompressible flow, PS + ½ ρV2 is constant. Therefore, as velocity increases, static pressure decreases. This is called the Bernoulli effect. Finally, because the lower static pressure over the top surface of the airfoil is less than the static pressure on the lower surface, the airfoil experiences a lift force in the upward direction (Figure 2-3). 47 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-4 Lift in a Loop Although lift is often thought of as an upward force opposing weight, it can act in any direction. It is always perpendicular to the relative wind, not the horizon. In Figure 2-4, the relative wind and lift vectors are shown for an airfoil during a loop maneuver. Note that the lift vector is always perpendicular to the relative wind. Increasing the angle of attack results a continued reduction of the cross-sectional area of the streamtube flowing over the top surface of an airfoil, resulting in more lift being created (Figure 2- 5). Angle of attack will be discussed in greater detail later. It is sometimes written that a fluid element that comes to the leading edge splits into two elements, one of which flows over the top surface and the other over the bottom surface. It is then assumed that the two elements meet at the trailing edge and since the distance over the top of the surface is longer than over the bottom that the top element needed to move faster. This is a common misconception and not true. Experimental results and complex mathematics show that the element traveling over the top surface of the airfoil departs the surface long before the companion element reaches the trailing edge. A symmetric airfoil at zero angle of attack produces identical velocity increases and static pressure decreases on both the upper and lower surfaces. Since there is no pressure differential perpendicular to the relative wind, the airfoil produces zero net lift. Figure 2-5 Pressure Distribution Around Symmetric Airfoil at Zero and Positive AOA 48 Fundamentals of Aerodynamics: Part 1 – M06-042 2 2 FACTORS AFFECTING LIFT L qSCL 1 V SCL There are eight factors that affect lift: density (ρ), velocity (V), surface area (S), compressibility, aspect ratio (AR), viscosity (μ), angle of attack (α), and camber. Compressibility, aspect ratio, viscosity, angle of attack, and camber are accounted for in the coefficient of lift. When an airfoil is exposed to greater dynamic pressure (q), it encounters more air particles and thus produces more lift. Therefore, lift is dependent upon the density of the air (i.e., the altitude) and the velocity of the airflow. An increase in density or velocity will increase lift. Since lift is produced by pressure, which is force per unit area, it follows that a greater area produces a greater force. Therefore, an increase in wing surface area produces greater lift. The pilot has no control over viscosity and compressibility. Viscosity and compressibility will vary with altitude, airspeed, air composition and many other factors. Because there is no way to control viscosity, or compressibility, they will be ignored in this discussion unless specifically addressed. Aspect ratio deals with the shape of the wing. In swing-wing aircraft wing sweep can be controlled. However, since these aircraft are no longer common, aspect ratio will also be ignored in this discussion unless specifically addressed. The two aspects of the coefficient of lift that can be controlled are the shape of the airfoil and the AOA. AOA is the most important factor in the coefficient of lift, and the easiest for the pilot to change. As angle of attack increases, the coefficient of lift initially increases. In order to maintain level flight while increasing angle of attack, velocity must decrease. Otherwise, lift will be greater than weight and the airplane will climb. Velocity and angle of attack are inversely related in level flight. 1 2 L 2 V S CL As angle of attack increases, the coefficient of lift increases up to a maximum value (CLmax). The AOA at which CLmax is reached is called CLmax AOA (Figure 4-6). Any increase in angle of attack beyond CLmax AOA causes a decrease in the coefficient of lift. Since CLmax is the greatest coefficient of lift that can be produced, we call CLmax AOA the most effective angle of attack. Note that as long as the shape of an airfoil remains constant, CLmax AOA will remain constant, regardless of weight, dynamic pressure, bank angle, etc. 49 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-6 plots CL as it varies with AOA. These curves are for three different airfoils: One symmetric, one negative camber and one positive camber. The shape of the CL curve is similar for most airfoils. At zero angle of attack, the positive camber airfoil has a positive CL, and the negative camber airfoil has a negative CL. The point where the curves cross the horizontal axis is the AOA where the airfoil produces no lift (CL = 0). At zero AOA the symmetric airfoil has CL = 0. The positive camber airfoil must be at a negative AOA, and the negative camber airfoil must be at a positive AOA for the CL to equal zero. -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 ANGLE OF ATTACK Figure 2-6 Camber vs. AOA 50 Fundamentals of Aerodynamics: Part 1 – M06-042 2 2 DRAG Drag is the component of the aerodynamic force that is parallel to the relative wind, and acts in the same direction. The drag equation is the same as the aerodynamic force equation, except that that the coefficient of drag (CD) is used. D 1 V SCD CD may be plotted against angle of attack for a given aircraft with a constant configuration (Figure 2-7). Note that CD is low and nearly constant at very low angles of attack. As angle of attack increases, CD rapidly increases. Since there is always some resistance to motion, drag will never be zero, so CD will never be zero. Drag is divided into parasite drag 0 2 4 6 8 10 12 14 16 18 20 22 ANGLE OF ATTACK Figure 2-7 Coefficient of Drag and induced drag. By independently studying the factors that affect each type, we can better understand how they act when combined. DT DP DI PARASITE DRAG Parasite drag (DP) is drag that is not associated with the production of lift. It is composed of form drag, friction drag and interference drag. Form drag, also known as pressure drag or profile drag, is caused by airflow separation from a surface and the low pressure wake that is created by that separation. It is primarily dependent upon the shape of the object. In Figure 2-8, the flat plate has a leading edge stagnation point at the front with a very high static pressure. There is also a low static pressure wake area behind the plate. This pressure differential pulls the plate backward and retards forward motion. Conversely, streamlines flow smoothly over a smooth shape (Figure 2-9 and Figure 2-10) and less form drag is developed. To reduce form drag, the fuselage and other surfaces exposed to the airstream are streamlined (shaped like a teardrop). Streamlining reduces the size of the high static pressure area near the leading edge stagnation point and reduces the size of the low static pressure wake. Because of the decreased pressure differential, form drag is decreased. 51 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-8 Flat Plate Figure 2-9 Sphere Figure 2-10 Streamlining Due to viscosity, a retarding force called friction drag is created as air interacts with the surfaces of an object. Friction drag is usually small per unit area, but since it covers the entire surface of the airplane, friction drag can become significant in larger airplanes. Rough surfaces create irregularities that increase skin friction. Friction drag can be reduced by smoothing the exposed surfaces of the airplane through painting, cleaning, waxing or polishing. Since irregularities of the wing’s surface cause the boundary layer to become turbulent, using flush rivets on the leading edges also reduces friction. Greater friction drag can be useful in decreasing form drag. Recall that form drag increased with the size of the low pressure wake that resulted from airflow separation around an object. Delaying the separation Figure 2-11 Dimples on a Golf Ball of airflow reduces the form drag encountered. Friction drag allows airflow to adhere to a surface longer, delaying airflow separation and reducing the size of the turbulent wake. The dimples on a golf ball (Figure 2-11) reduce the overall parasite drag on the ball by reducing the size of the turbulent wake. The reduction in form drag is greater than the increase in friction drag and allows the dimpled golf ball to travel further than a smooth golf ball. Interference drag is generated by the mixing of streamlines between components. An example is the air flowing around the fuselage mixing with air flowing around an external fuel tank. We know the drag of the fuselage and the drag of the fuel tank individually. The total drag after we attach the fuel tank will be greater than the sum of the fuselage and the fuel tank separately. Roughly 5 to 10 percent of the total drag on an airplane can be attributed to interference drag. Interference drag can be minimized by proper fairing and filleting, which allows the streamlines to meet gradually rather than abruptly. 52 Fundamentals of Aerodynamics: Part 1 – M06-042 2 PARASITE DRAG (DP) Total parasite drag (DP) can be found by multiplying dynamic pressure by an area. Equivalent parasite area (f) is the area of a flat plate perpendicular to the relative wind that would produce the same amount of drag as form drag, friction drag and interference drag combined. It is not the cross-sectional area of the airplane. The equation for DP is: VELOCITY (TAS) DP 1 V 2 f qf Figure 2-12 DP vs. Velocity Parasite drag varies directly with velocity squared (V2), so a doubling of speed will result in four times as much parasite drag (Figure 2-12).
INDUCED DRAG INFINITE WING Consider a wing placed in a wind tunnel with the tips flush against the walls. For all practical purposes it has no wingtips and is called an infinite wing. The relative wind on an infinite wing can only flow chordwise, and therefore produces lift. As the relative wind flows around the infinite wing, the high pressure air under the leading Figure 2-13 Upwash and Downwash edge attempts to equalize with the low pressure air above the wing. The shortest route is around the leading edge. This results in some of the air that otherwise would have passed under the wing flowing up and over the leading edge. This flow is called upwash. Upwash increases lift because it increases the average angle of attack on the wing. Some of the air on top of the wing also flows down and under the trailing edge. This flow is called downwash. Downwash decreases lift by reducing the average angle of attack on the wing. For an infinite wing, the upwash exactly balances the downwash resulting in no net change in lift. Upwash and downwash exist any time an airfoil produces lift. 53 Fundamentals of Aerodynamics: Part 1 – M06-042 INDUCED DRAG (DI) FINITE WING Upwash and downwash are not equal on a finite wing. Not only does air flow up around the leading edge of a finite wing producing upwash, it also flows around the wingtips. Some of the high pressure air in the leading edge stagnation point flows spanwise to the wingtips instead of chordwise over the upper surface of the wing. Once it reaches the wingtips it flows around the wingtips and up to the upper surface of the wing. There, it combines with the chordwise flow that has already produced lift and adds to the downwash. Downwash approximately Figure 2-14 Finite Wing doubles by this process due to the spanwise airflow moving around the wingtip. The circular motion imparted to the increased downwash also results in the formation of wingtip vortices. Induced drag (DI) is that portion of total drag associated with the production of lift. We can add the airflow at the leading edge and the airflow at the trailing edge of the wing in order to determine the average relative wind in the immediate vicinity of the wing. Since there is twice as much downwash as upwash near the wingtips of a finite wing, the average relative wind has a downward slant compared to the free airstream relative wind. The total lift vector will now be inclined aft, in order to remain perpendicular to the average relative wind. The total lift vector has components that are perpendicular and parallel to the free airstream relative wind. The perpendicular component of total lift is called effective lift. Because total lift is inclined aft, effective lift will be less than total lift. The parallel component of total lift is called induced drag since it acts in the same direction as drag and tends to retard the forward motion of the airplane. VELOCITY (TAS) Figure 2-15 Induced Drag Figure 2-16 DI vs. Velocity 54 Fundamentals of Aerodynamics: Part 1 – M06-042 The DI equation is derived from the aerodynamic force equation and the assumption that weight equals lift in equilibrium level flight: DI k L 2 V 2b2 kW2 V 2b2 Analyzing the equation shows that increasing the weight of an airplane will increase induced drag, since a heavier airplane requires more lift to maintain level flight. Induced drag is reduced by increasing density (ρ), velocity (V), or wingspan (b). In level flight where lift is constant, induced drag varies inversely with velocity, and directly with angle of attack. Another method to reduce induced drag is to install devices that impede the spanwise airflow around the wingtip. These devices include winglets, wingtip tanks, and missile rails. GROUND EFFECT A phenomenon, known as ground effect, significantly reduces induced drag and increases effective lift when the airplane is within one wingspan of the ground. Because takeoffs and landings are conducted at low airspeeds, induced drag makes up a large portion of the total drag on the airplane. As an airplane nears the ground, the downwash at the trailing edge of the wing is unable to flow downward. The decrease in downwash allows the total lift vector to rotate forward, increasing effective lift and decreasing induced drag. When the aircraft is one wingspan above the ground (about 33 feet for T-6B) induced drag is reduced by only 1.4%, at one-fourth the wingspan, induced drag is reduced by 23.5%, and a maximum reduction of 60% occurs just prior to touchdown or after liftoff (Figure 2-17). Because of the increased lift, it is possible to get airborne at an airspeed below normal flying speed. As an airplane takes off and leaves ground effect, induced drag increases and lift decreases, which could cause an altitude loss, possibly resulting in an unintentional gear-up landing. Entering ground effect (during landing) increases effective lift and decreases induced drag by preventing the aft inclination of the lift vector. When the plane enters ground effect it will float down the runway if the pilot does not reduce thrust. The timing of the flare and power reduction when in ground effect is the most difficult aspect of the landing phase for most students. 55 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-17 Ground Effect TOTAL DRAG Parasite and Induced drag can be added together to create a total drag curve. By superimposing both drag curves on the same graph, and adding the values of induced and parasite drag at each velocity, the total drag curve of Figure 2-18 is derived. The numbers 1, 4.4, and 17 depicted near the curve are reference values in the angle of attack scale. Note that they decrease as TAS increases. The drag curve depicted is particular to one aircraft at one weight, one altitude and one configuration. As weight, altitude and configuration change, the total drag curve will shift. Figure 2-18 DT vs. Velocity 56 Fundamentals of Aerodynamics: Part 1 – M06-042 2 2 LIFT TO DRAG RATIO An airfoil is designed to produce lift, but drag is unavoidable. An airfoil that produced the desired lift but causes excessive drag is not desirable. The lift to drag ratio (L/D) is used to determine the efficiency of an airfoil. A high L/D ratio indicates a more efficient airfoil. L/D is calculated by dividing lift by drag. All terms except CL and CD cancel out: 1 2 L 2 V SCL CL D 1 V SCD CD A ratio of the coefficients at a certain angle of attack determines the L/D ratio at that angle of attack. The L/D ratio can be plotted against angle of attack along with CL and CD (Figure 2-19). The maximum L/D ratio is called L/DMAX. Since angle of attack indicators are far less precise than airspeed indicators, pilots will typically fly an airspeed that corresponds to L/DMAX AOA. L/DMAX AOA produces the minimum total drag. L/DMAX is located at the bottom of the total drag curve. Any movement away from L/DMAX will increase drag. • At L/DMAX AOA, parasite drag and Figure 2-19 Lift to Drag Ratio induced drag are equal. At velocities below L/DMAX, the airplane is affected primarily by induced drag, while at velocities above L/DMAX, the airplane is affected primarily by parasite drag.
• L/DMAX AOA produces the greatest ratio of lift to drag. Note that this is not the maximum amount of lift that can be produced, nor does it correspond to the airplanes maximum speed.
• L/DMAX AOA is the most efficient angle of attack. Note that L/D is the efficiency of the wing, not the engine.
• An increase in weight or altitude will increase L/DMAX airspeed, but not affect L/DMAX or L/DMAX AOA. A change in configuration may have a large effect on L/DMAX and L/DMAX airspeed. The effect of configuration on L/DMAX AOA will depend on what causes the change (lowering landing gear or flaps, dropping external stores, speed brakes, etc.), and how much change is produced. This will be discussed more in the thrust and power section.
57 Fundamentals of Aerodynamics: Part 1 – M06-042 Thrust And Power Thrust and Power Curves This lesson makes the following assumptions: 1. Equilibrium flight on a standard day.
2. No afterburner for a turbojet.
3. Fixed pitch propeller for a turboprop.
Propeller Efficiency The opposing force to drag is called thrust. In propeller driven aircraft the thrust, or power available, is determined by the propeller/engine output. Thrust horsepower (THP) is the output from the propeller and shaft horsepower (SHP) is the output from the engine. The ability of the propeller to convert SHP into THP is determined by the propeller efficiency (PE). Propeller efficiency is defined as: PE THP SHP Under ideal conditions, THP = SHP. However, due to friction in the reduction gearbox and drag on the propeller, efficiency is never 100%. Fixed vs. Variable Pitch Propellers All propellers become less efficient as altitude and/or temperature increase due to the decrease in density. The loss of thrust can be minimized in one of two ways: fixed pitch or variable pitch propellers. Fixed pitch propellers have a constant blade angle and thrust is increased by increasing propeller RPM. Variable pitch propellers, also called constant speed propellers, keep propeller RPM constant and increase thrust by increasing blade angle of the propeller blades. The T-6B has a variable pitch, constant speed propeller that rotates at 2000 RPM. At sea level, the PT6A-68 engine in the T-6B is flat rated at 1100 SHP. There is no direct indication of engine power available to the pilot. Because the T-6B uses a constant speed propeller, the torque the engine exerts on the propeller shaft is directly proportional to engine output power. Shaft torque (referred to as Torque on the instruments) is therefore used as the instrument indication of engine power. 58 Fundamentals of Aerodynamics: Part 1 – M06-042 Thrust Required Total drag is the sum of parasite and induced drag. In equilibrium flight, thrust must equal drag, so the amount of thrust that is required to overcome drag can be found on the total drag curve. This amount of thrust is called thrust required (TR), and is expressed in pounds. As with the drag curve, the thrust required curve is for one specific weight, altitude and configuration. L/DMAX AOA is the point of minimum thrust required, and is obtained at some specific velocity. Flight at greater velocities requires a reduction in AOA (to maintain constant lift) and an increase in thrust (to match the increase in parasite drag). Flight at lower velocities requires an increase in AOA and an increase in thrust (to match the increase in induced drag).
Thrust Available Thrust available (TA) is the amount of thrust that the airplane’s engines actually produce at a given throttle setting, velocity, and density. The most important factor is the throttle, also called the power control lever (PCL) in turboprops. Maximum engine output occurs at full throttle. As the throttle is retarded, thrust available decreases. Since the propeller can only accelerate the air to a maximum velocity, as the velocity of the incoming air increases, the air is accelerated less through the propeller, and thrust available decreases (Figure 2-22). Turbojets do not suffer a decrease in thrust available with velocity because ram-effect overcomes the decreased acceleration (Figure 2-23). Therefore, TA is approximated by a straight line. For both turbojets and turboprops, as the density of the air decreases, thrust available decreases (Figures 2-35 and 2-37).
59 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-22 Thrust Available (Turboprop) Figure 2-23 Thrust Available (Turbojet) Power Required Power is the rate of doing work, and work is a force times a distance. Power required (PR) is the amount of power that is required to produce thrust required. PR is the product of TR and velocity (V). If V is expressed in knots, then the product of TR and V must be divided by 325 to give power in units of horsepower. Thus, thrust horsepower only depends on thrust and velocity (Figure 2-24). For simplicity, we will use the term power (P) rather than thrust horsepower (THP) or shaft horsepower (SHP) unless there is a significant difference. P TR V R 325 To find L/DMAX on the thrust required curve, draw a horizontal line tangent to the bottom of the curve. By applying the power equation to this line, the result is a straight line from the origin that is tangent to the power curve at L/DMAX. Unlike on the TR curve, L/DMAX is not at the bottom of the PR curve, but is to the right of the bottom of the curve. L/DMAX still represents minimum total drag, but minimum PR is to the left of L/DMAX. It should be noted that the velocity and AOA for L/DMAX are the same on the PR curve as on the TR curve. 17 4.4 Figure 2-24 Power Required 60 Fundamentals of Aerodynamics: Part 1 – M06-042 Power Available Power available (PA) is the amount of power that the airplane’s engines actually produce at a given throttle setting, velocity, and density. The most important factor is throttle setting. Maximum power available occurs at full throttle. P TA V A 325 As the pilot reduces the throttle setting, power available decreases. As velocity increases, power available for a jet will increase linearly. In a propeller driven aircraft, power available will initially increase, but will then decrease due to a decrease in thrust available (Figure 2-25 and Figure 2-26). As thrust available decreases with a decrease in density, power available will also decrease (Figures 2-36 and 2-38). Figure 2-25 Power Available (Turbojet) Figure 2-26 Power Available (Turboprop) Thrust Excess and Power Excess A comparison of the TR and TA curves on one graph allows one to predict airplane performance. To maintain equilibrium level flight, thrust available must equal thrust required for a specific angle of attack and velocity. This is depicted on a graph where the TR and TA curves cross. The right-hand point of equilibrium will produce the maximum velocity in level flight. This is the greatest airspeed that the aircraft can maintain without descending. It is approximately 255 KIAS at sea level for the T-6B.
61 Fundamentals of Aerodynamics: Part 1 – M06-042 A thrust excess (TE) occurs if thrust available is greater than thrust required at a particular velocity. A positive TE causes a climb, acceleration, or both, depending on angle of attack. A negative TE is called a thrust deficit and has the opposite effect. Maximum thrust excess occurs at a full throttle setting, and is depicted on a graph where the distance between the TR and TA curves is greatest. For a turbojet, max thrust excess occurs at L/DMAX (Figure 2-27). For a turboprop, max thrust excess occurs at a velocity less than L/DMAX (Figure 2-28). TE TA TR 17 17 4.4 4.4 Figure 2-27 Thrust Excess (Turbojet) Figure 2-28 Thrust Excess (Turboprop) Power excess (PE) is calculated in a similar manner as TE and will also produce a climb, acceleration, or both. Likewise, a power deficit will cause a decent, a deceleration, or both. For a turbojet, maximum power excess occurs at a velocity greater than L/DMAX (Figure 2-29). For a turboprop, max power excess occurs at L/DMAX (Figure 2-30). It is important to note that maximum power excess is achieved at a greater velocity and a lower angle of attack than maximum thrust excess. It should also be noted that a power excess cannot exist if thrust excess is zero. P E P A P R 17 4.4 17 4.4 Figure 2-29 Power Excess (Turbojet) Figure 2-30 Power Excess (Turboprop) 62 Fundamentals of Aerodynamics: Part 1 – M06-042 Factors Affecting TE and PE Weight If an airplane is in equilibrium level flight at a constant angle of attack, an increase in weight requires an increase in lift. In order to increase lift at a constant AOA, velocity must increase. 17 17 4.4 1 2 4.4 W L 2 V SCL A higher velocity and more lift increases both parasite and induced drag; therefore, total drag increases. The net result is the TR curve shifts up and to the right (Figure 2-31). Figure 2-31 Effect of Weight on TR 1 2 TR D 2 V SCD Power required (PR) is similarly affected by weight. An increase in weight requires an increase in velocity and a corresponding increase in thrust required (TR) at a specific angle of attack. Since PR is a function of thrust required and velocity, an increase in weight will result in an increase in power required. The net result of an increase in weight is that the TR and PR curves will shift up and right (Figure 2- 32). 17 4.4 17 T V 4.4 P R R 325 Figure 2-32 Effect of Weight on PR Weight changes have no effect on thrust available or power available, as they do not affect the engine. As weight increases, thrust required and power required increase while thrust available and power available remain constant. Thus thrust excess and power excess decrease at every AOA and velocity. TE TA TR P E PA P R 63 Fundamentals of Aerodynamics: Part 1 – M06-042 Altitude If an airplane weighs 5,000 lbs at sea level, it requires 5,000 lbs of lift. It will weigh 5,000 lbs and require the same lift at any higher altitude as well. Since density has decreased, velocity must increase to maintain 5,000 lbs of lift. Thus as altitude increases, the TR curve shifts to the right. 1 2 W L 2 V SCL The decrease in density is exactly offset by an increase in velocity to maintain constant lift for any given AOA, so the dynamic pressure felt by the airfoil remains constant. With no change in dynamic pressure as lift is maintained at a higher altitude for any fixed AOA, drag and TR remain constant. Thus as altitude increases, the thrust required curve shifts to the right, but not up (Figure 2-33). 17 17 4.4 4.4 1 2 TR D 2 V SCD Figure 2-33 Effect of Altitude on TR Since PR is the product of TR and velocity, the PR curve will shift to the right as altitude increases and the TR curve shifts to the right. Because the same thrust is multiplied by a higher velocity, the PR curve will move up as well (Figure 2- 34). T V P R R 325 Maximum engine output decreases with a reduction in air density. Thus, both TA and PA decrease at higher altitudes. Thrust excess will decrease with an increase in altitude due to the decrease in thrust available. Power excess will decrease Figure 2-34 Effect of Altitude on PR with an increase in altitude because power available decreases and power required increases. TE TA TR P E PA P R 4.4 4.4 17 17 64 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-35 Effect of Altitude on TA (Turbojet) Figure 2-36 Effect of Altitude on PA (Turbojet) Figure 2-37 Effect of Altitude on TA (Turboprop) Figure 2-38 Effect of Altitude on PA (Turboprop) Configuration Lowering the landing gear has no effect on the lift produced by the wing, so at any AOA no change in velocity is required to maintain lift. Lowering the landing gear does, however, dramatically increase parasite drag, which causes TR and PR to increase. Thus more thrust and power are required to maintain altitude for any given AOA and velocity, so both the TR and PR curves shift up. 1 2 TR D 2 V S CD The landing gear has no effect on the engine, so TA and PA are not affected. Thrust and power excess will decrease with deployment of the landing gear because TR and PR increase. 65 Fundamentals of Aerodynamics: Part 1 – M06-042 17 17 4.4 4.4 17 17 4.4 4.4 Figure 2-39 Effect of Landing Gear on TR Figure 2-40 Effect of Landing Gear on PR Lowering the flaps increases the coefficient of lift, allowing the aircraft to fly at a lower velocity to produce enough lift to offset weight, so the TR curve shifts left. 1 2 W L 2 V S CL While increasing lift, the flaps greatly increase parasite drag. Induced drag also increases. Thus total drag and thrust required increase. Viewed another way using the drag equation, the decrease in velocity is more than offset by the increase in the coefficient of drag, causing thrust required to increase. 1 2 TR D 2 V S CD The net effect of lowering flaps is to shift both the TR and PR curves up and to the left. More thrust and power are required to maintain altitude for any given velocity. 17 17 4.4 4.4 17 17 4.4 4.4 Figure 2-41 Effect of Flaps on TR Figure 2-42 Effect of Flaps on PR As with the landing gear, flaps have no effect on the engine, so TA and PA are not affected. Thrust and power excess will decrease with deployment of the flaps because thrust and power required increase. 66 Fundamentals of Aerodynamics: Part 1 – M06-042 Aircraft Control Systems Control Surfaces To maneuver, the pilot must redirect the forces acting on the airplane. Control surfaces allow the pilot to change the amount of lift of the airfoil to which they are attached and create different airplane motions, such as yaw, pitch, and roll. The elevator is attached to the trailing edge of the horizontal stabilizer, and controls the pitching moment around the lateral axis. Moving the control stick forward causes the elevator to move down. When the elevator moves downward, it increases the camber and produces more lift, forcing the tail of the airplane up and pitching the nose down. The opposite movements are used to pitch the aircraft up. Some airplanes move the entire horizontal stabilizer. This is called a stabilator and is used on the F-15 and F/A-18. The ailerons are attached to the outboard trailing edges of the wings, and produce a rolling moment. Ailerons move in unison in opposite directions. If the control stick is pushed left, the left aileron raises creating a negative camber on left wing, producing lift in the downwards direction. At the same time, the right aileron lowers increasing the camber of the right wing, producing more lift. The difference in lift between the wings causes the plane to roll left Figure 2-43 Aileron Operation During a Roll to the Left (Figure 2-43). As long as the ailerons are deflected the airplane will continue to roll. When the stick is centered, the airplane will stop rolling, and remain at that bank angle until the stick is deflected again. Spoilers may be attached to the wing’s upper surface to provide roll control on some aircraft. Spoilers disrupt the airflow over the top of the wing in order to decrease the lift on the wing and cause the wing to roll downward. Spoilers may be used in conjunction with ailerons and/or stabilators. The rudder is attached to the trailing edge of the vertical stabilizer, and produces a yawing moment. Stepping on the right rudder pedal moves the rudder to the right, creating an airfoil positively cambered and creating lift causing the tail to “fly” left and yawing the airplane’s nose to the right. Trim Tabs Trim tabs are attached to the trailing edge of each control surface and have two functions. The primary purpose of trim tabs is to trim. Trimming reduces the force required to hold control surfaces in a position necessary to maintain a desired flight attitude. Trim allows the pilot to fly virtually hands off, momentarily freeing the pilot’s hands for other tasks, such as tuning radios or folding charts. The second purpose of trim tabs is to provide artificial feel. 67 Fundamentals of Aerodynamics: Part 1 – M06-042 If the pilot pulls back on the control stick, the elevator is deflected up so that the nose of the airplane pitches up. The airflow around the horizontal stabilizer creates a downward force on the elevator that acts at a distance (moment arm) from the hinge line (Figure 2- 44). This creates a moment that tends to move the elevator back in line with the horizontal stabilizer. In order to keep the airplane’s nose up, the pilot must exert enough back pressure on the control stick to overcome the moment created by the elevator’s force. By moving the trim tab in the opposite direction as the control surface, a small force is created by the trim tab in the opposite direction. Since this small force has a greater moment arm, it creates a moment that exactly opposes the moment created on the elevator. Once the sum of the moments is zero, the elevator will remain in place until the pilot moves the control stick again. For trimming, Figure 2-44 Trim Tab Operation trim tabs must always be moved in the opposite direction as the control surface. If the pilot moves a control away from its trimmed position, and then releases it, the trim tabs will cause the control surface to move back to its trimmed position. If the pilot moves the control surface and wants it to remain in place, the control surface must be re-trimmed. T-6B trim settings are changed by adjusting the trim switches on the control stick (aileron/elevator) and on the PCL (Rudder). There are no aileron trim tabs that can be adjusted in flight on the T-6B. When the T-6B aileron trim switch is adjusted, the ailerons themselves actually move. T-6B rudder trim is automatically adjusted during flight by the trim aid device (TAD). Rudder trim compensates for prop wash and torque, which vary with power. Elevator trim is adjusted to maintain various angles of attack while changing airspeed. A general rule is: right rudder trim is required for power increases and slower airspeeds; left rudder trim is required for power reductions and faster airspeeds (power changes take precedence at low speeds). Elevator trim is adjusted up at slower speeds and down at higher speeds (Figure 2-45). Control Balancing Figure 2-45 Rudder and Elevator Trim The forces that act at the control surface’s center of gravity and aerodynamic center must be balanced around the hinge line in order to regulate control pressure, prevent control flutter, and provide control-free stability. Control-free is the situation where the controls are not being manipulated by the pilot (hands off). Aerodynamic balance concerns balancing the forces that act at the aerodynamic center. Mass balance concerns balancing the forces that act at the center of gravity. 68 Fundamentals of Aerodynamics: Part 1 – M06-042 Aerodynamic balance is used to keep control pressures associated with higher velocities within reasonable limits. As the trailing edge of the control surface is deflected in one direction, the leading edge deflects into the airstream forward of the hinge line (Figure 2-46). The force on the leading edge creates a moment that reduces the force required to deflect the control surface, so the pilot may control the airplane more easily. For aerodynamic balance, the T-6B uses shielded horns on the elevator and rudder (Figure 2-47). The relationship of the control surface CG to the hinge line will determine the control-free stability of the airplane. The hinge line is the point around which control surfaces are balanced in the T-6B. Stability is more desirable in transport and bomber type airplanes and therefore the control surface CG is usually located forward of the hinge line. This keeps the control surface aligned with the fixed surface ahead of it when struck by gusts from turbulence. For high performance airplanes, the CG is located Figure 2-46 Aerodynamic Balance on or aft of the hinge line. With the CG aft of the hinge line, the control tends to float into the relative wind and cause a greater displacement which allows a faster response to control action and makes the airplane more maneuverable. To gain a balance between control response and stability, the T-6B control surfaces CGs are located on the hinge line. To locate the CG on the hinge line, weights are placed inside the control surface in the area forward of the hinge line (shielded horn and leading edges). This technique is called mass balancing. Mass balancing of the T-6B ailerons is achieved by placing weights in the overhang. Figure 2-47 Aerodynamic Balance on a Horizontal Stabilizer Control Feel There are several basic types of control systems used to move the control surfaces: conventional, power- boosted, and full power (fly-by-wire). In conventional controls the forces applied to the stick and rudder pedals are transferred directly to the control surfaces via push-pull tubes, pulleys, cables and levers. If an external force moves the control surfaces, the stick or rudder pedal will move in the cockpit. This action is called reversibility and gives the pilot feedback. HINGELINE 69 Fundamentals of Aerodynamics: Part 1 – M06-042 Feedback is the force that the pilot feels in his hands or feet for a given deflection of the stick or rudder pedals. Without feedback the pilot would tend to over control and possibly overstress the airplane. The T-6B uses conventional controls. Power-boosted controls have mechanical linkages with hydraulic, pneumatic, or electrical boosters to assist the pilot in moving the controls in the same way power steering assists a car driver. The degree to which the controls are boosted varies depending upon the mission and design of the airplane. These systems have some reversibility, and the pilot receives some control feel through the cockpit controls. If the boost system fails, the pilot can still control the airplane, although the control forces will be greatly increased. With a full-power or fly-by-wire control system, the pilot has no direct connection with the control surfaces. The controls of a full power system are connected to hydraulic valves or electrical switches which control the movement of the control surfaces. The fly-by-wire system uses computer commands to displace the controls. These systems are not reversible. Movement of the control stick causes the control surfaces to move, but movement of the control surfaces will not cause the control stick to move. Since these systems are not reversible, they require an artificial means of producing control feel. Artificial feel is the use any device used to create or enhance control feedback under various flight conditions such as airspeed and acceleration changes. The T-6B uses trim tabs, a bobweight and 2 downsprings to provide artificial feel to the pilot. There are three types of trim tabs that provide artificial feel: servo, anti-servo, and neutral. Servo trim tabs move in the opposite direction of the control surface, helping the pilot deflect the control surface and making the airplane easier to maneuver (Figure 2-48). Servo trim tabs are generally found on ailerons. The T-6B uses neutral trim tabs on the ailerons. Anti-servo trim tabs move in the same direction, requiring more force to hold the control surface at full deflection. Artificial feel is provided in the T-6B rudder by an anti-servo trim tab. In the T-6B, when the rudder is displaced, the anti-servo trim tab moves in the same direction at a faster rate (Figure 2-49). The more that a rudder pedal is pressed, the greater the resistance that the pilot will feel. Because trim tabs do not provide the desired type of artificial feel, the T-6B elevator uses a neutral trim tab that maintains a constant angle to the elevator when the control surface is deflected (Figure 2-50). The elevator uses both a bobweight, 2 downsprings and a neutral trim tab to provide the pilot with some artificial feel (Figure 2-51). The downsprings increase the force required to pull the stick aft at low airspeeds and the bobweight increases the force required to pull the stick aft during maneuvering flight. Figure 2-48 Servo Trim Tab Figure 2-49 Anti-Servo Trim Tab Figure 2-50 Neutral Trim Tab 70 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 2-51 Elevator Artificial Feel Historical Note The problem of balancing stability and maneuverability has been around since flight began. It is interesting to note that the Wright brothers fully recognized this problem. They were the first to understand the need for positive roll control, such as ailerons. The Wright brothers were the first to demonstrate the use of ailerons with rudder for producing a coordinated turn (no sideslip). Interestingly they made their airplanes highly maneuverable by designing them to be statically unstable. Since then, most airplanes have been statically stable and relatively easy to fly. 71 Fundamentals of Aerodynamics: Part 1 – M06-042 ASSIGNMENT SHEET 2-2-3 LIFT PRODUCTION AND DRAG REVIEW A.
INTRODUCTION This lesson is a continuation of aerodynamic principles from Lesson 2.1. Definitions of lift, weight, thrust and drag and how each relates to one another on an aircraft will be presented. Basic configuration principles and effects on lift production will also be discussed.
B.
ENABLING OBJECTIVES C.
STUDY ASSIGNMENT 1.
Review Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 2 2.
Read Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 3 D.
STUDY QUESTIONS 1. Aerodynamic force is the . a. force that is perpendicular to the relative wind b. leading edge stagnation point c. result of pressure and friction distribution over an airfoil d. force that is parallel to the relative wind 2. The component of aerodynamic force that acts parallel to the relative wind is . a. parasite drag b. lift c. drag d. angle of attack 3. A airfoil produces lift at zero degrees angle of attack. a. symmetric b. tapered c. dihedral d. positively cambered 72 Fundamentals of Aerodynamics: Part 1 – M06-042 4. L/DMAX is the point where . a. drag is the greatest b. lift is the greatest c. the ratio of lift to drag is the greatest d. the stall angle of attack is reached 5. Total drag is composed of . a. parasite and induced drag b. form and interference drag c. thrust and lift d. upwash and downwash 6. Parasite drag is _. a. the component of drag associated with the production of lift b. minimized by installing winglets, wingtip fuel tanks or missile rails c. greatest at low airspeeds d. made up of form, friction, and interference drag 7. Maximum excess power for a propeller-driven aircraft occurs a. at an airspeed below b. at c. at an airspeed above d. independent of L/DMAX. 8. As altitude increases, power available . a. increases b. remains the same c. decreases d. is equal to thrust available 9. A a. servo b. neutral trim tab will increase the force required to move a control surface to full deflection. c. anti-servo d. balanced 73 Fundamentals of Aerodynamics: Part 1 – M06-042 Additional Aero I Review Questions: 1. What are the differences between true airspeed (TAS), groundspeed (GS), indicated airspeed (IAS), calibrated airspeed (CAS), and equivalent airspeed (EAS)?
2. What is the difference between a symmetric and a cambered airfoil?
3. What are the definitions for angle of attack, pitch angle, and angle of incidence?
4. Why does parasite drag increase and induced drag decrease with increased airspeed?
5. When does maximum excess thrust occur for any aircraft in-flight?
6. Does the maximum power excess for a prop aircraft occur at a velocity slower than, equal to, or faster than L/DMAX airspeed?
74 Fundamentals of Aerodynamics: Part 1 – M06-042 ANSWERS: 1.C 6. D 2.C 7. B 3.D 8. C 4.C 9. C 5.A ADDITIONAL REVIEW ANSWERS: 1. a. Indicated airspeed (IAS): Read on airspeed indicator; measure of dynamic pressure b. Calibrated airspeed (CAS): IAS corrected for instrument error c. Equivalent airspeed (EAS): CAS corrected for compressibility; speed at sea level on a std day d. True airspeed (TAS): EAS corrected for density; speed relative to the air mass e. Groundspeed (GS): TAS corrected for wind; speed relative to the ground 2. a. Symmetric: MCL and chord line are the same b. Positively cambered airfoil: MCL above chord line 3. a. Angle of attack (AOA): Angle between relative wind and wing chord line b. Pitch angle: Angle between the horizon and aircraft longitudinal axis c. Angle of incidence: Preset angle between wing chord line and aircraft longitudinal axis 4. a. Parasite drag is dependent upon the amount of air flowing over, around, and between surfaces. When airflow increases, parasite drag increases.
b. Induced drag is created by the production of lift and is proportional to aircraft AOA.
Unless the aircraft is in high G flight, AOA decreases with increased airspeed, therefore induced drag decreases.
5. a. Maximum thrust excess occurs when the difference between thrust available and thrust required (total drag) is the greatest.
b. Max thrust excess for a propeller-driven aircraft occurs at an airspeed slower than L/DMAX airspeed.
6. a. Maximum power excess occurs when the difference between power available and power required (total drag X velocity) is the greatest.
b. Max power excess for a propeller-driven aircraft occurs at L/DMAX airspeed.
75 Fundamentals of Aerodynamics: Part 1 – M06-042 OUTLINE SHEET 2-3-1 STALLS A.
INTRODUCTION This lesson covers fundamental information on the aerodynamic causes and characteristics of aircraft stalls. Factors affecting stalls, anti-stall devices, and T-6 stalls are also discussed.
B.
ENABLING OBJECTIVES 3.1 DEFINE the boundary layer, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.2 DESCRIBE the different types of flow within the boundary layer, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.3 DESCRIBE boundary layer separation, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.4 DEFINE CL MAX AOA, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.5 DEFINE stall, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.6 EXPLAIN how a stall occurs, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.7 IDENTIFY the aerodynamic parameters causing a stall, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.8 COMPARE power-on and power-off stalls, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.9 DESCRIBE the order of losing control effectiveness approaching a stall in the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 3.10 EXPLAIN the difference between true and indicated stall speed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 76 Fundamentals of Aerodynamics: Part 1 – M06-042 3.11 EXPLAIN the effects of gross weight, altitude, load factor and maneuvering on stall speed, given the stall speed equation, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.12 STATE the purpose of using high lift devices, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.13 DESCRIBE how different high lift devices affect the values of CL, CL MAX, and CL MAX AOA, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.14 DESCRIBE devices used to control boundary layer separation, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.15 DESCRIBE devices used to change the camber of an airfoil, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.16 DESCRIBE methods of stall warning used in the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.17 DESCRIBE the stall tendency of the general types of wing planforms, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.18 DESCRIBE the various methods of wing tailoring, including geometric twist, aerodynamic twist, stall strips, and stall fences, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 C.
TOPIC OUTLINE 1.
Introduction 2.
This Lesson Topic 3.
Stall Aerodynamics 4.
Stall Characteristics 5.
Factors Affecting Stall Speed 6.
High Lift Devices 7.
T-6B Stalls 8.
Stall Pattern / Wing Design 9.
Summary and Review 10.
Application 11.
Assignment 77 Fundamentals of Aerodynamics: Part 1 – M06-042 INFORMATION SHEET 2-3-2 STALLS A.
INTRODUCTION This lesson covers fundamental information on the aerodynamic causes and characteristics of aircraft stalls. Factors affecting stalls, anti-stall devices, and T-6 stalls are also discussed.
B.
REFERENCES 1.
Aerodynamics for Naval Aviators, NAVAIR 00-80T-80 2.
Introduction to the Aerodynamics of Flight, NASA SP-367 3.
T-6 Joint Primary Pilot Training, AETC / CNATRA Syllabus P-V4A-J NATOPS Flight Manual C.
INFORMATION THE BOUNDARY LAYER In the preceding discussion of lift, it was shown that when air flows across any surface, friction develops. The air immediately next to the surface slows to near zero velocity as it gives up kinetic energy to friction. As a viscous fluid resists flow or shearing, the adjacent layer of air is also slowed. Succeeding streamlines are slowed less, until eventually some outer streamline reaches the free airstream velocity. The boundary layer is that layer of airflow over a surface that demonstrates local airflow retardation due to viscosity. It is usually no more than 1mm thick (the thickness of a playing card) at the leading edge of an airfoil, and grows in thickness as it moves aft over the surface. The boundary layer has two types of airflow. In laminar flow, the air moves smoothly along in streamlines. A laminar boundary layer produces very little friction, but is easily separated from the surface. In turbulent flow, the streamlines break up and the flow is disorganized and irregular. A turbulent boundary layer produces higher friction drag than a laminar boundary layer, but adheres better to the upper surface of the airfoil, delaying boundary layer separation. Any object that moves through the air will develop a boundary layer that varies in thickness according to the type of surface. The type of flow in the boundary layer depends on its location on the surface. The boundary layer will be laminar only near the leading edge of the airfoil. As the air flows aft, the laminar layer becomes turbulent. The turbulent layer will continue to increase in thickness as it flows aft (Figure 9-1). Air flows aft from the leading edge (high static pressure) of the airfoil towards the point of maximum thickness (low static pressure) resulting in a favorable pressure gradient assisting the boundary layer in adhering to the surface by maintaining its high kinetic energy. As the air flows aft from the point of 78 Fundamentals of Aerodynamics: Part 1 – M06-042 maximum thickness (lower static pressure) toward the trailing edge (higher static pressure), it encounters an adverse pressure gradient which impedes the flow of the boundary layer. The adverse pressure gradient is strongest at high lift conditions and at high angles of attack in particular. If the boundary layer does not have sufficient kinetic energy to overcome the adverse pressure gradient, the lower levels of the boundary layer will stagnate. The boundary layer will then separate from the surface, and airflow along the surface aft of the separation point will be reversed. Aft of the separation point, the low static pressure that produced lift is replaced by a turbulent wake. Figure 3-1 Boundary Layer Separation If the separation point moves forward enough close to the leading edge, the net suction on the top of the airfoil will decrease and a decrease in CL will occur, resulting in a stall. The angle of attack beyond which CL begins to decrease is CLmax AOA. Even at low angles of attack there will be a small adverse pressure gradient behind the point of maximum thickness, but it is insignificant compared to the kinetic energy in the boundary layer until CLmax AOA is approached. Figure 3-1 shows the boundary layer attached at a normal AOA. The point of separation remains essentially stationary near the trailing edge of the wing, until AOA approaches CLmax AOA. The separation point then progresses forward as AOA increases, eventually causing the airfoil to stall. At high angles of attack the airfoil is similar to a flat plate being forced through the air; the airflow simply cannot conform to the sharp turn. Note that the point where stall occurs is dependent upon AOA and not velocity. A stall is a condition of flight in which an increase in AOA results in a decrease in CL. In Figure 3-3 CL increases linearly over a large range of angles of attack then reaches a peak and begins to decrease. The highest value of CL is Figure 3-2 Progression of Separation Point Forward with Increasing AOA referred to as CLmax, and any increase in AOA beyond CLmax AOA produces a decrease in CL. Therefore, CLmax AOA is known as the stalling angle of attack or critical angle of attack, and the region beyond CLmax AOA is the stall region. Regardless of the flight conditions or airspeed, the wing will always stall at the same AOA, CLmax AOA. The only cause of a stall is excessive AOA. Stalls result in decreased lift, increased drag, and an altitude loss. They are particularly dangerous at low altitude or when allowed to develop into a spin. The only action necessary for stall recovery is to decrease AOA below CLmax AOA. 79 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 3-3 CL vs. AOA STALL INDICATIONS Numerous devices may give the pilot a warning of an impending stall. They include AOA indicators, rudder pedal shakers, stick shakers, horns, buzzers, warning lights and other devices. Some of these devices receive their input from attitude gyros, accelerometers, or flight data computers, but most receive input from an AOA probe. The AOA probe is mounted on the fuselage or wing and has a transmitter vane that remains aligned with the relative wind. The vane transmits the angle of attack of the relative wind to a cockpit AOA indicator or is used to activate other stall warning devices. Most U.S. military airplanes have standardized AOA indicators graduated in arbitrary units of angle of attack, or graduated from zero to 100 percent. The T-6B AOA indicator is calibrated so that the airplane stalls at approximately 18 units angle of attack regardless of airspeed, nose attitude, weight or altitude. The AOA system in the T-6B is self-adjusting to account for differences in full-flap or no-flap stall angles. The T-6B also uses AOA indexer and stick shakers that receive their input from an AOA probe on the left wing. The stick shakers are activated at 15.5 units AOA, followed by airframe buffeting. Stalls at idle in a clean configuration are characterized by a nose down pitch with a slight rolling tendency to the right at near full aft stick. The effect of the landing gear and speed brake position on stalls is negligible, but extending the flaps will aggravate the stall characteristics by increasing the rolling tendency. Loss of control effectiveness progresses from ailerons to elevator to rudder in the T-6B. STALL SPEED As angle of attack increases, up to CLmax AOA, true airspeed decreases in level flight (Figure 3-3). Since CL decreases beyond CLmax AOA, true airspeed cannot be decreased any further. Therefore, the minimum airspeed required for level flight occurs at CLmax AOA. Stall speed (VS) is the minimum true airspeed required to maintain level flight at CLmax AOA. Although the stall speed may vary, the stalling AOA remains constant for a given airfoil. Since lift and weight are equal in equilibrium flight, weight (W) can be substituted for lift (L) in the lift equation. By solving for velocity (V), we derive a basic equation for stall speed. 𝑉 𝑆= √ 2𝑊 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 By substituting the stall speed equation into the true airspeed equation and solving for indicated airspeed, we derive the equation for the indicated stall speed (IASS). The greatest factors in stall speed 80 Fundamentals of Aerodynamics: Part 1 – M06-042 are weight, altitude, power, maneuvering, and configuration. Increased stall speed due to maneuvering (accelerated stall speed) will be discussed later. 𝐼𝐴𝑆𝑆= √ 2𝑊 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 As airplane weight decreases, stall speed decreases because the amount of lift required to maintain level flight decreases. When an airplane burns fuel or drops ordnance, stall speeds decrease. Carrier pilots often dump fuel before shipboard landings in order to reduce stall speed and approach speed. A comparison of two identical airplanes at different altitudes illustrates the effect of altitude on stall speed. The airplane at a higher altitude encounters fewer air molecules. In order to create sufficient dynamic pressure to produce the required lift, it must fly at a higher velocity (TAS). Therefore, an increase in altitude will increase stall speed. Since ρ0 is constant, indicated stall speed will not change as altitude changes. The stall speed discussed up to this point assumes that aircraft engines are at idle, and is called power- off stall speed. Power-on stall speed will be less than power-off stall speed because at high pitch attitudes, part of the weight of the airplane is actually being supported by the vertical component of the thrust vector (Figure 3-4). For propeller driven airplanes the portion of the wing immediately behind the propeller produces more lift because the air is being accelerated by the propeller (Figure 3-4). 𝑉 𝑆= √ 2(𝑊−𝑇sin 𝜃) 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 𝐼𝐴𝑆𝑆= √ 2(𝑊−𝑇sin 𝜃) 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 Figure 3-4 Power-On Stall 81 Fundamentals of Aerodynamics: Part 1 – M06-042 High Lift Devices High lift devices also affect stall speeds since they increase CL at high AOA. The primary purpose of high lift devices is to reduce takeoff and landing speeds by reducing both indicated and true stall speeds. The increase in CL allows a decrease in airspeed. For example, an airplane weighing 20,000 pounds flying at 250 knots develops 20,000 pounds of lift. As the airplane slows to 125 knots for landing, high lift devices can increase CL so that 20,000 pounds of lift can still be produced at the lower velocity. There are two common types of high lift devices: Those that delay boundary layer separation, and those that increase camber. BOUNDARY LAYER CONTROL DEVICES The maximum value of CL is limited by the AOA at which boundary layer separation occurs. If airflow separation can be delayed to an AOA higher than normal stalling AOA, a higher CLmax can be achieved. Both CLmax and CLmax AOA increase with the use of Boundary Layer Control (BLC) devices (Figure 3-5). Slots operate by allowing the high static pressure air beneath the wing to be accelerated through a nozzle and injected into the boundary layer on the upper surface of the airfoil (Figure 3-6). As the air is accelerated through the nozzle, its potential energy is converted to kinetic energy. Using this extra kinetic energy, the turbulent boundary layer is able to overcome the adverse pressure gradient and adhere to the airfoil at higher AOAs. There are generally two types of slots, fixed slots and automatic slots. Figure 3-5 Effect of BLC Figure 3-6 Slat and Slot increase in drag. Fixed slots are gaps located at the leading edge of a wing that allow air to flow from below the wing to the upper surface. High pressure air from the vicinity of the leading edge stagnation point is directed through the slot, which acts as a nozzle converting the static pressure into dynamic pressure. The high kinetic energy air leaving the nozzle increases the energy of the boundary layer and delays separation. This is very efficient and causes only a small Slats are moveable leading edge sections used to form automatic slots. When the slat deploys, it opens a slot. Some slats are deployed aerodynamically. At low AOA, the slat is held flush against the leading edge by the high static pressure around the leading edge stagnation point. When the airfoil is at a high AOA, the leading edge stagnation point and associated high pressure area move down away from the leading edge and are replaced by a low (suction) pressure which creates a chordwise force forward and actuates the slat. Other automatic slots are deployed mechanically, hydraulically or electrically. Since slats and slots on their own have no effect on camber, there is no change to CL at low AOA. The higher value of CLmax is achieved at higher AOAs, i.e. the stall is delayed to a higher AOA. (Figure 3-5). A simple form of BLC is achieved by vortex generators, which are small vanes installed on the upper surface of an airfoil to disturb the laminar boundary layer and induce a turbulent 82 Fundamentals of Aerodynamics: Part 1 – M06-042 boundary layer. Because turbulent flow adheres to a surface more than laminar flow, vortex generators ensure the area behind them benefit from airflow that adheres better to the wing, delaying separation. CAMBER CHANGE The most common method of increasing CLmax is increasing the camber of the airfoil. There are various types of high lift devices that increase the camber of the wing and increase CLmax. Trailing edge flaps are the most common type of high lift devices, but leading edge flaps are not unusual. The change in CL and AOA due to flaps is shown in Figure 3-7. Note the value of CL for this airfoil before and after flaps are deployed. Extending the flaps increases the airfoil’s positive camber, shifting its zero lift point to the left. Note that the stalling AOA (CLmax AOA) decreases. Although stalling AOA decreases, visibility on takeoff and landing improves due to flatter takeoff and landing attitudes made possible by these devices. Since boundary layer control devices increase stalling AOA, many modern designs utilize BLC with camber change devices to maintain low pitch attitudes during approach and landing. Flaps also increase the drag on the airplane, enabling a steeper glideslope and higher power setting during approach without increasing the airspeed. This allows an airplane such as an EA-6B to carry more thrust throughout the landing phase and not significantly increase the approach speed (a higher throttle setting results in less spool-up time in case of a wave-off or go-around). For many airplanes, the first 50 percent Figure 3-7 Effect of Flaps of flap down movement produces most of the desired lift increase with less than half of the unwanted drag increase. Thus, raising flaps from 100 to 50 percent reduces drag significantly without a large loss of lift. This is especially important during engine failures on multi-engine airplanes. A plain flap is a simple hinged portion of the trailing edge that is forced down into the airstream to increase the camber of the airfoil. A split flap is a plate deflected from the lower surface of the airfoil. This type of flap creates a lot of drag because of the turbulent air between the wing and deflected surface. A slotted flap is similar to the plain flap, but moves away from the wing to open a narrow slot between the flap and wing for boundary layer control. A slotted flap may cause a slight increase in wing area, but the increase in lift is insignificant. The fowler flap is used extensively on larger airplanes. When extended, it moves down, increasing the camber, and aft, causing a significant increase in wing area as well as opening one or more slots for boundary layer control. Because of the larger area created on airfoils Figure 3-8 Types of Flaps with fowler flaps, a large twisting moment is developed. This requires a structurally stronger wing to withstand the 83 Fundamentals of Aerodynamics: Part 1 – M06-042 increased twisting load and precludes their use on high speed, thin wings. Leading edge flaps are devices that change the wing camber at the leading edge of the airfoil. They may be operated manually with a switch or automatically by computer. Leading edge plain flaps are similar to a trailing edge plain flap. Leading edge slotted flaps are similar to trailing edge slotted flaps, but are sometimes confused with automatic slots. Often the terms are interchangeable since many leading edge devices have some characteristics of both flaps and slats. The exact stall speed for various airplane conditions are given in stall speed charts in an airplane’s flight manual. The directions on how to use the stall speed chart are on the chart itself and are self- explanatory. Stall Pattern And Wing Design The most desirable stall pattern on a wing is one that begins at the root. The primary reason for a root first stall pattern is to maintain aileron effectiveness until the wing is fully stalled. Additionally, turbulent airflow from the wing root may buffet the empennage, providing an aerodynamic warning of impending stall. Different planforms have characteristic stall patterns (Figure 3-9). The lift distribution on the rectangular wing (λ = 1.0) is due to low lift coefficients at the tip and high lift coefficients at the root. Since the area of the highest lift coefficient will stall first, the rectangular wing has a strong root stall tendency. This pattern provides adequate stall warning and aileron effectiveness. This planform is limited to low speed, light-weight airplanes where simplicity of construction and favorable stall characteristics are the predominating requirements. A highly tapered wing (λ = 0.25) is desirable from the standpoint of structural weight, stiffness, and wingtip vortices. Tapered wings produce most of the lift toward the tip and have a strong tip stall tendency. Figure 3-9 Stall Patterns Swept wings are used on high speed aircraft because they reduce drag and allow the airplane to fly at higher Mach numbers. They have a similar lift distribution to a tapered wing, and therefore stall easily and have a strong tip stall tendency. When the wingtip stalls, the stall rapidly progresses over the remainder of the wing. The elliptical wing has an even distribution of lift from the root to the tip and produces minimum induced drag. An even lift distribution means that all sections stall at the same angle of attack. There is little advanced warning and aileron effectiveness may be lost near a stall. It is also more difficult to manufacture than other planforms, but is considered the ideal subsonic wing due to its lift to drag ratio. Moderate taper wings (λ = 0.5) have a lift distribution and stall pattern that is similar to the elliptical wing. The T-6B uses tapered wings because they reduce weight, improve stiffness, and reduce wingtip vortices. However, the even stall progression is undesirable for the same reasons as with the elliptical wing. As a stall progresses, the pilot will lose lateral control of the airplane. 84 Fundamentals of Aerodynamics: Part 1 – M06-042 Wing Tailoring Although stalls cannot be eliminated, they can be made more predictable by having the wing stall gradually. Since most airplanes do not have rectangular wings, they tend to stall with little or no warning. Wing tailoring techniques are used to create a root-to-tip stall progression and give the pilot some stall warning while ensuring that the ailerons remain effective up to a complete stall. Trailing edge flaps decrease the stalling angles of attack in their vicinity, causing initial stall in the flap area. BLC devices generally delay stall in their vicinity. Propeller-driven airplanes may have a tip stall tendency during power-on stalls due to the increased airflow over the wing root. Geometric twist is a decrease in angle of incidence from wing root to wingtip (Figure 3- 10). The root section is mounted at some angle to the longitudinal axis, and the leading edge of the remainder of the wing is gradually twisted downward. This results in a decreased AOA at the wingtip due to its lower angle of incidence. The root stalls first because of its higher AOA. The T-6B wing is geometrically twisted. Figure 3-10 Wing Tailoring Aerodynamic twist, also called section variation, is a gradual change in airfoil shape that increases CLmax AOA to a higher value near the tip than at the root (Figure 3-10). This can be accomplished by a decrease in camber from the root to the tip and/or by a decrease in the relative thickness of the wing (as compared to chord) from the root to the tip. Since thicker and more positively cambered airfoils stall at lower angles of attack, the wing root stalls before the wingtip. The T-6B wings are aerodynamically twisted. The spanwise flow on a swept wing is not accelerated over the wing and does not contribute to the production of lift. Instead, it induces a strong tip stall tendency. Stall fences redirect the airflow along the chord, thereby delaying tip stall and enabling Figure 3-11 Stall Fences the wing to achieve a higher AOA without stalling (Figure 3- 11). The T-6B uses a sharply angled piece of metal called a stall strip mounted on the leading edge of the root section to induce a stall at the wing root (Figure 3-12). Since subsonic airflow cannot flow easily around sharp corners, it separates the boundary layer at higher angles of attack, ensuring that the root section stalls first. Figure 3-12 Stall Strip 85 Fundamentals of Aerodynamics: Part 1 – M06-042 STALL RECOVERY To produce the required lift at slow airspeeds, the pilot must fly at high angles of attack. Because flying slow at high angles of attack is one of the most critical phases of flight, pilots practice recovering from several types of stalls during training. The steps in a stall recovery involve simultaneously adding power, relaxing back stick pressure and rolling wings level and using rudder to ensure coordinated flight (“RELAX, MAX, LEVEL, BALL”). 1. Reduce angle of attack. This may require a reduction in back stick pressure, moving stick progressively towards neutral, or moving stick forward of the trim position.
2. Advance PCL as required to maintain flying airspeed. Anticipate engine power effects, applying aileron and rudder as necessary to maintain or achieve wings level.
3. Use aileron and rudder control as necessary to maintain wings-level, coordinated flight throughout the recovery.
4. As flying speed is regained, smoothly increase back pressure on the control stick to stop the altitude loss and return to level flight, taking care to avoid entering a secondary, accelerated stall during recovery.
86 Fundamentals of Aerodynamics: Part 1 – M06-042 ASSIGNMENT SHEET 2-3-3 STALLS REVIEW A.
INTRODUCTION This lesson covers fundamental information on the aerodynamic causes and characteristics of aircraft stalls. Factors affecting stalls, anti-stall devices, and T-6 stalls are also discussed.
B.
ENABLING OBJECTIVES C.
STUDY ASSIGNMENT 1.
Review Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 4 2.
Read Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 5 D.
STUDY QUESTIONS 1. What is the definition of boundary layer separation?
2. What effect does increasing AOA have on the kinetic energy of the relative wind?
3. What is the definition of a stall?
4. What are the two major classifications of stalls?
5. What are the factors associated with increased thrust that impact stall characteristics?
6. Which aircraft would stall at a higher indicated airspeed?
87 Fundamentals of Aerodynamics: Part 1 – M06-042 7. What are the benefits provided by Boundary Layer Control (BLC) devices?
8. Which type of stall will have the lower stall airspeed?
9. What is the T-6B stall AOA?
10. What is the only reason the T-6B will stall?
88 Fundamentals of Aerodynamics: Part 1 – M06-042 Answers: 1. Point in streamline where airflow no longer adheres to the airfoil 2. Decreases it 3. Condition in flight where increase in AOA results in decrease in CL 4. Power-on stall and power-off stall 5. Vertical component of thrust and propeller acceleration factor 6. 6000 pound aircraft 7. Increased maneuvering capabilities, decreased landing speed and distance 8. Power-on stall 9. 18 units 10. Exceeding T-6B stall AOA (18 units) 89 Fundamentals of Aerodynamics: Part 1 – M06-042 OUTLINE SHEET 2-4-1 PERFORMANCE & MANEUVERING A.
INTRODUCTION Performance and Maneuvering is the third aerodynamics lesson in a series presenting aerodynamic theories and operational principles. Upon completion of this lesson, you will understand aircraft performance with respect to temperature, maximum range and endurance, and aircraft speeds relating to the lift and thrust curves. You will also understand maneuvering characteristics relating to velocity, angle of attack, load factor, and aircraft stability.
B.
ENABLING OBJECTIVES 2.124 DEFINE takeoff and landing airspeed in terms of stall speed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.125 STATE the various forces acting on an airplane during the takeoff and landing transition, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM- SG-200 2.126 STATE the factors that determine the coefficient of rolling friction, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.127 DESCRIBE the effects on takeoff and landing performance, given variations in weight, altitude, temperature, humidity, wind, and braking, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.128 DESCRIBE the effects of outside air temperature (OAT) on airplane performance characteristics, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.129 DEFINE maximum angle of climb and maximum rate of climb profiles, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.130 EXPLAIN the performance characteristics profiles that yield maximum angle of climb and maximum rate of climb for turboprops, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.131 DESCRIBE the effect of changes in weight, altitude, configuration, and wind on maximum angle of climb and maximum rate of climb profiles, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 90 Fundamentals of Aerodynamics: Part 1 – M06-042 2.132 DESCRIBE the performance characteristics and purpose of the best climb profile for the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.133 DEFINE absolute ceiling, service ceiling, cruise ceiling, combat ceiling, and maximum operating ceiling, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.134 STATE the maximum operating ceiling of the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.135 STATE the relationship between fuel flow, power available, power required, and velocity for a turboprop airplane in straight and level flight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.136 DEFINE maximum range and maximum endurance profiles, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.137 EXPLAIN the performance characteristics profiles that yield maximum endurance and maximum range for turboprops, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.138 DESCRIBE the effect of changes in weight, altitude, configuration, and wind on maximum endurance and maximum range performance and airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.139 DEFINE Mach number, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.140 DEFINE critical Mach, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.141 STATE the effects of altitude on Mach number and critical Mach number, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.142 DEFINE maximum glide range and maximum glide endurance profiles, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.143 EXPLAIN the performance characteristics profiles that yield maximum glide range and maximum glide endurance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.144 DESCRIBE the effect of changes in weight, altitude, configuration, wind, and propeller feathering on maximum glide range and maximum glide endurance performance and 91 Fundamentals of Aerodynamics: Part 1 – M06-042 airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.145 DESCRIBE the locations of the regions of normal and reverse command on the turboprop power curve, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.146 EXPLAIN the relationship between power required and airspeed in the regions of normal and reverse command, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.147 DEFINE nosewheel liftoff/touchdown speed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.148 STATE the pilot speed and attitude inputs necessary to control the airplane during a crosswind landing, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.149 STATE the crosswind limits for the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.150 DEFINE hydroplaning, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.151 STATE the factors that affect the speed at which an airplane will hydroplane, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.152 DESCRIBE the effects of propeller slipstream swirl, P-factor, torque, and gyroscopic precession as they apply to the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.153 DESCRIBE what the pilot must do to compensate for propeller slipstream swirl, P-factor, torque, and gyroscopic precession as they apply to the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.154 DESCRIBE the effect of lift on turn performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.155 DESCRIBE the effect of weight on turn performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.156 DESCRIBE the effect of thrust on turn performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 92 Fundamentals of Aerodynamics: Part 1 – M06-042 2.157 DESCRIBE the effect of drag on turn performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.158 DEFINE turn radius and turn rate, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.159 DESCRIBE the effects of changes in bank angle on turn performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.160 DESCRIBE the effects of changes in airspeed on turn performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.161 DESCRIBE the effects of aileron and rudder forces during turns, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.162 EXPLAIN the aerodynamic principle that requires two G's of back stick pressure to maintain level, constant airspeed flight, at 60 degrees angle of bank, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.163 DESCRIBE the relationship between load factor and angle of bank for level, constant- airspeed-flight, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.164 DEFINE load, load factor, limit load factor, and ultimate load factor, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.165 DEFINE static strength, static failure, fatigue strength, fatigue failure, service life, creep, and overstress/over-G, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.166 DEFINE maneuvering speed, cornering velocity, redline airspeed, accelerated stall lines, and the safe flight envelope, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.167 DESCRIBE the boundaries of the safe flight envelope, including accelerated stall lines, limit load factor, ultimate load factor, maneuver point, and redline airspeed, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.168 DEFINE asymmetric loading and state the associated limitations for the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 93 Fundamentals of Aerodynamics: Part 1 – M06-042 2.169 DEFINE static stability and dynamic stability, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.170 DESCRIBE the characteristics exhibited by aircraft with positive, neutral, and negative static stabilities, when disturbed from equilibrium, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.171 DESCRIBE the characteristics exhibited by aircraft with positive, neutral, and negative dynamic stabilities, when disturbed from equilibrium, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.172 DESCRIBE the characteristics of damped, undamped, and divergent oscillations, and the combination of static and dynamic stabilities that result in each, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.173 EXPLAIN the relationship between stability and maneuverability, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.174 STATE the methods for increasing an airplane's maneuverability, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.175 STATE the effects of airplane components on an airplane's longitudinal static stability, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.176 EXPLAIN the criticality of weight and balance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.177 STATE the effects of airplane components on an airplane's directional static stability, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.178 STATE the effects of airplane components on an airplane's lateral static stability, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.179 STATE the static stability requirements for, and the effects of, directional divergence, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.180 STATE the static stability requirements for, and the effects of, spiral divergence, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 94 Fundamentals of Aerodynamics: Part 1 – M06-042 2.181 STATE the static stability requirements for, and the effects of, Dutch roll, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.182 DEFINE proverse roll, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.183 DEFINE adverse yaw, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.184 EXPLAIN how an airplane develops phugoid oscillations, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.185 EXPLAIN how an airplane develops pilot induced oscillations, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.186 DEFINE asymmetric thrust, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 C.
TOPIC OUTLINE 1.
Introduction 2.
This Lesson Topic 3.
Takeoff Performance 4.
Climb Performance 5.
Cruise Performance 6.
Glide Performance 7.
Reverse Command 8.
Landing Performance 9.
Propeller 10.
Turn Performance 11.
Slips and Skids 12.
Load Factor 13.
V-n Diagram 14.
Factors Affecting the Safe Flight Envelope 15.
Static vs Dynamic Stability 16.
Contributors to Longitudinal, Directional, and Lateral Stability 17.
Dynamic Factors Affecting Stability 18.
Summary and Review 19.
Application 20.
Assignment 95 Fundamentals of Aerodynamics: Part 1 – M06-042 INFORMATION SHEET 2-4-2 PERFORMANCE AND MANEUVERING A.
INTRODUCTION Performance and Maneuvering is the fourth aerodynamics lesson in a series presenting aerodynamic theories and operational principles. Upon completion of this lesson, you will understand aircraft performance with respect to temperature, maximum range and endurance, and aircraft speeds relating to the lift and thrust curves. You will also understand maneuvering characteristics relating to velocity, angle of attack, load factor, and aircraft stability.
B.
REFERENCES 1.
Aerodynamics for Naval Aviators, NAVAIR 00-80T-80 2.
Introduction to the Aerodynamics of Flight, NASA SP-367 3.
T-6 Joint Primary Pilot Training, AETC / CNATRA Syllabus P-V4A-J NATOPS Flight Manual C.
INFORMATION Takeoff Speed Takeoffs are a transitional maneuver during which the weight of the airplane is shifted from the landing gear to the wings. The minimum airspeed for takeoff is 20 percent above the power off stall speed. Thus, it is affected by the same factors that affect stall speed. This 20 percent safety margin minimizes operation in the region of reverse command and allows for shallow turns after takeoff, especially during an engine failure. High lift devices are often used to decrease takeoff speeds. Note that the below equation is expressed in terms of true airspeed: 𝑉𝑇𝑂≈1.2√ 2𝑊 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 Indicated airspeed for takeoff will not be affected by changes in air density: 𝐼𝐴𝑆𝑇𝑂≈1.2√ 2𝑊 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 96 Fundamentals of Aerodynamics: Part 1 – M06-042 TAKEOFF FORCES Figure 4-1 shows the forces acting on an airplane during takeoff. During ground roll, rolling friction (FR) accounts for the effects of friction between the landing gear and the runway. Like any frictional force, it is the product of a coefficient of friction (μ) and a perpendicular force (weight-on-wheels or WOW). WOW is the difference between weight and lift. The coefficient of friction (μ) is dependent upon runway surface, runway condition, tire type and degree of brake application. Note that brake application should be negligible during takeoff. FR W L Takeoff performance is dependent upon acceleration. According to Newton’s Second Law, a body accelerates in the direction of the unbalanced force acting on it. Thrust is the most out of balance force on takeoff. For an airplane to accelerate from zero to its takeoff speed, it must generate enough thrust to overcome rolling friction (FR) and drag (D). T - D - FR is called the net accelerating force. Although thrust and weight may change slightly Figure 4-1 Takeoff and Landing Forces during our takeoff, we will consider them to remain nearly constant. As velocity increases during takeoff, the aerodynamic force increases, increasing both lift and drag (drag is primarily parasitic during a takeoff or landing). The increase in lift during the takeoff roll decreases the weight on wheels and rolling friction. TAKEOFF PERFORMANCE An equation for determining minimum takeoff distance is: 2 STO W gSCL max T D FR where g = gravity. Weight is the greatest factor in determining takeoff distance. Looking at the takeoff distance formula, we see that doubling the weight will increase the takeoff distance by a factor of four. Increasing weight requires greater lift and a higher takeoff velocity. It also increases rolling friction which decreases the net accelerating force. Most takeoff and landing performance charts use density altitude (DA) to account for air density. Increasing DA (decreasing air density) requires a higher takeoff velocity and decreases the amount of thrust our engine can provide. This will decrease the acceleration on the takeoff roll and increase the minimum takeoff distance. There are three major factors that decrease density: increasing airfield 97 Fundamentals of Aerodynamics: Part 1 – M06-042 elevation, increasing air temperature and increasing humidity. Note that indicated takeoff airspeed remains constant, regardless of temperature, humidity, and elevation. Along with weight, these three density factors are the worst conditions for takeoff and landing. A helpful mnemonic device is the “4-H Club,” where the members of the club are high, hot, heavy and humid. Whenever three or more of the 4-H Club are present, expect extended takeoff and landing distances. Under extreme circumstances, two or even one of the factors may cause longer takeoff and landing distances. Using high lift devices such as flaps or boundary layer control devices will decrease the takeoff distance. High lift devices decrease both the indicated and true takeoff speeds. Since true airspeed for takeoff decreases, the ground speed during takeoff will decrease, thus decreasing takeoff distance. A headwind will decrease the takeoff distance by reducing the ground speed associated with the takeoff velocity. Conversely, a tail wind will increase takeoff distance since it increases ground speed. CLIMB PERFORMANCE A “steady climb” is defined as a climb in which the airplane is not accelerating; the airplane is in equilibrium. However, the altitude is no longer constant. In this discussion, the same thrust and power curves are used to analyze level flight to discuss and locate the different climb performance parameters of an airplane. Both types of climbs discussed are performed at maximum power. Angle of climb (γ, AOC) is a comparison of altitude gained to distance traveled. For maximum angle of climb, we want maximum vertical velocity (altitude increase) for a minimum horizontal velocity (ground speed). Maximum AOC is commonly used when taking off from a short Figure 4-2 Max AOC vs. Max ROC airfield surrounded by high obstacles, such as trees, or power lines. The objective is to gain sufficient altitude to clear the obstacle with the least horizontal distance traveled. Rate of climb (ROC) is a comparison of altitude gained relative to the time needed to reach that altitude. Flying at maximum rate of climb yields a maximum vertical velocity. Maximum rate of climb is used to expedite a climb to an assigned altitude. The greatest vertical distance must be gained in the shortest time possible. In a maximum angle of climb profile, a certain airplane takes 30 seconds to reach 1000 feet AGL, but covers only 3000 feet over the ground. Using its maximum rate of climb profile, the same airplane climbs to 1500 feet in 30 seconds, but covers 6000 feet across the ground. It should be noted that both climb profiles are executed at maximum throttle setting, and that differences between max rate and max angle of climb lie solely in differences of angle of attack and velocity. 98 Fundamentals of Aerodynamics: Part 1 – M06-042 ANGLE OF CLIMB The equations that represent equilibrium in a climb are (Figure 4-3): L W cos T D W s in By rearranging the bottom equation, we see that: s in T D W TA TR W TE W Thus, angle of climb performance depends upon thrust excess. Essentially, the greater the force that pushes the airplane upwards, the steeper it can climb. Maximum angle of climb occurs at the velocity and angle of attack that produce the maximum thrust Figure 4-3 Climb Forces excess. Therefore, maximum angle of climb for a turbojet occurs at L/DMAX AOA and velocity. Maximum angle of climb for a turboprop occurs at a velocity less than L/DMAX and an angle of attack greater than L/DMAX AOA. At max angle of climb, an aircraft can be operating near stall speed. Therefore, for the T-6B, the recommended best climb speed is 140 KIAS. This speed will meet or exceed any obstacle clearance requirements while providing a greater safety margin. Max angle of climb is not flown in the T-6B. Figure 4-4 Turbojet Angle of Climb Figure 4-5 Turboprop Angle of Climb 99 Fundamentals of Aerodynamics: Part 1 – M06-042 RATE OF CLIMB Rate of climb (ROC) is simply the vertical component of velocity (Figure 4-6): ROC V sin sin By substitution: TA TR W TE W ROC V sin VTE P E W W Figure 4-6 Climb Velocity Vectors Thus, rate of climb performance depends upon power excess. Since climbing is work and power is the rate of doing work, any power that is not used to maintain level flight can increase the rate of climbing. Maximum rate of climb occurs at the velocity and angle of attack that produce the maximum power excess. Therefore, maximum rate of climb for a turbojet occurs at a velocity greater than L/DMAX and an angle of attack less than L/DMAX AOA. Maximum rate of climb for a turboprop occurs at L/DMAX AOA and velocity. Best climb speed, not max rate of climb, is flown in the T-6B. Figure 4-7 Turbojet Rate of Climb Figure 4-8 Turboprop Rate of Climb Climb Performance Factors Since weight, altitude, and configuration changes affect thrust and power excess, they will also affect climb performance. Climb performance is directly dependent upon the ability to produce either a thrust excess or a power excess. In the previous lesson, it was determined that an increase in weight, an increase in altitude, lowering the landing gear, or lowering the flaps will all decrease both maximum thrust excess and maximum power excess in all airplanes. Therefore, maximum angle of climb and maximum rate of climb performance will decrease under any of these conditions. 100 Fundamentals of Aerodynamics: Part 1 – M06-042 Consider an airplane that has a maximum angle of climb TAS of 160 knots, a ground speed of 160 knots, and no wind. If this airplane flies into a headwind of 30 knots, its ground speed is reduced to 130 knots. The headwind has increased the airplane’s maximum angle of climb, because it reaches the same altitude as before with a smaller distance covered over the ground. A tailwind has the opposite effect (Figure 4-9). Wind does not affect rate of climb performance. LEVEL FLIGHT PERFORMANCE Fuel flow is the rate of fuel consumption by the Figure 4-9 Effect of Wind on Max AOC engine, measured in pounds per hour (pph). Since the supply of fuel onboard is limited, the engine’s fuel flow is a critical determinant of how long and how far the airplane can fly. A turbojet engine directly produces thrust through its exhaust. Therefore, the fuel consumed by a turbojet engine is proportional to its thrust available (TA). In order to maintain equilibrium flight, thrust available must be set equal to thrust required (TR), therefore we say that minimum fuel flow for a turbojet is found on the thrust required curve. The thrust provided by a propeller is not produced directly by the engine, so there is no direct relationship between thrust and fuel flow. The engine turns a shaft that turns the propeller that produces the thrust. In turning the shaft, the engine produces power. Therefore, for a turboprop, fuel flow varies directly with the power output of the engine (PA). Minimum fuel flow for equilibrium flight will be found on the power required (PR) curve. Maximum endurance and maximum range are both achieved in equilibrium, level flight. Any thrust or power excess would cause the airplane to either climb or accelerate. We will look on the thrust required or power required curve to determine the velocity that our airplane must fly. Once the velocity is determined, the pilot must adjust the throttle to eliminate any thrust or power excess. Maximum endurance is the maximum amount of time that an airplane can remain airborne on a given amount of fuel. The slower an engine burns fuel, the longer the airplane can remain airborne. Minimum fuel flow occurs at minimum TR for a turbojet and minimum PR for a turboprop. Therefore, maximum endurance is found at L/DMAX AOA and velocity for a turbojet and at a velocity less than L/DMAX, and an angle of attack greater than L/DMAX AOA for a turboprop. For the T-6B, maximum endurance is achieved at 8.8 units AOA. 101 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 4-10 Turbojet Maximum Endurance Figure 4-11 Turboprop Maximum Endurance Maximum range is the maximum distance traveled over the ground for a given amount of fuel. To find maximum range we must minimize fuel flow per unit of velocity. Any straight line drawn from the origin represents a constant ratio of fuel flow to velocity. The minimum ratio that allows the airplane to remain airborne occurs where the line from the origin is tangent to the TR curve for jets or the PR curve for props. Maximum range for a turbojet is found at a velocity greater than L/DMAX and an angle of attack less than L/DMAX AOA. Maximum range for a turboprop is found at L/DMAX AOA and velocity. Maximum range with no wind is achieved in the T-6B is at 4.4 units AOA. Note that maximum range is faster than maximum endurance. Figure 4-12 Turbojet Maximum Range Figure 4-13 Turboprop Maximum Range As explained previously, if the weight of an airplane increases, the thrust required curve (Figure 4-12) and the power required curve (Figure 4-13) will both shift to the right and up. Since thrust represents fuel flow for a turbojet, as TR increases so will fuel flow for a turbojet. As PR increases, fuel flow for a turboprop will also increase. Higher fuel flow means maximum endurance performance will decrease with an increase in weight and max endurance airspeed will increase. The increased fuel flow will also decrease maximum range performance and increase max range airspeed.
102 Fundamentals of Aerodynamics: Part 1 – M06-042 An increase in altitude moves the thrust required curve to the right (Figure 4-12) and the power required curve to the right and up (Figure 4-13). However, as altitude increases (sea level to 36,000 ft MSL), the temperature rapidly decreases (to –56.5ºC). Decreased temperatures make turbine engines more fuel efficient, requiring less fuel for a given amount of thrust or power. Although the pilot physically increases the throttle setting as altitude increases, fuel flow decreases. Since the airplane is burning less fuel to remain airborne, maximum endurance performance increases with an increase in altitude. An airplane at a higher altitude will fly at a greater TAS while burning less fuel. Since the fuel consumed per mile flown has decreased, an increase in altitude increases maximum range performance. With the same increase in altitude, turbojet airplane will notice a greater gain in performance than turboprop airplane. This is due in part to the loss of propeller efficiency with altitude. Configuration changes will affect both max endurance and max range. Lowering the landing gear or flaps causes the thrust required and power required curves to shift up (Figure 4-12 through Figure 4- 13). Max endurance and max range will decrease with landing gear and/or flaps extended. Since range is distance over the ground, ground speed must be considered when determining the effect of wind on maximum range. When flying into a headwind, ground speed is less than true airspeed. Therefore, the range of the airplane decreases since less ground will be covered in a given time. Headwinds will decrease maximum range performance while tailwinds will increase maximum range performance. Winds will have no effect on maximum endurance performance. Figure 4-14 Turbojet Maximum Range (Corrected for Headwind) Figure 4-15 Turboprop Maximum Range (Corrected for Headwind) To partially make up for the decreased performance with a headwind, the pilot can increase the TAS of the airplane. One cannot totally make up for the distance lost, but some of the headwind effect can be overcome. Consider the extreme case of an airplane flying into a headwind that equals TAS. Ground speed and range are zero. Any increase in true airspeed would increase range. The straight line drawn from the origin tangent to the TR or PR curve represents a ratio of fuel flow to true airspeed. To make the tangent line represent a ratio of fuel flow to ground speed, one must subtract headwind or add a tailwind to true airspeed. With a headwind, one moves the base of the tangent line to the right of the origin by the amount of the headwind velocity (Figure 4-14 and Figure 4-15). The airspeed at the new 103 Fundamentals of Aerodynamics: Part 1 – M06-042 tangent point is the velocity needed to fly maximum range with the headwind. With a tailwind, one moves the base of the tangent line to the left of the origin by the amount of the tailwind velocity. CEILINGS As an airplane climbs and PE decreases, the rate of climb will also decrease. The altitude where maximum power excess allows only 500 feet per minute rate of climb is called the combat ceiling. The cruise ceiling is the altitude at which an airplane can maintain a maximum climb rate of only 300 feet per minute. The service ceiling is the altitude at which an airplane can maintain a maximum rate of climb of only 100 feet per minute. Eventually, the airplane will reach an altitude where maximum power excess is zero. At this altitude, the airplane can no longer perform a steady climb, and its maximum rate of climb is zero. The altitude at which this occurs is called the absolute ceiling. If the airplane flies at its maximum rate of climb velocity, it will only be possible to maintain level equilibrium flight. At any velocity other than this, PR will exceed PA, and the airplane will descend. The operational ceiling for the T-6B is 31,000 ft. GLIDE PERFORMANCE Gliding is a condition of flight without any operating engine. It does not refer to a single engine failure in a multi-engine airplane. When our engine fails, we may need to glide as far as possible to reach a safe landing area. This is a maximum glide range profile. If we lose power within easy reach of a safe runway, we may decide to fly a maximum glide endurance profile while the runway is being cleared. The equations that represent equilibrium in a glide are (Figure 4- 16): GLIDE RANGE L W cos D W s in Figure 4-16 Glide Forces To achieve maximum glide range, a pilot should maintain the minimum glide angle. Rearranging the above equation: s in D TR W TA W TD W 104 Fundamentals of Aerodynamics: Part 1 – M06-042 Thus, the angle of descent is directly related to the thrust deficit, TD. To achieve the minimum angle of descent, we must minimize the thrust deficit, which occurs at L/DMAX (Figure 4-17). Therefore, maximum glide range occurs at L/DMAX. Maximum glide range velocity (VBEST) is L/DMAX for any airplane regardless of engine type. Since the L/D ratio is determined by angle of attack, any change away from L/DMAX AOA would result in a decreased L/D ratio and a decrease in glide range. By holding a constant AOA, we can maintain a constant L/D ratio, regardless of weight or velocity. VBEST is 125 KIAS for the T-6B. Glide range is often expressed as a ratio of horizontal distance to vertical distance. The glide ratio for the T-6B (clean) is Figure 4-17 Max Glide Range 11:1. A glide ratio of 11:1 indicates that an airplane will move forward 11 feet for every foot of altitude lost. One might feel a tendency to try to “stretch out” the glide by increasing the angle of attack. If the angle of attack is increased beyond L/DMAX AOA, the horizontal distance the plane will travel will actually decrease. The minimum glide angle obtained at L/DMAX will not produce the minimum sink rate, but will produce the greatest horizontal distance for a given altitude. GLIDE ENDURANCE Maximizing glide endurance is simply a matter of minimizing rate of descent (ROD) or negative vertical velocity (Figure 4-18): ROD V sin By substituting: s in D TD W W Figure 4-18 Glide Velocity Vectors ROD V sin VTD P D W W 105 Fundamentals of Aerodynamics: Part 1 – M06-042 To minimize the rate of descent, the pilot must fly at the velocity where the minimum power deficit occurs. This is at the bottom of the PR curve (Figure 4-19). Maximum glide endurance velocity is less than L/DMAX velocity, and the angle of attack for max glide endurance is greater than L/DMAX AOA. Glide Performance Factors As the airplane’s weight is increased, the TR and PR curves shift up and to the right. The lowest point on each curve will shift as well, increasing the velocity at which it occurs. As long as the pilot maintains L/DMAX AOA, the L/D ratio and angle of descent remain constant. Therefore, an increase in Figure 4-19 Max Glide Endurance weight will not affect maximum glide range. An increase in the velocity during a descent will cause the rate of descent to increase, and glide endurance to decrease. Increasing the weight will cause the airplane to fly faster and descend faster, but still glide the same distance. An increase in altitude will increase the maximum glide range and maximum glide endurance of an airplane (Figure 4-20). Wind has the same effect on maximum glide range that it has on maximum range (Figure 4-21). Since a headwind decreases groundspeed, it causes a decrease in the maximum glide range. Conversely, a tailwind will increase the maximum glide range. Wind has no effect on rate of descent or on glide endurance. Figure 4-20 Effect of Altitude on Glide Figure 4-21 Effect of Wind on Glide During power off flight, airplane configuration plays a major role in determining glide performance. If the pilot alters the configuration by extending the landing gear and/or flaps, the sink rate will increase and glide range will decrease. 106 Fundamentals of Aerodynamics: Part 1 – M06-042 The greatest effect of configuration on glide performance deals with the propeller. In normal flight, the propeller blades are almost flat to the relative wind, but create no drag since the engine is driving the prop. When the engine fails, if the propeller blades stay flat to the relative wind, the wind will drive the propeller blades around, a situation called windmilling. Windmilling drastically increases the drag on the airplane and adversely affects glide performance. In order to stop the propeller from windmilling, the individual propeller blades can be turned so they are aligned with the relative wind. This procedure is called feathering the propeller. THE REGIONS OF NORMAL AND REVERSE COMMAND Velocities above maximum endurance are referred to as the region of normal command. The region of normal command is characterized by airspeed stability. Assume an airplane is in equilibrium at point B (Figure 4-22 or Figure 4-23). A decrease in airspeed (for example: a headwind gust) results in a thrust or power excess that will eventually accelerate the airplane back to the original airspeed at point B. An increase in airspeed from point B (for example: a tailwind gust) results in a thrust or power deficit that slows the airplane back to the original airspeed.
In the region of normal command, velocity and throttle setting for level flight are directly related. To fly in equilibrium at a faster airspeed, more TA/PA is needed than at a slower airspeed. To fly slower, less TA/PA is needed. 107 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 4-22 Turbojet Reverse Command Figure 4-23 Turboprop Reverse Command Velocities below maximum endurance are referred to as the region of reverse command. The region of reverse command is characterized by airspeed instability. Assume an airplane is in equilibrium at point A (Figure 4-22 or Figure 4-23). A decrease in airspeed (for example: a headwind gust) results in a thrust or power deficit that will eventually slow the airplane to the point of stalling (assuming a level flight attitude is being maintained). An increase in airspeed (for example: a tailwind gust) from Point A results in a thrust or power excess that accelerates the airplane away from point A. The airplane will eventually reach equilibrium at point B. In the region of reverse command, velocity and throttle setting for level flight are inversely related. Once stabilized at a faster airspeed in equilibrium flight, TA/PA will be lower than when stabilized at a slower airspeed. Simply stated, the slower an airplane flies in the region of reverse command, the more thrust and power is needed. A complete knowledge of this flight region is particularly important because most aviation accidents occur while operating in the region of reverse command. Whenever an airplane is taking off or landing, it is flying in or near this region. A very dangerous situation for an inexperienced pilot is trying to slow down in the region of reverse command. If the pilot increases back pressure to increase angle of attack and decrease velocity, this will causes thrust and power required to increase, creating a deficit. Once the airspeed bleeds off, the deficit causes the airplane to descend. The inexperienced pilot tends to pull back on the control stick in order to keep from descending. This causes the airplane to move further into the region of reverse command, creating a greater deficit. Eventually the deficit will be so great that even full throttle is not able to overcome it. Since this usually occurs during landing, there is not enough altitude to recover. This is the origin of the phrase “behind the power curve.” An experienced pilot knows that in order to maintain level flight as an airplane slows down in the region of reverse command, the throttle must be increased. Increasing angle of attack will only aggravate the situation. LANDING SPEED Landing is a transitional maneuver during which the weight of the airplane is shifted from the wings to the landing gear. As with takeoff speed, landing speeds also build in a safety margin above stall speed. 108 Fundamentals of Aerodynamics: Part 1 – M06-042 Landing speed is 30 percent higher than stall speed. The extra safety margin is due to the operation at low altitudes with a low power setting. High lift devices are often used to decrease landing speeds. The equations for landing speed are almost identical to takeoff speed: 𝑉 𝐿𝐷𝐺≈1.3√ 2𝑊 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 𝐼𝐴𝑆𝐿𝐷𝐺≈1.3√ 2𝑊 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 LANDING FORCES The same forces that are present during takeoff are present during landing. During the landing roll, thrust and weight still remain nearly constant (reverse thrust is discussed later). Lift and drag are functions of airspeed, so they are greatest immediately upon touchdown and decrease over the remaining landing roll. As lift decreases, weight on wheels increases causing rolling friction to increase. Drag and rolling friction will decelerate the airplane to a safe taxi speed. D + FR - T is the net decelerating force. LANDING PERFORMANCE Landing is essentially the reverse of takeoff. The takeoff distance equation requires only slight modifications to be applicable to landing: 𝑆𝐿𝐷𝐺= 𝑊2 𝑔𝜌𝑆𝐶𝐿 𝑚𝑎𝑥(𝐹 𝑅+ 𝐷−𝑇) The primary consideration in landing is dissipation of the airplane’s kinetic energy. Any factor affecting velocity must be considered when trying to reduce the landing distance. Final approach is flown at the lowest velocity feasible. Notice that in the landing distance equation the net accelerating forces are reversed. Drag and rolling friction are now desirable and of course, thrust is not. An increase in weight will increase landing distance since a greater airspeed is required to support the airplane. An increase in elevation, temperature or humidity will increase landing distance since the reduced density results in a higher landing velocity. High lift devices decrease landing distance because they reduce the ground speed during the landing. A headwind reduces landing distance because it reduces ground speed. A tailwind increases landing distance since it increases ground speed. Charts for predicting takeoff and landing distance are located in the NATOPS manual for each US Navy aircraft (“Dash-1” for USAF aircraft). The net decelerating force can be increased by use of three different techniques. Aerodynamic braking is accomplished by increasing the parasite drag on the airplane by holding a constant pitch attitude after touchdown and exposing more of the airplane’s surface to the relative wind. This method of braking helps to reduce wear on the brakes. Drag chutes, spoilers, and speed brakes are also considered aerodynamic braking. Aerodynamic braking is used at the beginning of the landing roll 109 Fundamentals of Aerodynamics: Part 1 – M06-042 Mechanical braking (also called frictional or wheel braking) is effective only after enough weight is transferred to the wheels and the airplane has slowed sufficiently. A common procedure is to raise flaps or use spoilers to decrease lift and transfer the airplane’s weight to the wheels when transitioning from aerodynamic to mechanical braking. Mechanical braking is used toward the end of the landing roll. Some airplanes use reverse thrust or beta to shorten the landing roll. Thrust is usually negligible after touchdown, but in the case of reverse thrust or “beta” equipped airplanes, thrust increases the net decelerating force. HYDROPLANING Hydroplaning causes the airplane’s tires to skim atop a thin layer of water on a runway. If there is standing water in excess of 0.1 inches, hydroplaning may occur. Deeper tread or “channels” that allow water to escape while the tire contacts the runway may require as much as 2 inches of water before hydroplaning occurs. The speed for normal dynamic hydroplaning in mph, can be found using the following formula: 𝑉 ℎ𝑦𝑟𝑑𝑜𝑝𝑙𝑎𝑛𝑒 = 9√𝑡𝑖𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 For knots, divide Vhyrdoplane by 1.15 At first thought, one might think that a heavier airplane would require a faster speed before hydroplaning could occur, but experiments have shown this speed to be independent of weight. Weight only determines the size of the “footprint” that the tire makes. A heavier airplane makes a larger footprint, but the weight supported per square inch of the tire is the same. Weight has no effect on the velocity that an airplane will hydroplane, but a heavier airplane must take off and land at higher speeds which increases the possibility of hydroplaning. If hydroplaning is suspected, the use of frictional brakes should be minimized, since their use may cause loss of directional control. To minimize the effects of hydroplaning, aircrews should consider factors such as tire condition, touchdown speeds and runway condition when operating on a wet runway. Table 4-1 gives the tire pressure and approximate hydroplaning speeds for the landing gear of the T-6B. Pressure Vhydroplane Nosewheel 120±5 psi 85 kts Main Landing Gear 225±5 psi 115 kts Table 4-1 T-6B Hydroplaning Speeds 110 Fundamentals of Aerodynamics: Part 1 – M06-042 CROSSWINDS Since winds do not always blow directly down the runway, the possibility of a crosswind takeoff or landing exists. The rudder is the primary means of maintaining directional control in order to compensate for the crosswind during takeoff or landing. Since the rudder loses effectiveness at low airspeeds, the self-centering feature of the T-6B nosewheel provides additional directional stability if the nosewheel is contacting the runway. This enables the pilot to maintain directional control until the rudder becomes effective at higher airspeeds. The pilot must also place the ailerons into the wind during a crosswind takeoff or landing. The ailerons are not used to maintain directional control, but to overcome the lateral stability that is trying to roll the airplane away from the sideslip relative wind (crosswind). NATOPS and the Dash-1 both contain a Takeoff/Landing Crosswind chart which allows the pilot to determine the minimum safe airspeed that the nosewheel may leave the runway during takeoff, or the minimum airspeed at which the nosewheel must return to the runway following a landing. Lifting the nosewheel below the minimum nosewheel liftoff/touchdown (NWLO/TD) speed may cause the airplane to weathercock or weathervane into the wind and possibly run off the runway. Many airplanes have maximum crosswind limits that are based upon minimum nosewheel liftoff/touchdown speed. The major consideration for determining maximum authorized crosswind components is the ability to maintain directional control at low speeds. Maximum crosswind component for a takeoff or landing in the T-6B is 25 knots. For variable or gusting winds, always use the maximum wind angle and the maximum gust velocity given to determine the crosswind component. PROPELLER TORQUE Torque is a reactive force based on Newton’s Third Law of Motion. A force must be applied to the propeller to cause it to rotate clockwise. A force of equal magnitude, but opposite direction, is produced which tends to roll the airplane’s fuselage counter-clockwise. In the T-6B, rudder and the automatic Trim Aid Device (TAD) are the primary means of compensating for engine torque.
A turbojet aircraft will not experience torque from its engines. Jet engines do not push against the airframe in order to rotate, they rest on bearings and push against the airflow to rotate. The torque in a turboprop is applied through its gearbox, not its engine. 111 Fundamentals of Aerodynamics: Part 1 – M06-042 P-FACTOR Propeller factor (P-factor) is the yawing moment caused by one prop blade creating more thrust than the other. The angle at which each blade strikes the relative wind will be different (Figure 4-24), causing a different amount of thrust to be produced by each blade. For practical purposes, only the up- going and down-going blades are considered. If the relative wind is above the thrust line, the up-going propeller blade on the left side creates more thrust since it has a larger angle of attack with the relative wind. This will yaw the nose to the right (Figure 4-25). Note that this right yaw will result at high airspeeds due to the slight nose-down attitude required in level flight. If the relative wind is below the thrust line, such as in flight near the stall speed, the down-going blade on the right side will create more thrust and will yaw the nose to the left. Figure 4-24 Propeller Side View Figure 4-25 Relative Wind (Nose-Down Attitude) There are two basic requirements for P-factor to have a noticeable effect: The engine must be set to a high power condition, and the thrust axis must be displaced from the relative wind. Since airplane designers want P-factor to be minimized during the majority of flight, they align the thrust axis with the relative wind for cruise airspeeds. Thus, P-factor will be most prevalent at AOAs significantly different from cruise AOA, such as very high speed level or descending flight, and high angle of attack climbs. 112 Fundamentals of Aerodynamics: Part 1 – M06-042 SLIPSTREAM SWIRL The propeller imparts a corkscrewing motion to the air called the slipstream swirl. This corkscrewing air flows around the fuselage until it reaches the vertical stabilizer where it increases the AOA on the vertical stabilizer (Figure 4-26). When a propeller driven airplane is at a high power setting and low airspeed (e.g., during takeoff), the increased angle of attack creates a horizontal lifting force that pulls the tail to the right and causes the nose to yaw left. Right rudder and lateral control stick inputs are required to compensate for slipstream swirl. GYROSCOPIC PRECESSION Figure 4-26 Propeller Slipstream Swirl Gyroscopic precession is a consequence of the properties of spinning objects. When a force is applied to the rim of a spinning object (such as a propeller) parallel to the axis of rotation, a resultant force is created in the direction of the applied force, but occurs 90° ahead in the direction of rotation (Figure 4- 27). Pitching the nose of the T-6B down produces an applied force acting forward on the top of the propeller disk. The resultant force would act 90° ahead in the direction of propeller rotation (clockwise), and cause the T-6B to yaw left. Gyroscopic precession often plays a large role in determining an airplane’s entry characteristics into a spin. Figure 4-27 Gyroscopic Precession 113 Fundamentals of Aerodynamics: Part 1 – M06-042 MANEUVERING FORCES Turning flight is described as changing the direction of the airplane’s flight path by reorienting the lift vector in the desired direction. During a turn, the lift vector is divided into two components, a horizontal component (LH) and a vertical component (LV) (Figure 4-28). The horizontal component of lift, called centripetal force, accelerates the airplane toward the inside of the turn. In straight and level flight (constant altitude, constant direction) total lift is equal to weight, but in a turn, only the vertical component of the lift vector opposes weight. If the pilot does not increase the total lift vector, the airplane will lose altitude because weight will be greater than LV. The increased lift is normally obtained by increasing the angle of attack, i.e. pulling back on the stick.
Figure 4-28 Turning Flight Increasing the lift produced by the wings increases the load on the airplane. Load factor (n) is the ratio of total lift to the airplane’s weight. It is sometimes called Gs since it is measured as the number of times the earth’s gravitational pull felt by the pilot. For example, a 3,000 pound airplane in a 60º angle of bank turn must produce 3,000 pounds of vertical lift to maintain altitude. Therefore, the wings must produce 6,000 pounds of total lift so the airplane experiences a load on its wings that is twice the force due to gravity, or 2 Gs. One “G” is what we experience just sitting or walking. n L W or L W n In maneuvering flight, the amount of lift produced by an airplane is equal to its weight (W) multiplied by its load factor (n). By substituting W ∙ n into the lift equation and solving for V, we can derive an equation for stall speed during maneuvering flight. This is called accelerated stall speed because it represents the stall speed at velocities greater than minimum straight and level stall speed, and load factors greater than one. Phi (φ) is the angle of bank associated with the load factor (n). VS 2Wn SCL max IAS S 2Wn 0 SCL max Maneuvering the airplane will significantly affect stall speed. Stall speed increases when we induce a load factor greater than one on the airplane. Figure 4-29 is a generic chart that can be used for any fixed wing aircraft and assumes a constant altitude turn. It lists the load factors and percent increase in stall speed for varying angles of bank. Notice that above 45º angle of bank the increase in load factor and stall speed is rapid. This emphasizes the need to avoid steep turns at low airspeeds. An airplane in a 60º angle of bank experiences 2 Gs, but has an accelerated stall speed that is 40% greater than wings level stall speed. 114 Fundamentals of Aerodynamics: Part 1 – M06-042 2 A quick method for calculating accelerated stall speed is to round your normal stall speed off to a higher, round number and multiply it by the square root of the number of Gs sustained. For example, if stall speed is 92 kts and a 2 G maneuver is performed, accelerated stall speed can be estimated by rounding 92 kts up to 100 kts, then multiplying by the square root of two (1.4). As the load factor approaches three Gs, the pilot will notice a sensation of blood draining from the head and a tendency for his or her face to sag. Further increases in G loading may cause the pilot to gray out or even temporarily lose consciousness if the pilot does not correctly apply the anti-G straining maneuver. As the load factor approaches negative three Gs, the Figure 4-29 Stall Speed vs. AOB pilot will notice a sensation of blood rushing to his head, and the face and eyeballs will feel like they are bulging out. Exceeding negative three Gs may cause one to “red out” or suffer from bursting blood vessels. TURN PERFORMANCE Turn performance is measured using two different parameters, turn rate and turn radius. Turn rate (ω) is the rate of heading change, measured in degrees per second. Turn radius (r) is a measure of the radius of the circle the flight path scribes. Turn performance in a level coordinated turn is controlled only by airspeed and angle of bank. Weight, altitude, load factor, stalling angle of attack, engine performance, and wing loading may limit either the airspeed or angle of bank. This would limit maximum turn rate or minimum turn radius, however, the actual performance would still be determined using only airspeed and angle of bank. The formulas for determining the turn rate and turn radius for an airplane in coordinated flight are: g tan r V V g tan where ω = turn rate, r = turn radius, V = velocity, θ theta = angle of bank and g = gravity. If velocity is increased for a given angle of bank, turn rate will decrease, and turn radius will increase. An example of this would be turning a very sharp corner on a bicycle at 5 mph versus trying to turn the same corner at 30 mph. If angle of bank is increased for a given velocity, turn rate will increase, and turn radius will decrease. 115 Fundamentals of Aerodynamics: Part 1 – M06-042 The maximum turn rate and minimum turn radius would be achieved in a 90º angle of bank turn, at the airplane’s minimum velocity. However, there are limits on angle of bank and velocity. Minimum velocity, stall speed, is determined by CLmax AOA. Maximum turn performance will be achieved at the accelerated stall speed for whatever angle of bank is being flown. An increase in angle of bank increases the accelerated stall speed, and vice versa. If an airplane’s limit load is 2 Gs, the maximum angle of bank that it could maintain will be 60 degrees (Figure 4-29). With a limit load factor of 7 Gs, the T-6B is limited to about 83 degrees angle of bank in level flight. An airplane’s thrust limit may also limit its turn performance. Since induced drag is directly proportional to lift squared, an airplane pulling 5 Gs would produce 25 times as much induced drag as in level flight. If the maximum thrust available can only overcome 16 times as much induced drag, then the airplane can only maintain level flight at 4 Gs. Of the three factors that limit turn performance, CLmax AOA and the limit load factor are found on the V-n diagram at the maneuver point. Assuming the airplane’s angle of bank is not thrust limited, this is where maximum turn performance is achieved. Any deviation from the maneuver point produces an undesired result. If velocity increases at a constant load factor, turn rate will decrease and turn radius will increase. If velocity decreases at a constant load factor, the airplane will stall. If angle of bank (limit load) increases at a constant velocity, the airplane will stall. If angle of bank (limit load) decreases at a constant velocity, turn radius will increase and turn rate will decrease.
Turn rate and turn radius are independent of weight. Any two airplanes capable of flying at the same velocity and same angle of bank can fly in formation, regardless of their weights. The load factor and turn performance for both airplanes will be the same, although the heavier airplane will be producing more lift. Instrument flight requires that turns be made at a standard rate. A Standard Rate Turn (SRT) is one in which 3º of turn are completed every second. As airspeed increases, the turn radius of a SRT will 300 AOB 1:00 TIME: 0:30 1:30 2:00 116 Fundamentals of Aerodynamics: Part 1 – M06-042 increase. A Standard Rate Turn in the T-6B corresponds to two needle widths’ deflection on the turn needle. COORDINATED TURNS The turn-and-slip indicator gives the pilot a visual indication of coordinated flight. It consists of a turn needle and a ball suspended in fluid. If the ball is centered, the aircraft is in coordinated flight (Figure 4-30). If the ball is displaced in the same direction as the turn, the aircraft is in a slip. If the ball is displaced in the opposite direction as the turn, the aircraft is in a skid. Whenever the aircraft becomes uncoordinated during flight, the corrective action is to alter the amount of rudder being used. This simply means to apply rudder in the direction the ball is displaced. Therefore, if the ball is displaced to the right, apply right rudder. A useful mnemonic device for proper rudder correction is “step on the ball.” Figure 4-30 Coordinated Turn A skid is caused by using too much rudder in the desired direction of turn (Figure 4-31). The yawing movement is toward the inside of the turn and the balance ball is deflected toward the outside due to centrifugal force. In a skid, turn radius will decrease and turn rate will increase. Skids are dangerous because the airplane will roll inverted if stall occurs (a skidded turn stall). Such a stall will probably be fatal at low altitude. A slip is caused by opposite or insufficient rudder in the desired direction of turn (Figure 4-32). The yawing movement is toward the outside of the turn, and the balance ball is deflected toward the inside, due to gravitational pull. In a slip, turn radius will increase and turn rate will decrease. Slips are useful for crosswind landings (commonly described as “wing down, top rudder”), or when trying to increase the airplane rate of descent without increasing airspeed. A stall while in a slip will cause the airplane to roll toward wings level (a safer reaction than in a skid). Still, any stall at low altitude could be fatal. Figure 4-31 Slip Figure 4-32 Skid 117 Fundamentals of Aerodynamics: Part 1 – M06-042 THE APPROACH TURN At this point, it is important to understand how the aircraft’s turning limitations affect flying the approach turn. The approach turn is a critical phase of flight where disregard for aerodynamic limitations can result in disaster. Increased Gs are required as bank angle increases during a level turn. This also applies during constant descents such as the approach turn. Increased load factors result in increased stall speed (Figure 4-30). Many pilots have made the fatal mistake of excessively increasing bank and back pressure during the approach turn in an effort to avoid overshooting the runway. They stall the aircraft, and depending upon the nature of the aircraft, the power response time, altitude, and the stall recognition point, the aircraft may not be recoverable prior to ground impact. This situation can easily be avoided. First, pattern winds should be analyzed and the pattern planned so that an excessively tight approach turn will not be required. Second, if faced with an overshooting final approach, the pilot should initiate a go-around/off and re-attempt the turn with safe amounts of bank and back pressure. DEFINITIONS A load is a stress-producing force that is imposed upon an airplane or component. Strength is a measure of a material’s resistance to load. There are two types of strength: Static strength and fatigue strength. Static strength is a measure of a material’s resistance to a single application of a steadily increasing load or force. Static failure is the breaking or serious permanent deformation of a material due to a single application of a steadily increasing load or force. For instance, a pencil breaks when too much force is applied and its static strength is exceeded. Fatigue strength is a measure of a material’s ability to withstand a cyclic application of load or force, i.e., numerous small applications of a small force over a long period of time. Fatigue failure is the breaking (or serious permanent deformation) of a material due to a cyclic application of load or force.
Breaking a wire coat hanger by bending it back and forth demonstrates fatigue failure. Airplanes may experience fatigue failure on many components (landing gear struts, tailhooks, and mounting brackets) due to the numerous arrested landings, catapult shots, and high G maneuvers performed in normal operation. The components are designed to withstand repeated loads, but not forever. Service life is the number of applications of load or force that a component can withstand before it has the probability of failing. Fatigue strength plays a major role in determining service life. Service life may apply to an individual component, or to the entire airframe.
When a metal is subjected to high stress and temperature it tends to stretch or elongate. This is called plastic deformation or creep. Engine turbine blades are periodically monitored for creep damage due to high heat and stress. Modern supersonic aircraft may also suffer from creep damage on the skin of the airplane, especially on the leading edge of the wings. The structural limits of an airplane are primarily due to the metal skeleton or airframe. Any time a wing produces lift, it bends upward. The wing may permanently deform if lift becomes too great. Airframe components, particularly the wings, determine the maximum load that the airplane can withstand. The two greatest loads on an airplane are lift and weight. Since weight doesn’t vary greatly from one 118 Fundamentals of Aerodynamics: Part 1 – M06-042 moment to the next, lift will be the force that causes the maximum load to be exceeded. It is difficult to measure the amount of lift produced by the airplane, but it is relatively easy to measure acceleration. Since acceleration is proportional to force (Newton’s Second Law), and we know the weight of the airplane, we can determine the amount of lift by monitoring the airplane’s acceleration. Since load factor is a ratio of an airplane’s lift to its weight, and the mass being accelerated by lift and weight is the same mass, load factor is actually the acceleration due to lift expressed as a multiple of the earth’s acceleration, and can easily be measured by an accelerometer. Structural considerations determined by the airplane’s mission and desired service life force a manufacturer to meet certain limits, such as maximum load factor, airspeed and maneuvering limitations. These design limits include the limit load factor, ultimate load factor, redline airspeed and maneuvering parameters. Limit load factor is the greatest load factor an airplane can sustain without any risk of permanent deformation. It is the maximum load factor anticipated in normal daily operations. If the limit load factor is exceeded, some structural damage or permanent deformation may occur. Aircraft will have both positive and negative limit load factors. The T-6B’s limit load factor is at +7 Gs and -3.5 Gs. Overstress/Over-G is the condition of possible permanent deformation or damage that results from exceeding the limit load factor. This type of damage will reduce the service life of the airplane because it weakens the airplane’s basic structure. Overstress/over-g may occur without visibly damaging the airframe. Inside the airplane are a variety of components, such as hydraulic actuators and engine mounts, which are not designed to withstand the same loads that the airframe can. Before the airframe experiences static failure these components may break if overstressed. The wing will not depart the airplane if the limit load factor is exceeded, but if an engine mount breaks, a fire could result from fuel spewing on hot engine casing. Any time an airplane experiences an overstress, maintenance personnel must inspect to determine whether damage or permanent deformation actually occurred. Always report an overstress/over-G to maintenance. Whether or not deformation or damage occurs depends on the elastic limit of the individual components. If a rigid metal object, such as a wing, is subjected to a steadily increasing load, it will bend or twist. When the load is removed, the component may return to its original shape. The elastic limit is the maximum load that may be applied to a component without permanent deformation. When a component is stressed beyond the elastic limit, it will experience some permanent deformation, but may still be usable. If the force continues to increase, the component will break. To ensure the airplane may operate at its limit load factor without permanent deformation, the limit load factor is designed to be less than the elastic limit of individual components. This virtually guarantees the airplane will reach its expected service life. Ultimate load factor is the maximum load factor that the airplane can withstand without structural failure. There will be some permanent deformation at the ultimate load factor, but no actual failure of the major load-carrying components should occur. If you exceed the ultimate load factor, structural failure is imminent (something major on the airplane will break). The ultimate load factor should be avoided since the typical airplane is rather difficult to fly after its wings tear off. The ultimate load 119 Fundamentals of Aerodynamics: Part 1 – M06-042 factor is 150% of the limit load factor. V-N / V-G DIAGRAM The V-n diagram or V-G diagram is a graph that summarizes an airplane’s structural and aerodynamic limitation. The horizontal axis is indicated airspeed, since this is what we see in the cockpit. The vertical axis of the graph is load factor, or Gs. The V-n diagram represents the maneuvering envelope of the airplane for a particular weight, altitude, and configuration. Accelerated stall lines, or lines of maximum lift, represent the maximum load factor that an airplane can produce based on airspeed. The accelerated stall lines are determined by CLmax AOA. They are the curving lines on the left side of the V-n diagram (Figure 4-33). If one tries to maintain a constant airspeed and increase lift beyond the accelerated stall lines, the airplane will stall because we have exceeded the stalling angle of attack. As airspeed increases, more lift can be produced without exceeding the stalling angle of attack. Figure 4-33 Example V-n Diagram 120 Fundamentals of Aerodynamics: Part 1 – M06-042 The limit load factors and ultimate load factors, both positive and negative, are plotted on the diagram. These lines represent the manufacturer’s and the military’s structural limitations. Any G load above the limit load factor will overstress the airplane. Any G load above the ultimate load will cause structural failure. Notice that the positive and negative limit load factors are different. Since the pilot cannot sustain a negative acceleration much greater than three Gs, the designer can save some structural weight by reducing the airplane’s ability to sustain negative Gs. For this reason, most maneuvers are performed with positive accelerations. The point where the accelerated stall line and the limit load factor line intersect is called the maneuver point. The IAS at the maneuver point is called the maneuver speed (Va) or cornering velocity. It is the lowest airspeed at which the limit load factor can be reached. Below the maneuver speed, we can never exceed the limit load factor because the airplane will stall before the limit load factor is reached. The T-6B’s maneuver speed is 227 KIAS at maximum gross weight. The vertical line on the right side is called the redline airspeed, or VNE (Velocity never-to-exceed). Redline airspeed is the highest airspeed that an airplane is allowed to fly. Flight at speeds above VNE can cause structural damage. VNE is determined by one of several methods: Critical Mach Number (MCRIT), airframe temperature, excessive structural loads, or controllability limits. MACH NUMBER 𝑀= 𝑇𝐴𝑆 𝐿𝑆𝑂𝑆 Mach number represents the ratio of an airplane’s speed through a given air mass to sound’s speed through the same air mass. An airplane traveling at Mach 1 is moving exactly as fast as sound is moving through the same air. An airplane’s Critical Mach Number is the lowest Mach number that an airplane can travel and create sonic (Mach 1) airflow somewhere on the aircraft. Because not all airplanes are designed to deal with stresses of breaking the sound barrier, care must be taken to ensure that airplanes not intended for supersonic (Mach 1+) flight remain below their critical Mach number. Since airplanes typically climb at a constant indicated airspeed, this becomes very important at higher altitudes. Recall the TAS equation. As an airplane climbs, air density decreases, and a pilot must fly at a higher true airspeed to maintain the same indicated airspeed. As altitude increases, temperature also decreases which causes sound to travel slower. Both of these changes will cause Mach number to increase with an increase in altitude. Since Critical Mach Number is based on an airplanes design, which does not change with altitude, as a pilot climbs they may eventually reach a point where they must transition from a constant indicated airspeed climb to a constant Mach number climb to avoid catastrophic airframe damage. If an airplane reaches its critical Mach number (MCRIT), and is not designed to withstand supersonic airflow, the shock waves generated may damage the structure of the airplane. Redline airspeed for these aircraft will be slightly below the airspeed at which they will achieve MCRIT. Turboprop aircraft must also take MCRIT for the propeller into account as well. It is crucial that propeller blades not exceed the LSOS, otherwise, catastrophic failure of the blades can occur. Redline airspeed may also be used to set limits on airframe temperature. As airspeed increases, the airplane encounters more air particles producing friction which heats up the airframe. This heating 121 Fundamentals of Aerodynamics: Part 1 – M06-042 can be extreme and hazardous at high speeds. Once the temperature becomes excessive, the airframe may suffer creep damage. Excessive structural loads may be encountered on components other than the main structural members. Control surfaces, flaps, stabilizers, and other external components are often not able to withstand the same forces that the wings or fuselage can withstand. Deflecting control surfaces at very high airspeeds may create sufficient forces to twist or break the wing or stabilizer on which they are located. The maximum operating airspeed for the T-6B is 316 KIAS or .67 Mach, whichever is less Controllability may determine the redline airspeed on aircraft with conventional control systems. At high airspeeds, dynamic pressure may create forces on the control surfaces which exceed the pilot’s ability to overcome. Or, due to the aeroelasticity of the controls surfaces, full deflection of the cockpit controls may cause only small deflection of the control surfaces. In either case, the pilot will be unable to provide sufficient control input to safely maneuver the airplane. 122 Fundamentals of Aerodynamics: Part 1 – M06-042 REDLINE AIRSPEED FACTORS AFFECTING THE SAFE FLIGHT ENVELOPE The portion of the V-n diagram that is bounded by the accelerated stall lines, the limit load factors and redline airspeed is called the safe flight envelope. The five major factors affecting the safe flight envelope are gross weight, altitude, configuration, asymmetric loading, and gust loading. The gross weight of an airplane will affect the airplane’s limit load factor and ultimate load factor. Consider an airplane whose wing is built to withstand 20,000 pounds of static load; this will determine how many Gs can be pulled. If the airplane takes off with a weight of 5,000 pounds, it could withstand 4 Gs (20,000 / 5,000 = 4). If the airplane weight decreases by burning fuel or expending ordnance, the limit load factor will increase. If the same airplane decreased its weight to 4,000 pounds, it could now withstand 5 Gs. An increase in weight will also cause the accelerated stall lines to sweep to the right since an increase in weight increases an airplane’s stall speed. This causes the maneuver speed to increase (Figure 4-34). Weight MANEUVER POINT ULTIMATE LOAD LIMIT LOAD ACCELERATED STALL LINES LIMIT LOAD ULTIMATE LOAD generally does not affect redline airspeed. Since its weight changes are small compared to other Figure 4-34 Effect of Increased Gross Weight aircraft, they are not accounted for in the T-6B’s safe flight envelope. 123 Fundamentals of Aerodynamics: Part 1 – M06-042 REDLINE AIRSPEED REDLINE AIRSPEED As altitude increases, the speed of sound will decrease and TAS will increase for a given IAS. With an increase in altitude the indicated redline airspeed must decrease in order to keep a subsonic airplane below MCRIT TAS. Since the limit and ultimate load factors are structural limits, they do not change with altitude. Since the horizontal axis is indicated airspeed, the accelerated stall lines will not change (Figure 4-35). Another factor that affects the safe flight envelope is configuration. When the landing gear and high MANEUVER POINT ACCELERATED STALL LINES LIMIT LOAD lift devices are extended; the envelope is substantially reduced in size. This is mainly due to the relatively weak structure of the landing gear doors and the deployed high lift devices. High airspeeds could possibly tear the landing gear doors off or bend the flaps. An airplane in the landing Figure 4-35 Effect of Increased Altitude configuration does not need to maneuver at high speeds and create high G loading. Changing the configuration by adding external stores, such as weapons or drop tanks, may also reduce redline airspeed because the higher air loads imposed may tear them from the airplane (Figure 4-36). Asymmetric loading refers to uneven production of lift on the wings of an airplane. It may be caused by a rolling pullout, trapped fuel, or hung ordnance. The V-n diagram may reflect limits that are imposed because of this condition (Figure 4-37). When an airplane is rolling, the up-going wing is producing more lift than the down-going wing. If the airplane performs a MANEUVER POINT ACCELERATED STALL LINES ULTIMATE LOAD rolling pullout, the up-going wing may become overstressed even though the accelerometer in the cockpit shows a G load at or below the limit load factor. This would be aggravated even further if there were an imbalance of ordnance or fuel on the wings. For this reason, in all maneuvers requiring a pullout at higher than normal loading, one of the first steps is to always level the wings. If the pilot were to experience an asymmetric load after a bombing LIMIT LOAD LIMIT LOAD ULTIMATE LOAD Figure 4-36 Effect of Configuration run, e.g., hung ordnance, special attention must be paid to the amount of Gs and angle of bank. Because asymmetric loading is cumulative with pilot induced loading, the limit load factor due to pilot induced loads should be reduced to approximately two-thirds of the normal limit load factor. This will ensure that one wing is not overstressed. In the T-6B, the maximum load factor during asymmetric loading is +4.7 to -1.0 Gs. 124 Fundamentals of Aerodynamics: Part 1 – M06-042 Gust loading refers to the increase in the G load due to vertical wind gusts. The load imposed by a gust is dependent upon the velocity of the gust: the higher the velocity, the greater the increase in load. If an airplane were generating the limit load factor during a maximum performance turn and hit a vertical gust, the gust would instantaneously increase the angle of attack of the airfoils and increase the lift on the wings enough to raise the G load above the limit load factor. For this reason, intentional flight through forecasted severe or reported moderate turbulence is prohibited in the T-6B. Vertical gusts of up to 30 feet per second may be encountered in moderate turbulence. This could produce up to 2 Gs of acceleration on the airplane. Because gust loading is cumulative with pilot induced loading, the limit load factor due to pilot induced Figure 4-37 Effect of Asymmetric Loading or Gust Loading loads should be reduced to two-thirds of the normal limit load factor. Turbulence penetration also requires that you slow the airplane to a speed that will reduce the effects of stress caused by gust loading. NATOPS states that the maximum airspeed for the T-6B in turbulence is 207 KIAS, and the recommended airspeed is 180 KIAS. When entering severe or extreme turbulence, the pilot should slow to an airspeed less than this. Maneuver speed is normally what the manufacturer recommends because the airplane cannot be overstressed for positive Gs at that airspeed. At a slower airspeed the aircraft will stall more easily; it makes no sense to spend more time than necessary in the turbulence. 125 Fundamentals of Aerodynamics: Part 1 – M06-042 Stability Types Of Stability Stability is the tendency of an object to return to its state of equilibrium once disturbed from it. There are two kinds of stability: static and dynamic. Static stability is the initial tendency of an object to move toward or away from its original equilibrium position. Dynamic stability is the position with respect to time, or motion of an object after a disturbance. Static Stability If an object has an initial tendency toward its original equilibrium position after a disturbance, it is said to possess positive static stability. Consider a ball inside a bowl (Figure 4-38). The ball’s equilibrium position is at the bottom of the bowl. If the ball is moved from this position toward the rim of the bowl, its initial tendency, when released, is to roll back toward the bottom of the bowl. It is therefore said to possess positive static stability.
Negative static stability is the initial tendency to continue moving away from equilibrium following a disturbance. Consider the bowl upside down with the ball on top as in Figure 4-39. Observe the ball’s new equilibrium position. If the ball is moved away from its equilibrium position and released, its initial tendency is to roll farther away from equilibrium. The ball exhibits negative static stability. Neutral static stability is the initial tendency to accept the displacement position as a new equilibrium. If we place the ball on a flat surface, it is again in equilibrium. If it is moved away from its original spot, the ball adopts the new equilibrium position (Figure 4-40). It does not have a tendency to move toward or away from the original equilibrium position. The ball now demonstrates neutral static stability. Figure 4-38 Positive Static Stability Figure 4-39 Negative Static Stability Figure 4-40 Neutral Static Stability 126 Fundamentals of Aerodynamics: Part 1 – M06-042 Dynamic Stability Static stability reveals nothing about whether the object ever settles back to its original equilibrium position. To study dynamic stability, we will first assume the object to possess positive static stability. Consider a ball at the top of Figure 4-38. After it is released, it will roll back to the bottom and up the other side. It will roll back and forth, and due to friction will oscillate less and less about the equilibrium position until it finally comes to rest at the bottom of the bowl. It possesses positive dynamic stability. Note that although the ball passes through the equilibrium position, it is not Figure 4-41 Positive Static and Positive Dynamic Stability in equilibrium again until it has stopped moving. The motion described is damped oscillation (Figure 4-41).
Figure 4-42 Positive Static and Neutral Dynamic Stability If the ball oscillates about the equilibrium position and the oscilla- tions never dampen out, in a frictionless environment, it possesses neutral dynamic stability. Figure 4-42 depicts its displacement relative to equilibrium over time. This motion is undamped oscillation. If, somehow, the ball did not slow down, but continued to climb to a higher and higher position with each oscillation, it would never return to its original equilibrium position. Figure 4-43 depicts negative dynamic stability. This motion is impossible with a ball, but occasionally aircraft behave this way. This motion is divergent oscillation. If an object does not have positive static stability, it cannot have positive dynamic stability. If an object has positive static stability, it can have any type of dynamic stability. In other words, static stability does not ensure dynamic stability, but static instability ensures dynamic instability. If an object is dynamically stable, the displacement from equilibrium will be reduced until the object is again at its original equilibrium. It must have both positive static and positive dynamic stability. If an object is dynamically unstable, the displacement may or may not increase, but the object will never return to its original equilibrium. Airplane Static Stability And Maneuverability Figure 4-43 Positive Static and Negative Dynamic Stability Equilibrium occurs when the sum of the forces and moments around the center of gravity (CG) are equal to zero. An aircraft in equilibrium will travel in a constant direction at a constant speed, developing no moments that would cause it to rotate around the CG. Since an airplane can rotate around three different axes, we must consider its stability around each of these axes. Longitudinal stability is stability of the longitudinal axis around the lateral axis (pitch). Lateral stability is stability 127 Fundamentals of Aerodynamics: Part 1 – M06-042 of the lateral axis around the longitudinal axis (roll). Directional stability is stability of the longitudinal axis around the vertical axis (yaw). Each motion requires a separate discussion. Any discussion of airplane stability requires an explanation on how the wings, fuselage, vertical stabilizer, horizontal stabilizer, etc, affect the longitudinal, lateral, and directional stability of the airplane. This lesson considers only conventional airplanes, that is, airplanes with their wings, fuselage and stabilizers in their normal positions. We’ll make some basic assumptions to simplify our study. First, we assume a constant TAS. The disturbances that cause the airplane to pitch, yaw, or roll will be small enough that it does not affect the airplane’s forward velocity. The disturbances will also be small enough to keep the change in pitch attitude, and degree of yaw and roll small enough so that we do not approach any stalling AOAs or unusual attitudes. These disturbances are external and not caused by the pilot. The pilot applies no inputs to correct the displacement from equilibrium. Any moment that corrects the airplane’s attitude results from the design of the airplane. An airplane’s maneuverability is the ease with which it will move out of its equilibrium position. Maneuverability and stability are opposites. A stable airplane tends to stay in equilibrium and is difficult for the pilot to move out of equilibrium. The more maneuverable an airplane is, the easier it departs from equilibrium, and the less likely it is to return to equilibrium. There are two ways to increase an airplane’s maneuverability. If we want an airplane to move quickly from its trimmed equilibrium attitude, we can give it weak stability. Of course, this means the airplane will be more difficult to fly in equilibrium and will require more of the pilot’s attention. Our other option is to give the airplane larger control surfaces. If the control surfaces are large, they can generate large moments by producing greater aerodynamic forces. The airplane designer must decide how to compromise between stability and maneuverability. The mission of a specific airplane dictates the compromises the designer will have to make. A transport plane must be relatively stable for long range flights and ease in landing. A fighter must possess great maneuverability for high performance turning. Now that we have a basic understanding of static stability, we can discuss each component and its individual contribution to static stability. Afterwards, we’ll combine all the components and discuss the overall static stability of the airplane. 128 Fundamentals of Aerodynamics: Part 1 – M06-042 Longitudinal Static Stability Longitudinal static stability is stability of the longitudinal axis around the lateral axis. An airplane has longitudinal stability if, after some disturbance causes it to pitch up or down, it generates forces and moments that tend to move the pitch attitude to a level flight condition. The Flying Wing Model Each individual component may have its own aerodynamic center, and thus its own effect on static stability. These individual components create moments around the CG of our airplane that can be either stabilizing or destabilizing. To examine stability in greater detail, we will first take a simplified approach using a "flying wing" model. By choosing the flying wing we have essentially eliminated the stability effect of any component except the wing itself. An airplane experiences four main forces in equilibrium flight: lift, weight, drag, and thrust. Recall Figure 4-44 Flying Wing Model that these forces act around the center of gravity. For our discussion of longitudinal stability we only need to address lift and weight. Figure 4-44 shows these two forces in equilibrium on our airplane. Notice in Figure 4-44 that lift is acting through the aerodynamic center (AC), which is at a distance from our CG. This creates a moment around the CG. It should be understood that our flying wing in Figure 4-44 is in equilibrium, and that trim devices are preventing the wing from rotating to a nose- down attitude. Consider how the flying wing reacts to a disturbance that increases the AOA sensed by the airfoil. The increased AOA will increase lift. If the CG is ahead of the AC, the increase in lift at the AC develops a moment that pitches the nose of the airplane down in the direction of its original equilibrium AOA. Our flying wing has positive longitudinal static stability because of its initial tendency to return to equilibrium (Figure 4-45). If a component’s aerodynamic center is behind the airplane’s center of gravity the component will be a positive contributor to longitudinal static stability. Figure 4-45 Positive Longitudinal Static Stability Figure 4-46 Negative Longitudinal Static Stability Next, examine a flying wing where the AC is ahead of the CG. When the disturbance increases AOA, the wing produces more lift and rotates the flying wing further away from equilibrium (Figure 4-46). Any disturbance would soon lead to stall and possibly out of control flight. We can generalize this and 129 Fundamentals of Aerodynamics: Part 1 – M06-042 say that if a component’s aerodynamic center is in front of the airplane’s center of gravity the component will be a negative contributor to longitudinal static stability. Straight Wings The wing’s contribution to longitudinal static stability depends mainly on the location of the wing’s AC with respect to the airplane’s CG. Most airplanes have straight wings with the AC forward of the airplane’s CG. Like the second flying wing example, having the AC forward of the CG causes longitudinal static instability. The wings of most conventional airplanes are negative contributors to longitudinal static stability. Figure 4-47 illustrates the location of the wing’s AC and the airplane’s CG. Figure 4-47 Straight Wings Wing Sweep Figure 4-48 Wing Sweep When an F-14’s wings are swept forward, they have a strong longitudinally destabilizing effect. This is because the wing’s AC is well forward of the airplane’s CG. This increases the F-14’s maneuverability. As sweep angle increases (i.e. the wings move aft), the wings’ AC moves aft, closer to the airplane’s CG (Figure 4-48), making the airplane more longitudinally stable. Sweeping an airplane’s wings back is a positive contributor to longitudinal static stability. The Fuselage The fuselage acts as an airfoil and thus produces lift. The fuselage’s AC is usually located ahead of the airplane’s CG (Figure 4-49). If a disturbance causes an increase in angle of attack, the fuselage will produce greater lift that produces a destabilizing effect. The fuselage is a negative contributor to longitudinal stability. THE HORIZONTAL STABILIZER The horizontal stabilizer is a symmetric airfoil designed to stabilize the airplane around the lateral axis. Its contribution to longitudinal static stability is determined by the moment it produces around the CG. Since its AC is well behind the airplane’s CG (Figure 4-50), the horizontal stabilizer has the greatest positive effect on longitudinal static stability. The pitching moment can be increased by increasing the distance between the airplane’s CG and the stabilizer’s AC, or by enlarging the horizontal stabilizer. Thus, for a short moment arm between the airplane’s CG and the horizontal stabilizer’s aerodynamic center, a large horizontal stabilizer is needed. For an airplane with a longer moment arm, a smaller horizontal stabilizer will suffice. 130 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 4-49 Fuselage Figure 4-50 Horizontal Stabilizer THE NEUTRAL POINT The longitudinal static stability provided by the horizontal stabilizer must overcome the instabilities of the wings and fuselage in order to produce a stable airplane. Figure 4-51 shows the AC for each individual component. The neutral point (NP) is the location of the center of gravity along the longitudinal axis that would provide neutral longitudinal static stability. It can be thought of as the aerodynamic center for the entire airplane. The location of the NP is fixed on conventional airplanes, but we can change the location of the CG by moving around cargo or Figure 4-51 AC for each component mounting ordnance and fuel in various locations. As the CG is moved aft, the airplane’s static stability decreases. The NP defines the farthest aft CG position without negative stability. Once the CG is aft of the NP the airplane becomes unstable and the pilot may have difficulty controlling it in flight. Directional Static Stability Directional static stability is stability of the longitudinal axis around the vertical axis. When an airplane yaws, its momentum keeps it moving along its original flight path for a short time. This condition is known as a sideslip. The angle between the longitudinal axis and the relative wind is called the sideslip angle (β) (Figure 4-52). The component of the relative wind that is parallel to the lateral axis is called the sideslip relative wind. Reaction to the sideslip will determine a component’s contribution to directional static stability. We will examine the effects of the wings, wing Figure 4-52 Sideslip Angle β sweep, fuselage, and the vertical stabilizer on directional static stability. Straight Wings During a sideslip, the advancing wing on a straight winged airplane has a momentary increase in airflow velocity as it moves forward. This increases parasite drag on that wing and pulls it back to its equilibrium position. The retreating wing has less velocity and less parasite drag, which helps to bring the nose back into the relative wind. Therefore, straight wings have a small positive effect on directional static stability. 131 Fundamentals of Aerodynamics: Part 1 – M06-042 SWEPT WINGS The swept design of a wing will further increase directional stability. The advancing wing not only experiences an increase in parasite drag, but also an increase in induced drag due to the increased chordwise flow. Remember that lift and induced drag are produced by the wings when air flows chordwise over them. The retreating wing experiences more spanwise flow. Figure 4-53 depicts this phenomenon with the left wing experiencing greater induced and parasite drag. The result is an airplane that comes back into the relative wind. Figure 4-53 Swept Wings Figure 4-54 Fuselage THE FUSELAGE The fuselage is a symmetric airfoil with its aerodynamic center forward of the airplane’s CG. At zero angle of attack or zero sideslip it produces no net lift. When the airplane enters a sideslip, an angle of attack is created on the fuselage. The lift produced at the fuselage AC pulls the nose away from the relative wind, thus causing an increase in the sideslip angle. Therefore, the fuselage is a negative contributor to the airplane’s directional static stability (Figure 4-54). The Vertical Stabilizer The vertical stabilizer is the greatest positive contributor to the directional static stability of a conventionally designed airplane. The vertical stabilizer is a symmetric airfoil mounted far behind the airplane’s CG. A sideslip causes the vertical stabilizer to experience an increased angle of attack. This creates a horizontal lifting force on the stabilizer that is multiplied by the moment arm distance to the airplane’s CG (Figure 4-55). The moment created will swing the nose of the airplane back into the relative wind. This is identical to the way a weather-vane stays oriented into the wind. There is an inverse relationship between tail Figure 4-55 Vertical Stabilizer size and moment arm length. The smaller the distance to the CG, the larger the vertical stabilizer must be and vice versa. It is not always desirable to have a large vertical stabilizer because of limited storage room aboard aircraft carriers and the large radar signatures. Designers often use two or more smaller vertical stabilizers (A-10, F-15, F/A-18, and E-2), to accomplish the same stability effects as one large stabilizer. 132 Fundamentals of Aerodynamics: Part 1 – M06-042 Lateral Static Stability Lateral stability is stability of the lateral axis around the longitudinal axis. An airplane has lateral stability if, after some disturbance causes it to roll, it generates forces and moments that tend to reduce the bank angle and restore the airplane to a wings level flight condition. When an airplane rolls, the lift vector points to the inside of the turn, reducing the vertical component of lift. Since weight still acts downward with the same force (Figure 4-56), the plane descends. The horizontal component of lift pulls the airplane to the side, thus creating a sideslip relative wind. This sideslip relative wind acts on the various components of the airplane causing stability or instability. DIHEDRAL EFFECT When an airplane is laterally sideslipping, dihedral wings cause an increase in angle of attack and lift on the down-going wing. The up-going wing has a reduced angle of attack and a decrease in lift (Figure 4-57). This difference in lift creates a rolling moment that rights the airplane and stops the sideslip. Wings that are straight have neutral lateral static stability. Dihedral wings are the greatest positive contributors to lateral static stability. Anhedral wings are the greatest negative contributors to lateral static stability. Figure 4-56 Sideslip Relative Wind Figure 4-57 Dihedral Wings WING PLACEMENT ON THE VERTICAL AXIS During a sideslip the fuselage of a high-winged airplane impedes the airflow generated by the sideslip. This increases the upwash at the wing root on the down-going wing which increases the AOA and lift. The up-going wing receives downwash which decreases the AOA, and lift. The lift imbalance rolls the airplane back to wings level. A low-mounted wing has the opposite effect. Thus, a high mounted wing is a positive contributor, and a low mounted wing is a negative contributor to lateral static stability (Figure 4-58). 133 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 4-58 High- and Low-Mounted Wings Figure 4-59 Wing Sweep WING SWEEP Another way to affect lateral stability is to sweep the wings aft. As an airplane begins to sideslip in the direction of roll, the wing toward the sideslip has more chordwise flow than the wing away from the sideslip (Figure 4-59). The wing toward the sideslip (the lower wing) generates more lift, which levels the wings. Swept wings are laterally stabilizing. These effects are cumulative. High-mounted, swept dihedral wings are much more stable than low-mounted, straight wings with the same dihedral. The Vertical Stabilizer The only other major effect on lateral stability comes from the vertical stabilizer. When in a lateral sideslip, the vertical stabilizer senses an angle of attack, so it produces lift. Since the tail is above the airplane’s center of gravity, this lift produces a moment that tends to right the airplane (Figure 4- 60). Dynamic Stability Figure 4-60 Vertical Stabilizer Our discussion thus far has centered on static stability. When we discuss dynamic stability, we must realize that lateral and directional stability are interrelated. This interrelationship is called cross- coupling. The motions of an airplane are such that a rolling motion causes a yawing motion and vice versa. This cross-coupling between directional static stability and lateral static stability causes several dynamic effects including spiral divergence, Dutch roll, proverse roll, and adverse yaw. Directional Divergence Directional divergence is a condition of flight in which the reaction to a small initial sideslip results in an increase in sideslip angle (Figure 4-61). Directional divergence is caused by negative directional static stability. If the vertical stabilizer becomes ineffective for some reason (battle damage, mid-air collision), directional divergence could cause out of control flight. Most airplanes have very strong directional stability to prevent this from occurring. 134 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 4-61 Directional and Spiral Divergence SPIRAL DIVERGENCE Spiral divergence occurs when an airplane has strong directional stability and weak lateral stability (Figure 4-61). For example, an airplane is disturbed so that its wing dips and starts to roll to the left. Because the airplane has weak lateral stability it cannot correct itself and the flight path arcs to the left. The airplane senses a new relative wind from the left and aligns itself with the new wind by yawing into it (strong directional stability). The right wing is now advancing and the increased airflow causes the airplane to roll even more to the left. The airplane will continue to chase the relative wind and will develop a tight descending spiral. This is easily corrected by control input from the pilot. 135 Fundamentals of Aerodynamics: Part 1 – M06-042 DUTCH ROLL Dutch roll is the result of strong lateral stability and weak directional stability. The airplane responds to a disturbance with both roll and yaw motions that affect each other. For example, a gust causes the airplane to roll left, producing a left sideslip. The strong lateral stability increases lift on the left wing and corrects it back to wings level. At the same time, the nose of the airplane yaws left into the sideslip relative wind. This leaves the airplane wings level, with the nose cocked out to the left. The weak directional stability now swings the nose to the right to correct the nose back into the relative wind. This causes the left wing to advance faster than the right wing, a situation which produces more lift on the left wing and rolls the airplane to the right, creating a right sideslip. The strong lateral stability corrects the wings back to level, while the nose yaws right into the sideslip relative wind. This leaves the airplane wings level, with the nose cocked out to the right. As the nose yaws left into the relative wind, the wings will roll left which starts the entire process again. The airplane appears to be “tail wagging” (Figure 4-62). This Figure 4-62 Dutch Roll condition can be tolerated and may eventually dampen out. However, it is not acceptable in a fighter or attack airplane when the pilot is trying to aim at a target. Phugoid Oscillations Phugoid oscillations are long period oscillations (20 to 100 seconds) of altitude and airspeed while maintaining a nearly constant angle of attack. Oscillations of pitch attitude do occur, but are often minor. Phugoid oscillations are not a result of cross-coupling. Upon being struck by an upward gust, an airplane would gain altitude and lose airspeed. A large but gradual change in altitude and airspeed occurs. When enough airspeed is lost, the airplane will nose-over slightly, commencing a gradual descent, gaining airspeed and losing altitude. When enough airspeed is regained, the nose will pitch up, starting the process over. The period of this oscillation is long enough that the pilot can easily correct it. Often, due to the almost negligible changes in pitch, the pilot may make the necessary corrections while being completely unaware of the oscillation. Pilot / Airplane Interaction A complete discussion of an airplane’s stability characteristics is not limited to how the airplane reacts to various external forces, but must also consider the interaction of the pilot and the airplane. Proverse Roll Proverse roll is the tendency of an airplane to roll in the same direction as it is yawing. When an airplane yaws, the yawing motion causes one wing to advance and the other wing to retreat. This increases the airflow on the advancing wing and decreases airflow over the retreating wing. A difference in lift is created between the two wings, and the airplane rolls in the same direction as it 136 Fundamentals of Aerodynamics: Part 1 – M06-042 yawed. Proverse roll is even more pronounced on swept wing airplanes since the advancing wing will have more chordwise flow and will produce more lift. Figure 4-63 Proverse Roll Figure 4-64 Adverse Yaw Adverse Yaw Adverse yaw is the tendency of an airplane to yaw away from the direction of aileron roll input. When an airplane rolls, it has more lift on the up-going wing than on the down-going wing. This causes an increase in induced drag on the up-going wing that will retard that wing’s forward motion and cause the nose to yaw in the opposite direction of the roll. The aircraft produces adverse yaw each time the ailerons are deflected (rolling into and out of a turn). We can do three things to overcome this problem. The first method is to use spoilers instead of ailerons. The spoiler is deflected into the airstream from the upper surface of the wing. This spoils the airflow and reduces lift, causing the airplane to roll. The spoiler increases the parasite drag on the down-going wing, offsetting the induced drag on the up-going wing and helps reduce or eliminate adverse yaw. The second method is to use a rudder input to offset adverse yaw. The third is actually a design method of building the aircraft with differential ailerons. PILOT INDUCED OSCILLATIONS Pilot induced oscillations (PIO) are short period oscillations of pitch attitude and angle of attack. PIO or porpoising occurs when a pilot is trying to control airplane oscillations that happen over approximately the same time span as it takes to react. For example, a gust of wind causes the nose to pitch up. The natural longitudinal stability of the airplane will normally compensate. However, if the pilot tries to push the nose-down, his input may coincide with the stability correction, causing the nose to over correct and end up low. The pilot then pulls back on the stick causing the nose to be high again. Since the short period motion of PIO is of relatively high frequency, the amplitude of the pitching could reach dangerous levels in a very short time. If PIO is encountered, the pilot must rely 137 Fundamentals of Aerodynamics: Part 1 – M06-042 on the inherent stability of the airplane and immediately release the controls, if altitude permits. If not, the pilot should “freeze” the stick slightly aft of neutral. ASYMMETRIC THRUST Any airplane with more than one engine can have directional control problems if one engine fails. This is known as asymmetric thrust. If an airplane with its engines located far from the fuselage, such as an S-3, E-2 or KC-10, has an engine failure, the thrust from the operating engine(s) will create a yawing moment toward the dead engine (Figure 4-65). This can happen even if the engines are relatively close, such as with the F/A-18. The farther from the longitudinal axis that the engines are located the greater the moment created by the operating engine. The yawing motion may be sufficient to cause proverse roll. Full opposite rudder may be required to compensate for the yawing moment, while opposite aileron should be used to correct the proverse roll. Every multi-engine airplane has a minimum directional control speed that must be flown to ensure maximum effectiveness of the vertical stabilizer following an engine failure. Figure 4-65 Asymmetric Thrust 138 Fundamentals of Aerodynamics: Part 1 – M06-042 ASSIGNMENT SHEET 2-4-3 PERFORMANCE & MANEUVERING REVIEW A.
INTRODUCTION Performance and Maneuvering is the third aerodynamics lesson in a series presenting aerodynamic theories and operational principles. Upon completion of this lesson, you will understand aircraft performance with respect to temperature, maximum range and endurance, and aircraft speeds relating to the lift and thrust curves. You will also understand maneuvering characteristics relating to velocity, angle of attack, load factor, and aircraft stability.
B.
ENABLING OBJECTIVES C.
STUDY ASSIGNMENT 1.
Review Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 3 2.
Read Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 4 D.
STUDY QUESTIONS 1. Takeoff distance is . a. decreased by a tailwind b. increased by a tailwind c. decreased by higher elevation d. increased by use of flaps 2. Max range is defined as . a. the farthest distance an airplane can travel with a headwind component of 50 knots or less b. the farthest distance an airplane is allowed to travel c. the most time an airplane can remain airborne for a given amount of fuel d. the farthest distance an airplane can travel on a given amount of fuel 3. Max endurance is . a. the farthest distance an airplane can travel on a given amount of fuel b. the maximum time that an airplane can remain airborne on a given amount of fuel c. the most amount of time an airplane can spend at cruise altitude d. not affected by configuration 139 Fundamentals of Aerodynamics: Part 1 – M06-042 4. Max glide range (flown at Best Glide Speed) is . a. obtained by flying at L/DMAX airspeed b. decreased by a tailwind c. not affected by altitude d. flown at an airspeed less than L/DMAX airspeed 5. Gaining the most altitude in a given amount of time is . a. based on amount of excess thrust available b. used for obstacle clearance c. Max Angle of Climb d. Max Rate of Climb 6. Slipstream swirl is most noticeable . a. at low power settings and high airspeed b. at high power settings and low airspeed c. in a spin d. when the aircraft displays positive directional stability 7. Ultimate load factor for the T-6B is defined as . a. the least load factor an airplane can sustain without any risk of permanent deformation b. +6.0 Gs c. -1.0 G d. the maximum load factor that the airplane can withstand without structural failure 8. Turn radius is . a. a factor of velocity and bank angle b. a factor of weight, airspeed, and temperature c. a factor of velocity, wingspan, and bank angle d. measured in degrees per second 9. Critical Mach is defined as . a. the ratio of true airspeed to the local speed of sound b. the speed at which the airplane goes supersonic c. the Mach number that produces the first evidence of local supersonic flow d. aircraft’s maximum allowable Mach number 10. Sideslip angle is . a. the angle between the relative wind and the lateral axis b. the difference between the flightpath and the longitudinal axis c. greater in aircraft that have positive static stability d. smaller in fighter/attack aircraft 140 Fundamentals of Aerodynamics: Part 1 – M06-042 11. When flying at maximum endurance airspeed in the T-6B, increasing AOA to maintain level flight will result in .
a. higher airspeed requiring less power b. higher airspeed requiring more power c. slower airspeed requiring less power d. slower airspeed requiring more power ADDITIONAL LESSON REVIEW QUESTIONS 1. Of the factors affecting takeoff and landing performance, what is the single largest contributor to increasing takeoff or landing distance?
2. Why is it impossible to exceed the aircraft load limit at airspeeds below cornering velocity?
3. What is the difference between positive and negative static stability?
4. Which airframe component is the largest positive contributor to aircraft directional static stability?
141 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 1 – Takeoff Factors Summary Chart Figure 2 – Climb Factors Summary Chart 142 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 3 – Cruise Factors Summary Chart Figure 4 – Turn Performance Summary Chart 143 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 5 – Landing Factors Summary Chart 144 Fundamentals of Aerodynamics: Part 1 – M06-042 Answers: 1.B 7. D 2.D 8. A 3.B 9. C 4.A 10. B 5.D 11. D 6.B Answers to Additional Lesson Review Questions: 1. Weight is the greatest factor. Doubling weight quadruples both takeoff and landing distance.
2. Cornering velocity is the minimum airspeed at which the aircraft G-load limit can be obtained. Below this airspeed the aircraft will stall, making an over-G condition impossible.
3. a. Positive static stability: Initial tendency is for the object to return to its original equilibrium position b. Negative static stability: Initial tendency is for the object to move away from its original equilibrium position.
4. The vertical stabilizer is the greatest positive contributor to directional static stability due to its weathervane effect.
145 Fundamentals of Aerodynamics: Part 1 – M06-042 OUTLINE SHEET 2-5-1 SPINS A.
INTRODUCTION This lesson covers fundamental information on the aerodynamic causes and characteristics of aircraft spins. At the completion of this lesson, you will be able to define and state the causes of a spin and describe the aerodynamic factors affecting spin development.
B.
ENABLING OBJECTIVES 3.19 DEFINE a spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.20 DEFINE autorotation, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.21 DESCRIBE the aerodynamic forces affecting a spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.22 STATE the characteristics of erect, inverted, and flat spins, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.23 DESCRIBE the factors contributing to aircraft spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.24 DISCUSS the effects of weight, pitch attitude, and gyroscopic effects on spin characteristics, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.25 STATE how empennage design features change spin characteristics, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.26 STATE the cockpit indications of an erect and inverted spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.27 DESCRIBE the pilot actions necessary to recover from a spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 3.28 DESCRIBE a progressive spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 146 Fundamentals of Aerodynamics: Part 1 – M06-042 3.29 DESCRIBE an aggravated spin, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 C.
TOPIC OUTLINE 1.
Introduction 2.
This Lesson Topic 3.
Spin Development 4.
Spin Aerodynamics 5.
Spin Indications and Recovery 6.
Summary and Review 7.
Application 8.
Assignment 147 Fundamentals of Aerodynamics: Part 1 – M06-042 INFORMATION SHEET 2-5-2 SPINS A.
INTRODUCTION This lesson covers fundamental information on the aerodynamic causes and characteristics of aircraft spins. At the completion of this lesson, you will be able to define and state the causes of a spin and describe the aerodynamic factors affecting spin development.
B.
REFERENCES 1.
Aerodynamics for Naval Aviators, NAVAIR 00-80T-80 2.
T-6 Joint Primary Pilot Training, AETC / CNATRA Syllabus P-V4A-J NATOPS Flight Manual C.
INFORMATION Spin Aerodynamics A spin is an aggravated stall that results in autorotation. Autorotation is a combination of roll and yaw that propagates itself due to asymmetrically stalled wings. For an aircraft to spin two conditions are required: the aircraft must be stalled, and yaw must be present. Without both conditions present, an aircraft will not enter a spin. Yaw can be introduced intentionally, such as during training, or unintentionally. Unintentional yaw inputs can be pilot induced, such as misapplication of controls, or non-pilot induced, such as fuel loading, or wake turbulence. To help you understand the aerodynamics of the spin, consider the motions an aircraft undergoes during a spin. Every aircraft exhibits different spin characteristics, but they all have stall and yaw about the spin axis. The spin axis is the aerodynamic axis around which stall and yaw forces act to sustain spin rotation. For the T-6B, the spin axis is a vertical axis through the cockpit, forward of the center of gravity. There are four phases to a spin: post-stall gyration, incipient, developed, and recovery phases. The poststall gyration phase begins at the instant the airplane stalls, and is where the pilot provides the necessary elements for the spin, either unintentionally or intentionally. Poststall gyrations are determined by the aerodynamic forces during a stall that result in movement around the pitch, roll, and yaw axes. Poststall gyrations characteristics are dependent on airspeed. The higher the airspeed at stall entry, the greater poststall gyration severity, while the lower the airspeed at stall entry, the less poststall gyration severity. During a stall, lift and drag of both wings are balanced unless yaw is introduced. The introduction of yaw creates an AOA difference between the left and right wings. Poststall gyrations can result in introduction of yaw and can result in spin entry. 148 Fundamentals of Aerodynamics: Part 1 – M06-042 Examine the AOA and relative wind on each wing (Figure 5-1). In this example, the airplane stalls and begins a roll to the left. The left wing now becomes the down-going wing and senses a roll relative wind from beneath. This roll relative wind is added to the existing relative wind and creates an average relative wind that is further from the chord line; therefore, the down-going wing has a higher AOA. This wing has become more stalled. Conversely, the right wing becomes the up-going wing and senses a roll relative wind from above. When added to the original relative wind, the up-going wing has a lower AOA and is less stalled. Remember that while both wings are stalled, they do not lose all their lift, nor are they equally stalled. Figure 5-1 Roll Relative Wind The AOA differential results in two cases: lower AOA, more lift and less drag on the up-going wing, and higher AOA, less lift and more drag on the down-going wing (Figure 5-2). The higher AOA on the down-going wing decreases the CL generated by that wing. The up-going wing has a greater CL due to a lower AOA, and therefore has greater total lift. The lift differential results in a continued rolling motion of the airplane around the spin axis. 149 Fundamentals of Aerodynamics: Part 1 – M06-042 The increased AOA on the down-going wing increases the CD generated by that wing. The greater drag on the down-going wing results in a drag differential and causes a yawing motion around the spin axis. The combined effects of roll and yaw cause the airplane to continue its autorotation. After the plane is stalled and poststall gyrations have introduced yaw, the incipient stage begins. The incipient stage begins after the poststall gyrations and ends when the spin is fully developed. This can take up to two rotations in most airplanes. Figure 5-2 CL and CD in a Spin 150 Fundamentals of Aerodynamics: Part 1 – M06-042 Factors Affecting Spins Conservation of angular momentum plays a significant role in determining how an airplane will spin. In the T-6B, the spin axis will be forward of the center of gravity. As the nose of the aircraft pitches down, the moment arm between the spin axis and the center of gravity shortens. This results in more mass closer to the moment of inertia and increases the rotation rate. An example of this is a spinning ice skater bringing their arms closer to their body and increasing their rate of rotation. Conversely, as the nose of the aircraft pitches up, the moment arm between the spin axis and the center of gravity increases, resulting in a more mass away from the moment of inertia and therefore, a slower rotation rate. Continuing the example from above, the ice skater is now moving their arms away from their body and slowing their rotation. Pitch attitudes can be affected by control inputs, aircraft weight, pitch attitude at stall entry, and spin direction Ailerons Depending on wing shape and the aerodynamic properties of a wing, ailerons applied in the direction of spin will cause increased roll and yaw oscillations, while ailerons applied in the direction opposite of spin rotation will tend to dampen roll and yaw oscillations. However, ailerons are not used to recover from a spin in the T-6B since they rarely assist in the recovery. This is a result of the wing being in a deep stall and little useful air flowing over the ailerons. Therefore, during a spin, apply neutral ailerons. Rudder The rudder is the principal control for stopping autorotation in the T-6B. During normal operation a rudder increases the angle of attack on the vertical stabilizer and produces lift, in the horizontal direction, that creates a yawing moment about the center of gravity. Due to the direction of the relative wind in a spin, the vertical stabilizer acts as a flat plate, instead of as an airfoil (Figure 5-3) and the rudder is used to create drag, not lift, to create the yawing moment. The direction of the rudder has a significant effect on the amount of drag created. If the rudder is deflected in the same direction as the spin (pro-spin rudder), the amount of rudder exposed to the relative wind will be minimized and result in less drag. A rudder deflected in the opposite direction as the spin (anti-spin rudder), will maximize the amount of rudder exposed to the relative wind, and the amount of drag. The drag created by the vertical stabilizer can be divided into a horizontal and vertical component. The horizontal component creates a force that opposes the yawing of the airplane slowing rotation rate. The vertical component creates a force that pulls the tail up and pitches the nose-down, reducing the AOA on both wings. 151 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 5-3 Rudder Forces During a Spin The design of the vertical stabilizer and rudder and the placement of the horizontal control surfaces will significantly affect spin recovery. In the case of a swept vertical fin (Figure 5-4), airflow to the rudder is almost entirely blocked by the horizontal surfaces and is therefore not effective at stopping the autorotation. With the T-6B tail design, airflow to the rudder is not blocked and the horizontal stabilizer is farther aft, exposing more of the rudder in a spin increasing spin-recovery characteristics. It is interesting that although an inverted spin in a T-6B is difficult to enter and very disorienting, it is easy to recover from. This is due to the entire vertical stabilizer is in an inverted position and therefore completely exposed to the relative wind The T-6B uses a dorsal fin, strakes, and ventral fin to decrease the severity of spin characteristics (Figure 5-5). The dorsal fin is attached to the front of the vertical stabilizer to increase its surface area. This decreases the spin rate and aids in stopping the autorotation. A ventral fin on the T-6B is located beneath the empennage. Ventral fins decrease the spin rate and aid in maintaining a nose down attitude. The T-6B has strakes located in front of the horizontal stabilizer. These strakes increase the surface area of the horizontal stabilizer in order to keep the nose pitched down and prevent a flat spin. Figure 5-4 Effect of a Swept Vertical Fin Figure 5-5 Empennage 152 Fundamentals of Aerodynamics: Part 1 – M06-042 Elevator In a stabilized spin, the horizontal stabilizer and elevator are fully stalled due to an angle of attack in excess of 50°. This results in very little lift and a great amount of drag. The drag will be maximized with full down elevator and minimized with full up elevator (Figure 5-6). The increased drag on the horizontal stabilizer and elevator will cause a more nose- down pitch attitude. Similar to the vertical stabilizer, this drag will also have a vertical and horizontal component. Figure 5-6 Effect of Elevator Full aft stick results in the flattest pitch attitude and therefore the slowest spin rate. This is referred to as an unaccelerated spin. Any stick position other than full aft will result in a steeper pitch attitude and an increase in rotation rate. This is referred to as an accelerated spin. As the pitch attitude is slowly lowered, the increase in spin rate causes the center of gravity to experience a greater force away from the spin axis. The acceleration resists the airplane moving to a steeper pitch attitude. This is why brisk control inputs are emphasized in a recovery. Abrupt forward stick will drive the pitch attitude down before rotation rate can increase appreciably and build up the nose down resistance. Aircraft Weight The aircraft’s weight varies primarily due to fuel usage, but can also vary if items are dropped off the airplane (e.g., ordnance, fuel tanks, etc.). If an aircraft carries fuel in the wings, a large portion of the weight of the airplane is away from the center of gravity. This creates a large moment of inertia for a spin to overcome. A heavier airplane will have a slower spin entry with lesser oscillations due to this large moment of inertia. A lighter airplane will enter a spin more quickly, with greater oscillations possible, but will also recover from a spin faster. Pitch Attitude The pitch attitude will have a direct impact on the speed the aircraft stalls. For a given power setting, stall speed varies inversely with pitch attitude. As an airplane increase its pitch attitude, a larger portion of the thrust vector is in the vertical, in effect, adding lift. This additional lift reduces the load seen by the wings allowing for a slower stall speed. Slower stall speeds make the spin entry slower and with lesser oscillations. At lower pitch attitudes, the aircraft stalls at a higher airspeed and entries are faster and more oscillatory. Spin Direction Gyroscopic precession is a phenomenon that occurs when a gyroscope experiences a force. A gyroscopic mass reacts to a disturbance (force) along the rotational axis at a point 90° further in the rotation cycle. The propeller of the T-6B is a clockwise rotating gyroscope (as viewed from cockpit). If an airplane is in a right spin (nose yawing right), the nose of the T-6B will tend to pitch down due to gyroscopic precession. Conversely, if the T-6B is in a left spin, the nose will tend to pitch up. The T-6B will therefore have a flatter attitude when spinning to the left than to the right. This makes for a 153 Fundamentals of Aerodynamics: Part 1 – M06-042 more nose-high pitch attitude, slower rotation rate and smoother entries into spins that stabilize quicker. A T-6B in a right spin will have a more nose down pitch attitude, higher rotation rate, and a more oscillatory entry. Spin Indications After the incipient stage, a spin is considered to be in the developed phase. The developed phase occurs when the airplane’s angular rotation rate, airspeed, and vertical speed are stabilized while in a flight path that is nearly vertical. This is where airplane aerodynamic forces and inertial forces are in balance, and the attitude, angles, and self-sustaining motions about the vertical axis are constant or repetitive. The T-6B will spin either erect (upright) or inverted. Erect spins result from positive-G stall entries. Inverted spins occur from a negative-g stall. The type of spin is independent of aircraft attitude at entry. In case of spatial disorientation during a spin, the pilot must be aware of what the cockpit instrument indications are for each type of spin. The instruments used to confirm a spin are the turn needle, AOA indicator, and airspeed indicator. The turn needle is the only reliable indicator of spin direction. The balance ball (slip indicator) gives no useful indication of spin direction and should be disregarded. The altimeter is monitored to ensure compliance with bailout/ejection criteria. Gauge Spin Indications Remarks Altimeter Rapidly decreasing May indicate up to 1000 feet above actual altitude. AOA 18+ Units (pegged) Stalled Airspeed 120-135 KIAS Stable Turn Needle Pegged in direction of spin Spin rate: 100–170° per sec. VSI 6000 fpm (pegged) 8,000–15,000 fpm Attitude Gyro May be tumbling 60° Nose down Table 5-1 T-6B Indications of an Erect Spin (characterized by nose-down, upright attitude, and positive Gs) An inverted spin is characterized by an inverted attitude and negative Gs on the airplane. Stabilized inverted spins are uncommon because the positioning of the vertical stabilizer in this spin causes the airplane to recover easily. Inverted spins are very disorienting to the aircrew and difficult to enter. The T-6B is prohibited from performing intentional inverted spins. 154 Fundamentals of Aerodynamics: Part 1 – M06-042 Gauge Spin Indications Remarks Altimeter Rapidly decreasing May indicate up to 1000 feet above actual altitude. AOA 0 units (pegged) Stalled Airspeed 40 KIAS Stable Turn Needle Pegged in direction of spin Spin rate: 120° per sec. VSI 6000 fpm (pegged) Approx 9000 fpm Attitude Gyro May be tumbling 30° Nose down Table 5-2 T-6B Indications of an Inverted Spin (characterized by nose-down, upside down attitude, and negative Gs) A flat spin is characterized by a flat attitude and transverse or “eyeball out” Gs. Since the relative wind is from directly below the airplane, the control surfaces are ineffective. The cockpit indications will be similar to an erect spin, except airspeed may vary depending on how flat the spin is. The T-6B is aerodynamically incapable of entering a flat spin. 155 Fundamentals of Aerodynamics: Part 1 – M06-042 Recovery Phase The spin recovery is the most positive recovery available and is 100% effective when properly applied. 1. Gear, flaps, and speed brake – Retracted 2. PCL – IDLE 3. Rudder – Full opposite to turn needle deflection 4. Control stick – Forward of neutral with ailerons neutral 5. Smoothly recover to level flight after spin rotation stops If the pilot is hanging in the straps, he is in an inverted spin. Inverted spin direction is hard to determine visually, so the turn needle should be referenced. Recovery instructions can be found in the T-6 Flight Manual (NATOPS). The spin will recover to a steep, inverted, nose down dive. Roll or split-S out of the dive to level flight in a timely manner as airspeed will build rapidly.
If proper recovery procedures are not followed, a progressive or aggravated spin could result. A progressive spin will result if, during the recovery phase, the pilot puts in full opposite rudder but inadvertently maintains full aft stick. After one or two more turns in the initial spin direction, the nose will pitch steeply down and the airplane will snap into a reversed direction of rotation. The spin reversal is disorienting and significantly more violent than a normal erect spin entry. No matter how disorienting and violent the entry may be, remember to look at the turn needle to determine spin direction. To recover, apply full rudder opposite the turn needle and stick slightly forward of neutral. When rotation stops, the horizon should be referenced to maintain a wings level attitude during the pull-out. An aggravated spin is caused by maintaining pro-spin rudder while moving the control stick forward of the neutral position. Neutralizing rudder while advancing the stick may also be sufficient to enter an aggravated spin. Aggravated spins are characterized by a steep nose-down pitch attitude (approximately 70° nose down) and an increase in spin rate (approximately 280° per second). In addition, aggravated spins tend to induce severe pilot disorientation. Recovery procedures from an aggravated spin are the same as from a progressive spin. 156 Fundamentals of Aerodynamics: Part 1 – M06-042 ASSIGNMENT SHEET 2-5-3 SPINS REVIEW A.
INTRODUCTION This lesson covers fundamental information on the aerodynamic causes and characteristics of aircraft spins. At the completion of this lesson, you will be able to define and state the causes of a spin and describe the aerodynamic factors affecting spin development.
B.
ENABLING OBJECTIVES C.
STUDY ASSIGNMENT 1.
Review Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 5 2.
Read Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 6 D.
STUDY QUESTIONS 1. What two conditions must be present in order for an aircraft to spin?
2. How does the introduction of yaw affect the lift and drag of a stalled wing?
3. What conditions must be present to enter an inverted spin?
4. What is the main aerodynamic factor affecting spins?
5. What causes an accelerated spin?
6. In a right spin, which rudder position will provide the greatest anti-spin forces?
7. How can a progressive spin be entered?
157 Fundamentals of Aerodynamics: Part 1 – M06-042 8. Which aircraft will enter a spin slower, with less oscillation, and take longer to recover?
9. Which aircraft will spin at a lower pitch attitude and higher rotation rate?
158 Fundamentals of Aerodynamics: Part 1 – M06-042 Answers: 1. Stall and yaw 2. Causes asymmetrical lift and drag between the outside and inside wing 3. Negative G stall and yaw introduced 4. Conservation of angular momentum 5. Spinning with the control stick anywhere other than full aft 6. Full left rudder 7. By maintaining full aft stick while applying and holding anti-spin rudder 8. 6000 pound aircraft 9. Right spin a. Stabilizes at lower pitch b. Stabilizes more slowly with increased oscillations c. Rotation rate increased 159 Fundamentals of Aerodynamics: Part 1 – M06-042 OUTLINE SHEET 2-6-1 WAKE TURBULENCE AND WIND SHEAR A.
INTRODUCTION This lesson discusses the causes of, hazards created by, and procedures to be followed for wake turbulence and wind shear. When the lesson is completed, you will have a basic understanding of these flying hazards and the procedures used to avoid them.
B.
ENABLING OBJECTIVES 2.187 DESCRIBE wake turbulence, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.188 DESCRIBE the effects of changes in weight, configuration, and airspeed on wake turbulence intensity, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.189 DESCRIBE the effects of wake turbulence on aircraft performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.190 STATE the takeoff and landing interval requirements for the T-6B, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.191 DESCRIBE procedure for wake turbulence avoidance during takeoff, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.192 DESCRIBE procedure for wake turbulence avoidance during landing, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.193 DEFINE wind shear, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.194 STATE the conditions that will lead to an increasing performance wind shear, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.195 STATE the conditions that will lead to a decreasing performance wind shear, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG- 200 2.196 DESCRIBE the effects of wind shear on aircraft performance, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 160 Fundamentals of Aerodynamics: Part 1 – M06-042 2.197 DESCRIBE procedures for flying in and around wind shear, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 2.198 DESCRIBE wind shear avoidance techniques, in a classroom, in accordance with Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 C.
TOPIC OUTLINE 1.
Introduction 2.
This Lesson Topic 3.
Wake Turbulence 4.
Wind Shear 5.
Summary and Review 6.
Application 7.
Assignment 161 Fundamentals of Aerodynamics: Part 1 – M06-042 INFORMATION SHEET 2-6-2 WAKE TURBULENCE AND WIND SHEAR A.
INTRODUCTION This lesson discusses the causes of, hazards created by, and procedures to be followed for wake turbulence and wind shear. When the lesson is completed, you will have a basic understanding of these flying hazards and the procedures used to avoid them.
B.
REFERENCES 1.
Aerodynamics for Naval Aviators, NAVAIR 00-80T-80 2.
Introduction to the Aerodynamics of Flight, NASA SP-367 3.
T-6 Joint Primary Pilot Training, AETC / CNATRA Syllabus P-V4A-J NATOPS Flight Manual C.
INFORMATION Wake Turbulence The spanwise airflow that moves around the wingtip does more than just create induced drag, it also creates wingtip vortices. Wingtip vortices are spiraling masses of air that are formed at the wingtip when an airplane produces lift (Figure 6-1). This disturbance is often called “jetwash” or “wake turbulence”. Flying into another aircraft’s wing vortex can lead to a variety of dangerous situations including structural damage. Vortices may instantly change the direction of the relative wind and cause one or both wings of the trailing airplane to stall, or disrupt airflow in the engine inlet inducing a compressor stall. The strength of a vortex depends on three main factors: airplane weight, airplane speed, and wing shape. To maintain level flight, a heavier airplane must produce more lift, and will therefore have a greater pressure differential at the wingtip where the vortex is created. Because weight is the most significant factor in the strength of wingtip vortices, the FAA has divided aircraft into three weight classes: Small aircraft (up to 41,000 lbs), large aircraft (41,000 to 255,000 lbs), and heavy aircraft (255,000 lbs or more). The FAA requires heavy aircraft to use the word “heavy” in terminal area radio communications. Vortex strength has a direct correlation to induced drag, the greater the induced drag the stronger the vortex. Since induced drag is dominant at lower Figure 6-1 Wingtip Vortices airspeeds, a slower aircraft will have stronger vortices. Also, a faster aircraft will spread the vortices 162 Fundamentals of Aerodynamics: Part 1 – M06-042 energy over a greater distance, reducing the effect of the vortex. Configuration also plays a significant role in vortex strength. If the flaps are lowered, more lift is created at the wing root, which decreases the pressure differential at the wingtip. The greatest vortex strength occurs when the generating airplane is heavy, slow, and clean. It is important to note that all aircraft, regardless of size, speed, or configuration generating lift produce a vortex hazard. The diameter of the core of the vortex is about one fourth of the generating aircraft’s wingspan. As vortices trail behind the aircraft, they remain within about three-fourths of the generating aircraft’s wingspan. Because of the close proximity, the vortices tend to merge creating a larger field of influence called the wake turbulence zone. The wake turbulence zone is the region behind the aircraft containing the trailing vortices, and the region between them. They sink at a rate of 400 to 500 feet per minute and level off about 900 feet below the flight path of the generating airplane. Vortices will lose strength and break up after a few minutes. Atmospheric turbulence will accelerate this breakup. Hazards The primary hazard to aircraft is loss of control caused because of induced roll. Wing vortices generate sufficient airflow to exceed the roll control capability of an airplane flying into the vortex. In cases where the wingspan and ailerons of the encountering airplane extend beyond the rotational flow of the vortex counter control is usually effective against induced roll. It is more difficult for airplanes with short wingspans (compared with the vortex generating airplane) to counter the induced roll (Figure 6-2). Pilots of short wingspan airplanes, even of the high performance type, must be especially alert to vortex encounters. The most significant factor affecting your ability to counteract the roll induced by the vortices is the relative wingspan between the two airplanes. Figure 6-2 Figure 6-3 Along with induced roll, a second hazardous condition exists called the induced flow field. An induced flow field is created by the interactions of both vortices resulting in a downwash, between the vortices, of up to 1500 feet per minute. This can be disastrous to an aircraft that are already descending at a low power setting (Figure 6-3). 163 Fundamentals of Aerodynamics: Part 1 – M06-042 Helicopters also create unique hazards in the airfield environment. When a helicopter is in a hover it creates a tremendous downwash similar to other prop/jet blasts from aircraft. Because of the extreme downwash, small aircraft should avoid operating within 3 rotor diameters of any hovering helicopter (Figure 6-4). When a helicopter is in forward flight, it produces twin vortices similar to wingtip vortices. Therefore, pilots should give helicopters the same spacing consideration as conventional airplanes of similar size and weight. Figure 6-5 Lateral Vortex Movement Figure 6-4 Avoidance Most accidents involved with wake turbulence occur during takeoff and landing operations, due to the limited altitude with which to recover. Since vortices are a by-product of lift, they are generated from the moment an airplane rotates for takeoff until the airplane nosewheel touches down for landing. Once the vortex comes in contact with the ground, it will move laterally outward at about 5 knots. A crosswind of 4 to 6 knots will cause the upwind vortex to remain stationary while the downwind vortex will move at a rate of 6 to 10 knots. This may result in the upwind vortex remaining in the touchdown zone, and the downwind vortex drifting over a parallel runway (Figure 6-5). Use caution when operating on parallel runways less than 2,500 feet apart. The most important technique for dealing with wing tip vortices/wake turbulence is avoidance. To help avoid unexpected encounter with vortices created by landing and departing aircraft, the T-6B’s minimum takeoff spacing requirement is 2 minutes behind a heavy aircraft (over 255,000 pounds). The same spacing is recommended for large aircraft (41,000-255,000 pounds) as well. For landing, the minimum landing spacing requirement is 3 minutes behind a heavy aircraft (over 255,000 pounds). 164 Fundamentals of Aerodynamics: Part 1 – M06-042 In addition to the minimum spacing requirements, pilots can adjust their rotation and landing points to avoid wake turbulence. Because wingtip vortices are not created until the departing aircraft’s nosewheel lift from the runway, there are no hazards associated with wake turbulence before the liftoff point. Upon landing, once the nosewheel contacts the runway, the hazard ceases as well. When taking off behind a larger airplane that is departing ahead of you, ensure your rotation is complete at least 300 feet prior to the larger airplane’s point of rotation and conduct your climb-out to remain above his flight path (Figure 6-6C). If departing after a larger aircraft has landed, plan to rotate at a point forward of where the larger aircraft’s nosewheel touched down (Figure 6-6D). These techniques will ensure you will not be climbing through any wake turbulence after takeoff. When landing behind a larger airplane, besides observing the minimum spacing requirement, stay at or above the larger airplane’s final approach path and land beyond its nosewheel touchdown point (Figure 6-6A). If a larger airplane performs a touch-and-go or low approach, observe the required spacing interval. When landing behind a larger aircraft that has just departed, ensure that your touchdown point is prior to the larger aircraft’s rotation point (Figure 6-6B). This will ensure that you avoid flying through wake turbulence on short final or in the flare.
Figure 6-6 Vortex Avoidance During formation flying and in-flight refueling, airplanes close to one another produce a mutual interference especially when the trailing airplane is slightly aft and below the lead airplane. The leading airplane experiences an effect that is similar to ground effect because of a reduction in downwash and induced drag. For the second airplane, this mutual interference of the flow pattern can instantaneously alter the direction of the relative wind that the airfoils are sensing. Flying through lead’s flightpath will place you in wake turbulence, which could result in an over-G or a flameout. 165 Fundamentals of Aerodynamics: Part 1 – M06-042 When operating on intersecting runways, or parallel runways within 2500 feet of each other, you must be alert to adjacent large aircraft operations, particularly upwind of your runway (Figure 6-5). A large aircraft executing a takeoff, low approach, or landing can severely affect your runway and flight path. Adjust your takeoff or landing point to avoid possible wake turbulence and assure that an interval of at least 2 minutes has elapsed before you attempt to takeoff or land. Wind Shear Wind shear is defined as a sudden change in wind direction and/or speed over a short distance in the atmosphere. Wind shear boundaries can be vertical or horizontal and vary in intensity. Weak shears distribute the wind change from one body of air to the other gradually. A pilot flying through this type of shear may not even notice a change in aircraft performance. On the other hand, strong shears distribute the wind abruptly creating rapid changes in aircraft performance. Wind shear is most often caused by jet streams, land or sea breezes, fronts, inversions and thunderstorms. Wind shears can be very complex combinations of wind velocities. Usually the more complex the wind shear, the more difficult it is for the pilot to react correctly. To simplify things we will limit our discussion in this section to horizontal wind shears so that we may gain a basic understanding of how wind shear will affect aircraft performance. Wind shears change airflow over the aircraft. The velocity of the relative wind can be altered causing immediate changes in the indicated airspeed and/or angle attack of the aircraft. Once the aircraft is stabilized in the body of air it behaves as if nothing happened. You have probably experienced this effect while riding on a moving sidewalk or escalator. As you step onto these moving surfaces you feel a little unstable for a few seconds. Shortly thereafter, you stabilize and function normally. The only difference is your “groundspeed” is now a little faster. If we confine ourselves to the horizontal plane, we can say wind shear either causes an increases or decrease in aircraft performance. With ample airspeed and altitude, wind shear does not pose a serious threat. However during slow airspeed and low altitude operations, such as during takeoffs and landings, wind shear becomes hazardous. Wind Shear During Takeoff Increasing Performance Wind Shear: Figure 6-7 shows an aircraft passing through a wind shear which increases headwind by 20 knots resulting in an increase of dynamic pressure by 20 knots. Because IAS is directly proportional to dynamic pressure, IAS will instantly increase by 20 knots. The increase in IAS results in an increase in lift and therefore causes an initial increase in performance. Also, a 20 knot increase in headwind will reduce our ground speed by 20 knots, resulting in a steeper angle of climb. As long as a proper climb attitude is maintained, a wind shear with an increasing headwind component on takeoff does not pose a serious threat. 166 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 6-7 Increasing Performance Wind Shear on Takeoff Decreasing Performance Wind Shear: Figure 6-8 shows an aircraft entering a shear which decreases the indicated airspeed by 25 knots. This causes a significant decrease in performance. The drop in dynamic pressure reduces lift and also results in a shallower angle of climb. A rapid drop of airspeed can place an aircraft near stall speed. An increase in angle of attack in this situation will probably result in an approach to stall indication and possibly a stall. This example illustrates how shear can be dangerous during a decreasing performance wind. Figure 6-8 Decreasing Performance Wind Shear on Takeoff Wind Shear During Landing An aircraft established on a glidepath for landing is usually trimmed for a constant airspeed descent. Any change in indicated airspeed will cause a change in pitch due to trim and a change in the rate of descent. The pilot will have to make quick control inputs to maintain the desired glidepath. Increasing Performance Wind Shear: Figure 6-9 shows an aircraft descending through a shear which increases its indicated airspeed. Notice that the transition from a tailwind to zero wind causes an increase in performance. This shear increases lift and causes the aircraft to pitch up and rise above the glidepath. If the pilot is slow to recognize the wind shear, a steep fast approach can quickly develop. This may result in significantly longer landing distances, a hazard that will vary depending on the runway environment (runway length, icy, wet, etc.). An overly aggressive correction can lead to a high rate of descent with a dangerously low power setting. 167 Fundamentals of Aerodynamics: Part 1 – M06-042 Figure 6-9 Increasing Performance Wind Shear on Landing For the purpose of comparing relative power settings, assume a hypothetical thrust scale of from 1 to 10, and that the normal no-wind power setting is 6. Because of the tailwind (higher groundspeed) above the shear, the pilot needs a power setting of 5 to maintain the glidepath. When the aircraft crosses the shear and begins to rise above glidepath, the pilot reduces power to 4. As the aircraft returns to glidepath, a power setting of 6 will be required to maintain the glidepath. This is due to the lower rate of descent required by a slower groundspeed. Notice that a higher power setting is required after the shear than before the shear to maintain the glidepath. In other words, you must eventually add more power than was removed to stabilize on the glidepath. If too much power is removed, combined with an overly aggressive correction back to the original glidepath, there may be insufficient altitude for the pilot to recover, resulting in landing short of the runway (Figure 6-10). If a strong wind shear is encountered at low altitudes a wave-off/go-around can be executed if necessary. Figure 6-10 Increasing Performance Wind Shear on Landing Resulting in Landing Short Decreasing Performance Wind Shear: A decreasing performance wind shear on landing is one of the most hazardous situations a pilot can encounter. Figure 6-11 shows an aircraft descending through a shear which decreases its indicated airspeed. This shear causes the aircraft to pitch down and descend below the glidepath. The pilot counters this by adding power and raising the nose. However, the pilot may over correct and rise above the glidepath. Once back on glidepath, a lower power 168 Fundamentals of Aerodynamics: Part 1 – M06-042 setting will be required to compensate for the higher ground speed and new rate of descent within the new body of air. Figure 6-11 Decreasing Performance Wind Shear on Landing Again assuming a thrust scale of from 1 to 10, we can examine the power requirements during this type of shear. Let’s assume the normal no-wind power setting of 6. Because of the headwind above the shear, the pilot needs a power setting of 7 to maintain the glidepath. Then the aircraft crosses the shear and begins to descend below glidepath, the pilot increases power to 8 or 9. As the aircraft returns to glidepath, a power setting of 6 will be required to maintain it. Again, this is because of the higher rate of descent dictated by a higher groundspeed. The point to remember is that you must eventually reduce power by more than the amount added to stabilize on the glidepath. If a strong decreasing performance wind shear is encountered at very low altitude, or if a pilot is slow to recognize the situation, there may be insufficient time and power to overcome the resulting loss of lift (Figure 6-12) and landing short of the runway. 169 Fundamentals of Aerodynamics: Part 1 – M06-042 Lift Figure 6-12 Decreasing Performance Wind Shear on Landing Resulting in a Landing Short Wind Shear Avoidance The best technique for dealing with wind shear is to avoid it. If a moderate to strong wind shear is expected, delay your takeoff or landing until the shear condition no longer exists. Anytime wind shear is experienced, pilots should consider going around. If airborne and unable to delay, consider diverting to a place with more favorable conditions. Strong shears like those associated with microburst activity must be avoided at all cost. An aircraft encountering a microburst will initially experience a strong, increasing performance wind shear. This will cause the aircraft to pitch up and climb. A pilot’s natural reaction is to apply nose down stick. Soon after, the aircraft will experience a strong, decreasing performance wind shear. Combined with the nose-down stick applied by the pilot, the decreasing performance wind shear can result in impacting the terrain. These wind shears can be anywhere from 50 knots to 200 knots and can occur in a relatively short period and severely impair an aircraft’s ability to maintain controlled flight, especially at low altitude. Downdrafts exceeding 700 feet per minute have also been measured. Methods of Wind Shear Detection Because wind shear is such a dangerous phenomenon, early detection is vital to mishap prevention. In most wind shear accidents there have been warning signs that were ignored, misinterpreted or misunderstood. You must evaluate the warning signs and make a decision quickly and decisively. Fortunately, there are several methods to help pilots detect and avoid wind shear. Departure or, arrival weather reports calling for gusty winds, heavy rain or thunderstorms should be a clue that a high potential for microburst activity exists. When you receive your pre-flight briefing or call ahead for an en route update, be alert for any convective activity that might be forecast or building. It is important to remember that weather changes very quickly. The best briefing may not prepare you for every situation that you may encounter. Visual cues are very important because they help substantiate the information given by the weather guesser. In fact, in many fatal wind shear mishaps the pilot continued the approach or takeoff in 170 Fundamentals of Aerodynamics: Part 1 – M06-042 visible and known thunderstorm conditions. Visual cues include virga, localized blowing dust (especially in circular or elliptical patterns), rain shafts with rain diverging away from the core of the cell, and of course an indication of lightning or tornado-like activity. Wind Shear Alert Systems are another source of information about potential wind shear activity. There are several types in operation today at many civilian fields, especially those with a history of strong winds. For example the Low Level Wind Shear Alert System (LLWAS) measures the wind speed and direction at several points on the ground and compares them with a reference sensor located near the center of the airfield. Because of the small diameter of microbursts, many may go undetected. When they are detected, they are on the field and it may be too late. There are also some Doppler radar systems which show greater accuracy in wind shear warnings. A NEXRAD Doppler radar system is a ground based radar that can very accurately detect microburst activity. Systems onboard modern aircraft monitor changes in wind velocity and aircraft acceleration to provide wind shear warning and pitch guidance to help escape wind shear. Many modern commercial and military aircraft that have onboard system to monitor changes in wind velocity and aircraft acceleration to provide warning to the crew. In most onboard systems, the surface winds are loaded by the pilot. The computer is then able to constantly compare current winds with surface winds and alert the aircrew if the difference becomes significant. In more advanced onboard systems, aircraft pitch, airspeed, and other guidance is displayed to help pilots negotiate wind shear. Unfortunately, these systems are not on our trainer aircraft. PIREPS and Weather Alerts are one of the best sources of wind shear information. Because of the short lifespan and localized nature of wind shear and microbursts, PIREPS are sometime the only way to disseminate information in a timely manner. If you encounter wind shear, it is imperative that you make a PIREP to approach control or tower so they may notify other aircrews. Your PIREP should include the location where the shear was encountered, an estimate of its magnitude and most importantly a description of what was experienced, such as turbulence, airspeed gain or loss, glidepath problems, etc. Takeoff and Landing Procedures If wind shear cannot be avoided, here are some T-6B specific procedures to be used in areas of suspected wind shear. For takeoff: Use the longest suitable runway. Consider crosswind, obstacles, runway surface conditions, etc., when selecting the runway. Use takeoff flaps, but delay rotation (VROT) by the amount of predicted wind shear (up to 10 additional knots). Notice, this addition applies to increasing performance wind shear. Rotate to normal climb attitude at increased VROT and maintain attitude. If wind shear is encountered near VROT, abort if possible. For Landing: 171 Fundamentals of Aerodynamics: Part 1 – M06-042 Set flaps to takeoff and increase approach speed by the amount of wind shear potential (up to 10 knots above normal). Again, notice this addition applies to increasing performance wind shear. By setting the flaps to takeoff, there will be less drag and the aircraft will be able to accelerate more quickly. Establish the proper approach pitch, trim, and power settings by 1000 AGL. Resist the temptation to make large power reductions. Keep in mind that increased landing speed means longer landing distances. 172 Fundamentals of Aerodynamics: Part 1 – M06-042 ASSIGNMENT SHEET 2-6-3 WAKE TURBULENCE AND WIND SHEAR REVIEW A.
INTRODUCTION This lesson discusses the causes of, hazards created by, and procedures to be followed for wake turbulence and wind shear. When the lesson is completed, you will have a basic understanding of these flying hazards and the procedures used to avoid them.
B.
ENABLING OBJECTIVES C.
STUDY ASSIGNMENT 1.
Review Naval Aviation Fundamentals, NAVAVSCOLSCOM-SG-200 Unit 2, Chapter 6 D.
STUDY QUESTIONS 1. When viewed from behind the generating aircraft, wingtip vortices rotate from the right wing and from the left wing.
a. clockwise; clockwise b. counterclockwise; clockwise c. counterclockwise; counterclockwise d. clockwise; counterclockwise 2. Wake turbulence is generated . a. only by heavy aircraft b. only by helicopters at hover c. only when an aircraft’s flaps are extended d. by any aircraft whenever it is producing lift 3. When taking off behind a heavy aircraft in the T-6B, you should allow for a minimum spacing of a. 1 b. 2 c. 3 d. 4 minute(s).
4. When taking off behind a heavy aircraft that has just landed, what should you do to minimize the effects of its wake turbulence?
a. Plan to lift off before its nose gear touchdown point b. Plan to lift off after its nose gear touchdown point 173 Fundamentals of Aerodynamics: Part 1 – M06-042 5. Key hazards created by wake turbulence are a. low visibility; reduced engine performance b. slower climb rates; adverse yaw c. induced roll; induced flow field d. longer takeoff distances; induced pitch changes and . 6. The wake turbulence generated by a heavy aircraft taking off from a parallel or intersecting runway is not a factor for aircraft operating on another runway.
a. True b. False 7. How will a wind shear that decreases your headwind component by 20 knots affect your aircraft?
a. Indicated airspeed will decrease 20 knots thereby decreasing lift.
b. Indicated airspeed will increase 20 knots thereby increasing lift.
c. Indicated airspeed will decrease 20 knots thereby increasing lift.
d. There would be no appreciable effect.
8. During an approach, you pass through a shear that changes the wind from a 10 knot tailwind to a 10 knot headwind. How will this shear affect your aircraft?
a. Indicated airspeed will increase by 10 knots, lift will decrease, and aircraft will tend to descend below glidepath.
b. Indicated airspeed will increase by 20 knots, lift will decrease, and aircraft will tend to descend below glidepath.
c. Indicated airspeed will increase by 20 knots, lift will increase, and aircraft will tend to climb above glidepath.
d. Indicated airspeed will decrease by 10 knots, lift will decrease, and aircraft will tend to climb above glidepath.
9. What wind changes would an aircraft initially encounter if flown through a microburst?
a. A strong headwind b. A strong tailwind 10. An aircraft reported wind shear on final with a 10 knot loss of airspeed. How should you adjust your approach?
a. Subtract 10 knots from your approach speed.
b. Set flaps to “Takeoff” and maintain your normal approach speed.
c. Delay landing for 2 minutes.
d. Set flaps to “Takeoff” and add 10 knots to your approach speed.
174 Fundamentals of Aerodynamics: Part 1 – M06-042 ADDITIONAL REVIEW QUESTIONS 1. What is the definition of a stall?
2. What are the two major classifications of stalls?
3. Which type of stall will have lower stall airspeed?
4. What is the only reason the T-6B will stall?
5. What two conditions must be present in order for an aircraft to spin?
6. What causes an accelerated spin?
7. In a right spin, which rudder position will provide the greatest anti-spin forces?
8. Which aircraft will enter a spin slower, with less oscillation, and take longer to recover?
9. When taking off behind a heavy aircraft in the T-6B, you should allow for a minimum spacing of minute(s).
10. When landing behind a heavy aircraft that has just landed, what should you do to minimize the effects of its wake turbulence?
11. How will a wind shear that decreases your headwind component by 20 knots affect your aircraft?
12. An aircraft reported wind shear on final with a 10 knot loss of airspeed. How should you adjust your approach and what things should you look out for?
175 Fundamentals of Aerodynamics: Part 1 – M06-042 Answers: 1.B 6. FALSE 2.D 7. A 3.B 8. C 4.B 9. A 5.C 10. D Answers to Additional Lesson Review Questions: 1. Condition in flight where increase in AOA results in decrease in CL. 2. Power-on and power-off stalls 3. Power-on stall 4. Exceeding stall AOA (18 units) 5. Stall and yaw 6. Spinning with the control stick anywhere other than full aft 7. Full left rudder 8. 6000 pound aircraft 9. Two 10. Delay landing for three minutes. In addition, you can land beyond the landing aircraft’s nose gear touchdown point.
11. Indicated airspeed will decrease 20 knots, thereby decreasing lift.
12. a. Set flaps to “Takeoff” b. Add 10 knots to your approach speed c. Be alert for the tendency to land long 176 Fundamentals of Aerodynamics: Part 1 – M06-042 Glossary absolute altitude The aircraft’s height above the terrain directly beneath the aircraft, measured in feet above ground level (AGL). Absolute altitude is found by subtracting the terrain elevation from the true altitude. absolute ceiling The maximum altitude above sea level in a standard atmosphere that an airplane can maintain level flight. accelerated spin A spin in which the control stick is not held in the full aft position. An accelerated spin is characterized by steeper pitch attitudes and higher spin rates. accelerated stall A stall in which the load factor is greater than one, as in a pullout. Usually more violent and disorienting than a normal stall. accelerated stall line A curved line describing the number of g’s that can be generated at a given indicated airspeed as a function of CLMAX angle of attack for a particular airfoil. Also called line of maximum lift. acceleration A change in the velocity of a body with respect to magnitude or direction, or both. accelerometer An instrument that measures one or more components of the acceleration of a vehicle. adverse pressure gradient A pressure gradient of increasing static pressure in the direction of airflow. adverse yaw Yaw in the opposite direction of aileron roll input. aerodynamic balance The feature of a control surface that reduces the magnitude of the aerodynamic moment around the hinge line. See shielded horn. aerodynamic braking A technique for slowing an airplane to a speed suitable for frictional braking. Aerodynamic braking is accomplished by increasing the surface area exposed to the relative wind in order to increase parasite drag, primarily by holding the nose of the airplane in the landing attitude. aerodynamic center (AC) The point along the chord line of an airfoil where all changes in aerodynamic force effectively take place. It is normally located at the point of 25% chord. aerodynamic force (AF) A force acting on an airfoil that is the result of air pressure and friction distribution over the surface of the airfoil. aerodynamics The science that studies the motion of gaseous fluid flows, and of their actions against and around bodies, and of the forces acting on bodies within that flow. aerodynamic twist Form of wing tailoring that employs a decrease in camber and/or relative thickness from wing root to wingtip. The wing root is more positively cambered and/or thicker (relative to the chord) than the tip, resulting in a root first stall pattern. Also called section variation. aileron A movable control surface, attached to the wing of an airplane, used to produce a rolling moment around the longitudinal axis by creating unequal lifting forces on opposite sides of an airplane. aileron reversal Reversal of the control effect usually produced by an aileron, caused by a moment around the aerodynamic center twisting the wing and changing its angle of attack. 177 Fundamentals of Aerodynamics: Part 1 – M06-042 aircraft (A/C) Any device used or intended to be used for flight in the air. airflow A flow or stream of air. A rate of flow measured by mass per unit time. airfoil A streamlined shape designed to produce lift as it moves through the air. airframe The structural components of an airplane including the framework and skin of such parts as the fuselage, wings, empennage, landing gear, and engine mounts. airplane An engine driven, heavier-than-air, fixed-wing aircraft that is supported by the dynamic reaction of airflow over its wings. altimeter Any instrument for measuring altitude. An instrument similar to an aneroid barometer that uses the change of atmospheric pressure with altitude to indicate the approximate elevation above a given reference. altitude The height of a point, measured from a reference plane, such as mean sea level. ambient Pertaining to the air or air conditions around a flying aircraft but undisturbed or unaffected by it. aneroid barometer An instrument for measuring the pressure of the atmosphere which operates on the principle of having changing atmospheric pressure bend a metallic surface which, in turn, moves a pointer across a scale graduated in units of pressure. angle of attack (AOA, ) The angle formed between the relative wind and the chord line of the airfoil angle of bank (AOB, ) The angle between the horizon and the lateral axis of an aircraft. The angle of lateral displacement (roll) of an aircraft, especially in making a turn angle of climb (AOC, ) The angle between the horizon and the flightpath of a climbing aircraft. angle of descent () The angle between the horizon and the flightpath of a descending aircraft. angle of incidence The angle between the airplane’s longitudinal axis and the chord line of its wing. The root chord is commonly chosen to measure the angle of incidence. angular acceleration Rate of change of angular velocity. anhedral angle A negative dihedral angle. Also called cathedral angle. approach A specified flightpath and associated altitudes to be flown in preparation for a landing, especially a published instrument approach. artificial feel A method of simulating, altering, or otherwise enhancing the feedback or control feel that is transmitted to the cockpit controls by the forces acting on the control surfaces. aspect ratio (AR) The ratio of the wingspan to the average chord. attitude The orientation of an aircraft as determined by the relationship between its axes and some reference line or plane. Usually refers to nose attitude or pitch attitude. 178 Fundamentals of Aerodynamics: Part 1 – M06-042 automatic slot High lift device that consists of a movable vane attached to the leading edge of the wing that moves away from the body of the wing to allow airflow from below the wing to reach the upper surface and reenergize the boundary layer, delaying boundary layer separation. See slat. autorotation During a spin, a combination of roll and yaw that is self-sustaining. average chord (c) The geometric average of every chord from the wing root to the wingtip. Also called mean geometric chord. axis A reference line passing through a body, around which the body rotates. axis system A set of three mutually perpendicular axes, intersecting at the center of gravity of an aircraft, around which the motions, moments, and forces of roll, pitch, and yaw are measured. bank The position or attitude of an aircraft when its lateral axis is inclined from the horizontal. Bernoulli’s Equation PT = PS + q. (After Daniel Bernoulli, 1700-1782, Swiss scientist.) In aerodynamics, a law or theorem stating that in a flow of incompressible fluid, the sum of the static pressure and the dynamic pressure along a streamline is constant if gravity and frictional effects are disregarded. boundary layer The layer of airflow over the surface of an airfoil, which shows local airflow retardation caused by viscosity. The boundary layer is very thin at the leading edge of an airfoil (about 1 mm) and grows in thickness as it moves over a body. It is composed of laminar flow and turbulent flow. boundary layer control (BLC) The control of the airflow within the boundary layer in order to prevent its separation at high angles of attack. See also slot and slat. buffeting The beating, shaking, or oscillation of an aircraft’s structure or surfaces by an unsteady flow, gusts, turbulence, etc. cabin Compartment of an aircraft in which passengers, troops, or cargo are loaded. calibrated airspeed (CAS) Indicated airspeed corrected for instrument error. calibrated altitude Indicated altitude corrected for instrument error. camber The curvature of the mean line of an airfoil from leading edge to trailing edge; the amount of this curvature. cantilever A beam or object supported only at or near one end, or one point; without external bracing. cathedral See anhedral. center of gravity (CG) The point at which the weight of an object is considered to be concentrated. chord A measure of the chord line from the leading edge to the trailing edge of an airfoil. The chord may vary in length from the wingtip to wing root. The root chord, cR, is the chord at the wing centerline and the tip chord, cT, is measured at the wingtip. chord line An infinitely long, straight line drawn through the leading and trailing edges of an airfoil. chordwise flow Airflow perpendicular to the leading edge of an airfoil; airflow along the chord of an airfoil. Since chordwise flow is accelerated over a wing, it produces lift. 179 Fundamentals of Aerodynamics: Part 1 – M06-042 cockpit Compartment of an aircraft in which the flight crew, especially the pilot(s), are located. The cockpit is where the aircraft is controlled from. coefficient of aerodynamic force (CF) The dimensionless portion of the aerodynamic force that is a function of angle of attack, camber, aspect ratio, compressibility, and viscosity. coefficient of drag (CD) The dimensionless portion of the total drag on an airfoil that is dependent on the same variables that affect CF. coefficient of friction () A dimensionless number whose value depends primarily on the type of material and condition of the two surfaces that are in contact. coefficient of lift (CL) The dimensionless portion of the total lift on an airfoil that is dependent on the same variables that affect CF. compressibility The property of a substance that allows its density to increase as pressure increases. compressible flow Flow at speeds sufficiently high that density changes in the fluid can no longer be neglected. constant-speed propeller A propeller designed to maintain engine speed at a constant RPM, automatically increasing or decreasing pitch as engine speed tends to increase or decrease. continuity equation 1A1V1 = 2A2V2. Principle of physics that states that for fluids, the mass flow rate has the same value at every position along a closed tube. control feel The feel or impression of the stability and control of an aircraft that a pilot receives through the cockpit controls, either from aerodynamic forces acting on the control surfaces or from devices simulating these aerodynamic forces. control force A force, either aerodynamic or pilot induced, acting on a control surface. control horn A short lever or rigid post attached to a control surface, to which a control cable, wire, line, or rod is attached. controllability The capability of an aircraft to respond to control inputs, especially in direction or attitude. control stick A lever for controlling the movements of an aircraft in flight. On a fixed-wing airplane, the control stick operates the elevators by a fore-and-aft movement and the ailerons by a side-to-side movement. control surface A movable airfoil or surface, such as an aileron, elevator, rudder, or spoiler used to control the attitude or motion of an airplane and to guide it through the air. cosine (cos) In a right triangle, the function of an acute angle that is the ratio of the length of the adjacent side to the length of the hypotenuse. creep The gradual reduction in a material’s strength over time due to high temperature and stress. Also known as plastic deformation. critical altitude The maximum altitude at which, in the standard atmosphere, an engine produces its sea level rated horsepower or torque. 180 Fundamentals of Aerodynamics: Part 1 – M06-042 critical Mach number (MCRIT) The free airstream Mach number that produces the first evidence of local sonic flow. crosswind A wind blowing across the flightpath of an airplane. density () Mass per unit volume. density altitude (DA) Density altitude is pressure altitude corrected for nonstandard temperature. Density altitude is the pressure altitude on a standard day that has the same density as the ambient air. dihedral angle The angle between the spanwise inclination of a wing and the lateral axis. It is the upward slope of the wings when viewed from head on. A negative dihedral is called anhedral. directional control Control of the longitudinal axis around the vertical axis; yaw control. directional divergence A departure from equilibrium around the vertical axis caused by negative directional static stability. Condition of flight in which the reaction to a small initial sideslip is an increase in sideslip angle. This would result in the airplane yawing broadside to the relative wind. directional moment A moment created around an aircraft’s vertical axis. directional stability The stability of an aircraft around its vertical axis. The reaction of an aircraft to a sideslip. dive A steep descent, usually power on. downwash Chordwise airflow from the upper surface of an airfoil passing downward behind the trailing edge to the lower surface. Downwash decreases the amount of lift produced by the wing. Any downward moving airflow. drag (D) That component of the aerodynamic force acting parallel to, and in the same direction as the relative wind. It acts as a retarding force. Dutch roll Dynamic stability that is the result of strong lateral and weak directional static stability. An airplane prone to Dutch roll would appear to describe a figure eight on the horizon and would tail wag. dynamic pressure (q) The pressure of a fluid resulting from its motion, equal to one half the density times the velocity squared (q=1/2V2). dynamic stability The oscillatory motion of a body, beyond its initial tendency to move toward or away from equilibrium, after a disturbance. A measure of displacement with respect to time. elastic limit The maximum load that may be applied to a component without permanent deformation. elevator A control surface, attached to a horizontal stabilizer that produces a pitching moment around the airplane’s lateral axis. empennage The assembly of stabilizing and control surfaces at the tail of an airplane. endurance The length of time that an aircraft can fly under specified conditions without refueling. energy The ability or capacity to do work, expressed in foot-pounds. equilibrium Flight condition that exists when the sum of the forces and moments acting around the center of gravity equal zero. The absence of linear or angular acceleration. 181 Fundamentals of Aerodynamics: Part 1 – M06-042 equivalent airspeed (EAS) The true airspeed at sea level on a standard day that produces the same dynamic pressure as the actual aircraft condition. It is equal to calibrated airspeed corrected for the compressibility of air. equivalent parasite area (f) The total surface area of an airplane that contributes to parasite drag. Normally less than cross sectional area due to the effects of streamlining. erect spin A spin characterized by positive Gs and an upright attitude. fatigue failure The breaking (or serious permanent deformation) of a material due to a cyclic application of load or force. fatigue strength A measure of a material’s resistance to a cyclic application of load or force. feathered propeller A propeller whose blades have been rotated so that the leading and trailing edges are nearly parallel with the aircraft flightpath to minimize drag and to stop propeller rotation. feedback The transmission of forces initiated by aerodynamic action on control surfaces to the cockpit controls. The actual forces transmitted to the cockpit controls. fence A stall fence. final / final approach That portion or leg of an approach pattern after the last turn, in which the aircraft is in line with the runway in the landing direction. finite wing A wing with a finite span; a wing with wingtips. fixed slot A slot that remains open at all times. flap A high lift device consisting of a hinged, pivoted, or sliding airfoil or plate, or a combination of such objects regarded as a single surface, extended or deflected for increasing camber. Used primarily to decrease the takeoff or landing velocity. flat spin A spin characterized by transverse Gs and an attitude flatter than an erect spin. flightpath (FP) The path described by an airplane’s center of gravity as it moves through an air mass. flow separation The breakaway of the boundary layer airflow from a surface; the condition of a flow separated from the surface of a body and no longer following its contours. flutter A vibration or oscillation of a control surface or wing created and maintained by aerodynamic forces and the elastic and inertial forces of the object itself. force A vector quantity equal to the push or pull exerted on a body. By Newton’s Second Law, a force is a function of an acceleration and the mass of the body. form drag Drag resulting from airflow over a surface with some frontal area, often referred to as pressure drag, profile drag, or plate drag. fowler flap A high lift device that consists of a sliding airfoil attached to the trailing edge of a wing that increases camber, wing area, and uses BLC to increase the CL. friction Resistance due to the rubbing of one body or substance against another. Air friction results from the viscosity of the air, or its tendency to stick to a surface. 182 Fundamentals of Aerodynamics: Part 1 – M06-042 friction drag Drag arising from friction forces at the surface of an aircraft, due to the viscosity of the air. fuel flow The rate of fuel being consumed by an aircraft’s engine. fuselage The main structural component of an airplane. full-cantilever Supported at one point only, as in a full-cantilever wing, or a wing that is entirely internally supported, with no external bracing. G (gravitational acceleration) A constant, equal to 32.2 ft/sec2, representing the acceleration on an object due to the Earth’s gravity. General Gas Law P = R T. Law of physics that shows the relationship between properties of air: pressure (P), density (), and temperature (T). R is a constant for any given mixture of gases (such as dry air). geometric twist Form of wing tailoring that employs a decrease in the angle of incidence from the wing root to the wingtip. The wing root has a higher angle of incidence than the wingtip, causing it to stall first. glide A shallow descent, usually associated with power off flight. glide endurance (GE) The maximum time that an airplane can stay airborne in a glide as a function of weight, altitude, and angle of attack. glide range (GR) The maximum distance that can be traveled in a glide as a function of altitude, wind, and lift to drag ratio. glide ratio The ratio of the horizontal distance traveled to the vertical distance descended in a glide. Glide ratio is equal to the lift to drag ratio. gross weight The total weight of a fully loaded aircraft. ground effect The dramatic reduction of induced drag and thrust required that occurs within one wingspan of the ground or other surface. ground speed (GS) An airplane’s actual speed over the ground. gust A sudden and brief change of wind speed or direction. gust load A load imposed upon an aircraft or aircraft member by a gust. gyroscopic precession The resultant action or deflection of a spinning disc when a force is applied parallel to its axis. The resultant force occurs 90 ahead in the direction of rotation, and in the direction of the applied force. headwind A wind blowing from directly ahead, or blowing from a forward direction such that its principal effect is to reduce ground speed. helicopter A rotorcraft that, for its horizontal motion, depends principally on its engine driven rotors. high lift device Any device, such as a flap, or boundary layer control device, used to increase the lift of a wing by increasing the CL or area of the wing. The result is a reduction of takeoff and landing speeds. Increases in CL are achieved by increasing the camber of an airfoil, or by controlling the kinetic energy in the boundary layer. 183 Fundamentals of Aerodynamics: Part 1 – M06-042 hinge line The transverse axis around which a control surface moves. horizontal stabilizer The entire horizontal part of an airplane’s empennage comprising both fixed and movable surfaces. On most airplanes, the horizontal stabilizer is the greatest contributor to longitudinal stability. horsepower A unit of power equal to 550 ft-lbs/sec or 33,000 ft-lbs/min. humidity The amount of water vapor in the air. hypersonic Movement or flow at very high supersonic speeds, generally at a Mach number of 5 or greater. indicated airspeed (IAS) The instrument indication for the amount of dynamic pressure that the aircraft is creating during some given flight condition. Indicated airspeed is displayed in knots, abbreviated KIAS. indicated altitude The indication on a pressure altimeter when the Kollsman window is set to the current local altimeter setting. induced drag (DI) That portion of total drag resulting from the production of lift. infinite wing A wing with no wingtips; used in discussing airflow around an airfoil in ideal situations. interference drag Drag caused by the mixing of streamlines around aircraft components due to their proximity. It is a form of parasite drag. inverted spin A spin characterized by negative Gs and an inverted attitude. isothermal layer The layer of the atmosphere from approximately 36,000 through 66,000 feet, in which the air remains at a constant temperature of −56.5C. kinetic energy (KE) The ability of a body to do work because of its motion. laminar flow The portion of the boundary layer airflow that is smooth and unbroken and travels along well defined streamlines. laminar flow wing An airfoil specially designed to maintain a laminar flow boundary layer. lateral axis An axis going through an airplane’s center of gravity from side to side (wingtip to wingtip). Any movement developed around this axis is called pitch. lateral control Control of the lateral axis around the longitudinal axis; roll control. lateral moment A moment created around an airplane’s longitudinal axis. lateral stability The stability of an aircraft around its longitudinal axis. The reaction of an aircraft to an angle of bank. leading edge flaps A high lift device consisting of a hinged portion of the leading edge of a wing that moves down to increase the wing’s camber. leading edge radius The radius of a circle tangent to the leading edge, upper and lower surfaces of the airfoil. lift (L) The component of the aerodynamic force acting perpendicular to the relative wind. lift to drag ratio (L/D) The ratio of lift to drag, obtained by dividing the coefficient of lift by the coefficient of drag. A measure of the wing’s efficiency. The L/D ratio is also used as the glide ratio. 184 Fundamentals of Aerodynamics: Part 1 – M06-042 lift to drag ratio-maximum (L/DMAX) The greatest ratio of lift to drag. L/DMAX AOA is the most efficient AOA for that airfoil. limit airspeed See redline airspeed. limit load The maximum load factor an airplane can sustain without any possibility of perma- nent deformation. It is the maximum load factor anticipated in the normal operation of the airplane. linear acceleration Acceleration along a line or axis. load A stress-producing force. load factor (n) The ratio of the load applied by an airplane’s lift to the load applied by its weight. It is a multiple of the acceleration of gravity, commonly called “Gs.” local speed of sound The speed at which sound travels in a given medium under local ambient conditions. longitudinal axis An axis extending from the nose to the tail of an aircraft, passing through its center of gravity. Any movement developed around this axis is called roll. longitudinal control Control of the longitudinal axis around the lateral axis; pitch control. longitudinal moment A moment created around an airplane’s lateral axis. longitudinal stability The stability of an aircraft around the lateral axis. The reaction of an aircraft to changes in pitch. Mach number (M) (Pronounced “mock,” after Ernest Mach (1838-1916), Austrian scientist.) The ratio of the true airspeed of an object moving through the air to the local speed of sound in that air. maneuverability The ability of an airplane to readily alter its flightpath. The ease with which an airplane moves out of equilibrium. maneuver point The point on the V-n diagram at the intersection of the positive accelerated stall line and the positive limit load. It is the point where the limit load may be achieved without the possibility of overstress, or the lowest airspeed that the limit load is encoun- tered. maneuver speed (Va) The indicated airspeed that an airplane can achieve its maximum turn rate and minimum turn radius. The slowest velocity that an airplane can generate its limit load. It is usually the recommended turbulent air penetration airspeed. mass (m) The quantity of molecular material that comprises an object. mass balance The feature of a control surface that reduces the magnitude of the inertial and gravitational moments around the hinge line. mean aerodynamic chord (MAC) The chord of an imaginary rectangular airfoil that would have pitching moments throughout the flight range the same as those of an actual airfoil. mean camber line A line halfway between the upper and lower surface of an airfoil. minimum glide angle The smallest angle between the horizon and the flightpath of an airplane in a glide. 185 Fundamentals of Aerodynamics: Part 1 – M06-042 moment A tendency to cause rotation around a point or axis, as a control surface around its hinge or an airplane around its center of gravity; the measure of this tendency, equal to the product of the force and perpendicular distance between the point of rotation and the direction of the force., expressed as a vector. Also called torque. moment arm The distance from a point of rota- tion, perpendicular to the force, over which a force acts to create a moment. monocoque A type of construction, as an airplane fuselage, in which most or all the stresses are carried by the covering or skin. nacelle A streamlined structure or compartment on an aircraft, used as housing for an engine. negative camber airfoil An airfoil in which the mean camber line is below the chord line. neutral point (NP) The location of the center of gravity of an airplane that would produce neutral longitudinal static stability. The average aerodynamic center for the overall airplane. Newton’s First Law (The Law of Equilibrium.) “A body at rest tends to remain at rest and a body in motion tends to remain in motion in a straight line at a constant velocity unless acted upon by some unbalanced force.” Newton’s Second Law (The Law of Accelera- tion.) “The acceleration of a body is directly proportional to the force exerted on the body, is inversely proportional to the mass of the body, and is in the same direction as the force.” F = m a. Newton’s Third Law (The Law of Interaction.) “For every action, there is an equal and opposite reaction.” nosewheel liftoff / touchdown speed (NWLO/) The lowest speed that a heading and course along the runway can be maintained with full rudder and ailerons deflected when the nosewheel is off the runway. overstress The condition of possible permanent deformation or damage that results from exceeding the limit load. It also refers to the damage that may occur as a result of exceeding the limit load. Overstress damage will not cause structural failure of the airframe, but could result in internal damage to various components. parasite drag (DP) All drag not associated with the production of lift. phugoid oscillations Oscillations of altitude and airspeed that occur over relatively long periods of time, and are easily controlled by the pilot. Also called phugoid motion. pilot induced oscillations (PIO) Oscillations of attitude and angle of attack caused by the pilot trying to stop unwanted aircraft oscillations, or by the instability of the control surfaces. These inputs may result in an increase in the magnitude of the original oscillations. pitch The motion of an aircraft around its lateral axis. Pitch control is achieved through use of elevators or stabilators. pitch angle The angle between the chord line of the rotor blade and the rotor’s tip path plane; the angle between the propeller blade and the propeller tip path plane. pitch attitude () The angle between the longitudinal axis of the airplane and the horizon. pitching moment Any moment around the lateral axis of an airplane. 186 Fundamentals of Aerodynamics: Part 1 – M06-042 pitot static system A system consisting of a pitot tube, a static pressure port, and a device that determines the difference, used principally in order to calculate dynamic pressure. plain flap A high lift device consisting of a hinged airfoil attached to the leading or trailing edge of a wing that increases camber to increase the CL. planform The outline of an object, such as a wing, as viewed from above. positive camber airfoil An airfoil in which the mean camber line is above the chord line. potential energy (PE) The ability of a body to do work because of its position or physical state. power (P) The rate of doing work, or work per unit time, measured in ft-lbs/sec or horsepower. power available (PA) The power an engine is producing. Power available is a function of PCL setting, density altitude, and velocity. power control lever (PCL) Control on a propeller driven airplane or helicopter, that adjusts the fuel flow and therefore the power output of the engine(s). Similar to the throttle on a jet aircraft. power deficit (PD) The negative difference between power available and power required. power excess (PE) The positive difference between power available and power required. power required (PR) The power required to produce enough thrust to overcome drag in level equilibrium flight. pressure altimeter Aneroid barometers cali- brated to indicate altitude in feet instead of pressure. pressure altitude (PA) Height above the standard datum plane, i.e., altitude measured from standard sea level pressure by a barometric altimeter. pressure gradient A change in the pressure of a fluid per unit of distance. propeller efficiency (p.e.) A measure of the effectiveness of a propeller in converting shaft horsepower into thrust horsepower. Propeller wash The disturbed air produced by the passage of the propeller, usually making a corkscrew path around the airplane. pullout An act or instance of recovering from a dive. radar altimeter Specialized radar transmitter/ receiver used to indicate height above terrain. radius of turn (r) See turn radius. range The distance that an aircraft can travel without refueling. rate of climb (ROC) The rate at which an aircraft gains altitude, the vertical component of its airspeed in a climb. rate of descent (ROD) The rate at which an aircraft loses altitude, the vertical component of its airspeed in a descent. Also called sink rate. rate of turn () See turn rate. 187 Fundamentals of Aerodynamics: Part 1 – M06-042 redline airspeed (VNE) The maximum permissi- ble airspeed for an airplane. Beyond the redline airspeed, a pilot may experience control problems and structural damage to the aircraft due to aeroelastic effects. region of normal command The region of flight at velocities greater than maximum en- durance airspeed in which an airplane is in stable equilibrium. That is, if disturbed (slowed down), it tends to return to equilibri- um. region of reversed command The region of flight at velocities less than maximum endurance airspeed, in which a greater power setting is required to fly at a lower velocity, due to increased total drag caused by induced drag. Takeoff and landing normally take place while in this region. Also called the “back side of the power curve.” relative wind (RW) The airflow experienced by the aircraft as it flies through the air. It is always equal and opposite to the flightpath. The relative wind may arise from the motion of the body, from the motion of the air, or from both. reverse thrust Thrust applied to a moving object in a direction opposite to the direction of the object’s motion. reversibility The ability to transmit aerodynamic forces from the control surfaces to the cockpit controls. roll The motion of an airplane around its longi- tudinal axis. Roll is controlled by the use of ailerons or spoilers. rudder An upright control surface that is deflected to produce a yawing moment, rotating the airplane around its vertical axis. safe flight envelope The portion of the V-n diagram that is bounded on the left by the accelerated stall lines, on the top and bottom by the positive and negative limit loads, and on the right by redline airspeed. An aircraft my operate in its safe flight envelope without exceeding its structural or aerodynamic limits. scalar A quantity expressing only magnitude, e.g., time, amount of money, volume of a body. section A cross section of an airfoil taken at right angles to the span axis or some other specified axis of the airfoil. semi-monocoque A type of construction, as of a fuselage or nacelle, in which transverse members and stringers reinforce the skin and help carry the stresses. shaft horsepower The horsepower delivered at the rotating driveshaft of an engine. shielded horn The part of a control surface of longer chord than the rest of the surface, lying forward of the hinge line and partially shielded by the surface to which it is attached, used for aerodynamic balance. shockwave A surface or sheet of discontinuity set up in a supersonic field of flow, through which the fluid undergoes a finite decrease in velocity accompanied by a marked increase in pressure, density, temperature, and energy. sideslip A movement of an airplane such that the relative wind has a component parallel to the lateral axis. sideslip angle () The angle between the airplane’s longitudinal axis and the relative wind, as seen from above. 188 Fundamentals of Aerodynamics: Part 1 – M06-042 sideslip relative wind The component of the relative wind that is parallel to the airplane’s lateral axis. sine (sin) In a right triangle, the function of an acute angle that is the ratio of the length of the opposite side to the length of the hypotenuse. sink rate See rate of descent. skin friction The friction of a fluid against the skin of an aircraft or other body; friction drag. slat The vane used in a slot, especially an automatic slot. When the slat deploys it forms a slot. slot High lift device that consists of a fixed vane that forms a gap between the leading edge of the wing and the body of the wing that allows airflow from below the wing to reach the upper surface and reenergize the boundary layer, delaying boundary layer separation. Also called fixed slot. slotted flap A high lift device consisting of a hinged airfoil attached to the leading or trailing edge of a wing that increases camber and uses BLC to increase the CL. sonic Pertaining to sound or the speed of sound. sonic boom An explosion-like sound heard when a shock wave generated by a super- sonic airplane reaches the ear. sonic speed Speed equal to the speed of sound. sound barrier A popular term for the large increase in drag that acts upon an aircraft approaching the speed of sound. span See wingspan. spanwise flow Airflow that travels the span of the wing, parallel to the leading edge, normally root to tip. This airflow is not accelerated over the wing and therefore produces no pressure differential or lift. spar A principal spanwise beam in the structure of a wing. speed of sound The speed at which sound travels in a given medium under certain conditions. The speed of sound in air is primarily dependent on the temperature of the air mass. spin An asymmetrical aggravated stall resulting in autorotation. spiral A maneuver in which an airplane ascends or descends in a helical (corkscrew) path at an angle of attack within the normal range of flight angles. spiral divergence A motion resembling a spiraling descent, becoming steeper over time. Spiral divergence results from strong static directional stability and weak static lateral stability. split flap A high lift device consisting of a plate deflected from the lower surface of the trailing edge of a wing that increases camber to increase the CL. It produces a similar change in CL as a plain flap, but a much larger increase in drag due to the great turbulent wake produced. spoiler A movable control surface attached to the wing of an airplane, used to produce a rolling moment around the longitudinal axis by disturbing the flow of the boundary layer over one wing. stabilator A movable control surface that replaces the horizontal stabilizer and elevators. 189 Fundamentals of Aerodynamics: Part 1 – M06-042 stability The property of a body, such as an aircraft, to maintain its attitude or to resist displacement, and if displaced, to develop forces and moments that would return it to its original condition. stabilizer A fixed or adjustable airfoil or vane that provides stability for an aircraft, i.e., a fin, the horizontal or vertical stabilizer on an airplane. stagnation Loss of kinetic energy or velocity. Lack of motion. stall A condition of flight in which an increase in AOA will result in a decrease in CL. stall fence A plate or vane projecting from the upper surface of a wing, parallel to the airstream, used to prevent spanwise flow. stalling angle of attack The angle of attack on an airfoil beyond which a stall occurs, i.e., CLMAX. Beyond this angle of attack, the boundary layer is unable to remain attached to the wing, resulting in the decrease in CL. stall speed (VS) The minimum true airspeed required to maintain level flight at CLMAX AOA. stall strip A sharply angled device attached near the wing’s root on its leading edge to initiate a root first stall. standard atmosphere A reference set of average atmospheric conditions. standard datum plane (SDP) The actual elevation at which the barometric pressure is 29.92 in. Hg. standard rate turn (SRT) A turn in an aircraft with a three degree per second turn rate. static failure The breaking (or serious perma- nent deformation) of a material due to a steadily increasing, or sudden large applica- tion of force. This type of failure is often immediate and can occur without warning. static pressure (PS) The weight of a column of air over a given area; the pressure each air particle exerts on another due to the weight of all the particles above; the potential energy per unit volume. static stability The initial tendency of an object to either move toward or away from equilibrium after a disturbance. static strength A measure of a material’s resistance to a steadily increasing load or force. steady airflow Airflow in which at every point in the moving air mass, the pressure, density, temperature and velocity are constant. stiffness Resistance to deflection or deformation. straight horn See unshielded horn. streamline The path traced by a particle of air while in steady flow. streamtube An impenetrable tube formed by many streamlines. Streamtubes are closed systems. strength A measure of a material’s resistance to load or force. subsonic Movement or flow at speeds below the speed of sound, generally at a Mach number of 0.0 to 0.75. supersonic Movement or flow at speeds above the speed of sound, generally at a Mach number of 1.2 to 5.0. 190 Fundamentals of Aerodynamics: Part 1 – M06-042 sweep angle () The angle between the lateral axis and a line drawn 25% aft of the leading edge. symmetric Exhibiting a correspondence of parts on opposite sides of a boundary or axis symmetric airfoil An airfoil in which the mean camber line is coincident with the chord line. Also called a zero camber airfoil. tangent (tan) In a right triangle, the function of an acute angle that is the ratio of the length of the opposite side to the length of the adjacent side. A line, curve, or surface touching but not intersecting another line, curve or surface at only one point. taper A gradual reduction in the chord length of an airfoil from root to tip. taper ratio () The ratio of tip chord to root chord. The taper ratio affects the lift distribution and the structural weight of the wing. temperature A measure of the average kinetic energy of air particles, expressed in degrees Celsius (C), Fahrenheit (F), or Kelvin (K). terminal velocity The maximum velocity an airplane can attain under given conditions. A vertical (zero-lift) dive path, normal gross weight, zero engine thrust, and standard sea-level air density are assumed. thickness The cross sectional height of an airfoil measured perpendicular to the chord line. thrust available (TA) The thrust an engine produces under a specific velocity, density, and throttle setting. thrust axis The axis along which thrust is produced and the direction in which the force is generated. thrust deficit (TD) The negative difference between thrust available and thrust required. thrust excess (TE) The positive difference between thrust available and thrust required. thrust horsepower The actual amount of horsepower that an engine-propeller system transforms into thrust, equal to shaft horsepower multiplied by propeller efficiency. thrust required (TR) The thrust required to overcome drag to maintain level equilibrium flight. total pressure The pressure a moving fluid would have if it were brought to a rest without losses. transonic Movement or flow at speeds very near the speed of sound, generally at a Mach number of 0.75 to 1.2. trimmed flight A condition that exists when the sum of the moments acting around the center of gravity are equal to zero. The word “trim” often refers to the balance of control forces. trim tab A tab that is deflected to a position where it remains to keep the aircraft in the desired attitude. true airspeed (TAS) The velocity of an aircraft with respect to the air mass in which it is traveling. Airspeed value determined by correcting indicated airspeed for installation error, compressibility, and density. 191 Fundamentals of Aerodynamics: Part 1 – M06-042 true altitude The actual height above mean sea level. It is found by correcting calibrated altitude for temperature deviations from the standard atmosphere. turbulence An agitated condition of air in which random fluctuations in velocity and direction occur. Airflow in which the velocity at any point varies erratically in magnitude and direction. turbulent flow Boundary layer airflow characterized by turbulent, unsteady airflow. turn radius (r) One half the diameter of the circle an aircraft would fly if it completed a 360 degree turn. turn rate () The number of degrees of arc traversed per unit of time while turning, expressed in degrees/sec. ultimate load The maximum load factor that the airplane can withstand without structural failure. It is 1.5 times the limit load. unshielded horn The part of a control surface of longer chord than the rest of the surface, lying forward of the hinge line and entirely exposed to the relative wind, used for aerodynamic balance. upwash Chordwise airflow from the lower surface of an airfoil passing upward over the leading edge to the upper surface. Any upward airflow. V-n diagram A diagram describing the structural and aerodynamic limits within which an airplane must operate. vector A quantity that expresses both magni- tude and direction. A vector quantity is represented by an arrow that displays direction and has a length proportional to magnitude. velocity (V) Speed, as referenced to another plane, object, or system. A vector quantity equal to speed in a given direction. True airspeed. velocity never-to-exceed (VNE) See redline airspeed. vertical axis An axis passing from top to bottom through the aircraft’s center of gravity. Any movement developed around this axis is called yaw. vertical stabilizer A fin mounted approximately parallel to the plane of symmetry of an airplane, to which the rudder is attached. viscosity () A measure of a fluid’s resistance to flow and shearing. volume The size of the mass, or the amount of space occupied by an object. vortices / wingtip vortices A spiraling mass of air created at the wingtip, due to the airflow around the tip from the high-pressure region below the surface to the low-pressure region above it. Vortex strength is dependent upon the wing loading, gross weight, and speed of the generating airplane. Vortices from medium to heavy airplanes can be extremely hazardous to smaller airplanes. Also called wake turbulence, or jetwash. weight The force at which a mass is attracted toward the center of the earth by gravity. wing An airfoil that produces a pressure differential when air is forced over it, resulting in a lifting force. 192 Fundamentals of Aerodynamics: Part 1 – M06-042 wing area (S) The surface area of a wing from wingtip to wingtip. The area within the outline of a projection of a wing on the plane of its chord, including that area lying within the fuselage or nacelles. With a swept wing, the area within the fuselage is contained within lines having the same sweep angle as the leading and trailing edges, fairings or fillets being ignored. wing loading (WL) A ratio of airplane weight to the wing surface area. wing root The base of a wing, where it joins the fuselage or other main body of an airplane. wing section A cross section of a wing; the profile of a cross section or the area defined by a profile. wingspan (b) The length of a wing, measured from wingtip to wingtip. Also called span. work (W) Work is done when a force acts on a body and it moves. Work is a scalar quantity measured in ft-lbs. W = F x s yaw Rotation around the vertical axis of an airplane. Yaw is controlled by the rudder. 193 Fundamentals of Aerodynamics: Part 1 – M06-042 1 2 2 2 2 2 2 2 W = F s Useful Equations 2W 2Wn F = ma M = F d VS SCL max 2W VS SCL max 2Wn TE = PE + KE PE = mgh IASS 0 SCL max IASS 0 SCL max WL = W D 2 V SCD g tan V S DT DP DI V 2 KE 1 mV 2 DP qf 2 r g tan P = R T D kL 2W S bc I V 2b2 P = TR V VTO 1.2 SCL max = CT C R AR = b R 325 P = TA V A 325 IASTO 1.2 2W 0 SCL max c 1 A1 V 1 = 2 A2 V 2 THP = SHP p.e. VLDG 1.3 2W SCL max PT = PS + q TE TA TR IAS LDG 1.3 2W SC q 1 V 2 P E P A P R 0 L max M = TAS s in TE W FR W L 2 LSOS ROC P E STO gSC W T D F TAS = IAS AF 1 V 0 SCAF W ROD P D W S LDG L max gSCL max W 2 FR R D T L 1 V SCL n L W Vhydroplane 9 tire pressure 𝑇𝐴𝑆= 𝐼𝐴𝑆√ 𝜌0 𝜌 𝑉 ℎ𝑦𝑟𝑑𝑜𝑝𝑙𝑎𝑛𝑒 = 9√𝑡𝑖𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐼𝐴𝑆𝑆= √ 2𝑊 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 𝑉 𝑆= √ 2𝑊 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 𝐼𝐴𝑆𝑆𝜙= √ 2𝑊𝑛 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 𝑉 𝑆𝜙= √ 2𝑊𝑛 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 𝑉𝑇𝑂≈1.2√ 2𝑊 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 𝐼𝐴𝑆𝑇𝑂≈1.2√ 2𝑊 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 𝑉 𝐿𝐷𝐺≈1.3√ 2𝑊 𝜌𝑆𝐶𝐿 𝑚𝑎𝑥 𝐼𝐴𝑆𝐿𝐷𝐺≈1.3√ 2𝑊 𝜌0𝑆𝐶𝐿 𝑚𝑎𝑥 194 Fundamentals of Aerodynamics: Part 1 – M06-042 Standard Day Conditions Altitude (feet) Temperature LSOS (knots) Pressure (in. Hg) Density (g / L) °C °F 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 16,000 17,000 18,000 19,000 20,000 25,000 30,000 35,000 36,000 40,000 45,000 50,000 55,000 60,000 65,000 15.0 13.0 11.0 9.1 7.1 5.1 3.1 1.1 -0.8 -2.8 -4.8 -6.8 -8.8 -10.7 -12.7 -14.7 -16.7 -18.7 -20.6 -22.6 -24.6 -34.5 -44.4 -54.2 -56.2 -56.5 -56.5 -56.5 -56.5 -56.5 -56.5 59.0 55.4 51.9 48.3 44.7 41.2 37.6 34.0 30.5 26.9 23.4 19.8 16.2 12.7 9.1 5.5 2.0 -1.6 -5.1 -8.7 -12.3 -30.0 -47.8 -65.6 -69.2 -69.7 -69.7 -69.7 -69.7 -69.7 -69.7 661.7 659.4 657.1 654.8 652.5 650.2 647.9 645.6 643.2 640.9 638.5 636.2 633.8 631.4 629.1 626.7 624.3 621.8 619.4 617.0 614.6 602.2 589.6 576.8 574.1 573.7 573.7 573.7 573.7 573.7 573.7 29.921 28.856 27.821 26.817 25.843 24.897 23.980 23.090 22.228 21.391 20.581 19.795 19.035 18.298 17.584 16.893 16.225 15.578 14.952 14.346 13.761 11.118 8.903 7.060 6.732 5.558 4.375 3.444 2.712 2.135 1.682 1.225 1.190 1.155 1.121 1.088 1.056 1.024 0.993 0.963 0.934 0.905 0.877 0.849 0.823 0.797 0.771 0.746 0.722 0.698 0.676 0.653 0.550 0.459 0.380 0.366 0.303 0.238 0.188 0.148 0.116 0.092 195 Fundamentals of Aerodynamics: Part 1 – M06-042 PHYSICAL CHARACTERISTICS T-6B Data FUSELAGE: Construction: Semi-monocoque, Length: 33 ft 4 in, Height: 10 ft 8 in at tail LANDING GEAR: Tricycle WING: Construction: Full cantilever, Wingspan: 33 ft 5 in, Flap type: Split PROPELLER: Type: Variable pitch, Rotation: clockwise, Prop arc: 8 ft 1 in CONTROLS: Type: conventional – reversible, trim – electric, with trim aid system for directional assisted trim Aerodynamic Balance Artificial Feel Aileron Overhang Neutral trim tabs Rudder Shielded Horn Anti-servo trim tabs Elevator Shielded Horn Bobweights, Downsprings, and neutral trim tabs FLIGHT CHARACTERISTICS CLmax AOA: 18 units Stall warning: Buffet, stick shaker, AOA display on PFD/HUD, AOA indexers (one in each cockpit) Spin indications: Erect Inverted Altimeter Decreasing Decreasing AOA 18+ Units 0 Units Airspeed 120–135 KIAS 40 KIAS Turn Needle Fully deflected Fully deflected Crosswind limits: 25 kts (Flaps UP or TO, and dry runway), 10 kts (Flaps LDG, or wet runway) Redline airspeed: 316 KIAS, 0.67 MACH (>19,020’ MSL), 150 KIAS (gear down and/or flaps down) Maneuver airspeed: 227 KIAS, Maximum turbulence airspeed: 207 KIAS (recommended 180 KIAS) Limit load factor: +7.0 Gs, −3.5 Gs; Gear down and/or Flaps down: +2.5 Gs, 0.0 G Standard rate turn: 2 needle widths; AOB = 15–20% indicated airspeed 196 Fundamentals of Aerodynamics: Part 1 – M06-042 PERFORMANCE CHARACTERISTICS Engine: PT6A-68, Sea level flat rated 1100 SHP (2900 ft-lbs torque) Critical altitude: 16,000 ft Operational ceiling: 31,000 ft Max endurance is achieved at 125 KIAS and 8.8 units AOA Max range is achieved at 4.4 units AOA (KIAS varies) Max AOC is not used in the T-6B, max ROC airspeed / best climb speed: 140 KIAS Max level airspeed: 255 KIAS, normal climb airspeed: 160 KIAS Max glide range / best glide speed at 125 KIAS for 11:1 glide ratio 197 Fundamentals of Aerodynamics: Part 1 – M06-042 CONTRIBUTIONS TO STATIC STABILITY Feature Longitudinal Directional Lateral Straight Wings - + Swept / Delta Wings + + + Fuselage - - Horizontal Stabilizer ++ Neutral Point. aft of C.G. + Vertical Stabilizer ++ + Dihedral Wings ++ Anhedral Wings -- High-Mounted Wings + Low Mounted Wings -198 Fundamentals of Aerodynamics: Part 1 – M06-042 Performance Study Guide Weight Increase Altitude Increase Gears Down Flaps Down TA TR TE PA PR PE Performance Characteristic Goal Curve Referenced & V vs L/DMAX Effect of Factors on Performance Jet Prop WeightAltitudeTailwindGear/Flap Endurance Min Fuel Flow Thrust, = Power, < Range Min Fuel Flow / Velocity Thrust, > Power, = AOC Max TE Thrust, = Thrust, < ROC Max PE Power, > Power, = Glide Endurance Min PD Power, < Power, < Glide Range Min TD Thrust, = Thrust, = 199 Fundamentals of Aerodynamics: Part 1 – M06-042 Thrust Required • VELOCITY (T AS) Figure 1 Thrust Available Qet) Thrust Available (prop) IF\IUPClJ (Ful Th"'111o) T · (11111-) TA it----.. IHlff PCLJ -.CCITY (TAS) IIB.Oa'IY (TAS) Figure 2 Figure 3 Max Thrust Excess Oet) Max Thrust Excess (prop (F,.PCLI VELOCITY (TAS) VELOCITY (TAS) Figure 4 Figure 5 200 Fundamentals of Aerodynamics: Part 1 – M06-042 201 Fundamentals of Aerodynamics: Part 1 – M06-042 Symbols & Abbreviations angle of attack sideslip angle climb or glide angle pitch attitude taper ratio sweep angle viscosity or coefficient of friction air density angle of bank rate of turn < less than > greater than approximately equal to a acceleration A area AC aerodynamic center AF aerodynamic force AOA angle of attack AOC angle of climb AR aspect ratio b wing span c average chord CAS calibrated airspeed CD coefficient of drag CDi coefficient of induced drag CDp coefficient of parasite drag CF coefficient of aerodynamic force CG center of gravity CL coefficient of lift CLMAX maximum coefficient of lift cR root chord cT tip chord d moment arm distance D drag D Density altitude DI induced drag Dp parasite drag DT total drag EAS equivalent airspeed f equivalent parasite area F force FF fuel flow FR rolling friction G acceleration of gravity GS groundspeed 202 Fundamentals of Aerodynamics: Part 1 – M06-042 IAS indicated airspeed IASLDG indicated landing speed IASS indicated stall speed IASS indicated accelerated stall speed IASSP indicated power-off stall speed IASTO indicated takeoff speed k constant KE kinetic energy L/D lift to drag ratio L/DMAX maximum lift to drag ratio L lift LEFF effective lift LSOS local speed of sound m mass M Mach number MCRIT critical Mach number n load factor NP neutral point NWLO nosewheel liftoff speed NWTD nosewheel touchdown speed P pressure p.e.
propeller efficiency PE potential energy PA pressure altitude PA power available PD power deficit PE power excess PR power required PS static pressure PT total pressure q dynamic pressure Q torque r radius of turn ROC rate of climb RW relative wind s distance of displacement sLDG landing distance sTO takeoff distance S wing surface area SHP shaft horsepower SRT standard rate turn T temperature or thrust TA thrust available TD thrust deficit 203 Fundamentals of Aerodynamics: Part 1 – M06-042 Bibliography Anderson, John D. Fundamentals of Aerodynamics, Fourth Edition. New York: McGraw Hill, 2007. Anderson, John D., Introduction to Flight, Sixth Edition. New York: McGraw Hill, 2008. Hurt, Hugh. Aerodynamics For Naval Aviators (NAVWEPS 00-80T-80). Washington D.C.: U.S. Government Printing Office, 1960. National Oceanic and Atmospheric Administration. U.S. Standard Atmosphere, 1976, Washington, D.C.: U.S. Government Printing Office, 1976.
Naval Aviation Schools Command. Fundamentals of Aerodynamics. Center for Naval Aviation Technical Training, 2008. Office of the Chief of Naval Operations. Flight Manual, NATOPS, USAF/USN Series T-6B Aircraft (NAVAIR A1-T6BAA-NFM-100). Commander, Naval Air Systems Command, 2012. Talay, Theodore. Introduction to the Aerodynamics of Flight. Washington D.C.: Scientific and Technical Information Office, National Aeronautics and Space Administration, 1975. 204 Fundamentals of Aerodynamics: Part 1 – M06-042 |
189437 | https://math.stackexchange.com/questions/2244248/is-logarithm-of-a-matrix-a-monotone-function | Stack Exchange Network
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Is logarithm of a matrix a monotone function
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Let $A$ be a positive definite matrix and let \begin{align} B \succeq \log(A) \end{align}
Is it true that $e^{B} \succeq A$?
Also, if $A\succeq C$ is it true that \begin{align} \log(A) \succeq \log(C). \end{align}
In the last one we do not claim that $C$ is psd.
I would also appreciate if someone can point me to a good reference on this subject.
linear-algebra
logarithms
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asked Apr 20, 2017 at 22:35
BobyBoby
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$\begingroup$ How do you define the logarithm? Is it a unique logarithm of some kind, or are the results supposed to hold for all possible logarithms? $\endgroup$
Joonas Ilmavirta
– Joonas Ilmavirta
2017-04-20 22:41:59 +00:00
Commented Apr 20, 2017 at 22:41
$\begingroup$ @JoonasIlmavirta This the usual logarithm of a matrix. See here: en.wikipedia.org/wiki/Logarithm_of_a_matrix $\endgroup$
Boby
– Boby
2017-04-20 22:45:59 +00:00
Commented Apr 20, 2017 at 22:45
$\begingroup$ That page says that the logarithm is typically not unique. And it's certainly not unique for all positive definite matrices. $\endgroup$
Joonas Ilmavirta
– Joonas Ilmavirta
2017-04-20 22:49:54 +00:00
Commented Apr 20, 2017 at 22:49
$\begingroup$ a reference with relevant proofs is dx.doi.org/10.7153/jmi-07-08. Also kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1893-10.pdf might be of interest $\endgroup$
glS
– glS
2022-11-04 08:19:48 +00:00
Commented Nov 4, 2022 at 8:19
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The key phrase to look up is "operator monotone". The (principal branch) logarithm on positive definite matrices is operator monotone, but exponential is not. For example, try
$$A = \pmatrix{e & 0\cr 0 & 1\cr},\ \log(A) = \pmatrix{1 & 0\cr 0 & 0\cr}, \ B = \pmatrix{2 & 1\cr 1 & 1\cr}$$
You will find that $B \succeq \log(A)$ but $\exp(B) \not\succeq A$.
EDIT:
$$ \exp(B) \approx \pmatrix{ 10.3247023472592 & 5.47549688301169\cr 5.47549688301169 & 4.84920546424751\cr} $$
$$\exp(B) - A \approx \pmatrix{7.60642051925920 & 5.47549688301169\cr 5.47549688301169 & 3.84920546424751\cr}$$ which is not p.s.d. as it has determinant $\approx -.702390689773839$.
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edited Apr 21, 2017 at 5:35
answered Apr 20, 2017 at 22:55
Robert IsraelRobert Israel
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$\begingroup$ Thanks. Any references that you might suggest for me to look up? $\endgroup$
Boby
– Boby
2017-04-20 23:20:29 +00:00
Commented Apr 20, 2017 at 23:20
$\begingroup$ I think one of the standard references is Horn and Johnson, "Topics in Matrix Analysis". $\endgroup$
Robert Israel
– Robert Israel
2017-04-20 23:31:20 +00:00
Commented Apr 20, 2017 at 23:31
$\begingroup$ @RobertIsrael: Is the above obvious? It took me a while to verify that $e^B \not \ge A$ $\endgroup$
copper.hat
– copper.hat
2017-04-21 02:41:15 +00:00
Commented Apr 21, 2017 at 2:41
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Passive Components – Technologies and Dimensioning
4.3.1 Notion of mutual inductance
Mutual inductance M12 is the link that exists between the flux circulating in a coil 1 generated by the circulation of a current i2 in a coil 2:
[4.47]
This is more general than cases of self-inductance, and is essential for understanding the operation of multiple coils on a shared magnetic circuit, such as those used in flyback or forward transformers. One important point when establishing coupled inductance equations is the reciprocal nature of actions between two coils 1 and 2, leading to the equality of mutual inductances (M12 and M21, the action of 2 toward 1 and 1 toward 2, respectively):
[4.48]
In the following section, we will consider the case of two coupled coils in a magnetic circuit with or without an air gap. For this study, as in the case of a single coil, we will use a reluctance diagram model of the magnetic circuit.
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Chapter
Inductance and Transformers
MUTUAL INDUCTANCE
Although the total inductance of coils connected in series or parallel can be determined using the same technique used for determining the total resistance of resistors connected in series or parallel, inductors have a property not at all similar to resistive circuits. This property is called mutual inductance. When the current in an inductor increases or decreases, the magnetic flux field surrounding that inductor changes. The varying flux can cut across any other coil which may be nearby, causing induced voltage to exist in it.
For example, series–connected coils L1and L2in the circuit of Figure 8.31 are close to one another. An ac voltage source is connected to L1and L2. The current through L1produced by the ac source causes a magnetic field to exist around L1which also cuts through the windings of L2, inducing a voltage in its windings. Thus, a voltage exists across L2as a result of the magnetic field of L1cutting through its windings. At the same time L1is producing this voltage across L2, the current flowing through L2produces a field around L2that induces a voltage in L1. This effect is known as mutual inductance.
Mutual inductance is measured in henrys and is designated by the symbol LM. The amount of mutual inductance between two coils, L1and L2, can be calculated by this equation:
(8–7)
where
Coefficient of Coupling
The coefficient of coupling is the fraction of total magnetic flux lines produced by both coils that is common to both coils. Stated as an equation:
(8–8)
where
For example, in a circuit there are two coils, L1and L2, which produce 10,000 flux lines, all of which link both coils. Therefore, using equation 8–8 the coefficient of coupling can be calculated:
In this case, there is unity coefficient of coupling and k = 1. However, if L1and L2produce 10,000 lines of flux and only 5,000 are common, the coefficient of coupling is
Therefore, in this case, the coefficient of coupling is 0.5. There are no units for k since it is a ratio of two amounts of magnetic flux.
A low value of k is called “loose coupling”; a high value of k is called “tight coupling.” Coils wound on a common iron core can be considered to have a coefficient of coupling of one or “unity coupling.” Figure 8.32 summarizes the types of coupling and range of coefficients of coupling.
The value of k can be varied by factors relative to the coils. For example, as shown in Figure 8.33, k becomes larger (with a maximum of one) as the coils are placed closer together; are placed parallel to one another, as opposed to perpendicular; or are both wound on a common iron core. Conversely, k becomes smaller as the coils are placed farther apart; are placed perpendicular to one another; or are both wound on separate iron cores.
Calculating Mutual Inductance
Mutual inductance is calculated using equation 8–7Figure 8.34 shows a circuit with two inductors, L1and L2. L1equals 20 henrys and L2equals 5 henrys. The coefficient of coupling is 0.4. The coils' mutual inductance is calculated as follows:
Series–Connected Inductors with Mutual Inductance
When two series–connected inductors are separated by a sufficient distance, are mounted at right angles to one another, or are well–shielded, their mutual inductance is negligible. Their equivalent total inductance is the sum of the inductance values as described previously. However, if series–connected inductors are situated so that their magnetic fields interact, the mutual inductance between the two coils must be considered. Their total inductance, then, is determined by this equation:
(8–9)
When the inductors are situated such that their magnetic fields aid each other the plus sign is used.
(8–10)
When the fields oppose, the minus sign is used.
(8–11)
The 2 is used because magnetic fields of L1and L2interact and affect each other.
For example, a 10–henry and 20–henry coil are connected in series as shown in Figure 8.35 such that their magnetic fields aid one another, and they have a mutual inductance of 5 henrys. Their total inductance is calculated:
If the same two coils are oriented such that their magnetic fields oppose one another, their total inductance is calculated:
Parallel–Connected Inductors with Mutual Inductance
When two coils are parallel–connected without any mutual inductance, their equivalent total inductance is found by the reciprocal addition or product–over–sum method discussed earlier. However, if their magnetic fields interact, their equivalent total inductance is found by one of the following equations:
Parallel–aiding connection — If the two coils are oriented such that their magnetic fields aid each other.
(8–12)
Parallel–opposing connection — If the two coils are oriented such that their magnetic fields oppose each other.
(8–13)
Using the same 10 henry and 20 henry coils with a mutual inductance of 5 henrys due to an aiding field, but connected in parallel, the total inductance using equation 8–12 is
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Book2000, Basic AC Circuits (Second Edition)John Clayton Rawlins M.S.
Chapter
Electrical and electronics principles
1994, Mechanical Engineer's Reference Book (Twelfth Edition)Charles J. Fraser
2.1.17 Mutual inductance
Two coils possess mutual inductance if a current in one of the coils produces a magnetic flux which links the other coil. Figure 2.8 shows two such coils sharing a common magnetic flux path in the form of a toroid. The mutual inductance between the two coils is
(2.33)
where N2 is the number of turns on coil 2,
I1 is the current through coil 1, and
ϕ is the magnetic flux linking coils 1 and 2.
The mutual inductance effect finds great application both to electrical transformers and to rotating electrical machines.
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Book1994, Mechanical Engineer's Reference Book (Twelfth Edition)Charles J. Fraser
Chapter
Inductance and Transformers
2000, Basic AC Circuits (Second Edition)John Clayton Rawlins M.S.
Calculating Mutual Inductance
Mutual inductance is calculated using equation 8–7Figure 8.34 shows a circuit with two inductors, L1and L2. L1equals 20 henrys and L2equals 5 henrys. The coefficient of coupling is 0.4. The coils' mutual inductance is calculated as follows:
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Book2000, Basic AC Circuits (Second Edition)John Clayton Rawlins M.S.
Chapter
Variable Frequency Operation of Induction Motors
2013, Electric Motors and Drives (Fourth Edition)Austin Hughes, Bill Drury
4.7 Self and mutual inductance
The self and mutual flux linkages produced by a winding are proportional to the current in the winding. The ratio of flux linkage to the current that produces it is therefore a constant, and is defined as the inductance of the winding. The self inductance (L) is given by
while the mutual inductance (M) is defined as
The self and mutual inductances therefore depend on the design of the magnetic circuit and the layout of the windings.
We can now recast the expressions for e.m.f. derived above so that they involve the rates of change of the currents and the inductances, rather than the fluxes. This is a very important simplification because it allows us to represent the distributed effects of the magnetic coupling in single lumped-parameter electric circuit terms. The self-induced and mutually induced e.m.f.s are now given by
and
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Book2013, Electric Motors and Drives (Fourth Edition)Austin Hughes, Bill Drury
Chapter
Printed Circuit-Board Design Issues
2008, Linear Circuit Design HandbookHank Zumbahlen, with the engineering staff of Analog Devices
Mutual Inductance
Another consideration regarding inductance is the separation of outward and return currents. As we shall discuss in more detail later, Kirchoff's law tells us that current flows in closed paths—there is always an outward and return path. The whole path forms a single-turn inductor.
This principle is illustrated by the contrasting signal trace routing arrangements of Figure 9-10. If the area enclosed within the turn is relatively large, as in the upper “nonideal” picture, then the inductance (and hence the AC impedance) will also be large. On the other hand, if the outward and return paths are closer together, as in the lower “improved” picture, the inductance will be much smaller.
Note that the nonideal signal routing case of Figure 12-19 has other drawbacks—the large area enclosed within the conductors produces extensive external magnetic fields, which may interact with other circuits, causing unwanted coupling. Similarly, the large area is more vulnerable to interaction with external magnetic fields, which can induce unwanted signals in the loop.
The basic principle is illustrated in Figure 12-20 and is a common mechanism for the transfer of unwanted signals (noise) between two circuits.
As with most other noise sources, as soon as we define the working principle, we can see ways of reducing the effect. In this case, reducing any or all of the terms in the equations in Figure 12-20 reduces the coupling. Reducing the frequency or amplitude of the current causing the interference may be impracticable, but it is frequently possible to reduce the mutual inductance between the interfering and interfered with circuits by reducing loop areas on one or both sides and, possibly, increasing the distance between them.
A layout solution is illustrated in Figure 12-21. Here two circuits, shown as Z1 and Z2, are minimized for coupling by keeping each of the loop areas as small as is practical.
As also illustrated in Figure 12-22, mutual inductance can be a problem in signals transmitted on cables. Mutual inductance is high in ribbon cables, especially when a single return is common to several signal circuits (A). Separate, dedicated signal and return lines for each signal circuit reduces the problem (B). Using a cable with twisted pairs for each signal circuit as in (C) is even better (but is more expensive and often unnecessary).
Shielding of magnetic fields to reduce mutual inductance is sometimes possible, but is by no means as easy as shielding an electric field with a Faraday shield (following section). HF magnetic fields are blocked by conductive material provided the skin depth in the conductor at the frequency to be screened is much less than the thickness of the conductor, and the screen has no holes (Faraday shields can tolerate small holes, but magnetic screens cannot). LF and DC fields may be screened by a shield made of mu-metal sheet. Mu-metal is an alloy having very high permeability, but it is expensive, its magnetic properties are damaged by mechanical stress, and it will saturate if exposed to too high fields. Its use, therefore, should be avoided where possible.
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Book2008, Linear Circuit Design HandbookHank Zumbahlen, with the engineering staff of Analog Devices
Review article
Low-power near-field magnetic wireless energy transfer links: A review of architectures and design approaches
2017, Renewable and Sustainable Energy ReviewsAkaa Agbaeze Eteng, ... Beng Wah Chew
5.1.3 Improving coupling and operating range
Coupling, as characterized in circuit theoretic analysis, is a function of self-inductances of the coupled coils, and the mutual inductance between them. The mutual inductance is proportional to the magnetic flux enclosed by the receiving coil as a consequence of the magnetic field excited by current flow in the transmitter coil. Also, the mutual inductance is influenced by the geometry of the interacting coils, since a larger receiving coil encloses more flux than a smaller coil. It also follows that mutual inductance is a function of the strength of the magnetic field excited by the transmitter coil. Consequently, research undertakings to enhance coupling between paired coils focus on two areas, namely field enhancement, and coil geometry.
Magnetic field enhancement is a widely employed coupling enhancement strategy for short-range direct-fed links. Typical techniques involve the use of ferrite materials to alter the distribution of the excited magnetic fields from the transmitter coil. Popular methods include the use of ferrite cored transmitter coils [139–143] ferrite plates [141,144], and ferrite sheet support for receiver coils [145,146]. Ferrites are typically characterized by high magnetic permeability, and relatively low eddy current loss potentials. Ferrite materials can therefore by used to redirect excited magnetic fields towards the intended coupling directions .
Generally, coupling strength is directly related to the sizes of the interacting coils . This implies that coupling between a pair of coils can be enhanced by using larger coils. However, most practical scenarios place restrictions on the coil sizes. Coil design studies, consequently, place more emphasis on optimal geometries within the available footprint [3,53,125]. Usually, the adopted methods maximize conductor lengths for a given coil geometry, through an appropriate number of turns, spacing between turns, and width of turns. The realized coil structures with high coupling potentials are often characterized by high Q-factors as well. Alternatively, Zierhofer and Hochmair demonstrate a method of distributing coil turns away from the coil edge, resulting in an increase in the coupling coefficient between paired coils. However, since the coil self-inductance and resistance also depend on the geometrical layout of coil turns, gains in coupling may be offset by a reduction in the coil Q-factor. Consequently, the method discussed in has been modified to take into cognisance the impact of turn distribution on the Q-factor as well [148–150].
Coupling also depends on the intervening distance between interacting coils, since the strength of the excited magnetic field decays with distance. Range concerns are, therefore, intertwined with coupling issues. A typical focus is on the ability of an energy transfer link to achieve high levels of transfer efficiency at extended operational distances. The major attraction to indirect-fed links is the facilitation of mid-range energy transfers over relatively greater distances than are possible with two-coil links. CMT-based analysis of two-terminal energy transfer links asserts that efficient mid-range energy transfer is contingent on the operation of a link within a “strong-coupling” regime, in which the coupling rate is much greater than the intrinsic decay rates of the coupled resonant coils [9,10]. By implication, the desired operating condition for the energy transfer link – strong-coupling – is defined by , where is regarded as the link figure-of-merit [9,76]. However, the term “strong-coupling” does not seem to have found widespread acceptance in the electrical engineering community. This may be due to the wider adoption of circuit theoretic analysis, and differences between its characterizations of coupling and coil quality (i.e. “coupling coefficient” and “Q-factor”), from the CMT approach (i.e. “coupling rate” and “decay rate”). Kurs et al. , and Bou et al. , however, note the following relationships between coupling rates and coupling coefficients, on one hand, and between Q-factors and intrinsic decay rates on the other:
(23)
Bearing in mind that the circuit theoretic coupling coefficient is a distance-dependent variable, one can appreciate that mid-range links operate at lower coupling coefficients relative to short-range links. Consequently, by circuit theory analysis, the CMT “strong-coupling” regime is actually characterized by “loose-coupling” [5,38,151,152] and low coupling coefficients.
However, Sample et al. note that even with low coupling coefficients, high efficiencies can be obtained by employing high Q-factor coils, since coupling and Q-factors equivalently determine transfer performance. This is basically how indirect-fed links sustain high transfer efficiencies at mid-range. In addition, Ramrakhyani and Lazzi explain that an indirect feed enables the decoupling of high Q-factor transmitter and receiver coils from finite source and load resistances, thereby boosting the end-to-end power transfer efficiency. In addition, the authors note that the availability of a wider reflected load range in indirect-fed links eases the tuning required to achieve maximum power transfer to a load for a given source voltage. This facilitation of an easier impedance transformation has been attributed in to additional degrees-of-freedom made available by inter-coil mutual inductances between driver and transmitter coils, or receiver and load coils. Furthermore, the indirect feeding arrangement is an implementation of the mini-loop impedance match technique, which provides low-loss, high-ratio impedance transformation. The required impedance transformation ratio typically grows with the distance of wireless energy transfer, making indirect-fed link configurations more appropriate for longer distances .
Table 4 summarizes the design approaches to improve coupling and operating range.
Table 4. Comparison of methods to improve coupling and operating range.
| Primary Goal | Method | Typical Transfer Efficiency |
---
| Increased strength of magnetic field | Use of ferrite cored transmitter coils [137–141] ferrite plates [139,142], and ferrite sheet support for receiver coils [143,144]. | >70% . |
| Increased coupling coefficient | Optimal geometries within the available space constraint [3,50,123]; distribution of coil turns away from coil circumference [48,148]. | >70% . |
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Journal2017, Renewable and Sustainable Energy ReviewsAkaa Agbaeze Eteng, ... Beng Wah Chew
Chapter
Faraday's Law of Electromagnetic Induction
2002, Electricity, Magnetism, and LightWayne M. Saslow
12.6 Mutual Inductance
Since, by the Biot–Savart law, is proportional to the current IP in the primary, so is ΦB . Hence it is convenient to write, for the flux through the secondary,
(12.12)
where the proportionality constant M is called the mutual inductance (mutual because one circuit affects the other). (Context will determine whether M refers to mutual inductance or magnetization.) Inductance is given in units of Wb/A = V−s/A= H, called the henry, in honor of Joseph Henry.
Using (12.12), we can calculate M in a number of important cases. The mutual inductance is important because it can be used to calculate the induced emf. Specifically, using (12.12), Equation (12.4) yields for the induced emf
(12.13)
Note that the mutual inductance between the primary and the secondary (the flux in the secondary, per current in the primary) is the same as between the secondary and the primary (the flux in the primary, per current in the secondary). M can be of either sign, according to the relative position of the primary and the secondary. If M is positive when two identical coils are one on top of another, then M is negative when they are side by side. This is because when they are one on top of another a clockwise current for the primary makes in the secondary a field that is into the page, but when they are side by side the primary makes in the secondary a field that is out of the page.
Example 12.8
Loop within a solenoid
Consider a rectangular loop within a solenoid, the loop axis aligned with the axis of the solenoid. Let the loop have N turns, area Aloop, and let the solenoid have n turns per unit length and carry electric current IP . See Figure 12.14(a). (The final results do not depend either on the shape of the loop or on the cross-section of the solenoid, so long as the solenoid completely encloses the loop.) (a) Find the mutual inductance M in algebraic form; (b) evaluate M for N = 200, n = 6 × 105/m, and Aloop = 4 × 10−4 m2; (c) evaluate the induced emf for dIP /dt = 500 A/s; (d) discuss the direction of the emf and the induced forces on the loop.
Solution:
(a)
: Figure 12.14(b) gives our and A conventions, following the circuit-normal right-hand rule. Since the field is uniform within the solenoid, and the normal to the loop is along the field , by (12.3) the flux ΦB is simply BAloop . By (11.40), the solenoid field is B = 4πkm nIP . Each of the N turns picks up this flux, so by (12.12)
(12.14)
(b)
: Using N = 200, n = 6 × 105/m, and Aloop = 4 × 10−4 m2, (12.14) gives M = 0.060 H. (c) If dIP /dt = 500 A/s (corresponding to an increasing field), then by (12.4) and (12.12), the emf has magnitude MdIP /dt = 30 V. (d) Since the field is increasing into the page, the induced field is out of the page, necessitating (by Ampère's right-hand rule) a counterclockwise-induced current in the rectangular loop. Because the field is uniform, there is no net force on the loop, but there is a compressive force, by the motion statement of Lenz's law.
Food for Thought:
: How would the mutual inductance of Example 12.8 change if the axis of the loop were at an angle θ to the axis of the solenoid?
Food for Thought:
: Qualitatively describe the induced current and the torque, if the loop were rotated about the x-axis, so that the area exposed to the field were to decrease. The rotation can be described using a nonzero value of θ.
If the loop were partly in and partly out of the solenoid, only the part of the coil area in the solenoid would contribute to (12.14). Since n varies inversely with length and Aloop varies as length squared, M is proportional to length. Like capacitances, inductances scale linearly with the length. Note that the larger the number of turns in the solenoid (making a larger field) or the coil (making a larger effective area), the larger the M.
Example 12.9
Mutual inductance in Faraday's electromagnet experiment
Figure 12.4 presents Faraday's electromagnet. The iron toroid has thickness d and diameter D ≫ d. The primary coil (P) has nP turns per unit length, and the secondary coil (S) has NS turns. Each covers about half of the iron toroid. Find the mutual inductance M between primary and secondary.
Solution: From (9.23) or (11.51), the solenoid produces a field within the core that is larger than 4π km nP IP by a factor of (1 + χ), where the magnetic susceptibility χ is a measure of the Amperian current of the iron core relative to the current of the solenoid. Thus, instead of B = 4πkm nPIP, we have
(12.15)
where χ ≈ 5000 for soft iron. (This enhancement may be thought of as due to Amperian currents circulating around the iron.) Because the iron nearly completely holds the flux, it doesn't matter how the turns of wire are distributed: only the number of turns and the perimeter π D of the toroid matter. Wrapping the wire of the primary around half the iron toroid hence gives nP = NP /π D. The secondary (S) has area AS = πd2/4 and NS turns. Thus the mutual inductance is obtained by modifying (12.14), replacing N by NS , n by nP = NP /π D, Aloop by AS, and multiplying by (1 + χ). Hence
(12.16)
Since M varies as d2/D, for fixed numbers of turns, M doubles if all length scales double (i.e., d and D double); as noted above, like capacitance, inductance scales linearly with length.
Faraday's description permits an estimate of many of the parameters of his electromagnet. The primary consisted of about 72 feet of copper wire wrapped around an iron bar with d = 7/8 inches that had been welded into a ring with D= 6 inches, so it had about NP ≈ [(72)(12)/π(0.875)] ≈ 314 turns. The secondary consisted of about 60 feet of copper wire, so it had about NS ≈ [(60)(12)/π(0.875)] ≈ 251 turns. Inserting the parameters of Faraday's
On the Source of Magnetic Fields
If the magnetism of soft magnets were due to magnetic poles, then, because for the toroidal geometry there are no magnetic poles, would be due only to of the solenoid. Then the factor of 1 + χ in (12.16) would be replaced by 1. However, experiments support the factor of 1 + χ. This indicates that the magnetism of soft magnets is due to microscopic Amperian currents rather than magnetic poles. Indeed, the operation of every iron core electrical transformer, throughout the world, is testimony that (due to currents, both macroscopic and microscopic), rather than µ0 (due to macroscopic currents and microscopic magnetic poles), applies in Faraday's law.
circuit into (12.16) gives M = 0.401 H. If dIP /dt = 100 A/s, the emf would have been, by (12.13), of magnitude 40.1 V.
Note. Even if we cannot compute M, we may be able to measure it. If εS = 12 V, and dIP /dt = 200 A/s, then (12.16) gives |M| = 0.06 H.
Example 12.10
Rectangular loop and a long wire
Consider a long wire and a rectangular loop of sides a and b, with side a parallel to the long wire. See Figure 12.15. They are separated by a distance s. (a) Find their mutual inductance M in algebraic form. (b) Evaluate M for a = 0.02 m, b = 0.04 m, s = 0.04 m. (c) Evaluate M for 200 turns. (d) Evaluate M with a soft iron core and 200 turns.
Solution:
(a)
: M can be obtained either from the flux due to the long wire acting on the loop or from the flux on the long wire (considered to be part of a circuit that connects at infinity) due to the loop. It is much easier to calculate the first. We use (12.12) with Bwire = 2km I/r, taking the circuit normal into the paper, so that and are collinear. With dA= drdz, (12.12) yields
(12.17)
(b)
: For the stated values of a, b, and s, (12.17) yields M= 2.89 nH. (c) A rectangular loop of N turns has a mutual inductance N times larger than given by (12.17). For N= 200 turns, the mutual inductance M becomes (200)(2.89 nH) = 577 nH. This is still rather small. (d) Including a soft iron core causes M to be multiplied by a factor of (1 + χ), as in (12.16), where for soft iron χ ≈ 5000. Thus, with soft iron core and N = 200 turns of wire, M becomes (5000)(200)(2.89 nH) = 2.89 mH, a significant value. This enhancement of the mutual inductance explains the presence of soft iron in many electrical devices.
Example 12.11
Loop pulled through a field, via time-varying mutual inductance
Again consider the rectangular loop moving with constant velocity into the region of a uniform magnetic field , as in Figure 12.13. Find the emf by thinking of Faraday's law in terms of a changing mutual inductance M.
Solution: As the loop enters the field, with A pointing out of the paper, as before, we have
(12.18)
where B/I is some constant, independent of the current. (For example, if B were due to a solenoid of n turns per unit length, so B = 4πkm nI, we would haveB/I = 4πkmn.) From (12.4) and (12.18),
(12.19)
in agreement with (12.7). This approach, using a time-varying mutual inductance M, is sometimes taken by mechanical engineers. Note that M is negative in this case, due to our opposite sign conventions for positive currents in the loop and in the solenoid. For A pointing into the page, M would be positive.
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Book2002, Electricity, Magnetism, and LightWayne M. Saslow
Chapter
Magnetic couplers for automotive WPT systems
2024, Wireless Power Transfer for E-MobilityMauro Feliziani, ... Francesca Maradei
5.3.3 Mutual inductance between square loops
Given two parallel square loops of side 2a and 2c, parallel and located at a distance h from each other as shown in Fig. 5.11, the mutual inductance is given by (Cheng and Shu, 2014):
(5.69)
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Book2024, Wireless Power Transfer for E-MobilityMauro Feliziani, ... Francesca Maradei
Chapter
Field oriented control of induction motors
2019, Electric Motors and Drives (Fifth Edition)Austin Hughes, Bill Drury
8.3.2 Self and mutual inductance
The self and mutual flux linkages produced by a winding are proportional to the current in the winding: the ratio of flux linkage to the current that produces it is therefore a constant, and is defined as the inductance of the winding. The self inductance (L), is given by
while the mutual inductance (M) is defined as
The self and mutual inductances therefore depend on the design of the magnetic circuit and the layout of the windings. In an induction motor the self inductances are constant, but the mutual inductance between a stator and a rotor winding varies with the angular position of the rotor.
We can now re-cast the expressions for e.m.f. derived above so that they involve the rates of change of the currents and the inductances, rather than the fluxes. This is a very important simplification because it allows us to represent the distributed effects of the magnetic coupling in single lumped-parameter electric circuit terms. The self-induced and mutually-induced e.m.f.’s are now given by
and
As mentioned above, the mutual inductance between stator and rotor windings varies with the rotor position, so MSR is a function of θ, and the rotor e.m.f. has to be expressed as
The first term of the equation is the ‘transformer e.m.f.’ that results from changes in the stator current, while the second is present even when the stator current is constant, and it is proportional to the speed of the rotor. We have already seen the vital role that this ‘motional’ e.m.f. plays in the energy conversion process. When the term occurs in a circuit model, it is often referred to as a ‘speed’ voltage.
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Energy Engineering
Antenna
Amplifier
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Magnetic Field
Binary Digit
Electric Lines
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Transducer
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Law of Sines
The Law of Sines is a useful identity in a triangle, which, along with the law of cosines and the law of tangents can be used to determine sides and angles. The law of sines can also be used to determine the circumradius, another useful function.
Contents
[hide]
1 Statement
2 Proof
2.1 Method 1
2.2 Method 2
3 Method 3
4 Problems
4.1 Introductory
4.2 Intermediate
4.3 Olympiad
5 See Also
Statement
In triangle , where is the side opposite to , opposite to , opposite to , and where is the circumradius:
Proof
Method 1
In the diagram above, point is the circumcenter of . Point is on such that is perpendicular to . Since , and . But making . We can use simple trigonometry in right triangle to find that
The same holds for and , thus establishing the identity.
Method 2
This method only works to prove the regular (and not extended) Law of Sines.
The formula for the area of a triangle is .
Since it doesn't matter which sides are chosen as , , and , the following equality holds:
Assuming the triangle in question is nondegenerate, . Multiplying the equation by yields:
Method 3
We can circumvent some of the work in Method 1 by setting up the circle in a different way. Let be a diameter and be the center of the circle, and let be on . Furthermore, let , and let , , and .
We have that is a right angle, as is a diameter. Therefore, , so, rearranging, we have , or . Likewise, . Finally, we observe that , so evidently . Combining all three equalities,
Problems
Introductory
If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?
(Source)
Intermediate
Triangle has sides , , and of length 43, 13, and 48, respectively. Let be the circlecircumscribed around and let be the intersection of and the perpendicular bisector of that is not on the same side of as . The length of can be expressed as , where and are positive integers and is not divisible by the square of any prime. Find the greatest integer less than or equal to .
(Source)
Olympiad
Let be a convex quadrilateral with , , and let be the intersection point of its diagonals. Prove that if and only if .
(Source)
See Also
Trigonometry
Trigonometric identities
Geometry
Law of Cosines
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189440 | https://mathinsight.org/polar_coordinates | Polar coordinates - Math Insight
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Polar coordinates
Suggested background
Cartesian coordinates
In two dimensions, the Cartesian coordinates(x,y)(x,y) specify the location of a point P P in the plane. Another two-dimensional coordinate system is polar coordinates. Instead of using the signed distances along the two coordinate axes, polar coordinates specifies the location of a point P P in the plane by its distance r r from the origin and the angle θ θ made between the line segment from the origin to P P and the positive x x-axis. The polar coordinates (r,θ)(r,θ) of a point P P are illustrated in the below figure.
As r r ranges from 0 to infinity and θ θ ranges from 0 to 2 π 2 π, the point P P specified by the polar coordinates (r,θ)(r,θ) covers every point in the plane. Adding 2 π 2 π to θ θ brings us back to the same point, so if we allowed θ θ to range over an interval larger than 2 π 2 π, each point would have multiple polar coordinates. Hence, we typically restrict θ θ to be in the interal 0≤θ<2 π 0≤θ<2 π. However, even with that restriction, there still is some non-uniqueness of polar coordinates: when r=0 r=0, the point P P is at the origin independent of the value of θ θ.
The following applet allows you to explore how changing the polar coordinates r r and θ θ moves the point P P around the plane. You change the polar coordinates using sliders and observe how the point moves in the Cartesian plane. You can also move the point in the Cartesian plane and observe how the polar coordinates change. Notice the non-uniqueness of polar coordinates when r=0 r=0.
Polar coordinates. When you change the values of the polar coordinates r r and θ θ by dragging the red points on the sliders, the blue point moves to the corresponding position (x,y)(x,y) in Cartesian coordinates. Alternatively, you can move the blue point in the Cartesian plane directly with the mouse and observe how the polar coordinates on the sliders change. The coordinate r r is the length of the line segment from the point (x,y)(x,y) to the origin and the coordinate θ θ is the angle between the line segment and the positive x x-axis.
More information about applet.
Conversion formulas
We can calculate the Cartesian coordinates of a point with polar coordinates (r,θ)(r,θ) by forming the right triangle illustrated in the below figure. The hypotenuse is the line segment from the origin to the point, and its length is r r. The projection of this line segment on the x x-axis is the leg of the triangle adjacent to the angle θ θ, so x=r cos θ x=r cosθ. The y y-component is determined by the other leg, so y=r sin θ y=r sinθ. Our conversion formula is
x y=r cos θ=r sin θ.(1)x=r cosθ(1)y=r sinθ.
To go the other direction, one can use the same right triangle. Since r r is the distance from the origin to (x,y)(x,y), it is the magnitude r=x 2+y 2−−−−−−√r=x 2+y 2. Alternatively, from the equation (1)(1), one can calculate directly that
x 2+y 2=r 2 cos 2 θ+r 2 sin 2 θ=r 2(cos 2 θ+sin 2 θ)=r 2.x 2+y 2=r 2 cos 2θ+r 2 sin 2θ=r 2(cos 2θ+sin 2θ)=r 2.
Taking the ratio of y y and x x from equation (1)(1), one can obtain a formula for θ θ,
y x=r sin θ r cos θ=tan θ.y x=r sinθ r cosθ=tanθ.
One can also see this relationshp from the above right triangle. We can set θ=arctan y x θ=arctany x, but a problem is that arctan arctan gives a value between −π/2−π/2 and π/2 π/2. One might neet to add π π or 2 π 2 π to get the correct angle. With this caveat (and also mapping points where x=0 x=0 to θ=π/2 θ=π/2 or −π/2−π/2), one obtains the following formula to convert from Cartesian to polar coordinates
r θ=x 2+y 2−−−−−−√=arctan y x.(2)r=x 2+y 2(2)θ=arctany x.
What about the point where (x,y)=(0,0)(x,y)=(0,0)? In this case, the angle θ θ isn't well defined. You could just take it to be θ=0 θ=0 to be concrete.
Another perspective on polar coordinates
In the above applet, we represented polar coordinates by two separate points on a r r-axis and a θ θ-axis (the two sliders). We could also put those two axes together into a single (r,θ)(r,θ) plane (just like the (x,y)(x,y) plane of Cartesian coordinates). This simple change of representing the polar coordinates as a single point on the polar (r,θ)(r,θ) plane is reflected in the following applet. Such a slight shift in perspective has more consequences than you might imagine, as you can read about in a page about polar coordinates as a mapping from the (r,θ)(r,θ) plane to the (x,y)(x,y) plane.
Polar coordinates with polar axes. The red point in the inset polar (r,θ)(r,θ) axes represent the polar coordinates of the blue point on the main Cartesian (x,y)(x,y) axes. When you drag the red point, you change the polar coordinates (r,θ)(r,θ), and the blue point moves to the corresponding position (x,y)(x,y) in Cartesian coordinates. Alternatively, you can move the blue point in the Cartesian plane directly with the mouse and observe how the polar coordinates represented by the red point change. The coordinate r r is the length of the line segment from the point (x,y)(x,y) to the origin and the coordinate θ θ is the angle between the line segment and the positive x x-axis.
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189443 | https://www.quora.com/How-can-3-thirds-of-1-ever-equal-1-if-theyre-represented-as-333-repeating | How can 3 thirds of 1 ever equal 1 if they're represented as .333 repeating? - Quora
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Mathematics
Recurring Decimals
Dividing Fractions
Decimal Representation of...
Numbers (mathematics)
Arithmetic
Multiplying Decimals
Fractions (mathematics)
Fractions and Decimals
5
How can 3 thirds of 1 ever equal 1 if they're represented as .333 repeating?
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Martin Brilliant
Former MTS at Bell Labs (1966–1989) · Author has 13.6K answers and 12.5M answer views
·4y
To begin with, 0.333… is an awkward representation of 1/3. Algebraically, 3 x 1/3 = 1, unquestionably.
But you want to add 0.333… + 0.333… + 0.333…. You will get 0.999…. I will now prove that 0.999… equals 1.
Understand that when I write 0.999… I can assume as many 9s as I want. Now I argue that 0.999… must be either greater than 1, less than 1, or equal to 1. There are no other possibilities.
It can’t be greater than 1 because any decimal greater than 1 must start with 1 or some higher digit, not with 0. So I only need to prove that it isn’t less than 1. If I can prove that, it must be equal to
Continue Reading
To begin with, 0.333… is an awkward representation of 1/3. Algebraically, 3 x 1/3 = 1, unquestionably.
But you want to add 0.333… + 0.333… + 0.333…. You will get 0.999…. I will now prove that 0.999… equals 1.
Understand that when I write 0.999… I can assume as many 9s as I want. Now I argue that 0.999… must be either greater than 1, less than 1, or equal to 1. There are no other possibilities.
It can’t be greater than 1 because any decimal greater than 1 must start with 1 or some higher digit, not with 0. So I only need to prove that it isn’t less than 1. If I can prove that, it must be equal to 1.
If 0.999… is less than 1, then it must be different from 1 by some non-zero amount. Call that amount D, where D > 0. But by extending 0.999… with enough 9s, I can make it as close to 1 as I want, in particular, closer than D. So 0.999… can’t be less than 1.
So 0.999… must be equal to 1.
How can 3 thirds of 1 ever equal 1 if they're represented as .333 repeating?
Upvote ·
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9 1
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999 485
999 103
99 17
Related questions
More answers below
If 1/3= 0.333 then 3/3=0.999. That means 1 isn't actually equal to 1. Why does this happen?
Why is 1 equal to 3/3 when 1/3 equal 0.333? Doesn’t that mean 3/3 equal to 0.999? Are we just rounding to the nearest 1?
Is 0 greater than 3 1/2?
Why is 1/3 equal to 0.33333333333333333 repeating?
Why doesn't 0.333 (repeating) plus 0.666 (repeating) equal 1, since 1/3 + 2/3 =1?
Chase Aaron
B.A. in Physics (college major), The College of Wooster (Graduated 2019) · Author has 55 answers and 75.4K answer views
·5y
Related
Why is 1 equal to 3/3 when 1/3 equal 0.333? Doesn’t that mean 3/3 equal to 0.999? Are we just rounding to the nearest 1?
Something like that.
It turns out 1 3=0.¯3=0.333...1 3=0.3¯=0.333...
The 3 repeats forever. Literally forever. That’s what the bar means in this context. So when you take the sum
1 3+1 3+1 3=0.¯9,1 3+1 3+1 3=0.9¯,
which, as you noted, also means
1 3+1 3+1 3=3 3=1.1 3+1 3+1 3=3 3=1.
But we know the left side above is equal to 0.¯9 0.9¯!
So it’s not that we’re rounding to the nearest whole number. It’s true that
0.¯9=1 0.9¯=1.
There’s a very heated discussion around this fact actually. The idea is that at the end of the repeating nines, isn’t there always a 0.¯0 1 0.0¯1 difference b
Continue Reading
Something like that.
It turns out 1 3=0.¯3=0.333...1 3=0.3¯=0.333...
The 3 repeats forever. Literally forever. That’s what the bar means in this context. So when you take the sum
1 3+1 3+1 3=0.¯9,1 3+1 3+1 3=0.9¯,
which, as you noted, also means
1 3+1 3+1 3=3 3=1.1 3+1 3+1 3=3 3=1.
But we know the left side above is equal to 0.¯9 0.9¯!
So it’s not that we’re rounding to the nearest whole number. It’s true that
0.¯9=1 0.9¯=1.
There’s a very heated discussion around this fact actually. The idea is that at the end of the repeating nines, isn’t there always a 0.¯0 1 0.0¯1 difference between 0.¯9 0.9¯ and 1? The conceptual argument is that you’ll never reach the end of the repeating nines and similarly that oddly written, perhaps sloppily written, 0.¯0 1 0.0¯1 will simply have zeroes forever which makes the statement identically equal to zero.
There are more rigorous proofs than my outline, so feel free to peruse them at your leisure.
Upvote ·
99 10
9 5
Charles Holmes
Studied Financial Markets&Mathematics (Graduated 1990) · Author has 16.6K answers and 13.6M answer views
·4y
y=1 (decimal fraction: 1.0)
PREMISES
y=3/3(1)
ASSUMPTIONS
Let 3 thirds=3 of three equal parts, i.e., 3/3
CALCULATIONS
3 thirds of 1 can be denoted by:
y=3/3(1)
y=(31)/3
y=3/3
y=
1 (decimal fraction: 1.0)
C.H.
Upvote ·
9 1
Assistant
Bot
·
1y
The concept of 0.333 0.333 repeating (often written as 0.¯¯¯3 0.3¯) can indeed be a bit confusing, but it’s important to understand how it relates to fractions and whole numbers.
Understanding 0.333...0.333...:
The notation 0.¯¯¯3 0.3¯ represents a repeating decimal where the digit 3 3 continues infinitely. Mathematically, this can be expressed as:
x=0.333...x=0.333...
If we multiply both sides of this equation by 10 10:
10 x=3.333...10 x=3.333...
Now, subtract the original x x from this new equation:
10 x−x=3.333...−0.333...10 x−x=3.333...−0.333...
9 x=3 9 x=3
Dividing both sides by 9 9:
x=3 9=1 3 x=3 9=1 3
So, 0.¯¯¯3=1 3 0.3¯=1 3.
Add
Continue Reading
The concept of 0.333 0.333 repeating (often written as 0.¯¯¯3 0.3¯) can indeed be a bit confusing, but it’s important to understand how it relates to fractions and whole numbers.
Understanding 0.333...0.333...:
The notation 0.¯¯¯3 0.3¯ represents a repeating decimal where the digit 3 3 continues infinitely. Mathematically, this can be expressed as:
x=0.333...x=0.333...
If we multiply both sides of this equation by 10 10:
10 x=3.333...10 x=3.333...
Now, subtract the original x x from this new equation:
10 x−x=3.333...−0.333...10 x−x=3.333...−0.333...
9 x=3 9 x=3
Dividing both sides by 9 9:
x=3 9=1 3 x=3 9=1 3
So, 0.¯¯¯3=1 3 0.3¯=1 3.
Adding Up Three Thirds:
When you say "3 thirds of 1," you're referring to:
3×1 3=1 3×1 3=1
This means that three times one-third equals one whole.
Conclusion:
Thus, when you consider 0.¯¯¯3 0.3¯ as 1 3 1 3, three of those (3×0.¯¯¯3 3×0.3¯) indeed equals 1 1:
3×0.¯¯¯3=3×1 3=1 3×0.3¯=3×1 3=1
In summary, 0.¯¯¯3 0.3¯ is mathematically equal to 1 3 1 3, and when you take three of those, you get one whole, consistent with the basic principles of fractions and decimals.
Upvote ·
Related questions
More answers below
Is one third equal to 1/3?
Is 3/4 equal to 3 times 1/4?
What does 1/3 + 1/3 + 1/3 equal?
If 1/3=.333 repeating and 2/3=.666 repeating, does that make 3/3=.999 repeating and not equal 1?
Why is 1/3 equal to 0.3333, but 0.3333 is not equal to 1?
MJ
Former Math Tutor (2015–2021) · Author has 64 answers and 64.4K answer views
·4y
0.333 repeating is an imperfect decimal answer. When we say repeating, we mean there are an infinite number of threes after the decimal. But we can’t write that out infinitely, so if you just add up 0.333 + 0.333 + 0.333 = 0.999, which isn’t exactly 1. As you add more threes, your answer will get closer and closer to 1, but you can never technically type it to be exactly one, simply because it’s an infinite decimal. You can get very very close: think 0.999999999999999999 and so on. Because it gets so close, we tend to just call it 1, even though we can’t necessarily exactly perfectly prove it
Continue Reading
0.333 repeating is an imperfect decimal answer. When we say repeating, we mean there are an infinite number of threes after the decimal. But we can’t write that out infinitely, so if you just add up 0.333 + 0.333 + 0.333 = 0.999, which isn’t exactly 1. As you add more threes, your answer will get closer and closer to 1, but you can never technically type it to be exactly one, simply because it’s an infinite decimal. You can get very very close: think 0.999999999999999999 and so on. Because it gets so close, we tend to just call it 1, even though we can’t necessarily exactly perfectly prove it with decimal addition.
However, a fraction, because it represents division, is an exact answer. 1/3 is an exact number that we don’t have to use repeating decimals to express, so three thirds is always exactly equal to 1 (which is why we know 0.3333 repeating plus itself three times is exactly equal to 1)
This is kind of the problem with decimals vs fractions, because decimals will always have a degree of approximation if they are infinite decimals, or irrational numbers.
Upvote ·
9 1
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20K 20K
1.6K 1.6K
999 446
James Rong
4y
Originally Answered: Why is 1/3=.33333333, but 3/3=1? ·
A similar answer to this problem is why does 1/9 = 0.1111111…, and 2/9 = 2.222222…, … and 8/9 = 0.888888888888, but 9/9 doesn’t = 0.9999999… ? Well, take a look at 1/3 = 0.33333…, 2/3 = 0.666666666, and 3/3 = 0.99999999999… this way: The difference between 1 and 0.99999… is equivalent to 0.000000…, which means that the difference between the two numbers is equivalent to the number 0 followed after the decimal point by an infinite amount of zero’s. Which equals 0, because you’ll never reach the end of that number because of the infinite amount of zero’s. So, using this formula, we can conclude
Continue Reading
A similar answer to this problem is why does 1/9 = 0.1111111…, and 2/9 = 2.222222…, … and 8/9 = 0.888888888888, but 9/9 doesn’t = 0.9999999… ? Well, take a look at 1/3 = 0.33333…, 2/3 = 0.666666666, and 3/3 = 0.99999999999… this way: The difference between 1 and 0.99999… is equivalent to 0.000000…, which means that the difference between the two numbers is equivalent to the number 0 followed after the decimal point by an infinite amount of zero’s. Which equals 0, because you’ll never reach the end of that number because of the infinite amount of zero’s. So, using this formula, we can conclude that 0.999999… = 1, so 3/3 is equivalent to both 0.999999… and 1. It isn’t too difficult, is it? :)
Upvote ·
9 3
9 3
Hilmar Zonneveld
Translator (1985–present) · Author has 58.5K answers and 19.4M answer views
·4y
If you divide something in three equal parts, and join the three parts, you get one whole again. How you write the thirds is irrelevant for the issue.
By the way, an argument similar to the question proves, precisely, that 0.99999… repeating is equal to one.
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Khenan Mak
M.Eng.Sc. in Mathematical Sciences&Engineering, MMU Cyberjaya (Graduated 2009) · Author has 2.6K answers and 1.2M answer views
·4y
Without going into the mathematics of infinite series required to prove this, perhaps this simple demonstration might be convincing.
1 9=0.111…1 9=0.111…
9×0.111…=0.999…=1 9×0.111…=0.999…=1
1 3=0.333…1 3=0.333…
3×0.333…=0.999…=1 3×0.333…=0.999…=1.
Upvote ·
9 1
Peter Vanroose
Ph.D in Mathematics, KU Leuven (Graduated 1989) · Author has 1.1K answers and 2.1M answer views
·7y
Related
Why is 1/3 equal to 0.33333333333333333 repeating?
Clearly, 1/3 lies between 0.3333 and 0.3334, since 3 times 1/3 is 1, while 3 times 0.3333 is 0.9999 and 3 times 0.3334 is 1.0002
Clearly, 1/3 lies between 0.33333 and 0.33334, since … (well, you can write that down yourself, isn’t it?)
Clearly, 1/3 lies between 0.333333 and 0.333334.
Moreover, the “error” when representing 1/3 by 0.333333 is less than 0.000001 (one millionth!), since that’s the distance between our two bounds, a lower bound and an upper bound.
4. Clearly, 1/3 lies between 0.3333333 and 0.3333334.
Now, the error is even 10 times smaller than in the previous approximation. And “error”
Continue Reading
Clearly, 1/3 lies between 0.3333 and 0.3334, since 3 times 1/3 is 1, while 3 times 0.3333 is 0.9999 and 3 times 0.3334 is 1.0002
Clearly, 1/3 lies between 0.33333 and 0.33334, since … (well, you can write that down yourself, isn’t it?)
Clearly, 1/3 lies between 0.333333 and 0.333334.
Moreover, the “error” when representing 1/3 by 0.333333 is less than 0.000001 (one millionth!), since that’s the distance between our two bounds, a lower bound and an upper bound.
4. Clearly, 1/3 lies between 0.3333333 and 0.3333334.
Now, the error is even 10 times smaller than in the previous approximation. And “error” means: “difference between the real thing and the approximation”.
5. Clearly, 1/3 lies between 0.33333333 and 0.33333334.
6. I guess you can continue from here on.
And I hope you see that you can keep continuing without ever having an ending decimal expression that exactly equals 1/3. But on the other hand, you certainly get closer and closer at each step: the difference between the “real thing” 1/3 and each subsequent approximation becomes as small as you want, just by continuing as long as needed.
Now, the rest of the story is a matter of what you define
0.3333333333333 repeating
to be. I would interpret it as “the one and only number that is as close as I want to one of these approximations with as many 3s as I want”.
Of course, maybe there is no unique such number. We only know that 1/3 is such a number. That’s where mathematicians come in (and actually also our intuition) to “prove” that no other real number comes arbitrarily close to zero point repeated threes.
So that notation “0.33333 repeating”, or equivalently “0.33333…”, uniquely refers to a single number. Which has many other ways of being written, like e.g. 1/3; or 1 3 1 3; or 2/6; or 3/9; (etc.) Nothing bizarre about something having many names, yet being a single thing, is it? We can “prove” that 2/6=1/3 2/6=1/3, so we can also prove that 1/3 equals one of those other “names”, including 0.3333[rep].
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Charlee Anthony
Go-to Resource for Realistic, Side Hustle Ideas
·Sep 22
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played games and completed offers
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This one good method to make extra income brought me over $3,000, and I still make $150–$200 every single week from Freecash. As a gig hunter, this app came natural to me.
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Upvote ·
99 19
9 5
Graham Waltermire
6y
Originally Answered: Why is 1/3=.33333333, but 3/3=1? ·
1/3 is equal to 0.333333… It repeats forever. A calculator will only show so many digits, because nothing can process to infinity. If you multiply it by three, you get 0.999999… Since the difference between 1 and 0.999999… is infinitely small, they equal the same thing, and since they equal the same thing, instead of dividing 1 by 3, 0.999999… can be divided by 3, resulting in 0.333333…
Upvote ·
9 1
Bob Sampson
Years of love of maths through school and county maths teams · Author has 274 answers and 153K answer views
·6y
Originally Answered: Why is 1/3=.33333333, but 3/3=1? ·
0.3333333…. (3 reoccurring so the 3’s go to infinity)
0.3333333… x 10 = 3.33333333… (3 point 3 reoccurring)
3.333333… - 0.3333333….. = 3 (because everything after the equals sign must disappear as it has been subtracted)
0.3333… x 10 - 0.3333… is 0.3333 x 9
So if 9 x 0.333333… = 3 then 3 x 0.33333… must equal 1.
Upvote ·
Ella Bella
none in Math (NOT Meth)&Mathematics, Hogwarts School of Witchcraft and Wizardry (fictional school) (Graduated 2020)
·7y
Related
Why does 1 times 1 equal to 1?
Imagine this:
You have a pack of pencils. Lets say a pack of 6 pencils. You can arrange the group of pencils in “groups”. In one set, you have 2 pencils, in another set you have 2 pencils, and the third one you have 2 pencils. All groups have 2 pencils! And how many groups are there? There are 3 groups of 2 pencils!
So to tell what the total amount of pencils we have is, we can say 3(groups) times 2(pencils in each group) is 6 total pencils. You can also add up 2+2+2=6, or 3+3=6.
Now lets make the problem less with another example. Imagine you have 1 pencil. You put 1 pencil in 1 group, since tha
Continue Reading
Imagine this:
You have a pack of pencils. Lets say a pack of 6 pencils. You can arrange the group of pencils in “groups”. In one set, you have 2 pencils, in another set you have 2 pencils, and the third one you have 2 pencils. All groups have 2 pencils! And how many groups are there? There are 3 groups of 2 pencils!
So to tell what the total amount of pencils we have is, we can say 3(groups) times 2(pencils in each group) is 6 total pencils. You can also add up 2+2+2=6, or 3+3=6.
Now lets make the problem less with another example. Imagine you have 1 pencil. You put 1 pencil in 1 group, since that is the max amount of even groups you can make becasue there is one pencil. We can do what we did in the previous example: 1(group) times 1(pencil) is still 1 pencil. 1=1 or 1=1.
Hope that makes sense!
Upvote ·
99 41
99 14
9 1
David Reichard
Studied at Cleveland State University (Graduated 1975) · Author has 486 answers and 143.2K answer views
·6y
Originally Answered: Why is 1/3=.33333333, but 3/3=1? ·
The first quotient, 1/3, gives a decimal fraction when done by long division. That system cannot exactly express the answer, only give a closer and closer estimate the more places it is carried to. There's a math term for that situation.It's because it is a fraction, unlike the second whole number example.
Upvote ·
Related questions
If 1/3= 0.333 then 3/3=0.999. That means 1 isn't actually equal to 1. Why does this happen?
Why is 1 equal to 3/3 when 1/3 equal 0.333? Doesn’t that mean 3/3 equal to 0.999? Are we just rounding to the nearest 1?
Is 0 greater than 3 1/2?
Why is 1/3 equal to 0.33333333333333333 repeating?
Why doesn't 0.333 (repeating) plus 0.666 (repeating) equal 1, since 1/3 + 2/3 =1?
Is one third equal to 1/3?
Is 3/4 equal to 3 times 1/4?
What does 1/3 + 1/3 + 1/3 equal?
If 1/3=.333 repeating and 2/3=.666 repeating, does that make 3/3=.999 repeating and not equal 1?
Why is 1/3 equal to 0.3333, but 0.3333 is not equal to 1?
What is 1/3 +1/3?
What is ((1/333) x 333)- 333/333=?
How is 1-(1/3) equal to (1+1) /3? What are the steps?
What should be added to -1/3 to get 1?
If (1/3) +(1/3) +(1/3) =3/3=1, doesn't 0.333 repeating plus 0.333 repeating plus 0.333 repeating also equal 1?
Related questions
If 1/3= 0.333 then 3/3=0.999. That means 1 isn't actually equal to 1. Why does this happen?
Why is 1 equal to 3/3 when 1/3 equal 0.333? Doesn’t that mean 3/3 equal to 0.999? Are we just rounding to the nearest 1?
Is 0 greater than 3 1/2?
Why is 1/3 equal to 0.33333333333333333 repeating?
Why doesn't 0.333 (repeating) plus 0.666 (repeating) equal 1, since 1/3 + 2/3 =1?
Is one third equal to 1/3?
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189444 | https://leo.phys.unm.edu/~lwa/memos/memo/lwa0065a.pdf | Use of NEC-2 to Calculate Collecting Area Steve Ellingson∗ December 27, 2006 Contents 1 Introduction 2 2 Problem Statement 2 3 Proposed Method 2 4 Results and Comparison to Theory 3 5 Discussion 4 ∗Bradley Dept. of Electrical & Computer Engineering, 302 Whittemore Hall, Virginia Polytechnic Institute & State University, Blacksburg VA 24061 USA. E-mail: ellingson@vt.edu 1 1 Introduction This report describes a simple method for determining the collecting area of an antenna using the Numerical Electromagnetics Code (NEC-2) code. This technique determines the collecting area of the antenna under test directly in “receive mode”, as opposed to an alternative technique in which the collecting area is determined from the pattern gain in “transmit mode.” The technique is used to estimate the collecting area of a thin straight half-wave dipole, for which theoretical results are also available.
Without too much effort to optimize accuracy, it is found in this case that the predicted collecting area is about 6% larger than the theoretical value. Greater accuracy might be possible with additional effort. This approach is easily extended to facilitate modeling of complete LWA stations, assuming sufficient memory is available. The more familiar indirect (transmit mode) method yields accuracy of better than 1%, but relies on reciprocity arguments which may be difficult to apply for complex arrays in realistic scenarios.
2 Problem Statement We wish to determine the collecting area – referred to here as the effective aperture, Ae – of a thin straight half-wave dipole. For simplicity, we consider only one frequency f = 38 MHz, for which the free space wavelength λ = 7.895 m, and the dipole is exactly λ/2 long. The dipole is constructed from perfectly-conducting material of circular cross section having a radius of 0.05 mm. Its terminals are located at the origin of the coordinate system and it is aligned along the z-axis, such that its pattern maximum is in the x −y plane and it’s nulls are along the z axis. For simplicity we shall consider only Ae in the direction of maximum gain; in this case, in the x −y plane.
A simple technique by which to determine Ae for any antenna is simply to use the relationship : Ae = λ2 4π G , (1) where G is the pattern gain. The theoretical value of G for a vanishingly-thin half-wave dipole is 2.15 dB, yielding Ae = 8.137 m2 at 38 MHz. G can be obtained for any antenna by the principle of reciprocity . For example, one might apply a voltage source to the antenna terminals, then use NEC or some other technique to calculate the transmit pattern gain, and exploit reciprocity to interpret this value as the receive pattern gain. This is quite straightforward for simple antennas in free space; however the reciprocity argument is more difficult to apply for arrays of complex elements in the vicinity of complex media, such as the interface between lossy ground and free space. Thus, it is desireable to have an independent method by which the calculation can be done “directly” with the antenna system operating in “receive mode.” 3 Proposed Method The alternative approach proposed here uses directly the definition of Ae as the ratio of Pr, the power successfully received by the antenna, to the incident power density Si, typically having units of W/m2. Of course, Pr depends on the impedance presented by the load attached to the terminals of the antenna under test. Typically the maximum value of Ae is of interest, which occurs when the load impedance ZL is equal to the complex conjugate of the antenna’s terminal impedance ZA; i.e., ZL = Z∗ A. Thus, it is necessary to know ZA before proceeding.
Determination of ZA using NEC-2 is a simple matter. A NEC “deck” (input file describing the problem) for this is shown in Figure 1. This deck models the antenna using 11 segments, with a volt-age source with magnitude 1 V applied to the center segment, which corresponds to the terminals.
When this deck is run, NEC computes the resulting currents including the current in the segment containing the voltage source. From this, ZA is computed as the ratio of voltage to current in the segment corresponding to the terminals; NEC reports ZA = 77.41 + j45.09 Ω. Thus, we choose 2 ZL = 77.41 −j45.09 Ω.
At this point it is noted that a slight modification to NEC deck shown in Figure 1 – specifically, the addition of an “RP” (“radiation pattern”) “card” (line) – can be used to obtain G, the pattern gain in transmit mode. In this case, I obtain G = 2.16 dB, which corresponds to Ae = 8.156 m2, or just 0.2% higher than the theoretical value.
To compute Ae using the alternative method, it is proposed to illuminate the antenna under test with a plane wave having a known Si and then to measure the power Pr delivered to the load.
NEC-2 provides a mechanism for plane wave excitation of antennas, however I find this feature to be poorly documented and I am reluctant to trust it. An equally effective way to generate the desired incident wave is simply to use another antenna located in the far field of the antenna under test. As I would ultimately like to use this same method to study LWA stations, I wish the source dipole to be also in the far field of a station. Assuming a maximum dimension D = 100 m for a station, the far field criterion is R > 2D2/λ = 2.53 km; and I choose R = 25 km to be sure.
For simplicity, the source dipole is chosen to be identical to the antenna under test, except the load is replaced by a voltage source Vt = 1 + j0 V applied to the center segment. Because we are interested in Ae in the x −y plane, the terminals of the source dipole are located at x = R, y = 0, z = 0. The source dipole is oriented in the same direction so as to be co-polarized with the antenna under test.
For this problem, Ae can be determined as follows: 1. The electric field radiated by a vanishingly-thin half-wave dipole is known to be : E(θt, R) = ˆ θt cos( π 2 cos θt) sin θt jωµ 2πk It e−jkR R , (2) where It = Vt/ZA is the current at the terminals, θt is the angle measured from the +z direc-tion with associated unit vector ˆ θt, j = √−1, ω = 2πf, k = 2π/λ, and µ = 4π × 10−7 H/m (the permeability of free space).
2. In the far field of the source, the radiated wave appears locally as a plane wave. Thus the power density of the plane wave incident on the antenna under test is : Si = 1 2η |E|2 , (3) where η = 376.7 Ω, the impedance of free space.
3. The power delivered to the matched load of the antenna under test is determined by first calculating the current Ir generated at the terminals of the antenna under test using NEC.
The NEC deck used in this case is shown in Figure 2.
4. The power delivered to the load is then: Pr = Re ½1 2 |Ir|2 ZL ¾ (4) 5. Finally, we calculate Ae = Pr/Si.
4 Results and Comparison to Theory Using the procedure described in the preceding section, I obtained Ir = 0.3340−j0.3185 µA, leading to an estimate of Ae = 8.64 m2.
3 One reason for selecting this particular antenna to evaluate is because theoretical results are available for comparison. The effective aperture of any vanishingly-thin co-polarized linear antenna attached to matched load is : Ae = ηl2 e 4Rrad , (5) where Rrad = Re {ZA} for a lossless antenna and le is the “effective length”, given in our case by: le = 1 Ir Z I(z)dz , (6) where I(z) is the current at the indicated position on the dipole. For a half-wave dipole, le = λ/π, and if the dipole is vanishingly thin, Rrad = 73 Ω. Thus we obtain the well-known formula Ae = 0.13λ2, which in our case is 8.16 m2. Our computed result is 5.9% greater than this well-known theoretical result; perhaps reasonable agreement.
An important distinction between the actual problem and the theoretical problem is that the actual radius of the dipole is finite.
This has a small but significant effect on Rrad, which we found to be 77.41 Ωas opposed 73 Ω.
This changes the result obtained using Equation 5 to Ae = 7.69 m2, making our computed result now 12.3% greater than theoretical. At first glance, this seems counterintuitive as one would expect that actions that make the theoretical model conform more closely to the actual situation would result in an improvement in agreement, which is clearly not the case here. However, if Rrad is modified then perhaps le should also be modified. Using Equation 6 with the dipole current reported by NEC, we find a significant increase in le, now yielding Ae = 8.19 m2.
Thus the computed results are just 5.5% greater than the theoretical result computed using Equation 5, if we take into account both the actual load impedance and the NEC-derived dipole currents. This is a slight improvement in agreement.
5 Discussion The above results indicate that the method proposed yields an estimate of collecting area that is about 6% higher than that expected from theory. This level of uncertainty is probably acceptable for most applications, since the magnitudes of other errors are likely to have similar values. Increasing the number of segments in the NEC calculations might help; however, given that the intended use of this procedure is to analyze complete LWA stations, an increase in the number of segments required per dipole will greatly increase the computation time and thus is not desired. In any event, this method provides a useful independent check of Ae calculations based on reciprocity arguments.
4 CM thin half-wave dipole with unit voltage source CE GW 1 11 0.0000 0.0000 -1.9737 0.0000 0.0000 1.9737 0.00005 GE 0 EX 0 1 6 0 0 FR 0 1 0 0 38.0 0.0 EN Figure 1: NEC deck used to determine ZA for the antenna under test.
CM thin half-wave dipole with matched load illuminated by another dipole in far field CE GW 1 11 +25000.0000 0.0000 -1.9737 +25000.0000 0.0000 +1.9737 0.00005 GW 2 11 0.0000 0.0000 -1.9737 0.0000 0.0000 +1.9737 0.00005 GE 0 EX 0 1 6 0 0 LD 4 2 6 6 +77.4100 -45.0900 FR 0 1 0 0 38.0 0.0 EN Figure 2: NEC deck used to determine Ir for the antenna under test (with matched load), using another dipole in the far field as the source.
5 References W.L. Stutzman and G.A. Thiele, Antenna Theory and Design, 2nd Ed., Wiley, 1997.
6 |
189445 | https://byjus.com/temperature-conversion-formula/ | Temperature conversion formulas refer to changing the value of temperature from one unit to another. They are many temperature conversion methods. Among them Kelvin, Celsius and Fahrenheit are the most commonly used methods. According to the Kelvin scale, the freezing point of water is 273.15K and the boiling point is 373.15K. According to the Fahrenheit scale, the freezing point of water is 32°F and the boiling point is 212°F. According to the Celsius scale, the freezing point of water is 0°C and the boiling point is 100°C.
What are Temperature Conversion Formulas?
There can be three main conversions of temperature which are:
Conversion of Temperature Between Celsius and Kelvin
| |
| K = C + 273.15 |
| |
| C = K − 273.15 |
Conversion of Temperature Between Fahrenheit and Celsius
Fahrenheit to Celsius and Celsius to Fahrenheit formulas are listed below. These formulas will help to convert the temperature into required units.
| |
| C = (F − 32) × 5⁄9 |
| |
| F = C(9⁄5) + 32 |
Conversion of Temperature Between Fahrenheit and Kelvin
| |
| K = (F − 32) × 5⁄9 + 273.15 |
| |
| F = (K – 273.15) × 9⁄5 + 32 |
Temperature Conversion Formula Table
| Unit | To Celsius | To Fahrenheit | To Kelvin |
--- --- |
| Celsius (C) | C (°) | C(9⁄5) + 32 | C + 273.15 |
| Fahrenheit | (F − 32) × 5⁄9 | F | (F − 32) × 5⁄9 + 273.15 |
| Kelvin | K – 273.15 | (K – 273.15) 9/5 + 32 | K |
Solved Examples
Question: What is 30°C in Kelvin?
Solution:
The temperature conversion formula from Celsius to Kelvin is, K = C + 273.15
K = 30 + 273.15 K = 303.15
Question 2: Convert 50°C to °F.
Solution:
Celsius to Fahrenheit conversion formula is given by:
°F = °C(9⁄5) + 32
= 50 (9/5) + 32
= 90 + 32
= 122
Therefore, 50°C = 122°F
Question 3: Convert 113°F to Kelvin.
Solution:
Fahrenheit to Kelvin conversion formula is given by:
K = (F − 32) × 5⁄9 + 273.15
= (113 – 32) × (5/9) + 273.15
= 81 × (5/9) + 273.15
= 45 + 273.15
= 318.15
Therefore, 113°F = 318.15 K
Question 4: Convert 225 K to Celsius.
Solution:
Kelvin to Celsius conversion formula is given by:
C = K – 273.15
= 225 – 273.15
= -48.15
Therefore, 225 K = -48.15°C
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189446 | https://leo.uniandes.edu.co/queismo-y-dequeismo/ | Lectura
Escritura
Oralidad
Queísmo y dequeísmo
Queísmo y dequeísmo
Recurso elaborado por Alejandra Ramírez Becerra
¿Cuántas veces hemos escuchado a alguien decir cosas como “Tienes que darte cuenta de que cometiste un error” e inmediatamente pensamos que esa persona no sabe hablar? Más aún, nuestro pensamiento frente a la expresión de que suele ser generalizado: “decir de que suena muy feo”, y entonces resultamos cometiendo errores gramaticales en los que afirmamos que lo correcto es decir “Tengo que darme cuenta que cometí un error”. Sin notarlo, aquí caímos en el llamado queísmo. Esta es una guía que quiere mostrar y enseñar a evitar los casos más usuales de queísmo y dequeísmo, de forma tal que usemos el de que y el que de manera apropiada, y no porque hay una percepción de gusto o disgusto frente a cómo suena.
Tabla de contenido
1. Queísmo
Según la RAE, es la supresión indebida de una preposición antes de la conjunción que, cuando la preposición es exigida por alguna palabra del enunciado. Es decir, incurrimos en queísmos cuando eliminamos preposiciones como de, a o en que necesariamente deben estar dentro de la formulación. Miremos los casos más usuales en los que el queísmo se presenta:
1.1 Sustantivos
No hay una regla establecida que nos permita identificar cuáles sustantivos exigen una preposición antes del que, por lo que tendremos que apelar a nuestra memoria para recordar en qué casos la preposición es demandada:
En este caso decimos que había una noticia (ellos se comieron los perros calientes) y decimos de esta noticia que nunca llegó. Sabemos entonces que noticia es un sustantivo que exige de la preposición de. Pero como solemos pensar que la expresión “de que” es incorrecta, el queísmo nos lleva a eliminar la preposición “de”:
En este caso, eliminando la preposición, decimos que la noticia no llegó porque se la comieron los perros. Este error altera por completo el sentido de la oración.
Otros ejemplos comunes en los que solemos caer en queísmo son:
En este caso, la expresión correcta e inalterable es a pesar de que, no a pesar que. Asimismo, debemos fijarnos en que la palabra sensación es uno de los sustantivos que exigen de la preposición de.
1.2 Adjetivos
Al igual que los sustantivos, tampoco hay una regla que nos permita identificar cuáles adjetivos exigen una preposición antes del que. Por ello, lo más sencillo es tratar de recordar cuáles son aquellos que tienen esta característica: convencido y condenado son unos de los casos más usuales.
Otros adjetivos que exigen la preposición de son: ansioso, asombrado, ávido, capaz, celoso, contento, culpable, deseoso, diferente, difícil, digno, distinto, imposible, satisfecho, orgulloso, partidario, pendiente, preocupado, proveniente, rápido, representativo, seguro, separado.
Otros adjetivos que exigen la preposición a son: acostumbrado, adicto, aficionado, alérgico, atento, atribuible, condenado, decidido, equivalente, fiel, insensible, propenso, propicio, relativo, reticente.
1.3 Verbos intransitivos y pronominales
Los verbos intransitivos son aquellos que no tienen un complemento directo pues por sí mismos dan sentido a la oración. Por ejemplo: vivir, jugar, nadar, dormir, hablar.
Por otro lado, los verbos pronominales son aquellos que distinguimos porque siempre los encontramos en el diccionario con la terminación –se. Por ejemplo: alegrarse, acordarse, arrepentirse, olvidarse.
Algunos de estos verbos pueden tener un complemento, el cual se conoce como objeto prepositivo y, como es de esperarse, a un objeto prepositivo no se le puede quitar la preposición:
Otros verbos que exigen esta preposición –tanto intransitivos como pronominales– son: acordarse, acusar, asombrarse, avergonzarse, cansarse, cesar, cuidar, darse cuenta, dejar, despedirse, dudar, enamorarse, enterarse, hartarse, maravillarse, pasar, preocuparse, quejarse, tener ganas, tratar. En todos estos casos, la preposición de debe ir después del verbo y antes del que.
2. Dequeísmo
Según la RAE, es el uso indebido de la preposición de delante de la conjunción que cuando la preposición no es exigida, es decir, cuando de no se usa correctamente. Miremos los casos más usuales en los que el dequeísmo se presenta:
2.1 Locuciones conjuntivas
Las locuciones son un conjunto de palabras que funcionan como si fuesen una sola. Este conjunto no debe ser alterado de ninguna forma, pues son expresiones que han sido aceptadas de manera excepcional dada su particularidad. En este caso, tal como su nombre lo dice, este conjunto de palabras funciona como una conjunción, es decir, como un elemento que une palabras.
Las locuciones conjuntivas más usuales en las que se comente dequeísmo son:
2.2 Oraciones subordinadas
Las subordinadas son aquellas oraciones secundarias que dependen estructuralmente de una oración principal, para tener sentido. Usualmente, estas oraciones se introducen con que. Los casos más comunes del dequeísmo se presentan así:
Nunca se debe utilizar la preposición de cuando la oración subordinada está precedida por verbos de «pensamiento» (pensar, opinar, creer, considerar), de «habla» (decir, comunicar,exponer), de «temor» (temer) y de «percepción» (ver, oír, escuchar, sentir).
Nunca se debe utilizar la preposición de cuando la oración subordinada está funcionando como el sujeto de la oración, pues las normas de la sintaxis no permiten ninguna preposición antes de un sujeto.
3. Recomendaciones para no incurrir en el queísmo y dequeísmo
Como procedimiento para determinar si la preposición es necesaria, se puede construir el enunciado en forma interrogativa. La preposición será necesaria si aparece dentro de la pregunta. Por ejemplo:
En estos casos, la preposición es usada correctamente pues la forma interrogativa del enunciado tiene en su misma formulación la preposición de.
Pero, si pensamos en otros enunciados, la forma interrogativa correcta de estos no requiere de la preposición:
4. Bibliografía
Nota
Este recurso pedagógico le pertenece a la Universidad de los Andes y está protegido por derechos de autor. Así mismo, el recurso se encuentra amparado bajo la licencia de Atribución – No comercial – Sin derivar de Creative Commons.
Bajo los términos de esta licencia, se permite descargar este recurso y compartirlo con otras personas, siempre y cuando se reconozca su autoría. No obstante, la licencia impide modificar este material y prohíbe utilizarlo con fines comerciales. Para reconocer la autoría de este recurso le recomendamos citarlo y referenciarlo según las normas del formato que rija su disciplina o su publicación.
© - Derechos Reservados Universidad de los Andes |
189447 | https://byjus.com/maths/venn-diagrams/ | Venn diagrams are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way. Venn diagram, introduced by John Venn (1834-1883), uses circles (overlapping, intersecting and non-intersecting), to denote the relationship between sets. A Venn diagram is also called a set diagram or a logic diagram showing different set operations such as the intersection of sets, union of sets and difference of sets. It is also used to depict subsets of a set.
For example, a set of natural numbers is a subset of whole numbers, which is a subset of integers. The relation between the sets of natural numbers, whole numbers and integers can be shown by the Venn diagram, where the set of integers is the universal set. See the figure below.
Here, W represents whole numbers and N represents natural numbers
The universal set (U) is usually represented by a closed rectangle, consisting of all the sets. The sets and subsets are shown by using circles or oval shapes.
Table of Contents:DefinitionSymbolsHow to DrawOperationsComplementIntersectionUnionComplement of UnionComplement of IntersectionDifferenceSymmetric DifferenceExampleFAQs
What is a Venn Diagram?
A diagram used to represent all possible relations of different sets. A Venn diagram can be represented by any closed figure, whether it be a Circle or a Polygon (square, hexagon, etc.). But usually, we use circles to represent each set.
In the above figure, we can see a Venn diagram, represented by a rectangular shape about the universal set, which has two independent sets, X and Y. Therefore, X and Y are disjoint sets. The two sets, X and Y, are represented in a circular shape. This diagram shows that set X and set Y have no relation between each other, but they are a part of a universal set.
For example, set X = {Set of even numbers} and set Y = {Set of odd numbers} and Universal set, U = {set of natural numbers}
We can use the below formula to solve the problems based on two sets.
n(X ⋃ Y) = n(X) + n(Y) – n(X ⋂ Y)
Venn Diagram of Three Sets
Check the Venn diagram of three sets given below.
The formula used to solve the problems on Venn diagrams with three sets is given below:
n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) – n(A ⋂ B) – n(B ⋂ C) – n(A ⋂ C) + n(A ⋂ B ⋂ C)
Venn Diagram Symbols
The symbols used while representing the operations of sets are:
How to draw a Venn diagram?
To draw a Venn diagram, first, the universal set should be known. Now, every set is the subset of the universal set (U). This means that every other set will be inside the rectangle which represents the universal set.
So, any set A (shaded region) will be represented as follows:
Figure 1:
Where U is a universal set.
We can say from fig. 1 that
A ∪ U = U
All the elements of set A are inside the circle. Also, they are part of the big rectangle which makes them the elements of set U.
Venn Diagrams of Set operations
In set theory, there are many operations performed on sets, such as:
etc. The representations of different operations on a set are as follows:
Complement of a set in Venn Diagram
A’ is the complement of set A (represented by the shaded region in fig. 2). This set contains all the elements which are not there in set A.
Figure 2:
It is clear that from the above figure,
A + A’ = U
It means that the set formed with elements of set A and set A’ combined is equal to U.
(A’)’= A
The complement of a complement set is a set itself.
Properties of Complement of set:
Intersection of two sets in Venn Diagram
A intersection B is given by: A ∩ B = {x : x ∈ A and x ∈ B}.
This represents the common elements between set A and B (represented by the shaded region in fig. 3).
Figure 3:
Intersection of two Sets
Properties of the intersection of sets operation:
Union of Two Sets in Venn Diagram
A union B is given by: A ∪ B = {x | x ∈A or x ∈B}.
This represents the combined elements of set A and B (represented by the shaded region in fig. 4).
Figure 4:
Union of two sets
Some properties of Union operation:
Complement of Union of Sets in Venn Diagram
(A ∪ B)’: This is read as complement of A union B. This represents elements which are neither in set A nor in set B (represented by the shaded region in fig. 5).
Figure 5:
Complement of A U B
Complement of Intersection of Sets in Venn Diagram
(A ∩ B)’: This is read as complement of A intersection B. This represents elements of the universal set which are not common between set A and B (represented by the shaded region in fig. 6).
Figure 6:
Complement of A ∩ B
Difference between Two Sets in Venn Diagram
A – B: This is read as A difference B. Sometimes, it is also referred to as ‘relative complement’. This represents elements of set A which are not there in set B(represented by the shaded region in fig. 7).
Figure 7:
Difference between Two Sets
Symmetric difference between two sets in Venn Diagram
A ⊝ B: This is read as a symmetric difference of set A and B. This is a set which contains the elements which are either in set A or in set B but not in both (represented by the shaded region in fig. 8).
Figure 8:
Symmetric difference between two sets
Related Articles
Venn Diagram Example
Example: In a class of 50 students, 10 take Guitar lessons and 20 take singing classes, and 4 take both. Find the number of students who don’t take either Guitar or singing lessons.
Solution:
Let A = no. of students who take guitar lessons = 10.
Let B = no. of students who take singing lessons = 20.
Let C = no. of students who take both = 4.
Now we subtract the value of C from both A and B. Let the new values be stored in D and E.
Therefore,
D = 10 – 4 = 6
E = 20 – 4 = 16
Now logic dictates that if we add the values of C, D, E and the unknown quantity “X”, we should get a total of 50 right? That’s correct.
So the final answer is X = 50 – C – D – E
X = 50 – 4 – 6 – 16
X = 24 [ Answer]
Venn’s diagrams are particularly helpful in solving word problems on number operations that involve counting. Once it is drawn for a given problem, the rest should be a piece of cake.
Venn Diagram Questions
Out of 120 students in a school, 5% can play Cricket, Chess and Carroms. If so happens that the number of players who can play any and only two games are 30. The number of students who can play Cricket alone is 40. What is the total number of those who can play Chess alone or Carroms alone?
Draw the diagram that best represents the relationship among the given classes:
Animal, Tiger, Vehicle, Car
At an overpriced department store, there are 112 customers. If 43 have purchased shirts, 57 have purchased pants, and 38 have purchased neither, how many purchased both shirts and pants?
In a group, 25 people like tea or coffee; of these, 15 like tea and 6 like coffee and tea. How many like coffee?
To learn more from sets and other topics, download BYJU’S – The Learning App and explore the visual power of learning.
Frequently Asked Questions on Venn DiagramQ1 What do you mean by venn diagram?Venn diagram is a diagram used to represent all possible relations of different sets.Q2 How do Venn diagrams work?A Venn diagram is an illustration that uses circles to exhibit the relationships among things or finite groups of items. Circles that overlap have a commonality, while circles that do not overlap do not share those attributes. Thus, Venn diagrams help to describe the similarities and differences between the given concepts visually.Q3 How do you do Venn diagrams in math?In maths, Venn diagrams are used to represent the relationship between two or more sets. This way of expressing sets or certain situations helps in better understanding the scenario to conclude easily. Thus, we generally use circles to exhibit the relationships among things or finite groups of items.Q4 What does A ∩ B mean?In mathematics, the intersection of two sets A and B, denoted by A ∩ B, contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A).Q5 What are the different types of Venn diagrams?The Venn diagram types are defined based on the number of sets or circles involved in the Universal set or the rectangular space. They are:
Two-set Venn diagram
Three-set Venn diagram
Four-set Venn diagram
Five-set Venn diagramQ6 What are the four benefits of using Venn diagrams?The four benefits of using Venn diagrams include the following:
These are used for the classification and comparison of two things.
Different parameters can be obtained without much calculation, such as intersection, union, difference, etc.
Due to its pictorial representation, one can understand the given situation quickly.
These will make the complex calculation much simpler that involves sets.
Venn’s diagrams are particularly helpful in solving word problems on number operations that involve counting. Once it is drawn for a given problem, the rest should be a piece of cake.
Venn Diagram Questions
To learn more from sets and other topics, download BYJU’S – The Learning App and explore the visual power of learning.
Frequently Asked Questions on Venn DiagramQ1 What do you mean by venn diagram?Venn diagram is a diagram used to represent all possible relations of different sets.Q2 How do Venn diagrams work?A Venn diagram is an illustration that uses circles to exhibit the relationships among things or finite groups of items. Circles that overlap have a commonality, while circles that do not overlap do not share those attributes. Thus, Venn diagrams help to describe the similarities and differences between the given concepts visually.Q3 How do you do Venn diagrams in math?In maths, Venn diagrams are used to represent the relationship between two or more sets. This way of expressing sets or certain situations helps in better understanding the scenario to conclude easily. Thus, we generally use circles to exhibit the relationships among things or finite groups of items.Q4 What does A ∩ B mean?In mathematics, the intersection of two sets A and B, denoted by A ∩ B, contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A).Q5 What are the different types of Venn diagrams?The Venn diagram types are defined based on the number of sets or circles involved in the Universal set or the rectangular space. They are:
Two-set Venn diagram
Three-set Venn diagram
Four-set Venn diagram
Five-set Venn diagramQ6 What are the four benefits of using Venn diagrams?The four benefits of using Venn diagrams include the following:
These are used for the classification and comparison of two things.
Different parameters can be obtained without much calculation, such as intersection, union, difference, etc.
Due to its pictorial representation, one can understand the given situation quickly.
These will make the complex calculation much simpler that involves sets.
Frequently Asked Questions on Venn Diagram
What do you mean by venn diagram?
How do Venn diagrams work?
How do you do Venn diagrams in math?
What does A ∩ B mean?
What are the different types of Venn diagrams?
What are the four benefits of using Venn diagrams?
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189448 | https://laryngopedia.com/glottis/ | Glottis - Laryngeal Anatomy (with photos)
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Glottis
The glottis is often described as the triangular space between the vocal cords, during breathing, when a person is said to have an “open glottis.” In common usage, the term “glottis” often includes the cords as well. A “glottal stop” is the sound made when the vocal cords close abruptly to stop the voice.
The true vocal cords that define the lateral boundaries of the glottis are comprised of two main parts: The membranous cord/glottis (anterior 2/3 of its length) and the cartilaginous cord/glottis (posterior 1/3).
Membraneous Glottis vs. Cartilaginous Glottis
The membranous glottis/cords (M) are what vibrate to produce voice. The cartilaginous glottis/cords (C) do not vibrate, because this part of each vocal cord is inhabited by the arytenoid cartilage with only a thin covering of tissue. The inside of the larynx is often looked at with a mild fisheye camera, and depending on the angle of viewing, the proportions of membranous and cartilaginous segments of the cords may seem different from the usual two-thirds vs. one-third.
Glottis IllustrationCartilaginous + Membranous glottis-paralyzedHeel and Toe of arytenoid cartilages
Disorders of the Glottis
How does disorder of the membranous glottis come to attention? Through a change or loss of voice, most commonly due to overuse vibratory injury such as swelling, nodules, or polyps. Or voice change may result from infection (viral, bacterial, or fungal) causing laryngitis. Or, a person might become hoarse due to growths like papillomas, or cancer due to tobacco use.
If the cartilaginous glottis cords make no positive contribution to voice, how does it come to the attention of a patient or doctor? Mainly when it suffers injury, most commonly due to a longterm breathing tube in a mechanically ventilated patient. When injured, the perichondrium (covering layer of the cartilage) may respond by generating granulation tissue. See also contact granuloma. And when severe, movement of the cricoarytenoid joint may be impaired.Tumors of the cartilaginous glottis/cords are rare. See also post-intubation phonatory insufficiency.
Bilateral Vocal Cord Fixation
Bilateral vocal cord fixation (1 of 2)
After bilateral posterior cordotomies, “keyhole” shaped glottis. Size of opening at posterior commissure is exaggerated by the wide-angle lens of the distal-chip video-endoscope.
Bilateral vocal cord fixation (1 of 2)
After bilateral posterior cordotomies, “keyhole” shaped glottis. Size of opening at posterior commissure is exaggerated by the wide-angle lens of the distal-chip video-endoscope.
Voice is still good (2 of 2)
Persistent posterior opening, but musculomembranous cords come into good contact and have not been damaged by either the original endotracheal tube or the posterior cordotomies, and voice is very serviceable.
Voice is still good (2 of 2)
Persistent posterior opening, but musculomembranous cords come into good contact and have not been damaged by either the original endotracheal tube or the posterior cordotomies, and voice is very serviceable.
Introduction to Larynx, Pharynx, and Airway Anatomy
In this presentation, Dr. Robert Bastian provides an introduction to laryngeal anatomy. This information is meant to serve as an educational resource.
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189449 | https://www.frontiersin.org/journals/medicine/articles/10.3389/fmed.2024.1399429/full | REVIEW article
Front. Med., 03 May 2024
Sec. Gastroenterology
Volume 11 - 2024 |
Acute upper and lower gastrointestinal bleeding management in older people taking or not taking anticoagulants: a literature review
Acute upper and lower gastrointestinal (GI) bleeding may be a potentially life-threatening event that requires prompt recognition and an early effective management, being responsible for a considerable number of hospital admissions. Methods. We perform a clinical review to summarize the recent international guidelines, helping the physician in clinical practice. Older people are a vulnerable subgroup of patients more prone to developing GI bleeding because of several comorbidities and polypharmacy, especially related to an increased use of antiplatelet and anticoagulant drugs. In addition, older patients may have higher peri-procedural risk that should be evaluated. The recent introduction of reversal strategies may help the management of GI bleeding in this subgroup of patients. In this review, we aimed to (1) summarize the epidemiology and risk factors for upper and lower GI bleeding, (2) describe treatment options with a focus on pharmacodynamics and pharmacokinetics of different proton pump inhibitors, and (3) provide an overview of the clinical management with flowcharts for risk stratification and treatment. In conclusion, GI is common in older patients and an early effective management may be helpful in the reduction of several complications.
Introduction
Gastrointestinal (GI) bleeding is one of the most frequent gastroenterological conditions that require medical attention. Its incidence varies with age, with older patients being more frequently affected. The severity of gastrointestinal bleeding can vary from a mild form to a potentially threatening life condition. The estimated prevalence for overall GI bleeding is approximately 3.1% (1). The origin of the bleeding defines its clinical presentation and definition, with hemorrhages originating before the Treitz ligament being classified as upper GI bleeding and those originating after this landmark as lower GI bleeding.
Upper GI bleeding in the older population is frequently associated with gastric and duodenal ulcer or esophagitis, being responsible for the episode in 80% of the cases (2). The incidence of upper GI bleeding can also vary depending on the geographical region. In Northern Europe, the annual incidence ranges from 213 to 570 per 100,000 patients (3); in a UK cohort of older patients affected with acute upper GI bleeding, the rate was 63% in those above 60 years of age (4); and in a North American cohort, this rate ranged between 35 and 45% (5). Although the incidence of upper GI bleeding in older patients is high, there is an even higher incidence of lower GI bleeding (6).
The incidence of hospitalizations for upper GI bleeding increased with age, being 197.4 per 100,000 population between 66 and 75 years of age and rising to 425.2 per 100,000 in people older than 75 years in a North American Cohort (7). In addition, the risk of readmittance after a first hospitalization remains higher than that of the general population.
As the risk of complications increases, so does the risk of mortality in a patient aged over 60 years experiencing an upper GI bleeding episode, estimated to be between 12 and 25%, compared to the 10% of the general population (2). In a study based on the Welsh population, the case fatality for an upper GI bleeding ranged from 11.2 to 21.5% in men above 65 years of age and from 9.1 to 20.7% in women, increasing with age (8).
Lower GI bleeding is a condition that most frequently appears in older patients, with an incidence that also depends on geographic and socio-economic factors and comorbidities (3). The most common etiologies are diverticulosis, ischemic colitis, colitis, hemorrhoids, and colorectal cancer (3). In Northern Europe, the incidence rate for lower GI bleeding ranges from 2.41 to 3.64 in male patients and from 1.72 to 3.10 in female patients, increasing with age (9). In a Spanish study, the incidence of new lower GI bleeding was between 100 and 150 per 100,000 patients in 2005 (6).
With the higher frequency of comorbidities, this cohort of patients also has an increased risk of hospitalization and longer in-hospital stay (10), with an estimated rate between 127.7 and 380.1 per 100,000 population, increasing with age (7). A multicentric study in a European cohort shows that this cohort of patients also suffers from an increased mortality rate, where the hospital mortality for lower GI bleeding was 2.5 and 1.17% in the following 3 months (1).
Pharmacological history
Medication history should be carefully reviewed upon admission, as many drugs may be associated with gastrointestinal bleeding.
First, anticoagulants and antiplatelets increase bleeding risk (11, 12). Their prescription is very common, as it has been reported that about half of the patients presenting for an upper gastrointestinal bleeding (UGIB) were treated using antithrombotic drugs (13). Clinicians should assess the risk–benefit ratio of antithrombotic administration in this setting. Indeed, as shown by the ASPREE trial enrolling 19,114 healthy older people (>65 years) patients without previous cardiovascular disease, the administration of low-dose aspirin increased the risk of major bleeding (hazard ratio [HR] 1.38, 95% confidence interval [95%CI] 1.18–1.62) and UGIB (HR 1.87, 95%CI 1.32–2.66) without an improvement of cardiovascular disease prevention (14). Furthermore, recent European guidelines advise that the use of aspirin for primary cardiovascular prophylaxis should be discontinued in patients who have a confirmed UGIB, although it should be continued for secondary prevention (15). Similarly, people taking warfarin should have the medication stopped along with anticoagulant reversal in situations of severe UGIB, while continued anticoagulation must be evaluated in cases of less severe UGIB (15).
Second, non-steroidal anti-inflammatory drugs (NSAIDs) are commonly prescribed in adult people; they represent a common risk factor for gastric hemorrhage. Their primary mechanism is the inhibition of cyclooxygenase 1 (COX-1); as a consequence, there is a great reduction of prostaglandin production, which leads to poorer protection of gastric mucosa. Coadministration of aspirin and other NSAIDs have been associated with greater gastric damage because of combined inhibition of both COX-1 and COX-2 mucosae-protective pathways (16).
Beyond these medications, other drugs have been associated with a potentially increased risk of GI bleeding, such as serotonin-selective reuptake inhibitors and calcium channel blockers; therefore, patients should be questioned whether they have been prescribed these drugs as well (17, 18).
Moreover, drugs per se may induce upper bleeding by causing pill esophagitis. A large number of medications have been associated with this phenomenon, with antibiotics, NSAIDS, and bisphosphonates being the most common (19). This occurrence is more frequent in older adults with a reduced esophageal transit; therefore, it should always be excluded in case of retrosternal pain, odynophagia, and drug assumption before sleeping up to 3 days before and whether the real cause of bleeding has not been identified yet (19).
Finally, the patient should be asked about recent consumption of products that may change stool appearance and make it look similar to real bleeding (i.e., iron, bismuth, liquorice, and charcoal), in order to avoid a wrong diagnosis in the initial stages (20, 21).
Pharmacological treatment
Anti-acid treatment
Pharmacological suppression of gastric acid production is routinely performed in patients with UGIB upon admission. This effective strategy is based on the reason that the coagulation process benefits from a higher than normal gastric pH, which, therefore, leads to a better control of hemorrhage (22).
However, despite this physio-pathological mechanism, not all acid-suppressive strategies have proved to be equal; a meta-analysis of 11 studies highlighted that H2 receptor antagonists are less efficient at stopping recurrent or persistent hemorrhage than proton pump inhibitors (PPIs), in particular for Forrest Ia, Ib, or IIa ulcers (23). For this reason, although there is evidence for a potential role of this drug class in ulcer prevention (24), it is not routinely prescribed in this setting.
There is no clear evidence on the best schedule of PPI use in terms of time, type of administration, and type of PPI. Intravenous (IV) formulation is usually preferred; however, if not available, the oral formulation may be used as well, as it has been suggested to have a similar effectiveness (25).
A recent Cochrane meta-analysis of RCTs has investigated the pre-endoscopic PPI role. There are insufficient data to determine high-certainty evidence; however, it seems that PPI administered before endoscopic procedures may not reduce mortality and need for surgery (26). On the contrary, they may reduce rebleeding and the need for hemostatic treatment performed at index endoscopic procedures with low and moderate certainty of evidence, respectively (26).
Several PPIs are available and widely used (Table 1); however, there are no recommendations on the best one to administer in this setting. As reported, they have similar pharmacokinetic parameters such as high bioavailability, volume of distribution, and protein binding. However, some differences may induce a preference according to specific situations.
Table 1. Comparison of proton pump inhibitors: pharmacokinetics and special populations (27, 28).
Half-lives range from 0.5 to 3 h; given this short period and the fact that not all protonic pump inhibitors are targetable at the same time, a three-day period has been estimated to reach a steady-state inhibition of gastric acid (29). Despite the slightly different half-lives of PPIs, the drug of choice should not be determined on this feature as they all cause irreversible and durable inhibition of protonic pumps. Considering that preclinical models have shown that H+, K+-ATPase has a half-life of approximately 54 h (29), it must be considered that the effects of PPIs do not cease right after the last dose administration.
All the PPIs undergo hepatic metabolism; in particular, CYP2C19 plays a relevant role, followed by CYP3A4. This aspect should be taken into account as CYP2C19 genetic polymorphisms may give an extensive or poor-metabolizer phenotype, which reflects in a lower or higher PPI exposure, respectively (29); in case of genetic variants, esomeprazole should be preferred to omeprazole as its metabolism is less influenced by CYP2C19 polymorphisms (30).
PPIs are eliminated by the kidney; this is less true for lansoprazole, as the kidney accounts only for 14–23% of its elimination. In any case, no dosage adjustment for kidney function is required, regardless of PPI type and severity of the renal disease.
PPI treatment is usually chronic after the acute bleeding episode; older people commonly have concomitant diseases, which require other treatments as well. It has been estimated that 55–98% of people over 65 years old have at least two comorbidities (31). A higher disease burden implies a higher number of medications prescribed, thus leading to a higher risk of interactions among drugs. PPIs may interact with other medications in several ways.
First, PPI-induced modulation of gastric pH may result in reduced bioavailability of certain drugs administered orally; as an example, it has been reported that coadministration of omeprazole may lower bioavailability of methotrexate, ketoconazole, mycophenolate mofetil, and protease inhibitors by affecting their solubility; for this reason, their pharmacokinetic profile may be altered (32).
Second, a certain potential of PPIs to interact with intestinal P-glycoprotein (P-gp) cannot be excluded; this may be an issue, as many P-gp substrates, such as digoxin, nifedipine, amitriptyline, and tacrolimus, are widely administered in people over 65 years old; for this reason, coadministration must be carefully monitored (33).
As previously stated, PPIs are mainly metabolized by CYP2C19; this may lead to drug–drug interactions (DDI) with pharmacological agents that are substrates of the same enzymes. Among the PPIs, omeprazole has a higher DDI potential given its strong affinity for CYP2C19 and CYP3A4; it has been reported to have interaction with diazepam, moclobemide, phenytoin, and warfarin (32). In case of concomitant drugs, which may have an interaction, there is some evidence that pantoprazole, rabeprazole, and lansoprazole may carry a weaker risk of interactions (32).
Interaction between omeprazole/esomeprazole and clopidogrel is of high clinical relevance in this setting and should be monitored: It has been demonstrated that concomitant administration of omeprazole is associated with a lower exposure to active clopidogrel metabolite, regardless of a possible double dose of clopidogrel or a 12-h time period between the administrations of the two drugs (32, 34). In this case, pantoprazole should be preferred given its lower influence on clopidogrel metabolism.
Beyond pharmacological interactions, numerous studies have shown that PPIs are linked to an increased risk of a variety of negative effects, including Clostridium difficile infection, osteoporotic-related fractures, renal impairment, community-acquired pneumonia, vitamin B12 deficiency, and dementia (35). Given the range of potential side effects linked to long-term use in older people, an assessment for the need to continue PPI therapy should be routinely conducted.
This is the reason why guidelines for PPI deprescribing have been developed. A periodic assessment of PPI indications should be conducted regularly, to lower exposure; however, discontinuation is not indicated for patients with severe gastro-esophageal diseases (36).
Vasoactive drugs
Pharmacological agents acting on vasoconstriction (e.g., terlipressin, octreotide, and somatostatin) are recommended in addition to endoscopy in patients with UGIB from varices (or at risk for varices) (37); therapy should last from presentation to 3–5 days after bleeding cessation.
As a class, they are associated with a better hemostasis, lower need of blood transfusions, and a lower risk of 1-week mortality (38).
Among the others, terlipressin (2 mg IV q4h and then 1 mg IV q4h) has proved a 34% relative risk reduction in mortality (39). From a pharmacological point of view, terlipressin is a vasopressin analog, which acts by constricting mesenteric artery; this leads to a lower portal venous flow and, therefore, to a lower portal pressure.
Octreotide is a synthetic analog of somatostatin, a hormone which reduces release of vasodilators, thus causing a reduced portal inflow. It is administered by bolus 50 mcg IV, followed by continuous infusion (CI) 50 mcg IV each hour, and is the most widespread choice in USA in these cases, as terlipressin is not available in this country. Compared to somatostatin, octreotide has a longer half-life, but causes a similar prompt reduction of variceal pressure; however, despite adding continuous infusion, these effects only last some minutes, probably due to a pharmacodynamic desensitization (40). Nevertheless, a longer term effect mediated by other pathways may not be excluded (41).
When compared with octreotide, in a randomized controlled trial (RCT) of cirrhotic patients, terlipressin has shown a longer effect in reducing portal pressure (42); therefore, it should be preferred if available.
No vasoactive treatments should be used in place of endoscopic variceal ligation.
Prokinetics
Prokinetic drugs may be administered as they help in cleaning the stomach from blood clots and other residues, thus allowing endoscopist have a better visualization of active bleeding sources.
Erythromycin has been studied in this setting due to its role as motilin receptor agonists. A meta-analysis of RCT has proved that it may improve visualization of gastric mucosa (43), while another showed that it is statistically associated with a lower rate of second-look endoscopies and a shorter length of stay in hospital (44). There is no clear evidence on whether adding erythromycin has a further benefit compared to nasogastric tube lavage only (43).
Notably, erythromycin is a strong CYP3A4 inhibitor: In older patients, this may represent an issue as concomitant drugs are often administered and their metabolism may be altered (45). Similarly, it has been associated with QTc prolongation and a higher risk of torsades de pointes, which should be taken into account in this population (46). Obviously, the shorter the exposure to this drug, the lower the risk of clinically significant DDI, which, however, may not be excluded.
It may be argued that, given the similar mechanism, metoclopramide may have a role as well. However, previous evidence already discussed metoclopramide role and found no effect. For this reason, despite a similar function, erythromycin is currently preferred to metoclopramide in UGIB (47).
Antibacterial treatment during GI bleeding
Patients with cirrhosis and GI bleeding are frequently diagnosed with bacterial infections; approximately 22% of patients develop an infection in the first 2 days of hospital stay, while this incidence peaks up to 66% considering the first 2 weeks (48). For this reason, an antibiotic prophylaxis is usually administered in cirrhotic patients with gastrointestinal hemorrhage. A large spectrum antibiotic prophylaxis has been associated with a lower mortality, rate of bacterial infections, rebleeding rate, and length of hospitalization (49). Broad-spectrum antibiotics should be started before endoscopy and administered for up to 7 days (37).
A usual choice may be ceftriaxone 1 g once daily (OD) IV; If the patient is discharged before a week, change to ciprofloxacin 500 mg bis in die (BID) may be an alternative, although ceftriaxone has proven to be superior (37). However, drug should be selected considering patients characteristics, such as comorbidities and previous exposure to antibiotics, as people above the age of 65 years may have hepatic or renal impairments, which may affect drug metabolism and elimination; for instance, a pharmacokinetic study on older patients with moderate-to-severe impairment in renal function has highlighted a greater ceftriaxone exposure with a 48 h dosing schedule (50). Similarly, bacterial characteristics should be considered as well, as a local pattern of ceftriaxone and quinolone resistance in cirrhotic patients have been reported (51, 52).
However, despite antibiotic administration, bacterial infections still occur in approximately one-fifth of cirrhotic patients admitted for variceal bleeding; therefore, it still remains a crucial issue that should be carefully taken into account (53).
Management of acute upper GI bleeding
Upper gastrointestinal bleeding (UGIB) is defined as hemorrhage proximal to the Treitz ligament involving the esophagus, stomach, and duodenum (54). The most common symptoms of UGIB are melena, hematemesis, and coffee ground vomiting (54). Hematochezia, instead, is a rare manifestation of UGIB and is commonly a presentation of lower gastrointestinal bleeding (LGIB) (55, 56). Systemic manifestations, in major and life-threatening gastrointestinal (GI) bleeding (both UGIB and LGIB), include hemodynamic instability, hypotension, abdominal pain associated with lethargy, fatigue, syncope, and angina (56, 57).
The most common cause of UGIB is peptic ulcer disease, involving approximately 32–36% of all hospitalized patients; then esophagitis, duodenitis, and gastritis (until 24% of hospitalization); and finally variceal bleeding (approximately 11% of hospitalization, but 90% of UGIB in patients with liver cirrhosis) (4, 56, 58, 59).
The incidence of UGIB is widely different among countries ranging between 67 and 172/100.00 person with similar rates between Europe and the United States (59–61). Although hospitalizations for UGIB have declined due to H. pylori eradication, the use of proton pump inhibitors, and increased access to endoscopy (59, 62, 63), the mortality rate of UGIB is approximately 2–10% (54).
Antiplatelet and anticoagulant use, non-steroidal inflammatory drugs, corticosteroids, liver cirrhosis, the presence of multiple comorbidities, and older age are common risk factors for gastrointestinal bleeding, especially of UGIB (1, 6, 56, 62).
Several guidelines suggest general recommendations for initial management of UGIB (a flowchart of UGIB management is proposed in Figure 1). First, patients with UGIB should be guaranteed an IV access by cannula (≤18 G) in each antecubital fossa and an early fluid resuscitation (37) should be started, reducing the risk of mortality and myocardial infarction (54), achieving 90–100 mmHg systolic blood pressure as target (64, 65). In particular, a first approach with 500 mL of crystalloids infused in less than 15 min are suggested as first choice in hemodynamically unstable patients (66), although studies showed no difference between colloids and crystalloids in fluid resuscitation during UGIB (54).
Figure 1. Flowchart of upper gastrointestinal bleeding management. IV, intravenous; NSBB, non-selective beta-blockers; PLTs, platelets; PPI, proton pump inhibitors; RBC, red blood cells; SBC, systolic blood pressure; TIPS, transhepatic intrajugular portosystemic shunt; UGIB, upper gastrointestinal bleeding.
During resuscitation, transfusion of packed red cells should be performed with a restrictive approach using a threshold of 7 g/dL (<8 g/dL in patients with cardiovascular disease) for hemoglobin (Hb) (37, 54) and transfusion of platelets should be performed using a threshold of 50 × 109/L (67, 68). After resuscitation, treatment of UGIB is divided into three phases: pre-endoscopic, endoscopic, and post-endoscopic phases.
Pre-endoscopic phase
General recommendations
In this phase (Figure 1), if patients are hemodynamically stable, erythromycin (250 mg IV infusion approximately 30–120 min before endoscopic procedures) was administered: Indeed, erythromycin, as a prokinetic agent, improves visualization during the endoscopy procedure resulting in a lower length of hospital stay, a lower rate of re-intervention, and less need for blood transfusions (37, 54). Furthermore, a large meta-analysis conducted by the Cochrane Institute showed that PPI may be useful (26) and performed on 2,223 patients included in six RCTs. Indeed, there is moderate-certainty evidence that PPI started before endoscopy for UGIB likely reduces the requirement for endoscopic hemostatic treatment. However, there is insufficient evidence to conclude whether PPI had a role on mortality, rebleeding, and need for surgery.
On the other hand, all guidelines recommended against the use of tranexamic acid in UGIB due to high risk of venous thromboembolism without an improvement on mortality (37, 54, 69).
Variceal bleeding
Patients with compensated advanced chronic liver disease and clinically significant portal hypertension defined as hepatic venous pressure gradient>10 mmHg and/or liver stiffness by transient elastography >25 kPa should be treated for nonselective beta blocker (NSBB) as carvedilol to prevent the development of variceal bleeding (37). For patients unsuitable for NSBB and with high-risk esophageal varices, endoscopic band ligation is the endoscopic prophylactic treatment of choice (37). Of note, in patients with advanced chronic liver disease and portal vein thrombosis, an anticoagulation treatment, if not contraindicated, may be helpful to prevent variceal bleeding: indeed, in a large meta-analysis (70), there were no differences in major or minor bleeding in patients treated or not treated with anticoagulants (11% for both groups), but a lower rate of variceal bleeding was observed in patients taking anticoagulants, maybe due to thrombus resolution in portal vein (70).
In patients with suspected variceal bleeding (37, 54), the use of vasoactive agents such as terlipressin, octreotide, or somatostatin at hospital admission is recommended and continued for a duration of up to 5 days. Furthermore, antibiotic prophylaxis is also recommended in patients with UGIB by suspected esophageal varices (37, 54). In particular, the European Society of Gastrointestinal Endoscopy (ESGE) suggests the use of ceftriaxone 1 g/day for up to 7 days for all patients with suspected variceal bleeding (or in accordance with local antibiotic resistance and patient allergies) (37). In addition, all patients should be stratified according to CHILD-PUGH and MELD scores and the endoscopic evaluation should take place within 12 h from the time of patient presentation/fluid resuscitation.
Endoscopic phase
Predictive pre- and post-endoscopic scores were developed during several years. In particular, pre-endoscopic score may help the physician to choose the optimal management of patients with UGIB evaluating an outpatient approach and estimating the risk of complications and death. In Table 2, we summarize the items of pre-endoscopic risk assessment scores recommended and validated in clinical practice (pre-endoscopic Rockall score, AIMS-65, and Glasgow Blatchford score) (54, 71–73). Recently, a simple ABC score (74) was proposed (Table 2), but not sufficient data are available to recommend it in clinical practice (54).
Table 2. Pre-endoscopic risk score for upper gastrointestinal bleeding.
In particular, a recent multicenter study involving 3,012 consecutive patients with UGIB showed that the Glasgow-Blatchford score has high accuracy at predicting need of hospitalization or death. Furthermore, a score of ≤1 is the optimum threshold for choose an outpatient management (75). For this reason, international guidelines recommend this score as first choice (37, 69, 76, 77).
In patients with UGIB candidate to endoscopy in emergency setting, this should be performed within 24 h of presentation (within 12 h if variceal bleeding is suspected) (37) and hemostatic endoscopic treatment is recommended only for ulcers with active spurting, active oozing, and non-bleeding visible vessels (37), while it is unclear whether endoscopic hemostatic treatment is useful for ulcers with adherent clot resistant to vigorous irrigation (37). No endoscopic treatment indicated whether only flat pigmented spots or clean base is found during endoscopy (69).
Finally, in patients with recurrent bleeding, after previous successfully endoscopic procedure, a new endoscopic treatment with clips is recommended, although with low quality of evidence (37).
Post-endoscopic phase
In the post-endoscopic phase, medical therapy should be administered to reduce the risk of rebleeding and death. All guidelines recommend the use of high dose of PPIs (54) without the difference between continuous and intermittent regimen (37) (Figure 1).
The American College of Gastroenterology (ACG) guidelines recently suggest a medical therapy for UGIB based on endoscopic features (69). While the treatment of active ulcers or adherent clot findings is coherent with other guidelines (a high-intensity PPIs: for continuous regimen, 80 mg bolus followed by 8 mg/h infusion for 3 days and for intermittent regimens, 40 mg 2–4 times daily for 3 days, orally if feasible, after an initial bolus of 80 mg) (69), ACG guidelines for flat pigmented spot or clean base suggest standard dose-regimen PPI (69).
After high-dose PPIs, in patients undergoing hemostatic treatment, a further 2-week treatment with twice-daily PPIs is recommended to reduce rebleeding risk (37).
In addition, in case of proven variceal bleeding, vasoactive drugs should be continued for 2 to 5 days (54) (Figure 1).
In patients with variceal bleeding at high risk of recurrent bleeding following successful endoscopic hemostasis, pre-emptive transjugular intrahepatic portosystemic shunt (TIPS) within 72 h (preferably within 24 h) must be considered (37). NSBBs (propranolol or carvedilol) in combination with endoscopic therapy for secondary prophylaxis should be continued in patients with advanced chronic liver disease and/or and previous esophageal variceal bleeding.
Management of acute lower GI bleeding
Lower gastrointestinal bleeding (LGIB) represents the 3% of emergency surgical referrals (78), and its incidence is estimated to be 33–87 for 100.000 patients (79). The mortality is 3.4% rising 18–20% in patients with LGIB during hospitalization (79). The most common cause of LGIB is diverticular bleeding, followed by benign anorectal conditions such as hemorrhoids, fissures, and rectal ulcers (79). Other common causes are telangiectasia in multiple sites of GI tract, colitis, and colorectal cancer. Of note, 23% of hospitalized patients with LGIB in the UK are discharged without a diagnosis (80). Patients with LGIB should be clinically evaluated to establish the hemodynamic stability. Clinical history (bleeding history and comorbidities), clinical evaluation (including digital-rectal exploration), laboratory test, and concomitant therapy are needed to establish the hemodynamic status of patients (81). In particular, shock index (heart rate [HR] and systolic blood pressure [SBP] ratio) is recommended by current guidelines (79). A shock index >1 defined the patient as hemodynamically unstable (79).
Similar to UGIB, a resuscitation strategy (previous described) should be performed in unstable patients (79). In these patients, a computed tomography (CT) scan with angiography should be performed to evaluate the focus of bleeding; then, patients should undergo to interventional radiology (preferably <60 min from hospital admission) or endoscopy (79). Although endoscopy treatment represents the first line of treatment from international guidelines (79, 81, 82), only 2.1% of cases of LGIB undergo endoscopic treatment and the most common intervention is red blood cell transfusions (79). If a treatment failure occurred during endoscopy or radiological intervention, surgery should be evaluated in selected cases. If no focus of bleeding is identified during CT scan with angiography, patients should be considered stable with major bleeding.
If shock index is <1, LGIB should be considered stable and Oakland score should be performed to establish whether major or minor bleeding occurred and whether hospitalization is required (Oakland score ≤ 8 suggests a possible outpatient management) (79).
Fluid resuscitation is needed, and a restricted red blood cell transfusion regimen should be preferred with a threshold of Hb <70 g/L with a target of 70–90 g/L after transfusion, except for patients with previous history of cardiovascular disease with a threshold of Hb <80 g/L with a target of 10.0 g/L (79), and platelets transfusion should be performed using a threshold of 50 × 109/L (82). Figure 2 summarizes the management of LGIB.
Figure 2. Flowchart of lower gastrointestinal bleeding management. CT, computed tomography; HR, heart rate; LGIB, lower gastrointestinal bleeding; SBP, systolic blood pressure.
In major bleeding, if patient is hemodynamically stable, colonoscopy should be performed after adequate colon cleansing (a nasogastric tube may help colon preparation in patients with a low risk of aspiration and ongoing bleeding) (82). In total, 4–6 liters of a polyethylene glycol (PEG)-based solution or the equivalent should be administered over 3–4 h until the rectal effluent is clear of blood and stool (82).
In patients with high-risk clinical features or ongoing bleeding, endoscopy should be performed within 24 h of patient’s admission to the emergency department, after an adequate colon preparation; otherwise, a colonoscopy should be performed next available after a colon purge (82). Further investigations are needed if no bleeding focus was found, such as CT scan with angiography repetition or use of video capsules (79).
Management of GI bleeding in the older patients
In older people, the severity and prognosis of GI bleeding are influenced by medical comorbidities (1) and therapies as well as the use of antiplatelet and anticoagulants medication (83). Indeed, 70% of UGIB occurred in patients >60 years old and its incidence and mortality risk rise according to age (83). Similarly, patients with LGIB are more common in older patients, with a mean onset age between 63 and 77 years, with higher mortality risk (83). In addition, GI bleeding incidence seems to be reduced only in patients <70 years old (61). For these reasons, GI hemorrhage management is a backbone in the older care and there are some peculiarities of old age that should be addressed.
First, endoscopy, the first line of diagnosis and treatment for GI hemorrhage, has similar mortality risk in older patients compared to general population and old age is not a contraindication to endoscopy (83). Of note, older patients had an increased risk to developing adverse events and oxygen desaturation, especially if benzodiazepines (BDZ) are administered during endoscopy for sedation (83, 84); for this reason, a lower dose of BDZ with careful titration is suggested (83).
In addition, older patients are more likely to be treated with antiplatelets, anticoagulants, especially with complex antithrombotic therapy (CAT) resulting in an increased risk of hospitalization and transfusion as shown in a large cohort study of 78,133 old veterans aged >60 years treated with antiplatelets and/or anticoagulants, with the highest risk of hospitalization and transfusion in patients treated with dual antiplatelet agents and anticoagulant (85).
In patients treated with antiplatelets, a GI bleeding incidence rate of 0.7–1.3% for aspirin and 1.2–2% for aspirin and clopidogrel combination during a follow-up of 1–2 years was observed in Western countries (86–88). In older patients, the GI bleeding incidence rate rise to 2.7% as shown by an observational study on 1852 patients undergone the implantation of drug-eluted stent (DES) with a mean age of 70.9 years (89).
Antiplatelet management had a fundamental role in older patients. If antiplatelet is administered in primary prophylaxis should be permanently discontinued, while antiplatelet in secondary prevention should not be stopped, but if suspension is needed, it should be restarted when hemostasis is guaranteed (79, 82).
In patients treated with dual antiplatelet therapy with aspirin and a P2Y12 inhibitor, P2Y12 should be stopped only in unstable hemorrhage and restarted within 5 days, especially if recent coronary stenting is performed (54, 79, 81).
In older patients, comorbidities such as atrial fibrillation (AF) that require indefinite anticoagulation are common. The most common oral anticoagulant prescribed in older patients is vitamin K antagonists (VKAs) and direct oral anticoagulants (DOACs).
In older patients (≥75 years), the incidence rate of GIB was 2.19% per year for dabigatran 110 mg and 2.80% per year for dabigatran 150 mg (90). Instead, an incidence GIB rate of 1.51 and 0.83% per year was observed for edoxaban 60 mg and 30 mg, respectively (90). The incidence of GIB rate of 2.0% for rivaroxaban and 0.76% for apixaban, but no data according to dose were found (90). The incidence of GIB for warfarin ranged between 0.86 and 1.59% per year in phase III clinical trials (90). A large cohort meta-analysis performed on 129.357 patients by Miller et al. summarized the risk of GIB in patients treated with DOACs and VKAs (pooled rate: 1.5% versus 1.3%, respectively; odds ratio [OR]: 0.98; 95%CI: 0.80–1.21) (91).
Recently, a large network meta-analysis on 605.771 AF patients showed a reduced risk of GIB in patients treated with apixaban compared to ones treated with dabigatran or rivaroxaban (92).
VKAs, as warfarin, should be discontinued during hemorrhage and should be restarted at 7 days after GI bleeding if low thrombotic risk (54, 79, 81). In patients with high thrombotic risk as well as patients with mechanical prosthetic heart valve, AF with prosthetic heart valve or mitral stenosis, <3 months after venous thromboembolism, low molecular weight heparin treatment should be started at 48 h after hemorrhage (54, 79). In patients on warfarin, although the optimal international normalized ratio (INR) to perform endoscopy is <1.3, endoscopy may be also considered when INR is <2.5 without a significantly increased risk of rebleeding (54).
On the other hand, DOAC should be stopped at presentation of GI bleeding and restarted within 7 days after GI bleeding (54, 79, 81).
In life-threatening GI bleeding and hemodynamically unstable patients, the interruption of oral anticoagulants is not enough, and a reversal agent is needed (Figure 3) (12, 54, 79, 81).
Figure 3. Reversal strategies for anticoagulants in major/life threatening gastrointestinal bleeding. 4F-PCC, 4-factor prothrombin complex concentrate; aPCC, activated prothrombin complex concentrate; FXa-I, inhibitors of FXa; GI, gastrointestinal; INR, international normalized ratio; PCC, prothrombin complex concentrate; VKA, vitamin K antagonist.
In VKAs, Vitamin K is a specific reversal agent in a dose-dependent manner (1 to 10 mg). Slow intravenous administration (in 25 to 50 mL normal saline over 15 to 30 min) causes a rapid reduction in the INR (4–6 h) (93). However, the administration of vitamin K does not result in immediate correction of coagulopathy, and in life threatening bleeding, vitamin K administration must be accompanied by the administration of 4-factor prothrombin complex concentrate (4F-PCCs), or, if not available, plasma (93). 4F-PCCs should be administered according to INR range and body weight (INR 2–4 at 25 U/kg, INR 4–6 at 35 U/kg, and INR >6 at 50 U/kg; max dose 5,000 U if 100 kg body weight) (93).
In DOACs, in dabigatran users, a reversal agent, idarucizumab is suitable (5gr + 2.5gr, IV) (93). However, in patients taking apixaban, rivaroxaban, and edoxaban, andexanet alfa may be useful. If these drugs are not available, 4F-PCC or activated PCC (aPCC) may be an alternative (50 U/kg) (12, 93).
Furthermore, a large cohort study of 3,166 patients treated with antiplatelet and without routine PPI use, due to previous myocardial infarction or cerebrovascular event, showed that the long-term risk of bleeding is higher in older patients than in younger patients with a substantial risk of disabling or fatal UGIB, suggesting that a co-prescription of PPI should be encouraged (94).
On the other hand, the use of PPI is associated with several adverse effects, as well as increased risk of fractures, osteoporosis, higher risk of Clostridium difficile (CD) infection and community-acquired pneumonia (CAP), especially in older patients (95).
In particular, a large meta-analysis including 2,181,546 individuals taking or not taking PPI showed that patients not taking PPI, those taking PPI, had an increased risk of developing any-site fractures, hip fractures, spine fracture, and osteoporosis (96), this evidence is confirmed independently from dose and duration of therapy, as suggested by a large meta-analysis on 2,103,800 patients showing a high risk of hip fracture in patients with long- and short-term therapy and in low, medium, and high dosage of PPI (97).
In addition, PPI is also associated with an increased risk of developing CD infection as shown in a large meta-analysis of 56 studies (40 case control and 16 cohort) involving 356,683 patients (98). This risk is estimated approximately 64% compared to ones not taking PPI (99) and probably is related to PPI-gut dysbiosis (100, 101) that increases also all-cause mortality (101).
Finally, several studies showed an increased risk of developing CAP in patients taking PPI compared to ones who are not taking these drugs, particularly within 30 days (102–104). A pathogenic mechanism has been proposed to explain the association between PPI use and the incidence of CAP: PPIs may increase the gastric pH altering also normal oropharyngeal flora, which could increase susceptibility to respiratory infections by permitting survival of pathogens that lead to CAP (104).
Osteoporosis, bone fractures, CD infection, and CAP may be deadly for older patients, and these complications should be avoided. For this reason, PPI in older patients should be used only according to clinical indications, with long-term treatments only for selected cases (36).
Conclusion
In conclusion, UGIB and LGIB represent a severe common complication, especially in older patients with comorbidities and on treatment with antiplatelet and/or anticoagulant drugs. Several drugs are available to reduce bleeding complications, especially for UGIB. Current evidence and guidelines suggest a clinical approach based on hemodynamic status with endoscopy as the first line for diagnosis and treatment.
Author contributions
DM: Conceptualization, Writing – original draft. GG: Writing – original draft. FS: Writing – original draft. AP: Writing – review & editing. PP: Supervision, Writing – review & editing. DP: Conceptualization, Supervision, Visualization, Writing – review & editing.
Funding
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
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Keywords: endoscopy, older population, anticoagulants, gastrointestinal bleeding, proton pump inhibitors
Citation: Menichelli D, Gazzaniga G, Del Sole F, Pani A, Pignatelli P and Pastori D (2024) Acute upper and lower gastrointestinal bleeding management in older people taking or not taking anticoagulants: a literature review. Front. Med. 11:1399429. doi: 10.3389/fmed.2024.1399429
Received: 11 March 2024; Accepted: 11 April 2024; Published: 03 May 2024.
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Reviewed by:
Copyright © 2024 Menichelli, Gazzaniga, Del Sole, Pani, Pignatelli and Pastori. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
Correspondence: Daniele Pastori, ZGFuaWVsZS5wYXN0b3JpQHVuaXJvbWExLml0
†These authors have contributed equally to this work
Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.
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189450 | https://math.answers.com/basic-math/Factors_of_1000000000 | Factors of 1000000000? - Answers
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Factors of 1000000000
Anonymous ∙ 13 y ago
Updated: 4/28/2022
The factors of 1000000000 are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1250, 1280, 1600, 2000, 2500, 2560, 3125, 3200, 4000, 5000, 6250, 6400, 8000, 10000, 12500, 12800, 15625, 16000, 20000, 25000, 31250, 32000, 40000, 50000, 62500, 64000, 78125, 80000, 100000, 125000, 156250, 160000, 200000, 250000, 312500, 320000, 390625, 400000, 500000, 625000, 781250, 800000, 1000000, 1250000, 1562500, 1600000, 1953125, 2000000, 2500000, 3125000, 3906250, 4000000, 5000000, 6250000, 7812500, 8000000, 10000000, 12500000, 15625000, 20000000, 25000000, 31250000, 40000000, 50000000, 62500000, 100000000, 125000000, 200000000, 250000000, 500000000, and 1000000000.
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### What is 5 percent of 1000000000? 5 percent of 1000000000 = 50000000 5% of 1000000000 = 5% 1000000000 = 5%/100% 1000000000 = 50000000
### What are the factors and prime factors of 1000000000? The factors of 1000000000 are the numbers that can be multiplied together to result in 1000000000. These factors include 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 800, 1000, 1250, 1600, 2000, 2500, 3125, 4000, 5000, 6250, 8000, 10000, 12500, 20000, 25000, 40000, 50000, 100000, 125000, 200000, 250000, 500000, and 1000000. The prime factors of 1000000000 are the prime numbers that can be multiplied together to result in 1000000000, which are 2, 5.
### What is 1 percent of 1 billion? 1 percent of 1 billion = 100000001% of 1000000000= 1% 1000000000= 0.01 1000000000= 10000000
### What are all prime numbers 1- 1000000000? -999999999
### What is 1000000000 more than 5438724022? 1000000000 more than 5438724022 = 6438724022
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Exercise 9: Define the vector v = [1 3 5 7]. Then use the vector in a mathematical expression to create the following vectors: a = [ 3 9 15 21] b = [ 1 9 25 49] c = [1 1 1 1] d = [6 6 6 6] Exercise 10: For the function, calculate the value of y for the following values of x using element-by-element operations: –2, –1.5, –1, –0.5, 0, 0.5, 1, 1.5, 2. y
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Question: Exercise 9: Define the vector v = [1 3 5 7]. Then use the vector in a mathematical expression to create the following vectors: a = [ 3 9 15 21] b = [ 1 9 25 49] c = [1 1 1 1] d = [6 6 6 6] Exercise 10: For the function, calculate the value of y for the following values of x using element-by-element operations: –2, –1.5, –1, –0.5, 0, 0.5, 1, 1.5, 2. y
Exercise 9:
Define the vector v = [1 3 5 7]. Then use the vector in a mathematical expression to create the following vectors:
a = [ 3 9 15 21]
b = [ 1 9 25 49]
c = [1 1 1 1]
d = [6 6 6 6]
Exercise 10:
For the function, calculate the value of y for the following values of x using element-by-element operations: –2, –1.5, –1, –0.5, 0, 0.5, 1, 1.5, 2.
y = (x+x x+3 )(1+2 x 2 )- x 3
solve using Matlab (typed answer would be perfect)
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For the first step of Exercise 9, define the vector v as [1,3,5,7]. % matlab script for exercise 9 clc clear all close all % defining vector v v = [ 1 3 5 7 ] %determing the vector v size [m,n ] =size(v); % performing mathematical expression on vector v for resulting a vector for i = 1:n if i == 1 a(i) = v(i)…
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189452 | https://www.youtube.com/watch?v=AQ6tUHOHiKs | How to Switch Between Radians and Degrees on the TI-84 Plus CE Calculator
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I will show you how to change the mode to radians and to degrees. We also do an example with a trigonometric function.
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Transcript:
Introduction hello in this video I'm going to show you how to switch back and forth between radian mode and degree mode in the ti84 plus C graphing calculator so I'm going to go ahead and turn it on by pressing the on button and in order to access the Accessing the Menu menu where you can go back and forth between radian and degrees you want to press uh basically mode right here and then here it says radian and here it says degrees so it's already in radian so let's go to degree and let's just press enter and then we want to leave this screen so you have to hit quit quit is a blue key though so you want to hit second first so second quit all right so now we're in Degrees degree mode so let's just do a computation let's do like the cosine of 90° so cosign there's cosine and then 9 0 okay and then parentheses and then enter and it gives you zero right the cosine of 90° is zero um now let's let's go back to radian mode and press enter okay and then again second quit Cosine and now let's do the cosine of pi over 2 we should get the same answer Pi is a blue key so you have to hit second first and then the fraction key so divided by and then two and then parentheses press enter and we get the same answer both times so again to switch between radian and degrees you just basically go to mode and then you can access it here and then to get out of this menu you hit second quit that's it I hope this video has been helpful and hopefully now you know how to switch back and forth between radian and degree mode in the ti84 plus CE graphing calculator good luck |
189453 | https://www.chemeurope.com/en/encyclopedia/Land%C3%A9_g-factor.html | Landé_g-factor
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Encyclopedia
Landé_g-factor
Landé g-factor
In physics, the Landé g-factor is a particular example of a g-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921.
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In atomic physics, it is a multiplicative term appearing in the expression for the energy levels of an atom in a weak magnetic field. The quantum states of electrons in atomic orbitals are normally degenerate in energy, with the degenerate states all sharing the same angular momentum. When the atom is placed in a weak magnetic field, however, the degeneracy is lifted.
The factor comes about during the calculation of the first-order perturbation in the energy of an atom when a weak uniform magnetic field (that is, weak in comparison to the system's internal magnetic field) is applied to the system. Formally we can write the factor as,
Here, J is the total electronic angular momentum, L is the orbital angular momentum, and S is the spin angular momentum. Because S=1/2 for electrons, one often sees this formula written with 3/4 in place of S(S+1). The quantities g L and g S are other g-factors of an electron, as defined in the article on g-factors.
If we wish to know the g-factor for an atom with total atomic angular momentum F=I+J,
This last approximation is justified because g I is smaller than g J by the ratio of the electron mass to the proton mass.
Category: Atomic physics
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Landé_g-factor". A list of authors is available in Wikipedia.
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189454 | https://www.quora.com/How-do-you-calculate-the-dot-product-of-two-vectors-in-R-n | How to calculate the dot product of two vectors in R^n - Quora
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How do you calculate the dot product of two vectors in R^n?
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Alexandru Carausu
Former Former University Associate Professor at Universitatea Tehnica "Gh. Asachi" Iasi (1966–2010) · Author has 3K answers and 875.5K answer views
·3y
If x = [ x 1 x _2 . . . x _ n ]^T and y = [ y _1 y_2 . . . y n ]^T are two vectors in R^n then
x • y = x _1 y _1 + x _2 y _2 + . . . + x _ n y_n ] . (1)
This dot product is a particular case of the more general inner product , for two vectors in the vector space V defined over a field F , which satisfies 3 axioms :
( IP_1 ) : ≥ 0 & = 0 ==>x = 0 .
( IP_2 ) : = and = conj if F = C = the complex field .
( IP_3 ) : ( ∀ λ_1 , . . . , λ_ m∈ F ) ( ∀ x 1 , . . . , x _ m ∈ V ) < ∑_1^m λ i x i , y > = ∑_1^m λ i .
Notes. In ( IP_1 ) , 0 is
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If x = [ x 1 x _2 . . . x _ n ]^T and y = [ y _1 y_2 . . . y n ]^T are two vectors in R^n then
x • y = x _1 y _1 + x _2 y _2 + . . . + x _ n y_n ] . (1)
This dot product is a particular case of the more general inner product , for two vectors in the vector space V defined over a field F , which satisfies 3 axioms :
( IP_1 ) : ≥ 0 & = 0 ==>x = 0 .
( IP_2 ) : = and = conj if F = C = the complex field .
( IP_3 ) : ( ∀ λ_1 , . . . , λ_ m∈ F ) ( ∀ x 1 , . . . , x _ m ∈ V ) < ∑_1^m λ i x i , y > = ∑_1^m λ i .
Notes. In ( IP_1 ) , 0 is the zero vector of space V ; if V = R^n then 0 = [ 0 0 . . . 0 ]^T .
In ( IP_2 ) , the first equation mean that the IP is commutative if F = R , while it is “conjugate-commutative” if F = C . We could not a bar on with the editing possibilities for answers to questions from Quora. ( IP_3 ) is the property of extended linearity in the first argument, and it also holds in the second argument if F = R . If F = C , the scalars of < ∑1^n x , μ i y _j > “get out” conjugated, as factors on . By the way, this property of linearity can be postulated for linear combinations of two vectors only, and its extended versions can be proved by induction / m , or / n . It can be easily checked that the definition (1) of the IP in R^n s atisfies the three axioms of a general IP.
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What does the dot product of 2 vectors represent?
How can I calculate the magnitude of a vector's product?
How do you calculate the angle between two vectors using the dot product?
How can you tell by using the dot product if your vectors are parallel?
How do you calculate the angle between two vectors using the dot product of matrices?
Marc Sulfridge
Engineering Member of Technical Staff - Process at ON Semiconductor (company) (2014–present) · Author has 59 answers and 142.5K answer views
·7y
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What exactly does the dot product of two vectors represent?
Think of it as the product of the portion of the two vectors that is shared in the direction of either of the two vectors. The following illustration shows how this looks for the two vectors A and B, that intersect at an angle of θ θ :
Clearly, if we consider the direction of B, then all of the length of B , |B|, is in that direction, and the portion of A in that direction is just|A|cos(θ θ). Similarly, if we consider the direction of A, then all of the length of A, |A|, is in that direction, and the portion of B in that direction is just|B|cos(θ θ). In either case, the product comes o
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Think of it as the product of the portion of the two vectors that is shared in the direction of either of the two vectors. The following illustration shows how this looks for the two vectors A and B, that intersect at an angle of θ θ :
Clearly, if we consider the direction of B, then all of the length of B , |B|, is in that direction, and the portion of A in that direction is just|A|cos(θ θ). Similarly, if we consider the direction of A, then all of the length of A, |A|, is in that direction, and the portion of B in that direction is just|B|cos(θ θ). In either case, the product comes out the same:
A⋅B=|A||B|cos(θ)A⋅B=|A||B|cos(θ)
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Alex Jones
B.S. in Applied Mathematics, University of Central Florida (Graduated 2016) · Author has 1.2K answers and 2.2M answer views
·3y
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How do I calculate the average of the dot product of six unit vectors?
The only way “the dot product of six … vectors” makes any sense is if I assume that’s we’re taking all pairwise dot products. Whether we should include self-products is kind of in the air, I’m going to do them since it makes the math a bit nicer. Let’s look at this for any n n.
What we’re essentially trying to do is calculate the value
M=2 n(n+1)∑1≤i≤j≤n x i⋅x j M=2 n(n+1)∑1≤i≤j≤n x i⋅x j
This is very similar to the sum of the product of a matrix and its transpose. Define the matrix X X as the matrix whose columns are x i x i. Then, we have
∑X T X=∑lim∑X T X=∑lim
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The only way “the dot product of six … vectors” makes any sense is if I assume that’s we’re taking all pairwise dot products. Whether we should include self-products is kind of in the air, I’m going to do them since it makes the math a bit nicer. Let’s look at this for any n n.
What we’re essentially trying to do is calculate the value
M=2 n(n+1)∑1≤i≤j≤n x i⋅x j M=2 n(n+1)∑1≤i≤j≤n x i⋅x j
This is very similar to the sum of the product of a matrix and its transpose. Define the matrix X X as the matrix whose columns are x i x i. Then, we have
∑X T X=∑(i,j)∈Z 2 n x i⋅x j∑X T X=∑(i,j)∈Z n 2 x i⋅x j
where ∑A∑A means to sum the entries of the matrix A A. Aside from the coefficient, these summations also differ in which pairs (i,j)(i,j) are summed over. In the latter sum, all pairs of numbers 1 1 to n n are done, while in the former, (j,i)(j,i) is never done when (i,j)(i,j) is. This means that the second sum is double counting all the non-diagonal entries of X T X X T X compared to the first. We can compensate for this by adding the trace (to double count the diagonal as well) and then dividing by 2 2. This gives us
M=1 n(n+1)(Tr+∑)X T X M=1 n(n+1)(Tr+∑)X T X
If you want to use the fact that the x i x i are unit vectors, we can say Tr X T X=n Tr X T X=n, since each diagonal element will be x i⋅x i=|x i|2=1 x i⋅x i=|x i|2=1. This reduces the expression to
M=1 n+1+1 n(n+1)∑X T X M=1 n+1+1 n(n+1)∑X T X
Lastly, we can get a very simple expression if we happen to know the angle between every pair of vectors θ i j θ i j. If this is the case, then we can use
x i⋅x j=|x i||x j|cos θ i j=cos θ i j x i⋅x j=|x i||x j|cosθ i j=cosθ i j
because the x i x i are unit length. This gives
M=2 n+1+2 n(n+1)∑1≤i<j≤n cos θ i j M=2 n+1+2 n(n+1)∑1≤i<j≤n cosθ i j
For any of these expressions, simply choose n=6 n=6 and substitute your given vectors into X X or their angles into θ θ.
As a bonus, one quick and easy thing we can say about the average is that it’s absolute value is at most 1 1. This is simply because the dot product of two unit vectors is certainly at most 1 1 in absolute value:
|x i⋅x j|=|cos θ i j|≤1|x i⋅x j|=|cosθ i j|≤1
and so the absolute value of the average will also be at most 1 1. However, if we restrict our attention to n>1 n>1 non-parallel unit vectors, we can say more strongly that the absolute value of the average will be strictly less than 1 1.
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Tom Capizzi
Former College Math Tutor at University of Massachusetts, Boston (1998–2000) · Author has 3.5K answers and 721.7K answer views
·8mo
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What is the relationship between the dot product of two vectors and their orthogonality?
This is straightforward, and I’m absolutely certain that Wikipedia has a perfectly good explanation, in great detail. I will just hit the high points. First, I’m assuming that we are talking about Euclidean geometry, since the Euclidean definitions to not apply in hyperbolic geometry. Specifically, orthogonality refers to being perpendicular in the Euclidean sense. All that being said, the dot product is defined geometrically as the product of the magnitudes of the vectors with the cosine of the included angle. It is the included angle which determines the degree of orthogonality. Two such vec
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This is straightforward, and I’m absolutely certain that Wikipedia has a perfectly good explanation, in great detail. I will just hit the high points. First, I’m assuming that we are talking about Euclidean geometry, since the Euclidean definitions to not apply in hyperbolic geometry. Specifically, orthogonality refers to being perpendicular in the Euclidean sense. All that being said, the dot product is defined geometrically as the product of the magnitudes of the vectors with the cosine of the included angle. It is the included angle which determines the degree of orthogonality. Two such vectors that are parallel have zero included angle, and a cosine of 1. The dot product is the same as the simple product of the magnitudes. If they are, in fact, perpendicular, the cosine of the included angle is zero, and the dot product is also zero. So, that’s it. The dot product of any pair of vectors can be positive, negative or zero, depending on whether they are parallel, anti-parallel, perpendicular or anywhere between. The arccos of the ratio of the dot product to the simple product of the magnitudes is the angle between the vectors. If the product of the magnitudes is the same as the dot product, they are parallel, it it is the negative of the dot product, they are anti-parallel, and if it is 0, they are perpendicular.
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What is the dot product of two vectors in two dimensions?
How do I do a dot product of vectors?
Does the Geometric interpretation of dot product held for vectors in R^n?
What is the minimum value of the dot product of two vectors?
How do you calculate the dot product between two vectors if they have different numbers of dimensions?
Alexandru Carausu
Former Former University Associate Professor at Universitatea Tehnica "Gh. Asachi" Iasi (1966–2010) · Author has 3K answers and 875.5K answer views
·2y
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How do you find the dot product of two vectors?
I gave (earlier) an answer to a similar question on and I’m posting it below.
What is the relationship between the dot product of two vectors and the vectors' angle?
➀ The answer is rather simple. If u and v are two (non-zero) vectors, then their dot product is
u • v = u v cos θ where u = | u | , v = | v | and θ = ∢ ( u , v ) . (1)
It follows from (1) that the dot ( • ) maps a pair of vectors onto a scalar in ℝ . It also follows that the real value of u • v needs information on the two vectors’ magnitudes (or lengths) and on the angle between them. This angle θ should fall in the compact interval
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I gave (earlier) an answer to a similar question on and I’m posting it below.
What is the relationship between the dot product of two vectors and the vectors' angle?
➀ The answer is rather simple. If u and v are two (non-zero) vectors, then their dot product is
u • v = u v cos θ where u = | u | , v = | v | and θ = ∢ ( u , v ) . (1)
It follows from (1) that the dot ( • ) maps a pair of vectors onto a scalar in ℝ . It also follows that the real value of u • v needs information on the two vectors’ magnitudes (or lengths) and on the angle between them. This angle θ should fall in the compact interval [ - π , π ] , on which the cosine function is bijective. This operation is defined among free vectors , which are equivalence classes among bound (or fixed) vectors defined as directed line segments and wholly determined by their initial points (or tails) and endpoints (or tips). They form the object of the VECTOR ALGEBRA which is included - in many textbooks / lecture notes / monographs - as an introductory chapter to the ANALYTIC GEOMETRY .
Coming back to the question, the defining formula (1) with the numerical data on u , v gives just the required relationship between u • v and θ = ∢ ( u , v ) . If their dot product and their lengths are known, then
θ = ∢ ( u , v ) = cos^(-1) [ ( u • v ) / ( u v ) ] . (2)
The argument of cos^(-1) in (2) is included in the compact interval [ -1 , 1 ] , as we are going to prove in what follows. It follows that the value of θ , in radians or degrees, can be easily found by means of a pocket calculator or by tables with the values of trigonometric functions and their inverses.
➁ If the two vectors are expresses in the “usual” (and most convenient) standard orthonormal basis of the space of free vectors (in the 3D) space, that is ( i , j , k ) , then their dot product and their magnitudes can be easily determined, implicitly their dot product according to formula (2) .
u = x _1 i + y _1 j + z _1 k and v = x _2 i + y _2 j + z _2 k ==>
u • v = x _1 x _2 + y _1 y _2 + z _1 z _2 , (3)
u = sqrt ( x _1^2 + y _1^2 + z _1^2 ) , v = sqrt ( x _2^2 + y _2^2 + z _2^2 ) . (4)
These real values, taken to formula (2) , will render the value of θ = ∢ ( u , v ) . The expression of (3) follows from the linearity of the dot product in each argument, and from the “multiplication table” with the unit vectors i , j , k : i • i = 1 , i • j = 0 , . . . . The expressions (4) of u and v are justified below.
➂ The dot product, with its expression (3) , is also defined in the n-dimensional real Euclidean space ℝ^3 , in particular in ℝ^n : if X = [ x _1 x _2 . . . x _n ]^T and Y = [ y _1 y _2 . . . y_n ]^T, then
X • Y = ∑ _1^n x _i y _i . (5)
Clearly, n = 3 in (5) renders just an expression like (3) for the dot product, with the respective notational differences for the coordinates in (3) , for the components in (5) . As a technical detail, the elements in the space ℝ^n are ordered n-tuples of real numbers, and they can be equivalently written in three ways : as in Cartesian products X = ( x _1 , x _2 , . . . , x_n ) , as row vectors X = [ x _1 x _2 . . . x_n ] or as column vectors , like above. This third way to write the vectors of ℝ^n is the most convenient, both theoretically and practically.
➃ It is important to outline that the dot product(s), just considered in the above points ➀ , ➁ , ➂ is (are) particular particular cases of a more general operation : the inner product. This operation < ,> , defined on a general vector (or linear) space V turns it into an IP space, also called an Euclidean space or a pre-Hilbert space. I use the notation 〈 V ; + , λ x , < , > 〉 . The three axioms of an inner product are :
( IP_1 ) ( ∀ x ∈ V ) ≥ 0 & = 0 ==>x = 0 ∈ F ; (6)
( IP_2 ) ( ∀ x , y ∈ V ) = , if F = ℝ ; (7)
( IP_2 ) ( ∀ λ_1 , λ_2 ∈ F ) ( ∀ x _1 , x _2 , y ∈ V )
< λ_1 x _1 + λ_2 x _2 , y> = λ_1 + λ_2 < x_2 , y >. (8)
If the field of scalars is F = C - the complex field , axioms in (7) and (8) look a little different.
Any IP space is a normed vector space, NVS , with the norm || || induced by the IP
( ∀ x ∈ V ) || x || = sqrt ( < x , x > ) . (9)
It can be easily checked that the dot products, above considered, satisfy these three axioms. In any IP space, a famous inequality holds : ( C - S - B ) , the Cauchy -Schwarz - Bunjakovski inequality :
( ∀ x , y ∈ V ) | < x , y > | ≤ sqrt ( || x || || y || ) <==>
<==>< x , y >^2 ≤ . (10)
This inequality, applied to the dot product of (1) and (2) , implies
| u • v | / | u v | ≤ 1 ==>u • v / ( u v ) ∈ [ -1 , 1 ] = the domain of cos^(-1) : see (2) .
Another important relation between vectors, characterized in terms of the IP, is their orthogonality :
u ⊥ v <==> θ = ∢ ( u , v ) = π /2 <==>u • v = 0 . (11)
Final remarks. It seems that my information / explanations, above given, go (much) beyond the expected answer to this question submitted by QUORA. But I have given them in view of the importance of this operation of dot / inner or scalar product. Let us only mention its implication in the definition of orthogonal / orthonormal bases of vector spaces. And the same inner product and its induced notion of norm are essential notions in the FUNCTIONAL ANALYSIS and in HILBERT SPACES. To give a simplified definition, a Hilbert space in an infinite-dimensional inner product space over the complex field C .
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Jörg Straube
M.Sc. in Computer Science, ETH Zurich (Graduated 1987) · Author has 6.2K answers and 1.7M answer views
·3y
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What does the vector dot product represent?
Let‘s assume you sit in a trolley on a rail track and this trolley has a huge sail mounted in front of you perpendicular to the rail track. The rail track points eastward.
If there is a wind blowing north to south, the trolley does not move at all as the sail does not catch any of this north-south wind.
If the wind direction changes (eg blowing from NW to SE, the trolley starts moving eastwards, as a little bit of wind is catched by the sail. In an ideal situation the wind blows from west to east, and the sail catches all of the tail wind.
The dot product tells you how fast your trolley moves eas
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Let‘s assume you sit in a trolley on a rail track and this trolley has a huge sail mounted in front of you perpendicular to the rail track. The rail track points eastward.
If there is a wind blowing north to south, the trolley does not move at all as the sail does not catch any of this north-south wind.
If the wind direction changes (eg blowing from NW to SE, the trolley starts moving eastwards, as a little bit of wind is catched by the sail. In an ideal situation the wind blows from west to east, and the sail catches all of the tail wind.
The dot product tells you how fast your trolley moves eastward.
A is the direction vector of the rail track
B is the direction vector of the wind
If both are pointing in the same direction (full tail wind), the dot product is the largest. If both vectors are at 90°, the dot product of the two vectors is 0.
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If you’ve ever stood in front of the sunscreen aisle confused by the terms “mineral” and “chemical”, you’re not alone. Both types of sunscreens protect against harmful UV rays, but they work differently, and knowing the difference can help you choose the right one for your skin.
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Great for everyday wear under makeup.
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Garrett Thomas
CS PhD student at Stanford · Author has 355 answers and 1.3M answer views
·9y
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How do I calculate dot product of 4 different vectors?
You’re going to have to be more specific about what you’re trying to achieve.
Standard multiplication relates two elements of some set to another element of that set. Because of this property, you can multiply as many things as you want.
The dot product, on the other hand, results in a different type of object than that of the two things that are multiplied; specifically, the dot product of two vectors is not a vector but a scalar. Thus it’s not clear (to me, at least) how one would generalize to the case of more than two vectors.
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Deb P. Choudhury
Former Professor at University of Allahabad · Author has 10K answers and 8M answer views
·1y
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How do I calculate the dot product of two vectors when the second has 2 components only?
You can define scalar(dot) product of two vectors only when they have the same number of components.
Thus (a,b).(c,d) = ac+bd, and
(a,b,c).(a',b',c') = aa'+bb'+cc'.
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Pritam Ghoshal
GATE 2020 AIR 400 Mechanical Engineering · Author has 188 answers and 633.9K answer views
·8y
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If the magnitudes of two vectors and the magnitude of their cross product is given, how do I find the magnitude of their dot product?
Let the two vectors be A and B
The magnitude of their cross product is given by |A||B|sin(theta) where theta is the angle between the two vectors.
Magnitude of their dot product is given by |A||B|cos(theta)
Therefore the sum of the squares of the magnitude of the cross product and the dot product must be equal to (|A||B|)^2
as [sin(theta)]^2+[cos(theta)]^2=1
So the magnitude of the dot product is
[(magnitude of Amagnitude of B)^2-(magnitude of cross product)^2]^0.5
or,
-[(magnitude of Amagnitude of B)^2-(magnitude of cross product)^2]^0.5
Hope that was useful
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Terry Moore
M.Sc. in Mathematics, University of Southampton (Graduated 1968) · Author has 16.6K answers and 29.4M answer views
·6y
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Why does the dot product of two vectors have no direction?
Why does the dot product of two vectors have no direction?
The short answer is that we chose to define it that way.
However, the definition was not made in a vacuum, it was defined because of its physical uses. There are many of these, but I’ll give you just one: the work done by a force. This is the dot product of the force and the displacement. Work doesn’t have a specific direction.
Others have asked what it means. Well, it doesn’t mean work, that’s just one of its uses. The only meaning mathematical concepts have is through their definitions. Applications can’t supply meaning because there ar
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Why does the dot product of two vectors have no direction?
The short answer is that we chose to define it that way.
However, the definition was not made in a vacuum, it was defined because of its physical uses. There are many of these, but I’ll give you just one: the work done by a force. This is the dot product of the force and the displacement. Work doesn’t have a specific direction.
Others have asked what it means. Well, it doesn’t mean work, that’s just one of its uses. The only meaning mathematical concepts have is through their definitions. Applications can’t supply meaning because there are many applications.
Try to get used to the idea that mathematics is abstract. If several real life problems suggest a useful mathematical definition, we make it. But the mathematics is then independent of the application. It has its own relationships to other mathematical definitions, but it is not tied to any specific application.
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Brett Schmidt
Author has 2K answers and 3.2M answer views
·5y
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When are two vectors dot products equal to their magnitude?
QUESTION:
When are two vectors dot products equal to their magnitude?
ANSWER:
The Question is somewhat ambiguous, but hopefully this Answer will tell you what you want to know.
Let A and B be two vectors. Then the dot product of A and B, written as A•B, is defined as the product of the magnitudes of A and B, multiplied by the cosine of the angle between them. So if θ is the angle between A and B (i.e. the angular difference between the directions of A and B), then:
A•B = |A|×|B|cos(θ).
From this it follows that:
If either A or B or both A and B is/are the zero vector, then |A|×|B| = 0, and A•B = |A|
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QUESTION:
When are two vectors dot products equal to their magnitude?
ANSWER:
The Question is somewhat ambiguous, but hopefully this Answer will tell you what you want to know.
Let A and B be two vectors. Then the dot product of A and B, written as A•B, is defined as the product of the magnitudes of A and B, multiplied by the cosine of the angle between them. So if θ is the angle between A and B (i.e. the angular difference between the directions of A and B), then:
A•B = |A|×|B|cos(θ).
From this it follows that:
If either A or B or both A and B is/are the zero vector, then |A|×|B| = 0, and A•B = |A|×|B|×cos(θ) = 0. So in this case, the dot product of A and B is equal to zero and the product of the magnitudes of A and B is equal to zero, so the dot product of A and B is equal to the product of the magnitudes of A and B.
If neither A nor B is the zero vector, and A and B point in the same direction, then θ = 0, so A•B = |A|×|B|×cos(0) = |A|×|B|×1 = |A|×|B|. So in this case, the dot product of A and B is equal to the product of the magnitudes of A and B.
If neither A nor B is the zero vector, and A and B point in the exactly opposite directions, then θ = 180°, so A•B = |A|×|B|×cos(180°) = |A|×|B|×(-1) = -(|A|×|B|). So in this case, the dot product of A and B is equal to negative the product of the magnitudes of A and B.
If neither A nor B is the zero vector, and A and B are perpendicular to each other, then θ = 90°, so A•B = |A|×|B|×cos(90°) = |A|×|B|×0 = 0. So in this case, the dot product of A and B is equal to zero.
If neither A nor B is the zero vector, and |B|×cos (θ) = 1, then A•B = |A|×|B|×cos(θ) = |A|×1 = |A|. So in this case, the dot product of A and B is equal to the magnitude of A.
I hope that this answers your question.
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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
Related
Is the dot product of 3 vectors possible? If so, why?
No, it is not possible.
Let us try to do a dot product for three vectors a, b and c. This would be something like (a . b) . c
We would need to do a dot product for two vectors at first and then do a dot product between the result from a.b and c
But this is not possible because a.b gives a scalar. a and b are vectors but the dot product between a and b would give a scalar.
So for performing the operation of dot product, we need two vectors and since a.b is a scalar , this result cannot be involved in a dot product with vector c.
Thus, dot product of three vectors is not possible but cross product is
Continue Reading
No, it is not possible.
Let us try to do a dot product for three vectors a, b and c. This would be something like (a . b) . c
We would need to do a dot product for two vectors at first and then do a dot product between the result from a.b and c
But this is not possible because a.b gives a scalar. a and b are vectors but the dot product between a and b would give a scalar.
So for performing the operation of dot product, we need two vectors and since a.b is a scalar , this result cannot be involved in a dot product with vector c.
Thus, dot product of three vectors is not possible but cross product is possible.
Upvote ·
99 15
9 1
Thomas Bell
Ph.D. in Mathematics, University of Oregon (Graduated 2013) · Author has 3.9K answers and 687.2K answer views
·1y
Related
How do I calculate the dot product of two vectors when the second has 2 components only?
The dot product is only defined for vectors of the same length, i.e. the same number of components.
Computers often add extra zeros to the shorter vector, essentially placing them in a higher dimension, before computing the product. On your homework, you should not do this.
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189455 | http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/electpot.html | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
| Standard Electrode Potentials in Aqueous Solution at 25°C | | | --- | | Cathode (Reduction)Half-Reaction | Standard PotentialE° (volts) | | Li+(aq) + e- -> Li(s) | -3.04 | | K+(aq) + e- -> K(s) | -2.92 | | Ca2+(aq) + 2e- -> Ca(s) | -2.76 | | Na+(aq) + e- -> Na(s) | -2.71 | | Mg2+(aq) + 2e- -> Mg(s) | -2.38 | | Al3+(aq) + 3e- -> Al(s) | -1.66 | | 2H2O(l) + 2e- -> H2(g) + 2OH-(aq) | -0.83 | | Zn2+(aq) + 2e- -> Zn(s) | -0.76 | | Cr3+(aq) + 3e- -> Cr(s) | -0.74 | | Fe2+(aq) + 2e- -> Fe(s) | -0.41 | | Cd2+(aq) + 2e- -> Cd(s) | -0.40 | | Ni2+(aq) + 2e- -> Ni(s) | -0.23 | | Sn2+(aq) + 2e- -> Sn(s) | -0.14 | | Pb2+(aq) + 2e- -> Pb(s) | -0.13 | | Fe3+(aq) + 3e- -> Fe(s) | -0.04 | | 2H+(aq) + 2e- -> H2(g) | 0.00 | | Sn4+(aq) + 2e- -> Sn2+(aq) | 0.15 | | Cu2+(aq) + e- -> Cu+(aq) | 0.16 | | ClO4-(aq) + H2O(l) + 2e- -> ClO3-(aq) + 2OH-(aq) | 0.17 | | AgCl(s) + e- -> Ag(s) + Cl-(aq) | 0.22 | | Cu2+(aq) + 2e- -> Cu(s) | 0.34 | | ClO3-(aq) + H2O(l) + 2e- -> ClO2-(aq) + 2OH-(aq) | 0.35 | | IO-(aq) + H2O(l) + 2e- -> I-(aq) + 2OH-(aq) | 0.49 | | Cu+(aq) + e- -> Cu(s) | 0.52 | | I2(s) + 2e- -> 2I-(aq) | 0.54 | | ClO2-(aq) + H2O(l) + 2e- -> ClO-(aq) + 2OH-(aq) | 0.59 | | Fe3+(aq) + e- -> Fe2+(aq) | 0.77 | | Hg22+(aq) + 2e- -> 2Hg(l) | 0.80 | | Ag+(aq) + e- -> Ag(s) | 0.80 | | Hg2+(aq) + 2e- -> Hg(l) | 0.85 | | ClO-(aq) + H2O(l) + 2e- -> Cl-(aq) + 2OH-(aq) | 0.90 | | 2Hg2+(aq) + 2e- -> Hg22+(aq) | 0.90 | | NO3-(aq) + 4H+(aq) + 3e- -> NO(g) + 2H2O(l) | 0.96 | | Br2(l) + 2e- -> 2Br-(aq) | 1.07 | | O2(g) + 4H+(aq) + 4e- -> 2H2O(l) | 1.23 | | Cr2O72-(aq) + 14H+(aq) + 6e- -> 2Cr3+(aq) + 7H2O(l) | 1.33 | | Cl2(g) + 2e- -> 2Cl-(aq) | 1.36 | | Ce4+(aq) + e- -> Ce3+(aq) | 1.44 | | MnO4-(aq) + 8H+(aq) + 5e- -> Mn2+(aq) + 4H2O(l) | 1.49 | | H2O2(aq) + 2H+(aq) + 2e- -> 2H2O(l) | 1.78 | | Co3+(aq) + e- -> Co2+(aq) | 1.82 | | S2O82-(aq) + 2e- -> 2SO42-(aq) | 2.01 | | O3(g) + 2H+(aq) + 2e- -> O2(g) + H2O(l) | 2.07 | | F2(g) + 2e- -> 2F-(aq) | 2.87 | | | | Discussion of Standard Electrode Potentials | | IndexTablesReferenceEbbingAppendix I |
| HyperPhysics Chemistry | Go Back | |
189456 | https://ae.iitm.ac.in/~nidish/courses/as3020/files/mod1/jointdesign.pdf | 8–12 Bolted and Riveted Joints Loaded in Shear10 Riveted and bolted joints loaded in shear are treated exactly alike in design and analysis. Figure 8–23a shows a riveted connection loaded in shear. Let us now study the various means by which this connection might fail. Figure 8–23b shows a failure by bending of the rivet or of the riveted members. The bending moment is approximately M 5 Fty2, where F is the shearing force and t is the grip of the rivet, that is, the total thickness of the connected parts. The bend-ing stress in the members or in the rivet is, neglecting stress concentration, s 5 M Iyc (8–52) where Iyc is the section modulus for the weakest member or for the rivet or rivets, depending upon which stress is to be found. The calculation of the bending stress in this manner is an assumption, because we do not know exactly how the load is distrib-uted to the rivet or the relative deformations of the rivet and the members. Although this equation can be used to determine the bending stress, it is seldom used in design; instead its effect is compensated for by an increase in the factor of safety. In Fig. 8–23c failure of the rivet by pure shear is shown; the stress in the rivet is t 5 F A (8–53) where A is the cross-sectional area of all the rivets in the group. It may be noted that it is standard practice in structural design to use the nominal diameter of the rivet rather than the diameter of the hole, even though a hot-driven rivet expands and nearly fills up the hole.
Point E For the Gerber criterion, from Eq. (8–46), the safety factor is Answer nf 5 1 2saSe [Sut2S2 ut 1 4Se(Se 1 si) 2 S2 ut 2 2siSe] 5 1 2(3.10)(18.6) [12021202 1 4(18.6)(18.6 1 63.72) 2 1202 2 2(63.72)(18.6)] 5 3.65 which is greater than np 5 3.43 and contradicts the conclusion earlier that the danger of failure is fatigue. Figure 8–22 clearly shows the conflict where point D lies between points C and E. Again, the conservative nature of the Goodman criterion explains the discrepancy and the designer must form his or her own conclusion.
Screws, Fasteners, and the Design of Nonpermanent Joints 443 10The design of bolted and riveted connections for boilers, bridges, buildings, and other structures in which danger to human life is involved is strictly governed by various construction codes. When designing these structures, the engineer should refer to the American Institute of Steel Construction Handbook, the American Railway Engineering Association specifications, or the Boiler Construction Code of the American Society of Mechanical Engineers.
bud98209_ch08_401-466.indd Page 443 10/18/13 2:12 PM f-496 bud98209_ch08_401-466.indd Page 443 10/18/13 2:12 PM f-496 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles 444 Mechanical Engineering Design Rupture of one of the connected members or plates by pure tension is illustrated in Fig. 8–23d. The tensile stress is s 5 F A (8–54) where A is the net area of the plate, that is, the area reduced by an amount equal to the area of all the rivet holes. For brittle materials and static loads and for either ductile or brittle materials loaded in fatigue, the stress-concentration effects must be included. It is true that the use of a bolt with an initial preload and, sometimes, a rivet will place the area around the hole in compression and thus tend to nullify the effects of stress concentration, but unless definite steps are taken to ensure that the preload does not relax, it is on the conservative side to design as if the full stress-concentration effect were present. The stress-concentration effects are not considered in structural design, because the loads are static and the materials ductile. In calculating the area for Eq. (8–54), the designer should, of course, use the combination of rivet or bolt holes that gives the smallest area. Figure 8–23e illustrates a failure by crushing of the rivet or plate. Calculation of this stress, which is usually called a bearing stress, is complicated by the distribution of the load on the cylindrical surface of the rivet. The exact values of the forces act-ing upon the rivet are unknown, and so it is customary to assume that the components of these forces are uniformly distributed over the projected contact area of the rivet. This gives for the stress s 5 2F A (8–55) where the projected area for a single rivet is A 5 td. Here, t is the thickness of the thinnest plate and d is the rivet or bolt diameter.
Figure 8–23 Modes of failure in shear loading of a bolted or riveted connection: (a) shear loading; (b) bending of rivet; (c) shear of rivet; (d) tensile failure of members; (e) bearing of rivet on members or bearing of members on rivet; (f) shear tear-out; (g) tensile tear-out.
(a) (e) ( f ) (g) (b) (c) (d) bud98209_ch08_401-466.indd Page 444 10/18/13 2:12 PM f-496 bud98209_ch08_401-466.indd Page 444 10/18/13 2:12 PM f-496 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles Screws, Fasteners, and the Design of Nonpermanent Joints 445 Edge shearing, or tearing, of the margin is shown in Fig. 8–23f and g, respectively. In structural practice this failure is avoided by spacing the rivets at least 11 2 diameters away from the edge. Bolted connections usually are spaced an even greater distance than this for satisfactory appearance, and hence this type of failure may usually be neglected. In a rivet joint, the rivets all share the load in shear, bearing in the rivet, bearing in the member, and shear in the rivet. Other failures are participated in by only some of the joint. In a bolted joint, shear is taken by clamping friction, and bearing does not exist. When bolt preload is lost, one bolt begins to carry the shear and bearing until yielding slowly brings other fasteners in to share the shear and bearing. Finally, all participate, and this is the basis of most bolted-joint analysis if loss of bolt preload is complete. The usual analysis involves • Bearing in the bolt (all bolts participate) • Bearing in members (all holes participate) • Shear of bolt (all bolts participate eventually) • Distinguishing between thread and shank shear • Edge shearing and tearing of member (edge bolts participate) • Tensile yielding of member across bolt holes • Checking member capacity EXAMPLE 8–6 Two 1- by 4-in 1018 cold-rolled steel bars are butt-spliced with two 1 2- by 4-in 1018 cold-rolled splice plates using four 3 4 in-16 UNF grade 5 bolts as depicted in Fig. 8–24. For a design factor of nd 5 1.5 estimate the static load F that can be carried if the bolts lose preload. Solution From Table A–20, minimum strengths of Sy 5 54 kpsi and Sut 5 64 kpsi are found for the members, and from Table 8–9 minimum strengths of Sp 5 85 kpsi, Sy 5 92 kpsi, and Sut 5 120 kpsi for the bolts are found.
F F F 4 in F 1in (a) (b) 3 4 1 1 2 in 1 2 in 1 2 in 1 1 2 in 1 1 4 in 1 1 4 in 1 1 2 in 1 1 2 in 1 1 2 in in-16 UNF SAE grade 5 Figure 8–24 bud98209_ch08_401-466.indd Page 445 21/10/13 11:33 AM user-f-w-198 bud98209_ch08_401-466.indd Page 445 21/10/13 11:33 AM user-f-w-198 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles 446 Mechanical Engineering Design Fy2 is transmitted by each of the splice plates, but since the areas of the splice plates are half those of the center bars, the stresses associated with the plates are the same. So for stresses associated with the plates, the force and areas used will be those of the center plates.
Bearing in bolts, all bolts loaded: s 5 F 2td 5 Sy nd F 5 2td Sy nd 5 2(1)(3 4)92 1.5 5 92 kip Bearing in members, all bolts active: s 5 F 2td 5 (Sy)mem nd F 5 2td(Sy)mem nd 5 2(1)(3 4)54 1.5 5 54 kip Shear of bolt, all bolts active: If the bolt threads do not extend into the shear planes for four shanks: t 5 F 4pd2y4 5 0.577 Sy nd F 5 0.577pd 2 Sy nd 5 0.577p(0.75)2 92 1.5 5 62.5 kip If the bolt threads extend into a shear plane: t 5 F 4Ar 5 0.577 Sy nd F 5 0.577(4)ArSy nd 5 0.577(4)0.351(92) 1.5 5 49.7 kip Edge shearing of member at two margin bolts: From Fig. 8–25, t 5 F 4at 5 0.577(Sy)mem nd F 5 4at0.577(Sy)mem nd 5 4(1.125)(1)0.577(54) 1.5 5 93.5 kip Tensile yielding of members across bolt holes: s 5 F [4 2 2(3 4)]t 5 (Sy)mem nd F 5 [4 2 2(3 4)]t(Sy)mem nd 5 4 2 2(3 4)54 1.5 5 90 kip bud98209_ch08_401-466.indd Page 446 10/18/13 2:12 PM f-496 bud98209_ch08_401-466.indd Page 446 10/18/13 2:12 PM f-496 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles Shear Joints with Eccentric Loading In the previous example, the load distributed equally to the bolts since the load acted along a line of symmetry of the fasteners. The analysis of a shear joint undergoing eccentric loading requires locating the center of relative motion between the two members. In Fig. 8–26 let A1 to A5 be the respective cross-sectional areas of a group of five pins, or hot-driven rivets, or tight-fitting shoulder bolts. Under this assumption the rotational pivot point lies at the centroid of the cross-sectional area pattern of the pins, rivets, or bolts. Using statics, we learn that the centroid G is located by the coordinates x and y, where xi and yi are the distances to the ith area center: x 5 A1x1 1 A2x2 1 A3x3 1 A4x4 1 A5x5 A1 1 A2 1 A3 1 A4 1 A5 5 ^n 1 Aixi ^n 1 Ai y 5 A1y1 1 A2y2 1 A3y3 1 A4y4 1 A5y5 A1 1 A2 1 A3 1 A4 1 A5 5 ^n 1 Ai yi ^n 1 Ai (8–56) On the basis of bolt shear, the limiting value of the force is 49.7 kip, assuming the threads extend into a shear plane. However, it would be poor design to allow the threads to extend into a shear plane. So, assuming a good design based on bolt shear, the limiting value of the force is 62.5 kip. For the members, the bearing stress limits the load to 54 kip.
Screws, Fasteners, and the Design of Nonpermanent Joints 447 Bolt d a Figure 8–25 Edge shearing of member.
Figure 8–26 Centroid of pins, rivets, or bolts.
y y _ x x _ O G A1 A2 A4 A3 A5 bud98209_ch08_401-466.indd Page 447 21/10/13 11:27 AM user-f-w-198 bud98209_ch08_401-466.indd Page 447 21/10/13 11:27 AM user-f-w-198 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles 448 Mechanical Engineering Design In many instances the centroid can be located by symmetry. An example of eccentric loading of fasteners is shown in Fig. 8–27. This is a portion of a machine frame containing a beam subjected to the action of a bending load. In this case, the beam is fastened to vertical members at the ends with spe-cially prepared load-sharing bolts. You will recognize the schematic representation in Fig. 8–27b as a statically indeterminate beam with both ends fixed and with moment and shear reactions at each end. For convenience, the centers of the bolts at the left end of the beam are drawn to a larger scale in Fig. 8–27c. Point O represents the centroid of the group, and it is assumed in this example that all the bolts are of the same diameter. Note that the forces shown in Fig. 8–27c are the resultant forces acting on the pins with a net force and moment equal and opposite to the reaction loads V1 and M1 acting at O. The total load taken by each bolt will be calculated in three steps. In the first step the shear V1 is divided equally among the bolts so that each bolt takes F9 5 V1yn, where n refers to the number of bolts in the group and the force F9 is called the direct load, or primary shear. It is noted that an equal distribution of the direct load to the bolts assumes an absolutely rigid member. The arrangement of the bolts or the shape and size of the members sometimes justifies the use of another assumption as to the division of the load. The direct loads F9 n are shown as vectors on the loading diagram (Fig. 8–27c). The moment load, or secondary shear, is the additional load on each bolt due to the moment M1. If rA, rB, rC, etc., are the radial distances from the centroid to the center of each bolt, the moment and moment loads are related as follows: M1 5 F– ArA 1 F– BrB 1 F– CrC 1 p (a) where the F0 are the moment loads. The force taken by each bolt depends upon its radial distance from the centroid; that is, the bolt farthest from the centroid takes the greatest load, while the nearest bolt takes the smallest. We can therefore write F– A rA 5 F– B rB 5 F– C rC (b) Figure 8–27 (a) Beam bolted at both ends with distributed load; (b) free-body diagram of beam; (c) enlarged view of bolt group centered at O showing primary and secondary resultant shear forces.
w lbf in (a) (c) (b) F ' F " F " F ' F " F ' F ' F " rB rD B A D C O O Beam V1 V2 M2 M1 O rA rC A B B A C C D D + ⁄ w lbf in ⁄ bud98209_ch08_401-466.indd Page 448 10/18/13 2:12 PM f-496 bud98209_ch08_401-466.indd Page 448 10/18/13 2:12 PM f-496 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles Screws, Fasteners, and the Design of Nonpermanent Joints 449 where again, the diameters of the bolts are assumed equal. If not, then one replaces F0 in Eq. (b) with the shear stresses t0 5 4F0ypd2 for each bolt. Solving Eqs. (a) and (b) simultaneously, we obtain F– n 5 M1rn r 2 A 1 r 2 B 1 r 2 C 1 p (8–57) where the subscript n refers to the particular bolt whose load is to be found. Each moment load is a force vector perpendicular to the radial line from the centroid to the bolt center. In the third step the direct and moment loads are added vectorially to obtain the resultant load on each bolt. Since all the bolts or rivets are usually the same size, only that bolt having the maximum load need be considered. When the maximum load is found, the strength may be determined by using the various methods already described. EXAMPLE 8–7 Shown in Fig. 8–28 is a 15- by 200-mm rectangular steel bar cantilevered to a 250-mm steel channel using four tightly fitted bolts located at A, B, C, and D. Assume the bolt threads do not extend into the joint.
For the F 5 16 kN load shown find (a) The resultant load on each bolt (b) The maximum shear stress in each bolt (c) The maximum bearing stress (d) The critical bending stress in the bar Solution (a) Point O, the centroid of the bolt group in Fig. 8–28, is found by symmetry. If a free-body diagram of the beam were constructed, the shear reaction V would pass through O and the moment reactions M would be about O. These reactions are V 5 16 kN M 5 16(300 1 50 1 75) 5 6800 N ? m In Fig. 8–29, the bolt group has been drawn to a larger scale and the reactions and resultants are shown. The distance from the centroid to the center of each bolt is r 5 2(60)2 1 (75)2 5 96.0 mm 250 10 15 200 60 60 75 75 50 300 C D B A O M16 2 bolts F = 16 kN Figure 8–28 Dimensions in millimeters.
bud98209_ch08_401-466.indd Page 449 10/18/13 2:12 PM f-496 bud98209_ch08_401-466.indd Page 449 10/18/13 2:12 PM f-496 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles 450 Mechanical Engineering Design The resultants are found as follows. The primary shear load per bolt is F¿ 5 V n 5 16 4 5 4 kN Since the rn are equal, the secondary shear forces are equal, and Eq. (8–57) becomes F– 5 Mr 4r 2 5 M 4r 5 6800 4(96.0) 5 17.7 kN The primary and secondary shear forces are plotted to scale in Fig. 8–29 and the resultants obtained by using the parallelogram rule. The magnitudes are found by measurement (or analysis) to be Answer FA 5 FB 5 21.0 kN Answer FC 5 FD 5 14.8 kN (b) Bolts A and B are critical because they carry the largest shear load. The problem stated to assume that the bolt threads are not to extend into the joint. This would require special bolts. If standard nuts and bolts were used, the bolts would need to be 46 mm long with a thread length of LT 5 38 mm. Thus the unthreaded portion of the bolt is 46 2 38 5 8 mm long. This is less than the 15 mm for the plate in Fig. 8–28, and the bolts would tend to shear along the minor diameter at a stress of t 5 FyAs 5 21.0(10)3y144 5 146 MPa. Using bolts not extending into the joint, or shoulder bolts, is preferred. For this example, the body area of each bolt is A 5 p(162)y4 5 201.1 mm2, resulting in a shear stress of Answer t 5 F A 5 21.0(10)3 201.1 5 104 MPa A FA ' FA " F A B FB ' rB rC rD rA FB " F B C FC ' FC " F C D FD ' F D " F D x y M V O Figure 8–29 bud98209_ch08_401-466.indd Page 450 21/10/13 11:28 AM user-f-w-198 bud98209_ch08_401-466.indd Page 450 21/10/13 11:28 AM user-f-w-198 /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles /204/MH01996/bud98209_disk1of1/0073398209/bud98209_pagefiles (c) The channel is thinner than the bar, and so the largest bearing stress is due to the pressing of the bolt against the channel web. The bearing area is Ab 5 td 5 10(16) 5 160 mm2. Thus the bearing stress is Answer s 5 2 F Ab 5 221.0(10)3 160 5 2131 MPa (d) The critical bending stress in the bar is assumed to occur in a section parallel to the y axis and through bolts A and B. At this section the bending moment is M 5 16(300 1 50) 5 5600 N ? m The second moment of area through this section is obtained as follows: I 5 Ibar 2 2(Iholes 1 d 2A) 5 15(200)3 12 2 2 c 15(16)3 12 1 (60)2(15)(16) d 5 8.26(10)6 mm4 Then Answer s 5 Mc I 5 5600(100) 8.26(10)6 (10)3 5 67.8 MPa Screws, Fasteners, and the Design of Nonpermanent Joints 451 PROBLEMS 8–1 A power screw is 25 mm in diameter and has a thread pitch of 5 mm.
(a) Find the thread depth, the thread width, the mean and root diameters, and the lead, provided square threads are used.
(b) Repeat part (a) for Acme threads. 8–2 Using the information in the footnote of Table 8–1, show that the tensile-stress area is At 5 p 4 (d 2 0.938 194p)2 8–3 Show that for zero collar friction the efficiency of a square-thread screw is given by the equation e 5 tan l 1 2 f tan l tan l 1 f Plot a curve of the efficiency for lead angles up to 45°. Use f 5 0.08. 8–4 A single-threaded power screw is 25 mm in diameter with a pitch of 5 mm. A vertical load on the screw reaches a maximum of 5 kN. The coefficients of friction are 0.06 for the collar and 0.09 for the threads. The frictional diameter of the collar is 45 mm. Find the overall efficiency and the torque to “raise” and “lower” the load. 8–5 The machine shown in the figure can be used for a tension test but not for a compression test. Why? Can both screws have the same hand?
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189457 | https://courses.lumenlearning.com/intermediatealgebra/chapter/identify-the-domain-of-a-logarithmic-function/ | Quotient and Power Rules for Logarithms
Learning Outcome
Define and use the quotient and power rules for logarithms
For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to simplify division of like bases raised to powers by subtracting the exponents: xaxb=xa−b. The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.
The Quotient Rule for Logarithms
The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.
logb(MN)=logbM−logbN
We can show logb(MN)=logb(M)−logb(N).
Given positive real numbers M, N, and b, where b>0 we will show
logb(MN)=logb(M)−logb(N).
Let m=logbM and n=logbN. In exponential form, these equations are bm=M and bn=N. It follows that
logb(MN)=logb(bmbn)Substitute for M and N.=logb(bm−n)Apply the quotient rule for exponents.=m−nApply the inverse property of logs.=logb(M)−logb(N)Substitute for m and n.
Example
Expand the following expression using the quotient rule for logarithms.
log(2x2+6x3x+9)
Show Solution
Factoring and canceling, we get:
log(2x2+6x3x+9)=log(2x(x+3)3(x+3))Factor the numerator and denominator.=log(2x3)Cancel the common factors.
Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.
log(2x3)=log(2x)−log(3)=log(2)+log(x)−log(3)
In the previous example, it was helpful to first factor the numerator and denominator and divide common terms. This gave us a simpler expression to use to write an equivalent expression. It is important to remember to subtract the logarithm of the denominator from the logarithm of the numerator. Always check to see if you need to expand further with the product rule.
Example
Expand log2(15x(x−1)(3x+4)(2−x)).
Show Solution
First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.
log2(15x(x−1)(3x+4)(2−x))=log2(15x(x−1))−log2((3x+4)(2−x))
Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5
log2(15x(x−1))−log2((3x+4)(2−x))=[log2(3)+log2(5)+log2(x)+log2(x−1)]−[log2(3x+4)+log2(2−x)]=log2(3)+log2(5)+log2(x)+log2(x−1)−log2(3x+4)−log2(2−x)
Analysis of the Solution
There are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for x=−43 and x=2. Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm, that x>0, x>1, x>−43, and x<2. Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises.
In the following video, we show more examples of using the quotient rule for logarithms.
Using the Power Rule for Logarithms
We have explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as x2? One method is as follows:
logb(x2)=logb(x⋅x)=logbx+logbx=2logbx
Notice that we used the product rule for logarithms to simplify the example above. By doing so, we have derived the power rule for logarithmswhich says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,
100=102√3=3121e=e−1
The Power Rule for Logarithms
The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.
logb(Mn)=nlogbM
Example
Expand log2x5.
Show Solution
The argument is already written as a power, so we identify the exponent, 5, and the base, x, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.
log2(x5)=5log2x
The power rule for logarithms is possible because we can use the product rule and combine like terms. In the next example, you will see that we can also rewrite an expression as a power in order to use the power rule.
Example
Expand log3(25) using the power rule for logs.
Show Solution
Expressing the argument as a power, we get log3(25)=log3(52).
Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.
log3(52)=2log3(5)
Now let us use the power rule in reverse.
Example
Rewrite 4ln(x) using the power rule for logs to a single logarithm with a leading coefficient of 1.
Show Solution
Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression 4ln(x), we identify the factor, 4, as the exponent and the argument, x, as the base, and rewrite the product as a logarithm of a power:
4ln(x)=ln(x4).
Summary
You can use the quotient rule of logarithms to write an equivalent difference of logarithms in the following way:
Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.
Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.
Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.
To use the power rule of logarithms to write an equivalent product of a factor and a logarithm, consider the following:
Express the argument as a power, if needed.
Write the equivalent expression by multiplying the exponent times the logarithm of the base.
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2017 AMC 12B Problems/Problem 24
Contents
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1 Problem
2 Diagram
3 Solution 1
4 Solution 2
5 Solution 3
6 Solution 4
7 Solution 5
8 Solution 6
9 Video Solution by MOP 2024
10 Notes
11 See Also
Problem
Quadrilateral has right angles at and , , and . There is a point in the interior of such that and the area of is times the area of . What is ?
Diagram
Solution 1
Let , , and . Note that . By the Pythagorean Theorem, . Since , the ratios of side lengths must be equal. Since , and . Let F be a point on such that is an altitude of triangle . Note that . Therefore, and . Since and form altitudes of triangles and , respectively, the areas of these triangles can be calculated. Additionally, the area of triangle can be calculated, as it is a right triangle. Solving for each of these yields:
[B E C]=[C E D]=[B E A]=x 3 2(x 2+1)[A B C D]=[A E D]+[D E C]+[C E B]+B E A(A B+C D)2=17∗[C E B]+[C E B]+[C E B]+[C E B]x 3+x 2=20 x 3 2(x 2+1)x x 2+1=20 x 3 x 2+1(x 2+1)2=20 x 2 x 4−18 x 2+1=0⟹x 2=9+4 5=4+2(2 5)+5
Therefore, the answer is
Solution 2
Draw line through , with on and on , . WLOG let , , . By weighted average .
Meanwhile, . This follows from comparing the ratios of triangle DEG to CFE and triangle AEG to FEB, both pairs in which the two triangles share a height perpendicular to FG, and have base ratio .
. We obtain , namely .
The rest is the same as Solution 1.
Solution 3
Let , ,
Note that cannot be the intersection of and , as that would mean
Let ,
~isabelchen
Solution 4
Let . Then from the similar triangles condition, we compute and . Hence, the -coordinate of is just . Since lies on the unit circle, we can compute the coordinate as . By Shoelace, we want Factoring out denominators and expanding by minors, this is equivalent to This factors as , so and so the answer is .
Solution 5
Let where . Because . Notice that the diagonals are perpendicular with slopes of and . Let the intersection of and be , then . However, because is a trapezoid, and share the same area, therefore is the reflection of over the perpendicular bisector of , which is . We use the linear equations of the diagonals, , to find the coordinates of . The y-coordinate of is simply The area of is . We apply shoelace theorem to solve for the area of . The coordinates of the triangle are , so the area is
1 2|a 3 a 2+1+1 a−a a 2+1−a a 2+1|=1 2|a 3−2 a a 2+1+1 a|=1 2|a 4−2 a 2 a(a 2+1)+a 2+1 a(a 2+1)|=1 2(a 4−a 2+1 a(a 2+1)).
Finally, we use the property that the ratio of areas equals :
~Zeric
Solution 6
This solution involves proving .
Let be the intersection of and . Label points and the same way as .
. Additionally, , so by SAS. Therefore, .
Next, because . Also, , so . Therefore, by AA. Since , .
Given , we deduce that the ratio of corresponding side lengths of to must be . Now, we set , , and . Using the Pythagorean Theorem, . Thus, . Solving gives .
Finally, .
~Zhixing
Video Solution by MOP 2024
~r00tsOfUnity
Notes
1) is the most relevant answer choice because it shares numbers with the givens of the problem.
2) It's a very good guess to replace finding the area of triangle AED with the area of the triangle DAF, where F is the projection of D onto AB(then find the closest answer choice).
See Also
2017 AMC 12B (Problems • Answer Key • Resources)
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Problem 25
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189459 | https://www.quora.com/How-much-amount-in-percent-is-left-after-four-half-lives-of-a-radioactive-substance | How much amount in percent is left after four half lives of a radioactive substance? - Quora
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How much amount in percent is left after four half lives of a radioactive substance?
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The amount of a radioactive substance remaining after a certain number of half-lives can be calculated using the formula:
Remaining Amount=(1 2)n Remaining Amount=(1 2)n
where n n is the number of half-lives.
After four half-lives (n=4 n=4):
Remaining Amount=(1 2)4=1 16 Remaining Amount=(1 2)4=1 16
To express this as a percentage:
Percentage Remaining=1 16×100≈6.25%Percentage Remaining=1 16×100≈6.25%
So, after four half-lives, approximately 6.25% of the original amount of the radioactive substance remains.
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After 1 year, 80% of the initial amount of a radioactive substance remains. What is the half-life of the substance?
What percentage of original radioactive atoms left after 15 half lives?
What percentage of a given mass of a radioactive substance will be left un-decayed after four half life periods?
After 7 days, a particular radioactive substance decays to half of its original amount. What is the decay rate of this substance?
If a radioactive substance that weighs one kilogram has a half life of 100 years, what would be the percentage of the substance left after 300 years?
Ted Dannemiller
Former Armor, Cav, & Infantry officer. Armor Archaeology. at U.S. Army (1970–1997) · Author has 3.2K answers and 3.9M answer views
·2y
See here:
BTW- the term “half-life” is not a “unit of time”. Instead, the half-life is defined in terms of probability: "Half-life is the time required for exactly half of the entities to decay on average". In other words, the probability of a radioactive atom decaying within its half-life is 50%.
Continue Reading
See here:
BTW- the term “half-life” is not a “unit of time”. Instead, the half-life is defined in terms of probability: "Half-life is the time required for exactly half of the entities to decay on average". In other words, the probability of a radioactive atom decaying within its half-life is 50%.
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Bob Siegerdt
Quality Assurance Manager · Author has 3.4K answers and 8.6M answer views
·5y
Originally Answered: What percentage of radioactive sample remains after five half lives? ·
Take the total of fresh sample (100%) and divide by 2 for each step;
1st HL - 50%
2nd - 25%
3rd - 12.5%
4th - 6.25%
5th - 3.125%
The radiation levels may remain high because the transmutation of material into other elements may produce other radioactive elements. The above number shows how much radiation would remain from the primary element.
The mass of the material will change as some transmutations may generate gasses or other volatiles.
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9 1
Dennis Mosebey
Former Retired From 39 Years Nuclear Plant Ops (1976–2014) · Author has 1.5K answers and 250.2K answer views
·5y
Originally Answered: What percentage of a radioactive isotope remains after 4 half-lives? ·
After first half life 1/2 then next half life 1/2 x1/2 is 1/4then next half I’ve is 1/4x1/2 is1/8 then final half life is 1/8 x1/2is 1/16 convert to percent is 16 divided into 1 which is .0625 or 6 .3 percent
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A radioactive substance has a half life of four months. 3/4 of the substance will decay in?
If the half-life of a radioactive sample is 24 hours, then can we calculate how much of the radioactive sample will be left in 12 hours?
For radioactive elements, which one of the time durations is higher, average lifetime or half-life period, or does it depend?
What is the half-life of a substance that decays to 1/3 of its original radioactive amount in 5 years?
What percentage of a radioactive species would be found as daughter material after five half-lives?
Robert Heaton
Medical Physicist, Radiation Therapy (1997–present)
·3y
This is really a simple answer — the answer is actually in the question. Recall that a half-life is the time it takes for half the substance to “disappear” or transmute into something different. So 4 half-lives reduce the original material by a factor of 1/2 x 1/2 x 1/2 x 1/2 = (1/2)^4 = 1/16. This is just over 6% of the original quantity left (6.25% is the precise value).
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David M Browne
Self-employed Since 1991, Business Trial Attorney · Author has 9.1K answers and 6.4M answer views
·7y
The most likely amount is 1/16 or 6.25%, but there is also a probability curve as to other outcomes. By definition, a half life does not guarantee that exactly half of the radioactive atoms will always undergo decay in the time period. But when speaking about very large numbers of atoms, the amount of deviation becomes very small due to the law of large numbers. For example, if you have just four atoms, what are the various probabilities that 1, 2 or 3 will have undergone decay in one half life period? It is not 0%, 50% and 0%, but rather 25%, 37.5% and 25% (with zero or four each at 6.25%). B
Continue Reading
The most likely amount is 1/16 or 6.25%, but there is also a probability curve as to other outcomes. By definition, a half life does not guarantee that exactly half of the radioactive atoms will always undergo decay in the time period. But when speaking about very large numbers of atoms, the amount of deviation becomes very small due to the law of large numbers. For example, if you have just four atoms, what are the various probabilities that 1, 2 or 3 will have undergone decay in one half life period? It is not 0%, 50% and 0%, but rather 25%, 37.5% and 25% (with zero or four each at 6.25%). But if we are talking about billions of atoms, the likelihood that the amount that have decayed in a half life will vary measurably from 50% is vanishingly small.
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Satya Parkash Sud
Ph.D. in Nuclear Physics, Panjab University, Chandigarh (Graduated 1970) · Author has 8.1K answers and 27.4M answer views
·5y
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What percentage of radioactive nuclei is left after 3 half lives?
Let at time t = 0, the number of radioactive nuclei be N•. After one half-life, t = T, the number of radioactive nuclei left undecayed would be N(t =T) = N•/2.
So radioactive nuclei N(t =T) left undecayed after one half-life: N (t = T) = N•/2
After another half-life, ie two half-lives from t = 0, the number of nuclei left undecayed:
N(t= 2 T) = (N•/2) × ½ = N•/4
After completion of three half-lives the number of nuclei left undecayed:
N(t = 3 T) = (N•/4) × ½ = N•/8
The number nuclei left undecayed after 3 half-lives
N (t = 3 T) = N•/8
The fraction of undecayed nuclei left undecayed after 3 half-lives
Continue Reading
Let at time t = 0, the number of radioactive nuclei be N•. After one half-life, t = T, the number of radioactive nuclei left undecayed would be N(t =T) = N•/2.
So radioactive nuclei N(t =T) left undecayed after one half-life: N (t = T) = N•/2
After another half-life, ie two half-lives from t = 0, the number of nuclei left undecayed:
N(t= 2 T) = (N•/2) × ½ = N•/4
After completion of three half-lives the number of nuclei left undecayed:
N(t = 3 T) = (N•/4) × ½ = N•/8
The number nuclei left undecayed after 3 half-lives
N (t = 3 T) = N•/8
The fraction of undecayed nuclei left undecayed after 3 half-lives = N(t = 3 T)/N• = ⅛ = ⅛ × 100 = 12.5%.
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Andrew Alder
First worked on solar energy (hot water service) in 1962 · Author has 8.7K answers and 3.8M answer views
·5y
Originally Answered: What percentage of radioactive sample remains after five half lives? ·
One over two to the fifth power, or one thirty-second or about 3% (3.125 to be exact). Every half-life the amount is reduced to half, that’s what half-life means. After five half-lives, that’s happened five times, so one thirty-second of the original is left.
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Steven Moss
BS in Physics, The Cooper Union for the Advancement of Science and Art · Author has 569 answers and 87K answer views
·3y
1 half-life leaves 50%
2 half-lives leaves 25%
3 half lives leaves 12.5%
4 half-lives leaves 6.25%.
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JH Simon
Author of 'How To Kill A Narcissist'
·Updated Fri
How do you overcome narcissistic mental enslavement? I grew up with a narc mother and dated a narc that I just cannot get out of my head.
Reclaim your True Self.
Narcissistic mental enslavement begins with but goes well beyond the mind. Every narcissist mentally overwhelms their target, but what gives the narcissist’s attacks teeth is the emotional reaction. When a narcissist puts down their target, shame arises. When a narcissist threatens their target, fear arises. When a narcissist questions their target’s morality, guilt arises. These attacks bind together over time and produce a corresponding complex, making the target easier to trigger as a ripple becomes an emotional tsunami. A psychological cage is built with layer upon l
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Reclaim your True Self.
Narcissistic mental enslavement begins with but goes well beyond the mind. Every narcissist mentally overwhelms their target, but what gives the narcissist’s attacks teeth is the emotional reaction. When a narcissist puts down their target, shame arises. When a narcissist threatens their target, fear arises. When a narcissist questions their target’s morality, guilt arises. These attacks bind together over time and produce a corresponding complex, making the target easier to trigger as a ripple becomes an emotional tsunami. A psychological cage is built with layer upon layer of emotional pain, which in time splits from the target’s consciousness and reduces the target to a helpless and compliant object. For example, when guilt becomes a complex, a person acts, thinks and believes as though they are guilty, regardless of the situation. Any judgement, whether justified or not, triggers a guilt attack and cripples the person’s capacity for independent thought and action. Reaction becomes the only option.
Overcoming mental enslavement involves facing all emotions and integrating them into consciousness. This means having the courage to fall below the realm of thought. Through mindful exploration, you come to notice your triggers, and therefore welcome the emotional intensity into awareness. Rather than reacting to or dissociating from your emotions, you create space and allow them to rise up, and most importantly, to release. Meditation, therapy and bodywork are all potent tools to help you weave together a connection between your Higher Self and your True Self. In doing so, you create a state of wholeness and power which eclipses the mental enslavement you experience after narcissistic abuse.
If you have just started your narcissistic abuse recovery journey, check out How To Kill A Narcissist. Or if you wish to immunise yourself against narcissists and move on for good, take a look at How To Bury A Narcissist.
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Saeed Khan
Visting Lecturer at COMSATS Institute of Information Technology (2019–present)
·6y
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A radioactive substance has a half life of four months. 3/4 of the substance will decay in?
i
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Mark Laris
I am a nuclear engineer · Author has 2.1K answers and 3.5M answer views
·7y
1/16 as much as you had at the start.
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9 3
Adam Jacholkowski
Former experimental particle physicist at CERN (organization) (1983–2015) · Author has 32.6K answers and 6.6M answer views
·5y
Originally Answered: What percentage of radioactive sample remains after five half lives? ·
Each half-life time period reduces the sample by 1/2 so in case of 5 half-lifes period only (1/2)^5 of the initial sample will survive. On average as statistical fluctuations are possible , specially in case of a small sample.
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David Levinson
Studied Nuclear Physics at Self Education · Author has 300 answers and 378.7K answer views
·Apr 29
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The half life of a radioactive substance is 10 days. What does that mean?
After 10 days, you have exactly half of the sample you started with. That's what that means.
Practical example: You start with 10 kg of Actinium-225. I'm going with that one, since its 1/2-life is indeed 10 days! After 10 days, you will have exactly 5 kg left of your Actinium. The rest will have decayed first into Francium-221, via alpha decay(two neutrons plus two protons; 225–4 total nucleons is 221, and the two protons drop the atomic number by 2, so Actinium at #89 knock off two protons in the alpha decay gives us #87, Francium). But your resulting Francium will hit its own half life after
Continue Reading
After 10 days, you have exactly half of the sample you started with. That's what that means.
Practical example: You start with 10 kg of Actinium-225. I'm going with that one, since its 1/2-life is indeed 10 days! After 10 days, you will have exactly 5 kg left of your Actinium. The rest will have decayed first into Francium-221, via alpha decay(two neutrons plus two protons; 225–4 total nucleons is 221, and the two protons drop the atomic number by 2, so Actinium at #89 knock off two protons in the alpha decay gives us #87, Francium). But your resulting Francium will hit its own half life after a mere 5 minutes. Continued alpha decay will give us Astatine-217(shorter-lived still), then Bismuth-213. Then we get a beta-minus decay, which actually increases the atomic number by one by kicking off a high energy electron plus an anti-neutrino. This makes a neutron transition into a proton. Note that we haven't lost any nucleons, but we now have Polonium in place of our Bismuth due to one more proton—with the same atomic mass number! Another alpha decay, and we get Lead-209. The final step to a stable isotope is Bismuth-209, with another beta-minus decay.
So, to summarize, after 10 days, you will have half your Actinium, and the rest of your sample will be mostly Bismuth, with the intermediary elements rapidly fading due to their short half-lives.
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Related questions
After 1 year, 80% of the initial amount of a radioactive substance remains. What is the half-life of the substance?
What percentage of original radioactive atoms left after 15 half lives?
What percentage of a given mass of a radioactive substance will be left un-decayed after four half life periods?
After 7 days, a particular radioactive substance decays to half of its original amount. What is the decay rate of this substance?
If a radioactive substance that weighs one kilogram has a half life of 100 years, what would be the percentage of the substance left after 300 years?
A radioactive substance has a half life of four months. 3/4 of the substance will decay in?
If the half-life of a radioactive sample is 24 hours, then can we calculate how much of the radioactive sample will be left in 12 hours?
For radioactive elements, which one of the time durations is higher, average lifetime or half-life period, or does it depend?
What is the half-life of a substance that decays to 1/3 of its original radioactive amount in 5 years?
What percentage of a radioactive species would be found as daughter material after five half-lives?
What percentage of a given mass of a radioactive substance will be left undeclared after four periods?
Why half of life of a radioactive element is not affected by any condition or its original value?
After how many half-lives will there be one gram of any radioactive element left behind (no matter what form)?
How long does it take for radioactive carbon 14 to complete four half lives?
An unknown radioactive element decays into non-radioactive substances. In 580 days the radioactivity of a sample decreases by 45 percent. What is the half-life of the element?
Related questions
After 1 year, 80% of the initial amount of a radioactive substance remains. What is the half-life of the substance?
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A radioactive substance has a half life of four months. 3/4 of the substance will decay in?
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Question 1
Coordinate Geometry
MCQ
Read the question, then carefully choose the correct answer(s).
In the adjoining figure, point P the intersection of lines l1 and l2 has coordinates (a, b) as shown here. Which of the following could be the ordered pair (a, b)?
point P the intersection of lines l1 & l2 has coordinates (a, b)
Choices
| Choice (4) | Response |
--- |
| a. | (2 3,1 3) | - [x] |
| b. | (0,4) | - [x] |
| c. | (5,−5 2) | - [x] |
| d. | (−1,5 8) | - [x] |
✔️ Check Answer!
Question 2
Coordinate Geometry
MCQ
Read the question, then carefully choose the correct answer(s).
Two points with coordinate (5,3), (5,6) and (5, -4) lie on a line, parallel to which axis?
Choices
| Choice (4) | Response |
--- |
| a. | X-axis | - [x] |
| b. | Y-axis | - [x] |
| c. | None of the above. | - [x] |
| d. | All of the above. | - [x] |
✔️ Check Answer!
Question 3
Coordinate Geometry
MCQ
Read the question, then carefully choose the correct answer(s).
The distance between the pointsA (1,2) and B (4,6) is calculated as
Choices
| Choice (4) | Response |
--- |
| a. | 5 | - [x] |
| b. | 10 | - [x] |
| c. | 15 | - [x] |
| d. | 20 | - [x] |
✔️ Check Answer!
Question 4
Coordinate Geometry
MCQ
Read the question, then carefully choose the correct answer(s).
The coordinate axes divides the plane into __.
Choices
| Choice (4) | Response |
--- |
| a. | Four parts | - [x] |
| b. | Three parts | - [x] |
| c. | One part | - [x] |
| d. | Two parts | - [x] |
✔️ Check Answer!
Question 5
Coordinate Geometry
MCQ
Read the question, then carefully choose the correct answer(s).
If each individual square has a side length of 4 cm in the following figure, what is the perimeter of the cross.
Image shows the perimeters of the cross.
Choices
| Choice (4) | Response |
--- |
| a. | 58 cm | - [x] |
| b. | 38 cm | - [x] |
| c. | 48 cm | - [x] |
| d. | 28 cm | - [x] |
✔️ Check Answer!
Question 6
Coordinate Geometry
MCQ
Read the question, then carefully choose the correct answer(s).
On plotting the points A (2,9), B (6,11), C (10,9), D (8,5), E (4,5) and joining AB, BC, CD, CE, EA, Which of the following figure is obtained?
Choices
| Choice (4) | Response |
--- |
| a. | Pentagon | - [x] |
| b. | Hexagon | - [x] |
| c. | Parallelogram | - [x] |
| d. | Trapezoid | - [x] |
✔️ Check Answer!
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189461 | https://breakingnewsenglish.com/2412/241205-delta-airlines-stowaway.html | Airplane Stowaway - ESL Lesson Plan - Breaking News English Lesson
Breaking News English Lesson: Airplane Stowaway
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Stowaway found on New York to Paris flight (5th December 2024)
PRINT
27-page lesson(40 exercises)
2-page MINI lesson
All 4 graded readings
LISTEN
North American & British English
20 questions
5-speeds
Listen & spell
Dictation
READ
3-speed reading
Jumble 1
Jumble 2
No spaces
Text jumble
GRAMMAR
Gap fill
The / An / A
Prepositions
Word order
SPELL
Consonants
Vowels
Missing letters
Initials only
No letters
WORDS
Word pairs
Missing words
Gap fill
The Reading / Listening - Airplane Stowaway - Level 3
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A woman was found hiding on an airplane from New York to Paris. She flew as a stowaway – a person who gets onto a plane without paying. She secretly got onto the plane without airport staff seeing her. The woman had a Russian passport and was around 60 years old. She boarded the Delta Air Lines aircraft at JFK International Airport in New York. She managed to get past many security checks to board the plane. She had no boarding pass. Police say she hid in the plane's toilets throughout the seven-hour flight. She was discovered when a flight attendant noticed the woman was making frequent, long trips to different toilets on the airplane.
The woman is now being held by police in Paris. Airline officials tried to fly her back to New York on Saturday. However, she caused a disturbance while the plane was on the runway waiting to take off. A man who sat near her on the plane as it waited to depart from Paris spoke to CNN. He said the woman looked very worried. He said she kept repeating: "I do not want to go back to the USA. Only a judge can make me go back." Investigators are currently looking into how she got on the plane in New York. A Delta spokesperson said: "Nothing is of greater importance than matters of safety and security." He said the airline would work hard to find out what happened.
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Try the same news story at these easier levels:
Airplane Stowaway - Level 0,Airplane Stowaway - Level 1orAirplane Stowaway - Level 2
Sources
Make sure you try all of the online activities for this reading and listening - There are dictations, multiple choice activities, drag and drop activities, sentence jumbles, which word activities, text reconstructions, spelling, gap fills and a whole lot more. Please enjoy :-)
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News
"Much has been said and written on the utility of newspapers; but one principal advantage which might be derived from these publications has been neglected; we mean that of reading them in schools."
The Portland Eastern Herald (June 8, 1795)
"News is history in its first and best form, its vivid and fascinating form, and...history is the pale and tranquil reflection of it."
Mark Twain, in his autobiography (1906)
"Current events provide authentic learning experiences for students at all grade levels.... In studying current events, students are required to use a range of cognitive, affective, critical thinking and research skills."
Haas, M. and Laughlin, M. (2000) Teaching Current Events: It's Status in Social Studies Today.
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Warm-ups
1. AIRPLANES:Students walk around the class and talk to other students about airplanes. Change partners often and share your findings.
2. CHAT: In pairs / groups, talk about these topics or words from the article. What will the article say about them? What can you say about these words and your life?
airplane / New York / Paris / airport / passport / security checks / flight attendant /
flight / police / disturbance / runway / judge / safety / investigators / work hard
Have a chat about the topics you liked. Change topics and partners frequently.
3. SECURITY:Students A strongly believe governments should increase security at all airports and train stations; Students B strongly believe the opposite. Change partners again and talk about your conversations.
4. AIRPORTS:What are the problems with airports. How can we improve these things? Complete this table with your partner(s). Change partners often and share what you wrote.
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ProblemsImprovements
Security
Check-in
The boarding gate
Shopping
Restaurants
Parking
MY e-BOOK
See a sample
5. FLIGHT:Spend one minute writing down all of the different words you associate with the word "flight". Share your words with your partner(s) and talk about them. Together, put the words into different categories.
6. FLYING:Rank these with your partner. Put the best things about flying at the top. Change partners often and share your rankings.
Airplane food
Duty free
The departure lounge
The excitement
The view
The movies
Landing
Taking off
Vocabulary
Paragraph 1
1.hiding a.Staying in a place where no one can see you.
2.fly b.In every part or during all of something.
3.stowaway c.To move through the air, like a bird or a plane.
4.boarded d.Tried hard and was able to do something.
5.managed e.A person who hides on a ship or plane to travel without paying.
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6.throughout f.Got on a ship, train, or plane.
7.frequent g.Happening many times.
Paragraph 2
8.officials h.To leave a place, especially by train, bus, or plane.
9.disturbance i.When a plane leaves the ground and starts to fly.
10.runway j.People with important jobs who work for the government or a company.
11.take off k.A person who decides if someone is right or wrong in court.
12.depart l.Something that makes noise or causes problems.
13.judge m.Happening now or at the present time.
14.currently n.The long road where planes take off or land.
Before reading / listening
1. TRUE / FALSE: Read the headline. Guess if 1-8 below are true (T) or false (F).
A stowaway was found on a flight to New York.T / F
The stowaway paid less than half price for her ticket.T / F
The stowaway did not have a boarding pass.T / F
The stowaway spent the whole flight in a toilet.T / F
The woman was put on a flight back to New York.T / F
The woman created a disturbance and was removed from the flight. T / F
The woman said only a judge could make her return to the USA.T / F
Delta said safety and security was the most important thing for them.T / F
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2. SYNONYM MATCH: Match the following synonyms from the article.
secretly
boarded
checks
throughout
frequent
officials
disturbance
depart
looking into
importance
executives
endless
leave
investigating
got on
for the whole of
value
under cover
annoyance
controls
3. PHRASE MATCH: (Sometimes more than one choice is possible.)
a person who gets onto a plane
She had no boarding
throughout the
flight
the woman was making
The woman is now being held
while the plane was on the runway
Only a judge can
Investigators are currently looking
the airline would work hard to
into how
find out what happened
without paying
waiting to take off
pass
attendant
frequent, long trips
by police in Paris
make me go back
seven-hour flight
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Gap fill
Put these words into the spaces in the paragraph below.
_boarded
paying
checks
flight
hiding
frequent
staff
pass_
A woman was found (1) _______ on an airplane from New York to Paris. She flew as a stowaway - a person who gets onto a plane without (2) ______. She secretly got onto the plane without airport (3) _______ seeing her. The woman had a Russian passport and was around 60 years old. She (4) ______ the Delta Air Lines aircraft at JFK International Airport in New York. She managed to get past many security (5) _______ to board the plane. She had no boarding (6) ______. Police say she hid in the plane's toilets throughout the seven-hour (7) _______. She was discovered when a flight attendant noticed the woman was making (8) ______, long trips to different toilets on the airplane.
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Put these words into the spaces in the paragraph below.
_looking
runway
held
judge
caused
greater
hard
depart_
The woman is now being (9) _______ by police in Paris. Airline officials tried to fly her back to New York on Saturday. However, she (10) ______ a disturbance while the plane was on the (11) _______ waiting to take off. A man who sat near her on the plane as it waited to (12) ______ from Paris spoke to CNN. He said the woman looked very worried. He said she kept repeating: "I do not want to go back to the USA. Only a (13) _______ can make me go back." Investigators are currently (14) ______ into how she got on the plane in New York. A Delta spokesperson said: "Nothing is of (15) _______ importance than matters of safety and security." He said the airline would work (16) ______ to find out what happened.
Listening — Guess the answers. Listen to check.
1) A woman was found hiding ______
a. on a airplane
b. on the airplane
c. on an airplane
d. inner airplane
2) a person who gets onto a ______
a. plane without paying
b. plane without pain
c. plane without pay in
d. plane without payee
3) She secretly got onto the plane without airport ______
a. stuff see in her
b. stuff seeing her
c. staff seen her
d. staff seeing her
4) She managed to get past many security checks to ______
a. bored the plane
b. board the plane
c. aboard the plane
d. broad the plane
5) Police say she hid in the plane's toilets throughout the ______
a. seven-hours flight
b. seven-hour flights
c. seven-hour fight
d. seven-hour flight
6) However, she caused a disturbance while the plane was ______
a. on the runaway
b. on the runs away
c. on the run way
d. on the runway
7) A man who sat near her on the plane as it ______
a. waited to impart
b. waited to the part
c. waited to dip part
d. waited to depart
8) He said the woman ______
a. looks very worried
b. looked verily worried
c. looked very worried
d. looked very worry
9) Only a judge can make ______
a. me gone back
b. me go back
c. me going back
d. me goes back
10) Nothing is of greater importance than matters of ______
a. safely and security
b. safety and securely
c. safe tea and secure tea
d. safety and security
Listening — Listen and fill in the gaps
A woman was (1) _______ an airplane from New York to Paris. She flew as a stowaway - a person who gets onto a plane without paying. She (2) _______ the plane without airport staff seeing her. The woman had a Russian passport and was around 60 years old. (3) _______ Delta Air Lines aircraft at JFK International Airport in New York. She managed to get (4) _______ checks to board the plane. She had no boarding pass. Police say she (5) _______ plane's toilets throughout the seven-hour flight. She was discovered when a flight attendant noticed the woman (6) _______, long trips to different toilets on the airplane.
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The woman is now (7) _______ police in Paris. Airline officials tried to fly her back to New York on Saturday. However, she caused a disturbance while the plane was on (8) _______ to take off. A man who sat near her on the plane as it waited (9) _______ Paris spoke to CNN. He said the woman looked very worried. He said she kept repeating: "I do not want to go back to the USA. Only (10) _______ make me go back." Investigators are currently looking into how she got on the plane in New York. A Delta spokesperson said: "Nothing (11) _______ importance than matters of safety and security." He said the airline would work hard to find (12) _______.
Comprehension questions
Where did the woman get on the airplane?
What nationality is the woman?
What did the woman manage to get past?
How long did the plane take to get to Paris?
Who noticed that the woman spent a long time in the toilets?
Where is the stowaway now?
What did the stowaway create on the airplane as it waited to take off?
How did the woman look as she sat on the plane?
To where did the stowaway not want to return?
What did Delta say was the most important thing?
Multiple choice quiz
1) Where did the woman get on the airplane?
a) Paris
b) London
c) Tokyo
d) New York
2) What nationality is the woman?
a) Swedish
b) Russian
c) Chilean
d) Australian
3) What did the woman manage to get past?
a) sniffer dogs
b) immigration
c) a fear of flying
d) security checks
4) How long did the plane take to get to Paris?
a) 8 hours
b) 6 hours
c) 7 hours
d) 9 hours
5) Who noticed that the woman spent a long time in the toilets?
a) the co-pilot
b) the pilot
c) a passenger
d) a flight attendant
6) Where is the stowaway now?
a) Tokyo
b) London
c) Paris
d) New York
7) What did the stowaway create on the airplane as it waited to take off?
a) a disturbance
b) a fight
c) a bad atmosphere
d) a mess
8) How did the woman look as she sat on the plane?
a) beautiful
b) bored
c) worried
d) afraid
9) To where did the stowaway not want to return?
a) Tokyo
b) New York
c) Paris
d) London
10) What did Delta say was the most important thing?
a) safety and security
b) customer feedback
c) passenger satisfaction
d) profits
Role play
Role A – Airplane Food
You think airplane food is the best thing about flying. Tell the others three reasons why. Tell them what is wrong with their things. Also, tell the others which is the worst of these (and why): the view, the movies or taking off.
Role B – The View
You think the view is the best thing about flying. Tell the others three reasons why. Tell them what is wrong with their things. Also, tell the others which is the worst of these (and why): airplane food, the movies or taking off.
Role C – The Movies
You think the movies are the best things about flying. Tell the others three reasons why. Tell them what is wrong with their things. Also, tell the others which is the worst of these (and why): the view, airplane food or taking off.
Role D – Taking Off
You think taking off is the best thing about flying. Tell the others three reasons why. Tell them what is wrong with their things. Also, tell the others which is the worst of these (and why): the view, the movies or airplane food.
After reading / listening
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1. WORD SEARCH: Look in your dictionary / computer to find collocates, other meanings, information, synonyms … for the words...
'airplane'
and 'flight'.
• Share your findings with your partners.
• Make questions using the words you found.
• Ask your partner / group your questions.
2. ARTICLE QUESTIONS: Look back at the article and write down some questions you would like to ask the class about the text.
•Share your questions with other classmates / groups. •Ask your partner / group your questions.
3. GAP FILL: In pairs / groups, compare your answers to this exercise. Check your answers. Talk about the words from the activity. Were they new, interesting, worth learning…?
4. VOCABULARY: Circle any words you do not understand. In groups, pool unknown words and use dictionaries to find their meanings.
5. TEST EACH OTHER: Look at the words below. With your partner, try to recall how they were used in the text:
found
seeing
past
pass
seven
long
held
caused
sat
worried
looking
hard
Student survey
Write five GOOD questions about this topic in the table. Do this in pairs. Each student must write the questions on his / her own paper. When you have finished, interview other students. Write down their answers.
(Please look at page 12 of the PDF to see a photocopiable example of this activity.)
Discussion - Airplane Stowaway
STUDENT A’s QUESTIONS (Do not show these to student B)
What did you think when you read the headline?
What images are in your mind when you hear the word 'airplane'?
What do you think of airplanes?
What do you think of stowaways?
Have you ever not paid on a bus or train?
What do you think of airports?
Do you think airport security checks are good enough?
Why do you think the woman stowed away?
What should Delta do about this issue?
What advice do you have for the stowaway?
STUDENT B’s QUESTIONS (Do not show these to student A)
Did you like reading this article? Why/not?
What do you think of when you hear the word 'flight'?
What do you think about what you read?
What do you think of flying?
How can we improve airports?
What should happen to the stowaway?
Do you worry about safety and security?
What do you think of your nation's airline?
What should happen to Delta?
What questions would you like to ask the airport security staff?
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Discussion — Write your own questions
STUDENT A’s QUESTIONS (Do not show these to student B)
(a) ____
(b) ____
(c) ____
(d) ____
(e) ____
STUDENT B’s QUESTIONS (Do not show these to student A)
(f) ____
(g) ____
(h) ____
(i) ____
(j) ____
Language — Cloze (Gap-fill)
A woman was found (1) _ on an airplane from New York to Paris. She flew as a stowaway - a person who gets onto a plane without (2) . She secretly got onto the plane without airport staff seeing her. The woman had a Russian passport and was around 60 years old. She (3) _ the Delta Air Lines aircraft at JFK International Airport in New York. She managed to get (4) many security checks to board the plane. She had no boarding pass. Police say she hid in the plane's toilets (5) _ the seven-hour flight. She was discovered when a flight attendant noticed the woman was making (6) _, long trips to different toilets on the airplane.
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The woman is now being (7) _ by police in Paris. Airline officials tried to fly her back to New York on Saturday. However, she caused a disturbance while the plane was on the (8) waiting to take off. A man who sat near her on the plane as it waited to depart from Paris spoke to CNN. He said the woman looked very worried. He said she (9) _ repeating: "I do not want to go back to the USA. Only a judge can make me (10) back." Investigators are currently looking into how she got on the plane in New York. A Delta spokesperson said: "Nothing is of (11) _ importance than matters of safety and security." He said the airline would work hard to find out (12) _ happened.
Which of these words go in the above text?
(a) hides (b) hiding (c) hide (d) hideout
(a) pay (b) paying (c) payee (d) paid
(a) bored (b) boredom (c) boarding (d) boarded
(a) post (b) past (c) paste (d) passed
(a) thou (b) though (c) thorough (d) throughout
(a) frequented (b) frequency (c) frequent (d) frequently
(a) held (b) holding (c) hold (d) holds
(a) runaway (b) run away (c) runway (d) catwalk
(a) leapt (b) adept (c) wept (d) kept
(a) go (b) went (c) going (d) goes
(a) greater (b) greatly (c) great (d) greeter
(a) what (b) that (c) if (d) so
Spelling
Paragraph 1
She lfwe as a stowaway
She ctlserye got onto the plane
airport ffsat
She oedarbd the Delta Air Lines aircraft
get past many tsiyeruc checks
feqneurt, long trips to different toilets
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Paragraph 2
Airline oilfaicfs
she caused a uisatncdber
Only a egudj can make me go back
erclytrnu looking into how
Nothing is of greater cneaptomri
matters of syaetf
Put the text back together
(...) than matters of safety and security." He said the airline would work hard to find out what happened.
(...) woman looked very worried. He said she kept repeating: "I do not want to go back to the USA. Only a
(...) onto a plane without paying. She secretly got onto the plane without airport staff seeing
(...) throughout the seven-hour flight. She was discovered when a flight attendant noticed the woman was
(...) checks to board the plane. She had no boarding pass. Police say she hid in the plane's toilets
(...) off. A man who sat near her on the plane as it waited to depart from Paris spoke to CNN. He said the
(...) to New York on Saturday. However, she caused a disturbance while the plane was on the runway waiting to take
(...) aircraft at JFK International Airport in New York. She managed to get past many security
(...) on the plane in New York. A Delta spokesperson said: "Nothing is of greater importance
(...) making frequent, long trips to different toilets on the airplane.
(...) The woman is now being held by police in Paris. Airline officials tried to fly her back
(...) her. The woman had a Russian passport and was around 60 years old. She boarded the Delta Air Lines
(...) judge can make me go back." Investigators are currently looking into how she got
(1 ) A woman was found hiding on an airplane from New York to Paris. She flew as a stowaway - a person who gets
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Put the words in the right order
hiding airplane . woman an was found A on
gets without paying . who Someone plane a onto
plane seeing . airport the onto Got without staff
past She managed many checks . security to get
different Frequent airplane . toilets trips the on to
woman is The held now police . by being
take runway the to on The waiting off . plane
can go back . me judge Only a make
importance Nothing of safety . is than greater
to would work airline The hard out . find
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Circle the correct word (20 pairs)
A woman was found / founded hiding on an airplane from New York to Paris. She flew as / was a stowaway - a person who gets onto a plane without paying / paid. She secretly got onto the plane without airport stuff / staff seeing her. The woman had a Russian passport and was around 60 years age / old. She bored / boarded the Delta Air Lines aircraft at JFK International Airport in New York. She managed to get passed / past many security checks to / for board the plane. She had no boarding pass. Police say she hid on / in the plane's toilets throughout the seven-hour flight. She was discovered when a flight attendant noticed the woman was making frequent, long trips to different / difference toilets on the airplane.
The woman is now being / been held by police in Paris. Airline officials tried to fly her forward / back to New York on Saturday. However, she caused a disturb / disturbance while the plane was on the runaway / runway waiting to take off. A man who sat near her on the plane as it waited to / at depart from Paris spoke to / at CNN. He said the woman looked very worried. He said she kept repeating / repeated: "I do not want to go back to the USA. Only a judge can make me go back." Investigators are current / currently looking into how she got on the plane in New York. A Delta spokesperson said: "Nothing is of greater important / importance than matters of safety and security." He said the airline would work hard / hardly to find out what happened.
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Talk about the connection between each pair of words in italics, and why the correct word is correct. Look up the definition of new words.
Insert the vowels (a, e, i, o, u)
_ w_m_n w_s f__nd h_d_ng n _n __rpl_n fr_m N_w Y_rk t_ P_r_s. Sh_ fl_w s _ st_w_w_y - _ p_rs_n wh g_ts nt _ pl_n_ w_th__t p_y_ng. Sh_ s_cr_tly g_t nt th_ pl_n_ w_th__t __rp_rt st_ff sng hr. Th_ w_m_n h_d _ R_ssn p_ssp_rt nd w_s _r__nd 60 y__rs _ld. Sh b__rd_d th_ D_lt_ __r L_n_s __rcr_ft t JFK _nt_rn_t__n_l __rp_rt _n N_w Y_rk. Sh m_n_g_d t_ g_t p_st m_ny s_c_r_ty ch_cks t_ b__rd th_ pl_n_. Sh_ h_d n_ b__rd_ng p_ss. P_l_c_ s_y sh_ h_d n th pl_n_'s t__l_ts thr__gh__t th_ s_v_n-h__r fl_ght. Sh_ w_s d_sc_v_r_d wh_n _ fl_ght tt_nd_nt n_t_c_d th w_m_n w_s m_k_ng fr_q__nt, l_ng tr_ps t_ d_ff_r_nt t__l_ts n th __rpl_n_.
Th_ w_m_n s n_w b__ng h_ld by p_l_c n P_r_s. __rl_n ff_c__ls tr__d t fly h_r b_ck t_ N_w Y_rk n S_t_rd_y. H_w_v_r, sh c__s_d _ d_st_rb_nc_ wh_l_ th_ pl_n_ w_s n th r_nw_y w__t_ng t_ t_k_ ff. _ m_n wh s_t n__r h_r n th pl_n_ s _t w__t_d t d_p_rt fr_m P_r_s sp_k_ t_ CNN. H_ s__d th_ w_m_n l__k_d v_ry w_rr__d. H_ s__d sh_ k_pt r_p__t_ng: " d n_t w_nt t_ g_ b_ck t_ th_ S. nly _ j_dg c_n m_k_ m_ g_ b_ck." nv_st_g_t_rs _r c_rr_ntly l__k_ng nt h_w sh_ g_t n th pl_n_ n N_w Y_rk. _ D_lt sp_k_sp_rs_n s__d: "N_th_ng s _f gr__t_r _mp_rt_nc th_n m_tt_rs f s_f_ty _nd s_c_r_ty." H s__d th_ __rl_n_ w__ld w_rk h_rd t_ f_nd __t wh_t h_pp_n_d.
Punctuate the text and add capitals
a woman was found hiding on an airplane from new york to paris she flew as a stowaway a person who gets onto a plane without paying she secretly got onto the plane without airport staff seeing her the woman had a russian passport and was around 60 years old she boarded the delta air lines aircraft at jfk international airport in new york she managed to get past many security checks to board the plane she had no boarding pass police say she hid in the planes toilets throughout the sevenhour flight she was discovered when a flight attendant noticed the woman was making frequent long trips to different toilets on the airplane
the woman is now being held by police in paris airline officials tried to fly her back to new york on saturday however she caused a disturbance while the plane was on the runway waiting to take off a man who sat near her on the plane as it waited to depart from paris spoke to cnn he said the woman looked very worried he said she kept repeating i do not want to go back to the usa only a judge can make me go back investigators are currently looking into how she got on the plane in new york a delta spokesperson said nothing is of greater importance than matters of safety and security he said the airline would work hard to find out what happened
Put a slash (/) where the spaces are
AwomanwasfoundhidingonanairplanefromNewYorktoParis.Sheflew
asastowaway-apersonwhogetsontoaplanewithoutpaying.Shesecr
etlygotontotheplanewithoutairportstaffseeingher.ThewomanhadaR
ussianpassportandwasaround60yearsold.SheboardedtheDeltaAirLi
nesaircraftatJFKInternationalAirportinNewYork.Shemanagedtoget
pastmanysecuritycheckstoboardtheplane.Shehadnoboardingpass.
Policesayshehidintheplane'stoiletsthroughouttheseven-hourfligh
t.Shewasdiscoveredwhenaflightattendantnoticedthewomanwasma
kingfrequent,longtripstodifferenttoiletsontheairplane.Thewomanis
nowbeingheldbypoliceinParis.AirlineofficialstriedtoflyherbacktoNew
YorkonSaturday.However,shecausedadisturbancewhiletheplanewa
sontherunwaywaitingtotakeoff.Amanwhosatnearherontheplaneasit
waitedtodepartfromParisspoketoCNN.Hesaidthewomanlookedvery
worried.Hesaidshekeptrepeating:"IdonotwanttogobacktotheUSA.O
nlyajudgecanmakemegoback."Investigatorsarecurrentlylookingint
ohowshegotontheplaneinNewYork.ADeltaspokespersonsaid:"Nothi
ngisofgreaterimportancethanmattersofsafetyandsecurity."Hesaidt
heairlinewouldworkhardtofindoutwhathappened.
Free writing
Write about stowaway for 10 minutes. Comment on your partner’s paper.
Academic writing
Security at airports, and railway and bus stations should increase security. Discuss.
Homework
1. VOCABULARY EXTENSION: Choose several of the words from the text. Use a dictionary or Google’s search field (or another search engine) to build up more associations / collocations of each word.
2. INTERNET: Search the Internet and find out more about this news story. Share what you discover with your partner(s) in the next lesson.
3. AIRPLANES:Make a poster about airplanes. Show your work to your classmates in the next lesson. Did you all have similar things?
4. SECURITY:Write a magazine article about increasing security at airports, train stations, bus stations and ferry ports. Include imaginary interviews with people who are for and against this.
Read what you wrote to your classmates in the next lesson. Write down any new words and expressions you hear from your partner(s).
5. WHAT HAPPENED NEXT?Write a newspaper article about the next stage in this news story. Read what you wrote to your classmates in the next lesson. Give each other feedback on your articles.
6. LETTER:Write a letter to an expert on airplanes. Ask him/her three questions about them. Give him/her three of your ideas on how to make airports better. Read your letter to your partner(s) in your next lesson. Your partner(s) will answer your questions.
A Few Additional Activities for Students
Ask your students what they have read, seen or heard about this news in their own language. Students are likely to / may have have encountered this news in their L1 and therefore bring a background knowledge to the classroom.
Get students to role play different characters from this news story.
Ask students to keep track of this news and revisit it to discuss in your next class.
Ask students to male predictions of how this news might develop in the next few days or weeks, and then revisit and discuss in a future class.
Ask students to write a follow-up story to this news.
Students role play a journalist and someone who witnessed or was a part of this news. Perhaps they could make a video of the interview.
Ask students to keep a news journal in English and add this story to their thoughts.
Also...
Buy my 1,000 Ideas and Activities for Language Teachers eBook. It has hundreds of ideas, activity templates, reproducible activities for:
News
Warm ups
Pre-reading / Post-reading
Using headlines
Working with words
While-reading / While-listening
Moving from text to speech
Post-reading / Post-listening
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Using lists
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189462 | https://m.fx361.com/news/2017/0927/2334645.html | 浅谈中学数学教学中的反证法_参考网
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浅谈中学数学教学中的反证法
2017-09-27 刘青
课程教育研究·新教师教学订阅 2015年8期 收藏
使用浏览器的分享功能,把这篇文章分享出去
关键词：中学数学
刘青
摘要：反证法是中学数学中常用的一种证明方法，在应用反证法的过程中要根据命题的性质进行推理论证。其基本的思路就是假设所要证明的结论不成立，再在这个假定条件下进行逻辑推理，得出一个矛盾结论。
關键词：中学数学 反证法 证明
中图分类号：G633.6
反证法是数学中常见的一种证明方法，它与一般证明方法不同，反证法又可分为归谬反证法和穷举反证法两种。当命题的结论的反面只有一种情况，只要推翻这一种情况就能肯定结论，这种反证法叫归谬法；当命题的结论的反面不只一种情况，则需要将反面情况一一推翻才能肯定结论，这种反证法叫穷举法。那么反证法的理论根据是什么？反证法是否就是证明原命题的逆否命题？怎样应用反证法？怎样的命题适合用反证法证明？本文拟就这些问题做初步探讨。
1.反证法的定义
反证法是从反面的角度思考问题的证明方法，属于“间接证明”的一类，即肯定题设而否定结论，从而导出矛盾，推理而得。
不妨设原命题为 是推出结论，s一般是条件、某公理、定义、定理或临时假设，用数学术语可以简单地表示为： 即 。
2.反证法的基本思路
首先假设所要证明的结论不成立，然后再在这个假定条件下进行一系列的正确逻辑推理，直至得出一个矛盾的结论来，并据此否定原先的假设，从而确认所要证明的结论成立，这里所说的矛盾是指与题目中所给的已知条件矛盾，或是与数学中已知定理、公理和定义相矛盾，还可以是与日常生活中的事实相矛盾，甚至还可以是从两个不同角度进行推理所得出的结论之间相互矛盾（即自相矛盾）。
3.反证法的一般步骤
首先假设命题的结论不成立，其次从这个假设出发，经过推理论证得出矛盾，最后由矛盾判定假设不正确，从而肯定命题的结论正确。
第一步：审题，弄清命题的前提和结论；
第二步：否定原命题，由假设条件及原命题构成推理的基础；
第三步：由假设出发根据公理、定义、定理、公式及命题的条件，正确逻辑推理，导出逻辑矛盾；
第四步：肯定原命题的正确性。
4.反证法的应用
反证法从否定命题的结论开始，然后推出矛盾，这就给证明原命题增加了条件。因此，如果“结论反面”较“结论”更具体、更简单，而直接证明又好下手，则可考虑应用反证法，具体来说，究竟什么样的命题用反证法证明比较方便呢？可归纳如下几个方面：
需要注意的是，我们只有用反证法证明一个正确的命题时才能导致矛盾，如果用反证法证明一个显然不成立的命题，在证明的过程中就不会产生矛盾。所以，我们应用反证法时，只要待证命题正确，只要推理正确，就一定会导致矛盾的出现，否则，不是待证命题错误，就是我们的推理有错误，本文就反证法的定义、逻辑原理、证明模式，以及解题的方法来说明反证法在中学数学中的应用，使大家对反证法有了更深刻的了解。
参考文献：
赵雄辉.证明的方法[M].湖南：湖南人民出版社，2001：85-92.
廉蒙.巧用反证法证明代数题[J].北京：思路·方法·技巧，2004.endprint
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19The Hydrogen Atom and The Periodic Table
19–1Schrödinger’s equation for the hydrogen atom
The most dramatic success in the history of the quantum
mechanics was the understanding of the details of the spectra of some
simple atoms and the understanding of the periodicities which are found
in the table of chemical elements. In this chapter we will at last bring
our quantum mechanics to the point of this important achievement,
specifically to an understanding of the spectrum of the hydrogen atom.
We will at the same time arrive at a qualitative explanation of the
mysterious properties of the chemical elements. We will do this by
studying in detail the behavior of the electron in a hydrogen atom—for
the first time making a detailed calculation of a distribution-in-space
according to the ideas we developed in Chapter 16.
For a complete description of the hydrogen atom we should describe the
motions of both the proton and the electron. It is possible to do this
in quantum mechanics in a way that is analogous to the classical idea of
describing the motion of each particle relative to the center of
gravity, but we will not do so. We will just discuss an approximation in
which we consider the proton to be very heavy, so we can think of it as
fixed at the center of the atom.
We will make another approximation by forgetting that the electron has a
spin and should be described by relativistic laws of mechanics. Some
small corrections to our treatment will be required since we will be
using the nonrelativistic Schrödinger equation and will disregard
magnetic effects. Small magnetic effects occur because from the
electron’s point-of-view the proton is a circulating charge which
produces a magnetic field. In this field the electron will have a
different energy with its spin up than with it down. The energy of the
atom will be shifted a little bit from what we will calculate. We will
ignore this small energy shift. Also we will imagine that the electron
is just like a gyroscope moving around in space always keeping the same
direction of spin. Since we will be considering a free atom in space the
total angular momentum will be conserved. In our approximation we will
assume that the angular momentum of the electron spin stays constant, so
all the rest of the angular momentum of the atom—what is usually
called “orbital” angular momentum—will also be
conserved. To an excellent approximation the electron moves in the
hydrogen atom like a particle without spin—the angular momentum of the
motion is a constant.
With these approximations the amplitude to find the electron at
different places in space can be represented by a function of position
in space and time. We let ψ(x,y,z,t)ψ(x,y,z,t) be the amplitude to find the
electron somewhere at the time tt. According to the quantum mechanics
the rate of change of this amplitude with time is given by the
Hamiltonian operator working on the same function. From
Chapter 16,
iℏ∂ψ∂t=ˆHψ,
iℏ∂ψ∂t=H^ψ,(19.1)
with
ˆH=−ℏ22m∇2+V(r).Here, m is the electron mass, and V(r) is the potential energy
of the electron in the electrostatic field of the proton. Taking V=0
at large distances from the proton we can write1
V=−e2r.The wave function ψ must then satisfy the equation
iℏ∂ψ∂t=−ℏ22m∇2ψ−e2rψ.
We want to look for definite energy states, so we try to find
solutions which have the form
ψ(r,t)=e−(i/ℏ)Etψ(r).
The function ψ(r) must then be a solution of
−ℏ22m∇2ψ=(E+e2r)ψ,where E is some constant—the energy of the atom.
Since the potential energy term depends only on the radius, it turns
out to be much more convenient to solve this equation in polar
coordinates rather than rectangular coordinates. The Laplacian is
defined in rectangular coordinates by
∇2=∂2∂x2+∂2∂y2+∂2∂z2.
We want to use instead the coordinates r, θ, ϕ shown in
Fig. 19–1. These coordinates are related to x, y,
z by
x=rsinθcosϕ;y=rsinθsinϕ;z=rcosθ.It’s a rather tedious mess to work through the algebra, but you can
eventually show that for any function f(r)=f(r,θ,ϕ),
∇2f(r,θ,ϕ)=1r∂2∂r2(rf)+1r2{1sinθ∂∂θ(sinθ∂f∂θ)+1sin2θ∂2f∂ϕ2}.
So in terms of the polar coordinates, the equation which is to be
satisfied by ψ(r,θ,ϕ) is
1r∂2∂r2(rψ)+1r2{1sinθ∂∂θ(sinθ∂ψ∂θ)+1sin2θ∂2ψ∂ϕ2}=−2mℏ2(E+e2r)ψ.
Fig. 19–1.
The spherical polar coordinates r, θ, ϕ of the
point P.
19–2Spherically symmetric solutions
Let’s first try to find some very simple function that satisfies the
horrible equation in (19.7). Although the wave
function ψ will, in general, depend on the angles θ and ϕ as
well as on the radius r, we can see whether there might be a special
situation in which ψ does not depend on the angles. For a
wave function that doesn’t depend on the angles, none of the
amplitudes will change in any way if you rotate the coordinate
system. That means that all of the components of the angular momentum
are zero. Such a ψ must correspond to a state whose total angular
momentum is zero. (Actually, it is only the orbital angular momentum
which is zero because we still have the spin of the electron, but we
are ignoring that part.) A state with zero orbital angular momentum is
called by a special name. It is called an “s-state”—you can
remember “s for spherically symmetric.”2
Now if ψ is not going to depend on θ and ϕ then the
entire Laplacian contains only the first term and
Eq. (19.7) becomes much simpler:
1rd2dr2(rψ)=−2mℏ2(E+e2r)ψ.
Before you start to work on solving an equation like this, it’s a good
idea to get rid of all excess constants like e2, m, and ℏ,
by making some scale changes. Then the algebra will be easier. If we
make the following substitutions:
r=ℏ2me2ρ,and
E=me42ℏ2ϵ,then Eq. (19.8) becomes (after multiplying through by ρ)
d2(ρψ)dρ2=−(ϵ+2ρ)ρψ.These scale changes mean that we are measuring the distance r and
energy E as multiples of “natural” atomic units. That is,
ρ=r/rB, where rB=ℏ2/me2, is called the “Bohr
radius” and is about 0.529 angstroms.
Similarly, ϵ=E/ER, with ER=me4/2ℏ2. This energy is
called the “Rydberg” and is about 13.6 electron volts.
Since the product ρψ appears on both sides, it is convenient
to work with it rather than with ψ itself. Letting
ρψ=f,
we have the more simple-looking equation
d2fdρ2=−(ϵ+2ρ)f.
Now we have to find some function f which satisfies
Eq. (19.13)—in other words, we just have to solve a
differential equation. Unfortunately, there is no very useful, general
method for solving any given differential equation. You just have to
fiddle around. Our equation is not easy, but people have found that it
can be solved by the following procedure. First, you replace f, which
is some function of ρ, by a product of two functions
f(ρ)=e−αρg(ρ).
This just means that you are factoring e−αρ out
of f(ρ). You can certainly do that for any f(ρ) at all. This
just shifts our problem to finding the right function g(ρ).
Sticking (19.14) into (19.13), we get the
following equation for g:
d2gdρ2−2αdgdρ+(2ρ+ϵ+α2)g=0.
Since we are free to choose α, let’s make
α2=−ϵ,and get
d2gdρ2−2αdgdρ+2ρg=0.
You may think we are no better off than we were at
Eq. (19.13), but the happy thing about our new equation is
that it can be solved easily in terms of a power series in ρ. (It
is possible, in principle, to solve (19.13) that way too,
but it is much harder.) We are saying that Eq. (19.17)
can be satisfied by some g(ρ) which can be written as a series,
g(ρ)=∞∑k=1akρk,
in which the ak are constant coefficients. Now all we have to do is
find a suitable infinite set of coefficients! Let’s check to see that
such a solution will work. The first derivative of this g(ρ) is
dgdρ=∞∑k=1akkρk−1,and the second derivative is
d2gdρ2=∞∑k=1akk(k−1)ρk−2.Using these expressions in (19.17) we have
∞∑k=1k(k−1)akρk−2−∞∑k=12αkakρk−1+∞∑k=12akρk−1=0.It’s not obvious that we have succeeded; but we forge onward. It will
all look better if we replace the first sum by an equivalent. Since
the first term of the sum is zero, we can replace each k by k+1
without changing anything in the infinite series; with this change the
first sum can equally well be written as
∞∑k=1(k+1)kak+1ρk−1.Now we can put all the sums together to get
∞∑k=1[(k+1)kak+1−2αkak+2ak]ρk−1=0.
This power series must vanish for all possible values of ρ. It
can do that only if the coefficient of each power of ρ is
separately zero. We will have a solution for the hydrogen atom if we
can find a set ak for which
(k+1)kak+1−2(αk−1)ak=0
for all k≥1. That is certainly easy to arrange. Pick any a1
you like. Then generate all of the other coefficients from
ak+1=2(αk−1)k(k+1)ak.With this you will get a2, a3, a4, and so on, and
each pair will certainly satisfy (19.21). We get a series
for g(ρ) which satisfies (19.17). With it we can make
a ψ, that satisfies Schrödinger’s equation. Notice that the
solutions depend on the assumed energy (through α), but for each
value of ϵ, there is a corresponding series.
We have a solution, but what does it represent physically? We can get
an idea by seeing what happens far from the proton—for large values
of ρ. Out there, the high-order terms of the series are the most
important, so we should look at what happens for large k. When
k≫1, Eq. (19.22) is approximately the same as
ak+1=2αkak,
which means that
ak+1≈(2α)kk!.But these are just the coefficients of the series
for e+2αρ. The function of g is a rapidly increasing
exponential. Even coupled with e−αρ to
produce f(ρ)—see Eq. (19.14)—it still gives a solution
for f(ρ) which goes like eαρ for large ρ. We
have found a mathematical solution but not a physical one. It
represents a situation in which the electron is least likely to
be near the proton! It is always more likely to be found at a very
large radius ρ. A wave function for a bound electron must
go to zero for large ρ.
We have to think whether there is some way to beat the game, and there
is. Observe! If it just happened by luck that α were equal
to 1/n, where n is any positive integer, then Eq. (19.22)
would make an+1=0. All higher terms would also be zero. We wouldn’t
have an infinite series but a finite polynomial. Any polynomial
increases more slowly than eαρ, so the
term e−αρ will eventually beat it down, and the function f
will go to zero for large ρ. The only bound-state solutions
are those for which α=1/n, with n=1, 2, 3, 4, and so on.
Looking back to Eq. (19.16), we see that the
bound-state solutions to the spherically symmetric wave equation can
exist only when
−ϵ=1,14,19,116,…,1n2,…
The allowed energies are just these fractions times the Rydberg,
ER=me4/2ℏ2, or the energy of the nth energy level is
En=−ER1n2.There is, incidentally, nothing mysterious about negative numbers for
the energy. The energies are negative because when we chose to
write V=−e2/r, we picked our zero point as the energy of an electron
located far from the proton. When it is close to the proton, its
energy is less, so somewhat below zero. The energy is lowest (most
negative) for n=1, and increases toward zero with increasing n.
Before the discovery of quantum mechanics, it was known from
experimental studies of the spectrum of hydrogen that the energy
levels could be described by Eq. (19.24), where ER
was found from the observations to be about 13.6 electron volts.
Bohr then devised a model which gave
the same equation and predicted that ER should be me4/2ℏ2.
But it was the first great success of the Schrödinger theory that it
could reproduce this result from a basic equation of motion for the
electron.
Now that we have solved our first atom, let’s look at the nature of
the solution we got. Pulling all the pieces together, each solution
looks like this:
ψn=fn(ρ)ρ=e−ρ/nρgn(ρ),
where
gn(ρ)=n∑k=1akρkand
ak+1=2(k/n−1)k(k+1)ak.So long as we are mainly interested in the relative probabilities of
finding the electron at various places we can pick any number we wish
for a1. We may as well set a1=1. (People often choose a1 so
that the wave function is “normalized,” that is, so that the
integrated probability of finding the electron anywhere in the atom is
equal to 1. We have no need to do that just now.)
For the lowest energy state, n=1, and
ψ1(ρ)=e−ρ.
For a hydrogen atom in its ground (lowest-energy) state, the amplitude
to find the electron at any point drops off exponentially with the
distance from the proton. It is most likely to be found right at the
proton, and the characteristic spreading distance is about one unit
in ρ, or about one Bohr radius, rB.
Putting n=2 gives the next higher level. The wave function for this
state will have two terms. It is
ψ2(ρ)=(1−ρ2)e−ρ/2.
The wave function for the next level is
ψ3(ρ)=(1−2ρ3+227ρ2)e−ρ/3.The wave functions for these first three levels are plotted in
Fig. 19–2. You can see the general trend. All of the wave
functions approach zero rapidly for large ρ after oscillating a few
times. In fact, the number of “bumps” is just equal to n—or, if
you prefer, the number of zero-crossings of ψn is n−1.
Fig. 19–2.
The wave functions for the first three l=0 states of the
hydrogen atom. (The scales are chosen so that the total probabilities
are equal.)
19–3States with an angular dependence
In the states described by the ψn(r) we have found that the
probability amplitude for finding the electron is spherically
symmetric—depending only on r, the distance from the proton. Such
states have zero orbital angular momentum. We should now inquire about
states which may have some angular dependences.
We could, if we wished, just investigate the strictly mathematical
problem of finding the functions of r, θ, and ϕ which
satisfy the differential equation (19.7)—putting in the
additional physical conditions that the only acceptable functions are
ones which go to zero for large r. You will find this done in many
books. We are going to take a short cut by using the knowledge we
already have about how amplitudes depend on angles in space.
The hydrogen atom in any particular state is a particle with a certain
“spin” j—the quantum number of the total angular momentum. Part
of this spin comes from the electron’s intrinsic spin, and part from
the electron’s motion. Since each of these two components acts
independently (to an excellent approximation) we will again ignore the
spin part and think only about the “orbital” angular momentum. This
orbital motion behaves, however, just like a spin. For example, if the
orbital quantum number is l, the z-component of angular momentum
can be l, l−1, l−2, …, −l. (We are, as usual, measuring
in units of ℏ.) Also, all the rotation matrices and other
properties we have worked out still apply. (From now on we will
really ignore the electron’s spin; when we speak of “angular
momentum” we will mean only the orbital part.)
Since the potential V in which the electron moves depends only
on r and not on θ or ϕ, the Hamiltonian is symmetric under
all rotations. It follows that the angular momentum and all its
components are conserved. (This is true for motion in any
“central field”—one which depends only on r—so is not a
special feature of the Coulomb e2/r potential.)
Now let’s think of some possible state of the electron; its internal
angular structure will be characterized by the quantum
number l. Depending on the “orientation” of the total angular momentum
with respect to the z-axis, the z-component of angular momentum
will be m, which is one of the 2l+1 possibilities between +l
and −l. Let’s say m=1. With what amplitude will the electron be found
on the z-axis at some distance r? Zero. An electron on the
z-axis cannot have any orbital angular momentum around that
axis. Alright, suppose m is zero, then there can be some nonzero
amplitude to find the electron at each distance from the proton. We’ll
call this amplitude Fl(r). It is the amplitude to find the electron
at the distance r up along the z-axis, when the atom is in the
state |l,0⟩, by which we mean orbital spin l and
z-component m=0.
If we know Fl(r) everything is known. For any state |l,m⟩, we
know the amplitude ψl,m(r) to find the electron
anywhere in the atom. How? Watch. Suppose we have the atom in the
state |l,m⟩, what is the amplitude to find the electron at the
angle θ,ϕ and the distance r from the origin? Put a new
z-axis, say z′, at that angle (see Fig. 19–3), and ask
what is the amplitude that the electron will be at the distance r
along the new axis z′? We know that it cannot be found along z′
unless its z′-component of angular momentum, say m′, is zero. When
m′ is zero, however, the amplitude to find the electron along z′
is Fl(r). Therefore, the result is the product of two factors. The
first is the amplitude that an atom in the state |l,m⟩ along the
z-axis will be in the state |l,m′=0⟩ with respect to the
z′-axis. Multiply that amplitude by Fl(r) and you have the
amplitude ψl,m(r) to find the electron at (r,θ,ϕ)
with respect to the original axes.
Fig. 19–3.
The point (r,θ,ϕ) is on the z′-axis of the
x′,y′,z′ coordinate frame.
Let’s write it out. We have worked out earlier the transformation
matrices for rotations. To go from the frame x,y,z to the
frame x′,y′,z′ of Fig. 19–3, we can rotate first around the
z-axis by the angle ϕ, and then rotate about the new
y-axis (y′) by the angle θ. This combined rotation is the
product
Ry(θ)Rz(ϕ).
The amplitude to find the state l,m′=0 after the rotation is
⟨l,0|Ry(θ)Rz(ϕ)|l,m⟩.Our result, then, is
ψl,m(r)=⟨l,0|Ry(θ)Rz(ϕ)|l,m⟩Fl(r).
The orbital motion can have only integral values of l. (If the
electron can be found anywhere at r≠0, there is some amplitude to
have m=0 in that direction. And m=0 states exist only for integral
spins.) The rotation matrices for l=1 are given in
Table 17–2. For larger l you can use the general
formulas we worked out in Chapter 18. The matrices for
Rz(ϕ) and Ry(θ) appear separately, but you know how to
combine them. For the general case you would start with the
state |l,m⟩ and operate with Rz(ϕ) to get the new
state Rz(ϕ)|l,m⟩ (which is just eimϕ|l,m⟩). Then you
operate on this state with Ry(θ) to get the
state Ry(θ)Rz(ϕ)|l,m⟩. Multiplying by ⟨l,0| gives the
matrix element (19.31).
The matrix elements of the rotation operation are functions of
θ and ϕ. The particular functions which appear
in (19.31) also show up in many kinds of problems which
involve waves in spherical geometries and so has been given a special
name. Not everyone uses the same convention; but one of the most common
ones is
⟨l,0|Ry(θ)Rz(ϕ)|l,m⟩≡aYl,m(θ,ϕ).
The functions Yl,m(θ,ϕ) are called the spherical
harmonics, and a is just a numerical
factor which depends on the definition chosen for Yl,m. For the
usual definition,
a=√4π2l+1.With this notation, the hydrogen wave functions can be written
ψl,m(r)=aYl,m(θ,ϕ)Fl(r).
The angle functions Yl,m(θ,ϕ) are important not only in
many quantum-mechanical problems, but also in many areas of classical
physics in which the ∇2 operator appears, such as
electromagnetism. As another example of their use in quantum
mechanics, consider the disintegration of an excited state of
Ne20 (such as we discussed in the last chapter) which decays by
emitting an α-particle and going into O16:
Ne20∗→O16+He4.
Suppose that the excited state has some spin l (necessarily an
integer) and that the z-component of angular momentum is m. We
might now ask the following: given l and m, what is the amplitude
that we will find the α-particle going off in a direction which
makes the angle θ with respect to the z-axis and the
angle ϕ with respect to the xz-plane—as shown in
Fig. 19–4.
Fig. 19–4.
The decay of an excited state of Ne20.
To solve this problem we make, first, the following observation. A
decay in which the α-particle goes straight up along z must
come from a state with m=0. This is so because both O16 and the
α-particle have spin zero, and because their motion cannot have
any angular momentum about the z-axis. Let’s call this amplitude a
(per unit solid angle). Then, to find the amplitude for a decay at the
arbitrary angle of Fig. 19–4, all we need to know is
what amplitude the given initial state has zero angular momentum about
the decay direction. The amplitude for the decay at θ and ϕ
is then a times the amplitude that a state |l,m⟩ with respect to
the z-axis will be in the state |l,0⟩ with respect to z′—the
decay direction. This latter amplitude is just what we have written
in (19.31). The probability to see the α-particle
at θ,ϕ is
P(θ,ϕ)=a2|⟨l,0|Ry(θ)Rz(ϕ)|l,m⟩|2.
As an example, consider an initial state with l=1 and various values
of m. From Table 17–2 we know the necessary
amplitudes. They are
⟨1,0|Ry(θ)Rz(ϕ)|1,+1⟩=−1√2sinθeiϕ,⟨1,0|Ry(θ)Rz(ϕ)|1,0⟩=cosθ,⟨1,0|Ry(θ)Rz(ϕ)|1,−1⟩=1√2sinθe−iϕ.
These are the three possible angular distribution
amplitudes—depending on the m-value of the initial nucleus.
Amplitudes such as the ones in (19.36) appear so often
and are sufficiently important that they are given several
names. If the angular distribution amplitude is proportional to any
one of the three functions or any linear combination of them, we say,
“The system has an orbital angular momentum of one.” Or we may say,
“The Ne20∗ emits a p-wave α-particle.” Or we say,
“The α-particle is emitted in an l=1 state.” Because there
are so many ways of saying the same thing it is useful to have a
dictionary. If you are going to understand what other physicists are
talking about, you will just have to memorize the language. In
Table 19–1 we give a dictionary of orbital angular
momentum.
Table 19–1
Dictionary of orbital angular momentum
(l=j= an integer)
| | | | | | |
--- --- --- |
| Orbital angular momentum, l | z-component,m | Angular dependence of amplitudes | Name | Number of States | Orbital parity |
| 0 | 0 | 1 | s | 1 | + |
| 1 | {+10−1 | −1√2sinθeiϕcosθ1√2sinθe−iϕ} | p | 3 | − |
| 2 | {+2+10−1−2 | √64sin2θe2iϕ−√62sinθcosθeiϕ12(3cos2θ−1)√62sinθcosθe−iϕ√64sin2θe−2iϕ} | d | 5 | + |
| 345 | }345{ | ⟨l,0|Ry(θ)Rz(ϕ)|l,m⟩∝Yl,m(θ,ϕ)∝Pml(cosθ)eimϕ} | fgh | 2l+1 | (−1)l |
| ⋮ | | | ⋮ | | |
If the orbital angular momentum is zero, then there is no change when
you rotate the coordinate system and there is no variation with
angle—the “dependence” on angle is as a constant, say 1. This is
also called an “s-state”, and there is only one such state—as
far as angular dependence is concerned. If the orbital angular
momentum is 1, then the amplitude of the angular variation may be
any one of the three functions given—depending on the value
of m—or it may be a linear combination. These are called
“p-states,” and there are three of them. If the orbital angular
momentum is 2 then there are the five functions shown. Any linear
combination is called an “l=2,” or a “d-wave” amplitude. Now
you can immediately guess what the next letter is—what should come
after s, p, d? Well, of course, f, g, h, and so on down
the alphabet! The letters don’t mean anything. (They did once mean
something—they meant “sharp” lines, “principal” lines,
“diffuse” lines and “fundamental” lines of the optical spectra of
atoms. But those were in the days when people did not know where the
lines came from. After f there were no special names, so we now just
continue with g, h, and so on.)
The angular functions in the table go by several names—and are
sometimes defined with slightly different conventions about the
numerical factors that appear out in front. Sometimes they are called
“spherical harmonics,” and written
as Yl,m(θ,ϕ). Sometimes they are
written Pml(cosθ)eimϕ, and if m=0, simply
as Pl(cosθ). The functions Pl(cosθ) are called the
“Legendre polynomials” in cosθ, and the
functions Pml(cosθ) are called the “associated Legendre
functions.” You will find tables of these functions in many books.
Notice, incidentally, that all the functions for a given l have the
property that they have the same parity—for odd l they change sign
under an inversion and for even l they don’t change. So we can write
that the parity of a state of orbital angular momentum l is (−1)l.
As we have seen, these angular distributions may refer to a nuclear
disintegration or some other process, or to the distribution of the
amplitude to find an electron at some place in the hydrogen atom. For
instance, if an electron is in a p-state (l=1) the amplitude to
find it can depend on the angle in many possible ways—but all are
linear combinations of the three functions for l=1 in
Table 19–1. Let’s take the case cosθ. That’s
interesting. That means that the amplitude is positive, say, in the
upper part (θ<π/2), is negative in the lower
part (θ>π/2), and is zero when θ is 90∘. Squaring this
amplitude we see that the probability of finding the electron varies
with θ as shown in Fig. 19–5—and is independent
of ϕ. This angular distribution is responsible for the fact that in
molecular binding the attraction of an electron in an l=1 state for
another atom depends on direction—it is the origin of the directed
valences of chemical attraction.
Fig. 19–5.
A polar graph of cos2θ, which is the relative
probability of finding an electron at various angles from the z-axis
(for a given r) in an atomic state with l=1 and m=0.
19–4The general solution for hydrogen
In Eq. (19.35) we have written the wave functions for
the hydrogen atom as
ψl,m(r)=aYl,m(θ,ϕ)Fl(r).
These wave functions must be solutions of the differential
equation (19.7). Let’s see what that means. Put
(19.37) into (19.7); you get
Yl,mr∂2∂r2(rFl)+Flr2sinθ∂∂θ(sinθ∂Yl,m∂θ)+Flr2sin2θ∂2Yl,m∂ϕ2=−2mℏ2(E+e2r)Yl,mFl.
Now multiply through by r2/Fl and rearrange terms. The result is
1sinθ∂∂θ(sinθ∂Yl,m∂θ)+1sin2θ∂2Yl,m∂ϕ2=−[r2Fl{1rd2dr2(rFl)+2mℏ2(E+e2r)Fl}]Yl,m.
The left-hand side of this equation depends on θ and ϕ,
but not on r. No matter what value we choose for r, the
left side doesn’t change. This must also be true for the
right-hand side. Although the quantity in the square brackets has
r’s all over the place, the whole quantity cannot depend on r,
otherwise we wouldn’t have an equation good for all r. As you can
see, the bracket also does not depend on θ or ϕ. It must
be some constant. Its value may well depend on the l-value of the
state we are studying, since the function Fl must be the one
appropriate to that state; we’ll call the constant Kl.
Equation (19.39) is therefore equivalent to two
equations:
1sinθ∂∂θ(sinθ∂Yl,m∂θ)+1sin2θ∂2Yl,m∂ϕ2=−KlYl,m,1rd2dr2(rFl)+2mℏ2(E+e2r)Fl=KlFlr2.
Now look at what we’ve done. For any state described by l and m,
we know the functions Yl,m; we can use Eq. (19.40)
to determine the constant Kl. Putting Kl into
Eq. (19.41) we have a differential equation for the
function Fl(r). If we can solve that equation for Fl(r), we have
all of the pieces to put into (19.37) to give ψ(r).
What is Kl? First, notice that it must be the same for all m
(which go with a particular l), so we can pick any m we want
for Yl,m and plug it into (19.40) to solve
for Kl. Perhaps the easiest one to use is Yl,l. From
Eq. (18.24),
Rz(ϕ)|l,l⟩=eilϕ|l,l⟩.
The matrix element for Ry(θ) is also quite simple:
⟨l,0|Ry(θ)|l,l⟩=b(sinθ)l,where b is some number.3 Combining the two, we obtain
Yl,l∝eilϕsinlθ.Putting this function into (19.40) gives
Kl=l(l+1).
Now that we have determined Kl, Eq. (19.41) tells us
about the radial function Fl(r). It is, of course, just the
Schrödinger equation with the angular part replaced by its
equivalent KlFl/r2. Let’s rewrite (19.41) in the form
we had in Eq. (19.8), as follows:
1rd2dr2(rFl)=−2mℏ2{E+e2r−l(l+1)ℏ22mr2}Fl.
A mysterious term has been added to the potential energy. Although we
got this term by some mathematical shenanigan, it has a simple
physical origin. We can give you an idea about where it comes from in
terms of a semi-classical argument. Then perhaps you will not find it
quite so mysterious.
Think of a classical particle moving around some center of force. The
total energy is conserved and is the sum of the potential and kinetic
energies
U=V(r)+12mv2=constant.
In general, v can be resolved into a radial component vr and a
tangential component r˙θ; then
v2=v2r+(r˙θ)2.Now the angular momentum mr2˙θ is also conserved; say it
is equal to L. We can then write
mr2˙θ=L,orr˙θ=Lmr,and the energy is
U=12mv2r+V(r)+L22mr2.If there were no angular momentum we would have just the first two
terms. Adding the angular momentum L does to the energy just what
adding a term L2/2mr2 to the potential energy would do. But this
is almost exactly the extra term in (19.46). The only
difference is that l(l+1)ℏ2 appears for the angular momentum
instead of l2ℏ2 as we might expect. But we have seen before
(for example, Volume II, Section 34-7) that this is just the substitution that is
usually required to make a quasi-classical argument agree with a correct
quantum-mechanical calculation. We can, then, understand the new term as
a “pseudo-potential” which gives the “centrifugal
force” term that
appears in the equations of radial motion for a rotating system. (See
the discussion of “pseudo-forces” in Volume I,
Section 12-5.)
We are now ready to solve Eq. (19.46) for Fl(r). It
is very much like Eq. (19.8), so the same technique
will work again. Everything goes as before until you get to
Eq. (19.19) which will have the additional term
−l(l+1)∞∑k=1akρk−2.
This term can also be written as
−l(l+1){a1ρ+∞∑k=1ak+1ρk−1}.(We have taken out the first term and then shifted the running
index k down by 1.) Instead of Eq. (19.20) we have
∞∑k=1[{k(k+1)−l(l+1)}ak+1−2(αk−1)ak]ρk−1−l(l+1)a1ρ=0.There is only one term in ρ−1, so it must be zero. The
coefficient a1 must be zero (unless l=0 and we have our previous
solution). Each of the other terms is made zero by having the square
bracket come out zero for every k. This condition replaces
Eq. (19.22) by
ak+1=2(αk−1)k(k+1)−l(l+1)ak.This is the only significant change from the spherically symmetric
case.
As before the series must terminate if we are to have solutions which
can represent bound electrons. The series will end at k=n if αn=1. We get again the same condition on α, that it must be
equal to 1/n, where n is some positive integer. However,
Eq. (19.50) also gives a new restriction. The index k
cannot be equal to l, the denominator becomes zero and al+1 is
infinite. That is, since a1=0, Eq. (19.50) implies
that all successive ak are zero until we get to al+1, which can
be nonzero. This means that k must start at l+1 and end at n.
Our final result is that for any l there are many possible solutions
which we can call Fn,l where n≥l+1. Each solution has the
energy
En=−me42ℏ2(1n2).
The wave function for the state of this energy with the angular
quantum numbers l and m is
ψn,l,m=aYl,m(θ,ϕ)Fn,l(ρ),with
ρFn,l(ρ)=e−αρn∑k=l+1akρk.The coefficients ak are obtained from (19.50). We
have, finally, a complete description of the states of a hydrogen
atom.
19–5The hydrogen wave functions
Let’s review what we have discovered. The states which satisfy
Schrödinger’s equation for an electron in a Coulomb field are
characterized by three quantum numbers n, l, m, all integers. The
angular distribution of the electron amplitude can have only certain
forms which we call Yl,m. They are labeled by l, the
quantum number of total angular momentum, and m, the
“magnetic” quantum number, which can range from −l to +l. For
each angular configuration, various possible radial
distributions Fn,l(r) of the electron amplitude are possible; they
are labeled by the principal quantum number n—which can range from l+1 to ∞. The energy of the state depends only on n, and increases with
increasing n.
The lowest energy, or ground, state is an s-state. It has
l=0, n=1, and m=0. It is a “nondegenerate” state—there is only one
with this energy, and its wave function is spherically symmetric. The
amplitude to find the electron is a maximum at the center, and falls
off monotonically with increasing distance from the center. We can
visualize the electron amplitude as a blob as shown in
Fig. 19–6(a).
Fig. 19–6.
Rough sketches showing the general nature of some of the
hydrogen wave functions. The shaded regions show where the amplitudes
are large. The plus and minus signs show the relative sign of the
amplitude in each region.
There are other s-states with higher energies, for n=2, 3,
4, … For each energy there is only one version (m=0), and they
are all spherically symmetric. These states have amplitudes which
alternate in sign one or more times with increasing r. There are
n−1 spherical nodal surfaces—the places where ψ goes through
zero. The 2s-state (l=0, n=2), for example, will look as
sketched in Fig. 19–6(b). (The dark areas indicate
regions where the amplitude is large, and the plus and minus signs
indicate the relative phases of the amplitude.) The energy levels of the
s-states are shown in the first column of Fig. 19–7.
Fig. 19–7.
The energy level diagram for hydrogen.
Then there are the p-states—with l=1. For each n, which must
be 2 or greater, there are three states of the same energy, one each
for m=+1, m=0, and m=−1. The energy levels are as shown in
Fig. 19–7. The angular dependences of these states are
given in Table 19–1. For instance, for m=0, if the
amplitude is positive for θ near zero, it will be negative
for θ near 180∘. There is a nodal plane coincident with the
xy-plane. For n>2 there are also spherical nodes. The n=2,
m=0 amplitude is sketched in Fig. 19–6(c), and the n=3,
m=0 wave function is sketched in Fig. 19–6(d).
You might think that since m represents a kind of “orientation” in
space, there should be similar distributions with the peaks of
amplitude along the x-axis or along the y-axis. Are these perhaps
the m=+1 and m=−1 states? No. But since we have three states with
equal energies, any linear combinations of the three will also be
stationary states of the same energy. It turns out that the
“x”-state—which corresponds to the “z”-state, or
m=0 state, of Fig. 19–6(c)—is a linear combination of the
m=+1 and m=−1 states. The corresponding “y”-state is another
combination. Specifically, we mean that
“z”=|1,0⟩,“x”=−|1,+1⟩−|1,−1⟩√2,“y”=−|1,+1⟩+|1,−1⟩i√2.
These states all look the same when referred to their particular axes.
The d-states (l=2) have five possible values of m for each
energy, the lowest energy has n=3. The levels go as shown in
Fig. 19–7. The angular dependences get more complicated.
For instance the m=0 states have two conical nodes, so the wave
function reverses phase from +, to −, to + as you go around from
the north pole to the south pole. The rough form of the amplitude is
sketched in (e) and (f) of Fig. 19–6 for the
m=0 states with n=3 and n=4. Again, the larger n’s have spherical
nodes.
We will not try to describe any more of the possible states. You will
find the hydrogen wave functions described in more detail in many books.
Two good references are L. Pauling and E. B. Wilson,
Introduction to Quantum Mechanics, McGraw-Hill (1935); and
R. B. Leighton, Principles of Modern Physics,
McGraw-Hill (1959). You will find in them graphs of some of the
functions and pictorial representations of many states.
We would like to mention one particular feature of the wave functions
for higher l: for l>0 the amplitudes are zero at the center. That
is not surprising, since it’s hard for an electron to have angular
momentum when its radius arm is very small. For this reason, the
higher the l, the more the amplitudes are “pushed away” from the
center. If you look at the way the radial functions Fn,l(r) vary
for small r, you find from (19.53) that
Fn,l(r)≈rl.
Such a dependence on r means that for larger l’s you have to go
farther from r=0 before you get an appreciable amplitude. This
behavior is, incidentally, determined by the centrifugal
force term in
the radial equation, so the same thing will apply for any potential
that varies slower than 1/r2 for small r—which most atomic
potentials do.
19–6The periodic table
We would like now to apply the theory of the hydrogen atom in an
approximate way to get some understanding of the chemist’s periodic
table of the elements. For an element with atomic number Z there are
Z electrons held together by the electric attraction of the nucleus
but with mutual repulsion of the electrons. To get an exact solution we
would have to solve Schrödinger’s equation for Z electrons in a
Coulomb field. For helium the equation is
−ℏi∂ψ∂t=−ℏ22m(∇21ψ+∇22ψ)+(−2e2r1−2e2r2+e2r12)ψ,
where ∇21 is a Laplacian which operates on r1, the
coordinate of one electron; ∇22 operates on r2;
and r12=|r1−r2|. (We are again neglecting the spin of
the electrons.) To find the stationary states and energy levels we
would have to find solutions of the form
ψ=f(r1,r2)e−(i/ℏ)Et.The geometrical dependence is contained in f, which is a function of
six variables—the simultaneous positions of the two electrons. No
one has found an analytic solution, although solutions for the lowest
energy states have been obtained by numerical methods.
With 3, 4, or 5 electrons it is hopeless to try to obtain exact
solutions, and it is going too far to say that quantum mechanics has
given a precise understanding of the periodic table. It is possible,
however, even with a sloppy approximation—and some fixing—to
understand, at least qualitatively, many chemical properties which
show up in the periodic table.
The chemical properties of atoms are determined primarily by their
lowest energy states. We can use the following approximate theory to
find these states and their energies. First, we neglect the electron
spin, except that we adopt the exclusion principle and say that
any particular electronic state can be occupied by only one electron.
This means that any particular orbital configuration can have up to
two electrons—one with spin up, the other with spin down. Next
we disregard the details of the interactions between the
electrons in our first approximation, and say that each electron moves
in a central field which is the combined field of the nucleus and
all the other electrons. For neon, which has 10 electrons, we say that
one electron sees an average potential due to the nucleus plus the other
nine electrons. We imagine then that in the Schrödinger equation for
each electron we put a V(r) which is a 1/r field modified by a
spherically symmetric charge density coming from the other electrons.
In this model each electron acts like an independent particle. The
angular dependence of its wave function will be just the same as the
ones we had for the hydrogen atom. There will be s-states,
p-states, and so on; and they will have the various possible
m-values. Since V(r) no longer goes as 1/r, the radial part of
the wave functions will be somewhat different, but it will be
qualitatively the same, so we will have the same radial quantum
numbers, n. The energies of the states will also be somewhat
different.
H
With these ideas, let’s see what we get. The ground state of hydrogen
has l=m=0 and n=1; we say the electron configuration is 1s. The
energy is −13.6 eV. This means that it takes 13.6 electron volts
to pull the electron off the atom. We call this the “ionization
energy”, WI. A large ionization energy means that it is harder to
pull the electron off and, in general, that the material is chemically
less active.
He
Now take helium. Both electrons can be in the same lowest state (one
spin up and the other spin down). In this lowest state the electron
moves in a potential which is for small r like a Coulomb field
for z=2 and for large r like a Coulomb field for z=1. The result is
a “hydrogen-like” 1s state with a somewhat lower energy. Both
electrons occupy identical 1s states (l=0, m=0). The observed
ionization energy (to remove one electron) is 24.6 electron
volts. Since the 1s “shell” is now filled—we allow only two
electrons—there is practically no tendency for an electron to be
attracted from another atom. Helium is chemically inert.
Li
The lithium nucleus has a charge of 3. The electron states will
again be hydrogen-like, and the three electrons will occupy the lowest
three energy levels. Two will go into 1s states and the third will
go into an n=2 state. But with l=0 or l=1? In hydrogen these
states have the same energy, but in other atoms they don’t, for the
following reason. Remember that a 2s state has some amplitude to be
near the nucleus while the 2p state does not. That means that a
2s electron will feel some of the triple electric charge of the Li
nucleus, but that a 2p electron will stay out where the field looks
like the Coulomb field of a single charge. The extra attraction lowers
the energy of the 2s state relative to the 2p state. The energy
levels will be roughly as shown in Fig. 19–8—which you
should compare with the corresponding diagram for hydrogen in
Fig. 19–7. So the lithium atom will have two electrons in
1s states and one in a 2s. Since the 2s electron has a higher
energy than a 1s electron it is relatively easily removed. The
ionization energy of lithium is only 5.4 electron volts, and it is
quite active chemically.
Fig. 19–8.
Schematic energy level diagram for an atomic electron with other
electrons present. (The scale is not the same as
Fig. 19–7.)
So you can see the patterns which develop; we have given in
Table 19–2 a list of the first 36 elements, showing the
states occupied by the electrons in the ground state of each atom. The
Table gives the ionization energy for the most loosely bound electron,
and the number of electrons occupying each “shell”—by which we mean
states with the same n. Since the different l-states have different
energies, each l-value corresponds to a sub-shell of
2(2l+1) possible states (of different m and electron spin). These
all have the same energy—except for some very small effects we are
neglecting.
Table 19–2The electron configurations of the first 36 elements
| | | | | | | | | | | | | | |
--- --- --- --- --- --- --- |
| Z | Element | | WI(eV) | Electron Configuration | | | | | | | | | |
| 1s | 2s | 2p | 3s | 3p | 3d | 4s | 4p | 4d | 4f |
| 1 | H | hydrogen | 13.6 | 1 | | | | | | | | | |
| 2 | He | helium | 24.6 | 2 |
| 3 | Li | lithium | 5.4 | FILLED (2) | 1 | |
| 4 | Be | beryllium | 9.3 | 2 |
| 5 | B | boron | 8.3 | 2 | 1 |
| 6 | C | carbon | 11.3 | 2 | 2 | Number of electrons in each state | | | | | | |
| 7 | N | nitrogen | 14.5 | 2 | 3 |
| 8 | O | oxygen | 13.6 | 2 | 4 | | | | | | | |
| 9 | F | fluorine | 17.4 | 2 | 5 |
| 10 | Ne | neon | 21.6 | 2 | 6 |
| 11 | Na | sodium | 5.1 | | | | 1 | | |
| 12 | Mg | magnesium | 7.6 | 2 |
| 13 | Al | aluminum | 6.0 | 2 | 1 | |
| 14 | Si | silicon | 8.1 | —FILLED— | | | 2 | 2 |
| 15 | P | phosphorus | 10.5 | 2 | 3 |
| 16 | S | sulfur | 10.4 | (2) | (8) | | 2 | 4 |
| 17 | Cl | chlorine | 13.0 | | | | 2 | 5 |
| 18 | Ar | argon | 15.8 | 2 | 6 |
| 19 | K | potassium | 4.3 | | | 1 | | | |
| 20 | Ca | calcium | 6.1 | 2 |
| 21 | Sc | scandium | 6.5 | 11 | 2 |
| 22 | Ti | titanium | 6.8 | 12 | 2 |
| 23 | V | vanadium | 6.7 | ——FILLED—— | | | | | 13 | 2 |
| 24 | Cr | chromium | 6.8 | 15 | 1 |
| 25 | Mn | manganese | 7.4 | (2) | (8) | | (8) | | 15 | 2 |
| 26 | Fe | iron | 7.9 | | | | | | 16 | 2 |
| 27 | Co | cobalt | 7.9 | 17 | 2 |
| 28 | Ni | nickel | 7.6 | 18 | 2 |
| 29 | Cu | copper | 7.7 | 10 | 1 |
| 30 | Zn | zinc | 9.4 | 10 | 2 |
| 31 | Ga | gallium | 6.0 | | | | | | | 2 | 1 | | |
| 32 | Ge | germanium | 7.9 | ——FILLED—— | | | | | | 2 | 2 |
| 33 | As | arsenic | 9.8 | 2 | 3 |
| 34 | Se | selenium | 9.7 | (2) | (8) | | (18) | | | 2 | 4 |
| 35 | Br | bromine | 11.8 | | | | | | | 2 | 5 |
| 36 | Kr | krypton | 14.0 | 2 | 6 |
Be
Beryllium is like lithium except that it has two electrons in the
2s state as well as two in the filled 1s shell.
B to Ne
Boron has 5 electrons. The fifth must go into a 2p state. There
are 2×3=6 different 2p states, so we can keep adding
electrons until we get to a total of 8. This takes us to neon. As we
add these electrons we are also increasing Z, so the whole electron
distribution gets pulled in closer and closer to the nucleus and the
energy of the 2p states goes down. By the time we get to neon the
ionization energy is up to 21.6 electron volts. Neon does not easily give up
an electron. Also there are no more low-energy slots to be filled, so
it won’t try to grab an extra electron. Neon is chemically
inert. Fluorine, on the other hand, does have an empty position
where an electron can drop into a state of low energy, so it is quite
active in chemical reactions.
Na to Ar
With sodium the eleventh electron must start a new shell—going into
a 3s state. The energy level of this state is much higher; the
ionization energy jumps down; and sodium is an active chemical. From
sodium to argon the s and p states with n=3 are occupied in
exactly the same sequence as for lithium to neon. Angular
configurations of the electrons in the outer unfilled shell have the
same sequence, and the progression of ionization energies is quite
similar. You can see why the chemical properties repeat with
increasing atomic number. Magnesium acts chemically much like
beryllium, silicon like carbon, and chlorine like fluorine. Argon is
inert like neon.
You may have noticed that there is a slight peculiarity in the
sequence of ionization energies between lithium and neon, and a
similar one between sodium and argon. The last electron is bound to
the oxygen atom somewhat less than we might expect. And sulfur is
similar. Why should that be? We can understand it if we put in just a
little bit of the effects of the interactions between individual
electrons. Think of what happens when we put the first 2p electron
onto the boron atom. It has six possibilities—three possible
p-states, each with two spins. Imagine that the electron goes with
spin up into the m=0 state, which we have also called the
“z” state because it hugs the z-axis. Now what will happen in carbon?
There are now two 2p electrons. If one of them goes into the
“z” state, where will the second one go? It will have lower energy if it
stays away from the first electron, which it can do by going into,
say, the “x” state of the 2p shell. (This state is, remember,
just a linear combination of the m=+1 and m=−1 states.) Next, when
we go to nitrogen, the three 2p electrons will have the lowest
energy of mutual repulsion if they go one each into the “x,” “y,”
and “z” configurations. For oxygen, however, the jig is
up. The fourth electron must go into one of the filled states—with
opposite spin. It is strongly repelled by the electron already in that
state, so its energy will not be as low as it might otherwise be, and
it is more easily removed. That explains the break in the sequence of
binding energies which appears between nitrogen and oxygen, and
between phosphorus and sulfur.
K to Zn
After argon, you would, at first, think that the new electrons would
start to fill up the 3d states. But they don’t. As we described
earlier—and illustrated in Fig. 19–8—the higher
angular momentum states get pushed up in energy. By the time we get to
the 3d states they are pushed to an energy a little bit above the
energy of the 4s state. So in potassium the last electron goes into
the 4s state. After this shell is filled (with two electrons) at
calcium, the 3d states begin to be filled for scandium, titanium, and
vanadium.
The energies of the 3d and 4s states are so close together that
small effects can shift the balance either way. By the time we get to
put four electrons into the 3d states, their repulsion raises the
energy of the 4s state just enough that its energy is slightly above
the 3d energy, so one electron shifts over. For chromium we don’t
get a 4, 2 combination as we would have expected, but instead a
5, 1 combination. The new electron added to get manganese fills up
the 4s shell again, and the states of the 3d shell are then
occupied one by one until we reach copper.
Since the outermost shell of manganese, iron, cobalt, and nickel have
the same configurations, however, they all tend to have similar
chemical properties. (This effect is much more pronounced in the
rare-earth elements which all have the same outer shell but a
progressively filling inner shell which has much less influence on
their chemical properties.)
In copper an electron is robbed from the 4s shell, finally
completing the 3d shell. The energy of the 10, 1 combination is,
however, so close to the 9, 2 configuration for copper that just
the presence of another atom nearby can shift the balance. For this
reason the two last electrons of copper are nearly equivalent, and
copper can have a valence of either 1 or 2. (It sometimes acts as
though its electrons were in the 9, 2 combination.) Similar things
happen at other places and account for the fact that other metals,
such as iron, combine chemically with either of two valences. By zinc,
both the 3d and 4s shells are filled once and for all.
Ga to Kr
From gallium to krypton the sequence proceeds normally again, filling
the 4p shell. The outer shells, the energies, and the chemical
properties repeat the pattern of boron to neon and aluminum to argon.
Krypton, like argon and neon, is known as “noble” gas. All three are
chemically “inert.” This means only that, having filled shells of
relatively low energy, there are few situations in which there is an
energy advantage for them to join in a simple combination with other
elements. Having a filled shell is not enough. Beryllium and magnesium
have filled s-shells, but the energy of these shells is too high to
lead to stability. Similarly, one would have expected another
“noble” element at nickel, if the energy of the 3d shell had been
lower (or the 4s higher). On the other hand, krypton is not
completely inert; it will form a weakly-bound compound with chlorine.
Since our sample has turned up most of the main features of the
periodic table, we stop our examination at element number 36—there
are still seventy or so more!
We would like to bring up only one more point—that we not only can
understand the valences to some extent but also can say something
about the directional properties of the chemical bonds. Take an atom
like oxygen which has four 2p electrons. The first three go into
“x,” “y,” and “z” states and the fourth will double one of
these states, leaving two—say “x” and “y”—vacant. Consider
then what happens in H2O. Each of the two hydrogens are willing to
share an electron with the oxygen, helping the oxygen to fill a
shell. These electrons will tend to go into the “x” and “y”
vacancies. So the water molecule should have the two hydrogen atoms
making a right angle with respect to the center of the oxygen. The
angle is actually 105∘. We can even understand why the angle is
larger than 90∘. In sharing their electrons the hydrogens end
up with a net positive charge. The electric repulsion “strains” the
wave functions and pushes the angle out to 105∘. The same
situation occurs in H2S. But because the sulfur atom is larger,
the two hydrogen atoms are farther apart, there is less repulsion, and
the angle is only pushed out to about 93∘. Selenium is even
larger, so in H2Se the angle is very nearly 90∘.
We can use the same arguments to understand the geometry of ammonia,
H3N. Nitrogen has room for three more 2p electrons, one each for
the “x,” “y,” and “z” type states. The three hydrogens
should join on at right angles to each other. The angles come out a
little larger than 90∘—again from the electric
repulsion—but at least we see why the molecule of H3N is not
flat. The angles in phosphene, H3P, are close to 90∘, and in
H3As are still closer. We assumed that NH3 was not flat when we
described it as a two-state system. And the nonflatness is what makes
the ammonia maser possible. Now we see that also that shape can be
understood from our quantum mechanics.
The Schrödinger equation has been one of the great triumphs of
physics. By providing the key to the underlying machinery of atomic
structure it has given an explanation for atomic spectra, for
chemistry, and for the nature of matter.
As usual, e2=q2e/4πϵ0.
↩
Since these
special names are part of the common vocabulary of atomic physics, you
will just have to learn them. We will help out by putting them
together in a short “dictionary” later in the chapter.
↩
You can with some work show that
this comes out of Eq. (18.35), but it is also easy to
work out from first principles following the ideas of
Section 18-4. A state |l,l⟩ can be made out of
2l spin one-half particles all with spins up; while the state |l,0⟩
would have l up and l down. Under the rotation the amplitude that an
up-spin remains up is cosθ/2, and that an up-spin goes down
is −sinθ/2. We are asking for the amplitude that l up-spins stay
up, while the other l up-spins go down. The amplitude for that
is (−cosθ/2sinθ/2)l which is proportional
to sinlθ.
↩ |
189464 | https://math.stackexchange.com/questions/942894/difference-between-linear-transformation-and-its-matrix-representation | matrices - difference between linear transformation and its matrix representation . - Mathematics Stack Exchange
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difference between linear transformation and its matrix representation .
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I can't understand this:
Given a matrix T ={T i j}∈M m n{T i j}∈M m n,define a transformation T:R n→R m T:R n→R m as follows:
If T(x 1,……,x n)=(y 1,……,y m)T(x 1,……,x n)=(y 1,……,y m) , then y i=∑n j=1 T i j x j y i=∑j=1 n T i j x j for all i=1,.....,m i=1,.....,m
I can't differentiate between a linear mapping and the matrix of transformation.Aren't they representing the same thing.Please help me understanding the difference between them...
matrices
linear-transformations
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edited Sep 23, 2014 at 16:28
hardmath
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asked Sep 23, 2014 at 13:18
spectraaspectraa
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The matrix T represents the linear transformation T T subject to the choice of an ordered basis for each the domain and the codomain. Change the bases and you induce a corresponding change in how the linear transformation is represented.
But the linear transformation T:R n→R m T:R n→R m is the same regardless of what basis you have in mind.
Often we want to pick a basis (or bases) so that the matrix T will be especially nice, say a diagonal or block diagonal form, or some other sparse representation (lots of zero entries).
But again, changing basis will affect only the matrix representation, not the linear transformation itself.
The case where n=m n=m is a little special in that ordinarily we prefer to use the same basis for the domain ("input") and the codomain ("output"). It is not mandatory to do so, but it imposes an extra burden on memory to keep track of whether R n R n is being used for the input or the output, and think in terms of the appropriate coordinate system for the purpose.
To simplify matters we will give an example of representing a linear transformation T:R 2→R 2 T:R 2→R 2 that uses the same basis for both. First consider the standard ordered basis for R 2 R 2:
E 2=[(1,0),(0,1)]E 2=[(1,0),(0,1)]
Note that I'm using square brackets, rather than a pair of curly brackets, to emphasize that we chose an ordering of the basis vectors. Curly brackets would mean a set, and the set does not entail any special order of its elements.
Consider now a linear transformation T:R 2→R 2 T:R 2→R 2 defined by swapping the x,y x,y coordinates:
T(x,y)=(y,x)T(x,y)=(y,x)
With respect to the standard basis we have a matrix M M such that multiplication by M M converts the standard coordinates of input vector (x,y)(x,y) as a column vector into standard coordinates of output vector T(x,y)=(y,x)T(x,y)=(y,x) as a column vector:
[0 1 1 0][x y]=[y x][0 1 1 0][x y]=[y x]
It's important to understand how the column vectors represent an input/output vector with respect to our chosen basis. Every vector can be represented uniquely as a linear combination of basis vectors, so in the case of the standard basis we have:
(x,y)=x⋅(1,0)+y⋅(0,1)(x,y)=x⋅(1,0)+y⋅(0,1)
But for the sake of making matrix multiplication work, we take those scalar multipliers and create a column vector with them as entries corresponding to the order of the respective basis vectors. In standard coordinates we represent (x,y)(x,y) as [x y][x y] and (y,x)(y,x) as [y x][y x]. Thus the multiplication by M M shown above converts the standard coordinates for (x,y)(x,y) to the standard coordinates for (y,x)(y,x).
Now let's try this with a "nonstandard" basis. Sometimes a better basis for representing a transformation may be as simple as just changing the order of the vectors in an existing basis, but we are going to do a more complicated example just to contrast steps that are easy from steps that are harder.
Choose nonstandard basis B=[(1,2),(3,4)]B=[(1,2),(3,4)]. To verify this is a basis for R 2 R 2, we only need to point out that the two vectors are linearly independent (neither is a multiple of the other, since there are only these two), and since the count (two vectors) matches the dimension (two), it must be a basis.
What we want is to find a 2×2 2×2 matrix T that represents T T with respect to this nonstandard basis. That is, multiplying the column vector of nonstandard coordinates for an input (x,y)(x,y) should give us the column vector of nonstandard coordinates for its output T(x,y)=(y,x)T(x,y)=(y,x).
The recipe I recommend is to make use of what we already know, the matrix M M which represents T T in standard coordinates. Specifically:
(1) Find a matrix B B whose multiplication converts columns of nonstandard coordinates into columns of standard coordinates.
(2) Find a matrix whose multiplication converts columns of standard coordinates into columns of nonstandard coordinates, name the inverse B−1 B−1 of the matrix found above.
Then T = B−1 M B B−1 M B, because multiplying by T accomplishes the properly sequenced operations to go from nonstandard coordinates of an input to nonstandard coordinates of its output.
Framing things in this way makes the exercise very tractable, because matrix B B turns out to be something we can write down "by inspection". Suppose that a vector (x,y)(x,y) has nonstandard coordinates [u v][u v]. This simply means:
(x,y)=u⋅(1,2)+v⋅(3,4)(x,y)=u⋅(1,2)+v⋅(3,4)
If we write down what this means in standard coordinates:
[x y]=u⋅[1 2]+v⋅[3 4][x y]=u⋅[1 2]+v⋅[3 4]
it becomes evident that B=[1 2 3 4]B=[1 3 2 4], with the columns of B B formed by standard coordinates of our nonstandard basis vectors:
B[u v]=[1 u+3 v 2 u+4 v]B[u v]=[1 u+3 v 2 u+4 v]
To carry out our construction of T as B−1 M B B−1 M B, we will need to invert B B, and this is certainly more effort than it took right down B B "by inspection". I leave it to the Reader to verify that applying the usual trick for 2×2 2×2 matrices:
B−1=−1 2[4−2−3 1]B−1=−1 2[4−3−2 1]
Finally we get (if I've not made a mistake) that T is:
−1 2[4−2−3 1][0 1 1 0][1 2 3 4]=[−5 2 3 2−7 2 5 2]−1 2[4−3−2 1][0 1 1 0][1 3 2 4]=[−5 2−7 2 3 2 5 2]
As a quick check, consider a nonstandard basis vector (1,2)(1,2), whose nonstandard coordinates are [1 0][1 0]. Multiplying by T gives the first column of T, namely [−5 2 3 2][−5 2 3 2], which should be the nonstandard coordinates of T(1,2)=(2,1)T(1,2)=(2,1). Sure enough:
(2,1)=−5 2⋅(1,2)+3 2⋅(3,4)(2,1)=−5 2⋅(1,2)+3 2⋅(3,4)
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edited Sep 23, 2014 at 16:00
answered Sep 23, 2014 at 13:22
hardmathhardmath
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thanks for the answer.please can you just help me being more clear about " Change the bases and you induce a corresponding change in how the linear transformation is represented." with the help of an example..spectraa –spectraa 2014-09-23 13:30:30 +00:00 Commented Sep 23, 2014 at 13:30
3 Yes, I'll be glad to do that. Sometimes the representation stays the same when you change basis, e.g. the identity transformation is always and only represented by the identity matrix. But this is a special case.hardmath –hardmath 2014-09-23 13:36:20 +00:00 Commented Sep 23, 2014 at 13:36
thanks the explanation was really helpful....spectraa –spectraa 2014-09-23 16:16:35 +00:00 Commented Sep 23, 2014 at 16:16
2 Here's an exercise to help cement the steps in your understanding. Using the same linear transformation T:R 2→R 2 T:R 2→R 2, find a basis B′B′ that give a diagonal matrix representation. Hint: eigenvectors.hardmath –hardmath 2014-09-23 16:20:49 +00:00 Commented Sep 23, 2014 at 16:20
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We have theorems and techniques for finite dimensional vector spaces that associate (a linear transformation + a basis) with a matrix. By mapping elements (via the choice of basis) to column vectors, we can compute the linear transformation's values by matrix-vector arithmetic. It is important to keep in mind that it is not simply linear transformation ~ matrix, but there is a choice of basis associated as well!
However matrices exist independent of linear transformations. They are just a funny way to write arrays of numbers, and they have their own rules and structures that can be studied.
Linear transformations likewise are simply special functions between vector spaces. They exist on their own, without regard for this special way to write them down.
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answered Sep 23, 2014 at 13:23
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Every linear transformation T:V(n)→W(m)T:V(n)→W(m) can be represented, with respect to two bases β∈V β∈V and γ∈W γ∈W, as a matrix A T A T of size m×n m×n. For each vector v∈V v∈V and w∈W w∈W, their coordinates in respective bases are written in column vectors as [v]β∈R n[v]β∈R n and [w]γ∈R m[w]γ∈R m, and the linear transformation u=T(v)u=T(v) is represented as a matrix multiplication [u]γ=A T[v]β[u]γ=A T[v]β.
Conversely, every matrix A m×n A m×n can be viewed as a linear transformation T A:R n→R m T A:R n→R m; indeed, all the matrix properties are derived from properties of linear transformations! As such, the matrix A A in the equation y=A x y=A x is by itself a linear transformation without the need of specifying the bases. (But you can always pick any two bases β,γ β,γ and derive its "matrix-representation"; the simplest ones are the standard bases of R n,R m R n,R m, in which [A]γ β[A]β γ = A)
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answered Dec 29, 2016 at 7:43
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Write down the definition in an equivalent way as follows,
T(x 1,……,x n)=(∑j=1 n T 1 j x j,∑j=1 n T 2 j x j,……,∑j=1 n T m j x j)T(x 1,……,x n)=(∑j=1 n T 1 j x j,∑j=1 n T 2 j x j,……,∑j=1 n T m j x j)
Then you should see that this is nothing but the Linear transformation defined by:
T⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜x 1 x 2 x 3..x n⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟=T⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜x 1 x 2 x 3..x n⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟T(x 1 x 2 x 3..x n)=T(x 1 x 2 x 3..x n)
Now as long as you fix two bases B 1 B 1 and B 1 B 1 for R n R n and R m R m this Linear Transformation is uniquely determined by the given matrix. Actually, fixing two bases we can define a bijection between the set M m n M m n and the set of Linear Transformations from R n→R m R n→R m. That means for each Linear Transformation T T "there exists one and exactly one" matrix T∈M m n T∈M m n such that T(x––)=T x––T(x _)=T x _ where x––x _ is a column vector in R n R n. Now all this is again as long as you fix two bases.
So essentially you're right. You need not differentiate between them at all as long as you are using the Standard Ordered Bases for the two Euclidean spaces, which I suspect you are.
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answered Sep 23, 2014 at 13:37
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Linear transformation Just talk about XY plane. If I transform (rotate) X axis to 90 degree I will get Y axis it is a one of the linear transformation. Another example just draw a real line in some interval after that draw another line for same inerwal in scaled version, it is also a linear transformation. If we condiser in two dimention First example draw a square and draw rectangle also. If I say I transformed somehow square in rectangle then it is a linear transformation of square into rectangle. Second example draw square and circle and then I say I transformed square into circle then it is not a linear transformation of square into circle because linear sides of former are not not linear (they are curved now).
Matrix transformation is nothing but a transformation caused by matrix. Matrix transformation is alway linear transformation.
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answered Dec 29, 2016 at 8:36
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189465 | https://math.stackexchange.com/questions/5033192/what-does-the-inverse-sine-function-actually-represent | Stack Exchange Network
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What does the inverse sine function actually represent?
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I know that the inverse of sine is used to find angle measures x when given a sine ratio (opposite/hypotenuse). However, my question is more conceptual. I understand how one can use the inverse sine function, or inverse cosine or inverse tangent, to find a missing angle, but I don't know exactly what any of these functions represent. Does it represent the value of the missing angle? I'm pretty confused about this, for I just started trig last week and these concepts are pretty new to me. Thanks!
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user386598user386598
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$\begingroup$ Wouldn't x just represent the length ratio because opposite/hypotenuse is the same as opposite/hypotenuse divided by 1? So the hypotenuse would be equal to one unit and the opposite side would also be equal to the length ratio? I do not really get your point, sorry. $\endgroup$
user386598
– user386598
2025-02-10 19:44:22 +00:00
Commented Feb 10 at 19:44
$\begingroup$ Yes, you're right, but, geometrically speaking, x is also the sine of the angle. If x is the sine function applied to the angle, then the angle is the inverse function of sine applied to x. This answers your question: What is does the inverse function represent? $\endgroup$
mins
– mins
2025-02-10 20:00:33 +00:00
Commented Feb 10 at 20:00
$\begingroup$ Doesn't this only apply to a unit circle though? $\endgroup$
user386598
– user386598
2025-02-10 21:23:13 +00:00
Commented Feb 10 at 21:23
$\begingroup$ Maybe I am overcomplicating this, I am just a little confused on how this example using the hypotenuse as one explains the meaning of the inverse. I might be missing something, sorry. $\endgroup$
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– user386598
2025-02-10 23:54:11 +00:00
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When you use any of these inverse functions, what you're essentially doing is reversing the trigonometric function. For example: If using sine, you might compute $\sin(30^\circ) = 0.5 $. If using inverse sine, you're saying, "I know the sine of an angle is 0.5, so what is the angle?" and $ \sin^{-1}(0.5) = 30^\circ $. These functions represent the angles whose trigonometric ratios you provide.
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answered Feb 8 at 22:55
AleAle
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$\begingroup$ Thank you! What would be the relationship between sine and its inverse, though? $\endgroup$
user386598
– user386598
2025-02-08 23:13:19 +00:00
Commented Feb 8 at 23:13
$\begingroup$ The sine function takes an angle $\theta$ and returns the sine of the angle, which is the ratio of the length of the opposite side to the hypotenuse in a right triangle. The output is constrained between -1 and 1. The arcsine function, denoted as $\sin^{-1}(x)$, takes a number $x$ (which must be within the range of -1 to 1, as these are the possible outputs of the sine function) and returns an angle $\theta$ such that $\sin(\theta) = x$. This angle is provided within the restricted range of $[-90^\circ, 90^\circ]$ or $[-\pi/2, \pi/2]$ radians. $\endgroup$
Ale
– Ale
2025-02-08 23:37:33 +00:00
Commented Feb 8 at 23:37
1
$\begingroup$ Technically, this is only true for certain angles. For example, $\sin 150^\circ=\frac12,$ too. $\endgroup$
Thomas Andrews
– Thomas Andrews
2025-02-08 23:50:59 +00:00
Commented Feb 8 at 23:50
$\begingroup$ So basically, the inverse sine represents what the angle is if given a specific ratio of opposite/hypotenuse? Therefore, for any given ratio, there is only one possible angle that fits within the given restrictions (-90 to 90 degrees)? I just want to make sure I understand this. $\endgroup$
user386598
– user386598
2025-02-09 06:11:40 +00:00
Commented Feb 9 at 6:11
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What is a mathematical expression for the sequence {1,1,−1,−1,1,1,−1,−1,…}{1,1,−1,−1,1,1,−1,−1,…}?
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What is a mathematical expression for the sequence {1,1,−1,−1,1,1,−1,−1,…}{1,1,−1,−1,1,1,−1,−1,…}, that is 1 1 and −1−1, two at a time alternating?
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edited Mar 13, 2016 at 21:44
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asked Mar 13, 2016 at 21:16
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1 Do you means the sequence of 1 1 s and −1−1 s, two at a time alternating?2'5 9'2 –2'5 9'2 2016-03-13 21:20:06 +00:00 Commented Mar 13, 2016 at 21:20
Try working with powers of −1−1. You have −1−1 to an even power is positive one, while −1−1 to an odd power is negative one. Now, try defining the exponent in a convenient way so that it goes even, even, odd, odd, etc... perhaps a quadratic might work well. (what does (n 2+n)/2(n 2+n)/2 look like in terms of even and odd numbers as a sequence?)JMoravitz –JMoravitz 2016-03-13 21:20:43 +00:00 Commented Mar 13, 2016 at 21:20
@alex.jordan Yes Postskjerm –Postskjerm 2016-03-13 21:22:12 +00:00 Commented Mar 13, 2016 at 21:22
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Starting with n=0 n=0, these all work.
a n a n a n a n=(−1)n(n−1)/2=(−1)⌊n 2⌋=cos(n π/2)+sin(n π/2)=2–√⋅sin((2 n+1)π 4)a n=(−1)n(n−1)/2 a n=(−1)⌊n 2⌋a n=cos(n π/2)+sin(n π/2)a n=2⋅sin((2 n+1)π 4)
If I'm being honest, being able to come up with these things for me comes from having seen them before. But in each case you can think about the pattern of even/odd exponents for −1−1 or the periodicity mod 4 if you were trying to build these having no prior knowledge.
U p d a t e:U p d a t e: And modified from this post so it starts a 0=1 a 0=1,
a n=2–√⋅cos((2 n−1)π 4)a n=2⋅cos((2 n−1)π 4)
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edited Apr 13, 2017 at 12:20
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answered Mar 13, 2016 at 21:20
2'5 9'22'5 9'2
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Wow, that was fast, thank you very much.Postskjerm –Postskjerm 2016-03-13 21:23:25 +00:00 Commented Mar 13, 2016 at 21:23
Third is the most intuitive IMHO Elimination –Elimination 2016-03-13 21:52:39 +00:00 Commented Mar 13, 2016 at 21:52
1 Now placed as second :)Elimination –Elimination 2016-03-13 21:58:02 +00:00 Commented Mar 13, 2016 at 21:58
1 I like the last one. :)Simply Beautiful Art –Simply Beautiful Art 2016-03-13 22:34:45 +00:00 Commented Mar 13, 2016 at 22:34
1 There seems to be a problem with the sine version, its absolute vaules cannot be one. Should it maybe be 2–√2 instead of 1 2√1 2?Matthias Klupsch –Matthias Klupsch 2016-12-14 12:09:11 +00:00 Commented Dec 14, 2016 at 12:09
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The sequence a 0=1 a 0=1, a 1=1 a 1=1, a 2=−1 a 2=−1, a 3=−1 a 3=−1, a 4=1 a 4=1 and so on satisfies the recursion
a 0=1,a 1=1,a n+2=−a n a 0=1,a 1=1,a n+2=−a n
so its characteristic equation is t 2+1=0 t 2+1=0. Thus the general solution is of the form
x i n+y(−i)n x i n+y(−i)n
The initial conditions tell that x+y=1 x+y=1 and x i−y i=1 x i−y i=1, thus
{x+y=1 x−y=−i{x+y=1 x−y=−i
that gives
x=1−i 2,y=1+i 2 x=1−i 2,y=1+i 2
Since i=cos π 2+i sin π 2 i=cosπ 2+i sinπ 2, we can write
a n=1 2((1−i)(cos n π 2+i sin n π 2)+(1+i)(cos n π 2−i sin n π 2))=cos n π 2+sin n π 2 a n=1 2((1−i)(cosn π 2+i sinn π 2)+(1+i)(cosn π 2−i sinn π 2))=cosn π 2+sinn π 2
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answered Mar 13, 2016 at 22:36
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+1 for showing a general approach to this class of problems AlexR –AlexR 2016-03-14 22:55:36 +00:00 Commented Mar 14, 2016 at 22:55
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A rather exotic formula uses tribonacci numbersT n T n,
a n=−(−1)T n=1,1,−1,−1,1,1,−1,−1,…a n=−(−1)T n=1,1,−1,−1,1,1,−1,−1,…
where,
T n=∑k=0 n∑j=0 n−k(n−k j)(j k−j)=1,1,2,4,7,13,24,44,81,149,274,504,…T n=∑k=0 n∑j=0 n−k(n−k j)(j k−j)=1,1,2,4,7,13,24,44,81,149,274,504,…
and we define T 0=1 T 0=1.
P.S. The above has period 4 4. For other periods which uses the Fibonacci numbers etc, see this post.
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answered Dec 14, 2016 at 12:07
Tito Piezas IIITito Piezas III
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You can simply interpolate(1,1,−1,−1)(1,1,−1,−1), and then use x=n mod 4 x=n mod 4:
a n=2 3(n mod 4)3−5(n mod 4)2+31 3(n mod 4)−5 a n=2 3(n mod 4)3−5(n mod 4)2+31 3(n mod 4)−5
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answered Dec 14, 2016 at 12:11
barak manosbarak manos
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I interpret your question as: "How can we define this sequence formally?"
A more general question is: how can we define x=(a,a,b,b,a,a,b,b,a,a,…)x=(a,a,b,b,a,a,b,b,a,a,…) formally?
If you prefer the convention that sequences start at 0 0, use:
x i=⎧⎩⎨⎪⎪⎪⎪⎪⎪a,b,if⌊i 2⌋∈2 Z if⌊i 2⌋∈2 Z+1 x i={a,if⌊i 2⌋∈2 Z b,if⌊i 2⌋∈2 Z+1
If you prefer the convention that sequences start at 1 1, use:
x i=⎧⎩⎨⎪⎪⎪⎪⎪⎪a,b,if⌊i 2⌋∈2 Z+1 if⌊i 2⌋∈2 Z x i={a,if⌊i 2⌋∈2 Z+1 b,if⌊i 2⌋∈2 Z
The above definitions are "set-theoretic." But we can also give an "algebraic" definition. This has the benefit that our numbering conventions become irrelevant. We begin by telling the reader that by ∑∑, we mean concatenation of sequences. Then we write:
x=∑n∈N(a,a,b,b)x=∑n∈N(a,a,b,b)
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edited Mar 13, 2016 at 22:24
answered Mar 13, 2016 at 22:19
goblin GONEgoblin GONE
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The sequence of exponents of −1−1 (i.e. 0,0,1,1,0,0,1,1,⋯0,0,1,1,0,0,1,1,⋯ but also 0,0,1,1,2,2,3,3⋯0,0,1,1,2,2,3,3⋯) can follow the recurrence
e n+2=e n+1,e 0=e 1=0.e n+2=e n+1,e 0=e 1=0.
The general solution is given by
e n=n 2+c 0+c 1(−1)n e n=n 2+c 0+c 1(−1)n
and you can determine the constants from the initial conditions, giving
e n=n 2−1−(−1)n 4.e n=n 2−1−(−1)n 4.
The method generalizes to runs of length k k, involving the k t h k t h roots of the unit.
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answered Dec 14, 2016 at 13:23
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The sequence {a 0,a 1,a 2,…}{a 0,a 1,a 2,…} with a n a n defined as the coefficient of the n n th power of x x in the expansion of (1+x)/(1+x 2)(1+x)/(1+x 2) will do.
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answered Dec 14, 2016 at 13:43
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189467 | https://math.stackexchange.com/questions/3332607/spherical-cap-area?rq=1 | calculus - Spherical Cap Area - Mathematics Stack Exchange
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Spherical Cap Area
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Find the Area of the upper Cap cut from the sphere x 2+y 2+z 2=2 x 2+y 2+z 2=2 by the cylinder x 2+y 2=1 x 2+y 2=1.
I got how to solve it after seeing solution using d S=∬d z d x 2+d z d y 2−1−−−−−−−−−−−√d x d y d S=∬d z d x 2+d z d y 2−1 d x d y
But in my attempt using spherical coordinates.. I got a different answer (which is wrong).. I don't understand what is the mistake. Please help me
∬r 2 sin θ d θ d ϕ∬r 2 sinθ d θ d ϕ
with r 2=2 r 2=2 and integrating with θ∈[π 4,3 π 4]θ∈[π 4,3 π 4] and ϕ∈[0,2 π]ϕ∈[0,2 π]
I was getting 4 2–√π 4 2 π
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edited Aug 24, 2019 at 9:13
Robert Z
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asked Aug 24, 2019 at 8:40
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There is an error in your specification of the θ θ range. The correct spherical integral for the upper cap cut should be
r 2∫2 π 0 d ϕ∫π/4 0 sin θ d θ=2 π(2−2–√)r 2∫0 2 π d ϕ∫0 π/4 sinθ d θ=2 π(2−2)
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edited Aug 24, 2019 at 14:11
answered Aug 24, 2019 at 9:11
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189468 | https://www.tiktok.com/@drstevegoudy/video/7172382083598388522 | Peritonsillar Abscess: Symptoms, Diagnosis, and Treatment | TikTok
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Had this in August, worst pain of my life. Took 4 doctors over 4 days to diagnose. Couldn’t open my mouth or swallow my own spit.
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😸
My throat hurts and it hurts when I swallow and it’s been like this for like 4 days
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Just let the hospital I feel so much better , the bad to drain the pus out they numb it . The pain was bearable during the drainage
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DaRealFazo
I’ve had 2 of these at the age of 19 worst pain of my life the drainage is the most relieving thing ever
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AL
I have this now. Thought was dealing with tonsil stone. Went to dentist today to see if they could get out. He said it was not a stone but an abscess. Gave amoxicillin and an antibiotic rinse.
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Oh man, I had one and it literally hurt worse than when I got run over by a horse 😂
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This happened to my 2yo. They did emergency surgery since the 2 abscesses were blocking most of her airway. Very sick toddler.
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Callsign: Falls
I just had it recently and i was a night away from a death sentence as my doctor said that it needed a immediate emergency drainage...😰😰
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I did this yesterday, abses was so big it cldnt be numbed wasnt the greatest feeling ever tbh. Before that intake i almost disnt sleep for 4 days and
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raerae444
almost died bc of it😭 had it over 2 years ago and had em taken out, it was the largest tonsil the surgeon ever saw
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kate
i think i have this again, i had it two weeks ago and got it drained. can i take my flight tonight or will the pressure be bad?
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Uncle Joe Nyota
that cardi b line 😅😅
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Alex-G88
We have similarly rare names! Long lost relative 😂😂
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😃
My son had many of these. Had to be hospitalized. Later got tonsils and adenoids out as well as deviated septum at same time. Miserable
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What if it’s not that big but it’s a lump behind tonsil wall?
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I’m alive!
Are you sore after the procedure?
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I want a video of the drainage
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Not your grandma’s Patriot
I have a knot on the back of my neck that goes down after messaged and then returns the next morning
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I’m about to have this done I’m scared
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Hi my girl 4 yr old put pomegranate seed in nose 2 days back. Now we cannot see it may go inside .. its ok or dangerous
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189469 | https://en.wikipedia.org/wiki/Barycentric_and_geocentric_celestial_reference_systems | Jump to content
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Contents
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1 Purpose and implementation
2 Relationship to other standards
2.1 ICRS
2.2 HCRF
3 Conversion of coordinates
4 See also
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6 Further reading
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Barycentric and geocentric celestial reference systems
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From Wikipedia, the free encyclopedia
Celestial coordinate system
The barycentric celestial reference system (BCRS) is a coordinate system used in astrometry to specify the location and motions of astronomical objects. Its center of coordinates as the center of mass of the entire Solar System, its barycenter. It was created in 2000 by the International Astronomical Union (IAU) to be the global standard reference system for objects located outside the gravitational vicinity of Earth: planets, moons, and other Solar System bodies, stars and other objects in the Milky Way galaxy, and extra-galactic objects.
The geocentric celestial reference system (GCRS), also created by the IAU in 2000, is a similar standard coordinate system used to specify the location and motions of near-Earth objects, such as satellites. Its center of coordinates is the center of mass of the Earth.
These systems make it easier for scientists and engineers to compile, share, compare, and convert accurate measurements worldwide, by establishing standards both of measure and of methodology, and providing a consistent framework of operations. The focus of the BCRS is on astronomy: exploration of the Solar System and the universe. The BCRS is the system currently used for expressing positional data in astronomical references, such as the Hipparcos star catalog.
The focus of the GCRS is somewhat more on the navigation of Earth satellites and the geophysical applications they support. The proper functioning of the Global Positioning System (GPS) is directly dependent upon the accuracy of satellite measurements as supported by the GCRS.
Purpose and implementation
[edit]
The BCRS was designed to support the extremely-high-precision measurements of position and motion required in astrometry. One critical factor in achieving that precision lies in how general relativistic effects are determined and measured. Both systems incorporate standards that enable the consistency and ready comparability of the resulting spacetime coordinates among astrometric measurements taken worldwide. They provide a metric tensor to establish a consistent frame of reference for observations. The tensor achieves consistency in part through its standardization of the reference point for gravity.
The geocentric system is simpler, being smaller and involving few massive objects: that coordinate system defines its center as the center of mass of the Earth itself. The barycentric system can be loosely thought of as being centered on the Sun, but the Solar System is more complicated. Even the much smaller planets exert gravitational force upon the Sun, causing it to shift position slightly as they orbit. Those shifts are very large in comparison to the measurement precisions that are required for astrometry. Thus, the BCRS defines its center of coordinates as the center of mass of the entire Solar System, its barycenter. This stable point for gravity helps to minimize relativistic effects from any observational frames of reference within the Solar System.
Relationship to other standards
[edit]
ICRS
[edit]
The orientation of the BCRS coordinate system coincides with that of the International Celestial Reference System (ICRS). Both are centered at the barycenter of the Solar System, and both "point" in the same direction. That is, their axes are aligned with that of the International Celestial Reference Frame (ICRF), which was adopted as a standard by the IAU two years earlier (1998). The motivation of the ICRF is to define what "direction" means in space, by fixing its orientation in relation to the Celestial sphere, that is, to deep-space background. Speaking casually, it does not move in relation to the stars and galaxies; it does not rotate.
Determining perfect immobilization of direction is not possible in practice, but we can get much closer than it is even possible for us to measure. The more distant an object is, the less its direction appears to move in relation to us (the parallax effect). The ICRF thus uses very distant objects, well outside our galaxy, to establish its directional points of reference. The chosen objects also emit radio wavelengths, which are less subject than other wavelengths to being obscured by celestial gas in front of them. The ICRF adopts coordinates for 212 defining objects, mostly quasars, fixing its orientation with respect to them.
HCRF
[edit]
The Hipparcos Celestial Reference Frame (HCRF) was similar to ICRF, but earlier, used in association with the Hipparcos satellite, which functioned between 1989 and 1993. That satellite took copious stellar parallax measurements at accuracies exceeding anything otherwise available at the time, thus producing a catalog of stars still in wide use today. No such extensive mapping has yet been completed based upon subsequent improvements in measurement capability. With lower precision then, and at optical wavelengths, the ICRS and BCRS can also be specified using the HCRF. That is the means by which BCRS can be used in relation to the Hipparcos star catalog.
Conversion of coordinates
[edit]
The BCRS and GCRS were also designed so as to make transformations of their coordinates between themselves and other reference systems possible, though the conversions are not by any means straightforward. There are two software libraries of IAU-sanctioned algorithms for manipulating and transforming among the BCRS and other reference systems: the Standards of Fundamental Astronomy (SOFA) system and the Naval Observatory Vector Astrometry Subroutines (NOVAS).
The orientation of the BCRS/ICRS axes also align within 0.02 arcsecond of the Earth's mean equator and equinox for the Fifth Fundamental Catalog (FK5) J2000.0 epoch.
See also
[edit]
Barycentric Coordinate Time
Geocentric Coordinate Time
Earth-centered inertial
Gaia Celestial Reference Frame
Topocentric coordinates
References
[edit]
^ a b c d Kaplan, George H. (20 Oct 2005), "The IAU Resolutions on Astronomical Reference Systems, Time Scales, and Earth Rotation Models", USNO Circular, vol. 179, Washington, DC: United States Naval Observatory, arXiv:astro-ph/0602086
^ Filippenko, Alex (2007), "Lecture 58", Understanding the Universe: An Introduction to Astronomy (DVD), The Great Courses (2nd ed.), Chantilly, VA, USA: The Teaching Company
Further reading
[edit]
IAU (2000), Resolution B1.3: Definition of barycentric celestial reference system and geocentric celestial reference system
External links
[edit]
Standards of Fundamental Astronomy (SOFA)
Naval Observatory Vector Astrometry Software (NOVAS) Archived 2015-06-28 at the Wayback Machine
SuperNOVAS is a maintained fork of the NOVAS library (above) for C/C++, providing bug fixes, improvements, new features, and online documentation.
Portals:
Astronomy
Stars
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Retrieved from "
Categories:
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Astrometry
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How can I find the lengths of a triangle if all the angles are given?
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1y
To find the lengths of the sides of a triangle when all the angles are known, you can use the Law of Sines. Here’s a step-by-step guide:
Steps to Find the Side Lengths of a Triangle Given the Angles
Identify the Angles: Let’s denote the angles of the triangle as A A, B B, and C C.
Choose a Reference Side Length: You need to know at least one side length to calculate the others. If no sides are given, you can assume a side length (often, it’s convenient to assume one side is 1 unit).
Use the Law of Sines: The Law of Sines states that:
a sin A=b sin B=c sin C a sinA=b sinB=c sinC
where:
Continue Reading
To find the lengths of the sides of a triangle when all the angles are known, you can use the Law of Sines. Here’s a step-by-step guide:
Steps to Find the Side Lengths of a Triangle Given the Angles
Identify the Angles: Let’s denote the angles of the triangle as A A, B B, and C C.
Choose a Reference Side Length: You need to know at least one side length to calculate the others. If no sides are given, you can assume a side length (often, it’s convenient to assume one side is 1 unit).
Use the Law of Sines: The Law of Sines states that:
a sin A=b sin B=c sin C a sinA=b sinB=c sinC
where:
a a, b b, and c c are the lengths of the sides opposite angles A A, B B, and C C, respectively.
Calculate the Side Lengths:
If you assume a=1 a=1 (or any value), you can find b b and c c as follows:
b=sin B sin A b=sinB sinA
c=sin C sin A c=sinC sinA
5. Example: If you have a triangle with angles A=30∘A=30∘, B=60∘B=60∘, and C=90∘C=90∘:
Assume a=1 a=1.
Calculate b b and c c:
b=sin 60∘sin 30∘=√3 2 1 2=√3 b=sin60∘sin30∘=3 2 1 2=3
c=sin 90∘sin 30∘=1 1 2=2 c=sin90∘sin30∘=1 1 2=2
Summary
To find the lengths of a triangle given only the angles, you need at least one side length or you can assume one side length. Then, use the Law of Sines to find the other sides. If you have specific angles or a side length, I can help you with those calculations!
Upvote ·
Vera Schroeder
Student at DeBakey High School for Health Professions (2014–present) · Upvoted by
Kevin Nunn
, M.Ed. Curriculum Development and Instruction & Mathematics, University of Louisiana at Lafayette (1995)
·8y
If all the angles are given, you can find the ratios of the side lengths to each other. You should plug in the known side lengths. If no lengths are given, I suppose you could assign any value you like to one of the side lengths and derive from there.
You can calculate these ratios using the law of sines
sin(a)/A = sin(b)/B = sin(c)/C
Continue Reading
If all the angles are given, you can find the ratios of the side lengths to each other. You should plug in the known side lengths. If no lengths are given, I suppose you could assign any value you like to one of the side lengths and derive from there.
You can calculate these ratios using the law of sines
sin(a)/A = sin(b)/B = sin(c)/C
Upvote ·
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99 15
Abhimanyu Bajaj
B Tech (Computer Science) from College of Engineering Roorkee
·8y
You can’t.
There exist infinitely many SIMILAR triangles having all three angles same.
But let there be a case when one of sides is known then there are many ways to find the rest two sides. According to me the simplest way would be to use sine rule.
Sine rule :
SinA/a=SinB/b=SinC/c
Where A,B,C,a,b,&c correspond to the following image.
Now with any one side and all three angles you can find the rest two sides.
Continue Reading
You can’t.
There exist infinitely many SIMILAR triangles having all three angles same.
But let there be a case when one of sides is known then there are many ways to find the rest two sides. According to me the simplest way would be to use sine rule.
Sine rule :
SinA/a=SinB/b=SinC/c
Where A,B,C,a,b,&c correspond to the following image.
Now with any one side and all three angles you can find the rest two sides.
Upvote ·
9 4
Related questions
More answers below
How do you use angles in a triangle to find the lengths of the sides?
How do you find the side lengths of a triangle given two angles and one side?
How can I find the lengths of the sides of a right-angled triangle given its angles?
How do I find the length of a triangle with angles?
How can we find out the corner angles from given side lengths of a triangle?
Alfred
Author has 89 answers and 94.8K answer views
·3y
Originally Answered: How do I find the length of a triangle with angles? ·
pythagorean theorem: If you know 2 length of a triangle (only works for right triangle). Formula: a square+ b square= c square. A and B are the legs.
You also can use trigonometric ratio (sin, cos, tan) and inverse trigonometric to find the angles and length of a triangle. Here are the formula
Sin= Opposite sides/hypotenuse
Cos= Adjacent sides/ hyopotenuse
Tan= Oppostie sides/ Adjacent sides.
Example: Find the hypotenuse
Angle A is about 33.56 and pretend we don’t know the hypotenuse of a triangle. We can use cos and cos-1. Make sure it’s in a deg mode.
Cos(33.56)=6750/h
h=8100.25
The hypotenuse is abo
Continue Reading
pythagorean theorem: If you know 2 length of a triangle (only works for right triangle). Formula: a square+ b square= c square. A and B are the legs.
You also can use trigonometric ratio (sin, cos, tan) and inverse trigonometric to find the angles and length of a triangle. Here are the formula
Sin= Opposite sides/hypotenuse
Cos= Adjacent sides/ hyopotenuse
Tan= Oppostie sides/ Adjacent sides.
Example: Find the hypotenuse
Angle A is about 33.56 and pretend we don’t know the hypotenuse of a triangle. We can use cos and cos-1. Make sure it’s in a deg mode.
Cos(33.56)=6750/h
h=8100.25
The hypotenuse is about 8100.25
To find the other length, use pythagorean theorem.
Special Cases:
If the angles of a triangle are 45,45, and 90, then the 2 legs are congruent and the hypotenuse is leg times the square root of 2.
If the angles of a triangle are 30,60,90 , then the smaller leg is half of the hypotenuse and the bigger leg is smaller leg times the square root of 3.
Hope this helps. If you see any mistake, let me know.
Upvote ·
9 3
9 3
Gary Russell
Former Professor at University of Iowa (1996–2025) · Author has 6K answers and 3.1M answer views
·Updated 4y
Unfortunately, you can’t. At least you can’t without more information.
All triangles with the same three angles are similar — meaning that they are different versions of the same triangle with different sizes (and perhaps, different orientations).
You always know one thing, however: the law of sines. If you know angles a, b and c, and then define the lengths of the side opposite each angle as A, B, and C respectively, then
A/sin(a) = B/sin(b) = C/sin(c)
Using this formula you can figure out the ratios B/A and C/A. Thus, if you know A and the angles, you can figure out B and C.
More generally, to so
Continue Reading
Unfortunately, you can’t. At least you can’t without more information.
All triangles with the same three angles are similar — meaning that they are different versions of the same triangle with different sizes (and perhaps, different orientations).
You always know one thing, however: the law of sines. If you know angles a, b and c, and then define the lengths of the side opposite each angle as A, B, and C respectively, then
A/sin(a) = B/sin(b) = C/sin(c)
Using this formula you can figure out the ratios B/A and C/A. Thus, if you know A and the angles, you can figure out B and C.
More generally, to solve a problem of this sort, you need one more piece of information aside from the angles. Examples are: the length of one side (as above), the area of the triangle, the perimeter of the triangle, or the area of a circle inscribed in the triangle. This additional piece of information allows you to determine the general “size” of the triangle and thereby to figure out the lengths of the sides.
Upvote ·
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Neal Schermerhorn
Lives in Massachusetts (1967–present) · Author has 7K answers and 1.8M answer views
·1y
Originally Answered: How do I find the length of a triangle with angles? ·
All triangles have angles. They are all a “triangle with angles.”
Triangles do not have lengths. Their sides have lengths.
Therefore you cannot find the length of a triangle, despite them all having angles.
If you meant to ask “How do I find the lengths of the sides of a triangle, given the measure of each angle”, then you still cannot. You can find the proportions of the sides, you can find the ratios, but without at least one given linear measurement you cannot determine the lengths.
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How can I find the side lengths of a triangle with the area and all three angles?
If we know all three interior angles of a triangle, can we find its missing side lengths?
How can we determine the values of all three angles of a triangle without knowing any side lengths, given that one side length and two angles are known?
How can you determine the sides and lengths of a triangle if you know all three angles?
How do you find the unknown side of a triangle when given lengths, angles, and points?
Mike Prichard
Former Vice President at Manatee, Inc. (1996–2015) · Author has 231 answers and 153.5K answer views
·3y
Originally Answered: How do I find the length of a triangle with angles? ·
You would need at least one dimension, for example the length of one of the triangle’s sides, to determine the lengths of the other sides.
If you were to assume that one of the sides was of unit length, you could derive the lengths of the other sides of a triangle ABC using the sine formula (AB/SinC=BC/SinA=AC/SinB). That triangle would be similar to every other triangle with the same internal angles and later, when you find out the length one of the sides, you could scale up your similar triangle to fit the new data.
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Alexander Mathey
Former Chemical Engineer, retired, lives in Athens, GR · Author has 5.6K answers and 10.9M answer views
·1y
Related
How can you determine the sides and lengths of a triangle if you know all three angles?
You can’t.
Triangles with the corresponding angles equal are similar, but to establish how big they are, you need at least one length, usually a side.
Continue Reading
You can’t.
Triangles with the corresponding angles equal are similar, but to establish how big they are, you need at least one length, usually a side.
Upvote ·
9 6
David Powers
CEO & President (2005–present) · Author has 206 answers and 236.2K answer views
·8y
Trigonometry
A = csind1
B= ccosd1
C = sqrt(a^2 +b^2)
A+B+C=180
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Klaus Ole Kristiansen
M.Sc. from University of Copenhagen (Graduated 1992) · Author has 4.7K answers and 2.4M answer views
·3y
Originally Answered: How do I find the length of a triangle with angles? ·
You can not find a length if you know only angles. If you know the angles and one side length, you can find the others by the sine formula (sin A)/|a| = (sin B)/ |b| = (sin C)/|c|
Upvote ·
9 2
Glenn Herman
M. D. in Doctor of Medicine&NONE!, New York Medical College (Graduated 1971) · Author has 7.9K answers and 39.7M answer views
·8y
You don't. All different size triangles have the same angles. Too figure out all the lengths of the triangle, at least one of the lengths must be given.
Upvote ·
9 2
9 1
Robert Harvey
Lived in Lincolnshire, UK · Author has 33.4K answers and 47.8M answer views
·2y
You can't.
A triangle can be scaled by any amount without changing the angles.
If you know any one side, you can find the other two by the cosine rule.
Upvote ·
George Ivey
Former Math Professor at Gallaudet University · Author has 23.7K answers and 2.6M answer views
·2y
Originally Answered: How can you find the length of a triangle if you know three angles of the triangle? ·
You DON’T. Given three angles that add to 180 degrees (or π π radians) there are infinitely many (similar) triangles with those angles.
Upvote ·
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How do you use angles in a triangle to find the lengths of the sides?
How do you find the side lengths of a triangle given two angles and one side?
How can I find the lengths of the sides of a right-angled triangle given its angles?
How do I find the length of a triangle with angles?
How can we find out the corner angles from given side lengths of a triangle?
How can I find the side lengths of a triangle with the area and all three angles?
If we know all three interior angles of a triangle, can we find its missing side lengths?
How can we determine the values of all three angles of a triangle without knowing any side lengths, given that one side length and two angles are known?
How can you determine the sides and lengths of a triangle if you know all three angles?
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How do you use angles in a triangle to find the lengths of the sides?
How do you find the side lengths of a triangle given two angles and one side?
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How do I find the length of a triangle with angles?
How can we find out the corner angles from given side lengths of a triangle?
How can I find the side lengths of a triangle with the area and all three angles?
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189471 | https://www.geeksforgeeks.org/maths/sum-and-difference-of-cubes/ | Tutorials
Courses
Sum and Difference of Cubes
The sum and difference of cubes are algebraic formulas used to factor expressions of the form a3+b3 and a3−b3 respectively. These formulas are particularly useful in simplifying and solving polynomial equations.
It is the basic formula of algebra used to solve the sum of the cubes and the difference of the cubes without actually calculating the values of the cubes. The sum of the cubes of the polynomial is represented as, a3 + b3 whereas the difference of the cubes is represented as a3- b3. These algebraic expressions are easily factorized using various algebraic expressions without actually calculating the cubes.
In this article, we will learn about Sum of Cubes, Sum of Cubes Formula, Factoring Sum of Cubes, Difference of Cubes, Difference of Cubes Formula with examples in detail below.
Sum and Difference of Cubes
Sum of Cubes
Sum of cubes is the formula that is used to find the sum of two cubes without actually finding their cubes arithmetically. The sum of cubes is very useful in solving various algebraic problems and is helpful in quickly solving various problems. The sum of cubes formula is the formula that is used to factorize the sum of cubes, The formula for the sum of cubes is discussed below:
Sum of Cubes Formula
Cube of a number is the number multiplied by itself twice. The sum of the cube is the formula that is the formula for a3 + b3 and its formula is added below,
a3 + b3 = (a + b)(a2 – ab + b2)
The above formula is algebraic identity and that is used to find the sum of cubes formula.
Sum of Cube Formula Proof
This identity can be proved by multiplying the expressions on the right side and getting equal to the left side expression. Here is the proof of this identity.
Given Identity:
a3 + b3 = (a + b) (a2 - ab + b2)
Proof:
= RHS
= (a + b)(a2 – ab + b2)
= a(a2 – ab + b2)) + b(a2 – ab + b2)
= a3 – a2b + ab2 + a2b – ab2 + b3
= a3 – a2b + a2b + ab2 – ab2 + b3
= a3 + b3
= LHS
Hence proved.
Factoring Sum of Cubes
We use the sum of cubes formula to easily factorize the cubes in polynomials. This is explained by the example added below,
For example, suppose we have to factorize, x3 + 27
Solution:
= x3 + 27
= x3 + 33
Using Identity, a3 + b3 = (a + b) (a2 - ab + b2)
where,
= (x + 3)(x2 -(x)(3) + 32)
= (x + 3)(x2 - 3x + 9)
Thus, the factors of x3 + 27 are easily found.
Difference of Cubes
When subtracting any two polynomials, a3 - b3, the difference of cubes formula is utilized. This formula is easy to memorize and may be completed in minutes. It is similar to how the sum of cubes formula works.
Difference of Cube Formula
a3 – b3 = (a – b) (a2 + ab + b2)
Difference of Cube Formula Proof
This identity can be proved by multiplying the expressions on the right side and getting equal to the left side expression. Here is the proof of this identity.
Given Identity:
a3 – b3 = (a – b) (a2 + ab + b2)
Proof:
= RHS
= (a - b)(a2 + ab + b2)
= a(a2 + ab + b2)) - b(a2 + ab + b2)
= a3 + a2b + ab2 - a2b - ab2 - b3
= a3 – a2b + a2b + ab2 – ab2 - b3
= a3 - b3
= LHS
Hence proved.
Factoring Difference of Cubes
We use the difference of cubes formula to easily factorize the cubes in polynomials. This is explained by the example added below:
For example, suppose we have to factorize, x3 – 343
Solution:
= x3 - 343
= x3 – 73
Using identity a3 – b3 = (a – b) (a2 + ab + b2)
where,
= (x – 7) (x2 + (x)(7) + 72)
= (x – 7) (x2 + 7x + 49)
Thus, the factors of x3 - 343 are easily found.
Read More
Examples on Sum and Difference of Cubes
Example 1: Factorize y3 – 125
Solution:
y3 – 125 = y3 – 53
Since, a3 – b3 = (a – b) (a2 + ab + b2),
here,
= (y – 5) (y2 + (y)(5) + 52)
= (y – 5) (y2 + 5y + 25)
Example 2: Evaluate 253 – 123
Solution:
Since, a3 – b3 = (a – b) (a2 + ab + b2),
where,
= 253 – 123
= (25 – 12) (252 + (25)(12) + 122)
= 13 (625 + 300 + 144)
= 13897
Example 3: Factorize 8p3 + 27
Solution:
8p3 + 27 = (2p)3 + (3)3
Since, a3 + b3 = (a + b)(a2 – ab + b2)
= (2p)3 + (3)3
= (2p + 3)[(2p)2 – (2p)(3) + (3)2]
= (2p + 3)[4p2 – 6p + 9]
Example 4: Factorize 512 + 729v3
Solution:
512 + 729v3 = (8)3 + (9v)3
Since, a3 + b3 = (a + b)(a2 – ab + b2)
= (8)3 + (9v)3
= (8 + 9v)[(8)2 – (8)(9v) + (9v)2]
= (8 + 9v)[64 – 72v + 729v2]
Example 5: Solve: 253 + 123
Solution:
Since, a3 + b3 = (a + b) (a2 – ab + b2)
where,
= 253 + 123
= (25 + 12) (252 - (25)(12) + 122)
= 37 (625 - 300 + 144)
= 17353
Practice Problems on Sum and Differences of Cubes
Q1. Factorize 64 + 343v3
Q2. Factorize 64 - 343v3
Q3. Evaluate 153 – 93
Q4. Evaluate 233 – 73
P
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189472 | https://www.legislation.gov.uk/ukpga/2006/46/notes | Companies Act 2006 - Explanatory Notes
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Companies Act 2006
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UK Public General Acts
2006 c. 46
Explanatory Notes
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Companies Act 2006
2006 CHAPTER 46
Introduction
1.These explanatory notes relate to the Companies Act 2006 (c.46) which received Royal Assent on 8 November 2006. They have been prepared by the Department of Trade and Industry (DTI) in order to assist the reader in understanding the Act. They do not form part of the Act and have not been endorsed by Parliament.
2.The notes need to be read in conjunction with the Act. They are not, and are not meant to be, a comprehensive description of the Act. So where a section or part of a section does not seem to require any explanation or comment, none is given. Further, where provisions in the Act restate what was in the Companies Act 1985 (the 1985 Act) an explanation is not always given, except to the extent required to explain changes to associated provisions.
Background
3.The UK was one of the first nations to establish rules for the operation of companies. Today our system of company law and corporate governance, setting out the legal basis on which companies are formed and run, is a vital part of the legal framework within which business is conducted. As the business environment evolves, there is a risk that the legal framework can become gradually divorced from the needs of companies, in particular the needs of smaller private businesses, creating obstacles to ways that companies want and need to operate.
4.In March 1998, the DTI commissioned a fundamental review of company law. An independent Steering Group led the “Company Law Review” (CLR) whose terms of reference required them to consider how core company law could be modernised in order to provide a simple, efficient and cost effective framework for British business in the twenty-first century. After extensive consultation with interested parties, the CLR presented its Final Report to the Secretary of State for Trade and Industry on 26 July 2001. The report contained a range of recommendations for substantive changes to many areas of company law, and a set of principles to guide the development of the law more generally, most notably that it should be as simple and as accessible as possible for smaller firms and their advisers and should avoid imposing unnecessary burdens on the ways companies operate.
5.Many of the provisions of the Act implement CLR recommendations. The Government set out and consulted on its intentions in this regard in the White Papers “Modernising Company Law” (July 2002) and “Company Law Reform” (March 2005). The 2005 White Paper included approximately 300 draft clauses and described in detail the policy intention for other areas. Further clauses were made publicly available for comment in July, September and October 2005. The Companies Bill, then titled the Company Law Reform Bill, was introduced to the House of Lords on 4 November 2005.
Overview of the Structure of the Act
6.The general arrangement of the Act is as follows:
| PART | SUMMARY |
--- |
| Parts 1 to 7 | The fundamentals of what a company is, how it can be formed and what it can be called. |
| Parts 8 to 12 | The members (shareholders) and officers (management) of a company |
| Parts 13 and 14 | How companies may take decisions |
| Parts 15 and 16 | The safeguards for ensuring that the officers of a company are accountable to its members |
| Parts 17 to 25 | Raising share capital, capital maintenance, annual returns, and company charges |
| Parts 26 to 28 | Company reconstructions, mergers and takeovers |
| Parts 29 to 39 | The regulatory framework, application to companies not formed under the Companies Acts and other company law provisions |
| Parts 40 to 42 | Overseas disqualification of directors, business names and statutory auditors |
| Part 43 | Transparency obligations |
| Parts 44 to 47 | Miscellaneous and general |
Summary of Legislative Changes
7.The company law provisions of the 2006 Act (Parts 1 to 39) restate almost all of the provisions of the 1985 Act, together with the company law provisions of the Companies Act 1989 (the 1989 Act) and the Companies (Audit, Investigations and Community Enterprise) Act 2004 (C(AICE) Act 2004). Paragraphs 9 and 10 below contain details of the provisions that remain in those Acts. The company law provisions also codify certain aspects of the case law.
8.Tables of origins and destinations are available that show the origins of the company law provisions of the Act by reference to enactments in force on 8 November 2006. The tables identify where provisions of the existing law have been re-enacted with or without changes and where provisions of the new law have no predecessor or are fundamentally different from their predecessors.
9.Of company law provisions in the Acts referred to in paragraph 77, the only ones that remain are those on investigations that go wider than companies (Part 14 of the 1985 Act) and the provisions on community interest companies in Part 2 of the C(AICE) Act 2004.
10.The non-company law provisions in those Acts that remain are:
a)
Part 18 of the 1985 Act (floating charges and receivers (Scotland)),
b)
Part 3 of the 1989 Act (powers to require information and documents to assist overseas regulatory authorities),
c)
Section 112 to 116 of the 1989 Act (provisions about Scottish incorporated charities)
d)
Part 7 of the 1989 Act (provisions about financial markets and insolvency)
e)
Schedule 18 of the 1989 Act (amendments and savings consequential upon changes in the law made by the 1989 Act)
f)
Sections 14 and 15 of the C(AICE) Act 2004 (supervision of accounts and reports), and
g)
Sections 16 and 17 of the C(AICE) Act 2004 (bodies concerned with accounting standards etc).
11.In non-company law areas the Act makes amendments to other legislation, in particular the Financial Services and Markets Act 2000, and also makes new provision of various kinds. The main areas in which provision of this kind is made are:
overseas disqualification of company directors (Part 40),
business names (Part 41) – replacing the Business Names Act 1985,
statutory auditors (Part 42) – replacing Part 2 of the Companies Act 1989, and
transparency obligations (Part 43) – amending Part 6 of the Financial Services and Markets Act 2000.
Territorial Extent and Devolution
Northern Ireland
12.Company law is a transferred matter. Currently, the provisions of GB company law are generally replicated, some time later, in separate Northern Ireland legislation. The Act provides for a single company law regime applying to the whole of the UK, so that companies will be UK companies rather than GB companies or Northern Ireland companies as at present. This does not affect the legislative competence of Northern Ireland: company law remains a transferred matter, and the Act could be separately amended or repealed in Northern Ireland if that were so desired.
13.Where a note describes a particular section as restating or replacing a provision in the 1985 Act, the 1989 Act or the C(AICE)Act 2004, this should be read as applying equally to the corresponding provision of the Companies (Northern Ireland) Order 1986, the Companies (Northern Ireland) Order 1990 or the Companies (Audit, Investigations and Community Enterprise) Order 2005.
Scotland
14.Company law is a reserved matter and Companies Acts extend to the whole of Great Britain. However, there are several areas where, in legislating about companies, the Act deals with matters that are devolved:
changes (in Part 41) to the regulation of business names (a devolved matter) – these correspond to changes (in Part 5) to the regulation of company names (a reserved matter);
statutory guidance to prosecutors and other enforcement authorities in relation to a new offence of knowingly or recklessly causing an audit report to be misleading, false or deceptive – although the offence itself is a reserved matter, guidance is to be issued by the Lord Advocate in Scotland (see section 509);
changes relating to exemptions from audit requirements for companies that are charities (see section 1175);
conferral of a power on the Auditor General for Scotland to specify public bodies for his audit (see section 483).
These were the subject of a legislative consent motion agreed to by the Scottish Parliament on 16 March 2006.
Wales
15.Company law is not transferred to the Welsh Assembly. There are no provisions that impact on devolved competences.
Crown Dependencies
16.Part 28 of the Act (takeovers) contains provision enabling it to be extended by Order in Council to the Isle of Man or any of the Channel Islands. This reflects the existing jurisdiction of the Takeover Panel (as the takeover regulator) and has been agreed by the relevant Island authorities. If the power were to be exercised, there would be further consultation with the Island authorities beforehand.
Part 1: General Introductory Provisions
Section 1: Companies
17.This section restates section 735(1)(a) and (b) of the 1985 Act. It defines “company” and provides signposts to provisions in the Act which relate to companies that are registered but not formed under the Act or former Companies Acts, to unregistered companies and to overseas companies.
Section 2: The Companies Acts
18.This section replaces section 744 of the 1985 Act. The Act does not restate or replace all existing companies legislation and section 2 makes it clear that any reference to “Companies Acts” in the Act includes those provisions of the Acts listed in subsection (1)(c) that remain in force as well as the company law provisions of the Act and Part 2 of the C(AICE) Act 2004.
19.The CLR recommended that the law should provide for the formation of new companies of each of the types that are currently available (Final Report, paragraph 9.2). This recommendation is taken forward in the following group of sections, which retains all of the current forms of companies.
Section 3: Limited and unlimited companies
20.This section restates section 1(2) of the 1985 Act. It updates the Companies Acts definitions of “limited company” and “unlimited company” to reflect changes to what is to be included in a company’s memorandum of association (see section 8). As now, a company may be limited by shares or by guarantee. Where there is no limit on the liability of the company’s members, a company is an “unlimited company.”
Section 4: Private and public companies
21.This section restates section 1(3) of the 1985 Act. It provides definitions of “private company” and “public company.”
22.A “private company” is any company that is not a public company.
23.A “public company” is a company whose certificate of incorporation states that it is a public company. To obtain this certificate the company will need to comply with the provisions of the Act (or former Companies Acts) as regards registration or re-registration as a public company. There is a minimum share capital requirement (the “authorised minimum”), which is currently set at £50,000, and remains unchanged under the Act. In future the authorised minimum will however be capable of being satisfied in sterling or the euro equivalent to the prescribed sterling amount (see Chapter 2 of Part 20).
24.Section 4 also provides a signpost to Part 20 of the Act, which sets out key differences between public and private companies, for example, a private company may not offer shares to the public.
Section 5: Companies limited by guarantee and having share capital
25.This section restates section 1(4) and section 15(2) of the 1985 Act. It makes it clear that a company can no longer be formed (or re-register) as a company limited by guarantee and with a share capital. This provision has been in force in Great Britain since 22nd December 1980 and in Northern Ireland since 1st July 1983.
Section 6: Community interest companies
26.The C(AICE) Act 2004 came fully into force on 1 July 2005. Part 2 of that Act created a new company vehicle, the “community interest company” or “c.i.c.”, which is designed for use by social enterprises.
27.This section provides a signpost to the provisions in the C(AICE) Act, which enable a company to be formed as or become a community interest company. Such companies are registered under the same legislation as other registered companies, but have to complete certain additional formalities and are subject to certain additional elements of regulation. Subsection (2) of this section highlights the fact that in some respects the requirements imposed on community interest companies are different from the requirements imposed on other registered companies.
Part 2: Company Formation
28.This Part of the Act is about how companies are formed. It replaces or, as the case may be, restates equivalent provisions in the 1985 Act.
Section 7: Method of forming company
29.This section replaces sections 1(1) of the 1985 Act. It retains the current requirement that individuals who wish to form a company must subscribe their names to the memorandum of association (“memorandum”). Subsection (1) introduces the new provisions about forming a company. In line with the recommendations of the CLR, it is provided that a single person is able to form any sort of company (not just a private company) (Final Report, paragraph 9.2).
30.Subsection (2) reproduces the existing requirement that a company may not be formed for an unlawful purpose.
Section 8: Memorandum of association
31.This section replaces section 3(1) of the 1985 Act.
32.Under the Act, the memorandum serves a more limited, but nonetheless important, purpose: it evidences the intention of the subscribers to the memorandum to form a company and become members of that company on formation. In the case of a company that is to be limited by shares, the memorandum will also provide evidence of the members’ agreement to take at least one share each in the company.
33.The memorandum of a company formed under the Act will, therefore, look very different from that of a company registered under the 1985 Act. In addition it will not be possible to amend or update the memorandum of a company formed under the Act.
34.These changes to the memorandum are based on the CLR’s recommendation that there should be a single constitution (Final Report, paragraph 9.4). In line with the principles behind this recommendation, in future key information regarding the internal allocation of powers between the directors and members of a company will be set out in one place: the articles of association (“articles”).
35.By virtue of section 28, provisions in the memoranda of existing companies will be treated as provisions in the articles if they are of a type that will not in be in the memoranda of companies formed under the Act. Existing companies will, therefore, not be required to amend their articles to reflect these changes, but they can do so if they wish. They will however be able to alter or update provisions in their constitution which are now set out in their memoranda by amending their articles, for example to reflect changes to the law made by the Act.
Section 9: Registration documents
36.This section replaces various provisions in sections 2 and 10 of the 1985 Act. It prescribes the types of information or “documents” that must be delivered to the registrar when an application for registration is made and the registrar to whom the information must be delivered.
37.The changes to the way in which certain information is delivered to the registrar are required as a result of the changes that have been made to the memorandum. In future, information which is currently set out in the memorandum will be provided to the registrar in accordance with the provisions of this section, which prescribes, amongst other things, the contents of the application for registration. In all cases this application must state:
the company’s proposed name;
whether the company’s registered office is to be situated in England and Wales (or Wales), in Scotland or in Northern Ireland;
whether the liability of the company’s members is to be limited and if so whether it is to be limited by shares or by guarantee;
whether the company is to be a private or a public company.
38.In the case of a company that is to have a share capital, the application must also contain a statement of capital and initial shareholdings (see section 10). In the case of a company that is to be limited by guarantee the application must also contain a statement of guarantee (see section 11).
39.In all cases the application must also contain a statement of the company’s proposed officers (see section 12) and a statement of the intended address of the company’s registered office (that is, the postal address of the company’s registered office as opposed to a statement confirming the jurisdiction in which the company’s registered office is to be situated – which is also required).
40.The application for registration must also contain a copy of any proposed articles (to the extent that the company does not intend to use the model articles (see sections 19 and 20) and must be accompanied by the memorandum (see subsection (1)) of this section and a statement of compliance (see section 13).
41.In future it will be possible to form a company on-line and the various types of information referred to in the section are, therefore, capable of being delivered as a series of data entries as well as in paper or such other form as the registrar may permit or prescribe. The registrar has power under section 1068 to prescribe the form and manner in which documents are to be delivered to her.
Section 10: Statement of capital and initial shareholdings
42.This section is a new provision. It sets out the contents of the statement of capital and initial shareholdings.
43.Currently, in the case of a limited company with a share capital the memorandum is required to state the amount of the share capital with which the company proposes to be registered and the nominal amount of each of its shares. This is known as the “authorised share capital” and acts as a ceiling on the amount of capital which can be issued (although this limit can be increased by ordinary resolution). The CLR recommended that the requirement for a company to have an authorised share capital should be abolished (Final Report, paragraph 10.6).
44.The Act gives effect to this recommendation and in future, information about the shares subscribed for by the subscribers to the memorandum, which is currently set out in the memorandum itself, will be provided to the registrar in the statement of capital and initial shareholdings.
45.Like the statement of guarantee (see section 11), the statement of capital and initial shareholdings must contain such information as may be prescribed by the Secretary of State, in regulations made under the Act, for the purpose of identifying the subscribers to the memorandum (i.e. the founder members of the company).
46.The statement of capital and initial shareholdings is essentially a “snapshot” of a company’s share capital at the point of registration. For public companies, this requirement is linked to the abolition of authorised share capital (see above). It implements (as far as public companies are concerned) Article 2 of the Second Company Law Directive (77/91/EC) (the “Second Directive”) which states:
““the statutes or instruments of incorporation of the company shall always give at least the following information…(c) when the company has no authorized capital, the amount of the subscribed capital….”.”
47.The statement of capital and initial shareholdings must contain the following information:
the total number of shares of the company to be taken on formation by the subscribers to the memorandum;
the aggregate nominal value of those shares;
for each class of shares: prescribed particulars of the rights attached to those shares, the total number of shares of that class and the aggregate nominal value of shares of that class; and
the amount to be paid up and the amount (if any) to be unpaid on each share (whether on account of the nominal value of the shares or by way of premium).
48.The reference to “prescribed particulars of the rights attached to the shares” in this section (and elsewhere in the Act where a statement of capital is called for), refers to such particulars as may be prescribed by the Secretary of State by statutory instrument (see section 1167).
49.Whilst the Second Directive only applies to public companies it is important that the information on the public register is up-to-date for both public and private companies. A statement of capital will, therefore, be required where it is proposed that a company formed under the Act will have a share capital on formation and, with limited exceptions (in particular, where there has been a variation of class rights which does not affect the company’s aggregate subscribed capital) whenever a limited company having a share capital makes an alteration to its share capital (and in certain cases where an unlimited company makes a return to the registrar).
Section 11: Statement of guarantee
50.This section replaces section 2(4) of the 1985 Act. It sets out the contents of the statement of guarantee that must accompany the application for registration where it is proposed that a company will be limited by guarantee on formation.
51.The statement of guarantee is essentially an undertaking, given by the founder members of the company, to contribute to the assets of the company up to a specified amount in the event of it being wound up. New members must also agree to make the same contribution.
52.A member of a company limited by guarantee is only liable to contribute to the assets of a company if it is wound up during the time that he is a member or within one year of him ceasing to be a member.
53.Like the statement of capital and initial shareholdings the statement of guarantee must contain such information as may be prescribed by the Secretary of State, in regulations made under the Act, for the purposes of identifying the subscribers to the memorandum (i.e. the founder members of the company).
Section 12: Statement of proposed officers
54.This section replaces section 10(2) and (3) of the 1985 Act and contains a new provision. Under section 10, details of the first director(s) and the secretary or joint secretaries must be given to the registrar at the time of application for registration. That requirement is carried forward but there are two changes:
firstly, to the required particulars. These are specified in relation to directors in sections 163 to 166. The main change is that a service address must be provided for each director who is a natural person. This is in addition to the requirement for the usual residential address;
secondly, as recommended by the CLR (Final Report, paragraph 4.7), there is no requirement for a private company to have a company secretary but it may do so if it wishes (see section 270(1)). As now, a company which proposes to be registered as a public company must have a company secretary (see section 271).
Section 13: Statement of compliance
55.This section replaces section 12(3) and (3A) of the 1985 Act. At present, where an application for registration of a company is made in paper form, the application must be accompanied by a statutory declaration (made before a solicitor or commissioner of oaths) confirming that the requirements of the 1985 Act in respect of registration, and of matters precedent and incidental to it, have been complied with (see section 12(1) of that Act). This statutory declaration must be made by one of the persons whom it is proposed will be a founder director or secretary of the company (that is, on registration) or a solicitor engaged in the formation of the company.
56.Where the application for registration is made in electronic form, in place of the statutory declaration required under section 12(3) of the 1985 Act, the same persons may, alternatively, deliver an “electronic statement” to the registrar. This statement must confirm that the requirements referred to in section 12(1) have been met.
57.Based on the recommendations of the CLR (Final Report, paragraph 9.5), the current requirement for a statutory declaration or electronic statement, here and elsewhere in the Act, is replaced by a requirement to make a statement of compliance. This statement does not need to be witnessed and may be made in paper or electronic form. It will be for the registrar’s rules under section 1068 to specify who may make this statement (and the form of it). As with all documents delivered to, or statements made to, the registrar, it is an offence to make a false statement of compliance – see section 1112.
Section 14: Registration
58.This section restates section 12(1) and (2) of the 1985 Act. As now, where the registrar is satisfied that all of the requirements of the Act as to registration have been met she will register the documents delivered to her and issue a certificate of incorporation under section 15.
Section 15: Issue of certificate of incorporation
59.This section restates section 13(1)(2) and (7)(a) of the 1985 Act and contains a new provision in subsection (2), which prescribes the contents of the certificate of incorporation issued by the registrar on registration of a company. The certificate of incorporation is conclusive evidence that the requirements of the Act as to registration have been met, that the company has been registered, and (where relevant) that the company has been registered as a limited company or a public company.
60.There is one change to what the certificate of incorporation is required to state: in future this will include details of whether the company’s registered office is situated in England and Wales (or in Wales), in Scotland or in Northern Ireland. The certificate will also state, where the company is limited, whether it is limited by shares or by guarantee.
Section 16: Effect of registration
61.This section replaces section 13(3) to (5) of the 1985 Act. It provides, amongst other things, that the subscribers to the memorandum, together with such other persons as may from time to time become members of a company, are a body corporate by the name stated in the certificate of incorporation and, in the case of a company having a share capital, that the subscribers to the memorandum become holders of the shares specified in the statement of capital and initial shareholdings. This means that on registration a company becomes a legal person in its own right, which is distinct from the people who own it (the members) and the people who manage it (the directors).
Part 3: a Company’s Constitution
62.This Part deals with various matters relating to a company’s constitution. It replaces similar provisions in the 1985 Act. It starts by defining (non-exhaustively) “a company’s constitution” and then makes provision about the main constituent parts of a company’s constitution (the articles of association and certain classes of members’ resolutions and agreements), including their legal effects, how they are to be notified to the registrar and made available to members, and how changes to them are to be dealt with.
Chapter 1: Introductory
Section 17: A company’s constitution
63.This section is a new provision. It sets out a definition of “a company’s constitution” which will apply throughout the Act, and the other “Companies Acts” (defined in section 2), unless the context requires a wider or more restricted meaning (see for example section 257, which expands the definition of a company’s constitution for the purposes of Part 10). The concepts of a company’s constitution and the rights and obligations arising under it are used both in this Part and elsewhere in the Act.
64.The definition is expressed to be non-exhaustive. In addition to the provisions of companies’ articles and the resolutions and agreements to which Chapter 3 of this Part applies (described in section 29), the contents of certain other documents are clearly of constitutional relevance for certain purposes. For example the certificate of incorporation summarises key information pertaining to the company such as whether it is public or private limited – see section 15.
Chapter 2: Articles of Association
65.A company’s articles are rules, chosen by the company’s members, which govern a company’s internal affairs. They form a statutory contract between the company and its members, and between each of the members in their capacity as members, and are an integral part of a company’s constitution. At present, companies may divide their constitutional rules between their memoranda and their articles, with the terms of their memoranda being capable of being altered after formation in some respects but not in others. In future, the memorandum will be a very simple document of purely historic significance, evidencing an intention to form a company, and all the company’s key internal rules on matters such as the allocation of powers between the members of a company and its directors will be set out in the articles – see notes on sections 8 and 28.
Section 18: Articles of association
66.This section replaces section 7(1) and (3) and section 744 of the 1985 Act. It carries forward the requirement that all registered companies must have articles. The provisions of this section have been updated to reflect the changes made by section 19, which gives the Secretary of State the power to prescribe “default” model articles for different descriptions of companies. As a result of this change, some types of company that are currently required to register articles with the relevant registrar of companies (for example, private companies limited by guarantee) will have the option of not registering articles but relying on the “relevant model articles” for that description of company.
67.As now, the articles must be contained in a single document and must be divided into consecutively numbered paragraphs.
68.Generally speaking, companies formed under the 1985 Act have freedom to make such rules about their internal affairs as they see fit, subject to the qualification that if a company’s articles contain anything that is contrary to the provisions of that Act, or against the general law, then it will have no effect. This principle will also apply to the articles of companies which are formed and registered under the Act.
Section 19: Power of Secretary of State to prescribe model articles
Section 20: Default application of model articles
69.Section 8 of the 1985 Act enables the Secretary of State to prescribe model forms of articles for companies registered under that Act (see the Companies (Tables A to F) Regulations 1985 (SI 1985/805)). Articles for certain special types of companies used in particular sectors, for example, commonhold associations, right to manage (“RTM”) companies, and right to enfranchise (“RTE”) companies are prescribed by regulations made under the Acts of Parliament that created these types of company.
70.Although sections 8 and 8A of the 1985 Act allow the Secretary of State to prescribe forms of articles (and memoranda) for a number of different types of company under section 8, he is only able to prescribe “default” model articles for companies limited by shares. “Default” model articles are model articles which apply to companies of a particular description where they have not registered any articles of their own, or have not made provision for a particular matter for which there is a corresponding model article. “Default” model articles apply to a company of the description for which they are prescribed only to the extent that it has not modified the default provision in question in its own registered articles or excluded it, or the model articles in their entirety, from the registered articles.
71.The rationale behind this is that the model articles should operate as a “safety net” which enables the members and directors of such companies to take decisions in circumstances where a company has failed to provide the appropriate authority in its registered articles (or failed to register articles at all).
72.These two sections replace section 8 of the 1985 Act. In line with the CLR’s recommendations (Company Formation and Capital Maintenance, paragraph 2.22), the Secretary of State will have the power to prescribe model articles, including “default” model articles, for different descriptions of companies formed under the Act.
73.For existing companies, there will be no change. The principle is maintained that the version of the model articles that was in force at the time that a particular company was originally registered will continue to apply to that company. For the majority of companies limited by shares on the register at the date that the Act comes into force, the “default” model articles will continue to be the Companies Act 1985 Table A (“Table A”).
74.Existing companies will be free to adopt, wholly or in part, the model articles prescribed for companies of a particular description formed under the Act (see subsection (3) of section 19). For example, an existing private company limited by shares may prefer to adopt the new model articles for private companies limited by shares, or indeed the new model articles for public companies formed under the Act (with or without modification) in place of the current Table A articles, or previous articles of its own devising.
75.As with Table A, the adoption of model articles by companies formed under the Act will be entirely a matter for individual companies. They will be able to incorporate (with or without amendment) provisions from the model articles, and/or add to those provisions, and/or exclude such provisions as they think fit.
76.They will also be able to adopt the provisions of model articles by reference. This is a common practice, which enables a company that wishes to incorporate specific provisions of the model articles into its own registered articles to do this without having to copy out the provision in question. To take an example, a company’s registered articles may say something to the following effect: “the model articles apply except for articles x, y and z”, or “the company’s articles are A, B and C, plus model articles g, p and q. Model article n applies but is amended as follows: …”. Companies have found such techniques useful in the past and they will continue to be permitted.
Section 21: Amendment of articles
77.Subsection (1) provides that, as now, a company’s articles can in general be amended by special resolution. This restates section 9(1) of the 1985 Act.
78.Subsections (2) and (3) make it clear that this general principle is subject to certain rules in charities legislation about the ability of companies which are charities to change their constitutions and the effects which such changes have. There are separate but broadly similar rules for English and Welsh, Scottish and Northern Irish charities.
Section 22: Entrenched provisions of the articles
79.Section 22 is a new provision. It replaces the current practice (provided for in section 17(2)(b) of the 1985 Act), whereby companies are able to entrench certain elements of their constitution by putting them in their memoranda and providing that they cannot be altered.
80.This section permits companies to provide in their articles that specified provisions may be amended or repealed only if conditions are met that are more restrictive than would apply in the case of a special resolution. Such a provision is referred to as a “provision for entrenchment”. As a result of this section companies formed under the Act will not be permitted to provide in their articles that an entrenched provision can never be repealed or amended.
Section 23: Notice to registrar of existence of restriction on amendment of articles
81.This is a new provision that requires a company to give notice to the registrar when an entrenching provision is included in its articles (whether on formation or subsequently) or where the company’s articles are altered by order of a court or other authority so as to restrict or exclude the power of the company to amend its articles. There is a corresponding requirement as to notice where the company amends its articles so as to remove a provision for entrenchment or where the articles are altered by order of a court or other authority so as to remove a provision for entrenchment or any other restriction on, or any exclusion of, the power of the company to amend its articles.
Section 24: Statement of compliance where amendment of articles restricted
82.This is a new provision. Where a company’s articles contain provision for entrenchment or where the articles are subject to an order of a court or other authority restricting or excluding the company’s power to amend its articles and the company subsequently amends its articles, it is required to send to the registrar the document making or evidencing the amendment. This document must be accompanied by a “statement of compliance” (see note on section 13).
83.The statement of compliance must certify that the amendment to the articles has been made in accordance with the company’s articles (including any provision for entrenchment) or, where relevant, in accordance with any order of the court or other authority that is in force at the time of the amendment.
84.The purpose of the provisions in sections 23 and 24 is to ensure that the registrar, and any person searching the public register, is on notice that the articles contain entrenching provisions and that special rules therefore apply to the company’s articles.
Section 25: Effect of alteration of articles on company’s members
85.This section restates section 16 of the 1985 Act. The only difference is that section 16 also applied to alterations of a company’s memorandum. A company formed under the Act will not be able to (or need to) alter its memorandum.
86.This section retains the principle that a member of a company is not bound by any alteration made to the articles subsequent to his becoming a member if the alteration has the effect of increasing his liability to the company or requires him to take more shares in the company. A member may however give his written consent to such an alteration and, where he does, he will be bound by it.
Section 26: Registrar to be sent copy of amended articles
87.The First Company Law Directive (68/151/EEC) requires Member States to take such measures as are required to ensure that companies disclose certain constitutional information which will then be made available to the public in a central register. In particular, companies are to be required to disclose (i) their “instrument of constitution, and the statutes if they are contained in a separate instrument”; (ii) any amendments to these instruments; and (iii) “after every [such] amendment…the complete text of the instrument or statutes as amended to date”. For UK companies, the “instrument of constitution” equates to the memorandum and the “statutes” equate to the articles. The central registers are those kept by the registrars of companies for England and Wales, Scotland and Northern Ireland.
88.This section replaces equivalent provisions in section 18(2) and (3) of the 1985 Act and Schedule 24 to that Act.
89.Where a company fails to comply with the provisions of this section, the company and every officer of the company who is in default commits an offence. The penalty for this offence is set out in subsection (4).
Section 27: Registrar’s notice to comply in case of failure with respect to amended articles
90.This section is a new provision. It gives the registrar a means of ensuring that companies comply with the obligation set out in section 26 without having to resort to criminal proceedings. (However, the offence of failing to file amended articles is retained: see of section 26(3)).
91.Where the registrar becomes aware of any default in complying with section 26 (or any similar provision of another enactment that was in force at the time of the default, for example, section 18(2) of the 1985 Act), she may give notice to the company requiring it to rectify the breach within 28 days. Where the company complies with the notice, the company will avoid prosecution for its initial failure to comply. If the company does not comply, it will be liable to a civil penalty of £200, recoverable by the registrar as a debt, in addition to any criminal penalty that may be imposed (see, for example, section 26(4)).
Section 28: Existing companies: provisions of memorandum treated as provisions of articles
92.For companies formed under the Act, the memorandum will contain limited information evidencing the intention of the founder members to form a company. The memoranda of existing companies, on the other hand, will contain key constitutional information of a type which will in future be set out in the articles or provided to the registrar in another format (see Part 2). Subsection (1) of this section provides that such material is to be treated for the future as part of the company’s articles.
93.Subsection (2) of this section makes it clear that where the memorandum of an existing company contains a provision for entrenchment (see note on section 22) at the date that this part of the Act comes into force, this will be deemed, with effect from that date, to be a provision for entrenchment in the company’s articles.
Chapter 3: Resolutions and Agreements Affecting a Company’s Constitution
94.This Chapter replaces equivalent provisions in the 1985 Act on the registration of resolutions and agreements and on making these available to members.
Section 29: Resolutions and agreements affecting a company’s constitution
95.This section replaces section 380(4) and (4A) of the 1985 Act. It lists the resolutions and agreements that must be forwarded to the registrar for registration (see section 30) and made available to members on request (see section 32).
Section 30: Copies of resolutions or agreements to be forwarded to registrar
96.This section restates section 380 (1), (5) and (7) of the 1985 Act and Schedule 24 to that Act. Where a company passes a resolution or enters into an agreement of the type listed in section 29, it must forward a copy of the resolution or agreement to the registrar for registration within 15 days of the date on which the resolution was passed. If a company fails to do this, the company, and every officer of it who is in default, commits an offence. For the penalty, see subsection (3).
97.Where a resolution or agreement which affects a company’s constitution is not in writing, the company is required to provide the registrar with a written memorandum setting out the terms of the resolution or agreement in question.
Chapter 4: Miscellaneous and Supplementary Provisions
Section 31: Statement of company’s objects
98.This section provides for a new approach to the question of a company’s objects. Under the 1985 Act all companies are required to have objects and these objects are required to be specified in the memorandum. The 1985 Act also makes specific provision for where a company states its objects to be to carry on business as a general commercial company (see section 3A of the 1985 Act).
99.Based on a recommendation of the CLR (Final Report, paragraph 9.10), under the Act a different approach is taken. Instead of companies being required to specify their objects, companies will have unrestricted objects unless the objects are specifically restricted by the articles (see subsection (1)). This will mean that unless a company makes a deliberate choice to restrict its objects, the objects will have no bearing on what it can do. Some companies will continue to restrict their objects. Companies that are charities will need to restrict their objects (under charities legislation) and some community interest companies may also choose to do so.
100.Subsection (2) provides that where a company changes its articles to add, remove or alter a statement of the company’s objects, it must give notice to the registrar. The registrar is to register that notice, and the alteration does not take effect until it has been so registered.
101.Subsection (3) ensures that such an amendment to the company’s articles will not affect any rights or obligations of the company or render defective any legal proceedings by or against it.
102.For companies which are charities, the effect of this section is subject to section 64 of the Charities Act 1993 in England and Wales and in Northern Ireland subject to the Charities (Northern Ireland) Order 1987 (SI 1987/2048 (NI 19)) (subsection (4)).
103.Subsection (5) makes equivalent provision for Scotland. These provisions impose additional requirements in the case of companies which are charities when changing certain aspects of their constitutions, including their objects.
104.The directors of a company are under a duty to observe the company’s constitution (see section 171) although restrictions in objects will, as now, have little effect outside of the internal workings of the company because of the effect of sections 39 and 40 (except in the case of charities where modified rules again apply – see section 42).
Section 32: Constitutional documents to be provided to members
105.This section replaces section 19 of the 1985 Act and Schedule 24 to that Act. It gives members the right to obtain from the company copies of the company’s articles and certain other documents of constitutional importance (see subsection (1)).
106.The provision in the 1985 Act which enables a company to charge its members 5p for a copy of its articles and/or memorandum has been removed. This information must in future be provided to the members (on request) free of charge.
107.Where a company fails to comply with the provisions of this section, every officer of the company who is in default commits an offence. For the penalty for this offence, see subsection (4).
Section 33: Effect of company’s constitution
108.Subsection (1) of this section replaces section 14(1) of the 1985 Act. Its effect is that the provisions of a company’s constitution constitute a special kind of contract, whose terms bind the company and its members from time to time. Like section 14(1), the provisions of this section are excepted from the general principle set out in section 1 of the Contracts (Rights of Third Parties) Act 1999, so that provisions of a company’s constitution will not confer any rights on persons other than the company and its members. Unlike section 14(1), section 34 refers to “a company’s constitution”, rather than its “memorandum and articles”. This reflects the new division of formation and constitutional information between the memorandum, articles and other constitutional documents noted above.
109.The language in subsection (1) has been updated but there is no change to the law (the provision continues to reflect what the law has always been: in particular a company’s constitution binds both the company and its members).
110.Subsection (2) replaces section 14(2) of the 1985 Act. It provides that amounts which a member of a company is obliged to pay to it under its constitution are debts due to the company. In England and Wales and Northern Ireland, such debts are ordinary contract debts.
Section 34: Notice to registrar where company’s constitution altered by enactment
111.This section replaces section 18 of the 1985 Act and Schedule 24 to that Act.
112.The provisions of a company’s constitution may be altered by legislation, rather than by a resolution or agreement of the company’s members. Such legislation will either be of general relevance to all companies (for example, a new Companies Act provision that provisions of a certain type in any company’s articles are void), or to all companies of a particular type (for example, new commonhold legislation changing the provisions prescribed for inclusion in the articles of all commonhold associations) or it will be relevant only to a particular company (for example, a private Act of Parliament amending the articles of a specific company established by an earlier Act).
113.In keeping with the principles underlying section 26, it is important that those searching the register of companies should be able to be made aware of the changes to companies’ articles which legislation may effect. However, there is a balance to be struck between maintaining transparency on the one hand and inundating the registrar and searchers with mountains of paper which will be of little practical use to persons searching the public register (and whose contents are generally available in any event). The section therefore does not require companies to send copies of most public general Acts which alter their articles (such as Companies Acts or new commonhold legislation) to the registrar. It does however require “special enactments” (as defined in subsection (4)) to be sent to the registrar by companies whose articles are altered by the enactment in question.
114.Where an enactment to which this section applies alters a company’s articles, or where such an enactment alters a resolution or agreement affecting the company’s constitution, the company is obliged to send a copy of the articles, or the resolution or agreement in question, as altered, to the registrar.
115.The procedural rules for sending such legislation to the registrar, and the penalties for non-compliance with them, are as for section 26.
Section 35: Notice to registrar where company’s constitution altered by order
116.This is a new provision which provides a mechanism for registering alterations which are made to a company’s constitution by an order of the court or other authority (for example, the Charity Commission). It obliges companies to give notice of such alterations to the registrar, and to supply a copy of the articles, or the resolution or agreement in question, as altered to the registrar.
Section 36: Documents to be incorporated in or accompany copies of articles issued by company
117.This section replaces section 380(2), (6) and (7) of the 1985 Act and Schedule 24 to that Act. It provides that every copy of a company’s articles which it issues must be accompanied by various documents: in particular resolutions, agreements, enactments or orders which affect or evidence alterations to the company’s constitution (see subsection (1)) unless the effect of the resolution, agreement, enactment or order has been incorporated into the company’s articles or is no longer in force.
118.The intention behind this provision is that information provided on a request for a copy of the company’s articles should be up-to-date but the company should not be obliged to provide the same information twice (i.e. in different forms).
Section 37: Right to participate in profits otherwise than as member void
119.This section restates section 15(1) of the 1985 Act. It provides that a company limited by guarantee without a share capital cannot, by means of a provision in its articles or a resolution of its members, confer on any person a right to participate in its divisible profits otherwise than as a member. As under the 1985 Act, there is no statutory restriction on the members of such companies participating in their profits, unless they have sought exemption from the use of the word “limited” in their names.
Section 38: Application to single member companies of enactments and rules of law
120.Under section 7 it will be possible for a single person to form any type of company. This section provides that in future any enactment or rule of law that is applicable to companies formed by two or more persons (or having two or more members) applies (with any necessary modifications) to companies formed with one member (or having only one person as a member). This is already the case in respect of private limited companies: see the Companies (Single Member Private Limited Companies) Regulations 1992 (SI 1992/1699).
Part 4: a Company’s Capacity and Related Matters
121.This Part replaces various provisions in the 1985 Act about a company’s capacity and related matters, including in particular those in Chapter 3 of Part 1 of that Act.
Section 39: A company’s capacity
122.This section provides that the validity of a company’s acts is not to be questioned on the ground of lack of capacity because of anything in a company’s constitution. It replaces the present section 35(1) and (4) of the 1985 Act, which made similar provision for restrictions of capacity contained in the memorandum.
123.The section does not contain provision corresponding to section 35(2) and (3) of the 1985 Act. It is considered that the combination of the fact that under the Act a company may have unrestricted objects (and where it has restricted objects the directors’ powers are correspondingly restricted), and the fact that a specific duty on directors to abide by the company’s constitution is provided for in section 171, makes these provisions unnecessary.
124.Subsection (2) indicates that the section, like section 35 of the 1985 Act, is modified in its application to charities.
Section 40: Power of directors to bind the company
125.This section provides safeguards for a person dealing with a company in good faith and restates section 35A and 35B of the 1985 Act. The power of the directors to bind the company, or authorise others to do so, is deemed not to be constrained by the company’s constitution. This means that a third party dealing with a company in good faith need not concern itself about whether a company is acting within its constitution.
126.Subsection (2)(b)(i) of the section replaces part of section 35B of the 1985 Act: an external party is not bound to enquire whether there are any limitations on the power of the directors. The first limb of section 35B (which refers to the memorandum) has not been carried forward. This is concerned with restrictions in a company's constitution that limit a company’s ability to act and consequently the powers of the directors to bind the company (the so called "ultra vires rule"). Under the Act, the objects no longer affect the company's capacity to act and so this limb is not necessary.
Section 41: Constitutional limitations: transactions involving directors or their associates
127.This section restates section 322A of the 1985 Act. It applies to a transaction if, or to the extent that, its validity depends on section 40 and provides that where the party to a transaction with a company is an “insider” (for example, a director of the company or person connected to such a director – see subsection (2)(b)(i) and (ii)), then the protection afforded by that section will not apply. Instead, the transaction will be voidable at the instance of the company.
128.Irrespective of whether the transaction is avoided, the “insider” and any director who authorised the transaction is liable to account to the company for any gain he has made as a result of the transaction and to indemnify the company for any loss or damage that the company has incurred (see subsection (3)). However, where the “insider” is not a director of the company, it may be possible for him to avoid liability if he can show that at the time he entered into the transaction with the company he was unaware that the directors were exceeding their powers (see subsection (5)).
129.As now, under subsection (4), a transaction will cease to be voidable in certain circumstances, for example, if restitution is no longer possible.
Section 42: Constitutional limitations: companies that are charities
130.This section restates section 65 of the Charities Act 1993. It is a qualification of the rules in sections 39 and 40.
131.It provides that the protection afforded to an external party by sections 39 and 40 will not apply where the company in question is a charity, unless:
the external party was unaware (at the time that the act was done) that the company was a charity; or
the company has received full consideration in respect of the act done, and the external party was unaware that the act in question was beyond the company’s capacity or beyond the powers of the directors.
132.Corresponding provisions for charities that are registered in Scotland can be found in section 112 of the Companies Act 1989 (see subsection (5)).
Section 43: Company contracts
133.This section restates the provisions of section 36 of the 1985 Act.
Section 44: Execution of documents
134.This section largely restates section 36A of the 1985 Act. It provides that a company may execute a document under the law of England and Wales or Northern Ireland by affixing the company seal or by signature by two directors or by one director and a secretary (or joint-secretary) or (for the first time) by a single director if that signature is witnessed and attested.
Section 45: Common seal
135.This section replaces the provisions of sections 36A(3) and 350 of the 1985 Act. It permits but does not require a company to have a common seal. If a company has a common seal, it requires the seal to include the company’s name: failure to do so is an offence.
Section 46: Execution of deeds
136.This section restates section 36AA, inserted into the 1985 Act by the Regulatory Reform (Execution of Deeds and Documents) Order 2005 (SI 2005/1906). The only change is to extend the application for the purposes of the law of Northern Ireland.
Section 47: Execution of deeds and other documents by attorney
137.This section replaces section 38 of the 1985 Act. The 1985 Act does not require the appointment of the attorney to be by deed nor does it say anything about deeds executed on behalf of the company in the United Kingdom. This section provides that a company may appoint, under the law of England and Wales or Northern Ireland, attorneys to execute deeds or other documents on its behalf, and that documents executed in this manner, whether in the UK or abroad, have effect as if executed by the company. It also makes clear that the method for a company appointing an attorney is by instrument executed as a deed, which is the same method by which an individual appoints an attorney.
Section 48: Execution of documents by companies
138.This section restates section 36B of the 1985 Act. It makes clear that no seal is required regardless of any other statutory provision. The only change is the addition of subsection (1) which makes clear that this section forms part of the law of Scotland only.
Section 49: Official seal for use abroad
139.This section replaces section 39 of the 1985 Act. It sets out the circumstances and manner in which a company may use its common seal outside the UK.
Section 50: Official seal for share certificates etc
140.This section restates section 40(1) of the 1985 Act. It enables a company that has a common seal to have an official seal for sealing securities issued by the company and for sealing documents creating or evidencing securities so issued.
Section 51: Pre-incorporation contracts, deeds and obligations
141.This section restates section 36C of the 1985 Act. A company is not bound by a contract purportedly made on its behalf before it came into existence unless the obligations are novated, i.e. a new contract must come into existence after incorporation on the same terms as the old one. Novation may be express or implied.
Section 52: Bills of exchange and promissory notes
142.This section restates section 37 of the 1985 Act. A bill of exchange is an unconditional order in writing, addressed by one person to another, signed by the person giving it, requiring the person to whom it is addressed to pay on demand or at a fixed or determinable future time a sum certain in money to or to the order of a specified person, or to its bearer. A promissory note is an unconditional promise in writing made by one person to another, signed by the maker, engaging to pay, on demand or at a fixed or determinable future time, a sum certain in money to, or to the order of, a specified person or to its bearer. Where someone acting under a company’s authority makes, accepts, or endorses such an instrument in the name of the company, or on its behalf, this section treats this as if these actions had been done by the company.
Part 5: a Company’s Name
143.This Part applies to the name under which a company is registered, sometimes called the "corporate name". This Part regulates the choice of name. The rules are primarily intended to ensure that third parties are not misled. There are no property rights in companies’ registered names as such. While there is no requirement for a company to use its registered name in the course of business, this Part also requires a company to disclose its name in specified circumstances.
144.Sections 70 to 74 provide for the appointment of adjudicators in cases where there is dispute over the registering of a company name. Section 71 safeguards the independence of the adjudicators and section 74 provides a right of appeal to the court.
Chapter 1: General Requirements
Section 53: Prohibited names
145.This section replaces section 26(1)(d) and (e) of the 1985 Act. It retains the existing prohibition of companies registering names that cannot be used without commission of an offence and of those that are offensive.
Section 54: Names suggesting connection with government or public authority
146.This section replaces section 26(2)(a) of the 1985 Act. It prevents a name being registered without the Secretary of State’s approval if it suggests a connection with Her Majesty’s Government, a local authority or – which represents a change from the 1985 Act – any part of the Scottish administration, or Her Majesty’s Government in Northern Ireland. A new power allows similar protection to be extended to other public authorities.
Section 55: Other sensitive words or expressions
147.This section replaces sections 26(2)(b), 29(1)(a) and 29(6) of the 1985 Act.
148.Subsection (1) requires prior approval for the adoption of a name that includes words or expressions specified in regulations. Subsection (2) provides for the procedure to be used for making the regulations. The words and expressions protected by the current Regulations (the Company and Business Names Regulations 1981, SI 1981/1685) include British, English, Scottish and Welsh; chamber of commerce, charity, Her Majesty, midwife, police, and university.
Section 56: Duty to seek comment of government department or other specified body
149.This section replaces section 29(1)(b) and (2) and (3) of the 1985 Act. It provides power for the Secretary of State to specify whose view must be sought when seeking approval for a name. For example, under the present Regulations, the approval of the General Dental Council is required for the use of either “dental” or “dentistry”. Regulations under the new power would be able to replicate this. They could also require the approval of, say, the House Authorities for names suggesting a connection with Parliament.
150.When a request is made under section 56 in connection with the registration or the change of name of a company, the registrar must be sent a statement that a request has been made, and a copy of the response (see subsections (3) and (4)). But the registrar must no make the response available for public inspections (see section 1087(1)(a)).
Section 57: Permitted characters etc
151.This section is a new provision. It provides power for regulations to specify what letters, symbols, etc may be used in a company’s registered name; the regulations may also specify a permitted format for a name (for example, to prevent the use of superscript or subscript).
Chapter 2: Indications of Company Type Or Legal Form
Section 58: Public limited companies
152.This section replaces section 25(1) of the 1985 Act (and also section 27(4)(b) and (d) in its application to public limited companies). It brings together in a single provision all the alternative statutory indicators of legal status that must be used by a public company as part of its registered name, i.e. “public limited company” or the Welsh equivalent or the specified abbreviations. This section does not apply to community interest companies.
Section 59: Private limited companies
153.This section replaces section 25(2) of the 1985 Act (and also section 27(4)(a) and (c) in its application to private limited companies). It brings together in a single provision all the alternative statutory indicators of legal status that must be used by a private company as part of its registered name, i.e. “limited” or the Welsh equivalent or the specified abbreviations. Certain companies are exempt (see section 61). This section does not apply to community interest companies.
Sections 60 to 62: Exemption from requirement as to use of “limited”
154.These sections replace section 30 of the 1985 Act. Section 30 exempts certain companies from the requirement for their names to conclude with “limited”. Exempt companies are also exempt under the 1985 Act from some of the requirements regarding publication of their name but they still have to disclose their limited status in correspondence. Those currently exempt are those with a licence granted under section 19 of the Companies Act 1948 which have delivered a statutory declaration to the Registrar that the company complies with the requirements for the exemption. These requirements are, in effect, that the company is non-profit-making and its objects are the promotion of commerce, art, science, education, religion, charity or any profession.
155.Section 60 continues the exemption for companies already exempt so long as they continue to meet the conditions and until they change their registered name. It also provides an exemption for charities and allows the Secretary of State to make regulations exempting other companies. Only private companies may be exempt
156.Sections 61 and 62, which replace section 30(2) and (3), specify the conditions that must be met for a company currently exempt to continue to qualify for the exemption: its objects must continue to satisfy the criteria for their exemption and its articles must both preclude distributions of dividends to its members and also, in the event of it being wound up, require its assets to be passed to a body with similar objects. For companies limited by shares benefiting from an exemption under the 1948 Act (or its Northern Irish equivalent), there is a new requirement that the articles prevent a distribution of capital. This is linked to the change in section 63(4) (see below).
Section 63: Exempt company: restriction on alteration of articles
157.This section replaces section 31(1) and (5). It prohibits a company benefiting from an exemption under the 1985 Act or the 1948 Act (or their Northern Irish equivalents) from changing its articles in such a way that it no longer meets the requirements for the exemption. It is an offence to change the company’s articles in such a way. Many companies with an exemption under the 1948 Act (or its Northern Irish equivalent) were made to include a provision in their memoranda preventing an amendment to their memoranda or articles without the consent of the Board of Trade (there were a number of variations on this theme). Subsections (4) and (5) make provision to remove this administrative burden.
Section 64: Power to direct change of name in case of company ceasing to be entitled to exemption
158.This section replaces section 31(2) to (6). It gives the Secretary of State power to withdraw a private company’s exemption from the requirement for its name to conclude with “limited” and to direct it to change its name if it no longer meets the criteria that applied when it was granted the exemption.
Section 65: Inappropriate use of indications of company type or legal form
159.This section replaces section 26(1)(a), (b), (bb) and (bbb) of the 1985 Act. These paragraphs restrict the use of various words, expressions and abbreviations that are indicators of legal status for various types of commercial entity, e.g. p.l.c., community interest company, open-ended investment company, etc. Some of the restrictions apply to the use of the particular indicator at the end of a company’s name; some anywhere other than the end of the name; and some anywhere in a company’s name.
160.This section provides power to make regulations prohibiting the inclusion in a company’s name of specified words, expressions and abbreviations. The only words etc that can be specified in the regulations are those associated with a particular type of company or form or organisation or those confusingly similar to such words and expressions. This section also provides power to require or prohibit the statutory indicators of legal status being used in conjunction with specified other words.
Chapter 3: Similarity to Other Names
Section 66: Name not to be the same as another in the index
161.This section replaces section 26(1)(c) and (3) of the 1985 Act.
162.Subsection (1) retains the present prohibition, in section 26(1)(c), on a company adopting a name that is already on the registrar’s index of company names – which includes not only the names of Companies Act companies but various other business entities (see section 1099). Subsections (2) and (3) provide power for the Secretary of State to make regulations to replace the detailed rules presently contained in section 26(3) of the 1985 Act as to:
what is to be disregarded; and
what words, letters and symbols are to be taken as the same, or as not the same,
when comparing a proposed and an existing name. At present only “and” and “&” are taken as the same.
163.The section provides power also to treat as the same:
currency symbols (e.g. £, $) and their respective English word equivalents;
“%” and “per cent”;
“1”, “2”, “3” etc and “one”, “two” “three”.
164.The prohibition of names that, under these rules, are the same as an existing name will not be discretionary. But in future, it will be possible for there to be exceptions: subsection (4) provides that the regulations may provide that names which would otherwise be prohibited as being the same may be permitted in specified circumstances, or with specified consent, and that a subsequent change of circumstances or withdrawal of consent will not affect the company’s registration.
Section 67: Power to direct change of name in case of similarity to existing name
165.This section replaces section 28(2) of the 1985 Act which provides power for the Secretary of State to direct a company to change its name if the name is the same as or too like a name already on the registrar’s index of company names (or one which should have been there). The objective is to prevent the public being confused by the simultaneous appearance on the register of two very similar names when the similarity is such that the later name was not caught by the non-discretionary prohibition of adopting a name effectively the “same as” an existing name (see section 66).
166.The section is intended to cover two circumstances. First, any delay in the entry on the index of company names of new names of entities that are not UK companies. Companies House enter all names immediately but there may be delays outside their control. If the name had already been taken by the other entity before the company adopted it, then the Secretary of State will direct the company to change its name. Second, the visual difference between the new name and an existing name being so small that third parties are likely to be confused by the simultaneous appearance of both names on the index of company names.
167.Subsections (2) and (3) provide power to make regulations, corresponding to that provided by section 66, to replace the detailed rules presently contained in section 26(3) of the 1985 Act as to:
what is to be disregarded; and
what words, letters and symbols are to be taken as the same
when comparing a proposed and an existing name. As in section 67 , subsection (4) provides for a power to make regulations permitting names that would otherwise be regarded as “too like” in certain circumstances or where consent is given.
Section 68: Direction to change name: supplementary provisions
168.This section replaces section 28(4) and (5) of the 1985 Act as they apply to section 28(2). It provides a deadline of 12 months for the Secretary of State to direct a change of name under section 67, and for the Secretary of State to specify a period for the company’s compliance. It makes failure by the company to comply an offence.
Similarity to other name in which person has goodwill
169.Sections 69 to 74 are new provisions. They respond to the CLR recommendation (Final Report, paragraph 11.50) that there be provision so that a person can apply for a company to be directed to change its name if the applicant can show that the name was chosen with the principal intention of seeking money from him or preventing him registering the name where it is one in which he has previously acquired reputation or goodwill.
Section 69: Objection to company’s registered name
170.This section provides for any person, not just a company, to object to a company names adjudicator if a company’s name is similar to a name in which the objector has goodwill. There is list of circumstances raising a presumption that a name was adopted legitimately. The respondent must show that one of these applies, or otherwise that he acted in good faith or that the interests of the applicant are not significantly affected (for example, where the applicant has hardly used the name at all). The objection will be upheld if the respondent cannot do so, or if the objector can show that the name was registered either to obtain money from him or to prevent him using the name.
Section 70: Company names adjudicators
171.This section provides power for the Secretary of State to appoint company names adjudicators and their staff and to finance their activities. One of the adjudicators is to be appointed Chief Adjudicator.
Section 71: Procedural rules
172.This section provides the Secretary of State with power to make rules for the proceedings before a company names adjudicator. The list of matters which the rules may cover is not exhaustive. It also enables the rule to confer on the Chief Adjudicator power to determine any matter that could be the subject of the rules made under this power.
Section 72: Decision of adjudicator to be made available to public
173.This section requires the adjudicator to publish his decision and his reasons for it, possibly through a website. The publication must be within 90 days of the decision.
Section 73: Order requiring name to be changed
174.This section is a new provision. If an objection made under section 69 is upheld, then the adjudicator is to direct the company with the offending name to change its name to one that does not similarly offend. A deadline must be set for the change. If the offending name is not changed, then the adjudicator will determine a new name for the company.
Section 74: Appeal from adjudicator’s decision
175.This section enables appeal to a court against the decision of the company names adjudicator. The court will either uphold or reverse the adjudicator’s decision, and may make any order that the adjudicator might have made.
Chapter 4: Other Powers of the Secretary of State
Section 75: Provision of misleading information etc
176.This section replaces section 28(3) of the 1985 Act and, insofar as they support that subsection, section 28(4) and (5). It provides power for the Secretary of State to direct a company to change its name within a specified period in two circumstances. First, if misleading information was given to enable the adoption of the name. Second, if an undertaking or assurance given to enable the adoption of the name has not been fulfilled. The direction can only be made up to five years after the adoption of the name. It is an offence not to comply with the direction.
Section 76: Misleading indication of activities
177.This section replaces section 32 of the 1985 Act. It provides power for the Secretary of State to direct a company to change its name, regardless of how long the company has had the name, in the specified circumstances. The circumstances are that, in his opinion, not only does the name give a misleading indication of the nature of the company’s activities but also that the public are likely to suffer harm as a result. The company may appeal to the court, who may either confirm the direction or set it aside. It is an offence not to comply with the direction.
178.The section also sets time limits for compliance with the direction (6 weeks) and the application to the court (3 weeks). If the court confirms the direction, it specifies the deadline for compliance.
Chapter 5: Change of Name
Section 77: Change of name
179.This section replaces section 28(1) of the 1985 Act. Under the existing provision, companies can only change their names:
by special resolution; or
following a direction by the Secretary of State in the restricted circumstances provided by section 31 of the 1985 Act, which apply only to companies exempt from their name concluding in “limited.”
180.This section also provides for the following means:
whatever means are provided in the company’s articles (this means that the company will be able to determine the procedures for changing its own name);
by an order of the company names adjudicator if an objection under section 73 is upheld, or by a court following an appeal against the adjudicator’s decision under section 74; and
under section 1033 on the company’s restoration to the register.
Section 78: Change of name by special resolution
181.This section is a new provision. It requires the company to notify the registrar of a change of name when it has been agreed by special resolution. This requirement is in addition to the obligation under Chapter 3 of Part 3 to forward a copy of the special resolution to the registrar. Subsections (2) and (3) address the particular situation where a company has passed a special resolution to change its name but the change is not to take place until some other event has occurred (e.g. a merger). The notice of change of name must say that the change is conditional and whether the event has occurred. If the event has not yet occurred, the registrar will not act on the notice to change the name until she has received a second notice stating that the specified event has occurred. The registrar may rely on that statement without further evidence.
Section 79: Change of name by means provided for in company’s articles
182.This section is a new provision, supplementing the new provision (section 77(1)(b)) whereby a company may change its name by any means provided for in its articles. Subsection (1) requires the company to provide the registrar with both a notice of the name change and a statement that the change has been made in accordance with the company’s articles. Subsection (2) ensures the registrar may rely on that statement without further evidence.
Section 80: Change of name: registration and issue of new certificate of incorporation
183.This section, which partly replaces sections 28(6) and 32(5) of the 1985 Act, provides for the procedures that the registrar must perform before a company’s proposed new name is effective. Subsection (2) provides for the checks both that the name meets all the requirements for a company’s name in this Part of the Act and that the necessary documents have been provided. Subsection (3) provides for the company to be issued with a certificate of incorporation with the new name.
Section 81: Change of name: effect
184.This section, which replaces sections 28(6) and 32(5) in part and, in total, section 28(7) of the 1985 Act, provides that the new name is effective as soon as the altered certificate of incorporation is issued. It also provides that the change of name does not affect the company’s rights or obligations or legal proceedings by or against it in its previous name.
Chapter 6: Trading Disclosures
Section 82: Requirement to disclose company name etc
185.This section replaces sections 348(1), 349(1), and 351(1) and (2) of the 1985 Act and, insofar as it applies to companies, section 4(1) of the Business Names Act 1985. It provides power for the Secretary of State to make regulations requiring every company:
to display a sign with its name and specified other information at specified locations;
to include its name and specified other information in specified documents and communications;
to provide its name and specified other information to those who request it in the course of business (this is a new provision insofar as it applies to companies doing business under their registered names).
Section 83: Civil consequences of failure to make required disclosure
186.This section replaces section 5 of the Business Names Act 1985, so far as it applies to companies. As recommended by the CLR (Final Report, paragraph 11.57), it follows the precedent of the Business Names Act as regards the civil consequences of failure to comply with the information requirements made in regulations under section 82: the provision for personal civil liability of officers in default in section 349(4) of the Companies Act 1985 is not included.
Section 84: Criminal consequences of failure to make required disclosures
187.This section replaces sections 348(2), 349(2) and (3) and 351(5) of the 1985 Act and, insofar as it applies to companies, part of section 7 of the Business Names Act 1985. It makes it an offence not to comply with the requirements, to be specified in regulations under section 82, for every company to disclose its name and specified other information.
Section 85: Minor variations in form of name to be left out of account
188.This section is a new provision. It means that the company’s name as used to comply with the disclosure requirements need not be exactly the same as the registered name. The permitted differences are the case of the letters, the use of punctuation, accents, etc and formatting. However the differences must not result in there being a risk of confusion.
Part 6: a Company’s Registered Office
Section 86: A company’s registered office
189.This section restates section 287(1) of the 1985 Act. It requires every company to have a registered office and provides for that office to be an address to which communications and notices may be sent. Section 1139 provides that the service of a document on a company is effective if it is sent to its registered office.
Section 87: Change of address of registered office
190.This section restates section 287(3) to (6) of the provisions of the 1985 Act. It provides the means by which a company may change the address of its registered office.
Section 88: Welsh companies
191.This section provides a definition of a Welsh company. A company can be set up as a Welsh company by delivering to the registrar a statement on formation that its registered office is to be situated in Wales (see section 9(2)(b)). Subsection (2) restates section 2(2) of the 1985 Act whereby a company may become a Welsh company by passing a special resolution (so that the register states that its registered office is to be situated “in Wales”). As recommended by the CLR, subsection (3) provides a mechanism whereby a company can cease to be a Welsh company (i.e. so that the register states that its registered office is to be situated in “England and Wales”). This is new. At present, while a company may choose to restrict the address of its registered office to Wales on formation or subsequently by special resolution, it is not possible under the 1985 Act for a Welsh company to drop the restriction so that its registered office address can be changed to anywhere in England and Wales.
192.Welsh companies may deliver documents to the registrar in Welsh (see section 1104). Welsh companies may also end their company name with Welsh versions of the statutory indicators of legal status. For example, “cyfyngedig” in place of “limited” or “c.c.c.“ in place of “p.l.c.” (see sections 58 and 59). When a company ceases to be a Welsh company using the procedure under this section, it may no longer take advantage of these provisions.
193.Where a company passes a special resolution under subsection (2) or (3) (and so becomes or ceases to be Welsh company) subsection (4) provides that the registrar will amend the register and issue the company with a new certificate of incorporation.
Part 7: Re-Registration as a Means of Altering a Company’s Status
194.This Part of the Act is about the re-registration of companies. It replaces equivalent provisions in Part 2 of the 1985 Act. There are some substantive changes as well as amendments reflecting the new provisions of the Act about registration which are carried through to the re-registration provisions.
Introductory
Section 89: Alteration of status by re-registration
195.This section provides for various ways under the Act by which a company may alter its status. As recommended by the CLR (Final Report, paragraph 11.6) it retains the current possibilities for re-registration, but there is one significant change to the 1985 Act regime: in line with the recommendations of the CLR (Final Report, paragraph 11.11), in future a public company will be able to re-register as an unlimited private company without first having to re-register as private limited – see section 109.
Private company becoming public
Section 90: Re-registration of private company as public
196.This section restates section 43(1) and (2), and section 48 of the 1985 Act. It enables a private company (whether limited or unlimited) to re-register as a public company providing that certain conditions are met. These conditions are set out in subsections (2) to (4). They include a requirement for the company to make such alterations to its name and articles of association (“articles”) as are necessary to reflect the fact that the company will be a public company. This will be particularly important for private companies formed under the Act who are using the model articles: in particular, the new model articles for private companies limited by shares formed under the Act will be written with such companies in mind and are unlikely to be suitable for use by a newly re-registered public company – see notes on sections 19 and 20.
197.As now (see section 48 of the 1985 Act), an unlimited private company with a share capital will be able to re-register as a public company and this is reflected in subsection (4) of this section.
198.Subsection (2)(e) retains the requirement that a private company may not re-register as a public company if it has previously re-registered as an unlimited company. The intention behind this provision is that a company should not be able to enjoy the benefits of limited liability or avoid the obligations that are attached to this, for example, the increased reporting requirements, by continually swapping from limited to unlimited status.
Section 91: Requirements as to share capital
199.This section restates sections 45, 47(3) and 48(5) to (7) of the 1985 Act. It sets out the requirements as to share capital of a company that it is proposing to re-register as a public company. These requirements carry forward the provisions of current companies legislation, for example, the company’s share capital must not be less than the authorised minimum (defined in section 763) and each of the company’s allotted shares must be paid up at least as to a quarter of the nominal value of that share and the whole of any premium on in.
200.Subsection (5) of this section replaces section 47(3) of the 1985 Act. It provides that the registrar must not issue a new certificate of incorporation on re-registration if the court has made an order confirming a reduction of capital which has the effect of bringing the company’s allotted share capital below the authorised minimum (which remains at £50,000 but may be satisfied in sterling or euros – see section 763) or if the company has reduced its capital via the new solvency statement procedure for capital reductions (see section 642) or in connection with a redenomination of share capital (see section 626).
Section 92: Requirements as to net assets
201.This section restates section 43(3)(b), (c) and (e), and (4), and section 46 of the 1985 Act. The requirements as to net assets for a public company remain unchanged (as now, these are determined by reference to the company’s most recent balance sheet).
Section 93: Recent allotment of shares for non-cash consideration
202.This section restates section 44 of the 1985 Act. As now, where there has been an allotment of shares for non-cash consideration between the date of the balance sheet required under section 92 and the date that the company passed the resolution to re-register as a public company, the registrar will not entertain an application for re-registration unless the consideration for the allotment has been valued in accordance with section 596.
Section 94: Application and accompanying documents
203.This section restates sections 43(3)(a) to (e) and 47(2) of the 1985 Act. It prescribes the contents of the application for re-registration where a private company is proposing to re-register as public. There is one important change, which is required as a result of the abolition of the current requirement for private companies to have a company secretary – see section 270. In future, where a private company is proposing to re-register as a public company the application for re-registration must include a statement of the company’s proposed secretary where the company does not already have a secretary. The contents of this statement are prescribed in section 95.
204.The application for re-registration must be accompanied by a statement of compliance – see section 90(1)(c)(ii) – which replaces the present requirement for a statutory declaration (or its electronic equivalent), contained in subsections 43(3)(e) and (3A) of the 1985 Act, with a requirement to make this statement (see note on section 13).
Section 95: Statement of proposed secretary
205.This section is a new provision, which is required as a result of the abolition of the requirement for private companies to have a company secretary – see section 270. Where a private company is proposing to re-register as a public company and the company does not already have a company secretary, the application for re-registration must include details of the person or persons who will act as company secretary or joint secretaries on re-registration. The statement of proposed secretary must also contain a consent, given by the person or each of the persons named in the statement, to act as company secretary or joint secretaries. If all the partners in a firm are to be joint secretaries, one partner in the firm may give consent to act on behalf of all of the partners.
Section 96: Issue of certificate of incorporation on re-registration
206.This section replaces section 47 of the 1985 Act. As now, where the registrar is satisfied that a company is entitled to be re-registered as a public company, she will issue a new certificate of incorporation (which must state that it is being issued on the re-registration of the company). On the issue of a new certificate of incorporation under this section: the company becomes a public company; the change to its name and any amendments that were made to the company’s articles take effect; and the person (or persons) named as secretary (or joint secretaries) in the statement of proposed secretary (see section 95) is deemed to have been appointed as such.
207.As now, the certificate of incorporation on re-registration is conclusive evidence that the company is now a public company and that the requirements of the Act as regards re-registration have been met.
Public company becoming private
Section 97: Re-registration of public company as private limited company
208.This section replaces section 53 of the 1985 Act. It enables a public company to re-register as a private limited company if the conditions specified in subsection (2) are met. The conditions are the same as those which are presently set out in section 53 but there are two important changes:
Consistent with the approach taken elsewhere in the Act, for example the sections on the re-registration of a private company as public, subsection (1)(c)(ii) of this section introduces a new requirement for a statement of compliance (see note on section 13).
Subsection (2) introduces new provisions which enable the registrar to process an application for the re-registration of a company from public to private limited within the 28-day period during which dissenting members may apply to the court, under section 98, for an order cancelling the resolution for re-registration, providing that she is satisfied that such an application cannot be made. This change reflects the registrar’s current practice.
209.As now, the company must make such changes to its name and articles as are necessary in connection with it becoming a private company limited by shares or, as the case may be, a private company limited by guarantee.
Section 98: Application to court to cancel resolution
210.This section restates section 54(1) to (3) and (5) to (6) of the 1985 Act. As now, where a public company has passed a special resolution to re-register as a private limited company, the requisite majority of the company’s members (see subsection (1)) may apply to the court for the cancellation of this resolution. Such an application to the court must be made within 28 days of the resolution to re-register being passed and on hearing the application the court may confirm or cancel the resolution or make such other order as it thinks fit.
Section 99: Notice to registrar of court application or order
211.This section replaces section 54(4) (7) and (10) of the 1985 Act. It makes it clear that, as now, where an application is made to the court under section 98 (that is, to cancel a resolution for re-registration as a private limited company), the company must immediately give notice to the registrar. Similarly, where the court has made an order in connection with such an application, the company must deliver a copy of that order to the registrar within 15 days of the order being made (or such longer time as the court may direct).
212.Subsection (1) of this section is a new provision which requires the dissenting members, on making an application to court seeking to cancel the resolution for re-registration from public to private, to give notice direct to the registrar. This ensures that the registrar is aware of any applications which have been made under section 98 and therefore will enable the registrar to process the application for re-registration without further delay where she is satisfied that no application to court may be made – see note on section 97.
213.Subsection (4) carries forward the offence in section 54(10) of the 1985 Act. Where the company fails to give notice to the registrar or fails to deliver a copy of the order made by the court under section 98 within the prescribed time limits (see subsections (2) and (3)), the company and every officer of the company who is in default commits an offence. The penalty for this offence is set out in subsection (5).
Section 100: Application and accompanying documents
214.This section replaces section 53(1)(b) of the 1985 Act and contains new provisions. It prescribes the documents/information that must accompany the application for re-registration where a company is proposing to re-register from public to private limited. Consistent with the approach taken in the Act with other forms of re-registration, in future the application for re-registration as a private limited company must be accompanied by a statement of compliance – see note on section 13. (There is currently no requirement for a statutory declaration (or electronic equivalent) where a public company re-registers as a private limited company).
Section 101: Issue of certificate of incorporation on re-registration
215.This section restates section 55 of the 1985 Act. As now, where the registrar is satisfied that a company is entitled to be re-registered as a private limited company, she will issue a new certificate of incorporation (which must state that it is being issued on the re-registration of the company). On the issue of a new certificate of incorporation under this section, the company becomes a private limited company and the change to its name and any amendments that were required to be made to the articles take effect.
216.As now, the certificate of incorporation on re-registration issued under this section is conclusive evidence that the company is now a private limited company and that the requirements of the Act as regards re-registration have been met.
Private limited company becoming unlimited
Section 102: Re-registration of private limited company as unlimited
217.This section replaces section 49, of the 1985 Act. As now, this section permits a private company that is limited by shares or, as the case may be, by guarantee, to re-register as an unlimited private company, providing that certain conditions are met (see subsection (2)) and all of the members have given their assent to the company being so re-registered. In the case of a deceased member, assent may be given by the personal representative of the deceased member’s estate. Where a member is bankrupt, assent may be given by his trustee in bankruptcy (to the exclusion of the member in question).
218.An “unlimited company” is a company not having any limit on the liability of its members.
219.As now, a company may not re-register as an unlimited company, if it has previously been re-registered as limited (having previously been unlimited) or as unlimited (having previously been limited).
220.The application for re-registration as an unlimited company must be accompanied by a statement of compliance (see note on section 13).
Section 103: Application and accompanying documents
221.This section replaces section 49(8) and (8A) of the 1985 Act. It prescribes the contents of the application for re-registration where a company is proposing to re-register from private limited to unlimited and the documents/information that must accompany this application. The current requirement for a statutory declaration made by the directors on application for re-registration as an unlimited company is replaced by a requirement for a statement of compliance. Unlike other statements of compliance made under the Act (see, for example, section 13) the statement of compliance made on application for re-registration as an unlimited company must contain a statement made by the directors confirming that:
the persons by whom or on whose behalf the form of assent is authenticated constitute the whole membership of the company; and
if any of the members have not authenticated that form themselves, that the directors have taken all reasonable steps to satisfy themselves that each person who authenticated it on behalf of a member was lawfully empowered to do so.
222.The contents of the directors’ statement carry forward the provisions of section 49(8) of the 1985 Act.
Section 104: Issue of certificate of incorporation on re-registration
223.This section restates section 50 of the 1985 Act. As now, where the registrar is satisfied that a company is entitled to be re-registered as an unlimited company, she will issue a new certificate of incorporation (which must state that it is being issued on the re-registration of the company). On the issue of a new certificate of incorporation under this section, the company becomes an unlimited company and the change to its name and any amendments that were required to be made to the articles take effect.
224.As now, the certificate of incorporation on re-registration is conclusive evidence that the company is now an unlimited company and that the requirements of the Act as regards re-registration have been met.
Unlimited private company becoming limited
Section 105: Re-registration of unlimited company as limited
225.This section replaces section 51(1) to (3) of the 1985 Act. As now, this section permits an unlimited company to re-register as a private limited company if certain conditions are met (see subsection (2)). As now, a re-registration from unlimited to limited requires a special resolution of the company’s members, (which must specify whether the company is to be limited by shares or limited by guarantee). The company must also make such changes to its name and articles as are required to reflect the change in the company’s status. As is the case under section 51(6) of the 1985 Act, this section does not permit the re-registration of an unlimited company as a public company (this section provides for the re-registration of an unlimited company as a private limited company). There is a new requirement for a statement of compliance (see note on section 13).
Section 106: Application and accompanying documents
226.This section replaces section 51(5) of the 1985 Act and contains new provisions. It prescribes the contents of the application for re-registration where an unlimited private company is proposing to re-register as private limited and the documents/information that must accompany this application. Where the company is to be limited by guarantee, subsection (2)(b) requires the application for re-registration to be accompanied by a “statement of guarantee” (see note on section 11). It should be noted that there is no requirement for a statement of capital and initial shareholdings where the company is to be limited by shares. This is unnecessary because the company will be required to make a return of allotments to the registrar, under section 555 as soon as it allots shares subsequent to its registration and the return must be accompanied by a statement of capital.
Section 107: Issue of certificate of incorporation on re-registration
227.This section restates section 52 of the 1985 Act. As now, it provides that, where the registrar is satisfied that a company is entitled to be re-registered as a private company, she will issue a new certificate of incorporation (which must state that it is being issued on the re-registration of the company). On the issue of a new certificate of incorporation under this section, the company becomes a private limited company and the change to its name and any amendments that were required to be made to the articles take effect.
228.As now, the certificate of incorporation on re-registration issued under this section is conclusive evidence that the company is now a private limited company and that the requirements of the Act as regards re-registration have been met.
Section 108: Statement of capital required where company already has share capital
229.This section is a new provision which requires a company that has re-registered from unlimited having a share capital to private limited by shares to file a statement of capital with the registrar in certain circumstances. The statement must be delivered to the registrar within 15 days of the company’s re-registration and, where the company fails to observe this requirement, the company and every officer of the company who is in default, commits an offence (see subsection (4)).
230.The provision is necessary because unlimited companies are required to provide a statement of capital to the registrar in a limited number of circumstances only: in particular, where the company has a share capital on formation (see section 10) or where an unlimited company having a share capital makes an annual return to the registrar under section 854. Consequently, where an unlimited company having a share capital re-registers as private limited by shares under section 107, in contrast to other companies that are limited by shares, the information on the public register pertaining to the company’s subscribed capital may be out of date (in particular if the company has allotted further shares subsequent to its formation or, as the case may be, its last annual return).
231.The requirement for a statement of capital in this section puts companies which have re-registered as private limited by shares under section 107 on the same footing as other companies limited by shares on the register and ensures that the information on the public register is up-to-date. The company will, however, be exempted from the requirement to provide a statement of capital on re-registration where there has been no change to the company’s total subscribed capital since it was first formed and registered or, as the case may be, since the company filed its most recent annual return (see subsection (2)).
Public company becoming private and unlimited
Section 109: Re-registration of public company as private and unlimited
232.This section is a new provision, which, as recommended by the CLR (Final Report, paragraph 11.11), enables a public company to re-register as a private unlimited company with a share capital without first having to re-register as a private limited company. The conditions specified in subsection (2) must be met and all of the members must give their assent to the company being so re-registered. In the case of a deceased member, assent may be given by the personal representative of the deceased member’s estate. Where a member is bankrupt, assent may be given by his trustee in bankruptcy (to the exclusion of the member in question).
233.A public company may not re-register as an unlimited private company under this section if it has previously been re-registered as limited or as unlimited (see subsection (2)). The intention behind this subsection (which is based on the provision in subsection (2)(e) of section 90) is that a company should not be able to enjoy the benefits of limited liability or avoid the obligations that are attached to this, for example, the increased reporting requirements, by continually swapping from limited to unlimited status.
Section 110: Application and accompanying documents
234.This section is a new provision. It prescribes the contents of the application for re-registration where a company is proposing to re-register from public to unlimited private and the documents/information that must accompany this application. There is a requirement for a statement of compliance (see note on section 13) and, in contrast to the statements of compliance that are required elsewhere in the Act, the statement of compliance that is required here must contain a statement made by the directors confirming that:
the persons by whom or on whose behalf the form of assent is authenticated constitute the whole membership of the company; and
if any of the members have not authenticated that form themselves, that the directors have taken all reasonable steps to satisfy themselves that each person who authenticated it on behalf of a member was lawfully empowered to do so.
235.This mirrors the requirements of the directors’ statement in section 103(4).
Section 111: Issue of certificate of incorporation on re-registration
236.This section is a new provision which requires the registrar to issue a new certificate of incorporation is she is satisfied that a public company is entitled to register as private and unlimited. On the issue of a new certificate of incorporation (which must state that it is being issued on the re-registration of the company), the company becomes a private unlimited company and the change to its name and any amendments that were made to the articles take effect.
237.The certificate of incorporation on re-registration is conclusive evidence that the company is now a private unlimited company and that the requirements of the Act as regards re-registration have been met.
Part 8: a Company’s Members
238.This Part of the Act defines who are a company’s members, provides rules relating to a company’s register of members and overseas branch registers and, subject to certain exceptions, prohibits a company from being a member of its holding company.
Chapter 1: the Members of a Company
Section 112: The members of a company
239.This section restates section 22 of the 1985 Act. There are additional words to make it clear that the subscribers to the memorandum become members on registration of the company, even if the company fails to enter their names in the register of members.
Chapter 2: Register of Members
Section 113: Register of members
240.This section restates section 352(1) to (5) of the 1985 Act. The only new provision is subsection (5) which makes it clear that, for the purposes of this Chapter, joint holders of a share fall to be treated as a single member, so the register need only show a single address although all their names must be stated in the register.
Section 114: Register to be kept available for inspection
241.This section replaces section 353 of the 1985 Act. Currently, the register of members is required to be kept at the registered office of the company, except that if the company has appointed a third party to maintain or update the register, it may be kept at the office where that work is done, subject to that office being in the jurisdiction where the company is registered. Under the Act, it is immaterial where the work of compiling or updating the register is carried out. Subsection (1) provides that the register must be kept available for inspection either at the company’s registered office or at a place permitted under regulations made under section 1136.
Section 115: Index of members
242.This section replaces section 354 of the 1985 Act. There is no change in the obligation of a company with more than 50 members to maintain an index of the names of the members (which the company is obliged to do unless the register itself is kept in such a form as to constitute an index).
Section 116: Rights to inspect and require copies
243.This section replaces section 356 of the 1985 Act. Under section 356, the obligation to make the register available for inspection is subject to an exception when the register is closed under section 358 of the 1985 Act. The power to close the register has not been carried forward in the Act and so the obligation in subsection (1) is absolute. The CLR recommended that information in a company’s register of members should be made available only for certain specified purposes (Final Report, paragraph 11.44). This section follows this recommendation. It modifies the rights of inspection and to be provided with copies of the register of members and its index. (Section 1137 provides power for the Secretary of State to make regulations about the inspection of records and provision of copies and to set fees.) Subsections (3) and (4), which are new, require those seeking to inspect or to be provided with a copy of the register of members to provide their names and addresses, the purpose for which the information will be used, and, if the access is sought on behalf of others, similar information for them.
Section 117: Register of members: response to request for inspection or copy
244.This is a new provision. This section provides a procedure by which the company can refer the matter to the court if it thinks that the request may not be for a proper purpose. It replaces the 10-day deadline for compliance with a request with a 5-day period within which the company must either comply with the request or apply to the court for relief from the obligation. If the company opts for the latter, then subsections (3), (4) and (5) apply. Under subsection (3), if the court is satisfied that the access to the register of members is not sought for a proper purpose, it will require the company of the obligation not to comply with the request and may require that the person who made the request pay the company’s costs. Under subsection (4), the court may also require the company not to comply with other requests made requests for similar purposes. If the court does not make an order under subsection (3), or the proceedings are discontinued, then, under subsection (5), the company must immediately comply with the request.
Section 118: Register of members: refusal of inspection or default in providing copy
245.This section replaces section 356(5) and (6) of and Schedule 24 to the 1985 Act. It retains the existing sanctions for the company’s failure to comply with a request. They do not apply if the court has directed that the company need not comply with the request.
Section 119: Register of members: offences in connection with request for or disclosure of information.
246.This is a new provision. It creates two offences. First, in relation to the new requirement in section 116 to provide information in a request for access, it is an offence knowingly or recklessly to make a statement that is misleading, false or deceptive in a material particular. Second, it is an offence for a person to disclose to another person information from a company’s register of members obtained under section 116 knowing or having reason to suspect that the other person may use the information for a purpose that is not a proper purpose.
Section 120: Information as to state of register and index
247.This is a new provision. It implements the CLR recommendation that companies be required to advise anyone exercising their right of inspection or right to demand a copy of the register or index whether the information is up-to-date and, if not, the date to which it has been made up (Final Report, paragraph 11.43). Failure to provide this information renders the company and any officer in default liable to a fine.
Section 121: Removal of entries relating to former members
248.This section replaces section 352(6) of the 1985 Act. Based on a recommendation by the CLR (Final Report, paragraph 11.40), it reduces the period for which the entry of a past member must be kept on the register from 20 years to 10 years.
Section 122: Share warrants
249.This section replaces section 355 of the 1985 Act and implements the CLR recommendation (Completing the Structure, paragraph 5.41) in making clear that shares need not first be issued in registered form, but can be issued directly in warrant to bearer form.
Section 123: Single member companies
250.This section replaces section 352A of the 1985 Act, which implements the Twelfth Company Law Directive (89/667/EEC) on single member private limited liability companies. This section requires a statement to be entered in a company’s register of members that it has only one member if that is the case on incorporation or at a later date – if the latter, the date on which it so became must also be entered. It also requires a statement that the company has ceased to have only one member together with the date of the increase. Section 352A of the 1985 Act applies to private companies alone, but this section applies to both private and public companies.
Section 124: Company holding its own shares as treasury shares
251.This section replaces section 352(3A) as regards the entries required to be made in the register of members where a company holds treasury shares. The effect of that provision is unchanged.
Section 125: Power of court to rectify register
Section 126: Trusts not to be entered on register
Section 127: Register to be evidence
252.These sections restate sections 359, 360 and 361 of the 1985 Act.
Section 128: Time limit for claims arising from entry in register
253.This section replaces section 352(7) of the 1985 Act. Based on a recommendation by the CLR (Final Report, paragraph 11.40), it reduces the time limit for claims relating to entries in the register from 20 years to 10 years.
Chapter 3: Overseas Branch Registers
254.This Chapter restates the 1985 Act’s provisions (section 362 and Parts I and II of Schedule 14) regarding overseas branch registers. It enables companies in specified circumstances to keep in a specified country or territory a register of those members resident in that country or territory. An overseas branch register is deemed to be part of the company’s register of members. It differs from the provisions of the 1985 Act in two respects.
First, rather than providing for an Order in Council, section 129(3) provides the Secretary of State with power to make regulations as to the circumstances in which a company is to be regarded as keeping an overseas branch register.
Second, section 131 provides power to modify the provisions of Chapter 2 of this Part (relating to the company’s register of members) in their application to overseas branch registers.
Chapter 4: Prohibition on Subsidiary Being Member of Its Holding Company
255.This Chapter is a restatement of the provisions of section 23 of the 1985 Act and Schedule 2 to that Act as it applies for the purposes of that section. Unless in circumstances covered by an exception, a company cannot be a member of its own holding company. There is no change of substance to the provision in the 1985 Act.
Part 9: Exercise of Members’ Rights
256.The CLR considered the rights of persons other than the registered shareholders, presenting their recommendations in Chapter 7 of the ‘Final Report’. The new provisions in Part 9 have been developed with these recommendations in mind and are designed to make it easier for investors to exercise their governance rights fully and responsibly. Nowadays when investors, whether major institutional investors or retail investors, buy shares in a listed company they are increasingly likely to hold their shares through an intermediary or a chain of intermediaries. This means that it is an intermediary’s name that appears on the company’s register of members. As a result investors typically have to rely on contractual arrangements with the intermediaries both to obtain information from the company and also to give any instructions they wish about how shares should be voted.
257.This Part of the Act introduces new provisions dealing with the ability of indirect investors to exercise governance rights. The first section in the Part removes any doubts as to the ability of companies to make provision in their articles for others to enjoy and exercise membership rights and enables indirect investors to enjoy information rights via the registered member. The next group of sections provides that indirect investors in traded companies can be nominated by the registered member to receive company documents and information. It is up to the registered member, typically a broker, to decide whether to nominate or not. The last two sections in the Part make it easier for registered members to exercise rights in different ways to reflect the underlying holdings and allow indirect investors to participate in, for example, requests for resolutions at the AGM. The provisions of this Part should be looked at together with sections 324 to 331 on proxies in Part 13, which enable the registered member to appoint indirect investors as proxies to exercise voting rights.
258.An important principle to note is that the information rights and exercise of other rights where shares are held on behalf of others can be initiated and enforced with the company only by the registered member. This Part does not compel the registered member to confer such rights on third parties. It will be for indirect investors, such as shareholders holding through a nominee, to choose a nominee operator who offers such rights as part of their service.
Effect of provisions in company’s articles
259.Section 145 protects arrangements through company articles to enfranchise indirect investors and provides that where a company makes provision, through its articles, to extend rights to those holding shares through intermediaries, the provision is legally effective in relation to various statutory requirements.
Section 145: Effect of provisions of articles as to enjoyment of exercise of members’ rights
260.This section (subsection (1)) allows a company’s articles to enable a member to identify another person or persons as entitled to enjoy or exercise all or any specified rights of a member. The articles may specify that this entitlement can apply only to certain rights or to all rights, except the right to transfer the shares. As subsection (4)(b) makes clear, the right to transfer shares must remain, as under the 1985 Act, with the member whose name is on the register.
261.Subsection (2) provides that where a company makes relevant provision in its articles, all the relevant references in the Companies Acts to ‘member’ should be read as if the reference to member was a reference to the person or persons nominated by the member. Subsection (3) provides a non-exhaustive list of the provisions in question.
262.Subsection (4)(a) makes clear that non-members do not have direct enforceable rights against the company. They should enforce their rights through the member whose name is on the register and who has the right to enforce the articles.
Information rights
263.Sections 146 to 150 introduce new provisions enabling indirect investors to be appointed by the registered member to receive information that is sent to members by the company. These provisions apply only to companies traded on a regulated market. The Secretary of State may extend or limit the classes of companies to which these provisions apply through the power provided under section 151 .
Section 146: Traded companies: nomination of persons to enjoy information rights
264.Subsections (1) and (2) provide new rights for members of companies whose shares are traded on regulated markets to nominate those on whose behalf they hold shares to receive information that is sent to members direct from the company and to exercise certain rights .
265.Subsection (3) sets out what is meant by ‘information rights’, namely the right to receive all communications that the company sends to members, the right to require copies of accounts and reports (as in section 431 or 432), and the right to require hard copy versions of documents (section 1145).
266.Subsection (4) refers to section 426, which allows under certain circumstances for summary financial statements rather than full accounts to be sent out as part of the general information. These must also be sent to nominated persons.
267.Subsection (5) provides that the company does not need to respond to a nomination that specifies only certain information rights.
Section 147: Information rights: form in which copies to be provided
268.This section deals with the way in which information is to be provided to a nominated person. Subsection (2) explains that if a nominated person wants communications to be in hard copy, they must ask the member, providing a postal address, before the nomination is made. If the member then passes this on to the company, under subsection (3) the nominated person will have the right to receive hard copy communications.
Section 148: Termination or suspension of nomination
269.This section provides that a nomination will stop having effect on the request of the nominated person or the member (subsection (2)), or on the death, bankruptcy or winding up of the nominated person or the member (subsection (3)).
270.Subsection (5) explains that all nominations made by member will be suspended if there are more nominations that the total number of shares, and subsection (6) makes similar provision where there are different classes of shares.
Section 149: Information as to possible rights in relation to voting
271.This section requires the company, when sending a meeting notice to nominated persons, to include a statement that the nominated person may have voting rights that he can exercise through the person who nominated him.
Section 150: Information rights: status of rights
272.This section deals with the rights arising from a nomination under section 146, and in particular provides that it is the member, rather than the nominated person, who can enforce the rights against the company.
Section 151: Information rights: power to amend
273.This section gives a power for the Secretary of State to amend the provisions of sections 146 to 150. The power allows for changes in the companies covered, the circumstances in which nominations can be made and the rights conferred by nomination.
Effect of rights where shares held on behalf of others
274.Sections 152 and 153 enable indirect investors, via the registered member, to exercise voting and requisition rights by making it easier for registered members to exercise rights in different ways to reflect underlying holdings and by allowing those on whose behalf they hold shares to participate in requisitions. These two sections apply to all companies.
Section 152: Exercise of rights where shares held on behalf of others: exercise in different ways
275.This section provides that a member can choose to split his holding and exercise rights attached to shares in different ways. This is to accommodate members who hold shares on behalf of more than one person, each of whom may want to exercise rights attaching to their shares in different ways. So, for example, it enables votes to be cast in different ways. Subsection (4) provides that if the member does not make it clear to the company in what way he is exercising his rights the company can assume that all rights are being dealt with in the same way.
Section 153: Exercise of rights where shares held on behalf of others: members’ requests
276.This section deals with four situations where the shareholder threshold required to trigger a right is 100 shareholders holding £100 each on average of paid-up capital. Indirect investors are able to count towards the total subject to certain conditions, intended to ensure that only genuine indirect investors are allowed to count towards the total, that the same shares cannot be used twice and that the indirect investor’s contractual arrangements with the member allow the former to give voting instructions.
Part 10: Company Directors
277.This Part replaces Part 10 of the 1985 Act (enforcement of fair dealing by directors), the provisions relating to directors in Part 9 of that Act and the provisions relating to confidentiality orders in Part 25 of that Act. It also introduces a statutory statement of directors’ general duties to the company.
Who is a director?
278.Section 250 defines a director as including any person occupying the position of director, by whatever name called. This is the same as the definition contained in section 741(1) of the 1985 Act. The Act does not attempt a more detailed definition of a director because it is important to ensure that the term is applied to anybody who exercises real power within the company, particularly in relation to decision taking. The term “director” therefore includes:
an executive director who has been properly appointed by the company;
a non-executive director who has been properly appointed by the company;
a de facto director (that is, a person who has assumed the status and functions of a company director even though he has not been properly appointed).
279.A “shadow director” is defined by section 251 as “a person in accordance with whose directions or instructions the directors of the company are accustomed to act”. The section provides that a person is not to be regarded as a shadow director by reason only that the directors act on advice given by him in a professional capacity. This definition is the same as the definition contained in section 741(2) of the 1985 Act.
Powers of directors
280.This Part of the Act does not generally directly give powers to the directors, but, under the draft model articles of association for private companies limited by shares, the directors’ functions are:
to manage the company’s business; and
to exercise all the powers of the company.
Chapter 1: Appointment and Removal of Directors
Section 154: Companies required to have directors
281.This section replaces section 282 of the 1985 Act. It distinguishes between private and public companies. It retains the requirement for a private company to have at least one director and requires all public companies to have at least two. There will no longer be an exception for public companies registered before 1st November 1929 (or before 1st January 1933 in Northern Ireland).
Section 155: Companies required to have at least one director who is a natural person
282.This section is a new provision. It introduces a requirement that every company have at least one director who is a natural person, ie an individual. Subject to this requirement being satisfied, any legal person, including one that is a company or a firm, can be a director but one company cannot be the sole director of another company. Subsection (2) provides that the requirement that the director be a natural person is met if the director is a corporation sole (for example, the Archbishop of Canterbury) or someone appointed on the basis of some other appointment that they hold.
Section 156: Direction requiring company to make appointment
283.This section is a new provision, enabling enforcement of the existing requirement for a private company to have at least one director and a public company to have at least two directors and of the new requirement for every company to have at least one director who is an individual. Where it appears to the Secretary of State that any of these requirements is not met, the Secretary of State will be able to direct the company to comply by issuing a notice. It will be an offence not to comply.
Section 157: Minimum age for appointment as director
284.This section is a new provision. It introduces a minimum age of 16 for a natural person to be a director. Subsection (2) provides that prohibition will not prevent the appointment of a younger person provided it is not to take effect until that person is 16. Subsection (3) provides that the age limit applies even if the director’s appointment is a consequence of some other appointment. Subsection (5) provides that this prohibition on under-age directors does not provide protection from criminal prosecution or civil liability if he or she were to act as director, i.e. as a de facto director, or if the company’s directors usually act on that young person’s instructions.
Section 158: Power to provide for exceptions from minimum age requirement
285.This section is a new provision. It provides for an exception from the prohibition in section 157 on anyone under 16 being appointed a director of a company. It provides a power for the Secretary of State to make regulations specifying circumstances in which a younger person may be a director. The regulations may differ for different parts of the UK.
Section 159: Existing under-age directors
286.This section is a new transitional provision. Subsections (1) and (2) provide that where a person under 16 has been appointed as director (or holds the office of director by virtue of another office or is a corporation sole) prior to the prohibition on under age directors coming into force, that person will cease to be a director when the prohibition in section 157 comes into force. Subsection (3) makes it the company’s responsibility to amend its register of directors accordingly but the company is not required to notify the registrar of the change. Subsection (4) gives the registrar power to amend the register without a notification by the company of the director’s removal but rather on the basis of information already held (i.e. the date of birth as provided when the appointment was notified).
Section 160: Appointment of directors of public company to be voted on individually.
287.This restates section 292 of the 1985 Act: the appointment of each proposed director of a public company must be voted on individually unless there is unanimous agreement to a block resolution. Without such consent, any appointment of a director that is not voted on individually is void. This ensures that members can express their disapproval of any particular director without having to reject the entire board.
Section 161: Validity of acts of directors
288.This section, which replaces section 285 of the 1985 Act, provides that a director’s actions are valid even if his or her appointment is subsequently found to have been defective or void.
Section 162: Register of directors
289.This section replaces part of section 288 of the 1985 Act. It imposes on every company a requirement to keep a register of its directors (secretaries are dealt with in Part 12). This register need not contain particulars of shadow directors.
290.This section requires the register to be kept available for inspection either at the company’s registered office or at a place specified in regulations made under section 1136. It must be available for inspection by members (without charge) or the public (for a prescribed fee, set under powers provided under section 1137). Refusal to permit inspection is an offence for which every officer in default (including a shadow director) can be liable. In addition, the court may compel immediate inspection of the register if the company has refused.
Section 163: Particulars of directors to be registered: individuals
291.This section replaces section 289 of the 1985 Act so far as it applies to individuals. It specifies the particulars that must be entered in the register of directors for each director who is an individual (as opposed to a company or similar entity). The most significant change is the requirement for companies to provide a service address for a director rather than, as now, the director’s usual residential address. A director may give the company’s registered office as his or her service address; the service address may also be the same as the director’s residential address – but this will not be apparent from the public record. In addition, in fulfilment of a Government commitment given in March 1998, the particulars no longer include details of other directorships held. There are also changes to the requirement to provide the director’s name. The requirement is now to include any name by which the individual was formerly known for business purposes. As recommended by the CLR (Final Report, paragraph 11.38), there is no longer an exception for a married woman’s former name. However the section retains a protective provision relating to the former names of peers.
Section 164: Particulars of directors to be registered: corporate directors and firms
292.This section replaces section 289(1)(b) of the 1985 Act. It retains the requirement for the corporate or firm name and the registered or principal office to be recorded where the director is either a body corporate or a firm that is a legal person under the law by which it is governed. In addition, as recommended by the CLR (Final Report, paragraph 11.38), it requires for EEA companies the register where the company is registered and its registration number; for all others, particulars of the legal form of the company or firm, the law by which it is governed, and, if applicable, where it is registered and its registration number.
Section 165: Register of directors’ residential addresses
293.This section is a new provision. It requires companies to keep a register of the usual residential addresses of directors who are individuals. Provided that a director’s service address is not the company’s registered office, if his/her residential address is the same as his/her service address, then the register need only contain an entry making that clear. This register is not to be open to public inspection, but can be used in accordance with Chapter 8 of this Part.
Section 166: Particulars of directors to be registered: power to make regulations
294.This section is a new provision. It provides power for the Secretary of State to make regulations that add or remove items from the particulars that have to be entered in a company’s register of directors and register of directors’ residential addresses.
Section 167: Duty to notify registrar of changes
295.This section replaces section 288(2) of the 1985 Act insofar as it applies to directors. It retains the requirement that the appointment of a director, or a director’s ceasing to hold office, and any change in an existing director’s particulars, be notified to the registrar within 14 days. This requirement does not extend to shadow directors. Default is an offence. This section also requires a notice of appointment to be accompanied by the appointee’s consent. This provision ensures that the public record is kept up to date. There is also a requirement to notify the registrar of changes to information in the register of directors’ residential addresses (but this information is not to be open to public inspection at Companies House).
Section 168: Resolution to remove director
296.This section replaces section 303 of the 1985 Act. Subsection (1) provides that an ordinary resolution is sufficient to remove a director, but requires that it be at a meeting so as to ensure the director’s right to be heard.
Section 169: Director’s right to protest removal
297.This section replaces section 304 of the 1985 Act. The only change is to be found in subsection (5); the court need no longer be satisfied that the rights conferred by the section are being abused to secure needless publicity for defamatory matter, so long as it is satisfied that they are being abused
Chapter 2: General Duties of Directors
Sections 170 to 181: General comments
298.The general duties form a code of conduct, which sets out how directors are expected to behave; it does not tell them in terms what to do. More particularly, the duties address:
the possibility that a director may put his own or other interests ahead of those of the company;
the possibility that he may be negligent.
299.The duties are derived from equitable and common law rules, and are not at the moment written down in statute.
300.The Law Commission and the Scottish Law Commission recommended that there should be a statutory statement of a director’s main fiduciary duties and his duty of care and skill in their joint report Company Directors: Regulating Conflicts of Interests and Formulating a Statement of Duties. The CLR’s main recommendations in respect of directors’ general duties are summarised in chapter 3 of the Final Report.
301.The CLR recommended that there should be a statutory statement of directors’ general duties, and that this should, with two exceptions, described in the next paragraph, be a codification of the current law. In particular they wanted:
to provide greater clarity on what is expected of directors and make the law more accessible. In particular, they sought to address the key question “in whose interests should companies be run?” in a way which reflects modern business needs and wider expectations of responsible business behaviour;
to make development of the law in this area more predictable (but without hindering development of the law by the courts);
to correct what the CLR saw as defects in the present duties relating to conflicts of interest.
The Government has accepted these recommendations.
302.There are two areas, both relating to the regulation of conflicts of interest, where the statutory statement departs from the current law:
under section 175, transactions or arrangements with the company do not have to be authorised by either the members or by the board; instead interests in transactions or arrangements with the company must be declared under section 177 (in the case of proposed transactions) or under section 182 (in the case of existing transactions) unless an exception applies under those sections;
section 175 also permits board authorisation of most conflicts of interest arising from third party dealings by the director (e.g. personal exploitation of corporate resources and opportunities). Such authorisation is effective only if the conflicted directors have not participated in the taking of the decision or if the decision would have been valid even without the participation of the conflicted directors. Board authorisation of conflicts of interest will be the default position for private companies, but public companies will need to make provision in their constitutions to permit this. Board authorisation is not permitted in respect of the acceptance of benefits from third parties (section 176).
303.Both reforms implement recommendations of the CLR, which noted that the basic principles in the current law relating to directors’ conflicts of interest are very strict:
they noted that in practice most companies permit a director to have an interest in a proposed transaction or arrangement with the company, provided that the interest is disclosed to his fellow directors. The statutory statement therefore reflects the current position in most companies;
they also took the view that the current strict rule relating to conflicts of interest in respect of personal exploitation of corporate opportunities fettered entrepreneurial and business start-up activity by existing company directors. The statutory statement therefore provides for board authorisation of such conflicts.
304.These reforms are modified for charitable companies in England and Wales and Northern Ireland by section 181.
Codification of common law rules and equitable principles
305.Codification is not a matter of transposing wording taken from judgments into legislative propositions. Judgments are, of necessity, directed at particular cases. Even when they appear to state general principles, they will rarely be exhaustive. They will be the application of (perhaps unstated) general principles to particular facts. In the company law field, the principles being applied will frequently be taken from other areas, in particular trusts and agency. It is important that these connections are not lost and that company law may continue to reflect developments elsewhere. Frequently the courts may formulate the same idea in different ways. In contrast legislation is formal. It is not easy to reconcile these two approaches but the draft sections seek to balance precision against the need for continued flexibility and development. In particular:
subsection (3) of section 170 provides that the statutory duties are based on, and have effect in place of, certain common law rules and equitable principles;
subsection (4) of section 170 provides that the general duties should be interpreted and applied in the same way as common law rules and equitable principles. The courts should interpret and develop the general duties in a way that reflects the nature of the rules and principles they replace;
subsection (4) of section 170 also provides when interpreting and applying the statutory duties, regard should be had to the common law rules and equitable principles which the general duties replace; thus developments in the law of trusts and agency should be reflected in the interpretation and application of the duties;
section 178 provides that the civil consequences of breach (or threatened breach) of the statutory duties are the same as would apply if the corresponding common law rule or equitable principle applied. It also makes clear that the statutory duties are to be regarded as fiduciary, with the exception of the duty to exercise reasonable care skill and diligence which is not under the present law regarded as a fiduciary duty.
306.The statutory duties do not cover all the duties that a director may owe to the company. Many duties are imposed elsewhere in legislation, such as the duty to file accounts and reports with the registrar of companies (section 441). Other duties remain uncodified, such as any duty to consider the interests of creditors in times of threatened insolvency.
Duties owed to the company
307.Section 170(1) makes it clear that, as in the existing law, the general duties are owed by a director to the company. It follows that, as now, only the company can enforce them. Part 11 (derivative claims and actions by members) describes the mechanism whereby members may be able to enforce the duties on behalf of the company.
Who are the duties owed by?
308.The duties are owed by every person who is a director of a company (as defined in section 250). They are therefore owed by a de facto director in the same way and to the same extent that they are owed by a properly appointed director.
309.Certain aspects of the duty to avoid conflicts of interest and the duty not to accept benefits from third parties continue to apply even when a person ceases to be a director; this is necessary to ensure that a director cannot, for example, exploit an opportunity of which he became aware while managing the company’s business without the necessary consent simply by resigning his position as director. The closing words of section 170(2) provide that these duties apply to a former director subject to any necessary adaptations. This is to reflect the fact that a former director is not in the same legal position as an actual director.
310.The statutory duties apply to shadow directors where, and to the extent that, the common law rules or equitable principles which they replace so apply (section 170(5)). This means that where a common law rule or equitable principle applies to a shadow director, the statutory duty replacing that common law rule or equitable principle will apply to the shadow director (in place of that rule or principle). Where the rule or principle does not apply to a shadow director, the statutory duty replacing that rule or principle will not apply either.
The relationship between the duties
311.Many of the general duties will frequently overlap. Taking a bribe from a third party would, for example, clearly fall within the duty not to accept benefits from third parties (section 176) but could also, depending on the facts, be characterised as a failure to promote the success of the company for the benefit of its members (section 172) or as an aspect of failing to exercise independent judgment (section 173).
312.The effect of the duties is cumulative, so that it is necessary to comply with every duty that applies in any given case. This principle is stated in section 179. One exception relates to the duty to avoid conflicts of interest (section 175). This particular duty does not apply to a conflict of interest arising in relation to a transaction or arrangement with the company. In such cases the duty to declare interests in proposed transactions or arrangements (section 177) or the requirement to declare interests in existing transactions or arrangements (section 182) will apply instead. Section 181 modifies these provisions for charitable companies in England and Wales and Northern Ireland.
313.The cumulative effect of the duties means that where more than one duty applies, the director must comply with each applicable duty, and the duties must be read in this context. So, for example, the duty to promote the success of the company will not authorise the director to breach his duty to act within his powers, even if he considers that it would be most likely to promote the success of the company.
314.As well as complying with all the duties, the directors must continue to comply with all other applicable laws. The duties do not require or authorise a director to breach any other prohibition or requirement imposed on him by law.
Relationship between the duties and the company’s constitution
315.Under section 171 a director must act in accordance with the company’s constitution.
316.Companies may, through their articles, go further than the statutory duties by placing more onerous requirements on their directors (e.g. by requiring shareholder authorisation of the remuneration of the directors). The articles may not dilute the duties except to the extent that this is permitted by the following sections:
section 173 provides that a director will not be in breach of the duty to exercise independent judgment if he has acted in a way that is authorised by the constitution;
section 175 permits authorisation of some conflicts of interest by independent directors, subject to the constitution;
subsection (4)(a) of section 180 preserves any rule of law enabling the company to give authority for anything that would otherwise be a breach of duty;
subsection (4)(b) of section 180 provides that a director will not be in breach of duty if he acts in accordance with any provisions in the company’s articles for dealing with conflicts of interest;
section 232 places restrictions on the provisions that may be included in the company’s articles. But nothing in that section prevents companies from including in their articles any such provisions as are currently lawful for dealing with conflicts of interest.
317.The company’s constitution may also set out the purposes of the company, especially in the case of an altruistic company which has purposes other than the benefit of the company’s members. It is very important that directors understand the purposes of the company, so that they are able to comply with their duty to promote the success of the company in section 172.
Relationship between the duties and the detailed rules requiring member approval of conflicts of interest
318.Under the provisions in Chapter 4 of this Part, the directors must sometimes obtain prior shareholder approval for the following types of transaction involving a director (or, in some cases, a person connected to a director): long-term service contracts; substantial property transactions; loans, quasi-loans and credit transactions; and payments for loss of office.
319.Section 180 provides that:
compliance with the general duties does not remove the need for member approval of such transactions (subsection (3));
(subject to the exception set out in the bullet point below) the general duties apply even if the transaction also falls within Chapter 4 (because it is a long-term service contract, substantial property transaction, loan, quasi-loan, credit transaction or payment for loss of office). So, for example, the directors should only approve a loan to a director if they consider that it would promote the success of the company. This is so, even if the loan does not require the approval of members under Chapter 4 because it falls within a relevant exception, such as the exception for expenditure on company business in section 204;
if the transaction falls within Chapter 4 (because it is a long-term service contract, substantial property transaction, loan, quasi-loan, credit transaction or payment for loss of office) and approval of the members is obtained to the transaction in accordance with that Chapter, or an exception applies, so that approval is not necessary under that Chapter, then the director does not need to comply with the duty to avoid conflicts of interest (section 175) or the duty not to accept benefits from third parties (section 176) in respect of that transaction. All other applicable duties will still apply. For example, a director would not be acting in breach of the duty to avoid conflicts of interests if he failed to obtain authorisation from the directors or the members for a loan from the company in respect of legal defence costs. Section 181 modifies this provision for charitable companies in England and Wales and Northern Ireland.
Relationship between the duties and the general law
320.Section 180(5) provides that the general duties have effect notwithstanding any enactment or rule of law except where there is an express or implied exception to this rule. For example, section 247 provides that directors may make provision for employees on the cessation or transfer of a company’s business even if this would otherwise constitute a breach of the general duty to promote the success of the company.
Consequences of breach
321.Section 178 preserves the existing civil consequences of breach (or threatened breach) of any of the general duties. The remedies for breach of the general duties will be exactly the same as those that are currently available following a breach of the equitable principles and common law rules that the general duties replace.
322.Subsection (2) of that section makes it clear that the duties are enforceable in the same way as any other fiduciary duty owed to a company by its directors (except for the duty to exercise reasonable care, skill and diligence, which is not considered to be a fiduciary duty). In the case of fiduciary duties the consequences of breach may include:
damages or compensation where the company has suffered loss;
restoration of the company’s property;
an account of profits made by the director; and
rescission of a contract where the director failed to disclose an interest.
Commentary on Individual Duties
Section 171: Duty to act within powers
323.This duty codifies the current principle of law under which a director should exercise his powers in accordance with the terms on which they were granted, and do so for a proper purpose. What constitutes a proper purpose must be ascertained in the context of the specific situation under consideration.
324.This duty codifies the director’s duty to comply with the company’s constitution. The constitution is defined for the purpose of the general duties in section 257. As well as the company’s articles of association it includes:
decisions taken in accordance with the company’s articles; and
other decisions taken by the members (or a class of them) if they are to be treated by virtue of any enactment or rule of law as decisions of the company, for example a decision taken by informal unanimous consent of all the members.
Section 172: Duty to promote the success of the company
325.This duty codifies the current law and enshrines in statute what is commonly referred to as the principle of “enlightened shareholder value”. The duty requires a director to act in the way he or she considers, in good faith, would be most likely to promote the success of the company for the benefit of its members as a whole and, in doing so, have regard to the factors listed.
326.This list is not exhaustive, but highlights areas of particular importance which reflect wider expectations of responsible business behaviour, such as the interests of the company’s employees and the impact of the company’s operations on the community and the environment.
327.The decision as to what will promote the success of the company, and what constitutes such success, is one for the director’s good faith judgment. This ensures that business decisions on, for example, strategy and tactics are for the directors, and not subject to decision by the courts, subject to good faith.
328.In having regard to the factors listed, the duty to exercise reasonable care, skill and diligence (section 174) will apply. It will not be sufficient to pay lip service to the factors, and, in many cases the directors will need to take action to comply with this aspect of the duty. At the same time, the duty does not require a director to do more than good faith and the duty to exercise reasonable care, skill and diligence would require, nor would it be possible for a director acting in good faith to be held liable for a process failure which would not have affected his decision as to which course of action would best promote the success of the company.
329.In requiring directors to have regard to the interests of employees, this provision replaces section 309(1) of the 1985 Act.
330.Subsection (2) addresses the question of altruistic, or partly altruistic, companies. Examples of such companies include charitable companies and community interest companies, but it is possible for any company to have “unselfish” objectives which prevail over the “selfish” interests of members. Where the purpose of the company is something other than the benefit of its members, the directors must act in the way they consider, in good faith, would be most likely to achieve that purpose. It is a matter for the good faith judgment of the director as to what those purposes are, and, where the company is partially for the benefit of its members and partly for other purposes, the extent to which those other purposes apply in place of the benefit of the members.
331.Subsection (3) recognises that the duty to promote the success of the company is displaced when the company is insolvent. Section 214 of the Insolvency Act 1986 provides a mechanism under which the liquidator can require the directors to contribute towards the funds available to creditors in an insolvent winding up, where they ought to have recognised that the company had no reasonable prospect of avoiding insolvent liquidation and then failed to take all reasonable steps to minimise the loss to creditors.
332.It has been suggested that the duty to promote the success of the company may also be modified by an obligation to have regard to the interests of creditors as the company nears insolvency. Subsection (3) will leave the law to develop in this area.
Section 173: Duty to exercise independent judgment
333.This duty codifies the current principle of law under which directors must exercise their powers independently, without subordinating their powers to the will of others, whether by delegation or otherwise (unless authorised by or under the constitution to do so).
334.The section provides that directors must not fetter the future exercise of their discretion unless they are acting:
a)
in accordance with an agreement which has been duly entered into by the company; or
b)
in a way authorised by the company’s constitution.
335.The duty does not confer a power on the directors to delegate, nor does it prevent a director from exercising a power to delegate conferred by the company’s constitution provided that its exercise is in accordance with the company’s constitution. Under the draft model articles of association for private companies limited by shares, the directors may delegate their functions in accordance with the articles.
Section 174: Duty to exercise reasonable care, skill and diligence
336.This duty codifies the director’s duty to exercise reasonable, care, skill and diligence. Traditionally, the courts did not require directors to exhibit a greater degree of skill than may reasonably be expected from a person with their knowledge and experience (a subjective test). More recently, the courts have said that the common law standard now mirrors the tests laid down in section 214 of the Insolvency Act 1986, which includes an objective assessment of a director’s conduct. This section is modelled on that section.
337.The section provides that a director owes a duty to his company to exercise the same standard of care, skill and diligence that would be exercised by a reasonably diligent person with:
a)
the general knowledge, skill and experience that may reasonably be expected of a person carrying out the same functions as the director in relation to that company (an objective test); and
b)
the general knowledge, skill and experience that the director actually has (a subjective test).
Section 175: Duty to avoid conflicts of interest
338.This duty replaces the no-conflict rule applying to directors. Under the current no-conflict rule, certain consequences flow if directors place themselves in a position where their personal interests or duties to other persons are liable to conflict with their duties to the company, unless the company gives its consent. A conflict of interest may, in particular, arise when a director makes personal use of information, property or opportunities belonging to the company or when a director enters into a contract with his company. Conflicts of interest may also arise whenever a director makes a profit in the course of being a director, in the matter of his directorship, without the knowledge and consent of his company.
339.This duty covers all conflicts, actual and potential, between the interests of the director and the interests of the company. This includes conflicts relating to the exploitation of the company’s property, information or opportunity for personal purposes. The only conflicts not covered by this duty, are those relating to transactions or arrangements with the company (interests in transactions or arrangements with the company must be declared under section 177 in the case of proposed transactions or under section 182 in the case of existing transactions unless an exception applies under those sections).
340.Section 180(4) preserves any current ability of the members of a company to authorise conflicts that would otherwise be a breach of this duty.
341.Under subsections (4) to (6) the duty is also not infringed if:
the situation cannot reasonably be regarded as likely to give rise to a conflict of interest;
in the case of a private company, unless its constitution prevents this, authorisation has been given by directors who are genuinely independent (in the sense that they have no direct or indirect interest in the transaction);
similarly, in the case of a public company, but only if its constitution expressly permits this, authorisation has been given by the independent directors.
342.The present law is that in all cases, conflicts of interest must be authorised by the members of the company, unless some alternative procedure is properly provided. The CLR were concerned that this strict requirement might stifle entrepreneurial activity; and therefore recommended that, in the case of a private company, it should be possible for conflicts to be authorised by independent directors unless the company’s constitution prevents this.
343.Under subsection (6), board authorisation is effective only if the conflicted directors have not participated in the taking of the decision or if the decision would have been valid even without the participation of the conflicted directors: the votes of the conflicted directors in favour of the decision are ignored and the conflicted directors are not counted in the quorum.
Section 176: Duty not to accept benefits from third parties
344.This section codifies the rule prohibiting the exploitation of the position of director for personal benefit. This duty prohibits the acceptance of benefits (including bribes). The acceptance of a benefit giving rise to an actual or potential conflict of interest will fall within the duty to avoid conflicts of interest (section 175) as well as this duty. This specific duty dealing with benefits from third parties is not subject to any provision for board authorisation.
345.Any current ability of the members of a company to authorise the acceptance of benefits which would otherwise be a breach of this duty is preserved by section 180(4).
346.The duty is not infringed if the acceptance of the benefit cannot reasonably be regarded as likely to give rise to a conflict of interest. Benefits conferred by the company (and its holding company or subsidiaries) do not fall within this duty.
Section 177: Duty to declare interest in proposed transaction or arrangement
347.The equitable rule that directors may not have interests in transactions with the company unless the interest has been authorised by the members is replaced by this duty. This section requires a director to disclose any interest, direct or indirect, that he has in relation to a proposed transaction or arrangement with the company. The director does not need to be a party to the transaction for the duty to apply. An interest of another person in a contract with the company may require the director to make a disclosure under this duty, if that other person’s interest amounts to a direct or indirect interest on the part of the director.
348.Under the current equitable rule, shareholder approval is required for transactions between a company and a director. Company articles often modify the equitable rule, requiring disclosure of the conflict instead. As proposed by the CLR, shareholder approval for the transaction is not a requirement of the statutory duty. The members of the company may however still impose requirements for shareholder approval in the articles.
349.The duty requires directors to disclose their interest in any transaction before the company enters into the transaction (subsection (4)). The duty does not impose any rules on how the disclosure of interest must be made, but subsection (2) allows the disclosure to be made by written notice, general notice or disclosure at a meeting of the directors.
350.Disclosure to the members is not sufficient. The director must declare the nature and extent of his interest to the other directors. It is not enough for the director to merely state that he has an interest.
351.If after he has disclosed his interest, he becomes aware that the facts have changed, or for some other reason the earlier disclosure is no longer accurate or complete, the director must make a further declaration, correcting the earlier one (subsection (3)). However, this is only necessary if the company has not yet entered into the transaction or arrangement at the time the director becomes aware of the inaccuracy or incompleteness of the earlier declaration (or ought reasonably to have become so aware).
352.As the duty requires disclosure to be made to the other directors, no disclosure is required where the company has only one director. There is no need to disclose anything the other directors already know about or ought reasonably to have known (subsection (6)(b)). A director will breach the duty if he fails to declare something he ought reasonably to have known, but the duty does not otherwise require a director to declare anything he does not know. Subsection (6)(c) makes special provision for service contracts that are considered by a meeting of the directors or a committee appointed for the purpose (such as a remuneration committee).
353.No declaration of interest is required if the director’s interest in the transaction cannot reasonably be regarded as likely to give rise to a conflict of interest (subsection (6)(a)). Currently regulation 85 of Table A imposes a materiality test.
354.Conflicted directors may, subject to the company’s articles of association, participate in decision-taking relating to such transactions with the company.
Section 181: Modification of provisions in relation to charitable companies
355.This section reverses certain relaxations made to the no-conflict rule as it applies to the directors of charitable companies in England and Wales and Northern Ireland.
356.Subsection (2)(a) replaces section 175(3) which excludes conflicts of interest arising out of transactions or arrangements with the company. The replacement excludes such conflicts of interest from the duty only if or to the extent that the charitable company’s articles so allow. The articles must describe the transactions or arrangements which are to be so excluded from the duty.
357.Subsection (2)(b) replaces section 175(5) which allows authorisation for conflicts of interest to be given by the directors. The replacement only allows authorisation to be given by the directors where the charitable company’s constitution expressly allows them to do so.
358.Subsection (3) restricts the application of section 180(2)(b) which disapplies sections 175 and 176 in relation to those matters excepted from the requirements for member approval under Chapter 4. Section 180(2)(b) is restricted so that it only applies if or to the extent that the charitable company’s articles allow the duties in sections 175 and 176 to be disapplied. The articles must describe the transactions or arrangements which are to be so excluded from those duties.
359.Subsection (4) amends the Charities Act 1993 to give the Charity Commission the power to authorise acts that would otherwise be in breach of the general duties. This is necessary to preserve the current power of the Charity Commissioners to do so, in the light of the statutory statement of the general duties.
Chapter 3: Declaration of Interest in Existing Transaction Or Arrangement
Section 182: Declaration of interest in existing transaction or arrangement
360.This section requires a director to declare the nature and extent of any direct or indirect interest that he has in any transaction or arrangement entered into by the company. It replaces the provision made by section 317 of the 1985 Act.
361.This chapter differs from the provisions of that section in a number of important respects. The main points are summarised below.
What should be declared?
362.Directors are required to declare any interest, direct or indirect, that they have in an existing transaction or arrangement entered into by the company. This section only applies to transactions or arrangements already entered into by the company. Section 177 (duty to declare interests) applies in the case of proposed transactions or arrangements with the company.
363.The director does not need to be a party to the transaction with the company in order for a declaration to be required under this section. For example, where the director’s spouse enters into a transaction with the company that may (but need not necessarily) give rise to an indirect interest on the part of the director in that transaction.
364.The declaration must be of the nature and extent of the director’s direct or indirect interest.
365.If the director has declared his interest in accordance with section 177 at the time the transaction was proposed, and before it was entered into by the company, the director does not need to repeat that declaration once the transaction becomes an existing transaction to which this section applies (subsection (1)).
366.Furthermore, a director need not declare any interest:
that cannot reasonably be regarded as likely to give rise to a conflict of interest;
that the other directors already know about, or ought reasonably to know about; or
that concerns the terms of his service contract, considered (or to be considered) by a meeting of directors or by the relevant committee of directors.
367.A director is regarded as failing to make the declarations required by this section if he fails to declare something that he ought reasonably to have known. But the director is not otherwise expected by this section to declare things he does not know (subsection (5)).
When should the declaration be made?
368.The declaration should be made as soon as is reasonably practicable. But even if the declaration is not made as soon as it should have been, it must still be made (subsection (4)). If after a declaration has been made the director’s interest in the transaction or arrangement changes, or the director realises that his interests were not as originally declared, the director must make another declaration of interest, correcting or updating the earlier one (subsection (3)).
How should the declaration be made?
369.The declaration of interest must be made to the other directors using one of the following three methods:-
at a meeting of the directors; or
by notice in writing (in accordance with the requirements of section 184); or
by general notice (in accordance with the requirements of section 185).
Section 183: Offence of failure to declare interest
370.This section restates section 317(7) of the 1985 Act. A director who fails to comply with the requirements of section 182 commits an offence. On conviction on indictment the maximum liability is an unlimited fine. On summary conviction the fine must not exceed the statutory maximum (currently £5,000). This section does not affect the validity of the transaction or impose any other civil consequences for a failure to make the declarations of interest required by section 182.
Section 184: Declaration made by notice in writing
371.This section provides a new written procedure for the declarations of interest required by section 182. A written notice declaring the nature and extent of the director’s interest must be sent to all the other directors. It may be sent in hard copy form or, if the recipient agrees, in electronic form. It may be posted, delivered by hand or, if the recipient agrees, by electronic means. When this is done, the notice is treated as forming part of the proceedings of the next meeting of the directors, and so should form part of the minutes of that meeting (subsection (5)).
Section 185: General notice treated as sufficient declaration
372.This section replaces section 317(3) and (4) of the 1985 Act. It enables a director to give a general notice of his interests. A general notice is a declaration that the director is interested in another body corporate or firm, or that the director is connected with another person. If the company enters into a contract with the body corporate, firm or other person named in the general notice, the director does not need to declare any direct or indirect interest that he has in that contract arising as a result of his interest in the body corporate or firm named in the general notice or arising as a result of his connection to the person named in the general notice.
373.In order to be effective, the general notice must state the nature and extent of the director’s interest in the body corporate or firm (for example, sole shareholder of the company) or the nature of his connection with the person (for example, spouse or other connected person as defined in section 253). The requirement to disclose the extent of the interest implements a recommendation of the Law Commissions.
Section 186: Declaration of interest in case of company with sole director
374.This is a new provision. Where a company has only one director, it is not possible for the director to declare his interests to the other directors, because there are no other directors. Therefore, a sole director does not need to comply with section 182 (declaration of interest in existing transaction or arrangement).
375.The section makes special provision where the company has only one director, when it should in fact have more than one director (for example, because it is a public company). In such a case, the sole director must record in writing the nature and extent of his interest in any transaction or arrangement that has been entered into by the company.
Section 187: Declaration of interest in existing transaction by shadow director
376.This section replaces section 317(8) of the 1985 Act. It extends this Chapter to shadow directors, so that a shadow director must also declare the nature and extent of his interest in any transaction or arrangement that has been entered into by the company, in accordance with section 182. The declaration must be made by notice in writing (section 184) or by general notice (section 185).
377.The declaration is not made at a meeting of the directors, as this is not appropriate in the case of a shadow director. If the shadow director makes the declaration by general notice, that notice must be given in accordance with the notice in writing procedure set out in section 184. This means that a general notice given by a shadow director must comply with both section 184 and the first three subsections of section 185.
378.Otherwise, apart from section 186 (declaration of interest in case of company with sole director), which is not relevant to a shadow director, all the other provisions of this chapter apply to a shadow director, including the exemptions in section 182.
General Introduction to Chapters 4 and 5
379.These Chapters contains several provisions designed to deal with particular situations in which a director has a conflict of interest. They replace provisions of Part 10 of the 1985 Act, but with a number of changes. The aim of the changes is:
to improve accessibility and consistency. The Law Commissions commented that Part 10 of the 1985 Act “is widely perceived as being extremely detailed, fragmented, excessive, and in some respects, defective, regulation of directors”; and
to implement various recommendations of the Law Commissions and the CLR (see in particular section B of the Law Commissions’ joint report Company Directors: Regulating Conflicts of Interests and Formulating a Statement of Duties, annex C of Developing the Framework, chapter 4 of Completing the Structure and chapter 6 of the Final Report).
380.Provisions regulating directors’ conflicts of interest fall into two main categories:
requirements for disclosure to members;
requirements for member approval.
381.The four types of transaction requiring the approval of members (long-term service contracts; substantial property transactions; loans, quasi-loans and credit transactions; and payments for loss of office) have been brought together within Chapter 4.
382.Provision for disclosure to members in respect of directors’ service contracts is contained in Chapter 5.
383.On the other hand, the requirements in Part 10 of the 1985 Act to disclose, and maintain a register of, share dealings by directors and their families are repealed (see section 1177).
Chapter 4: Transactions With Directors Requiring Approval of Members
Structure
384.This Chapter sets out requirements for member approval in relation to four different types of transaction by a company:
long-term service contracts;
substantial property transactions;
loans, quasi-loans and credit transactions;
payments for loss of office.
385.The rules relating to each type of transaction tend to adopt the following structure: they begin with the rule requiring member approval, followed by exceptions to that rule and finally the consequences of breaching that rule.
Alignment of provisions
386.The provisions of this Chapter have been aligned wherever appropriate so as to achieve greater consistency of approach. Particular examples of alignment are mentioned below.
Criminal penalties
387.This Chapter no longer imposes any criminal penalties for a failure to comply with its requirements.
Civil remedies
388.The civil consequences of a failure to comply with the requirements for member approval of substantial property transactions and loans, quasi-loans and credit transactions have been aligned.
Approval by holding company
389.This Chapter applies to long-term service contracts, substantial property transactions, loans etc and payments for loss of office entered into by a company and involving either a director of the company or a director of the company’s holding company. In the latter case, the transaction must be approved by both the company and the holding company (unless an exception applies).
Transactions between a company and the director of a fellow subsidiary
390.This Chapter does not normally apply to transactions entered into by a company that is neither the company of which the person is a director nor a subsidiary of the company of which the person is a director. The two exceptions are section 218 (payment for loss of office in connection with transfer of undertaking) and section 219 (payment for loss of office in connection with share transfer), where member approval is required for such a payment by any person to a director.
Exception for wholly-owned subsidiary
391.Approval is never required under this Chapter on the part of the member of a wholly-owned subsidiary or on the part of the members of an overseas company.
Shadow directors
392.Section 223 applies all the requirements of this Chapter to shadow directors (with a small modification in the case of payments for loss of office).
Approval required
393.Section 281(3) applies so that the member approval required is an ordinary resolution, but the company’s articles may require a higher majority or even unanimity
394.Where approval for a transaction or arrangement is required under more than one set of rules in Chapter 4, all relevant sets of rules should apply, unless otherwise provided (section 225). For example, if the matter involves both a substantial property transaction and a loan, approval should be required under section 190 and under section 197 unless in each case a relevant exemption applies. Approval may be given for both purposes by a single resolution.
Memorandum with details of the transaction
395.In the case of long-term service contracts, loans etc and payments for loss of office, a memorandum setting out certain particulars about the transaction requiring approval of the members must be made available to the members.
396.If the approval is to be given by way of written resolution, the memorandum must be sent to the members able to vote on the written resolution no later than when the written resolution is sent to them. Section 224 provides that any accidental failure to send the memorandum to one or more members will not invalidate the approval given by the members, unless the company’s articles state otherwise.
Requirement for Charity Commission consent for charitable companies
397.Section 66 of the Charities Act 1993 renders prior authorisation by the members for certain transactions invalid unless the Charity Commissioners have given their prior written consent. This reflects concern that, in some cases, the members of a charitable company are not independent of the directors, and that requiring their approval would not provide sufficient protection for the charity. Section 226 inserts two new sections into the Charities Act 1993 in place of section 66 of that Act to reflect the changes made by this Chapter.
Commentary
Sections 188 and 189: Service contracts
398.These sections replace section 319 of the 1985 Act and require member approval of long-term service contracts. In broad terms, these are contracts under which a director is guaranteed at least two years of employment with the company of which he is a director, or with any subsidiary of that company.
399.A director’s “service contract” is defined in section 227 to include a contract of service, a contract for services and a letter of appointment as director.
400.Failure to obtain approval allows the company to terminate the service contract at any time by giving reasonable notice. The purpose of this section is to limit the duration of directors’ service contracts, as a long-term contract can make it too expensive for the members to remove a director using the procedure in section 168 (ordinary resolution to remove director) while allowing the members to approve longer arrangements if they wish.
401.The length of service contract for which member approval is required has been reduced from those longer than five years to those longer than two years.
Sections 190 to 196: Substantial property transactions
402.These sections require member approval to substantial property transactions. These are transactions where the company buys or sells a non-cash asset (as defined in section 1163) to or from:
a director of the company;
a director of its holding company;
a person connected with a director of the company; or
a person connected with a director of its holding company.
Approval is only required where the value of the asset exceeds £100,000 or 10% of the company’s net assets (based on its last set of annual accounts or called-up share capital if it has not yet produced any accounts). No approval is required if the value of the asset is less than £5,000.
403.These sections replace sections 320 to 322 of the 1985 Act. The changes include:
permitting a company to enter into a contract which is conditional on member approval (section 190(1)). This implements a recommendation of the Law Commissions. In cases where the approval of the members of the holding company is also required, the company may enter into arrangements conditional on approval being obtained from the members of the holding company (section 190(2)). The company is not to be liable under the contract if member approval is not forthcoming (section 190(3));
providing for the aggregation of non-cash assets forming part of an arrangement or series of arrangements for the purpose of determining whether the financial thresholds have been exceeded so that member approval is required (section 190(5));
excluding payments under directors’ service contracts and payments for loss of office from the requirements of these sections (section 190(6)). This implements a recommendation of the Law Commissions;
raising the minimum value of what may be regarded as a substantial non-cash asset from £2,000 to £5,000 (section 191);
expanding the exception for transactions with members to include the acquisition of assets from a person in his character as a member of the company (section 192(a));
providing an exception for transactions made by companies in administration (section 193). This implements a recommendation of the Law Commissions;
not requiring approval on the part of the members of a company that is in administration or is being wound up (unless it is a members’ voluntary winding up) (section 193).
Sections 197 to 214: Loans, quasi-loans and credit transactions
404.In the case of a private company which is not associated with a public company, section 197 requires member approval for loans and related guarantees or security made by a company for:
a director of the company; or
a director of its holding company.
405.In the case of a public company, or a private company associated with a public company, sections 197, 198, 200 and 201 require member approval for loans, quasi-loans (as defined in section 199), credit transactions (as defined in section 202) and related guarantees or security made by the company for:
a director of the company; or
a director of its holding company;
a person connected with a director of the company; or
a person connected with a director of its holding company.
406.Section 256 explains what is meant by references to associated companies. A holding company is associated with all its subsidiaries, and a subsidiary is associated with its holding company and all the other subsidiaries of its holding company.
407.Member approval is not required by these sections for:
loans, quasi-loans, credit transactions and related guarantees or security to meet expenditure on company business. The total value of transactions under this exception made in respect of a director and any person connected to him must not exceed £50,000 (section 204);
money lent to fund a director’s defence costs for legal proceedings in connection with any alleged negligence, default, breach of duty or breach of trust by him in relation to the company or an associated company (section 205) or in connection with regulatory action or investigation under the same circumstances(section 206);
small loans and quasi-loans, as long as the total value of such loans and quasi-loans made in respect of a director and any person connected to him does not exceed £10,000 (section 207(1));
small credit transactions, as long as the total value of such credit transactions made in respect of a director and any person connected to him does not exceed £15,000 (section 207(2));
credit transactions made in the ordinary course of the company’s business (section 207(3));
intra-group transactions (section 208); and
loans and quasi-loans made by a money-lending company in the ordinary course of the company’s business (as long as the requirements of section 209 are met).
408.These sections replace sections 330 to 341 of the 1985 Act. The changes include:
abolishing the prohibition on loans, quasi-loans etc to directors and replacing it with a requirement for member approval. This implements a recommendation of the CLR;
abolishing the criminal penalty for breach;
replacing the concept of relevant company in section 331 of the 1985 Act with associated company, as defined in section 256;
removing some of the requirements currently imposed by section 337 of the 1985 Act on the exception for expenditure on company business (section 204);
widening the exception for expenditure on company business to include directors of the company’s holding company and connected persons (section 204);
creating a new exception specifically for expenditure in connection with regulatory action or investigations (section 206);
restricting the exceptions for expenditure on defending legal or regulatory proceedings to proceedings in connection with any alleged negligence, default, breach of duty or breach of trust by the director in relation to the company or an associated company (sections 205 and 206);
widening the exception for small loans to include small quasi-loans (section 207(1)) in place of the current exception for short-term small quasi-loans in section 332 of the 1985 Act;
widening the exception for small loans and quasi-loans to include transactions with connected persons (section 207(1));
widening the exception for “home loans” to include those connected persons who are employees (section 209(3));
raising the maximum amounts permitted under the exception for expenditure on company business (section 204), the exception for small loans and small quasi-loans (section 207(1)) and the exception for small credit transactions (section 207(2));
widening the exceptions for intra-group transactions (section 208);
abolishing the maximum amounts permitted under the exception for money-lending companies (section 209); and
allowing affirmation of loans, quasi-loans and credit transactions entered into by the company in line with the provision in respect of substantial property transactions (section 214).
Sections 215 to 222: Payments for loss of office
409.These sections require member approval for payments for loss of office. These are payments made to a director (or former director) to compensate them for ceasing to be a director, or for losing any other office or employment with the company or with a subsidiary of the company. They also include payments made in connection with retirement. In the case of loss of employment or retirement from employment, the employment must relate to the management of the affairs of the company.
410.Member approval is required under section 217 if a company wishes to make a payment for loss of office to:
one of its directors;
a director of its holding company.
411.Member approval is also required if any person (including the company or anyone else) wishes to make a payment for loss of office to a director of the company in connection with the transfer of the whole or any part of the undertaking or the property of the company or of a subsidiary of the company (section 218).
412.In the case of a payment for loss of office to a director in connection with the transfer of shares in the company or in a subsidiary of the company resulting from a takeover bid, approval is required of the holders of the shares to which the bid relates and of any other holders of shares of the same class (section 219).
413.These sections replace sections 312 to 316 of the 1985 Act. The changes include:
extending the requirements to include payments to connected persons (section 215(3));
extending the requirements to include payments to directors in respect of the loss of any office, or employment in connection with the management of the affairs of the company, and not merely loss of office as a director as such (section 215). This implements a recommendation of the Law Commissions;
extending the requirements to include payments by a company to a director of its holding company (section 217(2));
extending the requirements in connection with the transfer of the undertaking or property of the company to include transfers of the undertaking or property of a subsidiary (section 218(2));
extending the requirements in connection with share transfers so as to include all transfers of shares in the company or in a subsidiary resulting from a takeover bid (section 219(1));
excluding the persons making the offer for shares in the company and any associate of them from voting on any resolution to approve a payment for loss of office in connection with a share transfer (section 219(4)). This implements a recommendation of the Law Commissions;
setting out the exception for payments in discharge of certain legal obligations (section 220);
creating a new exception for small payments (section 221);
clarifying the civil consequences of breach of these sections (section 222(1) to (3)); and
resolving conflicts between the remedies where more than one requirement of these sections is breached (section 222(4) and (5)). For example, if the payment contravenes both section 217 and section 219 because it was a payment by a company to one of its directors and it was a payment in connection with a takeover bid, and none of the required member approvals have been obtained, then the payment is held on trust for the persons who have sold their shares as a result of the offer and not on trust for the company making the payment.
Chapter 5: Directors’ Service Contracts
Section 227: Directors’ service contracts
414.This section is a new provision. It defines what is meant in this Part by references to a director’s service contract. The term is used in sections 177, 182, 188 and 190 and in this Chapter. It includes contracts of employment with the company, or with a subsidiary of the company. It also includes contracts for services and letters of appointment to the office of director. The contract may relate to services as a director or to any other services that a director undertakes personally to perform for the company or a subsidiary.
Section 228: Copy of contract or memorandum of terms to be available for inspection
415.This section requires a company to keep available for inspection copies of every director’s service contract entered into by the company or by a subsidiary of the company. If the contract is not in writing, the company must keep available for inspection a written memorandum of its terms. This section, together with sections 229 and 230, replace section 318 of the 1985 Act.
416.Subsection (3) is new. It requires the service contracts to be retained and kept available for inspection by the company for at least one year after they have expired, but the subsection does not require the copies to be retained thereafter. As a result of the expanded definition of service contract in section 227, this section now applies to contracts for services and letters of appointment, as recommended by the Law Commissions.
417.As recommended by the Law Commissions, the exemption for contracts requiring a director to work outside the UK (section 318(5) of the 1985 Act) and the exemption for contracts with less than 12 months to run (section 318(11) of the 1985 Act) have not been retained.
418.Failure to comply with the requirements of this section is a criminal offence for which every officer of the company who is in default may be held liable on summary conviction to a fine not exceeding level 3 on the standard scale (currently £1,000) or in cases of continued contravention a daily default fine not exceeding one-tenth of that. In a change from the current position under section 318 of the 1985 Act, the company will no longer be liable under the criminal offence.
Section 229: Right of member to inspect and request copy
419.This section gives members a right to inspect without charge the copies of service contracts held by the company in accordance with section 228. Subsection (2) creates a new right for members to request a copy of the service contracts on payment of a fee set by regulations under section 1137.
Section 230: Directors’ service contracts: application of provisions to shadow directors
420.This section applies the requirements of this Chapter to service contracts with shadow directors.
Chapter 6: Contracts With Sole Members Who are Directors
Section 231: Contract with sole member who is also a director
421.Under this section, contracts entered into by a limited company with its only member must be recorded in writing if the sole member is also a director or shadow director of the company. This does not apply to contracts entered into in the ordinary course of the company’s business. The purpose of this section is to ensure that records are kept in those cases where there is a high risk of the lines becoming blurred between where a person acts in his personal capacity and when he acts on behalf of the company. This may be of particular interest to a liquidator should the company become insolvent.
422.This section replaces section 322B of the 1985 Act, which implements article 5 of the 12 th Company Law Directive (89/667/EEC). As the Act will permit public companies to have a single shareholder, this section applies to both private and public limited companies.
423.A failure to record the contract in writing will not affect the validity of the contract (subsection (6)) but other legislation or rules of law might (subsection (7)).
424.If there is a breach of this section, every officer of the company in default is liable on summary conviction to a fine not exceeding level 5 on the standard scale (currently £5,000). In a change from the current position under section 322B of the 1985 Act, the company will no longer be liable under the criminal offence.
Chapter 7: Directors’ Liabilities
425.The sections in this Chapter (sections 232 to 239) deal with two matters:
they restate sections 309A to 309C of the 1985 Act (provisions relating to directors’ liability). The only substantive changes to those sections are a new provision permitting companies to indemnify the directors of companies acting as trustees of occupational pension schemes (section 235), the creation of a right for members to request a copy of a qualifying third party indemnity provision (section 238(2)), the removal of criminal liability on the part of the company for failures to comply with the requirements of section 237 (copy of qualifying indemnity provision to be available for inspection), provision for regulations to specify places in addition to the registered office where inspection may take place (section 237(3)) and a requirement for all qualifying indemnity provisions to be retained by a company for at least one year after they have expired (section 237(4));
they introduce a substantive reform of the law on ratification of acts giving rise to liability on the part of a director (section 239).
Section 232: Provisions protecting directors from liability
426.This section prohibits a company from exempting a director from, or indemnifying him against, any liability in connection with any negligence, default, breach of duty or breach of trust by him in relation to the company. Subsection (2) prohibits indemnification by an associated company as well as by his own company. “Associated company” is defined in section 256 as, in effect, a company in the same group.
427.Any provision, whether in the company’s articles, in a contract or otherwise, attempting to exempt or indemnify a director in breach of this section is void. But this does not apply to lawful provisions in the articles for dealing with conflicts of interest.
Section 233: Provision of insurance
428.This section permits a company to purchase and maintain insurance for its directors, or the directors of an associated company, against any liability attaching to them in connection with any negligence, default, breach of duty or breach of trust by them in relation to the company of which they are a director.
Section 234: Qualifying third party indemnity provision
429.This section permits (but does not require) companies to indemnify directors in respect of proceedings brought by third parties (such as class actions in the US). It also permits (but does not require) companies to indemnify directors in respect of applications for relief from liability made under section 1157 (general power of the court to grant relief in case of honest and reasonable conduct) or under section 661(3) or (4)(power of court to grant relief in case of acquisition of shares by innocent nominee).
430.The indemnity may cover liability incurred by the director to any person other than the company or an associated company. This may include both legal costs and the financial costs of an adverse judgement. But the indemnity must not cover liabilities to the company or to any associated company (subsection (2)).
431.Another condition is that the indemnity must not cover criminal fines, penalties imposed by regulatory bodies (such as the Financial Services Authority), the defence costs of criminal proceedings where the director is found guilty, the defence costs of civil proceedings successfully brought against the director by the company or an associated company and the costs of unsuccessful applications by the director for relief (subsection (3)).
432.Subsections (4) and (5) explain when legal proceedings will be considered to have concluded for the purpose of the conditions imposed by subsection (3).
433.An indemnity that complies with these conditions is described as a qualifying third party indemnity provision.
Section 235: Qualifying pension scheme indemnity provision
434.This section permits (but does not require) companies to indemnify a director of a company acting as a trustee of an occupational pension scheme against liability incurred in connection with the company’s activities as trustee of the scheme. An indemnity that complies with the conditions set out in this section is described as a qualifying pension scheme indemnity provision.
Section 236: Qualifying indemnity provision to be disclosed in directors’ report
435.If a qualifying indemnity provision is in force for the benefit of one or more directors or was in force during the previous year, this must be disclosed by the company in the directors’ report (as to the directors’ report, see Chapter 5 of Part 15). Where the director is of one company but the qualifying indemnity provision is provided by an associated company, then it must be disclosed in the directors' reports of both companies. Companies which choose not to indemnify directors will not have to make any disclosure.
Section 237: Copy of qualifying indemnity provision to be available for inspection
436.This section requires a company to keep available for inspection copies of all the qualifying indemnity provisions it has made for its own directors, and also copies of all those it has made for directors of associated companies.
437.Subsection (4) is a new provision. It requires all qualifying indemnity provisions to be retained and made available for inspection for a further year after they have expired or terminated. But the company is not required by this section to retain copies of the indemnity provision thereafter.
438.Subsection (6) makes a failure to comply with the requirements of this section a criminal offence. The maximum penalty that can be imposed on summary conviction is a fine not exceeding level 3 on the standard scale (currently £1,000) or in cases of continued contravention a daily default fine not exceeding one-tenth of that. In a change from the current position under section 309C of the 1985 Act, the company will no longer be liable under the criminal offence.
Section 238: Right of member to inspect and request copy
439.This section gives members a right to inspect without charge the copies of the qualifying indemnity provisions (or where they are not in writing, the written memorandum of their terms) held by the company in accordance with section 237.
440.This section also creates a new right for members on payment of a fee to request a copy of the copy or memorandum held by the company. The fee will be set by regulations made under section 1137.
Section 239: Ratification of acts of directors
441.This section preserves the current law on ratification of acts of directors, but with one significant change. Any decision by a company to ratify conduct by a director amounting to negligence, default, breach of duty or breach of trust in relation to the company must be taken by the members, and without reliance on the votes in favour by the director or any connected person. Section 252 defines what is meant by a person being connected with a director. For the purposes of this section it may also include fellow directors (subsection (5)(d)).
442.If the ratification decision is taken by way of a written resolution (see Chapter 2 of Part 13) the director and his connected persons may not take part in the written resolution procedure (subsection (3)). This means that the company does not need to send them a copy of the written resolution, and they are not counted when determining the number of votes required for the written resolution to be passed.
443.If the ratification decision is taken at a meeting, those members whose votes are to be disregarded may still attend the meeting, take part in the meeting and count towards the quorum for the meeting (if their membership gives them the right to do so).
444.Subsection (6) makes clear that nothing in this section changes the law on unanimous consent, so the restrictions imposed by this section as to who may vote on a ratification resolution will not apply when every member votes (informally or otherwise) in favour of the resolution. The subsection also makes clear that nothing in this section removes any powers of the directors that they may have to manage the affairs of the company.
445.Subsection (7) explains that the requirements imposed by this section are in addition to any other limitations or restrictions imposed by the law as to what may or may not be ratified and when.
Chapter 8: Directors’ Residential Addresses: Protection from Disclosure
446.Under the 1985 Act (and previous Companies Acts), the usual residential address of every director must be entered on the public record held by:
the registrar; and
each company of which he is a director in its register of directors.
Access to the public record held by the registrar is made in a variety of ways, including daily bulk downloading by some subscribers. There is also a public right to inspect companies’ registers of directors.
447.There is an exception for directors at serious risk of violence or intimidation, e.g. from political activists and terrorists. Under sections 723B – 723E of the 1985 Act, introduced by the Criminal Justice and Police Act 2001, they may apply for a “confidentiality order”. A director with a confidentiality order provides a single service address in addition to his usual residential address. The service address is entered on the public record; the usual residential address is kept on a secure register to which access is restricted to specified enforcement authorities. The historic record is not affected by the confidentiality order. By October 2006, nearly 11,000 Confidentiality Orders had been issued of which, it is estimated that nearly 7,000 were to directors (certain other individuals, eg partners in Limited Liability Partnerships, are also eligible).
448.The CLR considered it essential that directors’ residential addresses be filed with the central register, so that enforcement and regulatory bodies as well as liquidators and, in some circumstances, creditors and shareholders can discover the individual’s residential address. However they were concerned that unrestricted public access to directors’ residential addresses had been abused. They considered that there should not be any discretion as to whether particular addresses should or should not be placed on the public record. Therefore, while welcoming the introduction of the confidentiality order regime, they recommended all directors be given the option of:
either, as now, providing their residential address for the public record;
or, providing both a service address and their residential address, with the service address being on the public record and the residential address being on a separate secure register to which access would be restricted. Access to the restricted register would be available to certain public authorities. Other parties, such as members and creditors, should have a right to apply to the court for access to a director’s residential address. (Final Report, paragraph 11.46)
449.This Chapter of the Act, together with the provisions on the register of directors’ residential addresses in Chapter 1 of this Part, is based on this recommendation. These provisions, which are all new, replace the confidentiality order regime.
Section 240: Protected information
450.This section sets out the information about directors’ usual residential addresses, recorded under Chapter 1 of this Part, that will be protected under the new provisions.
Section 241: Protected information: restriction on use or disclosure by company
451.This section provides for the protection that a company must give to the information covered by section 240. It prohibits the company from using or disclosing an individual director’s home address without his consent except for communicating with him, or to comply with an obligation to send information to the registrar or when required by a court.
Section 242: Protected information: restriction on use or disclosure by the registrar
452.This section provides for protection by the registrar of information that is covered by section 240. The registrar need only protect information where it is submitted on a form where directors’ usual residential addresses are required and entered in the appropriate place. The registrar is not obliged to check all documents submitted to her to ensure that an address has not been inadvertently disclosed. The protection is not retrospective: it does not apply to information on the public record when these provisions come into force. The Act makes separate provision, in section 1088 , for removal of addresses from the register in circumstances specified by regulations.
Section 243: Permitted use or disclosure by the registrar
453.This section provides for certain kinds of permitted use or disclosure of protected information, ie directors’ home addresses and whether a service address is a home address. Subsection (1) provides that the registrar may use the protected information for communicating with the director in question. Subsection (2) provides that the registrar may disclose protected information to a public authority or credit reference agency (the definition of the latter is drawn from the Consumer Credit Act 1974) but this should be read with subsections (3) and (4). Subsection (3) confers power on the Secretary of State to make regulations specifying conditions that must be met before the registrar may disclose protected information. The regulations may also provide for fees to be paid by the authority or agency seeking the address. Subsection (4) provides power to make regulations specifying the circumstances in which an application can be made for a director’s address not to be revealed to a credit reference agency.
Section 244: Disclosure under court order
454.This section provides for two circumstances in which the court may require the company to disclose protected information. The first circumstance is that the service address is not effective; the second is that the home address is needed for the enforcement of an order or decree of the court. If the company cannot provide the address, the court may require the registrar to reveal it. Subsection (3) provides that the application for the order may be made not only by a liquidator, creditor or member of the company but also by anyone with sufficient interest.
Section 245: Circumstances in which registrar may put address on the public record.
455.This section provides that if a service address is not effective, then the home address can be put on the public record. It provides for the registrar to send a warning notice, with a specified period for representations before the intended action, both to the director and to every company of which he is a director. The registrar must take account of any representations made within the specified period in deciding whether to proceed as provided by the next section.
Section 246: Putting the address on the public record
456.This section provides that, if the registrar is putting a director’s home address on the public record under the previous section, then the registrar updates the public record as if she had been notified that the service address is the director’s home address. She must also notify both the director and every company of which he/she is a director. The companies must each put the director’s home address on its register of directors as his/her service address. And for the next five years, the director may not register a service address other than his usual residential address.
Chapter 9: Supplementary Provisions
Section 247: Power to make provision for employees on cessation or transfer of business
457.This section confers a power on the directors to make provision for the benefit of employees (including former employees) of the company or its subsidiaries on the cessation or transfer of the whole or part of the undertaking of the company or the subsidiary (subsection (1)).
458.The directors may exercise this power, even if it will not promote the success of the company. The directors’ general duty under section 172 to act in the way they consider would be most likely to promote the success of the company for the benefit of its members as a whole, does not apply when the directors exercise this power to make provision for employees (subsection (2)).
459.There are a number of conditions to the exercise of this power. It must be authorised by a resolution of the members or, if the articles of the company allow it, by the board of directors. The company’s articles may also impose further conditions on its use (subsection (6)).
460.Any payments made by the directors using the power conferred by this section must be made before the commencement of the winding up of the company and can only be made out of profits available for dividend. Section 187 of the Insolvency Act 1986 confers power to make provision for employees once the company has commenced winding up.
461.This section replaces section 719 of the 1985 Act. In a change from that section, the directors can no longer use the power conferred by this section to make payments to themselves or to former directors or to shadow directors, unless the payments are authorised by the members. The CLR recommended that directors should be prevented from abusing the power by making excessive payments to themselves.
Section 248: Minutes of directors’ meetings
462.This section, together with section 249, replaces the provisions of section 382 of the 1985 Act relating to records of meetings of directors. The requirements of section 382 of the 1985 Act relating to records of meetings of managers have not been retained. This section requires a company to record minutes of all meetings of its directors.
463.Subsection (2) is new. The minutes must be kept for at least ten years.
464.Failure to make and keep minutes as required by this section is a criminal offence, applying to every officer of the company who is in default. In a change from section 382 of the 1985 Act, liability for the offence will no longer fall on the company.
465.Part 37 of the Act makes provision as to the form in which company records (including minutes) may be kept and imposes a duty to take precautions against falsification.
Section 249: Minutes as evidence
466.This section makes provision in respect of the evidential value of the minutes of directors’ meetings.
Section 250: “Director”
467.This section restates the definition of “director” in section 741(1) of the 1985 Act.
Section 251: “Shadow director”
468.This section restates the definition of “shadow director” in section 741(2) of the 1985 Act.
Section 252: Persons connected with a director
469.This section sets out the definition of “connected person” which is used in many of the sections in this Part in relation to the regulation of directors. The persons who are “connected” for this purpose with a director include:
certain family members (see section 253);
certain companies with which the director is connected (see section 254);
trustees of a trust under which the director or a relative mentioned in section 253 or a company with which the director is connected is a beneficiary (but not if the trust exists for the purposes of an employees’ share scheme as defined in section 1166 or a pension scheme);
certain partners; and
certain firms with legal personality (such as a Scottish firm in which the director is a partner).
470.This section, together with sections 253 to 255, replaces section 346 of the 1985 Act.
Section 253: Members of a director’s family
471.This section sets out those members of a director’s family who fall within the definition of persons connected with the director. The list includes all those family members currently falling within the definition of connected person in section 346 of the 1985 Act, and in addition it covers:
the director’s parents;
children or step-children of the director who are over 18 years old (those under 18 were already included under section 346 of the 1985 Act);
persons with whom the director lives as partner in an enduring family relationship; and
children or step-children of the director’s unmarried partner if they live with the director and are under 18 years of age.
472.This implements the Law Commissions’ recommendation that the definition of connected person be extended so as to include cohabitants, infant children of the cohabitant if they live with the director, adult children of the director and the director’s parents. The recommendation that the definition be extended to siblings has not been implemented.
Section 254: Director “connected with” a body corporate
473.This section determines whether a company or other body corporate is a person connected with a director. Broadly speaking, the director, together with any other person connected with him, must be interested in 20% of the equity share capital, or control (directly or indirectly through another body corporate controlled by them) more than 20% of the voting power exercisable at any general meeting.
474.Schedule 1 contains the rules for determining whether a person is “interested in shares” for this purpose.
Section 255: Director “controlling” a body corporate
475.This section defines the circumstances in which a director is deemed to control a body corporate for the purposes of section 254. These circumstances involve two cumulative hurdles. First, the director or any other person connected with him must be interested in the equity share capital or be entitled to control some part of the voting power exercisable at any general meeting. Secondly, the director, fellow directors and other persons connected with him must be interested in more than 50% of the equity share capital or be entitled to control more than 50% of the voting power exercisable at any general meeting.
476.Schedule 1 contains the rules for determining whether a person is “interested in shares” for this purpose.
Section 256: Associated bodies corporate
477.This section is a new provision. It explains what is meant by references in this Part to associated bodies corporate and associated companies. A holding company is associated with all its subsidiaries, and a subsidiary is associated with its holding company and all the other subsidiaries of its holding company.
Section 257: References to company’s constitution
478.This section is new. It makes provision as to the meaning of references to a company’s constitution in this Part.
479.The section is relevant to a number of provisions in this Part, including the duty to act within powers (section 171) and the duty to exercise independent judgment (section 173).
Section 258: Power to increase financial limits
480.This section confers power on the Secretary of State by order to increase financial limits in this Part of the Act. All the financial limits appear in Chapter 4 (provisions regulating transactions with directors requiring approval of members). This section restates section 345 of the 1985 Act.
Section 259: Transactions under foreign law
481.This section makes clear that the rules under this Part of the Act apply whether or not the proper law governing a transaction or arrangement is the law of the UK or a part of the UK.
482.This provision is necessary to prevent parties seeking to avoid the application of the rules relating to approval of long-term service contracts, substantial property transactions and loans and similar transactions by choosing a foreign law. This section restates section 347 of the 1985 Act.
Part 11: Derivative Claims and Proceedings by Members
483.Section 170 provides that directors’ general duties are owed to the company rather than to individual members (or third parties such as employees or pressure groups). It follows that, as now, only the company can enforce them. There are three main ways in which the company can take legal action against a director (or, more usually, a former director) for breach of duty:
if the board of directors decides to commence proceedings;
if the liquidator or administrator following the commencement of a formal insolvency procedure such as liquidation or administration decides to commence proceedings;
through a derivative claim or action brought by one or more members to enforce a right which is vested not in himself but in the company.
This part of the Act is concerned with the third of these types of action.
Existing Law
England and Wales or Northern Ireland
484.In England and Wales, it is possible as a matter of common law for a member to bring an action, in certain circumstances, on behalf of the company of which he is a member. This is known as a derivative claim. As noted above, a member may bring such an action to enforce liability for a breach by one of the directors of his duties to the company.
485.The law relating to the ability of a member to bring proceedings on behalf of the company is not written down in statute. The general principle – commonly known as the rule in Foss v Harbottle – is that it is for the company itself to bring proceedings where a wrong has been done to the company. However, where there has been conduct amounting to a “fraud on the minority”, an exception may be made to the rule, so that a minority shareholder may bring an action to enforce the company’s rights (for example, where there has been an expropriation of company property or dishonest behaviour by a director, and the company is improperly prevented from bringing proceedings against the director by the majority shareholders, perhaps because the wrongdoing director controls the majority of votes).
486.Under the current law, if a wrong has been effectively ratified by the company, this will be a complete bar to a derivative claim. In addition, if a wrong is capable of being ratified, then even if there has been no formal ratification, it may not be possible for a minority shareholder to bring a derivative claim.
487.The law in Northern Ireland in this area is the same as that in England and Wales.
Scotland
488.Under Scots law, the member’s right to raise an action is conferred by substantive law. Accordingly, a member has title as a matter of substantive law to raise proceedings in respect of a director’s breach of duty to obtain a remedy for the company. The action is raised in the name of the member but the remedy is obtained for the company and the rights which the member can enforce against a director or third party are those of the company.
489.The member’s right arises where the action complained of is fraudulent or ultra vires and so cannot be validated by a majority of the members of the company. This remedy is not available if the majority of members acting in good faith have validated or may validate the act complained of.
490.Two rules of substantive law apply to actions brought by the member to protect the company’s interests (as well as to actions brought to protect the shareholder’s personal interests such as enforcement of rights in the articles of association). First, the directors of a company owe duties to the company and not to the members. Second, the court will not interfere in matters of internal management which may be sanctioned by a majority of the members. The effect of these rules is similar to the first two legs of the rule in Foss v Harbottle.
Chapter 1: Derivative Claims in England and Wales Or Northern Ireland
491.The sections in this Part do not formulate a substantive rule to replace the rule in Foss v Harbottle, but instead reflect the recommendation of the Law Commission that there should be a “new derivative procedure with more modern, flexible and accessible criteria for determining whether a shareholder can pursue an action” (Shareholder Remedies, paragraph 6.15). In line with the recommendations of the Law Commission, the derivative claim will be available for breach of the duty to exercise reasonable care, skill and diligence, even if the director has not benefited personally, and it will not be necessary for the applicant to show that the wrongdoing directors control the majority of the company’s shares.
492.The sections in Chapter 1 of this Part introduce a two-stage procedure for permission to continue a derivative claim. At the first stage the applicant will be required to make a prima facie case for permission to continue a derivative claim and the court will be required to consider the issue on the basis of the evidence filed by the applicant only, without requiring evidence from the defendant. The courts must dismiss the application if the applicant cannot establish a prima facie case. At the second stage – but before the substantive action begins – the court may require evidence to be provided by the company. The sections set out a list of the matters which the court must take into account in considering whether to give permission and the circumstances in which the court is bound to refuse permission.
493.The sections will be supplemented by amended Civil Procedure Rules.
Section 260: Derivative claims
494.This section sets out the key aspects of a derivative claim.
Subsection (1) defines what is meant by a derivative claim. There are three elements to this: the action is brought by a member of the company; the cause of action is vested in the company; and relief is sought on the company’s behalf. (A “member” is defined in section 112. Subsection (5) provides that references to a member in this Chapter include a person who is not a member but to whom shares in the company have been transferred or transmitted by operation of law, for example where a trustee in bankruptcy or personal representative of a deceased member’s estate acquires an interest in a share as a result of the bankruptcy or death of a member).
Subsection (2) provides that the claim may only be brought either under this Chapter or in pursuance of an order of the court in proceedings under section 994 (proceedings for protection of members against unfair prejudice).
Subsection (3) provides that a derivative claim “may be brought only in respect of a cause of action arising from an actual or proposed act or omission involving negligence, default, breach of duty or breach of trust by a director of the company”. As such, a derivative claim may be brought in respect of an alleged breach of any of the general duties of directors in Chapter 2 of Part 10, including the duty to exercise reasonable care, skill and diligence (section 174).
Subsection (3) also provides that the cause of action may be against the director or against a third party, or both. Derivative claims against third parties would be permitted only in very narrow circumstances, where the damage suffered by the company arose from an act involving a breach of duty etc on the part of the director (e.g. for knowing receipt of money or property transferred in breach of trust or for knowing assistance in a breach of trust).
Subsection (4) provides that a derivative claim may be brought by a member in respect of wrongs committed prior to his becoming a member. This reflects the fact that the rights being enforced are those of the company rather than those of the member and is the position at common law.
Under subsection (5), the reference to a director in this Chapter includes a former director; and a shadow director is treated as a director.
Section 261: Application for permission to continue derivative claim
495.This clause provides that, once proceedings have been brought, the member is required to apply to the court for permission to continue the claim. This reflects the current procedure in England and Wales under the Civil Procedure Rules. The applicant is required to establish a prima facie case for the grant of permission, and the court will consider the issue on the basis of his evidence alone without requiring evidence to be filed by the defendant. The court must dismiss the application at this stage if what is filed does not show a prima facie case, and it may make any consequential order that it considers appropriate (for example, a costs order or a civil restraint order against the applicant). If the application is not dismissed, the court may direct the company to provide evidence and, on hearing the application, may grant permission, refuse permission and dismiss the claim, or adjourn the proceedings and give such directions as it thinks fit. This will enable the courts to dismiss unmeritorious claims at an early stage without involving the defendants or the company.
Section 262: Application for permission to continue claim as a derivative claim
496.This section addresses the possibility that, where a company has brought a claim and the cause of action on which the claim is based could be pursued by a member as a derivative action:
the manner in which the company commenced or continued the claim may amount to an abuse of process (e.g. the company brought the claim with a view to preventing a member bringing a derivative claim);
the company may fail to prosecute the claim diligently; and
it may be appropriate for a member to continue the claim as a derivative claim;
497.The section provides that, in these circumstances, a member may apply to the court to continue the claim as a derivative action.
Section 263: Whether permission to be given
498.This section sets out the criteria which must be taken into account by the court in considering whether to give permission to continue a derivative claim.
499.Subsection (2) provides that the court must refuse leave to continue a derivative claim if it is satisfied that:
a)
a person acting in accordance with the general duty of directors to promote the success of the company (section 172) would not seek to continue the claim; or
b)
the act or omission giving rise to the cause of action has been authorised or ratified by the company. Section 180(4) preserves any rule of law enabling the company to give authority for anything that would otherwise be a breach of duty. Section 239 preserves the current law on ratification of acts of directors, but with one significant change. Any decision by a company to ratify conduct by a director amounting to negligence, default, breach of duty or breach of trust in relation to the company must be taken by the members, and without reliance on the votes in favour by the director or any connected person.)
500.Subsection (3) sets out the criteria which the court must, in particular, take into account in considering whether or not to grant permission for the derivative claim to be continued.
501.Subsection (4) provides that, in considering whether to give permission, the court must have particular regard to any evidence before it as to the views of independent members of the company i.e. members who have no personal interest, direct or indirect in the matter.
502.Subsection (5) confers on the Secretary of State a power to make regulations with regard to the criteria to which the court must have regard in determining whether to grant leave to continue a derivative claim and where leave of the court must be refused. Subsection (6) provides that, before making any such regulations, the Secretary of State must consult with such persons as he considers appropriate. The power reflects a recommendation by the Law Commission in its 1997 report on shareholder remedies in respect of analogous shareholder actions in Scotland. Under subsection (7), the regulations will be subject to the affirmative resolution procedure.
Section 264: Application for permission to continue derivative claim brought by another member
503.This section addresses the possibility that, where the court has already decided that there is an appropriate case for a derivative claim and a member has commenced or continued a claim:
the manner in which the member commenced or continued the claim may amount to an abuse of the court (e.g. the member brought the claim with a view to preventing another member from bringing the claim);
the member may fail to prosecute the claim diligently;
it may be appropriate for another member to continue the claim (e.g. because the member who brought the claim has become very ill).
504.The section provides that, in these circumstances, another member may apply to the court to continue the claim as a derivative action.
Chapter 2: Derivative Proceedings in Scotland
505.Sections 265 to 269 seek to ensure maximum consistency between the position in England and Wales and Northern Ireland and the position in Scotland (although the clauses reflect the different procedural requirements which apply where proceedings are commenced in the Scottish courts, in particular the fact that the leave of court must be obtained before derivative proceedings may be raised). In view of this, they also put the rights of the member to raise actions on behalf of the company on a statutory footing.
506.Section 265 differs from section 260 in its approach in that it confers the right to bring the proceedings in the first place, and then, in the clauses which follow, regulate the proceedings. (By contrast, the sections relating to proceedings in England and Wales and Northern Ireland assume that there is already a right to bring such proceedings in England and Wales and Northern Ireland; they therefore regulate the proceedings rather than confer the right to bring them.)
507.Subsections (4) to (6) of section 268 confer on the Secretary of State a parallel power to that in section 263 to make regulations with regard to the criteria to which the court must have regard in determining whether to grant leave to continue a derivative claim and where leave of the court must be refused.
Part 12: Company Secretaries
Section 270: Private company not required to have secretary
508.This section replaces section 283(1) of the 1985 Act insofar as it applies to private companies. It implements the CLR recommendation (Final Report, paragraph 4.7) that the requirement for a private company to have a secretary be abolished. It defines a private company “without a secretary” for the purposes of the Act as a company which has taken advantage of the exemption provided by subsection (1) as opposed to one which normally has a secretary but for some reason (for example the death of the office holder) is without a secretary at a given time.Subsection (3) makes provision for private companies without a secretary.
Section 271: Public company required to have secretary
509.This section replaces section 283(1) of the 1985 Act insofar as it applies to public companies. It retains the requirement that a public company must have a secretary. The secretary may also be one of the directors.
Section 272: Direction requiring public company to appoint secretary
510.This section is a new provision, enabling enforcement of the continuing requirement for a public company to have a secretary. It does not apply to private companies. Where it appears that a public company does not have a secretary, the Secretary of State may give a direction to the company. The company must comply with the direction (by making the appropriate appointment and giving notice of it) within the period specified in the direction. The section provides for an offence for failure to comply with a direction.
Section 273: Qualifications of secretaries of public companies
511.This section updates section 286 of the 1985 Act. It makes it the duty of the directors of a public company to ensure that the secretary has both the necessary knowledge and experience and one of the qualifications listed in subsection (2). The qualifications specified in this section are the same as in the 1985 Act except that:
they do not include the qualification of having held the office of the company’s secretary (or assistant or deputy secretary) on 22 December 1980;
in subsection (3)(f), “Chartered Institute of Management Accountants” replaces “Institute of Cost and Management Accountants” as the Institute changed its name in 1986.
There is no requirement for the company secretary to be a natural person. (Compare the requirement in section 155 that a company must have at least one director who is a natural person.)
Section 274: Discharge of functions where office vacant or secretary unable to act
512.This section replaces section 283(3) of the 1985 Act. It provides for the situation where the office of secretary is vacant or there is no secretary capable of acting for any other reason. In these circumstances, if the company has an assistant or deputy secretary, then that person may fill the position of secretary; if not, any person authorised by the directors may do so. This section differs from section 283(3) of the 1985 Act by permitting the directors to authorise any person to act as secretary, rather than only an officer of the company.
Section 275: Duty to keep register of secretaries
513.This section replaces the requirement in section 288 of 1985 Act. It requires every company to keep a register of its secretaries containing specified details. Subsection (3) provides that the register must be kept available for inspection either at the company’s registered office or at a place specified in regulations made under section 1136. Subsections (5) to (8) retain the public right of inspection, sanctions and means of enforcement of the right of inspection.
Section 276: Duty to notify registrar of changes
514.This section replaces the requirement in section 288(2) of the 1985 Act. It requires notification to the registrar within 14 days of any change in the company’s secretary or any change in the particulars contained in the register of secretaries. The consent of the person having become a secretary or joint secretary of a company must accompany the notice. The section retains the existing sanction and ensures that the public record is kept up to date as regards the secretary of every company.
Section 277: Particulars of secretaries to be registered: individuals
515.This section replaces section 290 of the 1985 Act insofar as it applies to secretaries who are individuals. It requires a company to enter in its register of secretaries the name and address of any individual who is its secretary. The definition of name is the same as for directors (see section 163): in particular, the register must include any name used or in use for business purposes since the age of 16. The section retains an exception relating to the former names of peers but, as recommended by the CLR, not that for the former names of married women. The address to be registered is a service address: this implements the CLR recommendation (Final Report, paragraph 11.46) that the requirement for home addresses for company secretaries be abolished.
Section 278: Particulars of secretaries to be registered: corporate secretaries and firms
516.This section replaces section 290 of the 1985 Act insofar as it applies to secretaries who are not individuals. It sets out the details which must be registered where the secretary of a company is either a body corporate or a firm which is a legal person under the law by which it is governed. The requirements that apply in the case of an EEA company follow the recommendations of the CLR (Final Report, paragraph 11.39).
517.The section also makes provision about the details which must be registered where all the partners in a firm are joint secretaries.
Section 279: Particulars of secretaries to be registered: power to make regulations
518.This section is a new provision. It provides a power for the Secretary of State to make regulations that add or remove items from the particulars that have to be entered in a company’s register of secretaries. A similar power is provided by section 166 for directors’ particulars.
Section 280: Acts done by person in dual capacity
519.This section replaces section 284 of the 1985 Act. It provides that where a provision requires or authorises a thing to be done by or to both a director and a secretary of a company it will not be not be satisfied if done by the same person acting in both capacities.
Part 13: Resolutions and Meetings
520.The provisions in this Part replace most of Chapter 4 of Part 11 of the 1985 Act on meetings and resolutions. The changes in the law derive principally from the CLR’s consultation on “Company General Meetings and Shareholder Communications” and recommendations from Chapters 2, 6 and 7 of their “Final Report”, together with two subsequent consultations; the Modernising Company Law White Paper of July 2002 and the Company Law Reform White Paper of March 2005.
521.In addition to implementing detailed policy changes, Part 13 implements two general changes.
First, the law makes the current “elective” regime the default for private companies. This means, for instance, that private companies will no longer need to “elect” to dispense with the Annual General Meeting (AGM): they will not be required to hold an AGM in the first place.
Second, the current law is drafted on the basis that the main way in which shareholder decisions are taken is in general meetings. The new provisions proceed on the basis that in future this will not be the case for many private companies. Private companies will not be required in future to hold general meetings; instead provision is made for new procedures for decisions to be taken by written resolution.
522.The law relating to decisions has been restated in a way that deals first with private companies. Additional layers of requirements for public and quoted companies holding general meetings follow in subsequent provisions. There are provisions at the end of the Part about record keeping. In general, where this Part imposes an obligation or confers a power, it will apply notwithstanding anything in the articles unless otherwise indicated.
Chapter 1: General Provisions about Resolutions
Section 281: Resolutions
523.This section provides that members’ resolutions can only be passed in accordance with the provisions of this Part. There is no equivalent in the current legislation. Subsection (1) allows a private company to pass a resolution either as a written resolution or at a meeting of the members. Subsection (2) allows a public company to pass a resolution only at a meeting of the members. Subsection (3) ensures that where a resolution is required but the type of resolution is not specified, the default will be an ordinary resolution unless the articles require a higher majority. When a provision specifies that an ordinary resolution is required, the articles will not be able to specify a higher majority. Subsection (4) preserves the common law unanimous consent rule.
Section 282: Ordinary resolutions
524.This section provides a definition of an ordinary resolution, whether of the members generally or of a class of the members and whether as a written resolution or as a resolution passed at a meeting. A simple majority – that is, over 50% – is required.
Section 283: Special resolutions
525.This section provides a definition of a special resolution, whether of the members generally or of a class of the members and whether as a written resolution or as a resolution passed at a meeting. A 75% majority is required. If a resolution is proposed as a special resolution, there is a requirement to say so, either in the written resolution text or in the meeting notice. Where a resolution is proposed as a special resolution, it can only be passed as such. The main difference from the existing definition in section 378(2) of the 1985 Act is that there is no longer a requirement for 21 days’ notice where a special resolution is to be passed at a meeting. The subject matter of section 378(3) of the 1985 Act is now dealt with in section 307(4) to (6) (notice required of general meeting), while the subject matter of section 378(4) and (6) is dealt with in sections 320 and 301 respectively.
Section 284: Votes: general rules
526.This section sets out the general rules on votes of members taken by written resolution, on a show of hands at a meeting or on a poll taken at a meeting. These are adapted from section 370 of the 1985 Act and the default regulations in Table A. Subsection (4) allows these general rules to be varied by the company’s articles.
Section 285: Votes: specific requirements
527.This section sets out specific requirements on votes of members, which the company’s articles may not override.Subsections (1) and (2) provide for entitlement to vote where proxies have been appointed and ensure that the articles do not disadvantage a member voting by proxy or proxies. Subsection (3) makes new provision for voting rights on written resolutions, reflecting the fact that they will no longer need to be passed unanimously. A member will have the same number of votes whether passing a resolution on a poll in general meeting or on a written resolution.
Section 286: Votes of joint holders of shares
528.This section puts on a statutory footing what was a default regulation under article 55 of Table A on votes of joint holders of shares. The person whose vote counts is the “senior” holder, the joint holder whose name appears first in the register of members
Section 287: Saving for provisions of articles as to determination of entitlement to vote
529.This section makes new provision to preserve the right for a company to require objections to votes to be made in accordance with procedures in their articles. If an objection is overruled, the decision will be final except in cases of fraud and certain other kinds of misconduct detailed in case law where a court may intervene. This provision preserves the current law. The provision ensures, on the one hand, certainty for company by enabling the chairman to settle matters relating to the admissibility of votes in accordance with the articles and, on the other hand, sufficient remedies for members to challenge a decision if they have suffered unfair prejudice.
Chapter 2: Written Resolutions
530.The provisions of this Chapter replace the present rules on written resolutions of private companies. A key change (apparent from sections 282 and 283) is that where the statutory procedure under the 1985 Act requires unanimity, the procedure in this Act does not. Consequently, the sections are more detailed than sections 381A to 381C of the 1985 Act and set out the procedures for decisions taken outside of a general meeting framework. The use of the expression “written resolution” does not mean that there is a requirement for “writing” in the sense of hard copy.
General provision about written resolutions
Section 288: Written resolutions of private companies
531.This section introduces the written resolution provisions of this Chapter. They apply to private companies only. Subsection (2)(a) and (b) reproduce the two exceptions currently provided for in Part 1 of Schedule 15A to the 1985 Act: a resolution to remove a director or an auditor before the expiration of his term of office may not be passed as a written resolution. These are the only two exceptions to a private company’s right to pass resolutions using the written resolution procedure.
Section 289: Eligible members
532.The eligibility of members to vote on a written resolution is fixed on the day the resolution is circulated. Subsection (2) ensures that the same shares cannot be voted more than once on the same written resolution. If the person entitled to vote changes during the course of that day, the eligible member is the person entitled to vote at the time that the first copy of the resolution is sent or submitted to a member for his agreement.
Circulation of written resolutions
Section 290: Circulation date
533.This section provides that the circulation date of a written resolution means the date on which copies are sent or submitted to members (or if copies are sent on different days, the first of those days.
Section 291: Circulation of written resolutions proposed by directors
534.This section provides for the circulation of written resolutions by directors of the company. A company must circulate a written resolution either by sending it to all eligible members at the same time or, if it can be done without undue delay, submitting the same copy of the resolution to each eligible member in turn or a combination of these. The latter two options allow companies to pass round a document or email rather than sending out several copies.
Section 292: Members’ power to require circulation of written resolution
535.This section enables members to require a written resolution to be circulated. They may also require circulation of a statement about its subject matter. Like the members’ right to require a resolution to be moved at an AGM, the percentage needed is 5% of the total voting rights (or lower if specified in the company’s articles). Subsection (2) specifies some limits on the kind of resolution that may be circulated in this way, designed to stop the power being abused.
Section 293: Circulation of written resolution proposed by members
536.This section specifies what a company has to do when it is required under section 292 to circulate a resolution and accompanying statement. It must circulate the resolution and statement by sending it to all eligible members at the same time or, if it can be done without undue delay, by submitting the same copy of the resolution and statement to each eligible member in turn or a combination of these. The latter two options would allow companies to pass round a document or email rather than sending out several copies. Subsection (3) requires that the members’ written resolution be circulated within 21 days of the company being requested to do so by those members, except that if the written resolution is circulated to members on different days, then the first copy should be dispatched not more than 21 days after the request to circulate the resolution.
Section 294: Expenses of circulation
537.This section provides that the expenses of complying with section 293 are to be paid by the members who requested the circulation of the resolution unless the company resolves otherwise. The company can require the deposit of a sum to meet its expenses before it circulates the resolution, again subject to any resolution to the contrary.
Section 295: Application not to circulate members’ statement
538.This section enables the court, on application by the company or other aggrieved person, to relieve the company of an obligation to circulate a members’ statement under section 293 if in the court’s view the right to require circulation is being abused. This mirrors section 317 in the context of general meetings.
Agreeing to written resolutions
Section 296: Procedure for signifying agreement to written resolution
539.Under this section, a member may signify agreement to a written resolution in hard copy or electronic form, although if the company does not permit electronic form communications, or is not deemed to do so by virtue of section 298, the member will have to signify his consent in hard copy (see paragraph 6 (conditions for use of communications in electronic form) of Schedule 4 (documents and information sent or supplied to a company). Once a member has signified agreement to a written resolution, he cannot withdraw his agreement. This provides certainty for the company as to when the required majority of eligible members needed to agree the resolution has been reached.
Section 297: Period for agreeing to written resolution
540.This section puts a time limit of 28 days for passing a written resolution, unless the company’s articles specify a different period. This means that there will be a definite point when the company can say that a resolution with insufficient support has not been passed.
Supplementary
Section 298: Sending documents relating to written resolutions by electronic means
541.This clause needs to be read together with the provisions about electronic communications to companies in Part 3 (communications in electronic form) of Schedule 4. Taken together, these provisions allow a member to communicate with the company by electronic means where the company has given an electronic address in a document containing or accompanying a proposed written resolution.
Section 299: Publication of written resolution on website
542.This section should be read in conjunction with the provisions about communications by means of a website by a company other than a traded company in Part 4 (communications by means of a website) of Schedule 5 (communications by a company). This clause, together with those provisions, allow a company, provided certain conditions are met, to publish a written resolution on a website rather than send it to a member individually.
Section 300: Relationship between this Chapter and provisions of company’s articles
543.This section ensures that the company’s articles cannot remove the ability of a private company and its members to propose and pass a statutory resolution using the statutory written resolutions procedures of this Chapter.
Chapter 3: Resolutions at Meetings
544.This Chapter replaces sections 368 to 377, 379 and 381 of the 1985 Act and makes provision about resolutions passed in general meeting. The provisions apply equally to private and public companies. The new provisions reflect the fact that private companies will no longer have to hold AGMs. For example, the provisions about circulation of statements in sections 376 and 377 of the 1985 Act have been separated from the provisions on circulation of resolutions prior to an AGM – which are in Chapter 4. The Act repeals section 367 of the 1985 Act which gives the Secretary of State a power to call a meeting where there is no AGM.
General provisions about resolutions at meetings
Section 301: Resolutions at general meetings
545.This is a general provision about the circumstances in which resolutions at meetings are validly passed. It extends to all resolutions the principle in section 378(6) of the 1985 Act relating to special resolutions: that passing a resolution in a meeting is not just a question of obtaining the right majority but of using the correct procedures. An important difference from the position under section 378(6) is that, under this section, a resolution must be passed in accordance with the relevant provisions of the Bill and any additional requirements imposed by the company’s articles. So, where there are mandatory provisions in the Bill (like those about proxies’ rights to vote) these cannot be avoided by making alternative provision in the articles; and where provision is made about meetings in a company’s articles, these must also be complied with.
Calling meetings
Section 302: Directors’ power to call general meetings
546.This section puts into statute part of the default regulation at article 37 of Table A which allows the directors to call a general meeting. The company’s articles will set out how the directors act collectively.
Section 303: Members’ power to require directors to call general meeting
547.This section, together with sections 304 and 305 make provision similar to that in section 368 of the 1985 Act requiring the directors to call a general meeting if requested by the members. There are three main changes.
548.First, there is a change in the threshold required for a meeting request. For public companies this remains members with voting rights holding at least 10% of the paid-up capital. For private companies the threshold is 5% or 10% of the paid-up capital (or, in a company with no share capital, 5% or 10% of the total voting rights) depending on when there was last a meeting in advance of which members had a right – equivalent to the right under this clause (see below) – to circulate resolutions. The threshold is lower if there has been no such meeting in the last twelve months. Second, as indicated above, subsection (4)(b) extends the provisions of the 1985 Act by enabling members to include the text of a resolution to be moved at the requested meeting. Subsection (5) defines what type of resolution may be properly moved. For example, if the resolution would have no effect, then it cannot be properly moved. Third, requests in electronic form are permitted.
Section 304: Directors’ duty to call meetings required by members
549.This section sets time limits within which the directors must call and hold a meeting required by members. Subsection (2) requires that if the members’ request identifies a resolution to be moved at the meeting, notice of this resolution should be included in the notice of the meeting.
Section 305: Power of members to call meeting at company’s expense
550.This section enables the members to call a meeting at the company’s expense in the event that the directors fail to call a meeting on the members’ request. Subsections (6) and (7) provide for members to be reimbursed appropriately and that the directors are penalised directly by the reimbursement being taken out of the fees or other remuneration due to them.
Section 306: Power of court to order meeting
551.This section reproduces the effect of section 371 of the 1985 Act and gives the court power to order a meeting of the company and to direct the manner in which that meeting is called, held and conducted.
Notice of meetings
Section 307: Notice required of general meeting
552.This section replaces part of section 369 of the 1985 Act. It retains the current minimum notice period requirement of 21 days for public company AGMs, with 14 days’ notice required for all other general meetings (whether public or private company general meetings). A general meeting may be called on shorter notice if the requisite majority of members agree. The key substantive change from the position under existing legislation is that the requisite majority required to agree a short notice period has been reduced for private companies from 95% to 90% of the voting rights, although the articles may specify up to 95% if the company wishes. For public companies, the majority required to agree a short notice period remains at 95% of the voting rights.
Section 308: Manner in which notice to be given
553.This section should be read in conjunction with the general requirements for companies in sending and supplying information as set out in Part 37 and Schedule 5.
Section 309: Publication of notice of meeting on website
554.This section contains some specific provisions on communications by means of a website and needs to be read with the general provisions on communications referred to above. The overall effect is similar to that of the website provisions in the current section 369 of the 1985 Act.
Section 310: Persons entitled to receive notice of meetings
555.This section puts into statute part of article 38 of Table A. The new provision ensures that notice of meetings must be sent to all members, directors and any person entitled to a share as a consequence of the death or bankruptcy (or the equivalent in Scots insolvency law) of a member. The provision is subject to any enactment and to any provision in the articles. This means that a company may, for example, make provision in its articles to stop sending notice of meeting to members for whom the company no longer has a valid address.
Section 311: Contents of notices of meetings
556.This section puts into statute another part of article 38 of Table A. The new provision ensures that the notice of meeting must include the time, date and place of the meeting and, subject to the articles, the general nature of the business to be conducted at the meeting.
Section 312: Resolution requiring special notice
557.This section replaces section 379 of the 1985 Act setting out the requirements for special notice resolutions. It makes provision only in relation to resolutions passed at meetings. This is because the resolutions for which special notice is required are either resolutions that are not capable of being passed as written resolutions (in the case of sections 168 and 510) or in relation to which written resolutions have their own special procedure (see sections 514 and 515).
558.There is no change from the existing law whereby at least 28 days’ notice must be given to the company of the intention to move a resolution requiring special notice. Where it is not practicable for the company to give members notice of such a resolution at the same time as it gives notice of the meeting at which the resolution is to be moved, the company must in future give at least 14 days’ notice either by newspaper advertisement or by any other manner allowed by the articles.
Section 313: Accidental failure to give notice of resolution or meeting
559.This section expands on article 39 of Table A. It contains the rule that an accidental failure to give notice of a resolution or a general meeting is generally disregarded. Under subsection (2), this rule can be altered by the articles in some but not all cases.
Members’ statements
Section 314: Members’ power to require circulation of statements
560.This section, together with section 315, replaces sections 376 and 377 of the 1985 Act and provides a right for members to require the company to circulate a statement of up to 1,000 words. The key policy change is that where the statement relates to a resolution or other matter to be dealt with at a public company’s AGM and is received before the company’s financial year-end, the shareholders are not required to cover the costs of circulating the statement. There are two other notable changes. The first is that the shares relied on to trigger the circulation of a statement must in each case carry rights to vote on the relevant resolution rather than just at the meeting. The second is that requests in electronic form are permitted.
Section 315: Company’s duty to circulate members’ statement
561.This section replaces the remainder of sections 376 and 377 of the 1985 Act and specifies what the company is to do when it is required to circulate a members’ statement. The statement must be circulated in the same manner as notice of the meeting and at the same time, or as soon as reasonably practicable, after the company gives notice of the meeting. Where the company fails to comply with the provisions of this section an offence is committed by every officer of the company who is in default.
Section 316: Expenses of circulating members’ statement
562.This section provides that the expenses of complying with section 315 need not be paid by the members if the meeting to which the request relates is a public company AGM and a sufficient number of requests are received before the company’s year-end. Otherwise the company’s expenses will have to be met by the members who requested the circulation of the statement unless the company resolves otherwise. In this case, the members requesting the statement must deposit a sum to cover the company’s costs (unless the company has resolved otherwise).
Section 317: Application not to circulate members’ statement
563.This section replaces section 377(3) of the 1985 Act. It enables the court on application to relieve the company of an obligation to circulate a members’ statement if in its opinion the right to require circulation is being abused.
Procedure at meetings
Section 318: Quorum at meetings
564.This section replaces sections 370(4) and 370A of the 1985 Act. It sets a quorum for a meeting of one “qualifying person” in the case of a single member company and – as a default – two “qualifying persons” in any other case. Subsections (2) and (3) ensure that a member, corporate representative or proxy present at the meeting may all be “qualifying persons”, but excludes the possibility of two or more corporate representatives or proxies of the same member comprising a quorum. Under these provisions, proxies and corporate representatives do not count towards a quorum in companies with more than one member.
Section 319: Chairman of meeting
565.This section reproduces the effect of section 370(5) of the 1985 Act and provides a default provision where the company’s articles are silent, allowing any member to be elected as chairman of a general meeting by a resolution of the company passed at the meeting.
Section 320: Declaration by chairman on a show of hands
566.This section replaces section 378(4) of the 1985 Act and part of article 47 of Table A. This provision ensures that the chairman’s declaration of a vote taken on a show of hands is conclusive evidence of the resolution being passed or lost without further proof being provided, unless a poll is demanded on the resolution. There are two main differences from section 378(4), both of which are drawn from Table A. First, if the demand for a poll is withdrawn, then the chairman’s declaration will stand. Second, the minutes of the meeting also provide conclusive evidence of the chairman’s declaration. This section is intended to provide certainty by preventing members from challenging a declaration of the chairman as to the votes cast on a resolution at a meeting otherwise than by calling a poll.
Section 321: Right to demand a poll
567.This section replaces section 373 of the 1985 Act. It restricts companies’ ability, through their articles, to exclude members’ rights to call a poll. However, it allows articles to exclude the right to a poll on the election of the chairman of the meeting and the adjournment of the meeting. The section provides for three effective types of demands for a poll, including a demand made by at least 5 members with a right to vote on the resolution.
Section 322: Voting on a poll
568.This section replaces section 374 of the 1985 Act. This provision recognises that a member may hold shares on behalf of third parties and allows the member to cast votes in different ways according to instructions from his clients. The reference to class meetings in section 374 is dealt with by section 334.
Section 323: Representation of corporations at meetings
569.This section replaces section 375 of the 1985 Act. The section expressly provides for the appointment of multiple corporate representatives. This is possible under section 375 of the 1985 Act, although the effect of appointing multiple representatives under the existing law is in some cases unclear. The new section spells out the position. Any one of the corporate representatives will be entitled to vote and exercise other powers on behalf of the member at meetings, but in the event that representatives’ votes or other powers conflict, the corporation is deemed to have abstained from exercising its vote or power. If a corporation wishes to appoint people with different voting intentions or with authority to vote different blocks of shares, they should appoint proxies.
Proxies
Section 324: Rights to appoint proxies
570.This section sets out new provisions for the appointment of proxies, expanding on the existing rights given under section 372 of the 1985 Act and Table A. It puts on a statutory footing certain rights that under the 1985 Act are subject to the articles. In future, members of both private and public companies will have the right to appoint more than one proxy. All proxies will be able to attend, to speak and to vote at a meeting. As to the voting rights of a proxy on a show of hands, see sections 284(2)(b) and 285. The effect of those sections is that the default position will be that, where a member appoints more than one proxy, each proxy will have a vote. The articles will be capable of restricting the number of votes of the proxies, provided that they still have at least one vote between them.
Section 325: Notice of meeting to contain statement of rights
571.This section replaces sections 372(3) and 372(4) of the 1985 Act with changes consequential on the extended rights to appoint proxies under section 324. The new provision requires every notice calling a meeting to contain a statement informing the member of his rights to appoint one or more proxies and any more extensive rights conferred by the company’s articles. Failure to include such a statement will not invalidate the meeting, but is an offence attracting a fine for every officer of the company found in default.
Section 326: Company-sponsored invitations to appoint proxies
572.This section reproduces the effect of section 372(6) of the 1985 Act and requires a company to ensure that if it invites members to appoint a particular person or persons as proxy, such an invitation must be issued to all members entitled to vote at the meeting. Subsection (2) lists two exceptions to the requirement. Failure to comply attracts a fine for every officer in default.
Section 327: Notice required of appointment of proxy etc
573.This section replaces section 372(5) of the 1985 Act. There are two changes. The first relates to the timing required for a notice of proxy appointment. The new provision ensures that weekends, Christmas Day, Good Friday and any bank holiday are excluded from the time counting towards the minimum 48 hour notice required to appoint proxies. This means, for example, that for a meeting to be held at 3.00 pm on a Tuesday after a bank holiday Monday, the cut-off point for proxy appointment will be 3.00 pm the previous Thursday, not 3.00 pm on Sunday as under the 1985 Act. The second is that polls which are not taken immediately are covered by the rules as well as meetings and adjourned meetings.
Section 328: Chairing meetings
574.This section provides as a default rule, subject to the articles, that a proxy may be elected as chairman of a general meeting by resolution of the company passed at the meeting.
Section 329: Right of proxy to demand a poll
575.This section sets out the way in which a proxy may participate in a demand for a poll.
Section 330: Notice required of termination of proxy’s authority
576.This section provides a default regulation to replace article 63 of Table A. This ensures that, subject to the articles, an appointed proxy’s actions at a meeting are valid unless notice of termination of the proxy’s authority is given before the meeting starts. The company’s articles may specify a longer advance notice period but this cannot be more than 48 hours in advance of the meeting (excluding weekends, Christmas Day, Good Friday and bank holidays).
Section 331: Saving for more extensive rights conferred by articles
577.This section makes clear that the company’s articles may confer more extensive rights than are provided for under the provisions of the Bill on members and their proxies.
Adjourned meetings
Section 332: Resolution passed at adjourned meeting
578.This section reproduces the effect of part of section 381 of the 1985 Act as it applies to members’ meetings. It ensures that a resolution of the members of the company passed at an adjourned meeting is treated as passed on that date and not on any earlier date. The reference to class meetings in section 381 is dealt with by section 334.
Electronic communications
Section 333: Sending documents relating to meetings etc in electronic form
579.This section needs to be read together with the provisions about electronic communications to companies in Part 3 of Schedule 4. Taken together these provisions allow a member to communicate with the company by electronic means where the company has given an electronic address in a notice calling a meeting or in an instrument of proxy or proxy invitation.
Application to class meetings
Section 334: Application to class meetings
580.This section applies the provisions of this Chapter with some modifications to meetings of holders of a class of shares in companies having a share capital.
Section 335: Application to class meetings: companies without a share capital
581.This section applies the provisions of this Chapter with some modifications to meetings of classes of members of companies without a share capital.
Chapter 4: Public Companies: Additional Requirements for Agms
582.The requirements for public companies relating to annual general meetings are set out in this Chapter. The main substantive changes to the 1985 Act are, as the CLR recommended, that:
private companies will no longer be required to hold an AGM. The provisions of this Chapter therefore do not apply to private companies; and
public company AGMs must be held within six months of their financial year-end.
Section 336: Public companies: annual general meeting
583.This section replaces section 366 of the 1985 Act but will apply only to public companies since private companies are no longer to be required to hold an AGM. Where section 366 required an AGM to be held each year and not more than 15 months after the previous AGM, a public company will now be required to hold an AGM within 6 months of its financial year-end. This new requirement is intended to ensure that shareholders have a more timely opportunity to hold the directors of a public company to account.
Section 337: Public companies: notice of AGM
584.This section reproduces the effect of parts of section 369 of the 1985 Act relating to the AGM notice. The minimum notice period for calling a public company AGM is 21 days as set out in subsection (2) of section 307 or longer if provided for in the company’s articles. An AGM may be called at shorter notice if all members of the company agree.
Section 338: Public companies: members’ power to require circulation of resolutions for AGMs
585.This section, with section 339, replaces sections 376 and 377 of the 1985 Act (to the extent that they relate to resolutions proposed by members to be moved at an AGM). Members holding at least 5% voting rights or at least 100 members holding on average £100 paid-up capital have the right to propose a resolution for the AGM agenda and to require the company to circulate details of the resolution to all members. A change from the existing legislation is that the shares must in each case carry rights to vote on the relevant resolution. The key policy change is that, if the members’ request is received before the financial year-end, then the members are not required to cover the costs of circulation.
Section 339: Public companies: company’s duty to circulate members’ resolutions for AGMs
586.This section replaces the remaining parts of sections 376 and 377 of the 1985 Act (to the extent that they relate to resolutions proposed by members to be moved at an AGM). It specifies what a company has to do when it is required to circulate a members’ resolution for an AGM.
Section 340: Public companies: expenses of circulating members’ resolutions for AGM
587.This section provides that the expenses of complying with section 339 need not be paid by the members who requested the circulation of the resolution if requests sufficient to require the company to circulate it are received before the company’s year-end. Otherwise the company’s expenses will have to be met by the members who requested the circulation of the resolution unless the company resolves otherwise. In this case, the members requesting the statement must deposit a sum to cover the company’s costs (unless the company has resolved otherwise).
Chapter 5: Additional Requirements for Quoted Companies
588.This Chapter imposes new requirements on quoted companies relating to the disclosure on a website of the results of polls at general meetings, and an independent report on a poll if a sufficient number of members demand one. These two measures were recommended by the CLR (Final Report, paragraph 6.39(ii) and (iv)).
Website publication of poll results
Section 341: Results of poll to be made available on website
589.This section requires quoted companies to disclose on a website the results of all polls taken at a general meeting. Subsection (1) sets out the minimum information that must be disclosed. Companies may disclose additional information about the poll results if they wish. Subsection (4) imposes a penalty on every officer in default for non-compliance. Non-compliance however does not invalidate the poll, the resolution or other business to which the poll relates. Section 353(requirements as to website availability) sets out the requirements relating to the website on which the poll results must be published.
Independent report on poll
Section 342: Members’ power to require independent report on poll
590.This section gives members of a quoted company the right to require an independent report of any poll taken, or to be taken, at a general meeting. The minimum threshold required for the demand is the same as that for requiring the circulation of a resolution – that is members holding 5% of the voting rights or 100 members holding on average £100 of paid-up capital. The members’ request must be made within one week of the meeting where the poll is taken. This allows members to decide after a poll is taken whether they wish to require an independent report, for example on a controversial resolution or where there appears to be a problem relating to voting procedures. Members may make their request in advance of the meeting if they wish, but unless the company’s articles already require all votes to be taken on a poll, members may need to take steps to ensure that a poll is called.
Section 343: Appointment of independent assessor
591.The appointment of an independent assessor must be made within one week of the members’ request. This means that the appointment could be made either before or after the meeting depending on when the members’ request is made. The independent assessor must be independent (see section 344) and must not be someone already involved in the voting process for the company.
Section 344: Independence requirement
592.This section prevents a person acting as an independent assessor on a poll if he is too closely connected to the company or an associated undertaking of the company. The independence requirements are set out in subsection (1). They correspond to the independence requirements for a statutory auditor (see section 1214). Subsection (2) allows, but does not require, an auditor to be appointed as an assessor.
Section 345: Meaning of “associate”
593.This section defines “associate” for the purposes of the independence requirements in section 344.
Section 346: Effect of appointment of a partnership
594.This section provides for where a partnership that is not a legal person is appointed as an independent assessor on a poll.
Section 347: The independent assessor’s report
595.This section sets out the minimum information the independent assessor’s report must contain.
Section 348: Rights of independent assessor: right to attend meeting etc
596.This section gives the independent assessor rights to attend the meeting at which the poll or polls may be taken and to be provided with information relating to the meeting. He is to exercise these rights only to the extent he considers necessary for the preparation of his report.
Section 349: Rights of independent assessor: right to information
597.This section gives the independent assessor the right to access company records relating to any poll on which he is to report and to the meeting at which the poll or polls may be taken.
Section 350: Offences relating to provision of information
598.This section imposes a penalty on any person listed in subsection (2) of section 349 who fails to comply with the requirement to provide information or explanation relating to the poll on which the independent assessor is preparing a report.
Section 351: Information to be made available on website
599.This section requires the company to publish on a website the independent assessor’s report of the poll or polls and sets out the minimum information relating to the assessor’s appointment, his identity, the text of the resolution and the assessor’s report that must be made available. Subsections (3) and (4) impose a penalty on every officer in default for non-compliance with this requirement. Failure to comply, however, does not invalidate the poll or the resolution or other business to which the poll relates. Section 353 sets out the requirements relating to the website on which the independent report must be published.
Supplementary
Section 352: Application of provisions to class meetings
600.This section applies the provisions of this Chapter to meetings of holders of a class of shares of a quoted company.
Section 353: Requirements as to website availability
601.This section sets out the minimum requirements that should apply to information to be published on a quoted company’s website under section 341 and section 351. The website on which the information is made available must be maintained by or on behalf of the quoted company and must identify the company in question. This provides flexibility as to whether a website is the company’s own or one operated by a website service provider. Information published on a website must be kept available for a minimum of two years. Subsection (5) provides a let-out when a company’s failure to make the information available on a website for part of the period is wholly attributable to circumstances beyond the company’s control.
Section 354: Power to limit or extend the types of company to which provisions of this Chapter apply
602.At present the provisions of this Chapter apply to quoted companies as defined in section 385, which replaces the definition of “quoted company” in section 262 of the 1985 Act. This section confers on the Secretary of State a power to make regulations to limit or extend the types of company to which the provisions of this Chapter apply. The Parliamentary procedure that will apply to such regulations depends on whether they extend or limit the application of the Chapter.
Chapter 6: Records of Resolutions and Meetings
603.The following provisions replace sections 382, 382A, 382B and 383 of the 1985 Act relating to the records of company proceedings. They should be read in conjunction with the provisions on company records in Part 31. The main changes are the ten year minimum period for keeping records (the 1985 Act envisaged that records would be retained forever); that meetings of directors are dealt with elsewhere (in Part 10 of the Act); and that the new provisions apply to class meetings.
Section 355: Records of resolutions and meetings etc
604.This section requires all companies to maintain records comprising: copies of all resolutions passed otherwise than at general meetings (which would include all written resolutions), minutes of all proceedings of general meetings, and details of decisions of a sole member taken in accordance with section 357. All records must be kept for a minimum of 10 years. Subsections (3) and (4) impose a penalty on every officer in default for non-compliance.
Section 356: Records as evidence of resolutions etc
605.This section ensures that all records of resolutions or written resolutions and minutes of meetings, where signed off by a director or a company secretary or by the chairman in the case of a general meeting, are evidence of the passing of a resolution or the proceedings at the meeting. In legal proceedings, a litigant will have to accept that the records are accurate unless he can prove that they are not.
Section 357: Records of decisions by sole member
606.This section makes provision for the recording of decisions of a company with only one member.
Section 358: Inspection of records of resolutions and meetings
607.This section requires every company to keep its records available for inspection by members for 10 years. Subsection (5) enables a member to seek a court order to compel the company to make the records available for inspection or to provide copies of the records.
Section 359: Records of resolutions and meetings of class of members
608.This section applies the provisions of this Chapter to resolutions and meetings of holders of a class of shares in the case of a company with share capital or to classes of members in the case of a company without a share capital.
Chapter 7: Supplementary Provisions
Section 360: Computation of periods of notice etc: clear day rule
609.This is a new provision to ensure clarity and consistency in the calculation of time periods in relation to meetings and resolutions under Part 13. The section provides that in calculating periods of notice, or periods before a meeting by which a request must be received or sum deposited or tendered, the following are to be excluded –
the day of the meeting,
the day on which notice is given,
the day on which the request is received or the sum is deposited or tendered.
Section 361: Meaning of “quoted company”
610.This section provides that the definition for “quoted company” is as stated in Part 15 (Accounts and reports) of the Act.
Part 14: Control of Political Donations and Expenditure
Background and summary
611.In October 1998 the Committee on Standards in Public Life presented to the Prime Minister its report on the funding of political parties in the UK. The Report recommended that any company intending to make a donation (whether in cash or in kind, and including any sponsorship, or loans or transactions at a favourable rate) to a political party or organisation should be required to have the prior authority of its shareholders. The Government accepted this recommendation, and implemented it through the Political Parties, Elections and Referendums Act 2000 (“the PPERA”). The new regime for control of political donations and expenditure is in Part 10A of the 1985 Act, as inserted by section 139 of and Schedule 19 to the PPERA.
612.Part 14 of the Act restates the existing provisions in a style consistent with the other sections, but most of the key elements of the framework established by the PPERA remain. In particular:
companies will continue to be prohibited from making a donation to a political party or other political organisation or from incurring political expenditure unless the donation or the expenditure has been authorised, in a typical case by the members of the company;
a “political donation” will continue to be defined by reference to sections 50 to 52 of the PPERA, and for this purpose amendments made to the PPERA by the Electoral Administration Act 2006 (which remove from the definition of “donations” loans made otherwise than on commercial terms) will be disregarded;
an approval resolution may authorise the making of donations and incurring of expenditure for a period of not more than four years commencing with the date of the passing of the resolution up to a value specified in the resolution;
donations or expenditure by a subsidiary must, in general, be authorised by resolutions of the members of the subsidiary and of the holding company; and the directors of such a holding company will continue to be liable for unauthorised donations by the subsidiary company;
a company need not seek prior shareholder consent for a donation to a political party or organisation unless the aggregate amount of the donation together with any other relevant donations made by the company and other companies in the group of which it is a member in the previous 12 months exceeds £5,000;
there are no criminal sanctions in relation to the making of unauthorised donations or the incurring of unauthorised political expenditure;
civil remedies are available to a company in the event of breach of the prohibitions and may be pursued in the normal manner by the company. There will continue to be available an action under which shareholders may enforce on behalf of the company any of the remedies available to a company.
613.The main changes from Part 10A of the 1985 Act are that:
in line with the general approach in the Act, references to the general meeting are removed to make it clearer that private companies can authorise donations and/or expenditure by written resolution;
a holding company must authorise a donation or expenditure by a subsidiary company only if it is a “relevant holding company” (that is, the ultimate holding company or, where such a company is not a “UK-registered company”, the holding company highest up the chain which is a “UK-registered company”);
a holding company is permitted to seek authorisation of donations and expenditure in respect of both the holding company itself and one or more subsidiaries (including wholly-owned subsidiaries) through a single approval resolution (section 367(1));
companies are permitted to table separate approval resolutions in respect of donations to political parties and donations to other political organisations (section 367(3));
companies are required to seek authorisation for donations to independent candidates at any election to public office held in the UK or other EU member state and for expenditure by the company relating to independent election candidates;
the sections provide greater clarity for companies about the provision of facilities (for example, meeting rooms) for trade union officials by introducing a specific exemption for donations to trade unions (section 374). The Act does not introduce a specific exemption in relation to paid leave for local councillors because this does not constitute a political donation or political expenditure under Part 10A of the 1985 Act or this Act;
there are important changes to the rules on ratification and liability in cases of unauthorised donations or expenditure;
the special rules in respect of the parent company of a non-GB subsidiary undertaking (sections 347E and 347G of the 1985 Act) are not reproduced;
The new provisions apply to Northern Ireland.
Commentary
Section 362: Introductory
614.This section explains that this Part relates to political expenditure and to political donations made by companies to political parties, political organisations and independent election candidates.
Section 363: Political parties, organisations etc to which this Part applies
615.This section establishes the general scope of the provisions of this Part and introduces the concepts of:
political parties;
political organisations other than political parties;
independent election candidates at any election to public office.
Section 364: Meaning of “political donation”
616.This section defines a "political donation" for the purposes of this Part by reference to sections 50 to 52 of the Political Parties, Elections and Referendums Act 2000. For this purpose, amendments made to the 2000 Act by the Electoral Administration Act 2006 (which remove from the definition of “donation” loans made otherwise than on commercial terms) are disregarded. This section reproduces the effect of section 347A(4) of the 1985 Act, except that it includes donations to independent election candidates.
Section 365: Meaning of “political expenditure”
617.This section defines “political expenditure" for the purposes of this Part.
618.It reproduces the effect of section 347A(5) of the 1985 Act, except that it extends the definition to expenditure incurred by the company in relation to independent election candidates.
Section 366: Authorisation required for donations or expenditure
619.This section prohibits a company from making a donation or incurring political expenditure unless the transaction or the expenditure is authorised by a resolution of the members of the company. If the company is a subsidiary of another company, a resolution may instead, or in addition, be required from the members of the holding company. Sections 1159, 1160 and Schedule 6 provide the definition of “subsidiary”. This section reproduces the effect of section 347C(1) and (6) and section 347D of the 1985 Act, except that:
in line with the general approach in the Act, the section does not refer to the general meeting, to make it clear that private companies can authorise donations and/or expenditure by written resolution;
a donation or expenditure by a subsidiary company must be authorised by the members of the company and by members of a “relevant holding company” (rather than by the members of each holding company within a group). A “relevant holding company” is the ultimate holding company or, where such a company is not a “UK-registered company”, the holding company highest up the chain which is a “UK-registered company”;
a resolution is not required on the part of a company that is a wholly-owned subsidiary of a “UK-registered company” (rather than of any holding company, as in section 347D of the 1985 Act);
the section does not reproduce the prohibition (in section 347C(5) of the 1985 Act) on retrospective ratification of breaches of the rules.
Section 367: Form of authorising resolution
620.This section provides that an authorising resolution may identify the subsidiaries, the heads of donations or expenditure, and the amounts that it authorises. The section reproduces the effect of the 1985 Act, but with the following changes:
under subsection (1) and (2), a holding company may seek authorisation of donations and expenditure in respect of both itself and one or more of its subsidiaries (including wholly-owned subsidiaries) in a single approval resolution. The subsidiaries do not need to be named in the resolution if it applies to all of a holding company’s subsidiaries;
under subsection (3), a company may pass separate approval resolutions in respect of donations to political parties and donations to other political organisations.
Section 368: Period for which resolution has effect
621.This section provides that an approval resolution may seek authorisation for the making of donations and incurring of expenditure having effect over a period of not more than four years. It reproduces the effect of the 1985 Act.
Section 369: Liability of directors in case of unauthorised donation or expenditure
622.This section imposes civil liability on directors where unauthorised donations are made or unauthorised political expenditure is incurred. The liabilities are owed to the company and may be pursued in the normal manner by the company; that is they may be pursued by the directors in the exercise of the management powers conferred by the articles of association. The directors will be subject to the general duties set out in Chapter 2 of Part 10 in the conduct of the company's business. In addition, section 370 provides for enforcement by shareholder action.
623.The section largely reproduces the effect of section 347F of the 1985 Act, but:
only a director of the company and of a “relevant holding company” may be liable in respect of an unauthorised donation or unauthorised expenditure. This reflects the new rules relating to the authorisation of donations or expenditure by subsidiaries in section 366;
directors of the “relevant holding company” will not be liable for an unauthorised political donation or unauthorised political expenditure by a subsidiary if they took “all reasonable steps to prevent the donation being made or the expenditure being incurred”.
624.The conditions under which directors may be exempted from liability (currently set out in section 347H of the 1985 Act) are not reproduced in the new regime.
Section 370: Enforcement of directors' liabilities by shareholder action
625.This section provides a mechanism by which an authorised group of shareholders may enforce on behalf of the company any liability under section 369. In the case of a company limited by shares, an action may be brought by a group of shareholders if they are at least 50 in number, or hold at least 5% of the issued share capital. This section reproduces the effect of section 347I of the 1985 Act, except that, in a case where liability is owed by directors of a holding company in relation to a donation made by a subsidiary, the action may be brought by shareholders of the subsidiary or of the holding company.
Section 371: Enforcement of directors’ liabilities by shareholder action: supplementary
626.This section makes further provision in relation to proceedings brought under section 370. It reproduces the effect of section 347I of the 1985 Act. The group of shareholders wanting to take action under section 370 must give written notice to the company at least 28 days in advance of bringing the proceedings. Any director of the company has the right to apply to the court within 28 days of when the notice was given to request that the proceedings not be brought.
627.This section also provides that if the liability is already being pursued with due diligence by the company, the court may direct that the proceedings brought by the group of shareholders are either discontinued or brought on such terms and conditions as the court sees fit.
Section 372: Costs of shareholder action
628.This section provides that the authorised group of members are not entitled as of right to have the cost of the shareholder action met from the funds of the company, but have the right to apply to the court for an indemnity out of the company's assets in respect of costs incurred or to be incurred in a shareholder action. The court would have full discretion to grant such an indemnity on such terms as it thinks fit. The section reproduces the effect of section 347J of the 1985 Act.
Section 373: Information for purposes of shareholder action
629.This section provides that the authorised group of members is entitled, once the action is commenced, to be provided by the company in whose name it is brought with all information possessed by the company, or in its control or obtainable by it, relating to the subject matter of the action. It reproduces the effect of section 347K of the 1985 Act.
Sections 374 to 378: Exemptions
630.These sections set out five exemptions from the requirement for prior shareholder authorisation:
section 374 creates a new exemption in relation to donations to trade unions (including trade unions in countries other than the UK). The exemption covers donations such as the provision of company rooms for trade union meetings, the use of company vehicles by trade union officials and paid time off for trade union officials. However, a donation to a trade union’s political fund is not covered by the exemption;
section 375 restates the exemption in section 347B of the 1985 Act in respect of subscriptions paid to a trade association for membership of the association, except that it is not restricted to trade associations which carry out their activities mainly in the EU;
section 376 restates the exemption in section 347B of the 1985 Act in respect of donations to all-party parliamentary groups;
section 377 restates the exemption in section 347B of the 1985 Act for political expenditure that is exempt by virtue of an order by the Secretary of State. An order made by statutory instrument under this clause may confer an exemption on companies or expenditure of any description or category specified in the order. The parallel power in section 347B(8) to (11) of the 1985 Act was used in 2001 to exempt business activities such as the publication of newspapers which, by their very nature, involve the publication or dissemination of material which seeks to influence the views of members of the public;
section 378 restates the exemption in section 347B of the 1985 Act under which authorisation for donations is not required unless the donation or aggregate amount of the donations by the company exceeds £5,000 in a 12 month period. Donations by other group companies (including subsidiaries) must be taken into account in calculating whether the £5,000 threshold has been exceeded.
Section 379: Minor definitions
631.This section contains minor definitions for this Part.
Part 15: Accounts and Reports
632.The provisions of this Part replace the provisions of Part 7 of the 1985 Act relating to accounts and reports. The provisions of Part 7 of the 1985 Act relating to audit are replaced by provisions in Part 16 of the Act.
633.The provisions have been reordered and redrafted to make it easier for companies of whatever size to find the requirements relevant to them. In Part 7 of the 1985 Act the provisions applying to small companies are generally expressed as modifications of the provisions applying to large companies. These sections proceed on the opposite basis: where provisions do not apply to all kinds of company, provisions applying to small companies appear before the provisions applying to other companies.
634.A further change is to enable the Secretary of State to replace the detailed Schedules to Part 7 of the 1985 Act by regulations. This will give more flexibility to arrange the material currently in Schedules to make it easier to follow for different types of company. It is unnecessary and undesirable to have parallel and duplicative regimes on the detail for different types of company in primary legislation, but this could be done in parallel sets of regulations for different sizes and types of company.
635.The main substantive changes in this Part are:
a reduction in the time limit for private companies to file their accounts from ten months to nine months after the year end (section 442);
a reduction in the time limit for public companies to lay full financial statements before the company in general meeting and file them from 7 months to 6 months after the year end (section 442);
new requirements for quoted companies to publish their annual accounts and reports on a website (section 430); and
replacement of the general power of the Secretary of State to alter accounting requirements in section 257 of the 1985 Act by a general power of amendment by regulations (section 468) and more specific powers in relation to specific sections.
Chapter 1: Introduction
Section 380: Scheme of this Part
636.This introductory section indicates the main way in which the structure of this Part differs from that of Part 7 of the 1985 Act: provisions relating to small companies are set out before provisions relating to larger companies; provisions applying to private companies appear before those applying to public companies; and provisions applying to quoted companies appear after those applying to other companies.
Companies subject to the small companies regime
Section 381: Companies subject to the small companies regimeSection 382: Companies qualifying as small: generalSection 383: Companies qualifying as small: parent companiesSection 384: Companies excluded from the small companies regime
637.These sections set out which companies, parent companies or groups fall within the small companies regime – that is, those that qualify as small companies or groups and are not excluded from the regime for one of the reasons set out in section 384. With two small changes, the conditions for qualification as a small company are unchanged from the current regime (sections 247, 247A and 249 of the 1985 Act). Firstly, section 382(5) now contains a generalised definition of balance sheet total for both Companies Act and IAS individual accounts. Secondly, whereas section 247A(2) of the 1985 Act provides that a group is ineligible if any of its members is a body corporate having power to offer its shares or debentures to the public, the reference in section 384(2)(b) is now to a body corporate whose securities are admitted to trading on a regulated market in an EEA state. The definition of “regulated market” is to be found in section 1173. This reflects changes made by the Accounts Modernisation Directive (2003/51/EEC).
Section 385: Quoted and unquoted companies
638.The definitions of quoted and unquoted company in this section are equivalent to the definition of “quoted company” in section 262 of the 1985 Act. A power is conferred to amend the definition of “quoted company” by regulations. If the regulations extend the application of this Part then they will be subject to affirmative resolution procedure. Otherwise they are subject to negative resolution procedure.
Chapter 2: Accounting Records
Sections 386 to 389: Accounting records
639.These sections set out the general duty to keep accounting records and specify where and for how long records are to be kept. They replace equivalent provisions in sections 221 and 222 of the 1985 Act. Their purpose is to ensure that businesses record transactions to enable them to show the company’s financial position and to prepare accounts which comply with the Companies Act and, where relevant, with International Accounting Standards. “Accounting records” is a broad term and there is no specific definition as the records may differ depending on the nature and complexity of the business. For a simple business these may include, for example, bank statements, purchase orders, sales and purchase invoices, whilst a more sophisticated business may have integrated records, which it holds electronically.
640.Section 387 creates a criminal offence for every officer of a company who is in default, where the company has failed to keep adequate accounting records under section 386. The section replicates the existing penalties under section 221(5) and (6) of the 1985 Act (imprisonment or a fine).
641.Section 389 makes similar provision in relation to failure to comply with section 388, replacing section 222(4) and (6) of the 1985 Act.
Chapter 3: a Company’s Financial Year
Section 390: A company’s financial year
642.This section replaces section 223 of the 1985 Act. A company’s financial year is the period for which its accounts and reports must be prepared. A company’s financial year is the same as its accounting reference period (see section 391), subject to the directors’ decision to alter the last day of the period by plus or minus seven days.
Section 391: Accounting reference periods and accounting reference date
Section 392: Alteration of accounting reference date
643.These sections replace sections 224 and 225 of the 1985 Act.
644.Section 391(2) and (3) preserve the accounting reference dates of companies incorporated before 1st April 1996 (in the case of GB companies), and before 22nd August 1997 (in the case of Northern Irish companies). Otherwise, a company’s accounting reference date is the last day of the month in which the anniversary of its incorporation falls. Its first accounting reference period is a period of more than six months but not more than eighteen months beginning with the date of incorporation and ending with the accounting reference date unless the company changes its accounting reference date (the date on which the accounting reference period ends), in accordance with section 392. Subsequent accounting reference periods (financial years) are successive periods of 12 months, again subject to any alteration of the accounting reference date.
645.Section 392(4) provides that a company cannot change its accounting reference date if the period allowed for delivering accounts and reports to the registrar for that period has already expired. Under the corresponding provision in the 1985 Act, the company cannot change the date “if the period allowed for laying and delivering accounts and reports in relation to that period has already expired." Under the Act only public companies are obliged to lay their accounts at a general meeting (section 437).
Chapter 4: Annual Accounts
Section 393: Accounts to give true and fair view
646.Subsection (1) introduces an overarching obligation on directors (the preparers of accounts) not to approve accounts unless they give a true and fair view of the financial position of the company and, in the case of group accounts, the group. This provision reflects the underlying legal duty already expressed in Community law.
647.Subsection (2) in addition places a requirement on auditors to take this overarching duty to give a true and fair view into consideration when giving an opinion on the accounts. This requirement supplements the functions of an auditor set out in section 485.
Individual accounts
648.Sections 394 to 397, which replace sections 226, 226A and 226B of the 1985 Act, concern the duty of the directors to prepare individual accounts. The individual accounts may either be prepared under the Act (Companies Act individual accounts) or (unless the company is a charity) in accordance with international accounting standards adopted under the IAS Regulation (IAS individual accounts). The terms “IAS Regulation” and “international accounting standards” are defined in section 474. Once a company has switched to IAS individual accounts all subsequent individual accounts must be prepared in accordance with IAS unless there is a relevant change of circumstance (see section 395(3) to (5)). The provisions concerning the form and content of Companies Act accounts to be found in the Schedules to Part 7 of the 1985 Act will in future be contained in regulations to be made by the Secretary of State (section 396(3)). The Parliamentary procedure for such regulations is set out in section 473.
Group accounts: small companies
Section 398: Option to prepare group accounts
649.This section provides that a company that is subject to the small companies regime and is a parent company is not obliged to prepare group accounts in addition to its individual accounts, (restating section 248 of the 1985 Act), but it may opt to do so. The current exemption in section 248 of the 1985 Act from preparation of group accounts by parent companies heading medium sized groups has been abolished, following the substantial increase in the financial thresholds for medium sized groups in 2004.
Group accounts: other companies
Sections 399 to 402: group accounts: other companies
650.The sections relating to group accounts have been reorganised to make them easier to follow.
651.Sections 399 to 402 re-enact sections 227(1) and (8), 228, 228A and 229(5) of the 1985 Act. Section 399 concerns the requirements and exemptions from requirements in relation to group accounts. Parent companies not subject to the small companies regime have the duty to prepare consolidated accounts unless exempt from having to do so under sections 400 to 402. Section 400 provides an exemption from preparing group accounts for companies included in EEA group accounts of a larger group. Section 401 provides such an exemption for companies included in non-EEA group accounts of a larger group, and section 402 provides an exemption when all the company’s subsidiary undertakings could be excluded from consolidation in Companies Act group accounts (see section 405).
Group accounts: general
Section 403: Group accounts: applicable accounting framework
652.This section replaces section 227(2) to (7) of the 1985 Act. Parent companies whose securities are publicly traded must prepare group accounts in accordance with the IAS Regulation. Other parent companies (with the exception of charitable companies) have the choice whether to prepare group accounts under the Companies Act (Companies Act group accounts) or in accordance with adopted international accounting standards (IAS group accounts). Once a company has switched to IAS group accounts all subsequent group accounts must be prepared in accordance with IAS unless there is a relevant change of circumstance (see subsections (4) to (6)).
Section 404: Companies Act group accounts
653.For companies preparing Companies Act group accounts, this section gives the Secretary of State power to make provision by regulations as to the form and content of the consolidated balance sheet and consolidated profit and loss account and additional information to be provided by way of notes to the accounts. The regulations will replace the current requirements contained in Schedule 4A to the 1985 Act. These regulations are subject to the Parliamentary procedure in section 473.
Section 405: Companies Act group accounts: subsidiary undertakings included in the consolidation
654.This section replaces section 229 of the 1985 Act. It requires all subsidiary undertakings to be included in the consolidated accounts subject to certain permitted exclusions.
Section 406: IAS group accounts
655.This section re-enacts section 227B of the 1985 Act. A company may opt or may be required to prepare group accounts in accordance with international accounting standards. This section provides that where it does so, this must be stated in the notes to the accounts.
Section 407: Consistency of financial reporting within group
656.This section re-enacts section 227C of the 1985 Act. If the parent company prepares both consolidated and individual accounts under IAS, it is not required to ensure that all its subsidiary undertakings also use IAS. However, it must otherwise ensure that its individual accounts and those of all its subsidiary undertakings use the same financial reporting framework, unless there are good reasons for not doing so.
Section 408: Individual profit and loss account where group accounts prepared
657.This section replaces section 230 of the 1985 Act. A parent company that prepares group accounts and that meets the criteria in subsection (1)(a) and (b) may, subject to the profit and loss account being approved by the directors, dispense with the inclusion of a profit and loss account in the company’s accounts, for example when delivered to the registrar. The profit and loss account may also omit the information on employee numbers and costs required by section 411. The exemption currently provided for in section 230(2) of the 1985 Act for certain information required by provisions of Schedule 4 to the 1985 Act, will be provided for in regulations under section 396.
Information to be given in notes to the accounts
Section 409: Information about related undertakings
658.This section replaces section 231(1) to (4) of the 1985 Act. The requirement to disclose information about related undertakings in the notes to a company’s annual accounts applies whether or not the company has to produce group accounts but there are different disclosure requirements in each case. This section gives the Secretary of State a new power to make regulations requiring information about related undertakings to be given in notes to a company’s annual accounts. These regulations are subject to the Parliamentary procedure in section 473. The regulations will replace the provisions of Schedule 5 to the 1985 Act.
659.Subsection (3) enables regulations under the section to make provision corresponding to section 231(3) of the 1985 Act authorising the omission from the notes to the accounts of information in respect of undertakings established outside the UK, or carrying on business outside the UK where the directors consider that disclosure would be seriously prejudicial to the business of that undertaking, or to the business of the company or any of its subsidiary undertakings. The Secretary of State must agree to the omission. This exemption is sought by a very small number of companies each year.
Section 410: Information about related undertakings: alternative compliance
660.This section replaces section 231(5) to (7) of the 1985 Act. Where there are numerous related undertakings and the directors believe that full disclosure would result in information of excessive length in the notes to the accounts, they may give more limited information. As a minimum this must include information in subsection (2)(a) and (b). Subsection (3) provides that the full information on the related undertakings must be submitted with the next annual return.
Section 411: Information about employee numbers and costs
661.This section replaces section 231A of the 1985 Act concerning particulars of staff. Section 231A was inserted by the 1985 Act (International Accounting Standards and Other Accounting Amendments) Regulations 2004 (S.I. 2004/2947) re-enacting provisions previously in the Schedules to Part 7 of the 1985 Act so that they continued to apply both to companies preparing Companies Act accounts and to those preparing IAS accounts.
Section 412: Information about directors’ benefits: remuneration
662.This section, together with section 413, replaces section 232 of the 1985 Act. Section 232 of the 1985 Act, with Schedules 6 and 7A, provides for disclosure of specified information on directors’ remuneration in notes to a company’s annual accounts. Section 412 of the 2006 Act instead gives the Secretary of State a new power to make provision by regulations requiring information about directors’ remuneration to be given in notes to a company’s annual accounts. Regulations under this section are subject to the Parliamentary procedure in section 473.
Section 413: Information about directors’ benefits: advances, credit and guarantees
663.This section replaces section 232 of the 1985 Act as regards the disclosure of advances, credit and guarantees. Under section 232 of the 1985 Act, information on the following areas must be given in notes to a company’s annual accounts:
details of loans, quasi-loans, credit transactions and related guarantees and security between a company and its directors or persons connected with its directors;
details of any other transactions or arrangements in which a director, indirectly or directly, has a material interest.
This can be seen as an extension of the internal disclosure of directors’ interests required by section 317 of the 1985 Act.
664.Section 413 sets out the new disclosure requirements in respect of (a) advances and credits granted by the company to its directors, and (b) guarantees of any kind entered into by the company on behalf of its directors. The wording of section 413 is much closer to that of articles 43(1)(13) and 34(13) of the Fourth (78/660/EEC) and Seventh (83/349/EEC) Company Law Directives.
665.The powers under section 396(3)(b) (Companies Act individual accounts) and section 404(3)(b) (Companies Act group accounts) will be used to require the disclosure of information about certain related party transactions in the notes to Companies Act accounts. Companies will no longer be required to disclose transactions made between the company and officers other than directors.
666.Under section 413(8) banks and the holding companies of credit institutions need only state (a) the amount of an advance or credit, and (b) in relation to a guarantee, the amount of the maximum liability that may be incurred by the company (or its subsidiary). In the light of the simplified disclosure regime for advances, credit and guarantees, sections 343 and 344 of the 1985 Act, which make special provision for financial institutions, are repealed.
Section 414: Approval and signing of accounts
667.This section replaces section 233 of the 1985 Act. It provides that a company’s annual accounts (its individual accounts and any group accounts) must be approved by the board of directors and the balance sheet must be signed. Subsection (3), which requires the balance sheet of accounts prepared in accordance with the small companies regime to carry a statement to that effect, re-enacts section 246(8) of the 1985 Act. Subsections (4) and (5) re-enact the criminal offence in section 233 of the 1985 Act for approval of accounts that do not comply with the requirements of the Companies Act or, where applicable, of Article 4 of the IAS Regulation. Section 233 (4) of the 1985 Act, which required that a director of the company should sign the copy of the balance sheet delivered to the registrar, has not been reproduced. This requirement would have hampered developments in the electronic delivery of accounts.
Chapter 5: Directors’ Report
Sections 415 to 419: Directors’ report
668.These sections concern the duty to prepare a directors’ report, its content, approval and signature. They replace sections 234, 234ZZA, 234ZZB, 234ZA, 234A, 246(4)(a) and 246A(2A) and 246(8) of the 1985 Act.
669.Section 416(4) gives the Secretary of State power to make provisions by regulations as to other matters that must be disclosed in the directors’ report. These regulations replace the provision formerly made by Schedule 7 to the 1985 Act. The regulations are subject to the Parliamentary procedure in section 473.
670.Section 417 provides for what must be contained in the business review element of the directors’ report. All companies, other than small companies, will need to produce a business review, as required by the EU Accounts Modernisation Directive (2003/51/EEC). Subsection (2) sets out the purpose of the review, that is, to inform members of the company and help them assess how the directors have performed their duty under section 172 (duty to promote the success of the company). Subsections (3), (4), (6) and (8) specify the content of the review. Subsection (5) specifies information that quoted companies in particular must include in their review where necessary for an understanding of the company’s business. Where directors of quoted companies have nothing to report on environmental, employee, social and community matters or essential contractual or other arrangements, their review must say so. Subsection (7) exempts medium-sized companies from reporting non-financial key performance indicators – an exemption allowed by the EU directive. Subsection (9) provides that where the directors’ report is a group report, all references in section 417 to the company are to be read as references to the company and its consolidated subsidiary undertakings. Subsection (10) enables directors to omit from the business review information about impending developments or matters in the course of negotiation where in their opinion disclosure would be seriously prejudicial to the interests of the company. Subsection (11) enables directors to omit from the business review information about a third party otherwise required by subsection (5)(c) (essential contractual or other arrangements) where in the directors’ opinion it would be seriously prejudicial to that third party and contrary to the public interest.
Chapter 6: Quoted Companies: Directors’ Remuneration Report
Sections 420 to 422: Quoted companies: directors’ remuneration report
671.These sections replace sections 234B and 234C of the 1985 Act. Those sections, which were inserted into the Act by the Directors’ Remuneration Report Regulations 2002 (S.I. 2002/1986), require quoted companies to:
publish a report on directors’ remuneration as part of the company’s annual reporting cycle; and
disclose within the report details of individual directors’ remuneration packages, the company’s remuneration policy, and the role of the board and remuneration committee in this area.
672.Section 421 gives the Secretary of State power to make provision by regulations as to the information that must be contained in a directors’ remuneration report and how it should be set out. These matters are currently set out in Schedule 7A to the 1985 Act, and regulations made under section 421 will replace the provisions in Schedule 7A. The regulations will also specify the extent to which the directors’ remuneration report should be subject to audit. Regulations under section 421 are subject to the Parliamentary procedure in section 473.
Chapter 7: Publication of Accounts and Reports
Section 423: Duty to circulate copies of annual accounts and reports
673.This section replaces section 238 of the 1985 Act. Subsection (1) provides that a company must send a copy of its annual accounts and reports (as defined in section 471 and including any relevant auditor’s report) to specified persons. Subsection (2) restricts the general obligation of companies to send copies of accounts and reports. The obligation will in future be to send the accounts and reports only to persons for whom the company has a current address. This is to avoid companies having to send copies of the annual accounts and reports to addresses from which correspondence has previously been returned marked not known at this address (or its electronic equivalent). General provisions about how to supply copies to joint holders are in Part 6 of Schedule 5 (Communications by a company).
Section 424: Time allowed for sending out copies of accounts and reports
674.This section makes changes to the time for distributing accounts and reports for both private and public companies. Private companies (unless they opted out of the requirement) were previously required to lay their accounts at a general meeting and to send their accounts and reports to members 21 days before that meeting. They are no longer required to hold any general meeting and the requirement now is to send out their accounts and reports no later than the earlier of the date of actual delivery to the registrar or the deadline for delivery (see section 442 for the time limits for filing). Public companies must still send the annual accounts and reports out at least 21 days before the general meeting at which the accounts and reports are to be laid (defined as the “relevant accounts meeting”).
Section 425: Default in sending out copies of accounts and reports: offences
675.There is no change to these offences (in section 238(5) of the 1985 Act).
Sections 426 to 429: Option to provide summary financial statement
676.These sections restate section 251 of the 1985 Act. All companies have the option under section 426 to provide summary financial statements instead of copies of the full accounts and reports. This section reproduces the existing power for the Secretary of State to make provision by regulations:
as to the circumstances in which a company may send out summary financial statements; and
as to the manner in which it is to be ascertained whether a person wishes to receive a copy of the (full) accounts and reports.
It also makes new provision for persons nominated to enjoy information rights under section 146 (indirect investors) to be able to be provided with summary financial statements rather than the full accounts and reports.
677.Section 427 sets out the form and content requirements for summary financial statements prepared by unquoted companies, whilst section 428 sets out the form and content requirements for summary financial statements prepared by quoted companies. In both cases, the Secretary of State may make regulations as to the form and content of summary financial statements. There is also a new power for regulations to provide that any specified material be sent separately at the same time as the summary financial statement instead of being included in it. This is to cover the requirements of the Takeovers Directive as to necessary explanatory material (see section 992). As in the 1985 Act, these powers are subject to the negative resolution procedure. Section 429 restates the existing offences in section 251(6) of the 1985 Act.
Section 430: Quoted companies: requirements as to website publication
678.This section introduces a new requirement on quoted companies (as defined in section 385) to put the full annual accounts and reports on a website. A quoted company will still have to send the full accounts and reports to its members under section 423.
679.The annual accounts and reports must be made available as soon as is reasonably practicable on a website that is maintained by or on behalf of the company, and that identifies the company in question. Access to the website must be available to all members of the public and not just to members, and there must be continuous access to the website without charge. Access to the information on the website and the ability to obtain a hard copy of the information from the website, may be restricted by the company where necessary to comply with any statutory or regulatory requirement (e.g. of an overseas regulator).
680.The annual accounts and reports for a financial year must remain available until the accounts and reports for the next financial year are published on the website. Right of member or debenture holder to demand copies of accounts and reports
Sections 431 and 432: Right of member or debenture holder to demand copies of accounts and reports
681.These sections re-enact section 239 of the 1985 Act and entitle a member or debenture holder to demand a copy of the company’s last annual accounts and reports without charge. Section 431 lists the documents to which members or debenture holders of unquoted companies are entitled, while section 432 lists those to which members or debenture holders of quoted companies are entitled. The company must comply with a demand within seven days of receipt of the request by the company.
Sections 433 to 436: Requirements in connection with publication of accounts and reports
682.Section 433 brings together provisions scattered throughout Part 7 of the 1985 Act (in sections 233(3) and (6)(a), 234A(2) and (4)(a) and 234C(2) and (4)(a)) concerning statements of the name of the signatory in published accounts and reports. In the case of unquoted companies, every copy of the balance sheet and directors’ report that is published by or on behalf of the company must state the name of the director who signed it on behalf of the board. For quoted companies this applies to copies of the balance sheet, directors’ remuneration report and directors’ report.
683.Sections 434 and 435 re-enact section 240 of the 1985 Act concerning requirements in connection with the publication of statutory or non-statutory accounts.
684.“Publication” is defined in section 436.
Chapter 8: Public Companies: Laying of Accounts and Reports before General Meeting
Section 437: Public companies: laying of accounts and reports before general meeting
Section 438: Public companies: offence of failure to lay accounts and reports
685.These sections re-enact section 241 of the 1985 Act on the laying of accounts and reports before the company in general meeting, but restrict its application to public companies. Under the Act, private companies are under no statutory obligation to hold an AGM or to lay accounts and reports in general meetings. There is therefore no statutory link for them between the accounts and AGMs (although such a link might be provided for in the company’s articles). Any AGM that a private company may hold pursuant to its articles will not be a statutory meeting. Public companies will still be required to hold AGMs and they must now hold them within 6 months of the end of the accounting reference period.
Chapter 9: Quoted Companies: Members’ Approval of Directors’ Remuneration Report
Section 439: Quoted companies: members’ approval of directors’ remuneration report
Section 440: Quoted companies: offences in connection with procedure for approval
686.These sections restate the requirement under section 241A of the 1985 Act that a quoted company circulate a resolution approving the directors’ remuneration report for the preceding financial year to its shareholders prior to its annual general meeting. The vote is advisory: as such, it does not require directors to amend contractual entitlements, nor to amend their remuneration policy, but the result of the vote will send a very strong signal to directors about the level of support among shareholders for the board’s remuneration policy. In practice, directors will wish to take notice of the views of the company’s members, and to respond appropriately. All “existing directors” (that is, every person who, immediately before the general meeting, is a director of the company) have a responsibility to ensure that the resolution is put to the vote of the meeting. As such, the requirement does not apply to past directors (even if they served on the board or as members of the remuneration committee in the current financial year), but it does apply to “existing directors” who were, for whatever reason, not present at the general meeting.
Chapter 10: Filing of Accounts and Reports
Sections 441 to 443: Duty to file accounts and reports
687.These sections cover the general duty to file accounts and reports with the registrar of companies and the period allowed for filing accounts.
688.Section 442 reduces the period for filing accounts from ten months to nine months for private companies and from seven months to six months for public companies. These periods are calculated from the end of the relevant accounting reference period. The timetable for delivering accounts to the registrar was last amended in 1976. The periods have been reduced to reflect improvements in technology and the increased rate at which information becomes out of date. Filing timescales in other countries are generally less generous than in the UK. Under subsection (6), whether a company is private or public for the purpose of its filing obligations is determined by its status immediately before the end of the relevant accounting reference period.
689.Section 443 is a new provision defining how to calculate the periods allowed for filing accounts and reports. In general this is the same date the relevant number of months later. So, for example, if the end of the accounting reference period is 5th June, 6 months from then is 5th December. However, as months are of unequal length, there can be confusion as to whether 6 months from say 30th June is 30th December (exactly 6 months later) or 31st December (the end of the sixth month). Under the rule laid down in this section, 6 months from 30th June will be 31st December. This reverses the “corresponding date rule” laid down by the House of Lords in Dodds v Walker 1 WLR 1027 .
Sections 444 to 448: Filing obligations of different descriptions of company
690.These sections concern the filing obligations of different sizes of company. They restructure the provisions in sections 242, 246, 246A and 254 of the 1985 Act to make clearer what companies have to do.
691.Section 444 concerns the filing obligations of companies subject to the small companies regime. Such companies may file abbreviated accounts and this section gives the Secretary of State the power to make regulations concerning abbreviated accounts for such companies. Under subsection (5), small companies filing a full balance sheet with the registrar (whether prepared in accordance with international accounting standards or under the Act), but omitting a copy of the profit and loss account and/or the directors’ report, must include a statement on the balance sheet that they are delivered in accordance with the small companies regime. Subsection (7) requires the filed copy of the audit report to state the name of the auditor and, if there is one, of the senior statutory auditor, unless they are taking advantage of the exemption in section 506, in which case they must state that they are doing so.
692.Section 445 restates provisions in section 246A of the 1985 Act permitting medium-sized companies (as defined in section 465) to file abbreviated accounts and gives the Secretary of State the power to make regulations concerning abbreviated accounts for such companies.
693.Section 446 concerns the filing obligations of unquoted companies.
694.Section 447 concerns the filing obligations of quoted companies. This is a restatement of section 242 of the 1985 Act. Subsection (3) provides for the copies of the filed documents including the balance sheet to state the name of the person who signed the documents.
695.Section 448 replaces section 254 of the 1985 Act. It exempts unlimited companies from the obligation to file accounts. There are limitations on the exemption set out in subsections (2) and (3).
Sections 449 and 450: Requirements where abbreviated accounts delivered
696.Section 449 replaces the provision in section 247B of the 1985 Act. It requires a special auditor’s report in place of the auditor’s report required by section 495 where a company delivers abbreviated accounts to the registrar of companies. There is no requirement for the special auditor’s report where the company is entitled to exemption from audit and has taken advantage of that exemption.
697.Section 450 replaces sections 246(7) and (8) and 246A(4) of the 1985 Act concerning the approval and signing of abbreviated accounts.
Sections 451 to 453: Failure to file accounts and reports
698.Sections 451 and 452 re-enact sanctions in section 242(2) to (5) of the 1985 Act for failing to file accounts and reports within the required periods.
699.Section 453, which provides a civil penalty for failure to file accounts, restates section 242A of the 1985 Act with one change. Rather than setting out the table of penalties in the legislation, subsection (2) provides for the Secretary of State to make regulations specifying both the relevant periods and the amounts of the penalties. Regulations that have the effect of increasing the penalty will be subject to the affirmative resolution procedure. Otherwise, they will be subject to the negative resolution procedure.
Chapter 11: Revision of Defective Accounts and Reports
Section 454: Voluntary revision of accounts etc
700.This section restates section 245 of the 1985 Act providing for the voluntary revision of defective accounts and reports and summary financial statements. It replicates the existing power for the Secretary of State to make provision in regulations as to the application of the provisions of this Act to revised annual accounts and reports and summary financial statements. Regulations under this section are subject to the negative resolution procedure, which is consistent with the existing powers.
Section 455: Secretary of State’s notice in respect of accounts or reports
701.This section re-enacts section 245A of the 1985 Act. It concerns the Secretary of State’s giving notice to the directors of a company if there is or may be a question as to whether the annual accounts or directors’ report comply with the requirements of the Act or the IAS Regulation (Regulation (EC) 1606/2002 on the application of international accounting standards).
Sections 456 to 458: Application to court
702.Sections 456 and 457 concern applications to the court in respect of defective accounts or reports. They re-enact sections 245B and 245C of the 1985 Act. Section 457 gives the Secretary of State the power to authorise a person for the purposes of section 456 to apply to the courts to require the directors of companies to prepare revised accounts and reports where the original accounts or reports were defective. Authorisation is subject to the negative resolution procedure, which corresponds to the existing provision. The Financial Reporting Review Panel (FRRP) is the only authorised person under this provision to date (the Companies (Defective Accounts) (Authorised Person) Order 2005: SI 2005/699).
703.Section 458 re-enacts sections 245D and 245E of the 1985 Act. It provides for the disclosure of information by the Commissioners for Her Majesty’s Revenue and Customs to a person authorised under section 457 (currently the FRRP) to apply to the court in respect of defective accounts and reports. The provision contains important limitations, including criminal offences for use or disclosure of the information other than for permitted purposes. Section 458(5)(b)(ii) increases the term of imprisonment from three months to six months for a person convicted on summary conviction in Scotland or Northern Ireland for an offence of unlawful disclosure. Subsection (2) provides that personal data may not be disclosed in contravention of the Data Protection Act 1998.
Section 459: Power of authorised person to require documents, information and explanations
704.This section re-enacts section 245F of the 1985 Act. Subsections (1) to (3) provide the FRRP (as the person authorised under section 457) with a statutory power to require a company and its officers, employees and auditors to provide documents and information. Where a person refuses to provide information or documents to the FRRP, the FRRP may apply to the court for an order. The court may make an order requiring disclosure. Failure to comply with such an order would be contempt of court.
Section 460: Restrictions on disclosure of information obtained under compulsory powers
705.This section re-enacts section 245G of the 1985 Act. It ensures that information obtained by the FRRP under the powers in section 459 is subject to restrictions on onward disclosure. Information relating to the private affairs of an individual or to any particular business may not be disclosed by the FRRP without the consent of the individual or business in question, except for the purposes of carrying out the FRRP’s functions, or unless it is disclosed to specified persons or for specified purposes set out in section 461.
Section 461: Permitted disclosure of information obtained under compulsory powers
706.This section restates section 245G(3) of, and Schedule 7B to, the 1985 Act with modifications. It sets out the disclosures of information obtained by the authorised person under section 459 that are permitted. Subsection (3) lists the specified persons to whom disclosures are permitted and subsection (4) lists the specified purposes for which disclosure may be made. Subsections (5) and (6) set out the circumstances in which a disclosure to an overseas regulatory authority is permitted. Subsection (7) provides that nothing in the section authorises a disclosure in contravention of the Data Protection Act 1998.
Section 462: Power to amend categories of permitted disclosure
707.This section re-enacts section 245G(4) to (6) of the 1985 Act. It gives the Secretary of State power to amend the disclosure provisions relating to information obtained by the authorised person. As under the current law, an order under the section is subject to the negative resolution procedure.
Chapter 12: Supplementary Provisions
Section 463: Liability for false or misleading statements in reports
708.This section is concerned with the extent of directors’ liability in relation to the statutory narrative reporting requirements under this Part of the Act (accounts and reports). Subsection (1) specifies that the liability provision applies to statements made in the directors’ report (which includes the business review under section 417), the directors’ remuneration report (under section 420) or summary financial statements derived from them. Subsection (2) limits the directors’ liability to the company only in respect of loss suffered by it as a result of any untrue or misleading statement in a report, or the omission from a report of anything required to be included. Subsection (3) specifies that a director will only be liable in certain circumstances – that is, if an untrue or misleading statement is made deliberately or recklessly, or an omission amounts to dishonest concealment of a material fact. Subsection (4) ensures that third parties, such as auditors, will remain liable only to the company for negligence in preparing their own report. Subsection (6) ensures that these liability provisions do not affect any liability for a civil penalty or for a criminal offence.
Section 464: Accounting standards
709.This section re-enacts section 256 of the 1985 Act.
Sections 465 to 467: Companies qualifying as medium-sized
710.Medium-sized companies benefit from certain limited accounting and reporting exemptions. For example, section 417(7) exempts medium-sized companies from disclosing certain non-financial information in their directors’ reports.
711.Sections 465 to 467 set out which companies or parent companies qualify as medium-sized. The conditions for qualification as a medium sized company have been separated from those relating to small companies to make them easier to follow but are otherwise unchanged from the current regime (sections 247, 247A and 249 of the 1985 Act), save that, as in the case of the definition of small companies, the definition of balance sheet total in section 465(5) has been generalised.
Section 468: General power to make further provision about accounts and reports
712.This section gives the Secretary of State a general power to amend Part 15 by regulations in the areas specified in subsection (1)(a) to (d). This power, together with a number of specific powers in Part 15 to enable the form and contents of accounts and reports to be prescribed by regulations, replaces the wider general power in section 257 of the 1985 Act. Subsection (3) provides that the general power may not be used to amend the provisions of section 393 (accounts to give true and fair view) or Chapter 11 (revision of defective accounts and reports) other than consequentially. Subsections (4) and (5) enable regulations under the section to create criminal offences or provide for civil penalties in circumstances corresponding to those in Part 15. The regulations are subject to the Parliamentary procedure in section 473.
Section 469: Preparation and filing of accounts in euros
713.This section re-enacts section 242B of the 1985 Act, replacing references to ECUs with references to euros. It enables companies to show the amounts in their annual accounts additionally in euros, and to deliver to the registrar an additional copy of their accounts translated into euros.
Section 470: Power to apply provisions to banking partnerships
714.This section re-enacts section 255D of the 1985 Act. It gives the Secretary of State the power to apply the accounting and reporting provisions of this Act that apply to banking companies to banking partnerships. As under the current law, the regulations are subject to the affirmative resolution procedure.
Section 471: Meaning of “annual accounts” and related expressions
715.This section provides definitions of the terms “annual accounts” and “annual accounts and reports” for the purpose of this Part, the meaning being different for unquoted and quoted companies.
Section 472: Notes to the accounts
716.This section re-enacts section 261 of the 1985 Act. It concerns the notes to a company’s accounts.
Section 473: Parliamentary procedure for certain regulations under this Part
717.This section specifies the Parliamentary procedure that must be followed in connection with regulations made under the various provisions of this Part which replace the requirements as to the form and content of accounts and reports currently contained in Schedules to Part 7 of the 1985 Act, and in relation to the general regulation-making power in section 468. This section follows section 257 of the 1985 Act in requiring affirmative resolution procedure for regulations which add to the documents required to be prepared by companies, restrict the exemptions available to particular classes or types of company, add to the information to be included in any particular document or otherwise make the requirements more onerous. Other regulations are subject to negative resolution procedure.
Section 474: Minor definitions
718.This section contains other definitions for the purposes of this Part.
Part 16: Audit
719.This Part brings together various provisions on the audit of companies from the 1985 Act. It also introduces a number of significant changes to the law on auditing. Much of the law in this area reflects EU Company Law Directives, including parts of the Fourth (78/660/EEC), Seventh (83/349/EEC) and Eighth (84/253/EEC) Company Law Directives, and of the new Audit Directive (2006/43/EC), which will replace the Eighth.
Chapter 1: Requirement for Audited Accounts
720.This Chapter restates the existing requirement for company accounts to be audited, currently in section 235 of the 1985 Act, and the existing exemptions (except for the special provisions about charities).
721.The only changes from the existing law in this Chapter are the removal of special rules for the audit of the accounts of small charitable companies (see note on section 1175), and new provisions disapplying the requirement for audit in relation to certain companies in the public sector audited by public sector auditors.
Section 475: Requirement for audited accounts
722.This section restates the basic requirement for accounts to be audited, unless they are exempt. The obligation is now expressed as a duty on the company to have its accounts audited, whereas section 235 of the 1985 Act expressed it as a duty on the auditor to audit the accounts.
723.Directors must state in the balance sheet if they are taking advantage of an exemption. Unless the company is subject to a public sector audit, the statement must say that the members have not required an audit, and that the directors take responsibility for producing compliant accounts.
Section 476: Right of members to require audit
724.This section restates the right of members to require an audit, even if the company qualifies for one of the audit exemptions.
Sections 477 to 479: Exemption from audit: small companies
725.These sections restate the exemption from audit for small companies. Section 477 provides that a company must not only meet the general small company criteria in section 382, but its turnover and balance sheet totals must also fall below £5.6 million and £2.8 million respectively.
726.Section 478 excludes from the exemption various categories of company including public companies and some financial services companies. Section 479 sets out the conditions for qualification for the small company exemption of a company which is in a group.
Sections 480 and 481: Exemption from audit: dormant companies
727.These two sections restate the exemption from audit available to dormant companies. “Dormant” is defined in section 1169. Certain financial services companies are excluded from using the exemption even if they are dormant.
Sections 482 and 483: Companies subject to public sector audit
728.These two sections, the only wholly new provisions in this Chapter, are intended to ensure that certain non-commercial, public sector bodies constituted as companies that are audited by a public sector auditor are not required to be audited under the Act.
729.Section 482 exempts from Companies Act audit any non-departmental public body that is a company and is non-profit-making, if it is subject to public sector audit.
730.A UK body may be subject to public sector audit by virtue of an order under the Government Resources and Accounts Act 2000 The body in question will then be audited by the National Audit Office on behalf of the UK Comptroller and Auditor General. Under the Audit and Accountability (Northern Ireland) Order 2003, an order can make a body subject to audit by the Comptroller and Auditor General for Northern Ireland. Alternatively, a body may be subject to audit by the Auditor General for Wales under section 96 of the Government of Wales Act 1998, or an order under section 144 of that Act.
731.Some Scottish bodies are subject to public sector audit by the Auditor General for Scotland (AGS) under the Public Finance and Accountability (Scotland) Act 2000.
732.The companies exempted by this section are not subject to the Fourth Company Law Directive: the Directive is based on Article 44(2)(g) of the EC Treaty (formerly 54(3)(g) of the EEC Treaty), and Article 48 of the Treaty excludes from the scope of Article 44 undertakings that are non-profit-making. That is why subsection (3) gives “non-profit-making” the same meaning as in the Treaty.
733.Subsection (2) provides that a group company can benefit from this exemption only if every company in the group is non-profit-making. The effect of s ubsection (4) is that the exemption is not available unless the balance sheet contains a statement that the company is entitled to it.
734.Section 483 confers a new power on Scottish Ministers to provide that a company should have its accounts audited by the Auditor General for Scotland (AGS). This is available for companies depending on their functions or their funding. The Scottish Ministers can designate a company under this power if its functions are public functions that are all covered by the Scottish Parliament’s responsibilities, or if the company receives all or most of its funding from a public body already audited by the AGS. In the latter case, the funding body may be audited by the AGS because it is covered by the Public Finance and Accountability (Scotland) Act 2000, or because it is itself a company that Scottish Ministers have made auditable by the AGS by a previous order under this section.
735.If an order is made under this section providing that a company should have a public sector audit by the AGS, and if that company is non-profit-making, then it will benefit from the exemption from audit in the preceding section.
Section 484: General power of amendment by regulations
736.This section provides a power for the Secretary of State to amend the provisions of this Chapter. Taken together with section 468, it broadly restates the power in section 257 of the 1985 Act. Subsection (2) enables the regulations to make consequential changes to other legislation. The power is subject to affirmative resolution if it is extending the requirement for audit, or otherwise making requirements more onerous; and to negative resolution otherwise.
Chapter 2: Appointment of Auditors
737.This Chapter broadly restates the existing law in sections 384 to 388A of the 1985 Act on the way in which shareholders appoint a company’s auditors, with some minor changes (as explained below). The provisions are reorganised to deal with private and public companies separately. The Chapter also restates the rules in sections 390A and 390B of the 1985 Act on auditors’ remuneration and the disclosure required of services provided by auditors and introduces a new power for the Secretary of State to require disclosure of the terms of audit appointments.
Private companies
738.Sections 485 to 488 restate the law on appointment of auditors of private companies, providing that auditors are generally to be appointed by shareholders by ordinary resolution. For any financial year other than the first, this will generally be done within 28 days of the circulation to a company’s shareholders of the accounts for the previous year.
739.There are two changes: firstly, an auditor’s term of office will typically run from the end of the 28 day period following circulation of the accounts until the end of the corresponding period the following year. This will apply even if the auditor is appointed at a meeting where the company’s accounts are laid. The second change is that an auditor is now deemed to be re-appointed unless the company decides otherwise.
Section 485: Appointment of auditors of private company: general
740.This section provides for a private company’s obligation to appoint an auditor, unless it is taking advantage of an exemption from audit. The appointment is to be made by the shareholders by ordinary resolution, except that the directors can appoint the company’s first auditor (or the first after a period of audit exemption), and can fill a casual vacancy.
Section 486: Appointment of auditors of private company: default power of Secretary of State
741.This section requires a company to inform the Secretary of State if it has failed to appoint an auditor within 28 days of circulation of its accounts. The Secretary of State has power to appoint an auditor in those circumstances.
Section 487: Term of office of auditors of private company
742.This section provides that the end of the term of office of the auditor of a private company is to be the end of the next period for appointing auditors. At the end of his term an auditor will automatically be deemed to be re-appointed except in five cases:
if he was appointed by the directors;
if the company’s articles require actual re-appointment;
if enough members have given notice to the company under section 488;
if there has been a resolution that the auditor should not be reappointed; or
if the directors decide that they do not need auditors for the following year.
743.When there is a change of auditor the term of office of the incoming auditor does not begin before the end of the previous auditor’s term. This means that a new auditor’s term will typically begin immediately after the end of the 28-day period for appointing auditors.
Section 488: Prevention by members of deemed re-appointment of auditor
744.This section enables members with at least 5% of the voting rights in a private company to prevent an auditor being automatically re-appointed by giving notice to the company. The company’s articles can enable members to do this with less than 5% of the voting rights, but cannot increase the required percentage.
745.Subsection (3) provides that the deadline for a notice preventing the deemed reappointment of an auditor is the end of the financial year for the accounts he is auditing.
Public companies
746.Sections 489 to 491 restate the law on appointment of auditors of public companies, providing that auditors are generally to be appointed by shareholders by ordinary resolution in the general meeting before which the company’s accounts are laid.
Section 489: Appointment of auditors of public company: general
747.This section restates a public company’s obligation to appoint auditors, unless it is taking advantage of exemption from audit. This is to be done by the shareholders by ordinary resolution, normally at the general meeting at which the accounts are laid. The directors can appoint the company’s first auditors (or the first after a period of audit exemption), and can fill a casual vacancy.
Section 490: Appointment of auditors of public company: default power of Secretary of State
748.This section restates the obligation of a company to inform the Secretary of State if it has failed to appoint an auditor at the general meeting that considers the previous year’s accounts. The Secretary of State has power to appoint an auditor in those circumstances.
Section 491: Term of office of auditors of public company
749.This section restates the rule that an auditor of a public company holds office until the end of the meeting at which the accounts are laid, unless re-appointed. Where there is a change of auditor, the term of office of the incoming auditor does not begin before the end of the previous auditor’s term. This means that a new auditor’s term will typically begin immediately after the end of the accounts meeting.
General provisions
750.These sections apply to both private and public companies.
Section 492: Fixing of auditor’s remuneration
751.This section restates the rule that it is the members of a company, by ordinary resolution, who determine the auditor’s remuneration, or decide the method by which it should be determined. If the auditor was appointed by someone other than the members, then it will be the directors or the Secretary of State as appropriate who will determine his remuneration.
Section 493: Disclosure of terms of audit appointment
752.This section creates a new power for the Secretary of State to require companies to disclose information about the terms on which they engage their auditors. Subsection (2) provides some examples of the detailed requirements that the Secretary of State could specify in regulations. Subsection (3) provides that regulations can require disclosure of changes in terms as well as the terms at the time of appointment. Subsection (4) specifies that the regulations are to be made by affirmative resolution procedure.
Section 494: Disclosure of services provided by auditor or associates and related remuneration
753.This section restates the existing power of the Secretary of State, in section 390B of the 1985 Act, to require disclosure of details of all the services supplied to a company by its auditor, and the remuneration involved. Subsections (2) to (4) give the detailed requirements that the Secretary of State can specify in regulations: subsection (2) relates to the level of disaggregation of different services and remunerations, and between the auditor and his associates; subsection (3) makes provision for some of the definitional issues that can be covered in regulations; and subsection (4) provides for where the information should be disclosed.
754.Under subsection (4), the regulations might require disclosure in a document compiled by the company rather than the auditor. Subsection (5) provides that, if so, the regulations can require the auditor to supply the directors with the information to be disclosed e.g. about the auditor’s associates. Subsection (6) specifies that the regulations are to be made by negative resolution procedure.
Chapter 3: Functions of Auditors
Sections 495 to 497: Auditor’s report
755.These sections restate, with modifications, the provisions of section 235 of the 1985 Act as to what the auditor should include in his report on the accounts.
756.Section 495 imposes the basic duty to produce an audit report and requires that it should set out the way the auditor has approached the audit. Subsection (3) requires the auditor in his report to state his opinion on three matters: (i) whether the accounts provide a true and fair view, (ii) whether they comply with the appropriate reporting framework, and (iii) whether the accounts comply with the requirements in Part 15 of the Act (and, where applicable, with article 4 of the IAS Regulation (Regulation (EC) 1606/2002 on the application of international accounting standards)). Subsection (4) requires the audit report to be either qualified or unqualified, though it is open to the auditor to draw attention to aspects of his audit without qualifying the report.
757.Sections 496 and 497 restate the law on what the auditor should include in relation to the directors’ report and the directors’ remuneration report.
Sections 498 to 502: Duties and rights of auditors
758.These sections bring together and restate the existing law on the auditor’s duties (currently in section 237 of the 1985 Act) in investigating, forming an opinion, and making his report; and on the auditor’s rights (sections 389A to 390 of that Act) to be provided with appropriate information.
759.Section 498 lists areas where an auditor must investigate and report on any problems: the company’s accounting records, and whether there is consistency between these and (i) the accounts and (ii) – where there is one – the appropriate part of the directors’ remuneration report. The auditor is also to report if he has not been able to get all the information he needs. If possible, he is to make good any gaps in the information relating to payments to directors. And he is to report if he believes that the company is taking advantage of the small companies accounts regime without being entitled to do so.
760.Section 499 restates the auditor’s rights to obtain information and explanations from the company and its UK subsidiaries, and from appropriate associated individuals. Section 500 sets out the corresponding right to require the company to obtain information or explanations from any subsidiaries that are not incorporated in the UK.
761.Section 501 sets out offences for those who supply inaccurate information to auditors or fail to respond to auditors’ requests for information without delay.
762.Section 502 requires a private company to send to its auditor all the information about any written resolutions that it sends to its shareholders. It also gives the auditor of any company – public or private – the right to attend any general meetings it may have, and to be allowed to speak on anything relevant to the audit. The auditor must also receive all communications relating to general meetings.
Section 503: Signature of auditor’s report
763.This section specifies who must sign the audit report submitted to a company by its auditor. The report must state the name of the audit firm, or if an individual has been appointed as auditor, his name. This is as currently required by section 236 of the 1985 Act.
764.For cases where the auditor is a firm, the section makes a change from the 1985 Act by requiring the senior statutory auditor, as defined in section 504, to sign the report in his own name on behalf of the firm. This implements a requirement of the Audit Directive (2006/43/EC). If the auditor is an individual, he must sign as under the 1985 Act.
Section 504: Senior statutory auditor
765.This section defines a new term – “the senior statutory auditor” – for the individual who will be asked to sign his name to an audit report carried out by a firm. The firm will identify this individual according to standards to be issued by the European Commission, or if there are no standards, to guidance issued either by the Secretary of State or by a body appointed by him by order subject to negative resolution. Subsection (2) specifies that to be identified as a senior statutory auditor of a company, an individual must be eligible himself to be appointed as auditor of the company. Subsection (3) ensures that for an individual to be nominated as senior statutory auditor will not affect his exposure to liability in any way.
Section 505: Names to be stated in published copies of auditor’s report
766.This section requires a company to ensure that the copies of its auditor’s report it sends out include the name of the auditor and of the senior statutory auditor if there is one, or to say that it is taking advantage of the exemption in the following section. Subsection (2) provides that this includes copies circulated to shareholders, as well as any others that would be expected to be seen by members of the public. It does not, however, cover copies sent to the registrar: these are dealt with by sections 444(7), 445(6), 446(4) and 447(4). Subsections (3) and (4) restate the offence, currently in section 236 of the 1985 Act, of not including the auditor’s name – and now also the senior statutory auditor’s name – as required.
Section 506: Circumstances in which names may be omitted
767.This section provides an exemption from the requirements to include the names of the auditor in both the published and filed copies of the audit report. This is available if the company passes a resolution not to reveal the names because it considers on reasonable grounds that revealing them would lead to a serious risk of violence or intimidation. It is also a condition of using the exemption that the company must inform the Secretary of State, giving details of the name of the auditor, and of the senior statutory auditor if there is one.
Section 507: Offences in connection with auditors’ report
768.This section creates a new criminal offence in relation to inaccurate auditors’ reports. The offence consists of knowingly or recklessly causing a report to include anything that is misleading, false or deceptive, or omitting a required statement of a problem with the accounts or audit.
769.Subsection (1) sets out the offence of commission, and subsection (2) that of omission. The items whose omission can be an offence are listed in paragraphs (a) to (c) of subsection (2): statements about accounting records not being properly reflected in the accounts, about the auditor having been unable to obtain all necessary information and explanations, and about the directors wrongly claiming the company is exempt from the requirement for group accounts.
770.Subsection (3) defines the individuals potentially caught by the offence as the auditor, if a sole practitioner, and his employees and agents; and the directors, members, employees and agents of an audit firm. But the offence only applies to such an individual if he is an accountant who would be qualified to act as auditor of the company in his own right. Subsection (4) sets out the maximum penalty as an unlimited fine.
Section 508: Guidance for regulatory and prosecuting authorities: England, Wales and Northern Ireland
771.This section enables the Secretary of State to issue guidance about handling matters where the same behaviour by an auditor could give rise both to disciplinary proceedings by a regulatory body, and to prosecution for the new offence. Subsection (2) requires the Secretary of State to obtain the Attorney’s General agreement to any guidance. Subsection (3) lists the regulatory and prosecuting authorities the guidance would be intended to help. The list comprises the accountancy supervisory bodies, recipients of grants under section 16 of the Companies (Audit, Investigations and Community Enterprise) Act 2004) (currently the Financial Reporting Council and its subsidiaries), the Director of the Serious Fraud Office and the Director of Public Prosecutions, as well as the Secretary of State himself. Under subsection (4), the Secretary of State’s guidance is limited to England, Wales and Northern Ireland.
772.It is likely that one of the most important aspects of the guidance would be to enable prosecutors to decide not to prosecute in a particular case that would be better handled through disciplinary proceedings.
Section 509: Guidance for regulatory authorities: Scotland
773.This section enables the Lord Advocate to issue guidance about handling matters in Scotland where the same auditor’s report could give rise both to disciplinary proceedings by a regulatory body, and to prosecution for the new offence. Subsection (2) requires the Lord Advocate to consult the Secretary of State before issuing guidance. Subsection (3) lists the regulatory bodies the guidance is intended to help. The list comprises the accountancy supervisory bodies, recipients of grants under section 16 of the C(AICE) Act 2004) (currently the Financial Reporting Council and its subsidiaries) and the Secretary of State.
Chapter 4: Removal, Resignation, Etc of Auditors
774.This Chapter restates the law on the ways in which auditors can cease to hold office. The current provisions are in section 388 and sections 391 to 394A of the 1985 Act. There are some changes to the existing law resulting from the changes elsewhere in the Act relating to written resolutions of private companies. There are also changes in the requirements when auditors leave office: increasing the range of cases in which there is a requirement for a statement explaining why they are leaving, and for copies of any statement to be sent to shareholders and to appropriate regulators.
Section 510: Resolution removing auditor from office
775.This section restates the rule that the shareholders in a company always have the right to dismiss its auditor by ordinary resolution. As at present, to remove the auditor before the end of his term of office, even a private company will need to hold a general meeting to pass such a resolution.
776.Subsection (2) requires special notice of the resolution (see note on section 511). Subsection (3) provides that shareholders’ right provided by this section does not prevent the auditor being entitled to being compensated for termination of his appointment. Subsection (4) specifies that the resolution described here is the only way in which an auditor can be removed before the end of his term of office.
Section 511: Special notice required for resolution removing auditor from office
777.This section restates the requirement that a resolution to dismiss an auditor needs special notice (i.e. 28 days before the general meeting, as provided in section 312). The company must send a copy to the auditor it is proposed to dismiss, and he has the right to make a statement of his case. The company then has to circulate his statement to the shareholders (or if time does not allow, the statement can be read out at the meeting).
778.Subsection (6) provides protection if the auditor it is proposed to dismiss is using the provision to have a statement circulated to secure needless publicity for defamatory material. It enables the company, or anyone else who is aggrieved by the statement, to apply to the court, and the court can then determine whether the auditor is using the provision in that way, in which case the company is not obliged to circulate the statement. The court can order the auditor to pay some or all of the costs of the proceedings.
Section 512: Notice to registrar of resolution removing auditor from office
779.This section restates the obligation on a company that has decided to dismiss its auditor to inform the registrar within 14 days.
Section 513: Rights of auditor who has been removed from office
780.This section restates the right of a dismissed auditor to attend certain meetings, namely, any meeting at which his term of office would have expired (i.e. a public company’s accounts meeting) and any meeting at which it is proposed to replace him.
Section 514: Failure to re-appoint auditor: special procedure required for written resolution
781.This section sets out the procedure for changing auditor from one financial year to the next by written resolution (a procedure only available to private companies). This may be done (i) during the term of office of the outgoing auditor, or (ii) afterwards, if no replacement has been appointed. But case (ii) will arise only if there is no automatic deemed reappointment for one of the five reasons in section 487(2).
782.Subsection (3) provides that the company must send a copy of the proposed resolution both to the outgoing auditor and to his proposed replacement; and subsection (4) provides that the former then has 14 days to make a statement setting out his views. Subsection (5) then provides that the company must send, to its shareholders, the resolution together with any statement from the outgoing auditor. Subsection (6) specifies how the general rules on written resolutions are to apply in this case.
783.Subsection (7) provides protection if the outgoing auditor is using the provision to have a statement circulated to secure needless publicity for defamatory material. It enables the company, or anyone else who is aggrieved by the statement, to apply to the court, and the court can then determine whether the auditor is using the provision in that way, in which case the company is not obliged to circulate the auditor’s representations. The court can order the auditor to pay some or all of the costs of the proceedings.
784.Subsection (8) provides that failure to comply with the rules in this section will make the resolution ineffective.
Section 515: Failure to re-appoint auditor: special notice required for resolution at general meeting
785.This section sets out the procedure for changing auditor between one financial year and the next at a general meeting. This may be done by resolution at the meeting, but special notice is required if no deadline for appointing auditors has passed since the outgoing auditor left, or if the deadline has passed when an auditor should have been appointed without one being appointed. So, for example, if a public company intends not to re-appoint an auditor at its accounts meeting, it would need to give special notice of the meeting to be able to appoint replacement auditors.
786.Subsection (3) provides that immediately it receives a proposed resolution for changing auditor, the company should send a copy of it both to the outgoing auditor and to his proposed replacement; and subsection (4) provides that the former may then send the company a written statement setting out his views. Subsections (5) and (6) provide that the company must send its shareholders any statement from the outgoing auditor, and that if it is received to late for this it should be read out at the meeting.
787.Subsection (7) provides protection if the outgoing auditor is using the provision to have a statement circulated to secure needless publicity for defamatory material. It enables the company, or anyone else who is aggrieved by the statement, to apply to the court, and the court can then determine whether the auditor is using the provision in that way, in which case the company is not obliged to circulate the auditor’s representations, nor need they be read out at the meeting. The court can order the auditor to pay some or all of the costs of the proceedings.
Section 516: Resignation of auditor
788.This section restates the right of an auditor to resign by written notice to the company. His resignation is effective from the date it is delivered to the company’s registered office, or from a later date specified in it. To be effective it must be accompanied by the statement required by section 519.
Section 517: Notice to registrar of resignation of auditor
789.This section restates the obligation on a company whose auditor resigns to inform the registrar. Default in complying is an offence.
Section 518: Rights of resigning auditor
790.This section restates the right of an auditor who resigns to require the directors to convene a general meeting of the company so that it can consider his explanation of the circumstances that led to his decision to resign. The auditor can ask the company to send out a written explanation either in advance of that meeting if he has requested one, or before the next appropriate general meeting. The directors have 21 days to send out a notice convening a meeting once a resigning auditor has asked for it, and it must then be held within 28 days of the notice.
791.Subsection (9) provides protection if the resigning auditor is using the provision to have a statement circulated to secure needless publicity for defamatory material. It enables the company, or anyone else who is aggrieved by the statement, to apply to the court, and the court can then determine whether the auditor is using the provision in that way, in which case the company is not obliged to circulate the statement. The court can order the auditor to pay some or all of the costs of the proceedings.
Section 519: Statement by auditor to be deposited with company
792.This section requires a departing auditor to make a statement when he stops being the auditor of a company and to deposit it with the company. For quoted companies, this statement should explain the circumstances surrounding his departure. For other public companies and all private companies, it should explain the circumstances unless the auditor thinks that there is no need for them to be brought to the attention of the shareholders or creditors. In that case, the statement should state that there are no such circumstances.
793.This changes the position under section 394 of the 1985 Act, where auditors were only required to make a statement if they considered there were relevant circumstances: auditors leaving quoted companies will now always be required to make a statement of the circumstances; and auditors leaving other companies must make a statement unless they think that there are no relevant circumstances.
794.Subsection (4) sets out the deadline for depositing such a statement with the company, namely:
if the auditor is resigning, the statement should accompany the resignation letter;
if the auditor is deciding not to seek re-appointment, the statement should be deposited at least 14 days before the end of the time allowed for appointing the next auditor; or
in any other case, no more than 14 days after the date on which he stops being the auditor.
Section 520: Company’s duties in relation to statement
795.Unless the departing auditor’s statement says that there are no circumstances to be brought to the attention of shareholders and creditors, this section obliges the company to circulate the statement to everyone to whom it needs to send the annual accounts. The company must do this within 14 days of receiving it.
796.If the company does not want to circulate the statement, it can apply to the court, and if the court decides that the departing auditor is trying to secure needless publicity for defamatory material, then the company need not circulate the statement, but instead must send an account of the court decision to those to whom it would have sent the statement. In the event of a successful application, the court can order the auditor to pay some or all of the costs. In the event of an unsuccessful application, the company must circulate the statement within 14 days of the end of the court proceedings.
Section 521: Copy of statement to be sent to registrar
797.This section provides that the departing auditor must send a copy of his statement to the registrar, unless within 21 days of depositing it he hears that the company has applied to the court. If he does not hear of an application to the court within that time, he must send the statement to the registrar within the next seven days; and if an application is made and the company lets him know that it was unsuccessful, he must send the statement to the registrar within seven days of being told.
Section 522: Duty of auditor to notify appropriate audit authority
798.This section introduces a new obligation on departing auditors to send copies of their leaving statements to an appropriate audit authority as defined in section 525. It contains different rules depending on whether the company the auditor is leaving is classified as a “major audit” as defined in section 525.
799.In relation to major audits, the departing auditor should always send a copy of his statement to the appropriate audit authority. He should do this as the same time as he deposits his statement with the company under section 519. In relation to other audits, the departing auditor is required to send his statement to the appropriate audit authority only if he is leaving before the end of his term of office, meaning only if he has resigned or has been dismissed; and he must do so at the time required by the authority.
800.Subsection (3) provides that where the auditor’s statement to the company said that there were no circumstances that needed to be brought to the attention of shareholders or creditors, that statement must have attached to it a statement of the auditor’s reasons for leaving when sending it to the audit authority.
801.Subsections (5) to (8) set out the offence of failure to comply with these requirements, and the maximum penalties.
Section 523: Duty of company to notify appropriate audit authority
802.This section introduces a new duty on a company to notify the appropriate audit authority whenever an auditor leaves office before the end of his term, that is when he has resigned or is dismissed. The company has the choice of sending in the statement of circumstances made by the auditor under section 519, or of sending in its own statement of the reasons. Subsection (3) sets the deadline for notification as 14 days after the auditor has deposited his statement with the company. Subsections (4) to (6) set out the offence of failure to comply with this requirement, and the maximum penalties .
Section 524: Information to be given to accounting authorities
803.This section sets out the duty of the audit authorities to give the accounting authorities information about auditors’ departure, and the power, if they think it right to do so, to pass on the statements which they receive from departing auditors under the section 522 or from companies under section 523. The accounting authorities are the Secretary of State and anyone the Secretary of State has authorised under Part 15 to apply to the court in respect of the revision of defective accounts. At present this is the Financial Reporting Review Panel, part of the Financial Reporting Council organisation.
804.Subsection (3) deals with the situation where the same body is both an audit authority and an accounting authority. If an accounting authority receives a statement that the court has determined need not be circulated to members, then subsection (4) provides that it must treat the statement as confidential, in the same way that authorities have to treat information obtained under compulsory powers under Part 15.
Section 525: Meaning of “appropriate audit authority” and “major audit”
805.This section defines two terms used in connection with the duty to inform the audit authority when an auditor leaves office, namely appropriate audit authority and major audit. The former means the Secretary of State, or the body to whom he has delegated functions in relation to the supervision of statutory auditors under Part 42, currently the Professional Oversight Board, part of the Financial Reporting Council organisation.
806.A major audit is defined as meaning the audit of a listed company, or of any other company where there is a major public interest. Whether there is a major public interest is to be determined by reference to guidance issued by any of the audit authorities. In practice, this will generally be guidance issued by the Financial Reporting Council.
Section 526: Effect of casual vacancies
807.This section applies when one out of two or more joint auditors ceases to be an auditor of the company. It enables the remaining auditors to continue in office. It restates section 388(2) of the 1985 Act.
Chapter 5: Quoted Companies: Right of Members to Raise Audit Concerns at Accounts Meeting
808.This Chapter introduces a new right for members of a quoted company to raise questions about the work of the auditors (all shareholders in a company limited by shares are members).
Section 527: Members’ power to require website publication of audit concerns
809.This section creates a new right for members of a quoted company – if they have a large enough holding in the company, or there are enough of them – to ask the company to publish on a website a statement raising questions about the accounts, or about the departure of an auditor, that they propose to bring up at the next meeting where the accounts are to be discussed.
810.Subsection (2) specifies the thresholds the members have to meet, which are the same as for shareholders who want to ask a company to circulate a statement under section 314: they must either have 5% of the total voting rights, or there must be at least 100 of them, holding shares on which there has been paid up an average sum per member of at least £100. Subsection (4) sets out the mechanics of transmitting the request to the company: it may be in hard copy or electronic.
811.Subsection (5) protects the company if members abuse the new right, e.g. by requesting a defamatory statement to be published. It enables the company, or someone else such as the auditor or a director, to apply to the court, and the court can then determine whether the right is being abused, in which case the company is not obliged to publish the statement. Subsection (6) provides that the court can order the shareholders who requested publication to pay some or all of the costs of the proceedings.
Section 528: Requirements as to website availability
812.This section sets out the requirements which the company must meet in making the shareholders’ statements available on a website, in the same way as section 353. Subsection (4) requires the company to get the statement onto a website within three days of receiving it, and to keep it available at least until after the meeting to which it relates.
Section 529: Website publication: company’s supplementary duties
813.This section requires quoted companies to draw attention to the possibility of a website statement in the notice of the accounts meeting. It also specifies the costs of publication are to be borne by the company. Subsection (3) requires the company to forward the statement to the auditor at the same time as it puts it on a website. Subsection (4) provides that a statement under this chapter can be dealt with at the accounts meeting.
Section 530: Website publication: offences
814.This section provides for offences when a company fails to comply with either of the preceding two sections, with maximum penalties of an unlimited fine.
Section 531: Meaning of “quoted company”
815.This section defines the phrase “quoted company” for the purposes of Chapter 5 of Part 16 as being the same as the definition in section 385 in Part 15, and that the power in Part 15 to amend the definition also applies in this Chapter.
Chapter 6: Auditors’ Liability
816.This Chapter makes it possible for auditors to limit their liability by agreement with a company, but the agreement will be effective only to the extent that it is fair and reasonable.
817.It achieves this by defining a “liability limitation agreement” – a contractual limitation of an auditor’s liability to a company, requiring member agreement – as a new exception to the general prohibition, restated here, on a company indemnifying its auditor. The court will be able to substitute its own limitation if the agreement purports to limit liability to an amount that is not fair and reasonable in all the circumstances.
Section 532: Voidness of provisions protecting auditors from liability
818.This section restates the existing general prohibition, currently in section 310 of the 1985 Act, against a company indemnifying its auditor against claims by the company in the case of negligence or other default. Any such indemnities are void and unenforceable except where permitted by sections 533 to 536.
Section 533: Indemnity for costs of successfully defending proceedings
819.This section contains the current exception from the prohibition in section 532 allowing the company to indemnify the auditor against the costs of successfully defending himself against a claim, though it does not repeat the current exception that allows the company to buy insurance for its auditor.
Section 534: Liability limitation agreements
820.This section defines a “liability limitation agreement” as an agreement that seeks to limit the liability of an auditor to a company whose accounts he audits. The agreement can cover liability for negligence, default, breach of duty or breach of trust by the auditor.
821.Subsection (2) provides that such an agreement is excepted from the general voidness of such agreements under section 532, provided that the agreement complies with the rules in section 535, and that it has been authorised by the members of the company in the way specified in section 536. Subsection (3) provides that the agreement’s effect is limited by section 537, which contains the test of fairness and reasonableness, and that certain provisions of the Unfair Contracts Terms Act 1977 do not apply.
Section 535: Terms of liability limitation agreement
822.This section contains rules about the terms of a liability limitation agreement. An agreement must relate to the audit of a specified financial year, and the limitation may be expressed in any terms, not necessarily as a fixed financial amount or a formula.
823.Subsection (2) confers on the Secretary of State a power to make regulations (subject to negative resolution) prescribing or proscribing specified provisions or descriptions of provisions; and subsection (3) provides that the power may be used to prevent adverse effects on competition.
Section 536: Authorisation of agreement by members of the company
824.This section specifies the way in which members of a company are to give their approval to a liability limitation agreement, without which approval the agreement will not be effective. The members of a private company can pass a resolution waiving the need for approval. The members in a private or a public company can pass a resolution before an agreement is signed approving its principal terms, or can approve the agreement after it is signed. The resolution may be an ordinary resolution, unless a higher threshold is set in the company’s articles.
825.Subsection (5) specifies what the principal terms of a liability limitation agreement are for this purpose, namely the terms that specify, or enable one to determine, (i) the sorts of faults by the auditor that are covered, (ii) the financial year in relation to which those faults are covered, and (iii) the limit on the auditor’s liability.
826.Subsection (6) provides that members, by passing an ordinary resolution, can withdraw their approval of a liability limitation agreement at any time before the agreement is entered into. If the company has already entered into the agreement, approval can be withdrawn, by ordinary resolution, only before the start of the financial year to which the agreement relates.
Section 537: Effect of liability limitation agreement
827.This section provides that a liability limitation agreement will not be effective to limit an auditor’s liability if the limitation would result in the company recovering an amount that was less that what was fair and reasonable, in all the circumstances of the case, having regard in particular to the auditor’s responsibilities, the auditor’s contractual obligations, and the standards expected of the auditor. If a court decides that a liability limitation agreement would limit the auditor’s liability to an excessive degree, the agreement will have effect as if it limited liability to the amount that the court determines is fair and reasonable.
828.Subsection (3) provides that in assessing what is fair and reasonable, the court should not take into account circumstances arising after the loss or damage in question has been incurred. Nor should it take into account the chances of the company successfully claiming compensation from any other people responsible for the loss or damage.
Section 538: Disclosure of agreement by company
829.This section requires companies to disclose any liability limitation agreement they have made with their auditor in accordance with any regulations made by the Secretary of State subject to negative resolution. Subsection (2) provides that the regulations may require this disclosure to be in a company’s annual accounts (or in any other manner in the case of group accounts), or in the directors’ report.
Chapter 7: Supplementary Provisions
Section 539: Minor definitions
830.This section defines a number of terms used in this Part.
Part 17: a Company’s Share Capital
831.This Part of the Act deals with various matters relating to a company’s share capital. It replaces Part 4 and (in part) Part 5 of the 1985 Act and contains a mixture of new sections which replace corresponding provisions in the 1985 Act and sections which restate corresponding provisions in that Act. Sections 541, 543 to 544, 547 to 548, 552 to 553, 558, 561, 563 to 568, 570 to 572, 574 to 577, 579 to 582, 584 to 588, 590 to 605, 607 to 609, 611 to 616, 645 to 648 and 655 to 656 restate various provisions in the 1985 Act but do not make any changes to those provisions.
Chapter 1: Shares and Share Capital of a Company
Section 540: Shares
832.As now, generally speaking, references to a “share” in the Companies Acts (defined in section 2) includes stock. However, as recommended by the CLR, in future it will no longer be possible for a company to convert its shares into stock (see subsection (2)) but a company that has stock at the date that this provision comes into force will be able to reconvert its stock back into shares (see note on section 620).
Section 542: Nominal value of shares
833.This is a new provision, which is required as a result of the changes to the requirements with respect to the memorandum (see note on section 8).
834.Currently, section 2(5)(a) of the 1985 Act (requirements with respect to memorandum) requires that, in the case of a company having a share capital, the memorandum of a limited company must state the amount of the share capital with which the company proposes to be registered and the division of that share capital into shares of a fixed amount. This capital figure as stated in the memorandum (known as the “authorised share capital”) acts as a ceiling on the amount of shares that a company may issue. Such authorised share capital may, however, be increased by ordinary resolution under section 121 of that Act. The CLR recommended that the requirement for a company to have an authorised share capital should be abolished (Final Report, paragraph 10.6), and so the Act does not require a company to state in its memorandum the amount of its authorised share capital.
835.This section is required as a consequence of the repeal of this requirement. It does two key things:
it makes it clear that the shares in a limited company having a share capital must have a fixed nominal value, e.g. 1p, £1, $1 or 1 euro and therefore prevents a company from issuing shares of no par value (thereby implementing for public companies, Article 8 of the Second Company Law Directive (77/91/EEC)); and
it places in statute the common law rule that shares may be denominated in any currency and that different classes of shares may be denominated in different currencies. However, this is subject to the requirement in section 765 that a public company may only satisfy its initial authorised minimum share capital requirement if its shares are denominated in either sterling or euros.
836.Where a company purports to allot shares without a fixed nominal value, every officer of the company who is in default commits an offence and is liable to a fine (see subsections (4) and (5)). Moreover, such a purported allotment is void (see subsection (2)).
837.This section needs to be read alongside section 9, which requires the application for registration of a company that is to be formed with a share capital to include a “statement of capital and initial shareholdings”. The contents of this statement are prescribed in section 10 and this includes a requirement to set out the total number of shares and the aggregate nominal value of the shares which are to be taken by the subscribers to the memorandum on formation.
Section 545: Companies having a share capital
838.Section 545 is a new provision which makes it clear that references in the Companies Acts (defined in section 2) to a company having a share capital are to company that has power under its constitution to issue shares.
Section 546: Issued and allotted share capital
839.Section 546 is a new provision which makes it clear that references in the Companies Acts (defined in section 2) to issued or allotted shares include the shares taken by the subscribers to the memorandum on the formation of a company.
Chapter 2: Allotment of Shares: General Provisions
840.Generally speaking, the directors of a company may currently only allot shares (or grant rights to subscribe for shares or to convert any security into shares) if they are authorised to do so by ordinary resolution of the company’s members or by the articles.
841.Such an authority may be general or specific (that is, it may, for example, be restricted to a specified allotment, an allotment of shares up to a specified value, or an allotment of shares of a particular class). In either case, the authority must state the “maximum amount of relevant securities that may be allotted under it” and the date when the authority will expire, (which must not be more than five years from the date on which the authority is given). The authority may be renewed for further periods not exceeding five years.
842.There is a relaxation for private companies from the requirement to state the date on which the authority will expire and so such companies may, by elective resolution under section 379A of the 1985 Act, give such authority either for an indefinite period or a fixed period of the company’s choice.
843.The Act removes for private companies the requirement for prior authorisation in certain circumstances (described in section 550). It also abolishes the concept of authorised share capital (see note on section 542) and a company’s constitution will therefore no longer have to contain a ceiling on the number of shares that the directors are authorised to allot.
Section 549: Exercise by directors of power to allot shares etc
844.This section replaces section 80(1), (2), (9) and (10) of the 1985 Act. It provides that the directors may not allot shares (or grant rights to subscribe for shares or to convert any security into shares) except in accordance with one of the following two sections.
845.Subsection (2) of this section provides that directors may allot shares in pursuance of an employees’ share scheme without having to comply with one of the following two sections. This mirrors the current position (see section 80(2) of the 1985 Act).
846.Similarly, where a right to subscribe for, or to convert any security into, shares already exists, then the directors may allot shares pursuant to that right without having to comply with one of the following two sections (see subsection 3).
847.A director who knowingly allots shares in contravention of the requirements imposed by this section commits an offence. Such an allotment is not, however, invalid.
Section 550: Power of directors to allot shares etc: private company with only one class of shares
848.In line with the recommendations of the CLR (Final Report, paragraph 4.5), this is a new provision which empowers the directors to allot shares (or to grant rights to subscribe for or convert any security into shares) where the company is a private company which will have only one class of shares after the proposed allotment and removes the current requirement, contained in section 80 of the 1985 Act, for the directors to have prior authority from the company’s members for such an allotment of shares. In addition, it provides that the members may, if they wish, restrict or prohibit this power through the articles. The definition of “class of shares” is contained in section 629.
Section 551: Power of directors to allot shares etc: authorisation by company
849.This section replaces section 80(1) and (3) to (8) of the 1985 Act and applies both to private companies which will have more than one class of shares after a proposed allotment and to public companies. It provides that the directors may only allot shares (or grant rights to subscribe for shares or to convert any security into shares) if they have been given prior authorisation for the proposed allotment by ordinary resolution of the company’s members or by the articles.
850.Subsections (2) to (5) set out details of the way in which prior authorisation (or a renewal of such authorisation) may be given and, in particular, provides that the authority may not be given for a period of more than five years. An authority given to the directors under this section, and any resolution of the company renewing such an authority, must state “the maximum amount of shares” to be allotted pursuant to the authority. This mirrors the formulation of words used in section 80 of the 1985 Act and enables the members to limit the authority to a specific number of shares or shares up to a given maximum nominal value.
851.Subsection (8) makes it clear that an ordinary resolution of the company’s members will suffice for the purposes of giving authority to the directors, even where the effect of the resolution is to alter the company’s articles of association (which would normally require a special resolution of the company’s members).
Section 554: Registration of allotment
852.This is a new provision which requires the directors to register an allotment of shares as soon as practicable (but in any event within two months of the date of allotment). Whereas the 1985 Act imposes a duty on the company to issue certificates within two months after the allotment of its shares it does not stipulate a timescale relating to the step which is anterior to this, namely the registration of the allotment.
853.Subsection (2) makes it clear that the requirement to register an allotment of shares does not apply if the company has issued a share warrant in respect of the shares in question (see section 779).
854.Where a company fails to comply with this section, the company and every officer of the company who is in default commits an offence. The penalty for this offence is set out in subsection (4).
Section 555: Return of allotment by limited company
855.This section replaces section 88 of the 1985 Act. As now, within one month of an allotment of new shares in a limited company, the company is required to make a return of allotments to the registrar. This return must contain “prescribed information” relating to the allotment (that is, prescribed by the Secretary of State by order or by regulations made under the Act).
856.A return of allotments made under this section must be accompanied by a statement of capital. A statement of capital is in essence a “snapshot” of a company’s total subscribed capital at a particular point in time (in this context, the date to which the return of allotments is made up).
857.The requirement for a statement of capital when an allotment of new shares is made is new. It is based on a recommendation by the CLR (Final Report, paragraph 7.30) and for public companies, this implements a requirement in the Second Company Law Directive (77/91/EEC) which states:
“the statutes or instruments of incorporation of the company shall always give at least the following information…(c) when the company has no authorized capital, the amount of the subscribed capital….”.
“Statutes” and “instruments of incorporation” equate to the articles and memorandum and the need to disclose information pertaining to the aggregate of a company’s subscribed capital flows from the abolition of the requirement for a company to have an authorised share capital (see note on section 542).
858.Whilst this Directive only applies to public companies, the requirement to provide a statement of capital, here and elsewhere in the Act, has been extended to private companies limited by shares (and in certain cases to unlimited companies having a share capital, for example, where such companies make their annual return to the registrar). This will mean that the public register will contain up-to-date information on a company’s share capital (the requirement for a statement of capital supplements existing provisions which require a company to give notice to the registrar when it amends its share capital in any way).
859.The information which will in future be set out in the statement of capital includes prescribed particulars of the rights attached to each class of shares. Again this information will be prescribed in regulations or by order made under the Act. Such information is currently required to be filed under either section 123 of the 1985 Act (which relates to increases in authorised share capital) or section 128(1) and (2) of that Act (which relates to allotments of a new class of shares).
860.Currently, if shares are allotted as fully or partly paid up otherwise than in cash, the company must deliver the contract that it has with the allottee (or details of this contract if it is not in writing) to the registrar. Such a contract may contain commercially sensitive information which the company would not normally want to disclose. This section does not reproduce this requirement. It should be noted, however, that, in prescribing the information which must be included in the return of allotments, the Secretary of State may require details of any consideration received in respect of shares which are allotted as fully or partly paid up otherwise than in cash.
Section 556: Return of allotment by unlimited company allotting new class of shares
861.This section requires unlimited companies to make a return of allotments to the registrar where the directors allot a new class of shares. This carries forward the provisions of section 128(1) and (2) of the 1985 Act as they apply to unlimited companies. The return must contain “prescribed particulars of the rights attached to the shares”, that is such information as may be prescribed by the Secretary of State in regulations or by order made under the Act.
Section 557: Offence of failure to make return
862.This section replaces section 88(5) and (insofar as it relates to a requirement for an unlimited company to register particulars of an allotment of a new class of shares) section 128(5) of the 1985 Act. Where a company fails to comply with the requirements to make a return of allotments to the registrar, every officer of the company who is in default commits an offence.
863.As now under section 88(6), where there is a default in making a return of allotments within the specified time (one month after the allotment) a person who is liable for the default may apply to the court for relief (see subsection (3) which extends the right to apply for relief to a person liable under section 556).
Section 559: Provisions about allotment not applicable to shares taken on formation
864.this provision replicates the effect of section 80(2)(a) of the 1985 Act and provides that the allotment provisions in Chapter 2 of this Part do not apply to the shares taken by the subscribers to the memorandum on the formation of a company. Such persons become members of the company in respect of the shares that are taken by them on formation by virtue of section 16 and the provisions of the Act on share allotments do not apply to them.
Chapter 3: Allotment of Equity Securities: Existing Shareholders’ Right of Pre-Emption
Section 560: Meaning of “equity securities” and related expressions
865.This section sets out a definition of “equity securities” for the purposes of Chapter 3 of this Part (which is concerned with the allotment of equity securities and existing shareholders’ right of pre-emption). It partially restates section 94(2), (3), (3A) and (5) of the 1985 Act. The exception for shares taken by a subscriber to the memorandum and for bonus shares provided in section 94(2) of the 1985 Act is contained in sections 577 and 564. The exclusion of the allotment of shares pursuant to the grant of a right to subscribe for such shares contained in section 93(3) of the 1985 Act is contained in section 561(3).
Section 561: Existing shareholders’ right of pre-emption
866.Subject to some exceptions, under section 89(1) of the 1985 Act, a company that is proposing to allot equity securities (defined in section 560) must offer them to existing shareholders first (that is, on a pre-emptive basis). The basic principle (which is unchanged by the Act) is that a shareholder should be able to protect his proportion of the total equity of a company by having the opportunity to subscribe for any new issue of equity securities. This is subject to various exceptions and subsection (5) provides a pointer to these exceptions.
Section 562: Communication of pre-emption offers to shareholders
867.This section replaces section 90(1), (5) and (6) of the 1985 Act. Section 90(6) of the 1985 Act provides that where a company communicates a pre-emption offer to its existing shareholders the offer must state a period of not less than 21 days during which it may be accepted and it may not be withdrawn before the end of that period. This section contains a new provision which gives the Secretary of State the power to vary, in regulations made under the Act, the period of 21 days (but not so as to reduce it to fewer than 14 days) – see subsection (6).
868.It also updates the 1985 Act provision to ensure that the communications of pre-emption offers to shareholders continue to be compatible with EU law: in particular, in future companies will be required to give individual notice (which may be in hard copy or electronic form) to all shareholders who have a registered address in the EEA or who have given an address for service of notices in the EEA (under the 1985 Act, a company is only required to give individual notice to shareholders who have given a service address in the UK). As now, where no relevant address for service has been provided, the company may discharge its obligation by causing notice of the offer to be published in the London, Edinburgh or Belfast Gazette as appropriate.
Sections 569 to 573: Disapplication of pre-emption rights
869.This group of sections deals with the circumstances in which the statutory pre-emption requirements may be disapplied or modified by a power under the articles or by special resolution in accordance with the detailed rules in these sections. The rules replace or restate equivalent provisions in section 95 of the 1985 Act.
870.Section 569 is a new provision which sets out how members of a private company with only one class of shares may authorise the directors to allot shares without complying with the statutory pre-emption provisions.
871.Section 573 is concerned with the disapplication of pre-emption rights in connection with a sale of treasury shares. Generally speaking, where a company buys back its own shares, it is normally required to cancel those shares (see section 706(b)). Certain companies (principally those which are listed or those whose shares are traded on the Alternative Investment Market and equivalent companies in the EEA) may however elect not to cancel shares which have been bought back but may hold the shares “in treasury”. A share which is held in treasury may be sold at a future point in time and this facility enables such companies to raise capital more quickly than they would otherwise be able to do, as the directors do not have to obtain prior authority from the company’s members before selling treasury shares. However, the provisions of section 561 do apply to sales of treasury shares as they apply to allotments of shares (see section 560(2)(b)).
872.This section applies to a sale of shares which have been held in treasury by the company. It replaces section 95(2A) of the 1985 Act and reproduces the effect of that section by enabling a company’s members to give a general power to the directors (through the company’s articles or by special resolution of the company’s members) to sell such shares as if statutory pre-emption rights did not apply, or applied with modifications.
873.This section also permits the members to confer upon the directors (by special resolution) a specific power which enables them to sell treasury shares as if statutory pre-emption rights did not apply to a specified sale, or applied with modifications.
Chapter 4: Public Companies: Allotment Where Issue Not Fully Subscribed
Section 578: Public companies: allotment where issue not fully subscribed
874.The provisions of this section restate section 84 of the 1985 Act and relate to the allotment of shares by public companies, and apply where not all the shares offered are taken up. A public company must not allot shares following an offer to subscribe for shares unless all the shares offered are taken up or the offer is made on the basis that it will go ahead even if all the shares offered are not taken up or if other conditions specified in the offer are met. It is not possible for the terms of the offer to override the requirements of this section (subsection 6)).
875.The purpose of this rule is to protect persons who apply for shares, by ensuring that if the increase in capital is not fully subscribed, the capital will be increased by the amount of the subscriptions received only if the conditions of the issue so provide (Article 28 of the Second Company Law Directive (77/91/EEC)).
876.If 40 days after first making the offer, the offer is unsuccessful because not enough shares have been applied for under the offer, any money or other consideration received from those that did apply for shares under the offer must be repaid or returned (subsection (2)). Interest becomes payable after the expiration of the 48th day after the offer was first made (subsection (3)). The rate of interest will be as specified at the time under section 17 of the Judgments Act 1838 (currently 8%). This is a change from section 84(3) of the 1985 Act which sets the interest rate at 5% per annum.
877.The 40 day and 48 day time limits imposed by subsections (2) and (3) now run from the making of the offer rather than from the issue of any prospectus (as was the case under section 84 of the 1985 Act) given that the requirement or otherwise for a prospectus is a matter of securities law.
878.The regulation of public offers, especially requirements relating to prospectuses, is generally a matter of securities law. Sections 82 and 83 of the 1985 Act are, therefore, not restated in this Act.
Section 583: Meaning of payment in cash
879.This section replaces section 738(2) to (4) of the 1985 Act. It provides a definition of “payment in cash” for the purposes of the Companies Acts and is relevant to a number of provisions (for example section 593 requires public companies to obtain an independent valuation of any non-cash consideration where it allots shares otherwise than for cash).
880.Subsection (3) provides a definition of “cash consideration” which lists the items currently contained in section 738(2) of the 1985 Act. It is generally accepted that certain forms of payment, in addition to those listed in subsection (3), constitute “payment in cash” where shares in a company are deemed to be paid up or allotted for cash, for example an assured payment obligation under the CREST assured payment system, but this matter is not beyond doubt. (An assured payment obligation is the creation of an obligation to make payment to or for the account of the company in accordance with the rules and practices of the operator of a relevant system as defined by regulation 2(1) of the Uncertificated Securities Regulations 2001). The power contained in subsection (4) will enable the Secretary of State to make provision for other forms of payment to be regarded as falling within the definition of “payment in cash”. This will eradicate the uncertainty which currently surrounds certain forms of payment and will also “future proof” the current definition should other settlement systems be developed in the future (or should other settlement systems within the EU be identified).
Section 589: Power of court to grant relief
881.Section 589 restates section 113(1) to (7) of the 1985 Act. It enables the court to grant relief, to the applicant, from a liability to the company which has arisen as a result of a contravention of section 585, 587(2) or (4) or 588. There is a minor change in the restatement insofar as the matters to which the court must have regard in applying the just and equitable test in subsection (3) also apply where the liability relates to the payment of interest (under section 113 (2)(b)) of the 1985 Act the court is not required to have regard to those matters in applying the just and equitable test).
Section 606: Power of court to grant relief
882.Section 606 restates section 113(1) to (8) of the 1985 Act. It enables the court to grant relief, to the applicant, from a liability to the company which has arisen (under any provision of Chapter 6) in relation to payment in respect of shares in a company or an undertaking given to the company in, or in connection with, payment for any shares in it. There is a minor change in the restatement insofar as the matters to which the court must have regard in applying the just and equitable test in subsection (2) also apply where the liability relates to the payment of interest (under section 113 (2)(b)) of the 1985 Act the court is not required to have regard to those matters in applying the just and equitable test).
Chapter 7: Share Premiums
883.Under section 130 of the 1985 Act, where shares in a company are issued at a premium, (that is, at a price which is greater than their nominal value), an amount equal to the premium paid on those shares must be transferred to a non-distributable reserve: the share premium account. This account can only be used in a limited number of circumstances described in section 130.
Section 610: Application of share premiums
884.In line with the recommendations of the CLR (Completing the Structure, paragraph 7.8), this section further restricts the application of the share premium account and in the future, companies will not be able to use the share premium account to write off preliminary expenses (that is, expenses incurred in connection with the company’s formation). Companies will continue to be able to use the share premium account to write off any expenses incurred, or commission paid, in connection with an issue of shares but the application of the share premium account in these circumstances will be limited so that the company will only be able to use the share premium account arising on a particular issue of shares to write off expenses incurred or commission paid in respect of that issue. As now, companies will also be able to use the share premium account to pay up new shares to be allotted to existing members as fully paid bonus shares.
885.A further change is that in future companies will not be able to use the share premium account to write off any expenses incurred, commission paid or discount allowed in respect of an issue of debentures or in providing for the premium payable on a redemption of debentures.
Chapter 8: Alteration of Share Capital
Section 617: Alteration of share capital of limited company
886.This section prohibits a limited company from altering its share capital except in the ways permitted under the Act. It includes a signpost to a new provision which will enable companies limited by shares easily to convert (or “redenominate”) their share capital from one currency to another (see section 622).
Section 618: Sub-division or consolidation of shares
887.Consolidation of a company’s share capital involves combining a number of shares into a new share of commensurate nominal value: for example, ten £1 shares may be combined to make one £10 share. Sub-division of a company’s share capital involves dividing a share into a number of new shares with a smaller nominal value: for example, a £10 share may be sub-divided into ten £1 shares.
888.Section 618 replaces section 121(2)(b) and (d) of the 1985 Act. It sets out the circumstances and manner in which a limited company may consolidate or sub-divide its share capital. Where shares in a company are sub-divided or consolidated, the proportion between the amount paid and the amount unpaid (if any) on the original share(s) must remain the same in relation to the share(s) resulting from the sub-division or consolidation. If, for example, £2 is unpaid on a £10 share that is subsequently sub-divided into ten £1 shares, there will now be 20p unpaid on each of those ten shares.
889.A company may exercise a power conferred on it under this section only if the members have passed a resolution authorising it to do so, which may be an ordinary resolution or a resolution requiring a higher majority (as the articles may require). Such a resolution may authorise a company to exercise more than one of the powers conferred on it under this section, for example, the resolution may authorise a sub-division of one class of the company’s shares and a consolidation of another. It may also authorise the company to exercise a power conferred on it under this section on more than one occasion or at a specified time or in specified circumstances. This avoids the directors having to obtain authorisation from the company’s members on each and every occasion that a company alters its share capital under this section (which may be inconvenient to the directors and members alike or impractical due to timing constraints).
890.The flexibility to pass a conditional resolution (that is, a resolution that will only take effect if certain conditions are met) given in subsection (4)(c) is necessary as a sub-division or consolidation of share capital (or any class of it) may form part of a wider re-organisation of a company’s share capital, for example, a reduction of share capital following a redenomination of share capital. It may, therefore, not be appropriate, or necessary, for a company’s share capital to be altered in this way if the reorganisation of share capital that the sub-division or consolidation is linked to does not go ahead.
891.Under the 1985 Act a company may only sub-divide or consolidate its share capital if it is authorised to do so by the company’s articles (see section 121 of that Act). This restriction has not been retained.
Section 619: Notice to registrar of sub-division or consolidation
892.The section replaces a similar requirement to notify the registrar contained in section 122(1)(a) and (d) of the 1985 Act. Where a company sub-divides or consolidates its share capital under section 618, it will continue to be required to give notice of this alteration to its share capital to the registrar within one month. However, there is a new requirement to file a statement of capital (see subsections (2) and (3)), which is in essence a “snap-shot” of the company’s total share capital at a particular point in time: in this case following the consolidation/sub-division.
893.For public companies, the requirement for a statement of capital is linked to the abolition of authorised share capital: it implements Article 2 of the Second Company Law Directive (77/91/EEC) which states:
““the statutes or instruments of incorporation of the company shall always give at least the following information…(c) when the company has no authorized capital, the amount of the subscribed capital…”.”
894.The statement of capital will require the following information to be provided:
the total number of shares of the company,
the aggregate nominal value of those shares,
for each class of shares, prescribed particulars of the rights attached to the shares, the total number of shares of that class and the aggregate nominal value of shares of that class, and
the amount paid up and the amount (if any) unpaid on each share (whether on account of the nominal value of the share or by way of premium).
895.Whilst this Directive applies only to public companies it is important that the information on the public register is up-to-date. A statement of capital will, therefore, be required where it is proposed that a company formed under the Act will have a share capital on formation and, with limited exceptions (in particular, where there has been a variation of class rights which does not affect the company’s aggregate subscribed capital) whenever a limited company makes an alteration to its share capital. A statement of capital is also called for in certain circumstances where an unlimited company having a share capital makes a return to the registrar (see, section 856).
896.In making a statement of capital, a company is required to provide “prescribed particulars of the rights attached to the shares”. Here, and elsewhere in the Act where a statement of capital is called for, “prescribed” means prescribed by the Secretary of State in regulations or by order made under the Act.
897.The power conferred on the Secretary of State under this section enables the Secretary of State to specify the particular detail of the information which he requires to be filed with the registrar by a company. A statutory instrument made pursuant to this power will not be subject to any form of Parliamentary scrutiny.
898.Criminal liability for any failure to comply with the procedural requirements as to notice is retained (see subsection (4)). The penalty for this offence is set out in subsection (5).
Section 620: Re-conversion of stock into shares
899.Stock cannot be issued directly by a company but arises from a conversion of fully paid up shares into stock under section 121(2)(c) of the 1985 Act. This ability to convert shares into stock has not been retained. A company that currently has stock may, however, wish to re-convert this stock back into fully paid shares, and this is permitted by the following section.
900.Section 620 replaces section 121(2)(c) of the 1985 Act. It retains the ability to re-convert stock back into fully paid shares but removes the requirement for prior authorisation in the articles (currently a company may only re-convert stock back into shares if provision for this is made in its articles).
901.A re-conversion of stock into shares will require an ordinary resolution of the company’s members. Such a resolution may give the directors power to convert stock into fully paid shares on more than one occasion; at a specified time; or only if certain conditions are met (see subsection (3)). The flexibility to pass a conditional resolution (that is, a resolution that will only take effect if certain conditions are met) is necessary as a re-conversion of stock into shares may form part of a wider re-organisation of a company’s share capital.
Section 621: Notice to registrar of reconversion of stock into shares
902.Where a company re-converts stock into shares it must give notice of the alteration to its share capital to the registrar under the provisions in section 621. This requirement replaces a similar provision in section 122(1)(c) of the 1985 Act.
903.A statement of capital is required (see note on section 619).
904.Criminal liability for any failure to comply with the procedural requirements as to notice is retained (see subsection (4)). The penalty for this offence is set out in subsection (5).
Section 622: Redenomination of share capital
905.Where a public company applies for a trading certificate under section 117 of the 1985 Act it must satisfy a minimum share capital requirement (known as the “authorised minimum”). There is a similar requirement where a private company re-registers as a public company under section 43 of that Act. The authorised minimum is currently set at £50,000 and must be expressed in sterling. This implements Article 6 of the Second Company Law Directive (77/91/EEC) which requires that, in order that a company may be incorporated or obtain authorisation to commence business, a minimum capital shall be subscribed the amount of which shall be not less than 25000 ECU (expressed in the domestic currency of the Member State). Under section 763, in future the authorised minimum will be capable of being satisfied in sterling (£50,000) or the euro equivalent to the sterling amount. Subject to this change, the Act retains the effect of the 1985 Act provisions on the authorised minimum (see, for example, section 91, section 650 and section 761).
906.Subject to the above qualification (and any restriction in a company’s articles) a company is free to allot shares in any currency that it wishes (see section 542(3)). It may also have its share capital made up of shares of a mixture of denominations, for example, one class of a company’s shares may be denominated in sterling, whereas another class may be denominated in dollars, euros or some other currency of the company’s choosing. What a company cannot currently do is easily redenominate its share capital (or any class of it) from one currency to another, for example, from dollars to sterling or vice versa. The current procedure involves cancelling existing shares under the court approved procedure for capital reductions set out in section 135 of the 1985 Act or, in the case of private companies only, buying back or redeeming shares out of capital under section 171 of that Act, and then issuing new shares in the desired currency.
907.Section 622 introduces a new procedure that will allow a company limited by shares to redenominate its share capital easily. This requires a resolution of the company’s members. (Unlimited companies having a share capital are already free to redenominate their share capital as they see fit and no change to the legislation is required in respect of such companies).
908.Subsection (2) of this section provides that the spot rate used when converting a company’s share capital from one currency to another must be specified in the resolution to redenominate the company’s share capital. There is a choice of spot rates and this is set out in subsection (3).
909.A company is free to pass a conditional resolution under this section (see subsection (4)). A resolution will, however, lapse if the redenomination of share capital has not taken effect within 28 days of the date on which the resolution is passed (see subsection (6)). Where a resolution lapses, the company will not be able to redenominate its capital unless it passes a new resolution and the redenomination is effected in accordance with the new resolution.
910.Subsection (7) makes it clear that, if it wishes, a company may restrict or prohibit a redenomination of its share capital by incorporating a provision to this effect in the company’s articles.
911.It should be noted that this section does not make provision for the authorised minimum to continue to be denominated in sterling (or the euro equivalent). This means that once a public company has obtained a trading certificate under section 761 (or previously under section 117 of the 1985 Act) or where a private company has re-registered as a public company, such a company is free, if it wishes, to redenominate all of its share capital, including the authorised minimum into any currency of its choosing.
Section 623: Calculation of new nominal values
912.This section explains how the new nominal value of a share which has been redenominated from one currency to another should be calculated.
Section 624: Effect of redenomination
913.This section makes it clear that a redenomination of a company’s share capital (or any class of it) does not affect any rights or obligations that the members may have under the company’s constitution or any restrictions affecting members under the company’s constitution. In particular, it does not affect entitlement to dividends, voting rights or any liability in respect of amounts unpaid on shares. If, for example, a dividend of 20p was declared on a £1 share prior to a redenomination of that share, and that £1 share is subsequently converted into a $1.5 share, the member who now owns a $1.5 share in the company will still be entitled to a 20p dividend (albeit that the company and the member in question may agree that the 20p dividend can be paid in cents – or indeed in some other currency). Similarly, where a company has issued partly paid shares, the member’s liability to the company will remain in the currency in which the share was originally denominated.
Section 625: Notice to registrar of redenomination
914.This section sets out the requirements as to notice where a company redenominates its share capital (or any class of it). Notice must be given to the registrar in accordance with subsections (1) and (2) of this section and there is a requirement for a statement of capital (see note on section 619).
915.A copy of the resolution to redenominate the company’s share capital must be forwarded to the registrar within 15 days after it is passed notwithstanding that it may be an ordinary resolution (see section 622(8) which provides that Chapter 3 of Part 3 applies to the resolution and in particular section 30)
916.If a company fails to comply with the procedural requirements as to notice the company and every officer of the company commits an offence. The penalty for this offence is set out in subsection (5).
Section 626: Reduction of capital in connection with redenomination
917.Following a redenomination of a company’s share capital, it is likely that the company will be left with shares expressed in awkward fractions of the new currency, for example, 0.997 dollars or 1.01 euros. The company may therefore wish to renominalise the value of the shares affected (that is, alter the nominal value of these shares) to obtain share values in whole units of the new currency. It can do this in one of two ways: if the company has distributable reserves it may capitalise those reserves to increase the nominal value of the shares affected; alternatively, it may reduce its share capital using the procedure set out in section 626.
918.This section enables a company to renominalise the value of its shares by cancelling part of its share capital. A special resolution of the company’s members is required but there is no need for the directors to make a solvency statement or for the company to go to court (as required where a company reduces its share capital under Chapter 10 of this Part).
919.Under subsection (3), a resolution to reduce capital in connection with a redenomination must be passed within 3 months of the resolution to redenominate the company’s share capital.
920.Subsection (4) provides that the amount by which a company can reduce its share capital using this new provision is capped at 10% of the nominal value of the company’s share capital immediately after the reduction. This 10% cap is required by the Second Company Law Directive (77/91/EEC) and applies to any reduction of capital in a public company which is not approved by the court.
921.Where a company reduces its share capital under this section, the amount by which the company’s share capital is reduced must be transferred to a new non-distributable reserve (see section 628).
Section 627: Notice to registrar of reduction of capital in connection with redenomination
922.This section sets out the requirements as to notice where a company reduces its share capital in connection with a redenomination of its share capital (that is, to renominalise the value of its shares). Notice must be given to the registrar in accordance with subsection (1) of this section. This notice must be accompanied by a statement of capital (see note on section 619).
923.The resolution to reduce the share capital must be filed with the registrar in accordance with section 30.
924.The reduction of capital will not take effect until the documents that are required to be delivered to the registrar under subsections (1) and (2) are registered by the registrar (see subsection (5)).
925.In addition to delivering the above documents to the registrar, within 15 days of the date that a resolution to reduce capital in connection with a redenomination is passed, under subsection (6) the company must also deliver to the registrar a statement made by the directors confirming that the reduction of share capital was made in accordance with subsection (4) of section 626.
926.If a company fails to comply with the procedural requirements as to notice the company and every officer of the company commits an offence. The penalty for this offence is set out in subsection (8). In addition, where the statement made by the directors under subsection (6) is misleading, false or deceptive in a material particular, the directors are liable to an offence under section 1112.
Section 628: Redenomination reserve
927.Where a company reduces its share capital under section 626 it must transfer an amount equal to the value of the reduction to a non-distributable reserve known as the redenomination reserve.
928.This section provides that amounts transferred to the redenomination reserve may be used by the company in paying up shares to be allotted to existing members as fully paid bonus shares. Subject to this, the provisions of the Companies Acts relating to the reduction of a company’s share capital, apply to the redenomination reserve as if it were paid-up share capital. These provisions mirror those contained in section 733 (which restates section 170 of the 1985 Act).
Chapter 9: Classes of Shares and Class Rights
929.“Classes of shares” (or “class rights”) is not defined in the 1985 Act but at common law this term is normally used where the rights that attach to a particular share relate to matters such as voting rights, a right to dividends and a right to a return of capital when a company is wound-up. Rights attach to a particular class of shares if the holders of shares in that class enjoy rights that are not enjoyed by the holders of shares in another class.
Section 629: Classes of shares
930.This section provides that for the purposes of the Act, shares are of one class if the rights attached to them are in all respects uniform. It reproduces the provision in section 128(2) of the 1985 Act. It is particularly relevant to the provisions of section 550 and section 569. This definition of “classes of shares” also applies in determining the extent to which shares constitute different classes for the purposes of the statement of capital required to be filed under various provisions of the Bill (see note on section 619).
Section 630: Variation of class rights: companies having a share capital
931.On variation of class rights the CLR recommended (Final Report, paragraph 7.28) that the current provisions should be retained with some simplification, and extended to companies without a share capital (see section 631).
932.Section 630 replaces section 125 of the 1985 Act. It is concerned with the manner in which rights attached to a class of shares may be varied. Class rights typically cover matters such as voting rights, rights to dividends and rights to a return of capital on a winding up.
933.Currently class rights may be set out in the memorandum or articles or elsewhere, and provision may or may not be made for their alteration. Under the Act it will not be possible for class rights to be set out in the memorandum (see section 8) and where class rights attaching to shares in an existing company are specified in the memorandum these will be deemed, by virtue of section 28, to be a provision in the company’s articles.
934.Class rights are “attached to a class of shares” (see subsection (1)). Where all the shares in a company fall within the one class, there are no class rights, only shareholder rights. What amounts to a class is not defined either in the current law or the Act (other than in section 629) and remains a matter for case law.
935.The current requirement for an extraordinary resolution where a company is proposing to vary the rights attached to a class of its shares is replaced with a requirement for a special resolution (see subsection (4)(b)). The Act abolishes the concept of an extraordinary resolution. Special resolution is defined in section 283.
936.Subsections (2) and (4) provide that rights may be varied in accordance with the company’s articles or, where the articles make no provision for a variation of class rights, if the holders of at least three-quarters in nominal value of the issued shares of that class consent in writing or a special resolution passed by the holders of that class sanctions the variation. This means that the articles may specify a less demanding procedure for a variation of class rights than the statutory scheme (for example, that the holders of 51% by nominal value of the class consent in writing), or may permit a simple majority of the class at a class meeting.
937.The provisions of section 630 are expressed to be without prejudice to any other restriction on the variation of rights (see subsection (3)). This has two important effects. First, if and to the extent that the company has adopted a more onerous regime in its articles for the variation of class rights, for example requiring a higher percentage than the statutory minimum, the company must comply with the more onerous regime. Second, if and to the extent that the company has protected class rights by making provision for the entrenchment of those rights in its articles (see section 22), that protection cannot be circumvented by changing the rights attached to a class of shares under this section.
Section 631: Variation of class rights: companies without a share capital
938.This section extends the statutory provisions on variation of class rights to companies without a share capital. Companies limited by guarantee (which since December 1980 cannot be formed with a share capital) may, for example, have different classes of members with different voting rights.
939.At present the question of how members’ rights may be varied will depend to a large extent on whether provision has been made, either in the memorandum or articles, for their variation. Under the Act class rights may also be varied in accordance with this section, which contains new provisions, comparable to those for companies with a share capital. Thus there is a minimum requirement that class rights may be varied if three-quarters of that class consent in writing or a special resolution of those members sanctions the variation, unless the company has made provision for a less onerous regime to apply in its articles. Again, a company may also make provision, in its articles, for a more onerous regime to apply than that provided in this section and where they do the company must comply with the regime set out in the articles.
Section 632: Variation of class rights: saving for court’s power under other provisions
940.This section preserves the court’s powers under various other provisions of the Act and substantially restates section 126 of the 1985 Act.
Sections 633 and 634: Right to object to variation
941.Section 633 replaces section 127 (which confers a right on shareholders to object to a variation of class rights). It sets out the procedure that must be followed where there is an objection to a variation of the rights attached to a class of a company’s shares and enables shareholders holding not less in aggregate than 15% of the issued shares of the class in question (being persons who did not consent to or vote in favour of the resolution approving the variation) to apply to the court for the variation to be cancelled.
942.Section 634 makes similar provision in respect of a variation of class rights in companies not having a share capital. This is a new provision which enables members, amounting to not less than 15% of the members of the class affected (being persons who did not consent to or vote in favour of the resolution approving the variation), to apply to the court to have the variation cancelled and gives the court the power to confirm the variation, or disallow it if the court is satisfied that it would unfairly prejudice the members in that class.
Section 635: Copy of court order to be forwarded to the registrar
943.Section 635 sets out the procedural requirements as to notice where the court has made an order on an application under section 633 or 634. Where the court has made an order on application under these sections, the company must forward a copy of that order to the registrar within 15 days of the date on which the order is made. Where a company fails to comply with the provisions of this section, the company, and every officer of the company who is in default, commits an offence (see subsections (2) and (3)).
Section 636 to 640: Matters to be notified to the registrar
944.These sections replace various provisions in sections 128 and 129 of the 1985 Act which are concerned with notification to the registrar of the creation of, and variations to, rights attached to a class of a company’s shares (section 128 of the 1985 Act) or class rights of members (section 129 of that Act).
945.Under the Act, where a limited company creates a new class of shares, it will be required to provide details of the rights attached to the shares in the return of allotment and statement of capital required under section 555. There is a similar requirement in section 556 where an unlimited company allots a new class of share. Those provisions replace section 128(1) and (2) of the 1985 Act.
946.In addition, where a company varies the rights attached to any of its shares (or assigns a name or other designation, or a new name or other designation to any class or description of its shares) it will in future be required to register particulars of the rights affected under section 637 (or 636) irrespective of how the variation in rights was achieved. Currently companies are not required to provide this information if the rights attached to a particular share or class of shares are varied by an amendment to the company’s memorandum or articles or by special resolution or agreement of the company’s members which is required to be filed under section 380 of the 1985 Act. Sections 638 to 640 make similar changes to the disclosure requirements which apply to companies limited by guarantee not having a share capital and unlimited companies not having a share capital which may, nevertheless, have different classes of members.
947.It should be noted, that, in contrast to other alterations to a company’s share capital, there is no requirement in section 637 for a statement of capital (see note on section 619). Such a requirement would be superfluous, as a variation of class rights will not result in a change to the aggregate amount of a company’s subscribed capital.
Chapter 10: Reduction of Share Capital
948.Section 135 of the 1985 Act lays down a statutory procedure under which a limited company may, if authorised by its articles, reduce its share capital. This requires a special resolution of the company’s members and the reduction must be confirmed by the court. Companies limited by shares may also reduce their share capital under section 171 (private company redemption or purchase of own shares out of capital) and sections 146 to 147 of the 1985 Act (which require a public company which acquires shares in any of the specified ways, for example, through forfeiture for failure to pay up, to cancel those shares within a specified period), the provisions of both of which are carried forward by the Act. A reduction of capital may also occur as a result of the court making an order for the purchase by a company of its members’ shares.
949.A company may wish to reduce its share capital for a variety of reasons, for example, where its capital is in excess of the company’s wants or where the value of the company’s net assets has fallen below the amount of its capital (as stated in the company’s accounts) and the position is likely to be permanent.
Section 641: Circumstances in which a company may reduce its share capital
950.This section replaces section 135(1) and (2) of the 1985 Act. It sets out the circumstances and manner in which a company limited by shares may reduce its share capital. As recommended by the CLR (Final Report, paragraph 10.6), in future a private company limited by shares will be able to reduce its share capital using a new solvency statement procedure for capital reductions (see section 642).
951.A company may only reduce its share capital under section 135 of the 1985 Act if it is authorised to do so by its articles. In line with the recommendations of the CLR (Completing the Structure, paragraph 2.15), the requirement for prior authorisation in the articles has not been retained but, if it wishes, a company may restrict or prohibit a reduction of capital by making provision to this effect in its articles (see subsection (6)).
952.Subsection (1)(a) contains a signpost to a new provision, which will enable a private company limited by shares to reduce its share capital using the new solvency statement procedure (see above). In addition, private companies and public companies alike will continue to be able to use the current court approved procedure for capital reductions – which is retained in subsection (1)(b).
953.In the case of a private company limited by shares which is proposing to use the new solvency statement procedure to effect a reduction of capital, the company may only reduce its share capital under subsection (1)(a) if it will have at least one member remaining after the proposed reduction (see subsection (2)). That member need only hold one share in the company but that share must not be a redeemable share. The principle behind this requirement is that a private company limited by shares should not be capable of reducing its share capital to zero unless the reduction of capital is sanctioned by the court. This mirrors the existing equivalent provision in section 162(3) of the 1985 Act – which applies to a purchase of own shares.
954.Both the solvency statement procedure for capital reductions and the court-approved procedure require a special resolution of the company’s members. Under subsection (5) a special resolution to reduce a company’s share capital may not provide for the proposed reduction to take effect on a date later than the date on which the resolution to reduce capital takes effect. Under the solvency statement procedure a resolution to reduce capital will take effect when the documents referred to in section 644 have been registered by the registrar (see section 644(4)). This would operate to prevent a company passing a resolution on, say, 1st January stating that the reduction is to take effect on 1st October. Under the court approved procedure, the resolution will take effect on the registration of the court order and statement of capital or, in the context of a reduction forming part of a compromise or arrangement under Part 26, on delivery of those documents to the registrar (unless the court orders otherwise) (see section 649).
Section 642: Reduction of capital supported by solvency statement
955.This section sets out the conditions that must be satisfied in order for a private company limited by shares to reduce its share capital using the new solvency statement procedure.
956.The procedural requirements that the directors must follow when they propose a capital reduction using the solvency statement route are set out in subsections (1) to (3) which provides that the solvency statement made in connection with a reduction of capital by a private company cannot be made more than 15 days before the date on which the resolution to reduce capital is passed. It also provides that both the resolution and the solvency statement must be filed with the registrar in accordance with the provisions of section 644.
957.The solvency statement must also be made available to the company’s members when they vote on the resolution to reduce capital and the procedure for providing a copy of the solvency statement to the members varies according to whether the resolution to reduce capital is proposed as a written resolution or at a meeting of the company’s members (see subsections (2) and (3)). Whilst a failure to observe these procedural requirements will not affect the validity of the resolution to reduce capital, if a solvency statement which has not been provided to the company’s members in accordance with the provision of this section is subsequently filed with the registrar, every officer of the company who is in default commits an offence (see section 644).
Section 643: Solvency statement
958.A solvency statement made under section 643 must be made by all of the directors. If one or more of the directors is unable or unwilling to make this statement, the company will not be able to use the solvency statement procedure to effect a reduction of capital unless the dissenting director or directors resign (in which case the solvency statement must be made by all of the remaining directors).
959.The solvency statement must be in the “prescribed form” and “prescribed” in this context means prescribed by the Secretary of State in regulations or by order made under the Act.
960.The solvency statement must state the date on which it is made and the name of each director of the company but there is no requirement that the directors must all be in the same location when they make this statement. The registrar will be able to make rules under section 1068 as to the form of the solvency statement.
961.In forming their opinions, the directors must take account of all the company’s liabilities including contingent and prospective liabilities (see subsection (2)). So, in circumstances where a company holds redeemable preference shares which, for the purposes of the accounting standards that applied to the company on the date that the directors made the solvency statement, are treated as liabilities, a proposed redemption or purchase of these shares in the relevant period should be treated as a contingent or prospective liability.
962.If the directors make a solvency statement without having reasonable grounds for the opinions expressed in it, and that statement is subsequently delivered to the registrar, every director who is in default commits an offence (see subsection (4)). The penalty for this offence is set out in subsection (5).
Section 644: Registration of resolution and supporting documents
963.This section sets out the requirements as to delivery of the solvency statement and other key documents to the registrar. The resolution to reduce capital itself must be filed with the registrar within the same time period as currently applies – that is, within 15 days of the date that it is passed (see section 30) and it will not take effect until the solvency statement and statement of capital (see subsections (1) and (2)) are registered by the registrar. As with all circumstances where the company makes an alteration to its subscribed capital, the company is required to deliver a statement of capital to the registrar (see note on section 619).
964.In addition to making a solvency statement in accordance with section 643, the directors must also make a statement confirming that the solvency statement was made not more than 15 days before the date on which the resolution to reduce capital was passed and that this statement was provided to the company’s members in accordance with section 642 – see subsection (5).
965.In addition to the new offences which are set out in sections 643(4) (directors making solvency statement without reasonable grounds for the opinion expressed in it) and subsection (7) (company delivering solvency statement that was not provided to members to registrar), where a company fails to comply with any of the filing requirements under section 644, an offence is committed by the company and every officer of the company who is in default (see subsection (8)). The penalty for this offence is set out subsection (9).
Sections 645 to 649: Reduction of capital confirmed by the court
966.These sections replace or restate various provisions in the 1985 Act that are concerned with reductions of capital confirmed by order of the court.
967.Sections 645 and 646 restate section 136 of the 1985 Act which is concerned with the procedure for making an application to court to confirm a reduction of capital (including the creditors’ right to object). If, on such an application, an officer of the company intentionally or recklessly conceals a creditor or misrepresents the nature or amount of a debt owed by the company, or is knowingly concerned in any such concealment or misrepresentation he commits an offence (see section 647). As now the court may make an order confirming the reduction of capital on such terms and conditions as it thinks fit (see section 648 which restates section 137 of the 1985 Act).
968.Section 649 replaces section 138(1) to (4) of the 1985 Act. Under section 138 of that Act, a resolution to reduce capital using the existing court approved scheme takes effect when the court order confirming the reduction and minute of the reduction are registered by the registrar. The minute (which must be approved by the court) sets out key information regarding the company’s share capital immediately after the reduction. Section 649 updates the 1985 Act provisions by replacing the current requirement for a minute of the reduction with a statement of capital (see note on section 619). Like the minute confirming the reduction, this statement must be approved by the court.
969.In line with the CLR’s recommendations (Final Report, paragraph 13.11), subsection (3)(a)(i) of this section provides that a reduction of capital that forms part of a compromise or arrangement under Part 26 of the Act will take effect at the same time as other aspects of that compromise or arrangement: namely on delivery of the court order confirming the reduction (and statement of capital approved by the court) to the registrar (unless the court orders that it should take effect on the registration of these documents) (see new subsection (3)(a)(ii)).
970.In all other cases, that is, where the reduction of capital does not form part of a compromise or scheme of arrangement under Part 26, where a company reduces its share capital using the court approved procedure the reduction will, as now, take effect on registration of the court order confirming the reduction (and statement of capital) by the registrar. Subsection (5) requires the registrar to certify the registration of the order and statement of capital. Subsection (6) restates section 138(4) of the 1985 Act in relation to such certificate.
Section 651: Expedited procedure for re-registration as a private company
971.This section, together with section 650, substantially restates section 139 of the 1985 Act and provides for the consequences where the court confirms the reduction by a public company of its share capital below the authorised minimum (defined in section 763): in particular they facilitate the re-registration of the company as private.
972.Subsection (3) replaces section 139(4) of the 1985 Act. It introduces a requirement to send a copy of the court’s order (that is, the order authorising the company to be so re-registered without its having passed a special resolution) to the registrar, together with an application for re-registration. The current requirement for the application to be signed by a director (or secretary) has not been retained.
Sections 652 and 653: Effect of reduction of capital
973.These sections restate section 140 of the 1985 Act (with the exception of references to the “minute” being replaced with references to the statement of capital) which is concerned with the liability of a company’s members in respect of any amounts unpaid on its shares following a reduction of capital. As now, there are special rules where a creditor was omitted from the list of creditors settled by the court.
Chapter 11: Miscellaneous and Supplementary Provisions
Section 654: Treatment of reserve arising from reduction of capital
974.This is a new provision which enables the Secretary of State, by order, to specify the circumstances in which a reserve arising from a reduction of capital will be distributable.
975.Whilst there is no requirement in the Act (or indeed the 1985 Act) to create a statutory reserve following such a reduction, we understand that it is usual for companies to create an accounting reserve in these circumstances to “balance the books” (that is, the section relates to reserves that arise as a result of generally accepted accounting treatments). Currently, the question whether a reserve arising from a reduction of capital (which, for a limited company, may currently only be made pursuant to a court order) may be treated as a realised profit for the purposes of computing whether a company has sufficient distributable profits to make a distribution, is the subject of technical guidance issued by the Institutes of Chartered Accountants. The Act introduces a new procedure which enables private companies to reduce their share capital without going to court (see section 641) which is not on all fours with the court approved route (in particular there is no requirement to settle a list of creditors or to provide security for the company’s debts) and in the circumstances it is desirable to deal with the question of when amounts credited to such a reserve should be treated as a realised profit in statute. Owing to the technical nature of the rules that will need to be made this issue will be dealt with in secondary legislation. An order made under section 654 will however be subject to the affirmative resolution procedure – that is, the regulations will need to be approved by both Houses of Parliament.
Section 656: Public companies: duty of directors to call meeting on serious loss of capital
976.Section 656 restates section 142(1) and (3) of the 1985 Act. It sets out the procedure that must be followed where the net assets of a public company fall below half (or less) of the company’s called up share capital. Subsection (4) imposes liability on any director who knowingly authorised or permitted a failure to call a meeting as required by this section.
Section 657: General power to make further provision by regulations
977.This is a new provision which enables the Secretary of State, in regulations made under the Act, to modify various provisions in Part 17 of the Act (see subsection (1)).
978.Regulations made under this section may amend or repeal any of the specified provisions or make such other provision as appears to the Secretary of State appropriate in place of those provisions. This will enable the Secretary of State to future-proof the specified provisions in this Part of the Act.
979.Regulations made pursuant to the power in this section are subject to the affirmative resolution procedure.
Part 18: Acquisition by Limited Company of Its Own Shares
980.This Part replaces various provisions in Chapters 6 and 7 of Part 5 of the 1985 Act and makes substantive changes to some of those provisions. Sections 658 to 659, 662, 666 to 677, 680 to 683, 687, 691, 693, 695 to 701, 704 to 706, 710 to 713, 716 to 719, 721 to 726, 728 to 729, 731 and 733 to 736 restate various provisions in the 1985 Act but do not make any changes to those provisions.
Chapter 1: General Provisions
Sections 660 and 661: Shares held by company’s nominee
981.These sections restate sections 144 and 145(1) and (2) of the 1985 Act, but with the clarification that they apply to shares taken by a subscriber to the memorandum as nominee of the company.
Section 663: Notice of cancellation of shares
982.This section restates section 122(1)(f) and (2) of the 1985 Act and Schedule 24 to that Act. Subsections (2) and (3) update the current notice requirements to require a company that has cancelled shares in order to comply with section 662 to provide the registrar with a statement of capital (see note on section 619) at the time of giving notice of the cancellation.
Section 664: Re-registration as private company in consequence of cancellation
983.This section replaces section 147(2) and (3) of the 1985 Act. These provisions have been updated to reflect the fact that in future it will not be possible to alter the memorandum and that key information of a type which was previously in the memorandum will now be in the articles (see note on section 8). The resolution to re-register as a private company in consequence of a duty to cancel shares will however still need to be filed with the registrar under Chapter 3 of Part 3 of the Act.
984.There is also a new requirement, in subsection (3), for the application for re-registration to be accompanied by a statement of the company’s proposed name on re-registration, Subsections (5) and (6) are also new. Consistent with the approach taken where a company is formed as a private company under the Act (see section 9), where a public company applies to re-register as private under this section the application for re-registration must be accompanied by a statement of compliance (see note on section 13).
Section 665: Issue of certificate of incorporation on re-registration
985.This section replaces section 147(4)(a) of the 1985 Act and restates section 147(4) and (4)(b) of that Act. As with the previous section, the provision has been updated to reflect the fact that in future companies will not be capable of, and will not need to, alter their memoranda. Subsection (3) is new. Consistent with the approach taken in Part 7 of the Act, a certificate of incorporation issued on the re-registration of a company under section 664 will need to specify that it is being issued on the re-registration of the company and the date on which it is issued.
Chapter 2: Financial Assistance for Purchase of Own Shares
986.The following sections replace Chapter 6 of Part 5 of the 1985 Act which contains a prohibition on the giving of financial assistance (broadly defined) by a company or any of its subsidiaries for the purpose of the acquisition of shares in itself. There are exceptions which apply to all companies, contained in section 153 of the 1985 Act, and a relaxation of the general rule for private companies in sections 155 to 158 of that Act.
987.As recommended by the CLR (Final Report, paragraph 10.6), the Act abolishes the prohibition on private companies giving financial assistance for a purchase of own shares and, as a consequence, the relaxation for private companies (sometimes referred to as the “whitewash” procedure) is no longer required. The provisions in sections 155 to 158 have therefore been repealed and are not carried forward in the Act.
Section 678: Assistance for acquisition of shares in public company
988.This section replaces section 151(1) and (2) of the 1985 Act and restates section 153(1) and (2) of that Act. The key change is that the prohibition on private companies providing financial assistance for a purchase of own shares is not carried forward.
989.The general prohibition on the giving of financial assistance by a public company is required by the Second Company Law Directive (77/91/EEC) and this prohibition is retained in subsection (1). As under the current law, the prohibition extends to post-acquisition assistance (see subsection (3)).
990.The prohibition on the giving of post-acquisition assistance only applies if the company in which the shares were acquired is a public company at the time that the assistance is given (see subsection (3)). It follows that where a company has re-registered as a private company since the shares were acquired and is a private company at the time the post-acquisition assistance is given, the prohibition in this section will not apply. However, if at the time the shares were acquired the company was a private company, but at the time the post-acquisition assistance is given it has re-registered as a public company, the prohibition will apply.
991.The provisions of section 153(1) and (2) of the 1985 Act are retained in subsections (2) and (4) which carry forward the current exemption from the prohibition on the giving of financial assistance: namely, that such assistance is not prohibited if the principal purpose of the assistance is not to give it for the purpose of an acquisition of shares, or where this assistance is incidental to some other larger purpose of the company and (in either case) where the assistance is given in good faith in the interests of the company. As now, in these circumstances no offence is committed by the company or its officers (see section 680). The changes introduced by section 678 also give statutory effect to the decision in Arab Bank plc v. Mercantile Holdings Ltd 2 All ER 74: namely, that the statutory prohibition on a company giving financial assistance for the purpose of acquiring its own shares or shares in its holding company does not apply to the giving of assistance by a subsidiary incorporated in an overseas jurisdiction.
992.In the Arab Bank case, Millett J considered the geographical scope of section 151 of the 1985 Act and concluded that this had inadvertently been altered during the consolidation of UK companies legislation in 1985. In particular, the 1985 Act appears to go further than the 1980 Act and, in interpreting the current provision, Millett J applied the presumption that, in the absence of a contrary intention, section 151 could not have extra-territorial effect. The difficulty with the 1985 Act provision arises as a result of how the prohibition is framed: in particular, the prohibition applies “to the company or any of its subsidiaries” and “subsidiary”, as defined in section 736 of the 1985 Act, includes foreign companies. The prohibition in the Act is restricted to UK public companies and their UK subsidiaries as a result of the definition of “company” in section 1. Subsection (1) of that section makes it clear that, unless the context otherwise requires, “company” means a company which is formed and registered under the Act or a former UK Companies Act.
Section 679: Assistance by public company for acquisition of shares in its private holding company
993.This section replaces sections 151(1) and (2) and 153(1) and (2) of the 1985 Act.
994.Like section 678 this section does not carry forward the prohibition on private companies providing financial assistance for a purchase of own shares, but the current prohibition on the giving of financial assistance by a public company subsidiary for the purpose of an acquisition of shares in its private holding company is retained. Section 679 also retains the prohibition on the giving of post-acquisition assistance by a public company subsidiary.
Chapter 3: Redeemable Shares
995.Under section 159 of the 1985 Act a company that is limited by shares, or limited by guarantee and having a share capital, may, if authorised to do so by its articles, issue shares which may be redeemed at a future point in time at the option of the company or the shareholder. The provisions of section 159 are carried forward in the following sections but there are changes to the ways in which companies may issue redeemable shares and redeem such shares (see below).
Section 684: Power of limited company to issue redeemable shares
996.This section replaces section 159(1) of the 1985 Act and restates section 159(2) of that Act.
997.For private companies only, it removes the requirement for prior authorisation in the company’s articles for a proposed allotment of redeemable shares. If they wish, the members may, however, restrict or prohibit the authority given to a company by this section, by including a provision to this effect in the company’s articles (see subsection (2)).
Section 685: Terms and manner of redemption
998.This replaces section 160(3) of the 1985 Act (which provides that the terms and manner of redemption must be set out in the company’s articles) and existing section 159A (also entitled “terms and manner of redemption”), which was inserted into the 1985 Act by section 133 of the Companies Act 1989 and remains uncommenced.
999.As recommended by the CLR (Final Report, paragraph 4.5), this section enables the directors of both private and public companies alike to determine the terms, conditions and manner of a redemption of redeemable shares. The power conferred on the directors by this section requires prior authorisation by the company’s members, either by resolution of the company or through the articles (see subsection (1)). As recommended by the CLR (Final Report, paragraph 7.30) the terms and conditions of redemption will have to be stated in the statement of capital required to be filed under section 555. If the directors are not authorised to set the terms of the redemption, then they must be set out in the company’s articles (see subsection (4)).
1000.Where the directors exercise this power they must do so before the shares in question are allotted (see subsection (3)).
Section 686: Payment for redeemable shares
1001.This section replaces section 159(3) of the 1985 Act (which requires that where a company issues redeemable shares, the terms of redemption must provide for payment on redemption). It removes the current requirement, in section 159(3), that the terms of redemption must provide for payment on redemption. This means that the terms of redemption may provide for the company and the holder of the shares to agree that payment may be made on a date later than the redemption date.
Section 688: Redeemed shares treated as cancelled
1002.This section restates section 160(4) of the 1985 Act but with the exception of the reference to the impact of the redemption on the authorised share capital of the company – the concept of which is not replicated under the Act.
Section 689: Notice to registrar of redemption
1003.This section restates section 122(1)(e) and 122(2) of the 1985 Act and Schedule 24 to that Act but there is a new requirement for a statement of capital where a company gives notice to the registrar of a redemption of redeemable shares (see subsections (2) and (3) and the note on section 619).
Chapter 4: Purchase of Own Shares
1004.Section 162 of the 1985 Act enables a company limited by shares or limited by guarantee and having a share capital to purchase its own shares, provided it is authorised to do so by the articles. It is common for the members to give authority for such a purchase of own shares through the articles, see, for example, regulation 35 of the Companies Act 1985, Table A.
Section 690: Power of limited company to purchase own shares
1005.This section replaces section 162(1) of the 1985 Act and restates section 162(3) of that Act.
1006.In line with the recommendations of the CLR (Completing the Structure, paragraph 2.15), section 690 removes the requirement for prior authorisation in a company’s articles for a purchase of own shares (including any redeemable shares) by the company but makes it clear that, if they wish, the members may restrict or prohibit a purchase of own shares by including a provision to this effect in the company’s articles.
Section 692: Financing of purchase of own shares
1007.This section restates, the provisions of section 160(1) and (2) and 162(2) of the 1985 Act on a redemption of own shares which are applicable to a purchase of own shares, and makes such adaptations to those provisions as are necessary to ensure that the restated provisions work in this context.
Section 694: Authority for off-market purchase
1008.This section replaces sections 164(2) and 165(2) of the 1985 Act and restates sections 164(1), (3) and (4) and 165(1) of that Act.
1009.Under the 1985 Act a company may only enter into a contract for an off-market purchase of shares if the shareholders approve the contract by special resolution before the contract is entered into. An off-market purchase of shares is a purchase that is not conducted through a recognised investment exchange (for example, the London Stock Market).
1010.Section 694 enables a company to enter into a contract for an off-market purchase of its own shares conditional on the contract being approved by the shareholders. This will save companies valuable time as it will be possible for the directors to negotiate and agree the terms of a contract for an off-market purchase of shares ahead of seeking shareholder approval. If, however, the shareholders do not subsequently pass a special resolution approving the contract, the company may not purchase the shares in question and the contract will lapse.
Section 702: Copy of contract or memorandum to be available for inspection
1011.This section replaces section 169(4) and (9) of the 1985 Act. Under the 1985 Act, where a company enters into a contract for a purchase of own shares it must make available for inspection a copy of the relevant contract or a memorandum of its terms at the company's registered office for a period of 10 years. Subsection (4) alters the current requirements by providing that the contract may, alternatively, be kept available for inspection at a place specified in regulations made under section 1136. If the company is a private company, the contract must be available for inspection by any of its members; otherwise it must be open to inspection by anyone. If default is made an offence is committed by the company and every officer in default.
1012.Subsection (5) is a new provision which requires companies to give notice to the registrar of the place where the contract is kept available for inspection. This is consequential on the choice conferred on companies under subsection (4) as to where such documents are kept.
Section 703: Enforcement of right to inspect copy or memorandum
1013.This section replaces section 169(7) of the 1985 Act and Schedule 24 to that Act. It retains the existing offences for failing to keep available/allow inspection of the contract/memorandum as required under section 702, but with the addition of a new offence for failure to notify the registrar of the place where such documentation is kept. It also restates section 169(8) of the 1985 Act.
Section 707: Return to registrar of purchase of own shares
1014.This section replaces section 169(1), (1A) and (1B) and restates section 169(2), (3) and (6) of the 1985 Act and Schedule 24 to that Act. It requires a company to make a return to the registrar within 28 days of a purchase of own shares stating, amongst other things, the number and nominal value of the shares purchased. The return must indicate whether the shares are of a type which could be held in treasury (see section 724), and, if so, whether the shares were cancelled forthwith or whether they are being held in treasury.
Section 708: Notice to registrar of cancellation of shares
1015.This section replaces sections 169(1), (1A) and (1B) and restates 169(6) of the 1985 Act and Schedule 24 to that Act. It updates the notice requirements so as to require that where shares are cancelled forthwith following a purchase (either in circumstances where the shares are treated as automatically cancelled or where treasury shares are cancelled) the company has to notify the registrar of such cancellation and provide the registrar with a statement of capital (see note on section 619).
1016.As now, where a company fails to comply with the procedural requirements as to notice, the company, and every officer of the company who is in default, commits an offence (see subsection (4)).
Chapter 5: Redemption Or Purchase by Private Company Out of Capital
1017.Sections 171 to 177 of the 1985 Act provide a statutory scheme for the redemption or purchase of own shares out of capital. This scheme is available to private companies only and the facility to redeem or purchase shares out of capital is carried forward in the following sections.
Section 709: Power of private limited company to redeem or purchase own shares out of capital
1018.This section replaces section 171(1) of the 1985 Act and restates section 171(2) of that Act. It removes the current requirement for prior authorisation in the articles where a private company makes a payment out of capital in respect of a redemption or purchase of its own shares. If they wish, the members may, however, restrict or prohibit such a payment by including a provision to this effect in the company’s articles.
Section 714: Directors’ statement and auditor’s report
1019.This section replaces section 173(3) to (5) of the 1985 Act.
1020.Currently, before a private company may make a payment out of capital in respect of a purchase of own shares, the directors must have made a full enquiry into the affairs and prospects of the company and are required, under section 173, to make a statutory declaration confirming that: as regards the company’s situation immediately after the date on which the payment out of capital is made, there will be no grounds on which the company could then be found unable to pay its debts; and as regards the company’s prospects for the year immediately following that date, the company will be able to continue to carry on business as a going concern and be able to pay its debts as they fall due in the year immediately following the date on which the payment out of capital is made. In forming their opinion on the company’s solvency and prospects, the directors must take into account the same liabilities (including contingent and prospective liabilities) as would be relevant under section 122 of the Insolvency Act 1986 (winding up by the court) to the question whether a company is unable to pay its debts.
1021.Consistent with the approach taken in respect of reductions of capital using the new solvency statement procedure (see sections 642 and 643 and in particular subsection (2) of section 643), this section requires a private company limited by shares that wishes to use this statutory scheme for a purchase or redemption of shares to take account of all contingent and prospective liabilities, not just those that are relevant for the purposes of section 122 of the Insolvency Act 1986 (see subsection (4)).
1022.Again, to achieve consistency with the approach taken elsewhere in the Act, the current requirement for a statutory declaration is replaced with a requirement for a simple statement. In contrast to a statutory declaration, the directors’ statement does not need to be sworn before a solicitor or Commissioner of Oaths.
Section 715: Directors’ statement: offence if no reasonable grounds for opinion
1023.This section restates section 173(6) of the 1985 Act and Schedule 24 to that Act (but substitutes the reference to “declaration” with “statement”). The offence that is currently contained in subsection (6) of section 173 (offence of making declaration without reasonable grounds) is replaced with an offence of making a statement under section 714 without having reasonable grounds for the opinion expressed in it. As now, the offence is committed by every director of the company who is in default.
Section 720: Directors’ statement and auditor’s report to be available for inspection
1024.This section replaces section 175(6)(a) and (7) and restates section 175(4), (6)(b), and (8) of the 1985 Act and Schedule 24 to that Act. Subsection (2) alters the current requirement, contained in section 175(6)(a), by providing that the directors’ statement and auditor’s report may, alternatively, be kept available for inspection at a place specified in regulations made under section 1136. There is a new requirement (in subsection (3)) for the company to give notice to the registrar of the place where the statement and report are kept available for inspection and of any change to that place. This change is consequential on the change introduced by subsection (2).
1025.Subsection (5) provides that if the company fails to give such notice to the registrar within 14 days or an inspection of the statement and report are refused, the company and every officer in default commit an offence.
Chapter 6: Treasury Shares
1026.Where a company buys back its own shares, it is normally required to cancel those shares. Certain companies (principally those which are listed or those which are traded on the Alternative Investment Market and equivalent companies in the EEA) may elect not to cancel shares which have been bought back but may hold the shares “in treasury”. A share which is held in treasury may be sold at a future point in time and this facility enables such companies to raise capital more quickly than they would otherwise be able to, as the directors do not have to obtain prior authority from the company’s members before selling treasury shares.
Section 727: Treasury shares: disposal
1027.This section replaces section 162D(2) of the 1985 Act and restates section 162D(1)(a) and (b) and 162D(3) of that Act. It defines (in subsection (2)) what is meant by “cash consideration” where treasury shares are sold and this mirrors, in part, the definition in section 583 (which restates section 738(2) to (4) of the 1985 Act and defines when a share is deemed to be paid up or allotted for cash).
1028.Subsections (2)(e), (3) and (5) are new. They enable the Secretary of State to specify, by order, what, in addition to the items specifically referred to in subsection (2), constitutes “cash consideration” for the purposes of subsection (1)(a).
1029.The power to make further provision in respect of what constitutes “cash consideration” for the purposes of a sale of treasury shares (that is, in addition to those already specified in subsection (2)) is intended to remove uncertainties surrounding other methods of settlement, for example, the CREST settlement system (see note on section 583) and will also act as a future proofing mechanism in the event that new methods of settlement are developed or identified.
Section 730: Treasury shares: notice of cancellation
1030.This section restates sections 169A(1)(b)(i), (2) to (4) of the 1985 Act and Schedule 24 to that Act. As now, where a company cancels shares which it has held in treasury it is required to give notice of this to the registrar within 28 days of the cancellation. The provision has, however, been updated to require companies to file a statement of capital in these circumstances (see note on section 619).
Section 732: Treasury shares: offences
1031.This section replaces section 162G of the 1985 Act. It renders both the company and every officer in default liable to an offence for any contravention of the provisions of this Chapter.
Section 737: General power to make further provision by regulations
1032.This section is a new provision which enables the Secretary of State, by regulations, to modify the provisions of Part 18 (see subsection (1)).
1033.Regulations made under this section may amend or repeal any of the provisions in this Part or make such other provision as appears to the Secretary of State appropriate in place of those provisions.
1034.The power to make regulations in this section will enable the Secretary of State to “future-proof” the provisions in Part 18 – which are primarily concerned with the maintenance of capital. This is desirable as many of these provisions are derived from EU law and may require amendment in the relatively near future (subject in part to the outcome of a fundamental study into alternatives to the current capital maintenance regime which is being carried out at EU level).
1035.Regulations made pursuant to the power in this section are subject to the affirmative resolution procedure, which means that they must be approved by both Houses of Parliament.
Part 19: Debentures
1036.This part restates the provisions of the 1985 Act relating to debentures. Sections 738-740, 742 and 749-754 make no change to the law.
Section 741: Registration of allotment of debentures
1037.This is a new section which obliges a company to register an allotment of debentures as soon as practicable, but in any event within two months after their allotment. It completes the picture as regards the existing requirement in section 185(1) of the 1985 Act (which is restated in section 769(1)) which obliges a company to complete and deliver certificates for debentures within two months after their allotment.
Section 743: Register of debenture holders
1038.This section replaces section 190 of the 1985 Act. There is no requirement for a company to keep a register of debenture holders but if such a register is kept, then it (or any duplicate) must be kept available for inspection at either the company’s registered office or a place permitted under regulations made under section 1136. (This is the same as for the obligatory registers of members, see section 114.)
Section 744: Register of debenture holders: right to inspect and require copy
1039.This section replaces part of section 191 of the 1985 Act. It modifies the existing right of public access to any register of debenture holders kept by a company. The changes mirror similar requirements in Part 8 relating to the register of members. Subsections (3) and (4) require those seeking to inspect or to be provided with a copy of the register to provide their names and addresses, the purpose for which the information will be used, and, if the access is sought on behalf of others, similar information for them.
Section 745: Register of debenture holders: response to request for inspection or copy
1040.This is a new provision. It provides a procedure by which the company can refer the matter to the court if it considers the request is not for a proper purpose. It specifies a 5-day period within which the company must either comply with the request or apply to the court for relief from the obligation. If the company opts for the latter, then subsections (3), (4) and (5) apply. Under subsection (3), if the court is satisfied that the access to the register of debenture holders is not sought for a proper purpose, it will require the company not to comply with the obligation to meet the request and may require that the person who made the request pay the company’s costs. Under subsection (4), the court may also require the company not to comply with other requests made for similar purposes. If the court does not make an order under subsection (3), or the proceedings are discontinued, then, under subsection (5), the company must immediately comply with the request.
Section 746: Register of debenture holders: refusal of inspection or default in providing copy
1041.This section retains the existing sanctions under section 191 of the 1985 Act for failure to comply with a request. They do not apply if the court has directed that the company need not comply with the request.
Section 747: Register of debenture holders: offences in connection with request for or disclosure of information
1042.This is a new provision. It creates two offences. First, in relation to the new requirement in section 744 to provide information in a request for access, it is an offence knowingly or recklessly to make a statement that is misleading, false or deceptive in a material particular. Second, it is an offence for a person having obtained information pursuant to an exercise of the rights in section 744 to do anything or fail to do anything which results in that information being disclosed to another person knowing or having reason to suspect that the other person may use the information for a purpose that is not a proper purpose.
Section 748: Time limit for claims arising from entry in register
1043.This section replaces section 191(7) of the 1985 Act. It amends the existing time limit for claims arising from errors in the register from twenty years to ten years. This mirrors equivalent provisions applicable to the register of members (see section 128).
Part 20: Private and Public Companies
1044.The provisions of this Part set out the two major differences between public and private companies.
1045.Chapter 1 replaces sections 58(3), 81 and 742A of the 1985 Act which provide that private companies are not allowed to offer their shares to the public.
1046.Chapter 2 replaces sections 117 and 118 of the 1985 Act which deal with the minimum share capital requirement for public companies (known as the “authorised minimum”). It contains new provisions that enable this requirement to be satisfied in euros as well as sterling. To facilitate this change it has been necessary to seek two new powers:-
a)
a power to prescribe the amount in euros that is to be treated as equivalent to the sterling amount of the authorised minimum (see section 763); and
b)
a power to prescribe how references to the authorised minimum in the Act are to be applied where a public company has shares denominated in more than one currency or redenominates its share capital (that is, converts its share capital from one currency to another) and to require that a company must re-register as a private company where the effect of redenomination is to bring the value of the company’s share capital below the authorised minimum (see section 766).
1047.The authorised minimum is relevant to all public companies, not just those that are incorporated as such, see for example, section 91.
Chapter 1: Prohibition of Public Offers by Private Companies
1048.The CLR considered the prohibition on private companies offering their shares to the public in paragraph 4.160 of Developing the Framework and then examined the dividing line between public and private companies in Chapter 2 of Completing the Structure. The CLR presented their conclusions in paragraphs 4.54 to 4.62 of the Final Report.
Section 755: Prohibition of public offers by private company
1049.Subsection (1) of this section continues the prohibition in section 81(1) of the 1985 Act on private companies offering their shares or debentures to the public, though the consequences of breaching the prohibition are changed. The prohibition applies only to private companies limited by shares or limited by guarantee and having a share capital. The prohibition does not apply to unlimited companies or to companies limited by guarantee and not having a share capital.
1050.Private companies are also prohibited from allotting their shares or debentures with the intention that they are offered to the public by someone else. Subsection (2) creates a presumption as to when shares or debentures have been allotted in this way. Similar provision was made in section 58(3) of the 1985 Act which this subsection replaces.
1051.A private company will no longer commit an offence if it offers its securities to the public. Instead, if a private company does breach the prohibition it will be compelled to re-register as a public company, unless it appears to the court that the company does not meet the requirements for re-registration and that it is impractical or undesirable to require it to take steps to do so, in which case the court may make a remedial order and/or an order for the compulsory winding up of the company.
1052.Subsection (3) contains an exemption to the prohibition on public offers. Where a private company intends to become a public company it will be able to make an offer before it has completed the formalities of re-registration as a public company. Acts done in good faith before allotment in anticipation of re-registration will not be treated as breaching the prohibition on offers to the public, even if the re-registration arrangements do not ultimately succeed. The exemption also applies if, as part of the terms of the offer, the company undertakes to re-register as a public company and then complies with that undertaking not later than 6 months after the day on which the offer is first made to the public.
Section 756: Meaning of “offer to the public”
1053.This section explains what is meant by “offer to the public” for the purposes of the prohibition on public offers contained in section 755. This section also sets out certain circumstances where an offer is not to be regarded as an offer to the public. It replaces section 742A of the 1985 Act.
1054.An offer will not be an offer to the public if it is not calculated to result in shares or debentures of the company becoming available to anyone other than those receiving the offer. An example would be where shares are offered to a particular person, with the intention that no one other than that particular person may take up the offer or acquire the shares as a result. Nor will an offer be an offer to the public if the offer is otherwise a private concern of the person receiving it and the person making it.
1055.Subsection (4) creates two further exemptions for offers to persons already connected with the company (as defined in subsection (5)) and for offers in respect of securities to be held under an employees’ share scheme (as defined in section 1166). Such offers are presumed to be the private concern of those involved and so not an offer to the public if the conditions set out in subsection (4) are met.
1056.The range of persons already connected with the company for the purposes of subsection (4) has been expanded slightly from the current provision in section 742A of the 1985 Act. Subsection (5) now includes a trustee of a trust where the principal beneficiary is an existing debenture holder of the company or the widow or widower, or surviving civil partner of a person who was a member or employee of the company.
1057.Subsection (6) explains what is meant by a member of a person’s family for the purposes of subsection (5).
Section 757: Enforcement of prohibition: order restraining proposed contravention
1058.This section enables members, creditors or the Secretary of State to apply to the court for an order restraining a private company from carrying out any proposed contravention of the prohibition on offering shares or debentures to the public. This is a new procedure which will enable the member, the creditor or the Secretary of State to prevent by civil action any further activity by the company towards making an offer in contravention of the public offer prohibition. The court must also make such an order if, in proceedings brought by a member under section 994 or by the Secretary of State under section 995, it appears to the court that the company is proposing to breach the public offer prohibition.
Section 758: Enforcement of prohibition: orders available to the court after contravention
1059.This section applies where a private company breaches the prohibition on offering securities to the public. It introduces a new enforcement procedure for breaches; it replaces the criminal offence currently imposed in section 81 of the 1985 Act with a civil enforcement procedure.
1060.If a company breaches the prohibition, certain members, certain creditors or the Secretary of State may apply to the court. In order to have standing to bring the application, the member or creditor must have been a member or a creditor at the time the offer was made in contravention of the public offer prohibition; in addition anyone who became a member as a result of the offer to the public may bring an application.
1061.On such an application, if the court decides the company has acted in contravention of the public offer prohibition then it must order the re-registration of the company as a public company, unless it appears to the court that the company does not meet the requirements for re-registration as a public company (see Part 7 of the Act), and it is impracticable or undesirable to require it to take steps to do so. If the court is unable to order re-registration for these reasons, it may instead make a remedial order or an order for the compulsory winding up of the company (see Chapter 6 of Part 4 of the Insolvency Act 1986). The court has discretion as to whether or not to make these orders. This might be appropriate for example where the company has breached the prohibition but has not allotted shares, and has withdrawn the offer and undertaken not to do it again.
Section 759: Enforcement of prohibition: remedial order
1062.A remedial order is an order for the purpose of putting anyone affected by the breach of the public offer prohibition back in the position they would have been in if the breach had not occurred. It may require any person knowingly concerned in the contravention, whether or not an officer of the company, to offer to purchase the shares or debentures that were the subject of the offer on such terms as the court thinks fit. The remedial order will override the terms of the company’s constitution, but no one holding the securities will be obliged to accept the offer made to purchase them. It may be made whether or not the holder of the securities subject to the order is the person to whom the company allotted or agreed to allot them.
Section 760: Validity of allotment etc not affected
1063.This section makes clear that any allotment or sale of securities or any agreement to allot or sell securities is not made void simply because there has been a breach of the prohibition on offers to the public. Equivalent provision was made in section 81(3) of the 1985 Act.
Chapter 2: Minimum Share Capital Requirement for Public Companies
1064.Under the 1985 Act, a public company which is incorporated as such may not do business without first obtaining a trading certificate from the registrar. There is a minimum allotted share capital requirement, known as the “authorised minimum”, which is currently set at £50,000 and which must be denominated in sterling. The same minimum share capital requirement applies where a private company re-registers as a public company under Part 7 of the Act.
1065.If a public company reduces its share capital below the authorised minimum it will no longer meet the minimum share capital requirement for a public company and must re-register as a private company.
1066.The requirement for public companies to have a minimum share capital is derived from EU law. Article 6 of the Second Company Law Directive (77/91/EEC) provides that:
““The laws of member states shall require that in order that a company may be incorporated or obtain authorisation to commence business, a minimum share capital shall be subscribed the amount of which shall not be less than 25,000 euros.””
1067.The Department’s interpretation of this Directive is that it permits the authorised minimum to be denominated in euros, or the national currency of the Member State, but not in other currencies (so for example, the authorised minimum could not be satisfied in dollars).
1068.When this requirement was implemented in the Companies Act 1980 the amount of the authorised minimum was set at £50,000 (a figure considerably higher than the minimum required under the Directive). The CLR considered whether the authorised minimum should be maintained, increased or reduced. Their recommendation was to maintain it at £50,000.
Section 761: Public company: requirement as to minimum share capital
1069.This section replaces section 117(2) of the 1985 Act and restates section 117(1), (4) and (6) of that Act. Like the provisions of the 1985 Act, it only applies to public companies that are formed as such on their original incorporation (as opposed to companies that re-register from private limited to public under the provisions of Part 7 of the Act).
1070.The current requirement for a statutory declaration to be filed with an application for a trading certificate, contained in section 117(2) of the 1985 Act, has not been carried forward. This is replaced by a requirement for a statement of compliance (see section 762 and the note on section 13).
1071.As now, the registrar will only issue a trading certificate if she is satisfied that certain conditions are met: in particular the company must satisfy the minimum share capital requirement for public companies – known as the “authorised minimum” (see subsection (2)).
1072.A trading certificate has effect from the date that it is issued and is conclusive evidence that the company is entitled to do business as a public company.
Section 762: Procedure for obtaining certificate
1073.This section replaces section 117(3) of the 1985 Act. It prescribes the contents of the application for a trading certificate (see subsection (1)), which, amongst other things, must include a statement that the nominal value of the company’s share capital is not less than the authorised minimum.
1074.The current requirement for a statutory declaration (or “electronic statement”) when an application is made for a trading certificate is replaced by a requirement to make a statement of compliance. This statement does not need to be witnessed and may be made in paper or electronic form. It will be for the registrar’s rules to specify who may make this statement (and the form of it).
Section 763: The authorised minimum
1075.This section replaces section 118(1) of the 1985 Act. Under section 118 the authorised minimum is £50,000. This implements Article 6 of the Second Company Law Directive which requires that in order that a public company may be incorporated or obtain authorisation to commence business, a minimum capital shall be subscribed (see above). As recommended by the CLR (Completing the Structure, paragraph 7.6), this section retains the authorised minimum at £50,000. In contrast to the 1985 Act provisions, the section also enables the minimum share capital requirement for public companies to be satisfied in sterling or the prescribed euro equivalent (see subsection (1)).
1076.Once a company has obtained a trading certificate under section 762 or section 117 of the 1985 Act (in the case of companies that obtain a trading certificate before these provisions of the Act come into force), there is no requirement for the authorised minimum to remain denominated in sterling or euro, and if it wishes a public company may subsequently redenominate all of its share capital (including the authorised minimum) under the provisions of Chapter 8 of Part 17 of the Act (which contains new provisions which permit companies easily to redenominate or convert their share capital from one currency to another).
1077.Subsection (2) of section 763 contains a new power which enables the Secretary of State, in regulations made under the Act, to prescribe the amount in euros that is for the time being to be treated as equivalent to the sterling amount of the authorised minimum. This is required in order to achieve parity between the prescribed sterling and euro amounts (which may become necessary due to exchange rate fluctuations). The amount prescribed will be determined by applying an appropriate spot rate of exchange to the prescribed sterling amount and rounding to the nearest 100 euros (see subsection (4)).
1078.The power to alter the authorised minimum, contained in section 118 of the 1985 Act, is carried forward in section 764.
Section 764: Power to alter authorised minimum
1079.This section replaces section 118(1) of the 1985 Act and restates section 118(2) and (3) of that Act. The power to alter the authorised minimum, contained in section 118, is carried forward but this has been updated to reflect the fact that in future companies will be able to satisfy the authorised minimum in sterling or the prescribed euro equivalent of the sterling amount (see section 763).
1080.Section 764 also contains a new provision which enables the Secretary of State, in regulations made under the Act, to alter both the sterling amount of the authorised minimum and to make a corresponding alteration to the prescribed euro equivalent (which is to be determined by applying an appropriate spot rate of exchange to the sterling amount and rounding up to the nearest 100 euros – see subsection (2)).
1081.As now, the power in this section will enable the Secretary of State to alter the sterling amount of the authorised minimum, for example from £50,000 to £60,000, (and the prescribed euro equivalent) should it become necessary to do so (for example, because of changes to the prescribed minimum capital requirement for public companies at EU level) or desirable (for example, if it was considered appropriate for business reasons to raise or lower the minimum share capital requirement for public companies within the limits permitted by the Second Company Law Directive).
Section 765: Authorised minimum: application of initial requirement
1082.This section is a new provision which prescribes how the authorised minimum is to be met. Subsection (1) makes it clear that the authorised minimum may be satisfied by reference to allotted share capital denominated in sterling or euros (but not a combination of both).
1083.Where a company has allotted sterling and euro shares, the question of whether the authorised minimum has been satisfied will be determined firstly by reference to the total sterling amount of the company’s allotted share capital and then by reference to the company’s euros shares (see subsection (2)). To take an example, if a company has allotted sterling shares to the total value of £25,000 and euro shares to the equivalent of £60,000, the authorised minimum will have been satisfied in euros. If the same company had allotted sterling shares to the total value of £10,000 and euro shares to the equivalent of £40,000 it would not have satisfied the minimum share capital requirement for a public company as the authorised minimum may be satisfied in sterling or euro but not partly in sterling and partly in euro.
Section 766: Authorised minimum: application where shares denominated in different currencies etc
1084.This section is a new provision which enables the Secretary of State to prescribe, in regulations made under the Act, how references to the authorised minimum are to be applied where a public company has its share capital denominated in more than one currency, or where it redenominates (converts) its share capital from one currency to another (see section 622).
1085.There are various provisions in the Act (for example where a public company applies to court to reduce its share capital) which provide that a company must re-register as a private company where the nominal value of its allotted share capital falls below the authorised minimum. It is therefore necessary to make provision for how references to the authorised minimum in the Act are to be applied where a public company has its shares denominated in different currencies, or currencies other than those in which the authorised minimum may be satisfied, and to require that a company must re-register as a private company where the effect of a redenomination of its share capital is to bring the value of the company’s share capital below the authorised minimum.
1086.To take an extreme example, the type of scenario that the power in this section is intended for is the situation where a public company incorporates with a share capital of £50,000 (expressed in sterling), allots additional dollar shares, subsequently redenominates part of its share capital into euros and then applies to the court to reduce its share capital. It will be necessary in such circumstances to determine what test should be applied to ascertain whether the company’s allotted share capital has fallen below the authorised minimum (in other words what exchange rates must be applied, as at what date they should be applied and as between what currencies). Regulations made pursuant to this section will need to deal with this type of issue. They will be subject to the negative resolution procedure due to their highly technical nature.
Section 767: Consequences of doing business etc without a trading certificate
1087.This section restates section 117 (7) and (8) of and Schedule 24 to the 1985 Act.
1088.As now, where a public company which is required to have a trading certificate enters into a transaction without first obtaining such a certificate, the directors are jointly and severally liable for any loss or damage caused to the other party to the transaction as a result of the company failing to meet its obligations. A director will only be jointly and severally liable with the company if he was a director at the time that the transaction was entered into and if the company has failed to meet its obligations under the transaction in question within 21 days of being called on to do so (see subsection (3)).
1089.Notwithstanding the fact that the company should not have entered into the transaction, the transaction itself is valid.
1090.Where a public company that is formed under this section, or under section 117 of the 1985 Act, has not obtained a trading certificate within a year of its incorporation, it may be wound up by the court (see section 122(1)(b) of the Insolvency Act 1986).
Part 21: Certification and Transfer of Securities
Chapter 1: Certification and Transfer of Securities: General
1091.Sections 768 to 770, 772 to 779 and 781 to 782 in this Chapter restate the provisions in Part 5 of the 1985 Act (sections 183 to 189) relating to the certification and transfer of shares and other securities.
Section 771: Procedure on transfer being lodged
1092.Under section 183(5) of the 1985 Act, if a company refuses to register a transfer of shares (or debentures), it must, within two months of receipt of the transfer, send to the transferee notice of its refusal to register the transfer of shares. Such a refusal will not affect the transferee’s beneficial interest in a share, for example, he will still be entitled to any dividend declared on that share, and a return of capital on winding-up, but the transferee will not be able to exercise all of the rights of a member of the company, for example, he may not vote at meetings, until such time as the transfer is registered and his name is entered in the register of members.
1093.Section 771 is a new provision which amends the law on the registration of transfers. As recommended by the CLR, it requires the directors to either register a transfer of shares or debentures or provide the transferee with reasons for their refusal to register (see Final Report, paragraphs 7.44 and 7.45).
1094.In either case, this must be done as soon as practicable, but in any event within two months of the transfer being lodged with the company.
1095.Under subsection (2), where the directors refuse to register the transfer of a share, the transferee is entitled to receive such information as he may reasonably require regarding the reasons for the directors’ refusal to register the transfer. Such information does not extend to minutes of meetings of the directors.
1096.Where a company fails to comply with this section, the company and every officer of the company who is in default commits an offence (see subsection (3)).
1097.Subsection (5) makes it clear that this section does not apply to a transfer of shares if the company has issued a share warrant in respect of the shares under section 779 or in relation to a transmission of shares by operation of the law (for example, where a bankrupt member’s trustee in bankruptcy or a deceased member’s personal representative becomes entitled to shares).
Section 780: Duty of company as to issue of certificates on surrender of share warrant
1098.This provision is new and makes it clear that a company must issue a share certificate where a share warrant is subsequently surrendered for cancellation. It gives a company two months from the date of surrender to complete and have ready for delivery a certificate of the shares specified in the warrant and failure to do so is a criminal offence (see subsection (3)). This requirement is subject to any contrary provisions in the company’s articles, which may give the company more or less time to deliver such certificates to the transferee (see subsection (2)).
Chapter 2: Evidencing and Transfer of Title to Securities Without Written Instrument
1099.This Chapter replaces the existing power under section 207 of the 1989 Act relating to transfer of securities without a written instrument but goes beyond it in that it can be used to require, as well as to permit, the paper-free holding and transfer of shares or other securities. The provisions that make changes to the power are in sections 784, 786, 787 and 789.
Section 784: Power to make regulations
1100.This section provides for the power to make regulations about the transfer of title to securities without written instrument to be exercisable by the Secretary of State or the Treasury. Responsibility for section 207 of the 1989 Act and the regulations made under it passed from the Department of Trade and Industry to HM Treasury by virtue of article 2(1) of the Transfer of Functions (Financial Services) Order 1992 (S.I. 1992/1315) as part of a general transfer of responsibility for financial services matters. Dual responsibility is considered more appropriate for the making of regulations under the new power as the extension of paperless holding and transfer to new classes of shares or other securities involve matters which are part of company law. Exercise of the power will continue to be subject to the affirmative procedure.
Section 786: Provision enabling or requiring arrangements to be adopted
1101.This section provides that regulations under this Chapter may require, as well as permit, the paper-free holding and transfer of securities. The effect of subsections (1) and (2) is that regulations made under section 207 may:
enable members of companies, or of designated classes of company, by ordinary resolution, to adopt a new form of paperless holding and transfer of shares and abandon paper-based forms of holding and transfer in relation to all existing and new securities of that company, or to specified types of securities; or
make the adoption of a form of paperless transfer and the abandonment of paper-based forms of transfer mandatory for all securities, or specified types of securities, issued by companies generally or by designated classes of company.
1102.Regulations do not need to make it obligatory both to hold and to transfer securities in a paper-free way: the new arrangements could relate just to holding or just to transfer.
1103.Subsection (3) is designed to protect the right of individual investors to continue to hold shares in their own names rather than through nominees. It ensures that the new arrangements prescribed in the regulations will not mean that:
people who would have been entitled to have their names entered in the company’s register of members will lose that entitlement; or
people who are entitled to exercise rights in respect of securities will lose that right.
1104.Subsection (4) provides that the regulations will be able to:
prohibit the issue of share certificates or certificates for other types of security. Holders of securities to which any such prohibition applies will lose the option of continuing to hold certificates and transfer their shares by paper-based methods;
ensure that such holders of securities are sent periodic statements of their holdings;
make provision about the evidential value of certificates or statements.
Section 787: Provision enabling or requiring arrangements to be adopted: order-making powers
1105.This section provides additional flexibility by enabling Ministers to designate, by order (subject to negative resolution procedure), companies or classes of company to which the regulations are to apply, or to modify the effect of the regulations (or disapply them) in relation to a designated class of companies or specified companies.
Section 789: Duty to consult
1106.Ministers will be obliged to consult such persons as they consider appropriate before making regulations or designating a class of companies by order under the new powers. This obligation reflects the breadth of the proposed new powers, as well as the technical nature of some of the regulations which could be made under it.
Part 22: Information about Interests in Company’s Shares.Background
1107.The provisions of this Part concern a public company’s right to investigate who has an interest in its shares. They replace equivalent provisions in Part 6 of the 1985 Act. These are purely domestic provisions, and are not required by European Community Law.
1108.The automatic disclosure obligations currently contained in sections 198 to 211 of Part 6 of the 1985 Act will be replaced by regulations under the Financial Services and Markets Act 2000, as amended by Part 43 of this Act, in implementation of the Transparency Obligations Directive. In the regulations, a different concept of “interest in voting rights” will be adopted in order to implement the Transparency Obligations Directive.
1109.This Part re-enacts, with certain modifications, the disclosure obligations pursuant to a notice issued by the company contained in sections 212 to 219 of the 1985 Act. There is no change to the definition of “interest in shares” for this purpose.
1110.The main changes to section 212 of the 1985 Act and related provisions are:
making clear that notices are not required to be in hard copy, and therefore can be given in electronic form (section 793 read in conjunction with the provisions in Part 37 on the sending or supplying of documents or information);
providing for how information is to be entered on the register of interests disclosed when the name of the present holder of the shares is not known or there is no present holder (section 808);
removing the requirement on the company to verify third party information supplied in response to a section 793 notice before putting it on the register (section 817);
requiring a company to refuse a request to inspect the register if it is not satisfied that the request is made for a proper purpose (section 812);
removing the requirement for a company to keep information on the register in relation to entries made more than six years previously (section 816).
Section 791: Companies to which this Part applies
1111.This section provides that this Part only applies to public companies (as section 212 of the 1985 Act provides currently).
Section 792: Shares to which this Part applies
1112.This section re-enacts in part the definition in section 198(2) of the 1985 Act of the type of shares concerning which a section 793 notice may be issued, namely shares carrying rights to vote in all circumstances at general meetings. However, shares held by a company “in treasury” following a purchase of its own shares (as an alternative to cancelling such shares on purchase) are now included in the definition.
Section 793: Notice by company requiring information about interests in its shares
1113.This section re-enacts section 212(1) to (4) of the 1985 Act. It allows a public company to issue a notice requiring a person it knows, or has reasonable cause to believe, has an interest in its shares (or to have had an interest in the previous three years) to confirm or deny the fact, and, if the former, to disclose certain information about the interest, including information about any other person with an interest in the shares.
1114.Subsections (3) and (4) enable the company to require details to be given of a person’s past or present interests and to provide details of any other interest subsisting in the shares of which he is aware. This provision allows the company to pursue information through a chain of nominees by requiring each in the chain to disclose the person for whom they are acting. Under subsection (6), where the addressee’s interest is a past one, a company can ask for information concerning any person by whom the interest was acquired immediately subsequent to their interest. Particulars may also be required of any share acquisition agreements, or any agreement or arrangement as to how the rights attaching to those shares should be exercised (sections 824 and 825).
1115.This section serves a different purpose to the automatic disclosure obligations currently contained in sections 198 to 211 of Part 6 of the 1985 Act. It enables companies to discover the identity of those with voting rights (direct or indirect) that fall below the thresholds for automatic disclosure, and it also enables companies (and members of the company) to ascertain the underlying beneficial owners of shares.
1116.The notice is not required to be in hard copy (see the general provisions on sending or supplying documents or information in Part 37 of the Bill). Notices, and responses thereto, may be given in electronic form. A response must be given in a reasonable time. What is reasonable has not been defined so as to allow flexibility according to the circumstances, but if the time given is not reasonable, the company will not have served a valid notice.
Sections 794 and 795: Notice requiring information: order imposing restrictions on shares and offences
1117.These sections re-enact section 216(1) to (4) of the 1985 Act. They specify the penalties for failure to provide information within the specified time when served with a notice under section 793. There are criminal penalties (although a person does not commit an offence if he can show that the requirement to give information was frivolous or vexatious).
1118.Additionally, application may be made to the court for a direction that the shares in question are to be subject to the restrictions specified in section 797.
Section 796: Notice requiring information: persons exempted from obligation to comply
1119.This section re-enacts section 216(5) of the 1985 Act. It provides that the Secretary of State may exempt a person from complying with a notice. The Secretary of State must consult the Governor of the Bank of England, and must be satisfied that there are special reasons for exempting the person (taking account of any undertaking given).
Sections 797 to 802: Orders imposing restrictions on shares
1120.These sections restate Part 15 of the 1985 Act without substantive change in so far as its provisions apply in relation to Part 22 of this Act. They set out the effect of a court order made under section 794 imposing restrictions on shares, and the penalties for attempted evasion of the restrictions. They also make provision for the relaxation or removal of restrictions, or for an order for the sale of shares.
Section 803: Power of members to require company to act
1121.This section re-enacts section 214(1) and (2) of the 1985 Act. It requires a company to exercise its powers under section 793 on the request of members holding at least 10% of such of the paid up capital of the company as carries the right to vote at general meetings (other than voting rights attached to shares held in treasury). This provision, which has rarely been used under the 1985 Act, recognises that members of a company may have a legitimate reason for wanting the company to exercise its statutory powers to demand information even if the management does not want to. For example, the members might want to act where they suspect that the directors are involved in building a holding from behind the shelter of nominees.
1122.Provision is made as to the form and the procedure in relation to requests. In contrast to section 214, the 10% threshold may be met by a series of requests from members that the company act, rather than one collective request. Those making a request must not only specify the manner in which they require the powers to be exercised, but must also give reasonable grounds for requiring the company to exercise the powers in the manner specified (subsection (3)(b)(ii) and (iii)).
Section 804: Duty of company to comply with requirement
1123.This section re-enacts section 214(4) and (5) of the 1985 Act. It specifies the criminal penalties arising if the company fails to act as required. In contrast to section 214, every officer in default is liable to a fine, but the company itself is not.
Section 805: Report to members on outcome of investigation
1124.This section re-enacts section 215 of the 1985 Act. It specifies that on the conclusion of an investigation required by members it is the duty of the company to prepare a report of the information received. The report must be available within a reasonable period (not exceeding 15 days) after the conclusion of the investigation. Where the company’s investigation exceeds three months, it must make interim reports available at three monthly intervals. Those making the request must be notified of reports being available. In contrast to section 215, the report may be kept at the company’s registered office or at a place specified in regulations made under section 1136.
Section 806: Report to members: offences
1125.This section re-enacts section 215(8) of the 1985 Act. It specifies the criminal penalties arising if the company fails to report as required on the outcome of the investigation or to keep the report under section 805. Unlike section 215, every officer in default is liable to a fine for a failure to report, but the company itself is not.
Section 807: Right to inspect and request copy of reports
1126.This section re-enacts provisions in section 219 of the 1985 Act. It requires the company to allow reports to members to be inspected by anyone without charge. Any person can request a copy of a report, on payment of the prescribed fee. Subsections (3) to (5) specify the criminal penalties arising if the company fails to disclose a report as required and make provision for the courts to compel disclosure.
Section 808: Register of interests disclosed
1127.The register required to be kept by section 211 of the 1985 Act covers all interests notified, whether under the automatic disclosure rules or in response to a notice served under section 212 of that Act (company investigations). The latter are kept as a separate part of the register of interests in shares. In future it will be for regulations made under the Financial Services and Markets Act 2000 (as amended by Part 43 of this Act) to make provision as to how interests notified under the automatic disclosure rules will be made public.
1128.This section provides that if, as a result of a section 793 investigation, the company receives information relating to interests held by any person in relevant shares, it must within three days enter in a register of interests disclosed:
the fact that the requirement (to disclose information under the notice) was imposed and the date on which it was imposed; and
the information received in response to the notice under section 793.
1129.The section provides that the information must be entered either against the name of the present holder of the shares in question (as under the 1985 Act), or if the present holder is not known or there is no present holder, then against the name of the person holding the interest. Subsections (5) and (6) provide for criminal penalties for any default in complying with this section. Subsection (7) makes clear that information that a company receives under this Part does not mean that the company needs to be concerned with the existence of any trust over the shares.
Sections 809 and 810: Register to be kept available for inspection; and associated index
1130.These sections re-enact section 211(6) and (8) (as applied by section 213(3)) of the 1985 Act. Section 809 provides that the register of interests disclosed must be kept available for inspection at the company’s registered office or at a place specified in regulations made under section 1136. The company must advise the registrar where the register is kept (unless it has always been kept at the registered office). Subsections (4) and (5) provide for criminal penalties for any default in complying with this section.
1131.Section 810 provides that the register should have an index unless it is in a form that itself constitutes an index (for example a searchable database).
Section 811: Right to inspect and require copy of entries
1132.This section re-enacts section 219 of the 1985 Act. It provides that the register and index must be open to inspection by any person without charge. For a prescribed fee, any person is entitled to a copy of any entry on the register. A person seeking access to the register under this section must provide the information specified in subsection (4), including his name and address and the purpose for which the information is to be used.
Section 812: Court supervision of purpose for which rights may be exercised
1133.This section provides that the company must only allow the inspection of the register or provide the copy requested if satisfied that it is for a proper purpose. If it refuses, the person concerned may apply to the court for it to allow the inspection or require the copy to be provided. If an application to the court is made, the person must notify the company, and the company must use its best endeavours to notify any persons whose details might be disclosed.
Section 813 Register of interests disclosed: refusal of inspection or default in providing copy
1134.This section provides for court enforcement and criminal penalties for any default in complying with section 811.
Section 814 Register of interests disclosed: offences in connection with request for or disclosure of information
1135.This section provides for criminal penalties for misleading, false or deceptive statements given when making a request under section 811. It also makes it a criminal offence for the person who receives information under section 811 to disclose it to another person, if he knows or has reason to suspect that it may be used for an improper purpose.
Section 815: Entries not to be removed from register
1136.This section re-enacts section 218 of the 1985 Act. It provides that entries can only be removed from the register in accordance with sections 816 and 817, and if wrongly deleted must be restored as soon as reasonably practicable. Subsections (3) and (4) provide for criminal penalties for any default in complying with this section.
Section 816: Removal of entries from register: old entries
1137.Section 217(1) of the 1985 Act provides that a company may remove an entry against a person’s name from the register of interests in shares if more than six years have elapsed since the date of the entry being made, and either:
the entry recorded the fact that the person in question had ceased to have an interest notifiable under Part 6 in the company’s relevant share capital (in which case the person’s name may also be removed from the register); or
the entry has been superseded by a later entry against the same person’s name.
1138.By contrast, this section simply provides that a company is not required to keep information on the register if more than six years have elapsed since the entry was made.
Section 817: Removal of entries from register: incorrect entry relating to third party
1139.This section re-enacts in part section 217 of the 1985 Act but does not include the requirement for the company to verify information relating to third parties supplied in response to a section 793 notice. However, the third party retains the right to apply to have his name removed from the register if the information is incorrect. Subsection (4) provides for the courts to enforce removal of incorrect information.
Section 818: Adjustment of entry relating to share acquisition agreement
1140.This section re-enacts section 217(4) and (5) of the 1985 Act. It provides that a person identified in the register as being party to a section 824 share acquisition agreement (this may include a concert party agreement) may when he ceases to be party to the agreement, request that the register should be amended to record that information. Such entries may appear in several places on the register, as each member of the concert party is required in their individual notification to identify the other members of the concert party. If the company refuses an application, the court may order the company to comply if it thinks fit.
Section 819: Duty of company ceasing to be public company
1141.This section re-enacts provisions in section 211(7) and (10) (as applied by section 213(3)) of the 1985 Act. It provides that a company ceasing to be a public company must continue to keep any register it has kept under section 808 and any associated index for six years after it ceases to be a public company.
Section 820: Interest in shares: general
1142.This section re-enacts the definition of “interest in shares” in section 208 (as applied by section 212(5)) of the 1985 Act for the purposes of Part 22 of this Act. An “interest in shares” is widely defined as an interest of any kind whatsoever in the shares, and includes beneficial ownership as well as direct ownership. The courts have described this wide definition as being designed “to counter the limitless ingenuity of persons who prefer to conceal their interests behind trusts and corporate entities” (re TR Technology Investment Trust plc BCLC 256 at 261).
Section 821: Interest in shares: right to subscribe for shares
1143.This section re-enacts section 212(6) of the 1985 Act. It provides that a notice under section 793 applies in relation to rights to subscribe for shares.
Sections 822 and 823: Interest in shares: family and corporate interests
1144.These sections re-enact section 203 (as applied by section 212(5)) of the 1985 Act. They provides for certain family interests to be attributed to persons for the purpose of disclosure, as well as certain interests held indirectly through a corporate body.
Sections 824: Interest in shares: agreement to acquire interests in a particular company
1145.This section re-enacts section 204 (as applied by section 212(5)) of the 1985 Act concerning the obligation to give details of certain share acquisition arrangements in response to a notice under section 793. It covers any agreement or arrangement, whether or not legally binding, which involves undertakings, expectations or understandings that interests in shares will be acquired and that they will be subject to relevant restrictions while the agreement subsists. This may include groups of persons acting in concert to prepare the way for a takeover offer for the company or to support a pending takeover offer.
Section 825: Extent of obligation in case of share acquisition agreement
1146.This section re-enacts section 205 (as applied by section 212(5)) of the 1985 Act. It attributes interests in shares held by a party to a section 824 agreement to the other parties to the agreement.
Section 826: Information protected from wider disclosure
1147.This section re-enacts provisions in section 211(9) (as applied by section 213(3)) and 215(4) of the 1985 Act. Under section 409 the Secretary of State may make regulations exempting a company from the need to disclose information relating to related undertakings in notes to its accounts in certain circumstances. The Secretary of State must agree that the information need not be disclosed. Where advantage is taken of this exemption, the fact must be stated in the company’s annual accounts. This section provides that this same information must not be included in a section 805 report, (though its omission must be noted in the report), and must not be available for inspection under section 811.
Section 827: Reckoning of periods for fulfilling obligations
1148.This provision re-enacts the provision of section 220(2) of the 1985 Act concerning the calculation of periods in the Part expressed as a number of working days (as defined in section 1173). In contrast to section 220(2), the definition of “working days” excludes bank holidays only in the part of the UK where the company is registered.
Section 828: Power to make further provision by regulations
1149.This section re-enacts section 210A of the 1985 Act. It confers power on the Secretary of State to make regulations to amend the definition of shares to which this Part applies (subsection (1)(a) re-enacting section 210A(1)(a)). Power is also conferred to amend the provisions in section 793 as to notice by a company requiring information about interests in its shares, (subsection (1)(b) re-enacting section 210A(1)(e)), and the provisions as to what is to be taken to be an interest in shares, (subsection (1)(c) re-enacting section 210A(1)(d)).
Part 23: Distributions
1150.This Part restates the provisions on distributions in Part 8 of the 1985 Act. The only substantive change is to the rules on distributions in kind, and the new provisions are in sections 845, 846 and 851.
Sections 845 and 846: Distributions in kind
1151.In Capital Maintenance: Other Issues (paragraphs 24 to 43) the CLR explored the difficulties created by the decision in Aveling Barford Ltd v. Perion Ltd BCLC 626 and made a number of suggestions as to how these difficulties might be overcome. Section 845 is a new provision which removes doubts to which the decision in this case has given rise: in particular when a transfer of an asset to a member amounts to a distribution. The concern behind this section is that, following the decision in the Aveling Barford case, it is unclear when intra-group transfers of assets can be conducted by reference to the asset’s book value rather than its market value (which will frequently be higher than the book value).
1152.The decision in Aveling Barford concerned the sale of a property by a company (which had no distributable profits) at a considerable undervalue to another company controlled by the company’s ultimate sole beneficial shareholder. The transaction was held to be void as an unauthorised return of capital. Whilst this case decided nothing about the situation where a company that has distributable profits makes an intra-group transfer of assets at book value, there was a concern that, as such a transfer of an asset at book value may have an element of undervalue, the transaction would constitute a distribution thereby requiring the company to have distributable profits sufficient to cover the difference in value. The result has been that companies are often required either to abandon a transfer or to structure it in a more complex way, for example, having the assets revalued and then sold (or distributed under section 276 of the 1985 Act) so that the distributable reserves are increased by the “realised profit” arising on the sale/distribution followed by a capital contribution of the asset to the relevant group member.
1153.Section 845 does not disturb the position in the Aveling Barford case such that where a company which does not have distributable profits makes a distribution by way of a transfer of assets at an undervalue, this will be an unlawful distribution contrary to Part 23 of the Act.
1154.It clarifies, however, the position where a company does have distributable profits and provides that where the conditions referred to in subsection (1)(a) and (b) are met, the amount of any distribution consisting of or arising from the sale, transfer or other disposition by a company of a non-cash asset to a member of the company should be calculated by reference to the value at which that asset is included in the company’s accounts, that is, its “book value”. Thus, if an asset is transferred for a consideration not less than its book value, the amount of the distribution is zero, but if the asset is transferred for a consideration less than its book value, the amount of the distribution is equal to that shortfall (which will therefore need to be covered by distributable profits) – see subsection (2)(a) and (b). This avoids the potential need for many companies to carry out asset revaluations requiring professional advice and incurring fees to advisors prior to making a distribution of a non-cash asset.
1155.The conditions that must be satisfied for subsection (2)(a) and (b) to apply are that at the time of the disposition of the asset, the company must have profits available for distribution and that if the amount of such a distribution were to be determined in accordance with this section, it could be made without contravening any of the provisions of this Part (for example, section 830 and section 831).
1156.Under subsection (3), in determining whether it has profits available for distribution (as defined in section 830), a company may treat any profit that would arise on the proposed disposition of the non-cash asset (that is, the amount (if any) by which the consideration received exceeds the book value of the asset) as increasing its distributable profits.
1157.Section 846 replaces section 276 of the 1985 Act which applies where a company “makes a distribution of or including a non-cash asset” and allows a company which has revalued assets showing an unrealised profit in the accounts, to treat that profit as a realised profit where the distribution is one of, or including, a non-cash asset. Section 846 tracks the drafting of section 845 so that it applies not only where the company makes a distribution consisting of or including a non-cash asset, but also where a company makes a distribution arising from the sale, transfer or other disposition by it of a non-cash asset, in other words in the same circumstances that are described in section 845.
Section 851: Application of rules of law restricting distributions
1158.This section is a new provision which preserves the existing common law rules on unlawful distributions (see subsection (1)) – which continue to be an essential component in determining what amounts to an unlawful distribution.
1159.Subsection (2) makes an exception to this: in particular, it provides that the lawfulness and amount of distributions in kind are established by the statutory rules in sections 845 and 846 and not by any applicable common law rules.
Part 24: a Company’s Annual Return
1160.This Part restates Chapter 3 of Part 11 of the 1985 Act. It requires every company to deliver to Companies House an Annual Return with the specified information. The only significant difference from the 1985 provisions is that section 857 confers power on the Secretary of State not only to amend or repeal sections 855 and 856 but also, in subsection (2)(b), to make exceptions from the requirements of those sections. (The comparable power in the 1985 Act is only to amend or repeal.)
Part 25: Company Charges
Chapters 1 and 2: Companies Registered in England and Wales Or in Northern Ireland; and Companies Registered in Scotland
1161.This Part of the Act provides a scheme for the registration of charges created by a company. Part 12 of the 1985 Act was prospectively repealed and replaced by the 1989 Act, but these amendments and repeals were never brought into force, and they are now themselves being repealed (see Schedule 16).
1162.Chapters 1 and 2 of Part 25 restate Part 12 of the 1985 Act, with a few changes. The principal differences between the restated provisions and those of the 1985 Act are described below.
Changes have been made to provisions in this Part as a result of other provisions in the Act. So, for example, references to a statutory declaration in sections 403 and 419 of the 1985 Act are replaced by references to a statement in sections 872 and 887. Additionally changes have been made to sections 408 and 423 of the 1985 Act (now sections 877 and 892 respectively) to enable a company to keep its instruments creating charges and its register of charges in a place other than its registered office, bringing these provisions into line with provisions elsewhere in the Act relating to availability of documents for inspection.
The provisions relating to charges created by an overseas company in sections 409 and 424 of the 1985 Act have not been restated. Instead section 1052 provides a new regulation-making power for the Secretary of State to make provision about the registration of charges over property in the United Kingdom of an overseas company that has registered its particulars with the registrar under section 1046.
There are no longer to be daily default fines for the offences under this Part of failure to register a new charge under section 860 or 878 (compare sections 399 and 415 of the 1985 Act) and failure to register an existing charge over acquired property under section 862 or 880 (compare sections 400 and 416).
Chapter 3: Powers of the Secretary of State
Section 893: Power to make provision for effect of registration in special register
1163.This is a new provision, which provides power for the Secretary of State to make an order providing that, if a charge is registered in another register (for example, the register of floating charges to be established under the Bankruptcy and Diligence etc (Scotland) Bill), then the registrar may not register it, but it will be treated as if it had been registered in accordance with the requirements of Part 25. The power may only be exercised if appropriate information-sharing arrangements have been made between the registrar and the person responsible for the other register. This is to ensure that a person searching of the register will have access to information about charges registered in that other register.
Section 894: General power to make amendments to this Part
1164.This is a new provision providing the Secretary of State with a power to amend the provisions of Part 25.
Part 26: Arrangements and Reconstructions
1165.The provisions of this Part enable companies to apply to the court for an order sanctioning an arrangement or reconstruction agreed with a majority of members or creditors. They restate sections 425 to 427 of the 1985 Act. In addition to drafting changes resulting from the re-arrangement of the provisions, there are two changes of substance.
1166.Section 899(2) makes clear that the persons who may apply for a court order sanctioning a compromise or arrangement are the same as those who may apply to the court for an order for a meeting (under section 896(2));
1167.Section 901 requires a company to deliver to the registrar a court order that alters the company’s constitution. It also requires that every copy of the company’s articles subsequently issued must be accompanied by a copy of the order, unless the effect of the order has been incorporated into the articles by amendment. These changes are included for consistency with other provisions in the Act concerning such orders.
Part 27: Mergers and Divisions of Public Companies
1168.The provisions of this Part enable a public company – under certain conditions – to apply to the court for an order under Part 26 sanctioning an arrangement or reconstruction which concerns the merger or division of a public company. They implement the Third Council Directive 78/855/EEC concerning mergers of public limited liability companies and the Sixth Council Directive 82/891/EEC concerning the division of public limited liability companies.
1169.The provisions of this Part restate section 427A and Schedule 15B to the 1985 Act. The opportunity has been taken to put the provisions in a form more closely corresponding to that of the Directives. Chapters 2 and 3 deal separately with mergers and divisions and the provisions within those Chapters broadly follow the order of the provisions of the relevant Directive.
1170.The independence requirements for experts and valuers in sections 936 and 937 are new and correspond to the new independence requirements for a statutory auditor (see section 1214). They include a new power for the Secretary of State to define a disallowed connection for the purposes of determining whether a person is sufficiently independent to be an expert or valuer under this Part. This is consistent with the approach taken in sections 344 and 1151 of the Act.
Part 28: Takeovers Etc
Introduction
1171.This Part implements the European Directive on Takeover Bids (2004/25/EC, the “Takeovers Directive”) which was adopted on 21 April 2004 and had to be implemented by 20 May 2006. It also contains a few minor amendments to the existing law not required by the Directive.
Summary and background
1172.With the exception of Chapter 3 which restates, with amendments, Part 13A of the 1985 Act, the provisions in Part 28 are new. The principal body of provisions emerged from the consultation document, “Company Law – Implementation of the European Directive on Takeover Bids” published by the DTI in January 2005. Additionally, the CLR considered issues related to “squeeze-out” and “sell-out” (concerning the problems of, and for, residual minority shareholders following a successful takeover bid) in Chapter 11 and Annex B of “Completing the Structure” and presented their conclusions in Chapter 13 of the Final Report. Certain provisions in this Part have been developed in the light of these conclusions.
Overview of the Part
1173.The Part is divided into 4 Chapters:
Chapter 1 deals with matters related to the Takeover Panel and its takeover regulatory functions;
Chapter 2 concerns matters related to barriers to takeovers;
Chapter 3 contains provisions relating to “squeeze-out” and “sell-out” (concepts designed to address the problems of, and for, residual minority shareholders following a successful takeover bid); and
Chapter 4 amends the provisions in Part 7 of the 1985 Act about the content of annual reports of companies traded on a regulated market.
Note: It is intended that certain provisions of Part 28 will be extended to unregistered companies with shares traded on a regulated market. This will be achieved by the regulation-making power at section 1043 of the Act. It is necessary to ensure compliance with the Takeovers Directive.
Chapter 1: the Takeover Panel.Summary and Background
1174.Since 1968, takeover regulation in the UK has been overseen by the Panel on Takeovers and Mergers (“the Panel”) which administered rules and principles contained in the non-statutory City Code on Takeovers and Mergers. In order to bring UK takeover regulation within the requirements laid down in the Takeovers Directive, Chapter 1 places it within a statutory framework.
1175.The Panel will supervise takeover activity and similar types of transactions. The Panel will retain considerable autonomy to provide for its own constitution and appointment procedures. However, a minimum constitutional structure is laid down, providing for the Panel to make arrangements for carrying out its functions and, in particular, to function through committees, sub-committees, officers and members of staff. It is envisaged that the Panel will continue to carry out its day-to-day activities through its Executive. Provisions underpinning the funding of the Panel’s regulatory activities are also included.
1176.Principally, the Panel is placed under an obligation to make statutory rules giving effect to certain Articles of the Directive. It is also given a statutory rule-making power to make rules in relation to takeover activity and similar types of transactions, reflecting the current field of activity over which the existing Code lays down rules.
1177.Sections 945, 951, 955, 956 and 961 of the Act are intended to limit litigation by: (a) channelling parties to seek decisions of the Panel (including the Panel’s Hearings Committee and the independent Takeover Appeal Board) before having recourse to the courts; (b) excluding new rights of action for breach of statutory duty; (c) protecting concluded transactions from challenge for breach of the Panel’s rules; and (d) exempting the Panel and its individual members, officers and staff from liability in damages for things done in, or in connection with, the discharge of the regulatory functions of the Panel.
1178.The Bill does not affect the availability of judicial review by the courts. In the takeovers field, in the Datafin case (R v Panel on Takeovers, ex parte Datafin plc QB 815) the Court of Appeal concluded that generally the courts should limit themselves only to reviewing the Panel’s decision-making processes after the bid has been concluded.
1179.The Bill confers on the Panel powers to make rulings and directions and to enforce these through the courts, to obtain information and documents from those involved in regulated activities and to impose sanctions on those who transgress its rules.
The Panel and its rules
Section 942: The Panel
1180.This section confers on the Panel the takeover regulatory functions set out in Chapter 1. The Panel is empowered to do anything that it considers necessary or expedient in relation to its prescribed functions and it may also make arrangements for such functions to be carried out on its behalf by a committee or sub-committee of the Panel or an officer or member of staff of the Panel or a person acting as such.
1181.This Chapter does not confer on the Panel the status of a statutory body. The Panel will remain an unincorporated body, as constituted from time to time, and, as such, having rights and obligations under the common law. Those rights and obligations will be supplemented by the specific provisions set out in the Bill.
Sections 943 and 944: Rules; Further provisions about rules
1182.The Panel is given the power to make rules in relation to takeover regulation. The rule-making power is broadly drawn to ensure that the Panel can continue to make rules on the range of matters presently regulated by the City Code on Takeovers and Mergers. The following provisions are included:
a)
The Panel is placed under an obligation to make rules as required by specified Articles of the Takeovers Directive. These are the general principles (Article 3.1 of the Directive), jurisdictional rules (Article 4.2), matters related to the protection of minority shareholders, mandatory bid and equitable price (Article 5), contents of the bid documentation (Article 6.1 to 6.3), time allowed for acceptance of a bid and publication of a bid (Articles 7 and 8), obligations of the management of the target company (Article 9) and other rules applicable to the conduct of bids (Article 13). In making rules in relation to these Articles, the Panel will be entitled to exercise Member State options where these are provided for in the Directive. The Panel’s rules will not, however, deal with certain matters contained in the Directive such as barriers to takeovers (Article 11), squeeze-out and sell-out (Articles 15 and 16), and information to be published by companies in their annual reports (Article 10) which are more appropriately dealt with in company legislation (and are the subject of further provision at Chapters 2, 3 and 4 of this Part).
b)
The Panel is permitted to make rules on takeover bids (including, but not limited to, those which are the subject of the Directive), mergers and other transactions affecting the ownership or control of companies. The power is designed to be broad enough to cover the existing scope of the Code and sufficiently flexible to take account of future market developments. Types of matters currently covered by the Code but not covered by the Directive include the takeovers of companies not traded on a regulated market and transactions involving a change of control of a like nature to takeovers.
1183.When making rules under this section, the Panel must do so by a committee of the Panel, except in the case of rules for fees and charges under section 957 which must be made either by a committee of the Panel or by the Panel itself.
1184.The further provisions about rules that may be made by the Panel under section 944 include the power to grant derogations and waivers, which by virtue of section 943(1) must respect the general principles laid down in Article 3.1 of the Directive.
1185.Section 944(2) to (7) makes provision as to the form, public availability and verification of rules made by the Panel.
Section 945: Rulings
1186.This section enables the Panel, including (by virtue of section 942(3)) its Executive, to make rulings on the interpretation, application or effect of the rules made by the Panel.
1187.To the extent and in the circumstances specified in the rules, a ruling of the Panel has binding effect unless reviewed by the Hearings Committee or successfully appealed to the Takeover Appeal Board in accordance with rules made under section 951. It is envisaged that rules made under sections 943 and 944 will address matters such as notice to parties and right of representation of persons who might be bound by a Panel ruling.
Section 946: Directions
1188.This section allows the Panel to make provision in its rules for it to give a direction preventing a person from breaching the rules (including a direction operating on an interim basis whilst a matter is awaiting determination by the Panel) or otherwise to ensure compliance with the rules.
Information
Section 947: Power to require documents and information
1189.The Panel has historically had no formal power to require those involved in takeover activity to provide it with the information the Panel requires to carry out its functions. Persons authorised under the Financial Services and Markets Act 2000 are required by the rules of the Financial Services Authority to provide information and assistance to the Panel. In relation to others, the Panel has relied on the voluntary co-operation of market participants to provide explanations and documents which are not publicly available.
1190.This section enables the Panel to require the production of such documents and information as it may reasonably require in the exercise of its functions. The Panel may also authorise a person to exercise the powers under the section on its behalf, for example, if the Panel were to appoint a law or accountancy firm to help it collect and analyse documents.
1191.Subsection (7) provides that a lien on a document is not affected by the production of that document in compliance with a requirement imposed by the Panel or someone authorised on its behalf. A lien is a legal right to keep possession of a document belonging to someone else until a claim is satisfied – for example, a claim for payment of professional fees. This subsection does not entitle a professional to refuse to hand over a document to the Panel but preserves his rights over those documents.
1192.The section provides that the Panel may require the production of information in hard copy where it is held in some other form (for instance, electronically on a floppy disk).
1193.The Panel may not compel the production of documents which would be protected from disclosure in legal proceedings on the grounds of legal professional privilege or confidentiality of communications.
Sections 948 and 949: Restrictions on disclosure; Offence of disclosure in contravention of section 948
1194.These sections provide that information obtained by the Panel in the course of exercising its functions will be subject to restrictions on onward disclosure. Aside from the desirability of such provisions, so that those providing information to the Panel can do so knowing that it will not be subject to improper further disclosure, these provisions also meet a requirement under Article 4.3 of the Directive that Member States shall ensure that information provided to those employed, or formerly employed, by takeover supervisory authorities shall not be further divulged: “to any person or authority except under provisions laid down by law.” Section 948, accordingly, prescribes the conditions under which such information can be released.
1195.Information concerning the private affairs of an individual or a business provided to the Panel in connection with its functions may not be disclosed during the individual’s lifetime or while the business is carried on without the consent of the individual or business in question except for the purposes of carrying out the Panel’s functions or unless it is disclosed to a person or for a purpose set out in Schedule 2.
1196.Schedule 2 sets out the “gateways” for disclosure of information obtained by the Panel in the exercise of its functions which are permitted under section 948, including the circumstances in which a disclosure to an overseas regulatory authority is permitted. Under section 948(4) and (5), the Secretary of State has the power to amend the Schedule, but only to specify persons exercising functions of a public nature or descriptions of disclosure where the purpose for which the disclosure is permitted is likely to assist in the exercise of a function of a public nature.
1197.Section 948(6)(a) provides that certain authorities mentioned in subsection (7) are not bound by the restrictions on disclosure imposed by subsection (2). These bodies are those other takeover supervisory authorities and financial services regulators with which the Panel has a duty to co-operate. Subsection (6)(b) provides that persons or bodies obtaining information from those authorities (whether directly or indirectly) are also not bound by the restrictions on disclosure imposed by subsection (2). These provisions are necessary to implement fully Article 4.4 of the Directive. Those bodies mentioned in subsection (7), and persons and bodies receiving information from them, will themselves be subject to restrictions on disclosure that will mirror those imposed by section 948, and so information originating from the Panel will still be protected from improper further disclosure.
1198.Section 949 makes it an offence to disclose information in contravention of section 948. A person guilty of such an offence is liable on conviction on indictment to two years’ imprisonment or a fine or both; and on summary conviction to twelve months’ imprisonment (six months in Scotland and Northern Ireland) or a fine or both. Section 949(1) provides a person with a defence if he can prove that he did not know, and had no reason to suspect, that the information in question had been provided to the Panel in the exercise of its functions; or that he took reasonable steps to prevent wrongful disclosure.
Co-operation
Section 950: Panel’s duty of co-operation
1199.Article 4.4 of the Directive requires that takeover supervisory authorities and financial services regulators provide reasonable assistance to other such authorities within the EEA for the purposes of the Directive. This section is designed to give effect to this requirement by obliging the Panel to co-operate with overseas takeover and financial services regulatory authorities.
1200.The form and manner of co-operation will be as the Panel considers appropriate in the light of the circumstances (in particular, its power to require documents and information may be exercised to support such an authority) and may include sharing information which the Panel is not prevented from disclosing. The section mirrors similar co-operation obligations imposed on the Financial Services Authority by section 354 of the Financial Services and Markets Act 2000.
Hearings and Appeals
Section 951: Hearings and Appeals
1201.This section ensures that proper procedures for review of and appeal against decisions taken by the Panel in connection with its regulatory functions are provided. Section 951(1) requires that the rules made by the Panel provide for a decision of the Panel to be subject to review by a “Hearings Committee” when requested by affected persons specified in the rules. Section 951(3) provides that the rules must give a right of appeal to an independent tribunal (the “Takeover Appeal Board”) against a decision of the Hearings Committee. The rules may make provision in relation to the Hearings Committee as to procedural matters, evidence and the powers of the Committee. Further, rules may contain provisions related to enforcement of decisions of the Hearings Committee and the Takeover Appeal Board.
1202.The section also requires the rules to provide that:
a)
when appearing before the Hearings Committee or the Takeover Appeal Board, the Panel must act through an officer or member of staff of the Panel (who must not be a member of the rule-making committee referred to in section 943(5), the Hearings Committee or the Appeal Board); and
b)
no person who is, or has been, a member of the rule-making committee can be a member of the Hearings Committee or the Takeover Appeal Board.
1203.The general rules of natural justice will preclude a person who had taken part in a decision from later considering a review or appeal in relation to that decision.
1204.This approach is designed to ensure a clear and transparent division of responsibilities between the various organs of the Panel in its executive, judicial and rule-making roles.
Contravention of rules, etc
Section 952: Sanctions
1205.This section confers on the Panel the power to make rules for imposing sanctions for breach of its rules or directions given under section 946. The Panel’s current sanctions regime, which is set out in the Introduction to the City Code on Takeovers and Mergers and which it is envisaged will remain in place under the Act, provides for private and public statements of censure of persons in breach of the Code.
1206.Particularly flagrant breaches may lead to the Panel publishing a statement indicating that the offender is someone who is not likely to comply. The rules of the Financial Services Authority and certain professional organisations contain provisions obliging their members, in certain circumstances, not to act for a person named in such a statement. This is referred to as “cold-shouldering”. The provisions in question cover transactions that are subject to the City Code on Takeovers and Mergers, such as transactions in relevant securities requiring disclosure under rule 8 of the Code. Under section 952, it will continue to be possible, in the case of transactions that are subject to the Panel rules, for the Panel to issue “cold-shouldering” statements in appropriate cases. (The Panel will also be able to pass information concerning breaches of rules to other regulatory authorities and professional bodies by virtue of the statutory “gateways” set out at section 948 and Schedule 2.)
1207.Should future rules made by the Panel confer a power on the Panel to impose a sanction of a kind not contained in the City Code on Takeovers and Mergers as it has effect immediately before the passing of the Act, the Panel must prepare a policy statement in respect of the sanction. The policy statement must set out the policy of the Panel with regard to imposition of the sanction and, for financial penalties, the penalty that may be imposed. An element of the policy must be that the Panel, in making a decision about any such matter, have regard to the seriousness of the breach or failure, the extent to which the breach or failure was deliberate or reckless and whether the person on whom the sanction is to be imposed is an individual.
Section 953: Failure to comply with rules about bid documentation
1208.This section creates new offences in relation to takeover bid documentation (i.e. offer documents prepared by the bidder and documents in response to the bid prepared by the board of the target company). Provisions related to bid documentation are laid down in particular by Articles 6.3 and 9.5 of the Directive which are to be implemented by rules which the Panel are obliged to make under section 943. Consequently, in each case an offence will be committed where the document in question does not comply with rules designated by the Panel as giving effect to those provisions. The offence relating to offer documents may be committed by the bidder and any of its directors, officers or members who caused the offer document to be published. The offence relating to response documents may be committed by directors or other officers of the target company. Where either offence is committed by a corporate body (for instance, a corporate director), provisions are also included dealing with liability of directors, officers or members of that body. In each case, an offence will be committed only where the relevant person knew that the document did not comply (or was reckless as to whether it did so) and failed to take all reasonable steps to ensure that it did comply.
1209.A person guilty of an offence under this provision is liable on conviction to a fine (on summary conviction limited to the statutory maximum).
Section 954: Compensation
1210.This section confers on the Panel the power to make rules providing for financial redress (together with interest (including compound interest)) in consequence of a breach of rules which require monetary payments to be made (for instance, a payment by the bidder to shareholders of any difference between the price actually paid and any higher price for shares that the bidder should have paid under the rules).
Section 955: Enforcement by the court
1211.This section provides a mechanism by which the Panel may, if necessary, apply to the court in order to enforce Panel rule-based requirements as well as requests for documents and information under section 947. The Panel may apply to the court either where there is reasonable likelihood that a person will contravene a requirement imposed by or under the rules or where a person has failed to comply with such a requirement or with a requirement imposed under section 947.
1212.It is expected that in accordance with usual practice, the court will not, in exercising its jurisdiction under this section, rehear substantively the matter or examine the issues giving rise to the ruling or, as the case may be, the request for documents or information except on “judicial review principles”, where there has been an error of law or procedure.
1213.The court is given a broad discretion as to the order it may make to secure compliance with the requirement; but aside from the power granted to the Panel by this section, there is no right to seek an injunction (or interdict) to prevent a person contravening, or continuing to contravene, a rule-based requirement or disclosure requirement.
Section 956: No action for breach of statutory duty etc
1214.Compliance with the rules made by the Panel is a matter solely for the Panel. This section does two things:
a)
it excludes new rights of action for breach of statutory duty for contraventions of requirements imposed by or under rules or a requirement imposed under section 947; and
b)
in order to ensure certainty, it provides that a contravention of a rule based requirement does not make a transaction void or unenforceable and (subject to any provision of the rules) does not affect the validity of any other thing. As currently, transactions will be capable of being set aside or unravelled in cases of, for example, misrepresentation or fraud.
Funding
Section 957: Fees and charges
1215.This section enables the Panel to make rules for the payment of fees or charges to the Panel for the purposes of meeting the Panel’s expenses incurred in exercising its functions. Such fees and charges may be imposed to meet expenses of the Takeover Appeal Board; the cost of repaying capital and paying interest on loans; and the cost of maintaining adequate reserves. The rules under this section must be made by the Panel itself or by a committee of the Panel (section 943(4)).
Section 958: Levy
1216.This section gives the Secretary of State the power to make regulations imposing a levy for meeting the costs of the Panel. In determining the appropriate rate of the levy, the Secretary of State must take account of other income received, or expected to be received, by the Panel (which would include fees and charges under section 957) and may take account of estimated as well as actual costs of the Panel.
1217.It is anticipated that a levy would only be imposed if the existing voluntary levy funding arrangements (contributions collected by member firms of the London Stock Exchange and Ofex currently set at a flat rate charge of £1 on contract notes on all chargeable transactions with a consideration in excess of £10,000) were no longer viable. The categories of persons or bodies to which the levy would apply may include only those capable of being directly affected by the exercise of the Panel’s functions or otherwise having a substantial interest in the exercise of those functions.
1218.The first regulations made in respect of the levy power – and any further regulations which change the persons or bodies by whom, or the transactions on which, the levy is payable – will be subject to the affirmative resolution procedure in both Houses of Parliament. A draft of an instrument containing such regulations will not be treated as being hybrid even if otherwise it would be. Any other subsequent regulations will be subject to the negative resolution procedure.
Section 959: Recovery of fees, charges or levy
1219.This section provides that an amount payable by a person as a consequence of fees and charges imposed by the Panel under section 957 or as a result of any levy fixed by the Secretary of State under section 958 will constitute a debt owed by that person to the Panel and be recoverable by the Panel as a debt.
Miscellaneous and supplementary
Section 960: Panel as party to proceedings
1220.This section provides that, notwithstanding its unincorporated status, the Panel may in its own name bring proceedings under this Chapter and bring or defend other proceedings.
Section 961: Exemption from liability in damages
1221.This section confers limited immunity on the Panel and those involved in carrying out its regulatory activities. The immunity provisions are consistent with those recently extended to the Financial Services Authority and the Financial Reporting Council in the exercise of their duties under financial services and companies legislation.
1222.The section exempts the Panel, its members, officers and staff (which would include secondees), and persons authorised under section 947(5) by the Panel to exercise its powers in relation to requiring documents and information, from liability in damages for things done or omitted in relation to the Panel’s regulatory activities. (The Takeover Appeal Board benefits from a common law immunity on account of its exercise of judicial functions.)
1223.Section 961(3) sets out the circumstances where the exemption will not apply – that is to say, where the act or omission was in bad faith or where it was unlawful under section 6(1) of the Human Rights Act 1998.
Section 962: Privilege against self-incrimination
1224.This section provides that a statement made by a person to the Panel, or a person authorised on its behalf, in compliance with a requirement to provide information under section 947 (or a court order made to secure compliance with such a requirement under section 955) cannot be used against that person in most types of criminal proceedings. Such statements can, however, be used in proceedings for offences of making false statements otherwise than on oath under section 5 of the Perjury Act 1911 and its Scottish and Northern Irish equivalents. These offences exist to deter and punish the making of false statements and it would not be possible to prosecute such offences if the false statements themselves could not be used in evidence against those by whom they were made.
Section 963: Annual reports
1225.As is the Panel’s existing practice, the Panel will be required to publish an annual report containing annual accounts, setting out how the Panel’s functions were discharged and including other matters considered by the Panel to be of relevance. Annual reports published by the Panel are available on the Panel’s website.
Section 964: Amendments to Financial Services and Markets Act 2000
1226.This section repeals section 143 of the Financial Services and Markets Act 2000 (“FSMA”) which, by endorsing the City Code on Takeovers and Mergers, presently provides a mechanism for the Financial Services Authority to bring disciplinary and enforcement action against authorised persons for misconduct in relation to the Code. Given that the Code will be replaced by rules which have legal force as a consequence of the Act, it is considered that there is no longer a need to maintain section 143.
1227.This will not, however, preclude the Panel from reporting breaches of the Code by authorised persons in relation to takeover bids to the Financial Services Authority, as at present, and any such breaches will still be taken into account by the FSA, for example, in assessing whether such persons are fit and proper to be authorised for business of that kind or have otherwise complied with their regulatory obligations (for example, whether they are meeting proper standards of market conduct).
1228.A consequential amendment is made by section 964(3) to preserve the definition of “consultation procedures” currently contained at section 143(7) for the purposes of the provisions in section 144 of FSMA relating to price stabilising rules. Additionally, to ensure consistency with the requirements of Article 4.4 of the Directive as regards the duties of takeover regulatory and financial services authorities within the EU to co-operate with each other, the existing disclosure and regulatory co-operation obligations of the Financial Services Authority under sections 349 and 354 of FSMA are amended to include cooperation with relevant authorities referred to by the Directive and to remove restrictions on disclosure to such authorities. These duties reflect the disclosure and co-operation provisions in sections 948 and 950 (including provisions related to the rules on disclosure that apply where information is passed to other takeover supervisory authorities and financial services regulators described in relation to section 948(6)).
Sections 965 and 973: Power to extend to Isle of Man and Channel Islands
1229.These sections allow any provisions of Chapters 1 and 2 to be extended to the Isle of Man or any of the Channel Islands by Order in Council, with any specified modifications.
Chapter 2: Impediments to Takeovers
Summary and background
1230.Article 11 of the Takeovers Directive seeks to override, in certain circumstances relating to a takeover, a number of defensive devices that may be adopted by companies prior to the bid, such as: differential share structures under which minority shareholders may exercise disproportionate voting rights; restrictions on transfer of shares in the company articles or in contractual agreements; and limitations on share ownership.
1231.There are currently no restrictions on the way that UK companies which are admitted to trading on a regulated market can structure their share capital and control. However, market pressure brought to bear, in particular, by institutional investors has ensured that there are now few UK listed companies with differential voting structures.
1232.As permitted by Article 12 of the Directive, it has been decided not to apply the provisions of Article 11 in all cases but instead to include in the Act (sections 966 to 972) provision for companies with voting shares traded on a regulated market to opt in to its provisions should they choose to do so.
Sections 966, 967 and 970: Opting in and opting out; Further provision about opting-in and opting-out resolutions; Communication of decisions
1233.A company may pass a special resolution opting in to Article 11 (an “opting-in resolution”) provided that three conditions are met:
a)
it has voting shares admitted to trading on a regulated market (it is not considered necessary to extend this provision to other types of companies which are not covered by the Directive);
b)
the company’s articles of association do not contain restrictions of the kind mentioned in Article 11 (or other provisions which would be incompatible with Article 11) or, if they do contain such restrictions, the restrictions will not apply in circumstances related to a takeover bid as described by Article 11. Article 11 relates to both the takeover bid period and the time following the bid when the bidder has acquired 75% or more of the company’s capital carrying voting rights. It provides that restrictions both on the rights to transfer shares and on voting rights that are contained in the articles of the company should not apply. It also provides that, in certain circumstances, shares carrying multiple voting rights shall only have one vote and extraordinary rights of shareholders concerning the appointment or removal of board members should be disapplied; and
c)
no shares are held by a Minister conferring special rights in the company and no such special rights are provided for in law. The Directive expressly provides that Article 11 does not apply to shares held by Member States conferring special rights on the Member State which are compatible with the Treaty, or to special rights provided for in national law which are compatible with the Treaty. The UK Government holds a number of so-called “golden shares” in formerly publicly-owned businesses which have been privatised to ensure that essential public interest considerations are protected. This provision will exclude all such companies where the Government holds the beneficial ownership of a golden share (since holdings by nominees and subsidiaries are also covered). The concept of Minister is broadly defined in section 966(7) of the Act to include Scottish Ministers and Northern Ireland Ministers under section 7(3) of the Northern Ireland Act 1998. (As a result of the Government of Wales Act 2006, the definition of Minister will be changed to include Welsh Ministers). Under section 966(8), a power is provided to the Secretary of State by the negative resolution procedure to apply the provision in section 966(4) (Minister holding golden shares) to persons or bodies exercising functions of a public nature as it applies in relation to a Minister.
1234.Section 966(5) enables a company to revoke an opting-in resolution by means of a further special resolution (an “opting-out resolution”).
1235.Section 967 sets down provisions relating to the date on which the opting-in and opting-out resolutions will take effect. Generally, this will be the date stated in the resolution.
1236.Section 970 requires companies, within 15 days of an opting-in or opting-out resolution being passed, to notify the Panel and any other takeover supervisory authority in a Member State in which the company has shares admitted to trading on a regulated market or has requested such admission. Where a company fails to comply with this requirement, the company and every officer in default will be guilty of an offence and be liable on summary conviction to a fine not exceeding level 3 on the standard scale (and to a daily default fine for continued contravention).
Section 968: Effect on contractual restrictions
1237.This section provides that agreements entered into between shareholders in the company on or after 21 April 2004 (the date on which the Takeovers Directive was adopted), and agreements entered into between a shareholder and the company before as well as on or after that date, are invalid in so far as they impose any of the restrictions set out in subsection (2).
1238.Those restrictions relate both to the bid period and to the time following a takeover bid when the bidder holds 75% or more in value of all the voting shares in the company. Types of restrictions overridden are those imposing restrictions on the transfer of shares and on rights to vote at general meetings of the company to decide on action to frustrate the bid and at the first meeting to be held after the end of the offer period. For the purposes of determining when the bidder holds 75% or more in value of all the voting shares in the company, both debentures and shares which do not normally carry rights to vote at a general meeting (such as preference shares) held by the bidder are to be disregarded (see subsection (8)).
1239.The provisions related to the types of contractual agreements to which the override will apply (including the date at which such contracts were entered into) and the restrictions which are made invalid are designed to replicate the provisions of Article 11 of the Directive.
1240.Section 968(6) provides that a person who suffers loss as a result of a contractual agreement being overridden can apply to the court for compensation. It is expected that, in the first instance, such compensation will be offered by the bidder in making the takeover offer. Where, however, the compensation offered by the bidder is not acceptable to the person whose rights are being overridden, there is a right to apply to the court. The court will award compensation to the person who suffers loss on a just and equitable basis to be paid by any person (which could include the bidder or the other party to the contract which has been overridden) who would have been liable to him for committing or inducing the breach of contract which would have been committed had the restriction in question not been made invalid by this section.
Sections 969 and 972: Power of offeror to require general meeting to be called; Transitory provision
1241.Section 969 provides the bidder with the special right to require the directors of an opted-in company to call a general meeting of the company when he holds 75% in value of all the voting shares in the company (excluding debentures and shares that do not normally carry rights to vote at a general meeting (such as preference shares)). Section 969(3) applies sections 303 to 305 of the Act, which deal with the calling of meetings, to such a request (with the necessary modifications). But as those sections may not be in force at the time when section 969 comes into force, section 972 makes the same sort of adaptations in relation to the equivalent provisions of the 1985 Act. In particular, section 972(3) alters the application of section 378(2) so that a special resolution may still be passed at a general meeting called at only 14 days’ notice (normally at least 21 days’ notice would have to be given of the meeting for it to be able to pass a special resolution).
Chapter 3: “Squeeze-Out” and “Sell-Out”
Summary and background
1242.The concepts of “squeeze-out” and “sell-out” are designed to address the problems of, and for, residual minority shareholders following a successful takeover bid. Squeeze-out rights enable a successful bidder to compulsorily purchase the shares of remaining minority shareholders who have not accepted the bid. Sell-out rights enable minority shareholders, in the wake of such a bid, to require the majority shareholder to purchase their shares. Because they involve the compulsory purchase or acquisition of shares against the will of the holder of the shares or the acquirer, high thresholds apply to the exercising of such rights and there are protective rules on the price that must be paid for the shares concerned.
1243.Squeeze-out and sell-out provisions have been a feature of national company law for many years (and were previously contained in Part 13A (Takeover Offers) of the 1985 Act). Articles 15 and 16 of the Takeovers Directive, however, introduce EU-wide rules requiring all Member States to put appropriate provisions in place for the first time. The provisions at sections 974 to 991 of the Act restate Part 13A of the 1985 Act in a clearer form. However, in doing so they also make important changes to reflect the need to ensure compliance with the Directive and the decision to accept some recommendations of the CLR. These are described below.
Detail of changes made to the operation of provisions previously contained in Part 13A of the 1985 Act
1244.The rules laid down in the Directive in relation to squeeze-out and sell-out are broadly consistent with provisions of Part 13A (sections 428 to 430F) of the 1985 Act. The restated and amended provisions will apply equally to all companies and all bids within the ambit of Part 13A of the 1985 Act, regardless of whether or not the Directive is required to be applied to such companies and bids.
1245.The following changes are made in implementation of the Directive:
Calculation of Squeeze-out Threshold (section 979) – there is a dual test imposed: in order to acquire the minority shareholder’s shares, the bidder must have acquired both 90% of the shares to which the offer relates, and 90% of the voting rights carried by those shares. Where the offer relates to shares of different classes, then, in order to acquire the remaining shares in a class, the bidder must have acquired 90% of the shares of that class to which the offer relates, and 90% of the voting rights carried by those shares. Under section 429 of the 1985 Act, in each case only the first limb of that test applied.
Calculation of Sell-out Threshold (section 983) – mirroring the change to be made in relation to the squeeze-out threshold, a dual test is similarly imposed in relation to the sell-out threshold, so that a minority shareholder may force a bidder to acquire his shares (i) when the bidder holds 90% of the shares in the company, and 90% of the voting rights attached to those shares, or (ii) when the bidder holds 90% of the shares in the class to which the minority shareholder’s shares belong, and 90% of the voting rights attached to those shares. Under section 430A of the 1985 Act, the test was that the bidder should have acquired 90% of all shares in the company (or in the class concerned).
Revised Period during which Squeeze-Out and Sell-Out Rights may be Exercised (section 980(2)) – the Directive provides (Articles 15.4 and 16.3) that squeeze-out and sell-out rights must be exercisable within a three month period following the time allowed for acceptance of the bid. Section 429(3) of the 1985 Act provided that squeeze-out could be exercised within a period of four months beginning with the date of the offer and had to be exercised within two months of reaching the 90% threshold. Accordingly, the rule provided by the Directive is substituted for the rule in the 1985 Act. An exception to this rule is provided where takeover bids are not subject to the Directive, for instance takeovers of most private companies. In these cases, the squeeze-out notices must be given within six months of the date of the offer if this is earlier than the period ending three months after the end of the offer. This is intended to prevent offerors in such circumstances continually extending the offer period. A change is also made as regards the period during which sell-out may be exercisable so that this period is to be either three months from the end of the offer or, if later, three months from the notice given to the shareholder of his right to exercise sell-out rights (section 984(2)). An extended period during which the sell-out right can be exercised where notice of such a right is only given after the end of the offer period is consistent with provisions of the Directive allowing more stringent provisions to be put in place (in this case to ensure the proper protection of minority shareholders).
The court will no longer be able to reduce the consideration in relation to squeeze-out or sell-out following a takeover bid to below the consideration offered in the bid (which the Takeovers Directive presumes to be fair in all cases). Again utilising provisions of the Directive which allow more stringent provisions to be included to protect minority shareholders, minority shareholders will continue to be able to apply to the court to request that consideration higher than that offered in the bid be paid in exceptional circumstances (section 986(4)).
1246.In most instances, it is considered that the first and second changes above will make no practical difference as the percentage of total capital carrying voting rights in a company (or class of shares) and the percentage of voting rights will normally be the same. The provisions about voting rights will not apply where the shares being squeezed out or sold out are non-voting shares.
1247.The CLR also considered the issue of squeeze-out and sell-out and the scope for improving the provisions in the 1985 Act. Its Final Report (chapter 13, pages 282 – 300), made a number of recommendations in relation to the reform of the squeeze-out and sell-out regime. Some of these recommendations are closely related to implementation of the Takeovers Directive. For instance, the CLR questioned whether, in calculating the relevant squeeze-out and sell-out thresholds, only shares that had been unconditionally acquired should be taken into account or whether shares acquired subject to contract should also be included.
1248.In implementing the Takeovers Directive, the opportunity is being taken to adopt recommendations of the CLR, whether or not related to implementation of the Directive, except to the extent that they are not consistent with Articles 15 and 16 of the Directive or are no longer appropriate as a consequence of the Directive. The recommendations made by the CLR implemented by Chapter 3 of Part 28 are set out below:
Meaning of takeover offer and entitlement to dividends (sections 974 and 976)
1249.In order to be a takeover offer for the purposes of Part 13A of the 1985 Act, an offer to acquire shares had to be on terms which were the same in relation to all the shares to which the offer related. One problem with the 1985 legislation was how to treat any variations in value between shares of the same class that were attributable to the fact that some of the shares, because they were allotted later, do not yet carry a dividend. Section 976(2) rectifies this problem by providing that, even if the offeror offers to pay more for shares that carry a dividend than for those in the same class which do not, the offer will be treated as being made on the same terms in relation to those shares.
Meaning of a takeover offer and communication of that offer (sections 974 and 978)
1250.To deal with issues arising from an increasingly globalised market in shares and different legislative regimes outside the EEA, it is made clear that an offer is not prevented from being a takeover offer for the purposes of Chapter 3 of Part 28 merely because there are some offerees who will be unable to accept it (for instance, where the offeree cannot accept the offer because of restrictions on the cross-border transfer of cash or securities in the country in which the offeree resides). It is also provided that an offer can still be a takeover offer for the purposes of the squeeze-out and sell-out provisions if a shareholder has no registered address in the UK and the offer is not communicated to him to avoid contravening the law of another country as long as either the offer itself is published in the Gazette or a notice is published in the Gazette stating that a copy of the offer document can be obtained from a place in the EEA or on a website.
Shares that the offeror has “contracted to acquire” (section 975)
1251.Clarificatory amendments are made on this issue. Section 428(5) of the 1985 Act dealt with the offeror’s position at the start of the bid, for the purpose of determining which shares could not be counted towards the achievement of the 90% threshold (at which point shares may be compulsorily purchased). It was unclear as to whether the phrase “contracted to acquire” in section 428(5) covered conditional as well as unconditional contracts. It is, therefore, clarified that, in ascertaining the offeror’s position at the start of the bid, the shares he has conditionally contracted to acquire (other than those subject to irrevocable undertakings (see below), as under the 1985 Act) should be treated as being shares already held by the offeror. This means that only shares that the offeror has either acquired or unconditionally contracted to acquire will count towards the 90% total needed to exercise squeeze-out. Consequential changes are also made to the provisions on joint offers and associates of the offeror to bring these into line with the above.
1252.Under the 1985 Act, the registered holder of shares could give an irrevocable undertaking to accept a takeover bid, and if he did this for no consideration or only in exchange for a promise to make the bid, his shares were still treated for the purposes of squeeze-out as included within the offer. This is extended to include undertakings given for only negligible consideration and undertakings the effect of which is to require the registered holder to accept the offer (where the undertaking is given by a person who is not the registered holder of the shares but can contract to bind the registered holder, such as the manager of shares held by a bare nominee). (“Irrevocable undertakings” are contractual agreements entered into by a bidder usually with major shareholder(s) of a proposed target company. Such agreements aim to give the bidder certainty – he will know that support for the offer can be guaranteed from shareholders party to the contract – so that his bid has a greater prospect of success. Such undertakings would normally prevent the giver of the undertaking from selling their shares or exercising voting rights to prevent the takeover from becoming successful.)
Date of the offer (section 991(1))
1253.The “date of the offer” is defined to mean either the date of publication, or if the offer is not published or notice of the offer is sent out earlier, the date on which the offeror first sends notice of the offer to the offerees.
Right of offeror to buy out minority shareholders: treatment of options etc (section 979(5))
1254.Where an offeror makes an offer for all the target company’s allotted shares and all or any shares subsequently allotted, it is provided that (a) in deciding whether the offeror has reached the 90% threshold for the purposes of section 979, the offeror need only bring into the calculation shares which are actually in issue (i.e. allotted) at the relevant time; (b) if the offeror serves squeeze-out notices and more shares are subsequently allotted which take the percentage of acceptances then received below 90%, that will not invalidate squeeze-out notices already served; and (c) if the offeror wishes to serve further squeeze-out notices, he must have at least 90% acceptances of shares (or shares in a class) then in issue and subject to the offer at the time he sends the notices out.
Consideration not exclusively in cash (section 981(5))
1255.It is clarified that where an offer of shares, or a mixture of shares and cash, is made, and it is no longer possible when the offeror exercises his right of squeeze-out to give the consideration in shares, the offeror should pay the cash equivalent irrespective of whether the shareholders had previously been offered a choice (i.e. whether the offer was “mix and match” or not). Parallel changes are made as regards sell-out (section 985(5)).
Shares that the offeror has “contracted to acquire” (section 983)
1256.It is clarified that, in addition to shares acquired by the offeror, shares subject to both conditional and unconditional contracts of acquisition are included in calculating whether the sell-out threshold has been reached. As a result of this change, there might be circumstances where the 90% threshold required for sell-out to be exercised was reached only because of shares which the offeror had conditionally contracted to acquire. However, if the conditions of such contracts were not fulfilled, the offeror could in fact find that he was being required to buy a minority shareholder’s shares even though the offeror had not actually acquired 90% of the shares. So section 983 also provides that, if that is the case at the time when the minority shareholder exercises his right of sell-out, the offeror does not have to purchase the shares unless he has acquired or unconditionally contracted to acquire 90% or more of the shares by the time the period referred to in section 984(2) (the period within which shareholders can exercise sell-out rights) ends. (A corresponding change is made in section 979(6) and (7) to prevent minority shareholders in this situation who have to wait to see if they can exercise sell-out from being squeezed out in the meantime.)
Applications to the Court (section 986)
1257.This section provides that a shareholder receiving a squeeze-out notice may make an application to the court (within six weeks of receiving the notice) seeking to overturn an offeror’s intention to purchase his shares compulsorily (or the terms of that purchase). A requirement that the offeror be promptly notified of such an application is now included (this was not previously required by section 430C of the 1985 Act). As a consequence of this requirement, it is also required that the offeror is obliged, at the earliest opportunity, to notify shareholders who are being squeezed out or who are exercising their rights of sell-out, and are not party to a section 986 application, that proceedings have been initiated.
Chapter 4: Amendments to Part 7 of the Companies Act 1985
Section 992: Matters to be dealt with in directors’ report
1258.This section implements Article 10 of the Takeovers Directive. Article 10.1 and 10.2 require companies admitted to trading on a regulated market to provide in their annual reports detailed information relating to matters such as the control and share structures of the company. It is, therefore, provided by amendment to Part 7 of the 1985 Act that the information required by the Directive must be set out in the directors’ report.
1259.Additionally, Article 10.3 of the Directive requires boards of companies to present an explanatory report to shareholders on the issues referred to in Article 10.1 and 10.2 at the company’s annual general meeting. This section requires this additional explanatory material to be contained in the directors’ report submitted to the annual meeting of shareholders.
1260.Section 992(5) amends section 251 of the 1985 Act on summary financial statements. It provides for the explanatory material required by Article 10.3 of the Takeovers Directive either to be included in the summary financial statement or to accompany it.
1261.Failure to include either the information concerning control and share structures or explanatory material in the annual report will attract existing criminal sanctions under section 234(5) of the 1985 Act (directors responsible for the failure to comply with provisions related to the directors’ report are to be liable to a fine).
1262.Section 992(6) provides that these new provisions will apply in relation to directors’ reports for financial years beginning on or after 20 May 2006 (the date by which the Directive had to be implemented).
1263.These are general requirements designed to bring greater transparency to the market and apply to all relevant companies whether or not they are involved in a takeover. Accordingly, the requirements will apply to all companies registered in the UK which have voting shares traded on a regulated market, whether or not that includes an official listing on the London Stock Exchange.
Under Part 15 of the Act (section 416), the Secretary of State may in future make regulations as to the contents of the directors’ report and those regulations will be able to incorporate the provisions introduced by section 992(2) to (4). Regulations under sections 427 and 428 will be able to make provision for the additional explanatory material when a summary financial statement is sent out rather than the full accounts and report.
Part 29: Fraudulent Trading
Section 993: Offence of fraudulent trading
1264.This section restates section 458 of the 1985 Act, but in doing so increases the maximum sentence for the offence from seven years’ imprisonment to ten years.
Part 30: Protection of Members Against Unfair Prejudice
1265.Sections 994-998 restate sections 459, 460 and 461 of the 1985 Act, which provide a remedy where a company’s affairs are being conducted in a manner which is unfairly prejudicial to the interests of its members.
Section 999: Supplementary provisions where company’s constitution altered
1266.Section 999 is a new section which ensures that, if the court makes an order under Part 30 amending the company’s articles, updated articles are registered and a copy of the court order is supplied with any copies of the articles that are issued by the company, unless they already incorporate the amendments.
Part 31: Dissolution and Restoration to the Register
Chapters 1 and 2: Striking Off and Property of Dissolved Company
1267.These Chapters restate sections 652 to 654 and sections 656 to 658 of the 1985 Act with no changes of effect, except as described below for sections 1003 and 1013.
Section 1003: Striking off on applications by company
1268.Section 652A of the 1985 Act currently provides that, in certain circumstances, a company may apply to the registrar to be struck off the register. The provision is limited to private companies. This section, in restating that provision, no longer includes the limitation, with the effect that public companies too may now apply for voluntary strike-off.
Section 1013: Crown disclaimer of property vesting as bona vacantia
1269.Where a company is dissolved while still holding property, that property passes to the Crown. Section 656 of the 1985 Act provides that the Crown’s title to the property may be disclaimed by means of a notice signed by the Crown representative.
1270.At present this disclaimer must generally be executed within 12 months of the date on which vesting of the property came to the notice of the Crown representative or within three years of the Crown representative receiving an application from an interested party. This section, in restating the relevant provisions, extends the 12-month period to three years, and provides that, if the ownership of the property is not established when the Crown representative first has notice that the property may have vested, that period runs from the end of the period reasonably necessary for the Crown representative to establish ownership.
1271.Section 656 of the 1985 Act also provides that a disclaimer may be made within three years of the Crown representative receiving an application from an interested party. This section changes this to 12 months.
Chapter 3: Restoration to the Register
1272.Under the 1985 Act, where a company has been struck off the register, mechanisms are available (sections 651 and 653) by which the company may be restored to the register following a court order. The CLR (Final Report, pages 227 to 229) recommended that an alternative, administrative restoration procedure should be available in certain circumstances. Sections 1024 to 1028 make provision implementing that recommendation.
Section 1024: Application for administrative restoration to the register
1273.This section provides that an application may be made to restore a company that has been struck off under sections 1000 or 1001; that it can be made whether or not the company has also been dissolved; that the application must be made by a former director or former member of the company, and that it must be made within six years of the date of dissolution.
Section 1025: Requirements for administrative restoration
1274.This section sets out the requirements for restoration, including the conditions that the company was carrying on business or in operation at the time of its striking off; that the Crown representative has given any consent that may be necessary; and that the applicant has delivered any documents necessary to bring the registrar’s records up to date and has paid any penalties due at the date of dissolution or striking off.
Section 1026: Application to be accompanied by statement of compliance
1275.This section provides that an application for restoration must be accompanied by a statement that the applicant has the necessary standing to make the application and that the requirements for administrative restoration have been met. The registrar may accept the statement of compliance as sufficient evidence of the matters stated in it.
Section 1027: Registrar’s decision on application for administrative restoration
1276.This section provides that the registrar must give notice of her decision, and that (if the decision is that the company is restored) the restoration takes effect from the date that notice is sent. The section also sets out the consequential actions the registrar must take.
Section 1028: Effect of administrative restoration
1277.This section provides that the effect of restoration is that the company is deemed to have continued in existence as if it had not been struck off, and that application may be made to the court within three years of restoration for the court to make such directions as may be needed to place the company itself, and other persons, in the same position as they would have been had the company not been struck off.
Restoration to the register by the court
1278.These sections bring together what are currently two separate procedures for a company to be restored to the register by court order (under sections 651 and 653 of the 1985 Act). The CLR recommended (Final report pages 227 to 229) that the two separate procedures be replaced by a single new procedure, which should largely be based on the precedent of the procedure under section 653 of the 1985 Act.
Section 1029: Application to court for restoration to the register
1279.The section sets out that an application may be made for the restoration of companies which have been dissolved, are deemed to be dissolved, or have been struck off under the various provisions set out in subsection (1). Subsection (2) sets out the persons who may make such an application.
Section 1030: When application to the court may be made
1280.At the moment, applications to the court under section 651 of the 1985 Act must be made within two years, and under section 653 within twenty years. This new section provides that the time limit for the new single procedure will generally be six years, although special provision is made (subsection (5)) for situations where an application for administrative restoration has been made and refused.
1281.Subsection (1) makes clear that there is no time limit where the application is for the purpose of bringing proceedings against the company for damages for personal injury.
Section 1031: Decision on application for restoration by the court
1282.This section sets out the circumstances in which the court may order restoration (including any case in which the court considers it just to do so) and provides that restoration takes effect when the court’s order is delivered to the registrar. It also requires the registrar to give appropriate public notice of the restoration.
Section 1032: Effect of court order for restoration to the register
1283.This section provides that the effect of restoration is that the company is deemed to have continued in existence as if it had not been struck off, and that the court may make such directions as are needed to place the company itself, and other persons, in the same position as they would have been had the company not been struck off. The court may also make directions as to the issues set out in subsection (4) to do with the company’s file at Companies House and costs.
Restoration to the register: supplementary provisions
Section 1033: Company’s name on restoration
1284.This section establishes the fundamental position that a company is restored to the register with the name it had before it was struck off, but also makes new provision for circumstances where restoration of a company would have the effect that two companies with the same or very similar names would appear in the registrar’s index. There is a procedure for the restored company to change its name.
Section 1034: Effect of restoration to the register where property has vested a bona vacantia
1285.This section replaces section 655 of the 1985 Act with little change of substance. However, it makes new provision (subsection (3)) that, where a company’s property has passed to the Crown and been disposed of, the Crown, in reimbursing the newly restored company, may deduct the reasonable costs of sale which were incurred.
Part 32: Company Investigations: Amendments
Powers to appoint inspectors
1286.The 1985 Act gives the Secretary of State the power to appoint competent inspectors to carry out inspections, and report the result to him, in a number of circumstances. There are three categories of inspections at present:
investigations into the affairs of companies;
investigations into the membership or control of companies; and
investigations of dealings in share options by company directors and their families and failure to disclose interests in shares.
1287.Investigations by inspectors into the affairs of companies and certain other bodies corporate can be initiated under sections 431 and 432. Such inspections can be launched on the application of a company or a proportion of its members, or on the Secretary of State’s own initiative, and must be carried out where the court orders it.
1288.Investigations by inspectors into the membership or control of companies can be initiated under section 442. The Secretary of State can launch such an inspection on his own initiative under section 442(1), and is obliged to do so where the requisite number of members of a company apply.
1289.Inspections in the third category, under section 446, relate to suspected contraventions of certain provisions of Part 10 of the 1985 Act. The 2006 Act repeals the relevant provisions of Part 10 (see section 1177) and section 446 is repealed in consequence.
1290.Two inspectors are generally appointed to carry out an inspection – usually a QC and a partner in one of the leading accountancy firms.
1291.Inspectors are appointed to investigate and to report the results of their investigations to the Secretary of State. At the end of an inspection, the inspectors generally have a duty to make a final report to the Secretary of State. The inspectors may also make interim reports during the course of the inspection, and the Secretary of State can direct them to do so.
1292.Unless the appointment was made under section 432 of the 1985 Act on terms that any report is not for publication (section 432(2A)) interim and final reports are publishable; the Secretary of State has discretion to publish an interim or final report under section 437(3). The availability of a published report is a crucial aspect of the inspection system.
Changes brought in by the 2006 Act
1293.The 2006 Act confers new powers on the Secretary of State to bring to an end an investigation when it is no longer in the public interest to continue with it, to revoke the appointment of an inspector and to issue directions about the scope of an investigation, its duration and certain other matters.
1294.The main purpose of these sections is to give the Secretary of State power to take appropriate action where an investigation appears to be taking too long. The sections also provide for situations not currently explicitly provided for, such as the resignation or death of inspectors, and the ability to appoint replacement inspectors.
1295.The details of these changes and the circumstances in which the changes will apply are set out below.
Section 1035: Powers of Secretary of State to give directions to inspectors
1296.Subsection (1) inserts new sections 446A and 446B into the 1985 Act, which provide new powers for the Secretary of State to give directions to inspectors with which they are obliged to comply (new sections 446A(1) and 446B(5)).
1297.The power in new section 446A(2) is exercisable by the Secretary of State in relation to inspectors appointed under sections 431, 432(2) and 442(1). Directions under new section 446A(2) can either relate to the investigation itself or the inspectors’ reports of the results of their investigations. Regarding the former, such directions can take two forms:
a direction as to the subject matter of an investigation (whether by reference to a specified area of a company’s operation, a specified transaction, a period of time or otherwise); or
a direction which requires an inspector to take or not to take a specified step in his investigation.
1298.As regards inspectors’ reports the Secretary of State will have a power to secure that any report (new section 446A(3)):
includes the inspector’s views on a specified matter;
does not include any reference to a specified matter;
is made in a specified form or manner; or
is made by a specified date.
1299.New section 446A(4) enables directions by the Secretary of State to be capable of being given on an inspector’s appointment. It also provides that directions may vary or revoke a direction previously given and may be given at the request of an inspector.
1300.New section 446A(5) confirms that the scope of the term “investigation” will include any investigation undertaken under section 433(1) into the affairs of the company’s holding company or subsidiary (or a subsidiary of its holding company or a holding company of its subsidiary).
1301.New section 446B(1) will give the Secretary of State power to direct an inspector to take no further steps in an investigation, and the inspector shall comply with any direction given to him under this section (new section 446B(5)). However, if the appointment of inspectors is one that the Secretary of State is obliged to make (either because a court orders that a company’s affairs ought to be so investigated or because the requisite number of its members has applied for an investigation into its ownership), such a direction can only be given if matters have come to light in the course of the investigation which suggest that a criminal offence has been committed and those matters have been referred to the appropriate prosecuting authority (new section 446B(2)).
1302.Under new section 446B(3), any direction given to the inspector under section 437(1) to produce an interim report, and any direction under new section 446A(3) in relation such a report, shall cease to have effect.
1303.If the Secretary of State directs an inspector to take no further steps in an investigation then the inspector shall not make a final report to him unless:
the requisite number of a company’s members has applied for an investigation into its ownership, matters have come to light in the course of the investigation which suggest that a criminal offence has been committed, those matters have been referred to the appropriate prosecuting authority and the Secretary of State directs the inspectors to make a final report; or
the inspector was appointed following a court order that a company’s affairs ought to be investigated (new section 446B(4)).
1304.New section 446B(6) confirms that the scope of the term “investigation” will include any investigation undertaken under section 433(1) into the affairs of the company’s holding company or subsidiary (or a subsidiary of its holding company or a holding company of its subsidiary).
1305.Subsections (2) to (5) concern consequential changes to other sections within the 1985 Act.
Section 1036: Resignation, removal and replacement of inspectors
1306.This section inserts new sections 446C and 446D which provide for the resignation or revocation of an inspector’s appointment and the provision to replace an inspector.
1307.New section 446C(1) and (2) provides not only that an inspector may resign but also that the Secretary of State has the power to revoke his appointment.
1308.New section 446D(1) provides that, if an inspector resigns, dies or has his appointment revoked, the Secretary of State has the power to appoint a replacement inspector to continue the investigation. Any appointment which takes place under new subsection (1) will be treated as though it were made under the provision under which the former inspector were appointed (new section 446D(2)).
1309.The Secretary of State is obliged to ensure that at least one inspector continues the investigation (new section 446D(3)) unless such a step would be pointless because he could direct the termination of the investigation in circumstances which would result in a final report not being made (new section 446D(4)).
1310.New section 446D(5) confirms that the scope of the term “investigation” will include any investigation undertaken under section 433(1) into the affairs of the company’s holding company or subsidiary (or a subsidiary of its holding company or a holding company of its subsidiary).
Section 1037: Power to obtain information from former inspectors etc
1311.Subsection (1) inserts new section 446E into the 1985 Act.
1312.New section 446E(1) provides that, where an inspector resigns or has his appointment revoked or is given a direction under section 446B (termination of an investigation) (new section 446E(2)), the Secretary of State can direct him to hand over documents that he has obtained or generated during the course of his investigation, either to the Secretary of State or to another inspector appointed under this Part (new section 446E(3)).
1313.A requirement under new subsection (3) includes the power to ensure that the production of a copy of the document is made in hard copy or in a form from which a hard copy can be obtained (new section 446E(4)). A document includes information recorded in any form (new section 446E(7)(b)). New section 446E(5) enables the Secretary of State to direct any person to whom this section applies to inform him of any matters that came to that person’s attention as a result of the investigation. New section 446E(6) confirms that a person shall comply with any direction given to him under this section.
1314.New section 446E(7)(a) confirms that the scope of the term “investigation” will include any investigation undertaken under section 433(1) into the affairs of the company’s holding company or subsidiary (or a subsidiary of its holding company or a holding company of its subsidiary).
1315.Subsections (2) and (3) concern consequential changes to other sections within the 1985 Act.
Section 1038: Power to require production of documents
1316.The 2006 Act introduces a new definition for documents in hard copy form, (see section 1135). This section ensures that there is a consistent approach in existing investigation provisions where there is a requirement to produce documents.
Section 1039: Disqualification orders: consequential amendments
1317.Subsections (a) and (b) extend the Company Directors Disqualification Act 1986 so that decisions on whether to take action to disqualify directors can be taken on the basis of information that was obtained or generated by an inspector (or came to his knowledge) as a result of his investigation, notwithstanding whether such information is or will be included in any formal report. In some cases this may speed up the ability to seek to disqualify directors.
Part 33: Uk Companies Not Formed under Companies Legislation
1318.The CLR considered the position of unregistered companies in Chapter 9 of Completing the Structure and presented their recommendations in paragraphs 11.34 to 11.37 of the Final Report. The provisions in this Part have been developed with these recommendations in mind.
Chapter 1: Companies Not Formed under Companies Legislation but Authorised to Register
Section 1040: Companies authorised to register under this Act
1319.This section replaces section 680 of the 1985 Act. It applies to companies incorporated within the UK but not formed under the Companies Acts (or certain earlier companies legislation). It enables such companies to register under the Act. The types of company that can take advantage of this provision are listed in subsection (1). They include companies formed before 2 November 1862; companies formed by private Act of Parliament and companies incorporated by royal charter.
1320.The company may apply to register as a company limited by shares, a company limited by guarantee or as an unlimited company. Subsections (4) and (5) impose restrictions on this choice. So, a company with limited liability may not register as an unlimited company, and only a company with share capital may register as a company limited by shares.
1321.A company may wish to apply to register under the Act in order to take advantage of legislation applying to companies registered under the Companies Acts. Subsection (6) makes clear that a company may register even if it is in order to take advantage of certain provisions of the Insolvency Act 1986 not available to unregistered companies. For example, under section 221(4) of the Insolvency Act 1986, unregistered companies may not be wound up under that Act voluntarily (except in accordance with the EC regulation on insolvency proceedings).
Section 1041: Definition of “joint stock company”
1322.This section restates section 683 of the 1985 Act. It defines the joint stock companies that may register under section 1040.
Section 1042: Power to make provision by regulations
1323.This section is a new provision. It confers power on the Secretary of State to make regulations in connection with the registration of a company following an application under section 1040 (application by UK company not formed under the Companies Acts to register under the Companies Acts). Regulations made under this section will replace the provisions made by sections 681 to 682, 684 to 690 and Schedule 21 of the 1985 Act. The regulations will cover the procedural requirements for registration, the conditions to be satisfied before registration and the documents to be supplied on an application for registration. The regulations will also set out the consequences of registration, including the status of the company following registration and the application of the Companies Acts to such companies following registration. The regulations are subject to the negative resolution procedure.
Chapter 2: Unregistered Companies
Section 1043: Unregistered companies
1324.This section replaces section 718 of the 1985 Act. The section confers a power on the Secretary of State to apply provisions of the Companies Acts to certain unregistered companies. These are companies incorporated in the UK, and having their principal place of business in the UK, but not formed or registered under the Companies Acts or any other public general Act of Parliament. Examples include companies formed by letters patent or by private Act of Parliament. Subsection (1) exempts certain other companies from regulations under this section, including those exempted by direction of the Secretary of State.
1325.Regulations under this section will replace the provision made by Schedule 22 to the 1985 Act. The regulations may apply specified provisions of the Companies Acts to specified descriptions of unregistered company, and may make limitations, adaptations and modifications to the application of the Companies Acts to unregistered companies. The regulations are subject to the negative resolution procedure.
Part 34: Overseas Companies
1326.This Part applies to companies incorporated outside the UK (“overseas companies”). It enables various registration, reporting and disclosure requirements to be imposed on overseas companies.
1327.This Part, together with the regulations to be made under it, will replace the provisions made by Part 23 (including Schedules 21A to D) of the 1985 Act. Regulations made under this Part will continue to implement the requirements of the Eleventh Company Law Directive (89/666/EEC), which imposes disclosure requirements on overseas companies that set up branches in the UK.
1328.As originally enacted, Part 23 of the 1985 Act applied to companies incorporated outside Great Britain that established a place of business in Great Britain. Subsequently, the Eleventh Company Law Directive imposed a different set of disclosure requirements on those overseas companies with branches in the UK. The branch disclosure requirements also differ depending on whether or not the overseas company is incorporated within another EEA State. The result is that there are at present effectively two parallel regimes that apply to overseas companies.
1329.The CLR set out their initial analysis of the rules for regulating companies formed abroad in Chapter 5.6 of the Strategic Framework and then put forward their provisional detailed conclusions in their consultation document of October 1999 entitled "Reforming the law concerning overseas companies." The CLR presented their conclusions in paragraphs 11.21 to 11.33 of the Final Report.
Section 1044: Overseas companies
1330.This section explains that for the purposes of the Companies Acts an “overseas company” means a company incorporated outside the UK. This is wider than the definition of “oversea company” in section 744 of the 1985 Act which it replaces. The definition in section 744 refers to companies incorporated outside Great Britain that establish a place of business in Great Britain. Under the Act the regulations will be able to specify the connection with the UK that gives rise to the various disclosure obligations imposed under this Part.
Section 1045: Company contracts and execution of documents by companies
1331.This section restates section 130(6) of the Companies Act 1989 (company contracts and execution of documents by companies).
Section 1046: Duty to register particulars
1332.This section confers on the Secretary of State a new power to make regulations to require overseas companies to register with the registrar of companies. The regulations may require particular information to be included in the registration. For example, an address for the company and details of its directors. The regulations may also require particular documents to be sent to the registrar, such as a copy of the company’s constitution. Subsection (2) ensures that the regulations implement the requirements of the Eleventh Company Law Directive (89/666/EEC), under which an overseas company must register if the company opens a branch in the UK.
1333.Regulations may require the overseas company to inform the registrar of companies of any changes in the details or documents it has registered (subsection (4)). The regulations may set deadlines for sending the information to the registrar of companies. They may also determine whether the overseas company should register with the registrar for England and Wales, the registrar for Scotland or the registrar for Northern Ireland. For example, an overseas company that opens a branch in Scotland may be required to register with the registrar of companies for Scotland.
1334.The Eleventh Company Law Directive imposes different disclosure requirements depending on where the overseas company setting up the branch is incorporated. Different reporting requirements are imposed on credit and financial institutions. Therefore regulations under this section may make different provision according to the place where the company is incorporated and the activities carried on by it.
1335.Regulations made under this section will be subject to the affirmative resolution procedure.
Section 1047: Registered name of overseas company
1336.This section applies to overseas companies required to register with the registrar of companies by regulations made under section 1046 (duty to register particulars). Overseas companies registered under that section must be required to provide a name for registration. The name will be entered on the index of company names (see section 1099).
1337.The company may register its corporate name (by which is meant its registered or legal name in its place of incorporation) or another name. All companies are free to choose whether to register their corporate name or another name, subject to the restrictions imposed by subsections (4) and (5). A name other than the corporate name can be registered only if it complies with the requirements imposed on the names of companies formed and registered under the Act. Likewise, unless the overseas company is incorporated in an EEA State, its corporate name can only be registered if it complies with these requirements. The only requirements of Chapters 1 to 4 of Part 5 (a company’s name) that do not apply are the requirements for the names of certain types of company to end with certain words (sections 58 and 59). These rules are not appropriate for overseas companies as they are specific to the types of company formed under the Companies Acts.
1338.Where the overseas company is incorporated in an EEA State (defined in section 1170), it may always register its corporate name, even if it does not comply with the requirements imposed on the names of companies formed under the Act provided that it complies with the requirement relating to permitted characters (contained in section 57). This section, together with section 1048 (registration under alternative name) replaces section 694 of the 1985 Act.
Section 1048: Registration under alternative name
1339.This section enables an overseas company to be registered under a name other than its corporate name. It also enables an overseas company to change the name by which it is registered. To do so it must deliver a statement to the registrar of companies with its proposed new name for registration. As long as the proposed name complies with the requirements for registration (see section 1047) the registrar of companies will enter it on the index of company names in place of the name previously registered.
1340.The section also provides that whatever name an overseas company is registered under, whether its corporate name or another, it is treated as being its corporate name for the purposes of the law in the UK. The change of name will not affect any legal proceedings that are continued or commenced by or against the company.
Section 1049: Accounts and reports: general
1341.This section confers on the Secretary of State a power to make regulations requiring overseas companies to prepare accounts and directors’ reports, and to obtain an auditor’s report. The requirements must be like those imposed on companies formed and registered under the Act. The accounts, directors’ report and auditor’s report requirements applying to companies formed and registered under the Act appear in Part 15 (accounts and reports) and Part 16 (audit).
1342.Regulations under this section may require the overseas company to deliver to the registrar of companies copies of the accounts and reports prepared in accordance with the regulations; alternatively the overseas company may be required to deliver to the registrar a copy of the accounts and reports that it prepared and had audited in accordance with the law of the country in which it is incorporated. The registrar will place the accounts and reports on the public register.
1343.Regulations under this section will replace sections 699AA to 703 of, and Schedule 21D to, the 1985 Act. The regulations will be subject to the negative resolution procedure.
Section 1050: Accounts and reports: credit or financial institutions
1344.This section applies only to credit or financial institutions incorporated or formed outside the UK and Gibraltar, with their head office outside the UK and Gibraltar but having a branch in the UK (subsection (1)). This section confers on the Secretary of State a power to make regulations specifically in respect of accounts and directors’ reports by these credit or financial institutions.
1345.Credit institution and financial institution are both defined in section 1173.
1346.Regulations under this section will implement requirements of the Bank Branches Directive 89/117/EEC of the Council of 13 February 1989. The definition of “branch” for the purposes of this section (subsection (2)) is based on Article 1.3 of Directive 2000/12/EC of the European Parliament and of the Council of 20 March 2000 relating to the taking up and pursuit of the business of credit institutions. The power to make regulations under this section is similar to that in under section 1049 (accounts and reports: general).
1347.The regulations will replace section 699A of, and Schedule 21C to, the 1985 Act. The regulations will be subject to the negative resolution procedure.
Section 1051: Trading disclosures
1348.This section confers on the Secretary of State a power to make regulations as to the information that overseas companies must display in specified locations, include in specified documents or communications, or provide to those who make a request in the course of business. Regulations made under this section will replace the provision made by section 693 of the 1985 Act. This section complements the similar power under section 82 to make regulations imposing trading disclosure obligations on companies formed and registered under the Companies Acts. Regulations under this section may require an overseas company carrying on business in the UK:
to display particular information in particular places. For example, a sign with its name outside every branch;
to include particular information in certain documents. For example, its name and country of incorporation on every invoice;
to provide certain information, such as its name, to those who request it when doing business with the overseas company.
1349.They may also make provision, corresponding to that made in sections 83 and 84, in respect of a failure by a company formed and registered under the Companies Acts to comply with the trading disclosure requirements imposed on them by regulations under section 82.
1350.Regulations under this section, like those under section 82, are subject to the affirmative resolution procedure.
Section 1052: Company charges
1351.This section applies to overseas companies that are required to register particulars under section 1046. It confers on the Secretary of State a power to make regulations about the registration by those companies of charges they grant over property in the United Kingdom. Subsection (2) lists some of the matters that may be dealt with in such regulations, and subsections (3) and (4) enable the regime for overseas companies to mirror specified provisions in Part 25, with modifications. This section replaces sections 409 (charges on property in England and Wales created by an overseas company) and 424 (extension of Chapter 2) of the 1985 Act.
Section 1053: Other returns etc
1352.This section applies to overseas companies that are required to register particulars under section 1046. It confers on the Secretary of State a power to make regulations requiring those companies to deliver returns to the registrar if they are being wound up or subjected to insolvency proceedings. The regulations may also require the liquidator of such a company to deliver returns to the registrar. They may specify the circumstances in which a return is to be made to the registrar. For example, on termination of the winding-up. The regulations may specify the information to be included in the return and set deadlines for sending it to the registrar (subsection (3)). They may require notice to be given to the registrar of certain appointments (subsections (4) and (5)).
1353.The regulations will replace sections 703P and 703Q of the 1985 Act. The regulations will be subject to the affirmative resolution procedure.
Section 1054: Offences
1354.This section ensures that the regulations will be able to specify the person or persons who would be responsible for complying with any specified requirement of the regulations. It allows regulations to provide for offences, including who would be liable in the event of any contravention and what might be considered a defence should a charge be brought. The maximum level of penalty permissible under the regulations on indictment is an unlimited fine and on summary conviction a fine not exceeding level 5 on the standard scale, or for continued contravention, a daily default fine not exceeding one-tenth of that.
Section 1055: Disclosure of individual’s residential address: protection from disclosure
1355.If the regulations under section 1046 require an overseas company to register an individual’s usual residential address, then the regulations must also provide for its protection on the same basis as is provided for directors’ residential addresses in Chapter 8 of Part 10.
Section 1056: Requirement to identify persons authorised to accept service of documents
1356.Every overseas company required by regulations under section 1046 to register with the registrar of companies must register particulars identifying every person resident in the UK who is authorised to accept service of documents on the company’s behalf or make a statement that there is no such person.
1357.As to how a document may be served on a registered overseas company, see section 1139(2) and the company communications provisions (sections 1144 to 1148 and Schedules 4 and 5).
1358.This section replaces the provision made by section 691(1)(b)(ii) and paragraph 3(e) of Schedule 21A to the 1985 Act.
Section 1057: Registrar to whom returns, notices etc to be delivered
1359.This section makes provision for regulations in respect of overseas companies that are required to register, or have registered, particulars under section 1046 in more than one part of the United Kingdom. The regulations may set out what should happen, for example, if the overseas company has registered branches in Scotland and in Northern Ireland; the regulations may require the returns or notices to be delivered to each registrar with whom the company is registered, or to the registrar for such part or parts of the United Kingdom as the regulations may specify.
1360.Regulations made under this section will be subject to the negative resolution procedure.
Section 1058: Duty to give notice of ceasing to have registrable presence
1361.Where an overseas company has registered particulars with the registrar following the opening of a branch in the United Kingdom, this section will enable regulations to require the overseas company to give notice to the registrar if it subsequently closes that branch. In addition, an overseas company that has registered particulars in other circumstances specified by regulations under section 1046 may be required by regulations to give notice to the registrar if those circumstances cease to obtain.
1362.The regulations will require the notice to be delivered to the registrar for the part of the United Kingdom in which the overseas company was registered and may set deadlines for sending the information to the registrar.
1363.Regulations made under this section will be subject to negative resolution procedure.
Section 1059: Application of provisions in case of relocation of branch
1364.This section provides that the relocation of a branch from one part of the UK to another is to be treated as the closing of the branch in one part and the opening in another. For example, if an overseas company moves a branch from Scotland to Wales, it must tell the registrar for Scotland that it is closing the branch. It must also tell the registrar for England and Wales that it is opening a branch in Wales. This section replaces the provision made by section 695A(4) of the 1985 Act.
Part 35: the Registrar of Companies
1365.This Part largely replaces Part 24 of the 1985 Act, and sets out the basic functions of the registrar of companies (these functions are currently carried out by Companies House for England and Wales and in Scotland and by the equivalent registry in Northern Ireland). The new sections implement a number of recommendations of the CLR.
Sections 1060 and 1061: The registrar and the registrar’s functions
1366.Section 1060 carries forward the approach of section 704 of the 1985 Act as to the appointment and status of the registrar of companies but omits some of the more antiquated parts of that provision. It specifies that there shall continue to be a registrar for England and Wales, for Scotland and for Northern Ireland.
1367.Section 1061 the registrar continues to have the functions conferred by the Companies Acts and in other legislation as specified in the section. The Secretary of State also has power to confer functions on the registrar, in relation to the registration of companies or other matters.
Section 1062: The registrar’s official seal
1368.This section, replacing section 704(4) of the 1985 Act, provides that the registrar must have an official seal for the authentication of documents.
Section 1063: Fees payable to the registrar
1369.This section gives the Secretary of State a power to set fees by regulations in relation to any function of the registrar and in relation to the provision of services and facilities incidental to the registrar’s functions. It replaces section 708 of the 1985 Act, but is more specific about the types of things for which fees may be charged, although this list is not exhaustive.
1370.As now, fees relating to the normal statutory obligations of companies under companies legislation are to be set by regulations made by the Secretary of State. It is also possible for fees to be charged for any ad hoc or bespoke services that Companies House provides. The 1985 Act (section 708(5)) provides that the registrar determines fees for services for which there is no direct legal obligation. Subsection (6) of this section replaces this with a more general power for the registrar to determine fees where no fee has been set in regulations by the Secretary of State. Such fees might relate for example to the introduction of new services (e.g. those made possible by new technologies) which could not have been anticipated when the Secretary of State last made fees regulations; or for services such as seminars and road shows which Companies House arranges.
Section 1064 to 1065: Certificates of incorporation
1371.Section 1064 replaces section 711(1)(a) of the 1985 Act and provides for notice of the issue of certificates of incorporation to be published in the Gazette. The publication must include the company’s registered number as well as its name. Section 1065 replaces section 710 of that Act and allows any person to obtain a certificate of incorporation of a company. These sections cover all certificates of incorporation (including, for example, certificates of incorporation on change of name).
Section 1066 and 1067: Registered numbers
1372.Section 1066 replaces section 705 of the 1985 Act on companies’ registered numbers, without change of substance. Section 1067 replaces the provisions of section 705A of that Act relating to registered numbers of branches of overseas companies.
Delivery of documents to the registrar
Section 1068: Registrar’s requirements
1373.This section gives the registrar power to make rules about form, authentication and manner of delivery of documents, including the physical form and means of communication, the format, and the address to which they are to be sent, and where appropriate, technical specification. The power conferred by this section does not authorise the registrar to require documents to be delivered in electronic form.
Section 1069: Power to require delivery by electronic means
1374.This section provides that the Secretary of State (not the registrar) has a new power to provide for electronic-only delivery of classes of document. The Secretary of State may only exercise this power in respect of classes of document which are authorised or required to be delivered and for which the registrar has published rules relating to electronic delivery (in other words where it is clear precisely what mechanism is to be used for the electronic communication).
Section 1070: Agreement for delivery by electronic means
1375.This section sets out the power of the registrar to make agreements with companies to deliver information only electronically. The agreements could cover all documents (to the extent that electronic means of filing are available) or just selected documents. It is envisaged that the agreements will be in a standard form and contain detailed provisions for communications between the registrar and the company (including possible use of codes and encryption). The agreements need not be available to be entered into by everyone in the same form or at all.
Section 1071: Document not delivered until received
1376.This section provides that “delivery” obligations go beyond an obligation simply to send or post information to the registrar, and that the registrar may make rules governing what it means for a document to be “received” (which might include, for example, setting out which offices of the registrar should receive a document).
Requirements for proper delivery
Section 1072: Requirements for proper delivery
1377.This section provides that, in order for a document to be properly delivered to the registrar, various conditions (specified in subsection (1)(a) to (g)) must be met. Where those conditions are not satisfied, and the document is therefore not “properly” delivered, it is not to be treated as having been delivered for the purposes of the underlying provision which authorises or requires it.
Section 1073: Power to accept documents not meeting requirements for proper delivery
1378.The registrar may still accept (and register) a document that does not comply with the requirements for proper delivery, although such acceptance does not (subsection (4)) exempt the filer from any consequence attaching to failure to comply with the original requirements for delivery.
Section 1074: Documents containing unnecessary material
1379.Documents are sometimes delivered to the registrar which contain “unnecessary” material, i.e. material for which there was no legal requirement or authorisation . Where the unnecessary material cannot readily be separated from the necessary material, then the document as a whole is treated as not properly delivered. Where it is separable, the registrar has the option of either registering the entire document as delivered, or excising the unnecessary material and registering the remainder.
Section 1075: Informal correction of document
1380.This is a new provision, giving the registrar power to correct information in a document by informal means (for example, by taking revisions or supplementary information from the company over the telephone) but only in very limited circumstances. It can be used as an alternative to rejecting or removing information:
on the grounds that it is incomplete (e.g. empty fields within the document);
on the grounds that it is internally inconsistent (e.g. the company number does not correspond to the company name);
1381.This ability to make informal corrections only applies where companies have informed the registrar that it should apply. The registrar needs to initiate the correction and be satisfied that the person is authorised to give the information sought. In order to be satisfied as to the authority of the person she is telephoning, the registrar may provide for identification numbers or other checks on identity.
Section 1076: Replacement of document not meeting requirements for proper delivery
1382.This section sets out how the registrar may accept a replacement document that was not properly delivered in the first place. In essence, the registrar must be satisfied that the replacement document is delivered by the original filer, or by the company to which the original document relates, and that the replacement is “properly delivered” (as defined in section 1072)). It also allows the registrar to impose requirements to ensure that the replacement can clearly be associated with a particular original.
Sections 1077 and 1078: Public notice of receipt of documents subject to Directive disclosure requirements
1383.These sections replace section 711 of the 1985 Act, which provides that certain notices must be published in the Gazette, and lists the documents to which that requirement relates. This list derives from Community legislation, principally the First Company Law Directive (68/151/EEC) as amended by Directive 2003/58/EC.
1384.Section 1077 provides that notice of receipt of these documents must be published either in the Gazette, or by some other means (as may be specified under section 1116). Section 1078 lists the documents subject to Directive disclosure requirements.
Section 1079: Effect of failure to give public notice
1385.This section effectively replaces section 42 of the 1985 Act. It sets out how a company, in its dealings with third parties, may not rely on the consequences of certain events (those which are set out in subsection (2)) unless notice of the event has duly appeared in the Gazette or been published in some other way provided for in section 1116.
The register
Section 1080: The register
1386.This section imposes an obligation on the registrar to keep a record of the material received. It gives a discretion as to the form in which the record is kept. S ubsection (3) provides that this discretion is subject to the terms of the amended First Company Law Directive (68/151/EEC), Article 3.2 of which requires any documents and particulars it covers which are delivered on after 1 January 2007 to be retained in electronic form. The documents covered by this obligation are those set out in section 1078.
Section 1081: Annotation of the register
1387.It is important that the register is as useful and transparent a source of information as possible for users. Hence, this section sets out certain circumstances in which the registrar is obliged to annotate the information on the register to gloss it or provide supplementary information. Annotations must for example be provided to show the date of delivery of information; and the fact that information has been replaced, corrected or removed. This section responds to a recommendation of the CLR (summarised in their Final Report at paragraph 11.48). Subsection (5) provides that the court can dispense with the need for annotation in certain circumstances.
1388.The Secretary of State has power to make provision by regulations extending the circumstances where the registrar can or should make annotations.
Section 1082: Allocation of unique identifiers
1389.This section is a new provision. It gives the Secretary of State a power to make regulations so that unique identifiers can be allocated to company officers such as directors. This provision supports those that provide for the home addresses of directors no longer to be kept on the public record. The unique identifier enables searchers to distinguish between different persons of the same name.
Sections 1083 and 1084: Preservation of original documents and records relating to companies that have been dissolved etc
1390.Section 1083 replaces section 707A(2) of the 1985 Act. However, the obligation on the registrar to keep the originals of documents received now only applies for three years (as opposed to ten in the existing provision). This section also provides that the obligation to retain originals does not extend to an original document provided electronically (provided that the information itself has been placed on the register).
1391.Section 1084 replaces sections 707(3) and (4) of the 1985 Act, and provides that records may be transferred to the Public Records Office two years after a company has been dissolved. It also makes equivalent provision for certain overseas companies which, for example, by ceasing to have any connection with the UK, are no longer caught by UK regulatory requirements.
Inspection etc of the register
Section 1085: Inspection of the register
1392.This section provides that any person may inspect the register. Searchers however have a right to inspect the original of a hard copy document only where the registrar still retains it and where the public record kept by the registrar and derived from it is illegible or unavailable.
Section 1086: Right to copy of material on the register
1393.This section provides that any person is entitled to a copy of material on the register. Consistent with the provisions of the amended First Company Law Directive (68/151/EEC), subsection (2) provides that the fee for a copy may not exceed the administrative cost of providing the service.
Section 1087: Material not available for public inspection
1394.This section sets out a number of exceptions to the above rights to inspect and copy material on the register. These are listed in subsection (1)(a) to (k) and include, for example, “protected information”, i.e. information about directors’ home addresses. Subsection (2) provides that the fact that certain material (for example, an address), which has been placed on the register as a result of the filing of two or more different types of document, is confidential in one of those contexts, does not mean that it cannot be made public in its other context.
Section 1088: Application to registrar to make address unavailable for public inspection
1395.This section is a new provision. It confers power on the Secretary of State to make regulations providing for applications to remove addresses from the public record held by Companies House. The regulations may set out the details of who can apply and on what grounds and the procedure involved. They are subject to the affirmative resolution procedure.
Sections 1089 to 1091: Provision of copies of material on the register
1396.These sections enable the registrar to specify the form and manner in which applications for inspection of the register, or for copies of material on it, must be made, and to determine the form and manner in which copies are provided. They are subject to important exceptions, arising from the amended First Company Law Directive (68/151/EEC), in respect of the documents listed at section 1078.
1397.Section 1089 provides that applications must be capable of being submitted in hard copy or in electronic form, as the applicant chooses. Section 1090 similarly provides that the applicant is entitled to insist on receiving the copies themselves in hard copy or in electronic form (subject to an exception in respect of documents delivered before 1 January 2007).
1398.Section 1091, again responding to provisions of the amended First Company Law Directive (68/151/EEC), provides that, unless the applicant chooses otherwise, copies of information provided in hard copy must be certified as true copies but electronic copies must not be so certified. Subsection (3) provides for the evidential status of certified hard copies in legal proceedings. The Secretary of State has power to prescribe by regulations methods of certification for copies provided by electronic means.
Section 1092: Issue of process for production of records kept by the registrar
1399.This section restates section 709(5) of the 1985 Act and provides that no-one can take proceedings against the registrar for production of records without first obtaining the permission of the court.
Correction or removal of material on the register
Section 1093: Registrar’s notice to resolve inconsistency on the register
1400.This section enables the registrar to notify a company of an apparent inconsistency in the information on the register. An example might be where a document is received notifying the removal of a director where there is no record of his appointment. In such circumstances, the registrar may give notice to the company requiring them to resolve the inconsistency within 14 days by providing additional or replacement documents. Failure to do so on the company’s part is an offence (subsection (3)).
Section 1094: Administrative removal of material from the register
1401.The registrar has a power to remove from the register information that there was a power but no duty to enter. Under subsection (4), the registrar will need to send a notice to the presenter of the information in question, or to the company to which the material relates, on or before removing the material.
1402.The registrar may not however remove information from the register where registration has had legal consequences for the company as regards certain key events, as set out in subsection (3), including for example its formation or a change of registered office.
Section 1095: Rectification of register on application to registrar
1403.This section gives the Secretary of State a power to make regulations under which, following a successful application, the registrar may be required to remove certain kinds of material from the register. The procedure may only cover certain types of document. It may operate in respect of material that derives from something that is invalid or ineffective or from something that was done without the authority of the company (this would cover forms filed without authority); and material that is factually inaccurate or forged or derives from something that is factually inaccurate or forged.
1404.The registrar may only act as a result of an application, and regulations may provide for matters such as who may make the application and what information will need to be provided with it. Where the material removed is of a kind whose registration has had legal consequences, subsection (5) provides that interested parties have the right to go to court to obtain an order as to the material’s legal effect.
Section 1096: Rectification of the register under court order
1405.The registrar is also required to remove material from the register where there is a court order to that effect. The court’s rectification power operates in the same circumstances as the registrar’s power following regulations made under section 1094. However, the court’s power is of general application. For example, there is no limit on the types of document covered. The court may make an order to remove material from the register where its presence on the register has caused damage or may cause damage to the company and the company’s interests in removing the material outweigh the interests of others in it continuing to be on the register. The court may make such consequential orders as appear just regarding the period that the information was on the register and the effect of the information being on the register during that period. The court’s rectification power does not operate where the court has other rectification powers (e.g. in relation to accounts or charges).
Section 1097: Powers of court on ordering removal of material from the register
1406.This section provides that where a court decides that certain information should be removed from the public register, the court may also make directions as to annotations (removing notes that are already there or directing that now new notes appear as a result of its order – or that notes appear in a restricted form) and as to whether its own order should be available for public inspection.
Section 1098: Public notice of removal of certain material from the register
1407.Section 1077 provides for the registrar to give public notice that she has received certain documents relating to a company in the Gazette or through some other form of publication. This section creates a corresponding obligation for her to give notice where she removes such material.
Section 1099: The registrar’s index of company names
1408.This section replaces section 714 of the 1985 Act with changes. It provides for the registrar of companies to keep an index of the names not only of companies incorporated under Companies Acts but also of business entities formed under other legislation and of overseas companies with a UK branch.
1409.The section provides power for regulations to update the categories of business entities that are included in the index. This power is subject to negative resolution procedure.
Section 1100: Right to inspect index
1410.This section retains the public right to inspect the index. (It can be searched online, without charge, at www.companieshouse.gov.uk.) The index of company names is important not only as the means of access to the information on the public record of companies incorporated in the United Kingdom but also as the list of names with which a proposed new name is compared to ensure that a new entity is not registered in a name that is the same or similar to that of an existing entity.
Section 1101: Power to amend enactments relating to bodies other than companies
1411.This section provides power for the Secretary of State to amend the rules for the names that can be adopted by other business entities on the index of company names. This power is subject to affirmative resolution procedure.
1412.Each category of business entity is subject to its own rules which include various safeguards to minimise the risk of public confusion. These rules differ from those that apply to companies in particular as regards the adoption of a name the same or similar to one already on the index. This lack of reciprocity is a weakness of the existing system which this section provides power to address.
Section 1102: Application of language requirements
Section 1103: Documents to be drawn up and delivered in English
1413.These sections set out language requirements. Section 1103 sets out the general rule that all documents must be in English (subject to the exceptions in the following sections). Section 1102 provides that this general rule, and its exceptions, apply automatically to documents required under the Companies Acts and Insolvency Act 1986 (and its Northern Ireland equivalent).
1414.There are however a variety of other pieces of legislation which may require companies in certain circumstances to supply material to the registrar. Depending on the nature of the particular requirement and its origin (for example, whether it responds to European Community law), it may or may not be appropriate to apply the language provisions of this Act unchanged to such material. Sub s ection (2) of section 1102 therefore enables the Secretary of State to make regulations to apply specified requirements to documents filed under other legislation.
Section 1104: Documents relating to Welsh companies
1415.This section provides an exception to the general rule in section 1103: documents relating to Welsh companies may be drawn up and filed in Welsh (and sometimes only in Welsh). It replaces, without any substantive change, section 710B of the 1985 Act.
Section 1105: Documents that may be drawn up and delivered in other languages
1416.This section sets out the circumstances in which documents may be drawn up and filed in other languages, but requires them to be accompanied by a certified translation into English. These documents are listed in subsection (2): agreements affecting the company’s constitution, documents relating to group accounts for companies in a group, and instruments relating to company charges. For some companies, documents of these sorts may well originate in languages other than English, and there may be an interest in ensuring that the original version is registered with the registrar. Subsection (2)(d) also allows the Secretary of State to extend the categories of documents to which this section applies.
Section 1106: Voluntary filing of translations
1417.The main purpose of this section is to implement aspects of the amended First Company Law Directive (68/151/EEC). It provides that companies may send the registrar certified translations of documents relating to the company. Subsection (2) enables the Secretary of State to set out in regulations the languages and documents in relation to which this facility is available. Subsection (3) provides that these regulations must as a minimum specify the official languages of the EU, and the documents covered by the amended First Company Law Directive (68/151/EEC) (see section 1078), to ensure compliance with that Directive. However, other languages (and categories of document) may be covered by the regulations.
Section 1107: Certified translations
1418.This section provides that a “certified translation” is one that has been certified in a manner prescribed by the registrar. It also provides that, where there is a discrepancy between an original and a translation, the company may not rely on the translation as against a third party, but the third party may rely on the translation (unless the company can show that the third party had knowledge of the original). This implements article 3a.4 of the amended First Company Law Directive (68/151/EEC).
Section 1108: Transliteration of names and addresses: permitted characters
1419.This section is a new provision. It deals with the possibility that the name and address of a director or of an overseas company may use a character set (for example, that of Urdu or Japanese) which is different from those with which the bulk of Companies House’s users are familiar. This section restricts the characters that are permitted for names and addresses in a document delivered to the registrar to those specified in regulations. The regulations, which are subject to negative resolution procedure, may also provide for names and addresses to be delivered in their original form.
Section 1109: Transliteration of names and addresses: voluntary transliteration into Roman characters
1420.This section is a new provision. It provides for the possibility that the Regulations made under section 1108 may permit letters and characters that are not drawn from the Roman alphabet, for example Greek letters. This section permits these names to be transliterated provided that certain requirements are met.
Section 1110: Transliteration of names and addresses: certification
1421.This section is a new provision. It confers power on the Secretary of State to make regulations relating to the certification of the transliteration of names and addresses. The regulations, which are subject to negative resolution procedure, may distinguish between compulsory transliteration under section 1108 and voluntary transliteration under section 1109.
Section 1111: Registrar’s requirements as to certification or verification
1422.Documents delivered to the registrar are sometimes required to be certified or verified in some way, for example to the effect that they are an accurate translation. This section allows the registrar to impose requirements as who must provide the relevant certification or verification. Subsection (2) provides that the registrar’s general powers to specify requirements in relation to documents submitted to her (section 1068) extends to the certification or verification as if it were a separate document.
Section 1112: General false statement offence
1423.This section provides a new offence of knowingly or recklessly delivering to the registrar information which is misleading, false or deceptive in a material particular. It responds to a recommendation of the CLR (Final Report, paragraph 11.48). This new general offence makes it unnecessary to reproduce specific offences covering false information or false statements in respect of specific legislative requirements that were a feature of the 1985 Act.
Section 1113: Enforcement of company’s filing obligations
1424.This section, which restates section 713 of the 1985 Act, provides the mechanism for ensuring that companies can be compelled to comply with their obligations to file documents or give notices to the registrar. Where a company has defaulted on an obligation, the registrar herself, any member of the company, or any creditor, may serve a notice on the company requiring it to file. If the company continues the breach after 14 days, the applicant may apply to the court for an order requiring the company, or any specified officer of it, to make good the default. The court may order that the costs of the proceedings are borne by the company or its officers. Subsection (5) provides that this process does not affect any offence or civil penalty arising from the company’s failure to comply with the original requirement.
Section 1114: Application of provisions about documents and delivery
1425.This section, which replaces section 715A of the 1985 Act, provides that “document” means information recorded in any form, and that “delivering” a document includes forwarding, lodging, registering, producing or submitting it, or giving a notice. It also provides that requirements relating to documents also apply (unless otherwise provided for) to information passed to the registrar in some other way. This caters for the possibility that information may not be in documentary form, for example when it is sent via a link to a website.
Section 1115: Supplementary provisions relating to electronic communications
1426.This section, which replaces section 710A of the 1985 Act, allows the registrar to require those who choose to file electronically to accept electronic communications from the registrar. It also provides that, where a document is required to be signed by the registrar, or authenticated by seal, she may determine by rules how it is to be authenticated when it is sent by electronic means.
Section 1116: Alternative to publication in the Gazette
1427.The registrar is required under the 1985 Act to publish certain statutory notices in the Gazette. The objective of that requirement is to ensure that such notices are well-publicised and made available to all those who might wish to take notice of them. The Gazette is a long-established and well-understood mechanism for ensuring such publicity. However, it is possible that developments, in particular in electronic publishing, will mean over time that alternative mechanisms are equally or more appropriate as ways of meeting the underlying policy objective. The CLR envisaged that the registrar should be able to make use of such mechanisms (Final Report, paragraph 11.48). This section therefore provides a power for the Secretary of State to specify alternative means which the registrar may then approve for use. To ensure that any such change is itself well-publicised in advance, subsection (5) provides that the change must itself be announced in the Gazette.
Section 1117: Registrar’s rules
1428.Other provisions in this Part enable the registrar to impose requirements in relation to certain matters. For example, section 1068 enables the registrar to specify the form, authentication and manner of delivery of documents to her; and section 1075 similarly enables her to determine the form and manner of any company instructions as to informal correction of the register. This section provides that the registrar may set out these requirements in registrar’s rules. The rules can make different provision for different cases, and may allow the registrar to modify or disapply the rules. The registrar must publicise any rules in a way designed to make sure that those who will need to know about them get to hear of them (which might in practice, for example, be by using the Companies House website); and must make copies of the rules publicly available.
Section 1118: Payments into the Consolidated Fund
1429.This section ensures that nothing in this or other companies legislation affects the continued operation in relation to the registrar of the Government Trading Funds Act 1973. (Companies House is and remains a Trading Fund.)
Section 1119: Contracting out of registrar’s functions
1430.This section largely restates subsections (7) and (8) of section 704 of the 1985 Act. The Deregulation and Contracting Out Act 1994 envisages that some of the registrar’s functions may be contracted out. This section provides for this possibility by saying that where a contractor is processing documents the registrar can provide for them to be sent directly to the contractor.
1431.The Deregulation and Contracting Out Act 1994 does not permit the function of making subordinate legislation to be delegated. Subsection (3) provides that registrar’s rules are not regarded as subordinate legislation for this purpose, permitting the contractor to make rules about form and manner of delivery, for example.
Section 1120: Application of Part to overseas companies
1432.This section provides that, except where the context otherwise requires, the provisions of this Part of the Act apply equally to overseas companies.
Part 36: Offences under the Companies Acts
1433.The CLR sought to draw out the basis on which criminal liability for a breach of Companies Act requirements is allocated under existing legislation to companies and to officers of companies. It stated in its final report that a reformed Companies Act must be underpinned by effective and proportionate sanctions and enforcement.
1434.The key changes in the Act are:
refinements to the “officer in default” provisions to make it clearer which individuals in which circumstances may be liable for a breach; and
removal of criminal liability from the company itself in certain circumstances.
1435.The general principle adopted as to whether a company should be liable for a breach of requirements of the Companies Acts is that where the only victims of the offence are the company or its members, the company should not be liable for the offence. On the other hand, where other persons may be victims of the offence, then the company should be potentially liable for a breach, whether or not the offence may also harm the company or its members.
Section 1121: Liability of officer in default
1436.This section specifies which persons may be liable as an officer of a company for an offence committed by the company under the Act (or the other Companies Acts). It only applies where another provision expressly states that an offence is committed by every officer of a company who is in default.
1437.An “officer” of a company is defined as including a director, manager or (company) secretary, and any person who is to be treated as an officer of the company for the purposes of the provisions in question. An officer is liable for an offence when he is “in default”, meaning he authorises or permits, participates in, or fails to take all reasonable steps to prevent the offence being committed.
Section 1122: Liability of company as officer in default
1438.Under this provision, where a company is an officer of another company, liability for an offence can be fixed upon the company as an officer in default only if one of its officers is in default.
Section 1123: Application to bodies other than companies
1439.This clause provides that section 1121 applies to persons in bodies other than companies where their role is equivalent to that of an officer of a company. It makes specific provision for bodies corporate, partnerships and unincorporated bodies.
Section 1124: Amendments of the Companies Act 1985
Schedule 3: Amendments of remaining provisions of the Companies Act 1985 relating to offences
1440.Section 1124 introduces Schedule 3, which contains amendments to certain provisions relating to offences which remain in Parts 14 and 15 of the 1985 Act.
1441.Many of these amendments are necessary due to the repeal by the Act of Schedule 24 to the 1985 Act. Schedule 24 set out the level of punishment for offences under the 1985 Act. The provisions in Parts 14 and 15 are amended so that the applicable punishments are now included alongside the description of the offence instead of in Schedule 24.
1442.Schedule 3 also makes amendments to the offences provisions remaining in Parts 14 and 15 of the 1985 Act to reflect the (non-textual) changes made to that Act by the Criminal Justice Act 2003. Section 282 of the Criminal Justice Act increases from 6 months to 12 months the maximum term of imprisonment to which a person is liable on summary conviction of an offence triable either way, and section 154(1) of that Act gives power to magistrates to impose a 12 month term of imprisonment. The increased penalties only apply in England and Wales; in Scotland and Northern Ireland the maximum term of imprisonment that may be imposed on summary conviction remains 6 months. When the Act received Royal Assent, neither section 282 nor section 154(1) of the Criminal Justice Act 2003 had come into force (which is the reason for the transitory provision in section 1131).
1443.A number of the amendments make reference to “the statutory maximum fine”. This was set at £5,000 at the time the Act received Royal Assent.
1444.Schedule 3 makes only one substantive change to the offence provisions in Parts 14 and 15. This is to include a daily default fine, of one-fiftieth of the statutory maximum, for continued contravention of section 444(3) (failure to provide information about interests in shares).
Section 1125: Meaning of “daily default fine”
1445.This section defines “daily default fine”. It replaces provision currently in section 730(4) of the 1985 Act.
Section 1126: Consents required for certain prosecutions
1446.This section provides that certain proceedings can only be brought with the consent of specified persons. It replaces section 732(1) and (2) of the 1985 Act.
Section 1127: Summary proceedings: venue
1447.This section restates section 731(1) of the 1985 Act. It specifies the possible venues for summary proceedings for any breach of Companies Acts requirements. For a body corporate the venue may be any place at which the body corporate has a place of business, and for any other person, it may be at any place that the person is.
Section 1128: Summary proceedings: time limit for proceedings
1448.This section restates section 731(2) to (4) of the 1985 Act. It sets out time limits for summary proceedings. The prosecution must be commenced within three years of the offence being committed, and within one year of the prosecuting authorities receiving sufficient evidence to justify the prosecution.
Section 1129: Legal professional privilege
1449.This section restates section 732(3) of the 1985 Act and applies its provisions to all offences prosecuted under the Companies Acts, rather than just those instituted by the Director of Public Prosecutions or the Secretary of State. It provides that the Companies Acts provisions on offences are not to be read as requiring any person to disclose information that is protected by legal professional privilege.
Section 1130: Proceedings against unincorporated bodies
1450.This section restates section 734(1) to (4) of the 1985 Act. It provides for proceedings for offences under the Companies Acts committed by unincorporated bodies to be brought against such bodies as if they were corporate bodies.
Section 1131: Imprisonment on summary conviction in England and Wales: transitory provision
1451.This section provides for the period before the commencement of section 154(1) of the Criminal Justice Act 2003, which makes new provision about the powers of magistrates’ courts in England and Wales to impose sentences of imprisonment on summary conviction. For offences committed before section 154(1) is brought into force, the maximum term of imprisonment in England and Wales for a person guilty of an offence on summary conviction under the Companies Acts is to be 6 months (as it is at present) instead of 12 months.
Section 1132: Production and inspection of documents where offence suspected
1452.This section restates section 721 of the 1985. It makes provision about orders for the production of documents where there is reasonable cause to believe that an offence has been committed.
Section 1133: Transitional provision
1453.This section provides that this Part of the Act (with the exception of section 1132) does not apply to offences committed before the commencement of the provision relevant to the offence.
Part 37: Companies: Supplementary Provisions
Sections 1134 and 1135: Company records
1454.These sections replace sections 722(1) and 723(1) and (2) of the 1985 Act and enable a company to use electronic storage or paper for its records provided that there can be paper printouts of electronic records. Section 1135(1) also enables the contents to be arranged as the directors see fit provided that it is adequately recorded. Section 1138 (duty to take precautions against falsification) also applies if the records are not kept in bound books.
Section 1136: Regulations about where certain company records to be kept available for inspection
1455.This section is a new provision. It provides power for the Secretary of State to make regulations to provide an alternative to the company’s registered office as the location for inspection of specified records. The regulations are subject to negative resolution procedure.
Section 1137: Regulations about inspection of records and provision of copies
1456.This section replaces section 723A of the 1985 Act. It allows the Secretary of State to make regulations about the inspection and provision of copies of company records.
Section 1138: Duty to take precautions against falsification
1457.This section replaces section 722(2) and (3) of the 1985 Act. It makes it an offence not to take measures that protect records from falsification or help trace any falsification. This provision applies to records that are not kept in bound books, whether or not they are kept electronically, but does not apply either to companies’ copies of the directors’ service contracts or to copies of any qualifying third party indemnity provisions benefiting a director.
Section 1139: Service of documents on company
1458.This section replaces section 725 of the 1985 Act. It ensures that there is a place at which a document may be served on companies registered under the Act. It also applies to overseas companies registered in accordance with regulations made under section 1046. Subsection (4) enables court service on a company registered in Scotland or Northern Ireland at the company’s principal place of business in England and Wales. (Since this is a provision about court service, the rules on companies registered in England and Wales carrying on business elsewhere in the UK will be found in the relevant legislation in those parts of the UK.) Subsection (5) makes clear that this section is supplemented by the “company communications provisions” referenced by section 1143.
Section 1140: Service of documents on directors, secretaries and others
1459.This section is a new provision. It ensures that the address on the public record for any director or secretary is effective for the service of documents on that person. Subsection (3) provides that the address is effective even if the document has no bearing on the person’s responsibilities as director or secretary. This provision also applies to the address on the public record of various other persons for whom the Act requires an address on the public record. Subsection (7) similarly makes clear that this section is supplemented by the “company communications provisions” referenced by section 1143.
Section 1141: Service addresses
1460.This section defines “service address” for the purposes of the Companies Acts as an address at which documents may be effectively served and provides a power for regulations to specify conditions with which a service address must comply.
Section 1142: Requirement to give service address
1461.This section is a new provision. It qualifies requirements elsewhere to give an "address". Unless the requirement is for a particular kind of address (say, the usual residential address), the section makes clear that a service address (as defined in the previous Section) is what is required.
Section 1143: The company communications provisions
1462.Section 1143 introduces sections 1144 to 1148 and Schedules 4 and 5. These make new provision for communications with or from companies, typically with their members but also with debenture-holders and others.
1463.The Companies Act 1985 (Electronic Communications) Order 2000 facilitated the use of electronic and website communications in certain contexts, but there has been uncertainty as to whether other provisions under the 1985 Act for information to be communicated ‘in writing’ required the use of paper or could be satisfied by electronic communications. The Act makes new general provision about communications, including electronic and website communications for the Companies Acts as a whole.
1464.The general principle behind the company communications provisions is that companies should be able to use hard copy or electronic communications in all cases. However, these provisions are subject to anything in or under any other enactment, and, as regards communications between companies and Companies House, they are also subject to the provisions of Part 35 of the Act.
Section 1144: Sending or supplying documents or information
1465.This section introduces two Schedules, which apply irrespective of the company’s articles:
Schedule 4 deals with documents and information sent or supplied to a company;
Schedule 5 deals with communications by a company and between companies
1466.Sections 1146 (requirement of authentication), 1148 (interpretation of company communications provisions) and 1168 (hard copy and electronic form and related expressions) contain provisions necessary for the interpretation of Schedules 4 and 5.
Schedule 4: Documents and information sent or supplied to a company
1467.This Schedule brings together the rules on communications to a company. In some cases a company will have other companies as members, debenture holders etc, and paragraph 1(2) of the Schedule makes clear that communications by such members etc is governed by Schedule 5.
1468.Part 2 of the Schedule sets out the position for communications sent or supplied in hard copy form. These are very similar to the present position. Part 3 sets out rules for communications in electronic form. Part 4 is a more general provision and means that unless the Companies Acts specify a means of communication, for example under section 291(3)(a) (circulation of written resolutions proposed by directors), then a communication to the company can be sent or supplied in any way agreed by the company.
Schedule 5: Communications by a company
1469.This Schedule sets out provisions on how companies are to communicate with their members, debenture holders etc. These rules are subject to, for example, additional rules which the FSA may require of companies traded on a regulated market.
1470.Part 2 of the Schedule sets out standard rules for communications in hard copy. Paragraph 4 lists the addresses to which the company may send or supply documents or information. Where the company has no address for the intended recipient, the company may use the recipient’s last known address. By virtue of section 310(4), this provision does not prevent a company making provision in its articles not to send notice of a general meeting to members for whom the company no longer has a valid address. Section 423(2) makes similar provision for the annual report and accounts.
1471.Part 3 of the Schedule relates to communications by e-mail or other electronic methods. Paragraph 6 allows a company to send a document or information in electronic form to a person who has agreed (either generally for all communications or specifically for a particular document or piece of information) and where such person has not revoked that agreement.
1472.Part 4 deals with publication of documents or information on a website. Paragraph 10 allows a company to pass a members’ resolution or make provision in its articles about website communication. If it has done so, members (or their nominees) are taken to have agreed to receiving information from the company via a website if they have also been asked individually for their acceptance and have either agreed or not responded within 28 days of the company’s request. Where a member has not agreed to communications in this way, the company may not ask the member again within a period of twelve months. In addition, section 1145 (right to hard copy version) gives a member the right to request a hard copy of the communication. Paragraph 11 and section 1145 make equivalent provision for debenture holders. Paragraph 13 requires companies to notify intended recipients each time material is published on a website. Paragraph 15 enables the company and a member to agree alternative methods of communication, for example other than website communication where a company has defaulted to website communications.
Section 1145: Right to hard copy version
1473.This section provides individual members or debenture holders the right to require information to be sent in paper copy form. A company is required to send a paper copy of the document or information within 21 days of receiving a member’s request at no charge to the member. Subsections (4) and (5) impose a penalty on every officer in default if the company fails to comply.
Section 1146: Requirement of authentication
1474.This section operates where provisions of the Companies Acts require information sent or supplied to a company to be authenticated. This is chiefly the case in the context of written resolutions and requests (formerly requisitions) for meetings, etc. The rule is that a signature on a document in hard copy form by the person sending it is always sufficient authentication. The company can make its own rules in respect of documents in electronic form (but there is a default where no such rules exist).Subsection (4) makes it clear that, where someone purports to authenticate a document on another's behalf, the company can require proof of the former's authority to do so.
Section 1147: Deemed delivery of documents and information
1475.This section sets out when communications from the company are deemed to have been delivered, but it can be excluded by contrary provision elsewhere (for example, in other legislation, in contracts or in the articles). Subsection (5) provides that the 48-hour period for deemed delivery is counted during normal working days only. For example, a document posted on a Friday at 3.00 pm is deemed to have been delivered the following Tuesday at 3.00 pm, unless it is a bank holiday weekend, in which case deemed delivery is the Wednesday at 3.00 pm.
Section 1148: Interpretation of company communications provisions
1476.This section sets out defined terms used in the company communications provisions and, in particular, makes clear that references in the Companies Acts to “sending” and "supplying” documents or information include all such similar expressions.
Sections 1149 to 1153: Requirements as to independent valuation
1477.These sections restate the requirements in sections 108 and 110 of the 1985 Act for the independent valuation of non-cash assets accepted by a public company. The independence requirements in sections 1151 and 1152 correspond to the independence requirements for a statutory auditor (see section 1214). They include a new specific power for the Secretary of State to define a disallowed connection for the purposes of determining whether a person is sufficiently independent to be a valuer. This is consistent with the approach taken in sections 344 and 936 of the Act.
Sections 1154 and 1155: Duty to notify registrar of certain appointments etc and failure to give such notice
1478.These sections are new provisions. The requirement to give notice of the appointment of a judicial factor (in Scotland) gives effect to a recommendation by the CLR (Final Report, paragraph 11.39). The section also requires the Charity Commissioners to notify their appointment of a receiver and manager and the regulator of community interest companies to give notice of the appointment of a manager. These officers displace directors. Section 1155 makes failure by a judicial factor to give notice an offence.
Section 1156: Meaning of "the court"
1479.This section defines the term "the court" for the purposes of the Companies Acts. The effect of this definition is that, except where an enactment or rule of law provides otherwise, cases under the Companies Acts can be heard either in the High Court or the county court in England and Wales, in the Court of Session or the sheriff court in Scotland, and, in Northern Ireland, in the High Court of Northern Ireland.
1480.The allocation of cases between the county court (or sheriff court) and the High Court (or Court of Session) will be determined partly by the courts' general powers and partly by subordinate legislation. The allocation of cases between county courts will, as now, be determined by orders made by the Lord Chancellor (see subsection (3)).
Section 1157: Power of court to grant relief in certain cases
1481.Under this section, which restates section 727 of the 1985 Act without substantive amendment, an officer of a company (such as a director) or a person employed by a company as auditor may apply to the court for relief from liability for negligence, default, breach of duty or breach of trust. A court may grant relief if it appears to the court that:
the director (or other officer or auditor) has acted honestly and reasonably; and
having regard to all the circumstances, he ought fairly to be excused.
Part 38: Companies: Interpretation
1482.This Part gives an interpretation of definitions used throughout the Act. Most are based on corresponding definitions in the 1985 Act. Those that are changed or new are described below.
Section 1158: Meaning of “UK-registered company”
1483.The expression “UK-registered company” is used as a drafting device to refer to any company registered under this Act. The expression includes companies treated as so registered (for instance, by virtue of having been registered under earlier legislation). It does not include an overseas company that is not itself registered in the UK but has registered particulars in the UK under section 1046.
Section 1168: Hard copy and electronic form and related expressions
1484.Subsections (2) to (4) of this section contain new definitions of the terms “hard copy”, “electronic form” and related expressions for all purposes of the Companies Acts. Subsection (5) requires that documents or information be sent in electronic form must be in a form that is capable of being read and retained for future reference.
Section 1171: The former Companies Acts
1485.This section defines “former Companies Acts”. The list includes the companies legislation listed in section 735 of the 1985 Act and the later enactments repealed by the Act.
Section 1172: References to requirements of this Act
1486.This section provides that requirements to be imposed under the Act (by regulations or orders to be made under a power contained in the Act) are included in references in the company law provisions of the Act to “the requirements of this Act”.
Section 1173: Minor definitions: general
1487.The definitions in subsection (1) that are new or changed are as follows.
1488.The definitions of “body corporate” and “corporation”, and of “firm”, are new in part. They clarify the position of corporations sole and of partnerships that are legal persons but are not regarded as bodies corporate (as under Scots law).
1489.The definitions of “credit institution” and “regulated market” are changed to follow the definitions in more recent EU Directives. Subsection (2) makes provision to deal with the postponement of the Directive on markets in financial instruments and the fact that it may be implemented in different member States on different dates.
1490.The definition of “working day” is new. This expression replaces a variety of expressions in the existing legislation; there is no change of substance.
Section 1174 and Schedule 8: Index of defined expressions
1491.Section 1174 introduces Schedule 8 to the Act, which provides an index setting out where the definitions of terms used in the Companies Acts are to be found.
Part 39: Companies: Minor Amendments
Section 1175 and Schedule 9: Removal of special provisions about accounts and audit of charitable companies
1492.This section and Schedule remove from company law special rules about the audit of companies that are charities. Under section 249A of the 1985 Act, certain companies are subject to audit, or to an accountant’s report, because they are charities, when they would be exempt as small companies if they were not charities.
1493.This section is part of achieving the objective of changing the treatment of small charitable companies so that, as far as their accounts scrutiny is concerned, they will be required to comply with the requirements of charity law rather than those of company law. The Charities Act 2006 introduced a power (section 77 of that Act) to enable the Office of the Third Sector to bring forward regulations, subject to the affirmative resolution procedure, applying charity law rules about scrutiny of financial records to charitable companies.
Section 1176: Power of Secretary of State to bring civil proceedings on company’s behalf
1494.This section repeals the power of the Secretary of State, under section 438 of the 1985 Act, to bring civil proceedings on behalf of a company. Subsections (2) and (3) are consequential amendments to sections 439 and 453 of the 1985 Act respectively. This repeal does not affect any proceedings begun before this section comes into force.
Section 1177: Repeal of certain provisions about company directors
1495.This section repeals various provisions of Part 10 of the 1985 Act.
1496.Section 311 of the 1985 Act prohibits a company from paying director remuneration free of income tax. The Law Commissions recommended its repeal as the tax which the company agreed to pay is itself taxed as part of the emoluments of a director, and as the company is required to disclose in its annual accounts an estimate of the tax which it has undertaken to pay.
1497.Section 323 of the 1985 Act prohibits directors (including shadow directors) from buying “put” and “call” options in listed shares or debentures in the company or another in the same group. This prohibition is extended to spouses and minor children of directors by section 327 of the 1985 Act. The Law Commissions recommended its repeal.
1498.Sections 324 to 326 and 328 to 329 of, and Parts 2 to 4 of Schedule 13 to, the 1985 Act deal with the duty of a director to notify interests in shareholdings to his company and impose an obligation on the company to record those interests in a register and to disclose them to the relevant exchanges. They are repealed.
1499.Sections 343 and 344 of the 1985 Act make special provision for banking companies and the holding companies of credit institutions, allowing them to disclose in their annual accounts abbreviated particulars of loans, quasi-loans and credit transactions with directors or their connected persons. Section 413 of the Act, which replaces the annual accounts disclosure requirements of the 1985 Act in respect of loans, quasi-loans and credit transactions, makes its own special provision in subsection (8) of that section for banking companies and the holding companies of credit institutions.
Section 1178: Repeal of requirement that certain companies publish periodical statement
1500.This section repeals section 720 of, and the related Schedule 23 to, the 1985 Act. Section 720 requires certain insurers and deposit, provident or benefit societies to publish a periodical statement in the form set out in the Schedule. The statement contains basic information about certain liabilities and assets and, in the case of a company with shares, basic information about its share capital and issued shares. This general disclosure requirement has been superseded by specialised regulatory developments in particular fields of financial services. The application of the section is now very limited as it does not apply to any UK insurance company which is regulated by the FSA under FSMA and which complies with its rules as to the publication of annual accounts and balance sheet. Nor does it apply to any insurer authorised in any other EEA State carrying on business in the UK if it complies with equivalent rules of its home State.
Section 1179: Repeal of requirement that Secretary of State prepare annual report
1501.This section repeals the requirement, under section 729 of the 1985 Act, for the Secretary of State to cause a “general annual report on matters within the Companies Acts” to be prepared and laid before both Houses of Parliament.
Section 1180: Repeal of certain provisions about company charges
1502.This section repeals the provisions in Part 4 of the 1989 Act relating to company charges. These provisions have not been brought into force.
Section 1181: Access to constitutional documents of RTE and RTM companies
1503.This section enables the Secretary of State to make an order amending certain provisions of the Commonhold and Leasehold Reform Act 2002 and the Leasehold Reform, Housing and Urban Development Act 1993 so as to make it easier to ascertain the contents of the articles and other constitutional documents of Right To Manage (“RTM”) and Right to Enfranchise (“RTE”) companies (two new types of company provided for in the 2002 Act – in the case of RTE companies, by amendment to the 1993 Act).
1504.Under the Commonhold and Leasehold Reform Act 2002 and the Leasehold Reform, Housing and Urban Development Act 1993 as amended by it, the Secretary of State may make regulations prescribing model memoranda and articles of association for RTM and RTE companies, and the provisions of the model memoranda and articles so prescribed may have effect notwithstanding contrary provision in the memoranda and articles of such companies as registered at Companies House. As the legislation stands currently, a person consulting the Companies House record of an RTM or RTE company’s memorandum or articles may not be aware of the company’s RTM or RTE status, and therefore may also be unaware that its registered memorandum and articles have to be read in the light of any relevant regulations prescribing model memoranda and articles for RTM or RTE companies. Since the prescribed memoranda and articles may invalidate provisions of the registered documents and apply in place of them, this may cause problems.
1505.The RTM and RTE legislation is likely to be adjusted to reflect the new status of the memorandum under sections 8 and 28 in particular. Reference is made to “other constitutional documents” because it is possible that under the new constitutional arrangements, the RTM and RTE legislation should make provision about the contents of constitutional documents other than articles.
Part 40: Company Directors: Foreign Disqualification Etc
1506.Part 40 addresses a gap in the present law. Persons who have been disqualified from being a director, or from holding an equivalent position, or engaging in the management of a company in another State, are currently able to form a company in the UK, to appoint themselves a director of that company and then operate that company either in the UK or in the State where they have been disqualified. The provisions in this Part give the Secretary of State a power to close the gap by making regulations to disqualify from being a director of a UK company, persons who have been disqualified in another State.
1507.Part 40 is the first Part which is outside the company law provisions of the Act. It does not, therefore, form part of the Companies Acts. This is due to the fact that the provisions in this Part are linked with those of the Company Directors’ Disqualification Act 1986. That Act is not part of the Companies Acts because it has implications beyond companies to other bodies (such as NHS foundation trusts) and extends beyond persons covered by the Companies Acts to persons such as insolvency practitioners. The fact that Part 40 is not part of the Companies Acts has the consequence that the definitions in the earlier Parts of the Act do not apply – hence the need to define the term “the court” in section 1183. Similarly, the definitions for Part 40 are not listed in Schedule 8 to the Act.
Section 1182: Persons subject to foreign restrictions
1508.Section 1182 defines what is meant by “a person being subject to foreign restrictions.” Only persons falling into this category may be disqualified by regulations made under this Part. This category is intended to include those who have been disqualified under (or otherwise fallen foul of) a foreign law equivalent to that in the Company Directors’ Disqualification Act 1986.
Section 1183: Meaning of “the court” and “UK company”
1509.Section 1183 sets out the meaning of “the court” and “UK company” for the purposes of this Part.
Section 1184: Disqualification of persons subject to foreign restrictions
1510.Section 1184 provides a power for the Secretary of State to make regulations disqualifying a person subject to foreign restrictions from being the director of a UK company, acting as a receiver of a UK company’s property, or, in any way, taking part in the promotion, formation or management of a UK company.
1511.The power is subject to affirmative resolution procedure.
Section 1185: Disqualification regulations: supplementary
1512.Section 1185 states that the regulations under section 1184 may make different provision for different types of case, and sets out some examples. If the regulations provide for application to the court (either by the Secretary of State for a disqualification order under section 1184(2)(b), or by a disqualified person seeking relief under section 1184(5)), the section requires the regulations to specify the grounds on which an application to the court may be made. It also allows the regulations to set out matters to which the court should have regard when considering an application.
Section 1186: Offence of breach of disqualification
1513.Section 1186 provides that regulations made under section 1184 may provide that a person disqualified under this Part who acts in breach of the disqualification commits an offence.
Section 1187: Personal liability for debts of company
1514.Section 1187 provides for the Secretary of State to make regulations to the effect that a person disqualified under this part who acts as a director of a UK company, or is involved in the management of a UK company is personally responsible for all debts and liabilities of the company incurred during the time that he or she is subject to foreign restrictions.
Section 1188: Statements from persons subject to foreign restrictions
1515.Section 1188 provides a power for the Secretary of State to make regulations providing that a person not disqualified under this Part but subject to foreign restrictions must send a statement to the registrar if he or she does anything that, if done by a person disqualified under this Part, would be a breach of the disqualification.
1516.The power is subject to affirmative resolution procedure.
Section 1189: Statements from persons disqualified
1517.Section 1189 provides a power for the Secretary of State to make provisions by regulation that would require a disqualified director to provide an additional statement where he or she has received approval from the court to act in a capacity that would otherwise be in breach of the disqualification.
1518.The power is subject to affirmative resolution procedure.
Section 1190: Statements: whether to be made public
1519.Section 1190 provides for regulations under sections 1188 and 1189 to state whether statements made under those regulations shall be on the public register, and the circumstances in which they may withheld from public inspection or removed from the register.
Section 1191: Offences
1520.Section 1191 provides for regulations to apply criminal sanctions for a failure to comply with any requirements on statements under sections 1188 and 1189.
Part 41: Business Names
1521.The provisions of this Part replace the Business Names Act 1985.
Chapter 1: Restricted Or Prohibited Names
Section 1192: Application of this Chapter
1522.This section partly replaces section 1 of the Business Names Act 1985. It ensures that the restrictions on the use of names in the course of business apply to all persons carrying on business in the UK, other than certain individuals or partnerships (see below). In particular, the restrictions:
apply to all companies (and not as in the Business Names Act, just to any company capable of being wound up under the Insolvency Act 1986 which trades under a name other than that under which it is registered); and
apply to any partnership whose members include a company (and not, as in the Business Names Act, only if the name under which such a partnership does business includes names for the corporate partners other than those under which they are registered).
1523.As in the Business Names Act, the restrictions do not apply to individuals if they trade either alone or in partnership under their surnames augmented only by their forenames and/or initials. Sole traders and individuals carrying on business in partnership are also excluded from the scope of the Chapter if the only addition to their name shows the business’s previous ownership
1524.The main effect of the wider coverage is that controls apply to all overseas companies carrying on business in the UK. It also removes any uncertainty as to whether the controls apply to business entities other than companies incorporated under the Companies Acts.
Section 1193 to 1196: Sensitive words or expressions
1525.Sections 1193 to 1195 replace sections 2, 3, 6 and 7 of the Business Names Act 1985. Section 1199 (see below) contains savings equivalent to those currently in section 2(2) of the Business Names Act.
1526.These sections require prior approval for the use of any name for carrying on business for which a company would require approval before it could be registered under it. (Sections 54 to 56, replacing sections 26(2) and 29 of the 1985 Act, apply corresponding restrictions to company names.) The differences between the requirements under these sections and the requirements under the Business Names Act are:
Section 1193(1)(a) requires prior approval for names likely to give the impression that the business is connected with Her Majesty’s Government in Northern Ireland;
Section 1193(1)(c) provides a power for the Secretary of State to specify in regulations the public authorities such that prior approval will be required for names likely to give the impression that the business is connected with it;
The definition of local authority in section 1193(2) is brought up to date for Scotland and includes a district council in Northern Ireland;
Section 1195(4) provides that the Secretary of State may refuse to consider an application for approval that is not compliant with the statutory requirements.
1527.Section 1196 provides that approval for the use of a name may be withdrawn in appropriate circumstances.
Section 1197: Name containing inappropriate indication of company type or legal form
1528.This section replaces sections 33, 34 and 34A of the 1985 Act. Rather than making it an offence on the face of the Act to use prohibited words, this section provides that the Secretary of State may by regulations make it an offence to carry on business under names using indicators of particular legal status, or similar words, unless entitled to do so. It complements sections 58 and 59, which control the use of statutory indicators of legal status (eg “ltd.” and “p.l.c.”) in companies’ registered names.
Section 1198: Name giving misleading indication of activities
1529.This section makes it an offence to use a business name that gives so misleading an indication of the nature of the activities of the business as to be likely to cause harm to the public. This section complements section 76 which gives the Secretary of State power to direct a company to change its registered name in these circumstances.
Section 1199: Savings for existing lawful business names
1530.This section provides exemptions for those continuing to use a name that was lawful before the Act comes into force. The exemption is both from the requirement for prior approval and from using names that include a protected indicator of company status. It also retains the existing provision for when a business is transferred: providing the name was previously lawful, the business may continue under that name for 12 months even if otherwise it would not be lawful for whoever is now carrying on the business (see subsection (3)).
Chapter 2: Disclosure Required in Case of Individual Or Partnership
1531.This Chapter re-enacts for individuals and partnerships the Business Names Act provisions relating to information which must be displayed at places of business and in correspondence. These sections ensure that a business’s suppliers and customers can discover the legal identity of the person with whom they are doing business and can serve documents upon it. Section 1203 makes special provision for large partnerships so that not all the partners’ names are required in all business documents, provided certain conditions are met.
Section 1200: Application of this Chapter
1532.This section partly replaces section 1 of the Business Names Act 1985. It provides that Chapter 2 applies to:
sole traders if they trade under any name other than their true surnames augmented only by their forenames and/or initials. (Section 1208 defines initial to include any recognised abbreviation of a name);
partnerships unless their name is the surnames of all its human partners (augmented only by their forenames and/or initials) and the registered names of its other partners.
1533.It also excludes sole traders and partnerships if the only addition to their name shows the business’s previous ownership.
1534.This section ensures that the coverage of this Chapter is the same as the Business Names Act except that, unlike that Act, it does not apply to any companies. The comparable requirements for companies are in Part 5, Chapter 6.
Section 1201: Information required to be disclosed
1535.This section replaces section 4(1)(a)(i), (ii) and (iv) of the Business Names Act 1985. It specifies the information that is to be the subject of disclosure under this Chapter (ie names and addresses for service).
Sections 1202 to 1204: Disclosure requirements
1536.Sections 1202 and 1203 replace section 4(1)(a) and (2) to (7) of the Business Names Act 1985. They are designed to ensure that customers and suppliers:
of sole traders know the true identity of the person with whom they are dealing and have an address for him/her which is effective for the service of documents relating to the business;
of partnerships with 20 or fewer partners know the identity of every partner and the address which is effective for the service of documents relating to the business;
of larger partnerships know the address which is effective for the service of documents relating to the business and either the identity of every partner or the address at which they can discover the identity of every partner.
1537.Large partnerships are not permitted to choose which partners’ names are included in documents: they must either include the names of all the partners or none (except in the text or as a signatory) (see subsection (2)(b)).
1538.Section 1202 also provides power for regulations relating to the form of a notice giving the trader’s or partners’ name(s) and address in response to any person who asks for the information in the course of business. For companies’ registered names, equivalent provision may be made in regulations under section 82.
1539.Section 1204 replaces section 4(1)(b) of the Business Names Act 1985 so far as it applies to sole traders and partnerships. It makes provision to enable customers and suppliers to discover the name(s) and the address for service of documents when visiting any business premises of the trader or partners.
Section 1205: Criminal consequences of failure to make required disclosure
1540.This section provides that certain provisions in Part 36 (offences under the Companies Acts) also apply to offences under this Part. It replaces and expands upon section 7 of the Business Names Act 1985 so far as it applies to sole traders and partnerships. It retains the existing offences of failure to comply with the requirements relating to disclosure of name and address in documents and notices.
Section 1206: Civil consequences of failure to make required disclosure
1541.This section replaces section 5 of the Business Names Act 1985 so far as it applies to sole traders and partnerships. It provides legal rights to anyone who has sustained losses as a result of failure to comply with this Chapter’s requirements by a sole trader or partnership.
Chapter 3: Supplementary
Section 1207: Application of general provisions about offences
1542.This section replaces section 7(6) of the Business Names Act 1985.
Section 1208: Interpretation
1543.This section replaces section 8 of the Business Names Act 1985. In particular, the definition of “initial” means that the restrictions on names in Chapter 1 not only would not apply to James Alexander Scotland if he were to trade as “James Alexander Scotland” or “J. A. Scotland” but also if he were to trade as “Jimmy Scotland” or “Jim A. Scotland”. However, the restrictions would apply if he were to trade as “Scotland Bakers” or “John Scotland”.
Part 42: Statutory Auditors
1544.The provisions of this Part concern the regulation of auditors. The effects of this Part are:
To replace Part 2 of the 1989 Act and equivalent Northern Ireland provisions, by restating those provisions with some modifications.
To extend the category of auditors that are subject to regulation and to make provision for the registration and regulation of auditors (whether based in the UK or not) who audit companies which are incorporated outside the EU but listed in the UK;
To provide that the Comptroller and Auditor General and the regional Auditors General are eligible to be appointed to perform statutory audits and provide a mechanism for the regulation and supervision of their functions as statutory auditor.
1545.Many of the provisions in this Part implement obligations contained in the Updated Eighth Company Law Directive on Audit (2006/43/EC) that was published on 9 June 2006. The provisions relating to Auditors General implement recommendations contained in Lord Sharman’s report, “Holding to Account, The Review of Audit and Accountability for Central Government”, published in 2001.
Chapter 1: Introductory
Sections 1209 to 1211: Introductory
1546.Part 2 of the 1989 Act regulates only the auditors of companies. Section 1210(1) defines the meaning of statutory auditor more broadly. Persons within subsection (1)(a) to (g) are ‘statutory auditors’. This list includes those persons who audit companies (as required under Part 16 of the Act) and those who audit building societies, insurers and banks. In addition, the Secretary of State has a power to add auditors of other persons to this list. Section 1211 cross-refers the eligibility for appointment as a statutory auditor to the requirements contained in Chapter 2 or Chapter 3 of this Part of the Act.
Chapter 2: Individuals and Firms
Sections 1212 and 1213: Eligibility for appointment
1547.These sections are restatements of sections 25 and 28 of the 1989 Act adapted so as to apply in relation to statutory auditors. The sections provide that for a person or firm (defined in section 1261) to be eligible for appointment as a statutory auditor, the person must be a member of a recognised supervisory body and be eligible for appointment under the rules of that body. Section 1217(2) clarifies that references to such members include references to persons who are not members but who are subject to the body’s rules. (Section 1217 and Schedule 10 address the recognition of supervisory bodies, and lay down the requirements they must meet to be recognised.)
1548.Section 1213 provides that no person may act as a statutory auditor if he is ineligible. It specifies that, on becoming ineligible, the auditor must resign his office and give notice in writing. Failure to comply with this requirement is an offence, conviction of which can result in a fine (subsections (3) and (4)). If the auditor continues to act as a statutory auditor after conviction (subsection (5)), or continues to fail to give notice that he is ineligible for appointment as a statutory auditor (subsection (6)), he commits a further offence for which a daily fine may be imposed after conviction (subsection (7)). Subsection (8) provides a defence if the person did not know or had no reason to believe that he was, or had become, ineligible.
Section 1214: Independence requirement
1549.This section restates section 27 of the 1989 Act and indicates circumstances where a person may not act as a statutory auditor on grounds of lack of independence. Under subsection (2) this includes persons who are officers or employees of the audited entity, or the partner or employee of such a person. Under subsection (3), this includes where the person is an officer or employee of a subsidiary of the audited entity. Subsection (4) allows the Secretary of State to make regulations regarding other connections between the audited entity and the statutory auditor by virtue of which a person will be regarded as lacking independence.
Section 1215: Effect of lack of independence
1550.This section sets out the consequences of the prohibition from acting as a statutory auditor on grounds of lack of independence, as defined in section 1214. They replicate the effect of ineligibility as explained for section 1213.
Section 1216: Effect of appointment of a partnership
1551.This section is a restatement of section 26 of the 1989 Act. The effect of the section is to ensure that when a partnership constituted in England and Wales, Northern Ireland, or any other country or territory in which a partnership is not a legal person, is appointed as a statutory auditor under this Part, the appointment may continue even if a partner leaves the partnership. For a partnership or other person to be considered as appropriate for the appointment to continue, they must be eligible for appointment as a statutory auditor and not be prohibited (as indicated in section 1214(1)). Without this provision, the appointment would cease every time the membership of the partnership changed.
Section 1217: Supervisory bodies
1552.This section restates section 30 of the 1989 Act and defines a supervisory body as a body established in the UK which maintains and enforces rules regarding the eligibility of persons appointed as statutory auditors and the conduct of statutory audit work. Subsection (4) introduces Schedule 10, which specifies the requirements supervisory bodies must meet in order to be recognised, and the process for doing so.
Schedule 10: Recognised supervisory bodies
Part 1: Grant and Revocation of recognition of a Supervisory Body
1553.This Schedule restates provisions in Schedule 11 to the 1989 Act. Paragraph 1 identifies the steps a body is required to take to become recognised by the Secretary of State. Paragraph 3 specifies the steps that the Secretary of State is required to take if the recognition of the body is revoked. Paragraph 5 provides that recognition (and revocation) orders are not statutory instruments. Paragraph 4 is a transitional provision that allows bodies recognised under the 1989 Act or the Companies (Northern Ireland) Order 1990 to continue to be recognised.
Part 2: Requirements for recognition of a supervisory body
1554.Paragraphs 6 and 7 require a recognised supervisory body to ensure that persons eligible for appointment as a statutory auditor hold appropriate qualifications (as defined in section 1219). They require a firm that is a statutory auditor to be controlled by qualified persons. Paragraphs 8 to 11 require the bodies to have rules and practices which ensure that auditors are fit and proper persons, that professional integrity and independence is maintained, that technical standards for audits are assured and that there are procedures for maintaining appropriate levels of competence. Paragraphs 12 to 16 specify the requirements for monitoring, enforcement, discipline and investigation of complaints.
Part 3: Arrangements in which recognised supervisory bodies are required to participate
1555.Paragraphs 21 to 27 specify the arrangements with independent bodies that recognised supervisory bodies must enter into in order to meet the requirements of this Schedule described above.
Section 1218: Exemption from liability for damages
1556.This section is a restatement of section 48 of the 1989 Act. It sets out those bodies and individuals that are exempt from liability for damages arising from the discharge or claimed discharge of supervisory functions as specified in this Part of the Act (these include the effects of rules, practices, powers and arrangements of the body). It applies to recognised supervisory bodies (see sections 1217 and Schedule 10) and their officers, employees and members of their governing bodies. The exemption does not apply if they have acted in bad faith, or if it would prevent an award of damages because the act was unlawful under the Human Rights Act.
Section 1219: Appropriate qualifications
1557.This section restates section 31 of the 1989 Act. It provides that a person holds an appropriate audit qualification if he holds a professional qualification obtained in the UK which is recognised in accordance with section 1220 and Schedule 11. Qualifications recognised under Part 2 of the 1989 Act or the Companies (Northern Ireland) Order 1990 will continue to be recognised.
1558.Persons whose qualifications from other EU Member States are recognised under the European Communities (Recognition of Professional Qualifications) (First General System) Regulations 2005 to practise as statutory auditors are also considered to hold an appropriate qualification. So too are overseas qualifications from non-EU countries if approved under section 1221. Subsection (2) restates a transitional provision from the 1989 Act for those persons who began a course of study in accountancy before 1 January 1990 and obtained a qualification between 1 January 1990 and 1 January 1996, enabling them to apply to the Secretary of State for approval of their qualification. The transitional provisions contained in section 31(2) and (3) of the 1989 Act have not been restated.
Section 1220: Qualifying bodies and recognised professional qualifications
1559.This section is a restatement of section 32 of the 1989 Act. It defines the term “qualifying body” as a body that offers a professional qualification in accountancy and introduces Schedule 11 which sets out the requirements that qualifying bodies must impose. Only a qualification recognised in accordance with these provisions can be considered a recognised professional qualification within the meaning of section 1219(1)(a).
Schedule 11: Recognised Professional Qualifications
Part 1: Grant and revocation of recognition of a professional qualification
Paragraph 1: Application for recognition of professional qualification
1560.This Schedule restates provisions in Schedule 12 to the 1989 Act. Paragraph 1 identifies the steps a body is required to take for a qualification it offers to be recognised by the Secretary of State. Paragraph 3 specifies the steps that the Secretary of State is required to take if the recognition is revoked. Paragraph 5 provides that recognition (and revocation) orders are not statutory instruments. Paragraph 4 is a transitional provision that allows qualifications recognised under the 1989 Act or the Companies (Northern Ireland) Order 1990 to continue to be recognised.
Part 2: Requirements for recognition of a professional qualification
1561.Paragraph 6 sets the minimum academic standards that a person must have attained before he can attempt to gain the professional qualification. Paragraph 7 requires that the qualification is restricted to persons who have either completed a relevant academic course or have seven years’ professional experience. Paragraph 8 requires that an examination must be passed (part of which has to be in writing) for the person to achieve the qualification. This examination must be in subjects of theoretical knowledge prescribed by the Secretary of State; or a university or equivalent level examination; or by practical demonstration of knowledge to examination or diploma level that is recognised by the Secretary of State. Paragraph 9 requires persons to carry out at least three years’ practical training.
Section 1221: Approval of overseas qualifications
1562.This section restates section 33 of the 1989 Act as regards the approval of overseas qualifications from non-EU countries. It sets out the conditions that will need to be satisfied, relating to the assurance of professional competence. The section provides for approval of all those in a specified country who are qualified to audit accounts, or only those who hold specified qualifications in that country. In the case of the latter, the Secretary of State may specify any additional requirements to be satisfied. The section allows the Secretary of State to recognise an overseas qualification only if there is comparability of treatment of UK qualifications in the country in question.
Section 1222: Eligibility of individuals retaining only 1967 Act authorisation
1563.This section restates section 34 of the 1989 Act. Prior to 1967 auditors of an unquoted company were exempt from the statutory qualification requirements placed on other company auditors. The Companies Act 1967 abolished this exemption but allowed an auditor with sufficient practical experience to apply to the Secretary of State for authorisation to practise. Past authorisations will continue to be valid by virtue of the transitional provision in section 1219(1)(b). Section 1222 provides that auditors authorised under the 1967 Act may not be treated as statutory auditors for any purpose other than to perform the statutory audit of an unquoted company (as defined in section 385(2)).
Section 1223: Matters to be notified to the Secretary of State
1564.This section is a restatement of section 37 of the 1989 Act and allows the Secretary of State to identify events that must be notified to him if they occur. It requires that recognised supervisory and qualifying bodies must provide information, either in writing or some other specified manner, that is reasonably required for the Secretary of State to carry out his functions – this might include annual reports, or notification of rule or bye-law changes. This information might relate to specific time periods or specific occurrences.
Section 1224: The Secretary of State’s power to call for information
1565.This section restates section 38 of the 1989 Act. It provides the Secretary of State with the power to require information from a recognised supervisory body, a recognised qualifying body or an individual statutory auditor. For example, as a result of a report provided under section 1223, the Secretary of State may request further information on a specific point to clarify if a recognised supervisory body is complying with the requirements in Schedule 10. The Secretary of State can specify the time period in which this information has to be provided.
Section 1225: Compliance orders
1566.This section is a restatement of section 39 of the 1989 Act. If a recognised supervisory or qualifying body fails to meet the requirements in Schedule 10 or 11, or it fails to comply with another requirement contained in this Part of the Act, then the Secretary of State may apply to the court for an order to make the body comply. The ultimate sanction for non-compliance by a body would be revocation of its status as a recognised body under Schedule 10 or 11.
Chapter 3: Auditors General
Section 1226: Auditors General: eligibility for appointment as a statutory auditor
1567.Subsection (1) defines an “Auditor General” for the purposes of this Part as the Comptroller and Auditor General, the Auditor General for Scotland, the Auditor General for Wales or the Comptroller and Auditor General for Northern Ireland. Subsections (2) and (3) explain that an Auditor General is eligible for appointment as a statutory auditor, unless his eligibility has been suspended by the Independent Supervisor under section 1234.
Section 1227: Individuals responsible for audit work on behalf of Auditors General
1568.This section provides that an Auditor General must ensure that the individuals within his charge, who are carrying out statutory audits on the Auditor General’s behalf, are, in their own right, eligible for appointment as a statutory auditor by virtue of the qualifications and requirements that are set out in Chapter 2.
Section 1228: Appointment of the Independent Supervisor
1569.Subsections (1) and (2) provide that the Secretary of State must appoint a body to be the Independent Supervisor of Auditors General in respect of the exercise of statutory audit functions. Subsection (3) provides for the appointment of the Independent Supervisor to have the effect of making it subject to the obligations of the Freedom of Information Act 2000. Subsections (4), (5) and (6) provide that a body may be appointed as Independent Supervisor of an Auditor General if it is a corporate body or unincorporated association that is willing to carry out the function, that has arrangements in place that will ensure the supervision is carried out effectively, and that will exercise such functions and requirements that may be laid down in the Secretary of State’s order appointing it. The appointed Independent Supervisor must perform its function on a UK-wide basis for all four Auditors General in accordance with section 1229(1).
Section 1229: Supervision of Auditors General by the Independent Supervisor
1570.This section sets the framework for the supervision arrangements to be carried out by the Independent Supervisor. Subsection (2) provides that the Independent Supervisor must establish arrangements with one or more third parties to carry out aspects of the supervisory function. Subsection (3) provides that the arrangements with a third party cover standards on professional integrity and independence, as well as the technical standards for statutory audit work; monitoring performance; investigating matters arising from that performance; and as necessary holding disciplinary hearings and deciding whether any disciplinary action should be taken. Subsections (6) and (7) make provisions relating to the payment of fines under the disciplinary arrangements.
Section 1230: Duties of Auditors General in relation to supervision arrangements
1571.Subsection (1) makes it a duty for each Auditor General to comply with the standards set by, as well as the monitoring arrangements and decisions of, the independent supervision arrangements. It also provides in subsection (2) for each Auditor General to pay the proportion of the costs of the independent supervisory arrangements that may be notified to the Auditor General in writing. Subsection (3) provides that the payment of such costs is to be regarded as expenditure of the National Audit Office in the case of the Comptroller and Auditor General, and as expenditure of the Northern Ireland Audit Office in the case of the Comptroller and Auditor General for Northern Ireland. In the case of the Auditor General for Scotland, under section 13 of the Public Finance and Accountability (Scotland) Act 2000 (asp 1) the expenses of the Auditor General are paid by Audit Scotland. In the case of the Auditor General for Wales, under section 93 of the Government of Wales Act 1998 the expenses of the Auditor General are met by the Assembly.
Section 1231: Reports by the Independent Supervisor
1572.This section provides that the Independent Supervisor must provide at least one report each calendar year to the Secretary of State and to the First Minister in Scotland, The First Minister and the Deputy First Minister in Northern Ireland and the Assembly First Minister in Wales. The Secretary of State must then lay the report before each House of Parliament.
Section 1232: Matters to be notified to the Independent Supervisor
1573.This section makes it a legal requirement for an Auditor General to notify the Independent Supervisor in writing of events that the Independent Supervisor may specify and is consistent with the requirement for other statutory auditors as contained in section 1223.
Section 1233: The Independent Supervisor’s power to call for information
1574.This section makes provision enabling the Independent Supervisor to require an Auditor General to provide information. It enables the Independent Supervisor to specify the period within which the information must be provided and how the information must be verified. This section is consistent with the requirement for other statutory auditors as contained in section 1224.
Section 1234: Suspension notices
1575.This section provides the Independent Supervisor with the power to suspend an Auditor General’s eligibility for appointment as a statutory auditor if, for example, he falls short of the standards laid down for performance of statutory audit work. It also sets out the provisions as to how the suspension will be effected, and the considerations pertaining to the decision to suspend. It provides for a process leading up to the issuing of a suspension notice, including the hearing of representations from the Auditor General in question.
Section 1235: Effect of suspension notices
1576.This section provides that an Auditor General must not act as a statutory auditor of a particular person if he is suspended in relation to that person. If the suspension starts during his term of office, the Auditor General must resign as a statutory auditor immediately, and tell the audited person that he has resigned. Subsection (3) makes it clear that the criminal offences in section 1213 (ineligibility for appointment as a statutory auditor) do not apply to an Auditor General who is ineligible by virtue of a suspension notice.
Section 1236: Compliance orders
1577.This section provides the power for the Independent Supervisor to take an Auditor General to court if he fails to comply with any obligation imposed by or by virtue of this Part of the Act. The court may direct the Auditor General to take such steps as it thinks fit to ensure compliance.
Section 1237: Proceedings involving the Independent Supervisor
1578.This section provides that where the Independent Supervisor is an unincorporated association it may take proceedings in the name of the body corporate under which it is constituted.
Section 1238: Grants to the Independent Supervisor
1579.This section amends section 16(2) of the C(AICE) Act 2004. The effect of the amendment is that the body that carries out the functions of the Independent Supervisor is eligible for grants from the Secretary of State under section 16 of that Act to meet the expenditure of the body and any subsidiary. It also means that the body may be exempt from liability in damages under section 18 of the Act.
Chapter 4: the Register of Auditors Etc
Section 1239: The register of auditors
1580.This section restates section 35 of the 1989 Act but extends the provision to cover other statutory auditors (as defined in section 1210) and third country auditors (as defined in section 1241). It requires the Secretary of State to make regulations that require the keeping of a register of those persons eligible to be a statutory auditor and third country auditors. Subsection (2) sets out the information that must be included on the register and includes the person’s name and address and the name of the relevant supervisory body for the person. If an individual statutory auditor works for a firm that is a statutory auditor, both must be entered separately on the register and cross-referenced. In subsection (3) additional information, namely the name and address of directors, members or partners, is required from bodies corporate (including limited liability partnerships), corporations sole and partnerships. The section allows for certain parts of the register to be kept by different persons, for example an oversight body may keep the information regarding third country auditors, whilst the recognised supervisory bodies may keep information regarding other statutory auditors. Subsection (6) confers a power to provide that information in the register, or a certified copy of it, is to be made available to the public upon request. A charge for access to this information is permitted. Subsection (7) allows the Secretary of State to disapply some or all of the requirements of subsections (2)(e) and (3) in relation to third country auditors (for example, if they are already subject to equivalent supervision in their home country).
Section 1240: Information to be made available to public
1581.This new provision gives the Secretary of State the power to make regulations placing an obligation on statutory auditors to make information regarding their ownership, governance, internal controls with respect to quality and independence of audit work, turnover and names of persons for whom the person has acted as statutory auditor, available to the public. Any such obligations are additional to those referred to in section 1239.
Chapter 5: Registered Third Country Auditors
Section 1241: Meaning of “third country auditor”, “registered third country auditor” etc
1582.This is a new provision that sets out the definition of a third country auditor and a registered third country auditor. The section provides that a third country auditor is an auditor (whether based in the UK or not) of the accounts of a company incorporated or formed in a non-EU country, whose shares are admitted for trading on a UK market such as the London Stock Exchange.
Section 1242: Duties of registered third country auditors
1583.Subsections (1) to (3) require registered third country auditors to be subject to systems of independent monitoring and discipline in the UK in accordance with Schedule 12. These provisions are similar to supervision arrangements for statutory auditors contained in section 1212(1) (membership of a Recognised Supervisory Body) and section 1217 (Supervisory Bodies) and Schedule 10. Subsection (4) empowers the Secretary of State to disapply the requirements in subsections (1) to (3). For example, he may disapply the requirements if satisfied that the third country auditor is already subject to equivalent supervision arrangements in his home country.
Schedule 12: Arrangements in which registered third country auditors are required to participate
1584.The requirements in this Schedule are new. They describe the independent monitoring and investigation arrangements which third country auditors must participate in.
Sections 1243 and 1244: Information
1585.These sections replicate for registered third country auditors the requirements in section 1223 and 1224 for the notification of information to the Secretary of State. Third country auditors may be required to provide any information that might reasonably be required for the Secretary of State to carry out his functions.
Sections 1245 and 1246: Enforcement
1586.The provisions in section 1245 enable the Secretary of State to apply to the court for an order to make a registered third country auditor comply with its obligations under the Part. The provisions in section 1246 empower the Secretary of State to make provision as to the removal of the third country auditors from the register of auditors in certain circumstances. In doing so, regard must be had to whether the third country auditor has complied with his obligations under this Part.
Section 1247: Grants to bodies concerned with arrangements under Schedule 12
1587.This section amends section 16(2) of the C(AICE) Act 2004. The effect of the amendment is that the body that carries out the monitoring and investigation functions in relation to third country auditors is eligible for grants from the Secretary of State under section 16 of that Act. It also means that the body may be exempt from liability in damages under section 18 of the Act.
Chapter 6: Supplementary and General
Sections 1248 and 1249: Power to require second company audit
1588.These sections restate section 29 of the 1989 Act empowering the Secretary of State to require a second audit of a company in circumstances where the person appointed as statutory auditor was not eligible for appointment or was not independent of the company audited. Subsection (2) permits the Secretary of State to direct either that a second audit is performed or that a review of the first audit is carried out (which will inform whether a second audit is required). Subsections (5) to (8) set out the criminal sanctions on the company should it fail to comply with that order. Section 1249 allows the audited person to recover the costs of the second audit from the first auditor, if the first auditor knew when he acted that he was not eligible or not independent.
Section 1250: Misleading, false and deceptive statements
1589.This section is a restatement of the offences in section 41 of the 1989 Act but also extends these offences to third country auditors. Subsection (1) sets out offences in respect of persons who provide information that they know to be misleading, false or deceptive. Subsection (2) makes it an offence for a person to hold himself out as a registered auditor where he is not registered as such in accordance with section 1239. Subsection (3) makes a similar provision for third country auditors. Subsection (4) makes it an offence for either a supervisory or qualifying body to hold itself out as recognised when it is not so recognised. Subsection (8) provides a defence if the person took all reasonable precautions and exercised due diligence to avoid committing the offence.
Section 1251: Fees
1590.This provision is based on section 45 of the 1989 Act and extends the powers of the Secretary of State to make regulations to prescribe periodical fees which must be paid by the Auditors General and registered third country auditors as well as recognised supervisory bodies and recognised qualifying bodies.
Section 1252 and 1253: Delegation of Secretary of State’s functions
1591.These provisions replace sections 46 and 46A of the 1989 Act and empower the Secretary of State to establish a body, or appoint an existing body, to exercise his functions relating to statutory auditors and the recognition of bodies that supervise auditors and/or provide professional qualifications. To do so, the Secretary of State must make a delegation order that is in accordance with Schedule 13. However, subsection (6) provides that some delegated functions must remain exercisable concurrently by the Secretary of State: namely the power to call for information (sections 1224 and 1244) and the power to issue directions to comply with international obligations (section 1254). Subsection (7) also provides that certain delegated functions concerning the approval of overseas qualifications (section 1221) can be exercised only with the consent of the Secretary of State. Subsection (3) provides for the delegation of the body to have the effect of making it subject to the obligations of the Freedom of Information Act 2000. The Professional Oversight Board is currently appointed under section 46 of the 1989 Act to exercise the Secretary of State’s functions.
1592.Section 1253 specifies the conditions for delegating functions to an existing body. It ensures that an existing body is not precluded from exercising any delegated function on the basis of its involvement with the monitoring, investigation or disciplinary arrangements that are set out in Schedule 10.
Schedule 13: Supplementary provisions with respect to delegation order
1593.This Schedule restates the provisions of Schedule 13 to the 1989 Act. Paragraph 2 provides that the delegated body is not to be regarded as acting on behalf of the Crown. Paragraphs 7 to 9 provide for the delegated body to exercise any legislative functions by instrument in writing and not by statutory instrument. Instruments must be made available to the public and the Secretary of State may require the body to consult prior to the making of regulations. Paragraph 10 requires the delegated body to report annually to the Secretary of State on the performance of its functions.
Section 1254: Directions to comply with international obligations
1594.This provision restates section 40 of the 1989 Act and empowers the Secretary of State to direct recognised supervisory or qualifying bodies, or any body delegated under section 1252, to comply with Community or other international obligations. If the body fails to comply with a direction, the Secretary of State can apply to the court for his direction to be enforced.
Section 1255: Offences by bodies corporate, partnerships and unincorporated associations
1595.This provision restates section 42 of the 1989 Act and deals with offences committed by bodies corporate, partnerships and other unincorporated associations. Where an offence committed by such a body is committed with the consent or connivance of, or is attributable to the neglect of, an officer (in the case of a body corporate), a partner (in the case of a partnership) or an officer or member (in the case of an unincorporated association), that officer, partner or member is also guilty of the offence.
Section 1256: Time limits for prosecution of offences
1596.This provision restates section 43 of the 1989 Act and sets a twelve-month time limit for the prosecution of offences within each of the jurisdictions. Subsections (1) to (4) identify that the date on which knowledge of sufficient evidence of the offence to justify prosecuting becomes known to either the Secretary of State or Director of Public Prosecutions (for England and Wales), the Lord Advocate (for Scotland) or Director of Public Prosecutions for Northern Ireland is taken as the date from which the twelve month time limit commences. In any event, the prosecution may not be commenced if three years have passed since the date on which the offence was committed.
Section 1257: Jurisdiction and procedure in respect of offences
1597.This provision restates section 44 of the 1989 Act and deals with the jurisdiction and procedure in respect of offences. It specifies that the jurisdiction is that in which a body corporate or unincorporated association has its place of business or, in the case of an individual, where he is located. It also provides for an unincorporated association to be treated in the same way as a body corporate.
Section 1258: Service of notices
1598.This provision restates section 49 of the 1989 Act and states how notices and other documents may be served under this Part of the Act on any person other than the Secretary of State. The three permitted methods of service are: delivery to the person, leaving the document at the person’s address, or sending it by post to the person’s address.
Section 1259: Documents in electronic form
1599.This is a new provision to allow delivery of notices, directions or other documents in electronic form. It allows the use of e-communications where existing provisions in this Part impose requirements on the giving or sending of notices, directions or other documents, provided the recipient indicates he is prepared to accept this form of delivery.
Section 1260: Meaning of “associate”
1600.This provision restates section 52 of the 1989 Act and defines the meaning of “associate”. This definition is particularly relevant for the independence requirement for statutory auditors set out in section 1214.
Section 1261: Minor definitions
1601.This provision is a restatement of section 53 of the 1989 Act with certain extra definitions. Subsection (3) empowers the Secretary of State, by regulations, to make amendments to this Part which are needed in relation to the application of the Part to a “firm” (as defined by subsection (1)) which is not a partnership or body corporate.
Section 1262: Index of defined expressions
1602.This provision contains an index to the defined terms used in the Part.
Section 1263: Power to make provision in consequence of changes affecting accountancy bodies
1603.This provision restates section 51 of the 1989 Act. The provision empowers the Secretary of State to amend by regulation legislation (including this Act) that refers to accountancy bodies in the event of a name change, merger or transfer of engagements affecting the bodies.
Section 1264 and Schedule 14: Consequential amendments
1604.Section 1264 introduces Schedule 14, which contains amendments consequential on this Part to the C(AICE) Act 2004.
Part 43: Transparency Obligations and Related Matters
Section 1265: The transparency obligations directive
1605.Section 1265 inserts a definition of the “transparency obligations directive” at the appropriate place in Part 6 of the Financial Services and Markets Act 2000(“FSMA”).
Section 1266: Transparency rules
1606.Section 1266 inserts seven new sections into Part 6 of FSMA: sections 89A, 89B, 89C, 89D, 89E, 89F and 89G. Part 6 of FSMA deals with certain aspects of the regulation of securities that are traded on regulated markets in the UK. These new sections make provision about rules that may be made by the “competent authority” (which is the Financial Services Authority (“the Authority”)) for the purposes of the Transparency Directive (2004/109/EC)“transparency rules”.
New Section 89A: Transparency rules
1607.Subsection (1) of new section 89A of FSMA enables the Authority to make transparency rules to implement the Transparency Directive in the UK. Subsection (2) enables the rules to include provision for any matter arising out of or related to the Directive provisions.
1608.The Transparency Directive itself covers issuers whose securities are traded on regulated markets and people who hold voting rights attached to shares in such issuers. The scope of the rule-making power allows the rules to address other matters arising from the Directive’s implementation, for example, to ensure that secondary legislation adopted by the Commission can be incorporated into the transparency rules, and that optional aspects of the Directive can be implemented, where the Authority considers this appropriate.
1609.It is expected that rules made under section 89A(1) will implement the Transparency Directive by-
requiring holders of votes attached to shares in issuers to make disclosure about their holdings at certain thresholds (see new section 89B);
requiring issuers to make public their annual accounts and reports, prepared in accordance with the EU International Accounts Standards Regulation (Regulation (EC) 1606/2002), and, where appropriate, half-yearly and interim management statements about their business(see new section 89C);
requiring issuers to make notification about voting rights held by themselves in respect of their own voting shares (see new section 89D);
requiring issuers to notify the Authority and the market of any proposed change to their constitution (see new section 89E).
1610.Subsection (3)(a) enables the Authority to make rules about disclosures of voteholdings to UK markets that are not regulated markets (within the meaning of section 103(1) of FSMA) (such as the AIM). Subsection (3)(b) enables the Authority to make rules about disclosure in relation to certain comparable instruments in respect of voting shares. These are instruments that give the holder a level of economic, as opposed to legal, control over votes attached to shares. An example of the type of instrument that the rules could extend to cover is a contract for difference, known as a “CFD”.
1611.Subsection (4) specifies further matters that the rules may cover. These include: how the proportion of voting rights held by an issuer is to be determined; when voting rights held by one person may be regarded as being held by another; the nature, form, timing and presentation of any notification; and the circumstances in which any of the requirements of section 89A may not apply.
New Section 89B: Provision of voteholder information
1612.New s ection 89B sets out provisions for notifications by voteholders under transparency rules. Subsection (1) specifies that notification can be required to be made to the issuer or to the public or to both. Under subsection (2), rules may provide for such information to be notified at the same time to the Authority.
1613.Subsection (5) sets out the circumstances in which voteholders may be required to notify of a change in the proportion of voting rights (i.e. when a proportion crosses above or below, or reaches, a proportion designated in the rules).
New Section 89C: Provision of information by issuers of transferable securities
1614.New s ection 89C sets out provisions for issuers of transferable securities to provide information under transparency rules. Subsection (1) clarifies that information can be required to be given to the public or the Authority or both.
1615.The rules cover annual financial reports (both financial statements and management reports) and, for certain issuers, half-yearly financial reports and interim management statements, as required by the Transparency Directive. The rules can also require issuers to disclose certain other information relating to voteholder information, information about the different classes of share they have issued and the total number of voting rights attached to each class, their own voteholdings, their capital, and information about new loan issues.
New Section 89D: Notification of voting rights held by issuer
1616.New s ection 89D enables the rules to provide for issuers to make notification of the proportion of voting rights they hold in respect of their own voting shares. Subsection (1)(a) permits rules to set the initial notification period in accordance with the requirements of the Transparency Directive at Article 30.2. Subsections (1)(b), (2) and (3) set out the circumstances under which issuers of transferable securities must notify of a change in the proportion of voting rights (i.e. when a proportion crosses above or below, or reaches, a proportion designated in the rules).
New Section 89E: Notification of proposed amendment of issuer’s constitution
1617.New section 89E enables the rules to provide that an issuer of transferable securities admitted to trading on a regulated market must notify a proposed amendment to its constitution to the Authority and the market.
New Section 89F: Transparency rules: interpretation etc
1618.New section 89F defines a number of terms used in the sections 89A to 89G.
New Section 89G: Transparency rules: other supplementary provisions
1619.New section 89G sets out further supplementary provisions relating to the transparency rules. Subsection (1) enables the Authority to make rules imposing the same obligations on a person who has applied for the admission of transferable securities to trading on a regulated market without the issuer’s consent as they impose on an issuer of transferable securities. Subsection (2) enables the Authority to make rules to allow it to make public information that voteholders or issuers are required to make public, where they fail to do so themselves. Subsection (3) will enable the Authority to make public information notified to it in accordance with transparency rules.
1620.There is some overlap between notifications required by the Panel on Takeovers and Mergers in the rules made under Part 28, and notifications required by the Transparency Directive. Subsection (4) enables transparency rules to cross-refer to rules made by the Panel under Part 28, which will enable greater alignment between the two sets of rules.
Section 1267: Competent Authority’s power to call for information
1621.Section 1267 inserts three new sections into Part 6 of FSMA: sections 89H to 89J.
1622.New section 89H permits the Authority to call for information from specified persons, set out in subsection (2), including issuers of shares and their auditors and directors, and voteholders and their auditors, directors and persons controlling or controlled by voteholders.
1623.Subsection (3) limits the Authority to requesting information and documents reasonably required in connection with the transparency rules. Subsection (4) enables the Authority to determine the timeframe for production and provision of information, and the location for the information to be provided. Subsection (5) makes it clear that the production of the material as required by this section does not affect any lien on a document.
1624.New section 89I sets outs the requirements connected with the Authority’s power to call for information. The Authority will be empowered to specify the form of the information or documents it calls for under section 89H (1), and may require its authentication or verification (s ubsection (2)). The Authority is permitted, under subsection (3), to take copies of and extracts from the documentation provided, and may also require the persons providing the information, or any “relevant person” within the meaning of subsection (4)(which includes directors, auditors, actuaries, accountants, lawyers and employees), to submit an explanation of any documentation produced.
1625.If a person fails to comply with the requirement to produce a document, the Authority is permitted under subsection (5) to require a person to state where the document is.
1626.New section 89J sets out the supplementary provisions in relation to the competent authority’s power to call for information in sections 89H and 89I.
Section 1268: Powers exercisable in case of infringement of transparency obligation
1627.Section 1268 inserts four new sections into Part 6 of FSMA: sections 89K to 89N.
1628.The four new sections set out the Authority’s powers in case of infringement of transparency obligations. Section 89K enables the Authority to make a public statement if an issuer is failing or has failed to comply with its obligations. It may only do so after it has issued a warning notice to the issuer (subsection (2)), and after any representations from the issuer, it has provided the issuer with a decision notice (subsection (3)). Subsection (4) requires the Authority to provide the issuer with notice that it has a right to refer the matter to the Tribunal.
1629.New section 89L gives the Authority the power, in certain circumstances, to suspend or prohibit trading of securities admitted to trading on a regulated market, or to request the market operator to suspend or prohibit such trading. The powers are to be used where the Authority suspects (subsections (2) and (3)) or finds (subsection 4) applicable breaches of transparency obligations. The Authority’s powers to request a market operator to prohibit trading could be used where and issuer whose home member State is the UK is listed in another EEA State.
1630.Section 89M sets out the procedures relating to the suspension and prohibition powers of the Authority set out in section 89L.
1631.New section 89N sets out the right for those who receive a decision notice or a notice under section 89M to refer matters to the Tribunal.
Section 1269: Corporate governance rules
1632.Section 1269 inserts new section 89O into FSMA which gives the Authority a power (under Part 6 of FSMA) to make rules implementing, enabling the implementation of or dealing with matters arising out of Community obligations on corporate governance of issuers on a regulated market.
1633.This rule-making power will enable the Authority to make corporate governance rules to cover issuers for whom the UK is the home member State, and whose securities are traded on a regulated market in the UK or elsewhere in the EEA.
1634.Subsection (2) sets out the type of corporate governance provision covered by this rule making power. These include:
the nature, constitution or functions of the organs of issuers;
the manner in which organs of the issuer conduct themselves;
the requirements imposed on organs of the issuer;
the relationship between the different organs of the issuer;
the relationship between the organs of the issuer and the members of the issuer (or holders of the issuer's securities).
1635.Subsection (3) provides that greater burdens must not be imposed by corporate governance rules on issuers whose securities are traded outside the UK than those imposed by corporate governance rules or listing rules on issuers with securities on UK markets.
Section 1270: Liability for false or misleading statements in certain publications
1636.Section 1270 inserts sections 90A and 90B into FSMA and establishes a regime for civil liability to third parties by issuers admitted to trading on a regulated market in respect of disclosures made public in response to provisions implementing obligations imposed by the Transparency Directive.
1637.Although no issuer has been found liable in damages under English law in respect of statements made in narrative reports or financial statements, the law relating to financial markets and to the obligations of issuers to investors on those markets has been developing, in the light of increased regulation of both domestic and European origin. The Transparency Directive has continued that process and increased the level of uncertainty as to whether any actionable duty is owed by an issuer and its directors to investors.
1638.The Transparency Directive sets out the periodic financial disclosures that must be made by issuers admitted to trading on a regulated market. Articles 4 and 5 of the Transparency Directive provide for annual and half-yearly reports, including management statements, to be made public, and requires statements made by persons responsible within the issuer for these disclosures (the directors in the case of a public company) that these give a true and fair view, and that the management report includes a fair review of certain matters. Article 6 requires the disclosure of interim management statements.
1639.The Transparency Directive also sets out the minimum requirements for a liability regime that must be adopted by the UK at Article 7, and recital (17) states “Member States should remain free to determine the extent of the liability”.
1640.These provisions give considerable flexibility to Member States in the liability regime they choose to adopt in respect of disclosures under the Directive. The Government has established an exhaustive regime in relation to ensuring the delivery and accuracy of these reports including criminal offences, administrative penalties and actions for civil damages. The provisions in this section relate only to the position in respect of the civil liability of issuers on regulated markets to investors in their securities. The liability regime does not cover issuers on exchange-regulated markets. Their position remains unchanged by implementation of the Transparency Directive.
1641.While it is intended that there be no additional liability under the Directive in respect of the disclosures to which it relates, the regime leaves undisturbed any other liability owed by directors to the issuer and to members of the company under UK and other national law, and any liability under other FSA rules. It also leaves undisturbed any liability of the issuer in respect of any loss or damage arising otherwise than as a result of acquiring securities in reliance on the relevant statement or report.
1642.The primary liability of directors and issuers for the accuracy of the required disclosures comprises criminal offences and administrative penalties under the provisions of Part 15 of this Act and Part 6 of FSMA. The provisions in Part 6 require compliance with FSA rules giving effect to the obligations in the Directive and provide for penalties in respect of failure to comply with the rules. In addition, restitution can potentially be ordered by the court, on application of the Authority or Secretary of State, under section 382 of FSMA or by the Authority directly under section 384 of FSMA.
1643.The Government’s intention in developing a civil liability regime has been to provide certainty in an uncertain area and to ensure that the potential scope of liability is reasonable, in relation both to expectations and the likely state of the law after the implementation of the Transparency Directive. In particular, the Government was anxious not to extend unnecessarily the scope of any duties which might be owed to investors or wider classes of third parties, in order to protect the interests of company members, employees and creditors. However, as the state of the law after the implementation of the Transparency Directive is not certain, the Government has taken a power, at new section 90B, that will enable the provision introduced by section 1270 to be added to or amended if a wider or narrower civil liability regime is deemed appropriate.
New section 90A: Compensation for statements in certain publications
1644.Subsection (1)(a) of new section 90A provides that the civil liability regime set out in that section applies to those reports and statements required by provisions implementing Articles 4 to 6 of the Transparency Directive. Depending on transparency rules, we would expect this to include annual and half yearly financial statements and management reports, the sign-off by directors or other responsible parties, as well as interim management statements.
1645.Subsection (1)(b) adds to the scope of the regime the information included in preliminary announcements of results made in advance of the reports and statements required by provision implementing Article 4 of the Transparency Directive, but only to the extent that it is intended that the information will appear in the final report or statement and be presented in substantially the same form as that in which it is presented in the preliminary announcement.
1646.Subsection (2) sets the scope of the civil liability regime to cover securities of all issuers for which the UK is the home Member State (whether the regulated market on which they are traded is situated in or outside the UK), as well as to cover those issuers whose securities are traded on a regulated market situated in the UK and for whom the UK is the host Member State. UK holders of securities of other issuers (i.e. those for whom the UK is neither a host nor a home State) will not be able to rely on the rights of action set out.
1647.Subsection (3) provides that issuers of such securities are liable to pay compensation to a person who has acquired those securities and has suffered loss in respect of them as a result of any untrue or misleading statement in a publication to which this section applies, or an omission of a required statement from such a statement. Subsection (4) however limits the liability of the issuer to circumstances where a “person discharging managerial responsibilities” in relation to the publication within the issuer (see subsection (9)) knows the statement to be untrue or misleading, or is reckless as to whether the statement is untrue or misleading, or, in the case of omissions, where it is known to be a dishonest concealment of a material fact.
1648.Subsection (5) provides that loss will not be regarded as having been suffered for the purposes of subsection (3) unless the person suffering it acquired the relevant securities in reliance on the information in the publication and at a time when and in circumstances where it was reasonable to rely on that publication.
1649.Subsection (6) limits the liability with regard to untrue or misleading statements, or omissions, in documents to which the section applies. It sets out that issuers are not liable for any liability other than that provided for by the section and that any person who is not the issuer is not liable, other than to the issuer.
1650.Subsection (8) clarifies that the section does not affect Part 6 of FSMA conferring liability for a civil penalty, liability for a criminal offence or the right to seek restitution.
1651.Subsection (9) sets out the persons who are to be considered as discharging managerial responsibilities for the purposes of the section. This is any director of the issuer, or where the issuer’s affairs are managed by the members, a member of the issuer. In the case where the issuer does not have directors, or members, any senior executive with responsibilities in relation to the publication is considered as discharging managerial responsibilities.
New section 90B: Power to make further provision about liability for published information
1652.Subsection (1) of new section 90B establishes a power to make further provision about liability for published information. The new section allows the Treasury by regulations to amend any primary or subordinate legislation relating to the liability of issuers and others in respect of information, including the regime set out in new section 90A of FSMA. The exercise of the proposed power could, for example, result in that regime or some other appropriate regime applying to other classes of information, such as information that is required to be disclosed by issuers to shareholders or markets under the Market Abuse Directive (MAD).
1653.Regulations made under the section would be made using the affirmative procedure (see the amendment to section 429(2) of FSMA made by paragraph 12 of Schedule 15).
Section 1271: exercise of powers where UK is host member State
1654.Section 1271 inserts a new section into Part 6 of FSMA: section 100A.
1655.New section 100A sets out the Authority’s ability to exercise powers in relation to infringements of prospectus rules and transparency rules or related provisions where issuers’ home State is not the UK. Subsection (2) clarifies that the enforcement powers extend only to cover infringements required by the relevant directive. Subsection (3) sets out the process by which the Authority must engage with the home State competent authority when it finds there has been an infringement. Subsection (4) sets out limitations on the Authority’s ability to act in those circumstances, but subsection (5) provides that, in the appropriate circumstances, it must take all appropriate measures to protect investors.
1656.Subsection (6) imposes an obligation on the Authority to inform the Commission where it takes action to protect investors.
Section 1272 and Schedule 15: Transparency obligations and related matters: minor and consequential amendments
1657.Section 1272 introduces Schedule 15, which makes minor and consequential amendments to FSMA related to the provision in sections 1265 to 1271. The Schedule also makes amendments to the C(AICE)Act 2004.
Part 1: Amendments of the Financial Services and Markets Act 2000
1658.Part 1 of Schedule 15 makes minor and consequential amendments to FSMA.
1659.Paragraph 2 amends section 73 of FSMA to extend, for the purposes of the transparency rules (which can apply to non-regulated UK markets), the factors to which the Authority must have regard when making rules under Part 6 of FSMA, so that these extend to effects on markets other than regulated markets.
1660.Paragraph 3 amends section 73A of FSMA to provide that transparency rules and corporate governance rules are “Part 6 rules” for the purposes of Part 6 of FSMA. But paragraph 3 also makes clear that these rules are distinct and separate from other Part 6 rules, such as the listing rules, disclosure rules, and prospectus rules. These different kinds of rules impose different, and sometimes overlapping, obligations on different groups of issuers.
1661.Paragraph 6 amends the penalty regime for breaches of Part 6 rules in section 91 of FSMA, so that it applies also to non-compliance with transparency rules, provisions made under the Transparency Directive, and corporate governance rules.
1662.Paragraph 8 amends section 97 of FSMA to enable the Authority to appoint a person to carry out investigations into breaches of the transparency rules or related provisions or the corporate governance rules.
1663.Paragraph 9 amends section 99 of FSMA, which relates to fees, so as to enable the Authority to levy fees under the transparency rules.
1664.Paragraphs 10 and 11 amend two definitions in Part 6 of FSMA (“transferable securities” in section 102A and “regulated market” in section 103) to refer to the up–to-date Community legislation (i.e. the Markets in Financial Instruments Directive (2004/39)). These paragraphs also add definitions for the purposes of the provisions on transparency rules.
1665.Paragraph 12 adds regulations made under new section 90B of FSMA to the list of statutory instruments subject to the affirmative procedure in section 429(2) of FSMA.
Part 2: Amendments of Companies (Audit, Investigations and Community Enterprise) Act 2004
1666.Paragraphs 14 and 15 amend section 14 and 15 of the C(AICE) Act 2004. The amendments mean that periodic accounts and reports of issuers required under corporate governance rules or transparency rules may be examined by the FRRP.
Section 1273: Corporate governance regulations
1667.Section 1273 provides the Secretary of State with a regulation-making power similar to the power given to the Authority in new section 89O of FSMA inserted by section 1269.
1668.The Secretary of State may make regulations for the purposes of implementing, enabling the implementation of or dealing with matters arising out of Community obligations on corporate governance for UK companies whose securities are traded on a regulated market in the UK or elsewhere in the EEA.
1669.Subsection (3)(a) allows for regulations to be made by reference to any code regulating corporate governance. This could include, for example, the Combined Code on Corporate Governance (issued by the Financial Reporting Council).
1670.Subsection (4) specifies that any criminal offence created by the regulations may not impose a greater penalty than an unlimited fine.
1671.Subsection (5) provides for regulations to be made by way of negative resolution. However, by virtue of section 1292(4), it will also be possible to make regulations under this power by affirmative procedure.
Part 44: Miscellaneous Provisions
Regulation of actuaries etc
1672.These provisions are the first step in implementing the central recommendation of the Morris Review of the Actuarial Profession: that the Financial Reporting Council (the “FRC”) take on a similar role in relation to the oversight of the actuarial profession to the one it currently exercises in relation to accountancy and the auditors’ profession.
1673.The Government announced in Budget 2005 its intention to legislate in due course to put the oversight regime onto a full statutory footing. It has not been possible to develop such a regime in time for inclusion in this Act. It was therefore agreed with the FRC and the Institute and Faculty of Actuaries that, pending the introduction of a full statutory regime, the FRC would begin voluntary oversight of the actuarial profession at the earliest possible opportunity. The FRC assumed this responsibility for actuarial standards and oversight of the profession in April 2006.
1674.The aim of these provisions is to provide the minimum necessary statutory underpinning for a voluntary regime. They amend the C(AICE) Act 2004 in two ways:
they extend the statutory immunity conferred on the FRC and its companion bodies so that it covers acts or omissions relating to oversight of the actuarial profession;
they allow the Secretary of State, if necessary, to make regulations to require beneficiaries of the actuarial oversight to contribute towards the funding costs of the proposed regime.
1675.This latter is a reserve power. It is proposed, as is currently the case with accountancy and the auditors’ professions, to fund this activity on a non-statutory basis by agreement with the insurance and pensions industries and the actuarial profession. The FRC published its final funding proposals in March 2006.
Section 1274: Grants to bodies concerned with actuarial standards etc
1676.This section amends section 16(2) of the C(AICE) Act 2004 so as to include in the list of matters carried on by bodies eligible for grants activities concerned with the setting of actuarial standards, compliance with those standards, oversight of the actuarial profession and related matters.
1677.A body to which a grant has been paid under section 16 is protected by section 18 of that Act from certain liabilities in connection with its section 16(2) activities.
Section 1275: Levy to pay expenses of bodies concerned with actuarial standards etc
1678.This section amends section 17 of the C(AICE) Act 2004 so as to include amongst those by whom a levy may be payable—
the administrators of a public service pension scheme, and
the trustees and managers of an occupational or personal pension scheme.
1679.The effect of the amendments is to enable the Secretary of State to make regulations specifying such persons as liable to pay a levy if he considers that the oversight activities of the FRC are relevant to them to a significant extent.
1680.Subsection (4) enables regulations under section 17 to make different provision for different cases so that, for example, they can provide for different rates of levy to be payable by different kinds of bodies or persons.
1681.Subsection (5) prevents the first regulations under section 17, and any other regulations under that section that would result in any change in the bodies or persons by whom the levy is payable, from being treated as hybrid instruments for the purposes of the standing orders of either House of Parliament. The effect is that such regulations are not subject to the special procedures in the House of Lords that apply to such instruments.
1682.Subsection (7) amends Schedule 3 to the Pensions Act 2004 to enable the Pensions Regulator to disclose restricted information to the Secretary of State to enable or assist him in the exercise of his functions under section 17 of the C(AICE) Act 2004.
Section 1276: Application of provisions to Scotland and Northern Ireland
1683.This section amends the C(AICE) Act 2004 as regards the application of certain provisions to Scotland and Northern Ireland.
1684.Subsection (2) amends section 16 of that Act so that paragraphs (a) to (t) of subsection (2) of that section, which list matters carried on by bodies eligible for grants, only apply to Scotland insofar as they relate to matters for which provision would be outside the legislative competence of the Scottish Parliament. This is necessary because, whilst section 16 and the provisions of this Act amending it extend to Scotland, some of the matters listed in paragraphs (a) to (t) are not reserved matters for the purposes of section 30 of the Scotland Act 1998 and are therefore within the legislative competence of the Scottish Parliament.
1685.Subsections (3) to (5) amend section 16(2) and (5) and section 66(2) of the C(AICE) Act 2004 so that sections 16 and 18, as well as section 17, of that Act extend to Northern Ireland.
Exercise of voting rights by institutional investors
1686.Institutional investors own and manage assets on behalf of and for the benefit of clients or members and have an obligation to manage those assets in their interests. In some cases there is a trustee-beneficiary relationship between the institution and the client, and in all cases there are contractual and regulatory requirements imposing duties of asset management on the institution. Voting is central to the exercise of ownership control. However, the ability of ultimate beneficiaries (e.g. members of a pension fund) to monitor the way in which institutional investors exercise voting rights is limited in practice.
1687.The CLR (Final Report, paragraph 6.39) concluded that disclosure of voting by institutional shareholders was a desirable objective. There has been a growing trend internationally to require disclosure. There has also been an increasing trend by UK fund managers towards voluntary disclosure.
Section 1277: Power to require information about exercise of voting rights
1688.This section confers a power on the Secretary of State and the Treasury to make regulations requiring certain categories of institutional investor to provide information about the exercise of their voting rights. The power is drawn intentionally widely to enable any mandatory disclosure regime to respond to varied corporate governance arrangements and to capture a range of institutions investing in different markets. Exercise of the power is subject to affirmative resolution procedure.
1689.Subsection (4) provides that the obligation imposed by regulations under this section is enforceable by civil proceedings brought either by the person to whom the information should have been provided or by a regulatory authority specified in the regulations (which could, for example, be the FSA).
Section 1278: Institutions to which information provisions apply
1690.This section lists the categories of institutions in relation to which the power conferred by section 1277 is exercisable. Subsection (2) enables the Treasury or Secretary of State to add to or amend the categories. Subsection (3) requires that the regulations specify by whom the duty imposed by the regulations is to be fulfilled.
Section 1279: Shares to which the information provisions apply
1691.This section confers power to specify by regulations the descriptions of shares in relation to which the information provisions apply. They will apply wherever a listed institution has an interest in such shares. Subsections (2) to (4) provide that an institution is taken to have an interest in shares in certain cases.
Section 1280: Obligations with respect to provision of information
1692.This section specifies the information that can be required. This covers the exercise or non-exercise of voting rights, instructions given by the institution and any delegation of a function related to the exercise or non-exercise of voting rights.
1693.Subsection (1) contains a power to require institutional investors to procure disclosure of voting or of any instructions given by any person acting on the institution’s behalf. Institutional investors would need to make sure that their investment contracts required such information to be passed on to them or disclosed on their behalf.
1694.Under s ubsection (4), the regulations may specify how and to whom the disclosure is to be made. This would allow the regulations to both specify the manner of disclosure and require disclosure to (for example) clients and members only, or to the public generally.
Disclosure of information under the Enterprise Act 2002
1695.Part 9 of the Enterprise Act applies to information which public authorities receive in connection with competition and consumer functions under certain Parts of the Enterprise Act 2002 and under other specified competition and consumer protection legislation. Information relating to the affairs of an individual or business must be kept confidential unless Part 9 permits its disclosure.
1696.This provision amends Part 9 so as to enable public authorities to disclose information for the purposes of civil proceedings or otherwise for the purpose of establishing, enforcing or defending legal rights.
Section 1281: Disclosure of information under the Enterprise Act 2002
1697.The new section 241A allows a public authority to disclose prescribed information to any person for the purposes of prescribed civil proceedings in the United Kingdom or elsewhere. Prescribed means prescribed by the Secretary of State by order. The new provision extends to prospective proceedings, taking legal advice about proceedings and other ways of establishing, enforcing or defending legal rights (such as alternative dispute resolution schemes).
1698.Information obtained by a public authority in connection with competition functions is excluded from the new provision.
Expenses of winding up
1699.The House of Lords decided in Buchler and another v Talbot and others, in re Leyland Daf UKHL 9 that property subject to a floating charge is not available to fund the general expenses of winding up. This provision is intended to reverse that decision.
Section 1282: Payment of expenses of winding up (England and Wales)
1700.Subsection (1) inserts a new section 176ZA in the Insolvency Act 1986 under which property subject to a floating charge may, where necessary, be used to fund the general expenses of winding up in priority to the floating charge holder and to any preferential creditors entitled to be paid out of that property. There is power to make provision by rules requiring the authorisation or approval of the floating charge holder, or any preferential creditors, or the court, in certain circumstances.
1701.Subsection (2) makes a corresponding amendment of the Insolvency (Northern Ireland) Order 1989 (S.I. 1989/2405 (N.I.19)).
Commonhold associations
1702.Commonhold associations are a new form of company limited by guarantee established under the Commonhold and Leasehold Reform Act 2002. Commonhold associations must register their memorandum and articles of association both with Companies House (on formation) and with HM Land Registry (on registration of the commonhold).
1703.At present paragraph 3(1) of Schedule 3 to the Commonhold and Leasehold Reform Act 2002 provides that an alteration of a commonhold association's memorandum or articles is of no effect if it is not registered with the Land Registry. The purpose of the provision is to ensure that the version of those documents held by the Land Registry is up to date. An unintended consequence of it, however, is that it effectively prohibits any change of an association’s memorandum or articles before the land which the commonhold association is established to manage is registered as commonhold land or after it has stopped being commonhold land.
Section 1283: Amendment of memorandum or articles of commonhold association
1704.This section amends paragraph 3(1) of Schedule 3 to the Commonhold and Leasehold Reform Act 2002 so as to limit the application of the provision to alterations made at a time when the land the association is established to manage is commonhold land.
Part 45: Northern Ireland
1705.Companies Acts since 1929 have extended to Great Britain only. But Northern Ireland companies legislation has followed changes in GB companies legislation very closely. The principal piece of current Northern Irish companies legislation, the Companies (Northern Ireland) Order 1986, is effectively a copy of the 1985 Act, with only very minor modifications to fit the Northern Irish context.
1706.Following public consultation, it was decided that the new Act should extend directly to Northern Ireland, along with certain other closely related areas of law. Company law would remain in formal terms a transferred matter, and a future Northern Ireland Assembly could for example decide to enact separate Northern Ireland companies legislation if it considered it desirable. In the meantime, companies in Northern Ireland would experience the regulatory effects of new companies legislation at the same time as their GB counterparts. The Act gives effect to these arrangements.
Section 1284: Extension of Companies Acts to Northern Ireland
1707.This section provides that the Companies Acts extend Northern Ireland. The Companies Acts are defined in section 2 of the Act: in essence, they include the company law provisions of this Act, the remaining provisions of the 1985 Act, and Part 2 of the C(AICE) Act 2004 (which relates to community interest companies). Section 1284 also repeals the principal pieces of separate Northern Ireland companies legislation The other (non-company law) provisions of this Act extend to Northern Ireland by virtue of section 1299.
Sections 1285 to 1287: Extension of certain other GB enactments to Northern Ireland
1708.These sections similarly extend to Northern Ireland other GB legislation in various areas related to company law, and repeal the separate Northern Ireland legislation in these areas. This is the case in relation to:
SEs (European Public Limited-Liability Companies);
certain other forms of business organisation where the law is partly modelled on, and closely relates to, company law; namely limited liability partnerships, limited partnerships, open-ended investment companies, and European Economic Interest Groupings; and
business names.
Part 46: General Supplementary Provisions
Sections 1288 to 1292: Regulations and orders
1709.These sections provide how regulations and orders made under the Act are to be made.
1710.Section 1288 provides that, unless the provision in the Act creating the power states otherwise, all regulations and orders are to be made by statutory instrument.
1711.Most of the powers to make regulations or orders are exercisable by the Secretary of State and are to be made by statutory instrument. The Act also confers powers on the registrar of companies to make rules, which are not required to be made by statutory instrument (section 1117(3) requires appropriate publicity). Other non-statutory instrument powers are conferred on the Takeover Panel (see Part 28) and the Financial Services Authority (see Part 43, which inserts new sections into the FSMA).
1712.Virtually all the provisions of the Act conferring power to make regulations or orders by statutory instrument specify one or other of the following three types of Parliamentary procedure:
negative resolution procedure (defined in section 1289): the statutory instrument containing the regulations or order is laid before Parliament and must be revoked if either House passes a resolution against it within 40 Parliamentary days. An instrument subject to the negative procedure is normally laid at least 21 days before it is to come into effect to ensure scrutiny of the instrument before its provisions come into force;
affirmative resolution procedure (defined in section 1290): the statutory instrument containing the regulations or order is laid before Parliament in draft and can only be made when approved by affirmative resolution in each House. This means that they are always subject to debate in each House;
approval after being made (defined in section 1291): the statutory instrument containing the regulations or order is laid before Parliament after being made. It ceases to have effect after 28 Parliamentary days unless it is approved by resolution of each House during the 28 day period. Should the regulations or order cease to have effect at the end of the 28 days, anything done under them during the period remains effective and new regulations or a new order may be made.
1713.Section 1292(1) provides that regulations or orders may make different provision for different cases or circumstances, may include supplementary, incidental and consequential provision, and may make transitional provision and savings.
1714.Subsections (2) to (4) of section 1292 enable orders or regulations to be made combining provisions in relation to which different procedural requirements apply. A power to make regulations can be exercised by making an order, and a power to make an order can be exercised by making regulations. Provisions subject to the affirmative resolution procedure, provisions subject to the negative resolution procedure and provisions subject to no Parliamentary procedure at all may be included in a single instrument, and subsections (3) and (4) clarify which procedure applies when powers are combined.
Section 1293: Meaning of “enactment”
1715.This clause explains what the term “enactment” includes, when used in the Act, to make it clear that it goes beyond the definition in the Interpretation Act 1978 (c.30). Unless the context in which it is used dictates otherwise, “enactment” includes:
an enactment contained in subordinate legislation within the meaning of the Interpretation Act 1978 (c.30);
an enactment contained in, or in an instrument made under, an Act of the Scottish Parliament;
an enactment contained in, or in an instrument made under, Northern Ireland legislation within the meaning of the Interpretation Act 1978.
Section 1294: Power to make consequential amendments etc
1716.This section confers on the Secretary of State, and on the Treasury, order-making powers to amend enactments in consequence of any provision in the Act. Such amendments and repeals are additional to those made by any other provision of the Act. Orders under this section are subject to the affirmative resolution procedure.
1717.Orders may be made to amend, repeal or revoke any enactment that is:
passed or made before the passing of the Act;
contained in the Act or in subordinate legislation made under it;
passed or made before the end of Parliamentary session 2006-7.
1718.In particular, orders may make provision corresponding to that made in the Act in relation to companies, and may extend provision to other forms of organisation. The provisions of the Act may be applied with any adaptations or other modifications that appear to be necessary or expedient.
Section 1295: Repeals
1719.This section introduces Schedule 16, the repeal Schedule. The repeals include, in addition to purely consequential repeals, repeals of the restated provisions of the 1985 Act and repeals of enactments that are no longer of practical utility.
1720.Schedule 16 repeals a number of Parts and/or sections of a number of pieces of legislation, including most of the 1985 Act, most of the Companies Act 1989 and the whole of the Business Names Act 1985.
Section 1296: Power to make transitional provision and savings
1721.This section gives the Secretary of State and the Treasury order-making powers to make transitional provision and savings in connection with the commencement of any provision made in the Act. Orders are subject to the negative resolution procedure.
Section 1297: Continuity of the law
1722.This section provides that things done under the provisions in the 1985 Act that are repealed and replaced by the Act will continue to be legally effective. Similarly, references to the repealed provisions in enactments, instruments or documents are to be construed as including references to the corresponding new provision.
1723.Articles of association, company resolutions and contracts are all likely to refer to provisions of the Companies Acts or to rely for their effect on the way in which those provisions work. Except where a change is intended, those articles, resolutions and contracts should continue to have effect, not only with old references converted into new but also with their legal effect capable of continuing despite verbal differences between the old and the new.
1724.The section applies automatically in all cases in which it is capable of applying. It is in addition to any more specific transitional provisions, which may be included in commencement orders by use of the power in section 1296.
Part 47: Final Provisions
Section 1298: Short title
1725.This section sets out the short title of the Act.
Section 1299: Extent
1726.This section provides that, except where otherwise provided for or the context requires otherwise, the Act extends to the whole of the United Kingdom (in other words, including Northern Ireland, as provided for in Part 45 and discussed in the notes to that Part).
Section 1300: Commencement
1727.Subsection (1) provides for commencement on Royal Assent of Part 43 (except for a definition not yet in force) and of sections 1274 and 1276, so that the provisions on transparency obligations and related matters, those conferring a statutory immunity from liability in damages in relation to the oversight of the actuarial profession and those relating to the extension of provisions of the C(AICE) Act 2004 to Scotland and Northern Ireland came into force on that date (8 November 2006). It also provides for Parts 46 (general supplementary provisions) and this Part to come into force on Royal Assent, except for the repeals Schedule.
1728.Subsection (2) provides for the Secretary of State or the Treasury to appoint by order the timing of commencement of the other provisions of the Act.
Commencement
1729.As explained in the note on section 1300 immediately above, certain provisions came into force on the day of Royal Assent. The other provisions will be the subject of commencement orders, and the Government has announced its intention to commence all parts of the Act by October 2008.
Parliamentary History
1730.The following table sets out the dates and Hansard references for each stage of this Act’s passage through Parliament.
| Stage | Date | Hansard reference |
---
| House of Lords |
| First Reading | 1 November 2005, HL Bill 34 | Vol. 675, Col. 127 |
| Second Reading | 11 January 2006 | Vol. 677, Cols. 180-249 |
| Grand Committee | 30 January, 1, 6, 9, 27 February, 1, 7, 14, 15, 20, 28, 30 March, 25 April 2006, reprinted HL Bill 98 | Vol. 678, Cols. GC1-GC64, GC119-GC176, GC237-GC296, GC 321-GC382, Vol. 679, Cols. GC1-GC62, GC121-GC188, GC263-GC324, GC395-GC466, GC467-GC532, Vol. 680, Cols. GC1-GC70, GC285-GC356, GC357-GC440, Vol. 681, Cols. GC61-GC100 |
| Report | 9, 10, 16 May 2006, reprinted HL Bill 108 | Vol. 681, Cols. 777-898, 912-1034, Vol. 682, Cols. 141-253 |
| Third Reading | 23 May 2006 | Vol. 682, Cols. 709-797 |
| House of Commons |
| First Reading | 24 May 2006, reprinted Bill 190 | Votes and proceedings |
| Second Reading | 6 June 2006 | Vol. 447, Cols. 122-223 |
| Committee | 15, 20, 22, 27, 29 June, 4, 6, 11, 13, 18, 20 July 2006, reprinted Bill 218 | Hansard Standing Committee D |
| Report and Third Reading | 17, 18, 19 October 2006 | Vol. 450, Cols. 743-838, 881-980, 1030-1108 |
| House of Lords |
| Consideration of Commons Amendments | 3 November 2006, HL Bill 155 | Vol. 686, Cols. 428-510 |
| House of Commons |
| Lords Reasons for insisting on certain of their amendments to which the Commons have disagreed, considered | 6 November 2006, Bill 245 | Vol. 451, Cols. 667-676 |
Royal Assent –8 November 2006 House of Lords Hansard Vol 686 Col 750
House of Commons Hansard Vol 451 Col 825
ANNEX A:: TRANSPOSITION NOTES
Part 28: Takeovers, etc. – Directive on Takeovers Bids (2004/25/EC)
The Takeovers Directive
1731.Part 28 of the Act implements Directive 2004/25 EC of the European Parliament and of the Council of 21 April 2004 on Takeover Bids (OJ L142, 30 April 2004).
1732.The Takeovers Directive lays down, for the first time, minimum EU rules concerning the regulation of takeovers of companies whose shares are traded on a regulated market. The Directive was one of the measures adopted under the EU Financial Services Action Plan and aims to strengthen the Single Market in financial services by facilitating cross-border restructuring and enhancing minority shareholder protection.
1733.The Takeovers Directive contains general principles that Member States must adhere to in regulating takeover activity and a framework relating to the functions and jurisdiction of takeover regulatory authorities. It also lays down provisions relating to the mandatory bid (a requirement whereby a party gaining control of a company must make an offer to all shareholders at an equitable price), takeover bid documentation, time allowed for acceptance of the bid, the obligations of the board of the offeree company and other matters related to the bid.
1734.Additionally, the Takeovers Directive has provisions addressing barriers to takeovers (such as action that might be taken by a company or its board before or during a bid to prevent a takeover), requiring disclosure of certain information by companies traded on a regulated market and dealing with the problems of, and for, residual minority shareholders following a successful takeover bid (so-called ‘squeeze-out’ and ‘sell-out’ provisions).
The Takeovers Directive (Interim Implementation) Regulations 2006
1735.In view of the fact that the Takeovers Directive was required to be implemented by 20 May 2006, by which date the Act had not completed Parliamentary passage and received Royal Assent, interim implementation provisions were introduced under section 2(2) of the European Communities Act 1972 (ECA 1972). These provisions are contained in The Takeovers Directive (Interim Implementation) Regulations 2006 (S.I. 2006 No.1183). A copy of those Regulations together with the accompanying Explanatory Memorandum, Regulatory Impact Assessment and Transposition Notes is available on the website of the Office of Public Sector Information ( The Regulations will be repealed and replaced on commencement of Part 28 of the Act.
Part 28 – Takeovers etc
1736.Since 1968, takeover regulation in the UK has been overseen by the Takeover Panel administering rules and principles contained in the “City Code on Takeovers and Mergers”. In order to bring UK takeover regulation within the requirements laid down in the Directive, Part 28 of the Act is designed to place it within a complete and coherent statutory framework.
1737.The detailed rules relating to takeover regulation in compliance with the Directive will be prescribed by the Panel in its Takeover Code, under a statutory rule-making obligation imposed upon the Panel by the Act (section 943(1)). The Takeover Code has already been revised with effect from 20 May 2006, on an interim basis under the 2006 Regulations, to make it wholly consistent with the requirements of the Takeovers Directive.
1738.‘Squeeze-out’ and ‘sell-out’ provisions were previously prescribed by Part 13A of the 1985 Act. Chapter 3 of Part 28 of the Act replaces those provisions in their entirety with certain amendments which ensure they are wholly consistent with the Takeovers Directive requirements.
1739.Provisions related to disclosures by companies are contained in Part 7 of the 1985 Act and amendments to that Part are made in Chapter 4 of Part 28 to give effect to the additional disclosure requirements imposed by the Takeovers Directive on companies traded on a regulated market.
1740.Responsibility for the measures, described in this transposition note, taken to implement the Takeovers Directive lies with the Secretary of State for Trade and Industry.
1741.The table below describes the substantive provisions implementing the Takeovers Directive.
Part 24: Takeovers etc: Transposition Measures
| Article | Objective | Implementation |
---
| 1 | Defines the scope of Directive in terms of transactions and types of company to which it applies (“takeover bids for the securities of companies governed by the laws of Member States, where all or some of those securities are admitted to trading on a regulated market”). | No specific implementing provision necessary. |
| 2 | Contains key definitions for the purposes of the Directive (such as, “takeover bid”, “offeree company”, and “securities”). | No specific implementing provision necessary. |
| 3.1 | Lays down general principles which Member States shall ensure are adhered to for the purpose of implementing the Directive. | Section 943(1) requires that the Panel give effect to the general principles set out at Article 3.1 of the Directive in the exercise of their statutory rule-making duty. |
| 3.2 | Provides that Member States may, in ensuring that the minimum requirements laid down by the Directive are adhered to, lay down additional conditions and provisions more stringent than those of the Directive. | No specific implementing provision necessary. |
| 4.1 | Requires Member States to designate supervisory authorities (which must act independently of parties to a bid). | This will be achieved by administrative designation of the Takeover Panel as supervisory authority for the purposes of the Directive. |
| 4.2 | Lays down jurisdictional rules in relation to takeover regulation | Section 943(1) requires that the Panel give effect to the jurisdictional provisions of the Directive in the exercise of their statutory rule-making duty. |
| 4.3 | Requires Member States to ensure that persons employed or formerly employed by takeover regulatory authorities are bound by professional secrecy (information covered by this obligation should not be disclosed other than under conditions laid down by national law). | Section 949 makes it a criminal offence to disclose information provided to the Takeover Panel other than under the circumstances and gateways laid down in section 948 and Schedule 2. |
| 4.4 | Lays down cooperation obligations in relation to EU takeover and financial markets supervisory authorities. | Section 950 requires the Takeover Panel to cooperate with EU takeover and financial services regulators. The existing cooperation duties of the Financial Services Authority under section 354 of the Financial Services and Markets Act 2000 are extended to include relevant authorities (section 964). |
| 4.5 | Requires that takeover supervisory authorities be provided with all powers necessary for carrying out their duties and provides that Member States may, provided that the general principles are respected, permit derogation from the rules of the Directive in certain circumstances and grant supervisory authorities the power to grant waivers. | In addition to the rule-making duty at section 943(1) and rule-making powers at section 943(2), the following powers are provided to the Takeover Panel: Section 945 – power to make rulings Section 946 – power to give directions Section 947 – power to require documents and information Section 952 – power to set down sanctions by rules Section 954 – power to order compensation in certain circumstances Section 955 – power to apply to the court for enforcement Section 960 – power to bring and defend proceedings. Section 944(1) authorises the Takeover Panel to provide for derogations and waivers in certain circumstances from rules made under section 943. |
| 4.6 | Makes provision for certain Member States’ powers to be unaffected by the Directive (for instance, designation of judicial or other authorities responsible for dealing with disputes, the circumstances in which parties may bring administrative or judicial proceedings, any capacity of the courts to decline to hear legal proceedings and the liability of supervisory authorities). | Section 945(2) provides that a ruling of the Takeover Panel is to have binding effect (subject to provisions in the Panel’s rules and any review or appeal). |
| Section 951 provides for matters relating to reviews of and appeals from Takeover Panel decisions to be contained in the rules made by the Panel. |
| Section 956 provides that there shall be no action for breach of statutory duty, or any voidness or unenforceability of transactions, as a result of breach of rules made by the Panel. |
| Section 961 provides for exemption of the Takeover Panel (and those involved in its functions) from liability in damages in certain circumstances related to the regulatory activities of the Panel. |
| 5 | Requires that a “mandatory bid rule” is introduced requiring a person acquiring “control” of a company to make a bid to all holders of securities at an equitable price | Section 943(1) requires that the Panel give effect to the “mandatory bid” and “equitable price” provisions in the exercise of their statutory rule-making duty. |
| Contains rules related to the calculation of the equitable price. |
| 6 | Requires that the decision to make a takeover bid is made public. Contains detailed provision related to the contents of the takeover offer document. Requires that the parties to a bid are obliged to provide supervisory authorities with information related to the bid. | Section 943(1) requires that the Panel give effect to the “bid” disclosure and documentation provisions in the exercise of its statutory rule-making duty. Section 947 provides the Takeover Panel with power to require documents and information. |
| 7 | Lays down rules related to the time allowed for acceptance of the takeover bid. | Section 943(1) requires that the Panel give effect to the offer “acceptance” period provision in the exercise of its statutory rule-making duty. |
| 8 | Requires that takeover bids are made public so as to ensure market transparency. It also provides for the disclosure of bid documentation to shareholders and employees’ representatives (or, where there are no such representatives, the employees directly). | Section 943(1) requires that the Panel give effect to the bid disclosure provisions in the exercise of their statutory rule-making duty. |
| 9 | Imposes obligations on the board of the offeree company, including the obligation not to take action to frustrate the bid without the approval of shareholders at the time of the bid and to draw up and make public a statement containing their views on the effects of implementation of the bid. | Section 943(1) requires that the Panel give effect to the provisions relating to the obligations of the board of the offeree company in the exercise of their statutory rule-making duty. |
| 10 | Requires that companies shall publish detailed information on their share and control structures, etc. in their annual report and present an explanatory report on such matters to the annual general meeting of shareholders. | Section 992 (amending Part 7 of the 1985 Act) requires that the relevant information, including necessary explanatory material, is set out in the annual report of companies. |
| 11 | “Breakthrough” – This provision overrides, in certain circumstances connected with a takeover, provisions in the articles of companies and contractual arrangements related to restrictions on transfer and voting rights of shares, etc. It does not apply to special shares held by Member States or to cooperative. This provision may be made optional by Member States for companies under the provisions of article 12. | The right to make these provisions optional for companies is exercised in the implementing provisions. Sections 966 and 967 define the types of companies, circumstances and mechanisms by which a company may opt-in to “breakthrough”. Section 968 lays down the effect on contractual restrictions overridden by “breakthrough”. |
| 12.1 | Provides that Member States may make optional the provisions of articles 9(2) and (3) and/or Article 11. | Exercise of this option has been taken only in relation to the provisions of Article 11 (the relevant implementing provisions of which are described above). |
| 12.2 (and 12.4) | Requires, where optional arrangements are in place, that companies have the right to voluntarily opt-in to the provisions of the relevant articles. Such a decision must be communicated to the supervisory authorities and be disclosed. | Section 970 requires that any opting-in decision be communicated to the Takeover Panel without delay. |
| The opting-in resolution passed by the company must be filed with the Registrar of Companies under section 30. |
| 12.3 (and 12.5) | Permits Member States to provide that the effects of Articles 9(2) and (3) and/or Article 11 only apply on a “reciprocal” basis, i.e. where the takeover bid is made by a company also subject to the effects of the relevant articles. Such restrictions on the application of Articles 9(2) and (3) and Article 11 shall be subject to the authorisation of the general meeting of shareholders of the offeree company. | The Member State option to provide for “reciprocity” has not been exercised. |
| 13 | Requires that rules relating to the lapsing or revision of bids, competing bids, disclosure of results of bids and irrevocability of bids be put in place. | Section 943(1) requires that the Panel give effect to the requirement that such rules be put in place in the exercise of its statutory rule-making duty. |
| 14 | Provides that the Directive shall be without prejudice to various provisions relating to information and consultation of employees and their representatives. | No specific implementing provision necessary. |
| 15 | Requires Member States to put in place rules enabling a bidder to compulsorily purchase the shares of minority shareholders following a successful takeover bid (“squeeze-out” rights). The circumstances in which such a right must apply (including time periods and relevant thresholds) and relating to the price that must be paid are set out. | “Squeeze-out” rights were previously contained in the 1985 Act (Part 13A (sections 428-430F)). These have been replaced by Chapter 3 of Part 28 of the Act (necessary amendments to ensure these provisions are consistent with Article 15 have been made). |
| 16 | Requires Member States to put in place rules enabling minority shareholders to require a bidder to compulsorily purchase their shares following a successful takeover bid (“sell-out” rights). The circumstances in which such a rule must apply (including time periods and relevant thresholds) and relating to the price that must be paid are set out. | Sell-out” rights were previously contained in the 1985 Act (Part 13A). These have been replaced by Chapter 3 of Part 28 of the Act (necessary amendments to ensure these provisions are consistent with article 16 have been made). |
| 17 | Requires that effective, proportionate and dissuasive sanctions be put in place. | Sections 952 and 954 provide that the rules made by the Takeover Panel may confer power on the Panel to impose sanctions on those who transgress its rules or order compensation in certain circumstances. |
| Section 949 makes it an offence to contravene the provisions of section 948 (relating to the restrictions on disclosure of information provided to the Takeover Panel). |
| Section 953 provides an offence where takeover bid documentation does not comply with Panel rules giving effect to Articles 6.3 and 9.5 of the Directive. |
| Misconduct in relation to takeover activity also needs to be viewed in the wider context of the overall regulatory framework and the protections available to shareholders and others. |
| A robust market regulatory regime and company law framework is in place in the UK to investigate and pursue misconduct in relation to takeover activity (for instance, sanctions with stringent sanctions are already in place to deter fraudulent misrepresentation or market abuse). |
| 18 | Lays down a Committee procedure whereby the Commission may adopt rules related to the application of Article 6.3 (contents of takeover bid documentation). | No implementing provision necessary (no such rules have been adopted). |
| 19 | Requires the EU Commission to establish a Contact Committee to facilitate the harmonised application of the Directive and advise the Commission, if necessary, on any additions or amendments to the Directive. | No implementing provision necessary. |
| 20 | Provides for the review of the Directive by the EU Commission five years after its entry into force. | No implementing provision necessary. |
| Requires that Member States provide the Commission annually with certain information related to takeover bids. | Such information will be provided to the EU Commission as an administrative process. |
| 21 | Requires that the relevant provisions of the Directive be transposed no later than 20 May 2006. Details of transposition measures shall be communicated to the Commission. | No specific implementing provision necessary (NB paragraph 5 above regarding the Takeovers Directive (Interim Implementation) Regulations 2006 which came into force on 20th May 2006). Details of the transposition measures will be communicated to the EU Commission by administrative process. |
| 22 | Provides that the Directive enters into force on 20 May 2004 | No implementing provision necessary. |
| 23 | Addresses the Directive to the Member States. | No implementing provision necessary. |
Part 35: The Registrar of Companies – Directive 2003/58/EC (which amends Council Directive 68/151/EEC as regards disclosure requirements in respect of certain types of companies)
1742.Various provisions within Part 35 of the Act serve to implement Directive 2003/58/EEC, which amends section I of the First Company Law Directive (68/151/EEC), primarily to enable companies to register certain documents electronically and searchers to access them electronically.
1743.Section I of the First Company Law Directive requires that basic company documents be disclosed via filing with a company registry, and by publication in the national gazette either of the full or partial text of the document or by reference to the document deposited in the company registry. It also requires that those documents be available for inspection. In addition, the First Company Law Directive specifies minimum information that companies must include on their letters and order forms. The First Company Law Directive assumes the use of paper documents. The amended directive reflects the use of information technology and electronic communications.
1744.In practice, the 1985 Act already allows the Registrar to accept electronic filing of all documents covered by the First Company Law Directive, although specific directions as to the form and manner of filing any particular document electronically have to be given by the Registrar. At present, the Registrar has mechanisms for the electronic filing of many of those documents.
1745.The 1985 Act also already allows the Registrar to keep documents in electronic form, and to provide for inspection by electronic means. Legislation is however necessary to impose formal obligations on the Registrar in relation to electronic filing, so as to transpose the amending Directive properly.
1746.Responsibility for the transposition of the amending Directive lies with the Secretary of State for Trade and Industry. The table below describes the substantive provisions in the Act which implement it.
Part 35: The Registrar of Companies
| Article | Objective | Implementation |
---
| 2. | (Which amends Article 2.1(f) of the First Company Law Directive) requires certain accounting documents to be filed. | Part 35 of the Act contains the relevant filing requirements (which are restated from the current legislation). |
| 3. | (So far as it amends Article 3.2.) Company Registries must allow companies to file electronically all basic documents (those specified in Article 2 of the First Company Law Directive, and those to which Article 3 of the First Company Law Directive is applied by other legislation). | Section 1078 lists the documents which are now subject to the Directive disclosure requirements under Article 2 of the First Company Law Directive as amended. |
| Section 1068(5) provides that all such documents may be delivered to the Registrar in electronic form. |
| 3. | ( So far as it also amends Article 3.2.) Company Registries must allow requests for inspection of such documents to be made electronically. | Section 1089(2) provides that applications in respect of such documents may be submitted electronically. |
| 3. | ( So far as it also amends Article 3.2.) Company Registries must offer electronic copies of such documents to those inspecting the register (subject to a permitted derogation in respect of documents filed before 1 January 2007). | Section 1090(2) provides that copies of such documents must be provided in electronic form if the applicant so chooses (subject to section 1090(3) which takes advantage of the permitted derogation). |
| 3. | ( So far as it also amends Article 3.2.) Company Registries must keep all such documents (whether submitted electronically or in hard copy) in electronic form. | Section 1080(3) provides that information from such documents must be kept in electronic form. |
| 3. | ( So far as it amends Article 3.3) In the case of electronic copies, Company Registries need only provide certified copies if they are asked to do so. Member States need to take measures to ensure the authenticity of electronic certified copies. | Section 1091 contains provision about certifying copies and allows the Secretary of State to make regulations about how electronic copies are certified. |
| 3. | ( So far as it amends Article 3.4.) The option is provided to Members States of using an alternative to publication in the National Gazette as a means of publicising information received. | Section 1077 specifies that notices must be published either in the Gazette, or in accordance with section 1116. The latter section enables the Secretary of State to make regulations specifying alternative means of publication. |
| 4. | (Which inserts a new Article 3a.) This provides that, while documents must be submitted in a language permitted by the language rules of the member state in question, voluntary translations in other Community languages must also be accepted. | Sections 1106 and 1107 provide that companies may deliver certified translations of documents. The languages and types of document in respect of which this facility is available will be specified in regulations made by the Secretary of State, but subsection (3) of section 1106 provides that these regulations must as a minimum cover the documents subject to the directive disclosure requirements and the official languages of the EU. |
| 5. | (Which replaces the previous Article 4). This provides that certain information (already currently required on hard copy letters and order forms) must be stated in documents in any form and displayed on websites. | This will be implemented by regulations under section 82. |
| 6. | This provides that that there must be appropriate penalties for breach of new Articles 2(1)(f) and 4 of the First Company Law Directive. | See entries for new Articles 2(1)(f) and 4 above. |
Part 43: Transparency obligations and related matters – Transparency Directive (2004/109/EC)
1747.Sections 1265, 1266, 1267, 1268, 1270, 1271, and 1272 in Part 43 (Transparency obligations and related matters) of the Act implement Directive 2004/109/EC on the harmonisation of transparency requirements in relation to information about issuers whose securities are admitted to trading on a regulated market and amending Directive 2001/34/EC. Section 1266 inserts seven new sections into Part 6 of the Financial Services and Markets Act 2000; sections 89A, 89B, 89C, 89D, 89E, 89F and 89G. The new sections give power to the competent authority (at present the Financial Services Authority (“FSA”)) to make rules for the purposes of the Transparency Directive (2004/109/EC) (“Transparency Directive”) and connected regulatory purposes. Sections 1267 and 1268 insert three and four sections, respectively, into Part 6 of FSMA (89H to 89N) setting out the regulatory powers of the FSA in connection with the Directive. Section 1270 inserts new sections 90A and 90B into FSMA, which set out the issuers’ liability in damages for disclosures required under the Transparency Directive, and section 1271 inserts a new section 100A into FSMA setting out provisions in relation to the exercise of the FSA’s powers where the UK is a host member state.
1748.The Transparency Directive imposes minimum harmonisation requirements on the information to be provided to the public about issuers whose securities are traded on a regulated market and the control of votes attached to shares in those issuers. It permits Home Member States to impose more stringent requirements on entities that they regulate but Host Member States, i.e. those states in which the issuers securities are traded on a regulated market but whose competent authority are not responsible for primary oversight of that issuer, are not permitted to impose any requirements more stringent than those contained in the Transparency Directive.
1749.There are three main categories of obligation that are imposed under the Transparency Directive and that the FSA’s transparency rules will implement in respect of UK markets and issuers:
a)
requirements for issuers to make public, at regular intervals, information about their financial position and the progress and management of the business of the issuer;
b)
requirements for holders of votes attached to shares of issuers to notify the issuers when the number of votes they control reaches specified proportions of the total votes available; and
c)
requirements for issuers to treat the holders of the same securities equally.
1750.The detailed and technical provisions about the required notifications, disclosures and treatment of security-holders will be prescribed in rules made by the FSA under the new rule-making power at section 89A of the Financial Services and Markets Act 2000. The FSA is required by that Act to carry out consultation and a cost benefit analysis when making any rules under this power.
1751.Having the power to make these rules will promote the harmonisation of practice with other EU jurisdictions, and help enhance investor confidence through increased transparency of the financial markets.
1752.Responsibility for the transposition of the Transparency Directive lies both with HM Treasury and with the FSA. The measures in the Companies Act that implement the Transparency Directive are the responsibility of HM Treasury.
1753.The table below describes the substantive provisions in the Act implementing the Transparency Directive.
Part 43: Transparency obligations and related matters
| Article | Objective | Implementation |
---
| 1. | Sets out scope of the Directive and two derogations from the requirements of the Directive. The Member States may apply the derogations in respect of securities issued by the government, local government or a state’s national central bank. | Part 43 of the Act inserts new provisions into the Financial Services and Markets Act 2000 (“FSMA”) to give the Financial Services Authority power to make Transparency Rules. |
| Most provisions in the Transparency Directive will be implemented by the FSA’s Transparency Rules. Other provisions in the Act or in FSMA implement the other requirements. |
| If the derogations are to be implemented, the FSA’s Transparency Rules will do this. |
| 2. | Provides various definitions used in the Directive. | These will be applied in Transparency Rules, or apply in relation to the implementation of the Article to which they relate. |
| 3. | Limits the circumstances in which Member States may impose more stringent requirements than those contained in the Directive on issuers of securities and holders of interests in those issuers’ shares. | Transparency Rules and new FSMA section 100A(2) introduced by section 1271 of the Act. |
| 4. | Requires issuers of securities which are traded on regulated markets to make public its annual financial report consisting of its audited financial statements and the management report. | Transparency Rules: see in particular new sections 89A and 89C of FSMA, inserted by section 1266 of the Act. |
| 5. | Requires issuers of shares or debt securities which are traded on a regulated market to make public a half-yearly financial report. | Transparency Rules: see in particular new sections 89A and 89C of FSMA, inserted by section 1266 of the Act. |
| 6. | Requires issuers whose shares are traded on a regulated market to make public an interim quarterly statement. | Transparency Rules: see in particular new sections 89A and 89C of FSMA, inserted by section 1266 of the Act. |
| 7. | Requires Member States to ensure that responsibility for the information to be drawn up and made public in accordance with Articles 4, 5, 6 and 16 lies at least with the issuer or its administrative, management or supervisory bodies and to ensure that their laws, regulations and administrative provisions on liability apply to the issuers, the bodies referred to in this article or the persons responsible within the issuers. | Provisions relating to liability inserted into FSMA as new sections 90A and 90B by section 1270 of the Act. |
| 8. | Provides various exemptions from the requirements of articles 4, 5 and 6 including to optional exemptions. | Transparency Rules. |
| 9. | Provides that where a shareholder with a significant level of holding acquires or disposes of shares of an issuer whose shares are admitted to trading on a regulated market and to which voting rights are attached, such shareholder notifies the issuer of the proportion of voting rights in the issuer held by the shareholder as a result of the acquisition or disposal where that proportion reaches, exceeds or falls below the thresholds of 5%, 10%, 15%, 20%, 25%, 30%, 50% and 75%. | Transparency Rules: see in particular new sections 89A and 89B of FSMA, inserted by section 1266 of the Act. |
| 10. | The notification requirements in Article 9 shall also apply to a natural person or legal entity to the extent it is entitled to acquire, to dispose of, or to exercise voting rights in any of the cases set out in the Article or a combination of them. | Transparency Rules. |
| (Voting rights acquired through agreement or interest). |
| 11. | Exempts shares provided to or by the members of the ESCB in certain circumstances from the notification requirements imposed by Articles 9 and 10. | Transparency Rules. |
| 12. | Sets out the information that must be included in the notification under Articles 9 and 10 and includes provision on the timing of the notification and when aggregation of holdings required. Paragraph (6) requires the issuer to make public all information contained within a notification within 3 days. | Transparency Rules. |
| 13. | Requires the holders of financial instruments, which are to be specified by the Commission, to notify the issuer of their control of votes in accordance with the requirements in Article 9. | Transparency Rules. |
| 14. | Requires an issuer of shares admitted to trading on a regulated market to make public the proportion of its own shares that it holds when those proportions reach, exceed or fall below the thresholds of 5% or 10%. | Transparency Rules: see in particular new sections 89A and 89C of FSMA, inserted by section 1266 of the Act. |
| 15. | Requires the Member State to ensure that an issuer of shares traded on a regulated market, makes public the total number of voting rights and capital at the end of each month during which the number changes. | Transparency Rules. |
| 16. | Requires issuers of securities to make public information about any changes in the rights attached to their securities and any new loan issues and any guarantee or security in respect of such loans. | Transparency Rules: see in particular new sections 89A and 89C of FSMA, inserted by section 1266 of the Act. |
| 17. | Requires issuers of shares admitted to trading on a regulated market to treat their shareholders, who are in the same position, equally. It provides for information to be distributed in particular ways and for shareholders to be able to exercise their rights in specified ways. | Transparency Rules. |
| 18. | Makes similar provision as that contained in Article 17 but in respect of issuers whose debt securities are admitted to trading on a regulated market. | Transparency Rules. |
| 19. | Requires issuers to file information that they are required to make public under the Directive, with the FSA and permits the FSA to publish that information itself. It also requires issuers to inform the FSA and the regulated market to which its securities are admitted of any proposed change to its instrument of incorporation. | Transparency Rules. |
| 20. | Sets out the rules for determining which language the issuer must use to disclose regulated information in various circumstances. | Transparency Rules. |
| 21. | Requires issuers to disclose regulated information in a manner ensuring fast access to such information on a non-discriminatory basis. | Transparency Rules. |
| Also requires each Member State to have an officially appointed mechanism for the central storage of regulated information. |
| 22. | Requires the competent authorities of the Member States (for the UK it is the FSA) to draw up guidelines to create an electronic network at national level to share information between the various competent authorities, operators of regulated markets and national company registers. Such guidelines must aim to further facilitate public access to be disclosed under this Directive, Directive 2003/6/EC (the Market Abuse Directive) and Directive 2003/71/EC (Prospectus Directive). | The FSA will draw up guidelines in accordance with the obligations under this Article. |
| 23. | Enables the FSA to exempt issuers based in third countries from certain disclosure requirements if there are equivalent provisions in the third country. Requires the FSA to ensure that where a third country issuer is regulated in the UK for EU purposes, any information which may be important to the public in the Community is disclosed in accordance with Articles 20 and 21. | Transparency Rules. |
| 24. | Requires each Member State to designate a central competent authority responsible for ensuring that the Directive is applied and to give that competent authority specified powers which are necessary for the performance of its functions. Permits each Member State to designate a competent authority for examining that information is drawn up in accordance with the relevant reporting framework. | The central competent authority in the UK will be the FSA, by virtue of the amendments being inserted into Part 6 of FSMA. The FSA already has various powers under FSMA. Other powers for the FSA to perform its functions are contained in new FSMA sections 89H to 89N inserted by sections 1267 and 1268 of the Act. The Act provides power to designate a competent authority for reporting framework purposes by amending the Companies (Audit, Investigations and Community Enterprise) Act 2004. See Schedule 15 (Part 2) of the Act. |
| 25. | Imposes a requirement for professional secrecy on those who work for the competent authority and requires cooperation between the competent authorities of the various Member States. | FSMA already contains provisions relating to professional secrecy for those who work for the FSA and the Companies (Audit, Investigations and Community Enterprise) Act 2004 also contains provisions in relation to authorities appointed under that Act. |
| 26. | Provides for host Member States to take action in relation to infringements where an issuer or security holder continues to infringe the requirements of the Directive. | New section 100A of FSMA introduced by section 1271 of the Act. |
| 27. | Sets out the committee procedure for the Commission to make implementing measures required by the Directive. | No implementing provision required. |
| 28. | Requires, without prejudice to the right of Member States to impose criminal penalties, Member States to ensure, in conformity with their national law that at least the appropriate administrative measure may be taken or civil and/or administrative penalties imposed in respect of the persons responsible. | Schedule 15 (Part 1) of the Act amends section 91 of FSMA to enable the FSA to impose financial penalties for breach of the Transparency Rules. |
| 29. | Requires a right of appeal to the courts to be in place. | No further implementation is required. FSMA already makes provision for appeals of FSA decisions to the Financial Services and Markets Tribunal and to the Court of Appeal. |
| 30 – 35 | These articles contain transitional and final provisions, including the date by which the Directive must be transposed – 20 January 2007. | No specific implementation is required for most of these provisions. New sections 89B(4) and 89D(1) introduced by section 1266 of the Act make provision for transitional arrangements. |
ANNEX B:: GLOSSARY
1985 Act The Companies Act 1985
AGM Annual General Meeting
BCLCL Butterworths Company Law Cases
C(AICE) Act 2004 Companies (Audit, Investigations and Community Enterprise) Act 2004
CFD Contract for Difference
c.i.c.Community Interest Company
CLR Company Law Review
DTI Department of Trade and Industry
EC European Community
ECU European Currency Unit
EEA European Economic Area
EEC European Economic Community
ESCB European System of Central Banks
EU European Union
FRC Financial Reporting Council
FRRP Financial Reporting Review Panel
FSA Financial Services Authority
FSMA Financial Services and Markets Act 2000
GB Great Britain
IAS International Accounting Standards
NI Northern Ireland
PPERA Political Parties, Elections and Referendums Act 2000
PSM Professional Securities Market
QC Queen’s Counsel
RTE Right to Enfranchise
RTM Right to Manage
SE Societas Europaea (European Public Limited-Liability Company)
SI Statutory Instrument
UK United Kingdom
UKHL UK House of Lords
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189473 | https://www.youtube.com/watch?v=_B3XdXYutpk | Intro to Algebra: Balance Method
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Solving an addition problem using the balance method.
A simple problem demonstrates the method.
I have included sync'd closed captions. Please note that since I have a lot of literacy graphics sync'd to my speech, I have not entirely transcribed the video. Only the parts that don't have text support on the regular video. I hope that makes sense.
Please let me know if this helped you. Thanks so much. Ms Steele
Transcript:
It's a classic puzzle idea! Here's an equation we need to solve... It's a 'curly' kind of X Remember: 'equation' is just the math word for... Algebra uses a letter... The unknown amount is called... Any number value, like you see here is called a... So, X + 10 = 12 Of course, you can just count on from 10 to 12 But we're going to do something a bit different We want it to say... In textbooks they usually tell you that's one side of the = as long as you do the same thing to the other side! Here, I want to get rid of this 10 to get X on its own I'm going to put -10 after it When I subtract 10 on THIS side I have to subtract 10 on the other side of the equals sign too I make this SAME change to BOTH SIDES of the equation on this line of math so it's BALANCED. On the next line you can see we get It's a straightforward approach... Whatever you do to one side you MUST do to the other See you soon |
189474 | https://brainly.com/question/37433526 | [FREE] When you roll 2 dice, there are 36 possible combinations. What is the probability of each possible outcome? - brainly.com
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When you roll 2 dice, there are 36 possible combinations. What is the probability of each possible outcome?
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When you roll two dice, there are 36 possible outcomes because each die has 6 sides, and there are 6 6 = 36 total combinations.
To find the probability of each possible outcome, you can create a table showing all the possible combinations and their probabilities. Here are the outcomes and their probabilities:
Rolling a sum of 2: There's only one way to get a sum of 2 (1 on the first die and 1 on the second die). So, the probability is 1/36.
Rolling a sum of 3: There are two ways to get a sum of 3 (1+2 and 2+1). So, the probability is 2/36, which simplifies to 1/18.
Rolling a sum of 4: There are three ways to get a sum of 4 (1+3, 2+2, and 3+1). So, the probability is 3/36, which simplifies to 1/12.
Rolling a sum of 5: There are four ways to get a sum of 5 (1+4, 2+3, 3+2, and 4+1). So, the probability is 4/36, which simplifies to 1/9.
Rolling a sum of 6: There are five ways to get a sum of 6 (1+5, 2+4, 3+3, 4+2, and 5+1). So, the probability is 5/36.
Rolling a sum of 7: There are six ways to get a sum of 7 (1+6, 2+5, 3+4, 4+3, 5+2, and 6+1). So, the probability is 6/36, which simplifies to 1/6.
Rolling a sum of 8: There are five ways to get a sum of 8 (2+6, 3+5, 4+4, 5+3, and 6+2). So, the probability is 5/36.
Rolling a sum of 9: There are four ways to get a sum of 9 (3+6, 4+5, 5+4, and 6+3). So, the probability is 4/36, which simplifies to 1/9.
Rolling a sum of 10: There are three ways to get a sum of 10 (4+6, 5+5, and 6+4). So, the probability is 3/36, which simplifies to 1/12.
Rolling a sum of 11: There are two ways to get a sum of 11 (5+6 and 6+5). So, the probability is 2/36, which simplifies to 1/18.
Rolling a sum of 12: There's only one way to get a sum of 12 (6+6). So, the probability is 1/36.
These are the probabilities of each possible outcome when rolling two dice.
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When rolling two dice, there are 36 possible outcomes. The probabilities for each possible sum range from 36 1 for sums of 2 and 12 to 6 1 for a sum of 7. Each sum's probability varies based on the number of combinations that can produce that sum.
Explanation
When you roll two dice, there are 36 possible outcomes because each die has 6 sides, leading to 6×6=36 combinations. To find the probability of each possible sum outcome, we need to know how many combinations can lead to each sum from 2 to 12. Here’s how it breaks down:
Sum of 2: There’s only one way to get a sum of 2 (1+1). Probability = 36 1.
Sum of 3: Two combinations exist (1+2 and 2+1). Probability = 36 2=18 1.
Sum of 4: Three combinations (1+3, 2+2, 3+1). Probability = 36 3=12 1.
Sum of 5: Four combinations (1+4, 2+3, 3+2, 4+1). Probability = 36 4=9 1.
Sum of 6: Five combinations (1+5, 2+4, 3+3, 4+2, 5+1). Probability = 36 5.
Sum of 7: Six combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Probability = 36 6=6 1.
Sum of 8: Five combinations (2+6, 3+5, 4+4, 5+3, 6+2). Probability = 36 5.
Sum of 9: Four combinations (3+6, 4+5, 5+4, 6+3). Probability = 36 4=9 1.
Sum of 10: Three combinations (4+6, 5+5, 6+4). Probability = 36 3=12 1.
Sum of 11: Two combinations (5+6, 6+5). Probability = 36 2=18 1.
Sum of 12: One combination (6+6). Probability = 36 1.
The probabilities vary because different sums can be achieved in different ways. For instance, a sum of 7 has the highest probability because it can be made from many combinations, while sums like 2 and 12 have the lowest probabilities because there is only one way to achieve them.
Examples & Evidence
For example, there are 6 different combinations that give a sum of 7: (1+6), (2+5), (3+4), (4+3), (5+2), (6+1), while there is only one combination each for sums of 2 and 12. This demonstrates how the probabilities differ depending on the achievable combinations.
The calculations are based on the principles of probability, where the total number of outcomes is the basis for determining the likelihood of each specific outcome, verified by counting all potential combinations.
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Movies Plus charges its customers a $10 monthly service fee plus $2 for each movie the customer rents. Movies For Less charges $3 for each movie but does not have a monthly service fee. The costs of renting movies during the same month is represented in the table below. \begin{table} \captionsetup{labelformat=empty} \caption{Movie Rentals} | Number of Movies | Movies Plus (Cost in $) | Movies For Less (Cost in $) | :--- | 0 | 10 | 0 | | 2 | 14 | 6 | | 4 | 18 | 12 | | 10 | 30 | | | 15 | | | \end{table} Which statement is true if 15 movies are rented from each company? A. Movies For Less costs $5 more. B. Movies Plus costs $5 more.
Arrange the following set of decimal numbers in order from smallest to largest: 0.22, 0.21, 0.158, 0.2296. Separate each number with a comma.
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189475 | https://wwwmpa.mpa-garching.mpg.de/~komatsu/cmb/lecture_cosmo_iucaa_2011.pdf | Some Basics of the Expansion of the Universe, Cosmic Microwave Background, and Large-scale Structure of the Universe Originally Developed for Lecture Notes on AST396C/PHY396T Elements of Cosmology Spring 2011 Eiichiro Komatsu Texas Cosmology Center & Department of Astronomy The University of Texas at Austin Chapter 1 Expansion of the Universe One of the main goals of cosmology is to figure out how the universe expands as a function of time.
1.1 Expansion and Conservation To describe the evolution of the average universe, one needs only two kinds of equations: 1. The equation that relates the density and pressure of constituents of the universe (such as baryons, cold dark matter, photons, neutrinos, dark energy) to the expansion of the universe, and 2. The equation that describes the energy conservation of the constituents.
Consider a line connecting two arbitrary points in space (which is expanding), and call it L. As the universe expands, L changes with time. As you will derive in homework using General Relativity, the equation of motion for L is given by ¨ L(t) = −4πG 3 L(t) X i [ρi(t) + 3Pi(t)] , (1.1) where ρi(t) and Pi(t) are the energy and pressure of the ith component of the universe, respectively.
Here, note that the absolute value of L does not affect the equation of motion for L. Therefore, one may define a dimensionless “scale factor,” a(t), such that L(t) ≡a(t)x, where x is a time-independent separation called a “comoving” separation, which is in units of length. In cosmology, 1 we often encounter the Hubble expansion rate, H(t), which is defined by H(t) ≡˙ a(t) a(t).
(1.2) The dimension of this quantity is 1/(time). The age of the universe can be calculated from the above definition of H, which gives H(t)dt = da/a. Now, if we know H as a function of a instead of t, we obtain t = Z da aH(a).
(1.3) Another interpretation of H is found by writing ˙ L(t) = H(t)L(t), which tells us that H(t) gives a relation between the distance, L, and the recession velocity, ˙ L. For this reason, it is often convenient to write H(t) in the following peculiar units: H(t) = 100 h(t) km/s/Mpc, where h is a dimensionless quantity. The current observations suggest that the present-day value of h is h(ttoday) ≈0.7.∗ Dividing both sides of equation (1.1) by L and using L(t) = a(t)x, we find one of the key equations connecting the energy density and pressure to the expansion of the universe: ¨ a(t) a(t) = −4πG 3 X i [ρi(t) + 3Pi(t)] (1.4) As expected, positive energy density and positive pressure slow down the expansion of the universe.† This equation cannot be solved unless we know how ρi and Pi depend on time. How ρi depends on time is given by the energy conservation equation, while how Pi depends on time is usually given by the equation of state relating Pi to ρi and other quantities.
As you will derive in homework, the energy conservation equation is given by X i ˙ ρi(t) + 3 ˙ a(t) a(t) X i [ρi(t) + Pi(t)] = 0 (1.5) Equation (1.5) is general and does not assume presence or absence of possible interactions between different components. If we assume that each component is conserved separately, then we have ˙ ρi(t) + 3 ˙ a(t) a(t) [ρi(t) + Pi(t)] = 0, (1.6) ∗The most precise value of h(ttoday) to date from the direct measurement using low-z supernovae and Cepheid variable stars is h(ttoday) = 0.742 ± 0.036 (Riess, Macri, et al., ApJ, 699, 539 (2009)).
†If we ignore the effect of pressure relative to that of the energy density (which is always a good approximation for non-relativistic matter), and write ρ(t) in terms of the total mass enclosed with a radius L, P i ρi(t) = 3M 4πL3 , then equation (1.1) becomes ¨ L = −GM L2 , which is the familiar Newtonian inverse-square law.
Although one must not apply the Newtonian mechanics to describe the evolution of space (because Newtonian mechanism assumes static space), this is a convenient way to understand equations (1.1) and (1.4).
2 for each of the ith component. Note that the second term contains the pressure, and thus how the energy density evolves depends on the pressure.‡ Looking at equations (1.4) and (1.5), one might think that we cannot solve for a(t) unless we have the equation of state giving Pi(t) as a function of ρi(t) etc. While in general that would be true, for these equations a little mathematical trick lets us combine equations (1.4) and (1.5) without knowing the evolution of P(t)!
First, rewrite equation (1.4) as ¨ a(t) a(t) = 8πG 3 X i ρi(t) −4πG X i [ρi(t) + Pi(t)] .
(1.7) Using equation (1.5) on the second term of the right hand side, we get ¨ a(t) a(t) = 8πG 3 X i ρi(t) + 4πG 3 a(t) ˙ a(t) X i ˙ ρi(t) ˙ a(t)¨ a(t) = 8πGa(t)˙ a(t) 3 X i ρi(t) + 4πGa2(t) 3 X i ˙ ρi(t) 1 2(˙ a2)· = 4πG(a2)· 3 X i ρi(t) + 4πGa2(t) 3 X i ˙ ρi(t).
(1.8) As this has the form of ˙ A = ˙ BC + B ˙ C = (BC)·, it is easy to integrate and obtain: ˙ a2(t) = 8πGa2(t) 3 X i ρi(t) −κ, (1.9) where κ is an integration constant, which is in units of 1/(time)2. (A negative sign is for a historical reason.) Dividing both sides by a2(t), we finally arrive at the so-called Friedmann equation: ˙ a2(t) a2(t) = 8πG 3 X i ρi(t) − κ a2(t).
(1.10) ‡While it is a wrong explanation, it is useful to compare this equation to the first law of thermodynamics: TdS = dU + PdV, where T, S, U, and V are the temperature, entropy, internal energy, and volume, respectively.
To a very good accuracy, the entropy is conserved in the universe, dS = 0. The internal energy is U ∝ρa3 and the volume is V ∝a3, and thus d(ρa3) + Pd(a3) = 0, which gives ˙ ρ + 3 ˙ a a(ρ + P) = 0.
This is a wrong explanation because it assumes that the pressure is doing work as a increases. However, in the average universe, the pressure is the same everywhere, and thus there is no under-pressure region against which the pressure can do work. Equation (1.5) must be derived using GR, which you will do in homework, but the above thermodynamic argument is an amusing way to arrive at the same equation. Also, this gives us some confidence that it is not crazy to think that the evolution of ρ depends on P.
3 A beauty of this equation is that it is easy to solve, once a time dependence of ρi(t) is known, which is usually the case.
General Relativity tells us that the integration constant, κ, is equal to ±c2/R2 where R is the curvature radius of the universe (in units of length) and c the speed of light. When the geometry of the universe is flat (as suggested by observations), R →∞(giving κ →0), and thus one can ignore this term. Since we have so much to learn, to save time we will not consider the curvature of the universe throughout (most of) this lecture: ˙ a2(t) a2(t) = 8πG 3 X i ρi(t) (1.11) 1.2 Solutions of Friedmann Equation In order to use solve equation (1.11) for a(t), one must know how ρi(t) depends on time.
To find solutions for a(t), let us first assume that the universe is dominated by one energy component at a time, i.e., ˙ a2(t) a2(t) = 8πG 3 X i ρi(t) ≈8πG 3 ρi(t), (1.12) and further assume that ρi depends on a(t) via a power-law: ρi(t) ∝ 1 ani(t).
(1.13) 4 Finding the solution is straightforward: a(t) ∝t2/ni.
(1.14) This is usually an excellent approximation, except for the transition era where two energy compo-nents are equally important. There are 3 important cases: 1. Radiation-dominated (RD) era. A radiation component (photons, massless neutrinos, or any other massless particles) has a large pressure, PR = ρR/3,§ which gives ρR(t) ∝1/a4(t), or nR = 4. We thus obtain aRD(t) ∝t1/2.
(1.15) The expansion of the universe decelerates. With this solution, we can relate the age of the universe to the Hubble expansion rate: H(t) = ˙ a(t) a(t) = 1 2t.
(1.16) 2. Matter-dominated (MD) era. A matter component (baryons, cold dark matter, or any other non-relativistic particles) has a negligible pressure compared to its energy density, PM ≪ ρM, which gives ρM(t) ∝1/a3(t), or nM = 3. We thus obtain aMD(t) ∝t2/3.
(1.17) §Again, a “wrong” derivation, but there is an intuitive way to get this result using the equation of state for non-relativistic ideal gas (this is obviously a wrong derivation because we are about to apply non-relativistic equation of state to relativistic gas!): P = nkBT = ρkBT ⟨E⟩, where n is the number density, T the temperature of gas, kB the Boltzmann constant, and ⟨E⟩the mean energy per particle. For relativistic particles in thermal equilibrium, ⟨E⟩≈3kBT, which gives P ≈ρ/3. Now, actually, it turns out that the error we are making by using non-relativistic equation of state for relativistic gas cancels out precisely the error we are making by using an approximate relation ⟨E⟩≈3kBT. This gives us the exact relation, P = ρ/3 for relativistic particles. More precisely, the equation of state for relativistic gas takes on the form P = (1 + ϵ)ρ kBT ⟨E⟩ with ⟨E⟩= 3(1 + ϵ)kBT, giving P = ρ/3. Here, ϵ ≃0.05 and −0.10 for Fermions and Bosons, respectively.
5 The expansion of the universe decelerates. With this solution, we can relate the age of the universe to the Hubble expansion rate: H(t) = ˙ a(t) a(t) = 2 3t.
(1.18) 3. Constant-energy-density-dominated (ΛD) era. A hypothetical energy component (let’s call it Λ) whose energy density is a constant over time, nΛ = 0. In this case we cannot use equation (1.14). Going back to equation (1.12) and setting ρΛ = constant, we get ˙ a/a = constant, whose solution is aΛD(t) ∝eHt, (1.19) where an integration constant, H, is the same as the Hubble expansion rate (which is a constant for this model). The expansion of the universe accelerates, which must mean that, according to the acceleration equation (1.4), the pressure of this energy component is negative.
The conservation equation (1.5) tells us that such a component indeed has an enormous negative pressure given by PΛ = −ρΛ.
(1.20) While this looks quite strange, we now know that something like this may actually exist in our universe, as the current observations suggest that the present-day universe is indeed accelerating.
6 1.3 Equation of State of “Dark Energy” and Density Parameters The matter has PM ≪ρM; the radiation has PR = ρR/3; and Λ has PΛ = −ρΛ. This motivates our writing the equation of state of the ith component in the following simple form: Pi = wiρi.
(1.21) Here, wi is called the “equation of state parameter,” and can depend on time (although it is usually taken to be constant).
Why this form? It is important to keep in mind that there is no fundamental reason why we should use this form. This form is often used either just for convenience, or simply for parametrizing something we do not know. At the very least, this form is exact for radiation, wR = 1/3, and for Λ, wΛ = −1. For matter, since wM ≪1, the exact value does not affect the results very much.
The equation of state parameter is almost exclusively used for parametrizing “dark energy,” which is supposed to cause the observed acceleration of the universe. If we assume that w for dark energy, wDE, is constant, then the current observations suggest that (Komatsu, et al., ApJS, 192, 18 (2011)) wDE = −0.98 ± 0.05 (68% CL).
(1.22) In other words, the energy density of dark energy is consistent with being a constant (wDE = wΛ = −1).
Determining wDE with better accuracy may tell us something about the nature of dark energy, especially if wDE ̸= 1 is found with high statistical significance, as it would tell us that dark energy is something dynamical (time-dependent).
Ignoring a potential interaction between dark energy and other components in the universe (e.g., dark matter), the energy density of dark energy obeys (see equation (1.6)) ˙ ρDE(t) + 3 ˙ a(t) a(t) (1 + wDE) ρDE(t) = 0, (1.23) whose solution is ρDE(t) ∝[a(t)]−3(1+wDE). On the other hand, if we do not assume that wDE is a constant, then the energy density of dark energy obeys ˙ ρDE(t) + 3 ˙ a(t) a(t) [1 + wDE(t)] ρDE(t) = 0, (1.24) whose solution is ρDE(t) ∝e−3 R d ln a[1+wDE(a)].
(1.25) Putting these results together, we obtain the Friedmann equation for our Universe containing radiation, matter, and dark energy (but not curvature) as ˙ a2(t) a2(t) = H2(t) = 8πG 3 ρM(t0)a3(t0) a3(t) + ρR(t0)a4(t0) a4(t) + ρDE(t0)e−3 R a(t) a(t0) d ln a[1+wDE(a)] , (1.26) where t0 is some epoch, which is usually taken to be the present epoch.
7 Now, taking t →t0, we find the present-day expansion rate H2 0 ≡H2(t0) = 8πG 3 [ρM(t0) + ρR(t0) + ρDE(t0)] ≡8πG 3 ρc(t0), (1.27) which has been determined to be H0 ≈70 km/s/Mpc. Here, ρc(t0) is the so-called “critical density” of the universe, which is equal to the total energy density of the universe when the universe is flat.
The numerical value of the critical density is ρc(t0) ≡3H2 0 8πG = 2.775 × 1011 h2 M⊙Mpc−3.
(1.28) The critical density provides a natural unit for the energy density of the universe, and thus it is convenient to measure all the energy densities in units of ρc(t0). Defining the so-called density parameters, Ωi, as Ωi ≡ρi(t0) ρc(t0), (1.29) one can rewrite the Friedmann equation (1.26) in a compact form: H2(t) H2 0 = ΩM a3(t0) a3(t) + ΩR a4(t0) a4(t) + ΩDEe−3 R a(t) a(t0) d ln a[1+wDE(a)] (1.30) Basically, most of the literature on cosmology (within the context of General Relativity) use this equation as the starting point.¶ Taking z = 0, one finds that all the density parameters must sum to unity: P i Ωi = 1.
In summary, the Friedmann equation is a combination of two key equations: (1) the equation describing how the universe decelerates/accelerates depending on the energy density and pressure of the constituents, and (2) the equation describing the energy conservation of the constituents. Once the Friedmann equation is given with the proper right hand side containing the energy densities of the relevant constituents of the universe, we can find a(t) as a function of time easily.
¶An interesting possibility is that General Relativity may not be valid on cosmological scales. There are scenarios in which the form of the Friedmann equation is modified.
One widely-explored example is the so-called Dvali-Gabadadze-Porrati (DGP) model (Dvali, Gabadadze & Porrati, Phys. Lett. B485, 208 (2000)). In this scenario, the Friedmann equation is modified to: H2(t) −H(t) rc = 8πG 3 X i ρi(t), where rc is some length scale below which General Relativity is restored. (For r ≪rc, the potential is given by −GNm/r where GN is the ordinary Newtonian gravitational constant. For r ≫rc, the potential is modified to −G5m/r2 and decays faster. G5 is the gravitational strength in the 5th dimension.) This model has attracted a huge attention of the cosmology community, as it was shown that this modified Friedmann equation gives an accelerating expansion without dark energy. Namely, even when the right hand side contains only matter, the solution for this equation can still exhibit an accelerating expansion. As this is a quadratic equation for H(t), we can solve it and find H(t) = 1 2 1 rc ± r 1 r2 c + 32πG 3 ρM(t) .
At late times when ρ(t) becomes negligible compared to the other term, one of the solutions is given by a(t) ∝et/rc, i.e., an exponential, accelerated expansion.
8 At present, the radiation is totally negligible compared to matter, ΩR/ΩM ≃1/3250, and the dark energy density is about 3 times as large as the matter density, ΩDE/ΩM ≃2.7 (with ΩM ≃0.27 and ΩDE ≃0.73).
1.4 Redshift As the universe expands, the wavelength of light, λ, is stretched linearly: λ(t) ∝a(t), (1.31) which implies that photons lose energy as E(t) ∝1/a(t).
This is something one can observe, by comparing, for example, the observed wavelength of a hydrogen line to the rest-frame wavelength that we know from the laboratory experiment. We often use the redshift, z, to quantify the stretching of the wavelength: 1 + z ≡ λ(t0) λ(temitted).
(1.32) The present-day corresponds to z = 0.
Using equation (1.31), we can relate the observed redshift to the ratio of the scale factors: 1 + z = a(t0) a(temitted).
(1.33) Using this result in the Friedmann equation (1.30), we obtain the most-widely-used form of the Friedmann equation: H2(z) H2 0 = ΩM(1 + z)3 + ΩR(1 + z)4 + ΩDEe3 R z 0 d ln(1+z)[1+wDE(z)] (1.34) 9 From this result, it follows that the best way to determine the equation of state of dark energy is to measure H(z) over a wide range of z. If we can only measure the expansion rates at z ≪1, then Taylor expansion of equation (1.34) with ΩR ≪ΩM and ΩDE ≃1 −ΩM gives H2(z ≪1) H2 0 ≈1 + 3ΩMz + 3(1 + wDE)(1 −ΩM)z.
(1.35) As we know from observations that |1 + wDE| is small (of order 10−1 or less), the third term is tiny compared to other terms, making it difficult to measure wDE. This is why we need to measure H(z) over a wide redshift range.
1.5 Alcock-Paczy´ nski Test We have learned that, in order to determine wDE, we need to measure H(z) over a wide redshift range. But, how? In principle, one can measure H(z) in the following way.
Consider two points A and B, which are separated by LAB along the line of sight. Both points are on the Hubble flow. The tip (A) and tail (B) emit light, which we observe to be at redshifts of zA and zB, respectively. These are our observables. Now we show that the redshift difference, ∆z ≡zB −zA, is somehow related to the Hubble expansion rate at ¯ z = (zA + zB)/2.
Using a(tA) = a(¯ t + ∆tAB/2) ≈a(¯ t) + ˙ a(¯ t)∆tAB/2 and similarly a(tB) = a(¯ t) −˙ a(¯ t)∆tAB/2, we find ∆z = zB −zA = a0 a(tB) − a0 a(tA) ≈a0 ˙ a(¯ t) a2(¯ t)δtAB, where ∆tAB is the time the light takes to go from B to A, which is equal to LAB/c. Therefore ∆z = H(¯ z) c a0LAB a(¯ z) .
(1.36) 10 Here, a0LAB/a(¯ z) = xAB is a comoving separation (which is time-independent; a0 is the scale factor at present). Rewriting the result in terms of H(¯ z) and xAB, we finally find the relation between what we want to determine, H(¯ z), and the observable, ∆z, as H(¯ z) = c∆z xAB (1.37) This is a beautiful result, but has one problem. In order to use this method, we need to know the intrinsic comoving separation, xAB, which is not always known. (As a matter of fact, xAB is not known for most cases.) In other words, this method works if we have the standard ruler, for which the intrinsic size is known.
There is another way, which does not require the prior knowledge of the size. This was proposed first by Charles Alcock and Bohdan Paczy´ nski in 1979 (Alcock & Paczy´ nski, Nature, 281, 358 (1979)), and is known as the “Alcock-Paczy´ nski test.” While this method does not require the prior knowledge of the intrinsic size, it does still require an ideal situation: a collection of test particles (e.g., galaxies) which spatial distribution is spherically symmetric.
Consider a spherical distribution with a diameter of L. By measuring the redshift difference along the line of sight, we find H(z) = c∆z/[L(1 + z)]. On the other hand, the angular extension of this spherical distribution of the sky, θ, is related to the intrinsic physical size, L, as θ = L DA(z), (1.38) where DA(z) is the angular diameter distance. Therefore, by measuring the angular extension, θ, and the redshift difference, ∆z, and combining them, we obtain DA(z)H(z) = c∆z θ(1 + z) (1.39) 11 The right hand side only contains the observables, and thus the Alcock-Paczy´ nski test allows us to determine DAH.
A challenge for this method is to find objects whose distribution is spherically symmetric. There is one known example, which is the distribution of the large-scale structure. We will come back to this later.
1.6 Angular Diameter Distance In order to utilize the AP test (equation (1.39)), we need to relate the angular diameter distance, DA(z), to cosmological models. This can be done by realizing that the angular diameter distance is equal to the comoving radial distance times the scale factor: DA(z) = a(z)r = a0r 1 + z .
(1.40) Then, we can calculate r(z) as follows. Along the path of photons coming toward us in a flat universe, we have cdt = a(t)dr.∥Therefore, r = c Z t0 t dt′ a(t′) = c Z a0 a da′ (a′)2H(a′) = c Z z 0 dz′ a0H(z′), (1.41) with H(z) given by the Friedmann equation (1.34). The angular diameter distance is DA(z) = c 1 + z Z z 0 dz′ H(z′) (1.42) ∥In a curved space, we have cdt = a(t) dr √ 1−Kr2 where K = +1 and −1 for positively and negatively curved spaces, respectively.
12 Using this result in equation (1.39), we find that the Alcock-Paczy´ nski test provides (in a flat universe): H(z) Z z 0 dz′ H(z′) = ∆z θ .
(1.43) As the angular diameter distance is an integral of 1/H(z), it is less sensitive to the equation of state of dark energy. However, if we have many measurements of DA(z) at various redshifts, we can effectively differentiate DA(z) with respect to z, obtaining a measurement of 1/H(z). While we have not yet entered the era where we can do this with the angular diameter distance, we have been able to do this using the luminosity distances measured out to distant Type Ia supernovae, as described next.
1.7 Luminosity Distance Perhaps the best known method for measuring distances in cosmology is the luminosity dis-tance. This builds on a simple idea: the farther objects look dimmer. More specifically, the energy we receive per unit time per unit area, which is usually known as the “flux,” is related to the intrinsic luminosity of the light source as F = L 4πD2 L , where DL is the luminosity distance. This equation defines DL: DL ≡ r L 4πF .
(1.44) The flux F is our observable; thus, in order to use this method, we need to have the light sources whose intrinsic luminosity is known, i.e., the standard candles.
Type Ia supernovae, which are believe to be thermonuclear explosion of white dwarf stars, are known to exhibit similar peak luminosities (after a few corrections), and have been used as the 13 primary standard candles in the cosmology community. In fact, it was the observation of Type Ia supernovae which led to the discovery of the acceleration of the universe (Riess et al., AJ, 116, 1009 (1998); Perlmutter et al., ApJ, 517, 565 (1999)).
Now, we must relate DL to cosmological models. To do this, we first note that the energy emitted by a supernova is diluted by the surface area, which is 4πr2a2 0.
Second, each photon emitted by a supernova loses energy as E ∝a/a0 = 1/(1 + z). Third, the rate at which photons are received per unit time is dilated by a factor of a/a0 = 1/(1 + z) compared to the rate at which the light was emitted by a supernova. (I.e., we receive fewer photons per second at our location, relative to the number of photons emitted per second at the source). This leads to the cosmological inverse-square-law formula: F = L/(1 + z)2 4πr2a2 0 .
(1.45) Comparing this formula to the definition of DL above, we conclude that DL(z) = a0(1 + z)r = (1 + z)2DA(z) (1.46) This relation, DL(z) = (1 + z)2DA(z), is exact, and does not depend on cosmological models.
As of today, hundreds of distant Type Ia supernovae have been observed, and DL(z) has been determined out to z = 1.7. One can fit the data to DL = c(1 + z) R dz/H(z) and constrain the cosmological parameters such as ΩM and wDE.
Redshift 5Log10[DL(z)/Mpc]+25 Difference w.r.t. ΩΛ=0.72 & Ωm=0.28 Effect of w=–1±0.1 Year 2010 Amanullah et al. (2010) 557 Type Ia supernovae!
14 • Constraints on H(z) as determined by differentiating the luminosity distance data of 288 Type Ia supernovae and 69 Gamma-ray Bursts • Ref: Yun Wang, Phys. Rev. D, 80, 123525 (2009) One may also differentiate the DL data with respect to z, and see if one can measure 1/H(z).
1.8 Effects of Changing Effective Relativistic Degrees of Freedom The expansion rate during the matter era (well after the matter-radiation equality, but well before the dark energy domination) is given by H(z)/H0 = √ΩM(1+z)3/2. However, in general, we should be careful about applying this formula blindly to arbitrarily high redshifts, as some “matter” would start behaving as if they were radiation (massless particles) when the kinetic energy of the particles exceeds the rest mass energy. This can happen because the universe was hotter when it was younger.
1.8.1 Neutrinos A good example is the effect of massive neutrinos on the expansion rate of the universe. When the mass of neutrinos, mν, is larger than roughly 3kBTν (Tν is the neutrino temperature, which is equal to (4/11)1/3 of the photon temperature in the standard scenario for Tν ≪1 MeV), neutrinos behave as non-relativistic particles. In the opposite limit, they behave as relativistic particles.
While we do know that neutrinos have finite masses, we do not know the actual values of the masses. The current limit suggests that the sum of the masses of 3 neutrino species is greater than 0.05 eV, but smaller than 0.6 eV (or mν < 0.2 eV for each of the 3 species if we assume that all neutrino species have equal masses). As 1 eV corresponds to 1.16 × 104 K, neutrinos could have become non-relativistic when the neutrino temperature fell below 770 K, or the redshift less than 400. At the very least, one of the neutrino species must have become non-relativistic when the neutrino temperature fell below 190 K (or z < 100).
15 As the expansion rate is solely determined by the energy density of the constituents (in a flat universe), all we need to calculate is the energy density of neutrinos. As neutrinos are Fermions and were in thermal equilibrium in the early universe, their distribution function is given by the Fermi-Dirac distribution. Also, as they decoupled from the plasma when neutrinos were still highly relativistic (when the temperature of the universe was about 2 MeV∼20 billion K), their dis-tribution function will remain the Fermi-Dirac distribution for massless particles, even after neutrinos became non-relativistic.
With this information, we calculate the energy density of neutrinos (in natural units) by inte-grating the distribution function times energy per particle:∗∗ ρν(z) = (1 + z)4 Z q2dq π2 P i q q2 + m2 ν,i/(1 + z)2 eq/Tν0 + 1 .
(1.47) This can be evaluated numerically, and the result is shown for mν = 0.2 below.
∗∗This is derived as follows. The energy density of 1 neutrino species is given (in natural units) by ρν,i = 2 Z d3p (2π)3 Ei(p) ep/Tν + 1 = Z p2dp π2 q p2 + m2 ν,i ep/Tν + 1 .
Defining q ≡p/(1 + z), we rewrite this equation as ρν,i = (1 + z)4 Z q2dq π2 q q2 + m2 ν,i/(1 + z)2 eq(1+z)/Tν + 1 .
Finally, using (1 + z)/Tν = 1/Tν0, where Tν0 is the present-day temperature of neutrino, we obtain ρν,i = (1 + z)4 Z q2dq π2 q q2 + m2 ν,i/(1 + z)2 eq/Tν0 + 1 .
16 1.8.2 General Consideration As we go farther back in time, various other particles, such as electrons and positrons, become relativistic, and these effects must be taken into account when calculating the expansion rate.
More specifically, when the temperature of the universe was higher than above 1 MeV, but lower than 2 times the muon mass (105.7 MeV), the relativistic particles included photons, 3 neutrino species, electrons, and positrons.
And, they all shared the same temperature, T.
The energy density can be found easily by integrating the corresponding distribution functions times energy per particle. In natural units, we find ργ = 2 Z p3dp 2π2 1 ep/T −1 = π2 15T 4, (1.48) ρν = 6 Z p3dp 2π2 1 ep/T + 1 = 7π2 40 T 4, (1.49) ρe± = 4 Z p3dp 2π2 1 ep/T + 1 = 7π2 60 T 4.
(1.50) Here, “2” for photons is the number of helicity states (i.e., left and right circular polarization states); “6” for neutrinos is the number of helicity state (1; just left-handed neutrinos) times the number of neutrino species (3) times 2 because we count both neutrinos and anti-neutrinos; and “4” for electrons/positrons is the number of spin states (2; up and down) times 2 because we count both electrons and positrons.
It is more common to define the “effective number of relativistic degrees of freedom” by writing the total radiation energy as ρR = ργ + ρν + ρe± = π2 30g∗T 4, (1.51) where g∗= 2 + 7 8 (6 + 4) = 43 4 .
(1.52) With this, the expansion rate during the radiation era is given by H2 = 8πG 3 ρR = 4π3G 45 g∗T 4.
(1.53) Therefore, when we calculate the expansion rate during the radiation era, we must be careful about how many relativistic degrees of freedom we have in the universe at a given time. For g∗= 43/4, we obtain 1 H(T) = 1.48 1 MeV T 2 sec.
(1.54) As the age of the universe during the radiation era is t = 1/(2H), we also have t = 1 2H(T) = 0.74 1 MeV T 2 sec (1.55) Again, this formula is valid only for 1 MeV < T ≪200 MeV. Above this temperature, we will need to count muons as relativistic particles, etc.
17 PROBLEM SET 1 1.1 Expansion of the Universe In this section, we will use Einstein’s General Relativity to derive the equations that describe the expanding universe. Einstein’s General Relativity describes the evolution of gravitational fields for a given source of energy density, momentum, and stress (e.g., pressure). Schematically, [Curvature of Space-time] = 8πG c4 [Energy density, Momentum, and Stress] Here, the dimension of “curvature of space-time” is 1/(length)2, as the curvature is usually defined as the second derivative of a function with respect to independent variables, and for our application the independent variables are space-time coordinates: xµ = (ct, x1, x2, x3) for µ = 0, 1, 2, 3.
1.1.1 Space-time Curvature: Left Hand Side of Einstein’s Equation The coefficient on the right hand side, 8πG/c4, is chosen such that Einstein’s gravitational field equations reduce to the familiar Poisson equation when gravitational fields are weak and static, and the space is not expanding: ∇2φN = 4πGρM, where φN is the usual Newtonian potential, and ρM is the mass density. Let us rewrite it in the following suggestive form: ∇2 2φN c2 = 8πG c4 (ρMc2).
Here, as φN/c2 is dimensionless, and thus the left hand side has the dimension of curvature, i.e., 1/(length)2. The right hand side contains ρMc2, which is energy density; thus, G/c4 correctly converts energy density into curvature. Now, this equation tells us something Newton did not know but Einstein finally figured out: the second derivative of the dimensionless Newtonian potential times 2 with respect to space coordinates is the curvature of space, and mass deforms space.
In order to calculate curvature of space-time, we need to know how to calculate a distance be-tween two points. Of course, everyone knows that, in Cartesian coordinates, the distance between two points in flat space separated by dxi = (dx1, dx2, dx3) is given by dl = p (dx1)2 + (dx2)2 + (dx3)2, or dl2 = 3 X i=1 3 X j=1 δijdxidxj, (1.56) where δij = 1 for i = j and δij = 0 for i ̸= j. Since space is flat, the curvature of this space is zero.
This is a consequence of the coefficients of dxidxj on the right hand side of equation (1.56) being independent of coordinates. In general, when space is not flat but curved, the distance between two points can be written as dl2 = 3 X i=1 3 X j=1 gij(x)dxidxj, (1.57) 18 where gij(x) is known as the metric tensor. Schematically, the curvature of space is given by the second derivatives of the metric tensor with respect to space coordinates: Curvature of Space ∼ ∂2gij ∂xk∂xl .
In General Relativity, we extend this to the curvature of space-time. The distance between two points in space and time separated by dxµ = (cdt, dx1, dx2, dx3) is given by ds2 = 3 X µ=0 3 X ν=0 gµν(x)dxµdxν, (1.58) and Curvature of Space-time ∼ ∂2gµν ∂xµ∂xν .
Now, let us get into the gory details! The precise definition of space-time curvature, known as the Riemann curvature tensor, is given by†† Rµ νρσ ≡∂Γµ νσ ∂xρ −∂Γµ νρ ∂xσ + X α Γα νσΓµ αρ − X α Γα νρΓµ ασ, (1.59) where Γ is the so-called Christoffel symbol, also known as the affine connection: Γµ νρ ≡1 2 X α gµα ∂gαρ ∂xν + ∂gνα ∂xρ −∂gνρ ∂xα .
(1.60) The metric tensor with the superscripts, gµα, is the inverse of the metric tensor, in the sense that X α gµαgαν = δµ ν , where δµ ν = 1 for µ = ν and zero otherwise.
Question 1.1: For an expanding universe with flat space, the distance between two points in space is given by, perhaps not surprisingly, dl2 = a2(t) 3 X i=1 3 X j=1 δijdxidxj, (1.61) where x denotes comoving coordinates. The scale factor, a(t), depends only on time t. Then, the distance between two points in space-time is given by ds2 = −c2dt2 + dl2 = −c2dt2 + a2(t) 3 X i=1 3 X j=1 δijdxidxj.
(1.62) ††Different definitions of curvature are used in the literature. Here, we follow the definition used by Misner, Thorne, and Wheeler, “Gravitation” (1973). Steven Weinberg’s recent textbook, “Cosmology,” uses the opposite sign.
19 Non-zero components of the metric tensor are g00 = −1; gii = a2(t) for i = 1, 2, 3, and those of the corresponding inverse are g00 = −1; gii = 1 a2(t) for i = 1, 2, 3.
This metric is known as the Robertson-Walker metric (for flat space), and describes the distance between two points in space-time of a homogeneous, isotropic, and expanding universe. For this metric, non-zero components of the affine connection are Γi j0 and Γ0 ij. Calculate Γi j0 and Γ0 ij. The answers will contain a, ˙ a/c, and δij. Once again, our space-time coordinates are xµ = (ct, x1, x2, x3).
Question 1.2: Einstein’s field equations do not use all the components of the Riemann tensor, but only use a part of it. Specifically, they will use the so-called Ricci tensor: Rµν ≡ X α Rα µαν = X α ∂Γα µν ∂xα −∂Γα µα ∂xν + X αβ Γβ µνΓα βα −Γβ µαΓα βν , (1.63) and the Ricci scalar: R ≡ X µν gµνRµν.
(1.64) For the above flat Robertson-Walker metric, non-zero components of the Ricci tensor are R00 and Rij. Calculate R00, Rij, and R. The answers will contain a, ˙ a/c, ¨ a/c2, and/or δij.
Question 1.3: The left hand side of Einstein’s equation is called the Einstein tensor, denoted by Gµν, and is defined as Gµν ≡Rµν −1 2gµνR.
(1.65) Calculate G00 and Gij.
1.1.2 Stress-Energy Tensor: Right Hand Side of Einstein’s Equation The precise form of Einstein’s field equation is Gµν = 8πG c4 Tµν, (1.66) where Tµν is called the stress-energy tensor (also sometimes called “energy-momentum tensor”).
As the name suggests, the components of Tµν represent the following quantities: • T00: Energy density, 20 • T0i: Momentum, and • Tij: Stress (which includes pressure, viscosity, and heat conduction).
For a perfect fluid, the stress-energy tensor takes on the following specific form: Tµν = Pgµν + (ρ + P) (P α gµαuα)(P β gνβuβ) c2 , (1.67) where ρ and P are the energy density and pressure, respectively, and uµ is a four-dimensional velocity of a fluid element. The spatial components of a four velocity, ui, represent the usual 3-dimensional velocity of a fluid element, while the temporal component, u0, is determined by the normalization condition of uµ: gµνuµuν = −c2.
(1.68) Note that the 3-dimensional velocity, ui, does not contain the apparent motion due to the expansion of the universe, but only contains the true motion of fluid elements.
Question 1.4: In a homogeneous, isotropic, and expanding universe, fluid elements simply move along the expansion of the universe, and the 3-dimensional velocity vanishes.
(In other words, fluids are comoving with expansion.) Therefore, such a fluid element has ui = 0, and the normalization condition gives u0 = c. Non-zero components of the stress-energy tensor are T00 and Tij. Calculate T00 and Tij for the flat Robertson-Walker metric and comoving fluid.
Question 1.5: Now, we are ready to obtain Einstein’s equations. First, write down G00 = (8πG/c4)T00 and Gij = (8πG/c4)Tij for the flat Robertson-Walker metric and comoving fluid in terms of a, ˙ a/c, ¨ a/c2, and/or δij. Then, by combining these equations, obtain the right hand side of ˙ a2 a2 = ¨ a a = The first equation is the Friedmann equation, and the second one is the acceleration equation that we have learned in class (with c = 1).
1.1.3 Energy Conservation Combining the above equations for ˙ a/a and ¨ a/a will yield the energy conservation equation, ˙ ρ + 3 ˙ a a(ρ + P) = 0. In other words, the energy conservation is already built into Einstein’s equations.
Question 1.6: Alternatively, one can derive the energy conservation equation directly from the conservation of the stress-energy tensor. In General Relativity, the “conservation” means that the covariant derivative (rather than the partial derivative) of the stress-energy tensor vanishes.
0 = X αβ gαβTµα;β ≡ X αβ gαβ ∂Tµα ∂xβ − X λ Γλ αβTµλ − X λ Γλ µβTλα !
.
(1.69) 21 The energy conservation equation is P αβ gαβT0α;β = 0, while the momentum conservation equation is P αβ gαβTiα;β = 0. Reproduce ˙ ρ + 3 ˙ a a(ρ + P) = 0 from P αβ gαβT0α;β = 0.
1.1.4 Cosmological Redshift Consider a non-relativistic particle, which is moving in a gravitational field with a 3-dimensional velocity of ui ≪c.
The other external forces (such as the electromagnetic force) are absent.
According to General Relativity, the equation of motion of such a particle is dui dτ + X αβ Γi αβuαuβ = 0, (1.70) where dτ ≡ √ −ds2/c is called the proper time. The four-dimensional velocity is given by uµ = dxµ/dτ; thus, u0 = cdt/dτ and ui = dxi/dτ.
Question 1.7: Using the affine connection for the flat Robertson-Walker metric, rewrite the equation of motion in terms of ˙ ui = dui/dt, ˙ a/a and ui. Show how ui changes with the scale factor, a(t).
22 Chapter 2 Cosmic Microwave Background 2.1 Basic Properties The cosmic microwave background is the oldest light that one can ever hope to measure directly.
This light delivers the direct information of the physics condition of the universe when the universe was only 380,000 years old (which is z = 1090).
The important characteristics of the cosmic microwave background are: • The spectrum of the microwave background is a blackbody: Bν(TCMB) = 2hν3 c2 1 ehν/(kBTCMB)−1 , (2.1) with the temperature of TCMB = 2.725 K.
Spectrum of CMB 4K Blackbody 2.725K Blackbody 2K Blackbody Rocket Data (COBRA) Satellite (COBE/FIRAS) CN Rotational Excitation Ground-based Balloon-borne Satellite (COBE/DMR) Wavelength 3mm 0.3mm 30cm 3m Brightness, W/m2/str/Hz 23 • The photons of the microwave background are numerous: their number density is nCMB = 410 cm−3,∗which is about 2 billion times the number density of baryons. We do not quite know why baryons are so few compared to photons.
• The distribution of the microwave background on the sky is isotropic to the precision of 10−3.
Most of the residual anisotropy, at the level of a few mK, is due to the motion of our Solar system with respect to the rest frame of the cosmic microwave background, and is called the dipole anisotropy. After removing the dipole component, we are left with the primordial anisotropy at the level of 10−5: δTCMB ≈30 µK.
TCMB=2.725 K ΔTCMB=3.346 mK ΔTCMB=30 μK Monopole (l=0; mean temperature) Dipole (l=1; motion of Solar System) Primordial Anisotropy (l≥2) Galactic Plane T(θ,φ)=∑lm alm Ylm(θ,φ) ∗This number can be obtained by integrating the distribution function: nCMB = 2 Z d3p (2π)3ℏ3 1 epc/(kBTCMB) −1 = 2ζ(3) π2 kBTCMB cℏ 3 , where ζ(3) ≃1.202, TCMB = 2.725 K, and kB/(cℏ) = 4.367 cm−1 K−1.
24 • The cosmic microwave background is polarized, but only very weakly. The dominant polar-ization pattern is radial/tangential around temperature spots.
2.2 Evolution of Temperature and Entropy Conservation 2.2.1 Naive Consideration How does TCMB change with time? There are several ways of getting this.
1. The cosmological redshift reduces the energy of photons as E ∝1/a(t). The mean energy per particle, ⟨E⟩, is† ⟨E⟩= ρCMB nCMB = 2 R d3p (2π)3ℏ3 pc epc/(kBTCMB)−1 2 R d3p (2π)3ℏ3 1 epc/(kBTCMB)−1 = π2 15 2ζ(3) π2 (kBTCMB) ≃2.70(kBTCMB).
(2.2) Therefore, we obtain TCMB ∝1/a(t).
2. Use the conservation of the number of photons, nCMBV ∝nCMBa3 = constant. This gives nCMBa3 = 2ζ(3) π2(cℏ)3 (kBTCMB)3a3 = constant, giving TCMB ∝1/a(t).
3. Use the energy conservation, ρCMB = π2 15(cℏ)3 (kBTCMB)4 ∝1/a4(t), giving TCMB ∝1/a(t).
These results are valid as long as there is no net creation or destruction of photons.
†The mean particle energy can also be found from the blackbody formula: ⟨E⟩= R ∞ 0 dνBν(TCMB) R ∞ 0 dν Bν(TCMB) hν .
25 2.2.2 Entropy Conservation Is there a general formula that we can use for calculating the evolution of temperature, even when there is net creation or destruction of photons? The conservation of entropy provides such a formula. Roughly speaking, the entropy is proportional to the number of particles, i.e., S ≈ kBnV = constant. Because photons are much more numerous than matter particles, the entropy of the universe is completely dominated by that of photons (and neutrinos, whose number density is similar to the photon number density).
Let us calculate entropy. We begin with the first-law of thermodynamics, TdS = dU + PdV (where U is the internal energy), and another thermodynamic equation, V dP = HdT/T = (U + PV )dT/T (where H = U + PV is the enthalpy). By combining these equations, we obtain dS = d U + PV T .
(2.3) Integrating, we get S = U + PV T + constant (2.4) The integration constant should be chosen such that S = 0 for the absolute zero temperature, T = 0. We set the integration constant to be zero. Here, both U and P contain all the particles in the universe, including both radiation and matter: U = UR + UM and P = PR + PM.
• Radiation. For radiation, we have UR = ρRV and PR = ρR/3. We find SR = 4ρRV 3T .
(2.5) Using the mean particle energy, ⟨ER⟩= ρR/nR, one may rewrite this result as SR = kBnRV × 4⟨ER⟩ 3kBT ≈4kBnRV.
(2.6) Therefore, indeed the entropy is given by the number of particles (times kB). More precisely, by writing the radiation energy density as ρR = π2 30g∗ (kBT)4 (cℏ)3 , (2.7) we obtain SR = kB " π2 30g∗ kBT cℏ 3# V.
(2.8) As the effective number of relativistic degrees of freedom, g∗, can change with time, the entropy conservation, SR = constant, with V ∝a3(t) gives T ∝ 1 g1/3 ∗ a(t) (2.9) Therefore, a simple relation such as T ∝1/a(t) holds only when the effective number of relativistic species does not change, g∗= constant.
26 • Matter. For matter, we have UM = 3 2kBnMV T ‡ and PM = nMkBT. We find SM = 5 2kBnMV.
(2.10) Again, indeed the entropy is given by the number of particles (times kB)§, and nR ≫nM (e.g., the number density of photons is 2 billion times that of baryons) guarantees that we can safely ignore the matter contribution to entropy.
2.2.3 Photon Heating due to Electron-Positron Annihilation A good example for the temperature change due to the change in g∗is the electron-positron anni-hilation: e+ + e−→γ + γ.
When the temperature of the universe was greater than the rest mass energy of an electron, 0.511 MeV, the pair-creation, γ + γ →e+ + e−, also occurred; however, when the universe cooled down below 0.511 MeV, the pair creation no longer occurred.
In addition, particles behave as if they were relativistic when the temperature is greater than ≈m/3; thus, electrons and positrons were sufficiently relativistic when the temperature of the universe was greater than their rest mass energy.
Now, let us apply the entropy conservation: T2 = T1 g∗,1 g∗,2 1/3 , (2.11) where T1 and T2 are the photon temperatures before and after the annihilation, respectively. The effective numbers of relativistic degrees of freedom are g∗,1 = 2 + 7 8 × 4 = 11 2 , g∗,2 = 2, ‡Here, we do not include the mass energy in the internal energy.
§This expression, derived from thermodynamics of ideal gas, is only approximate. More rigorous derivation using the famous Boltzmann’s entropy formula, S = kB ln W, where W is the number of possible states, gives the so-called Sackur–Tetrode equation for non-relativistic, monatomic ideal gas: SM = kBnBV 5 2 + ln 1 nMΛ3 , where Λ ≡ℏ p 2π/(mkBT) is known as the thermal de Broglie length and m is the particle mass. Note that this formula is valid only when nMΛ3 ≪1 (which means that quantum effects are negligible).
27 before and after the annihilation, respectively. Therefore, we conclude that the annihilation in-creases the photon temperature by a factor of (11/4)1/3: T2 = T1 11 4 1/3 .
(2.12) After this, the photon temperature decreased as T ∝1/a(t).
As neutrinos decoupled from the plasma before the electron-positron annihilation epoch, the annihilation did not heat neutrinos. As a result, the annihilation creates a mismatch between the neutrino temperature and photon temperature, and the mismatch is given by the above factor.
Specifically, the neutrino temperature, Tν, is lower than the photon temperature, Tγ, by a factor of (4/11)1/3: Tν = Tγ 4 11 1/3 (2.13) The present-day neutrino temperature is 2.725 × (4/11)1/3 = 1.945 K.
2.3 Recombination and Decoupling 2.3.1 Opaque Universe While there were about equal numbers of electrons, positrons, and photons before the annihilation epoch, the number of electrons after the annihilation epoch is about 2 billion times smaller than that of photons, as most of electrons annihilated with positrons. (Why there was a tiny excess of electrons over positrons is still a mystery.) However, this tiny amount of electrons is enough to keep the universe “opaque,” as they efficiently scatter photons.
28 • Free electrons can scatter photons efficiently.
• Photons cannot go very far.
proton helium electron photon Whether the scattering is efficient or not can be quantified by the ratio of the mean free time of photons, 1/(σT nec), and the Hubble time, 1/H. Here, σT = 6.65 × 10−25 cm2 is the Thomson scattering cross section, and ne is the number density of free electrons. The scattering is efficient enough to keep the universe opaque if the mean free time is short compared to the Hubble time, i.e., H/(σT nec) < 1. In fact, the scattering is so efficient that the universe remains opaque when the universe is matter-dominated, for which the Hubble rate is given by H = H0 p Ωm(1 + z)3. Let us calculate H σT nec = H0 c p Ωm(1 + z)3 σT nCMB nCMB ne .
(2.14) Using nCMB = 410(1+z)3 cm−3, nCMB/ne ≈2×109, c/H0 = 2998 h−1 Mpc = 9.25 h−1 ×1027 cm, and Ωmh2 = 0.13, we obtain H σT nec ≃0.9 × 10−2 1000 1 + z 3/2 nCMB/ne 2 × 109 .
(2.15) Therefore, at z ≈103, the mean free time of photons was still only 1% of the Hubble time, and the universe was still quite opaque.
2.3.2 Neutral Hydrogen Formation and Decoupling However, at around this epoch (z ≈103, or TCMB ≈3000 K), the electron number density rapidly fell relative to nCMB, resulting in the decoupling of photons from the electron scattering. What happened? At this temperature, the universe was cool enough for electrons to be captured by protons, forming neutral hydrogen atoms: p + e−→H + γ.
Once started, this process rapidly eats electrons, reducing their number density and thus allowing for photons to propagate freely.
29 • [recombination] When the temperature falls below 3000 K, almost all electrons are captured by protons and helium nuclei.
• [decoupling] Photons are no longer scattered. I.e., photons and electrons are no longer coupled.
Time 1500K 6000K 3000K proton helium electron photon Why 3000 K?
As the ionization energy of hydrogen atoms is 13.6 eV, one might think that the neutral hydrogen begins to form when the temperature of photons falls below 13.6 eV ≃1.6 × 105 K. However, in reality, the formation of hydrogen atoms is delayed until T ≈3700 K.
When the temperature is T = 1.6 × 105 K, only 15% of photons have energies lower than 13.6 eV. When the temperature drops to T = 70, 000 K, about a half of photons have energies lower than 13.6 eV. Still, there are so many photons per hydrogen atom to begin with, and thus, roughly speaking, the ratio of the number of photons to the number of electrons, which is about a billion, gives a logarithmic correction to the temperature of the hydrogen formation epoch as T ≈70, 000 K/ ln(109) ≈3400 K. Finally, while a significant amount of hydrogen atoms are formed at this temperature, photons do not decouple from the plasma until the universe cools down to T ≈3000 K.
The first approximation would be to assume that protons, electrons, and hydrogen atoms are in thermal equilibrium.
At this temperature all of these species are non-relativistic, and their equilibrium densities are given by the non-relativistic limits of the Fermi-Dirac distribution: np = 2 Z d3p (2π)3ℏ3 exp − mpc2 + p2 2mp −µp kBT = 2e(µp−mpc2)/(kBT) mpkBT 2πℏ2 3/2 , (2.16) ne = 2 Z d3p (2π)3ℏ3 exp − mec2 + p2 2me −µe kBT !
= 2e(µe−mec2)/(kBT) mekBT 2πℏ2 3/2 , (2.17) nH = 4 Z d3p (2π)3ℏ3 exp − mHc2 + p2 2mH −µH kBT = 4e(µH−mpc2)/(kBT) mHkBT 2πℏ2 3/2 .
(2.18) Now, we also assume that protons, electrons, and hydrogen atoms are in ionization equilibrium, by which we mean that the reaction p + e−↔H + γ occurs fast enough to reach the chemical 30 equilibrium: µp + µe = µH.
(2.19) (Note that photon’s chemical potential is zero.) This condition lets us combine the above 3 number densities to obtain the so-called Saha equation: npne nH = mp mH mekBT 2πℏ2 3/2 e−(mp+me−mH)c2/(kBT).
(2.20) Here, the mass difference in the exponential is the binding energy of an hydrogen atom, which is of course equal to its ionization energy: BH ≡(mp + me −mH)c2 = 13.6 eV.
(2.21) Since me ≈mp/2000 and mp ≈1 GeV, we can set mp ≈mH in the parenthesis in front of the exponential factor. Finally, the charge neutrality demands ne = np. We thus obtain n2 p nH ≈ mekBT 2πℏ2 3/2 e−BH/(kBT).
(2.22) Now, define the ionization fraction: X ≡ np np + nH , (2.23) which goes from 1 (fully ionized hydrogen) to 0 (fully neutral hydrogen). The Saha equation now reads: X2 1 −X = 1 np + nH mekBT 2πℏ2 3/2 e−BH/(kBT).
(2.24) The goal here is to solve this equation for X as a function of the temperature, T. For this purpose, it is convenient to relate np + nH to the baryon mass density of the universe. We use the result from the Big Bang Nucleosynthesis (BBN): 76% of the baryonic mass in the universe after BBN is contained in protons (and the rest in helium nuclei). Therefore, mp(np + nH) = 0.76ρb. We then define the time-independent baryon-to-photon ratio: η ≡ ρb mpnCMB = 273.9(Ωbh2) × 10−10 (2.25) which takes on the value η = 6.30×10−10 for Ωbh2 = 0.023. Therefore, there are 1.6 billion photons per baryon. Note that we have used nCMB = 410 cm−3(T/T0)3 with T0 = 2.725 K for computing the numerical value of η. Putting all the numerical values in, we finally arrive at the following dimensionless form of the Saha equation: X2 1 −X = 2.50 × 106 η ˜ T −3/2e−1/ ˜ T (2.26) 31 where ˜ T ≡kBT/BH = T/(157894 K). This is a simple quadratic equation, which can be easily solved for X. The solution is X(T) = 2 1 + q 1 + (1.6 × 10−6η) ˜ T 3/2e1/ ˜ T (2.27) 0 0.2 0.4 0.6 0.8 1 2500 3000 3500 4000 4500 5000 Ionization Fraction, X Temperature [K] 2./(1.+sqrt(1.+1.6e-66.3e-10(x/157894.)1.5exp(157894./x))) Let us find an approximate temperature, Trec, at which the universe is half neutral, X = 1/2. Then we have ˜ T 3/2 rec e1/ ˜ Trec = 5 × 106/η, whose numerical solution is ˜ Trec = 0.0237, or Trec = 3740 K . ¶ It may be illustrative to find Trec for η = 1 (i.e., equal numbers of baryons and photons). We find Trec ≈7900 K; thus, even in the situation where there is one photon per baryon, the temperature of the universe at the hydrogen formation epoch (where the universe is half neutral) is significantly lower than the temperature corresponding to the hydrogen ionization energy, 1.6 × 105 K.
Now, with the ionization history calculated, we can re-calculate the ratio of the mean free time to the Hubble time to find the temperature of the epoch at which photons decouple from the ¶While it is not very accurate, we may solve this equation iteratively. Taking the logarithm of both sides, we get 3 2 ln ˜ Trec + 1 ˜ Trec = ln 5 × 106 η .
The zeroth-order iterative solution would then be obtained by ignoring the first term on the left hand side: ˜ Trec = 1/ ln(5 × 106/η), which gives Trec ≃4300 K for η = 6.3 × 10−10. One may improve accuracy of the solution by inserting this zeroth-order solution into the first term on the left hand side, and resolving for ˜ Trec. In any case, this analysis shows that the recombination temperature is reduced by a factor of ln(1/η).
32 electron scattering. We rewrite Eq. (2.14) as H σT nec = H0 c p Ωm(1 + z)3 σT nCMB 1 0.76ηX(z) = 0.94 × 10−2 X(z) 1000 1 + z 3/2 6.3 × 10−10 η = 0.94 × 10−2 X(T) 2725 K T 3/2 6.3 × 10−10 η . 0.01 0.1 1 10 100 2500 3000 3500 4000 4500 5000 (Mean Free Time) / (Hubble Time) Temperature [K] 0.94e-2/(2./(1.+sqrt(1.+1.6e-66.3e-10(x/157894.)1.5exp(157894./x))))(2725./x)1.5 1 Indeed, the mean free time becomes comparable to the Hubble time when the temperature of the universe is Tdec ≈3000 K , or zdec ≈1100 .
(The solution of H/(σT nec) = 1 from the above equation gives Tdec = 3065 K.) This is the epoch at which the universe became transparent, and photons began to propagate freely in space. We are detecting photons coming from this epoch as the cosmic microwave background. This epoch of often called the “decoupling epoch,” or the “last scattering surface.” Freeze-out of Recombination The above equilibrium calculation shows that all of electrons will eventually be captured by protons, leaving no free electrons at low temperatures. However, as the recombination rate is proportional to nenp, the rate falls rapidly as the number densities go down due to the expansion of the universe.
Eventually the recombination time becomes comparable to the Hubble time, and the recombination stops. This is the epoch of recombination freeze-out.
The recombination time per proton is given by 1/(⟨σrecv⟩ne), where ⟨σrecv⟩is given by ⟨σrecv⟩= 2.33 × 10−14 ln(1/ ˜ T) ˜ T 1/2 cm3 s−1.
(2.28) 33 It is convenient to divide this by the speed of light: ⟨σrecv⟩/c = 7.77 × 10−25 ln(1/ ˜ T) ˜ T 1/2 cm2, (2.29) which is the same order of magnitude as the Thomson-scattering cross section, σT = 6.65 × 10−25 cm2. Then, the ratio of the recombination time to the Hubble time is given by H ⟨σrecv⟩ne = H0 c p Ωm(1 + z)3 (⟨σrecv⟩/c)nCMB 1 0.76ηX(z) = 1.06 × 10−3 X(T) ln(157894/T) 2725 K T 6.3 × 10−10 η .
As this ratio is smaller than that for the decoupling (Eq. (2.28)) by a factor of ten, the recom-bination freeze-out occurs after photons decouple from the plasma.
The above ratio (Eq. (2.30)) crosses unity at Tfreeze−out = 2700 K , which is lower than the decoupling tempera-ture, ≈3000 K. 0.01 0.1 1 10 100 2500 3000 3500 4000 4500 5000 (Recombination Time) / (Hubble Time) Temperature [K] 1.06e-3/(2./(1.+sqrt(1.+1.6e-66.3e-10(x/157894.)1.5exp(157894./x))))(2725./x)/log(157894./x) 1 We can also calculate the residual ionization fraction of the recombination, i.e., the ionization fraction left after the recombination freeze-out, by evaluating X(T) at T = 2700 K. We find X(2700 K) = 2.7 × 10−4 (2.30) In other words, after the recombination freeze-out, there remains one free electron per about 4000 hydrogen atoms. This seems like a small amount: however, this small amount of residual electrons is necessary for forming hydrogen molecules via H +e−→H−+γ followed by H−+H →H2 +e−.
The hydrogen molecules formed in this way are expected to play an important role in cooling gas and forming the first generation of stars (Galli and Palla, A&A, 335, 403 (1998)).
34 2.4 Temperature Anisotropy 2.4.1 Dipole Anisotropy The dipole anisotropy at the level of mK is caused by the motion of Solar System with respect to the rest frame of the cosmic microwave background. Due to the Doppler effect, momentum of photons of the microwave background appears to be larger in the direction of our motion: pO(ˆ n) = p γ(1 −ˆ n · ⃗ v c) , (2.31) where pO(ˆ n) is the observed momentum of photons coming from a direction ˆ n,∥p is the momentum in the rest frame of the cosmic microwave background, and γ ≡(1 −v2/c2)−1/2 is the Lorentz factor. Expanding this expression to the first order in v/c, we obtain pO(ˆ n) ≈p 1 + ˆ n · ⃗ v c .
(2.32) As expected, when photons are coming from the direction of our motion, i.e., ˆ n·ˆ v = 1, the observed momentum takes on the maximum value, pO = p(1 + v/c).
Now, as the cosmic microwave background is a blackbody, we can relate the change in the momen-tum of photons to the change in the temperature as δT T ≡TO(ˆ n) −T T = pO(ˆ n) −p p = ˆ n · ⃗ v c .
(2.33) ∥Since photons are coming toward us, the propagation direction of photons, ˆ p, is opposite of the line of sight direction, i.e., ˆ p = −ˆ n.
35 This is the formula for the dipole anisotropy. The measured value of dipole in the direction of motion is δT = 3.355 ± 0.008 mK (Table 6 of Jarosik et al., ApJS, 192, 14 (2011)). The direction of motion in Galactic coordinates is (l, b) = (263.99 ± 0.14, 48.26 ± 0.03) (in degrees). This gives δT/T = 3.355 × 10−3/2.725 = 1.23 × 10−3. By equating this to v/c, we find∗∗ v = 368 km/s (2.34) This velocity should be the vector sum of various components: ⃗ v = (⃗ vSun −⃗ vMW) + (⃗ vMW −⃗ vLG) + ⃗ vLG, (2.35) where 1. ⃗ vSun −⃗ vMW is the orbiting velocity of Solar System with respect to the center of our Galaxy (Milky Way). This component is known (222.0±5.0 km/s in the direction of (l, b) = (91.1, 0) degrees), and thus can be subtracted.
2. ⃗ vMW −⃗ vLG is the velocity of our Galaxy (Milky Way) with respect to the center-of-mass of Local Group of galaxies. As the dominant masses of Local Group are given by Milky Way and Andromeda Galaxy (M31), which is a nearby galaxy, this component is small (≈80 km/s).
3. ⃗ vLG is the velocity of the center-of-mass of Local Group with respect to the rest frame of the cosmic microwave background. This component represents the cosmological velocity flow (called the “bulk flow”).
It turns out that the sum of the first two components, i.e., motion of Sun relative to the center-of-mass of Local Group, has a magnitude (307 km/s) comparable to the measured velocity, but is in nearly the opposite direction ((l, b) = (105 ± 5, −7 ± 4) degrees; Yahil, Tammann & Sandage, ApJ, 217, 903 (1997)). As a result, the inferred bulk flow component has a large velocity: vLG = 626 ± 30 km/s, (2.36) in the direction of (l, b) = (276 ± 2, 30 ± 2) degrees (Sandage, Reindl & Tammann, ApJ, 714, 1441 (2010)).
Who is pulling Local Group? One obvious nearby mass concentration is the Virgo clusters of galaxies (at 16.5 Mpc). After subtracting an estimate of the infall velocity to Virgo (220 km/s) in the direction of (l, b) = (283.8, 74.5) degrees, the velocity of Local Group corrected for the Virgo infall is vLG = 495 ± 25 km/s Corrected for Virgo infall (2.37) in the direction of (l, b) = (275 ± 2, 12 ± 4) degrees (Sandage, Reindl & Tammann, ApJ, 714, 1441 (2010)). Therefore, Virgo cannot be solely responsible for the motion of Local Group. We still do not know who is responsible for this velocity.
∗∗Rotation velocity of Earth around Sun, 30 km/s, has been removed from this value, as this component varies annually.
36 Recently, Nusser and David (arXiv:1101.1650) show that the measurements of peculiar velocities of nearby spiral galaxies within 100 h−1 Mpc give the velocity of 333 ± 38 km/s in the direction of (l, b) = (276 ± 3, 14 ± 3) degrees after correcting for the Virgo infall. This measurement accounts for most of the velocity inferred from the cosmic microwave background, but is still lower. This implies that mass concentrations on > 100 h−1 Mpc are partially responsible for the bulk flow of Local Group. It is encouraging that the directions inferred from both methods are in an excellent agreement.
2.4.2 Sachs–Wolfe Effect After removing the dipole anisotropy, what remains is the primordial anisotropy. It exhibits much richer angular distributions than dipole. This component can be divided into 2 contributions: 1. Gravitational effect (called the Sachs–Wolfe effect), and 2. Scattering effect.
This problem can be dealt with most intuitively by following the evolution of momentum of photons in a clumpy universe.
In a homogeneous universe, we know that the momentum just redshifts away as p ∝1/a; thus, the evolution equation would simply be: 1 p dp dt = −1 a da dt .
(2.38) However, in a clumpy universe, photons receive gravitational blue/redshifts. The evolution equa-tion, which you will derive in the homework question, is (with c = 1) 1 p dp dt = −1 a da dt − X i γi a ∂Ψ ∂xi −∂Φ ∂t .
(2.39) 37 Note that we carefully distinguish between the total derivatives and partial derivatives here. The γi is a unit vector satisfying X ij δijγiγj = 1, (2.40) which gives a direction of momentum. There is a factor of 1/a in the second term because xi denotes the comoving coordinates.
The perturbation variables, Ψ and Φ, are the Newtonian potential and the so-called curvature perturbation, respectively. They are defined by the following perturbed metric: ds2 = −[1 + 2Ψ(xi, t)]dt2 + a2(t)[1 + 2Φ(xi, t)] X ij δijdxidxj.
(2.41) For example, for a point mass with mass M, these variables reduce to the familiar forms: Ψ = −GM/r and Φ = −Ψ = GM/r (with c = 1).
The magnitude of momentum, p, is defined by p2 = X ij gijpipj = −g00(p0)2.
(2.42) The last equality follows from the normalization condition of momentum of massless particles, P µν gµνpµpν = 0. From this, one finds that p = p (1 + 2Ψ)p0 ≃(1+Ψ)p0. Note that it is p, rather than p0, that is directly related to the temperature, i.e., T ∝p. Finally, from the above definition 38 of p and the normalization condition for the unit vector γi (P ij δijγiγj = 1), one can derive the relation between pi and γi to the first order in perturbation: pi = pγi a (1 −Φ).
(2.43) It is convenient to rewrite equation (2.39) using γi a ∂Ψ ∂xi = ∂Ψ ∂t + γi a ∂Ψ ∂xi −∂Ψ ∂t = dΨ dt −∂Ψ ∂t .
(2.44) Then, we obtain 1 p dp dt = −1 a da dt −dΨ dt + ∂Ψ ∂t −∂Φ ∂t , (2.45) which can be readily integrated to give ln(ap)O = ln(ap)E + (ΨE −ΨO) + Z tO tE dt ∂ ∂t(Ψ −Φ), (2.46) where “O” and “E” denote the “observed epoch” and “emitted epoch,” respectively. Finally, we rewrite this result using the temperature anisotropy: ap ∝a ¯ T 1 + δT ¯ T .
(2.47) Here, ¯ T is the mean temperature and depends only on time. Taylor-expanding the logarithm to the first order in δT/ ¯ T, and recalling aO ¯ TO = aE ¯ TE for the mean temperature, we finally obtain: δT ¯ T O = δT ¯ T E + (ΨE −ΨO) + Z tO tE dt ∂ ∂t(Ψ −Φ).
(2.48) To this, we must add the Doppler terms due to the velocity at emission and observed location: δT ¯ T O = δT ¯ T E + (ΨE −ΨO) + Z tO tE dt ∂ ∂t(Ψ −Φ) + X i γi(vi E −vi O) (2.49) The last term, −P γivi O, is the dipole anisotropy discussed in the previous section.
39 This result has a simple interpretation.
1. There was an initial temperature anisotropy at the last scattering surface, δT/ ¯ T|E (which remains to be calculated), as well as the Doppler effect, P γivi E.
2. After the last scattering, photons escape from a potential well, losing energy: δT/ ¯ T|E + ΨE + P γivi E.
3. While photons are propagating toward us, photons gain or lose energy depending on how Ψ −Φ (≈2Ψ) changes with time, giving δT/ ¯ T|E + ΨE + R tO tE dt ∂ ∂t(Ψ −Φ) + P γivi E.
4. Finally, photons enter a potential well at our location, ΨO, gaining energy. Also, they receive the Doppler shift due to our local motion, giving δT/ ¯ T|E + ΨE −ΨO + R tO tE dt ∂ ∂t(Ψ −Φ) + P γi(vi E −vi O).
In particular, δT/ ¯ T|E + ΨE −ΨO is usually called the Sachs–Wolfe effect, and R tO tE dt ∂ ∂t(Ψ −Φ) is called the integrated Sachs–Wolfe effect. All of these terms were derived by Sachs and Wolfe in 1967 (Sachs and Wolfe, ApJ, 147, 73 (1967)).
Adiabatic Initial Condition How do we calculate the initial temperature fluctuation at the last scattering surface, δT/ ¯ T|E? To calculate this, we must specify the initial condition for perturbations. In principle, this cannot be known a priori without using the observational data. There are two widely explored initial conditions: • Adiabatic initial condition • Non-adiabatic initial condition The current observational data favor the adiabatic initial condition, and we have not yet found any evidence for non-adiabatic initial condition. Therefore, we shall focus on the adiabatic initial condition.
What is it? This is the initial condition in which radiation and matter are perturbed in a similar way. It is called adiabatic, as the entropy density per matter particle is constant (unperturbed): S/a3 nM ∝T 3 nM = constant, (2.50) whose variation gives ¯ T 3 ¯ nM 3δT ¯ T −δnM ¯ nM = 0.
(2.51) Therefore, the adiabatic initial condition corresponds to δT ¯ T = 1 3 δnM ¯ nM = 1 3 δρM ¯ ρM .
(2.52) 40 “Non-adiabatic initial conditions” would have δT/ ¯ T ̸= δρM/(3¯ ρM).
As this is the initial condition, it holds only on very large scales, much larger than the horizon size at the last scattering surface. While it is not obvious or intuitive, on such large scales, as you derive in the homework question, the density fluctuation during the matter-dominated era is related to the Newtonian potential as δρM ¯ ρM = −2Ψ (Matter-dominated & super-horizon).
(2.53) This gives, on large scales, the initial temperature fluctuation of δT ¯ T E = 1 3 δρM ¯ ρM E = −2 3ΨE.
(2.54) Then, the Sachs–Wolfe formula gives δT ¯ T O = 1 3ΨE + . . .
(2.55) Therefore, on large scales, an over-density region (i.e., a potential well) appears as a cold spot on the sky. While the temperature at the bottom of the potential well is hotter than the average (−2 3Ψ), photons lose more energy (Ψ) as they climb up the potential well, resulting in a cold spot (−2 3Ψ + Ψ = 1 3Ψ).
41 Observing the primordial perturbation via the Sachs–Wolfe effect As you have seen from the homework problem, Ψ = −Φ during the matter era, and both Ψ and Φ remain constant during the matter era. Therefore, after the decoupling, but before the dark energy dominated era, the integrated Sachs–Wolfe effect vanishes. In this case the observed temperature anisotropy toward a direction ˆ ni is given by δT ¯ T O (ˆ ni) = 1 3ΨE −ΨO − X i ˆ ni(vi E −vi O) during the matter era.
(2.56) Here, we have used the fact that the direction of photon, γi, is equal to −ˆ ni.
Now, let us consider temperature anisotropy on very large angular scales - the angular scale that is greater than the Hubble length at the decoupling epoch. As you have seen from the homework problem, the velocity perturbation vanishes in the large-scale limit, as it is proportional to ϵ, which is given by (with c = 1) ϵ ≡k ˙ a = k aH , (2.57) where k is the comoving wavenumber. Let us calculate the angular size of the Hubble length at z = 1090. The comoving wavenumber, λ, is related to k as λ = 2π/k. The angular size that corresponds to the half wavelength is then θ = λ/2 dA , (2.58) 42 where dA is the comoving angular diameter distance: dA ≡(1 + z)DA = c Z z 0 dz′ H(z′) = 14 Gpc for z = 1090.
(2.59) The numerator is the half-wavelength corresponding to the Hubble size: λH 2 = π kH = π aH .
(2.60) For ΩMh2 = 0.13 and ΩRh2 = 4.17 × 10−5, we find aH = p ΩMh2(1 + z) + ΩRh2(1 + z)2 3 Gpc = 4.6 Gpc−1 for z = 1090.
(2.61) Therefore, the angular size that corresponds to the Hubble length at z = 1090 is θ = π aHdA = 180◦ 4.6 × 14 = 2.8◦.
(2.62) This means that, for angular scales much greater than 3◦, we can ignore the contribution from the velocity perturbation at z = 1090, i.e., vE, and obtain δT ¯ T O (ˆ ni) = 1 3ΨE(r∗ˆ ni) −ΨO + X i ˆ nivi O on large angular scales.
(2.63) Here, we explicitly show that ΨE is a three-dimensional quantity, ΨE = ΨE(xi), and what we observe is the potential at the last scattering surface whose comoving distance is r∗= dA(z = 1090) = 14 Gpc. On the other hand, the second term, ΨO, is the value of Ψ at our location, which is just a number, and merely adds a constant to the value of δT/T over all sky; thus, this is a monopole term (l = 0). The third term, ˆ n · ⃗ vO, is the dipole anisotropy (l = 1) due to our local motion, which we have studied in the previous section.
Therefore, if we ignore the monopole and dipole and focus on the primordial anisotropy with l ≥2, we are left with the Sachs–Wolfe term: δT ¯ T O (ˆ ni) = 1 3ΨE(r∗ˆ ni) on large angular scales and l ≥2.
(2.64) This is an important result - since the angular size is greater than that of the Hubble length at z = 1090, the temperature anisotropy we observe on this scale is not altered by the physics at z > 1090. In other words, what we observe on large angular scales must reflect the initial, primordial perturbation (except for the integrated Sachs-Wolfe effect which we ignore here).
In order to characterize the observed temperature anisotropy, let us consider a patch of the sky whose center has the direction vector ˆ ni 0, and introduce the angular coordinates on this patch, ⃗ θ = (sin θ cos φ, sin θ sin φ). Furthermore, let us assume that the angular size is greater than 3◦, but is much less than 60◦, which corresponds to 1 radian. In this case, θ ≪1, and thus the angular coordinates become ⃗ θ = (θ cos φ, θ sin φ).
(2.65) 43 We then Fourier-transform the temperature anisotropy on this patch: δT ¯ T O (ˆ ni) = 1 ¯ T Z d2l (2π)2 f δT(⃗ l)ei⃗ l·⃗ θ.
(2.66) We also Fourier-transform Ψ in 3-d space: ΨE(xi) = Z d3k (2π)3 ˜ Ψ(⃗ k)ei⃗ k·⃗ x.
(2.67) Remember that Ψ is constant during the matter era.
Now, we wish to find the relation between f δT(⃗ l) and ˜ Ψ(⃗ k): f δT(⃗ l) ¯ T = Z d2θe−i⃗ l·⃗ θ 1 3 Z d3k (2π)3 ˜ Ψ(⃗ k)ei⃗ k·(r∗ˆ n) = Z d2θe−i⃗ l·⃗ θ 1 3 Z d3k (2π)3 ˜ Ψ(⃗ k)ei⃗ k⊥·(r∗⃗ θ)eik∥r∗cos θ = Z d2θ1 3 Z d3k (2π)3 ˜ Ψ(⃗ k)ei(⃗ k⊥r∗−⃗ l)·⃗ θeik∥r∗cos θ (2.68) Here, we have defined ⃗ k⊥and k∥such that ⃗ k = (⃗ k⊥, k∥), (2.69) where ⃗ k⊥is the wavenumber vector on the patch, and k∥is the wavenumber along the line of sight.
44 To proceed further, we use the fact that we consider the region in which θ ≪1 (so that we can treat a section of the sky as a flat surface): f δT(⃗ l) ¯ T ≈ Z d2θ1 3 Z d3k (2π)3 ˜ Ψ(⃗ k)ei(⃗ k⊥r∗−⃗ l)·⃗ θeik∥r∗ (cos θ ≈1) = 1 3 Z d3k (2π)3 ˜ Ψ(⃗ k) Z d2θei(⃗ k⊥r∗−⃗ l)·⃗ θ eik∥r∗ = 1 3 Z d3k (2π)3 ˜ Ψ(⃗ k) h (2π)2δ(2) D (⃗ k⊥r∗−⃗ l) i eik∥r∗ = 1 3 Z d2k⊥dk∥ (2π)3 ˜ Ψ(⃗ k) " (2π)2 δ(2) D (⃗ k⊥−⃗ l/r∗) r2 ∗ # eik∥r∗ = 1 3r2 ∗ Z dk∥ 2π ˜ Ψ ⃗ k⊥= ⃗ l r∗ , k∥ !
eik∥r∗.
(2.70) Finally, there is no way to predict the value of ˜ Ψ(⃗ k) for any given value of ⃗ k because Ψ is a random (stochastic) variable. However, what we can do it to calculate its variance, which is called the power spectrum: ⟨˜ Ψ(⃗ k)˜ Ψ∗(⃗ k)⟩ = (2π)3PΨ(k)δ(3) D (⃗ k −⃗ k′), (2.71) 1 ¯ T 2 ⟨f δT(⃗ l)f δT ∗(⃗ l′)⟩ = (2π)2Clδ(2) D (⃗ l −⃗ l′).
(2.72) The angular power spectrum of the temperature anisotropy, Cl, is an observable quantity.
Therefore, the remaining task is to relate the observable, Cl, to the power spectrum of Φ, PΨ(k).
45 The result is Cl = 1 9r2 ∗ Z dk∥ 2π PΨ s l2 r2 ∗ + k2 ∥ !
.
(2.73) Note that the small-angle approximation, θ ≪1, corresponds to l ≫1, as these are related via l = π θ .
(2.74) In terms of k and r∗, we have l = kr∗= 14 k 1 Gpc−1 .
(2.75) For example, the multipole that corresponds to the wavenumber of the Hubble horizon size at z = 1090, aH = 4.6 Gpc−1, is lH = 64.
(2.76) Therefore, the argument given here is valid only for 1 ≪l ≪64.
Now, we must make an assumption about the form of PΨ(k). We now believe that primordial fluctuations were generated during the period of inflation - an exponential expansion of the universe during a tiny fraction of a second after the birth of the universe. As you will learn from Bhaskar toward the end of this course, inflation predicts the following power-law form of the initial power spectrum: PΨ(k) ∝kns−4, (2.77) where ns is called the spectral tilt. The current data give (Komatsu et al., ApJS, 192, 18 (2011)) ns = 0.968 ± 0.012 (68% CL). As for the normalization of PΨ(k), we usually parametrize it as PΨ(k) = 2π2 k3 ∆2 Ψ(k0) k k0 ns−1 , (2.78) where k0 is some arbitrary pivot wavenumber which is often taken to be k0 = 2 Gpc−1 = 0.002 Mpc−1, and ∆2 Φ(k0) is the normalization.
The special case is ns = 1 (which is called the Harrison-Zel’dovich-Peebles spectrum, and is close to the observed value, ns = 0.968 ± 0.012), for which Cl = 2π l2 1 9∆2 Ψ(k0).
(2.79) This motivates our writing l2Cl 2π = 1 9∆2 Ψ(k0) = 1 9 k3PΨ(k) 2π2 for ns = 1 (2.80) Since this quantity does not depend on l, this spectrum (with ns = 1) is called the scale-invariant spectrum. Note also that k3PΨ(k) does not depend on k. For ns ̸= 1, we have †† l2Cl 2π = 1 9∆2 Ψ(k0) l k0r∗ ns−1 √π 2 Γ[(3 −ns)/2] Γ[(4 −ns)/2].
(2.84) ††This formula cannot be used for small l (such as l = 2) because we have treated our patch of the sky as a flat surface, which allowed us to use the familiar Fourier transform. For the full-sky treatment, we must take into account 46 We are not quite done yet. While Ψ (and hence Φ) is constant during the matter era, they change as the universe transitions from radiation-dominated to matter-dominated. They also change as the universe exits the inflationary period and becomes radiation dominated. Therefore, PΨ(k) that we determine from the observation of the microwave background, which is PΨ(k) during the matter era, cannot be directly compared with the prediction from inflation.
Fortunately, there is an easy solution for this problem. On very large scales, k ≪aH, there exists a conserved quantity called ζ, which is defined as ζ ≡Φ −aH k V = Φ −V ϵ (2.85) the fact that the sky is a sphere. For this purpose, we must use spherical harmonics decomposition rather than the Fourier transform. In any case, the exact result in the Sachs–Wolfe limit is Cl = 2 9π Z k2dkPΨ(k)j2 l (kr∗) = 2π 9 ∆2 Ψ(k0) 1 (k0r∗)ns−1 √π 2 Γ[(3 −ns)/2] Γ[(4 −ns)/2] Γ[l + (ns −1)/2] Γ[l + (5 −ns)/2].
(2.81) For ns = 1, Cl = 2π l(l + 1) 1 9∆2 Ψ(k0), (2.82) or l(l + 1)Cl 2π = 1 9∆2 Ψ(k0) for ns = 1 (2.83) For l ≫1, we indeed recover the flat-sky result, l2Cl/(2π) = ∆2 Ψ(k0)/9. This result explains why people tend to plot l(l + 1)Cl/(2π) against l.
47 This quantity remains constant on k ≪aH regardless of the contents of the universe. It is ζ that is predicted by theories of inflation. Now, if we plug in the large-scale solution for the velocity perturbation during the matter era that you find from the homework problem, V = −2 3ϵΦ, we find ζ = Φ + 2 3Φ = 5 3Φ = −5 3Ψ during the matter era.
(2.86) Therefore, the Sachs–Wolfe formula is modified to δT ¯ T O (ˆ ni) = −1 5ζ(r∗ˆ ni) (2.87) and l2Cl 2π = 1 25∆2 ζ(k0) l k0r∗ ns−1 √π 2 Γ[(3 −ns)/2] Γ[(4 −ns)/2].
(2.88) The current data give (Komatsu et al., ApJS, 192, 18 (2011)) ∆2 ζ(k0) = (2.43 ± 0.09) × 10−9.
(2.89) One should be impressed by these results! Using the observation of the cosmic microwave background, we were able to measure the amplitude and the scale-dependence of the initial perturbations generated during inflation. Studying the high-energy world before the Big Bang became a real science! 0 500 1000 1500 2000 2500 3000 0 10 20 30 40 50 60 Angular Power Spectrum, l(l+1)Cl/(2π) [µK2] Multipole, l ’bestfit_lensedCls.dat’ ’wmap_tt_spectrum_7yr_v4p1.txt’ ’wmap_binned_tt_spectrum_7yr_v4p1.txt’u 1:4:5 2.43e-9/25.(2.725e6)2(0.99(x/14/2)(0.968-1)) ’doppler.dat’ ’sachswolfe.dat’ 48 2.4.3 Gravitational Waves So far, we have studied how gravitational potential, Ψ, produces anisotropy in the cosmic microwave background. However, this is not the only source. Another source of temperature anisotropy is the gravitational waves. Gravitational waves stretch space, causing photons to redshift or blueshift depending on the phase of gravitational waves. This stretching of space is described by the ten-sor metric perturbation, hij(xi, t). The metric that includes hij(xi, t) on top of the Friedmann-Robertson-Walker background is given by (with c = 1): ds2 = −dt2 + a2(t) X ij [δij + hij(xi, t)]dxidxj.
(2.90) This metric perturbation, hij(xi, t), is the gravitational wave itself. In other words, it is hij(xi, t) that propagates as a wave. Gravitational waves have the following properties: 1. Gravitational waves are transverse (just like electromagnetic waves).
Therefore, they do not distort space along their propagation direction, but only distort space in the direction perpendicular to their propagation.
2. Gravitational waves have two polarization states (just like electromagnetic waves).
3. Gravitational waves (gravitons) are spin-2. (For comparison, electromagnetic waves (pho-tons) are spin-1.) Therefore, hij is a rank-2 tensor field, whereas electromagnetic waves are described by a vector potential, Ai.
As an example, let us take a single plane wave as a gravitational wave propagating in z direction (z ≡x3). We have hij ∝eik3x3. Because hij is transverse, we must have X j kjhij = k3hi3 = 0.
(2.91) As the metric is a symmetric tensor, hij is also symmetric, i.e., hij = hji. Using this information, we can write: hij = h11 h12 0 h12 h22 0 0 0 0 .
(2.92) However, since hij has only two polarization states, h11 and h22 must be related somehow. This relation can be found by noting that gravitational waves change the shape of space, but do not change the size. In other words, it shears space, but does not expand or contract it. This means that the determinant of δij + hij is unity: det(δij + hij) = (1 + h11)(1 + h22) −h2 12 = 1.
(2.93) To first order in hij, this condition gives h11 + h22 = 0.
(2.94) 49 Therefore, hij is a traceless tensor, P i hii = 0.
We conventionally write the components of hij propagating in z direction as hij = h+ h× 0 h× −h+ 0 0 0 0 , (2.95) where h+ and h× represent two linear polarization states of gravitational waves. As one may guess from this matrix, a wave would distort space along x and y directions when h+ ̸= 0 and h× = 0, and it would distort space along 45◦degrees when h+ = 0 and h× ̸= 0. This means that h+ and h× are not invariant under coordinate rotation. On the contrary, for clock-wise rotation of coordinates by an angle φ, h+ and h× transform as h+ h× !
→ h′ + h′ × !
= cos 2φ −sin 2φ sin 2φ cos 2φ ! h+ h× !
, (2.96) or equivalently h+ ± ih× →h′ + ± ih′ × = e±2iφ(h+ ± ih×). Therefore, gravitational waves are indeed a spin-2 field.
How would gravitational waves produce temperature anisotropy?
Suppose that we have a gravitational wave propagating in z direction.
1. If photons are propagating in the same direction (i.e., z direction), then there would be no change in temperature, as a gravitational wave does not distort space along its propagation direction.
2. If photons are propagating in x direction, then there would be redshift (δT < 0) if ˙ h+ > 0 (because space is stretching in x direction), and blueshift (δT > 0) if ˙ h+ < 0 (because space is contracting in x direction).
3. If photons are propagating in y direction, then there would be blueshift (δT > 0) if ˙ h+ > 0 (because space is contracting in y direction), and redshift (δT < 0) if ˙ h+ < 0 (because space is stretching in y direction).
4. If photons are propagating in 45◦direction, then there would be redshift (δT < 0) if ˙ h× > 0 (because space is stretching in 45◦direction), and blueshift (δT > 0) if ˙ h× < 0 (because space is contracting in 45◦direction).
5. If photons are propagating in 135◦direction, then there would be blueshift (δT > 0) if ˙ h× > 0 (because space is contracting in 135◦direction), and redshift (δT < 0) if ˙ h× < 0 (because space is stretching in 135◦direction).
In general, as you derive in the homework problem, ˙ hij changes momentum of photons as 1 p dp dt = −˙ a a −1 2 X ij ˙ hijγiγj, (2.97) where as usual γi is the unit vector for a propagation direction of photons.
50 51 Once again, by converting momentum to temperature anisotropy, and recalling ˆ ni = −γi where ˆ ni is the line-of-sight unit vector, we obtain δT ¯ T O (ˆ ni) = δT ¯ T E (ˆ ni) −1 2 X i′j′ ˆ ni′ˆ nj′ Z tO tE dt ˙ hi′j′(r(t)ˆ ni, t), (2.98) where we have made explicit that hij depends on spatial coordinates and time, and that xi = ˆ nir(t) where r(t) is the comoving distance to the time t. Note that this formula is valid for hij propagating in any directions (not just z direction).
We now need to know how hij changes with time. For this purpose, we need to solve Einstein’s equation for hij. This can be done in a straightforward way: Einstein’s equation is (with c = 1) Rµν −1 2gµνR = 8πGTµν.
(2.99) We simply calculate the left hand side of this equation using the metric given by equation (2.90), with the transverse (P j ∂hij/∂xj = 0) and traceless (P i hii = 0) conditions. The result is remark-ably simple: −1 2□hij = 8πGδTij, (2.100) where δTij is the linear perturbation to Tij that would affect gravitational waves, and □hij is □hij ≡gµνhij;µν.
(2.101) 52 For a perfect fluid, δTij = 0, and thus we have an equation describing waves propagating in vacuum: □hij = 0.
(2.102) For the Friedmann-Robertson-Walker metric, this equation becomes ¨ hij + 3 ˙ a a ˙ hij −1 a2 ∇2hij = 0 (2.103) By Fourier-transforming hij: hij(xi, t) = Z d3k (2π)3 ˜ hij(ki, t)ei P j kjxj, (2.104) the wave equation becomes ¨ ˜ hij + 3 ˙ a a ˙ ˜ hij + k2 a2 ˜ hij = 0.
(2.105) For a matter-dominated universe, a ∝t2/3, the solutions of this equation are ˜ hij(ki, t) = Aij(ki)3j1(kη) kη + Bij(ki)y1(kη) kη , (2.106) where Aij and Bij are constant matrices (which represent initial conditions) and η is defined as η ≡ Z dt a(t) = 3t2/3 0 t1/3, (2.107) which is, in a flat universe, related to the comoving distance as r(t) = η0 −η(t) (with c = 1 and η0 = 3t0 is the present-day value). The functions j1 and y1 are the spherical Bessel functions of the first and second kind, respectively: j1(x) = sin(x) x2 −cos(x) x , (2.108) y1(x) = −cos(x) x2 −sin(x) x .
(2.109) In order to determine the initial conditions, let us take the limit of t →0. We find ˜ hij(ki, t →0) →Aij(ki) −Bij(ki) (kη)3 .
(2.110) The second term blows up as t →0, which is unphysical. Therefore, we take Bij = 0 as the initial condition. The final form of the solution during the matter era is then ˜ hij(ki, t) = Aij(ki)3j1(kη) kη (2.111) and its time derivative is ˙ ˜ hij(ki, t) = −kAij(ki) a(t) 3j2(kη) kη (2.112) 53 where j2(x) ≡ 3 x3 −1 x sin(x) −3 x2 cos(x).
(2.113) With this result, we can finally write the temperature anisotropy as δT ¯ T O (ˆ ni) = δT ¯ T E (ˆ ni) + 3 2 X i′j′ ˆ ni′ˆ nj′ Z d3k (2π)3 Ai′j′(⃗ k) Z tO tE kdt a(t) j2(kη) kη ei⃗ k·ˆ nr(t) = δT ¯ T E (ˆ ni) + 3 2 X i′j′ ˆ ni′ˆ nj′ Z d3k (2π)3 Ai′j′(⃗ k) Z xO xE dxj2(x) x eiˆ k·ˆ n(xO−x), (2.114) where x ≡kη. Since j2(x)/x peaks at x ≈2, the integral over x is dominated by the modes with kη ≈2, with higher k modes highly suppressed. This will be reflected on the shape of the angular power spectrum of temperature anisotropy from gravitational waves.
-0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 3j1(kη)/(kη) [solid]; 3j2(kη)/(kη) [dashed] kη 3(sin(x)/x3-cos(x)/x2) 3((3/x4-1/x2)sin(x)-3cos(x)/x3) Calculation of Cl from equation (2.114) is a bit involved, so let us just give the result: Cl = (l + 2)!
(l −2)!
Z dk 2πPh(k) Z ηO ηE kdη3j2(kη) kη jl[k(ηO −η)] k2(ηO −η)2 2 (2.115) where Ph(k) is the power spectrum of each polarization state of the gravitational wave: ⟨h+(⃗ k)h∗ +(⃗ k′)⟩= ⟨h×(⃗ k)h∗ ×(⃗ k′)⟩= (2π)3Ph(k)δ(3) D (⃗ k −⃗ k′).
(2.116) Similarly to what we have done for the scalar perturbation (gravitational potential contribution), we usually parametrize Ph as Ph(k) = 2π2 k3 ∆2 h(k0) k k0 nt , (2.117) 54 and nt is called the tensor spectral tilt. We then define the so-called tensor-to-scalar ratio, r, defined by r ≡4∆2 h(k0) ∆2 ζ(k0) , (2.118) where a factor of four is there for a historical reason. 1 10 100 1000 0 50 100 150 200 250 300 Angular Power Spectrum, l(l+1)Cl/(2π) [µK2] Multipole, l ’bestfit_lensedCls.dat’ ’wmap_tt_spectrum_7yr_v4p1.txt’ ’wmap_binned_tt_spectrum_7yr_v4p1.txt’u 1:4:5 ’powerspectrum_from_gravitationalwave.dat’u 1:($20.24) Simple single-field inflation models predict a relation between the tensor tilt and the tensor-to-scalar ratio as r = −8nt (2.119) Therefore, using this relation, we have only three parameters for characterizing the primordial perturbation spectra produced by inflation: ∆ζ(k0), r, and ns. Among these, r is particularly important because a detection of non-zero r means a detection of primordial gravitational wave created during inflation. Many experts think that the detection of r would be a proof of inflation.
Currently, we have not detected r, and the latest limit on r is (Komatsu et al., Astrophysical Journal Supplement Series, 192, 18 (2011)) r < 0.24 (95% C.L.) (2.120) 55 This constraint comes from the temperature anisotropy spectra that we have learned so far. Since the gravitational wave spectrum adds power to lower multipoles (mainly l < 50), it tilts the total power spectrum. This effect can be absorbed by increasing ns (which will make the contribution from ζ at low multipoles smaller), and thus there is a positive correlation between ns and r. This gives a fundamental limit on r ≈0.1 we can reach by using the temperature power spectrum alone.
In order to break this correlation, one must use not only the temperature power spectrum, but only the power spectrum of polarization of the cosmic microwave background.
56 2.5 Polarization The cosmic microwave background is weakly polarized.
Polarization is generated by Thomson scattering, and thus it is generated at the last scattering surface (z = 1090) and during the epoch of reionization (z < 15).
The way Thomson scattering generates can be understood easily by recalling the dipole radia-tion.
57 58 Now, let us place an electron at the origin of coordinates, have some temperature anisotropy around it, and calculate the polarization pattern produced by that electron. It is easy to imagine this problem by thinking about it from a point of view of the electron at the origin.
Consider the Sachs–Wolfe effect, δT/T = 1 3Ψ, with Ψ being a plane wave going in z direction.
δT(ˆ n) ¯ T = 1 3A cos(kz cos θ), (2.121) where A > 0 is a constant representing the amplitude of Ψ, and θ is the usual polar angle measured from the z direction. The origin (z = 0) is hotter (δT > 0).
The polarization produced by a scalar perturbation (such as Ψ) scattered by an electron is given by Q + iU = − √ 6 10 2Y 0 2 (θ, φ) Z d˜ ΩδT(˜ θ, ˜ φ) ¯ T (Y 0 2 )∗(˜ θ, ˜ φ), (2.122) Q −iU = − √ 6 10 −2Y 0 2 (θ, φ) Z d˜ ΩδT(˜ θ, ˜ φ) ¯ T (Y 0 2 )∗(˜ θ, ˜ φ).
(2.123) Here, Y m l is the spherical harmonics, and 2Y m l is the spin-2 harmonics. For l = 2 and m = 0, we have Y 0 2 = r 5 16π(3 cos2 θ −1), (2.124) 2Y 0 2 = r 15 32π sin2 θ, (2.125) −2Y 0 2 = 2Y 0 2 .
(2.126) Note that, in general, −sY m l = (−1)m+s(sY −m l )∗.
The reason why Q ± iU is described by spin-2 harmonics is that Q ± iU is the spin-2 quantity.
For a clock-wise rotation of coordinates by an angle ϕ, Q and U transform as Q U !
→ Q′ U ′ !
= cos 2ϕ −sin 2ϕ sin 2ϕ cos 2ϕ ! Q U !
, (2.127) or equivalently Q ± iU →Q′ ± iU′ = e±2iϕ(Q ± iU). Therefore, gravitational waves are indeed a spin-2 field.
First of all, it follows from −2Y 0 2 = 2Y 0 2 that U = 0. Then, the Q polarization is given by Q = − √ 6 10 r 15 32π sin2 θ Z 1 −1 d cos θ1 3A cos(kz cos θ)Y 0 2 (θ) Z 2π 0 dφ = 1 4Aj2(kz) sin2 θ.
(2.128) Therefore, an observer at θ = 0 does not see any polarization, while an observer at θ = π/2 sees the maximum polarization with Q > 0 (polarization in the north-south direction). All of this can be understood graphically (see the next page).
59 60 PROBLEM SET 2 1.1 Cosmic Microwave Background - I While the speed of light is kept for completeness below, you may set c = 1 if you wish.
1.1.1 Propagation of photons in a clumpy universe How does the momentum of photons change as photons propagate through space? First, every photon suffers from the mean cosmological redshift, and thus its magnitude, p, will decrease as p ∝1/a.
In addition, as photons pass through potential wells and troughs, they gain or lose momentum.
Finally, not only the magnitude, p, but also the direction of momentum, γi, will change when photons are deflected gravitationally.
We can calculate the evolution of four-dimensional momentum, pµ ≡dxµ/dλ, using the following geodesic equation: dpµ dλ + X αβ Γµ αβpαpβ = 0.
(1.129) Here, λ is a parameter which gives the location along the path of photons. Using p0 = d(ct)/dλ, one may rewrite the geodesic equation in terms of the total time derivative of pµ: dpµ dt + c X αβ Γµ αβ pαpβ p0 = 0.
(1.130) In order to calculate Γµ αβ, we need to specify the metric. To describe a clumpy universe, we perturb the Robertson-Walker metric in the following way: ds2 = −[1 + 2Ψ(t, xi)]c2dt2 + a2(t)[1 + 2Φ(t, xi)] X ij δijdxidxj.
(1.131) Here, Ψ is the usual Newtonian potential (divided by c2 to make it dimensionless), and Φ is called the curvature perturbation. For this metric, all of the components of Γµ αβ are non-zero.
From now on, we will assume that the magnitudes of these variables are small: |Ψ| ≪1 and |Φ| ≪1, and calculate everything only up to the first order in these variables.
Question 1.1: Calculate Γ0 00, Γ0 0i, Γ0 ij, Γi 00, Γi 0j, and Γi jk, up to the first order in Φ and Ψ. You may use the short-hand notation such as ˙ Ψ ≡∂Ψ ∂t , Ψ,i ≡∂Ψ ∂xi .
The components of the metric and its inverse are given by g00 = −(1 + 2Ψ); g00 = −(1 −2Ψ); gij = a2(1 + 2Φ)δij; gij = 1 a2 (1 −2Φ)δij.
(1.132) 61 Question 1.2: Write down the geodesic equations in the following form: dp0 dt = . . . , dpi dt = . . . , up to the first order in Φ and Ψ. The final answers should not contain P ij δijpipj. You can eliminate this by using the normalization condition for momentum of massless particles, P αβ gαβpαpβ = 0, which gives, for the above perturbed metric, a2 X ij δijpipj = (1 −2Φ + 2Ψ)(p0)2.
(1.133) Question 1.3: Now, we want to derive the evolution equations for the magnitude of momentum, p, and its direction, γi. First, we define the magnitude as p2 ≡ X ij gijpipj.
(1.134) Also, we normalize the direction such that X ij δijγiγj = 1.
(1.135) Using this information, write p in terms of p0 and Ψ , and write γi in terms of p, pi, a, and Φ , up to the first order in Φ and Ψ.
Question 1.4: Write down the geodesic equations in the following form: dp dt = . . . , dγi dt = . . . , up to the first order in Φ and Ψ.
The answers should not contain p0 or pi.
Whenever you find them, replace them with p and γi, respectively. You can check the result for the deflection equation, dγi/dt, by making sure that the result satisfies P i γidγi/dt = 0. (You can derive this by differentiating the normalization condition, P ij δijγiγj = 1, with respect to time.) Note that the total time derivative of a variable is related to the partial derivatives as, e.g., dΦ dt = ˙ Φ + X i dxi dt Φ,i = ˙ Φ + X i cpi p0 Φ,i.
(1.136) 62 1.1.2 Perturbed Conservation Equations For A Pressure-less Fluid Consider the stress-energy tensor for a perfect fluid. We then take the limit that the pressure is much less than the energy density, which would be a good approximation for a non-relativistic fluid. The stress-energy tensor for such a pressure-less fluid is Tµν = ρ (P α gµαuα)(P β gνβuβ) c2 .
(1.137) As usual, uµ ≡dxµ/dτ is a four-dimensional velocity and τ is the proper time.
Suppose that the fluid is moving at a non-relativistic physical three-dimensional velocity of V i ≪c. By “physical” velocity, we mean V i ≡aui = adxi dτ .
(1.138) We also expand the energy density into the mean, ¯ ρ, and the fluctuation around the mean, δ: ρ = ¯ ρ(1 + δ).
(1.139) These perturbation variables, δ and V i/c, are small in the same sense that Φ and Ψ are small. There-fore, we shall expand everything only up to the first order in Φ, Ψ, δ, and V i/c. For example, Tij is of order (V/c)2, and thus can be ignored. On the other hand, T0i is of order (V/c), and thus cannot be ignored unless it is multiplied by other perturbation variables.
Question 1.5: Expand the following conservation equations up to the first order in Φ, Ψ, δ, and V i/c: 1. Energy conservation equation, P αβ gαβT0α;β = 0 2. Momentum conservation equation, P αβ gαβTiα;β = 0 Use the conservation equation for the mean density, ˙ ¯ ρ+3 ˙ a a ¯ ρ = 0, to eliminate the mean contributions from the above equations, and then rewrite these equations in the following form: ˙ δ = . . . , ˙ V i c = . . . .
1.1.3 Large-scale Solutions of Einstein Equations During Matter Era The energy and momentum conservation equations contain four unknown perturbation variables, δ, V i, Ψ, and Φ. Therefore, we cannot find solutions unless we have (at least) two more equations.
Such equations are provided by perturbed Einstein equations. Don’t worry - you are not asked to 63 derive them (though I would not stop you from deriving them). Here are the two equations that can be derived by combining perturbed Einstein equations:‡‡ k2 a2 ˜ Φ = 4πG c4 ¯ ρ ˜ δ + 3˙ a ˜ V kc2 !
, (1.140) ˜ Ψ = −˜ Φ.
(1.141) Here, ˜ Φ, ˜ Ψ, ˜ δ, and ˜ V are all in Fourier space, i.e., ˜ Φ = ˜ Φ(⃗ k, t), ˜ Ψ = ˜ Ψ(⃗ k, t), ˜ δ = ˜ δ(⃗ k, t), and ˜ V = ˜ V (⃗ k, t), and ⃗ k is the comoving wavenumber vector.
They are related to the original variables in position space by, e.g., ˜ Ψ(⃗ k, t) = Z d3xΨ(⃗ x, t)e−i⃗ k·⃗ x, (1.142) Ψ(⃗ x, t) = Z d3k (2π)3 ˜ Ψ(⃗ k, t)ei⃗ k·⃗ x.
(1.143) Here, ⃗ k · ⃗ x ≡P ij δijkixj. For example, the left hand side of the first perturbed Einstein equation, (k2/a2)Φ, came from the Laplacian of Φ: 1 a2 ∇2Φ(⃗ x, t) = 1 a2 Z d3k (2π)3 ˜ Φ(⃗ k, t) ∇2ei⃗ k·⃗ x = 1 a2 Z d3k (2π)3 ˜ Φ(⃗ k, t) −k2ei⃗ k·⃗ x , (1.144) where ∇2 ≡P ij δij ∂2 ∂xi∂xj , and k2 ≡P ij δijkikj.
Also, ˜ V in the right hand side of the first perturbed Einstein equation is defined as ˜ V (⃗ k, t) ≡iˆ k · ⃗ ˜ V (⃗ k, t) (where ˆ k ≡⃗ k/k is a unit vector), i.e., ⃗ ∇· ⃗ V (⃗ x, t) = Z d3k (2π)3 ⃗ ˜ V (⃗ k, t) · ⃗ ∇ei⃗ k·⃗ x = Z d3k (2π)3 ⃗ ˜ V (⃗ k, t) · i⃗ kei⃗ k·⃗ x (1.145) ≡ Z d3k (2π)3 k ˜ V (⃗ k, t)ei⃗ k·⃗ x.
(1.146) Here, ⃗ ∇· ⃗ V ≡P k V k ,k.
Now, let us use the above four equations to find solutions for Ψ, Φ, V , and δ. From now on, we shall drop the tildes on variables in Fourier space for simplicity. It is convenient to change the independent variable from t to the scale factor, a. Finally, let us define the following variable: ϵ(a) ≡ck ˙ a , (1.147) ‡‡To those who wish to derive these results: the first equation can be obtained by combining perturbed G00 = (8πG/c4)T00 and G0i = (8πG/c4)T0i, while the second equation can be obtained from the traceless part of Gij = (8πG/c4)Tij.
64 which goes as ϵ ∝√a during the matter-dominated era. This quantity is useful, as it is much less than unity for fluctuations whose wavelength is longer than the Hubble length (≈horizon size): ϵ ≪1 for super-horizon fluctuations, k ≪aH/c, where H = ˙ a/a is the Hubble expansion rate. Therefore, we can find large-scale (long-wavelength; super-horizon) solutions by consistently ignoring higher-order terms of ϵ.
Question 1.6: Using the Fourier-space variables and ϵ, show that the energy- and momentum-conservation equations can be re-written as follows : δ′ = −ϵ a V c −3Φ′, (1.148) V ′ c = −1 a V c + ϵ aΨ, (1.149) where the primes denote derivatives with respect to a.
Question 1.7: Using Φ = −Ψ, we now have the following three equations for three unknown variables: δ′ = −ϵ a V c −3Φ′, (1.150) V ′ c = −1 a V c −ϵ aΦ, (1.151) ϵ2Φ = 3 2 δ + 3V ϵc .
(1.152) Once again, during the matter era, ϵ ∝√a. Solve these equations on super-horizon scales, ϵ ≪1, and show that non-decaying solutions are given by δ = 2Φ, (1.153) V c = −2 3ϵΦ.
(1.154) By “non-decaying solutions” we mean the solutions that go as ∝an where n ≥0.
Finally, show that Φ (and hence Ψ) is a constant and does not depend on a in the super-horizon limit .
Hint: you cannot ignore ϵ when two different variables are involved, e.g., A + ϵB ̸= A, because you do not know a priori how A compares with B. You can ignore the terms of order ϵ only when you are sure that ϵ is compared to order unity, e.g., A′ + A a + ϵ A a ≈A′ + A a .
Do not use Mathematica to solve these coupled differential equations! Use your brain, please.
65 PROBLEM SET 3 1.1 Cosmic Microwave Background - II While the speed of light is kept for completeness below, you may set c = 1 if you wish.
1.1.1 Temperature Anisotropy From Gravitational Waves Gravitational waves stretch space as they propagate through space. This deformation of space is characterized by the following metric: ds2 = −c2dt2 + a2(t) X ij (δij + hij)dxidxj, where hij is the so-called tensor metric perturbation. (On the other hand, Φ and Ψ that we have dealt with before are called “scalar metric perturbations”.) The tensor metric perturbation is symmetric (hij = hji), traceless (P3 i=1 hii = 0), and transverse (P3 j=1 ∂hij ∂xj = 0).
At the first-order of perturbations, scalar and tensor perturbations are decoupled, and thus we can ignore the scalar perturbations when analyzing the tensor perturbations.
Question 1.1: Write down the geodesic equation for p ≡(P ij gijpipj)1/2 with the metric given above, up to the first order in hij. Then, by integrating the geodesic equation over time, derive the formula for the observed temperature anisotropy from gravitational waves as δT ¯ T O = δT ¯ T E + Z tO tE dt (. . . ) where (. . . ) should contain only ˙ hij and γi (where γi is the unit vector of the direction of photons, satisfying P ij δijγiγj = 1). Hint: you should check the result by making sure that you can recover a part of the scalar integrated Sachs–Wolfe effect, −˙ Φ, by using the scalar metric perturbation, hij = 2Φδij. (You cannot recover the terms containing Ψ because g00 = −1 for the above metric.) From now on, set δT ¯ T E = 0.
Question 1.2: Consider a gravitational wave propagating in the z (= x3) direction. For this special case, the components of the tensor metric perturbation are given by hij = h+ h× 0 h× −h+ 0 0 0 0 , 66 where h+ and h× denote two linear polarization states of a gravitational wave. Using polar coor-dinates for the propagation direction of photons with respect to the gravitational wave: γi = (cos φ sin θ, sin φ sin θ, cos θ), rewrite the equation for δT ¯ T O in terms of R ˙ h+dt, R ˙ h×dt, and trigonometric functions.
Question 1.3: A gravitational wave with ˙ h+ > 0 stretches space in x direction, while that with ˙ h× > 0 stretches space in 45◦direction (see the figure below).
This stretching of space causes gravitational redshifts and blueshifts in the corresponding directions. Using this picture, give physical explanations for the result obtained in Question 1.2. (In other words, now that you have an equation, how much physical interpretation can you get out of this equation?) For example: in which cases do you find hot (∆T > 0) or cold (∆T < 0), and why?; compare the results for θ = 0 and θ = π/2, and give a physical explanation for the difference; compare the results for φ = 0, π/4, π/2, and 3π/4, and give a physical explanation for the difference. Use graphics as needed. It is easier to think about this from a point of view of photons: if you were a photon, how would you experience redshift or blueshift, depending on the angle between your propagation direction and the direction of the gravitational wave, or depending on the azimuthal angle?
67 Question 1.4: As it is evident from the above figure, a gravitational wave produces a quadrupo-lar (l = 2) temperature anisotropy. To see this more clearly, it is convenient to define the following circular polarization amplitudes, hR (right-handed) and hL (left-handed), as h+ = 1 √ 2(hR + hL), (1.155) h× = i √ 2(hR −hL).
(1.156) Using hR and hL, and the definitions for spherical harmonics, Y m l , with l = 2: Y ±2 2 (θ, φ) = r 15 32π sin2 θe±2iφ, (1.157) Y ±1 2 (θ, φ) = (±1) r 15 8π sin θ cos θe±iφ, (1.158) Y 0 2 (θ, φ) = r 5 16π(3 cos2 θ −1), (1.159) rewrite the equation for δT ¯ T O in terms of R ˙ hRdt, R ˙ hLdt, and Y m 2 .
1.1.2 Polarization From Gravitational Waves Thomson scattering of a quadrupolar temperature anisotropy by an electron can produce linear polarization. In terms of the Stokes parameters produced by a scattering, Q(θ, φ) and U(θ, φ), 68 there is a formula relating the temperature quadrupole to polarization by a single scattering: Q + iU = − √ 6 10 X m=±2 2Y m 2 (θ, φ) Z d˜ ΩδT ¯ T O (˜ θ, ˜ φ)Y m∗ 2 (˜ θ, ˜ φ), (1.160) Q −iU = − √ 6 10 X m=±2 −2Y m 2 (θ, φ) Z d˜ ΩδT ¯ T O (˜ θ, ˜ φ)Y m∗ 2 (˜ θ, ˜ φ), (1.161) where d˜ Ω= d cos ˜ θd˜ φ, and 2Y m l is a spin-2 harmonics given by 2Y ±2 2 = r 5 64π(1 ∓cos θ)2e±2iφ, (1.162) −2Y ±2 2 = r 5 64π(1 ± cos θ)2e±2iφ.
(1.163) Note that an electron is at the origin, and photons are scattered by this electron at the origin into various directions, (θ, φ). In other words, these are the Stokes parameters of polarization that would be observed by observers at various directions from this electron.
Now, to simplify the analysis, let us assume that we have ∆hR ≡ R ˙ hRdt and ∆hL ≡ R ˙ hLdt at the origin, and similarly define the linear polarization amplitudes of gravitational waves: ∆h+ ≡ 1 √ 2(∆hR + ∆hL), (1.164) ∆h× ≡ i √ 2(∆hR −∆hL).
(1.165) Question 1.5: Calculate Q(θ, φ) and U(θ, φ) in terms of ∆h+,× and trigonometric functions.
Question 1.6: Give physical explanations for the results obtained in Question 1.5. For ex-ample: compare Q and U at θ = π/2 and φ = 0, and explain the origin of the difference; compare the results at different φ, and give a physical explanation for the behavior. Use graphics as needed.
69 For this problem, it is easier to think about this from a point of view of an electron at the ori-gin: if you were an electron scattering photons into various directions, what polarization would you produce depending on the scattering direction and the direction of the gravitational wave, or depending on the azimuthal angle?
70 Chapter 3 Large-scale Structure of the Universe In this chapter, we shall learn how matter density fluctuations grow and how structures are formed in an expanding universe.
3.1 Evolution of Density Fluctuations 3.1.1 Matter Era Super-horizon solutions As you saw in the homework problem, the evolution of matter density fluctuations during the matter era is given by the following 3 equations (with c = 1): • Energy (Mass) Conservation: δ′ = −ϵ aV −3Φ′, (3.1) where ϵ ≡k/˙ a = k/(aH) ∝√a during the matter era, and the primes denote partial deriva-tives with respect to a, i.e., δ′ ≡∂δ/∂a.
• Momentum Conservation: V ′ = −1 aV −ϵ aΦ.
(3.2) • Einstein Equation: ϵ2Φ = 3 2 δ + 3V ϵ .
(3.3) Then, we have seen that, on super-horizon scales (ϵ ≪1), the solutions are given by δ = 2Φ = constant, (3.4) V = −2 3ϵΦ ∝√a, (3.5) Φ = constant.
(3.6) 71 These super-horizon solutions were important when computing temperature anisotropy, as the large-scale temperature anisotropy (Sachs–Wolfe effect and Doppler effect on angular scales larger than about 3 degrees) is given by these solutions.
Sub-horizon solutions On the other hand, when studying the large-scale structure of the universe (such as the clustering of galaxies), we are always dealing with the scales much smaller than the horizon size. Therefore, let us take the opposite limit, ϵ ≫1, and find solutions. In this sub-horizon limit, the equations become δ′ = −ϵ aV, (3.7) V ′ = −1 aV −ϵ aΦ, (3.8) ϵ2Φ = 3 2δ.
(3.9) These equations can be solved easily, and the growing-mode solutions are δ = 2 3ϵ2Φ ∝a, (3.10) V = −2 3ϵΦ ∝√a, (3.11) Φ = constant.
(3.12) There are 3 important observations one can make: • The matter density perturbation, δ, is constant outside the horizon, but it grows linearly with a inside the horizon.
• The solution for the matter velocity perturbation, V , is the same in the super-horizon and sub-horizon limits, and grows as √a at all scales.
• Φ (= −Ψ) is constant both in the super-horizon and sub-horizon limits, and thus it is constant at all scales.
3.1.2 Radiation Era Super-horizon solutions How the matter density perturbations grow during the radiation era? As the energy density in radiation is much greater than that in matter during the radiation era, we need to take the radiation 72 energy density perturbation in Einstein equation: δ′ = −ϵ aV −3Φ′ (3.13) V ′ = −1 aV −ϵ aΦ, (3.14) ϵ2Φ = 3 2 δR + ¯ ρM ¯ ρR δ + 3V ϵ , (3.15) where ϵ ≡k/˙ a = k/(aH) ∝a during the radiation era. Here, δR is the fractional perturbation in the radiation energy density, ¯ ρR the mean radiation energy density, and ¯ ρM the mean matter energy density. As usual, δ is the fractional perturbation in the matter density.
First, let us look at the super-horizon (ϵ ≪1) solutions. On super-horizon scales, the radiation perturbation δR and the matter perturbation δ are related by the adiabatic initial condition: δ = 3 4δR (super horizon).
(3.16) Since ¯ ρM/¯ ρR ≪1 during the radiation era, this simply means that the term involving ¯ ρM/¯ ρR in Einstein’s equation can be ignored. Therefore, on super-horizon scales, we have δ′ = −ϵ aV −3Φ′ (3.17) V ′ = −1 aV −ϵ aΦ (3.18) ϵ2Φ = 3 2 4 3δ + 3V ϵ , (3.19) where we have used δR = 4 3δ.
Using the same technique we used for the matter era, we can solve these equations to find the super-horizon solutions during the radiation era: δ = 9 8Φ = constant, (3.20) V = −1 2ϵΦ ∝a, (3.21) Φ = constant.
(3.22) Therefore, similarly to the matter era, Φ and δ remain constant outside the horizon during the radiation era. However, the values of Φ and δ are not the same as those during the matter era. In other words, the values of Φ and δ change when the universe becomes matter dominated.
To see this, recall that, on super horizon scales, there is a conserved quantity ζ given by ζ ≡Φ −V ϵ .
(3.23) This quantity is given by ζ = Φ + 1 2Φ = 3 2Φ, (3.24) 73 during the radiation era. In other words, Φ = 2 3ζ during the radiation era. By comparing this to the matter-era relation, Φ = 3 5ζ, we find Φ(matter era) = 9 10Φ(radiation era), (3.25) and δ(matter era) = 8 5δ(radiation era), (3.26) on super horizon scales.
Sub-horizon solutions Next, let us consider the sub-horizon solutions. The relevant equations are δ′ = −ϵ aV, (3.27) V ′ = −1 aV −ϵ aΦ, (3.28) ϵ2Φ = 3 2 δR + ¯ ρM ¯ ρR δ .
(3.29) Now, on sub-horizon scales, the adiabatic condition does not have to be held because it is the initial condition, and the density perturbations can evolve away from the initial condition inside the horizon.
Interestingly, on sub-horizon scales during the radiation, we can ignore δR compared to ¯ ρM ¯ ρR δ, despite that ¯ ρM ¯ ρR ≫1. This is because radiation cannot cluster (cannot form clumps) inside the horizon due to a large amount of pressure it has (recall that the radiation pressure is given by PR = ρR/3, which is comparable to the energy density).
In other words, the distribution of radiation is quite smooth inside the horizon, and it does not contribute very much to Φ compared to the matter density perturbation. As a result, we can write δ′ = −ϵ aV, (3.30) V ′ = −1 aV −ϵ aΦ, (3.31) ϵ2Φ = 3 2 ¯ ρM ¯ ρR δ = 3 2 a aEQ δ, (3.32) where aEQ is the scale factor at which ¯ ρM = ¯ ρR. (Recall ¯ ρM ∝1/a3 and ¯ ρR ∝1/a4.) By combining these equations, one finds δ′′ + 1 aδ′ − 3 2aaEQ δ = 0.
(3.33) This is not quite straightforwardly solvable, so we use a trick: defining y ≡δ/a, we rewrite this equation in terms of y: y′′ + 3 ay′ + 1 a2 1 −3 2 a aEQ y = 0.
(3.34) 74 Since we are considering the radiation era, we can ignore a/aEQ ≪1.
y′′ + 3 ay′ + 1 a2 y = 0.
(3.35) The solution is y = A a + B a ln a. Therefore, δ = A + B ln(a/aEQ) (3.36) where A and B are integration constants. While the matter density perturbation grows linearly with a during the matter era, it grows only logarithmically during the radiation era. This has a very important implication, which we shall learn in a moment.
The other solutions are given by V = −B ϵ ∝1 a, (3.37) Φ = 3 2 a aEQϵ2 [A + B ln(a/aEQ)] ∝ln(a/aEQ) a .
(3.38) As Φ inside the horizon decays during the radiation era, the small-scale perturbation (which entered the horizon earlier) is suppressed relative to the primordial one. It is conven-tional to characterize this effect by using the so-called transfer function: T(k, a) ≡ Φ(k, a) Φprimordial .
(3.39) Since Φ becomes constant at all scales after the matter-radiation equality, the shape of T(k, a) gets frozen after the matter-radiation equality. Therefore, from now on, we shall simply write it as T(k) without time dependence.
75 3.2 Matter Density Power Spectrum 3.2.1 Shape In most cases, the large-scale structure is characterized by the power spectrum of matter density fluctuations, P(k, t): ⟨δ(⃗ k, t)δ∗(⃗ k, t)⟩= (2π)3δ(3) D (⃗ k −⃗ k′)P(k, t).
(3.40) Using Einstein’s equation during the matter era, ϵ2(k, t)Φ(⃗ k, t) = 3 2δ(⃗ k, t), (3.41) we can relate P(k, t) to the power spectrum of Φ: P(k, t) = 4 9 k aH 4 PΦ(k, t).
(3.42) As we have seen before, on super horizon scales, it is convenient to relate Φ to the conserved quantity ζ. During the matter era, we have Φ = 3 5ζ. Finally, we need to take into account the matter density evolution during the radiation era using the transfer function. The final result is P(k, t) = 4 9 k aH 4 × 9 25 2π2 k3 ∆2 ζ(k0) k k0 ns−1 T 2(k)D2(t) = 8π2k 25(aH)4 ∆2 ζ(k0) k k0 ns−1 T 2(k)D2(t).
(3.43) Here, D(t) is a time-dependent function giving the amount of growth of δ. During the matter era, D ∝a. Therefore, on very large scales where the transfer function is approximately unity, the scale-invariant power spectrum (ns = 1) yields the matter density power spectrum of P(k) ∝k.
Then, P(k) peaks at k = aEQHEQ ≈0.01 Mpc−1, and then decreases toward large values of k. The small-scale limit is given by P(k) ∝[ln(k)]2/k3 (for ns = 1).
Therefore, in principle, if one can measure the matter power spectrum accurately, one can determine the parameters such as ∆2 ζ and ns. As we learn later, the growth function D(t) also encodes important cosmological information. However, the accuracy of the measurement of P(k) is not yet good enough compared to the cosmic microwave background, and thus the information on ∆2 ζ, ns, etc., is dominated by the microwave background data.
76 3.2.2 Baryon Acoustic Oscillation Up until now, we have ignored interactions between matter and radiation.
Since photons and electrons interact efficiently via Thomson scattering, it is conceivable that this interaction leaves some signatures in the microwave background as well as in the matter power spectrum.
Once we include interactions between matter and radiation, we can no longer treat these com-ponents separately. As a result, the equation system becomes a bit more involved.
Since electrons and baryons (protons and helium nuclei) are also interacting efficiently via Coulomb interaction, we can treat photons and baryons as a coupled fluid. We should not forget also dark matter, which provides most of the gravitational potential during the matter era.
Then the relevant equations are 77 • Energy Conservation: ˙ δD = −k aVD −3 ˙ Φ, (3.44) ˙ δB = −k aVB −3 ˙ Φ, (3.45) ˙ δγ = −4 3 k aVγ −4 ˙ Φ, (3.46) where δD, δB, and δγ are the dark matter, baryon, and photon densities, respectively. Since we consider the matter-dominated era, we shall ignore ˙ Φ.
• Momentum Conservation: ˙ VD = −˙ a aVD −k aΦ, (3.47) ˙ VB = −˙ a aVB −k aΦ + σT ne R (Vγ −VB), (3.48) ˙ Vγ = 1 4 k aδγ −k aΦ + σT ne(VB −Vγ), (3.49) where R is the baryon-to-photon energy density ratio defined as R ≡3¯ ρB 4¯ ργ .
(3.50) • Einstein’s Equation: k2 a2 Φ = 4πG (¯ ρDδD + ¯ ρBδB) , (3.51) where we have ignored the radiation contribution in the right hand side because we are considering the matter-dominated era.
Now, while there are many equations, one can simplify the equation system considerably when the coupling between photons and baryons is very efficient. In such a case, baryons and photons basically move together, i.e., VB ≈Vγ.
We rewrite Eq. (5.48) as VB = Vγ − R σT ne ˙ VB + ˙ a aVB + k aΦ ≈ Vγ − R σT ne ˙ Vγ + ˙ a aVγ + k aΦ .
(3.52) Here, since we assume that the difference between VB and Vγ is small, we have replaced VB with Vγ in the right hand side. We then use the photon momentum conservation equation, Eq. (5.49), 78 in the right hand side: VB = Vγ − R σT ne 1 4 k aδγ + ˙ a aVγ + σT ne(VB −Vγ) (1 + R)VB = (1 + R)Vγ − R σT ne 1 4 k aδγ + ˙ a aVγ VB = Vγ − R 1 + R 1 σT ne 1 4 k aδγ + ˙ a aVγ σT ne(VB −Vγ) = − R 1 + R 1 4 k aδγ + ˙ a aVγ .
(3.53) Now, using this in the photon momentum conservation equation, and using the photon energy conservation equation Vγ = −3 4 a k ˙ δγ, we arrive at the following differential equation for the photon energy density: ¨ δγ + 1 + 2R 1 + R ˙ a a ˙ δγ + 1 3(1 + R) k2 a2 δγ = 4 3 k2 a2 Φ.
(3.54) This is a wave equation for δγ; thus, a coupling between baryons and photons results in the acoustic oscillations in the photon density perturbations. Since baryons and photons are coupled, the same oscillations must also be present in the baryon density perturbations as well. Indeed, the acoustic oscillations have been observed both in photons (microwave background) and the distribution of matter (galaxies).
79 Acoustic Oscillations in CMB To have a deeper understanding of the structures of the acoustic oscillation, let us focus on the regime where the oscillation frequency is much greater than the expansion rate of the universe. In this case, the wave equation simplifies to ¨ δγ + 1 3(1 + R) k2 a2 δγ = 4 3 k2 a2 Φ.
(3.55) Since ˙ Φ = 0 during the matter era, one may rewrite this equation in a suggestive way: ∂2 ∂t2 1 4δγ −(1 + R)Φ + k2c2 s a2 1 4δγ −(1 + R)Φ = 0, (3.56) where cs is the speed of sound: c2 s ≡ 1 3(1 + R) = 1 3(1 + 3¯ ρB 4¯ ργ ) .
(3.57) Note that this speed of sound is less than that for the relativistic fluid, c2 s = 1/3. This is due to the coupling to baryons: the inertia of baryons reduces the speed of sound of photon-baryon fluid relative to that of the relativistic fluid. The solution to the above wave equation is 1 4δγ = (1 + R)Φ + A cos(krs) + B sin(krs), (3.58) where rs is the sound horizon defined by rs ≡ Z t∗ 0 dt a cs(a) = 147 Mpc, (3.59) for the cosmological parameters best-fit to the WMAP data and t∗is the decoupling time.
How do we determine the integration constants, A and B? We determine these coefficients by noting that, on super horizon scales, these solutions should match the adiabatic initial condition: 1 4δγ = 1 3δm = 2 3Φ on super-horizon scales (kcs ≪1) (3.60) Therefore, A = 2 3Φ −(1 + R)Φ = − 1 3 + R Φ, (3.61) B = 0, (3.62) and 1 4δγ = (1 + R)Φ − 1 3 + R Φ cos(krs).
(3.63) 80 Since ργ ∝T 4, we can relate 1 4δγ to δT/T as 1 4δγ = δT/T. Moreover, since the observed temperature anisotropy is the sum of δT/T at the bottom of the potential well and the potential Ψ, we write, using Ψ = −Φ 1 4δγ + Ψ = −RΨ + 1 3 + R Ψ cos(krs).
(3.64) Since what we observe is the power spectrum, which is the temperature squared, we may plot this result squared as a function of krs. We then notice that the 1st peak to the 2nd peak ratio goes up as one increases R; thus, the 1st peak to the 2nd peak ratio can be used to determine the baryon density!
81 Acoustic Oscillations in Baryons We have seen that a coupling between photons and baryons induces acoustic oscillations in the distribution of photons. How about baryons?
By repeating a similar analysis, one can obtain the wave-like equation for baryon density fluc-tuations: ¨ δB + 1 + 2R 1 + R ˙ a a ˙ δB + 1 3(1 + R) k2 a2 3 4δγ = k2 a2 Φ.
(3.65) Note that we have δγ instead of δB in the third term on the left hand side, and thus it is not quite the wave equation for δB. However, we know that, on super horizon scales, the adiabatic initial condition gives 3 4δγ = δB. Therefore, with this initial condition, the baryons have the same acoustic oscillations as photons.
The most remarkable thing about this equation is that baryons acquire a significant speed of sound, cs = 1/ p 3(1 + R), via a coupling with photons. If baryons were not coupled to photons, their speed of sound would be simply given by cs = p T/mp. For the decoupling temperature, T ≈3000 K ≈0.26 eV, this is tiny: cs ≈2 × 10−5. So, the coupling between baryons and photons changes the behaviour of baryons completely.
This oscillation is imprinted on the power spectrum of galaxies today, and is often called the baryon acoustic oscillations (BAO). 1000 10000 0.001 0.01 0.1 Matter Power Spectrum, P(k) [Mpc3/h3] Wave Number, k [h Mpc-1] ’wmap5baosn_max_likelihood_matterpower.dat’ Primordial Power Spectrum P(k) ~ kns Horizon Size at the Matter-radiation Equality, kEQ=aEQHEQ Baryon Acoustic Oscillation Asymptotic Slope of the Transfer Function P(k) ~ kns(lnk)2/k4 82 Sloan Digital Sky Survey 10 Percival et al.
Fig. 12.— The redshift-space power spectrum recovered from the combined SDSS main galaxy and LRG sample, optimally weighted for both density changes and luminosity dependent bias (solid circles with 1-σ errors). A flat Λ cosmological distance model was assumed with ΩM = 0.24. Error bars are derived from the diagonal elements of the covariance matrix calculated from 2000 log-normal catalogues created for this cosmological distance model, but with a power spectrum amplitude and shape matched to that observed (see text for details).
The data are correlated, and the width of the correlations is presented in Fig. 10 (the correlation between data points drops to < 0.33 for ∆k > 0.01 h Mpc−1). The correlations are smaller than the oscillatory features observed in the recovered power spectrum. For comparison we plot the model power spectrum (solid line) calculated using the fitting formulae of Eisenstein & Hu (1998); Eisenstein et al. (2006), for the best fit parameters calculated by fitting the WMAP 3-year temperature and polarisation data, h = 0.73, ΩM = 0.24, ns = 0.96 and Ωb/ΩM = 0.174 (Spergel et al. 2006). The model power spectrum has been convolved with the appropriate window function to match the measured data, and the normalisation has been matched to that of the large-scale (0.01 < k < 0.06 h Mpc−1) data. The deviation from this low ΩM linear power spectrum is clearly visible at k > ∼0.06 h Mpc−1, and will be discussed further in Section 6. The solid circles with 1σ errors in the inset show the power spectrum ratioed to a smooth model (calculated using a cubic spline fit as described in Percival et al.
2006) compared to the baryon oscillations in the (WMAP 3-year parameter) model (solid line), and shows good agreement. The calculation of the matter density from these oscillations will be considered in a separate paper (Percival et al. 2006). The dashed line shows the same model without the correction for the damping effect of small-scale structure growth of Eisenstein et al. (2006). It is worth noting that this model is not a fit to the data, but a prediction from the CMB experiment.
Small Scale Large Scale Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) [expected] Small Scale Large Scale Since we know the value of rs (from measurements of the CMB anisotropy), we can use BAO of the matter power spectrum at a given redshift to determine the angular diameter distance and the expansion rate at that redshift. Namely: δθBAO = 1 1 + z rs DA(z), (3.66) δzBAO = rsH(z) c , (3.67) where δθBAO and δzBAO are the observed angular separations and redshift separations corresponding to rs. From these, it is clear that we can measure DA(z) and H(z) separately - BAO is the standard ruler that we discussed in Section 1.5! Therefore, in order to fully utilize the power of BAO, we must consider the power spectrum in 2-dimensional space: angular directions and redshift direction.
3.2.3 2-dimensional Power Spectrum: Alcock-Paczy´ nski test When discussing the power spectrum of matter density fluctuations (traced by, e.g., galaxies), P(k), it is important to realize that we cannot directly measure the wave numbers, k. In order to go to Fourier space, we first need to know 3-dimensional positions of galaxies; however, in order to know those, we must know the angular diameter distances and the expansion rates, as our observables are the angular coordinates and redshift coordinates, rather than the actual 3-dimensional positions.
83 As a result, the observed power spectrum would not be a function of k, but would always be a function of two wave numbers: k∥and k⊥. (Of course, k = q k2 ⊥+ k2 ∥.) However, the underlying matter power spectrum, P(k), must be isotropic and depend only on the magnitude of k, and thus we can use this property to determine DAH. This is precisely the Alcock-Paczy´ nski test that we studied in Section 1.5. Combining BAO and AP is a powerful method for measuring DA and H.
• DA: (RA,Dec) to the transverse separation, rperp, to the transverse wavenumber • kperp = (2π)/rperp = (2π)[Angle on the sky]/DA • H: redshifts to the parallel separation, rpara, to the parallel wavenumber • kpara = (2π)/rpara = (2π)H/(cΔz) The AP Test: How That Works If DA and H are correct: kpara kperp If DA is wrong: kperp If H is wrong: kperp kperp If DA and H are wrong: 84 BAO-only vs BAO+AP Test • BAO+AP improves upon the determinations of DA & H by more than a factor of two.
• On the DA-H plane, the size of the ellipse shrinks by more than a factor of four.
Shoji, Jeong & Komatsu, Astrophysical Journal, 693, 1404 (2009) 0.90 0.95 1.00 1.05 1.10 DA/DA,ref bestfit=1.000 0.90 0.95 1.00 1.05 1.10 H/Href bestfit=1.000 BAO only BAO+AP For HETDEX 1 3.2.4 2-dimensional Power Spectrum: Redshift Space Distortion However, things are not so simple. Motion of galaxies adds a complication. While we rely on the measured redshifts for inferring the locations of galaxies along the line of sight, the measured redshifts are in fact the sum of the cosmological redshifts and peculiar velocities. Namely, when galaxies moving toward us, they appear to be closer to us than they actually are, and when galaxies are moving away from us, they appear to be farther away than they actually are. This has an effect of increasing the clustering of galaxies (hence the power spectrum) along the line of sight on large scales, and decreasing the power spectrum on small scales. (See the diagram below.) The large-scale effect is called the Kaiser effect, while the small-scale effect is called the fingers-of-God effect. The latter is still too complicated to model reliably, so we shall focus only on the Kaiser effect.
As you derive in the homework, the observed power spectrum in redshift space is related to the underlying power spectrum in real space as Pobs(k, k∥) = 1 + f k2 ∥ k2 !2 P(k) (3.68) on large scales (Kaiser effect).
Here, f is the logarithmic derivative of the growth of density 85 fluctuations: f ≡d ln δ d ln a.
(3.69) For the directions perpendicular to the line of sight, k∥= 0, the observed power spectrum is equal to the underlying spectrum: Pobs(k, k∥= 0) = P(k).
(3.70) For the directions parallel to the line of sight, k∥= k, the observed power spectrum is enhanced relative to the underlying spectrum: Pobs(k, k∥= k) = (1 + f)2P(k).
(3.71) One can use this property to extract the information on the growth of structures. For a universe dominated by matter, f = 1; however, for a universe containing matter and dark energy, such as the universe that we live in, f decreases toward low redshifts, providing an important information on the effect of dark energy on the growth of structures.
Redshift Space Distortion This effect has been measured routinely from large-scale structure surveys. The measured values of f are consistent with the predictions from the standard ΛCDM model. The plots shown below are taken from the latest paper on “Wiggle Z” survey (Blake et al., arXiv:1104.2948).
86 87 3.3 Non-linear Evolution of Matter Density Fluctuations During the matter era, the linear matter density fluctuation, δ, grows as δ ∝a. However, this analysis is valid only for δ ≪1, and thus cannot be used to follow the evolution of non-linear density fluctuations that would eventually form objects such as galaxies.
Specifically, when δ ≫1, density fluctuations collapse gravitationally and form “halos” (i.e., gravitationally bound objects). Galaxies are hosted by these halos. Since the total matter must be conserved, the fact that some regions have δ ≫1 implies that other regions have δ < 1. From the definition, δ ≡ρ/¯ ρ −1, the minimum value of δ is δ = −1. These empty regions (or nearly empty regions) correspond to “voids.” The exact treatment of non-linear processes is difficult, and we usually use computer simulations (such as N-body simulations) to study the formation and evolution of halos. Before we go into some of the results obtained from simulations, it is useful to work out a simplified case known as the spherical collapse.
Consider a spherical region with mass M and radius r. Due to the expansion of the universe, initially ˙ r > 0. As the mass enclosed within r must be conserved, we have ˙ M = 0. During the matter era, the equation of motion is given as the usual Newtonian formula∗ ¨ r = −GM r .
(3.72) Multiplying both sides by ˙ r and integrating, we get 1 2 ˙ r2 = GM r + E, (3.73) where E is an integration constant.
This should be quite familiar to you: (kinetic energy) + (potential energy) = E, where E is the total energy. Now, since we wish to analyze the case where ∗Once again, the same result is obtained from General Relativity. There is a correction to this equation when we have components with a large pressure, such as radiation and dark energy.
88 the expansion of this region eventually stops, turns around, and collapses, we shall consider the case where E < 0. The solution to this equation is known as the cycloid and is given as r = A(1 −cos θ), (3.74) t = B(θ −sin θ), (3.75) A3 = GMB2, (3.76) where A and B are constants, and we have chosen the zero point of time such that t →0 as θ →0.
The evolution of matter density within this region is given as a function of a new parameter θ: ρ = M 4π 3 r3 = 3 4πGB2(1 −cos θ)3 .
(3.77) Now, in order to calculate δ = ρ/¯ ρ −1, we need to know how the mean density ¯ ρ depends on θ.
We do this by recalling that, from the Friedmann equation, H2 = 8πG 3 ¯ ρ = 4 9 1 t2 , (3.78) during the matter era. Therefore, ¯ ρ = 1 6πGt2 = 1 6πGB2(θ −sin θ)2 .
(3.79) By taking the ratio, δ = 9 2 (θ −sin θ)2 (1 −cos θ)3 −1 (3.80) This is the result. The collapse time corresponds to θ = 2π, at which δ goes to infinity. Does δ really go to infinity in practice? No. This is an artifact of spherical symmetry: in reality, a finite angular momentum makes it impossible for particles to go straight down to the center r = 0, and thus an object with a finite size would be formed.
It is instructive to take an early-time limit, θ ≪1. We find δ ≈3 20θ2 (θ ≪1).
(3.81) As δ ≪1 for this case, we should be able to recover the linear evolution, δ ∝a ∝t2/3. Looking at equation (5.75), t ∝θ3 for θ ≪1, and thus we indeed recover δ ∝t2/3 ∝θ2.
The time at which a density fluctuation collapses (θ = 2π; δ →∞) is given by tc = 2πB.
(3.82) While δ goes to infinity at t = tc, what would be the value of δ if we assume the linear evolution?
Using δ ≈(3/20)θ2 and t ≈(B/6)θ3 for θ ≪1, the linear evolution is given by δL = 3 20 6 B 2/3 t2/3.
(3.83) 89 Then, inserting the collapse time, tc = 2πB, into this result, we obtain δc ≡δL(tc) = 3(12π)2/3 20 ≃1.686 (3.84) Why is this result interesting?
While non-linear evolution of density fluctuations is generally quite complicated, the linear evolution is known. Now, suppose that we have some initial density fluctuations that are small, as a function of spatial coordinates (δini(⃗ x) ≪1). These fluctuations evolve in time. Some of them collapse, and some of them do not. More specifically, some density peaks collapse. In real-world picture, these collapsed regions have very high density. On the other hand, in the corresponding linear world, these collapsed regions have δL(⃗ x) ≥δc ≃1.686. This is a nice property, allowing us to calculate the number of collapsed objects at a given time.
90 3.4 Mass Function of Collapsed Halos How do we calculate the number of collapsed objects at a given time, as a function of masses? The idea is the following: Suppose that the distribution of initial density fluctuations, δini(⃗ x), is given by P(δini).
Then, the distribution of linearly-evolved density fluctuations, δL, should also obey the same probability distribution function, P(δ). Then, a fraction of the volume occupied by the collapsed regions is given simply by P(> δc) = Z ∞ δc dδ∞ L P(δL).
(3.85) This should be related to the number of collapsed objects at a given time. But how? To make progress, we must specify the form of P(δL). The current data (especially the cosmic microwave background) strongly suggest that the initial fluctuations obey a Gaussian distribution to high precision, which is consistent with the standard prediction of inflation. While it is possible that some level of non-Gaussianity (departure from a Gaussian distribution) were present, for this lec-ture we shall ignore non-Gaussianity and assume that the initial fluctuations obeyed a Gaussian 91 distribution. Then, the linearly-evolved density fluctuations also obey a Gaussian distribution: P(δL) = e−δ2 L/(2σ2 L) √ 2πσL , (3.86) with R ∞ −∞dδLP(δL) = 1. Here, σ2 L is the variance of density fluctuations.
We have to pause here. How are peaks related to objects? In order to answer this question, we must recall that the above discussion on δc relied upon the spherical collapse model - we started by discussing the evolution of a spherical overdensity region with mass M. When the fluctuation was linear, this region had the mass density that is close to the mean mass density of the universe.
92 Therefore, the initial comoving radius of this region was given by R = 3M 4π¯ ρM = 4.0 Mpc 0.135 ΩMh2 M 1013 M⊙ 1/3 , (3.87) where we have used ¯ ρM = 2.775 × 1011ΩMh2 M⊙Mpc−3 (which is the present-day mass density of the universe). Therefore, for galactic scales (M = 1012 M⊙), R = 1.9 Mpc; for clusters scales (M = 1015 M⊙), R = 19 Mpc. Again, note that this is not the real radius of objects observed today with a given mass. Rather, this is the radius that objects would have, if they had the mean mass density of the universe today. This is the most relevant radius when we talk about the linear density fluctuations. These regions then expand, turn around, and then contract to form objects with physical radii much smaller than R given above.
The next step is to find overdense regions that have a certain mass M. In order to do this, we need to “bin” the density fields with the corresponding radii R. Namely, we first average the density field as δR(⃗ x) ≡ 1 4πR3 3 Z |⃗ r|≤R d3rδL(⃗ x + ⃗ r), (3.88) where R = 3M 4π¯ ρM , and see if the averaged δR exceeds the critical overdensity δc. The Fourier transform of δR is related to that of the original linear density field δL as ˜ δR(⃗ k) = Z d3x δR(⃗ x)e−i⃗ k·⃗ x = Z d3x 1 4πR3 3 Z |⃗ r|≤R d3r Z d3k′ (2π)3 ˜ δL(⃗ k′)ei⃗ k′·(⃗ x+⃗ r)e−i⃗ k·⃗ x = 1 4πR3 3 Z |⃗ r|≤R d3r Z d3k′ δ(3) D (⃗ k −⃗ k′)˜ δL(⃗ k′)ei⃗ k·⃗ r = ˜ δL(⃗ k) 1 4πR3 3 Z |⃗ r|≤R d3r ei⃗ k·⃗ r = ˜ δL(⃗ k) 1 2R3 3 Z R 0 r2dr Z 1 −1 dµ eikrµ = ˜ δL(⃗ k) 3j1(kR) kR , (3.89) where j1(x) = sin(x)/x2 −cos(x)/x is a spherical Bessel function of order 1. Therefore, the Fourier transform of δR is the Fourier transform of δL times the “window function” given by 3j1(kR) kR .
93 The variance of the averaged density field δR is then given by σ2 R ≡ ⟨δ2 R(⃗ x)⟩ = Z d3k (2π)3 Z d3k′ (2π)3 ⟨˜ δR(⃗ k)˜ δ∗ R(⃗ k′)⟩ei(⃗ k−⃗ k′)·⃗ x = Z d3k (2π)3 Z d3k′ (2π)3 ⟨˜ δL(⃗ k)˜ δ∗ L(⃗ k′)⟩ 3j1(kR) kR 3j1(k′R) k′R ei(⃗ k−⃗ k′)·⃗ x = Z k2dk 2π2 P(k) 3j1(kR) kR 2 .
94 This integral can be estimated roughly as σ2 R ≈k3P(k) 2π2 k=1/R .
(3.90) This means that, for a power-law power spectrum of P(k) ∝km, the variance scales as σ2 R ∝R−(m+3) ∝M−(m+3)/3.
(3.91) For example, the large-scale limit of the power spectrum is P(k) ∝kns, and thus σ2 R ∝M−(ns+3)/3 ≈ M−4/3.
The small-scale limit of the power spectrum is P(k) ∝kns−4(ln k)2, and thus σ2 R ∝ M−(ns−1)/3 ≈M0 (except for a logarithmic factor). Finally, as σ2 is proportional to the power spectrum, its growth is given by the growth of mass density fluctuation squared, i.e., σ2 R ∝D2, where D ∝a during the matter era. 0.01 0.1 1 10 10000 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 Sigma(M,z=0) Mass [h-1 Solar Mass] ’sigma/Rh_sigma_z=0.txt’ u (4.3.141592.775e110.135$13/3.):2 ’sigma/Rh_sigma_z=10.txt’ u (4.3.141592.775e110.135$13/3.):2 The above figure shows σR as a function of M = 4π¯ ρMR3/3 in units of h−1 M⊙. The upper and lower curves correspond to z = 0 and z = 10, respectively. From this figure, one finds that, at z = 10, a 1-σ fluctuation corresponding to M = 104 h−1 M⊙has not yet reached the critical 95 overdensity, δc ≃1.686. This does not mean that these masses have not collapsed yet - according to a Gaussian distribution, there are fluctuations exceeding 1-σ fluctuations. They are just not very common. At z = 0, 1-σ fluctuations corresponding to M ≃1013 h−1 M⊙exceed δc, and thus typically collapsing halos at z = 0 have M ≃1013 h−1 M⊙, and halos more massive than that (such as clusters of galaxies) can be collapsing but are still rare.
With these, one can now calculate the mass function of halos, dn/dM, which is the “comoving number density of collapsed halos per unit mass interval at a given time.” This can be calculated as (Press & Schechter, Astrophysical Journal, 187, 425 (1974)) dn dM = −¯ ρM M d dM P(> δc).
(3.92) Plugging in a Gaussian form of the probability distribution function, we can calculate the derivative: dn dM = −¯ ρM M d dM Z ∞ δc dδR 1 √ 2πσR e−δ2 R/(2σ2 R) = −¯ ρM M d dM Z ∞ δc/σR dx 1 √ 2πe−x2/2 = ¯ ρM M δc dσ−1 R dM 1 √ 2πe−δ2 c/(2σ2 R).
(3.93) This is the mass function.
Now, let us check this mass function. Under the assumption that all the mass in the universe are enclosed in halos, the mass function times mass integrated over masses should be equal to the mean mass density of the universe, i.e., Z ∞ 0 dM M dn dM = ¯ ρM.
(3.94) Is this satisfied by the above mass function? A straightforward calculation shows that Z ∞ 0 dM M dn dM = 1 2 ¯ ρM, (3.95) and thus the above formula fails to account for a half of the mass in the universe!
Press and Schechter, who came up with the above formula, then arbitrarily multiplied the above formula by a factor of two, and came up with the formula now known as the “Press-Schechter mass function”: dn dM = ¯ ρM M δc dσ−1 R dM r 2 πe−δ2 c/(2σ2 R) (3.96) The arguments which have led to this formula are arguably simplistic - a spherical collapse - and even requires a fudge factor of two. However, a remarkable thing about this formula is that it gives more-or-less correct form of the mass function derived from N-body simulations.
Many research groups have been trying to find a better formula for the mass function. A big motivation for getting a correct mass function is that the mass function is an observable quantity, 96 and thus it can be used to infer the values of σ2 R. Since it is proportional to the growth rate, σ2 R ∝D2, the mass function can be used to infer D as a function of redshifts which, in turn, can be used to infer the nature of dark energy.
The latest fitting formula for the mass function is given by Tinker et al., Astrophysical Journal, 688, 709 (2008): dn dM = ¯ ρM M dσ−1 R dM A σR b −a + 1 e−c/σ2 R, (3.97) where A = 0.186, a = 1.47, b = 2.57, and c = 1.19. The comparison between Tinker et al.’s mass function and Press-Schechter mass function at z = 0 is given below. 1e-50 1e-45 1e-40 1e-35 1e-30 1e-25 1e-20 1e-15 1e-10 1e+12 1e+13 1e+14 1e+15 1e+16 1e+17 dn/dM Mass [h-1 Solar Mass] ’mf_pressschechter/Mh_dndlnMh_z=0.txt’ u 1:($2/$1) ’mf_tinker_redshift/Mh_dndlnMh_z=0.txt’ u 1:($2/$1) 97 The mass function has been derived from observations of the number of massive clusters of galaxies.
Chandra Cosmology Project led by Alexey Vikhlinin (Vikhlinin et al., 692, 1060 (2009)) has yielded an impressive agreement between the cluster number counts and the prediction from the standard ΛCDM model, as shown below.
z = 0.025−0.25 1014 1015 10−9 10−8 10−7 10−6 10−5 M500, h−1 M⊙ N(>M), h−3 Mpc−3 z = 0.35−0.90 98 PROBLEM SET 4 1.1 Large-scale Structure of the Universe 1.1.1 Growth of Linear Density Fluctuations Let us consider a universe containing matter and dark energy. Assuming that dark energy is a cosmological constant, the Friedmann equation gives H2 = 8πG 3 ρM0 a3 + ρΛ , (1.98) where ρM0 is the present-day value of the matter density, and ρΛ is the energy density associ-ated with a cosmological constant (which is, of course, constant). How would the matter density fluctuations evolve in such a universe?
On the sub-horizon scales, ϵ ≡ k aH ≫1, (1.99) the evolution of the matter density fluctuations obeys the following equations: δ′ = −ϵ aV, (1.100) V ′ = −1 aV −ϵ aΦ, (1.101) ϵ2Φ = 4πGρM0 a3H2 δ.
(1.102) The primes denote derivatives with respect to a. Here, the right hand side of Poisson’s equation (the third equation) contains only the matter density fluctuation, as a cosmological constant is spatially uniform and does not contribute to Φ.
Question 1.1: Combining equations (3), (4), and (5), obtain a single differential equation for δ.
The answer should contain only δ, H, a, and their derivatives with respect to a. Once you obtain the desired equation, you should check that the solutions to that equation in the matter-dominated limit are given by δ = C1a + C2/a3/2, where C1 and C2 are integration constants. Hint: You can relate ρM0 to H′.
Question 1.2: Show that one of the solutions to the equation obtained above is δ ∝H. This is a decaying solution.
Question 1.3: Show that another solution is given by δ ∝H Z da (aH)3 .
(1.103) This is a growing solution.
99 Question 1.4: Take the above growing solution, and define a new quantity, g ≡ δ a.
This quantity must approach a constant during the matter era. Adjust the integration constant such that g →1 during the matter era. Writing the expansion rate as H = H0 p ΩM(1 + z)3 + ΩΛ, (1.104) where 1 + z = 1 a, make a diagram showing the evolution of g as a function z from z = 0 to 3 for ΩM = 0.27 and ΩΛ = 0.73. The difference between g = 1 and g computed here is due to the effect of dark energy.
1.1.2 Redshift Space Distortion: Kaiser Effect While the underlying power spectrum of density fluctuations, P(k), should only depend on the magnitude of k owing to isotropy of the universe, the observed power spectrum can depend on directions of ⃗ k. This is due to the effect of peculiar velocity of matter (say, galaxies), and is called the Kaiser effect (N. Kaiser, Monthly Notices of Royal Astronomical Society, 277, 1 (1987)).
The Kaiser effect arises because we make observations in redshift space, rather than in real space. Specifically, we infer the location of galaxies along the line of sight from observed redshifts.
However, redshifts receive contributions from both the cosmological expansion and peculiar velocity along the line of sight: zobs = zreal + 1 a v∥ c , (1.105) where v∥≡ˆ n ·⃗ v. As a result, galaxies moving toward us appear to have smaller redshifts and to be closer than they actually are, while galaxies moving away from us appear to have larger redshifts and to be farther than they actually are.
As we learned in Section 1.5, the difference in redshifts is related to the comoving separation between two galaxies along the line of sight as δr∥= cδz H .
(1.106) Therefore, observationally inferred comoving separation along the line of sight is different from the real comoving separation by δr∥,obs = δr∥,real + v∥ aH .
(1.107) On the other hand, nothing would happen to the directions perpendicular to the line of sight.
This can be summarized as coordinate transformation. The coordinates in redshift space, si, and those in real space, xi, are related by s1 = x1, (1.108) s2 = x2, (1.109) s3 = x3 + v∥ aH , (1.110) where we have chosen the line of sight direction as the 3-direction. How does this affect the observed power spectrum?
100 Question 1.5: Since this is merely coordinate transformation, the mass within a unit volume must be conserved regardless of the choice of the coordinate system. We have ρsd3s = ρxd3x, (1.111) where ρs and ρx are the mass densities in redshift space and real space, respectively. Expanding these into perturbations, ¯ ρ(1 + δs)d3s = ¯ ρ(1 + δx)d3x.
(1.112) Note that the mean density, ¯ ρ, is the same in both real space and redshift space. From this, we obtain δs = 1 |J|(1 + δx) −1, (1.113) where |J| is the determinant of the Jacobian matrix: J ≡ ∂s1 ∂x1 ∂s1 ∂x2 ∂s1 ∂x3 ∂s2 ∂x1 ∂s2 ∂x2 ∂s2 ∂x3 ∂s3 ∂x1 ∂s3 ∂x2 ∂s3 ∂x3 .
(1.114) By expanding equation (16) up to the first order in perturbations (including density and velocity), find the relation between δs, δx, and v∥. Note that v∥depends on spatial coordinates xi, whereas H does not but it depends only on time. Hint: does the result you obtained make sense? What are the conditions for δs < δx or δs > δx? Can you explain why they are so?
Question 1.6: Now is the time to go to Fourier space. Use δs = Z d3k (2π)3 ˜ δs,⃗ k ei⃗ k·⃗ x, (1.115) δx = Z d3k (2π)3 ˜ δx,⃗ k ei⃗ k·⃗ x, (1.116) ⃗ v = Z d3k (2π)3 ⃗ ˜ v⃗ k ei⃗ k·⃗ x, (1.117) (1.118) and write the relation between δs and δx in Fourier space.† Here, ⃗ v = (v1, v2, v3) and v∥= v3.
We need to relate v∥to δx. For this, we can use the mass conservation equation: ˙ δx + 1 a ⃗ ∇· ⃗ v = 0.
(1.119) Fourier-transforming this, one finds ˙ ˜ δx,⃗ k + i⃗ k a · ⃗ ˜ v⃗ k = 0.
(1.120) †If you are careful, you might wonder why we can expand δs using x coordinates, rather than s coordinates. This is OK up to the first order - since δs is already a perturbation, the difference between x coordinates and s coordinates would appear only at the second order.
101 This equation is satisfied if ⃗ ˜ v⃗ k = ia ⃗ k k2 ˙ ˜ δx,⃗ k.
(1.121) As we have seen from the previous section, ˜ δx,⃗ k evolves by the same factor at all scales (all k), so we may write ˜ δx,⃗ k ∝D. Then, ˙ ˜ δx,⃗ k = ˙ D D ˜ δx,⃗ k = H d ln D d ln a ˜ δx,⃗ k.
(1.122) From now on, let us write f ≡d ln D d ln a , (1.123) so that ⃗ ˜ v⃗ k = iaHf ⃗ k k2 ˜ δx,⃗ k.
(1.124) By putting these altogether, show ˜ δs,⃗ k = 1 + f k2 ∥ k2 !
˜ δx,⃗ k, (1.125) where k∥= k3 for ⃗ k = (k1, k2, k3). The modification of density fluctuations in redshift space due to the peculiar velocity effect is known as the redshift space distortion, and is often called the Kaiser effect. Because of this, the observed power spectrum depends on k∥: Ps(k, k∥) = 1 + f k2 ∥ k2 !2 Px(k).
(1.126) This is a nice result, as one can use the dependence of the observed power spectrum on k∥to extract the information on the growth of structures via f. As we have seen in the previous section, D (hence f = d ln D/d ln a) changes if there is dark energy, and thus this information can be used to study the nature of dark energy.
102 |
189476 | https://simple.wikipedia.org/wiki/Acute_angle | Acute angle - Simple English Wikipedia, the free encyclopedia
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Acute Angle
An acute angle ("acute" meaning "small") is an angle smaller than a right angle. The range of an acute angle is between 0 and 90 degrees, or between 0 and π 2{\displaystyle {\frac {\pi }{2}}} radians. Angles larger than a right angle (90 degrees) are called obtuse. They are found exclusively in Quadrant I or a graph, due to the terminal arm not being able to leave this Quadrant without becoming too large.
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Right angle
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189477 | https://en.wikipedia.org/wiki/Hypergeometric | Jump to content
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Hypergeometric may refer to several distinct concepts within mathematics:
The hypergeometric function, a solution to the Gaussian hypergeometric differential equation, which provides a general solution to every second-order ordinary differential equation
Generalized hypergeometric functions, which generalize the hypergeometric function to specific higher orders
General hypergeometric functions, which provide general solution spaces for whole systems of classical hypergeometric functions
The hypergeometric distribution, a probability distribution which describes the probability of drawing a certain number of successes without replacement from a finite population
Higher-dimensional geometry, which generalizes three-dimensional geometry into an arbitrary number of dimensions; in this context, the prefix hyper- means "n-dimensional"
Hypergeometric motives, a form of motive in algebraic geometry
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189478 | http://www.matematicasvisuales.com/english/html/geometry/triangulos/steiner.html | Matematicas Visuales | Steiner deltoid
Geometry Real Analysis Complex Analysis Probability History Help Contact Links Site Map Updates Español Steiner Deltoid and Simson Lines Your browser does not support the video tag. Steiner proved that the envelope of the Wallace-Simson lines is a curve that has three cuspidal points that is called the Steiner deltoid. The Steiner Deltoid is tangent to the three sides of the triangle. In this case, can you say where is the red point in the circumcircle in relation with some vertex of the triangle? The Steiner Deltoid is tangent to the three altitudes of the triangle. REFERENCES Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer. de Guzmán, Miguel 'The envelope of the Wallace-Simson lines of a triangle. A simple proof of the Steiner theorem on the deltoid'. RACSAM, vol. 95, 2001. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: John Wiley and sons, 1969. NEXT Steiner deltoid is a hypocycloid Steiner deltoid is a hypocycloid related with the nine point circle of a triangle. PREVIOUS Wallace-Simson lines | Demonstration Interactive demonstration of the Wallace-Simson line. MORE LINKS Wallace-Simson lines Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line) The deltoid and the Morley triangle Steiner Deltoid and the Morley triangle are related. Central and inscribed angles in a circle Central angle in a circle is twice the angle inscribed in the circle. Central and inscribed angles in a circle | Mostration | Case I Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case I: When the arc is half a circle the inscribed angle is a right angle. Central and inscribed angles in a circle | Mostration | Case II Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case II: When one chord that forms the inscribed angle is a diameter. Central and inscribed angles in a circle | Mostration | General Case Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved. Drawing fifteen degrees angles Using a ruler and a compass we can draw fifteen degrees angles. These are basic examples of the central and inscribed in a circle angles property. Morley Theorem The three points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle (Morley's triangle) John Conway's proof of Morley's Theorem Interactive animation about John Conway's beautiful proof of Morley's Theorem Geometry| Analysis| Complex Analysis| Probability| History| Help| Contact| Links| Site map| Updates |
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Magic Square Worksheets
Welcome to our Magic Square Worksheets page.
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189480 | https://www.quora.com/F-x-ax-2-bx-c-What-is-the-domain-and-range-of-this-function | Something went wrong. Wait a moment and try again.
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5
F(x) = ax^2 + bx+c. What is the domain and range of this function?
·
The function F(x)=ax2+bx+c is a quadratic function. Let's analyze its domain and range:
The domain of any polynomial function, including quadratic functions, is all real numbers. Therefore, for F(x):
Domain: (−∞,∞)
The range of a quadratic function depends on the coefficient a:
If a>0:
The parabola opens upwards.
The minimum value occurs at the vertex, which can be found using the vertex formula x=−b2a.
The range is from the minimum value to infinity.
Range: [F(−b2a),∞)
If a<0:
The parabola opens d
The function F(x)=ax2+bx+c is a quadratic function. Let's analyze its domain and range:
Domain
The domain of any polynomial function, including quadratic functions, is all real numbers. Therefore, for F(x):
Domain: (−∞,∞)
Range
The range of a quadratic function depends on the coefficient a:
If a>0:
The parabola opens upwards.
The minimum value occurs at the vertex, which can be found using the vertex formula x=−b2a.
The range is from the minimum value to infinity.
Range: [F(−b2a),∞)
If a<0:
The parabola opens downwards.
The maximum value occurs at the vertex.
The range is from negative infinity to the maximum value.
Range: (−∞,F(−b2a)]
If a=0:
The function is linear (not quadratic), specifically F(x)=bx+c.
If b≠0, the range is all real numbers.
If b=0, then F(x)=c, and the range is simply {c}.
Domain: (−∞,∞)
Range:
If a>0: [F(−b2a),∞)
If a<0: (−∞,F(−b2a)]
If a=0: Depends on b and c.
Related questions
What is the domain and range of the rational function F(x) =x/x-3?
What is the domain and range of f(x)=6-x- x^2?
What is the domain & inverse function of (ax^2+bx+c) / (dx^2+ex+f)?
What is the domain and range of the function "f(x) =1-2x-x^2"?
How do I integrate
(
a
x
2
+
b
x
+
c
)
n
?
Mohd Saud
B.Sc(Hons.) in Mathematics, Aligarh Muslim University (Graduated 2023)
·
Author has 159 answers and 102.1K answer views
·
4y
F(x)=ax2+bx+c
Since, there is no restrictions on x
Domain :— x∈R or x∈(−∞,∞)
Range:—
y=F(x)=ax2+bx+c
y=ax2+bx+c
y=a(x2+bax+ca)
y=a(x2+2b2ax+ca)
y=a(x2+2b2ax+(b2a)2−(b2a)2+ca)
y=a(x+b2a)2−ab24a2+aca
y=a(x+b2a)2−b24a+c
y=a(x+b2a)2−(b2−4ac4a)
Since D=b2−4ac
y=a(x+b2a)2
F(x)=ax2+bx+c
Since, there is no restrictions on x
Domain :— x∈R or x∈(−∞,∞)
Range:—
y=F(x)=ax2+bx+c
y=ax2+bx+c
y=a(x2+bax+ca)
y=a(x2+2b2ax+ca)
y=a(x2+2b2ax+(b2a)2−(b2a)2+ca)
y=a(x+b2a)2−ab24a2+aca
y=a(x+b2a)2−b24a+c
y=a(x+b2a)2−(b2−4ac4a)
Since D=b2−4ac
y=a(x+b2a)2−(D4a)
If a>0
y=a(x+b2a)2minimum−0−(D4a)
If a<0
y=a(x+b2a)2maximum−0−(D4a)
So, Range:—
If a>0, then y∈[−D4a,∞)
If a<0, then y∈(−∞,−D4a]
Hope the answer helps.
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Johnny M
Master's Degree from Harvard University (Graduated 2011)
·
Updated Sep 9
Does switching car insurance really save you money, or is that just marketing hype?
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I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t.
This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars.
I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take.
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If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying.
Aaron Briseno
B.S in Mathematics & Teaching, University of California, Los Angeles (Graduated 2010)
·
Author has 1.3K answers and 3.1M answer views
·
4y
F(x)=ax2+bx+c is a quadratic, the visual representation of this function is a parabola, we know parabolas open up or down, and have either a min or max value associated with the parabola’s vertex. Since F(x) is a polynomial and given with no domain restrictions it is assumed the domain is
Domain: R
So we need two cases for our range, parabola opens up, and then down first up:
Case 1: a>0
If the lead coefficient is positive we know the parabola opens upward and then the vertex is the minimum and the parabola “goes up” forever thus the range “starts” at the vertex who’s y-coordinate is F(−b/2a)
Ran
F(x)=ax2+bx+c is a quadratic, the visual representation of this function is a parabola, we know parabolas open up or down, and have either a min or max value associated with the parabola’s vertex. Since F(x) is a polynomial and given with no domain restrictions it is assumed the domain is
Domain: R
So we need two cases for our range, parabola opens up, and then down first up:
Case 1: a>0
If the lead coefficient is positive we know the parabola opens upward and then the vertex is the minimum and the parabola “goes up” forever thus the range “starts” at the vertex who’s y-coordinate is F(−b/2a)
Range: [F(−b/2a),∞)
Case 2: a<0
If the lead coefficient is negative we know the parabola opens downward, and goes down for ever, and then also this means the vertex is the maximum, and so the y-value of the vertex is F(−b/2a)
Range: (−∞,F(−b/2a)]
John Malaspina
Former Chemical Refinery Facility Manager ‘76 to ‘11 (1976–2011)
·
4y
I hope someone can shed more light on this and educate me! In 1968, eighth grade, my school never taught the terms domain and range. I looked the definitions up, so my answer for the type equation is the domain and range are both “The infinite set of rational numbers“ for this type equation. This equation is called a quadratic equation.
Related questions
What is the domain and range of f(x) =|x-1|+|x - 2|, -1≤x≤3?
What is the domain and range of f(x) = x^2+1?
[F(x) is a one-to-one function with domain innocuous –2, 4) and range (–1, 2). What are the innocuous domain and range of f −1(x)?
What is the range of the function f(x) =(x-a) / (x^2-a^2)?
What’s the domain and range of (x^2+x+1)/x ?
Donald Hartig
PhD in Mathematics, University of California, Santa Barbara (Graduated 1970)
·
Author has 7.4K answers and 2.8M answer views
·
4y
The domain is all x.
SińceF(x)=a((x+b/(2a))2+c/a−b2/(4a2)),
if a>0 the range is [F(−b/(2a)),∞).
if a<0 the range is (−∞,F(−b/(2a))].
if a=0 the range is (−∞,∞)providedb≠0.
if a,b=0 the range is {c}.
Note. When a≠0,F(−b/(2a))=c−b2/(4a).
Promoted by Webflow
Metis Chan
Works at Webflow
·
Aug 11
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When it comes to AI website builders, there are a growing number of options, but a few stand out for their power, flexibility, and ability to grow with your needs.
Webflow’s AI Site Builder is a top choice for small businesses, in-house teams, and agencies who want the speed of AI and the freedom to fully customize every part of their site. With Webflow, you can:
Describe your business or idea and instantly generate a unique, production-ready website—no coding required.
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Ram Basnet
MA, B.Ed in English (language) & Mathematics, Swami Vivekanand Subharti University (Graduated 2024)
·
1y
Originally Answered: What are the characteristics of a polynomial function, f(x) = ax^2 + bx + c?
·
Characteristics of the function, f(x) = ax² + bx +c
i. It is a quadratic function.
ii. It represents the equation of a parabola.
iii. [(-b/2a),f(-b/2a)] are the vertices of parabola.
iv. Value of ‘a’determines the direction in which the parabola opens. Negative value shows a parabola opening downwards, where a positive value shows parabola opening upwards.
Daniel Zhang
5y
Related
How do I find the domain and range of the function f(x) = 5-(x+3) ²?
The domain corresponds to all valid x-values for the equation. Since this is a quadratic equation (highest degree is 2) all x-values are valid.
Domain: {x∈ℝ}
I’d like to point out that this equation is written in vertex form. Vertex formula is written as the following:
y = a(x - h)^2 + k, where (h, k) is the vertex.
y = -(x + 3)^2 + 5, where (-3, 5) is the vertex.
We know this is an upside down parabola because the “a” value in the equation is negative. When there is a positive value for “a”, it is right side up, but when there is a negative value, it is upside down. Because this is upside down, th
The domain corresponds to all valid x-values for the equation. Since this is a quadratic equation (highest degree is 2) all x-values are valid.
Domain: {x∈ℝ}
I’d like to point out that this equation is written in vertex form. Vertex formula is written as the following:
y = a(x - h)^2 + k, where (h, k) is the vertex.
y = -(x + 3)^2 + 5, where (-3, 5) is the vertex.
We know this is an upside down parabola because the “a” value in the equation is negative. When there is a positive value for “a”, it is right side up, but when there is a negative value, it is upside down. Because this is upside down, the vertex is at its maximum point, and no point on the parabola is higher than it. Therefore, the y-values all have to be lower than or equal to y = 5.
Range: {y∈ℝ | y ≤ 5}
Here is a graph of the equation:
Hope this helps!
Promoted by Network Solutions®
Zachary Tidwell
Senior Communications Manager at Network Solutions (2015–present)
·
Wed
Who is the best domain registrar and why? Who would you recommend for a new website owner, and why?
When you’re looking for the best domain provider, it’s important to take the full range of capabilities and options available through that provider into account. Why? Because an online journey, especially for a small business owner or entrepreneur, is rarely a straight line from A to B.
I’ve worked in the web presence industry for a decade now, and I’ve seen business owners who called in to buy the same product, who in fact had very different needs. I’ve seen sales agents help customers who didn’t even have a name for their business yet, and I’ve seen successful businesses with numerous locatio
When you’re looking for the best domain provider, it’s important to take the full range of capabilities and options available through that provider into account. Why? Because an online journey, especially for a small business owner or entrepreneur, is rarely a straight line from A to B.
I’ve worked in the web presence industry for a decade now, and I’ve seen business owners who called in to buy the same product, who in fact had very different needs. I’ve seen sales agents help customers who didn’t even have a name for their business yet, and I’ve seen successful businesses with numerous locations finally get around to their online presence, long after achieving brick-and-mortar success.
So, I’m going to focus on how to find the best domain provider for small businesses in particular, because that’s what my company, Network Solutions, specializes in, and that’s where my expertise lies.
With that in mind, here are some considerations to keep in mind when deciding where to purchase a domain.
Simplified Search
Anyone who’s spent time searching for a domain name can tell you that finding a great option isn’t always easy. In many cases, that shiny, perfect domain name you envisioned may already be taken. This is where a feature like an AI domain name generator can be extremely helpful.
I cover this in a lot more detail in this answer, but the main thing to know is that AI domain name generation helps you find helpful, pleasantly surprising alternatives to your preferred domain name, immensely simplifying the search process.
Reputation and Credibility
Whether you’re looking for a taco stand or a car dealership, you always want to make sure you choose an established company with a long-time presence and strong reputation. The same thing applies to the world of domains. In general, large, well-established companies will have the expertise and strong support to get you where you want to be online.
Look for a company with a strong score (definitely 4.0+) on Trustpilot, and check to see how long that company has been in business. In my view, longevity and established trust in the domain industry should be always be two major points of consideration when choosing a provider.
A Full Suite of Products
Maybe today you just want a domain name. But you really should try to work with a company that can support your end-to-end online journey, whatever direction it goes. Look for additional features like an AI-powered website builder, reliable web hosting, robust cybersecurity products and online marketing tools, so you can take that next step when you’re ready.
Another factor to consider here is professional services. DIY products are hugely popular, and for good reason, but it’s really nice to have the ability to turn to professional website designers and online marketing specialists if needed.
The Journey
So often, I see online journeys start with finding a great domain name, and progress from there to a truly expansive and successful digital endeavor. One of my favorite things is seeing small businesses make the most of their websites and eCommerce stores to get an edge on the competition and even compete with some of the bigger players out there, with the innovative tools available today really acting as an equalizer for so many businesses.
As you search for an exceptional domain provider, I recommend you keep your options open and leave yourself and your business room to grow. Wherever your online journey takes you, I wish you the best of luck, and I hope finding your perfect domain name is just the start for your web presence.
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
7y
Related
What is the domain and range of the function
f
(
x
)
=
x
2
−
x
+
1
x
2
+
x
+
1
?
I always like to finish off a question like this with a nice graph which clearly verifies my working.
Firstly the denominator does not factorise and its discriminant is negative so it cannot become zero. This means there are no vertical asymptotes.
Straight away we can say that the DOMAIN must be – ∞ < x < ∞ and this means x can be any number.
Differentiating we get…
The range is ⅓ ≤ y ≤ 3
Incidentally when x is large y ≈ 1 so there is a horizontal asymptote y = 1
Here is the graph:
I always like to finish off a question like this with a nice graph which clearly verifies my working.
Firstly the denominator does not factorise and its discriminant is negative so it cannot become zero. This means there are no vertical asymptotes.
Straight away we can say that the DOMAIN must be – ∞ < x < ∞ and this means x can be any number.
Differentiating we get…
The range is ⅓ ≤ y ≤ 3
Incidentally when x is large y ≈ 1 so there is a horizontal asymptote y = 1
Here is the graph:
BowTangey
PhD in Mathematics, Iowa State University (Graduated 1988)
·
Author has 1.2K answers and 250.4K answer views
·
Updated 2y
Related
What is the domain range of function 2^(x^x)?
2 raised to any real number is defined as a real number, so the issue is
x
x
. Ordinarily, this issue is evaded by claiming
x
x
=
e
x
ln
(
x
)
, so that
x
0
is required. But this is obviously sidestepping the issue, as any bright high-schooler knows that
(
−
1
)
−
1
=
−
1
.
There is some nice discussion of this at
math.stackexchange
. Google “ the domain of x^x”, and look at the May 3, 2019 posting of MSX.
Check out the graph of
y
=
x
x
from
Desmos
:
2 raised to any real number is defined as a real number, so the issue is
x
x
. Ordinarily, this issue is evaded by claiming
x
x
=
e
x
ln
(
x
)
, so that
x
0
is required. But this is obviously sidestepping the issue, as any bright high-schooler knows that
(
−
1
)
−
1
=
−
1
.
There is some nice discussion of this at
math.stackexchange
. Google “ the domain of x^x”, and look at the May 3, 2019 posting of MSX.
Check out the graph of
y
=
x
x
from
Desmos
:
Promoted by The Penny Hoarder
Lisa Dawson
Finance Writer at The Penny Hoarder
·
Updated Sep 16
What's some brutally honest advice that everyone should know?
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Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
Updated 1y
Related
What is the domain and range of the function f(x) = 2√ (x + 1)?
You just need to plot a few points and sketch a little graph…
NOTE: √4 = 2 and not ±2
The Domain is all x values ≥ – 1
and the Range is all y values ≥ 0
You just need to plot a few points and sketch a little graph…
NOTE: √4 = 2 and not ±2
The Domain is all x values ≥ – 1
and the Range is all y values ≥ 0
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
7y
Related
What is the domain and range of the function f (x) =x^2-1÷x-1?
Most students probably do this without a second thought:
It is true that:
but only if x does not equal 1
because then you would be saying that:
Because of this we have to exclude x = 1 by putting a “hole” in the l...
Most students probably do this without a second thought:
It is true that:
but only if x does not equal 1
because then you would be saying that:
Because of this we have to exclude x = 1 by putting a “hole” in the l...
Enrico Gregorio
Associate professor in Algebra
·
Upvoted by
David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 18.4K answers and 16M answer views
·
6y
Related
What is the domain and range of the real function
f
(
x
)
=
(
x
2
−
x
+
1
)
/
(
x
2
+
x
+
1
)
?
The denominator vanishes nowhere, so the function is defined for every real x.
In order to find the range, determine the maximum and minimum values (if they exist). Since
f(x)=1−2xx2+x+1
we can compute the derivative as
f′(x)=−2x2+x+1−x(2x+1)(x2+x+1)2=2(x2−1)(x2+x+1)2
so the function is increasing over (−∞,−1] and over [1,∞), while decreasing over [−1,1].
Note also that
which proves the function is bounded.
Since [math]f(-1)=3[/math] and [math]f(1)=1/3[/math], the range is [math][1/3,3][/math].
The denominator vanishes nowhere, so the function is defined for every real [math]x[/math].
In order to find the range, determine the maximum and minimum values (if they exist). Since
we can compute the derivative as
so the function is increasing over math[/math], while decreasing over [math][-1,1][/math].
Note also that
which proves the function is bounded.
Since [math]f(-1)=3[/math] and [math]f(1)=1/3[/math], the range is [math][1/3,3][/math].
Diane Evans
B.A. in Mathematics (college major) & Computer Science, University of Washington
·
Author has 5.7K answers and 12M answer views
·
4y
Related
What is the domain and range of f(x) =x^3?
The domain of a function is all of the values that you can use for x. In this case, you can use any number for x. (You generally will only have restrictions if x is in a denominator or radical.)
The domain is all real numbers.
The range is all values that can come out of the function. The range can be trickier to determine. In this case, since we have an odd power, we can get any value out of the function, so the range is all real numbers.
A quick way to check for domain and range is to put the function into a graphing calculator and see the result.
f(x) = x^3 looks like this:
But x^2 has obvious r
The domain of a function is all of the values that you can use for x. In this case, you can use any number for x. (You generally will only have restrictions if x is in a denominator or radical.)
The domain is all real numbers.
The range is all values that can come out of the function. The range can be trickier to determine. In this case, since we have an odd power, we can get any value out of the function, so the range is all real numbers.
A quick way to check for domain and range is to put the function into a graphing calculator and see the result.
f(x) = x^3 looks like this:
But x^2 has obvious range restrictions:
While 1/x has obvious domain and range restrictions:
Related questions
What is the domain and range of the rational function F(x) =x/x-3?
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What is the domain & inverse function of (ax^2+bx+c) / (dx^2+ex+f)?
What is the domain and range of the function "f(x) =1-2x-x^2"?
How do I integrate
(
a
x
2
+
b
x
+
c
)
n
?
What is the domain and range of f(x) =|x-1|+|x - 2|, -1≤x≤3?
What is the domain and range of f(x) = x^2+1?
F(x) is a one-to-one function with domain innocuous –2, 4) and range (–1, 2). What are the innocuous domain and range of f −1(x)?
What is the range of the function f(x) =(x-a) / (x^2-a^2)?
What’s the domain and range of (x^2+x+1)/x ?
What is the range of the function: f(x) =(x-2) ^2 +2?
[What is the domain and range of the real function
f
(
x
)
=
(
x
2
−
x
+
1
)
/
(
x
2
+
x
+
1
)
?
How do I find the domain and range of the function f(x) = 5-(x+3) ²?
What is the domain and range of the function f(x) =x^2?
What will be the range of the function f(x) = (x^2 + x + 2) / (x^2 + x + 1)?
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189481 | https://www.youtube.com/watch?v=AHDoc4K0wkA | Física I Lección 14B Órbitas circulares
EdwinLoaiza
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Posted: 17 Oct 2015
2. Órbitas circulares:
La fuerza gravitacional proporciona la aceleración centrípeta requerida para los satélites en órbita. Si se conoce el radio de una órbita circular de un satélite alrededor de la Tierra, se puede calcular la velocidad orbital y el período orbital. Se presentan algunos ejemplos tanto del movimiento de la luna alrededor de la Tierra como de la Tierra alrededor del sol. El periodo orbital es independiente de la masa del objeto en órbita.
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Let's pursue the idea of circular orbits. Later in the course we will cover elliptical orbits, but now let's exclusively talk about circular orbits. Now, this is the mass of the Earth, and in
a circular orbit is an object with mass m, a satellite, and m is way, way, way smaller
than the mass of the Earth. And the radius of the orbit is R, and this
object has a certain velocity v, tangential speed. The speed doesn't change, but the direction
changes, and there has to be a gravitational force to hold it in orbit, and the gravitational
force is exactly the same as the centripetal force-- we've discussed that many times before. And so the gravitational force which is necessary
to make it go around-- I could also say the centripetal force is
necessary to make it go around in a circle-- that gravitational force equals m M earth
G divided by R squared. This is now the distance from the Earth to
the satellite, and that must be equal to m v squared divided by R, and that is that tangential speed that you see here, which a little later in time, of course, would be here. I lose my m, and so you see now that the orbital
speed-- not to be mistaken for escape velocity-- the orbital speed is exactly the same what we have there except the square root of two divided by R. This is now R. And there it was R earth. If you know R, then you can calculate the
speed in orbit. If you know the speed in orbit, you can calculate
R. And so the period of going around in the orbit,
T equals two pi R divided by the orbital speed, and when you do that, you get two pi. You get an R to the power three-halfs, and
you have downstairs the square root of G M earth. Let me move this in a little. Two pi R to the power three-halfs. So again, if you know the radius, if you know how far you are away from the Earth, the period follows uniquely. If you know the period, then the distance
to the satellite follows uniquely. If we take the shuttle as an example of a
near-Earth orbit, so we have the shuttle. The shuttle may be 400 kilometers above the
Earth's surface. So we have to add to the radius of the Earth
400 kilometers, so you end up with about 6,800 kilometers for the radius of the orbit of the shuttle, and you substitute that in here, the mass of the Earth and the gravitational
constant, you'll find that T is about 90 minutes. It's about 1.5 hours. The shuttle takes about 1.5 hours to go around,
and the speed, that tangential speed, is very close to eight kilometers per second. And that holds for all near-Earth-orbit satellites. Whether they are 400 or 500 or 600 kilometers,
that doesn't change very much. If you take the moon-- the moon is much further away than the shuttle,
and you take the distance to the moon-- which is some 385,000 kilometers-- you substitute that in this equation, you will find that the period for the moon to go around the Earth is about 27.5 days. And its speed is only one kilometer per second. It's much further out. If it's much further out, R is much larger,
and so you see the speed will be much lower. If you take the Earth itself around the sun-- because we can use all these equations-- replace the mass of the Earth by the mass
of the sun, and then we can do this for planets. So if we take the Earth around the sun, then
we have to put in the mass of the sun, which is about two times ten to the 30 kilograms. And the distance from the Earth to the sun,
we have seen that before-- I call that the distance from the sun to the
Earth-- is about 150 million kilometers. Forgive me for mixing up meters with kilometers,
but you have to convert that, of course, to meters. And when you calculate how long it takes the
Earth to go around the sun, no surprise-- you will find 365.5 days. So that's simply substitution of these two quantities in the equation that I have here and that I have here. The velocity of the Earth in orbit is about
30 kilometers per second. That's a substantial speed, by the way, that
the Earth is going around the sun-- 30 kilometers per second-- way higher than the speed that the shuttle
is in around the orbit... around the Earth, which is only eight kilometers per second. Jupiter is five times further away than the
Earth, and so the time for Jupiter to go around goes with five to the power 1.5. That's about 12, so it takes Jupiter about
12 years to go around the sun. Notice that this period is independent of
the mass of the little satellite, and that was very unfortunate for the Americans when
on October 4, 1957, Sputnik was launched. They could find the radius very easily, because
they knew the period that it took Sputnik to go around the Earth. That was about 96 minutes. They could calculate the velocity, they could
calculate the radius, but they had no clue about the mass, and that was a key piece of ingredient that the Americans wanted, because if the mass was very large of Sputnik, that
would indicate, of course, that the Russians had very powerful rockets. You cannot tell the mass from the orbital
parameters-- it's independent of mass. Whether you have a very light object or a
very heavy satellite, they have the same velocity in orbit if they are at the same distance, and they have the same orbital period. I mentioned earlier, notice that the orbital
period and the escape velocity vary by a square root of two if you are at a particular position. For instance, you're at a particular position
around the Earth, here at a satellite. If you want to escape from this, you will
need a speed which is the square root of two times larger than that orbital velocity. And so if you wanted to escape from the Earth,
then you need your 11.2 kilometers-- we have it there. If you are near Earth in orbit, you are eight
kilometers per second, and eight times the square root of two is exactly that 11.2. So you see the connection is always through
this square root of two. There is something remarkable about these
numbers. The total mechanical energy-- and I will write that once more here-- which is one-half m v squared at a given radius
minus little m capital M G over r; whether M is the mass of the sun or the Earth is of
no concern to me now. This is the kinetic energy for something in
orbit at this radius, and this is the potential energy. But now notice, I can substitute now for this
v squared, I can substitute the square of this-- that is the orbital speed-- and then I get M G over R. And so this one equals one-half m MG over R. And now compare the two. They almost look like carbon copies of each
other, except that there is a minus sign here, which is crucial, and there is a half here,
which is missing here. And so the total energy E-- which I've called the mechanical energy-- always for a circular orbit is one-half U
and is the same as minus the kinetic energy. A remarkable coincidence, you would think. It is not as much a coincidence as you think,
of course. But if something is in orbit, this is the
orbital speed at radius r, then always is its total energy is half the potential energy. It's always negative. Subtitles by the Amara.org community |
189482 | https://math.stackexchange.com/questions/2891552/are-two-planes-perpendicular-to-a-third-plane-parallel-to-each-other | vector spaces - Are two planes perpendicular to a third plane parallel to each other? - Mathematics Stack Exchange
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Are two planes perpendicular to a third plane parallel to each other?
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Well, my intuition says they are always parallel, but the text book says they are not. I am quite shocked to be honest.
Does anyone know a case they are not parallel and perpendicular to a third plane at the same time?
Also, two planes that are parallel to a line must also parallel to each other in my opinion, and the text book says they are not always parallel.
Does anyone know why?
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asked Aug 22, 2018 at 23:45
강승태강승태
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In R 3 R 3, a simple counterexample for the first assertion is the three standard coordinate planes, which are pairwise perpendicular.
For the second assertion, consider the z z-axis. The planes x=1 x=1 and y=1 y=1 are both parallel to the z z-axis, but are perpendicular to each other. Being parallel to a line still leaves a degree of freedom in choosing the direction of the plane.
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answered Aug 23, 2018 at 0:06
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No. Of course not. Place a hardback book standing up on your desktop and open the two hard covers to some arbitrary angle. The planes of the covers are each perpendicular to the desk top, but not parallel to each other (or necessarily perpendicular to each other).
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answered Aug 22, 2018 at 23:47
David G. StorkDavid G. Stork
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Take any two planes π 1 π 1 and π 2 π 2 that are not parallel to each other. The two planes π 1 π 1 and π 2 π 2 intersect in a line. Call that line ℓ ℓ.
Then any plane perpendicular to ℓ ℓ is also perpendicular both to π 1 π 1 and to π 2.π 2.
Also, any line parallel to ℓ ℓ (including ℓ ℓ itself) is parallel both to π 1 π 1 and to π 2.π 2.
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answered Aug 23, 2018 at 12:06
David KDavid K
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Thanks for help. It is clearer 강승태 –강승태 2018-08-24 02:54:47 +00:00 Commented Aug 24, 2018 at 2:54
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Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations.
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Home>Knowledge
How Many Amps Does A Toaster Use
Amps · Battery Backup
Updated July 31, 2025
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Table of Contents
How Many Amps Does a Toaster Use
How To Calculate Your Toaster Amps
Tips to Save Your Electricity Bill
Solar Generators for Toaster
FAQs
Conclusion
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A toaster normally uses 8 to 10 amps of amperage. However, specifically determining howmanyampsatoasterusesdepends on its size, voltage, daily consumption, type, and other conditions. The efficiency of a toaster depends highly upon its size, as the smaller it is, the faster it will heat up.
Usually, a toaster's impact on an electricity bill is almost negligible. In case of blackouts and/or off-grid circumstances, you can power your toaster using reliable solar-power generatorslike the Jackery Solar Generators. The solar-powered generator provides a steady, reliable, and eco-friendly power source, allowing electronic devices to be used indoors and outdoors.
Key Takeaways
The most commonly used pop-up toaster normally uses 8 to 10 amps of amperage.
The power rating, operating voltage, types, and size of the toaster are some of the primary factors determining its amps.
Whether you are using a commercial toaster, a pop-up, a panini press, a smart toaster, or an air fryer toaster, the startup wattage is almost equal to the running wattage.
A toaster with good sealing and insulation capabilities retains the heat inside the oven, reducing the need for excessive energy.
How Many Amps Does a Toaster Use
A two-slice toaster with a manual lift lever of 900 watts draws about 7 amps, whereas a rapid two-slice toaster of 1200 watts draws about 10 amps of current. A long-slot four-slice toaster of 1600 watts draws about 13 amps of current, and the one with extra-wide slots of 1000 watts draws 12 amps of current. The primary factors determining a toaster'sampsare its power rating, operating voltage, and design features affecting the current draw.
Definition Of Amps, Volts, Watts, Running Watts
As we deal with numerous electrical appliances in our day-to-day lives, it is crucial to understand basic terminology like amps, watts, and voltsto learn in detail about toaster usage. To understand how many amps a toaster oven uses, let's check out the basics of the electricity units.
Amps (Amperes):Ampere is the SI base unit of electric current, which measures the rate of electron flow in a system. In the context of a toaster, amps measure the flow of current that your toaster draws from the supply. You can calculate amps by using the formula:
Amps = Watts ÷ Volts.
Volts:In the MKS (meter–kilogram—second) system, a volt is the unit of electromotive force. It measures the potential difference between two points of a conducting wire in a toaster. The higher the voltage, the more electricity flows to a toaster, as voltage represents the pressure that pushes electricity.
Volts = Watts ÷ Amps
Watts:In simple terms, watts are the amount of energy a toaster consumes. Wattage represents the efficiency and functionality of a toaster. Understanding the wattage consumption is important as it determines the performance and safety of a toaster. You can quantify the rate of energy transfer by using the formula:
Watts = Volts × Amps.
Running Watts:The continuous power your toaster uses while operating at standard conditions refers to its running wattage. The measure of running wattage indicates how much continuous power a backup power source, such as a solar power generator, can supply. Running wattage can be calculated using the formula:
Running Watts = Volts × Running Amps.
To learn more about how many watts a toaster uses, refer to our detailed guide, which explains the amount of energy consumed by different-sized toasters in terms of categories, per-hour usage, toaster settings, and more.
How Many Amps Do A Toaster Oven, Pop-Up Toaster, Conveyor Belt Toaster, Panini Press Toaster, Smart Toaster, And Air Fryer Toaster Use?
Although pop-up toasters are the most common ones in every household, there are other types that align with your daily breakfast routine. Most household/pop-up toasters fall within a range of 800 to 1500 watts and draw 7 to 12 amps of current. A toaster oven, on the other hand, operates at 1000 to 1800 watts and draws 8 to 15 amps of amperage.
A commercial conveyor belt toaster for big households uses 2500 to 3000 watts and would draw 21 to 25 amps of current. Similarly, a panini press toaster would fall within 750 to 1000 watts and draw 6 to 8 amps. A smart toaster operating at 750 to 1400 wattage draws somewhat about 6 to 12 amps, and an air fryer toaster falling within a range of 800 to 1800 watts would draw 7 to 15 amps of current.
Here's a detailed comparison of the different types of toasters, the appliance wattage, and amp usage:
Toaster
TypesToaster WattageActual Energy Usage
(Running Wattage)Amps ( = Watts / Volts)
Pop-Up Toaster 800W-1500W 800W-1500W 7-12 Amps
Toaster Oven 1000W-1800W 1000W-1800W 8-15 Amps
Conveyor Belt Toasters 2500W-3000W 2500W-3000W 21-25 Amps
Panini Press Toasters 750W-1000W 750W-1000W 6-8 Amps
Smart Toasters 750W-1400W 750W-1400W 6-12 Amps
Air Fryer Toaster 800W-1800W 800W-1800W 7-15 Amps
How Many Amps Does a Toaster Draw on a Startup?
There are two wattages on which many electrical appliances run: starting and running wattage. Although many large appliances that come with compressors have a starting and running wattage, toasters do not have a starting/surge wattage. It is advisable to look at the running wattage of the toaster to determine how many amps it uses.
Starting Watts Vs. Running Watts
As the name suggests, running or rated wattage is the continuous power needed to keep the appliances operational. In motor-driven appliances, started wattage is the extra wattage needed to start the appliance, and this is the maximum wattage produced by the generator.
In toasters, basically, no initial current is required as there is no motor or compressor. Whether you are using a commercial toaster, a pop-up, a panini press, a smart toaster, or an air fryer toaster, the startup wattage is almost equal to the running wattage.
For example, if your toaster runs at 720 watts, 120 volts, and draws 6 amps of current, then:
Running Watts = Volts x Amps = 120V x 6A = 720W
Startup Amps Vs. Running Amps
Unlike motor-driven products, toasters have a low difference between started and running amps. That means if a toaster uses 6 amps of current while operating, the startup amp will be more or less the same.
For example, if you have a 1000W toaster plugged into a 120V outlet, then:
Running Amps = Watts ÷ Volts = 1000W ÷ 120V = 8.3 amps.
Even though the difference between the startup and running amp is nearly negligible, to understand the concept better, we can consider a power fluctuation where the toaster uses 1200W at startup and is plugged into a 120V outlet, then:
Startup Amps = Startup Watts ÷ Volts = 1200W ÷ 120V = 10 amps.
How To Calculate Your Toaster Amps
You need to understand how many amps a toaster draws to evaluate your energy requirements. Once you know your toaster amps, you can manage the electrical load of your kitchen and prevent unwanted circumstances.
Various easy-to-use methods are available to calculate your toasteramps, and they have different levels of accuracy. Determining your toaster's amperage is important before you can ensure its compatibility with power sources like solar power generators.
Method 1: Manufacturer Details: Energy Star Rating
This method is one of the simplest ways to determine the amps your toaster uses. Every product has the manufacturer's details on the label or user manual. Although toasters do not tend to fluctuate electricity bills, many households prefer energy-star-rated products.
The user manual or labels of the appliances mention the wattage and voltage information through which you can determine the amperage. For example, if your toaster is rated 1400W and 120V, then the user manual might indicate the amperage. This also lets you avoid confusion by not depending on any third-party device! The rating mentioned on the label can help you find the approximate amps value by making a simple calculation.
For example, if the toaster consumes 1200W of power and is plugged into a 120V outlet, then you can calculate the amps using the formula:
Amps = Watts ÷ Volts = 1200W ÷ 120V = 10 amps.
Method 2: Determine the Amps From the Wattage
In this method, you can determine the amperage if you know the wattage of your toaster, and the amperage is not listed in the manual. For example, if the running wattage of a pop-up toaster is 800W and operates at 120V, then using the below formula, you can calculate the amps:
Amps = Wattage ÷ Volts = 800W ÷ 120V = 6.6 amps.
Method 3: Toaster Power Meter
This method is efficient for situations when there is a power fluctuation, and you are looking for the most accurate measurement. A specialized toaster power meter is directly plugged into the outlet and provides an accurate measurement. To use the toaster power meter, connect the toaster power outlet to the wall outlet and plug your toaster into the meter. The meter will display the exact wattage, amperage, and voltage readings.
Tips to Save Your Electricity Bill
A toaster needs a few minutes to toast a few slices of bread. Hence, if the usage is more, the impact on the electricity bill can be significant. However, in case of power outages or blackouts, it is important to know the power requirement a toaster uses. Here are some tips to help you make efficient use of a toaster and reduce your electric bills significantly:
Choose a toaster according to your needs or the number of house residents. A small toaster has a small interior volume to heat and tends to consume less energy. A compact model will be energy-efficient if you use a toaster only for toasting purposes.
A toaster with good sealing and insulation capabilities retains the heat inside the oven. This helps the toaster maintain the desired temperature, reducing the need for excessive energy.
Some toaster ovens have cooking functions that are energy-saving and optimize energy consumption. One of the features is convection cooking, which allows food to be cooked evenly and faster. This helps to save cooking time and save much energy compared to traditional toasters.
Another feature that can help save energy is a built-in timer and shut-off feature. This feature turns off the toaster automatically once cooking is complete, reducing energy waste.
Toaster ovens with Energy Star certification are designed to consume less energy. These meet the guidelines set by the Environmental Protection Agency (EPA)and the Department of Energy (DOE).
Solar Generators for Toaster
There will be days when you are off-the-grid during camping, hiking, fishing, or RVing, or there is a prolonged power cut at home. In such scenarios, you can power your toaster using Jackery Solar Generators. With high efficiency and reliability, these solar generators can power up your toasters and other high-wattage appliances such as microwaves, televisions, refrigerators, etc.
Jackery Solar Generators combines solar panelsand power stations as a solar power system. The solar panels have specialized monocrystalline solar cells that absorb solar radiation when placed outside on a bright sunny day. Once absorbed, the solar panels convert solar energy into electrical energy. This direct current (DC) is then transferred to the power station, which is converted to alternating current (AC) to charge the toaster.
Jackery Solar Generator 2000 Plus
Jackery Solar Generator 2000 Plus features an expandable capacity and is compatible both for household and outdoor adventures. It can power various types of toasters, from pop-up to toaster ovens. The solar generator delivers mega power by adding multiple battery packs and solar panels. It is ideal for off-grid living, extended blackouts, and as a battery backup for home.
Jackery Solar Generator 2000 Plus
Jackery Solar Generator 2000 Plus
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Jackery Solar Generator 1000 Plus
The Jackery Solar Generator 1000 Plus is a compact, lightweight solar generator that supports 99% of devices. It is an ideal solar generator that can power pop-up toasters, panini presses, and even a smart toaster. With long-lasting reliability, you can even power your toaster while you are driving.
Jackery Solar Generator 1000 Plus
Jackery Solar Generator 1000 Plus
More Power in Smaller Size:With a capacity of 1264Wh and a 2000W output, the 1000 Plus supports 99% of devices.
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Jackery Solar Generator 3000 Pro
Boasting a large capacity, the Jackery Solar Generator 3000 Pro can power 99% of outdoor devices. It features a portable design with a pull rod and double wheels and ensures all-around safety. It can power conveyor toasters, toaster ovens, and pop-up toasters. The solar generator is ideal for glamping, camping, and home emergencies.
Jackery Solar Generator 3000 Pro
Jackery Solar Generator 3000 Pro
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Jackery ProductsCapacity & Output WattsToaster Running Time
Jackery Solar Generator 2000 Plus Capacity: 2042.8Wh
Rated Output: 3000W Pop-up Toaster (800W): 2.17H
Toaster Oven (1000W): 1.7H
Panini Press (750W): 2.3H
Smart Toaster (750W): 2.3H
Air Fryer (800W): 2.1H
Jackery Solar Generator 1000 Plus Capacity: 1264.64Wh
Rated Output: 2000W Pop-up Toaster (800W): 1.3H
Toaster Oven (1000W): 1.07H
Panini Press (750W): 1.4H
Smart Toaster (750W): 1.4H
Air Fryer (800W): 1.3H
Jackery Solar Generator 3000 Pro Capacity: 3024Wh
Rated Output: 3000W Pop-up Toaster (800W): 3.2H
Toaster Oven (1000W): 2.5H
Panini Press (750W): 3.4H
Smart Toaster (750W): 3.4H
Air Fryer (800W): 3.2H
Conveyor Belt Toaster (2500W): 1.02H
FAQs
How many amps does a 2 slice toaster use?
A two-slice toaster with a manual lift lever of 900 watts draws about 7 amps, whereas a rapid two-slice toaster of 1200 watts draws about 10 amps of current.
How many amps does a 4 slice toaster draw?
A long slot four-slice toaster of 1600 watts draws about 13 amps of current. A four-slice toaster with extra wide slots of 1000 watts draws 12 amps of current.
How many amps does a 1500-watt toaster oven use?
A 1500-watt toaster operating at 120V voltage will use around 12.5 amps current. The same can be calculated using the formula Amps = Watts ÷ Volts. Different types of toasters operating at around 1500 watts include pop-ups, panini presses, air-fryers, and smart toasters.
How many amps does a 1000-watt toaster use?
A 1000-watt toaster operating at 120V voltage will use around 8 amps current. The same can be calculated using the formula Amps = Watts ÷ Volts. Different types of toasters operating at around 1000 watts include pop-ups, toaster ovens, panini presses, smart toaster, and air fryer toaster.
Conclusion
Once you know how many amps a toaster uses,you can incorporate an energy-efficient toaster in your kitchen to make significant changes in terms of reducing energy waste and electric bills. There are many toasters in the market with advanced features such as automatic shutting off, toasting settings, and quick and even heating.
Choosing the one will help you optimize energy usage without compromising on toasting quality. Investing in solar power systems such as the Jackery Solar Generator is another way towards an energy-efficient solution. With reliable and long-lasting batteries, the solar generators can keep your toasters operational indoors and outdoors at any time of the day.
Disclaimer:
The runtime mentioned for appliances powered by Jackery is for reference only. Actual runtime may vary under different conditions. Please refer to real-world performance for accurate results.
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189485 | https://www.grc.nasa.gov/www/k-12/BGP/rktthsum.html | | | | | | | | | | | | | | | | | | |
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| | | | --- | | + Text Only Site + Non-Flash Version + Contact Glenn | | | | | | | | | | --- --- --- | | | | | | | | | | | | | On this slide, we have collected all of the equations necessary to calculate the thrust of a rocket engine. In a rocket engine, stored fuel and stored oxidizer are ignited in a combustion chamber. The combustion produces great amounts of exhaust gas at high temperature and pressure. The hot exhaust is passed through a nozzle which accelerates the flow. Thrust is produced according to Newton's third law of motion. The amount of thrust produced by the rocket depends on the mass flow rate through the engine, the exit velocity of the exhaust, and the pressure at the nozzle exit. All of these variables depend on the design of the nozzle. The smallest cross-sectional area of the nozzle is called the throat of the nozzle. The hot exhaust flow is choked at the throat, which means that the Mach number is equal to 1.0 in the throat and the mass flow rate m dot is determined by the throat area. mdot = (A pt/sqrt[Tt]) sqrt(gam/R) [(gam + 1)/2]^-[(gam + 1)/(gam - 1)/2] where A is the area of the throat, pt is the total pressure in the combustion chamber, Tt is the total temperature in the combustion chamber, gam is the ratio of specific heats of the exhaust, and R is the gas constant. The area ratio from the throat to the exit Ae sets the exit Mach number: A/A = {[(gam+1)/2]^-[(gam+1)/(gam-1)/2]} / Me [1 + Me^2 (gam-1)/2]^[(gam+1)/(gam-1)/2] We can determine the exit pressure pe and exit temperature Te from the isentropic relations at the nozzle exit: pe / pt = [1 + Me^2 (g-1)/2]^-[g/(g-1)] Te / Tt = [1 + Me^2 (g-1)/2]^-1 Knowing Te we can use the equation for the speed of sound and the definition of the Mach number to calculate the exit velocity Ve: Ve = Me sqrt (gam R Te) We now have all the information necessary to determine the thrust of a rocket. The exit pressure is only equal to free stream pressure at some design condition. We must, therefore, use the longer version of the generalized thrust equation to describe the thrust of the system. If the free stream pressure is given by p0, the rocket thrust equation is given by: F = m dot Ve + (pe - p0) Ae You can explore the design and operation of a rocket nozzle with our interactive nozzle simulator program which runs on your browser. The thrust equation shown above works for both liquid rocket and solid rocket engines. There is also an efficiency parameter called the specific impulse which works for both types of rockets and greatly simplifies the performance analysis for rockets. Activities: Guided Tours Navigation .. Beginner's Guide Home Page | | | | | | | | --- --- | | | + Inspector General Hotline + Equal Employment Opportunity Data Posted Pursuant to the No Fear Act + Budgets, Strategic Plans and Accountability Reports + Freedom of Information Act + The President's Management Agenda + NASA Privacy Statement, Disclaimer, and Accessibility Certification | | Editor: Tom Benson NASA Official: Tom Benson Last Updated: May 13 2021 + Contact Glenn | | | |
189486 | https://www.hilotutor.com/archives_hackneyed.html | Make Your Point > Archived Issues > HACKNEYED Send Make Your Point issues straight to your inbox.
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| | | | --- | | | | | | | pronounce HACKNEYED: | HACK need Hear it. | | | connect this word to others: | | | | | (Source) When Snoopy starts his novel with "It was a dark and stormy night," it's a hackneyed start: a boring, overdone, unoriginal start. You could even call it a tr___: an idea that gets reused in fiction, over and over. Can you recall that term? (To reveal any word with blanks, give it a click.) | | | definition: | |
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| pronounce HACKNEYED: |
HACK need
Hear it.
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| connect this word to others: |
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(Source)
When Snoopy starts his novel with "It was a dark and stormy night," it's a hackneyed start: a boring, overdone, unoriginal start. You could even call it a tr___: an idea that gets reused in fiction, over and over. Can you recall that term?
(To reveal any word with blanks, give it a click.)
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| definition: |
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| connect this word to others: |
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(Source)
When Snoopy starts his novel with "It was a dark and stormy night," it's a hackneyed start: a boring, overdone, unoriginal start. You could even call it a tr___: an idea that gets reused in fiction, over and over. Can you recall that term?
(To reveal any word with blanks, give it a click.)
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| definition: |
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(Source)
When Snoopy starts his novel with "It was a dark and stormy night," it's a hackneyed start: a boring, overdone, unoriginal start. You could even call it a tr___: an idea that gets reused in fiction, over and over. Can you recall that term?
(To reveal any word with blanks, give it a click.)
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| definition: |
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| definition: |
"Hackney" is the name of a place in England where, centuries ago, you could rent a small horse for the day. By the year 1575 or so, people were referring to this horse-renting as "hackneying:" you'd "hackney" a horse, or rent it for the day.
Over time, probably because these horses got overused and improperly treated, "to hackney" came to mean "to overuse anything."
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| grammatical bits: |
Part of speech:
| |
| how to use it: |
Pick the formal, semi-common word "hackneyed" when you want to insult or dismiss something as being boring, overdone, and unoriginal.
| |
| examples: |
"And we haven’t even gotten to the plot yet: It’s your basic bad-guy-wants-to-take-over-the-world kinda thing. But even Whedon seems to recognize what a hackneyed premise that is, so he has a little fun with it."
— Christy Lemire, Seattle Times, 30 April 2012
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| has this page helped you understand "hackneyed"? |
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| Awesome, I'm glad it helped! |
Awesome, I'm glad it helped!
Thanks for letting me know!
If you have any questions about this word, please message me at Liesl@HiloTutor.com.
| |
| study it: |
Explain the meaning of "hackneyed" without saying "done to death" or "banal."
| |
| try it out: |
Fill in the blank: "It's a hackneyed notion that (something has to be a certain way)."
Example 1: "It's a hackneyed notion that a song has to go 'verse, chorus, verse, chorus, bridge, chorus.'"
Example 2: "I had succumbed to the hackneyed notion that sparkling wines are only for celebrations."
— Eric Asimov, New York Times, 7 December 2020
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| If you like, use this space to write. (To save your work, copy and paste it into an email or a document.) |
| |
| try it out: |
Fill in the blank: "It's a hackneyed notion that (something has to be a certain way)."
Example 1: "It's a hackneyed notion that a song has to go 'verse, chorus, verse, chorus, bridge, chorus.'"
Example 2: "I had succumbed to the hackneyed notion that sparkling wines are only for celebrations."
— Eric Asimov, New York Times, 7 December 2020
| |
| If you like, use this space to write. (To save your work, copy and paste it into an email or a document.) |
| |
| If you like, use this space to write. (To save your work, copy and paste it into an email or a document.) |
| |
| before you review, play: |
Try to spend 20 seconds or more on the game below. Don’t skip straight to the review—first, let your working memory empty out.
Our game for this month is Palindromes in Poems: Yay!
Check out the snippet of a poem, and supply the missing palindrome: the word spelled the same backward and forward, like "yay," "sees," or "racecar." Highlight the hints if you need them, and see the answer by scrolling all the way down. Enjoy!
Try this one today:
"We fix you maternally in the mind, orient you in a case of '_____,'
'there there,' and 'you're young yet.'"
— CM Burroughs, "Gwendolyn as Lover," 2017
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| review this word: |
1.
Opposites of HACKNEYED include
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| Answers to the review questions: 1. B 2. A |
Answers to the review questions:
1. B
2. A
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| a final word: |
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| | I hope you're enjoying Make Your Point. It's made with love. I'm Liesl Johnson, a reading and writing tutor on a mission to explore, illuminate, and celebrate words. From my blog: On vocabulary... 36 ways to study words. Why we forget words, & how to remember them. How to use sophisticated words without being awkward. A disclaimer: When I write definitions, I use plain language and stick to the words' common, useful applications. If you're interested in authoritative and multiple definitions of words, I encourage you to check a dictionary. Also, because I'm American, I stick to American English when I share words' meanings, usage, and pronunciations; these elements sometimes vary across world Englishes. |
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| | I hope you're enjoying Make Your Point. It's made with love. I'm Liesl Johnson, a reading and writing tutor on a mission to explore, illuminate, and celebrate words. From my blog: On vocabulary... 36 ways to study words. Why we forget words, & how to remember them. How to use sophisticated words without being awkward. A disclaimer: When I write definitions, I use plain language and stick to the words' common, useful applications. If you're interested in authoritative and multiple definitions of words, I encourage you to check a dictionary. Also, because I'm American, I stick to American English when I share words' meanings, usage, and pronunciations; these elements sometimes vary across world Englishes. |
I hope you're enjoying Make Your Point. It's made with love. I'm Liesl Johnson, a reading and writing tutor on a mission to explore, illuminate, and celebrate words.
From my blog:
On vocabulary...
36 ways to study words.
Why we forget words, & how to remember them.
How to use sophisticated words without being awkward.
A disclaimer: When I write definitions, I use plain language and stick to the words' common, useful applications. If you're interested in authoritative and multiple definitions of words, I encourage you to check a dictionary. Also, because I'm American, I stick to American English when I share words' meanings, usage, and pronunciations; these elements sometimes vary across world Englishes.
Subscribe to "Make Your Point" for a daily vocabulary boost.
© Copyright 2023 | All rights reserved. |
189487 | https://www.chegg.com/homework-help/questions-and-answers/baseball-batting-average-probability-number-hits-divided-number-times-bat-batting-average--q106314766 | Solved In baseball, a batting average is the probability of | Chegg.com
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In baseball, a batting average is the probability of the number of hits divided by the number of times at bat. A batting average over 0.300 is very good. This means that a player will get a hit 300 times for every 1000 times at bat. So, assume the probability of getting a hit is 0.325 for each time a player is at bat. In a particular game, assume the batter
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Question: In baseball, a batting average is the probability of the number of hits divided by the number of times at bat. A batting average over 0.300 is very good. This means that a player will get a hit 300 times for every 1000 times at bat. So, assume the probability of getting a hit is 0.325 for each time a player is at bat. In a particular game, assume the batter
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In baseball, a batting average is the probability of the number of hits divided by the number of times at bat. A batting average over 0.300 is very good. This means that a player will get a hit 300 times for every 1000 times at bat. So, assume the probability of getting a hit is 0.325 for each time a player is at bat. In a particular game, assume the batter batted three times. a. This is an example of what type of probability? Jability of getting three hits in a particular game? (Round the final answer to 3 decimal places.) b. What is the probability of getting three hits in a particular game? (Round the final answer to 3 decimal places.) Probability c. What is the probability of not getting any hits in a game? (Round the final answer to 3 decimal places.) Probability d. What is the probability of getting at least one hit? (Round the final answer to 3 decimal places.) Probability
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189488 | https://mathoverflow.net/questions/332247/defining-the-standard-model-of-pa-so-that-a-space-alien-could-understand | set theory - Defining the standard model of PA so that a space alien could understand - MathOverflow
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Defining the standard model of PA so that a space alien could understand
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First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on conceptualism. Many of the ideas in those papers appeal to me, especially the idea (put in my own words, but hopefully accurate) that the power set of the natural numbers is a work in progress and not a completed infinity like N N.
In some of those papers the idea of a supertask is used to argue for the existence of the completed natural numbers. One could think of performing a supertask as building a machine that does an infinite computation in a finite amount of time and space, say by doing the n n th step, and then building a machine of half the size that will work twice as fast to do the (n+1)(n+1)th step and also recurse. (We will suppose that the concept of a supertask machine is not unreasonable, although I think this point can definitely be argued.)
The way I'm picturing such a machine is that it would be a Σ 1 Σ 1 oracle, able to answer certain questions about the natural numbers. I suppose we would also have machines that do "super-supertasks", and so forth, yielding higher order oracles.
To help motivate my question, suppose that beings from outer space came to earth and taught us how to build such machines. I suppose that some of us would start checking the validity of our work as it appears in the literature. Others would turn to the big questions: P vs. NP, RH, Goldbach, twin primes. With sufficient iterations of "super" we could even use the machines to start writing our proofs for us. Some would stop bothering.
Others would want to do quality control to check that the machines were working as intended. Suppose that the machine came back with: "Con(PA) is false." We would go to our space alien friends and say, "Something is wrong. The machines say that PA is not consistent." The aliens respond, "They are only saying that Con(PA) is false."
We start experimenting and discover that the machines also tell us that the shortest proof that "Con(PA) is false" is larger than BB(1000). It is larger than BB(BB(BB(1000))), and so forth. Thus, there would be no hope that we could ever verify by hand (or even realize in our own universe with atoms) a proof that 0=1 0=1.
One possibility would be that the machines were not working as intended. Another possibility, that we could simply never rule out (but could perhaps verify to our satisfaction if we had access to many more atoms), is that these machines were giving evidence that PA is inconsistent. But a third, important possibility would be that they were doing supertasks on a nonstandard model of PA. We would then have the option of defining natural numbers as those things "counted" by these supertask machines. And indeed, suppose our alien friends did just that--their natural numbers were those expressed by the supertask machines. From our point of view, with the standard model in mind, we might say that there were these "extra" natural numbers that the machines had to pass through in order to finish their computations--something vaguely similar to those extra compact dimensions that many versions of string theory posit. But from the aliens' perspective, these extra numbers were not extra--they were just as actual to reality as the (very) small numbers we encounter in everyday life.
So, here (finally!) come my questions.
Question 1: How would we communicate to these aliens what we mean, precisely, by "the standard model"?
The one way I know to define the standard model is via second order quantification over subsets. But we know that the axiom of the power set leads to all sorts of different models for set theory. Does this fact affect the claim that the standard model is "unique"? More to the point:
Question 2: To assert the existence of a "standard model" we have to go well beyond assuming PA (and Con(PA)). Is that extra part really expressible?
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asked May 22, 2019 at 21:04
Pace NielsenPace Nielsen
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23 We don't need aliens to create doubts about communicability and sensibility of the notion of a standard model.Andrej Bauer –Andrej Bauer 2019-05-22 22:16:35 +00:00 Commented May 22, 2019 at 22:16
3 @RodrigoFreire L o m e g a 1,ω L o m e g a 1,ω is indeed enough, and so are some other logics, like first-order logic augmented with the least-fixed-point operator. But I don't volunteer to explain to the aliens what those logics are and how to understand the formulas of those logics.Andreas Blass –Andreas Blass 2019-05-23 00:30:59 +00:00 Commented May 23, 2019 at 0:30
1 @RodrigoFreire To be more clear, I actually like your comment. If we need to assume the existence of an uncountable ordinal in order to talk about the "actual" natural numbers, so be it. I have to admit that this would be surprising to me.Pace Nielsen –Pace Nielsen 2019-05-23 00:42:03 +00:00 Commented May 23, 2019 at 0:42
2 @AndreasBlass And how do we understand "only finitely many" without access to the natural numbers?Noah Schweber –Noah Schweber 2019-05-23 05:03:53 +00:00 Commented May 23, 2019 at 5:03
2 @NoahSchweber If "finitely many free variables" were the only problem, we could get around it by saying that any formula must be a subformula of a sentence. But the real problem is "obtain" (along with "countable") because if the aliens understood "obtain" as we do, then we could define the natural numbers as the things obtainable from 0 by the operation of successor. That's one of the reasons I mentioned least-fixed-point logic in my earlier comment.Andreas Blass –Andreas Blass 2019-05-23 11:04:46 +00:00 Commented May 23, 2019 at 11:04
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These are fundamental questions. We know that any computable set of axioms which holds of the natural numbers must also have nonstandard models. But, paraphrasing Hilary Putnam, if axioms cannot capture the "intuitive notion of a natural number", what possibly could?
As I see it, there are two possible positions on this. One is that we do know what the natural numbers are, and the fact that axioms cannot capture them shows some limitation in the axiomatic method, not in the concept of number itself. The other is that our inability to capture the natural numbers axiomatically shows that we do not actually have a definite conception of them.
I hold the first view, but I admit that the second has its appeal. Asking how we could communicate the idea of a standard model to aliens brings home the difficulty of affirming that we have a clear conception of something while admitting that we are unable to communicate it through language (in particular, axioms).
This leads to deep philosophical questions about how we can communicate anything through language. Cf. Wittgenstein's "private language" argument and his ideas about rule-following.
Here are some things I would say in defense of the view that our conception of N N really is definite, despite the fact that we cannot capture it with (first order) axioms:
Skepticism about the natural numbers can be ramped up. What do you say to someone who denies that we have a clear conception of 10 100 10 100? There are serious people who would say we don't. Frankly, I think I have a clearer conception of N N than of 10 100 10 100.
My sense is that everyone accepts that our conception of the natural numbers is definite until they learn the incompleteness theorems, but some people are so impressed by these results that they abandon the idea that there even is a definite set of natural numbers. But Wittgenstein's rule-following paradox shows that even axioms may lack the definite character we ascribe to them. So why would we take them as the be-all end-all?
Taking the view that anything meaningful is captured by axioms, and thus that N N is indefinite, has some unpleasant consequences. Say you take the view that there is no distinguished "standard" model of PA: all that matters is what statements can be proven from the Peano axioms. Then you have to accept that "what statements can be proven from the Peano axioms" is itself indefinite. Because the length of a valid proof in PA is a natural number, so if we don't know what the natural numbers are then we don't know what are the possible lengths of proofs. There could be "proofs in PA" which are valid on one version of N N but not on another. Can you really swallow this?
I think the strongest argument in favor of the definiteness of N N, and against the idea that PA, or any other axiomatization, is the be-all end-all, is the evident fact that we have an open-ended ability to go beyond any computable set of axioms, for instance by affirming their consistency. If you accept PA you should accept Con(PA), and the process doesn't stop there: you can then accept Con(PA + Con(PA)), and so on. This goes on to transfinite levels. If our understanding of N N really were fully captured by some particular set of axioms then we would not feel we had a right to strengthen those axioms in any special way; the fact that we do feel we have this right shows that our understanding is not captured by any particular set of axioms.
This is my view.
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answered May 23, 2019 at 5:26
Nik WeaverNik Weaver
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1 I appreciate these ideas. For beings with access to an oracle of a model where Con(PA) is false, they would argue (presumably) that our conception of consistency is not reflected in reality. So to them, #4 might be question-begging. To them, #3 is moot because N N is not captured by the axioms, but is captured by their machines. But even assuming their machines did say that Con(PA) is true (and Con(PA+Con(PA)), and so on as far as we tested) could we be sure that the supertask machines are really getting the natural numbers? I'd argue no.Pace Nielsen –Pace Nielsen 2019-05-23 16:34:38 +00:00 Commented May 23, 2019 at 16:34
1 Point 4 is obvious in retrospect, given the second incompleteness theorem, but I hadn't quite looked at it that way. And it applies not just to PA but to any reasonably powerful theory T. Thus it seems to really drive home the limitations of the axiomatic method.user76284 –user76284 2019-05-26 04:32:05 +00:00 Commented May 26, 2019 at 4:32
1 @AndreasBlass: what makes me uncomfortable is the idea that one can "intuitively" understand something but not be able to express that understanding through language. Maybe there's no problem here, but I'm not sure there's no problem.Nik Weaver –Nik Weaver 2019-05-27 18:46:53 +00:00 Commented May 27, 2019 at 18:46
1 @NikWeaver, I think the aliens from the first-order land of P A+¬C o n(P A)P A+¬C o n(P A) say to us "you've agreed to accept arbitrarily long proofs, so you must accept this proof which you call non-standardly long; you have not articulated any criterion of length by which to reject it." In any case, I like this version of the question; "do you accept a non-standardly long proof of 0=1 0=1?" feels much sharper than "do you accept non-standard numbers?"user44143 –user44143 2019-05-29 11:27:17 +00:00 Commented May 29, 2019 at 11:27
1 "Then you have to accept that 'what statements can be proven from the Peano axioms' is itself indefinite. Because the length of a valid proof in PA is a natural number, so if we don't know what the natural numbers are then we don't know what are the possible lengths of proofs. There could be 'proofs in PA' which are valid on one version of N but not on another." This seems incorrect. The length of an actual proof in PA is not an object-language natural number. It is a metalanguage natural number. These can be thought of as corresponding to a sorites-like subcollection of N N.Mikhail Katz –Mikhail Katz 2023-11-16 10:12:44 +00:00 Commented Nov 16, 2023 at 10:12
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Here in the realm of mathematical fiction....
I imagine aliens who do not have the natural number that we call 13. They also lack 41 and 1681 and many other numbers. They do not have the successor or addition functions.
Instead...they regard multiplication and comparison as fundamental. They don't know how to add possibilities, but they know how to multiply and compare them. They rarely count, but they have a keen eye for when one rectangle is bigger than another. So they often arrange things into rectangles, and they regard it as the professional duty of bakers to sell goods in 3x4 boxes.
Instead of Peano arithmetic, these aliens have a first-order theory in the language (1,⋅,<)(1,⋅,<). Their << has the same properties as ours; their ⋅⋅ has the same universal properties as ours; and their << and ⋅⋅ are compatible.
They believe in the number 2 2, i.e. the unique number x x such that
1<x∧¬∃z(1<z<x)1<x∧¬∃z(1<z<x)
Similarly they believe in the numbers 3 3 and 7 7, and 12=2⋅2⋅3 12=2⋅2⋅3 and 14=2⋅7 14=2⋅7. But they can detect a gap between 12 12 and 14 14 because
∃k∃x(12 x<k x<14 x)∧∄k(12<k<14)∃k∃x(12 x<k x<14 x)∧∄k(12<k<14)
Their induction takes the form
P(1)∧∀x(∀y(y<x→P(y))→P(x))→∀x(P(x))P(1)∧∀x(∀y(y<x→P(y))→P(x))→∀x(P(x))
They may define x+y=z x+y=z by something like our formula
S(x z)S(y z)=S(z z S(x y))S(x z)S(y z)=S(z z S(x y))
but this does not play a large role in their arithmetic.
Finally, we observe an embedding of their natural numbers into ours, uniquely fixed by the desire to preserve both ⋅⋅ and <<. But if they believe that there is no number in between 12 12 and 14 14, and they have lived happily without any such need, what are we to say?
Meanwhile here in New York...I learned how to count from the numbers on an elevator in the apartment building, and there was no 13th floor.
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edited May 24, 2019 at 11:11
answered May 23, 2019 at 9:26
user44143 user44143
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1 "But if they believe that there is no number in between 12 and 14, and they have lived happily without any such need, what are we to say?" Nothing. Their number system is just fine for their purposes, and they can grok our notion of 13. The situation is different, I believe, when someone's notion of "finite" comes from access to an oracle (to a nonstandard model). How do we explain what a "standard" finite number is? That is, how do we define "finite" without an a priori understanding of it?Pace Nielsen –Pace Nielsen 2019-05-23 14:33:37 +00:00 Commented May 23, 2019 at 14:33
The bit about detecting the absence of 13 seems garbled, but if 169 is a number for the aliens, they might notice that it is a ratio of squares sitting between 12^2 and 14^2 without any corresponding number sitting between 12 and 14.Matt F. –Matt F. 2025-08-21 13:39:51 +00:00 Commented Aug 21 at 13:39
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The answer to this question depends on how skeptical you want to be.
Let's consider a simpler task. Can we communicate to the aliens what we mean by "3"? It's possible to imagine a scenario in which we are unable to do this. Imagine for example that the aliens don't look like human beings and don't communicate using anything like a human language, but instead resemble ants. We can imagine trying to train the aliens to understand "3" if we are able to identify something that they desire (food, for example) and putting food inside boxes marked with 3 copies of the same symbol, while leaving other boxes empty. After a sufficiently long training period, we might become convinced that the alien ants have mastered what we mean by "3", but how could we ever be sure? It's quite conceivable that the next time we put 3 symbols on a box, they might get confused. Perhaps we use a new type of symbol that they have trouble parsing. Or perhaps we use exactly the same type of symbol as in the past, but it turns out that the concept they have learned from us is, when translated into our language, something like "3 provided there is currently no syzygy of three planets, and 4 if there is currently a syzygy of three planets" (cf. Goodman's grue paradox). In principle we might detect such a thing if our training period were sufficiently long, but we would still be unsure whether some other clause is "hiding" in there, manifesting itself even more rarely than planetary syzygies do.
If "3" cannot be communicated, then certainly something infinite like the natural numbers cannot be communicated. But suppose we grant that somehow, "3" can be communicated. Could we communicate the standard model of PA as opposed to some nonstandard model of PA?
My view is that it is a very peculiar kind of skepticism that regards nonstandard models of PA as posing a particular problem for communication tasks. A nonstandard model of PA is, by any conceivable measure, a more complicated concept than the standard model of PA. For starters, what do we even mean by PA? PA contains an induction schema. Communicating what we mean by the induction schema for PA is no easier than communicating what the standard model is. If you think there is a difficulty disambiguating between PA and nonstandard models of PA, then how do you explain to someone that when you say "PA" you mean that the induction schema ranges over formulas whose length is a standard natural number as opposed to formulas that are nonstandardly long?
To be sure, you are free to be skeptical that we are able to communicate what we mean by PA. But this kind of communication difficulty is more fundamental than the difficulty of communicating the difference between a standard and a nonstandard model of PA, since the latter must at least presuppose that we know what PA is.
Another way to see that PA is a red herring is to replace PA with Robinson's arithmetic Q. Can we communicate to the aliens the distinction between a standard and a nonstandard model of Q? The same difficulty with infinity arises; specifics about PA are irrelevant.
It comes down to a question of whether we think we can communicate something infinite, and if so, why it is that we think we can do so. Technical details about logic and arithmetic are a distraction.
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answered May 23, 2019 at 15:17
Timothy ChowTimothy Chow
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Good points. This reminds me of that old Chinese room problem. Anyway, if it helps get rid of distractions, here is another formulation: You have (presumably) a picture in your head of the natural numbers, as do I. We try to express these pictures using finite symbols, sounds, words, etc... to each other. We seem to agree that we have the same picture. Are we justified in claiming that we in fact do have the same picture? If not, is anyone really justified in using the phrase "standard model of the natural numbers"? If so, what justifies it?Pace Nielsen –Pace Nielsen 2019-05-23 16:56:32 +00:00 Commented May 23, 2019 at 16:56
4 @PaceNielsen : Yes, that is a better way of phrasing the question. I would say that whatever it is that we appeal to for justification, it cannot be mathematical proof in the usual sense, because the claim that your internal mental picture matches my internal picture (in some sense at least) is not a mathematical statement. This observation may make some uncomfortable, because mathematicians like to think of themselves as relying solely on "proof" and not on any murky philosophical assumptions. But we are deluding ourselves if we think that mathematics is free of philosophical assumptions.Timothy Chow –Timothy Chow 2019-05-23 17:51:05 +00:00 Commented May 23, 2019 at 17:51
1 @PaceNielsen (cont'd): What we can say is that mathematicians who have been suitably trained in the linguistic usage of the term "standard model of the natural numbers" are able to reach agreement about which mathematical inferences employing that phrase are valid/invalid, in exactly the same way that they are able to reach agreement about inferences in any other area of mathematics. The empirical fact that the mathematical community is able to reach a high level of agreement about which inferences are valid is often taken to be a justification of the meaningfulness of mathematical activity.Timothy Chow –Timothy Chow 2019-05-23 22:16:21 +00:00 Commented May 23, 2019 at 22:16
@PaceNielsen Before asserting or denying that my mental picture of N N is the same as yours, I'd want a clear notion of what it means for pictures in two minds (or brains) to be the same. That notion might have something to do with homologous neurons firing in similar circumstances, but I think neuroscience today (and not merely my knowledge of it) is far from making such a notion precise. In other words, I don't even think it makes sense to ask whether two people have the same picture of N N (or of anything else).Andreas Blass –Andreas Blass 2019-05-27 17:53:45 +00:00 Commented May 27, 2019 at 17:53
I suspect that aliens if they have no prior notion of the number 3, there would be no reason for them to notice whether there are exactly three symbols or not to mark a box with a desirable content. Especially if they automatically perceive a combination of three different symbols as an entirety, or, conversely, if they perceive each symbol to be composed of a variety of subelements.Tommy R. Jensen –Tommy R. Jensen 2019-09-19 17:39:40 +00:00 Commented Sep 19, 2019 at 17:39
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Re: question 1, there is a precise sense in which this task is impossible, but we can arguably get a "dynamic approximation."
First the negative response. Since first-order logic isn't enough, a case can be made (which I would agree with incidentally) that there is no satisfactory way to communicate this. A relevant result for this general principle is Lindstrom's theorem, which intuitively says that any logic stronger than first-order logic must be fundamentally "infinitary," by virtue of not having a finitary proof system or by virtue of having to take into account uncountable structures even at the most basic level.
Now the positive response. Although prima facie it involves an appeal to a large fragment of the set-theoretic universe, we can think of the "from above" characterization (= the smallest model of PA) as a dynamic communication. The idea being that whenever you and I have possibly non-isomorphic models of PA, we each try to find embeddings of our own model into the other. If one of us succeeds and the other fails, then we tentatively agree that that person is holding the standard model. If both of us succeed, then we tentatively agree both are. And if we both fail, we tentatively agree that neither of us is holding on to the standard model. Note that while this process can be wildly fluctuating, it is stable in the following weak sense: if you are holding the standard model and I'm holding a nonstandard model, we won't possibly believe that my model is standard.
Of course, there is an appeal to quantification here. Namely, checking whether a purported embedding is actually an embedding involves a universal quantifier. But in this case we're only quantifying over things we already have; in particular, we can make perfect sense of this even if both of us are holding nonstandard models.
So I think this isn't entirely silly, although I do believe it's lightyears away from satisfying (and indeed that the actual answer is that no satisfying method exists).
Re: question 2, what do you mean by "expressible?"
We can easily write a formula in the language of set theory asserting "x x is the smallest inductive set," and so in that sense we have expressibility in the language of set theory.
On the other hand this definition is very model-dependent, with nonisomorphic models of set theory giving potentially non-isomorphic "smallest inductive sets." By the compactness theorem this is unavoidable as long as we stay in the realm of first-order logic, so that can be interpreted as a negative answer.
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answered May 23, 2019 at 5:16
Noah SchweberNoah Schweber
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By "expressible" I meant definable in a non-circular way. Your comment above, about understanding "only finitely many" only after having access to the natural numbers gets to the heart of it.Pace Nielsen –Pace Nielsen 2019-05-23 14:14:36 +00:00 Commented May 23, 2019 at 14:14
1 It seems to me to be pretty clear you can't define the natural numbers, in any reasonable sense, or in any sense agreeable to you, in a non-circular way. So - next question? I hope I'm not seeming to be flippant here, but still.abo –abo 2019-05-23 15:24:10 +00:00 Commented May 23, 2019 at 15:24
1 I think we can be absolutist about the notion “smallest inductive set.” It’s true that this notion can be interpreted differently in different models of set theory, but the concept of algebraic generation is so primitive and clear. Further, the standard numbers always embed into any nonstandard version, so there is some absolute, inter-universal sense in which it is “smallest.”Monroe Eskew –Monroe Eskew 2019-05-23 16:57:35 +00:00 Commented May 23, 2019 at 16:57
1 @PaceNielsen. I guess they talk about it because in a context such as ZF, or some such theory, it has a meaning (the one Monroe is talking about). That is, for mathematicians working in the context of ZF, it makes sense to talk about the standard model, and it helps them to do mathematics.This doesn't provide a "definition" in your sense, however.abo –abo 2019-05-23 17:58:59 +00:00 Commented May 23, 2019 at 17:58
1 @PaceNielsen : If you're complaining about the choice of the word "standard," I don't think it's any different from the way the adjective "normal" is overused in mathematics. What's "normal" about the "normal distribution"? Or is your complaint that if something can't be defined non-circularly then we can't know what it is? But in that case, I'd claim that the number "3" can't be defined non-circularly either, yet I don't believe we can't know what "3" is.Timothy Chow –Timothy Chow 2019-05-23 20:17:54 +00:00 Commented May 23, 2019 at 20:17
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Just an expanded comment. The standard model of PA can be characterized as the unique (up to isomorphism) model of PA such that all of its individuals are suitably named by numerals (like SSS...S0). Therefore, we can understand the standardness of the standard model as the matching between its individuals and the “metamathematical numbers.” Now, what if the metamathematics of aliens is nonstandard? As I see it, the metamathematics is not a model, so it is neither standard nor nonstandard. The metamathematical numbers are just subjects of the metamathematical discourse. Therefore, if we can explain the matching between the “metamathematical numbers” and the individuals of the standard model, then we can explain the standard model, because this matching can be seen as the essence of its standardness.
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answered May 23, 2019 at 14:24
Rodrigo FreireRodrigo Freire
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1 But "SSS...S0" is not actually a numeral. To define "numeral" don't you already need access to the natural numbers, thus making the definition of the standard model circular?Pace Nielsen –Pace Nielsen 2019-05-23 14:28:38 +00:00 Commented May 23, 2019 at 14:28
I am using SS...S0 as a numeral, a term of PA determined by a metamathematical number (the metamathematical number of S’s).Rodrigo Freire –Rodrigo Freire 2019-05-23 14:32:41 +00:00 Commented May 23, 2019 at 14:32
Are metamathematical numbers somehow prior to actual numbers? Seems circular to me.Pace Nielsen –Pace Nielsen 2019-05-23 14:36:17 +00:00 Commented May 23, 2019 at 14:36
1 Precisely. What are the actual numbers? You apparently need them to talk about formal systems, and metamathematical numbers. I might say "things that count". The aliens might say "the possible steps taken by a supertask machine" (or in other words, the numbers n n such that a supertask machine says that ∃n...∃n... gives a positive answer). The aliens might ask us how our notion differs from theirs. Could we express how they differ? (Do they differ?)Pace Nielsen –Pace Nielsen 2019-05-23 14:52:24 +00:00 Commented May 23, 2019 at 14:52
2 Yes, that is about it. The metamathematical numbers are not a model, therefore neither standard nor nonstandard. Models are mathematical objects living inside a mathematical theory (of sets) and modeling formal systems like PA. The metamathematics is not like that.Rodrigo Freire –Rodrigo Freire 2019-05-23 17:19:01 +00:00 Commented May 23, 2019 at 17:19
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So the aliens taught you how to build a device and interpret its outputs as statements about first-order number theory. They say that it is performing "supertask" computations, or at least that's how we've translated their claims into Scientific English. But since we cannot check for ourselves, with our dull senses and limited memories, how do we know that's what's happening?
This is a serious question, because the assumption of the OP is that the device is somehow checking against some model of PA. Why do we believe the outputs correspond to what is satisfied in an actual model of PA? And which model? My point is that the particular reasons we have for believing in the reliability of this infinitary computer, or our particular understanding of how it is working, would be key to shedding light on whether the outputs are about a standard or nonstandard model of PA.
It could also give us the vocabulary to explain to the aliens what we mean by "standard model." Because if they have any knowledge of models of PA, enough to actually "touch" one, then surely they know that are many different ones. So we should be able to ask them why the computer tracks truths about their preferred model of PA and not a different one. This will have to depend on how the particulars of this sci-fi story are elaborated.
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answered May 23, 2019 at 10:14
Monroe EskewMonroe Eskew
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"But since we cannot check for ourselves, with our dull senses and limited memories, how do we know that's what's happening?" Clearly, we don't. But we can see the idea of it (a machine building a smaller machine, etc...). So that gives us a mental picture. But the point is, as you have grasped, that our mental pictures of the infinite might not match what reality would do with an infinite process.Pace Nielsen –Pace Nielsen 2019-05-23 14:25:02 +00:00 Commented May 23, 2019 at 14:25
2 @PaceNielsen So it's doing some unknown thing, physically somehow. Why even believe this has to do with PA? It looks like it's computing some numbers, going faster and faster, and then, FLASH BANG, it sings a song about RH. Maybe the aliens just have a nice device for dazzling math fans. What do we even mean by "what reality would do with an infinite process?" I mean, what is it even doing? Does it have anything to do with number theory in the end? Or is it just a fireworks show? There's not enough information in the story.Monroe Eskew –Monroe Eskew 2019-05-24 08:07:47 +00:00 Commented May 24, 2019 at 8:07
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I think the answer to your question really depends on how we take ourselves to understand a canonical model of PA. Ultimately I think it comes down to the way language about mathematics is grounded in a kind of modal semantics for physical procedures.
Consider a human child learning arithmetic for the first time. The first thing they might learn in a contemporary mathematics curriculum is how to associate physical arrangements of marks/dots with arbitrary arrangements of distinguishable objects. A basket containing six oranges can be associated with six equally spaced dots on a line, like ∗∗∗∗∗∗, or six equally spaced dots in a rectangle, like:
∗∗∗∗∗∗or∗∗∗∗∗∗∗∗∗∗∗∗or∗∗∗∗∗∗
They learn that adding a single item to the basket corresponds to adding an additional dot on the end of a line of dots, and removing an single item from the basket corresponds to removing a dot from the end of a line of dots. They learn how to rearrange rectangles of dots into lines of dots. They then learn how to associate lines of dots with, say, arrangements of decimal numerals and verbal utterances. They then learn to associate addition with concatenation of lines of dots, and multiplication with creating a rectangle of dots whose sides are associated with the multiplicands. They learn a procedure for manipulating two arrangements of decimal numerals to to produce an arrangement of decimal numerals representing their sum or product (the usual addition and multiplication algorithms), and verify by many examples that these physical procedures on arrangements of numerals correspond to the previous physical procedures on arrangements of dots.
What are they actually learning here, though? It looks like they are learning certain modal claims about physical procedures, essentially by induction in the sense of Hume. They learn, for example, that given two equally spaced lines of dots, it is always possible to form their concatenation. They learn that the procedure for association "a+b" to an arrangement of dots and "b+a" to an arrangement of dots necessarily result in arrangements of dots which, if they were to verbally count them, end at the same verbalization. I would claim they are learning these modal claims in the same way a physicist might learn certain modal claims about the motions of physical objects, like "For any possible closed physical system, momentum is conserved". I think these are rightly considered modal claims, not just universal claims, because they are about what would happen, not what is: we are learning about what would happen if we were to do various physical procedures, not about what is (of course this is an anti-Platonist view).
So how does this relate to your original question about aliens? Well, numbers correspond to arrangements of equally spaced dots that we could create (in a somewhat liberal interpretation of "could"). As mathematicians, we often assume that there is no limit to the amount of time or resources we have to perform a manipulation of symbols (which is expressed by the totality of the successor function -- for any linear arrangement of equally spaced dots, we always could add another dot to the end). We believe that this modal language is unambiguous, because it seems to connect directly to our experience of the physical world we live in.
But there is a good argument this is not so, or at least the naive interpretation of these modal claims as physical possibility does not work: there is a finite amount of matter and energy in the universe with which to create these arrangements of dots.
However, there still may be something about our universe which allows these modal claims to be grounded in something "real". If our universe was Newtonian or Minkowskian, then in a physical line segment of unit length, there exist arbitrarily large finite collections of non-zero points along that segment whose spacing between adjacent triples pairs of points a<b<c a<b<c in that collection has the distance between b b and c c equal to half the distance between a a and b b. And any infinite such collection has the same order type. So we can find the order-type of the natural numbers, essentially, in the level of saturation of a unit interval of space (not quite in the model theoretic sense, but related). We might also believe time is unbounded, and that for any full time-like geodesic in our Minkowskian universe, any unbounded collection of equally-spaced points in time with an initial point is order-isomorphic to the natural numbers. On a theory of meaning based on the right notion of reference, our language could thus pick out a canonical model of arithmetic by reference to our actual universe.
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answered May 29, 2019 at 15:11
JamesJames
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If PA is inconsistent, then by Gentzen's consistency proof, this means that the logical system of primitive recursive arithmetic (PRA) augmented with the well-ordering of ϵ₀ is inconsistent. The consistency of PRA is, I'm pretty sure, beyond reasonable debate, so the more likely explanation is that ϵ₀ is not well-ordered, and thus has an infinite descending sequence beneath it. Given that every ordinal below ϵ₀ can be expressed as a finite rooted tree, there would have to be some "finite" rooted tree in the aliens' system which has a nonstandard number of nodes, and thus can be decremented from forever.
I think that the existence and well-orderedness of ϵ₀ would be something they could learn about and accept, and any model that doesn't prove it is not the standard model.
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answered May 25, 2019 at 22:04
ZemylaZemyla
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Not sure about aliens but if we had to approach the task of explaining N N to, say, the Pirahãs (who have presumably not been exposed to any modern mathematics), we would first have to answer Hamkins' question to the "mathematical oracle":
Do we have an absolute notion of "finite"?
Since the answer seems to be likely negative, the corresponding answer to your question is that the task would be impossible. Of course, the Pirahãs, who have the same cognitive capacity at birth as all humans, can be replaced here by anybody who has not been exposed to modern set theory, e.g., Leibniz, who would have rejected the formula N∈P(N)N∈P(N) as meaningless even after getting all the necessary explanations with regard to the notation, because he viewed "infinite wholes" as contradictory entities (namely, contradicting the part-whole principle, along the lines of Galileo's paradox with n↦n 2 n↦n 2). On the other hand, Leibniz would have agreed that infinitesimals are more complicated than 1,2,3,... Indeed, the counting numbers are assignable, whereas infinitesimals are inassignable, in Leibniz's terminology.
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edited Nov 17, 2023 at 11:16
answered Nov 14, 2023 at 14:19
Mikhail KatzMikhail Katz
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2 Irrespective of any deep philosophical musings, in practice we clearly communicate the idea of numbers to other humans all the time. It seems bizarre to think there is a group of humans who we couldn't, with some effort, communicate the idea of numbers to.Sam Hopkins –Sam Hopkins♦ 2023-11-15 13:26:49 +00:00 Commented Nov 15, 2023 at 13:26
2 @SamHopkins: While I see why you think it seems bizarre, here is a more advanced analogue that may explain why it may not be bizarre: Mathematicians of a certain era thought that infinitesimals are problematic and they avoided using them. Then came Robinson... One may say that Robinson would have hard time communicating the idea of consistently doing calculus with infinitesimals to, say, Bishop Berkeley. The only difference here is that N N seems more "basic" but that's just a habit of thinking. Is it really?Burak –Burak 2023-11-16 10:06:08 +00:00 Commented Nov 16, 2023 at 10:06
3 @SamHopkins: I think anecdotal and anthropological evidence is against you. "Understanding what strings are" is prior to counting: a child who asks her father "what does two mean?" has a grasp of grammar but not of arithmetic. The Pirahã people apparently speak a complex language but are wilfully against abstractions such as numbers. Mikhail's post seems to be to be very pertinent.Rob Arthan –Rob Arthan 2023-11-16 20:47:17 +00:00 Commented Nov 16, 2023 at 20:47
2 @SamHopkins: Counting numbers is easy, a child can do that. But understanding the smallest collection of objects that contains 0 0 and S S is difficult, even for professional mathematicians, as shown by numerous discussions on MO. So my comment still stands that the set of natural numbers is no easier to understand than imagining points on a line that are infinitely close to 0 0.Burak –Burak 2023-11-17 11:32:40 +00:00 Commented Nov 17, 2023 at 11:32
2 @Burak, in fact Leibniz considered such a "set" to be a contradictory notion (as explained in my answer), whereas infinitesimals are merely fictional (not contradictory).Mikhail Katz –Mikhail Katz 2023-11-17 11:37:50 +00:00 Commented Nov 17, 2023 at 11:37
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I'm going to have to tell you that describing your "standard model" is possible if and only if you believe that there is a true embedding of the naturals into the real world.
Specifically, if you do then you can simply define your "standard model" as precisely that embedding. For example, if you believe that one can build a true universal Turing machine that has an indefinitely extensible tape (not infinite but always can be extended) and that can run for an indefinitely long time without making a single error, then you can obviously define natural numbers as all finite unary strings that can be written by some instance of that Turing machine, and define the arithmetic operations on those unary encodings as the Turing machines that you can explicitly write down for them. There can then be no ambiguity at all what you mean by your "standard model".
However, if you hold a more 'physical realist' view, the finiteness of the observable universe strongly suggests that there is no embedding of naturals into the real world, and hence no Turing machine of the kind described above (for reasons of both space constraints and occasional infidelity). If so, then describing your "standard model" is impossible simply because it does not even exist! Think about it; you cannot even describe it uniquely to yourself, since no computable formal system can pin it down.
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edited May 27, 2019 at 16:35
answered May 26, 2019 at 10:50
user21820user21820
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To question whether one can "build a true Turing Machine" (in the "real world") does not make any real sense. A TM is an abstraction of how computation works. It is like asking whether one can "build the true number 0", or "build the true imaginary number i". Of course you can build a TM from examining the way it is defined as a tuple of sets in ZF and suggesting the appropriate sets to combine.Tommy R. Jensen –Tommy R. Jensen 2019-09-22 14:42:42 +00:00 Commented Sep 22, 2019 at 14:42
@TommyR.Jensen: Your comment does not make any sense. The fact that you can read this webpage (via HTTPS) rests on the correctness of Fermat's little theorem F when interpreted in the real world as proving a truth about some calculations in your computer involving 1024-bit integers or larger. This same interpretation can be used to interpret all theorems of PA (not just F). If such an interpretation actually works, then that is precisely what "true embedding of the naturals into the real world" means. Same for "true universal TM", which here clearly means a real-world embedding of a UTM.user21820 –user21820 2021-05-01 03:40:08 +00:00 Commented May 1, 2021 at 3:40
By "true universal TM" I did not mean "true" in the sense of "one true" but rather "not fake". Your computer is not a true UTM because of finite memory; you need it to have an indefinitely extensible memory for it to be a "true UTM".user21820 –user21820 2021-05-01 03:41:56 +00:00 Commented May 1, 2021 at 3:41
2 Your answer shows why describing a "standard model" is possible if there is a true embedding of the naturals into the real world. (I think "you believe" does not belong, because the way you propose to use that true embedding is to exhibit it physically, which you cannot do if you merely believe in it.) But why only if?LSpice –LSpice 2023-11-14 14:38:04 +00:00 Commented Nov 14, 2023 at 14:38
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General sum of n n th roots of unity raised to power m m comprime with n n
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I am trying to find a reference for the following proposition:
Let m m and n n be coprime. Then,
∑k=0 r−1 exp(i 2 π n k m)=0∑k=0 r−1 exp(i 2 π n k m)=0
if and only if r r is an integer multiple of n n.
Can anyone point a basic textbook or online material in which this basic fact is proven?
complex-numbers
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asked Apr 4, 2016 at 14:58
user809418user809418
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It can be proven easily using geometric summation. Observe that
∑k=0 r−1 exp(i 2 π k m n)=∑k=0 r−1 exp(2 π i m/n)k=1−exp(2 π i m/n)r 1−exp(2 π i m/n)∑k=0 r−1 exp(i 2 π k m n)=∑k=0 r−1 exp(2 π i m/n)k=1−exp(2 π i m/n)r 1−exp(2 π i m/n)
The sum is 0 0 if and only if exp(2 π i m/n)r=1 exp(2 π i m/n)r=1, which corresponds to n|r m n|r m.
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answered Apr 4, 2016 at 15:02
Henricus V.Henricus V.
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Reference: en.wikipedia.org/wiki/Root_of_unity#SummationHenricus V. –Henricus V. 2016-04-04 15:09:03 +00:00 Commented Apr 4, 2016 at 15:09
Neat :) Thanks.user809418 –user809418 2016-04-04 15:29:25 +00:00 Commented Apr 4, 2016 at 15:29
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189490 | https://www.deq.nc.gov/total-phosphorus-sm-4500-p-e-2021-checklist-pdf/open | NC DEQ/DWR WASTEWATER/GROUNDWATER LABORATORY CERTIFICATION BRANCH
LABORATORY NAME: CERT #:
PRIMARY ANALYST: DATE:
NAME OF PERSON COMPLETING CHECKLIST (PRINT):
SIGNATURE OF PERSON COMPLETING CHECKLIST:
Parameter: Total Phosphorus
Method: Standard Methods 4500 P E-2021 (Aqueous) Ascorbic Acid Method
With Standard Methods 4500 P B (5)-2021 Persulfate Digestion
EQUIPMENT:
Spectrophotometer w/infrared phototube and 2.5 cm light path or longer at 880 nm
Model:
Filter Photometer w/ red color filter and 0.5 cm light path or longer
Model:
Acid-washed glassware Hot plate
Autoclave Micro-Kjeldahl type digestion rack
Glass scoop Micro-Kjeldahl flasks
PERSULFATE DIGESTION REAGENTS: See last page for reagent recipes
Phenolphthalein Indicator Solution Sulfuric Acid Solution Ammonium Persulfate Solid
Potassium Persulfate Solid Sodium Hydroxide, NaOH, 1 N
ANALYSIS REAGENTS: See last page for reagent recipes
Sulfuric acid (H 2 SO 4 ), 5 N Ascorbic acid solution Standard phosphate solution
Antimony potassium tartrate solution Combined reagent
Ammonium molybdate solution Stock phosphate solution
PLEASE COMPLETE CHECKLIST IN INDELIBLE INK
Please mark Y, N or NA in the column labeled LAB to indicate the common lab practice and in the column labeled SOP to indicate whether it is addressed in the SOP.
GENERAL
L
A
B
S
O
P
EXPLANATION
Is the SOP reviewed at least every 2 years? What is the most recent review/revision date of the SOP? [15A NCAC 02H .0805 (a) (7)]
DATE:
Quality assurance, quality control, and
Standard Operating Procedure
documentation shall indicate the effective date of the document and be reviewed every two years and updated if changes in procedures are made.
Verify proper method reference. During
review notate deviations from the approved method and SOP.
Are all revision dates and actions tracked and documented? [15A NCAC 02H .0805 (a) (7)]
Each laboratory shall have a formal process to
track and document review dates and any
revisions made in all quality assurance, quality control, and Standard Operating Procedure documents.
Is there North Carolina data available for review? If not, review PT data
PRESERVATION and STORAGE
L
A
B
S
O
P
EXPLANATION
Are samples preserved at time of collection with H 2 SO 4 to pH of <2 S.U.? [40 CFR 136.3 Table II]
Preservation not required if analyzed within 15 minutes. Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 2 of 10
Revised 05/15/2024
Are samples iced to above freezing but ≤ 6 º C during shipment?
[40 CFR 136.3 Table II and footnote 18]
Is pH checked to document pH <2 S.U. upon receipt?
[15A NCAC 02H .0805 (a) (7) (M)]
What action is taken if pH is >2 S.U.? [15A NCAC 02H .0805 (a) (7) (M)]
Answer:
If another sample cannot be collected, analyze immediately or adjust pH to <2 S.U. and notify NC WW/GW LC Branch that a non-compliant sample was received.
Are samples refrigerated above freezing and ≤ 6°C during storage?
[40 CFR 136.3 Table II and footnote 18]
Are samples analyzed within 28 days of collection?
[40 CFR 136.3 Table II]
Persulfate Digestion
L
A
B
S
O
P
EXPLANATION
What volume of sample and standards is digested? [SM 4500-P B-2021 (5) (c)]
Answer:
The method says this volume is determined
by the analytical method, which says to use 50-ml for digestion. Following digestion, it says to dilute to the digested sample to a final volume of 100 ml.
Is 0.05 mL (1 drop) of phenolphthalein indicator solution added? [SM 4500-P B-2021 (5) (c)]
If a red color develops after the addition of phenolphthalein, is H 2 SO 4solution added dropwise to just discharge the color? [SM 4500-P B-2021 (5) (c)]
If no color change is observed, skip to question 14.
After discharging color, is one additional mL of H 2 SO 4 solution added? [SM 4500-P B-2021 (5) (c)]
Is 0.4 g of solid ammonium persulfate added? [SM 4500-P B-2021 (5) (c)]
Either ammonium persulfate or potassium persulfate is added, not both.
Is 0.5 g of solid potassium persulfate added? [SM 4500-P B-2021 (5) (c)]
Either ammonium persulfate or potassium persulfate is be added, not both.
If an autoclave or pressure cooker is used, skip to question 23
Is the sample gently boiled on a preheated hotplate? [SM 4500-P B-2021 (5) (c)]
Boil gently on a preheated hot plate for 30 to
40 min or until a final volume of 10 mL is reached. Some organophosphorus compounds may require as much as 1.5 to 2 h for complete digestion.
How long are samples digested? [SM 4500-P B-2021 (5) (c)]
Answer: See above.
To what volume are samples reduced? [SM 4500-P B-2021 (5) (c)]
Answer:
See above.
Are samples allowed to cool after digestion? [SM 4500-P B-2021 (5) (c)]
Are cooled samples diluted to 30 ml with reagent water? [SM 4500-P B-2021 (5) (c)]
Is 0.05 ml (1 drop) of phenolphthalein indicator solution added and neutralized to a faint pink color with NaOH? [SM 4500-P B-2021 (5) (c)] Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 3 of 10
Revised 05/15/2024
What volume are samples and standards diluted to after digestion?
[SM 4500-P B-2021 (5) (c)]
Answer:
At the end of the digestion, it says to dilute to the digested sample to a final volume of 100 ml.
Skip to question 28.
Is the sample heated for 30 minutes in an autoclave or pressure cooker at 98 to 137 kPa? [SM 4500-P B-2021 (5) (c)]
Are samples allowed to cool after digestion? [SM 4500-P B-2021 (5) (c)]
Is 0.05 ml (1 drop) of phenolphthalein indicator solution added and neutralized to a faint pink color with 1N NaOH? [SM 4500-P B-2021 (5) (c)]
What volume are samples and standards diluted to after digestion? [SM 4500-P B-2021 (5) (c)]
Answer:
At the end of the digestion, it says to dilute to the digested sample to a final volume of 100 ml.
Is information regarding sample ID, initial and final volume, digestion parameters (e.g., pressure, length of time, etc.) documented in a digestion log? [15A NCAC 02H .0805 (a) (7) (E)]
All analytical records, including original
observations and information necessary to facilitate historical reconstruction of the calculated results, shall be maintained for five years. All analytical data and records pertinent to each certified analysis shall be available for inspection upon request.
PROCEDURE – Calibration/Calibration Verification
L
A
B
S
O
P
EXPLANATION
Is a wavelength of 880 nm used? [SM 4500-P E-2021 (2) (a) (1)]
What is your laboratory’s lower reporting limit? [15A NCAC 02H .0805 (a) (7) (H)]
Answer:
For analytical procedures requiring analysis of a series of standards, the concentrations of these standards shall bracket the range of the sample concentrations measured. One of the standards shall have a concentration equal to or less than the laboratory’s lowest reporting concentration for the parameter involved.
Is the spectrophotometer zeroed with a digested calibration blank prior to calibration? [SM 4500-P E-2021 (4) (c)]
E (4) (a) states:
After at least 10 min but no more than 30 min, measure absorbance of each sample at 880-nm, using reagent blank as the reference
solution.
B (5) says to carry the standards through digestion, which would include the blank
4020 B 5 states:
reagent blank (method blank) consists of reagent water (see Section 1080) and all reagents (including preservatives) that typically are in contact with a sample during the entire analytical procedure. Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 4 of 10
Revised 05/15/2024
List the values of the standards used for the calibration.
[SM 4500-P E-2021 (4) (c)]
Standard Concentrations:
Prepare individual calibration curves from a series of 4 up to 6 standards within the phosphate ranges indicated below.
Plot absorbance vs. phosphate concentration to give a straight line passing through the origin. Test at least one phosphate standard with each set of samples.
Minimum detectable concentration:
Approximately 10 μg P/L. P ranges are as
follows:
Approximate
P Range Light Path
(mg/L) (cm) .
0.30 – 2.0 0.5
0.15 – 0.30 1.0
0.01 – 0.25 5.0
Are the calibration standards treated just like samples and carried through the digestion process? [SM 4500-P B-2021 (5) (c)]
Standards must be carried through the same digestion procedure as the samples.
Is the calibration performed for each analysis or is it stored until the calibration verification standard recovery no longer passes, but no longer than 1 year? [SM 4500-P E-2021 (4) (c)] [15A NCAC 02H .0805 (a) (7) (H)]
Answer:
If the calibration curve is stored, is the spectrophotometer zeroed with a digested calibration blank prior to sample analysis? [SM 4500-P E-2021 (4) (a)]
PROCEDURE – Sample Analysis
L
A
B
S
O
P
EXPLANATION
What sample volume is analyzed? [SM 4500-P E-2021 (4) (a)]
Answer:
Is 0.05 ml (1 drop) of phenolphthalein indicator solution added? [SM 4500-P E-2021 (4) (a)]
If a red color develops after the addition of phenolphthalein, is 5 N
H2 SO 4 solution added dropwise to just discharge the color? [SM 4500-P E-2021 (4) (a)]
Is 8.0 mL of combined reagent added and the sample mixed thoroughly? [SM 4500-P E-2021 (4) (a)]
Is the absorbance of the sample measured after at least 10 min, but no more than 30 min after adding the combined reagent? [SM 4500-P E-2021 (4) (a)]
What formula is used to calculate final results? [SM 4500-P E-2021 (5)]
Answer:
mg P/L = mg P (in ~ 58 ml final volume) X 1000
ml sample Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 5 of 10
Revised 05/15/2024
If necessary, is the sample concentration corrected for turbidity or interfering color with a sample blank? [SM 4500-P E-2021 (4) (b)]
Natural color of water generally does not
interfere at the high wavelength used. For highly colored or turbid waters, prepare a sample blank by adding all reagents except ascorbic acid and antimony potassium tartrate to the sample. Subtract blank absorbance from the absorbance of the sample.
QUALITY ASSURANCE
L
A
B
S
O
P
EXPLANATION
Is all glassware acid-washed as described in SM 4500-P C-2021 (2) (b)? [SM 4500-P E-2021 (2) (b)]
Use acid-washed glassware for determining low concentrations of phosphorus. Phosphate contamination is common because of its absorption on glass surfaces. Avoid using commercial detergents containing phosphate. Clean all glassware with hot dilute HCl and rinse well with distilled water. Preferably, reserve the glassware only for phosphate determination, after use, wash and keep filled with water until needed. If this is done, acid treatment is required only occasionally.
Has each analyst performing this analysis completed an Initial Demonstration of Proficiency? [15A NCAC 02H .0805 (a) (7) (P) (iii)]
Attach a copy of each analyst’s documentation to this checklist.
Each laboratory shall develop and implement a documented training program that includes documentation that: staff have obtained acceptable results on Proficiency Testing Samples pursuant to Rule .0803(1) of this Section or other demonstrations of proficiency (e.g., side- by-side comparison with a trained analyst, acceptable results on a single-blind performance evaluation sample, an initial demonstration of capability study prescribed by the reference method).
Is combined reagent made fresh daily & used within 4 hours? [SM 4500-P E-2021 (3) (e)] Combined reagent is stable for 4 hours.
Does each standard curve have a correlation coefficient ≥0.995?
[NC WW/GW LCB Correlation Coefficient for Linear Calibration Curves Policy]
Is the absorbance of each calibration standard compared to the curve and recalculated to determine its concentration? [SM 4020 B-2022 (1) (b)]
What are the acceptance criteria for the recalculated calibration standards? [SM 4020 B-2022 (1) (b)]
Answer: Up to twice the MRL ±50%
Between 3 and 5 times the MRL ±20%
Greater than 5 times the MRL ±10%
What corrective action is taken if the acceptance criteria are not met for the recalculated calibration standards? [SM 4020 B-2022 (1) (b)]
Answer: Identify the source of any outlier(s) and correct before sample quantitation
Does the laboratory analyze a second source standard with each initial calibration to verify standard preparation? [SM 4020 B-2022 (1) (b)] [15A NCAC 02H .0805 (a) (7) (H) (ii)]
List the second source standard concentration:
Rules: Laboratories shall analyze one known second source standard to verify the accuracy of standard preparation if an initial calibration is performed and in accordance with the referenced method requirements thereafter.
SM states: Verify the initial calibration by analyzing a standard prepared from adifferent stock standard than that used to Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 6 of 10
Revised 05/15/2024
create the calibration curve; its concentration
should be near the midpoint of the calibration range.
For a held calibration curve, a second source standard is not required each day samples are analyzed – only with the initial making or verification of the standard curve.
The required Laboratory Fortified Blank (LFB) may serve as the second source standard (refer to question #61)
Is the acceptance criterion for the second source standard ±10% recovery? [SM 4020 B-2022 (1) (b)]
The analytical results for this second source mid-range standard must be within 10% of its true value.
What corrective action is taken if the second source standard recovery is outside of established control limits? [SM 4020 B-2022 (1) (b)]
Answer:
If not, determine the cause of the error, take corrective action, and re-verify the calibration. If the re-verification passes, continue the analyses; otherwise, repeat the initial calibration.
Is a calibration blank analyzed prior to sample analysis, after every 10th sample and at the end of the sample group? [15A NCAC 02H .0805 (a) (7) (H)]
For this method, the calibration blank and reagent blank are the same
Are all the blank concentrations less than or equal to ½ of the lowest calibration standard concentration? [15A NCAC 02H .0805 (a) (7) (H) (i)]
What corrective action is taken if any blank is not less than or equal to ½ of the lowest calibration standard concentration? [15A NCAC 02H .0805 (a) (7) (B)]
Answer:
Reanalyze blank. If still not acceptable, repeat the initial calibration, etc. Once problem is resolved, repeat sample determinations since the last acceptable blank.
Is the calibration verified by analyzing a calibration verification standard (CVS) or Continuing Calibration Verification (CCV) prior to sample analysis, after every ten samples and at the end of the run? [SM 4020 B-2022 (1) (c) and 15A NCAC 02H .0805 (a) (7) (H)]
List value(s) of standard used:
Rules state: A calibration blank and calibration verification standard shall be analyzed prior to sample analysis, after every tenth sample, and at the end of each sample group, unless otherwise specified by the method, to check for carryover and calibration drift.
SM states: Verify calibration by analyzing one standard whose concentration is near the midpoint of the calibration range.
What is the acceptance criterion of the calibration verification standard? [15A NCAC 02H .0805 (a) (7) (A)]
Answer:
What corrective action is taken if the calibration verification standard does not meet the acceptance criterion? [15A NCAC 02H .0805 (a) (7) (B)]
Answer: Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 7 of 10
Revised 05/15/2024
Is a standard at the same concentration as the method reporting limit (MRL) analyzed with each analysis? [SM 4020 B-2022 (9)]
With each analytical batch, analyze a reagent-water sample spiked at MRL and ensure that it meets MRL acceptance criteria (generally ±50%).
What is the acceptance criterion of MRL standard? [SM 4020 B-2022 (9)] [15A NCAC 02H .0805 (a) (7) (A)]
ANSWER:
4020 B states “generally ±50%”. The laboratory must determine the criterion that will be used
What corrective action is taken if the MRL standard doesn’t meet the acceptance criterion? [SM 4020 B-2022 (9)]
Answer:
If not, re-analyze the entire batch or flag results for all samples in the batch. If the MRL is biased high, non-detect (ND) samples can be reported with flags if the method or regulation allows
Is a laboratory fortified blank (LFB) analyzed with each sample set or on a 5% basis, whichever is more frequent? [SM 4020 B-2022 (6)]
As a minimum, include one LFB with each sample set (batch) or on a 5% basis, whichever is more frequent. Depending on method requirements, prepare the addition solution from either the same reference source used for calibration or an independent source.
The LFB is a reagent blank (i.e., treated just like a sample including addition of the preservation acid) fortified with the analyte.
If the LFB is primary source , it may be equivalent to the CVS/CCV (refer to question #55). Analyze at least one daily or per batch of 20 or fewer samples. Use control charts to establish limits or default to the CVS acceptance criterion.
If the LFB is secondary source , it may be equivalent to the second source standard (refer to question #49). Analyze one daily or per batch of 20 or fewer samples. The acceptance criterion must be recovery within ± 10% of true value.
Is the LFB primary or secondary source?
Answer:
See explanation above
What is the concentration and acceptance criterion of the LFB? [15A NCAC 02H .0805 (a) (7) (A) and SM 4020 B-2022 (1) (b) and (6)]
Answer:
If used as the second source verification, the acceptance criterion must be recovery within ± 10% of true value.
Otherwise, evaluate the LFB for percent recovery of the added analytes by comparing results to method-specified limits, control charts, or other approved criteria.
What corrective action does the laboratory take if the LFB is outside of the acceptance criterion? [15A NCAC 02H .0805 (a) (7) (B)]
Answer:
Is a Laboratory Fortified Matrix (LFM) analyzed with each sample set, or on a 5% basis, whichever is more frequent? [SM 4020 B-2022 Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 8 of 10
Revised 05/15/2024
(7) ]
How is the LFM prepared? [NC WW/GW LCB Matrix Spike Technical Assistance] [ SM 4020 B-2022 (7)]
Answer:
See Matrix Spike Technical Assistance document.
SM states: Add a concentration that is at least 10 x MRL, less than or equal to the midpoint of the calibration curve, or method-specified level to the selected sample(s). The analyst should use the same concentration as for LFB (4020 B.6) to allow analysts to separate the matrix’s effect from laboratory performance. Prepare LFM from the same reference source used for LFB. Make the addition such that sample background levels do not adversely affect recovery (preferably adjust LFM concentrations if the known sample is more than 5 times the background level). At a minimum, the spike must at least equal the background concentration, unless the method specifies otherwise. For example, if the sample contains the analyte of interest, then add approximately as much analyte to the LFM sample as the concentration found in the known sample.
Is an LFM Duplicate (LFMD) analyzed with each sample set or on a 5% basis, whichever is more frequent? [SM 4020 B-2022 (8) and Table 4020:I]
SM (8) states: As a minimum, include one duplicate sample or one LFM duplicate with each sample set (batch) or on a 5% basis, whichever is more frequent, and process it independently through the entire sample preparation and analysis
Laboratory fortified matrix is the same as a matrix spike; that is, a spiked sample.
Note: Based on Table 4020:I, there is no option to perform an environmental sample duplicate and then spike separately – must perform MS/MSD for this method.
What is the acceptance criterion for LFM/LFMD recovery (accuracy)? [SM 4020 B-2022 (7) and 15A NCAC 02H .0805 (a) (7) (A)]
Answer:
There will be two % recovery calculations for accuracy from spike recoveries and one RPD calculation for precision from duplicate calculation.
SM states: Evaluate LFM results for percent recovery; if they are not within control limits, then take corrective action to rectify the matrix effect, use another method, use the method of standard addition, or flag the data if reported. See method for specific LFM acceptance criteria until the laboratory develops statistically valid, laboratory-specific performance criteria. If the method does not provide limits, use the calculated preliminary limits from the IDC (4020 B.3). LFM control limits may be wider than for LFB or LCS, and batch acceptance generally is not contingent upon LFM results.
What corrective action does the laboratory take if the LFM/LFMD results are outside of established control limits for accuracy? [15A NCAC 02H .0805 (a) (7) (B) and SM 4020 B-2022 (7)]
Rules state: If quality control results fall outside established limits or show an analytical problem, the laboratory shall identify the Root Cause of the failure. The problem shall be resolved through corrective action, the corrective action process documented, and any samples involved shall Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 9 of 10
Revised 05/15/2024
be reanalyzed, if possible. Compare to LFB
result and other QC. Reanalyze LFM. If it still fails, qualify the spiked sample result.
SM states: Evaluate LFM results for percent recovery; if they are not within control limits, then take corrective action to rectify the matrix effect, use another method, use the method of standard addition, or flag the data if reported.
What is the acceptance criterion for LFM/LFMD precision? [SM 4020 B-2022 (8) and 15A NCAC 02H .0805 (a) (7) (A)]
Answer:
SM states: See method for specific acceptance criteria for LFM duplicates or duplicate samples until the laboratory develops statistically valid, laboratory-specific performance criteria. If the method does not provide limits, use the calculated preliminary limits from the IDC.
Rule States: Each laboratory shall establish performance acceptance criteria for all quality control analyses. Each laboratory shall calculate and document the precision and accuracy of all quality control analyses with each sample set.
What corrective action does the laboratory take if the LFM/LFMD results are outside of established control limits for precision? [15A NCAC 02H .0805 (a) (7) (B) and SM 4020 B-2022 (8)]
Answer:
If quality control results fall outside established limits or show an analytical problem, the laboratory shall identify the Root Cause of the failure. The problem shall be resolved through corrective action, the corrective action process documented, and any samples involved shall be reanalyzed, if possible.
SM states: If duplicate results are out of control, then re-prepare and re-analyze the sample and take additional corrective action, as needed.
Is the data qualified on the Discharge Monitoring Report (DMR) or client report if Quality Control (QC) requirements are not met? [15A NCAC 02H (e) (5)]
Reported data associated with quality control failures, improper sample collection, holding time exceedances, or improper preservation shall be qualified as such.
Digestion Reagents:
Sulfuric Acid Solution: Carefully add 300 ml conc H 2 SO 4 to 600 mL reagent water and dilute to 1 L with reagent water.
Sodium Hydroxide, 1 N: Carefully add 40 g of NaOH to 800 ml reagent water. Sir until dissolved and allow to cool. Dilute to 1 L with
reagent water.
Sodium Hydroxide, 6 N: Carefully add 240 g of NaOH to 800 ml reagent water. Sir until dissolved and allow to cool. Dilute to 1 L with
reagent water.
Analytical Reagents & Standards Prep:
Sulfuric acid, H 2 SO 4 , 5 N: Dilute 70 mL conc H 2 SO 4 to 500 mL with reagent water.
Antimony potassium tartrate solution : Dissolve 1.3715 g K(SbO)C 4 H4 O6⋅½H 2 O in 400 mL reagent water in a 500-mL volumetric flask
and dilute to volume. Store in a glass-stoppered bottle.
Ammonium molybdate solution : Dissolve 20 g (NH 4 ) 6 Mo 7 O24 ⋅ 4H 2 O in 500 mL reagent water. Store in a glass-stoppered bottle.
Ascorbic acid, 0.1 M: Dissolve 1.76 g ascorbic acid in 100 mL reagent water. The solution is stable for about 1 week at 4°C. Total Phosphorus – SM 4500 P E-2021 (Aqueous) Page 10 of 10
Revised 05/15/2024
Combined reagent : Mix the above reagents in the following proportions for 100 mL of the combined reagent: 50 mL 5 N H2 SO 4 , 5 mL
antimony potassium tartrate solution, 15 mL ammonium molybdate solution, and 30 mL ascorbic acid solution. Mix after addition of each reagent. Let all reagents reach room temperature before they are mixed and mix in the order given. If turbidity forms in the combined reagent, shake and let stand for a few minutes until turbidity disappears before proceeding. The reagent is stable for 4 h.
Stock phosphate solution : Dissolve in reagent water 219.5 mg anhydrous KH 2 PO 4 and dilute to 1000 mL; 1.00 mL = 50.0 μg PO 43 – -P.
Standard phosphate solution : Dilute 50.0 mL stock phosphate solution to 1000 mL with reagent water; 1.00 mL = 2.50 μg P.
Additional Comments:
Inspector: _________ Date: _____ |
189491 | https://www.criticalcare.theclinics.com/article/S0749-0704(16)30059-8/pdf | Fluid Creep and Over-resuscitation - Critical Care Clinics
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Review articleVolume 32, Issue 4p587-598 October 2016
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Fluid Creep and Over-resuscitation
Jeffrey R.Saffle, MD
Jeffrey R.Saffle, MD
Correspondence
Jeffrey.saffle@me.com
Affiliations
University of Utah Health Center, PO Box 102, Lake Elmo, MN 55042, USA
Search for articles by this author
Jeffrey.saffle@me.com
Affiliations & Notes Article Info
University of Utah Health Center, PO Box 102, Lake Elmo, MN 55042, USA
Publication History:
Published online August 2, 2016
Footnotes:
The author has no commercial or financial conflicts of interest with any product or individual mentioned in this article.
DOI: 10.1016/j.ccc.2016.06.007 External LinkAlso available on ScienceDirect External Link
Copyright: © 2016 Elsevier Inc. All rights reserved.
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Fluid creep occurs when a patient requires more resuscitation fluid than is predicted by standard formulas.
Keywords
Burns
Fluid resuscitation
Fluid creep
Colloid
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References
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189492 | http://www.teachnlearnchem.com/Keys%20Worksheets/Gas%20Key/PDF/The%20Combined%20Gas%20Law%20key.pdf | Name: ___ Hour: _ Date: _____ Chemistry: The Combined Gas Law Solve the following problems. As always, include enough work and show the units to ensure full credit. 1. The pressure of a gas changes from 120 kPa to 50 kPa. The volume changes from 45 L to 40 L. If the initial temperature is 81oC, what is the final temperature in oC? 2. A sample of nitrogen goes from 21 m3 to 14 m3 and its pressure increases from 100 kPa to 150 kPa. The final temperature is 300 K. What was the initial temperature in Kelvins? 3. A sample of argon goes from 500 K to 350 K and its pressure changes from 280 kPa to 380 kPa. If the initial volume is 18 dm3, what is the final volume? 4. A sample of neon experiences a pressure drop from 75 kPa to 53 kPa. The temperature increases from 27oC to 93oC. If the initial volume is 12 L, what is the final volume? 5. The volume of a sample of helium increases from 5 L to 25 L and its temperature drops from 2000 K to 1750 K. If the initial pressure is 1500 mm Hg, what is the final pressure? 6. The temperature of a gas increases from 212oC to 380oC. The volume goes from 30 dm3 to 18 dm3. If the final pressure is 1.85 atm, what was the initial pressure? Answers: 1. –142oC 2. 300 K 3. 9.3 dm3 4. 20.7 L 5. 262.5 mm Hg 6. 0.82 atm Chemistry: The Combined Gas Law KEY Solve the following problems. As always, include enough work and show the units to ensure full credit. 1. The pressure of a gas changes from 120 kPa to 50 kPa. The volume changes from 45 L to 40 L. If the initial temperature is 81oC, what is the final temperature in oC? K x T K 354 273 C 81 T L 40 V L 45 V kPa 50 P kPa 120 P 2 o 1 2 1 2 1 1 1 1 T V P = 2 2 2 T V P K 354 L 45 kPa 120 = 2 T L 40 kPa 50 C 142 - C C 273 -K 131 C 273 -K K 131 T o o o o 2 2. A sample of nitrogen goes from 21 m3 to 14 m3 and its pressure increases from 100 kPa to 150 kPa. The final temperature is 300 K. What was the initial temperature in Kelvins? K 300 T K x T dm 14 V dm 21 V kPa 150 P kPa 100 P 2 1 3 2 3 1 2 1 1 1 1 T V P = 2 2 2 T V P 1 3 T dm 21 kPa 100 = K 300 dm 14 kPa 150 3 K 300 T 1 3. A sample of argon goes from 500 K to 350 K and its pressure changes from 280 kPa to 380 kPa. If the initial volume is 18 dm3, what is the final volume? K 350 T K 500 T dm x V dm 18 V kPa 380 P kPa 280 P 2 1 3 2 3 1 2 1 1 1 1 T V P = 2 2 2 T V P K 500 dm 18 kPa 280 3 = K 350 V kPa 380 2 dm 3 .
9 V 3 2 4. A sample of neon experiences a pressure drop from 75 kPa to 53 kPa. The temperature increases from 27oC to 93oC. If the initial volume is 12 L, what is the final volume? K 366 273 C 93 T K 300 273 C 27 T dm x V L 12 V kPa 53 P kPa 75 P o 2 o 1 3 2 1 2 1 1 1 1 T V P = 2 2 2 T V P K 300 L 12 kPa 75 = K 366 V kPa 53 2 L 7 .
20 V2 5. The volume of a sample of helium increases from 5 L to 25 L and its temperature drops from 2000 K to 1750 K. If the initial pressure is 1500 mm Hg, what is the final pressure? K 1750 T K 2000 T L 25 V L 5 V Hg mm x P Hg mm 1500 P 2 1 2 1 2 1 1 1 1 T V P = 2 2 2 T V P K 2000 L 5 Hg mm 1500 = K 1750 L 25 P2 Hg mm 5 .
262 P2 6. The temperature of a gas increases from 212oC to 380oC. The volume goes from 30 dm3 to 18 dm3. If the final pressure is 1.85 atm, what was the initial pressure? K 653 273 C 380 T K 485 273 C 212 T dm 18 V dm 30 V atm 1.85 P atm x P o 2 o 1 3 2 3 1 2 1 1 1 1 T V P = 2 2 2 T V P K 485 dm 30 P 3 1 = K 653 dm 18 atm 85 .
1 3 atm 82 .
0 P 1 The Combined Gas Law 1. K x T K 354 273 C 81 T L 40 V L 45 V kPa 50 P kPa 120 P 2 o 1 2 1 2 1 1 1 1 T V P = 2 2 2 T V P K 354 L 45 kPa 120 = 2 T L 40 kPa 50 C 142 - C C 273 -K 131 C 273 -K K 131 T o o o o 2 2. K 300 T K x T dm 14 V dm 21 V kPa 150 P kPa 100 P 2 1 3 2 3 1 2 1 1 1 1 T V P = 2 2 2 T V P 1 3 T dm 21 kPa 100 = K 300 dm 14 kPa 150 3 K 300 T 1 3. K 350 T K 500 T dm x V dm 18 V kPa 380 P kPa 280 P 2 1 3 2 3 1 2 1 1 1 1 T V P = 2 2 2 T V P K 500 dm 18 kPa 280 3 = K 350 dm x kPa 380 3 dm 3 .
9 V 3 2 4. K 366 273 C 93 T K 300 273 C 27 T dm x V L 12 V kPa 53 P kPa 75 P o 2 o 1 3 2 1 2 1 1 1 1 T V P = 2 2 2 T V P K 300 L 12 kPa 75 = K 366 L x kPa 53 L 7 .
20 V2 5. K 1750 T K 2000 T L 25 V L 5 V Hg mm x P Hg mm 1500 P 2 1 2 1 2 1 1 1 1 T V P = 2 2 2 T V P K 2000 L 5 Hg mm 1500 = K 1750 L 25 kPa x Hg mm 5 .
262 P2 6. K 653 273 C 380 T K 485 273 C 212 T dm 18 V dm 30 V atm 1.85 P atm x P o 2 o 1 3 2 3 1 2 1 1 1 1 T V P = 2 2 2 T V P K 485 dm 30 atm x 3 = K 653 dm 18 atm 85 .
1 3 atm 82 .
0 P 1 Answers: 1. –142oC 2. 300 K 3. 9.3 dm3 4. 20.7 L 5. 262.5 mm Hg 6. 0.82 atm |
189493 | https://www.facebook.com/ArchimedesLab/posts/sum-of-consecutive-cubes-visual-proofthe-sum-of-the-first-n-cubes-is-the-square-/572668591539993/ | Sum of Consecutive Cubes (Visual... - Archimedes Laboratory
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Archimedes Laboratory
January 11, 2023 ·
Sum of Consecutive Cubes (Visual Proof)
The sum of the first n cubes is the square of the nth triangular number:
1³ + 2³ + 3³ + 4³ + 5³ + . . . + n³ = (1 + 2 + 3 + 4 + 5 + . . . + n)².
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Sreeraj R Nair
Sir,
You could be gotton the sum of n terms of all cube as
(1/2n(n+1)^2 = 1/4n^2 (n^2 +2n + 1)
Thank You
2y
2 |
189494 | https://www.youtube.com/watch?v=HFbPciBAK3Q | Section 7.4: Optimizing Using Quadratic Forms (Video Lesson!)
Larry Choraszewski "Mr. C"
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[Music] do [Applause] [Music] [Applause] [Music] oh [Music] [Music] um [Music] you just gotta love that tune don't you so we're going to be looking at what are called constrained extremum problems and the idea is this whenever you graph a 3d surface whether it's a surface or a solid in three-dimensional space those extend indefinitely not not so much the case for 3d solids but 3d surfaces typically extend indefinitely and in function format they would be given usually as z equals f of x comma y and so you pick an ordered pair from the plane you plug it into your function involving the variables x and y and you get a z coordinate and this produces an ordered triple x comma y comma z and typically what you get is a 3d surface such as the one that you're seeing here in wireframe all right so this would be an example of at least a chunk of the 3d surface it would extend indefinitely in all of these directions but what we're doing is we're subjecting it to a particular what's called constraint and in this second section the constraint is going to be right here that the norm of the vector x is equal to one now if we assume the euclidean dot product to say that the norm of a vector x is equal to one let's assume that x comes from the plane so it has the components x comma y to say that the norm of the vector is equal to one is to say that the square root of x squared plus y squared is equal to 1 which if you square each side this implies that x squared plus y squared is equal to one and if you think geometrically about what this equation produces this produces the unit circle okay the circle of radius one centered at the origin long story short to say that the magnitude of a vector is equal to one means that we're constraining ourselves to the unit circle now geometrically what in the heck does that all mean well it's the reason for this chunk of a surface this chunk of the surface sits above the unit circle in the plane all right and we have the unit circle kind of sketched out in this diagram and our constraint of this unit circle allows us to find a point that's closest to it that's on the rim of this chunk of the surface and a point that's further furthest away from it on the rim of this 3d surface so if we focus ourselves on the rim which i just you know outlined in this pink color that rim is sitting right above the unit circle and just by its geometry there appears to be a point on this rim that is as close to the unit circle as possible and that appears to be let me use a different color you know one of those points appears to be right here and what i mean by as close as possible to the unit circle is that this is the minimum distance so that would be the minimum distance from z [Music] to the unit circle also known as a constrained minimum there appears to also be a point that is furthest away from the unit circle appears to be right there and by furthest away i mean it is the maximum distance from z to the unit circle okay so if we were to connect that point with unit circle the length of this line segment would be the longest of all such line segments that we could create that is known as a constrained maximum all right did i explain that okay is that making sense everything so far any questions about it all right well if we have a function in x and y that is a quadratic form we know that we can write the quadratic form as x transpose ax where a is a symmetric matrix and we will actually be doing that again in this section because we're going to try to maximize or minimize a quadratic form subject to this constraint of the unit circle and so we'll be working with symmetric matrices again and what's really cool about this section is that these the eigenvalues of the symmetric matrix actually give way to these minimum and maximum distances this theorem says that if a is an n by n symmetric matrix whose eigenvalues are in order of decreasing size given by this chain of inequalities so given this chain of inequalities lambda sub 1 would be the maximum eigenvalue so it'd be the largest of them all and then lambda sub n would be the smallest of all the eigenvalues if this is the case then our quadratic form x transpose ax will have a maximum value and a minimum value in plain old english that this means that max and min's exist furthermore the maximum value occurs at a vector corresponding to lambda sub 1. let me write that in this light blue so this eigenvalue lambda sub 1 the leading eigenvalue in this chain of inequalities that gives way to an eigenvector that produces the maximum value of that quadratic form the minimum value will occur at a vector that corresponds to the smallest of the eigenvalues lambda sub n one thing that i want to note alongside of this is that we're subject to the constraint that the magnitude of the vector x is equal to one geometrically we're subject to the constraint of the unit circle that means our surface is above the unit circle and so when we produce any of these eigenvectors which that's what they are we need to make sure that their norm is equal to one so i'm going to say the norm of each eigenvector has to equal one and the reason for that is because of our constraint another thing is that there might be more than one vector that produces the maximum minimum values okay so it it it's important to note what it doesn't say as well as what it does say the maximum value occurs at a vector it doesn't say it occurs at the vector all right so if we have multiple eigenvalues that we can form that have a normal one then we want to consider all those eigenvalues and the same thing would go for the minimum value right not meant to say eigenvectors let's get to an example show you what this is all about says find the maximum and minimum values of the quadratic form subject to the constraint x squared plus y squared is equal to one so again what this means is that the norm of a vector x is equal to one i.e the unit circle and to start this out this quadratic form is what i'm going to set z equal to so this is a particular function in two variables x and y that also happens to be a quadratic form equation and what we can do because it's a quadratic form is we can write it in the format x transpose ax and let me give you a moment to do that we'll go over it and then we'll proceed to the next step all right you guys ready so x and y x and y and then remember the coefficients of the square terms go along the main diagonal and then the coefficient of this cross term needs to be divided in half so that we can get a symmetric matrix now what we want to do to help us figure out the eigenvectors that produce the maximum value of z subject to this constraint is we want to find a basis for the eigenspaces of a okay so this is like a step one where we rewrite it in quadratic form step two is going to be to find and i'm gonna say this 5 find an on basis for each eigenspace that means we're going to have to find the eigenvalues of this matrix a which i'll provide to you and then we'll work on the basis lambda i minus a will look like lambda minus five along the main diagonal negative two is off the main diagonal and when you form the characteristic equation d t lambda i minus a equals zero you end up getting the eigenvalues lambda equals three and lambda equals seven let's start with lambda equals three and what we're starting here is the calculation of an orthonormal basis so when we plug 3 into lambda i minus a we're going to get negative 2 negative 2 and don't forget we augment it with the zero vector bottom row will be negative two negative two and zero as well the rref version of this matrix goes as follows one one zero in the top row and then zero zero zero in the bottom row first column associates with x second column for y and it looks like x corresponds to a leading one which means that y would have to be our parameter now this will give way to a basis vector with the first entry of negative t second entry of t which we can factor the t out to get negative one one and so a basis would consist of the vector negative 1 1. but is this the appropriate basis to use one so um this would this vector doesn't have a norm equal to one i'll say it this way the norm of that vector x does not equal one so because the norm of this factor doesn't equal one what we're going to have to do is unitize it which essentially turns it into an orthonormal basis and you would divide each component by this vector's magnitude which is radical two so the first entry would be negative one over radical two and the second actually would be positive one over radical two now i'm going to make mention of something else this actually isn't the only basis vector that has a norm equal to 1. alternatively what you could do and i'm going to include it with an and statement here if you were to choose t our parameter t as negative one then that would change the sign of our two entries of this basis vector but it would still have a norm equal to one so in other words this vector right here does have a norm equal to one so it satisfies our constraint of being in line with unit circle but there's another vector x that has a norm equal to one and that would be the vector that we can form by changing the signs of these two entries so positive one over radical two and negative 1 over radical 2. so in terms of the eigenvectors that satisfy the constraint we've got two of them we've got this basis vector and we also have this basis vector all right and i'll talk about the significance of each of these basis vectors but before that i would like you folks to take the next few minutes or so to find basis vectors for lambda equals seven that each have a norm of one all right i'll go ahead and flesh out the details of my basis vector solutions plugging 7 into lambda i minus a you get a first row of 2 negative 2 0 [Music] second row negative 2 to 0 our ref version of that matrix 1 negative 1 0 then row 0's at the bottom x and y x corresponds to the leading one again so y equals t this time the basis vector would both consist of entries of t factor that out and we get a basis vector with entries one one and so a basis vector for this eigenspace is one one but unfortunately it's a basis vector with the norm that is not equal to 1. so we proceed to form an orthonormal basis by unitizing this which amounts to taking each entry dividing it by radical two this will ensure that the norm of the vector is equal to one but there's another basis vector with a norm equal to one and that would be a vector that has the opposite sign of each of these entries so negative one over radical two and negative one over radical two okay that's another vector basis vector for the eigenspace that has a norm equal to one now you guys want to see yep go ahead sorry i just wanted to ask that oh where do you find that radical two from oh okay yep so this right here if we do the norm of one comma one we square the first component add it to the square of the second component and then we square root so it's the pythagorean theorem so we square each component so 1 squared plus 1 squared and that will give us two inside of the square root oh okay i've got it now yep thank you no problem all right so let me show you let me show you guys something cool this is the stuff that really excites math nerds like me i'd like to use one of these basis vectors and i'll make this a little note let's use the basis vector from lambda equals three which has the components negative one over radical two one over radical two if we plug this into our function z the original version of it we would have five times negative one over radical two squared plus five times one over radical two squared plus four times negative one over radical two times one over radical two you'll be amazed at what this simplifies to what i'm going to do is square each of the components and then i'll also form this product with the product of the two components this is going to be 5 times one half plus five times one half plus four times negative one half can you guys simplify this expression and type in the chat what it simplifies to any offers as to what that simplifies to folks three you got it now haven't we seen this three before what's the connection with our value of z and one of the results from earlier in this problem this is one of our eigenvalues isn't it in fact three is the smallest of the two eigenvalues z equals three represents our minimum value min value of z under the constraint of the unit circle now if you were to do this with one of the basis vectors from lambda equals seven you would get the maximum value let's take a look at that let's use one over radical two comma one over radical two that'd be five times one over radical two squared plus five times one over radical two squared plus four times the product of those two components go ahead and simplify this all and tell me what you get all right let's verify that this will be five times one-half plus five times one-half plus four times positive one-half so that's five over two plus five over two plus four over two that's fourteen over two which gives us seven and lo and behold with a basis vector for the eigenspace corresponding to lambda equals seven we plug that basis vector in after it's unitized we get the eigenvalue and this doubles up as the max value of z yep go ahead with your question um earlier you had um changed the sign so that there was uh a basis an orthonormal normal basis an alternative one with the sign changed what what exactly is the purpose of doing the the sign change it it wouldn't make a difference would it on the outcome given the polynomial equation by squaring it doesn't that sort of nullify the effect of changing the sign well what we're looking to achieve here is finding all the vectors that would give us these two values so you're exactly right if even when we change the sign to get the secondary basis vector it doesn't change the outcome of z um what we're doing by finding these basis vectors is we're trying to figure out like all the solutions that would give us the maximum and minimum value and so if we only keep the first unit ties basis vector we've only come up with half of the vectors that would give us the minimum of three and half the vectors that would give us the maximum of seven so we're trying to collect all possible solutions that would give us the same outcome okay in all that's involved is just changing the sign or is there more involved to identifying additional uh orthonormal basis vectors right so you would only need to change the sign and it's kind of a shortcut trick um think about think about our parameter t oh bad color views if this if this value of t was different from 1 or negative 1 the norm of our basis vectors would not equal one so like if i were to multiply the first basis vector by say um five then the second basis vector would no longer have a norm of one so so basically in order to you know get a second basis vector that still has a norm of one the only possibility of doing that is just to change the signs okay because any any value different from one or negative one is going to impact the norm of that second vector so by changing the signs we get a different basis vector but it still satisfies the criterion that the norm is equal to one okay thank you no problem i'm gonna go underneath the supplemental problems here and let's officially answer this problem it says find the maximum and minimum values i'll start with the minimum i'll say min value equals three okay that's the smallest of our eigenvalues and i'm going to say this occurs at two vectors x equals negative one over radical two comma one over radical two and x equals positive one over radical two negative one over radical two the maximum value is given by our largest eigenvalue which is seven and this occurs at two basis vectors the first one over radical two comma one over radical two and the other where the signs are changed and this is how we would officially answer this problem now joe i'm thinking about your question some more let's uh let's do a little drawing here i'm going to do a two-space diagram and on this diagram i'm going to include the unit circle so let's center at the origin and has a radius of one for lambda equals three we produce two vectors that would be points on the unit circle the first vector was negative one over radical two positive one over radical two and that would be right here at 3 pi over 4 on the unit circle now due to symmetry there's going to be another point that helps us find a minimum value and that would be on the opposite side of the unit circle so in in this first instance at 3 pi over 4 we've got negative 1 over radical two comma one over radical two and do due to symmetry across the other side on the unit circle we'll have another solution and that point right there would have the coordinates one over radical two negative one over radical two and that that's kind of like a geometric justification of changing the signs right because the first coordinate of each point are opposites of each other and so are the second coordinates okay appreciate the illustration does that help out yes alright cool well if you like that illustration i got something even cooler i'm going to drop you guys a link in the chat to this geogebra 3d calculator site hey hang tight my blackboard likes to freeze when i switch from tablet to laptop mode there we go follow me to this website folks and over on the left you should see a plus sign and the word input let's go ahead and click on that and we're going to type in our quadratic form there should be a keyboard at the bottom of your screen let's type z equals and then five x and right underneath x you have a button that puts a power of two on it it's already sketching a surface out for us plus 5 y squared plus 4 x y so that's our quadratic form and and you can click on the diagram and drag it around it kind of looks like a paraboloid but the rim of it looks more like looks more like an ellipse looks like half an ellipsoid of sorts but let's go down or go over to the left there's another plus sign which means that we have the capability of entering another equation in here and the equation that i'd like us to enter this time is the equation for the unit circle that'd be x squared plus y squared equals 1. now i believe you can change the color of your graphs i think if you just click the red circle no not that uh let's try this drop down yeah so so let's let's click on these three dots right here and we'll go to settings and there's a tab for color i like to use green for it so if you want to use green you can do so so i'm selecting that and i'm going to x out of it just to see if it changes the color of our yeah okay anybody need me to repeat those steps okay so we have both the quadratic form and the constraint entered in i'm going to get rid of this keyboard just to give me some more real estate here and you can see that the unit circle is actually drawn as a cylinder which is helpful because you can see that the cylinder intersects this quadratic form and i'm going to zoom into this area on the quadratic form and this is the this is the rim of that surface so you remember on page one of the notes let me go back to it i outlined the rim of the surface in this pink color this is where the unit circle would intersect the quadratic form you know should it be drawn as a cylinder okay so that's the rim and it's on the rim of that intersection that we get our minimum and maximum values we're going to our geogebra graphs this appears to be the lowest point of intersection between the cylinder and our quadratic form and if you look inside of our quadratic form and count the tick marks that seems to be parallel with the tick mark of z equals three and that actually is the case now if we look at the axes from the coordinate plane that point of intersection which gives us a minimum value let's see i gotta remember which one's x here because i have it labeled well either way one of the coordinates is positive and one of the coordinates is negative so we're either going to be in quadrant two or quadrant four in this one so there's another point of intersection that gives us the minimum value of three and that's on the exact opposite side okay so here's another instance of a lowest point on the rim of the intersection and that's at 3 as well okay so if we had forgotten to change the sign on our basis vectors we would only get um one vector that gives us the minimum value where in fact there's actually two vectors that give us this minimum value and you can zoom out a little bit and here's where the highest point of intersection is on the rim okay so the the unit circle uh transformed into a cylinder goes all the way up to here to form the maximum value so so this is where the highest point of intersection between the cylinder and our quadratic form occurs and that occurs at a point in quadrant one okay in this quadrant but there's another point in quadrant three you can kind of see it right here that there's another point or another vector that would give us the maximum value of seven and so again if we if we had forgotten to change sign on those basis vectors we would miss out on the vector that produces this high point and the coolest thing about this program is that if you click and just quickly scroll to the left and let go that's not doing it there we go you can have it rotate and uh get you so dizzy that you'll have to get up from uh your couch bad or chair wherever you're sitting for the zoom session and uh you know take care of business uh in the bathroom but at any rate i think that's where i'll leave it for here we can watch that introductory video on markov chains tomorrow do you guys have any questions for me about section 7.4 |
189495 | https://www.cs.umd.edu/class/fall2021/cmsc754/Lects/lect03-hulls-bounds.pdf | CMSC 754 Dave Mount CMSC 754: Lecture 3 Convex Hulls: Lower Bounds and Output Sensitivity Reading: Chan’s output sensitive algorithm can be found in T. Chan, “Optimal output-sensitive convex hull algorithms in two and three dimensions”, Discrete and Computational Geometry, 16, 1996, 361–368.
Lower Bound and Output Sensitivity: Last time we presented two planar convex hull algo-rithms, Graham’s scan and the divide-and-conquer algorithm, both of which run in O(n log n) time. A natural question to consider is whether we can do better.
Recall that the output of the convex hull problem a convex polygon, that is, a cyclic enumer-ation of the vertices along its boundary. Thus, it would seem that in order to compute the convex hull, we would “need” to sort the vertices of the hull. It is well known that it is not generally possible to sort a set of n numbers faster than Ω(n log n) time, assuming a model of computation based on binary comparisons. (There are faster algorithms for sorting small integers, but these are not generally applicable for geometric inputs.) Can we turn this intuition into a formal lower bound? We will show that in O(n) time it is possible to reduce the sorting problem to the convex hull problem. This implies that any O(f(n))-time algorithm for the convex hull problem implies an O(n + f(n))-time algorithm for sorting. Clearly, f(n) cannot be smaller than Ω(n log n) for otherwise we would obtain an immediate contradiction to the lower bound on sorting.
The reduction works by projecting the points onto a convex curve. In particular, let X = {x1, . . . , xn} be the n values that we wish to sort. We will map this into a 2-dimensional point set by projecting the points onto the boundary of a convex shape, so that the sorted order is preserved. For example, suppose that we project each point vertically onto the parabola y = x2, by mapping xi to the point pi = (xi, x2 i ) (see Fig. 1(a)). Let P denote the resulting set of points. It is easy to see that all the points of P lie on its convex hull, and the sorted order of points along the lower hull is the same as the sorted order X (see Fig. 1(b)). Once we obtain the convex hull as a cycle sequence of vertices, in O(n) additional time we can extract its lower hull from left to right, thus obtaining X in sorted order (see Fig. 1(c)).
p2 p4 p3 p5 p1 x2 x4 x3 x5 x1 y = x2 p2 p4 p3 p5 p1 Lift Compute hull p2 p4 p3 p5 p1 Get sorted sequence x2 x4 x3 x5 x1 (a) (b) (c) Fig. 1: Reduction from sorting to convex hull.
Theorem: Assuming computations based on comparisons (e.g., orientation tests) any algo-rithm for the convex hull problem requires Ω(n log n) time in the worst case.
Lecture 3 1 Fall 2021 CMSC 754 Dave Mount Is this the end of the story? Well, maybe not . . .
What if we don’t require that the points be enumerated in cyclic order? For example, suppose we just wanted to count number of points on the convex hull. Can we do better?
Suppose that we are not interested in worst-case behavior. For example, in many in-stances of convex hull, relatively few points lie on the boundary of the hull.
We will present three other results later in this lecture: We will present a convex hull algorithm, called Jarvis’s March, that runs O(nh) time, where h is the number of vertices on the hull. (This is beats Graham’s algorithm when-ever h is asymptotically smaller than O(log n).) We will present Chan’s algorithm, which computes the convex hull in O(n log h) time.
We will present a lower bound argument that shows that, assuming a comparison-based algorithm, even answering the question “does the convex hull have h distinct vertices?” requires Ω(n log h) time.
The last result implies that Chan’s algorithm is essentially the best possible as a function of h and n. An algorithm whose running time depends on the output size is called output sensitive. Both Jarvis’s March and Chan’s algorithm are output sensitive.
Jarvis’s March: Our next convex hull algorithm, called Jarvis’s march, computes the convex hull in O(nh) time by a process called “gift-wrapping.” In the worst case, h = n, so this is inferior to Graham’s algorithm for large h, it is superior if h is asymptotically smaller than log n, that is, h = o(log n).
Jarvis’s algorithm begins by identifying any one point of P that is guaranteed to be on the hull, say, the point with the smallest y-coordinate. (As usual, we assume general position, so this point is unique.) Call this v1. It then repeatedly finds the next vertex on the hull in counterclockwise order.
Given a triple of distinct points ⟨p, q, r⟩, define the turning angle of r with respect to p and q to be the (CCW) angle between the directed line pq and the directed line qr (see Fig. 2(a)).
v0 v1 (d) (c) v1 vi−2 vi−1 vi v2 v3 (b) (−∞, 0) p (a) q r Turning angle v2 v3 v4 v5 v6 Fig. 2: Jarvis’s march.
Jarvis’s march works by repeatedly computing the next hull vertex vi as the point of P that minimizes the turning angle with respect to the prior two, vi−2 and vi−1 (see Fig. 2(c)). Since Lecture 3 2 Fall 2021 CMSC 754 Dave Mount we need two points, to get the ball rolling, it is convenient to define an imaginary “sentinel point” v0 = (−∞, 0), which has the effect that the initial line v0v1 is directed horizontally to the right (see Fig. 2(d)).
Jarvis’s March (1) Given P, let v0 = (−∞, 0) and let v1 be the point of P with the smallest y-coordinate (2) For i ←2, 3, . . .
(a) vi ←the point of P \ {vi−1, vi−2} that minimizes the turning angle with respect to vi−2 and vi−1 (b) If vi == v1, return ⟨v1, . . . , vi−1⟩ The algorithm’s correctness follows from the fact that (by induction) vi−2vi−1 is a CCW-directed edge of the hull, and hence the next vertex of the hull is the one that minimizes the turning angle.
By basic trigonometry, turning angles can be computed in constant time. But it is interesting to note that it is possible to compare turning angles just using orientation tests. (Try this yourself.) This implies that if the input coordinates are integers, the vertices of the hull can be computed exactly (assuming double-precision integer computations).
To obtain the running time, observe that v1 can be computed in O(n) time, and each iteration can be implemented in O(n) time. After h iterations, the algorithm terminates, so the total running time is O(n + nh) = O(nh).
Chan’s Algorithm: Depending on the value of h, Graham’s scan may be faster or slower than Jarvis’ march. This raises the intriguing question of whether there is an algorithm that always does as well or better than these algorithms. Next, we present a planar convex hull algorithm by Timothy Chan whose running time is O(n log h).
While this algorithm is too small an improvement over Graham’s algorithm to be of significant practical value, it is quite interesting nonetheless from the perspective of the techniques that it uses: It combines two slower algorithms, Graham’s and Jarvis’s, to form a faster algorithm.
It employs a clever guessing strategy to determine the value of a key unknown parameter, the number h of vertices on the hull.
To gain some intuition behind Chan’s algorithm, let us first observe that in order to replace the O(log n) factor in Graham’s algorithm with O(log h), we cannot afford to sort any set whose size is (significantly) larger than h. This would seem impossible at first glance, since we don’t know the value of h until the algorithm terminates! We will get around this by playing a “guessing game” for the value of h. We’ll start low, and work up to successively guesses for what h is. Throughout, let h denote the true number of vertices on the hull. The algorithm will maintain a variable h∗, which is our current “guess” on the value of h. As we shall see, if we guess wrong, we will discover our error, and we will need to increase our guess.
For now, let us assume that a magical little bird has told us the value of h∗that works.
We will need to make use of a utility function, whose implementation we will leave as an exercise. Recall that a support line for a convex body is a line that contacts the boundary Lecture 3 3 Fall 2021 CMSC 754 Dave Mount of the body and the body lies entirely on one side of the line. Given a convex body Q and any point p external to Q, there are exactly two support lines of Q that pass through p. The next lemma shows that we can compute them logarithmic time.
Lemma: Given a convex polygon Q = ⟨q1, . . . , qm⟩, where the vertices are stored in an m-element array sorted in CCW order around Q’s boundary, and given any point p that is external to Q, in O(log m) time we can compute the two vertices q−and q+ of Q so that pq−and pq+ are support lines of Q.
p q− q+ Q q1 q2 q3 Fig. 3: Utility function - Computing tangent points.
We can now describe Chan’s algorithm, conditioned on the fact that we have a guess h∗on the size of the hull.
Step 1: (Mini-hulls) Partition P (arbitrarily) into k = ⌈n/h∗⌉groups, each of size at most h∗. Call these P1, . . . , Pk (see Fig. 4(b)). By Graham’s algorithm, compute the convex hull of each subset. Let H1, . . . , Hk denote the resulting mini-hulls (see Fig. 4(c)).
How long does this take? We can compute each mini-hull in time O(h∗log h∗). Applying this to each of the k groups, we have an overall time of O(k(h∗log h∗)) = O(n log h∗).
Note that if we guess the value of h correctly (that is, h∗= h) then this runs in time O(n log h), as desired.
Original point set (a) (b) Partition (h∗= 8) (c) Compute mini-hulls P1 P2 P3 P4 H1 H2 H3 H4 Fig. 4: Step 1: Partitioning and mini-hulls (for k = 4).
Step 2: (Merging) The high-level idea is to run Jarvis’s march on the mini-hulls (see Fig. 5(a)).
We will treat each mini-hull as if it is a “fat point”.
In particular, we take the most recent vertex vi−1 and for each mini-hull Hj employ the utility function of the above lemma to compute the vertices q− j and q+ j for the support lines for this Lecture 3 4 Fall 2021 CMSC 754 Dave Mount mini-hull (see Fig. 5(b)). As in Jarvis’s algorithm, among all of these support points, we take vi to be the one that minimizes the turn angle with respect to vi−2 and vi−1 (see Fig. 5(c)). (By the nature of Jarvis’s algorithm, we only need to compute the support line with the smaller turning angle, but computing both will not affect the asymptotic running time. Also note that we do this for the mini-hull containing vi−1 itself, but this is trivial.) (a) (b) Merge mini-hulls H1 H2 H3 H4 vi−2 vi−1 Hj q− j (c) Minimum turning angle H1 H2 H3 H4 vi vi−1 q+ j Turning angles for one mini-hull Fig. 5: Step 2: Merging the mini-hulls.
How long does this take? For each mini-hull, we can compute the two support lines in time O(log h∗) by the lemma. The number of support lines is twice the number of mini-hulls, so for each step of Jarvis’s algorithm, we can compute all the relevant turning angles in time O(k log h∗). Each iteration of Jarvis yields one more vertex of the final convex hull (of which there are h), so the overall running time is O(h(k log h∗)). Observe that if we managed to guess the right value of h (that is, if h∗= h), then this takes time O(h∗(k log h∗)) = O(n log h∗) = O(n log h), as desired.
In summary, we have argued above that, if we were lucky enough to guess the correct hull size (h∗= h), then the method outlined above will yield the convex hull in time O(n log h).
The Conditional Algorithm: We can now present a conditional algorithm for computing the convex hull.
The algorithm is given a point set P and an estimate h∗of the number of vertices on P’s convex hull. Letting h denote the actual number of vertices on the hull. We will see below that if the h∗is significantly larger or smaller than h, the algorithm will not be efficient. In the merge phase, if we ever see more than h∗hull vertices, we know that our estimate is too low, and we terminate the algorithm (returning “failure”) before the damage is too great.
Guessing the Hull’s Size: The conditional algorithm assumes that we have a good estimate in h∗for h. What are the consequences of guessing wrong?
Too large? If we guess a value of h∗> h, then the Graham scans, which together run in O(n log h∗) time may be too slow. Notice, however, that we pay only a constant factor even if h∗is polynomially larger than h. For example, if h < h∗≤h2, then the running time of Graham’s scan is O(n log(h2)) = O(2n log h) = O(n log h), which is okay for us.
Lecture 3 5 Fall 2021 CMSC 754 Dave Mount Chan’s Algorithm for the Conditional Hull Problem ConditionalHull(P, h∗) : (1) Let k ←⌈n/h∗⌉. Partition P (arbitrarily) into disjoint subsets P1, . . . , Pk, each of size at most h∗ (3) For j ←1 to k, compute Hj = conv(Pj) using Graham’s scan, storing each in an ordered array (4) Let v0 ←(−∞, 0), and let v1 be the bottommost point of P (5) For i ←1, 2, . . . , h∗: (a) For j ←1 to k, using the utility lemma, compute the tangents points q− j and q+ j for Hj with respect to vi−1.
(b) Set vi to be the tangent point that minimizes the turning angle with respect to vi−2 and vi−1 (c) If vi = v1 then return the pair (success, V = ⟨v1, . . . , vi−1⟩) (6) If we get here, we know that h∗< h, and we return (failure, ∅) Too small? If we guess a value of h∗< h, then the merge phase will be too slow. It is easy to verify that (even if we didn’t stop it because of the failure condition) it would run in time O(n(h/h∗) log h∗). If our estimate h∗is not within a constant factor of h, then we will not achieve our desired running time. This is why we chose to use the “failure” option. Since we never do more than h∗iterations, the running time of each failure phase is just O(n log h∗).
Here is what we’ll do. We’ll start with a low estimate for h∗(e.g., h∗= 3). If the algorithm returns “failure”, we increase h∗until we succeed. The question is how quickly we should step up the value of h∗?
It is easy to show that increasing h∗in an arithmetic progression (e.g., h∗= 3, 4, 5, . . .) will be way to slow. A smarter approach is to grow h∗through doubling (e.g. h∗= 4, 8, 16, . . . , 2i).
We will leave it as an exercise to show that this is also too slow. (It will lead to a running time of O(n log2 h).) Recall that we are allowed to overshoot the actual value of h by any polynomial. Let’s try repeatedly squaring the previous guess. In other words, let’s try h∗= 2, 4, 16, 256, . . . , 22i.
Clearly, as soon as we reach a value for which the restricted algorithm succeeds, we have h < h∗≤h2. Therefore, the running time for this last stage will be O(n log h), as desired.
But what about the total time for all the previous stages?
To analyze the total time, consider the ith guess, h∗ i = 22i.
The ith trial takes time O(n log h∗ i ) = O n log 22i = O(n2i).
We know that we will succeed as soon as h∗ i ≥h, that is if i = ⌈lg lg h⌉. (Throughout the semester, we will use “lg” to denote logarithm base 2 and “log” when the base does not matter.1) Thus, the algorithm’s total running time (up to constant factors) is T(n, h) = lg lg h X i=1 n2i = n lg lg h X i=1 2i.
The summation is a geometric series. It is well known that a geometric series is asymptotically 1When log n appears as a factor within asymptotic big-O notation, the base of the logarithm does not matter provided it is a constant. This is because loga n = logb n/ logb a. Thus, changing the base only alters the constant factor.
Lecture 3 6 Fall 2021 CMSC 754 Dave Mount dominated by its largest term. Thus, we obtain a total running time of T(n, h) < n · 2⌈lg lg h⌉< n · 21+lg lg h = n · 2 · 2lg lg h = 2n lg h = O(n log h), which is just what we want. In other words, by the “miracle” of the geometric series, the total time to try all the previous failed guesses is asymptotically the same as the time for the final successful guess. The final algorithm is presented in the code block below.
Chan’s Complete Convex Hull Algorithm Hull(P) : (1) h∗←2; status ←failure (2) while (status ̸= failure): (a) Let h∗←min((h∗)2, n) (b) (status, V ) ←ConditionalHull(P, h∗) (3) return V Lower Bound (Optional): We show that Chan’s result is asymptotically optimal in the sense that any algorithm for computing the convex hull of n points with h points on the hull requires Ω(n log h) time. The proof is a generalization of the proof that sorting a set of n numbers requires Ω(n log n) comparisons.
If you recall the proof that sorting takes at least Ω(n log n) comparisons, it is based on the idea that any sorting algorithm can be described in terms of a decision tree. Each comparison has at most three outcomes (<, =, or >). Each such comparison corresponds to an internal node in the tree. The execution of an algorithm can be viewed as a traversal along a path in the resulting ternary (3-way splitting) tree. The height of the tree is a lower bound on the worst-case running time of the algorithm. There are at least n! different possible inputs, each of which must be reordered differently, and so you have a ternary tree with at least n!
leaves. Any such tree must have Ω(log3(n!)) height. Using Stirling’s approximation for n!, this solves to Ω(n log n) height. (For further details, see the algorithms book by Cormen, Leiserson, Rivest, and Stein.) We will give an Ω(n log h) lower bound for the convex hull problem. In fact, we will give an Ω(n log h) lower bound on the following simpler decision problem, whose output is either yes or no.
Convex Hull Size Verification Problem (CHSV): Given a point set P and integer h, does the convex hull of P have h distinct vertices?
Clearly if this takes Ω(n log h) time, then computing the hull must take at least as long.
As with sorting, we will assume that the computation is described in the form of a decision tree. The sorts of decisions that a typical convex hull algorithm will make will likely involve orientation primitives. Let’s be even more general, by assuming that the algorithm is allowed to compute any algebraic function of the input coordinates. (This will certainly be powerful enough to include all the convex hull algorithms we have discussed.) The result is called an algebraic decision tree.
Lecture 3 7 Fall 2021 CMSC 754 Dave Mount The input to the CHSV problem is a sequence of 2n = N real numbers. We can think of these numbers as forming a vector in real N-dimensional space, that is, (z1, z2, . . . , zN) = ⃗ z ∈RN, which we will call a configuration. Each node branches based on the sign of some function of the input coordinates. For example, we could implement the conditional zi < zj by checking whether the function (zj −zi) is positive. More relevant to convex hull computations, we can express an orientation test as the sign of the determinant of a matrix whose entries are the six coordinates of the three points involved. The determinant of a matrix can be expressed as a polynomial function of the matrices entries. Such a function is called algebraic. We assume that each node of the decision tree branch three ways, depending on the sign of a given multivariate algebraic formula of degree at most d, where d is any fixed constant. For example, we could express the orientation test involving points p1 = (z1, z2), p2 = (z3, z4), and p3 = (z5, z6) as an algebraic function of degree two as follows: det 1 z1 z2 1 z3 z4 1 z5 z6 = (z3z6 −z5z4) −(z1z6 −z5z2) + (z1z4 −z3z2).
For each input vector ⃗ z to the CHSV problem, the answer is either “yes” or “no”. The set of all “yes” points is just a subset of points Y ⊂RN, that is a region in this space. Given an arbitrary input ⃗ z the purpose of the decision tree is to tell us whether this point is in Y or not. This is done by walking down the tree, evaluating the functions on ⃗ z and following the appropriate branches until arriving at a leaf, which is either labeled “yes” (meaning ⃗ z ∈Y ) or “no”. An abstract example (not for the convex hull problem) of a region of configuration space and a possible algebraic decision tree (of degree 1) is shown in the following figure. (We have simplified it by making it a binary tree.) In this case the input is just a pair of real numbers.
1 2 3 4 5 6 no Y Y no no no no Y Y 1 2 3 4 5 6 Y Y The set Hierarchical partition Decision tree (a) (b) (c) Fig. 6: The geometric interpretation of an algebraic decision tree.
We say that two points ⃗ u,⃗ v ∈Y are in the same connected component of Y if there is a path in RN from ⃗ u to ⃗ v such that all the points along the path are in the set Y . (There are two connected components in the figure.) We will make use of the following fundamental result on algebraic decision trees, due to Ben-Or. Intuitively, it states that if your set has M connected components, then there must be at least M leaves in any decision tree for the set, and the tree must have height at least the logarithm of the number of leaves.
Lecture 3 8 Fall 2021 CMSC 754 Dave Mount Theorem: Let Y ∈RN be any set and let T be any d-th order algebraic decision tree that determines membership in W. If W has M disjoint connected components, then T must have height at least Ω((log M) −N).
We will begin our proof with a simpler problem.
Multiset Size Verification Problem (MSV): Given a multiset of n real numbers and an integer k, confirm that the multiset has exactly k distinct elements.
Lemma: The MSV problem requires Ω(n log k) steps in the worst case in the d-th order algebraic decision tree Proof: In terms of points in Rn, the set of points for which the answer is “yes” is Y = {(z1, z2, . . . , zn) ∈Rn : |{z1, z2, . . . , zn}| = k}.
It suffices to show that there are at least k!kn−k different connected components in this set, because by Ben-Or’s result it would follow that the time to test membership in Y would be Ω(log(k!kn−k) −n) = Ω(k log k + (n −k) log k −n) = Ω(n log k).
Consider the all the tuples (z1, . . . , zn) with z1, . . . zk set to the distinct integers from 1 to k, and zk+1 . . . zn each set to an arbitrary integer in the same range. Clearly there are k! ways to select the first k elements and kn−k ways to select the remaining elements.
Each such tuple has exactly k distinct items, but it is not hard to see that if we attempt to continuously modify one of these tuples to equal another one, we must change the number of distinct elements, implying that each of these tuples is in a different connected component of Y .
To finish the lower bound proof, we argue that any instance of MSV can be reduced to the convex hull size verification problem (CHSV). Thus any lower bound for MSV problem applies to CHSV as well.
Theorem: The CHSV problem requires Ω(n log h) time to solve.
Proof: Let Z = (z1, . . . , zn) and k be an instance of the MSV problem. We create a point set {p1, . . . , pn} in the plane where pi = (zi, z2 i ), and set h = k. (Observe that the points lie on a parabola, so that all the points are on the convex hull.) Now, if the multiset Z has exactly k distinct elements, then there are exactly h = k points in the point set (since the others are all duplicates of these) and so there are exactly h points on the hull. Conversely, if there are h points on the convex hull, then there were exactly h = k distinct numbers in the multiset to begin with in Z.
Thus, we cannot solve CHSV any faster than Ω(n log h) time, for otherwise we could solve MSV in the same time.
The proof is rather unsatisfying, because it relies on the fact that there are many duplicate points. You might wonder, does the lower bound still hold if there are no duplicates? Kirk-patric and Seidel actually prove a stronger (but harder) result that the Ω(n log h) lower bound holds even you assume that the points are distinct.
Lecture 3 9 Fall 2021 |
189496 | https://frontierspecialtychemicals.com/applications/protoporphyrin-biosynthesis-functionalization-applications/?srsltid=AfmBOoqKiiWDZoQ-QlsvkD1TsyPGRUL6lbp7FVbUQ0q2gvG115CeiZ8J | Search for:
Protoporphyrin IX
Protoporphyrin IX Plays a Vital Role in Biological Systems
Life on earth is critically dependent on the existence of certain key molecules; among these critical molecules are a group of compounds containing tetra-pyrrole macrocycles as an important part of them.
Heme (Iron protoporphyrin IX), chlorophyll, and cobalamin (vitamin B12) are some of the tetra-pyrrole compounds vital to the survival of life — all of which require protoporphyrin IX to exist. Iron protoporphyrin IX plays a pivotal role in oxygen transportation when in the form of hemoglobin and is an active component of biologically essential molecules such as cytochrome p450, myoglobin, and numerous other metalloenzymes.1
Mense, S., Zhang, L. Heme: a versatile signaling molecule controlling the activities of diverse regulators ranging from transcription factors to MAP kinases. Cell Res 16, 681–692 (2006).
What is Protoporphyrin IX?
Protoporphyrin IX (also referred to as Protoporphyrin or PPIX) is the macrocyclic ring system of heme without the iron chelated into the center of the molecule. The enzyme ferrochelatase is responsible for facilitating the chelation of iron into the center of protoporphyrin to form heme. Interestingly, the aromatic porphyrin ring system in protoporphyrin IX strongly absorbs light in specific regions of the visible spectrum of light giving blood its characteristic red color. Of its many important roles, perhaps most notable is that protoporphyrin IX is a critical intermediate in the biosynthesis of heme, chlorophyll, and cobalamin.
Did you know?
Protoporphyrin IX’s chemical structure was first proposed by Kuster in 1912 and was confirmed when Fischer reported the chemical synthesis of heme in 1929.
Protoporphyrin IX contains functional groups (carboxylic acids and vinyl groups) that have been modified to generate many unique compounds of interest. The free base form of protoporphyrin IX has been metalated with a variety of metals other than iron, imparting unique properties that can be used to research biological mechanisms and may impact the activity of enzymes in various biological pathways. The introduction of metalo-protoporphyrins using metals that don’t occur in nature have been used to explore a multitude of applications such as photosensitizers used in photodynamic therapy (PDT) to treat cancer, catalysts in organic transformations, and as a unique antimicrobial agent.
It is protoporphyrin IX’s extensive customizability that gives it such a unique and impactful role in various fields of research and application. You can see how vastly it can be customized by checking out our exhaustive (but not comprehensive!) list of custom porphyrins that we offer.
The Biosynthesis of Protoporphyrin IX for Hemoglobin, Chlorophyll, and Cobalamin (Vitamin B12)
The biosynthetic pathway to produce protoporphyrin IX represents one of the most essential metabolic pathways in living organisms, serving as a critical juncture in cellular metabolism. Protoporphyrin IX is the key precursor for the biosynthesis of several vital biomolecules, including hemoglobin, chlorophyll, and cobalamin (vitamin B12).
Hemoglobin, an essential protein found in red blood cells, is responsible for oxygen transport throughout the body. The incorporation of iron into protoporphyrin IX forms heme, the functional component of hemoglobin, enabling efficient oxygen binding and release.
In plants and certain algae, protoporphyrin IX is the precursor for chlorophyll, the green pigment crucial for photosynthesis. Chlorophyll’s role in capturing light energy and converting it into chemical energy through the photosynthetic process is fundamental to the energy balance of our planet.
Additionally, protoporphyrin IX serves as a precursor in the synthesis of cobalamin, commonly known as vitamin B12. Cobalamin is an enzymatic cofactor essential for both mammals and bacteria. This vitamin is vital for various metabolic functions, including DNA synthesis, fatty acid metabolism, and neurological function.
Thus, the pathway leading to the production of protoporphyrin IX is indispensable for sustaining life, influencing a wide array of physiological processes across different biological kingdoms.
Protoporphyrin IX is synthesized in 7 steps
At last, the formation of protoporphyrin IX marks the final unifying phase in the biosynthesis of heme, chlorophyll, and cobalamin; from this point onward, the synthesis pathways diverge, each leading to the formation of these distinct but equally crucial biomolecules.
How is protoporphyrin IX used in further biosynthetic pathways?
The biosynthesis of heme only requires 1 additional step…
The biosynthesis of chlorophyll continues with 7 more steps…
The biosynthesis of cobalamin requires 17 steps of additional transformations…
Functionalization Chemistry of Protoporphyrin IX
Protoporphyrin IX is a versatile intermediate in the laboratory synthesis of many unique derivatives of protoporphyrin. The functionalization of protoporphyrin has allowed access to hundreds of new molecules with potential applications in medicine, materials science, and catalysis.12,13
Incorporation of Various Metals
Modification of Carboxylic Acids
Protoporphyrin IX dimethyl ester is useful for further derivatization since it is much more soluble in organic solvents than protoporphyrin IX itself. In addition to ester forms, peptide coupling conditions can be used to produce bis-amides from PPIX for use in the field of photodynamic therapy research14,15.
Need protoporphyrin IX dimethyl ester?
We offer it in research and bulk quantities, ready to be shipped worldwide!
Modification of Vinyl Groups
Functionalization of the bis-vinyl groups can be accomplished by several different synthetic transformations. Bromination of the vinyl groups followed by the addition of alcohols gives access to mesoporphyrin derivatives that contain ether groups. Reduction of the vinyl groups affords mesoporphyrin IX as the product.
A multistep procedure has been reported for the conversion of both vinyl groups into the bis-aldehyde species17.
Applications of Protoporphyrin IX
Heme oxygenase research & heme oxygenase inhibitors
G-quadruplex-Protoporphyrin IX-based sensors:
Ferrochelatase Inhibitors:
Antimicrobial activity:
Ga(III) protoporphyrin IX, In(III) protoporphyrin IX, Co(III) protoporphyrin IX, Mn(III) protoporphyrin IX, and Cu(II) protoporphyrin IX exhibit antimicrobial activity against several pathogenic microorganisms such as Staphylococcus aureus, Mycobacterium abscessus, and Haemophilus influenzae. This activity is attributed to reduced uptake of heme or adverse properties of the non-native metalloporphyrins38,39,40.
Reconstitution of Metalloproteins:
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Shankar Kumbar
Secretary in at Prarthana Education Society (2017–present) · Author has 123 answers and 122.9K answer views
·4y
Originally Answered: What is surface charge density? ·
Hi,
AccordingToMeOk……..
Surface charge density is charge distributed per unit area.
If a charge Q is uniformly distributed on a surface of area S, then the surface charge density σ is given by
σ = Q/S
The unit of σ is C/m2
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JH Simon
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How do you overcome narcissistic mental enslavement? I grew up with a narc mother and dated a narc that I just cannot get out of my head.
Reclaim your True Self.
Narcissistic mental enslavement begins with but goes well beyond the mind. Every narcissist mentally overwhelms their target, but what gives the narcissist’s attacks teeth is the emotional reaction. When a narcissist puts down their target, shame arises. When a narcissist threatens their target, fear arises. When a narcissist questions their target’s morality, guilt arises. These attacks bind together over time and produce a corresponding complex, making the target easier to trigger as a ripple becomes an emotional tsunami. A psychological cage is built with layer upon l
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Reclaim your True Self.
Narcissistic mental enslavement begins with but goes well beyond the mind. Every narcissist mentally overwhelms their target, but what gives the narcissist’s attacks teeth is the emotional reaction. When a narcissist puts down their target, shame arises. When a narcissist threatens their target, fear arises. When a narcissist questions their target’s morality, guilt arises. These attacks bind together over time and produce a corresponding complex, making the target easier to trigger as a ripple becomes an emotional tsunami. A psychological cage is built with layer upon layer of emotional pain, which in time splits from the target’s consciousness and reduces the target to a helpless and compliant object. For example, when guilt becomes a complex, a person acts, thinks and believes as though they are guilty, regardless of the situation. Any judgement, whether justified or not, triggers a guilt attack and cripples the person’s capacity for independent thought and action. Reaction becomes the only option.
Overcoming mental enslavement involves facing all emotions and integrating them into consciousness. This means having the courage to fall below the realm of thought. Through mindful exploration, you come to notice your triggers, and therefore welcome the emotional intensity into awareness. Rather than reacting to or dissociating from your emotions, you create space and allow them to rise up, and most importantly, to release. Meditation, therapy and bodywork are all potent tools to help you weave together a connection between your Higher Self and your True Self. In doing so, you create a state of wholeness and power which eclipses the mental enslavement you experience after narcissistic abuse.
If you have just started your narcissistic abuse recovery journey, check out How To Kill A Narcissist. Or if you wish to immunise yourself against narcissists and move on for good, take a look at How To Bury A Narcissist.
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Himanshu Bhandari
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Originally Answered: What is surface charge density? ·
Dear All,
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Please find attached video on Surface Charge Density : Surface Charge Density - Electric Field & Flux
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Greenwood Hansma
Emeritus Research Professor of Biophysics at University of California, Santa Barbara (2008–present) · Author has 3.8K answers and 2.2M answer views
·4y
Originally Answered: What is meant by the surface density of the charge? ·
I think of sodium ions, Na+, and potassium ions, K+. K+ is bigger than Na+, but they both have a charge of +1, so K+ has the charge spread over a larger area than Na+
Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. ... Charge density can be either positive or negative, since electric charge can be either positive or negative.
Charge density - Wikipedia Electric charge per unit length, area or volume In electromagnetism , charge density is the amount of electric charge per unit length , surface area , or volume . Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m −3 ), at any point in a volume. [ 1 ] [ 2 ] [ 3 ] Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m −2 ), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m −1 ), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative. Like mass density , charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position x {\displaystyle {\boldsymbol {x}}} , like a fluid, and ρ ( x ) {\displaystyle \rho ({\boldsymbol {x}})} , σ ( x ) {\displaystyle \sigma ({\boldsymbol {x}})} , and λ ( x ) {\displaystyle \lambda ({\boldsymbol {x}})} are usually regarded as continuous charge distributions , even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge , the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density ρ ( x ) {\displaystyle \rho ({\boldsymbol {x}})} and the current density J ( x ) {\displaystyle {\boldsymbol {J}}({\boldsymbol {x}})} . Since all charge is carried by subatomic particles , which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge. [ 4 ] [ page needed ] For example, the charge in an electrically charged metal object is made up of conduction electrons moving randomly in the metal's crystal lattice . Static electricity is caused by surface charges consisting of electrons and ions near the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space. The charge carrier density in a conductor is equal to the number of mobile charge carriers ( electrons , ions , etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the elementary charge on an electron is so small (1.6⋅10 −19 C) and there are so many of them in a macroscopic volume (there are about 10 22 conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macr
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Darpan Shah.
B.Sc in Physics&Mathematics, Christ (deemed university) (Graduated 2018)
·8y
Originally Answered: What is surface charge density? ·
Surface charge density in the context of electrostatics ( I assume ) refers to the density of the charges lying on the surface of the body under consideration. It is an average value, because different shapes of surfaces have different SCD in the same body. It is denoted by sigma and is given by the formula
Q/A ; Q - Total charge on the surface of the body, A - Total surface area.
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189498 | https://microbenotes.com/cori-cycle/ | Skip to content
Cori Cycle: Key Steps, Regulations, Importance
by Krisha Karki
Reviewed and Edited By: Sagar Aryal, PhD
The Cori Cycle, also known as the lactic acid cycle, is a vital metabolic pathway that operates within the human body to regulate glucose and lactate levels.
It plays a crucial role in maintaining glucose homeostasis during periods of high-intensity exercise or when the body requires rapid energy. The Cori Cycle involves the interconversion of lactate and glucose between muscle and liver cells.
Table of Contents
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Metabolic Overview
Metabolism encompasses all the biochemical reactions occurring within an organism, including processes that transform molecules into energy, building blocks for cells and tissues, and waste elimination. It is fundamental for the maintenance of life, enabling growth, repair, and the efficient functioning of the body’s systems.
Energy production and utilization are central to metabolism. Energy is essential for various bodily functions, from cellular activities to physical movement. Metabolism involves converting nutrients, such as glucose and fats, into usable energy in the form of adenosine triphosphate (ATP), providing the energy required for diverse biological processes.
Glucose, a simple sugar, is a primary energy source in metabolism. It can be oxidized through processes like glycolysis to produce ATP. Glucose metabolism is pivotal as it provides a readily available and efficient source of energy for the body’s cells and organs.
Glycolysis: The First Step
Glycolysis is the enzymatic pathway that breaks down glucose, a six-carbon sugar, into two molecules of pyruvate, a three-carbon compound. It occurs in the cytoplasm of cells and is the first step in both aerobic and anaerobic cellular respiration.
Glycolysis involves a series of enzymatic reactions, each catalyzed by specific enzymes. Key enzymes include hexokinase, phosphofructokinase, and pyruvate kinase. Reactions include phosphorylation, isomerization, oxidation, and substrate-level phosphorylation.
Likewise, it yields a net production of two ATP molecules and two NADH molecules for each glucose molecule. Pyruvate, a pivotal metabolite, is formed during glycolysis and serves as a precursor for various metabolic pathways.
Lactate Formation
In anaerobic conditions or during intense exercise, pyruvate generated from glycolysis can be converted to lactate through the action of lactate dehydrogenase. This conversion helps regenerate NAD+ necessary for glycolysis to continue in the absence of oxygen.
Lactate dehydrogenase is the enzyme responsible for the reversible conversion of pyruvate to lactate. It utilizes NADH to reduce pyruvate to lactate, allowing for the recycling of NAD+ in the glycolytic pathway.
Lactate production is a means to regenerate NAD+ and sustain glycolysis under anaerobic conditions. Lactate itself can be utilized as an energy source in various tissues, and its production is a vital component of the body’s adaptive response to varying metabolic demands.
Lactate Transport and the Cori Cycle
Lactate is transported from muscle tissue to the liver via the bloodstream. It is taken up by liver cells (hepatocytes) through specific lactate transporters.
Key Steps in the Cori Cycle
Lactate uptake by the liver: Lactate enters hepatocytes in the liver via lactate transporters.
Lactate conversion to glucose: In the liver, lactate is converted to glucose through gluconeogenesis.
Glucose release into the bloodstream: The newly synthesized glucose is released into the bloodstream to be used by other tissues for energy.
Liver’s Role and Significance in the Cori Cycle
The liver acts as a central hub in the Cori Cycle, playing a vital role in recycling lactate generated by muscles during intense exercise.
It converts lactate to glucose, replenishing the body’s glucose stores and ensuring a continuous energy supply during prolonged activity.
Regulation of the Cori Cycle
Regulatory Mechanisms Controlling Lactate Production
Enzyme activity: Enzymes involved in glycolysis and lactate production are regulated based on energy demands and metabolic conditions.
Hormonal control: Hormones like insulin and glucagon influence the rate of glucose and lactate production.
Factors Influencing Cori Cycle’s Activity
Exercise intensity: Higher intensity exercise results in increased lactate production and utilization by the Cori Cycle.
Glucose availability: The availability of glucose influences the Cori Cycle’s rate, as it directly affects the conversion of lactate to glucose in the liver.
Adaptive Responses of the Cori Cycle to Various Conditions
The Cori Cycle adapts to changes in energy demands, substrate availability, and metabolic conditions to maintain glucose homeostasis and support the body’s energy needs during different activities and conditions.
Conclusion
Cori Cycle emerges as a fundamental movement, ensuring a rhythmic supply of energy to our body’s cells.
As lactate travels from muscle to liver, transforming into glucose, this cycle embodies resilience and adaptability, shaping our response to varying levels of exertion. Appreciating the harmonious interplay of enzymes, hormones, and tissues within the Cori Cycle unveils promising avenues for understanding.
References
About Author
Krisha Karki
Krisha Karki is a B.Sc. microbiology student with a passion for writing, and expanding her knowledge is a unique blend of scientific curiosity and creative expression. Likewise, she deeply understands the microscopic world and its intricacies. With a deep-rooted curiosity and an insatiable thirst for knowledge, she eagerly explores the realms of microbiology. Her love for science and technology permeates the writing, infusing it with a unique blend of intellectual rigor and imaginative flair.
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Thanks to you I finally understand the Cori cycle and Glycolysis. The relationship between the two is just beautifully perfect.
BSc. Dietetics student.
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189499 | https://byjus.com/average-rate-of-change-formula/ | The Average Rate of Change function is defined as the average rate at which one quantity is changing with respect to something else changing. In simple terms, an average rate of change function is a process that calculates the amount of change in one item divided by the corresponding amount of change in another.
Formula for the Average Rate of Change of a Function
Using function notation, we can define the Average Rate of Change of a function f from a to b as:
Where,
f(a) and f(b) are the values of the function f(x) at a and b respectively. Also, a and be are the limits of the function.
Solved Examples
Question 1: Calculate the average rate of change of a function, f(x) = 3x + 12 as x changes from 5 to 8 .
Given,
f(x) = 3x + 12
a = 5
b = 8
f(5) = 3(5) + 12
f(5) = 15 + 12
f(5) = 27
f(8) = 3(8) + 12
f(8) = 24 + 12
f(8) = 36
The average rate of change is,
A(x) =
A(x) =
A(x) =
A(x) =
A(x) = 3
Question 2: Calculate the average rate of change of the function f(x) = x2 – 9x in the interval 2 ≤ x ≤ 7.
Solution:
From the given,
f(x) = x2 – 9x
a = 2
b = 7
f(a) = f(2) = (2)2 – 9(2) = 4 – 18 = -14
f(b) = f(7) = (7)2 – 9(7) = 49 – 63 = -14
The average rate of change is:
Therefore, A(x) = 0
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