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5
How do you prove that N/C is equal to V/M?
·
To prove that the units of electric field strength,
N
C
(newtons per coulomb), are equal to the units of voltage gradient,
V
m
(volts per meter), we can start by breaking down the definitions of these units:
Electric Field Strength (
E
)
:
The electric field strength is defined as the force (
F
) experienced by a charge (
q
) per unit charge:
E
=
F
q
- Here,
F
is measured in newtons (
N
), and
q
is measured in coulombs (
C
).
- Therefore, the unit of electric field strength is:
[
E
]
=
N
C
2. Voltage (Electric Potential Difference,
V
)
:
Voltage is defined as the
To prove that the units of electric field strength, NC (newtons per coulomb), are equal to the units of voltage gradient, Vm (volts per meter), we can start by breaking down the definitions of these units:
Electric Field Strength (E):
The electric field strength is defined as the force (F) experienced by a charge (q) per unit charge:
E=Fq
Here, F is measured in newtons (N), and q is measured in coulombs (C).
Therefore, the unit of electric field strength is:
[E]=NC
Voltage (Electric Potential Difference, V):
Voltage is defined as the work (W) done per unit charge (q):
V=Wq
Work is defined as force times distance, W=F⋅d, where d is measured in meters (m).
Thus, the unit of voltage can be expressed as:
[V]=WC=F⋅dC=N⋅mC
Voltage Gradient:
The voltage gradient (electric field strength) is the change in voltage per unit distance:
E=Vd
Here, V is in volts (V) and d is in meters (m).
Therefore, the unit of voltage gradient is:
[E]=Vm
Now, substituting the expression for voltage in terms of newtons and meters into the voltage gradient:
E=N⋅m/Cm=NC
Thus, we can conclude that:
NC=Vm
This shows that the electric field strength measured in newtons per coulomb is indeed equal to the voltage gradient measured in volts per meter.
Related questions
How do you show that 1 N/C = 1 V/M?
How can we convert E in the unit of V/m to N/C?
What are the best substances for N C Thinner?
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Joe Bak
Engineer
·
Author has 58 answers and 31.4K answer views
·
4y
Originally Answered: How do you show that 1 N/C = 1 V/M?
·
A little Wikipedia, a little algebra, and a little work will solve your problem.
In the SI system of measurement, the Volt (V) is defined as a Joule (J) of potential energy per Coulomb (C) of charge. (Volt - Wikipedia )
V = J/C
The Newton (N) is defined as the force required to move a kilogram (kg) of mass with an acceleration of one meter (m) per second, squared (s^2). (Newton (unit) - Wikipedia )
N = kg . m / s^2
The Joule (J) is defined as the energy transferred when a force of one Newton (N) is applied to move an object one meter (m). (Joule - Wikipedia )
J = N . m
So:
N/C = (J/m)/C = J / (m x C)
A little Wikipedia, a little algebra, and a little work will solve your problem.
In the SI system of measurement, the Volt (V) is defined as a Joule (J) of potential energy per Coulomb (C) of charge. (Volt - Wikipedia )
V = J/C
The Newton (N) is defined as the force required to move a kilogram (kg) of mass with an acceleration of one meter (m) per second, squared (s^2). (Newton (unit) - Wikipedia )
N = kg . m / s^2
The Joule (J) is defined as the energy transferred when a force of one Newton (N) is applied to move an object one meter (m). (Joule - Wikipedia )
J = N . m
So:
N/C = (J/m)/C = J / (m x C) = (J/C)/m = V/m
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N/c=v/m
R.h.s
V/m
:.1v=1j/1c
Putt this in above
J/c.m
:.j=N.M
N.m/c.m
=n/c
Which is left side
Roger Larson
Author has 5K answers and 4.7M answer views
·
3y
Originally Answered: How do you show that 1 N/C = 1 V/M?
·
Start with the definition of Volt
1 Volt = 1 Joule/ Coulomb
1 Joule = 1 Newton meter
So
1 Volt = 1 Newton meter/ Coulomb
divide the above by 1meter
1 Volt/meter = 1 Newton/Coulomb
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Shadow Tiger
Knows Middle English
·
5y
V = E.dx ,
Unit of E(electric field) = N/C ,Unit of V = V , Unit of x or dx = m
As V/dx = Ecosθ , V/m=N/C . (cosθ - unitless)
Chris Richardson
Studied Physics
·
Author has 7.2K answers and 4M answer views
·
Updated 2y
Originally Answered: How do you show that 1N/C=1V/m, where c is coulomb, N Newton, V volt, and m metre?
·
1 Coulomb is 1As
1 Newton is 1kgms−2
1 Volt is 1kgm2s−3A−1
V/m=(kgm2s−3A−1)/m
=kgms−3A−1
N/C=(kgms−2)/As
=kgms−3A−1
These are the same, so we are done.
If you don’t immediately know these conversions you can often work them out from equations you know:
A newton is a unit of force, and F=ma
Power, P=IV, so V=P/I and P=Fv (force x velocity)
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Roger Larson
Author has 5K answers and 4.7M answer views
·
May 19
Originally Answered: How do you show that 1N/C=1V/m, where c is coulomb, N Newton, V volt, and m metre?
·
Start with
1 Volt = 1 Joule/Coulomb
then
1V/m = 1 Joule/(( meter)(Coulomb))
But
1 Joule =(1Newton)(1 Meter)
1V/m = (1 Newton)(Meter)/(( Coulomb)(meter))
1 V/m = 1 Newton/ Coulomb
You can prove 1 Volt = 1 Joule per Coulomb
from P = VI
1 Watt =( 1 Volt)(1 Ampere)
note that
1 Watt=1 Joule/second
1 Ampere = 1 Coulomb/ second
(1 Joule/ second) =(1 Volt)(1 Coulomb/second)
1 Joule =(1 Volt)(1 Coulomb)
1 Volt = 1Joule/Coulomb
Rambabu Dontu
PhD in Physics, Tata Institute of Fundamental Research (Graduated 1986)
·
Author has 3.6K answers and 1.3M answer views
·
2y
Originally Answered: How do you show that 1 N/C = 1 V/M?
·
Consider a uniform electric field E=1N/C. On a charge of q=1 C, a force F= Eq= 1N acts. Now if we push the charge against this force to move it by s=1m, the work done by us on the charge is
W = Fs = (1N)(1m) =1J
Work done per unit charge is the potential difference V =W/q = 1 J/C == 1 volt
V = W/q = Fs/q
Potential difference per unit distance is
V/s = F/q = E
Hence (1 volt)/(1 m) = 1 N/C
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Roger Larson
Author has 5K answers and 4.7M answer views
·
Updated 4y
You’re asking about the units of the electric field (E) where the units N/C are Newton/ charge. Where Newton is force and charge is measured in Coulombs.
Voltage is defined as Ed, where d is distance, which means the units of E are also Volts/ meter.
The units of V are (Newton meter)/Coulomb = Joules/Coulomb since work is measured in Joules.
V/m = Newton/Coulomb
Peter Upton
BA in Physics & Mathematics, The Open University
·
Author has 14.5K answers and 10.8M answer views
·
4y
Originally Answered: How do you show that 1N/C=1V/m, where c is coulomb, N Newton, V volt, and m metre?
·
The electric field strength is defined as being the force per unit charge E=F/q
Now imagine parallel plates with a uniform field between them, separation x and pd is V.
Work done moving a charge across against the electric field is WD =Vq
However, the work done is also force x distance = F x = Eqx
Now we can equate the two equations for work done :
V q = E qx so V= E x and hence E= V/x (volts/ metre) but we defined E as F/q so it must have also units of newtons/coulomb.
Harry Ellis
PhD in Physics, Georgia Institute of Technology (Graduated 1974)
·
Author has 2.8K answers and 1.3M answer views
·
4y
Originally Answered: How do you show that 1N/C=1V/m, where c is coulomb, N Newton, V volt, and m metre?
·
There are several ways. Here is an easy one:
A Volt is defined as a Joule/Coulomb, and a Joule is a Newtonmeter.
So: 1 V/m = (1 J/C) / 1m = (1 Nm)/C) / 1 m = 1 N/C
Alf Salte
Works at Schlumberger (company)
·
Author has 3.8K answers and 2.6M answer views
·
4y
Originally Answered: How do you show that 1N/C=1V/m, where c is coulomb, N Newton, V volt, and m metre?
·
Let’s see, 1 N = 1 kgm/s^2 and 1 C = 1 As or ampere second alternatively 1 A = 1 C/s but ampere is considered the base unit in SI.
Volt can be seen as 1 V = 1 kg m^2/As^3
so 1 N/C = 1 kgm/s^2/As = 1 kgm/As^3 and 1 V/m = 1 kgm^2/As^3/m = 1 kgm/As^3 is the same.
Just consider the definitions of each unit.
M.S. in Physics, University of Minnesota - Twin Cities (Graduated 1971)
·
Author has 9K answers and 2.7M answer views
·
4y
Originally Answered: How do you show that 1N/C=1V/m, where c is coulomb, N Newton, V volt, and m metre?
·
1 V = 1 J/C. 1 C = 1 J/V
1 J = 1 Nm. 1 N = 1 J/m
1 N/C = (J/m) / (J/V)
N/C = (J/m) (V/J)
you got it from here?
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[n^2(n+1)^2]/4?](
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5701 | http://www.cs.umd.edu/class/fall2022/cmsc420-0201/Lects/lect04-union-find.pdf | CMSC 420 Dave Mount CMSC 420: Lecture 4 Disjoint Set Union-Find Equivalence relations: An equivalence relation over some set S is a relation that satisfies the following properties for all elements of a, b, c ∈S.
Reflexive: a ≡a.
Symmetric: a ≡b then b ≡a Transitive: a ≡b and b ≡c then a ≡c.
Equivalence relations arise in numerous applications. An example includes any sort of “group-ing” operation, where every object belongs to some group (perhaps in a group by itself) and no object belongs to more than one group. More formally these groups are called equivalent classes and the subdivision of the set into such classes is called a partition. For example, suppose we are maintaining a bidirectional communication network. The ability to commu-nicate is an equivalence relation, since if machine a can communicate with machine b, and machine b can communicate with machine c, then machine a can communicate with machine c (e.g. by sending messages through b). Now suppose that a new link is created between two groups, which previously were unable to communicate. This has the effect of merging two equivalence classes into one class.
We discuss a data structure that can be used for maintaining equivalence partitions with two operations: (1) union, merging to groups together, and (2) find, determining which group an element belongs to. This data structure should not be thought of as a general purpose data structure for storing sets. In particular, it cannot perform many important set operations, such as splitting two sets, or computing set operations such as intersection and complementation. And its structure is tailored to handle just these two operations. However, there are many applications for which this structure is useful.
As we shall see, the data structure is simple and amazingly efficient.
Union-Find ADT: We assume that we have an underlying finite set of elements S. We want to maintain a partition of the set. In addition to the constructor, the (abstract) data structure supports the following operations.
Set s = find(Element x): Return an set identifier of the set s that contains the element x. A set identifier is simply a special value (of unspecified type) with the property that find(x) == find(y) if and only if x and y are in the same set.
Set r = union(Set s, Set t): Merge two sets named s and t into a single set r containing their union. We assume that s, t and r are given as set identifiers. This operations destroys the sets s and t.
Note that there are no key values used here. The arguments to the find and union operations are pointers to objects stored in the data structure. The constructor for the data structure is given the elements in the set S and produces a structure in which every element x ∈S is in a singleton set {x} containing just x.
Inverted-Tree Implementation: We will derive our implementation of a data structure for the union-find ADT by starting with a simple structure based on a forest of inverted trees. You think of an inverted tree as a multiway tree in which we only store parent links (no child or Lecture 4 1 Fall 2022 CMSC 420 Dave Mount sibling links). A root’s parent pointer is null. There is no limit on how many children a node can have. The sets are represented by storing the elements of each set in separate tree. For example, suppose that S = {1, 2, 3, . . . , 13} and the current partition is: {1, 6, 7, 8, 11, 12}, {2}, {3, 4, 5, 13}, {9, 10}.
This might be stored in an inverted tree as shown in Fig. 1(a). Note that there is no particular order to how the individual trees are structured, as long as they contain the proper elements.
7 6 12 1 8 11 2 4 3 13 5 9 10 (a) (b) 13 12 11 10 9 8 7 6 5 4 3 2 1 7 0 0 0 0 7 7 6 6 4 4 3 9 set identifiers parent Fig. 1: (a) Partition stored as a forest of inverted trees and (b) array-based representation.
Array representation: You might wonder, “How do I find the node containing a given element?” Let’s assume that the set elements are integers {1, 2, . . . , n} as shown above. Rather than using a standard node and pointer structure, we will think of the elements as being entries in an array parent[1..n]. We define parent[i] to hold the index the parent of element i in our tree or zero if i is a root of a tree. (We will sometimes call this zero and sometimes call this null, but they mean the same thing.) Even though we will use the tree representations in our illustrations, the actual implementation of the data structure is the array.
Initially, every element is in its own set, which we can set up by setting every element of the parent array to zero.
Set identifiers: Using this representation, each Element of the set is just an integer x, where 1 ≤x ≤n. Each set is represented by a root in an inverted tree. A set identifier, called Set, is just an integer index x, where 1 ≤x ≤n and parent[i] == 0. For example, in Fig. 1(a) the set {3, 4, 5, 13} is represented by the set identifier (root) 4.
If we want to know whether two elements are in the same set, we simply find the roots of their associated trees (by following parent links) and then check whether these two roots are the same. For example, when we trace the parent links from 12 and 8 both stop at 7, implying that 12 and 8 are in the same set. On the other hand, when we trace the parent links from 6 and 10, they stop at the roots 7 and 9, respectively. Since these are different roots, these elements are in different sets.
Find Operation: As mentioned above, given any element x, we perform the operation find(x) by walking along parent links until reaching the root of its tree. We return the root of the tree as the desired set identifier. This root element is set identifier for the set containing x.
Notice that this satisfies the above requirement, since two elements are in the same set if and only if they have the same root. We call this simple-find(). (Later we will propose an improvement.) For example, in Fig. 1(b)), operation simple-find(11) would start at node 11 and trace the path up through 6 up to the root 7. It returns the index 7 as the answer.
Lecture 4 2 Fall 2022 CMSC 420 Dave Mount Find operation (without path compression) Set simple-find(Element x) { while (parent[x] != null) x = parent[x] // follow chain to root return x; // return the root } (a) 7 12 1 8 11 2 4 3 13 5 9 10 (b) simple-find(11) 13 12 11 10 9 8 7 6 5 4 3 2 1 7 0 0 0 0 7 7 6 6 4 4 3 9 simple-find(11) 6 Fig. 2: The operation simple-find(11).
Union Operation: A union is similarly straightforward to implement. To perform the union of two sets we just link the root of one tree into the root of the other tree. But here is where it pays to be smart. Recall that height of a tree is the maximum number of edges from any leaf to the root. In Fig. 3, we label each tree with its height. If we link the root of i as a child of b, the height of the resulting tree will be 2, whereas if we do it the other way the height of the tree will only be 1. Clearly, it is better to link the lower height tree under the other to keep the final tree’s height as small as possible. This will make future find operations run faster. In contrast, if we take the union of two trees of equal height, say g and d, we can make either the root. (You might wonder why we use rank rather than some other property, and this is a good question for further thought.) 7 6 12 1 8 11 2 4 3 13 5 9 10 7 6 12 1 8 11 2 4 3 13 5 9 10 union(2, 9) union(7, 4) 2 0 2 1 3 1 Fig. 3: Union-find with ranks.
In our later enhancements, we will compress paths, and this will change the tree height. So, we will invent an abstract concept, called rank, which behaves similar to tree height. We assume that the rank is stored as a field in each node, but we only use it for root nodes. The operation union(s, t) is given the roots of two trees. It swaps them if necessary so that t’s rank is at least as large as s’s. It then links s as a child of t and update’s t’s rank to be the maximum of t’s rank and 1 plus s’s rank. The implementation is shown in the following code block.
Storing Ranks: There is a cool trick for storing ranks. As suggested in the code block, we could just implement a separate array. But since ranks are only needed for tree roots, we can do Lecture 4 3 Fall 2022 CMSC 420 Dave Mount Union operation Set union(Set s, Set t) { if (rank[s] > rank[y]) { // s has higher rank?
swap s and t // swap so that t’s rank is larger } parent[s] = t // link s as child of t rank[t] = max(rank[t], 1 + rank[s]) // update t’s rank return t } all of this with just a single array. Rather than setting parent[x] = 0 to indicate a root, we can set parent[x] to be the negation of the rank of x. When searching for the root of a tree, rather than testing whether parent[x] == null (or equivalently zero), we can check whether parent[x] < 0. If so, then the rank can be extracted by negating its value.
Analysis of Running Time: Consider a sequence of m union-find operations, performed on a domain with n total elements. Observe that the running time of the initialization is propor-tional to n, the number of elements in the set, but this is done only once. Each union takes only constant time, O(1).
In the worst case, find takes time proportional to the height of the tree. The key to the efficiency of this procedure is the following observation, which implies that the tree height is never greater than lg m. (Recall that lg m denotes the logarithm base 2 of m.) Lemma: Using the union-find procedures described above any tree with height h has at least 2h elements.
Proof: Given a union-find tree T, let h denote the height of T, let n denote the number of elements in T. We will show that n ≥2h.
It will help to introduce an intermediate quantity to help drive the proof. Let u ≥0 denote the number of union operations used to build T. Our proof is based on induction on u. For the basis (no unions, or u = 0) we have a tree with n = 1 element of height h = 0. Since 1 = 20, we have n ≥2h, which establishes the basis case.
For the induction step, suppose that we form a tree T through u union operations by merging two trees T ′ and T ′′. Let n′ and n′′ be their respective numbers of elements, let h′ and h′′ be their respective heights, and let u′ and u′′ denote the number of union operations to build each. The number unions to build T is clearly u = 1 + u′ + u′′.
Thus, u′ and u′′ are both smaller than u, which means that we can apply the induction hypothesis to each of them, implying that n′ ≥2h′ and n′′ ≥2h′′.
Following the merge we have a total n = n′ + n′′ elements. The final height depends on h′ and h′′. We may assume without loss of generality that h′ ≤h′′. (If not, swap the trees T ′ and T ′′.) There are two possibilities. If h′ = h′′, then the resulting tree has height h = h′ + 1 = h′′ + 1 (see Fig. 4(a)). The number of elements in the final tree is n = n′ + n′′ ≥2h′ + 2h′′ = 2h−1 + 2h−1 = 2 · 2h−1 = 2h.
On the other hand, if h′ < h′′, then the final tree has height equal to the larger tree, that is, h = h′′ (see Fig. 4(b)). The number of elements is n = n′ + n′′ ≥n′′ ≥2h′′ = 2h.
Lecture 4 4 Fall 2022 CMSC 420 Dave Mount h′ h′′ h n′ n′′ (a) h′ h′′ h n′ n′′ (b) +1 +1 Fig. 4: Proof that n ≥2h.
In either case we obtain the desired conclusion.
Since the unions’s take O(1) time each, we immediately have the following.
Theorem: After initialization, any sequence of m union’s and find’s can be performed in time O(m log m). In other words, the amortized time for union-find operations is O(log m).
Path Compression: It is possible to apply a very simple heuristic improvement to this data structure which provides a significant improvement in the running time. Here is the intuition.
If the user of your data structure repeatedly performs find’s on a leaf at a very low level in the tree then each such find takes as much as O(log n) time. Can we improve on this?
Once we know the result of the find, we can go back and “short-cut” each pointer along the path to point directly to the root of the tree. This only increases the time needed to perform the find by a constant factor, but any subsequent find on this node (or any of its ancestors) will take only O(1) time. The operation of short-cutting the pointers so they all point directly to the root is called path-compression and an example is shown below. Notice that only the pointers along the path to the root are altered. We present a slick recursive version below as well. Trace it to see how it works.
5 4 3 2 1 find(1) find(1) 5 4 3 2 1 Fig. 5: Find using path compression.
The running time of find-compress is still proportional to the depth of node being found, but observe that each time you spend a lot of time in a find, you flatten the search path.
Thus the work you do provides a benefit for later find operations. (This is the sort of thing that we observed in earlier amortized analyses.) Does the savings really amount to anything? The answer is yes, but it is not easy to analyze.
In 1975, Robert Tarjan proved that the amortized running time is strictly more than O(1), Lecture 4 5 Fall 2022 CMSC 420 Dave Mount Find operation (with path compression) Set find-compress(Element x) { if (parent[x] == null) return x // return root else { mySet = find-compress(parent[x]) // find parent[x] = mySet // compress the link return mySet } } it is much less than O(log m).
Analyzing this algorithm is quite tricky. In order to create a bad situation you need to do lots of unions to build up a tree with some real depth. But as soon as you start doing finds on this tree, it very quickly becomes very flat again. In the worst case we need to an immense number of unions to get high costs for the finds.
To give the analysis (which we won’t prove) we introduce two new functions, A(i, j) and α(i). For i, j ≥0, the function A(i, j) is called Ackerman’s function (discovered by Wilhelm Ackerman way back in 1928).
A(i, j) = j + 1 if i = 0, A(i −1, 1) if i > 0 and j = 0, A(i −1, A(i, j −1)) otherwise.
It is famous for being just about the fastest growing function imaginable. It is not obvious from the definition, so here are a few examples as the value of i increases A(0, j) = j + 1 A(1, j) = j + 2 A(2, j) = 2j + 3 A(3, j) = 2j+3 −3 A(4, j) = 22···2 !
−3, where the tower of 2’s in A(4, j) is of height j + 3. As the value of i increases, the function increases to insanely large values very quickly. Even modest values, such as A(4, 5) ≈265,536, which is already much larger than the number of particles in the observable universe.
Ackerman’s function grows unbelievably fast. To create correspondingly slow growing func-tion, we will defined its inverse, which we call α. Define α(m, n) = min{i ≥1 | A(i, ⌊m/n⌋) > lg n}.
This definition is somewhat hard to interpret, but the important bottom line is that assuming ⌊m/n⌋≥1, we have α(m, n) ≤4 as long as m is less than the number of particles in the universe, which is certainly true for any input set your program will ever encounter!
The following result shows that an sequence of union-find operations take amortized time O(α(m, n)). While we cannot formally state that this is constant time, it is constant time for all practical purposes. (The proof is quite complicated, and we will not present the proof.) Lecture 4 6 Fall 2022 CMSC 420 Dave Mount Theorem: After initialization, any sequence of m union’s and find’s (using path compression) on an initial set of n elements can be performed in time O(m · α(m, n)) time. Thus the amortized cost of each union-find operation is O(α(m, n)).
Lecture 4 7 Fall 2022 |
5702 | https://www.youtube.com/watch?v=A1mQ9kD-i9I | Mean, Median, Mode, and Range - How To Find It!
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This central tendency statistics math video tutorial explains how to calculate the mean, median, mode, and range given a data set of odd numbers and even numbers. The average / arithmetic mean is the sum of all numbers divided by the number of terms in a sequence. The median is simply the middle number. In an even data set, the median is the midpoint or average of the two middle numbers. The mode is number with the highest frequency or the number that appears most often. The range is the difference between the highest number and the lowest number. This video contains plenty of examples and practice problems.
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in this video we're going to focus on calculating the mean median mode and range so let's begin let's say if we have the numbers 12 7 14 5 7 11 and 9 so how can we calculate the mean of this data set what would you do to find it to find the mean which is basically the average of the seven numbers you need to take the sum of the seven numbers and divide it by the numbers that you have which is seven and so that's how you can calculate the average or the arithmetic mean so let's go ahead and do that so let's add up 12 + 7 + 14 + 5 + 7 + 11 and + 9 and since there are seven numbers let's divide it by seven so the sum of the seven numbers is 65 65 / 7 is equal to 9. 285 or 286 if you round it so that's the arithmetic mean of those numbers now what about the median how can we calculate the median of this data set what do you think we need to do in order to find it to find a median it's going to help if you arrange numbers in increasing order the lowest number is five and then we have 27s next we have a 9 11 12 and 14 now in order to find the median we need to find the middle number the best way to find a middle number is to eliminate the first and the last number next eliminate the second number and the second to last number and then eliminate the third and the second or the third one from the left the one in the middle is the median so in this example the median is equal to 9 now what about the mode how can we calculate the mode the mode is simply the number that appears most most often or the number with the highest frequency every number appears once except seven seven appears twice so the mode is equal to S now the last thing that we need to calculate is the range so how can we find the range of this data set the range is simply the difference between the highest number and the low lowest number the highest number is 14 the lowest number is 5 14 - 5 is 9 and so 9 is equal to the range now let's work on another example so let's say if we have a data set that contains the numbers six 14 8 5 3 11 and 9 so go ahead pause the video and try this practice problem find a mean median mode and range when you finish just unpause it and see if you have the right answer so let's start with the mean so let's find a sum let's add up the seven numbers 6 + 14 + 8 + 5 + 3 + 11 + 99 adds up to 56 and there are seven numbers so we're going to divide it by 7 56 / 7 is 8 so the average or the arithmetic mean is equal to 8 so now let's go ahead and calculate the median but to to do that let's arrange it in increasing order the lowest number is three the next one is five and then we have 6 8 9 11 and 14 notice that the middle number is eight so therefore the median is equal to 8 now what about the mode what is the value of the mode in this problem which number occurs most frequently notice that every number occurs only once so therefore there's none there is no mode in this particular problem now the last thing that we need to figure out is the range which is the difference between the highest number and the lowest number the highest number is 14 the lowest number is three and so 14 - 3 is equal to 11 and so that's the range so far we've considered two examples in which we had an odd number of numbers in a data set in this case there was only seven numbers but what if we have an even number of numbers for example let's say if there are eight numbers in a data set let's try this example let's say the numbers are six 8 5 5 9 8 10 and 8 so go ahead take a minute calculate the mean median mode and range so let's start with the mean let's add up the numbers 6 + 8 + 5 + 5 + 9 + 8 + 10+ + 8 will give us a sum of 59 and there are eight numbers now 59 / 8 is equal to 7375 and so this is the average now let's calculate the mode now before we do that let's put the numbers in increas in order as we did before so the lowest number is five there's two of them and then we have a six there's three eights there's a nine and there is a 10 so notice that eight is the number that occurs most frequently there's three of them so the mode is eight now the median is going to be the middle number let's see we can get rid of 5 and 10 5 and 9 6 and six in this case the last two in the middle are eight because they're the same the median is going to be eight now the last thing that we need to calculate is the range and we know the range is simply the difference between the highest number and the lowest number so it's 10 minus 5 which is equal to 5 now let's work on one more example this time we're going to have 10 numbers in a data set instead of eight so the numbers are 12 15 21 4 36 15 11 48 29 and 38 so let's begin by by calculating the mean let's add up the 10 numbers so the sum of the 10 numbers is 229 229 / 10 will give us an average of 22.9 so that's the arithmetic mean now let's calculate the median so let's put everything in order the lowest number is four and then we have 11 and then 12 there's 2 15s next we have 21 29 36 and then 38 and 48 now let's calculate the mode 15 is the number that appears most frequently so the mode is going to be 15 now to find the median we need to calculate the middle number what do you think the middle number is going to be well we can eliminate 4 and 48 next we can get rid of 11 and 38 and then 12 and 36 and then 15 and 29 so at this point we're left with two numbers in the middle whenever you encounter in a situation like this what you need to do is take the average of the two numbers add up the two numbers and divide by two this will give you the midpoint between 15 and 21 the middle number is 18 15 + 21 is 36 half of 36 is 18 so that's the median that's the midpoint between 15 and 21 now lastly let's calculate the range the highest number is 48 the lowest number is 40 and that gives us a difference of 44 and so that's it for this video |
5703 | https://math.stackexchange.com/questions/763094/calculating-real-and-imaginary-part-of-a-complex-number | Stack Exchange Network
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Calculating real and imaginary part of a complex number
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Consider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$.
How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these parts?
I thought about reforming them to the form $x + i\cdot y$ which might be possible for a, but what about b?
I just started occupying with complex numbers and don't yet understand the whole context.
algebra-precalculus
complex-numbers
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edited Jun 19, 2014 at 9:26
Tunk-Fey
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asked Apr 21, 2014 at 15:07
muffelmuffel
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4 Answers 4
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Try to understand and prove each step:
$$\begin{align}\bullet&\;\;\frac{1+i}{1-i}=i\implies \frac{(1+i)^5}{(1-i)^3}=\left(\frac{1+i}{1-i}\right)^3(1+i)^2=i^3\cdot2i=(-i)(2i)=2\{}\\bullet&\;\;e^{b-\pi i}=e^be^{-\pi i}=e^b\left(\cos\pi-i\sin\pi\right)=-e^b\end{align}$$
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answered Apr 21, 2014 at 15:14
DonAntonioDonAntonio
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$\begingroup$ Just wondering since you said prove each step does $e^{b-\pi i}=e^be^{-\pi i}$ need to be proved(is there a catch apart from the usual law of exponents) $\endgroup$
Guy
– Guy
2014-04-21 16:11:01 +00:00
Commented Apr 21, 2014 at 16:11
$\begingroup$ Not really in this case as the exponent is definitely determined (of course, $\;b\in\Bbb R\;$ here), so we don't have to hesitate between different arguments... $\endgroup$
DonAntonio
– DonAntonio
2014-04-21 16:13:02 +00:00
Commented Apr 21, 2014 at 16:13
$\begingroup$ thank you for your answer. The first one is nearly clear to me, except for $\frac{1+i}{1-i} = i$, could you please help me with this one? For the 2nd one: $e^{b-\pi i}$ could be (afaik) turned into $\frac{e^b}{e^{\pi i}}$, but how do you get to $e^b e^{-\pi i}$? $\endgroup$
muffel
– muffel
2014-04-21 16:19:15 +00:00
Commented Apr 21, 2014 at 16:19
1
$\begingroup$ @muffel, multiply by the denominator's conjugate, as usual:$$\frac{1+i}{1-i}\frac{1+i}{1+i}=\frac{2i}2=i$$ About the second one: isn't $\;e^{-c}=\frac1{e^c}\;\;?$ . Just the same thing in two different dresses... $\endgroup$
DonAntonio
– DonAntonio
2014-04-21 16:22:28 +00:00
Commented Apr 21, 2014 at 16:22
$\begingroup$ Ah c'mon, you can have $e^{i\pi} = -1$ without writing out the sin and cos :-) $\endgroup$
Steve Jessop
– Steve Jessop
2014-04-21 19:53:28 +00:00
Commented Apr 21, 2014 at 19:53
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A useful way is the trigonometric one:
$a=\frac{(1+i)^5}{(1-i)^3}$.
Observe that \begin{align}1+i=&\frac{2}{\sqrt2}\left(\frac{\sqrt2}{2}1+\frac{\sqrt2}{2}i\right)\ =&\frac{2}{\sqrt2}(\cos(\pi/4)+i\sin(\pi/4))\ =&\frac{2}{\sqrt2}e^{\frac{i\pi}{4}}.\end{align}
Then $$ 1-i=\overline{1+i}=\overline{\frac{2}{\sqrt2}e^{\frac{i\pi}{4}}}=\frac{2}{\sqrt2}e^{\frac{-i\pi}{4}}. $$
Hence \begin{align} a=&\frac{(1+i)^5}{(1-i)^3}\ =&(1+i)^5(1-i)^{-3}\ =&\left(\frac{2}{\sqrt2}e^{\frac{i\pi}{4}} \right)^5\left(\frac{2}{\sqrt2}e^{\frac{-i\pi}{4}}\right)^{-3}\ =&\frac{2^2}{2}e^{2i\pi}=2\;. \end{align}
Finally $b=e^{3-\pi i}=e^3e^{-\pi i}=e^3(\cos(-\pi)+i\sin(-\pi))=-e^3$ hence $\Re b=-e^3$ and $\Im b=0$.
However the approach depends on the case you face.
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answered Apr 21, 2014 at 15:29
JoeJoe
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$\begingroup$ thank you for your answer. Could you please explain to me how you got from $1+i$ to $\frac{2}{\sqrt{2}}\left(\dots\right)$? $\endgroup$
muffel
– muffel
2014-04-21 16:23:12 +00:00
Commented Apr 21, 2014 at 16:23
$\begingroup$ Basically experience: you do a lot of complex numbers exercises in order to "form your eye". $z=1+i$ is a number in which $\Re z=\Im z$ and then I asked myself: "what is the value of $\alpha$ that makes $\cos\alpha=\sin\alpha$" (wlog $\alpha\in[0,2\pi[$)? It's $\alpha=\pi/4$ (even $\alpha=\frac{5\pi}{4}$), but $z$ and $\cos\alpha+i\sin\alpha$ differs by a multiplicative constant, that is $|z|=\sqrt{1^2+1^2}=\sqrt2=\frac{2}{\sqrt2}$. Then I adjusted all by using this constant... and this is all! $\endgroup$
Joe
– Joe
2014-04-21 16:53:21 +00:00
Commented Apr 21, 2014 at 16:53
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Consider any complex number $z=x+iy$, where $x$ and $y$ are the real and imaginary parts, respectively. The complex conjugate of $z$ will be $z^=x-iy$. Note that if we add $z$ and $z^$ together we get $z+z^=2x$, so the real part of $z$ may be written as $\Re(z)=\frac{z+z^}{2}$. This allows to circumvent decomposing a complex number into the form $x+iy$ in order to find the real part. All you need to do is compute the complex conjugate. You can similarly find that the imaginary part of $z$ can be written as $\Im(z)=\frac{z-z^}{2i}$.
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answered Apr 21, 2014 at 15:15
David HDavid H
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HINT : $$ \begin{align} a=\frac{(1+i)^5}{(1-i)^3}&=\frac{(1+i)^3(1+i)^2}{(1-i)^3}\ &=\left(\frac{1+i}{1-i}\right)^3(1+i)^2\ &=\left(\frac{1+i}{1-i}\cdot\frac{1+i}{1+i}\right)^3(1+2i-1)\ &=i^3\cdot2i \end{align} $$ and $$ e^{b-i\pi}=e^be^{-i\pi}=e^b(\cos\pi-i\sin\pi) $$
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answered Apr 21, 2014 at 15:18
Tunk-FeyTunk-Fey
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5704 | http://physics.gmu.edu/~isatija/Phys701/Sakurai.pdf | 153 CHAPTER 3 Theory of Angular Momentum This chapter is concerned with a systematic treatment of angular momen-tum and related topics. The importance of angular momentum in modern physics can hardly be overemphasized. A thorough understanding of angu-lar momentum is essential in molecular, atomic, and nuclear spectroscopy; angular-momentum considerations play an important role in scattering and collision problems as well as in bound-state problems. Furthermore, angu-lar-momentum concepts have important generalizations-isospin in nuclear physics, SU(3), SU(2)® U(l) in particle physics, and so forth. 3.1. ROTATIONS AND ANGULAR MOMENTUM COMMUTATION RELATIONS Finite Versus Infinitesimal Rotations We recall from elementary physics that rotations about the same axis commute, whereas rotations about different axes do not. For instance, a 30° rotation about the z-axis followed by a 60° rotation about the same z-axis is obviously equivalent to a 60° rotation followed by a 30° rotation, both about the same axis. However, let us consider a 90° rotation about the z-axis, denoted by R z( '7T/2), followed by a 90° rotation about the x-axis, denoted by R x< '7T/2); compare this with a 90° rotation about the x-axis 3.1. Rotations and Angular Momentum Commutation Relations followed by a 90° rotation about the z-axis. The net results are different, as we can see from Figure 3.1. Our first basic task is to work out quantitatively the manner in which rotations about different axes fail to commute. To this end, we first recall how to represent rotations in three dimensions by 3 X 3 real, orthogonal matrices. Consider a vector V with components V x ' VY ' and When we rotate, the three components become some other set of numbers, V;, V;, and Vz'. The old and new components are related via a 3 X 3 orthogonal matrix R: V' X Vx V' R Vy I, (3.1.1a) y V' z RRT = RTR =1, (3.1.1b) where the superscript T stands for a transpose of a matrix. It is a property of orthogonal matrices that /2 + V /2 + V /2 vV 2 + V2 y + Vz 2 =IV y (3.1.2) x Vx z is automatically satisfied. z Z z I I I I I /' x x x z Z z I I I I I I I I I I Rz(7T/2) a22 I 7 / ------! z "til' P" x x x FIGURE 3.1. Example to illustrate the noncommutativity of finite rotations. 154 155 Theory of Angular NlOmentUm To be definite, we consider a rotation about the z-axis by angle The convention we follow throughout this book is that a rotation opeptifl": affects a physical system itself, as in Figure 3.1, while the coordinate remain unchanged. The angle is taken to be positive when the rotation in question is counterclockwise in the xy-plane, as viewed from the positi z-side. If we associate a right-handed screw with such a rotation, a rotation around the z-axis means that the screw is advancing positive z-direction. With this convention, we easily verify that cos -sin R z (</»= cos</> ( o Had we adopted a different convention, in which a physical system reo mained fixed but the coordinate axes rotated, this same matrix with a positive would have represented a clockwise rotation of the x- and y-axes, when viewed from the positive z-side. It is obviously important not to mix the two conventions! Some authors distinguish the two approaches by using "active rotations" for physical systems rotated and "passive rotations" for the coordinate axes rotated. We are particularly interested in an infinitesimal form of R z : 2 1--e -e 0 2 Rz(e) = I e 1--e2 0 2 0 0 1 where terms of order e3 and higher are ignored. Likewise, we have 1 0 0 1---e RxCe) = I 0 e 2 2 2 0 e 1--e 2 and e2 1--0 e 2 Ry(e) = I 0 1 0 I, (3.1.5b) 2 -e 0 1--e 2 which may be read from (3.1.4) by cyclic permutations of x, y, z-that is, x y, y z, z x. Compare now the effect of a y-axis rotation followed by an x-axis rotation with that of an x-axis rotation followed by a y-axis II 1{"lations and Angular Momentum Commutation Relations fl1llll iOIl. Elementary matrix manipulations lead to 1--e2 0 e 2 2 (3.1.6a) RxCe)Ry(e) = I 2 e e 1---e 2 -e e 1- e2 2 2 1--e e e 2 2 (3.1.6b) Ry(e)Rx(e) = I e 0 1---e 2 -e e 1- e2 hOIll (3.1.6a) and (3.1.6b) we have the first important result: Infinitesimal IlIlations about different axes do commute if terms of order e2 and higher 11Il' ignored. The second and even more important result concerns the 1I111111lCr in which rotations about different axes fail to commute when terms III' urdcr e2 are kept: R.(,)R,(')- R,(,)R,(') ( o - e o 2 = R z ( e2 )-I, (3.1.7) where all terms of order higher than e2 have been ignored throughout this dnivation. We also have 1 = Rany(O) (3.1.8) wherc any stands for any rotation axis. Thus the final result can be written liS RxCe)Ry(e)-Ry(e)RxCe) . = Rz(e 2 )-Rany(O). (3.1.9) . Ihis is an example of the commutation relations between rotation oper-ations about different axes, which we will use later in deducing the angular-IIHllI1Cntum commutation relations in quantum mechanics. +Actually thcrc is a familiar example of this in elementary mechanics. The angular velocity \"','Ior w that characterizes an infinitesimal change in rotation angle during an infinitesimal IIIIll" interval follows the usual rule of vector addition, including commutativity of vector "ddilion. llowcvcr, we cannot ascribe a vectorial property to ajinite angular change. 156 157 Theory of Angular Momentum Infinitesimal Rotations in Quantum Mechanics So far we have not used quantum-mechanical concepts. The matrix R is just a 3 X 3 orthogonal matrix acting on a vector V written in column matrix form. We must now understand how to characterize rotations in quantum mechanics. Because rotations affect physical systems, the state ket corresponding to a rotated system is expected to look different from the state ket corre-sponding to the original unrotated system. Given a rotation operation R, characterized by a 3 X 3 orthogonal matrix R, we associate an operator in the appropriate ket space such that R) (3.1 where IO:)R and 10:) stand for the kets of the rotated and original system, respectively. Note that the 3 X 3 orthogonal matrix R acts on a column matrix made up of the three components of a classical vector, while the operator PIJ (R) acts on state vectors in ket space. The matrix representation of PIJ(R), which we will study in great detail in the subsequent sections, depends on the dimension1!:!ity N of the For _ N = 2, fo!.,..descrihing a spin t system with no other degrees of freedom, is represented by a 2x2 matrix; for a spin 1 system, the appropriate representation is a 3 X 3 unitary matrix, and so on. To construct the rotation operator it is again fruitful to examine first its properties under an infinitesimal rotation. We can almost guess how we must proceed by analogy. In both translations and time evolution, which we studied in Sections 1.6 and 2.1, respectively, the appropriate infinitesimal operators could be written as U. I-iGe (3.1.11) with a Hermitian operator G. Specifically, E -'> dx' (3.1.12) G-'>Ii' for an infinitesimal translation by a displacement dx' in the x-direction, and H E -'> dt G-'>Ii' for an infinitesimal time evolution with time displacement dt. We know from classical mechanics that angular momentum is the generator of rota-tion in much the same way as momentum and Hamiltonian are the generators of translation and time evolution, respectively. We therefore define the angular momentum operator Jk in such a way that the operator for an infinitesimal rotation around the kth axis by angle dq, can be H. Rotations and Angular Momentum Commutation Relations (lotained by letting Jk G-'> -E-'> Ii ' in (3.1.11). With Jk taken to be Hermitian, the infinitesimal rotation operator is guaranteed to be unitary and reduces to the identity operator in the limit dq, -'> O. More generally, we have PIJ(ii, dq,) = 1- i (h Jon) dq, (3.1.15) for a rotation about the direction characterized by a unit vector it by an infinitesimal angle dq,. We stress that in this book we do not define the angular-momentum operator to be x X p. This is important because spin angular momentum, to which our general formalism also applies, has nothing to do with Xi and IJut in another way, in classical mechanics one can prove that the angular momentum defined to be x X p is the generator of a rotation; in contrast, in 4l1antum mechanics we define J so that the operator for an infinitesimal rotation takes form (3.1.15). A finite rotation can be obtained by compounding successively inlinitesimal rotations about the same axis. For instance, if we are interested in a finite rotation about the z-axis by angle q" we consider I PlJz(q, ) = [ i ( ) ( t) r =1- iJzq, Ii -21i2 + ... (3.1 In order to obtain the angular-momentum commutation relations, we need one more concept. As we remarked earlier, for every rotation R represented by a 3 X 3 orthogonal matrix R there exists a rotation operator !7)(R) in the appropriate ket space. We further postulate that PIJ(R) has the group properties as R: Identity: R·1 R=;oPIJ(R)·l (3.1.17a) Closure: RIR2 = R3 =;0 = PIJ(R]) (3.1.17b) Inverses: RR- 1 l=;oPIJ(R)PlJl(R)=l R-1R =1 =;0 PIJ-1(R)PIJ(R) =1 Associativity: R1 (R 2 R 3 ) = (R)R2)R3 = R)R2 R 3 (3.l.17d) => PIJ(R1)[PIJ(R 2)PIJ(R )] 3 = [PIJ(R)PIJ(R 2 )]PIJ(R ) 3 Oi)( R )(jJ( R )(jJ( R ) Theory of Angular Momentum 158 Let us noW return to the fundamental commutation relations for rotation operations (3.1.9) written in terms of the R matrices. Its rotation operator analogue would read 2 2 iJxe J;e2) ( ifye Jye ) 1 --- 1----( Ii 21i2 , Ii 21i2 y Jy 2 x E I zE __ if'___ 13 ' e2 )(1 -J22)'J =1---1.
2 (3.1.18) non-Ii 21i2 21i2 Ii 2 Terms of order 13 automatically drop out. Equating terms of order 13 on both sides of (3.1.18), we obtain [J , J ] iIiJ,. (3.1.19) x y Repeating this kind of argument with rotations about other axes, we obtain {JpJ ] ilieijkJk' (3.1.20) j known as the fundamental commutation relations of angular momentum. In general, when the generators of infinitesimal transformations do not commute, the corresponding group of operations is said to be Abelian. Because of (3.1.20), the rotation group in three dimensions is non-Abelian. In contrast, the translation group in three dimensions is Abelian because Pi and Pj commute even with i' j.
We emphasize that in obtaining the commutation relations (3.1.20) we have used the following two concepts: 1. J is the generator of rotation about the kth axis. k 2. Rotations about different axes fail to commute. It is no exaggeration to say that commutation relations (3.1,20) summarize in a compact manner all the basic properties of rotations in three dimen-sions. 3.2. SPIN 1- SYSTEMS AND FINITE ROTATIONS Rotation Operator for Spin! The lowest number, N, of dimensions in which the angular-momen-tum commutation relations (3.1.20) are realized, is N = 2. The reader has already checked in Problem 8 of Chapter 1 that the operators defined by Sx=(%I{(I+)(-O+(-I)+(I-)(+I)}expl T = + )( -lei/2 + e- i/21 )( + • ! = [ { ( 1 + ) ( I) + ( 1- ) ( + ) } cos cp + i { ( 1 + ) ( - I) - (1- ) ( + I)} sin cp ] = Sxcos cp - SI,sincp. (3.2.6) Derivation 2: Alternatively we may use formula (2.3.47) to evaluate (3.2.5): ( exp ) ( - iSzcp ) ( icp Sxexp -li- = Sx + h [Sz, SJ iliSv ( 1 )(iCP)2 (1) (iCP)3 + 2! h + 3! h ... ,1i2Sx _ 1i 2Sx ili 3Sy = S[1 cp2 + ... ] S [A,. _ cp3 + ... ] x 2! y '1' 3! = Sxcoscp - S:vsincp. (3.2.7) 160 161 Theory of Angular Momentum (3.2.9) (3.2.10) (3.2.11) (3.2.12) Notice that in derivation 2 we used only the commutation relations for Si' so this method can be generalized to rotations of systems with angular momentum higher than For spin both methods give (Sx) -+ R(aISx\a)R = (Sx)cosq, (Sy)sinCP, where the expectation value without subscripts is understood to be taken with respect to the (old) unrotated system. Similarly, (Sy) -+ (Sy)cosCP + (S,)sinq,. As for the expectation value of Sz' there is no change because Sz commutes with .@.(q,): (S.) -+ (S.). Relations (3.2.8), (3.2.9), and (3.2.10) are quite reasonable. They show that rotation operator (3.2.3), when applied to the state ket, does rotate the expectation value of S around the z-axis by angle q,. In other words, the expectation value of the spin operator behaves as though it were a classical vector under rotation: (Sk) -+ 'ERk,(S,), I where Rkl are the elements of the 3 X 3 orthogonal matrix R that specifies the rotation in question. It should be clear from our derivation 2 that this property is not restricted to the spin operator of spin systems. In general, we have Uk) -+ 'ERk/(J1) , under rotation, where J are the generators of rotations satisfying the k angular-momentum commutation relations (3.1.20). Later we will show that relations of this kind can be further generalized to any vector operator. So far everything has been as expected. But now, be prepared for a surprisel We examine the effect of rotation operator (3.2.3) on a general ket, la) = +)( + la) + - >( -[a), (3.2.13) a little more closely. We see that ex ( - i;zq,) [a) = e-i<P/2+)( + la) + pi<P/2_)( -la). (3.2.14) p The appearance of the half-angle cp/2 here has an extremely interesting consequence.
Let us consider a rotation by 2'17. We then have (3.2.15) \a) ft -+ -la). .1.2. Spin! Systems and Finite Rotations So the ket for the 360 0 rotated state differs from the original ket by a minus sign. We would need a 720 0 (CP = 4'17) rotation to get back to the same ket with a plus sign. Notice that this minus sign disappears for the expectation value of S because S is sandwiched by la) and (ai, both of which change sign. Will this minus sign ever be observable? We will give the answer to this interesting question after we discuss spin precession once again. Spin Precession Revisited We now treat the problem of spin precession, already discussed in Section 2.1, from a new point of view. We recall that the basic Hamiltonian of the problem is given by wSz, (3.2.16) mec where B w::= (3.2.17) mec The time-evolution operator based on this Hamiltonian is given by iHt ) (iSzWI ) 'B! (1,0) = exp exp Ii .
( Comparing this equation with (3.2.3), we see that the time-evolution oper-ator here is precisely the same as the rotation operator in (3.2.3) with cP set equal to wi. In this manner we see immediately why this Hamiltonian causes spin precession. Paraphrasing (3.2.8), (3.2.9), and (3.2.10), we obtain (Sx)/ = (Sv)t=osinwt, (3.2.19a) (Sy)/ = + (Sx),=osinwt, (3.2.19b) (S,), = (3.2.19c) After t 2'17/w, the spin returns to its original direction. This set of equations can be used to discuss the spin precession of a muon, an electronlike particle which, however, is 210 times as heavy. The muon magnetic moment can be determined from other experiments--for example, the hyperfine splitting in muonium, a bound state of a positive muon and an electron-to be ehI2m",c, just as expected from Dirac's relativistic theory of spin! particles. (We will here neglect very small corrections that arise from quantum field theory effects). Knowing the magnetic moment we can predict the angular frequency of precession. So (3.2.19) can be and, in fact, has been checked experimentally. In practice, as the external magnetic field causes spin precession, the spin direction is analyzed by taking advantage of the fact that electrons from muon decay tend to be emitted preferentially in the direction opposite to the muon spin. 162 Theory of Angular Momentum Let us now look at the time evolution of the state ket . Assuming that the initial (t = 0) ket is given by (3.2.13), we obtain timet la, to = 0; t) e- iwt/ 2 1+)< + la) + e+ iwt / 2 1)< -\a). Expression (3.2.20) acquires a minus sign at t 2'11/w, and we must until t 4'11/w to get back to the original state ket with the same sign. sum up, the period for the state ket is twice as long as the period for precession 2'11 'Tprecession W 4'11 'Tstatc ket -W Neutron Interferometry Experiment to Study 2'IT Rotations We now describe an experiment performed to detect the minus sign in (3.2.15). Quite clearly, if every state ket in the universe is multiplied by minus sign, there will be no observable effect. The only way to detect the predicted minus sign is to make a comparison between an unrotated and a rotated state. As in gravity-induced quantum interference, discussed in Section 2.6, we rely on the art of neutron interferometry to verify extraordinary prediction of quantum mechanics. A nearly monoenergetic beam of thermal neutrons is split into parts-path A and path B; see Figure 3.2. Path A always goes through a magnetic-field-free region; in contrast, path B enters a small region where a static magnetic field is present. As a result, the neutron state ket going via. path B suffers a phase change e iwT/2, where T is the time spent in the B =1= 0 region and w is the spin-precession frequency eB gn ( ) w= --, g "" 1.91 mpc n for the neutron with a magnetic moment of gneli/2mpc, as we can see if we Interference region I1TfC.1IRF '.2. Exocriment to study the predicted minus sign under a 2", rotation, 3,2, Spin Systems and Finite Rotations 163 compare this with (3.2.17), which is appropriate for the electron with magnetic moment eli /2m eC' When path A and path B meet again in the InLerference region of Figure 3.2, the amplitude of the neutron arriving via puth B is iwT/2 c2 =cz(B= e (3.2.23) while the amplitude of the neutron arriving via path A is Ct , independent of D. So the intensity observable in the interference region must exhibit a Ninusoidal variation "+ wT ) cos -2-+8 , (3.2.24) ( where Ci is the phase difference between C1 and c2 (B 0). In practice, time spent in the B =1= 0 region, is fixed but the precession frequency w is vllried by changing the strength of the magnetic field. The intensity in the Interference region as a function of B is predicted to have a sinusoidal v"riation. If we call t:.B the difference in B needed to produce successive maxima, we can easily show that t:.B 4'11 lic (3.2.25) egnAI' where I is the path length. In deriving this formula we used the fact that a 4'11 rotation is needed ror the state ket to return to the original ket with the same sign, as required hy our formalism. If, on the other hand, our description of spin t systems were incorrect and the ket were to return to its original ket with the same Nign under a 2'11 rotation, the predicted value for t:.B would be just one-half of (3.2.25). Two different groups have conclusively demonstrated experimentally prediction (3.2.25) is correct to an accuracy of a fraction of a percent. This is another triumph of quantum mechanics. The nontrivial prediction (12.15) has been experimentally established in a direct manner. Pauli Two-Component Formalism Manipulations with the state kets of spin t systems can be conven-carried out using the two-component spinor formalism introduced by W. Pauli in 1926. In Section 1.3 we learned how a ket (bra) can be represented by a column (row) matrix; all we have to do is arrange the "xpansion coefficients in terms of a certain specified set of base kets into a -II. Rauch et aI., Phys. Leu. S4A, 425 (1975); S. A. Werner et aI., Phys. Rev. Lett. 3S (1 1 m), 1053. 164 165 Theory of Angular column (row) matrix. In the spin case we have 10) 1+) == X+ -) == \ 1 == x-(+ 1== (1,0) = xtj-( 1=(0,1) for the base kets and bras and = 1+)( + 10:) + 1-)(-10:) = ( + ) and ( 0:1 ( 0:1 + ) ( + 1+ ( 0:1- ) ( - I== « 0:1 + ), (0:1- ) ) for an arbitrary state ket and the corresponding state bra. Column (3.2.27a) is referred to as a two-component spinor and is written as ( + 10:) ) == (c + ) X ( (10:) c c+ X+ + c_x_, where c+ and c_ are, in general, complex numbers. For xt we have = «0:1+),(0:1-»= The matrix elements (± ISkl+) and (± ISkl-), apart from Ii12, to be set equal to those of 2 x 2 matrices ak' known as the Pauli 01<1....; ..010 We identify We can now write the expectation value (Sk) in terms of X and O'k: (Sk) = (o:ISklo:) = L L (o:la')(a'ISkla")(a"lo:) a'= +, a" +, Ii ) = \ I 2" XtakX, where the usual rule of matrix multiplication is used in the Explicitly, we see from (3.2.1) together with (3.2.30) that ( 0 1) a2 (0 -i) (1 0) alI 0' = i 0' 0'3 = \ 0 1 where the subscripts 1, 2, and 3 refer to x, y, and z, respectively. We record some properties of the Pauli matrices. First, a/ =1 aiO'j+ O'jO'i = 0, for i " j, where the right-hand side of (3.2.33a) is to be understood as the 2 X 2 Spin 1Systems and Finite Rotations tity matrix. These two relations are, of course, equivalent to the I:ommutation relations {ai' O'j} = 2l)ij' (3.2.34) !llso have the commutation relations [ai' O'j] = 2i£ijkak> ( 3.2.35) ell we see to be the explicit 2 x 2 matrix realizations of the angular- tum commutation relations (3.1.20). Combining (3.2.34) and (3.2.35), I:un obtain at0'2 = 0'20' 1 = i 0'3 •••• (3.2.36) 0'/ at, (3.2.37a) det( 0'; ) = 1, (3.2.37b) Tr( O'J O. (3.2.37c) We now consider aoa, where a is a vector in three dimensions. This is IWtuully to be understood as a 2x2 matrix. Thus aoa == LakO'k k + a 3 a 1 ia 2 ). (3.2.38) = ( a + ia -1 2 a 3 'There is also a very important identity, (aoa)( aob) = a·b+ jao(aXb). (3.2.39) prove this all we need are the anticommutation and commutation rdutions, (3.2.34) and (3.2.35), respectively: LO'jajLakbk = L L (-2 1 {O'j' O'd + } [O'j' ak1) ajbk j k j k = L L (l)jk + i£jkla,) ajbk j k = aob+ jao(axb). (3.2.40) If the components of a are real, we have (a·a)2=laI 2, (3.2.41) where lal is the magnitude of the vector a. Rotations in the Two-Component Formalism Let us now study the 2 x 2 matrix representation of the rotation operator §J(n, ef». We have ( -is.nef>)..!.. (-la'fief» exp Ii - exp 2 . (3.2.42) 166 Theory of Angular Momen", Using {I for n even, (ooit) n = ooit for n odd, which follows from (3.2.41), we can write exp( Hl- (a/ (H.. j ( + ... ) -i = lCOS( ) - iooitSin( ). Explicitly, in 2 X 2 form we have </» . . (</» ( in x n y) sin( i ) \ -"2 -mzsm "2 io 0 (- in", + ny)sin( i) )+inzsin(i) J Just as the operator exp( - iSoit</>/h) acts on a state ket matrix exp( io o it/2) acts on a two-component spinor x· Under rotations we change X as follows: 0 x - exp( to it ) 2 x· On the other hand, the (fk'S themselves are to remain unchanged rotations. So strictly speaking, despite its appearance, 0 is regarded as a vector; rather, it is XtOX which obeys the transformation property of a vector: Xt(fkX -L,RktXt(ftX' t An explicit proof of this may be given using = (flCOS-and so on, which is the 2 X 2 matrix analogue of (3.2.6). i(ft) (3.2 10:)' the 2x2 not to be II Spin Systems and Finite Rotations 167 In discussing a 2'IT rotation using the ket formalism, we have seen a spin t ket 10:) goes into - Ia). The 2 X 2 analogue of this statement is 0 - io it ) I exp( 2 1, for any it, (3.2.49) which is evident from (3.2.44). As an instructive application of rotation matrix (3.2.45), let us see how we can construct an eigenspinor of ooit with eigenvalue +1, where it is unit vector in some specified direction. Our object is to construct X .ul.isfying ooitx=x· (3.2.50) In other words, we look for the two-component column matrix representa-lion of IS· it; + ) defined by S°itISoo; +) ISoo; +). (3.2.51) this can be solved as a straightforward eigenvalue problem (see f'rohlem 9 in Chapter 1), but here we present an alternative method based 1)11 rotation matrix (3.2.45). Let the polar and the azimuthal angles that characterize 0 be /3 and n, respectively. We start with the two-component spinor that TP.l"1.TP<p..nt Ihe spin-up state. Given this, we first rotate about the y-axis by angle /3; we Hubsequently rotate by angle 0: about the z-axis. We see that the desired W I r. , Second rotation '\ , , / / / I First I rotation I I I I I FIGURE 3.3. Construction of 0"0 eigenspinor. 68 Theory of Angular Momentum pin state is then obtained; see Figure 3.3. In the Pauli spinor language this equence of operations is equivalent to applying exp( - ia2 f3/2) to (I ollowed by an application of exp( - ia3a/2). The net result is ( [cos( I)- ia3sin( I) J[cos( - ia2sin( 6) cos( I)- j sin( i) a O. a ) ( '"'( ) 'in( 1m ( o COs( 2 ) + 1 ) sin( ) cos( ) cos( -iaI 2) = . (f3). ' (3.2.52) ( SIn -e""/2 2 in complete agreement with Problem 9 of Chapter 1 if we realize that a phase common to both the upper and lower components is devoid of physical significance. 3.3. SO(3), SU(2), AND EULER ROTATIONS Orthogonal Group We will now study a little more systematically the group properties of the operations with which we have been concerned in the previous two sections. The most elementary approach to rotations is based on specifying the axis of rotation and the angle of rotation. It is clear that we need three real numbers to characterize a general rotation: the polar and the azimuthal angles of the unit vector" taken in the direction of the rotation axis and the rotation angle cp itself. Equivalently, the same rotation can be specified by the three Cartesian components of the vector "cp. However, these ways of characterizing rotation are not so convenient from the point of view of studying the group properties of rotations. For one thing, unless cp is infinitesimal or " is always in the same direction, we cannot add vectors of the form "cp to characterize a succession of rotations. It is much easier to work with a 3 X 3 orthogonal matrix R because the effect of successive rotations can be obtained just by multiplying the appropriate orthogonal matrices. How many independent parameters are there in a 3 X 3 orthogonal matrix? A real 3 X 3 matrix has 9 entries, but we have the orthogonality constraint RRT =1. (3.3.1) 3.3 SO(3), SU(2), and Euler Rotations 169 which corresponds to 6 independent equations because the product RRT, being the same as RTR, is a symmetrical matrix with 6 independent entries. As a result, there are 3 (that is, 9 6) independent in R, the same rlUmber previously obtained by a more elementary method. The set of all multiplication operations with orthogonal matrices forms a group. By this we mean that the following four requirements are satisfied: 1. The product of any two orthogonal matrices is another orthogonal matrix, which is satisfied because T T T ( RIR2 )( RIR2 ) = RIR2R2Rl = 1. (3.3.2) 2. The associative law holds: R1(R 2R 3 ) = (R 1R 2 )R · 3 (3.3.3) 3. The identity matrix I-physically corresponding to no rotation --defined by Rl lR = R (3.3.4) is a member of the class of all orthogonal matrices. 4. The inverse matrix corresponding to rotation in the opposite sense-defined by RR 1 RIR=l (3.3.5) is also a member. This group has the name SO(3), where S stands for special, 0 stands for orthogonal, 3 for three dimensions. Note only rotational operations are considered here, hence we have SO(3) rather than 0(3) (which can include the inversion operation of Chapter 4 later). Unitary unimodular group In the previous section we learned yet another way to characterize an arbitrary rotation-that is, to look at the 2 X 2 matrix: (3.2.45) that acts on the two-component spinor x. Clearly, (3.2.45) is unitary. As a result, for the ('. and c_, defined in (3.2.28), Ic+12+ 1 2 =1 (3.3.6) IS left invariant. Furthermore, matrix (3.2.45) is unimodular; that is, its determinant is 1, as will be shown explicitly below. We can write the most general unitary unimodular matrix: as a U(a, b) = ( b b), (3.3.7) \ - a where a and bare complex numbers satisfying the unimodular condition 2 2 ( ) + Ihl 1. 3.3.8 170 171 We can easily establish the unitary property of (3.3.7) as follows: U(a,b)tU(a,b) (a -b)( a b) 1, b a - b a where we have used (3.3.8). Notice that the number of independent real parameters in (3.3.7) is again three. We can readily see that the 2 X 2 matrix (3.2,45) that characterizes a rotation of a spin ! system can be written as U(a, b). Comparing (3.2,45) with (3.3.7), we identify Re(a) cos( t), Im(a) = nzSin(t), Re(b)= nySin(t), Im(b) nxsin(%), from which the unimodular property of (3.3.8) is immediate. Conversely, it is clear that the most general unitary unimodular matrix of form (3.3.7) can be interpreted as representing a rotation. The two complex numbers a and b are known as Cayley-Klein parameters. Historically the connection between a unitary unimodular ma-trix and a rotation was known long before the birth of quantum mechanics. In fact, the Cayley-Klein parameters were used to characterize complicated motions of gyroscopes in rigid-body kinematics. Without appealing to the interpretations of unitary unimodular matrices in terms of rotations, we can directly check the group properties of multiplication operations with unitary unimodular matrices. Note in par-ticular that U(a 1 ,b1 )U(a2,b2) U(ala2-blbi,alb2+a where the unimodular condition for the product matrix is lala2 - b1bi\2 + \a1b2+ aibl\2 = l. For the inverse of U we have U-1(a,b)=U(a, b). This group is known as SU(2), where S stands for special, U for unitary, and 2 for dimensionality 2. In contrast, the group defined by multiplication operations with general 2 X 2 unitary matrices (not necessarily constrained to be unimodular) is known as U(2). The most general unitary matrix in two dimensions has four independent parameters and can be written as (with '( real) times a unitary unimodular matrix: U eiY ( a b), \aI 2 +lbI2 =1, '( '(. (3.3.14) - b a SU(2) is called a subgroup of U(2). 3.3 SO(3), SU(2), and Euler Rotations Because we can characterize rotations using both the SO(3) language and the SU(2) language, we may be tempted to conclude that the groups 80(3) and SU(2) are isomorphic-that is, that there is a one-to-one cor-respondence between an element of SO(3) and an element of SU(2). This inference is not correct. Consider a rotation by 21T and another one by 41T. In the SO(3) language, the matrices representing a 21T rotation and a 41T rotation are both 3 x 3 identity matrices; however, in the SU(2) language the corresponding matrices are - 1 times the 2 x 2 identity matrix and the identity matrix itself, respectively. More generally, U(a,b) and U( a, - b) both correspond to a single 3 x 3 matrix in the SO(3) language. The cor-rcspondence therefore is two-to-one; for a given R, the corresponding U is double-valued. One can say, however, that the two groups are locally isomorphic. Euler Rotations From classical mechanics the reader may be familiar with the fact that an arbitrary rotation of a rigid body can be accomplished in three steps, known as Euler rotations. The Euler rotation language, specified by three Euler angles, provides yet another way to characterize the most general rotation in three dimensions. The three steps of Euler rotations are as follows. First, rotate the body counterclockwise (as seen from the positive z-side) about the z-axis by angle a. Imagine now that there is a body y-axis embedded, so to speak, in the rigid body such that before the z-axis rotation is carried out, the body coincides with the usual y-axis, referred to as the space-fixed .v-axis. Obviously, after the rotation about the z-axis, the body y-axis no longer coincides with the space-fixed let us call the former the v'-axis. To see how all this may appear for a thin disk, refer to Figure 3,4a. We now perform a second rotation, this time about the y'-axis by angle /3. As a result, the body z-axis no longer points in the space-fixed z-axis direction. We call the body-fixed z-axis after the second rotation the z'-axis; see Figure 3,4b. The third and final rotation is about the z'-axis by angle '(. The body y-axis now becomes the y"-axis of Figure 3.4c. In terms of 3 X 3 orthogonal matrices the product of the three operations can be written as R(a,/3,'() RA'()R (a). (3.3.15) A cautionary remark is in order here. Most textbooks in classical mechanics prefer to perform the second rotation (the middle rotation) about the body x-axis rather than about the body y-axis for example, Ooldstein 1980). This convention is to be avoided in quantum mechanics for n reason that will become apparent in a moment. In (3.3.15) there appear R y' and R z', which are matrices for rotations IIhout body axes. This approach 'to Euler rotations is rather inconvenient in 173 172 Theory of Angular Momentull\i z y' y x (a) z' z' z y" y' y' y y (c) (b) FIGURE 3.4. Euler rotations. quantum mechanics because we earlier obtained simple expressions for space-fixed (unprimed) axis components of the S operator, but not for body-axis components. It is therefore desirable to express the rotations we considered in terms of space-fixed axis rotations. Fortunatel), there is a very simple relation, namely, Ry'( /3) = Rz ( a)R y( /3)R; The meaning of the right-hand side is as follows. First bring the body. y-axis of Figure 3.4a (that is, the y'-axis) back to the original fixed-y-direction by rotating clockwise (as seen from the positive z-side) about z-axis by angle a; then rotate about the y-axis by angle /3. Finally, the body y-axis to the direction of the by rotating about fixed-space z-axis (not about the z'-axis!) by angle a. Equation (3.3.16) us that the net effect of these rotations is a single rotation about the y' by angle /3. .U SO(3), SU(2), and Euler Rotations To prove this assertion, let us look more closely at the effect of both Hides of (3.3.16) on the circular disc of Figure 3.4a. Clearly, the orientation of the body y-axis is unchanged in both cases, namely, in the y'-direction. Furthermore, the orientation of the final body z-axis is the same whether we upply R,.,(/3) or In both cases the final body z-axis makes a polar angle /3 with the fixed z-axis (the same as the initial z-axis), lind its azimuthal angle, as measured in the fixed-coordinate system, is just II. In other words, the final body z-axis is the same as the z'-axis of Figure JAb. Similarly, we can prove Rz'( y) = Ry,(/3)Rz ( y )R/(f3). (3.3.17) Using (3.3.16) and (3.3.17), we can now rewrite (3.3.15). We obtain RAY )Ry ,(/3)Rz (a) = Ry.(/3)R z ( y )R/(/3)Ry,(/3)RJa) = Rz(a)Ry(/3) l(a)Rz(y)Rz(a) = Rz(a)R y(f3)R ( y), (3.3.18) z where in the final step we used the fact that Rz(Y) and Rz(a) commute. To /ill mmarize, R(a,/3,y) R (a)R y(/3)R (y), (3.3 z z where all three matrices on the right-hand side refer to fixed-axis rotations. Now let us apply this set of operations to spin t systems in quantum mechanics. Corresponding to the product of orthogonal matrices in (3.3.19) Ihere exists a product of rotation operators in the ket space of the spin t system under consideration: !!iJ( a, /3, y) = !!iJz( a )!!iJy(/3 )!!iJz( y). (3.3.20) The 2 x 2 matrix representation of this product is - io a ) I - io /3) I -) exp( -2-3 - 2 2 (e;/2 o )( cos(/3/2) -sin (/3/2»)( e- iy/ 2 ) e ia / 2 sin(/3/2) cos(/3/2) 0 = ( ei(CI+Yl/2cos(/3/2) - e- i(a- Yl/2sin(/3/2) ) (3.3.21) e'(CI-yl/2sin( /3/2) ei(,,+Yl/2cos( /3/2) , where (3.2.44) was used. This matrix is clearly of the unitary unimodular form. Conversely, the most general 2 X 2 unitary unimodular matrix can be written in this Euler angle form. Notice that the matrix elements of the second (middle) rotation exp( - io,.</>/2) are purely real. This would not have been the case had we dlosen to rotate about the x-axis rather than the y-axis, as done in most 174 175 Theory of Angular Momentum textbooks in classical mechanics. In quantum mechanics it pays to stick our convention because we prefer the matrix elements of the rotation, which is the only rotation matrix containing off-diagonal elemem:SJ; to be purely real. >I< The 2 X 2 matrix in (3.3.21) is called the j = } irreducible represent tion of the rotation operator §1( a, fJ, y). Its matrix elements are denoted fJ, y). In terms of the angular-momentum operators we have 1 \ ( iJ,a) §1:"1,';;) (ex, fJ, y ) = j = 2' m' exp ----,:-( ( - iJyfJ) ( ilzy )\ ' 1 \ xexp -"- exp -h- J= 2,m r In Section 3.5 we will extensively study higher j-analogues of (3.3.21). 3.4. DENSITY OPERATORS AND PURE VERSUS MIXED ENSEMBLES Polarized Versus Unpolarized Beams The formalism of quantum mechanics developed so far makes tical predictions on an ensemble, that a collection, of identically physical systems. More precisely, in such an ensemble all members supposed to be characterized by the same state ket \a). A good example this is a beam of silver atoms coming out of an SG filtering apparat Every atom in the beam has its spin pointing in the same direction, the direction determined by the inhomogeneity of the magnetic field of filtering apparatus. We have not yet discussed how to describe mechanically an ensemble of physical systems for which some, say 60%, characterized by \a), and the remaining 40% are characterized by other ket \fJ)· To illustrate vividly the incompleteness of the formalism developed so far, let us consider silver atoms coming directly out of a hot oven, yet to be subjected to a filtering apparatus of the Stern-Gerlach type. On sym try grounds we expect that such atoms have random spin orientations; other words, there should be no preferred direction associated with such an ensemble of atoms. According to the formalism developed so far, the most general state ket of a spin t system is given by \a) c++)+c-), on our convention that the matrix clements of S" This, of course, lh, T l ,IT" taken to be imaginary, 1.4, Density Operators and Pure Versus Mixed Ensembles Is this equation capable of describing a collection of atoms with random Kpin orientations? The answer is clearly no; (3.4.1) characterizes a state ket whose spin is pointing in some definite direction, namely, in the direction of h. whose polar and azimuthal angles, fJ and a, respectively, are obtained by lIolving c+ cos( fJ/2) (3.4.2) = e i "'sin(fJ/2) ; Kee (3.2.52). To cope with a situation of this kind we introduce the concept of fractional population, or probability weight. An ensemble of silver atoms with completely random spin orientation can be viewed as a collection of Hilver atoms in which 50% of the members of the ensemble are characterized hy 1+) and the remaining 50% by 1-). We specify such an ensemble by ussigning w+=O.5, w 0.5, (3.4.3) where w+ and w _ are the fractional population for spin-up and -down, respectively. Because there is no preferred direction for such a beam, it is reasonable to expect that this same ensemble can be regarded equally well liS a 50-50 mixture of ISx; +) and ISx; ). The mathematical formalism needed to accomplish this will appear shortly. It is very important to note that we are simply introducing here two real numbers w+ and w . There is no information on the relative phase hctween the and the spin-down keto Quite often we refer to such a situation as an incoherent mixture of spin-up and spin-down states. What we are doing here is to be clearly distinguished from what we did with a coherent linear superposition, for example, (3.4.4) )1+)+( )1-), where the phase relation between 1+) and 1 ) contains vital information on the spin orientation in the xy-plane, in this case in the positive x-direc-tion. In general, we should not confuse w+ and w with Ic+12 and 12. The probability concept associated with w+ and w_ is much closer to that encountered in classical probability theory. The situation encountered in dealing with silver atoms directly from the hot oven may be compared with that of a graduating class in which 50% of the graduating seniors are male, the remaining 50% female. When we pick a student at random, the probabil-ity that the particular student is male (or female) is 0.5. Whoever heard of a student referred to as a coherent linear superposition of male and female with a particular phase relation? The beam of silver atoms coming directly out of the oven is an example of a completely random ensemble; the beam is said to be un-176 177 Theory of Angular Momentum polarized because there is no preferred direction for spin orientation. In contrast, the beam that has gone through a selective Stern-Gerlach-type measurement is an example of a pure ensemble; the beam is said to be polarized because all members of the ensemble are characterized by a single common ket that describes a state with spin pointing in some definite direction. To appreciate the difference between a completely random ensem-ble and a pure ensemble, let us consider a rotatable SG apparatus where we can vary the direction of the inhomogeneous B just by rotating the appara-tus. When a completely unpolarized beam directly out of the oven is subjected to such an apparatus, we always obtain two emerging beams of equal intensity no matter what the orientation of the apparatus may be. In contrast, if a polarized beam is subjected to such an apparatus, the relative intensities of the two emerging beams vary as the apparatus is rotated. For some particular orientation the ratio of the intensities actually becomes one to zero. In fact, the formalism we developed in Chapter 1 tells us that the relative intensities are simply COS2(/3/2) and sin2(/3/2), where /3 is the angle between the spin direction of the atoms and the direction of the inhomoge-neous magnetic field in the SG apparatus. A complete random ensemble and a pure ensemble can be regarded as the extremes of what is known as a mixed ensemble. In a mixed ensemble a certain fraction-for example, 70%-of the members are characterized by a state ket la), the remaining 30% by 1/3). In such a case the beam is said to be partially polarized. Here Ia) and 1/3) need not even be orthogonal; we can, for example, have 70% with spin in the positive x-direction and 30% with spin in the negative z-direction. Ensemble Averages and Density Operator We now present the density operator formalism, pioneered by J. von Neumann in 1927, that quantitatively describes physical situations with mixed as well as pure ensembles. Our general discussion here is not restricted to spin systems, but for illustrative purposes we return re-peatedly to spin systems.
A pure ensemble by definition is a collection of physical systems such that every member is characterized by the same ket la). In contrast, in a mixed ensemble, a fraction of the members with relative population WI are characterized by la(1», some other fraction with relative population w2 ' by la(2), and so on. Roughly speaking, a mixed ensemble can be viewed as a mixture of pure ensembles, just as the name suggests. The fractional 'In the literature what we call pure and mixed ensembles are often referred to as pure and mixed states. In this book. however, we use state to mean a physical system described by a detlnite state ket la). 1.4. Density Operators and Pure Versus Mixed Ensembles populations are constrained to satisfy the normalization condition LWi =1. (3.4.5) As we mentioned previously, la(1» and la(2) need not be orthogonal. Furthermore, the number of terms in the i sum of (3.4.5) need not coincide with the dimensionality N of the ket space; it can easily exceed N. For cxample, for spin systems with N = 2, we may consider 40% with spin in the positive z-direction, 30% with spin in the positive x-direction, and the remaining 30% with spin in the negative y-direction. Suppose we make a measurement on a mixed ensemble of some observable A. We may ask what is the average measured value of A when a large number of measurements are carried out. The answer is given by the ensemble average of A, which is defined by [A] == Lw;(a(i)IAla(i» = L L w il<a'la(i»1 2 a', (3.4.6) i a' where la') is an eigenket of A. Recall that <a(i)IAla(I» is the usual quantum mechanical expectation value of A taken with respect to state la(i». Equation (3.4.6) tells us that these expectation values must further be weighted by the corresponding fractional populations WI' Notice how prob-abilistic concepts enter twice; first in 1< a 'Ia(i» 12 for the quantum-mechani-cal probability for state la(i» to be found in an A eigenstate la'); second, in the probability factor Wi for finding in the ensemble a quantum-mechani-cal state characterized by la(i). We can now rewrite ensemble average (3.4.6) using a more general hasis, {I b') }: [A] = L Wi L L < a(i)lb')<b'IAIb")<b"la('» b' b" = L L ( L w;( b"la(il)<a(illb'))< b'IAlb"). (3.4.7) b' b" I The number of terms in the sum of the b' (b") is just the dimensionality of the ket space, while the number of terms in the sum of the i depends on how the mixed ensemble is viewed as a mixture of pure ensembles. Notice that in this form the basic property of the ensemble which does not depend on the particular observable A is factored out. This motivates us to define the density operator p as follows: p==Lwila(i)<a(i)I. (3.4.8) 'Quite often in the literature the ensemble average is also called the expectation value. II"wever, in this book, the term expectation value is reserved for the average measured value wilen measurements are carried on a pure ensemble. 178 179 Theory of Angular Momentum The elements of the corresponding density matrix have the following form: = Lwi(b"la(il)(aO)lb'). (3.4.9) The density operator contains all the physically significant information we can possibly obtain about the ensemble in question. Returning to (3.4.7), we see that the ensemble average can be written as L L (b"lplb')(b'IAlb") II 1/' = tr(pA). (3.4.10) Because the trace is independent of representations, tr(pA) can be evaluated using any convenient basis. As a result, (3.4.10) is an extremely powerful relation. There are two properties of the density operator worth recording. First, the density operator is Hermitian, as is evident from (3.4.8). Second, the density operator satisfies the normalization condition tr( p) = L L wi ( b'la(i)(a(i)lb') b' = L w ( a(i)la(i) i 1. (3.4.11) Because of the Hermiticity and the normalization condition, for spini systems with dimensionality 2 the density operator, or the corresponding density matrix, is characterized by three independent real parameters. Four real numbers characterize a 2 X 2 Hermitian matrix. However, only three are independent because of the normalization condition. The three numbers needed are [Sx]' [Sv]' and [Sz]; the reader may verify that knowledge of these three ensembie averages is sufficient to reconstruct the density oper-ator. The manner in which a mixed ensemble is formed can be rather involved. We may mix pure ensembles characterized by all kinds of with appropriate w/s; yet for spin systems three real numbers completely characterize the ensemble in question. This strongly suggests that a mixed ensemble can be decomposed into pure ensembles in many different ways. A problem to illustrate this point appears at the end of this chapter. A pure ensemble is specified by Wi = 1 for some la(l) -with i n, for instance-and Wi = 0 for all other conceivable state kets, so the corre-sponding density operator is written as p = la(I1)(a(I1)1 (3.4.12) with no summation. Clearly, the density operator for a pure ensemble is idempotent, that p (3.4.1 14. Density Operators and Pure Versus Mixed Ensembles or, equivalently, p(p-1) O. (3.4.14) Thus, for a pure ensemble only, we have tr{p2) = 1. (3.4.15) III addition to (3.4.11). The eigenvalues of the density operator for a pure ellsemble are zero or one, as can be seen by inserting a complete set of base kcts that diagonalize the Hermitian operator p between p and (p -1) of (.\,4.14). When diagonalized, the density matrix for a pure ensemble must Ihcrefore look like 0 o 0 0 I (diagonal form) 0 0 0 0 o (3.4.16) II can be shown that tr(p2) is maximal when the ensemble is pure; for a mixed ensemble tr(p2) is a positive number less than one. Given a density operator, let us see how we can construct the corresponding density matrix in some specified basis. To this end we first recall that = L Llb')(b'la)(alb")(b''J. (3.4.17) h' h" this shows that we can form the square matrix corresponding to la(l) combining, in the sense of outer product, the column matrix formed /I'la(i) with the row matrix formed by (a(i)\b"), which, of course, is equal to (b"la(i). The final step is to sum such square matrices with weighting factors Wi' as indicated in (3.4.8). The final form agrees with (3.4.9), as expected. It is instructive to study several examples, all referring to spin systems. '(xample 1. A completely polarized beam with Sz + . = (1 0) (3.4.18) o O· 180 181 Theory of Angular Momentum Example 2. A completely polarized beam with Sx ± . p )«+I±<-O + -2 1:.] (3.4.19) 1 .
== ( +--2 2 The ensembles of Examples 1 and 2 are both pure. Example 3. An unpolarized beam. This can be regarded as an incoherent mixture of a spin-up ensemble and a spin-down ensemble with equal weights (50% each): p '·;;II+'>{+ I I ) (3.4.20) == o 0 !' 2 which is just the identity matrix divided by 2. As we remarked earlier, the same ensemble can also be regarded as an incoherent mixture of an Sx + ensemble and an Sx ensemble with equal weights. It is gratifying that our formalism automatically satisfies the expectation (3.4.21) (1 O0)=1(1 1 1 1 1)+1( 2_1 1 -t), 2 2 2 2 2 where we see from Example 2 that the two terms on the right-hand side are the density matrices for pure ensemble with Sx + and S< . Because p in this case is just the identity operator divided by 2 (the dimensionality), we have (3.4.22) tr(pSx) = tr(pSy} = tr(pSz) = 0, where we used the fact that SI< is traceless. Thus for the ensemble average S we have (3.4.23) [S] = o. This is reasonable because there should be no preferred spin direction in a completely random ensemble of spin! systems. Example 4. As an example of a partially polarized beam, let us consider a 75-25 mixture of two pure ensembles, one with Sz + and the other with Sx +: (3.4.24) w{Sx +) = 0.25.
w(Sz +) 0.75, 3.4. Density Operators and Pure Versus Mixed Ensembles The corresponding p can be represented by • P=4 0 3 (1 ! n (3.4.25) (f n, from which follows n [Sz] = 3n [Sx1 = 8' [Sy] 0, 8 . (3.4.26) We leave as an exercise for the reader the task of showing that this ensemble can be decomposed in ways other than (3.4.24). Time Evolution of Ensembles How does the density operator p change as a function of time? Let us suppose that at some time to the density operator is given by w;la(i) > < aU)I. (3.4.27) If the ensemble is to be left undisturbed, we cannot change the fractional population Wi' So the change in p is governed solely by the time evolution of state ket la U»: laU» at (3.4.28) From the fact that la(i), to; t> satisfies the Schrodinger equation we obtain in ap = "w.(Bla(i) t . t>= O. (3.5.17) Stated another way, the eigenvalue of b cannot be increased beyond bmax • Now (3.5.17) also implies I 1+ la, bmax >= O. (3.5.18) 190 191 Theory of Angular Momentum But I _ I + = I} + I/ - i ( Iy Ix - IJy) = J2 - I z 2 - hlz • (3.5.19) So (J 2 - I/ - hlz )Ia, bmax ) = O. (3.5.20) Because la, b ) itself is not a null ket, this relationship is possible only if max a - b!ax - bmaxh = 0 (3.5.21) or (3.5.22) a = bmax(bmax + h). In a similar manner we argue from (3.5.13) that there must also exist a bmin such that I la, bmin ) = O. (3.5.23) By writing I + I_as I + I _ = J2 - I/ + hlz (3.5.24) in analogy with (3.5.19), we conclude that (3.5.25) a = benin(benin - h). By comparing (3.5.22) with (3.5.25) we infer that bmax = - benin' with bmax positive, and that the allowed values of b lie within - b bmax · bmax Clearly, we must be able to reach la, bmax ) by applying I + successively to la, bmin ) a finite number of times. We must therefore have = boon + nh, bmax where n is some integer. As a result, we get nh bmax = 2' It is more conventional to work with ), defined to be bmaxl h, instead of with bmax so that . n ]=2' The maximum value of the Iz eigenvalue is )h, where) is either an integer or a half-integer. Equation (3.5.22) implies that the eigenvalue of J2 is given by a = h 2)(j +1). .S. Eigenvalues and Eigenstates of Angular Momentum Let us also define m so that b == mh. (3.5.32) Ir ) is an integer, all m values are integers; if ) is a half-integer, all m values are half-integers. The allowed m-values for a given) are m= -),-)+1, ... ,)-1,). (3.5.33) 2) +1 states Instead of la, b) it is more convenient to denote a simultaneous cigenket of J2 and I by I), m). The basic eigenvalue equations now read z J 21), m) =)(j +1)h 21), m) (3.5.34a) lind Izl), m) = mhl), m), (3.5.34b) with) either an integer or a half-integer and m given by (3.5.33). It is very important to recall here that we have used only the commutation relations (J.J.20) to obtain these results. The quantization of angular momentum, manifested in (3.5.34), is a direct consequence of the angular-momentum rommutation relations, which, in turn, follow from the properties of rota-I ions together with the definition of Ik as the generator of rotation. Matrix Elements of Angular-Momentum Operators Let us work out the matrix elements of the various angular-momen-tllm operators. Assuming I), m) to be normalized, we obviously have from (U.34) (i', m 'IJ 21), m) =)(j + 1)h 2l)}'/>m'm (3.5.35a) lind (i', m'jIziJ, m) = mfzl)j'jl)m'm' (3.5.35b) To obtain the matrix elements of I ±' we first consider 2 (i, mill I + I), m) = (i, ml(J 2 - Iz - hIz)I), m) =h 2 [)(j+1)-m 2 -m]. (3.5.36) Now I+I),m) must be the same as 1),m+1) (normalized) up to a 11I1IItiplicative constant [see (3.5.12)]. Thus I+I),m)=c/;"I),m+1). (3.5.37) ( 'omparison with (3.5.36) leads to Icj:'I2 = h 2 (j(j +1)- m(m +1)] = h 2(j - m)(j + m +1). (3.5.38) 2 Theory of Angular MomenU ... necessarily ) we have determined c/ up to an arbitrary phase factor. It is \;U:slUlI1i:l.rJ m I choose c/;" to be real and positive by convention. So 1 + Ij, m) =V(j- m)(j + m +1) lilj, m + I). imilarIy, we can derive 1 _I i. m) =V(j + m)(j - m +1) lili, m -I). in ally, we determine the matrix elements of 1 ± to be (i', m'll± Ii, m) = V(j"+ m)(j ± m +1) 1i8j'j8m',m ±v Representations of the Rotation Operator Having obtained the matrix elements of lz and 1 ±' we are now in osition to study the matrix elements of the rotation operator §J(R). If )tation R is specified by iI and , we can define its matrix elements by rM (j) (R) _ (' 'I ( iJ· fl. ) I' ) ;;Um'm -j, m exp Ii j, m . 'hese matrix elements are sometimes called Wigner functions after E. Vigner, who made pioneering contributions to the group-theoretical nT'i'lnP....' .es of rotations in quantum mechanics. Notice here that the same ppears in the ket and bra of (3.5.42); we need not consider matrix f §J( R) between states with different j-values because they all :ivially. This is because £d(R)Ij, m) is still an eigenket of J2 with the igenvalue j(j + I)1i 2: J 2£d(R)\J, m) = £d(R)J 2Ij, n:) j(j + I)1i 2 [§J(R)Ij, m)], rhich follows directly from the fact that J 2 commutes with lk (hence with ny function of lk)' Simply stated, rotations cannot change the j-valueJ rhich is an eminently sensible result. Often in the literature the (2j +1) x (2j +1) matrix formed by is referred to as the (2j + I)-dimensional irreducible representation f the rotation operator £d(R).. This means that the matrix which corre-ponds to an arbitrary rotation operator in ket space not haracterized by a single j-value can, with a suitable choice of basis, be 3.5, Eigenvalues and Eigenstates of Angular Momentum 193 brought to block-diagonal form: o o o 0 (3.5.44) o 0 .0 "-o 0 o " where each shaded square is a (2j +1) x (2j + 1) square matrix formed by with some definite value of j. Furthermore, each square matrix itself cannot be broken into smaller blocks k ......-- 2j+1 ---10-k {B, 2j+.'-k , t 2jt1 2i+1-k{.
! (3.5.45) with any choice of basis. The rotation matrices characterized by definite j form a group. First, the identity is a member because the rotation matrix corresponding to no Malion (=0) is the (2j+l)X(2j+I) identity matrix. Second, the Inverse is also a member; we simply reverse the rotation angle ( __ </» without changing the rotation axis n. Third, the product of any two 194 195 Theory of Angular members is also a member; explicitly we have = tn' where the product RIR2 represents a single rotation, We also note that rotation matrix is unitary because the corresponding rotation unitary; explicitly we have To appreciate the physical significance of the rotation matrix start with a state represented by Ij, m). We now rotate it: m) m). Even though this rotation operation does not change }. we generally 0 states with m-values other than the original m. To find the amplitude being found in I}, m'), we simply expand the rotated state as follows: EP(R)I}, m) LI}, m')(j, m'IE&(R)I}, m) tn' LI}, tn' where, in using the completeness relation, we took advantage of the fact R) connects only states with the same}. So the matrix element is simply the amplitude for the rotated state to be found in jj, m ') when original unrotated state is given by I}, m). In Section 3.3 we saw how Euler angles may be used to the most general rotation. We now consider the matrix (3.3.20) for an arbitrary} (not necessarily t): -Ufl a (j) " __ y_ EPm'm ( a, fl , Y) - < J , m lexp( h )exp( h h - Uvfl) = e"'" - """( I. m'leXDI --h-'-Notice that the only nontrivial part is the middle rotation about the which mixes different m-values. It is convenient to define a d(j)(fl) as . (-Ufl) (j,m'lexp -t-I},m). Finally, let us turn to some examples. The} = t case has already been worked out in Section 3.3. See the middle matrix of (3.3.21), d(I/2) = sin ( cos( realization ) m) new matrix Ihe case} = \I) work out ,fl, Orbital Angular Momentum next simplest case is} 1, which we consider in some detaiL Clearly, ..." must first obtain the 3 X 3 matrix representation of Jl" Because U+-J ) "j -(3.5.53) Jv 'rom the defining equation (3.5.5) for J ±' we can use (3.5.41) to obtain m=l m=O m= 1 m' 1 o -Iii 0 (3.5.54) = oil m'=O. m' 1 Our next task is to work out the Taylor expansion of exp( - iJ"fll h). Unlike t, 1)]2 is independent of 1 and However, it is easy o o 1) ( h 1) r (3.5.55) h ('llnsequently, for) 1 only, it is legitimate to replace (JI')2 .(J" h ) h (l-cosfl) 111 (3.5.56) liN the reader may verify in detail. Explicitly we have ( )(1 + cos f3) (_1 )sin fl ( )(1- cos fl ) \If ' d(l)(fl) = I ( 1 ) sin fl cosfl ( )sin fl ( )(1 cos fl ) ()2 )sin fl +cosfl) (3.5.57) Clearly, this method becomes time-consuming for large }. In Section lX we will learn a much easier method for obtaining for any}. .l.6. ORBITAL ANGULAR MOMENTUM We introduced the concept of angular momentum by defining it to be the Jl,enerator of an infinitesimal rotation. There is another way to approach the of angular momentum when spin-angular momentum is zero or can Il$11vred. The angular momentum J for a single particle is then the same liS orbital angular momentum, which is defined as L=xxp. (3.6.1) In this section we explore the connection between the two approaches. 196 197 Theory of Angular Momentum Orbital Angular Momentum as Rotation Generator We first note that the orbital angular-momentum operator defined (3.6.1) satisfies the angular-momentum commutation relations [L;, Lj] = ie;jkliLk by virtue of the commutation relations among the components of x and p. This can easily be proved as follows: [Lx, Ly] = ypz - ZPy' zPx - xpzl = [YPz,zpx1+[zpy,xpzl = ypJpz, z]+ Pyxz, pzl = iii (xPy - ypJ = iliLz Next we let 1-1.( h Sep) Lz=I-1.( h Sep) (xPy- ypJ act on an arbitrary position eigenket lx', y', z') to examine whether it can be interpreted as the infinitesimal rotation operator about the z-axis by angle Sep. Using the fact that momentum is the generator of translation, we obtain [see (1.6.32)] [ 1 - i ( S: ) L z ] Ix " y', z ') = [1 - i ( ) ( S epx ' ) + i ( ) ( S epY')] Ix " y', z') =Ix'- y'Sep,y'+x'Sep,z'). (3.6.5) This is precisely what we expect if L generates an infinitesimal rotation z about the z-axis. So we have demonstrated that if p generates translation, then L generates rotation. Suppose the wave function for an arbitrary physical state of a spinless particle is given by (x', y', z'la). After an infinitesimal rotation about the z-axis is performed, the wave function for the rotated state is (x', y', Z'I[1- i( S: )Lz] la) = (x' + y'Sep, y' - x'Sep, z'la). (3.6.6) It is actually more transparent to change the coordinate basis (x', y', z'la) -> (r,O,epla). (3.6.7) For the rotated state we have, according to (3.6.6), (r, 0, epl [1- i ( S: ) Lz] la) = (r, 0, ep - Sepia) a =(r,O,epla)-Sep aep (r,O,epla). (3.6.8) 1II. Orbital Angular Momentum Ib;ause (r, 0, epl is an arbitrary position eigenket, we can identify . a (x'ILzla) = - Iii aep (x'la), (3.6.9) which is a well-known result from wave mechanics. Even though this rt'lillion can also be obtained just as easily using the position representation Ill' lhe momentum operator, the derivation given here emphasizes the role of I,. as the generator of rotation. We next consider a rotation about the x-axis by angle Sepx' In unalogy with (3.6.6) we have (x', y', Z'I[1- i ( S:x ) Lx] la) = (x', y' + z'Sepx' z' - y'Sepxla). (3.6.10) By expressing x', y', and z' in spherical coordinates, we can show that (x'ILxla) = - iii ( - sinep :0 - cot ° cos ep :ep )(x'la). (3.6.11) l.ikewise, (x'ILyla) = - iii (cosep :0 -cotO sinep a: )(x'la). (3.6.12) Iising (3.6.11) and (3.6.12), for the ladder operator L ± defined as in (3.5.5), we have .(a a) (x'IL± la)=-ilie± l ± iao-cotO (x'la). (3.6.13) aep I,'inally, it is possible to write (x'IVla) using L2 = L; +OJ(L+ L_ + L_ L+), (3.6.14) (.1.6.9), and (3.6.13), as follows: (x'IL2Ia) = -1i2 :;2 + :0 (sinO :0 ) } and the Iwo-dimensional spin space spanned by 1+) and 1-). Explicitly, we have for the base ket lx', ±) = Ix') ®I±), (3.7.1) whcre any operator in the space spanned by {Ix')} commutes with any llJ'lcrator in the two-dimensional space spanned by I ± ). 204 205 Theory of Angular The rotation operator still takes the form exp( - iJ . it/ n) but generator of rotations, is now made up of two parts, namely, J=L+S. It is actually more obvious to write (3.7.2) as J = L®l+ l®S, where the 1 in L ® 1 stands for the identity operator in the spin space, the 1 in 1 ® S stands for the identity operator in the infinite-dimensi space spanned by the position eigenkets. Because Land S commute, we write P)(R) = p)(orb)(R)®p)(spin)(R) _ (-iL.it</» (-is.it</>) - exp n ® exp n . The wave function for a particle with spin is written as (x', ± la) = 1/J ± (x'). The two components 1/J ± are often arranged in column matrix follows: 1/J+(X')), ( 1/J- (x') where 11/J ± (x')1 2 stands for the probability density for the particle to found at x' with spin up and down, respectively. Instead of Ix') as the kets for the space part, we may use In, I, m), which are eigenkets of L2 Lz with eigenvalues n2/(l + 1) and min, respectively. For the spin part, I are eigenkets of S2 and Sz with eigenvalues 3n 2/4 and ± n/2, respecti However, as we will show later, we can also use base kets which eigenkets of J2, fz' L2, and S2. In other words, we can expand a state ket a particle with spin in terms of simultaneous eigenkets of L2, S2, L z , and or in terms of simultaneous eigenkets of J2, fz' L2, and S2. We will study detail how the two descriptions are related. As a second example, we study two spin t particles-say electrons-with the orbital degree of freedom suppressed. The total operator is usually written as S = Sl +S2' but again it is to be understood as Sl®l + 1 ®S2' where the 1 in the first (second) term stands for the identity operator in the spin space of electron 2 (1). We, of course, have [SIx' S2Y] = 0 and so forth. Within the space of electron 1 (2) we have the usual , Addition of Angular Momenta utation relations [ SIx, SlY] = inSlz , [S2x, S2y] = inS2z , .... (3.7.10) 1\ direct consequence of (3.7.9) and (3.7.10), we have [Sx, Sy] = inSz (3.7.11) NO on for the total spin operator. The eigenvalues of the various spin operators are denoted as follows: S2 = (Sl + S2)2: s(s + 1)n 2 S=S +S2 z lz z :mn Slz : mIn (3.7.12) S2z : m 2n .n, we can expand the ket corresponding to an arbitrary spin state of dectrons in terms of either the eigenkets of S2 and Sz or the eigenkets It ."1: and S2z' The two possibilities are as follows: 1. The {ml' m 2} representation based on the eigenkets of Slz and S2z: 1+ + ), 1+ - ), 1- + ), and 1- - ), (3.7.13) where 1+ -) stands for ml = t, m 2 = - t, and so forth. 2. The {s, m} representation (or the triplet-singlet representation) based on the eigenkets of S2 and Sz: Is=l,m=± l,O),ls=O,m=O), (3.7.14) where s = 1 (s = 0) is referred to as spin triplet (spin singlet). Notice that in each set there are four base kets. The relationship h('tween the two sets of base kets is as follows: Is = 1, m = 1) = 1+ + ), (3.7.1Sa) Is = 1, m = 0) = ( ) (I + - ) + 1- + ) ), (3.7.1Sb) Is = 1, m = - 1) = 1- - ), (3.7.1Sc) )(1+-)'--1-+»). (3.7.1Sd) The right-hand side of (3.7.1Sa) tells us that we have both electrons with Npin up; this situation can correspond only to s = 1, m = 1. We can obtain (J.7.ISb) from (3.7.1Sa) by applying the ladder operator S_=Sl_+S2_ = (SlX- iSly)+(S2x- iS2y) (3.7.16) 207 :06 Theory of Angular Addition of Angular Momenta o both sides of (3.7.15a). In doing so we must remember that an )perator like SI affects just the first entry of 1+ +), and so on. We ¥rite S_ls=l,m 1)=(Sl_+S2)1++) lS V(l+l)(l I-tl)ls=l,m xI +) ), + hich immediately leads to (3.7.15b). Likewise, we can obtain Is == n = -I) by applying (3.7.16) once again to (3.7.15b). Finally, we )btain (3.7.15d) by requiring it to be orthogonal to the other three kets, ,articular to (3.7.15b). The coefficients that appear on the right-hand side of (3.7.15) are :implest example of Clebsch-Gordan coefficients to be discussed further at ater time. They are simply the elements of the transformation matrix :onnects the {m l' m 2} basis to the {s, m} basis. It is instructive to hese coefficients in another way. Suppose we write the 4 X 4 matrix :ponding to S2 S2+S2+2S.S I 2 I 2 Sf +s} + 2SlzS2z + Sli S2 + SI- S2+ sing the (mI' m z) basis. The square matrix is obviously not Ulagomu ecause an operator like Sl+ connects 1- +) with 1+ +). The natrix that diagonalizes this matrix carries the Im l , m 2) base kets into , m) base kets. The elements of this unitary matrix are precisely lebsch-Gordan coefficients for this problem. The reader is encouraged to ork out all this in detail. Formal Theory of Angular-Momentum Addition Having gained some physical insight by considering simple examples, e are now in a position to study more systematically the formal theory of ngular-momentum addition. Consider two angular-momentum operators J I nd J2 in different subspaces. The components of J I(J2) satisfy the usual mgular-momentum commutation relations: {JIi , JIj ] = ihcijkJIk nd J2i , we have ° [ J1k , J21 ] = (3.7.21) ween any pair of operators from different subspaces. The infinitesimal rotation operator that affects both subspace 1 and luhspace 2 is written as (1 iJ\' fi8<l» ® (1 iJ2 ·n8<1» =1 i(J1®1+1®J2)'n8<1> . h h h (3.7.22) We define the total angular momentum by J=J1®1+1®J2 , (3.7.23) which is more commonly written as J J 1 +J2. .7.24) The finite-angle version of (3.7.22) is exp( - )®exp(-----:=---(3.7.25) Notice the appearance of the same axis of rotation and the same angle of rotation. It is very important to note that the total J satisfies the angular-lIlomentum commutation relations {1;, = ihcijkJk (3.7.26) liN a direct consequence of (3.7.20) and (3.7.21). In other words, J is an momentum in the sense of Section 3.1. Physically this is reasonable flecause J is the generator for the entire system. Everything we learned in Section 3.5-for example, the eigenvalue spectrum of J2 and Jz and the matrix elements of the ladder operators-also holds for the total J. As for the choice of base kets we have two options. Option A: Simultaneous eigenkets of Jl, J{, JIz' and denoted fly IJIJ2; mIm2 ). Obviously the four operators commute with each other. The defining equations are J?IJIJ2; mIm2 ) JIUI + 1)h2IJIJ2; mIm2), (3.7.27a) JIz IJIJ2; mIm2 ) = m 1hIJIJ2; mIm2 ), (3.7.27b) JiIJIJ2; m1m2 ) = J2U2 + 1)h2 UIJ2; m1m 2), (3.7.27c) J2z IJIJ2; m1m 2) = m2hIJIJ2; mlm2)' (3.7.27d) Option B: Simultaneous eigenkets of J2, J I 2, Jf, and First, note that this set of operators mutually commute. In particular, we have =0, (3.7.28) 20R Theory of Angular Momentum which can readily be seen by writing J2 as .I2 = JI Z+J1 +2112 J22 + JI l z + 11_ JH . (3.7.29) We use 1)1' to denote the base kets of option B: Jl zl)I)2; )m) = )I(jl + 1) 1121)1)2; (3 .Jll)l)2; )m) = )2 (j2 +1)11 21J1)2; )m), (3.7.30b) ;)m), {3.7.30c} )m). (3.7.30d) )m) Quite often 12 are understood, and the base kets are written simply as m). It is very important to note that even though [.J 2 ,Jzl 0, (3.7.31) we have [.1 2 , Jlzl "" 0, [.J 2 ,1}2] 0, (3.7.32) as the reader may easily verify using (3.7.29). This means that we cannot add J 2 to the set of operators of option A. Likewise, we cannot add 1];: and/or to the set of operators of option B. We have two possible sets of base kets corresponding to the two maximal sets of mutually compatible observables we have constructed. Let us consider the unitary transformation in the sense of Section 1.5 that connects the two bases: )m)= L , m l m 2)Ul)Z; mlmzl)l)}; )m), (3.7.33) ml m2 where we have used L LIJI)z; m 1m 2 )UI12; m 1m 21 1 (3.7.34) 1'1'11 m2 and where the right-hand side is the identity operator in the ket space of given )] and The elements of this transformation matrix Ul)2; m 1m 21)112; Jm) are Clebsch-Gordan coefficients. There are many important properties of CIebsch-Gordan coefficients that we are now ready to study. First, the coefficients vanish unless m=m l +m 2 • (3.7.35) To prove this, first note that (lz lIZ 12z ) . Jm) O. (3.7.36) Multiplying m lm 2 1 on the we obtain (m ml - m 2 ; m1 m 21J1J2; 0, (3.7.37) Addition of Angular Momenta 209 which proves our assertion. Admire the power of the Dirac notation! It really pays to write the Clehsch-Gordan coefficients in Dirac's bracket form, as we have done. Second, the coefficients vanish unless IJI - Jzl ) )1 + )2' (3.7.38) This property may appear obvious from the vector model of angular-momentum addition, where we visualize J to be the vectorial sum of JI and .I,. However, it is worth checking this point by showing that if (3.7.38) holds, then the dimensionality of the space spanned by {lJli2; m l m 2 )} is Ihe same as that of the space spanned by {UI)2; im)}. For the (ml,m 1 ) way of counting we obtain N= +1)(2i2 + 1) (3.7.39) hecause for given JI there are 2il + 1 possible values of ml; a similar slatement is true for the other angular momentum J2' As for the (j, m) way or counting, we note that for each i, there are 2) + 1 states, and according It) (3.7.38), i itself runs from)1 i} to il + )2' where we have assumed, without loss of generality, that iI 2': We therefore obtain jj+h N= L (2i+ 1) - 12 H {2(ji - iz} +I} + {2(j1 + iz)+ I} 2 + +1)(2i2 + 1). (3.7.40) Ikcause both ways of counting give the same N-value, we see that (3.7.38) is quite consistent. The Clebsch-Gordan coefficients form a unitary matrix. Further-more, the matrix elements are taken to be real by convention. An immediate consequence of this is that the inverse coefficient (iti2; Jmlil)2; m lm 2) is Ihe same as UI m}m zlil)Z; im) itself. A real unitary matrix is orthogo-so we have the orthogonality condition LL( ; m I m 2 1)IJ2; ; )m) = 8", ""8,,, 1 2 m" 1 2 i m (3.7.41) which is obvious from the orthonormality of {IJI)2; m Im 2 )} together with he reality of the Clebsch-Gordan coefficients. Likewise, we also have L L UIi2; m jm 2lilJ2; )m)OjJ2; m l m 2IJI)1; J'm') = 8 ji ,8mm,· nl1 m2 (3.7.42) 'A l:ompletc proof of (3.7.38) is given in Gottfried 1966,215, and also in Appendix B of this hllok. 210 211 Theory of Angular ylS\,;IJ sometimes ). (3.7 and IVlOmentll As a special case of this we may set j'= J, m'= m = m + m2. We obtain i L LI<J[J2; mIm2liIJ2; Jm)1 2 =1, "'1 "'2 which is just the normalization condition for IJIJ2; Jm). Some authors use somewhat different notations for the Gordan coefficients. Instead of <Jti2; mIm 2 IJIJ2; Jm) we (JImIi2m2IiIJ2Jm), c(JIJ2J; m1m2m), Slh(jm; mIm2), and so on. can also be written in terms of Wigner's 3-j symbol, which is occasiunalll found in the literature: m;) IJIi2; J, m ± L L (lUI + mi)(J1 ± +1) IJ)i2; m1± 1, m;) n1i m2 + /(J2 +m;)(J2 ± m; + 1) IJIJ2; mi, m; ± 1») X (Jd2; mim;UIJ2; im). Our next step is to multiply by (Jd2; mlm2 1 on the left and use orthonor-mality, which means that nonvanishing contributions from the right-hand side are possible only with ml m1±1, m 2 m; for the first term and m 1 = mi, m2=m;± 1 "More-detailed discussion of Clebsch-Gordan and Racah coefficients. recoupling, and the like is given in A R. Edmonds 1960, for instance. n. Addition of Angular Momenta for the second term. In this manner we obtain the desired recursion relations: J(j + m)(j ±m + 1) (Jdz; m1m2 1i1J2; i, m ±1) = JUI + ml + 1)U} ± ml ) 0; i=t, 1=0 So for each I there are two possible i-values; for example, for 1=1 (p we get, in spectroscopic notation, P3/2 and PI/2' where the subscript to i. The mI m 2-plane, or better the m,ms-plane, of this problem is larly simple. The allowed sites form only two rows: the upper row for ms s and the lower row for ms = - t; see Figure 3.7. Specifically, we work m2=j2 m2=j I I A I • D t<----".A I I I", J+ I m,=-j, m,=j, I "I ----- +-----l J- ',' I • E I I, , J+ I I l " m,+m2=-j I J- ',' • F m2=-h (a) FIGURE 3.6. Use of the recursion relations to obtain the Clebsch-Gordan coeflkients, that the consider the problem " , I I J-" I I I (b) " I 2), Addition of Angular Momenta 213 x X X s m I' " , I', "-" I', I J-, I J-, I J-, .. m, ... -IRE 3.7. Recursion relations used to obtain the Clebsch-Gordan coefficients for 1/ =, I ,-, = s =,. Ihe case i = I + t. Because m s cannot exceed t, we can use the J _ recursion In NliCh a way that we always stay in the upper row (m2 = ms = t), while the ", rvalue changes by one unit each time we consider a new J _ triangle. luppressing il = I, i2 = t in writing the Clebsch-Gordan coefficients, we "hlain from (3.7.49) (lower sign) v(t+t+m+l)(t+t-m)(m-t,tIl+t,m) =V ( I+ m + t)(I- m + t) (m + t, til + t, m +1) , (3.7.54) where we have used mI=m,=m-t, m2=ms=t. (3.7.55) III this way we can move horizontally by one unit: I+m+t / 1 11 1 ) (3.7.56) (In - II + = \m+ 2 '2 / +2,m+l . We can in turn express (m + t, til + t, m +1) in terms of (m +t til + t, m and so forth. Clearly, this procedure can be continued until m, reaches I, the maximum possible value: (m-.!. .!.I/+.!. m)= 1+ m + / 3 1 I' 1 ) 2' 2 2' I+m+i \m+ 2 '2 / +2,m+2 I+m+i x ( m + %, II + , m + 3 ) I+ m + t ( I, II + + ). (3.7.57) •• J ..... 1 t. So 1m, l , m (3.7.63) """,u!:ula.r MOmf!ftr. Consider the angular-momentum configuration in which m, and are both maximal, that is, 1and t, respectively. The total m= m, + ms + 12 which is possible only for j =1+ t and not for j = ' . 1 m = t) must be equal to 11 = 1+ 2, m 1+ -1) up to a phase factor. take this phase factor to be real and positive by convention. With this we have (I,W+t,I+t)=L Returning to (3.7.57), we finally obtain 1 11 1 ) {!-+m+t ( m-2"2'[+2',m =21+1 But this is only about one-fourth of the story. We must still termine the value of the question marks that appear in the following: 1) /1+m+-11 . 1 1) j = 1+ 2' ' m = 21 +1 m1= m - 2" ms = 2' I +?\ml=m+},m s=-}), k I-t·m ) +,-m-i.m,-t)+?im,-m+t.m,--i). We note that the transformation matrix with fixed m from the (m basis to the (j, m) basis is, because of orthogonality, expected to have the form COSa Sina) ( sina Cosa' Comparison with (3.7.60) shows that COsa is (3.7.59) itself; readily determine sina up to a sign ambiguity: sin2a =1 (t + m + t) = (I - m :J) (21+1) (21+1) We claim that (ml = m + t, ms = = 1+ 1, m) must be positive because all j = [ + t states are reachable by applying the J _ operator successively to U = 1+ !. m = 1+ t), and the matrix elements of J" are always positive by convention. So the 2x2 transformation matrix (3.7.61) can be only I /I+m+l 21 +1 !I-m +1--1 2 1.7. AdllitiOil or Ansulur MOOlcntu We define spin-angular functions in two-component form as fonows: r1jfJ-I±1/2.m = ± Ii ± m + t ym-l/2(0 .f.)x I '" 1 I l ' ' 't' + I I+m+ t ym+l/2(0 ) + , 2t +1 I , x-l (± /I± m+tYim-l/2(0,</»1 (3.7.64) ,-----:-Yim + 1/ 2( 0, </» J. They are, by construction, simultaneous eigenfunctions of L2, S2, J2, and .I,. They are also eigenfunctions of L'S but L'S, being just L·S (t)(J 2 -L2-S2), (3.7.65) is not independent. Indeed, its eigenvalue can easily be computed as follows: tll2 for j = 1+ L ( 112 ) [ 2 j(j + 1(I+1)-!]= (1+.1)112 { for j 1- t. (3.7.66) Oebsch-Gordan Coefficients and Rotation Matrices Angular-momentum addition may be discussed from the point of view of rotation matrices. Consider the rotation operator EJ{Jt)( R) in the ket space spanned by the angular-momentum eigenkets with eigenvalue Likewise, consider EJ{h)(R). The product EJ(Jt)®EJ(h) is reducible in the sense that after suitable choice of base kets, its matrix representation can take the fonowing form: o I I r-i---..., (h+j2-1) EJ 1 2 ,+i.-2l l '" .... .......-----. (Ih -j21) EJ o (3.7.67) 216 217 Theory of Angular import""' 11' i2 ....; (3.7.72) JVlOmelllij In the notation of group theory this is written as !!2 (Jil®!!2 (h) = !!2(J, +h)(B!!2(J, +h- 1)(B ... (B!!2(ij,-hll. In terms of the elements of rotation matrices, we have an expansion known as the Clebsch-Gordan series: !!2(Jil ,(R)!!2(h) ,(R) = L L L<i1i2; m1 m21i1i2; im) mImI m2m2 j m m' ( .. "I .. . ') fiA (J) (R) X hh; m 1m 2 hh; Jm ::LImm, , where the i-sum runs from iiI - i21 to i1 + i2' The proof of this eq follows. First, note that the left-hand side of (3.7.69) is the same as <i1i2; m1 m21!!2(R)1i1i2; mim;) = <i1m11!!2(R)1i1mi)<i2 m21!!2(R)li2 m = !!2(J,) ,( R)!!2(h), (R).
mlml m2 m 2 But the same matrix element is also computable by inserting a complete of states in the (j, m) basis. Thus <i1i2; m1 m21!!2(R)li1i2; mim;) = L L L L<i1i2; m1 m21i1i2; im) = ( ) ( Ii + ; i - > - Ii - ;i + ) ), (3.9.1) ".,J....a....a. ""'..", hn'l(r""" ....;fl'lT ;_.-1; .............. A ............. , ......... .......... ! ...... ....... D""......... 11 ... J.... ....... 224 225 Theory of Angular Momentum Ii + ; i -) means that electron 1 is in the spin-up state and electron 2 is in the spin-down state. The same is true for Iz ; z+ ). Suppose we make a measurement on the spin component of one of the electrons. Clearly, there is a 50-50 chance of getting either up or down because the composite system may be in Ii +; z ) or Ii ; i + ) with equal But if one of the components is shown to be in the spin-up state, the other is necessarily in the spin-down state, and vice versa. When the spin component of electron 1 is shown to be up, the measurement apparatus has selected the first term, Iz+;z ) of a subsequent measurement of the spin component of electron 2 must ascertain that the state ket of the composite system is given by li+;2 ). It is remarkable that this kind of correlation can persist even if the two particles are well separated and have ceased to interact provided that as they fly apart, there is no change in their spin states. This is certainly the case for a J = 0 system disintegrating spontaneously into two spin particles with no relative orbital angular momentum, because angular-momentum conservation must hold in the disintegration process. An exam-ple of this would be a rare decay of the 1/ meson (mass 549 MeV/( 2 ) into a muon pair 1/ -> p. + + (3.9.2) which, unfortunately, has a branching ratio of only approximately 6 X 10 6. More realistically, in proton-proton scattering at low kinetic energies, the Pauli principle to be discussed in Chapter 6 forces the interacting protons to be in ISo (orbital angular momentum 0, spin-singlet state), and the spin states of the scattered protons must be correlated in the manner indicated by (3.9.1) even after they get separated by a macroscopic distance. To be more pictorial we consider a system of two spin \ particles moving in opposite directions, as in Figure 3.8. Observer A specializes in measuring Sz of particle 1 (flying to the right), while observer B specializes in measuring Sz of particle 2 (flying to the left). To be specific, let us assume that observer A finds Sz to be positive for particle 1. Then he or she can predict, even before B performs any measurement, the outcome of B's measurement with certainty: B must find S;; to be for particle 2. On the other hand, if A makes no measurement, B has a 50-50 chance of getting Sz + or Sz-' This by itself might not be so peculiar. One may say, "It is just like an urn known to contain one black ball and one white ball. When we blindly pick one of them, there is a 50-50 chance of black or white. But if the first ball we pick is black, then we can predict with certainty that the second ball will be white." It turns out that this analogy is too simple. The actual quantum-mechanical situation is far more sophisticated than that! This is because observers may choose to measure Sx in place of Sz' The same pair of 1.9. Spin Correlation Measurements and Bell's I Particle' 0 .1 A I i B It Particle 1 J FIGURE 3.8. Spin correlation in a spm-smgJel state. "quantum-mechanical balls" can be analyzed either in terms of black and white or in terms of blue and red! Recall now that for a single spin ! system the Sx eigenkets and Sz cigenkets are related as follows: )(IZ+)±IZ-»), iZ±) (1 )(lX+)±IX-». (3.9.3) Returning now to Our composite system, we can rewrite spin-singlet ket n.9.1) by choosing the x-direction as the axis of quantization: Ispin singlet> = ( ) ( Ix - ; x + ) Ix + ; x > ) . (3.9.4) Apart from the overall sign, which in any case is a matter of convention, we could have guessed this form directly from (3.9.1) because spin-singlet states have no preferred direction in space. Let us now suppose that observer A can choose to measure Sz or Sx of particle 1 by changing the orientation of his or her spin analyzer, while observer B always specializes in measuring Sx of particle 2. If A determines of particle 1 to be positive, B clearly has a 50-50 chance for getting Sx + or Sx ; even though of particle 2 is known to be negative with certainty, its Sx is completely undetermined. On the other hand, let us suppose that A also chooses to measure Sx; if observer A determines S, of particle 1 to be positive, then without fail, observer B will measure Sx of particle 2 to be negative. Finally, if A chooses to make no measurement, B, of course, will have a 50-50 chance of getting + or ,\', -. To sum up: 1. If A measures Sz and B measures there is a completely random correlation between the two measurements. 2. If A measures Sx and B measures there is a 100% (opposite sign) correlation between the two measurements. 3. If A makes no measurement, B's measurements show random results. Tahle 3.1 shows all possible results of such measurements when B and A are allowed to choose to measure Sx or Sz. 227 Theory of Angular TABLE 3.1. Spin-correlation Measurements Spin component Spin component measured by A A's result measured by B B's result z + z z x + x z x z + z + x x + x z + x + x x + z z + z x x + z + x + z These considerations show that the outcome of B's ars to depend on what kind of measurement A decides to perform: neasurement, an Sz measurement, or no measurement. Notice again nd B can be miles apart with no possibility of communications ual interactions. Observer A can decide how to orient his or -analyzer apparatus long after the two particles have separated. I t is gh particle 2 "knows" which spin component of particle 1 is sured. The orthodox quantum-mechanical interpretation of this situation allows. A measurement is a selection (or filtration) process. When Sz icle 1 is measured to be positive, then component Iz + ; z - ) is selected. t ubsequent measurement of the other particle's Sz merely ascertains system is still in Iz+; z-). We must accept that a measurement on what ears to be a part of the system is to be regarded as a measurement on the Ie system. Einstein's Locality Principle and Bell's Inequality Many physicists have felt uncomfortable with the preceding ortho-interpretation of spin-correlation measurements. Their feelings are 'fied in the following frequently quoted remarks by A. Einstein, which call Einstein's locality principle: "But on one supposition we should, in opinion, absolutely hold fast: The real factual situation of the system S2 ndependent of what is done with the system Sl' which is spatially arated from the former." Because this problem was first discussed in a ',U, Spin Correlation Measurements and Bell's Inequality paper of A. Einstein, B. Podolsky, and N. Rosei'l, it is sometimes known as the Einstein-Podolsky-Rosen paradox. Some have argued that the difficulties encountered here are inherent in the probabilistic interpretations of quantum mechanics and that the dynamic behavior at the microscopic level appears probabilistic only be-\':lluse some yet unknown parameters-so-called hidden variables- have not been specified. It is not our purpose here to discuss various alternatives to quantum mechanics based on hidden-variable or other considerations. Ruther, let us ask, Do such theories make predictions different from those of quantum mechanics? Until 1964, it could be thought that the alternative lheories could be concocted in such a way that they would give no predictions, other than the usual quantum-mechanical predictions, that could be verified experimentally. The whole debate would have belonged to the realm of metaphysics rather than physics. It was then pointed out by J. S. Bell that the alternative theories based on Einstein's locality principle IIdually predict a testable inequality relation among the observables of lipin-correlation experiments that disagrees with the predictions of quantum mechanics. We derive Bell's inequality within the framework of a simple model, conceived by E. P. Wigner, that incorporates the essential features of the various alternative theories. Proponents of this model agree that it is impossible to determine Sx and Sz simultaneously. However, when we have II large number of spin particles, we assign a certain fraction of them to have the following property: If Sz is measured, we obtain a plus sign with certainty. If Sx is measured, we obtain a minus sign with certainty. A particle satisfying this property is said to belong to type (z +,x - ). Notice that we are not asserting that we can simultaneously measure Sz and Sx to he + and ,respectively. When we measure Sz, we do not measure S., and vice versa. Weare assigning definite values of spin components in more than one direction with the understanding that only one or the other of the components can actually be measured. Even though this approach is funda-mentally different from that of quantum mechanics, the quantum-mechani-cal predictions for Sz and Sx measurements performed on the spin-up (Sz + ) state are reproduced provided there are as many particles belonging to type (z+ ,x+) as to type (z+ ,x Let us now examine how this model can account for the results of spin-correlation measurements made on composite spin-singlet systems. "To be historically accurate, the original Einstein-Podolsky-Rosen paper dealt with measurements of x and p. The use of composite spin} systems to illustrate the Einstein-Podolsky-Rosen paradox started with D. Bohm. 228 229 Theory of Angular Clearly, for a particular pair, there must be a perfect matching particle 1 and particle 2 to ensure zero total angular momentum: If 1 is of type (i + , x - ), then particle 2 must belong to type (i ,i + ), forth. The results of correlation measurements, such as in Table 3.1, reproduced if particle 1 and particle 2 are matched as follows: particlel particle2 ,i+), (i+ ,x+) (i-,i ), (z- ,x+) (i+,x ), (i- ,i-) (i+ ,x+) with equal populations, that is, 25% each. A very important assumption implied here. Suppose a particular pair belongs to type (3.9.5a) and A decides to measure Sz of particle 1; then he or she necessarily plus sign regardless of whether B decides to measure Sz or SX' It is in sense that Einstein's locality principle is incorporated in this model: result is predetermined independently of B's choice as to what to measuti In the examples considered so far, this model has been successful reproducing the predictions of quantum mechanics. We now more-complicated situations where the model leads to predictions from the usual quantum-mechanical predictions. This time we start three unit vectors a, b, and c, which are, in general, not mutually We imagine that one of the particles belongs to some definite type, (a - ,b +,c +), which means that if So ais measured, we obtain a minus with certainty; if Sob is measured, we obtain a plus sign with certainty; Soc is measured, we obtain a plus with certainty. Again there must be perfect matching in the sense that the other particle necessarily belongs type (a +,b - ,c-) to ensure zero total angular momentum. In any event, the particle pair in question must be a member of one of the types shown in Table 3.2. These eight possibilities are mutually exclu"'v, and disjoint. The population of each type is indicated in the first column. Let us suppose that observer A finds SI°8 to be plus and observer B finds Szob to be plus also. It is clear from Table 3.2 that the pair belong to either type 3 or type 4, s9 the number of particle pairs for which this situation is realized is N3 + NtJ.. Because Ni is positive semidefinite, we must have inequality relations like N3+ NA + N4 )+ (N3 + N7 )· Let P(a + ; b+ ) be the probability that, in a random selection, observer A measures S1° a to be + and observer B measures Sllob to be +, and so on. Spin Correlation Measurements and Bell's Inequality TABLE 3.2. Spin-component Matchinlt in the Alternative Theories Population Particle 1 Particle 2 (a+,h+,H) (A-,h- ,c-) N2 (a+,h+,c-) (A-,h-,c+) N3 ,h-,c (a- ,h+ ,c-) N4 (a+,h-,c-) (a-,h+,c+) Ns (a-,h+,c (a+,h-) N6 (a-,h+,c-) (iH,b-,c N7 (A-,h- ,c (a+,h+,c-) Ns (a-,h- ,c-) (a+,b+,c+) (,Iearly, we have + (3.9.7) P(a+;b+) In a similar manner, we obtain P(a+ ;c+) = (N2 + N4 ) and P(c+ ;b+) . (3.9.8) EfHr The positivity condition (3.9.6) now becomes P(a+ ;b+) P(a+ ;c+ )+p(c+ ;b+). (3.9.9) This is Bell's inequality, which follows from Einstein's locality principle. Quantum Mechanics and Bell's Inequality We now return to the world of quantum mechanics. In quantum mechanics we do not talk about a certain fraction of particle pairs, say belonging to type 3. Instead, we characterize all spin-singlet liystems by the same ket (3.9.1); in the language of Section 3.4 we are concerned here with a pure ensemble. Using this ket and the rules of 4uantum mechanics we have developed, we can unambiguously calculate eneh of the three terms in inequality (3.9.9). We first evaluate P(a + ; b+). Suppose observer A finds S1' a to be positive; because of the 100% (opposite sign) correlation we discussed carlier, B's measurement of Szoa will yield a Ininus sign with certainty. But 10 calculate P(a + ; b+) we must consider a new quantization axis b that makes an angle (jab with a; see Figure 3.9. According to the formalism of Section 3.2, the probability that the S2'b measurement yields + when particle 2 is known to be in an eigenket of S2°ft with negative eigenvalue is 230 231 Theory of Angular MomentUIII (3912) . . (3.9.13) (3.9.14) (3.9.15) adirection t )If b direction ' , /// , /. / <51> I i f I t -: / I // I <52> // I / / I t FIGURE 3.9. Evaluation of P(a+ ;1+). given by 2[('lI' Oab)]' 2(Oab) cos 2 - sm 2 . As a result, we obtain ' ' ) (1). 2(Oab) P ( a + ; b + "2 sm :2 ' where the factor arises from the probability of initially obtaining Sl'Ii with +. Using (3.9.11) and its generalization to the other two terms of (3.9.9), we can write Bell's inequality as . 2(Oab) . 2(Oa(') . 2(OCb) sm:2 sm:2 +sm :2 . We now show that inequality (3.9.12) is not always possible from a geometric point of view. For simplicity let us choose il, b, and c to lie in a plane, and let cbisect the two directions defined by aand b: 20, 00( 0".. =0. Inequality (3.9.12) is then violated for 'lI' 0<0<"2' For example, take 0 = 'lI'/4; we then obtain 0.500 S 0.292 ?? So the quantum-mechanical predictions are not compatible with Bell's inequality. There is a real observable-in the sense of being experimentally verifiable-difference between quantum mechanics and the alternative theo- ries satisfying Einstein's locality principle. I 'I Spin Correlation Measurements and Bell's Inequality Several experiments have been performed to test Bell's inequality. In Hill' of the experiments spin correlations between the final protons in low-energy proton-proton scattering were measured. In all other experi-IlIl'nts photon-polarization correlations between a pair of photons in a 1'1IS(;ade transition of an excited atom (Ca,Hg, ... ), (1= 0).1..(1=1).1..(1= 0), (3.9.16) III in the decay of a positronium (an e' e- bound state in Illeasured; studying photon-polarization correlations should be III view of the analogy developed in Section 1.1: Sz + e in x-direction, Sz e in y-direction, (3.9.17) Sx + e in 45 0 diagonal direction, Sx - e in 135° diagonal direction. The results of all recent preclSlon experiments have conclusively l'slablished that Bell's inequality was violated, in one case by more than nine standard deviations. Furthermore, in all these experiments the inequality relation was violated in such a way that the quantum-mechanical predict-ions were fulfilled within error limits. In this controversy, quantum mecha-nics has triumphed with flying colors. The fact that the quantum-mechanical predictions have been verified does not mean that the whole subject is now a triviality. Despite the experimental verdict we may still feel psychologically uncomfortable about many aspects of measurements of this kind. Consider in particular the following point: Right after observer A performs a measurement on particle I, how does particle 2-which may, in principle, be many light years away from particle I-get to "know" how to orient its spin so that the remark-ahle correlations apparent in Table 3.1 are realized? In one of the experi-ments to test Bell's inequality (performed by A. Aspect and collaborators) the analyzer settings were changed so rapidly that A's decision as to what to measure could not be made until it was too late for any kind of influence, traveling slower than light, to reach B. We conclude this section by showing that despite these peculiarities we cannot use spin-correlation measurements to transmit any useful infor-mation between two macroscopically separated points. In particular, super-luminal (faster than light) communications are impossible. Suppose A and B both agree in advance to measure S:; then, without asking A, B knows precisely what A is getting. But this does not mean that .. I t should be kept in mind here that by working with photons we are going outside the ,.ealm of nonrc1ativistic quantum mechanics, which is the subject of this book. 232 233 Theory of Angular Momentum A and B are communicating; B just observes a random sequence of positive and negative signs. There is obviously no useful information contained in it, B verifies the remarkable correlations predicted by quantum mechanics only after he or she gets together with A and compares the notes (or computor sheets). It might be thought that A and B can communicate if one of them suddenly changes the orientation of his or her analyzing apparatus. Let uS suppose that A agrees initially to measure Sz, and B, S,. The results of A's measurements are completely uncorrelated with the results of B's measure-ments, so there is no information transferred. But then, suppose A suddenly breaks his or her promise and without telling B starts measuring Sx' There are now complete correlations between A's results and B's results. However, B has no way of inferring that A has changed the orientation of his or her analyzer. B continues to see just a random sequence of + 's and's by looking at his or her own notebook only. So again there is no information transferred. 3.10. TENSOR OPERATORS Vector Operator We have been using notations such as x, p, S, and L, but as we have not systematically discussed their rotational properties. They are vector operators, but what are their properties under rotations? In this section we give a precise quantum-mechanical definition of vector operators based on their commutation relations with the angular-momentum operator. We then generalize to tensor operators with more-complicated transformation prop-erties and derive an important theorem on the matrix elements of vector and tensor operators. We all know that a vector in classical physics is a quantity with three components that transforms by definition like Vi -7 'ZjR,jVj under a rota-tion. It is reasonable to demand that the expectation value of a vector operator V in quantum mechanics be transformed like a classical vector under rotation. Specifically, as the state ket is changed under rotation according to la) -l> £ti( R) (3.10.1) the expectation value of V is assumed to change as follows: -l> (algt(R)V;g(R)la) LR;j(aWila). j This must be true for an arbitrary ket la). Therefore, (3.10.2) £tit(R)V;g(R) LRi}'i j (3.10.3) I 10. Tensor Operators lIIust hold as an operator equation, where R ij is the 3 X 3 matrix that \I IITcsponds to rotation R. Let us now consider a specific case, an infinitesimal rotation. When ,Ill' rotation is infinitesimal, we have £tieR) 1- ieJ·il (3.10.4) n We can now write (3.10.3) as V; + I: [V;,J·il] = LRi;(il; 1:) (3.10.5) j III particular, for il along the we have -I: 1 o n (3.10.6) so 1 : I: V, + iii [V x ,]= I = eVy (3.1O.7a) 2: Vv + iii I: [ (3.10.7b) e i = 3: Vz + ilz [V;, lz] V;. (3.10,7c) 'I'his means that V must satisfy the commutation relations [V;,.lj] = il:1jknVk· (3.10.8) Clearly, the behavior of V under a finite rotation is completely determined by the preceding commutation relations; we just apply the hy-now familiar formula (2.3.47) to iJ exp( (3.10.9) -t-) V; - ;<1». We simply need to calculate [JAJj ,[-· '[1i'VJ· (3.]0.10) Multiple commutators keep on giving back to us V; or V (k ' i, j) as in case (3.2.7). k We can use (3.10.8) as the defining property of a vector operator. Notice that the angular-momentum commutation relations are a special case of (3.10.8) in which we let V; -l>"", Vk -l> lk' Other special cases are [y, L ] = z (fix, rx, LJ = iny, [Px, Lz]= ilipy, [PY' Lz] ihpx; these can be proved 234 235 Theory of Angular Momentllll Cartesian Tensors Versus Irreducible Tensors In classical physics it is customary to define a tensor 'T;jk'" generalizing V, --+ Lj R;Fl as follows: 'T;jk'" --+ L L L ... R",Rjj' .. 'T;'j'k'''' i' j' k' under a rotation specified by the 3 X 3 orthogonal matrix R. The number indices is called the rank of a tensor. Such a tensor is known as a Cart_l_ tensor. The simplest example of a Cartesian tensor of rank 2 is a formed out of two vectors V and V. One simply takes a Cartesian compO! nent of V and a Cartesian component of V and puts them together: 'T;j == Cl;Jj. (3.10.12) Notice that we have nine components altogether. They obviously transform like (3.10.11) under rotation. The trouble with a Cartesian tensor like (3.10.12) is that it II reducible-that is, it can be decomposed into objects that transform dilfet-endy under rotations. Specifically, for the dyadic in (3.10.12) we have = V-V (Cl;Jj - t';v,) ( u-v ) Cl;Jj 3 8ij + 2 + 3 8ij . (3.10.13) The first term on the right-hand side, V· V, is a scalar product invariant under rotation. The second is an antisymmetric tensor which can be written as vector product eijk(VXVh. There are altogether 3 independent compo-nents. The last is a 3 x 3 symmetric traceless tensor with 5 ( 6 -1, where 1 comes from the traceless condition) independent components. The number of independent components checks: 3x3=1+3+5. We note that the numbers appearing on the right-hand side of (3.10.14) are precisely the multiplicities of objects with angular momentum 1= 0, 1=1, and 1=2, respec;tively. This suggests that the dyadic has been decomposed into tensors that Can transform like spherical harmonics with 1= 0, 1, and 2. In fact, (3.10.13) is the simplest nontrivial example to illustrate the reduc-tion of a Cartesian tensor into irreducible spherical tensors. Before presenting the precise definition of a spherical tensor, we first give an example of a spherical tensor of rank k. Suppose we take a spherical harmonic yt(O,4». We have already seen that it can be written as yt(ll), where the orientation of it is characterized by 0 and 4>. We now replace nby some vector V. The result is that we have a spherical tensor of rank k (in place of I) with 'magnetic quantum number q (in place of m), T(k) = y;m=q(v) q I=k . (3.10.15) Nprl'i fically, in the case k = 1 we take spherical harmonics with 1=1 and h'plncc (z / r) (0) z by Vz • and so on. y;O -cosO = {E --z --+ 1:(1)= {E V -{E 1 4'IT 4'IT r ° 4'IT z· (3.10.16) y;±l +J 3 x± iy --+T(l) / 3 (+ Vx± iV y ). 1 4'IT Iir ±l 4'IT Ii (.hviously this can be generalized for higher k, for example, Yz± 2 / (x )2 --+ = / (Vx ± iV)y. (3.10.17) 1;/ A) are irreducible, just as y;m are. For this reason, working with sphericul '('I\sors is more satisfactory than working with Cartesian tensors. To see the transformation of spherical tensors constructed in this IIIl1nner, let us first review how yt transform under rotations. First, we hllvc for the direction eigenket; --+ == In'), (3.10.18) which defines the rotated eigenket In'). We wish to examine how yt(n')'" (1.'1/, m) would look in terms of y;m(n). We can easily see this by starting wilh m) = LII, (3.10.19) rn' nlld contracting with (iii on the left, using (3.10.18): }/m(n') = L (3.10.20) rn' If there is an operator that acts like y;m(V), it is then reasonable to expect = L (3.10.21 ) rn' where we have used the unitarity of the rotation operator to rewrite (R -1). All this work is just to motivate the definition of a spherical tensor. We now consider spherical tensors in quantum mechanics. Motivated by Cl.IO.21) we define a spherical tensor operator of rank k with (2k + 1) {:omponents as k = L (3.1O.22a) q'-k 236 237 Theory of Angular Momentum or, equivalently, k PJ(R)Tq(kJPJt(R) = L (3.10.22b) k This definition holds regardless of whether Tq(k) can be written as Yi:;q(V); for example, (U + iUy)(V + is the q = + 2 component of a spherical x x tensor of rank 2 even though, unlike (Vx + iVJ2, it cannot be written as Yl(V). A more convenient definition of a spherical tensor is obtained by considering the infinitesimal form of (3.10.22b), namely, + iJ-.oe) (1+ iJ'i'lE)Tq(k)(l_ iJ.oe) t n k (3.10.23) or [J'n, L (3.10.24) q' By taking 0 in the z- and in the (x± iy) directions and using the non vanish-ing matrix elements of 1z and 1 ± [see (3.5.35b) and (3.5.41)]. we obtain [1 • Ttl] = nqTq(k) (3.10.25a) z and [1 • Tq(k)] = nv(k +q )(k ± q + i)Tq(;)j' (3.1O.25b) t These commutation relations can be considered as a definition of spherical tensors in place of (3.10.22). Product of Tensors We have seen how to form a scalar, vector (or antisymmetric tensor), and a traceless symmetric tensor out of two vectors using the Cartesian tensor language. Of course, spherical tensor language can also be used (Baym 1969, Chapter 17), for example, To(O)= -U'V = (Ut1V 1+ 1V+j-U OVo) 3 3 (UxV)q iii T(2) +2 (3.10.26) T(2) = ±1 Ii 3.10. Tensor Operators where UiVq) is the qth component of a spherical tensor of rank 1, corresponding to vector U(V). The preceding transformation properties can be checked by comparing with Yim and remembering that - (Ux + iUy)/li, U-1 = (Ux iUy)/li, U o A similar check can be made for V± 1,0' For instance, . vO_ 3z 2 -r2 12 -r2 where 3z 2 -r2 can be written as 2z 2 +2[- (x ) J ; hence, Y 20 is just a special case of TO(2) for U = V = r. A more systematic way of forming tensor products goes as follows. We start by stating a theorem: Theorem. Let X(k,) and q, be irreducible spherical tensors of rank k I and k 2' respectively. Then Tq(k) = L L(k1kz; q1q21k1k2; (3.10.27) q! q2 is a spherical (irreducible) tensor of rank k. Proof We must show that under rotation must transform according to (3.10.22) PJi'(R)T}k)PJ(R) L L(k1 k 2; qjq2lkjk2; kq) ql qz x = L L L L(k1kz; qlqzlk1k 2 ; kq) ql q2 qf q2 x R ) = L L L L L L L(k\k2; qlqzlk\k Z ; kq) k" q), q2 q{ qi q" q' x (k 1k z; k 2 ; k"q') X (k k . Ik k . k" ")to(k"J(R )X(k, \ 2, q\q2 1 2, q :;:LIq'q" qj where we have used the Clebsch-Gordan series formula (3.7.69). The preced-ing expression becomes = L L L L L8w ·8qq,,(k\kz; . k"q')t0(k"J(R I)X(k')Z(i')J , ;;;t/q'ql! ql ·"tll ' k" q{ q2 q" q' 238 239 Theory of Angular MomentUftli where we have used the orthogonality of Clebsch-Gordan (3.7.42). Finally, this expression reduces to = "'('" "'(k k . q'q'lk k . kq')X(k 1)Z(k2»)£ij(k)(R- 1 )' """ """ """ 1 2' 1 2 1 2' q; q'z q'q q' q{ = '" T(k)£ij(k)(R- 1 ) '"£ij(k)(R)T\k) """ q' q'q """ qq q q' q' The foregoing shows how we can construct tensor operators of higher or lower ranks by multiplying two tensor operators. Furthermore, the manner in which we construct tensor products out of two tensors is completely analogous to the manner in which we construct an angular-momentum eigenstate by adding two angular momentums; exactly the same Clebsch-Gordan coefficients appear if we let k 1,2 --'> Jl.2, ql,2 --'> m1,2' Matrix Elements of Tensor Operators; the Wigner-Eckart Theorem In considering the interactions of an electromagnetic field with atoms nd nuclei, it is often necessary to evaluate matrix elements of tensor perators with respect to angular-momentum eigenstates. Examples of this will be given in Chapter 5. In general, it is a formidable dynamic task to calculate such matrix elements. However, there are certain properties of these matrix elements that follow purely from kinematic or geometric onsiderations, which we now discuss. First, there is a very simple m-selection rule: m-selection Rule (a', j'm'ITq(k)la, jm) = 0, unless m'= q + m. (3.10.28) Proof Using (3.10.25a), we have (a', j'm'I([J , Tq(k)] IiqTq(k») la, jm) = [(m' m)1i liq] z x(a',j'm'ITJk)la,jm) 0; hence, ( a', = 0 unless m'= q + m. o Another way to see this is to note that transformation property of T?)la, jm) under rotation, namely, £ijTt)la, Jm) = £ijTt)£ijf£ijla, jm). If we now let £ij stand for a rotation operator around the z-axis, we get [see Tensor Operators (.UO.22b) and (3.1.16)] £ij(z, )Tp)la, jm) = e-iqe-imTq(k)la, jm), (3.10.30) which is orthogonal to la' ,j'm') unless q + m = m'. We are going to prove one of the most important theorems in quan-111m mechanics, the Wigner-Eckart theorem. The Wigner-Eckart Theorem. The matrix elements of tensor operators with respect to angular-momentum eigenstates satisfy (a', j'm'ITq(k)la, jm) = Ok; mqljk; j'm')-'--''-':':::=:':':::-''--"-where the douhle-bar matrix element is independent of m and m', and q. Before we present a proof of this theorem, let us look at its significance. First, we see that the matrix element is written as the product of two factors. The first factor is a Clebsch-Gordan coefficient for adding j and k to get jf. It depends only on the geometry, that is, the way the system is oriented with respect to the z-axis. There is no reference whatsoever to the particular nature of the tensor operator. The second factor does depend on the dynamics, for instance, a may stand for the radial quantum number and its evaluation may involve, for example, evaluation of radial integrals. On the other hand, it is completely independent of the magnetic quantum numbers m, m', and q, which specify the orientation of the physical system. To evaluate (a',j'm'ITq(k)la,jm) with various combina-tions of m, m', and q' it is sufficient to know just one of them; all others can be related geometrically because they are proportional to Clebsch-Gordan coefficients, which are known. The common proportionality factor is (a'j'IIT(k)l\aj), which makes no reference whatsoever to the geometric features. Tbe selection rules for the tensor operator matrix element can be immediately read off from the selection rules for adding angular momen-tum. Indeed, from the requirement that the Clebsch-Gordan coefficient be nonvanishing, we immediately obtain the m-selection rule (3.10.28) derived before and also the triangular relation jj-kl:Sj':Sj + k. (3.10.32) Now we prove the theorem. Proof Using (3.10.25b) we have , m, j it, m ml' k 12, and q -i> m2. Both recursion relations are of the form 'E j a ijXj = 0, that is, first-order linear homogeneous equations with the same coefficients a ij' Whenever we have I:aijxj = 0, I:aijYj = 0, j j we cannot solve for the Xj (or Yj) individually but we can solve for the ratios; so Yj x-Y or X·= cy.
k k J J' where C is a universal proportionality factor. Noting that (hj2;m l,m2± 11J1j2; jm) in the Clebsch-Gordan recursion relation (3.7.49) corresponds to (a', j'm'ITq<!\la, jm), we see that ( a', j'm'ITq<!\la, jm) = (universal proportionality constant independent of m, q, and m')Uk; m q ± 11Jk; j'm') , (3.10.37) which proves the theorem. 0 Let us now look at two simple examples of the Wigner-Eckart theorem. Example 1. Tensor of rank 0, that is, scalar To(O) = S. The matrix element of a scalar operator satisfies l"m'ISla l'm ) = 8,fj , 0 tranSItIOn is forbidden. This selection rule is of fundamental importance in the theory of radiation; it is the dipole selection rule obtained in the long-wavelength limit of emitted photons. For j = j' the Wigner-Eckart theorem-when applied to the vector operator-takes a particularly simple form, often known as the projection theorem for obvious reasons. The Projection Theorem (a', jm'lVqla, jm) = (a', jm) (jm'IJqljm), (3.10.40) where analogous to our discussion after (3.10.26) we choose J t 1 1= (Jx ± i.l y ) = + 1 Jo= Jz· (3.10.41) Proof Noting (3.10.26) we have (a',jmjJ'Vla,jm) (a',jml(JoVo Jt1V-l-J_lVtl)la,jm) = mli(a', jmlVola, jm) + - m +1) X(a',jm 11V_1Ia,jm) J(}-m)(}+m+1) (a',jm+ jm) = cjm(aiIlVllaj) (3.10.42) hy the Wigner-Eckart theorem (3.10.31), where cjm is independent of a, a', and V, and the matrix elements of Yo, ±l are all proportional to the double-bar matrix element (sometimes also called the reduced matrix ele-ment). Furthermore, cjm is independent of m because J.V is a scalar operator, so we may as well write it as cj • Because cj does not depend on V, 0.10.42) holds even if we let V -,> J and a' -,> a, that is, (a, jmiJ 2la, jm) = c/ajIIJllaj). (3.10.43) Returning to the Wigner-Eckart theorem applied to Vq and Jq, we have (a', jm'lVqla, jm) (aiIlVllaj) (3.10.44) (a, jm'llqla, jm) (ajIlJllaj)' "Additional parity selection rules are discussed in Chapter 4, Section 2. They lead to these I-: I d ipolc selection rules. .42 ,Theory of Angular Momentum lut the right-hand side of (3.10.44) is the same as (a', jmiJoVla, jm) '(a, jmiJ 2 la, jm) by (3.10.42) and (3.10.43). Moreover, the left-hand side .f (3.10.43) is just j(j +1)1i 2• So ( a' jm'lVla jm)= (a',jmiJoVla, jm) (}'m'IJ IJm) (31045) , q' 1i2j{j +1) q' • . vhich proves the projection theorem. o We will give applications of the theorem in subsequent sections. :JROBLEMS 1. Find the eigenvalues and eigenvectors of 0y = - Suppose an electron is in the spin state (;). If s.v is measured, what is the probability of the result Ii/2? 2. Consider the 2 X 2 matrix defined by ao+ iooa U= --'"---iO'·a ' ao -where ao is a real number and a is a three-dimensional vector with real components. a. Prove that U is unitary and unimodular. b. In general, a 2 x 2 unitary unimodular matrix represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for U in terms of ao, aI' a2' and a3· 3. The spin-dependent Hamiltonian of an electron-positron system in the presence of a uniform magnetic field in the z-direction can be written as H = ASVl·s(e+) + ( : )( S;n Sz(e')). Suppose the spin function of the system is given by a. Is this an eigenfunction of H in the limit A 40, eB/me =F O? If it is, what is the energy eigenvalue? If it is not, what is the expectation value of H? b. Same problem when eB/me 40, A =F O. 4. Consider a spin 1 particle. Evaluate the matrix elements of Sz(Sz+Ii)(Sz-li) and Sx(Sx+Ii)(Sx- Ii ). 5. Let the Hamiltonian of a rigid body be H ; ( 11 + 12 + 13 ), where K is the angular momentum in the body frame. From this Problems 243 expression obtain the Heisenberg equation of motion for K and then find Euler's equation of motion in the correspondence limit. 6. Let U eiGJiXeiG2fJeiGfY, where (a,/3, y) are the Eulerian angles. In order that U represent a rotation (a, /3, y), what are the commutation rules satisfied by the Gk? Relate G to the angular momentum operators. 7. What is the meaning of the following equation: U-1 AP= LRkIA" where the three components of A are matrices? From this equation show that matrix elements (mIAkln) transform like vectors. 8. Consider a sequence of Euler rotations represented by - i<1 a ) ( i02/3 ) ( - i<13Y) f2j) (l/2) ( a, /3, y) = exp -+ exp 2 exp -2-( /3 cos-e-'(0-"1' Si: ) . = 2 2 ei(a- y)/2' sm-/3 ei(a+ y)/2 cos "2 Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single rotation about some axis by an angle 8. Find 8. 9. a. Consider a pure ensemble of identically prepared spin systems. Suppose the expectation values (Sx) and (S,) and the sign of (Sy) are known. Show how we may determine the state vector. Why is it unnecessary to know the magnitude of (Sy)? b. Consider a mixed ensemble of spin t systems. Suppose the ensemble averages [Sx]' [Sy], and [S,] are all known. Show how we may construct the 2 X '2 density matrix. that characterizes the ensemble. 10. a. Prove that the time evolution of the density operator p (in the SchrOdinger picture) is given by p(t) = '1/(t, to)p(to)'1/t (t, to). b. Suppose we have a pure ensemble at t = O. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the SchrOdinger equation. 11. Consider an ensemble of spin 1 systems. The density matrix is now a 3 X 3 matrix. How many independent (real) parameters are needed to characterize the density matrix? What must we know in addition to [Sx], [Sv]' and [Sz] to characterize the ensemble completely? 12. An angular-momentum eigenstate Ij, m = mmax j) is rotated by an infinitesimal angle e about the y-axis. Without using the explicit form of the function, obtain an expression for the probability for the new rotated state to be found in the original state up to terms of order e2• 244 245 Theory of Angular Momentum 13. Show that the 3 X 3 matrices Gi (i = 1,2,3) whose elements are given by (Gi ) jk = - ihfijk' where i and k are the row and column indices, satisfy the angular momentum commutation relations. What is the physical (or geometric) significance of the transformation matrix that connects G; to the more usual 3 X 3 representations of the angular-momentum operator J, with J) taken to be diagonal? Relate your result to V....,V+D8xV under infinitesimal rotations. (Note: This problem may be helpful in understanding the photon spin.) 14. a. Let J be angular momentum. It may stand for orbital L, spin S, or Jtotal') Using the fact that Jx' Jy, Jz(J ± == Jx ± iJy) satisfy the usual angular-momentum commutation relations, prove J 2 =J/+J+J_-hlz' b. Using (a) (or otherwise), derive the "famous" expression for the coefficient c_ that appears in J-I/;jm=c-'I"j.m-l· 15. The wave function of a particle subjected to a spherically symmetrical potential V(r) is given by I/; (x) = (x+ y+3z)/(r). a. Is I/; an eigenfunction of V? If so, what is the I-value? If not, what are the possible values of I we may obtain when L 2 is measured? b. What are the probabilities for the particle to be found in various m, states? c. Suppose it is known somehow that I/;(x) is an energy eigenfunction with eigenvalue E. Indicate how we may find V(r). 16. A particle in a spherically symmetrical potential is known to be in an eigenstate of L2 and L with eigenvalues h 2/(l + 1) and mh, respec-z tively. Prove that the expectation values between 11m) states satisfy (Lx)=(Lv)=O, (L;)=(L;)= m 2 h 2 ] Interpret this result semiclassically. 17. Suppose a half-integer I-value, say 1. were allowed for orbital angular momentum. From L+ Y1/ 2,1/2(0, </» = 0, we may deduce, as usual, 0, </» ex: ei q,/2..[SIDii.
Yl/2 , I'mblcms Now try to construct Y1/ 2, -1/2(0, </»; by (a) applying L_ to Y1/ 2, 1/2(0, </»; and (b) using L_ Y 1/ 2, -1/2(0, </» = 0. Show that the two procedures lead to contradictory results. (This gives an argument against half-integer I-values for orbital angular momentum.) IX. Consider an orbital angular-momentum eigenstate 11= 2, m 0). Sup-pose this state is rotated by an angle p about the y-axis. Find the probability for the new state to be found in m = 0, ± 1, and ±2. (The spherical harmonics for 1= 0, 1, and 2 given in Appendix A may be useful.) 19. What is the physical significance of the operators K_,==ata! and K_=a+a in Schwinger's scheme for angular momentum? Give the non vanishing matrix elements of K +. lO. We are to add angularmomenta i1 = 1 and i2 = 1 to form i 2, 1, and ° states. Using either the ladder operator method or the recursion relation, express all (nine) {j, m} eigenkets in terms of !Jli2; m1 m2). Write your answer as 1J=I,m=l) ... , where + and 0 stand for m1,2 =1,0, respectively. ).1. a. Evaluate } L j for anyi (integer or half-integer); then check your answer for i =!. b. Prove, for any i, t = +1) sin2p + 1). J [Hint: This can be proved in many ways. You may, for instance, examine the rotational properties of J/ using the spherical (irreduci-ble) tensor language.] ,'.1 a. Consider a system with i = 1. Explicitly write (j 1, m'IJyii 1, m) in 3 X 3 matrix form. h. Show that for i = 1 only, it is legitimate to replace e- Uvf3/Ii by J) /J)2 1 i ( ; sinp - t; (1 cosf3). " Using (b), prove = (t)(1 +cosf3) ( ) sinf3 (t )(1-cos f3 ) Theory of Angular -( )sin f3 ( t ) (1- cos f3 ) cosf3 )sin f3 ( ) sinf3 (t)(1 +cosf3) Express the matrix element (a2f32Y2IJllalf3IYI) in terms of a series !»!nn(af3y) = (af3Yljmn). :onsider a system made up of two spin particles. Observer ;pecializes in measuring the spin components of one of the Slz,Slx and so on), while observer B measures the spin components :he other particle. Suppose the system is known to be in a spin- . ;tate, that is, Stotal = O. 1. What is the probability for observer A to obtain Siz = nl2 observer B makes no measurement? Same problem for SIx = n12. ). Observer B determines the spin of particle 2 to be in the S2z = n state with certainty. What can we then conclude about the outCOml of observer A's measurement if (i) A measures Slz and (ii) measures Sl) Justify your answer. :onsider a spherical tensor of rank 1 (that is, a vector) Vx ± iVy ViIi = + Ii Vorl) = v". Jsing the expression for given in Problem 22, evaluate q' md show that your results are just what you expect from the transfor-nation properties of Vx,y,z under rotations about the y-axis. 1. Construct a spherical tensor of rank lout of two different vectors U = (Ux' and V = (Vx' Vy' v,,). Explicitly write in terms of Ux,y,z and VX,y,z' . Construct a spherical tensor of rank 2 out of two different vectors U and V. Write down explicitly in terms of Ux , y,z and V<, y,I' :onsider a spinless particle bound to a fixed center by acentral force otential. 11"" Problems 247 a. Relate, as much as possible, the matrix elements (n', I', m'l+ ± iy )In, I, m) and (n', I', m'lzln, I, m) 2K. using only the Wigner-Eckart theorem. Make sure to state under what conditions the matrix elements are nonvanishing. b. Do the same problem using wave functions 1/1 (x) = R nl( r)yt( 0, <p ). a. Write xy, xz, and (x 2- y2) as components of a spherical (irreduci-ble) tensor of rank 2. b. The expectation value Q == e(a, j, m = jl(3z 2- r2)la, j, m = j) is known as the quadrupole moment. Evaluate e(a, j, m'l(x 2 - y2)la, j, m = j), (where m' = j, j -1, j - 2, ... ) in terms of Q and appropriate Clebsch-Gordan coefficients. 21). A spin nucleus situated at the origin is subjected to an external inhomogeneous electric field. The basic electric quadrupole interaction may by taken' to be H -eQ [( J 2 <P) S2 + ( J 2 <P) S2 + ( J 2 <P) S2] int- 2s(s-1)n2 Jx2 0 x Jy2 0 Y JZ 2 0 z , where <p is the electrostatic potential satisfying Laplace's equation and the coordinate axes are so chosen that II I ( J 2 <P) JxJy 0 = (J 2 <P) (J 2 <P) JyJz 0 = JxJz 0 =0. Show that the interaction energy can be written as A (3Sz 2- 8 2 ) + B(si + and express A and B in terms of (J 2 <pI Jx 2)o and so on. Determine the energy eigenkets (in terms of 1m), where m = ± ± and the corre-sponding energy eigenvalues. Is there any degeneracy? |
5705 | https://www.jamestanton.com/wp-content/uploads/2012/03/Curriculum-Newsletter_October-2013.pdf | TANTON’S TAKE ON … THALE’S THEOREM OCTOBER 2013 Let’s start with an action puzzle: PAPER PUSHING: Draw two dots on a white board and slide a piece of paper up between the dots as shown. Mark the position of the corner of the paper. Now slide the paper up between the dots at a different angle and mark where the top corner lies this time. Repeat at least 15 more times. What shape curve does the trace of dots seem to describe? Can you prove any claims you make? Thales of Ionia: Geometry, like all of mathematics, is an intensely human enterprise. Its name, from the Greek ge for “earth” and metria for “measure,” suggests that geometry evolved from practical concerns – the art of accurately measuring tracts of land, performing feats of navigation, and accomplishing success in construction and architectural design, and the like. But as soon as one starts a study of the subject, one is drawn from the products of the hand to beautiful and profound products of the mind. Geometry is a gateway to a universe of intellectual richness and surprise. Greek mathematician Thales of Ionia (ca 600 B.C.E.) is dubbed the “Father of Geometry” for his first attempts to identify and establish geometric theorems (some 300 years before Euclid). The fellow impressed the Egyptians by calculating the height of the Great Pyramid from measuring the length of its shadow. He did this on the day of the year that the Sun set directly behind the pyramid (so that the shape of the shadow was an isosceles triangle) at the exact time of day that the length of his own shadow matched his own height. Height = 2 B L + www.jamestanton.com and www.gdaymath.com Thales is also credited as the first to explicitly detail a logical proof of a geometric result. Thales’ theorem, as it is known today, states: Whenever an angle is drawn from the diameter of a circle to a point on its circumference, then the angle formed is sure to be a perfect right angle. Proving Thales’ Theorem: Thales was at the point of his understanding in geometry to believe two things: • The three interior angles of any given triangle sum to 180. • The two base angles of an isosceles triangle are congruent. If you are at a point in your understanding to believe (or be able to prove?) these two points too, then you are all set to establish Thales’ theorem. Draw in an extra radius and mark congruent base angles of the isosceles triangles formed. We see: ( ) 180 a a b b + + + = . Thus 2 2 180 a b + = giving 90 a b + = . We have a right angle. Back to Paper Pushing: If you conduct the paper-pushing experiment it is very tempting to believe that the curve emerging before your very eyes is a perfect semi-circle. Fortunately our intuition is correct! (That is not always the case in geometry.) This claim does not follow from Thales’ theorem. Thales starts with a circle and shows the construction of a right angle from it. The paper-pushing experiment starts with a right angle and constructs, allegedly, a circle from it. It is, in some sense, the converse of Thales’ theorem and requires its own proof. Creating (and notice I said “creating”) a proof of the paper-pushing claim makes for a great student discussion. If we think the curve is a perfect semi-circle, where do you think the center of that semicircle lies? Most everyone suspects the center to be the midpoint of line segment connecting the two original dots on the board. If we label those points A and B , and the midpoint M , we are led pondering the properties of a diagram like the following. (Here the point C is the corner of the paper pushed between A and B .) What would we like to be true in this diagram? What would establish that a plot of points like C is a semi-circle? We need to show that the point C is the same distance from M as each of A and B are. www.jamestanton.com and www.gdaymath.com So, can we prove MC equals MA and MB ? At this point I stop talking and wait for someone in the class to have a brilliant flash of insight. (And I assure you, that flash of insight has never failed to come – usually after about 20 minutes of silence or mostly silence!) Here’s the flash: Draw a rotated copy of triangle ABC like so. This figure looks like a rectangle with segment MC half of one of its diagonals. To prove it is a rectangle, notice that the two angles x and y in the original triangle sum to 90. Consequently all four interior angles of the quadrilateral we see are indeed right angles. Since the figure is a rectangle we know its diagonals bisect each other and are equal in length. Consequently, the length MC does have the same value as lengths MA and MB . (The Pythagorean Theorem shows the diagonals of a rectangle are the same length and congruent triangles – look for congruent alternate interior angles- establish that the diagonals bisect one another.) The properties of quadrilaterals establish a beautiful result about the construction of circles. How delightful! Here’s how you find the diameter of a drinking glass: Trace the rim of the drinking glass on a piece of paper. Lay the corner of the second piece of paper on the circle you just drew as shown. Mark where the edges of this paper intersect the circle and measure the distance between those two points. This is the diameter of your drinking glass. Exercise: Using nothing more than a piece of paper and a pencil mark, with high accuracy, the center of THIS VERY CIRCLE! www.jamestanton.com and www.gdaymath.com An extra result: Consider a right triangle constructed on the diameter of a circle as shown: Suppose the altitude of this right triangle divides the diameter into two parts of lengths x and y .Show that the altitude has length: h xy = . HINT: Explain why the two angles marked with a dot are congruent. Use similar triangles to establish / / h x y h = . Any mathematics discussion, including the ones each day in the classroom, should really attend to a story, with any one question and its resolution naturally pulling to a natural next question. Mathematics perpetually unfolds. So what is a natural next question to this discussion on Thales’ Theorem and its converse? One immediately comes to my mind … Why stick with 90 degrees? Cut a piece of paper so that its corner angle is not 90 degrees. What curve is traced by this corner if it is pushed up between two tacks on a wall at different angles? Watch your entire unit on circle theorems unfold before your very eyes with your students! © 2013 James Tanton tanton.math@gmail.com |
5706 | https://readingfeynman.org/tag/relativistic-energy/ | relativistic energy – Reading Feynman
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Field energy and field momentum
This post goes to the heart of the E = m c 2, equation. It’s kinda funny, because Feynman just compresses all of it in a sub-section of his Lectures. However, as far as I am concerned, I feel it’s a very crucial section. Pivotal, I’d say, which would fit with its place in all of the 115 Lectures that make up the three volumes, which is sort of mid-way, which is where we are here.So let’s get go for it.
Let’s first recall what we wrote about the Poynting vector S, which we calculate from the magnetic and electric field vectors E and B by taking their cross-product:
This vector represents the energy flow, per unit area and per unit time, in electrodynamical situations. If E and/or Bare zero (which is the case in electrostatics, for example, because we don’t have magnetic fields in electrostatics), then S is zero too, so there is no energy flow then. That makes sense, because we have no moving charges, so where would the energy go to?
I also made it clear we should think of S as something physical, by comparing it to the heat flow vectorh, which we presented when discussing vector analysis and vector operators. The heat flow out of a surface element da is the area times the component of h perpendicular to da, so that’s (h•n)·da = h n·da. Likewise, we can write(S•n)·da = S n·da. The units of S and h are also the same:joule per second and per square meter or, using the definition of the watt(1 W = 1 J/s), in watt per square meter.In fact, if you google a bit, you’ll find that both h and S are referred to as a flux density:
The heat flow vector h is the heat flux density vector, from which we get the heat flux through an area through the (h•n)·da = h n·da product.
The energy flow Sis the energy flux density vector, from which we get the energy flux through the (S•n)·da = S n·da product.
So that should be enough as an introduction to what I want to talk about here. Let’s first look at the energy conservation principle once again.
Local energy conservation
In a way, you can look atmy previous postas being all about the equation below, which we referred to as the ‘local’ energy conservation law:
Of course, it is not the complete energy conservation law. The local energy is not only in the field. We’ve got matter as well, and so that’s what I want to discuss here: we want to look at the energy in the field as well as the energy that’s in the matter. Indeed, field energy is conserved, and then it isn’t: if the field is doing work on matter, or matter is doing work on the field, then… Well… Energy goes from one to the other, i.e. from the field to the matter or from the matter to the field. So we need to include matter in our analysis, which we didn’t do in our last post. Feynman gives the following simple example: we’re in a dark room, and suddenly someone turns on the light switch. So now the room is full of field energy—and, yes, I just mean it’s not dark anymore. :-). So that means some matter out there must have radiated its energy out and, in the process, it must have lost the equivalent mass of that energy. So, yes, we had matter losing energy and, hence, losing mass.
Now, we know that energy and momentum are related. Respecting and incorporating relativity theory, we’ve got two equivalent formulas for it:
E 2− p 2 c 2= m 0 2 c 4
p c= E·(v/c)⇔ p = v·E/c 2= m·v
The E= m c 2 and m = ·m 0·(1−v 2/c 2)−1/2 formulas connect both expressions. So we can look at it in either of two ways. We could use the energy conservation law, but Feynman prefers the conservation of momentum approach, so let’s see where he takes us. If the field has some energy(and, hence, some equivalent mass) per unit volume, and if there’s some flow, so if there’s some velocity(which there is: that’s what our previous post was all about), then it will have a certain momentum per unit volume. [Remember: momentum is mass times velocity.] That momentum will have a direction, so it’s a vector, just like p = mv. We’ll write it as g, so we defineg as:
g is the momentum of the field per unit volume.
What units would we express it in? We’ve got a bit of choice here. For example, because we’re relating everything to energy here, we may want to convert our kilogram into eV/c 2 or J/c 2 units, using the mass-energy equivalence relation E= m c 2. Hmm… Let’s first keep the kg as a measure of inertia though. So we write: [g] = [m]·[v]/m 3= (kg·m/s)/m 3. Hmm… That doesn’t show it’s energy, so let’s replace the kg with a unit that’s got newton and meter in it, cf. the F = ma law. So we write:[g] = (kg·m/s)/m 3=(kg/s)/m 2= [(N·s 2/m)/s]/m 2=N·s/m 3. Well… OK. The newton·second is the unit of momentum indeed, and we can re-write it including the joule (1 J = 1 N·m), so then we get [g] = (J·s/m 4), so what’s that? Well… Nothing much. However, I do note it happens to be the dimension of S/c 2, so that’s [S/c 2] = [J/(s·m 2)]·(s 2/m 2) =(J·s/m 4). Let’s continue the discussion.
Now, momentum is conserved, and each component of it is conserved. So let’s look at the x-direction. We should have something like:
If you look at this carefully, you’ll probably say: “OK. I understood the thing with the dark room and light switch. Mass got converted into field energy, but what’s that second term of the left?”
Good. Smart. Right remark. Perfect. […] Let me try to answer the question. While all of the quantities above are expressed per unit volume, we’re actually looking at the same infinitesimal volume element here, so the example of the light switch is actually an example of a ‘momentum outflow’, so it’s actually an example of that second term of the left-hand side of the equation above kicking in!
Indeed, the first term just sort of reiterates the mass-energy equivalence: the energy that’s in the matter can become field energy, so to speak, in our infinitesimal volume element itself, and vice versa. But if it doesn’t, then it should get out and, hence, become ‘momentum outflow’. Does that make sense? No?
Hmm… What to say? You’ll need to look at that equation a couple of times more, I guess. But I need to move on, unfortunately. [Don’t get put off when I say things like this: I am basically talking to myself, so it means I’ll need to re-visit this myself. :-/]
Let’s look at all of the three terms:
The left-hand side (i.e. the time rate-of-change of the momentum of matter) is easy. It’s just the force on it, which we know is equal to F =q(E+v×B). Do we know that? OK… I’ll admit it.Sometimes it’s easy to forget where we are in an analysis like this, but so we’re looking at the electromagnetic force here. As we’re talking infinitesimals here and, therefore, charge density rather than discrete charges, we should re-write this as the force per unit volume which is ρE+j×B. [This is an interesting formula which I didn’t use before, so you should double-check it. :-)]
The first term on the right-hand side should be equally obvious, or… Well… Perhaps somewhat less so. But with all my rambling on the Uncertainty Principleand/or the wave-particle duality, it should make sense. If we scrap the second term on the right-hand side, we basically have an equation that is equivalent to the E = m c 2 equation. No? Sorry. Just look at it, again and again. You’ll end up understanding it.
So it’s that second term on the right-hand side. What the hell does that say? Well… I could say: it’s the local energy or momentum conservation law. If the energy or momentum doesn’t stay in, it has to go out. But that’s not very satisfactory as an answer, of course. However, please just go along with this‘temporary’ answer for a while.
So what is that second term on the right-hand side? As we wrote it, it’s an x-component – or, let’s put it differently, it is or was part of the x-component of the momentum density –but, frankly, we should probably allow it to go out in any direction really, as the only constraint on the left-hand side is a per second rate of change of something. Hence, Feynman suggest to equate it to something like this:
What a, b and c? The components of some vector? Not sure. We’re stuck. This piece really requires very advanced math. In fact, as far as I know, this is the only time where Feynman says: “Sorry. This is too advanced. I’ll just give you the equation.Sorry.” So that’s what he does. He explains the philosophy of the argument, which is the following:
On the left-hand side, we’ve got the time rate-of-change of momentum, so that obeys the F = dp/dt = d(mv)/dt law, with the force F,per unit volume, being equal to F(unit volume) =ρE+j×B.
On the right-hand side, we’ve got something that can be written as:
So we’d need to find a way to ρE+j×Bin terms of EandBonly– eliminating ρ andjby using Maxwell’s equations or whatever other trick – and then juggle terms and make substitutions to get it into a form that looks like the formula above, i.e. the right-hand side of that equation. But so Feynman doesn’t show us how it’s being done. He just mentions some theorem in physics, which says that the energy that’s flowing through a unit area per unit time divided by c 2– so that’s E/c 2 per unit area and per unit time– mustbe equal to the momentum per unit volume in the space, so we write:
g = S/c 2
He illustrates the general theorem that’s used to get the equation above by giving two examples:
OK. Two good examples. However, it’s still frustrating to not see how we get the g = S/c 2 in the specific context of the electromagnetic force, so let’s do a dimensional analysis at least. In my previous post, I showed that the dimension ofSmust be J/(m 2·s), so [S/c 2] = [J/(m 2·s)]/(m 2/s 2) = [N·m/(m 2·s)]·(s 2/m 2) = [N·s/m 3]. Now, we know that the unit of mass is 1 kg = N/(m/s 2). That’s just the force law: a force of 1 newton will give a mass of 1 kg an acceleration of 1 m/s per second, so 1 N = 1 kg·(m/s 2). So the[N·s/m 3] dimension is equal to [kg·(m/s 2)·s/m 3] = [(kg·(m/s)/m 3] =[(kg·(m/s)]/m 3, which is the dimension of momentum (p = mv) per unit volume, indeed.So, yes, the dimensional analysis works out, and it’s also in line with thep = v·E/c 2= m·v equation, but… Oh… We did a dimensional analysis already, where we also showed that [g]= [S/c 2] = (J·s/m 4). Well… In any case… It’s a bit frustrating to not see the detail here, but let us note the the Grand Resultonce again:
The Poynting vectorSgives us the energy flow as well as the momentum density g= S/c 2.
But what does it all mean, really? Let’s go through Einstein’s illustration of the principle. That will help us a lot. Before we do, however, I’d like to note something. I’ve always wondered a bit about that dichotomy between energy and momentum. Energy is force times distance: 1 joule is 1 newton× 1 meter indeed (1 J = 1 N·m). Momentum is force times time, as we can express it in N·s. Planck’s constant h combines all three in the dimension of action, which is force times distance times time: h≈6.6×10−34 N·m·s, indeed. I like that unity. In this regard, you should, perhaps, quickly review that post in which I explain that h is the energy per cycle, i.e. per wavelength or per period, of a photon, regardless of its wavelength. So it’s really something very fundamental.
We’ve got something similar here: energy and momentum coming together, and being shown as one aspect of the same thing: some oscillation. Indeed, just see what happens with the dimensions when we ‘distribute’ the 1/c 2 factor on the right-hand side over the two sides, so we write:c·g= S/c and work out the dimensions:
[c·g]= (m/s)·(N·s)/m 3= N/m 2= J/m 3.
[S/c] =(s/m)·(N·m)/(s·m 2) = N/m 2= J/m 3.
Isn’t that nice? Both sides of the equation now have a dimension like ‘the force per unit area’, or ‘the energy per unit volume’. To get that, we just re-scaledg and S, by c and 1/c respectively. As far as I am concerned, this shows an underlying unity we probably tend to mask with our ‘related but different’ energy and momentum concepts. It’s like E and B: I just love it we can write them together in our Poynting formula S= ε 0 c 2E×B. In fact, let me show something else here, which you should think about. You know that c 2= 1/(ε 0 μ 0), so we can write Salso as S =E×B/μ 0. That’s nice, but what’s nice too is the following:
S/c = c·g= ε 0 cE×B= E×B/μ0c
S/g= c 2= 1/(ε 0 μ 0)
So, once again, Feynman may feel the Poynting vector is sort of counter-intuitive when analyzing specific situations but, as far as I am concerned, I feel the Poyning vector makes things actually easier to understand. Instead of two E and B vectors, and two concepts to deal with ‘energy’ (i.e. energy and momentum), we’re sort of unifying things here. In that regard – i.e in regard of feeling we’re talking the same thing really – I’d really highlight the S/g= c 2 = 1/(ε 0 μ 0) equation. Indeed, the universal constant c acts just like the fine-structure constant here: it links everything to everything.
And, yes, it’s also about time we introduce the so-called principle of least action to explain things, because action, as a concept, combines force, distance and time indeed, so it’s a bit more promising than just energy, of just momentum. Having said that, you’ll see in the next section that it’s sometimes quite useful to have the choice between one formula or the other. But… Well…Enough talk. Let’s look at Einstein’s car.
Einstein’s car
Einstein’s car is a wonderful device: it rolls without any friction and it moves with a little flashlight. That’s all it needs. It’s pictured below. So the situation is the following: the flashlight shoots some light out from one side, which is then stopped at the opposite end of the car. When the light is emitted, there must be some recoil. In fact, we know it’s going to be equal to 1/c times the energy because all we need to do is apply the p c= E·(v/c) formula for v = c, so we know that p = E/c. Of course, this momentum now needs to move Einstein’s car. It’s frictionless, so it should work, but still… The car has some mass M, and so that will determine its recoil velocity: v = p/M. We just apply the general p = mv formula here, and v is not equal to c here, of course! Of course, then the light hits the opposite end of the car and delivers the same momentum, so that stops the car again. However, it did move over some distance x = vt. So we could flash our light again and get to wherever we want to get. [Never mind the infinite accelerations involved!] So… Well… Great! Yes, but Einstein didn’t like this car when he first saw it. In fact, he still doesn’t like it, because he knows it won’t take you very far.
The problem is that we seem to be moving the center of gravity of this car by fooling around on the inside only. Einstein doesn’t like that. He thinks it’s impossible. And he’s right of course. The thing is: the center of gravity did not change. What happened here is that we’ve got some blob of energy, and so that blob has some equivalent mass (which we’ll denote by U/c 2), and so that equivalent mass moved all the way from one side to the other, i.e. over the length of the car, which we denote by L. In fact, it’s stuff like this that inspired the whole theory of the field energy and field momentum, and how it interacts with matter.
What happens here is like switching the light on in the dark room: we’ve got matter doing work on the field, and so matter loses mass, and the field gains it, through its momentum and/or energy. To calculate how much, we could integrate S/c or c·gover the volume of our blob, and we’d get something in joule indeed, but there’s a simpler way here. The momentum conservation says that the momentum of our car and the momentum of our blob must be equal, so if T is the time that was needed for our blob to go to the other side – and so that’s, of course, also the time during which our car was rolling–then M·v = M·x/T must be equal to (U/c 2)·c=(U/c 2)·L/T. The 1/T factor on both sides cancel, so we write:M·x = (U/c 2)·L.Now, what is x? Yes. In case you were wondering, that’s what we’re looking for here. Here it is:
x = vT = vL/c = (p/M)·(L/c) = [U/c)/M]·(L/c) = (U/c 2)·(L/M)
So what’s next? Well… Now we need to show that the center-of-mass actually did not move with this ‘transfer’ of the blob. I’ll leave the math to you here: it should all work out. And you can also think through the obvious questions:
Where is the energy and, hence, the mass of our blob after it stops the car? Hint: think about excited atoms and imagine they might radiate some light back.
As the car did move a little bit, we should be able to move it further and further away from its center of gravity, until the center of gravity is no longer in the car. Hint: think about batteries and energy levels going down while shooting light out. It just won’t happen.
Now, what about a blob of light going from the top to the bottom of the car? Well… That involves the conservation of angular momentum: we’ll have more mass on the bottom, but on a shorter lever-arm, so angular momentum is being conserved. It’s a very good question though, and it led Einstein to combine the center-of-gravity theorem with the angular momentum conservation theorem to explain stuff like this.
It’s all fascinating, and one can think of a great many paradoxes that, at first, seem to contradict the Grand Principles we used here, which means that they would contradict all that we have learned so far. However, a careful analysis of those paradox reveals that they are paradoxes indeed:propositions which sound true but are, in the end,self-contradictory. In fact, when explaining electromagnetism over his various Lectures, Feynman tasks his readers with a rather formidable paradox when discussing the laws of induction, he solves it here, ten chapters later, after describing what we described above. You can busy yourself with it but… Well… I guess you’ve got something better to do. If so, just take away the key lesson: there’s momentum in the field, and it’s also possible to build up angular momentum in a magnetic field and, if you switch it off, the angular momentum will be given back, somehow, as it’s stored energy.
That’s also why the seemingly irrelevant circulation of Swe discussed in my previous post, where we had a charge next to an ordinary magnet, and where we found that there was energy circulating around, is not so queer. The energy is there, in the circulating field, and it’s real. As real as can be.
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Jean Louis Van BelleMathematics, Philosophy of science, Physics3 CommentsSeptember 29, 2015 September 30, 201514 Minutes
On (special) relativity: what’s relative?
Pre-scriptum (dated 26 June 2020): These posts on elementary math and physics have not suffered much the attack by the dark force—which is good because I still like them. While my views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. In fact, I find the simplest stuff is often the best.
Original post:
This is my third and final post about special relativity. In the previous posts, I introduced the general idea and the Lorentz transformations. I present these Lorentz transformations once again below, next to their Galilean counterparts. [Note that I continue to assume, for simplicity, that the two reference frames move with respect to each other along the x- axis only, so the y- and z-component of u is zero. It is not all that difficult to generalize to three dimensions (especially not when using vectors) but it makes an intuitive understanding of what’s relativity all about more difficult.]
As you can see, under a Lorentz transformation, the new ‘primed’ space and time coordinates are a mixture of the ‘unprimed’ ones. Indeed, the new x’ is a mixture of x and t, and the new t’ is a mixture as well. You don’t have that under a Galilean transformation: in the Newtonian world, space and time are neatly separated, and time is absolute, i.e. it is the same regardless of the reference frame. In Einstein’s world – our world – that’s not the case: time is relative, or local as Hendrik Lorentz termed it,and so it’s space-time– i.e. ‘some kind of union of space and time’ as Minkowski termed it – that transforms. In practice, physicists will use so-called four-vectors, i.e. vectors with four coordinates, to keep track of things. These four-vectors incorporate both the three-dimensional space vector as well as the time dimension. However, we won’t go into the mathematical details of that here.
What else is relative? Everything, except the speed of light. Of course, velocity is relative, just like in the Newtonian world, but the equation to go from a velocity as measured in one reference frame to a velocity as measured in the other, is different: it’s not a matter of just adding or subtracting speeds. In addition, besides time, mass becomes a relative concept as well in Einstein’s world, and that was definitely not the case in the Newtonian world.
What about energy? Well… We mentioned that velocities are relative in the Newtonian world as well, so momentum and kinetic energy were relative in that world as well: what you would measure for those two quantities would depend on your reference frame as well. However, here also, we get a different formula now. In addition, we have this weird equivalence between mass and energy in Einstein’s world, about which I should also say something more.
But let’s tackle these topics one by one. We’ll start with velocities.
Relativistic velocity
In the Newtonian world, it was easy. From the Galilean transformation equations above, it’s easy to see that
v’ = dx’/dt’ = d(x– ut)/dt = dx/dt– d(ut)/dt = v– u
So, in the Newtonian world, it’s just a matter of adding/subtracting speeds indeed: if my car goes 100 km/h (v), and yours goes 120 km/h, then you will see my car falling behind at a speed of (minus) 20 km/h. That’s it. In Einstein’s world, it is not so simply. Let’s take the spaceship example once again. So we have a man on the ground (the inertial or ‘unprimed’ reference frame) and a man in the spaceship (the primed reference frame), which is moving away from us with velocity u.
Now, suppose an object is moving inside the spaceship (along the x-axis as well) with a (uniform) velocity v x’, as measured from the point of view of the man inside the spaceship. Then the displacement x’ will be equal to x’ =v x’ t’. To know how that looks from the man on the ground, we just need to use the opposite Lorentz transformations: just replace u by –u everywhere (to the man in the spaceship, it’s like the man on the ground moves away with velocity –u), and note that the Lorentz factor does not change because we’re squaring and (–u)2 =u 2. So we get:
Hence,x’=v x’ t’ can be written as x = γ(v x’ t’ + ut’). Now we should also substitute t’, because we want to measure everything from the point of view of the man on the ground. Now, t = γ(t’ + u _v_ _x’_ _t’_/c 2). Because we’re talking uniform velocities, v x(i.e. the velocity of the object as measured by the man on the ground) will be equal to x divided by t(so we don’t need to take the time derivative of x), and then, after some simplifying and re-arranging (note, for instance, how the t’ factor miraculously disappears), we get:
What does this rather complicated formula say? Just put in some numbers:
Suppose the object is moving at half the speed of light, so 0.5 c, and that the spaceship is moving itself also at 0.5 c, then we get the rather remarkable result that, from the point of view of the observer on the ground, that object is not going as fast as light, but only at v x= (0.5 c + 0.5 c)/(1 + 0.5·0.5) = 0.8 c.
Or suppose we’re looking at a light beam inside the spaceship, so something that’s traveling at speed c itself in the spaceship. How does that look to the man on the ground? Just put in the numbers: v x= (0.5 c+ c)/(1 + 0.5·1) = c! So the speed of light is not dependent on the reference frame: it looks the same– both to the man in the ship as well as to the man on the ground. As Feynman puts it: “This is good, for it is, in fact, what the Einstein theory of relativity was designed to do in the first place–so it had better work!”
It’s interesting to note that, even if u has no y– or z-component, velocity in the y direction will be affected too. Indeed, if an object is moving upward in the spaceship, then the distance of travel of that object to the man on the ground will appear to be larger. See the triangle below: if that object travels a distance Δs’ = Δy’ = Δy = v’Δt’ with respect to the man in the spaceship, then it will have traveled a distance Δs = vΔt to the man on the ground, and that distance is longer.
I won’t go through the process of substituting and combining the Lorentz equations (you can do that yourself) but the grand result is the following:
v y=(1/γ)v y’
1/γ is the reciprocal of the Lorentz factor, and I’ll leave it to you to work out a few numeric examples. When you do that, you’ll find the rather remarkable result that v y is actually less than v y’. For example, for u = 0.6 c, 1/γ will be equal to 0.8, so v y will be 20% less than v y’. How is that possible? The vertical distance is what it is (Δy’ =Δy), and that distance is not affected by the ‘length contraction’ effect (y’ = y). So how can the vertical velocity be smaller?The answer is easy to state, but not so easy to understand: it’s the time dilation effect: time in the spaceship goes slower. Hence, the object will cover the same vertical distance indeed– for both observers –but, from the point of view of the observer on the ground, the object will apparently need more time to cover that distance than the time measured by the man in the spaceship: Δt > Δt’. Hence, the logical conclusion is that the vertical velocity of that object will appear to be less to the observer on the ground.
How much less? The time dilation factor is the Lorentz factor. Hence,Δt = γΔt’. Now, if u= 0.6 c, then γ will be equal to 1.25 and Δt = 1.25Δt’. Hence, if that object would need, say, one second to cover that vertical distance, then, from the point of view of the observer on the ground, it would need 1.25 seconds to cover the same distance. Hence, its speed as observed from the ground is indeed only 1/(5/4) = 4/5 = 0.8 of its speed as observed by the man in the spaceship.
Is that hard to understand? Maybe. You have to think through it. One common mistake is that people think that length contraction and/or time dilation are, somehow, related to the fact that we are looking at things from a distance and that light needs time to reach us. Indeed, on the Web, you can find complicated calculations using the angle of view and/or the line of sight (and tons of trigonometric formulas) as, for example, shown in the drawing below. These have nothing to do with relativity theory and you’ll never get the Lorentz transformation out of them. They are plain nonsense: they are rooted in an inability of these youthful authors to go beyond Galilean relativity. Length contraction and/or time dilation are not some kind of visual trick or illusion. If you want to see how one can derive the Lorentz factor geometrically, you should look for a good description of the Michelson-Morley experiment in a good physics handbook such as, yes :-), Feynman’s Lectures.
So, I repeat: illustrations that try to explain length contraction and time dilation in terms of line of sight and/or angle of view are useless and will not help you to understand relativity. On the contrary, they will only confuse you. I will let you think through this and move on to the next topic.
Relativistic mass and relativistic momentum
Einstein actually stated two principles in his (special) relativity theory:
The first is the Principle of Relativity itself, which is basically just the same as Newton’s principle of relativity. So that was nothing new actually: “If a system of coordinates K is chosen such that, in relation to it, physical laws hold good in their simplest form, then the same laws must hold good in relation to any other system of coordinates K’ moving in uniform translation relatively to K.” Hence, Einstein did not change the principle of relativity– quite on the contrary: he re-confirmed it–but he did change Newton’s Laws, as well as the Galilean transformation equations that came with them. He also introduced a new ‘law’, which is stated in the second ‘principle’, and that the more revolutionary one really:
The Principle of Invariant Light Speed: “Light is always propagated in empty space with a definite velocity [speed]c which is independent of the state of motion of the emitting body.”
As mentioned above, the most notable change in Newton’s Laws – the only change, in fact –is Einstein’s relativistic formula for mass:
m v= γm 0
This formula implies that the inertia of an object, i.e. its mass, also depends on the reference frame of the observer. If the object moves (but velocity is relative as we know: an object will not be moving if we move with it), then its mass increases. This affects its momentum. As you may or may not remember, the momentum of an object is the product of its mass and its velocity. It’s a vector quantity and, hence, momentum has not only a magnitude but also a direction:
pv= m vv= γm 0v
As evidenced from the formula above, the momentum formula is a relativistic formula as well, as it’s dependent on the Lorentz factor too. So where do I want to go from here? Well… In this section (relativistic mass and momentum), I just want to show that Einstein’s mass formula is not some separate law or postulate: it just comes with the Lorentz transformation equations (and the above-mentioned consequences in terms of measuring horizontal and vertical velocities).
Indeed, Einstein’s relativistic mass formula can be derived from the momentum conservation principle, which is one of the ‘physical laws’ that Einstein refers to. Look at the elastic collision between two billiard balls below. These balls are equal – same mass and same speed from the point of view of an inertial observer – but not identical: one is red and one is blue. The two diagrams show the collision from two different points of view: left, we have the inertial reference frame, and, right, we have a reference frame that is moving with a velocity equal to the horizontal component of the velocity of the blue ball.
The points to note are the following:
The total momentum of such elastic collision before and after the collision must be the same.
Because the two balls have equal mass (in the inertial reference frame at least), the collision will be perfectly symmetrical. Indeed, we may just turn the diagram ‘upside down’ and change the colors of the balls, as we do below, and the values w, u and v (as well as the angle α) are the same.
As mentioned above, the velocity of the blue and red ball and, hence, their momentum, will depend on the frame of reference. In the diagram on the left, we’re moving with a velocity equal to the horizontal component of the velocity of the blue ball and, therefore, in this particular frame of reference,the velocity (and the momentum) of the blue ball consists of a vertical component only, which we refer to as w.
From this point of view (i.e. the reference frame moving with, the velocity (and, hence, the momentum) of the red ball will have both a horizontal as well as a vertical component. If we denote the horizontal component by u, then it’s easy to show that the vertical velocity of the red ball must be equal to sin(α)v. Now, because u = cos(α)v, this vertical component will be equal to tan(α)u. But so what is tan(α)u? Now, you’ll say, that is quite evident: tan(α)u must be equal to w, right?
No. That’s Newtonian physics. The red ball is moving horizontally with speed u with respect to the blue ball and, hence, its vertical velocity will not be quite equal to w. Its vertical velocity will be given by the formula which we derived above:v y=(1/γ)v y’, so it will be a little bit slower than the w we see in the diagram on the right which is, of course, the same w as in the diagram on the left. [If you look carefully at my drawing above, then you’ll notice that the w vector is a bit longer indeed.]
Huh? Yes. Just think about it: tan(α)u=(1/γ)w. But then… How can momentum be conserved if these speeds are not the same? Isn’t the momentum conservation principle supposed to conserve both horizontal as well as vertical momentum? It is, and momentum is being conserved. Why? Because of the relativistic mass factor.
Indeed, the change in vertical momentum (Δp) of the blue ball in the diagram on the left or – which amounts to the same – the red ball in the diagram on the right (i.e. the vertically moving ball) is equal to Δp blue = 2m w w. [The factor 2 is there because the ball goes down and then up (or vice versa) and, hence, the total change in momentum must be twice the m w w amount.] Now, that amount must be equal to Δp red, which is equal to Δp blue= 2m v(1/γ)w . Equating both yields the following grand result:
m v/m _w_=γ⇔m v=γm w
What does this mean? It means that mass of the red ball in the diagram on the left is larger than the mass of the blue ball. So here we have actually derived Einstein’s relativistic mass formula from the momentum conservation principle !
Of course you’ll say: not quite. This formula is not the m u=γm 0 formula that we’re used to ! Indeed, it’s not. The blue ball has some velocity w itself, and so the formula links two velocities v and w. However, we can derive m v=γm 0 formula as a limit of m v=γm w for w going to zer0. How can w become infinitesimally small? If the angle α becomes infinitesimally small. It’s obvious, then, that v and u will be practically equal. In fact, if w goes to zero, then m w will be equal to m 0 in the limiting case, and m v will be equal to m u. So, then, indeed, we get the familiar formula as a limiting case:
m u=γm 0
Hmm… You’ll probably find all of this quite fishy. I’d suggest you just think about it. What I presented above, is actually Feynman’s presentation of the subject, but with a bit more verbosity. Let’s move on to the final.
Relativistic energy
From what I wrote above (and from what I wrote in my two previous posts on this topic), it should be obvious, by now, that energy also depends on the reference frame. Indeed, mass and velocity depend on the reference frame (moving or not), and both appear in the formula for kinetic energy which, as you’ll remember, is
K.E. = m c 2– m 0 c 2= (m–m 0)c 2= γm 0 c 2– m 0 c 2= m 0 c 2(γ– 1).
Now, if you go back to the post where I presented that formula, you’ll see that we’re actually talking the change in kinetic energy here: if the mass is at rest, it’s kinetic energy is zero (because m = m 0), and it’s only when the mass is moving, that we can observe the increase in mass. [If you wonder how, think about the example of the fast-moving electrons in an electron beam: we see it as an increase in the inertia: applying the same force does no longer yield the same acceleration.]
Now, in that same post, I also noted that Einstein added an equivalent rest mass energy(E 0= m 0 c 2) to the kinetic energy above, to arrive at the total energy of an object:
E =E 0+ K.E. =m c 2
Now, what does this equivalence actually mean? Is mass energy? Can we equate them really? The short answer to that is: yes.
Indeed, in one of my older posts (Loose Ends), I explained that protons and neutrons are made of quarks and, hence, that quarks are the actual matter particles, not protons and neutrons. However, the mass of a proton – which consists of two up quarks and one down quark – is 938 MeV/c 2(don’t worry about the units I am using here: because protons are so tiny, we don’t measure their mass in grams), but the mass figure you get when you add the rest mass of two u‘s and one d, is 9.6 MeV/c 2 only: about one percent of 938 ! So where’s the difference?
The difference is the equivalent mass (or inertia) of the binding energy between the quarks. Indeed, the so-called ‘mass’ that gets converted into energy when a nuclear bomb explodes is not the mass of quarks. Quarks survive: nuclear power is binding energy between quarks that gets converted into heat and radiation and kinetic energy and whatever else a nuclear explosion unleashes.
In short, 99% of the ‘mass’ of a proton or an electron is due to the strong force.So that’s ‘potential’ energy that gets unleashed in a nuclear chain reaction. In other words, the rest mass of the proton is actually the inertia of the system of moving quarks and gluons that make up the particle. In such atomic system, even the energy of massless particles (e.g. the virtual photons that are being exchanged between the nucleus and its electron shells) is measured as part of the rest mass of the system. So, yes, mass is energy. As Feynman put it, long before the quark model was confirmed and generally accepted:
“We do not have to know what things are made of inside; we cannot and need not justify, inside a particle, which of the energy is rest energy of the parts into which it is going to disintegrate. It is not convenient and often not possible to separate the total mc 2 energy of an object into (1) rest energy of the inside pieces, (2) kinetic energy of the pieces, and (3) potential energy of the pieces; instead we simply speak of the total energy of the particle. We ‘shift the origin’ of energy by adding a constant m 0 c 2 to everything, and say that the total energy of a particle is the mass in motion times c 2, and when the object is standing still, the energy is the mass at rest times c 2.” (Richard Feynman’s Lectures on Physics, Vol. I, p. 16-9)
So that says it all, I guess, and, hence, that concludes my little ‘series’ on (special) relativity. I hope you enjoyed it.
Post scriptum:
Feynman describes the concept of space-time with a nice analogy: “When we move to a new position, our brain immediately recalculates the true width and depth of an object from the ‘apparent’ width and depth. But our brain does not immediately recalculate coordinates and time when we move at high speed, because we have had no effective experience of going nearly as fast as light to appreciate the fact that time and space are also of the same nature. It is as though we were always stuck in the position of having to look at just the width of something, not being able to move our heads appreciably one way or the other; if we could, we understand now, we would see some of the other man’s time—we would see “behind”, so to speak, a little bit. Thus, we shall try to think of objects in a new kind of world, of space and time mixed together, in the same sense that the objects in our ordinary space-world are real, and can be looked at from different directions. We shall then consider that objects occupying space and lasting for a certain length of time occupy a kind of a “blob” in a new kind of world, and that when we look at this “blob” from different points of view when we are moving at different velocities. This new world, this geometrical entity in which the “blobs” exist by occupying position and taking up a certain amount of time, is called space-time.”
If none of what I wrote could convey the general idea, then I hope the above quote will. Apart from that, I should also note that physicists will prefer to re-write the Lorentz transformation equations by measuring time and distance in so-called equivalent units: velocities will be expressed not in km/h but as a ratio of c and, hence, c= 1 (a pure number) and so u will also be a pure number between 0 and 1. That can be done by expressing distance in light-seconds ( a light-second is the distance traveled by light in one second or, alternatively, by expressing time in ‘meter’. Both are equivalent but, in most textbooks, it will be time that will be measured in the ‘new’ units. So how do we express time in meter?
It’s quite simple: we multiply the old seconds with c and then we get:time expressed in meters= time expressed in seconds multiplied by 3×10 8 meters per second. Hence, as the ‘second’ the first factor and the ‘per second’ in the second factor cancel out, the dimension of the new time unit will effectively be the meter. Now, if both time and distance are expressed in meter, then velocity becomes a pure number without any dimension, because we are dividing distance expressed in meter by time expressed in meter, and it should be noted that it will be a pure number between 0 and 1 (0 ≤ u ≤ 1), because 1 ‘time second’ = 1/(3×10 8) ‘time meters’. Also, c itself becomes the pure number 1. The Lorentz transformation equations then become:
They are easy to remember in this form (cf. the symmetry between x–ut and t–ux) and, if needed, we can always convert back to the old units to recover the original formulas.
I personally think there is no better way to illustrate how space and time are ‘mere shadows’ of the same thing indeed: if we express both time and space in the same dimension (meter), we can see how, as result of that, velocity becomes a dimensionless number between zero and one and, more importantly, how the equations for x’ and t’ then mirror each other nicely. I am not sure what ‘kind of union’ between space and time Minkowski had in mind, but this must come pretty close, no?
Final note: I noted the equivalence of mass and energy above. In fact, mass and energy can also be expressed in the same units, and we actually do that above already. If we say that an electron has a rest mass of 0.511 MeV/c 2(a bit less than a quarter of the mass of the u quark), then we express the mass in terms of energy. Indeed, the eV is an energy unit and so we’re actually using the m = E/c 2 formula when we express mass in such units. Expressing mass and energy in equivalent units allows us to derive similar ‘Lorentz transformation equations’ for the energy and the momentum of an object as measured under an inertial versus a moving reference frame. Hence, energy and momentum also transform like our space-time four-vectors and – likewise – the energy and the momentum itself, i.e. the components of the (four-)vector, are less ‘real’ than the vector itself. However, I think this post has become way too long and, hence, I’ll just jot these four equations down – please note, once again, the nice symmetry between (1) and (2)– but then leave it at that and finish this post.
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Jean Louis Van BellePhysics3 CommentsMay 30, 2014 June 26, 202017 Minutes
Another post for my kids: introducing (special)relativity
Pre-scriptum (dated 26 June 2020): These posts on elementary math and physics have not suffered much the attack by the dark force—which is good because I still like them. While my views on the true nature of light, matter and the force or forces that act on them have evolved significantly as part of my explorations of a more realist (classical) explanation of quantum mechanics, I think most (if not all) of the analysis in this post remains valid and fun to read. In fact, I find the simplest stuff is often the best.
Original post:
In my previous post, I talked about energy, and I tried to keep it simple– but also accurate. However, to be completely accurate, one must, of course, introduce relativity at some point. So how does that work? What’s ‘relativistic’ energy? Well… Let me try to convey a few ideas here.
The first thing to note is that the energy conservation law still holds: special theory or not, the sum of the kinetic and potential energies in a (closed) system is always equal to some constant C.What constant? That doesn’t matter: Nature does not care about our zero point and, hence, we can add or subtract any(other) constant to the equation K.E. + P.E. = T + U = C.
That being said, in my previous post, I pointed out that the constant depends on the reference point for the potential energy term U: we will usually take infinity as the reference point (for a force that attracts) and associate it with zero potential (U = 0). We then get a function U(x) like the one below: for gravitational energy we have U(x) =–GMm/x, and for electrical charges, we have U(x) = q 1 q 2/4πε 0 x. The mathematical shape is exactly the same but, in the case of the electromagnetic forces, you have to remember that likes repel, and opposites attract,so we don’t need the minus sign: the sign of the charges takes care of it.
Minus sign? In case you wonder why we need that minus sign for the potential energy function, well… I explained that in my previous post and so I’ll be brief on that here: potential energy is measured by doing work against the force. That’s why. So we have an infinite sum (i.e. an integral) over some trajectory or path looking like this: U =– ∫F·ds.
For kinetic energy, we don’t need any minus sign: as an object picks up speed, it’s the force itself that is doing the work as its potential energy is converted into kinetic energy, so the change in kinetic energy will equal the change in potential energy, but with opposite sign: as the object loses potential energy, it gains kinetic energy. Hence, we write ΔT =–ΔU = ∫F·ds..
That’s all kids stuff obviously. Let’s go beyond this and ask some questions. First, why can we add or subtract any constant to the potential energy but not to the kinetic energy? The answer is… Well… We actually can add or subtract a ‘constant’ to the kinetic energy as well. Now you will shake your head: Huh? Didn’t we have that T = m v 2/2 formula for kinetic energy? So how and why could one add or subtract some number to that?
Well… That’s where relativity comes into play. The velocity v depends on your reference frame. If another observer would move with and/or alongside the object, at the same speed, that observer would observe a velocity equal to zero and, hence, its kinetic energy – as that observer would measure it – would also be zero. You will object to that, saying that a change of reference frame does not change the force, and you’re right: the force will cause the object to accelerate or decelerate indeed, and if the observer is not subject to the same force, then he’ll see the object accelerate or decelerate indeed, regardless of his reference frame is a moving or inertial frame. Hence, both the inertial as well as the moving observer will see an increase(or decrease) in its kinetic energy and, therefore, both will conclude that its potential energy decreases(or increases)accordingly. In short, it’s the change in energy that matters, both for the potential as well as for the kinetic energy. The reference point itself, i.e. the point from where we start counting so to say, does not: that’s relative. [This also shows in the derivation for kinetic energy which I’ll do below.]
That brings us to the second question. We all learned in high school that mass and energy are related through Einstein’s mass-energy relation, E = m c 2, which establishes an equivalence between the two: the mass of an object that’s picking up speed increases, and so we need to look at both speed and mass as a function of time. Indeed, remember Newton’s Law: force is the time rate of change of momentum: F = d(mv)/dt. When the speed is low (i.e. non-relativistic), then we can just treat m as a constant and write thatF= mdv/dt = ma(the mass times the acceleration). Treating m as a constant also allows us to derive the classical (Newtonian) formula for kinetic energy:
So if we assume that the velocity of the object at point O is equal to zero (so v o= 0), then ΔT will be equal to T and we get what we were looking for: the kinetic energy at point P will be equal to T = m v 2/2.
Now, y ou may wonder why we can’t do that same derivation for a non-constant mass? The answer to that question is simple: taking the m factor out of the integral can only be done if we assume it is a constant. If not, then we should leave it inside. It’s similar to taking a derivative. If m would not be constant, then we would have to apply the product rule to calculate d(mv)/dt, so we’d write d(m v)/dt = (dm/dt)v + m(dv/dt). So we have two terms here and it’s only when m is constant that we can reduce it to d(m v)/dt = m(dv/dt).
So we have our classical kinetic energy function. However, when the velocity gets really high – i.e. if it’s like the same order of magnitude as the velocity of light – then we cannot assume that mass is constant. Indeed, the same high-school course in physics that taught you that E = m c 2 equation will probably also have taught you that an object can never go faster than light, regardless of the reference frame. Hence, as the object goes faster and faster, it will pick up more momentum, but its rate of acceleration should (and will) go down in such way that the object can never actually reach the speed of light. Indeed, if Newton’s Law is to remain valid, we need to correct it such a way that m is no longer constant: m itself will increase as a function of its velocity and, hence, as a function of time. You’ll remember the formula for that:
This is often written as m =γm 0, with m 0 denoting the mass of the object at rest (in your reference frame that is) and γ = (1 –v 2/c 2)–1/2 the so-called Lorentz factor. The Lorentz factor is named after a Dutch physicist who introduced it near the end of the 19th century in order to explain why the speed of light is always c, regardless of the frame of reference (moving or not), or – in other words – why the speed of light is not relative. Indeed, while you’ll remember that there is no such thing as an absolute velocity according to the (special) theory of relativity, the velocity of light actually is absolute ! That means you will always see light traveling at speed c regardless of your reference frame. To put it simply, you can never catch up with light and, if you would be traveling away from some star in a spaceship with a velocity of 200,000 km per second, and a light beam from that star would pass you, you’d measure the speed of that light beam to be equal to 300,000 km/s, not 100,000 km/s. So c is an absolute speed that acts as an absolute speed limit regardless of your reference frame. [Note that we’re talking only about reference frames moving at a uniform speed: when acceleration comes into play, then we need to refer to the general theory of relativity and that’s a somewhat different ball game.]
The graph below shows how γ varies as a function of v. As you can see, the mass increase only becomes significant at speeds of like 100,000 km per second indeed.Indeed, for v = 0.3 c, the Lorentz factor is 1.048, so the increase is about 5% only. For v= 0.5 c, it’s still limited to an increase of some 15%. But then it goes up rapidly: for v= 0.9 c, the mass is more than twice the rest mass: m≈ 2.3m 0; for v= 0.99 c, the mass increase is 600%: m ≈ 7m 0; and so on. For v= 0.999 c – so when the speed of the object differs from c only by 1 part in 1,000 – the mass of the object will be more than twenty-two times the rest mass (m ≈ 22.4m 0).
You probably know that we can actually reach such speeds and, hence, verify Einstein’s correction of Newton’s Law in particle accelerators: the electrons in an electron beam in a particle accelerator get usually pretty close to c and have a mass that’s like 2000 times their rest mass. How do we know that? Because the magnetic field needed to deflect them is like 2000 times as great as their (theoretical) rest mass. So how fast do they go? For their mass to be 2000 times m 0, 1 –v 2/c 2 must be equal to 1/4,000,000. Hence, their velocity v differs from c only by one part in 8,000,000. You’ll have to admit that’s very close.
Other effects of relativistic speeds
So we mentioned the thing that’s best known about Einstein’s (special) theory of relativity: the mass of an object, as measured by the inertial observer, increases with its speed. Now, you may or may not be familiar with two other things that come out of relativity theory as well:
The first is length contraction: objects are measured to be shortened in the direction of motion with respect to the (inertial) observer. The formula to be used incorporates the reciprocal of the Lorentz factor: L = (1/γ)L 0. For example, a meter stick in a space ship moving at a velocity v = 0.6 c will appear to be only 80 cm to the external/inertial observer seeing it whizz past… That is if he can see anything at all of course: he’d have to take like a photo-finish picture as it zooms past !
The second is time dilation, which is also rather well known – just like the mass increase effect – because of the so-called twin paradox: time will appear to be slower in that space ship and, hence, if you send one of two twins away on a space journey, traveling at such relativistic speed, he will come back younger than his brother. The formula here is a bit more complicated, but that’s only because we’re used to measure time in seconds. If we would take a more natural unit, i.e. the time it takes light to travel a distance of 1 m, then the formula will look the same as our mass formula: t= γt 0 and, hence, one ‘second’ in the space ship will be measured as 1.25 ‘seconds’ by the external observer. Hence, the moving clock will appear to run slower – to the external (inertial) observer that is.
Again, the reality of this can be demonstrated. You’ll remember that we introduced the muon in previous posts: muons resemble electrons in the sense that they have the same charge, but their mass is more than 200 times the mass of an electron. As compared to other unstable particles, their average lifetime is quite long: 2.2 micro seconds. Still, that would not be enough to travel more than 600 meters or so– even at the speed of light (2.2 μs × 300,000 km/s = 660 m). But so we do detect muons in detectors down here that come all the way down from the stratosphere, where they are created when cosmic rays hit the Earth’s atmosphere some 10 kilometers up. So how do they get here if they decay so fast? Well, those that actually end up in those detectors, do indeed travel very close to the speed of light and, hence, while from their own point of view they live only like two millionths of a second, they live considerably longer from our point of view.
Relativistic energy: E = m c 2
Let’s go back to our main story line: relativistic energy. We wrote above that it’s the change of energy that matters really. So let’s look at that.
You may or may not remember that the concept of work in physics is closely related to the concept of power. In fact, you may actually remember that power, in physics at least, is defined as the work done per second.Indeed, we defined work as the (dot) product of the force and the distance. Now, when we’re talking a differential distance only (i.e. an infinitesimally small change only), then we can write dT = F·ds, but when we’re talking something larger, then we have to do that integral: ΔT = ∫F·ds. However, we’re interested in the time rate of change of T here, and so that’s the time derivative dT/dt which, as you easily verify, will be equal to dT/dt = (F·ds)/dt = F·(ds/dt) = F·vand so we can use that differential formula and we don’t need the integral. Now, that (dot) product of the force and the velocity vectors is what’s referred to as the power. [Note that only the component of the force in the direction of motion contributes to the work done and, hence, to the power.]
OK. What am I getting at? Well… I just want to show an interesting derivation: if we assume, with Einstein, that mass and energy are equivalent and, hence, that the total energy of a body always equals E = m c 2, then we can actually derive Einstein’s mass formula from that. How? Well… If the time rate of change of the energy of an object is equal to the power expended by the forces acting on it, then we can write:
dE/dt = d(m c 2)/dt = F·v
Now, we can not take the mass out of those brackets after the differential operator (d) because the mass is not a constant in this case (relativistic speeds) and, hence, dm/dt≠ 0. However, we can take out c 2(that’s an absolute constant, remember?) and we can also substitute F using Newton’s Law (F = d(m v)/dt), again taking care to leave m between the brackets, not outside. So then we get:
d(m c 2)/dt = c 2 dm/dt = [d(m v)/dt]·v = v· d(m v)/dt
In case you wonder why we can replace the vectors (bold face) v and d(mv) by their magnitudes (or lengths) v and d(m v):v and mvhave the same direction and, hence, the angle θ between them is zero, and so v·v =│v││v│cosθ =v 2. Likewise, d(mv) and v also have the same direction and so we can just replace the dot product by the product of the magnitudes of those two vectors.
Now, let’s not forget the objective: we need to solve this equation for m and, hopefully, we’ll find Einstein’s mass formula, which we need to correct Newton’s Law. How do we do that? We’ll first multiply both sides by 2m. Why? Because we can then apply another mathematical trick, as shown below:
c 2(2m)·dm/dt = 2m v· d(m v)/dt⇔d(m 2 c 2)/dt = d(m 2 v 2)/dt
However, if the derivatives of two quantities are equal, then the quantities themselves can only differ by a constant, say C. So we integrate both sides and get:
m 2 c 2=m 2 v 2+ C
Be patient: we’re almost there. The above equation must be true for all velocities v and, hence, we can choose the special case where v = 0 and call this mass m 0, and then substitute, so we get m 0 c 2=m 0 0 2+ C = C. Now we put this particular value for C back in the more general equation above and we get:
m c 2=m v 2+ m 0 c 2⇔m=m v 2/c 2+m 0⇔m(1–v 2/c 2) = m 0⇔m = m 0/(1–v 2/c 2)–1/2
So there we are: we have just shown that we get the relativistic mass formula (it’s on the right-hand side above) if we assume that Einstein’s mass-energy equivalence relation holds.
Now, you may wonder why that’s significant. Well… If you’re disappointed, then, at the very least, you’ll have to admit that it’s nice to show how everything is related to everything in this theory: from E =m c 2, we get m 0/(1–v 2/c 2)–1/2. I think that’s kinda neat!
In addition, let us analyze that mass-energy relation in another way. It actually allows us to re-definekinetic energy as the excess of a particle over its rest mass energy, or – it’s the same expression really – or the difference between its total energy and its rest energy.
How does that work? Well… When we’re looking at high-speed or high-energy particles, we will write the kinetic energy as:
K.E. = m c 2– m 0 c 2= (m–m 0)c 2= γm 0 c 2– m 0 c 2= m 0 c 2(γ– 1).
Now, we can expand that Lorentz factor γ = (1 –v 2/c 2)–1/2 into a binomial series (the binomial series is an infinite Taylor series, so it’s not to be confused with the (finite) binomial expansion: just check it online if you’re in doubt). If we do that, we we can write γ as an infinite sum of the following terms:
γ = 1 + (1/2)v 2/c 2+ (3/8)v 4/c 4+ (5/16)v 6/c 6+ …
Now, when we plug this back into our (relativistic) kinetic energy equation, we can scrap a few things (just do it) to get where I wanted to get:
K.E. = (1/2)m 0 v 2+ (3/8)m 0 v 4/c 2+ (5/16)m 0 v 6/c 4+ …
Again, you’ll wonder: so what? Well… See how the non-relativistic formula for kinetic energy (K.E. = m 0 v 2/2) appears here as the first term of this series and, hence, how the formula above shows that our ‘Newtonian’ formula is just an approximation. Of course,at low speeds, the second, third etcetera terms represent close to nothing and, hence, then our Newtonian ‘approximation is obviously pretty good of course !
OK… But… Now you’ll say: that’s fine, but how did Einstein get inspired to write E = m c 2 in the first place? Well, truth be told, the relativistic mass formula was derived first (i.e. before Einstein wrote his E = m c 2 equation),out of a derivation involving the momentum conservation law and the formulas we must use to convert the space-time coordinates from one reference frame to another when looking at phenomena (i.e. the so-called Lorentz transformations). And it was only afterwards that Einstein noted that, when expanding the relativistic mass formula, that the increase in mass of a body appeared to be equal to the increase in kinetic energy divided by c 2(Δm = Δ(K.E.)/c 2). Now, that, in turn, inspired him to also assign an equivalent energy to the rest mass of that body: E 0= m 0 c 2. […] At least that’s how Feynman tells the story in his 1965 Lectures… But so we’ve actually been doing it the other way around here!
Hmm… You will probably find all of this rather strange, and you may also wonder what happened to our potential energy. Indeed, that concept sort of ‘disappeared’ in this story: from the story above, it’s clear that kinetic energy has an equivalent mass, but what about potential energy?
That’s a very interesting question but, unfortunately, I can only give a rather rudimentary answer to that. Let’s suppose that we have two masses M and m. According to the potential energy formula above, the potential energy U between these two masses will then be equal to U =–GMm/r. Now, that energy is not interpreted as energy of either M or m, but as energy that is part of the (M, m)system, which includes the system’s gravitational field. So that energy is considered to be stored in that gravitational field. If the two masses would sit right on top of each other, then there would be no potential energy in the (M, m) system and, hence, the system as a whole would have less energy. In contrast, when we separate them further apart, then we increase the energy of the system as a whole, and so the system’s gravitational field then increases. So, yes, the potential energy does impact the (equivalent) mass of the system, but not the individual masses M and m. Does that make sense?
For me , it does, but I guess you’re a bit tired by now and, hence, I think I should wrap up here. In my next (and probably last) post on relativity, I’ll present those Lorentz transformations that allow us to ‘translate’ the space and time coordinates from one reference frame to another, and in that post I’ll also present the other derivation of Einstein’s relativistic mass formula, which is actually based on those transformations.In fact, I realize I should have probably started with that (as mentioned above, that’s how Feynman does it in his Lectures) but, then, for some reason, I find the presentation above more interesting, and so that’s why I am telling the story starting from another angle. I hope you don’t mind. In any case, it should be the same, because everything is related to everything in physics – just like in math. That’s why it’s important to have a good teacher.
Jean Louis Van BellePhysicsLeave a commentMay 24, 2014 June 26, 202015 Minutes
Re-visiting the matter wave(I)
Pre-scriptum(dated 26 June 2020): This post did not suffer from the DMCA take-down of some material. It is, therefore, still quite readable—even if my views on these matters have evolved quite a bit as part of my realist interpretation of QM. However, I now think de Broglie’s intuition in regard to particles being waves was correct but that he should have used a circular rather than a linear wave concept. Also, the idea of a particle being some wave packet is erroneous. It leads to the kind of contradictions I already start mentioning here, such as super-luminous velocities and other nonsense. Such critique is summarized in my paper on de Broglie’s wave concept. I also discuss it in the context of analyzing wavefunction math in the context of signal transmission in a crystal lattice.
Original post:
In my previous posts, I introduced a lot of wave formulas. They are essential to understanding waves – both real ones (e.g. electromagnetic waves) as well as probability amplitude functions. Probability amplitude function is quite a mouthful so let me call it a matter wave, or a de Broglie wave. The formulas are necessary to create true understanding – whatever that means to you –because otherwise we just keep on repeating very simplistic but nonsensical things such as ‘matter behaves (sometimes) like light’, ‘light behaves (sometimes) like matter’ or, combining both, ‘light and matter behave like wavicles’. Indeed: what does ‘like‘ mean? Like the same but different? So that means it’s different. Let’s therefore re-visit the matter wave (i.e. the de Broglie wave)and point out the differences with light waves.
In fact, this post actually has its origin in a mistake in a post scriptum of a previous post (An Easy Piece: On Quantum Mechanics and the Wave Function), in which I wondered what formula to use for the energy E in the (first) de Broglie relation E = hf(with f the frequency of the matter wave and h the Planck constant). Should we use (a) the kinetic energy of the particle, (b) the rest mass (mass is energy, remember?), or (c) its total energy? So let us first re-visit these de Broglie relations which, you’ll remember, relate energy and momentum to frequency (f) and wavelength (λ) respectively with the Planck constant as the factor of proportionality:
E = h f and p = h/λ
The de Broglie wave
I first tried kinetic energy in that E = h f equation. However, if you use the kinetic energy formula (K.E. = m v 2/2, with v the velocity of the particle), then the second de Broglie relation (p = h/λ) does not come out right. The second de Broglie relation has the wavelength λ on the right side, not the frequency f. But it’s easy to go from one to the other: frequency and wavelength are related through the velocity of the wave (v). Indeed,the number of cycles per second (i.e. the frequency f) times the length of one cycle (i.e. the wavelength λ) gives the distance traveled by the wave per second, i.e. its velocity v. So f λ = v. Hence, using that kinetic energy formula and that very obvious f λ=v relation, we can write E = h f as m v 2/2 = v/λ and, hence, after moving one of the two v’ s in v 2(and the 1/2 factor) on the left side to the right side of this equation, we get m v = 2h/λ. So there we are:
p = mv =2h/λ.
Well… No. The second de Broglie relation is just p = h/λ. There is no factor 2 in it. So what’s wrong?
A factor of 2 in an equation like this surely doesn’t matter, does it? It does. We are talking tiny wavelengths here but a wavelength of 1 nano meter (1×10–9 m) – this is just an example of the scale we’re talking about here – is not the same as a wavelength of 0.5 nm. There’s another problem too. Let’s go back to our an example of an electron with a mass of9.1×10–31 kg (that’s very tiny, and so you’ll usually see it expressed in a unit that’s more appropriate to the atomic scale), moving about witha velocity of2.2×10 6m/s (that’s the estimated speed of orbit of an electron around a hydrogen nucleus: it’s fast (2,200 km per second), but still less than 1% of the speed of light), and let’s do the math.
[Before I do the math, however, let me quickly insert a line on that ‘other unit’ to measure mass. You will usually see it written down as eV, so that’s electronvolt. Electronvolt is a measure of energy but that’s OK because mass is energy according to Einstein’s mass-energy equation: E = m c 2. The point to note is that the actual measure for mass at the atomic scale is eV/c 2, so we make the unit even smaller by dividing the eV (which already is a very tiny amount of energy) by c 2: 1 eV/c 2 corresponds to 1.782 6 62×10−36 kg, so the mass of our electron (9.1×10–31 kg) is about 510,000 eV/c 2, or 0.510 MeV/c 2. I am spelling it out because you will often just see 0.510 MeV in older or more popular publications, but so don’t forget that c 2 factor. As for the calculations below, I just stick to the kg and m measures because they make the dimensions come out right.]
According to our kinetic energy formula (K.E. = m v 2/2), these mass and velocity values correspond to an energy value of 22×10−19 Joule(the Joule is the so-called SI unit for energy – don’t worry about it right now). So, from the first de Broglie equation (f = E/h) – and using the right value for Planck’s constant (6.626 J·s), we get a frequency of 3.32×10 15 hertz(hertz just means oscillations per second as you know). Now, using v once again, and f λ =v, we see that corresponds to a wavelength of0.66 nanometer (0.66×10−9 m). [Just take the numbers and do the math.]
However, if we use the second de Broglie relation, which relates wavelength to momentum instead of energy, then we get0.33 nanometer (0.33×10−9 m), so that’s half of the value we got from the first equation. So what is it: 0.33 or 0.66 nm? It’s that factor 2 again. Something is wrong.
It must be that kinetic energy formula. You’ll say we should include potential energy or something. No. That’s not the issue. First, we’re talking a free particle here: an electron moving in space (a vacuum) with no external forces acting on it, so it’s a field-free space (or a region of constant potential). Second, we could, of course, extend the analysis and include potential energy, and show how it’s converted to kinetic energy (like a stone falling from 100 m to 50 m: potential energy gets converted into kinetic energy) but making our analysis more complicated by including potential energy as well will not solve our problem here: it will only make you even more confused.
Then it must be some relativistic effect you’ll say. No. It’s true the formula for kinetic energy above only holds for relatively low speeds (as compared to light, so ‘relatively’ low can be thousands of km per second) but that’s not the problem here: we are talking electrons moving at non-relativistic speeds indeed, so their mass or energy is not (or hardly) affected by relativistic effects and, hence, we can indeed use the more simple non-relativistic formulas.
The real problem we’re encountering here is not with the equations: it’s the simplistic model of our wave. We are imagining one wave here indeed, with a single frequency, a single wavelength and, hence, one single velocity – which happens to coincide with the velocity of our particle. Such wave cannot possibly represent an actual de Broglie wave: the wave is everywhere and, hence, the particle it represents is nowhere. Indeed, a wave defined by a specific wavelength λ(or a wave number k = 2π/λ if we’re using complex number notation) and a specific frequency f or period T (or angular frequency ω = 2π/T = 2π f) will have a very regular shape – such as Ψ= A e i(ωt-kx)and, hence, the probability of actually locating that particle at some specific point in space will be the same everywhere: |Ψ|2= |A e i(ωt-kx)|2=A 2. [If you are confused about the math here, I am sorry but I cannot re-explain this once again: just remember that our de Broglie wave represents probability amplitudes– so that’s some complex number Ψ = Ψ(x, t) depending on space and time – and that we need to take the modulus squared of that complex number to get the probability associated with some (real) value x (i.e. the space variable) and some value t (i.e. the time variable).]
So the actual matter wave of a real-life electron will be represented by a wave train, or a wave packet as it is usually referred to. Now, a wave packet is described by (at least)two types of wave velocity:
The so-called_group velocity_: the group velocity of a wave is denoted by_v_ gand is the velocity of the wave packet as a whole is traveling. Wikipedia defines it as “the velocity with which the overall shape of the waves’ amplitudes — known as the modulation or envelopeof the wave — propagates through space.”
The so-called phase velocity: the phase velocity is denoted by_v_ pand is what we usually associate with the velocity of a wave. It is just what it says it is: the rate at which the phase of the (composite) wave travels through space.
The term between brackets above – ‘composite’ – already indicates what it’s all about: a wave packet is to be analyzed as a composite wave: so it’s a wave composed of a finite or infinite number of component waves which all have their own wave number k and their own angular frequency ω. So the mistake we made above is that, naively, we just assumed that (i) there is only one simple wave (and, of course, there is only one wave, but it’s not a simple one: it’s a composite wave), and (ii) that the velocity v of our electron would be equal to the velocity of that wave. Now that we are a little bit more enlightened,we need to answer two questions in regard to point (ii):
Why would that be the case?
If it’s is the case, then what wave velocity are we talking about: the group velocity or the phase velocity?
To answer both questions, we need to look at wave packets once again, so let’s do that.Just to visualize things, I’ll insert – once more – that illustration you’ve seen in my other posts already:
The de Broglie wave packet
The Wikipedia article on the group velocity of a wave has wonderful animations, which I would advise you to look at in order to make sure you are following me here. There are several possibilities:
The phase velocity and the group velocity are the same: that’s a rather unexciting possibility but it’s the easiest thing to work with and, hence, most examples will assume that this is the case.
The group and phase velocity are not the same, but our wave packet is ‘stable’, so to say. In other words, the individual peaks and troughs of the wave within the envelope travel at a different speed (the phase velocity _v_ g), but the envelope as a whole (so the wave packet as such) does not get distorted as it travels through space.
The wave packet dissipates: in this case, we have a constant group velocity, but the wave packet delocalizes. Its width increases over time and so the wave packet diffuses – as time goes by – over a wider and wider region of space, until it’s actually no longer there. [In case you wonder why it did not group this third possibility under (2): it’s a bit difficult to assign a fixed phase velocity to a wave like this.]
How the wave packet will behave depends on the characteristics of the component waves. To be precise, it will depend on their angular frequency and their wave number and, hence, their individual velocities. First, note the relationship between these three variables: ω = 2π f and k = 2π/λ so ω/k = f λ = v. So these variables are not independent: if you have two values (e.g. v and k), you also have the third one (ω). Secondly, note that the component waves of our wave packet will have different wavelengths and, hence, different wave numbers k.
Now, the de Broglie relation p= ħ k (i.e. the same relation as p = h/λ but we replace λ with 2π/k and then ħ is the so-called reduced Planck constant ħ = h/2π) makes it obvious that different wave numbers k correspond to different values p for the momentum of our electron, so allowing for a spread in k (or a spread in λ as illustrates above) amounts to allowing for some spread in p. That’s where the uncertainty principle comes in – which I actually derived from a theoretical wave function in my post on Fourier transforms and conjugate variables. But so that’s not something I want to dwell on here.
We’re interested in the ω’s. What about them? Well…ω can take any value really – from a theoretical point of view that is. Now you’ll surely object to that from a practical point of view, because you know what it implies: different velocities of the component waves. But you can’t object in a theoretical analysis like this. The only thing we could possible impose as a constraint is that our wave packet should not dissipate – so we don’t want it to delocalize and/or vanish after a while because we’re talking about some real-life electron here, and so that’s a particle which just doesn’t vanish like that.
To impose that condition, we need to look at the so-called dispersion relation. We know that we’ll have a whole range of wave numbers k, but so what values should ω take for a wave function to be ‘well-behaved’, i.e. not disperse in our case? Let’s first accept that k is some variable, the independent variable actually, and so then we associate some ω with each of these values k. So ω becomes the dependent variable (dependent on k that is) and that amounts to saying that we have some function ω = ω(k).
What kind of function? Well… It’s called the dispersion relation – for rather obvious reasons: because this function determines how the wave packet will behave: non-dispersive or – what we don’t want here – dispersive. Indeed, there are several possibilities:
The speed of all component waves is the same: that means that the ratio ω/k = v is the same for all component waves. Now that’s the case only if ω is directly proportional to k, with the factor of proportionality equal to v. That means that we have a very simple dispersion relation: ω = αk with α some constant equal to the velocity of the component waves as well as the group and phase velocity of the composite wave. So all velocities are just the same (v =_v_ p=_v_ g=α) and we’re in the first of the three cases explained at the beginning of this section.
There is a linear relation between ω and k but no direct proportionality, so we write ω = αk + β, in which β can be anything but not some function of k. So we allow different wave speeds for the component waves. The phase velocity will, once again, be equal to the ratio of the angular frequency and the wave number of the composite wave (whatever that is),but what about the group velocity, i.e. the velocity of our electron in this example? Well… One can show – but I will not do it here because it is quite a bit of work – that the group velocity of the wave packet will be equal to _v_ g= dω/dk, i.e. the (first-order) derivative of ω with respect to k. So, if we want that wave packet to travel at the same speed of our electron (which is what we want of course because, otherwise, the wave packet would obviously not represent our electron), we’ll have to impose that dω/dk (or ∂ω/∂k if you would want to introduce more independent variables) equals v. In short, we have the condition that dω/dk = d(αk +β)/dk = α = k.
If the relation between ω and k is non-linear, well… Then we have none of the above. Hence, we then have a wave packet that gets distorted and stretched out and actually vanishes after a while. That case surely does not represent an electron.
Back to the de Broglie wave relations
Indeed, it’s now time to go back to our de Broglie relations – E = h f and p = h/λ and the question that sparked the presentation above: what formula to use for E? Indeed, for p it’s easy: we use p = mv and, if you want to include the case of relativistic speeds, you will write that formula in a more sophisticated way by making it explicit that the mass m is the relativistic mass m =γm 0: the rest mass multiplied with a factor referred to as the Lorentz factor which, I am sure, you have seen before: γ = (1 – v 2/c 2)–1/2. At relativistic speeds (i.e. speeds close to c), this factor makes a difference: it adds mass to the rest mass. So the mass of a particle can be written as m = γm 0, with m 0 denoting the rest mass. At low speeds (e.g. 1% of the speed of light – as in the case of our electron), m will hardly differ from m 0 and then we don’t need this Lorentz factor. It only comes into play at higher speeds.
At this point, I just can’t resist a small digression. It’s just to show that it’s not ‘relativistic effects’ that cause us trouble in finding the right energy equation for our E = h f relation. What’s kinetic energy? Well… There’s a few definitions – such as the energy gathered through converting potential energy – but one very useful definition in the current context is the following: kinetic energy is the excess of a particle over its rest mass energy. So when we’re looking at high-speed or high-energy particles, we will write the kinetic energy as K.E. = m c 2– m 0 c 2= (m–m 0)c 2= γm 0 c 2– m 0 c 2= m 0 c 2(γ– 1).Before you think I am trying to cheat you: where is the v of our particle? [To make it specific: think about our electron once again but not moving at leisure this time around: imagine it’s speeding at a velocity very close to c in some particle accelerator. Now, v is close to c but not equal to c and so it should not disappear. […]
It’s in the Lorentz factor γ = (1 –v 2/c 2)–1/2.
Now, we can expand γ into a binomial series (it’s basically an application of the Taylor series – but just check it online if you’re in doubt), so we can write γ as an infinite sum of the following terms: γ = 1 + (1/2)·v 2/c 2+ (3/8)·v 4/c 4+ (3/8)·v 4/c 4+(5/16)·v 6/c 6+ … etcetera. [The binomial series is an infinite Taylor series, so it’s not to be confused with the (finite) binomial expansion.] Now, when we plug this back into our (relativistic) kinetic energy equation, we can scrap a few things (just do it) to get where I want to get:
K.E. = (1/2)·m 0 v 2+ (3/8)·m 0 v 4/c 2+ (5/16)·m 0 v 6/c 4+ … etcetera
So what? Well… That’s it – for the digression at least: see how our non-relativistic formula for kinetic energy (K.E. = m 0 v 2/2 is only the first term of this series and, hence, just an approximation: at low speeds, the second, third etcetera terms represent close to nothing (and more close to nothing as you check out the fourth, fifth etcetera terms). OK, OK… You’re getting tired of these games. So what? Should we use this relativistic kinetic energy formula in the de Broglie relation?
No. As mentioned above already, we don’t need any relativistic correction, but the relativistic formula above does come in handy to understand the next bit. What’s the next bit about?
Well… It turns out that we actually do have to use the total energy – including(the energy equivalent to)the rest mass of our electron – in the de Broglie relation E = h f.
WHAT!?
If you think a few seconds about the math of this – so we’d use γm 0 c 2 instead of (1/2)m 0 v 2 (so we use the speed of light instead of the speed of our particle) – you’ll realize we’ll be getting some astronomical frequency (we got that already but so here we are talking some kind of trulyfantastic frequency) and, hence, combining that with the wavelength we’d derive from the other de Broglie equation (p = h/λ) we’d surely get some kind of totally unreal speed. Whatever it is, it will surely have nothing to do with our electron, does it?
Let’s go through the math.
The wavelength is just the same as that one given by p = h/λ, so we have λ = 0.33 nanometer. Don’t worry about this. That’s what it is indeed. Check it out online: it’s about a thousand times smaller than the wavelength of (visible) light but that’s OK. We’re talking something real here. That’s why electron microscopes can ‘see’ stuff that light microscopes can’t: their resolution is about a thousand times higher indeed.
But so when we take the first equation once again (E =h f) and calculate the frequency from f = γm 0 c 2/h, we get an frequency f in the neighborhood of 12.34×10 19 herz. So that gives a velocity of v = f λ = 4.1×10 10 meter per second (m/s).But…THAT’S MORE THAN A HUNDRED TIMES THE SPEED OF LIGHT.Surely, we must have got it wrong.
We don’t. The velocity we are calculating here is the phase velocity _v_ p of our matter wave – and IT’S REAL! More in general, it’s easy to show that this phase velocity is equal to _v_ p= f λ = E/p = (γm 0 c 2/h)·(h/γm 0 v) = c 2/v. Just fill in the values for c and v (3×10 8 and 2.2×10 6 respectively and you will get the same answer.
But that’s not consistent with relativity, is it? It is: phase velocities can be (and, in fact, usually are– as evidenced by our real-life example) superluminal as they say – i.e. much higher than the speed of light. However, because they carry no information – the wave packet shape is the ‘information’, i.e. the (approximate) location of our electron – such phase velocities do not conflict with relativity theory. It’s like amplitude modulation, like AM radiowaves): the modulation of the amplitude carries the signal, not the carrier wave.
The group velocity, on the other hand, can obviously not be faster than c and, in fact, should be equal to the speed of our particle (i.e. the electron). So how do we calculate that? We don’t have any formula ω(k) here, do we? No. But we don’t need one. Indeed, we can write:
v g = ∂ω/∂k =∂(E/ħ)/∂(p/ħ) = ∂E/∂p
[Do you see why we prefer the∂ symbol instead of the d symbol now?ω is a function of k but it’s – first and foremost – a function of E, so a partial derivative sign is quite appropriate.]
So what? Well… Now you can use either the relativistic or non-relativistic relation between E and p to get a value for∂E/∂p. Let’s take the non-relativistic one first (E = p 2/2m) :∂E/∂p = ∂(p 2/2m)/∂p = p/m = v. So we get the velocity of our electron! Just like we wanted. As for the relativistic formula (E = (p 2 c 2+ m 0 2 c 4)1/2), well… I’ll let you figure that one out yourself. [You can also find it online in case you’d be desperate.]
Wow! So there we are. That was quite something! I will let you digest this for now. It’s true I promised to ‘point out the differences between matter waves and light waves’ in my introduction but this post has been lengthy enough. I’ll save those ‘differences’ for the next post. In the meanwhile, I hope you enjoyed and – more importantly – understood this one. If you did, you’re a master! A real one!
Jean Louis Van BellePhysicsLeave a commentApril 14, 2014 June 26, 202016 Minutes
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What is a Subarray?
A subarray is a contiguous part of array, i.e., Subarray is an array that is inside another array.
In general, for an array of size n, there are n(n+1)/2 non-empty subarrays.
For example, Consider the array [1, 2, 3, 4], There are 10 non-empty sub-arrays. The subarrays are:
(1), (2), (3), (4), (1,2), (2,3), (3,4), (1,2,3), (2,3,4), and (1,2,3,4)
What is a Subsequence?
A subsequence is a sequence that can be derived from another sequence by removing zero or more elements, without changing the order of the remaining elements.
More generally, we can say that for a sequence of size n, we can have (2n– 1) non-empty sub-sequences in total.
For the same above example, there are 15 sub-sequences. They are:
(1), (2), (3), (4), (1,2), (1,3),(1,4), (2,3), (2,4), (3,4), (1,2,3), (1,2,4), (1,3,4), (2,3,4), (1,2,3,4).
What is a Subset?
If a Set has all its elements belonging to other sets, this set will be known as a subset of the other set.
A Subset is denoted as “⊆“. If set A is a subset of set B, it is represented as A ⊆ B.
For example, Let Set_A = {m, n, o, p, q}, Set_ B = {k, l, m, n, o, p, q, r}
Then, A ⊆ B.
Table of Content
What is a Subarray?
What is a Subsequence?
What is a Subset?
Easy Problems on Subarray
Medium Problems on Subarray
Hard Problems on Subarray
Easy Problems on Subsequence
Medium Problems on Subsequence
Hard Problems on Subsequence
Easy Problems on Subset
Medium Problems on Subset
Hard Problems on Subset
Easy Problems on Subarray:
Split an array into two equal Sum subarrays
Check if subarray with given product exists in an array
Subarray of size k with given sum
Sort an array where a subarray of a sorted array is in reverse order
Count subarrays with all elements greater than K
Maximum length of the sub-array whose first and last elements are same
Check whether an Array is Subarray of another Array
Find array such that no subarray has xor zero or Y
Maximum subsequence sum such that all elements are K distance apart
Longest sub-array with maximum GCD
Count of subarrays with sum at least K
Length of Smallest subarray in range 1 to N with sum greater than a given value
Sum of all subarrays of size K
Split array into K disjoint subarrays such that sum of each subarray is odd.
Find an array of size N having exactly K subarrays with sum S
Find the subarray of size K with minimum XOR
Length of the longest alternating even odd subarray
Count of subarrays which start and end with the same element
Count of subarrays having exactly K perfect square numbers
Split array into two subarrays such that difference of their maximum is minimum
Medium Problems on Subarray:
Print all K digit repeating numbers in a very large number
Length of longest subarray whose sum is not divisible by integer K
Min difference between maximum and minimum element in all Y size subarrays
Longest subarray of non-empty cells after removal of at most a single empty cell
First subarray with negative sum from the given Array
Largest subarray with frequency of all elements same
Bitwise operations on Subarrays of size K
Count subarrays having sum of elements at even and odd positions equal
Longest Subarray consisting of unique elements from an Array
Minimum Decrements on Subarrays required to reduce all Array elements to zero
Split array into two subarrays such that difference of their sum is minimum
Maximize count of non-overlapping subarrays with sum K
Smallest subarray which upon repetition gives the original array
Split array into maximum subarrays such that every distinct element lies in a single subarray
Maximize product of subarray sum with its minimum element
Sum of products of all possible Subarrays
Check if all subarrays contains at least one unique element
Length of smallest subarray to be removed such that the remaining array is sorted
Length of longest subarray having frequency of every element equal to K
Length of the longest increasing subsequence which does not contain a given sequence as Subarray
Hard Problems on Subarray:
Length of smallest subarray to be removed to make sum of remaining elements divisible by K
Maximum length of same indexed subarrays from two given arrays satisfying the given condition
Count ways to split array into two equal sum subarrays by changing sign of any one array element
Longest subarray in which all elements are smaller than K
Maximize product of a strictly increasing or decreasing subarray
Sum of maximum of all subarrays by adding even frequent maximum twice
Longest subarray of an array which is a subsequence in another array
Count of subarrays having product as a perfect cube
Minimize difference between maximum and minimum array elements by removing a K-length subarray
Maximum sum submatrix
Minimum removal of elements from end of an array required to obtain sum K
Check if any subarray of length M repeats at least K times consecutively or not
Minimize flips on K-length subarrays required to make all array elements equal to 1
Split array into K subarrays such that sum of maximum of all subarrays is maximized
Find minimum subarray sum for each index i in subarray [i, N-1]
Longest subarray with GCD greater than 1
Longest subsegment of ‘1’s formed by changing at most k ‘0’s
Lexicographically smallest Permutation of Array by reversing at most one Subarray
Find a subsequence which upon reversing gives the maximum sum subarray
Minimize steps to make Array elements 0 by reducing same A[i] – X from Subarray
Easy Problems on Subsequence:
Longest subsequence having equal numbers of 0 and 1
Powers of two and subsequences
Longest Subsequence where index of next element is arr[arr[i] + i]
Number of subsequences with zero sum
Longest sub-sequence with maximum GCD
Maximum Bitwise AND value of subsequence of length K
Length of the longest subsequence such that xor of adjacent elements is non-decreasing
Maximum product of bitonic subsequence of size 3
Length of Smallest Subsequence such that sum of elements is greater than equal to K
Longest subsequence of even numbers in an Array
Maximum length Subsequence with alternating sign and maximum Sum
Count of possible subarrays and subsequences using given length of Array
Maximum bitwise OR value of subsequence of length K
Count of subsequences consisting of the same element
Smallest occurring element in each subsequence
Length of the longest subsequence consisting of distinct elements
Minimize elements to be added to a given array such that it contains another given array as its subsequence
Maximize subsequences having array elements not exceeding length of the subsequence
Length of longest subsequence consisting of distinct adjacent elements
Maximum Sum Subsequence
Medium Problems on Subsequence:
Minimum removals required to make a given array Bitonic
Check if a non-contiguous subsequence same as the given subarray exists or not
Minimize the number of strictly increasing subsequences in an array
Count of unique subsequences from given number which are power of 2
Minimum number of insertions required such that first K natural numbers can be obtained as sum of a subsequence of the array
Length of longest subsequence such that prefix sum at every element remains greater than zero
Check if given Array can be divided into subsequences of K increasing consecutive integers
Longest Subsequence such that difference between adjacent elements is either A or B
Count subsequences of Array having single digit integer sum K
Shortest Subsequence with sum exactly K
Printing Longest Bitonic Subsequence
Sorted subsequence of size 3 in linear time using constant space
Count of subsequences having maximum distinct elements
Construct array having X subsequences with maximum difference smaller than d
Print all subsequences of a string using ArrayList
Longest Subsequence with at least one common digit in every element
Maximum Sum Subsequence of length k
Sum of minimum element of all sub-sequences of a sorted array
Find all combinations of two equal sum subsequences
Minimum cost of choosing 3 increasing elements in an array of size N
Hard Problems on Subsequence:
Number of subsequences with positive product
Longest subsequence having difference atmost K
Find all subsequences with sum equals to K
Maximize product of digit sum of consecutive pairs in a subsequence of length K
Count of subsequences of length atmost K containing distinct prime elements
Sum of all subsequences of length K
Minimize sum of smallest elements from K subsequences of length L
Unique subsequences of length K with given sum
Smallest subsequence with sum of absolute difference of consecutive elements maximized
Maximize product of same-indexed elements of same size subsequences
Longest Increasing Subsequence having sum value atmost K
Longest subsequence of a number having same left and right rotation
Maximize length of Non-Decreasing Subsequence by reversing at most one Subarray
Maximum subsequence sum possible by multiplying each element by its index
Generate all distinct subsequences of array using backtracking
Maximum subsequence sum such that no K elements are consecutive
Print all possible K-length subsequences of first N natural numbers with sum N
Longest subsequence having difference between the maximum and minimum element equal to K
Maximize difference between sum of even and odd-indexed elements of a subsequence
Convert an array into another by repeatedly removing the last element and placing it at any arbitrary index
Easy Problems on Subset:
Find if there is any subset of size K with 0 sum in an array of -1 and +1
Sum of sum of all subsets of a set formed by first N natural numbers
Count of subsets not containing adjacent elements
Sum of the sums of all possible subsets
Find whether an array is subset of another array
Total number of Subsets of size at most K
Check if it is possible to split given Array into K odd-sum subsets
Partition a set into two subsets such that difference between max of one and min of other is minimized
Sum of all possible expressions of a numeric string possible by inserting addition operators
Check if it’s possible to split the Array into strictly increasing subsets of size at least K
Largest subset of Array having sum at least 0
Medium Problems on Subset:
Number of subsets with sum divisible by m
Fibonacci sum of a subset with all elements <= k
Number of possible Equivalence Relations on a finite set
Largest divisible pairs subset
Recursive program to print all subsets with given sum
Subset Sum Queries in a Range using Bitset
Find all distinct subset (or subsequence) sums of an array | Set-2
Sum of (maximum element – minimum element) for all the subsets of an array
Count no. of ordered subsets having a particular XOR value
Sum of subsets of all the subsets of an array
Perfect Sum Problem
Count of subsets having sum of min and max element less than K
Split array into two equal length subsets such that all repetitions of a number lies in a single subset
Nth Subset of the Sequence consisting of powers of K in increasing order of their Sum
Largest possible Subset from an Array such that no element is K times any other element in the Subset
Check if an array can be split into subsets of K consecutive elements
Hard Problems on Subset:
Minimize count of divisions by D to obtain at least K equal array elements
Split array into K-length subsets to minimize sum of second smallest element of each subset
Median of all non-empty subset sums
Minimum removals required such that sum of remaining array modulo M is X
Sum of length of two smallest subsets possible from a given array with sum at least K
Reduce sum of any subset of an array to 1 by multiplying all its elements by any value
Sum of all subsets whose sum is a Perfect Number from a given array
Minimize sum of incompatibilities of K equal-length subsets made up of unique elements
Maximize sum of subsets from two arrays having no consecutive values
Product of the maximums of all subsets of an array
Count ways to place ‘+’ and ‘-‘ in front of array elements to obtain sum K
Count ways to split array into two subsets having difference between their sum equal to K
Find the subset of Array with given LCM
Count of subsets whose product is multiple of unique primes
Minimum count of elements to be inserted in Array to form all values in [1, K] using subset sum
Maximum subset sum having difference between its maximum and minimum in range [L, R]
Find all unique subsets of a given set using C++ STL
Subset sum problem where Array sum is at most N
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5708 | https://ocw.mit.edu/courses/8-20-introduction-to-special-relativity-january-iap-2021/ | Course Info
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8.20 | January IAP 2021 | Undergraduate
Introduction to Special Relativity
Course Description
The theory of special relativity, originally proposed by Albert Einstein in his famous 1905 paper, has had profound consequences on our view of physics, space, and time. This course will introduce you to the concepts behind special relativity including, but not limited to, length contraction, time dilation, the Lorentz …
The theory of special relativity, originally proposed by Albert Einstein in his famous 1905 paper, has had profound consequences on our view of physics, space, and time. This course will introduce you to the concepts behind special relativity including, but not limited to, length contraction, time dilation, the Lorentz transformation, relativistic kinematics, Doppler shifts, and even so-called “paradoxes.”
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Relativity
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Albert Einstein during a lecture in Vienna in 1921. (Image is in the public domain.)
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5709 | https://logeion.uchicago.edu/utor | ===============
ΛΟΓΕΙΟΝ
Nearby
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utinatio utinatio
utinizare utinizare
utiquam utiquam
utique utique
Utique Utique
utirum utirum
Utis Utis
utitare utitare
utlaare utlaare
utlaga utlaga
utlagare utlagare
utlagaria utlagaria
utlagatio utlagatio
utlageria utlageria
utlaghare utlaghare
utlagia utlagia
utlagiare utlagiare
utlagius utlagius
utlandiscus utlandiscus
utlandum utlandum
utlawyare utlawyare
utleg- utleg-
utleipa utleipa
utllagare utllagare
utlugaria utlugaria
utmultum utmultum
uto uto
Utopia Utopia
Utopianus Utopianus
Utopiensis Utopiensis
utor utor
utorica utorica
utplurimum utplurimum
utpote utpote
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utputa utputa
utpute utpute
utqui utqui
utquid utquid
utquod utquod
utquomque utquomque
utrae utrae
utralibet utralibet
utraque utraque
utrarii utrarii
utrarius utrarius
utrasque utrasque
Utrecht Utrecht
utres utres
utria utria
utribi utribi
utricida utricida
utricior utricior
utriclarius utriclarius
utricularius utricularius
utriculus utriculus
utrimlibet utrimlibet
utrimque utrimque
utrimquesecus utrimquesecus
utrinde utrinde
utrinque utrinque
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Parsed as a form of: utor,
See utor in Μορφώ
utor
Short Definition
utor, to use, make use of, employ, profit by, take advantage of, enjoy, serve oneself with
Frequency
utor is the 200th most frequent word
Search corpus for this lemma: utor
BWL
oratione utar aliena
I shall use a foreign mode of speaking
severitas qua tu in iis rebus usus es
the sternness which you have used in these matters
valetudine bona uti
to have good health
FriezeDennisonVergil
ūtor, ūsus sum, 3, dep. n.: to use, foll. by the abl.; employ, show, display, 5.192; address, 1.64; experience, enjoy, prove, meet with, 6.546.
LewisShort Georges Gaffiot 2016 LaNe Latino-Sinicum Lewis Examples from the corpus LewisShort Georges Gaffiot 2016 LaNe Latino-Sinicum Lewis Examples from the corpus
ūtor (old form oetor, oesus, etc., from oitor, oisus, Lex. Thor. lin. 11; inf. parag. oetier, Rogat. Tribun. ap. Fest. p. 246 Müll.; Cic.Leg. 3, 4), ūsus (inf. utier, Plaut.Cas. 2, 3, 4 (220); Ter.Phorm. 603), 3, v. dep. [etym. dub.].
1. I Prop., _to use._
2. A With abl.
1 _To make use of, employ_: cave ... ne tibi hoc scipione malum magnum dem. _Paeg._ Jam utere eo, Plaut._Pers._ 5, 2, 36 (812): _Th._ Oh Epidicumne ego conspicor? _Ep._ Certe oculis utere, Plaut._Ep._ 1, 1, 4 (4): hoc oculo, id._Mil._ 4, 7, 25 (1308): sola potest animi per se natura ... durare et sensibus uti, Lucr. 3, 560: de rebus ipsis utere tuo judicio, Cic._Off._ 1, 1, 2: utinam, quem ad modum oratione sum usurus alienā, sic mihi ore uti liceret alieno, id._Rep._ 3, 5, 8: utor neque perantiquis neque inhumanis ac feris testibus, _\_cite, appeal to\_,_ id. ib. 1, 37, 58: neque enim accusatore muto neque teste quisquam utitur eo, qui de accusatoris subsellio surgit, id._Rosc. Am._ 36, 104: num argumentis utendum in re ejus modi? id._Verr._ 2, 4, 6, § 11: mancipium, quo et omnes utimur, et non praebetur a populo, id. ib. 2, 4, 5, § 9: quo interprete non ad linguam Graecam, sed ad furta et flagitia uti solebat, id. ib. 2, 3, 37, § 84: ut postea numquam dextro (oculo) aeque bene usus sit, Nep._Hann._ 4, 3: si licet exemplis in parvo grandibus uti, Ov._Tr._ 1, 3, 25: viribus utendum est, quas fecimus, Luc. 1, 347.—With _ad_: ad eam rem usus est tuā mihi operā _Sa._ Utere, ut vis, Plaut._Pers._ 2, 5, 27 (328): earum (navium) materiā atque aere ad reliquas reficiendas utebatur, Caes._B. G._ 4, 31: administris ad ea sacrificia Druidibus, id. ib. 6, 16: ut eā potestate ad quaestum uteretur, Cic._Q. Fr._ 1, 1, 3, § 11: ad quam rem (deus) motu mentis ac ratione utatur, id._N. D._ 1, 37, 104.—With _pro_: utuntur aut aere aut taleis ferreis ad certum pondus examinatis pro nummo, Caes._B. G._ 5, 12.—
2 Esp.
a _To manage, control, wield_: bene ut armis, optime ut equis uteretur, Cic._Deiot._ 10, 28: nemo est quin eo ipso (equo), quo consuevit, libentius utatur quam intractato, id._Lael._ 19, 68.—
b _To spend, use_: velim cum illā videas ut sit qui utamur (sc. pecunia), Cic._Att._ 11, 11, 2: tantis vectigalibus ad liberalitatem utens, id._Fin._ 2, 26, 84: cum horis nostris nos essemus usi, _\_spent, exhausted\_,_ id._Verr._ 2, 1, 11, § 30.—_Absol._: notum et quaerere et uti, Hor._Ep._ 1, 7, 57.—
c _To wear_: pellibus aut parvis renonum tegimentis utuntur, magnā corporis parte nudā, Caes._B. G._ 6, 21 _fin._: ne insignibus quidem regiis Tullus nisi jussu populi est ausus uti, Cic._Rep._ 2, 17, 31.—
d _To accept, adopt_: eā condicione, quae a Caesare ferretur, se usuros ostendebant, Caes._B. G._ 4, 11: praeposteris enim utimur consiliis et acta agimus, Cic._Lael._ 22, 85.—
e _To resort to, consult_: neque Vectium ad se arcessit, quaestorem suum, cujus consilio uteretur, Cic._Verr._ 2, 5, 44, § 114: oraculo, Tac._A._ 2, 54.—
f Of a form or style of speech, sentiment, etc., _to make, adopt, employ_: sermonibus morologis utier, Plaut._Ps._ 5, 1, 21 (1266): si provincia loqui posset, hac voce uteretur, Cic._Div._ in Caecin. 5, 19: hac unā defensione, id._Verr._ 2, 4, 4, § 8: haec oratio, quā me uti res publica coëgit, id._Rosc. Am._ 49, 143: cum hortatione non egeas, non utar eā pluribus verbis, id._Fam._ 11, 5, 3: illa criminatio, quā in me absentem usus est, id._Agr._ 3, 1, 3.—
g _To perform, exercise, practise_, etc.: crucior, patrem ... nunc inprobi viri officio uti, Plaut._Stich._ 1, 1, 14 (14): eādem nos disciplinā utimur, id._As._ 1, 3, 49 (201); cf.: nec vero habere virtutem satis est quasi artem aliquam, nisi utare: etsi ars quidem, cum eā non utare, scientiā ipsā teneri potest, Cic._Rep._ 1, 2, 2: diuturni silentii, quo eram his temporibus usus, finem hodiernus dies attulit, _\_observed, kept\_,_ id._Marcell._ 1, 1: eos (senes) ego fortasse nunc imitor et utor aetatis vitio, id._Fam._ 2, 16, 6: ratione utuntur, _\_exercise moderation\_,_ Plaut._Cas._ prol. 27: ut anteponantur ... ratione utentia rationis expertibus, Cic._Top._ 18, 69: ne tu, leno, postules Te hic fide lenoniā uti: non potis, Plaut._Rud._ 5, 3, 30 (1386): viribus uteris per clivos, Hor._Ep._ 1, 13, 10.—With _adverb. acc._: ut hoc utimur maxime more moro multum, Plaut._Men._ 4, 2, 1 (571): ita aperte ipsam rem locutus nil circuitione usus es, Ter._And._ 202.—
h In gen., _to use, enjoy, profit by, take advantage of_, etc.: otio qui nescit uti plus negoti habet, quam, etc., Enn. ap. Gell. 19, 20, 12 (Trag. Rel. v. 252 Vahl.): sinite ... eodem ut jure uti senem Liceat, quo jure sum usus adulescentior, i. e. _enjoy, exercise_, Ter._Hec._ 10: commodius esse opinor duplici spe utier, id._Phorm._ 603: serius a terrā provectae naves neque usae nocturnā aurā in redeundo offenderunt, Caes._B. C._ 3, 8: commoda quibus utimur lucemque quā fruimur ab eo nobis dari, Cic._Rosc. Am._ 45, 131: in maximo meo dolore hoc solacio utor, quod, etc., id._Fam._ 11, 26 _init._: usus est hoc cupidine, tamdiu, dum, etc., _had the use of_, i. e. _borrowed_, id._Verr._ 2, 4, 3, § 6; cf. I. B. 2. infra: utatur suis bonis oportet et fruatur, qui beatus futurus est, id._N. D._ 1, 37, 103: propter nauticarum rerum scientiam plurimisque maritimis rebus fruimur atque utimur, id. ib. 2, 60, 152: si fortunā permittitis uti, _\_to try, take advantage of\_,_ Verg._A._ 9, 240: nostrā utere amicitiā, ut voles, Ter._Hec._ 764; cf.: decet hunc ordinem ... bene utier amicitiā, Plaut._Cist._ 1, 1, 24 (24): libertate modice utantur, Liv. 34, 49, 8: deorum Muneribus sapienter uti, Hor._C._ 4, 9, 48: Ofellam Integris opibus novi non latius usum Quam nunc accisis, id._S._ 2, 2, 113: quia parvo nesciet uti, id._Ep._ 1, 10, 41: temporibus sapienter utens, _\_taking advantage of\_,_ Nep._Epam._ 3, 1.—Prov.: foro uti, _to make one's market_, i. e. accommodate one's prices, actions, etc., to circumstances, take advantage of events: scisti uti foro, Ter._Phorm._ 79.—_Absol._: opportunae sunt divitiae ut utare (sc. eis), Cic._Lael._ 6, 22.— With _adverb. acc._: ne Silius quidem quicquam utitur (sc. suis hortis), Cic._Att._ 12, 22, 3. —
k Of passions, traits of character, etc., _to indulge, practise, exercise, yield to_, etc.: inter nos amore utemur semper subrepticio? Plaut._Curc._ 1, 3, 49 (206): alacritate ac studio, Caes._B. G._ 4, 24: severitas, quā tu in iis rebus usus es, Cic._Q. Fr._ 1, 1, 6, § 19: usus est ipse incredibili patientiā, id._Phil._ 1, 4, 9: ego pervicaciam (esse hanc) aio, et eā me uti volo, Att. ap. Non. 433, 1 (Trag. Rel. v. 5 Rib.): dementer amoribus usa, Ov._M._ 4, 259.—With _in_ and acc.: ut suā clementiā ac mansuetudine in eos utatur, Caes._B. G._ 2, 14.—
1 _To experience, undergo, receive, enjoy_, etc., ne simili utamur fortunā atque usi sumus, Quom, etc., Ter._Phorm._ 31: hoc honore usi togati solent esse, Cic._Phil._ 8, 11, 32: homines amplissimis usos honoribus, id._Fl._ 19, 45: nobiles amplis honoribus usi, Sall._J._ 25, 4: neminem curuli honore usum praeterierunt, Liv. 34, 44, 4: primus externorum usus illo honore quem majores Latio quoque negaverint, Plin. 7, 43, 44, § 136: quoniam semel est odio civiliter usus, Ov._Tr._ 3, 8, 41.—
m To use as food or medicine, _to take, drink_, etc.: lacte mero veteres usi memorantur et herbis, Ov._F._ 4, 369: aquis frigidis, Cels. 1, 1: antidoto, Scrib._Comp._ 171: medicamento, id. ib. 228: vino modice, Cels. 8, 11: ex altero (loco, i. e. ex lacu) ut pecus uti possit (sc. aquā), Varr._R. R._ 1, 11, 2.—
B With the thing used, etc., as direct obj. (class. only in _gerund._ constr.; v. infra): nuptias abjeci, amicos utor primoris viros, Turp. ap. Non. p. 497, 15 (Com. Rel. v. 164 Rib.): facilitatem vulgariam, Nov. ib. 481, 21 (Com. Rel. v. 98 ib.): res pulchras, quas uti solet, id. ib. 500, 16 (Com. Rel. v. 69 ib.): ita uti eum oportet libertatem, Titin. ib. 481, 19 (Com. Rel. v. 98 ib.): cetera quae volumus uti Graecā mercamur fide, Plaut._As._ 1, 3, 47 (199): dic mihi, an boni quid usquam'st, quod quisquam uti possiet, id._Merc._ 1, 2, 37 (147): diutine uti bene licet partum bene, id._Rud._ 4, 7, 15 (1241): profecto uteris ut voles operam meam, id._Poen._ 5, 2, 128 (1088): mea, quae praeter spem evenere, utantur sine, Ter._Ad._ 815: BALINEVM ... QVOD VSI FVERANT AMPLIVS ANNIS XXXX., _Inscr. Orell._ 202: si quid est, quod utar, utor: si non est, egeo, Cato ap. Gell. 13, 23, 1: oleam albam, quam voles uti, condito, id._R. R._ 118: quam rem etiam nomine eodem medici utuntur, Varr._R. R._ 3, 16, 23: ferrum, Aur. Vict._Caes._ 17, 4.—
2 Hence, esp. _gerund._ in phrases dare utendum, _to lend;_ recipere or rogare or petere utendum, _to borrow_, etc. (class.; freq. in Plaut.): quod datum utendum'st, Plaut._Trin._ 5, 2, 7 (1131): quae utenda vasa semper vicini rogant, id._Aul._ 1, 2, 18 (96); 2, 4, 32 (311); 2, 9, 4 (401); id._Pers._ 1, 3, 47 (127) sq.; id._Mil._ 2, 3, 76 (347); id._Rud._ 3, 1, 10 (602): auris tibi contra utendas dabo, Enn. ap. Non. 506, 1 (Trag. Rel. v. 364 Vahl.); Ter._Heaut._ 133: quae bona is Heraclio omnia utenda ac possidenda tradiderat, Cic._Verr._ 2, 2, 18, § 46: te, quod utendum acceperis, reddidisse, id._Tusc._ 3, 17, 36: multa rogant utenda dari, data reddere nolunt, Ov._A. A._ 1, 433.—
3. II Transf. (through the intermediate idea of having and using).
4. A Pregn., _to enjoy the friendship of_ any one; _to be familiar_ or _intimate with, to associate with_ a person.
a With _abl._: his Fabriciis semper est usus Oppianicus familiarissime, Cic._Clu._ 16, 46: quā (Caeciliā) pater usus erat plurimum, id._Rosc. Am._ 11, 27: Trebonio multos annos utor valde familiariter, id._Fam._ 1, 3, 1: Lucceius qui multum utitur Bruto, id._Att._ 16, 5, 3: utere Pompeio Grospho, Hor._Ep._ 1, 12, 22: quo pacto deceat majoribus uti, id. ib. 1, 17, 2: si sciret regibus uti, ib. ib. 14: ita me verebatur ut me formatore morum, me quasi magistro uteretur, Plin._Ep._ 8, 23, 2.—
b With acc.: vilica vicinas aliasque mulieres quam minimum utatur, Cato _R. R._ 143, 1.—
B _To be in possession of_ a thing, esp. _to have, hold_, or _find_ a thing in some particular mode or character; with abl.: mihi si unquam filius erit, ne ille facili me utetur patre, _\_he shall find an indulgent father in me\_,_ Ter._Heaut._ 217; cf.: patre usus est diligente et diti, Nep._Att._ 1, 2: bonis justisque regibus, Cic._Rep._ 1, 33, 50: quae (sc. libertas) non in eo est, ut justo utamur domino, sed ut nullo, id. ib. 2, 23, 43; cf. id._Fin._ 1, 1, 2: hic vide quam me sis usurus aequo, id._Verr._ 2, 5, 59, § 154: ut is illis benignis usus est ad commodandum, id. ib. 2, 4, 3, § 6: ne bestiis quoque immanioribus uteremur, id._Rosc. Am._ 26, 71: me Capitolinus convictore usus amicoque A puero est, Hor._S._ 1, 4, 95: uteris monitoribus isdem, id._Ep._ 2, 2, 154: valetudine non bonā, Caes._B. C._ 3, 49: quo (sc. Philoctete) successore sagittae Herculis utuntur, Ov._M._ 13, 52.—_Absol._: nam pol placidum te et clementem eo usque modo ut volui usus sum in alto (= placidum te esse ut volui, sic te usus sum), Plaut._Trin._ 4, 1, 8 (827).—Hence, _P. a._: ūtens, ntis, m., _possessing, that possesses_: utentior sane sit, _i. e. \_a larger possessor, richer\_,_ Cic._Off._ 2, 20, 71.
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etiam-269
multus-259
magnus-249
verbum-230
habeo-223
idem-220
enim-210
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ita-203
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5710 | http://clubztutoring.com/ed-resources/math/counterexample-definitions-examples-6-7-8/ | Home / Educational Resources / Math Resources / Counterexample: Definitions and Examples
Introduction:
A counterexample is a specific example that disproves a general statement. It is a powerful tool in mathematics and other fields where statements are made about general cases. A counterexample can be used to demonstrate that a statement is false, even if it appears to be true in some cases.
The concept of a counterexample is fundamental in mathematics. Many mathematical statements are made about general cases, such as all prime numbers, all even numbers, or all triangles. To prove such a statement, mathematicians must show that it is true for all possible cases. However, to disprove such a statement, it is enough to find a single case that does not fit the pattern. This is where counterexamples come in.
For example, consider the statement “All even numbers are divisible by 4.” This statement seems to be true, as many even numbers are indeed divisible by 4. However, a counterexample can easily be found: 6 is an even number, but it is not divisible by 4. Therefore, the statement is false.
Another example of a counterexample is the statement “All triangles have two acute angles.” This statement also seems to be true, as many triangles have two acute angles. However, a counterexample can be found: a right triangle has one right angle and two acute angles. Therefore, the statement is false.
Counterexamples can be used to disprove many different types of statements. For example, they can be used to disprove conjectures, which are statements that have not yet been proven. If a counterexample can be found for a conjecture, then the conjecture is false.
Counterexamples can also be used to disprove generalizations. A generalization is a statement that applies to many different cases. For example, the statement “All dogs like to play fetch” is a generalization, as it applies to many different types of dogs. However, if a counterexample can be found, such as a dog that does not like to play fetch, then the generalization is false.
In addition to mathematics, counterexamples are also used in other fields such as science and philosophy. In science, counterexamples can be used to disprove hypotheses or theories. For example, the theory that all living organisms require oxygen to survive can be disproved by finding a living organism that does not require oxygen.
In philosophy, counterexamples can be used to disprove arguments. For example, the argument “All humans are mortal, Socrates is human, therefore Socrates is mortal” is a valid argument. However, a counterexample can be found by imagining an immortal human. This counterexample shows that the argument is not always true.
Counterexamples are also used in computer science, particularly in testing software. A counterexample can be used to demonstrate that a program does not work correctly in a particular case. For example, if a program is designed to sort a list of numbers in ascending order, a counterexample would be a list that is not sorted correctly.
In conclusion, counterexamples are a powerful tool in many different fields. They are used to disprove statements, hypotheses, and theories. Counterexamples can be found for many different types of statements, including conjectures, generalizations, and arguments. In mathematics, counterexamples are fundamental to the process of proving and disproving statements. Without counterexamples, many false statements would be accepted as true, leading to incorrect results and conclusions.
Definition of a Counterexample:
A counterexample is a specific example or instance that disproves a general statement or hypothesis. More formally, a counterexample is an example that demonstrates the falsity of a proposition or conjecture. In mathematics, a counterexample is typically used to disprove a conjecture or to show that a theorem is false in some cases.
To provide an example, consider the statement “all even numbers are divisible by 3.” We know that this statement is false because the counterexample of 2 shows that not all even numbers are divisible by 3. Therefore, 2 is a counterexample to the statement.
Examples of Counterexamples in Mathematics:
Prime numbers: The statement “all prime numbers are odd” is false because the counterexample of 2 shows that there is at least one even prime number. Therefore, 2 is a counterexample to the statement.
Quadratic equations: The statement “all quadratic equations have two distinct roots” is false because the counterexample of x^2 = 0 shows that some quadratic equations can have only one root. Therefore, x^2 = 0 is a counterexample to the statement.
Geometry: The statement “all triangles have three sides” is false because the counterexample of a degenerate triangle (a triangle with zero area) shows that not all triangles have three sides. Therefore, a degenerate triangle is a counterexample to the statement.
Set theory: The statement “all sets have a greatest element” is false because the counterexample of the set of all negative integers shows that not all sets have a greatest element. Therefore, the set of all negative integers is a counterexample to the statement.
Calculus: The statement “all continuous functions are differentiable” is false because the counterexample of the function f(x) = |x| shows that some continuous functions are not differentiable at all points. Therefore, the function f(x) = |x| is a counterexample to the statement.
Importance of Counterexamples:
Counterexamples are important because they can help to identify the limits of a theory or concept. By providing specific examples that demonstrate the falsity of a general statement, counterexamples can help to refine the definition of a concept or theory. This refinement can lead to a better understanding of the concept or theory and can help to identify areas where further research is needed.
In addition, counterexamples can help to inspire new research questions and avenues of inquiry. By identifying the limits of a theory or concept, counterexamples can help to identify areas where further research is needed. For example, the counterexample of a degenerate triangle (a triangle with zero area) in geometry led to the development of a more precise definition of a triangle, which in turn led to further research on the properties of triangles.
Counterexamples can also be used to teach critical thinking skills. By providing examples that demonstrate the falsity of a general statement, counterexamples can help to teach students how to evaluate the validity of a statement and how to identify the limits of a theory or concept.
Conclusion
In conclusion, counterexamples play a crucial role in mathematics, logic, and other fields of study that rely on deductive reasoning. By providing evidence that a conjecture or statement is false, counterexamples help refine and improve our understanding of concepts and theories. They challenge our assumptions and force us to re-evaluate our reasoning, leading to deeper insights and a more accurate understanding of the world around us. As such, counterexamples are a valuable tool for researchers, educators, and anyone seeking to deepen their understanding of complex ideas.
Quiz
What is a counterexample in mathematics? A: A counterexample is an example that disproves a statement or conjecture.
True or false: A counterexample proves a statement or conjecture to be true. A: False. A counterexample shows that a statement or conjecture is false.
Can a single counterexample disprove a statement or conjecture? A: Yes, a single counterexample is sufficient to disprove a statement or conjecture.
True or false: A counterexample must always be a specific numerical value. A: False. A counterexample can also be a set, a function, or any other mathematical object.
What is the purpose of using counterexamples in mathematics? A: The purpose of using counterexamples is to disprove false statements or conjectures and to guide the development of new mathematical theories.
True or false: A counterexample can only be found by trying every possible case. A: False. A counterexample can sometimes be found by using logical reasoning or by constructing a specific example.
Can a statement or conjecture be proven by finding a counterexample? A: No, a statement or conjecture cannot be proven by finding a counterexample. It can only be disproven.
What is the difference between a counterexample and a proof? A: A counterexample disproves a statement or conjecture, while a proof shows that a statement or conjecture is true.
True or false: A counterexample is always unique. A: False. A statement or conjecture can have multiple counterexamples.
What is the significance of counterexamples in mathematical research? A: Counterexamples are essential tools in mathematical research, as they help mathematicians identify false statements or conjectures and develop new theories.
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Counterexample:
Definition
A counterexample is a form of counter proof.
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5711 | https://aviation.stackexchange.com/questions/88102/why-does-nasa-say-v2-is-velocity-squared-rather-than-speed-squared | terminology - Why does NASA say $v^2$ is velocity squared rather than speed squared? - Aviation Stack Exchange
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Why does NASA say v 2 v 2 is velocity squared rather than speed squared?
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Lift and drag forces scale as the square of the relative flow speed, v v. But speed is the magnitude of the velocity vector; that is, speed is the square root of the sum of the squared velocity vector components, by definition.
Why does NASA say that v 2 v 2 is velocity squared, rather than speed squared?
In aerodynamics, is the sum of squared velocity vector components referred to as velocity squared?
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edited Jul 9, 2021 at 21:26
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2 Where does NASA say that?Camille Goudeseune –Camille Goudeseune 2021-07-09 03:31:03 +00:00 Commented Jul 9, 2021 at 3:31
1 Cf. the generalized use of airspeed instead of air velocity. Yes it is sometimes found on Nasa site, e.g. "The air velocity is the relative speed between the kite and the air. When the kite is held fixed by the control line, the relative air velocity is the wind speed".mins –mins 2021-07-09 12:00:54 +00:00 Commented Jul 9, 2021 at 12:00
3 You really need to include a specific quote or a specific citation in the question itself. I'm sure that if you looked through the vast array of material NASA has published over the decades, you could find some instances of NASA "saying" that v represents "airspeed", or "speed".quiet flyer –quiet flyer 2021-07-09 14:10:49 +00:00 Commented Jul 9, 2021 at 14:10
1 Also, instead of phrasing your question "why does NASA say" (who knows why anyone does anything), it might perhaps be better to phrase it more like "would it be technically more correct to say that v represents airspeed rather than velocity in this equation"-- or something like that-- but maybe that's not the question that you are most interested in having answered--quiet flyer –quiet flyer 2021-07-09 14:11:19 +00:00 Commented Jul 9, 2021 at 14:11
1 I don't entirely understand what your concern is. The square of a vector is generally taken to mean the dot product of the vector with itself, and for velocity and speed v dot v = |v|^2 = speed^2. They're the same thing, just different names llama –llama 2021-07-09 17:35:17 +00:00 Commented Jul 9, 2021 at 17:35
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The proper technical term for the quantity is velocity, and it is a vector quantity. Speed is a colloquial name. If physicists use it, they only use it for the magnitude of velocity, but most of the time there is absolutely no reason to use it because the quantity is a vector one and acts as a vector one in all equations.
Now square of a vector quantity is the same as square of its magnitude, basically by definition, because magnitude is the square root of the square (in Euclidean vector spaces), so it does not really matter.
So given the magnitude is not actually use anywhere, why would they suddenly use the term speed if they don't use it anywhere else?
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answered Jul 9, 2021 at 5:18
Jan HudecJan Hudec
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re "given the magnitude is not actually use anywhere" -- I would suggest that the equation that appears in the link referenced in a comment by the asker of the question, grc.nasa.gov/www/k-12/airplane/lifteq.html , only works if you treat V and L as scalars, not vectors. If you treat V as a vector, don't you end up with L as a vector too, and pointing in the wrong direction, namely parallel to V?quiet flyer –quiet flyer 2021-07-09 14:17:45 +00:00 Commented Jul 9, 2021 at 14:17
@quietflyer It also works if you treat CI as a non-scalar, say a rotation and scaling matrix.Yakk –Yakk 2021-07-09 14:57:43 +00:00 Commented Jul 9, 2021 at 14:57
2 "Speed is a colloquial name." Pretty sure it's the technical name for a scalar value.nick012000 –nick012000 2021-07-09 16:26:50 +00:00 Commented Jul 9, 2021 at 16:26
1 @JanHudec "squaring" the velocity vector to get speed squared makes sense when thinking of squaring as taking the dot product of the velocity with itself. Llama's comment above clears it up for me. Thanks,user59327 –user59327 2021-07-09 18:30:42 +00:00 Commented Jul 9, 2021 at 18:30
1 @user59327 Squaring vectors in Euclidean spaces (that physics uses to model space) is indeed defined as taking dot product with itself. And magnitude as square root of that dot product with itself.Jan Hudec –Jan Hudec 2021-07-09 18:36:10 +00:00 Commented Jul 9, 2021 at 18:36
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Because velocity (being a vector) is a little more precise than pure (scalar) speed - so the angle between the object (aircraft, airfoil, ...) and the incoming air is also taken into consideration.
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answered Jul 9, 2021 at 3:36
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This doesn't sound right; you're talking about the angle of attack, which is accounted for in the lift and drag forces already, e.g. in the coefficient of lift / drag. But scaling as speed squared is different from scaling as velocity squared.user59327 –user59327 2021-07-09 03:48:55 +00:00 Commented Jul 9, 2021 at 3:48
2 @user59327 speed squared and velocity squared is the same thing, because speed is a square root of velocity squared, by definition!Jan Hudec –Jan Hudec 2021-07-09 05:20:11 +00:00 Commented Jul 9, 2021 at 5:20
I would suggest that the equation that appears in the link referenced in a comment by the asker of the question, grc.nasa.gov/www/k-12/airplane/lifteq.html , only works if you treat V and L as scalars, not vectors. If you treat V as a vector, don't you end up with L as a vector too, and pointing in the wrong direction, namely parallel to V?quiet flyer –quiet flyer 2021-07-09 14:20:14 +00:00 Commented Jul 9, 2021 at 14:20
@quietflyer no, it's a vector quantity: en.wikipedia.org/wiki/Lift_(force)eps –eps 2021-07-09 15:22:42 +00:00 Commented Jul 9, 2021 at 15:22
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It’s just because it is derived from a larger more complete theory of aerodynamic forces where it’s necessary to make the distinction. In this equation the coefficient is simplified but a more complete calculation of the coefficient uses the directional components of velocity.
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answered Jul 9, 2021 at 5:07
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TL:DR
Relative flow over a lifting body has to be described as a vector because Lift and Drag are defined wrt the relative flow vector, and Angle of Attack is a parameter.
Is the sum of squared velocity vector components referred to as velocity squared, in aerodynamics?
Yes. Velocity is used to refer to the vector as well as to it's scalar magnitude in scalar formula. This isn't sloppy, it's compact, and more importantly, it maintains the connection to the underlying vector domain.
The squaring-something operator actually represents a large number of different algorithms that are specific to the thing being squared.
Sometimes, there is more than one way to square a particular thing. For instance, a (scalar) distance squared might represent an area, or it might not. But the same symbol is used for the operation, and the associated units are often rendered identically. Sometimes it is important to keep track of the distinction and sometimes it isn't. An area has properties that the square of a distance does not have. Just because two things have the same units doesn't mean they have the same set of properties.
The same applies to velocities and speeds. Both speed and velocity have the same units, but they have different properties. Both can be squared, but the procedures are different.
So to get square of the speed of the relative flow, you take the magnitude of the relative flow, then square it. To get the square of the velocity of the relative flow, you square the velocity (a vector operation), and then take the magnitude if you want a scalar.
It's not luck that s c a l a r s q u a r e(m a g n i t u d e(v e c t o r))s c a l a r s q u a r e(m a g n i t u d e(v e c t o r)) and m a g n i t u d e(v e c t o r s q u a r e(v e c t o r)m a g n i t u d e(v e c t o r s q u a r e(v e c t o r) have the same numeric value and units, The operations were designed to work like that.
But do they have the same properties? Not really. When talking about velocity as a scalar, we are reminded that the domain is a vector domain and that different vectors can have the same magnitude.
Which brings us to a problem, different relative flow vectors with the same magnitude do not have the same Lift and Drag as each other. The only way the formula is valid is if the angle between the Lift and inflow remains fixed and the angle between the drag and the inflow remain fixed, which they do by definition. But all these have to remain fixed with respect to the lifting body as well. So the statement is only valid for constant angles of attack. To suddenly introduce a vector constraint on an entirely scalar chain of math is plain bad form.
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edited Jul 9, 2021 at 19:05
answered Jul 9, 2021 at 15:23
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5712 | https://math.libretexts.org/Courses/Cosumnes_River_College/Math_401%3A_Calculus_II_-_Integral_Calculus/01%3A_Applications_of_Integration/1.09%3A_The_Mean_Value_Theorem_for_Integrals | f(x)
[a,b]
m
M
[a,b]
x
[a,b]
m≤f(x)≤M
m(b−a)≤∫baf(x)dx≤M(b−a).
b−a
m≤1b−a∫baf(x)dx≤M.
1b−a∫baf(x)dx
m
M
f(x)
m
M
[a,b]
c
[a,b]
f(c)=1b−a∫baf(x)dx,
f(x)
14−0∫40(8−2x)dx.
1.9.1
x
y
A=12(base)(height).
A=12(4)(8)=16.
1/(4−0).
14(16)=4
f(c)
c
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1.9: The Mean Value Theorem for Integrals
Last updated
: Sep 18, 2024
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1.8E: Exercises
1.9E: Exercises
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Roy Simpson
Cosumnes River College
( \newcommand{\kernel}{\mathrm{null}\,})
Learning Objectives
Describe the meaning of the Mean Value Theorem for Integrals.
This short section is dedicated to an examination of another essential theorem, the Mean Value Theorem for Integrals.
The Mean Value Theorem for Integrals
Recall, for a discrete list of values, {a1,a2,…,an}{a1,a2,…,an}, we compute the average using the formulaaavg=a1+a2+⋯+ann.
aavg=a1+a2+⋯+ann.
We now have the "mathematical technology" to create the analog for a continuous function, f(x)f(x), over a closed interval [a,b][a,b].
We can approximate the average value of ff as followsfavg≈∑ni=1f(xi)n,
favg≈∑ni=1f(xi)n,
where xixi is defined using our traditional definitions from Riemann sums. However, we now perform a little manipulation to force the numerator to become a Riemann sum.favg≈Δx∑ni=1f(xi)nΔx=∑ni=1f(xi)ΔxnΔx=∑ni=1f(xi)Δxb−a,
favg≈Δx∑ni=1f(xi)nΔx=∑ni=1f(xi)ΔxnΔx=∑ni=1f(xi)Δxb−a,
where we used the fact that Δx=b−anΔx=b−an in our last step.
Taking the limit as n→∞n→∞, we obtain the exact value of the average for ff over the closed interval [a,b][a,b], simply called the average value of ff,favg=1b−a∫baf(x)dx.
favg=1b−a∫baf(x)dx.
This definition leads us to a new theorem to add to our "toolkit." The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. The theorem guarantees that if f(x)f(x) is continuous, a point cc exists in an interval [a,b][a,b] such that the value of the function at cc is equal to the average value of f(x)f(x) over [a,b][a,b]. We state this theorem mathematically with the help of the formula for the average value of a function that we presented in Calculus I.
Theorem: The Mean Value Theorem for Integrals
If f(x)f(x) is continuous over an interval [a,b][a,b], then there is at least one point c∈[a,b]c∈[a,b] such thatf(c)=1b−a∫baf(x)dx.
f(c)=1b−a∫baf(x)dx.
This formula can also be stated as∫baf(x)dx=f(c)(b−a).
∫baf(x)dx=f(c)(b−a).
Proof
: Since f(x) is continuous on [a,b], by the Extreme Value Theorem, it assumes minimum and maximum values - m and M, respectively - on [a,b]. Then, for all x in [a,b], we have m≤f(x)≤M. Therefore, by the Comparison Theorem, we havem(b−a)≤∫baf(x)dx≤M(b−a).Dividing by b−a gives usm≤1b−a∫baf(x)dx≤M.Since 1b−a∫baf(x)dx is a number between m and M, and since f(x) is continuous and assumes the values m and M over [a,b], by the Intermediate Value Theorem, there is a number c over [a,b] such thatf(c)=1b−a∫baf(x)dx,and the proof is complete.
Example 1.9.11.9.1: Finding the Average Value of a Function
Find the average value of the function f(x)=8−2xf(x)=8−2x over the interval [0,4][0,4] and find cc such that f(c)f(c) equals the average value of the function over [0,4].[0,4].
Solution
: The formula states the mean value of f(x) is given by14−0∫40(8−2x)dx.We can see in Figure 1.9.1 that the function represents a straight line and forms a right triangle bounded by the x- and y-axes. The area of the triangle is A=12(base)(height). We haveA=12(4)(8)=16.The average value is found by multiplying the area by 1/(4−0). Thus, the average value of the function is14(16)=4Set the average value equal to f(c) and solve for c.8−2c=4c=2At c=2,f(2)=4.
Figure 1.9.1: By the Mean Value Theorem, the continuous function f(x) takes on its average value at c at least once over a closed interval.
Checkpoint 1.9.1
Find the average value of the function f(x)=x2 over the interval [0,6] and find c such that f(c) equals the average value of the function over [0,6].
Answer
: The average value is 1.5 and c=3.
Example 1.9.2: Finding the Point Where a Function Takes on Its Average Value
Given ∫30x2dx=9, find c such that f(c) equals the average value of f(x)=x2 over [0,3].
Solution
: We are looking for the value of c such thatf(c)=13−0∫30x2dx=13(9)=3.Replacing f(c) with c2, we havec2=3c=±√3.Since −√3 is outside the interval, take only the positive value. Thus, c=√3 (Figure 1.9.2).
Figure 1.9.2: Over the interval [0,3], the function f(x)=x2 takes on its average value at c=√3.
Checkpoint 1.9.2
Given ∫30(2x2−1)dx=15, find c such that f(c) equals the average value of f(x)=2x2−1 over [0,3].
Answer
: c=√3
Key Concepts
The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that f(c) equals the average value of the function.
Key Equations
Mean Value Theorem for Integrals
If f(x) is continuous over an interval [a,b], then there is at least one point c∈[a,b] such that f(c)=1b−a∫baf(x)dx.
Glossary
Mean Value Theorem for integrals
: guarantees that a point c exists such that f(c) is equal to the average value of the function
1.8E: Exercises
1.9E: Exercises |
5713 | https://stackoverflow.com/questions/50608681/algorithm-behind-math-log-java | Algorithm behind Math.log - Java - Stack Overflow
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Algorithm behind Math.log - Java
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Asked 7 years, 4 months ago
Modified2 years, 2 months ago
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Lately I used the Math.log() to code some program and now I ask myself how does the method log() work. How does the computer calculate the logarithm? Thanks for solution.
java
algorithm
math
logarithm
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asked May 30, 2018 at 15:38
tzuxitzuxi
127 1 1 silver badge 9 9 bronze badges
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Possible duplicate of Calculate logarithm by handAlejandro –Alejandro 2018-05-30 15:54:34 +00:00 Commented May 30, 2018 at 15:54
Possible duplicate of Where to find algorithms for standard math functions?phuclv –phuclv 2018-05-30 16:43:49 +00:00 Commented May 30, 2018 at 16:43
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The definition of Math.log from openjdk is:
java
public static double log(double a) {
return StrictMath.log(a); // default impl. delegates to StrictMath
}
This lead me to look at the source for StrictMath, where log is declared as:
java
public static native double log(double a);
The native keyword indicates JNI. In other words, it's off to a libc source to find what we're looking for. Since I use NetBSD, I'll post its definition of log:
```java
/ __ieee754_log(x)
Return the logrithm of x
Method :
1. Argument Reduction: find k and f such that
x = 2^k (1+f),
where sqrt(2)/2 < 1+f < sqrt(2) .
2. Approximation of log(1+f).
Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
= 2s + 2/3 s3 + 2/5 s5 + .....,
= 2s + sR
We use a special Reme algorithm on [0,0.1716] to generate
a polynomial of degree 14 to approximate R The maximum error
of this polynomial approximation is bounded by 2-58.45. In
other words,
2 4 6 8 10 12 14
R(z) ~ Lg1s +Lg2s +Lg3s +Lg4s +Lg5s +Lg6s +Lg7s
(the values of Lg1 to Lg7 are listed in the program)
and
| 2 14 | -58.45
| Lg1s +...+Lg7s - R(z) | <= 2
| |
Note that 2s = f - sf = f - hfsq + shfsq, where hfsq = ff/2.
In order to guarantee error in log below 1ulp, we compute log
by
log(1+f) = f - s(f - R) (if f is not too large)
log(1+f) = f - (hfsq - s(hfsq+R)). (better accuracy)
3. Finally, log(x) = kln2 + log(1+f).
= kln2_hi+(f-(hfsq-(s(hfsq+R)+kln2_lo)))
Here ln2 is split into two floating point number:
ln2_hi + ln2_lo,
where nln2_hi is always exact for |n| < 2000.
Special cases:
log(x) is NaN with signal if x < 0 (including -INF) ;
log(+INF) is +INF; log(0) is -INF with signal;
log(NaN) is that NaN with no signal.
Accuracy:
according to an error analysis, the error is always less than
1 ulp (unit in the last place).
Constants:
The hexadecimal values are the intended ones for the following
constants. The decimal values may be used, provided that the
compiler will convert from decimal to binary accurately enough
to produce the hexadecimal values shown.
/
include "math.h"
include "math_private.h"
static const double
ln2_hi = 6.93147180369123816490e-01, / 3fe62e42 fee00000 /
ln2_lo = 1.90821492927058770002e-10, / 3dea39ef 35793c76 /
two54 = 1.80143985094819840000e+16, / 43500000 00000000 /
Lg1 = 6.666666666666735130e-01, / 3FE55555 55555593 /
Lg2 = 3.999999999940941908e-01, / 3FD99999 9997FA04 /
Lg3 = 2.857142874366239149e-01, / 3FD24924 94229359 /
Lg4 = 2.222219843214978396e-01, / 3FCC71C5 1D8E78AF /
Lg5 = 1.818357216161805012e-01, / 3FC74664 96CB03DE /
Lg6 = 1.531383769920937332e-01, / 3FC39A09 D078C69F /
Lg7 = 1.479819860511658591e-01; / 3FC2F112 DF3E5244 /
static const double zero = 0.0;
double
__ieee754_log(double x)
{
double hfsq,f,s,z,R,w,t1,t2,dk;
int32_t k,hx,i,j;
u_int32_t lx;
EXTRACT_WORDS(hx,lx,x);
k=0;
if (hx < 0x00100000) { / x < 2-1022 /
if (((hx&0x7fffffff)|lx)==0)
return -two54/zero; / log(+-0)=-inf /
if (hx<0) return (x-x)/zero; / log(-#) = NaN /
k -= 54; x = two54; / subnormal number, scale up x /
GET_HIGH_WORD(hx,x);
}
if (hx >= 0x7ff00000) return x+x;
k += (hx>>20)-1023;
hx &= 0x000fffff;
i = (hx+0x95f64)&0x100000;
SET_HIGH_WORD(x,hx|(i^0x3ff00000)); / normalize x or x/2 /
k += (i>>20);
f = x-1.0;
if((0x000fffff&(2+hx))<3) { / |f| < 2-20 /
if(f==zero) { if(k==0) return zero; else {dk=(double)k;
return dkln2_hi+dkln2_lo;}
}
R = ff(0.5-0.33333333333333333f);
if(k==0) return f-R; else {dk=(double)k;
return dkln2_hi-((R-dkln2_lo)-f);}
}
s = f/(2.0+f);
dk = (double)k;
z = ss;
i = hx-0x6147a;
w = zz;
j = 0x6b851-hx;
t1= w(Lg2+w(Lg4+wLg6));
t2= z(Lg1+w(Lg3+w(Lg5+wLg7)));
i |= j;
R = t2+t1;
if(i>0) {
hfsq=0.5ff;
if(k==0) return f-(hfsq-s(hfsq+R)); else
return dkln2_hi-((hfsq-(s(hfsq+R)+dkln2_lo))-f);
} else {
if(k==0) return f-s(f-R); else
return dkln2_hi-((s(f-R)-dkln2_lo)-f);
}
}
```
If you need more help, leave a comment and I'll elaborate.
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answered May 30, 2018 at 15:54
hd1hd1
34.8k 5 5 gold badges 82 82 silver badges 95 95 bronze badges
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cat
catOver a year ago
what are these Lg1...Lg7 constants? they are not log(1)...log(7), that is for sure
2018-06-14T20:52:57.613Z+00:00
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hd1
hd1Over a year ago
Results corresponding to successive invocations of the Reme algorithm for computing logs
2018-06-14T21:46:04.533Z+00:00
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When in doubt, look at the source code (working with Java 8).
java
public static double log(double a) {
return StrictMath.log(a); // default impl. delegates to StrictMath
}
You see it calls a static method from StrictMath.java.
java
static native double log(double a);
What native keyword means you can read here. Basically, it says the method is implemented in another language aside from Java. I have found the folder jdk/src/share/native/java/lang/fdlibm/src/ on GitHub with the math functions. The algorithm is implemented in e_log.c. What kind of sorcery happens inside, I don't know.
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edited Jul 24, 2023 at 15:11
answered May 30, 2018 at 16:08
NikolasNikolas
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tzuxi
tzuxiOver a year ago
So its all on the native referring to the c# code. Thanks for helping.
2018-05-30T16:11:59.633Z+00:00
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The log function (and many other functions such as sin(x)) can be approximated with a series:
If you need a good enough approximation (but not full precision), you can stop after the first few iterations of the series. That's how trigonometric functions were approached in early game/demoscene programming.
Look-up tables are a good approach as well, the key here is to find the appropriate distribution of the lookup values (linear may not be ideal in this case).
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answered Dec 18, 2019 at 10:18
Peter WalserPeter Walser
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5714 | https://www.youtube.com/watch?v=CX2OOQ89cwE | Ch02-03 Blue Ridge - Graphical Method for Linear Programming LP - Level Curves
Decision Making 101
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in the previous episode I showed you how to uh find a feasible region for a linear problem with two variables so we're considering the blue reach hot tubs problem and we determined that the feasible Solutions are points in this region and the points on the boundary of this region right and so now we're still interested we still haven't considered the objective function which will allow us to deter determine which of those feasible points is actually the best point which of those points is the optimal solution which of them gives us the highest total profit so to do this I'll um I I'm going to use um one of the two methods and this in this episode I'll show you just the first method which is called the level curves method and before I do this I'll I'll clean a little bit the picture so you see I removed some parts of the line just so that the picture is clearer if you if you solve it on a piece of paper you can actually draw it once again and um just indicating the boundaries the parts of the constraints which are active boundaries of the feasible region and actually it's a good idea to also keep the labels for the constraint so I know this line comes from the first constraint from the pumps and this line is label and this Lin is tubing the non- negativities are easy to to to identify so no need to have label for them so how do we use level curve method well level curve method right we're focusing on on this line will'll say let's assume some kind of value of the objective function 350 X1 + 300 X2 I'm going to assume equals to some constant value now what constant value is a good constant well you have to assume something for which right uh when let's say I assume 10,000 that gives me a linear equation and if if I want to find points that satisfy this linear equation there are some points on this graph so I have to choose such a constant value for which the straight line will be visible on this on this graph right so in this case for 10,000 you'll see if I try X1 is equal to zero and then solve for X2 well X2 will be 10,000 ided by 300 which is equal to 33.33 so I have a point0 33.33 and this point actually is somewhere here right if I assume for example 1 million I would get a point that is 3,300 and I wouldn't be able to draw it right the line would be off the chart so assume a value here that gives you that that gives you a straight line line level curve which will be you will be able to to draw if you assume a value that is too high you will have it off the chart if you assume a line that is too too uh small a value that is too small you'll just get a line that is very tiny that's next to the 0 okay so I found one point I just need one more Point let's say I assume X2 equals 0 and then find what X1 is that's 10,000 over 350 and that gives me 10,000 over 350 is 28.57 so I have a 28.57 0 X1 is 28.57 X2 0 and so the point is somewhere here so I need to draw a straight line that goes through those two and so you see uh the points that are feasible solutions that lie on this line are points that uh give me $10,000 profit right all points on this line are feasible Solutions so these are solutions I can use and all of them give me $10,000 profit so from just from this exercise I already I already know that I can obtain a profit of $10,000 now question is can I get more right I'm interested in maximizing so let me try another constant value here so let's try it so let's try 350 X1 + 300 X2 equals let me increase the 10,000 to something like three times larger 30,000 right so now if I assume X1 equal 0 X2 actually can be easily calculated is 30,000 over 300 that gives me exactly 100 so I have a 0 100 and the point is here and then if I assume X2 is equal to zero then X1 will be 30,000 over 350 which will be 87 8571 and this point is somewhere here it's 85 71 so I have a point I have two points again I need to draw a straight line that goes through this two point so what do I get from this well uh notice that whatever value I assume here I assumed 10,000 first first and now I assume 30,000 whatever value I assume I'm getting a straight line and those straight lines will be with the same slope right 10,000 30,000 line they have both the same slope all that changes is that the line shifts right and so on this line I have points uh that give me $30,000 profit and you can see there are feasible solutions that give me $30,000 profit and it looks like there will be solutions that can provide even more profit and uh from what we can see by from drawing two lines is that the profit will improve as I move in this direction right so I may ask myself what is the best solution where is the highest what is the solution with the highest profit so I can take this line now without assuming a constant value and just keep taking it to the right keep shifting it to the right and up right right at this point I still have some feasible Solutions I can shift it further but if I go here that's too far right there are no Solutions on this line that are feasible right even if I extend it further right so I I shifted it too far so what I have to do is I have to keep shifting this line in the direction where the profits improve until I reach a point like this where I am still crossing a feasible solution this line is still overlapping with the feasible region in this one point but if I take it any further I will not be overlapping with the feasible region right in this case uh this this way I will detect the solution that is the optimal solution because this line says right this is a certain profit I don't know what profit it is but it indicates that this point gives me the highest profit because if I go if I shift left I will have lower profits so I will have solutions that are worse and if I shift right I would have higher profits but there are no feasible Solutions so this must be the best solution right this must be then the best solution which means which we call the optimal solution or the highest profit solution in this case right so this solution is the optimal solution now you may ask yourself well uh what is what is it then how much how much is X1 how much is X2 and some of you might be tempted to actually read X1 or X2 from the graph and say oh this is somewhere 120 this must be 80 or 90 right reading the solution from the graph is not a good idea because you're not going to get an accurate value how can we get the accurate value well now now is the time when we can actually use the information we saved from before right this point is an intersection of two lines this and this this line corresponds to the constraint to the first constraint on pumps and this line corresponds to the boundary right these are boundaries of the second constraint labor hours so the straight lines are the same they have the equations that are the same as those formulas except that the same as those constraints except those uh those less than or equal signs have to be changed to equal so I can find the optimal Solution by solving a set of two equations right one of them will be the pumps constraint X1 + X2 = 200 and the other will be 9 X1 + 6 X2 = 1566 right if I solve this I will find the coordinates of this point so there are many ways of solving it of course I can try to uh get X1 from the first equation 200 - X2 and then plug it into the second equation so I'll have 9 X1 and X1 is 200 - X2 + 6 X2 = 1566 and so that is 1,800 - 9 X2 + 6 X2 = 1566 so that is - 3x2 = 234 - 284 34 sorry and that means X2 is equal to 234 / 3 78 so you see you will write that X2 is somewhere about 80 but it's not exactly 80 it is 78 and then X1 from this right X1 is 200 - 78 that means 122 so we we now determined the optimal solution of this problem is 12278 so that means we have to produce 122 aqua spass and 78 hydr luxes and still the interesting question is what is the total profit right so we can say first of all let's say optimal solution is we indicate it often as X1 star X2 star X2 star equals 120 278 and then we can say well what is the optimal objective [Music] function value well optimal objective function value is the value of the the total profit function right so what is the in other words optimal total profit well we can calculate it that's 350 time 122 plus 300 300 X2 and X2 optimal value is 78 so if you compute this you'll discover that it is actually $ 66,1 so we can write an interpretation of this is well optimal solution is produce 122 Aqua Spar and 78 hydro luxes and that will give you the optimal profit of $66,000 100 66100 $ 66,1 100 so that completes the first method the level curve method |
5715 | https://www.hansrajcollege.ac.in/hCPanel/uploads/elearning/elearning_document/Number_Theory_Course_Lesson_I.pdf | Mukund NUMBER THEORY COURSE LESSON : PRIMITIVE ROOTS-I 1. Introduction The additive group of the ring (Zn, +, ·) of integers modulo n is known to be cyclic but the multiplicative group of units in Zn namely U(n) may or may not be cyclic. The Lesson on primitive roots is aimed to determine the values of n for which U(n) is cyclic. Moreover, for a given n > 1, the problem of finding generators of U(n), whenever it is cyclic will also be settled.
We start with a simple result before giving the first definition.
Proposition 1.1. Let a, n ∈Z, n > 1.
Then ak ≡1 mod n for some k > 0 iff gcd (a, n) = 1.
Proof. If gcd (a, n) = 1 then by Euler’s theorem, aϕn ≡1 mod n. Conversely, if there is some k > 0 such that ak ≡1 mod n, we claim that ax ≡1 mod n has a solution. If k = 1, x = 1 is a solution. If k > 1, x = ak−1 is a solution. But we know that solvability of ax ≡1 mod n is equivalent to gcd (a, n) = 1.
□ Definitions 1.1. Let n > 1 and a be an integer such that gcd (a, n) = 1. k is called the order of a modulo n if k is the least postitve integer such that ak ≡1 mod n.
We write Ordna = k.
In view of the above proposition, the assumption gcd (a, n) = 1 is necessary as well as sufficient for the definition of order of a modulo n make sense.
By reducing down a modulo n, one may obtain an element of the group U(n).
Definition 1.1 simply defines order of this element in the group U(n).
Definitions 1.2 (Primitive root). An integer a, if exists, such that Ordna = φ(n) is called a primitive root of n.
2. Primitive roots for primes In most of the probes related to modular arithmatic, starting with primes hap-pens to be a nice idea. In this section we try to find whether there exists a primitive root for a prime or not, and if it does, what are the possible integers that may be primitive roots for the prime.
The first tool we need is a result due to Lagrange. The fundamental theorem of algebra tells that any polynomial with complex coefficients uses to have as many zeros as its degree. If a polynomial of degree n with rational coefficients is viewed as a polynomial with complex coefficients, one knows that it has n many (not necessarily distinct) complex zeross. Out of those complex zeross, some may be rational numbers and therefore the maximum number of rational zeros is n. A similar result can be stated for polynomials with coefficients from the field Zp.
1 Mukund 2 LESSON : PRIMITIVE ROOTS-I However, over an arbitrary ring, there are no bounds for number of zeros of a polynomial.
Example 2.1. Consider the ring Z8. The elements 1, 3, 5 and 7 are all zeros of x2 +7.
We now prove the result for the fields Zp.
Theorem 2.2. Let p be a prime and f(x) = anxn +an−1xn−1 +. . .+a0 be a polynomial with integral coefficients of degree n ≥1 and an . 0 mod p. Then the congruence f(x) ≡0 mod p has at most n incongruent solutions modulo p.
Proof. The proof is by induction on degree of f. If degree f is 1, then f(x) = a1x+a0.
Let x0 be the solution of ax ≡1 mod p then y = −x0a0 is the unique solution to f(x) ≡0 mod p. Now assume that the result is true for all polynomials with degree less than or equal to n and let deg f = n + 1. If f(x) ≡0 mod p does not have a solution then we are done. If it has a solution α then by division algorithm we may write f(x) = (x −α)g(x) + r(x) with deg r < 1 (Since deg (x −α)=1) i.e. r is an integer. Putting x = α, we get r ≡0 mod p and f(x) ≡(x −α)g(x) mod p. Now suppose β be a solution of f(x) ≡0 mod p that is incongruent to α modulo p.Then 0 ≡f(β) ≡(β −α)g(β) mod p Since β −α . 0 mod p, we have g(β) ≡0 mod p. Since deg g=n, by our induction hypothesis, there are atmost n such β that are incongruent modulo p. Hence f(x) ≡0 mod p has at most n + 1 many solutions incongruent modulo p.
□ Remark 2.1. Above theorem can be restated as ”A polynomial of degree n in Zp[x] can have at most n zeros in the base field Zp.” A close look at the proof indicates that the key part is availability of the division algorithm. Using the long division algorithm, one may conclude that division algorithm holds in F[x] for any field F.
Therefore it may be concluded that any polynomial of degree n in F[x] can have at most n zeros.
Proposition 2.3. If p is a prime then the equation xp−1 ≡1 mod p has precisely p −1 roots that are incongruent modulo p.
Proof. Let a ∈{1, 2, . . . , p −1} then by Fermat’s theorem, ap−1 ≡1 mod p.e So thte equation xp−1 ≡1 mod p has at least p −1 incongruent solutions. The result now follows by Lagrange’s theorem.
□ Proposition 2.4. If p is a prime and d p −1 then xd −1 ≡0 mod p has precisely d roots incongrunet modulo p.
Proof. When d p −1, (xd −1) (xp−1 −1) so that we may write xp−1 = xd −1 g(x) for some polynomial g(x) of degree p −1 −d. By Lagrange’s theorem, g can have at most p −1 −d incongruent zeros. Any zero of xp−1 is a zero of xd −1 or of g(x).
xp−1 −1 has p −1 incongruent zeros which is the maximum possible number of zeros. Therefore both the polynomials xd−1 and g(x) must have resp. d and p−1−d incongruent zeros.
□ Mukund NUMBER THEORY COURSE 3 Remark 2.2 (A new proof of Wilson’s theorem). Define the polynomial f(x) = (x −1)(x −2) . . . (x −p + 1) − xp−1 −1 .
This polynomial is of degree less than p −1 as xp−1 term gets cancelled out. We write f(x) = ap−2xp−2 + . . . + a1x + a0.
Let a ∈{1, 2, . . . , p−1}. Then (a−1)(a−2) . . . (a−p+1) = 0 and by Fermat’s theorem, xp−1 −1 ≡0 mod p. Therefore the equation f(x) ≡0 mod p has p −1 incongrunt solutions. This is possible only when f is the zero polynomial i.e. ap−2 = ap−3 = . . . = a0 ≡0 mod p. Therefore, for any integer a, (a −1)(a −2) . . . (a −p + 1) − ap−1 −1 ≡ 0 mod p i.e. (a −1)(a −2) . . . (a −p + 1) ≡ ap−1 −1 mod p Putting a = p −1 above, we get (p −1)! ≡−1 mod p The next objective is to prove the existence of primitive roots for prime numbers.
If that is assumed, for a prime p, the group U(p) would become cyclic of order p−1.
In that case, our knowledge of group theory tells that for every divisor d of p −1, there is exactly one subgroup of U(p) or order d which would contain ϕ(d) many elements of order d. The following theorem establishes all these facts about U(p) in one go.
Theorem 2.5. Let p be a prime and d (p −1) then there are exactly ϕ(d) many incongruent integers having order d modulo p.
Proof. The proof relies on the Lagrange’s theorem and the identity p −1 = X d|(p−1) ϕ(d).
(2.1) Since every element of the set A = {1, 2, . . . , p −1} is coprime to p, it has a finite order that divides ϕ(p) = p −1. Therefore, if ψ(d) denotes the number of elements of A having order d then we have p −1 = X d|(p−1) ψ(d).
(2.2) Comparing equations (2.1) and (2.2), we get X d|(p−1) ϕ(d) = X d|(p−1) ψ(d).
(2.3) We first prove that if for some d, ψ(d) is non-zero then ψ(d) must be equal to ϕ(d).
Let a be of order d in A. Then, for any 1 ≤k ≤p −1, Ordpak = d gcd (d, k) so that Ordpak = d iffgcd (d, k) = 1. So there are exactly φ(d) many integers in A whose order is d. Finally, all the elements of A satisfy xd −1 ≡0 mod p so by Lagrange’s theorem, any integer whose order is d must be congruent to some integer in A.
Thus the total number of incongruent integers having order d modulo p must be ϕ(d). Finally, it is easy to see that ψ(d) , 0 for any divisor d of p −1. For if ψ(d0) = 0 Mukund 4 LESSON : PRIMITIVE ROOTS-I for some divisor d0 of p −1 then the equation (2.3) can not hold, as vp(d0) > 0. This completes the proof.
□ Corollary 2.6. If p is a prime, then there are exactly ϕ(p −1) incongruent primitive roots of p.
Proof. Take d = p −1 in the above theorem.
□ Example 2.1. If p is a prime of type 4k + 1 then the congruence x2 ≡−1 mod p admits a solution.
Since p = 4k + 1, 4 is a divisor of p −1. Thus there exists an integer of order 4 modulo p. Let a be one such integer. Then a4 ≡1 mod p =⇒a4 −1 ≡0 mod p =⇒(a2 −1)(a2 + 1) ≡0 mod p.
Therefore, a2 −1 ≡0 mod p or a2 +1 ≡0 mod p. But a2 −1 ≡0 mod p contradicts to the assumption that order of a modulo p was 4. Hence a2 + 1 ≡0 mod p holds whereby, a becomes a solution to the congruence x2 ≡−1 mod p.
3. Exercises Exercise 1. If p is an odd prime, prove that (a) the only incongruent solutions of x2 ≡1 mod p are 1 and p −1.
(b) The congruence xp−2 +. . .+x2 +x+1 ≡0 mod p has exactly p−2 incongruent solutions and they are 2, 3, . . . , p −1.
Hint : (a) x2 ≡1 mod p =⇒ (x −1)(x + 1) ≡0 mod p whereby, x ≡±1 mod p. (b) Clearly, 1 does not sarisfy the congruence. If a . 1 mod p satisfies the congruence ap−2 + . . . + a2 + a + 1 ≡0 mod p if and only if (a −1)(ap−2 + . . . + a2 + a + 1) ≡0 mod p ⇐⇒ap−1 −1 ≡0 mod p.
Exercise 2. Determine all the primitive roots of 17.
Hint : By hit and trial, it turns out that 2 is not a primitive root of 17 but 3 is.
Now, ϕ(17) = 16. There must be ϕ(16) = 8 primitive roots of 7. These primitive roots are of type 3k where gcd (k, 16) = 1.
Exercise 3. Given that 3 is a primitive root of 43, find all positive integers less than 43 whose order is 6 modulo 43.
Hint : Order of an element of type 3k is 42 gcd (k, 42). Possible values of k are 7, 35 (there can be 2 elements only of order 6 as ϕ(6) = 2).
Exercise 4. Assume that r and r′ are primitive roots of an odd prime p. Show that rr′ can not be a primitive root of p.
Hint : First, observe that any primitve s of p satisfies s(p−1)/2 ≡−1 mod p.
For s(p−1)/2 is a root of x2 ≡1 mod p and s(p−1)/2 . 1 mod p. Now, (rr′)(p−1)/2 ≡ r(p−1)/2r′(p−1)/2 ≡1 mod p and thus rr′ can not be a primitive root of p. |
5716 | https://brainly.com/question/2497128 | [FREE] Calculate the molality of a sugar solution that contains 3.94 g of sucrose (C_{12}H_{22}O_{11}) dissolved - brainly.com
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Calculate the molality of a sugar solution that contains 3.94 g of sucrose (C 12H 22O 11) dissolved in 285 g of water.
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Answer : The molality of sugar solution is, 0.04038 mole/Kg
Solution : Given,
Mass of solute (sucrose) = 3.94 g
Mass of solvent (water) = 285 g
Molar mass of sucrose = 342.3 g/mole
Molality : It is defined as the number of moles of solute present in one kilogram of solvent.
Formula used :
M o l a l i t y=Molar mass of solute×Mass of solvent in g Mass of solute×1000=342.3 g/m o l e×285 g 3.94 g×1000=0.04038 m o l e/K g
Therefore, the molality of sugar solution is, 0.04038 mole/Kg
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Chemistry 2e - Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson
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The molality of the sugar solution is approximately 0.0404 m, derived from 3.94 g of sucrose dissolved in 285 g of water. This involves calculating moles of sucrose and converting the mass of water to kilograms. Finally, molality is calculated by dividing moles of solute by the mass of solvent in kg.
Explanation
To calculate the molality of the sugar solution containing 3.94 g of sucrose (C₁₂H₂₂O₁₁) dissolved in 285 g of water, we follow these steps:
Calculate the moles of sucrose:
First, we need the molar mass of sucrose, which is approximately 342.3 g/mol.
Next, we can calculate the number of moles using the formula:
Moles of sucrose=molar mass of sucrose(g/mol)mass of sucrose(g)=342.3 g/mol 3.94 g≈0.0115 moles
Convert the mass of water:
We need to convert the mass of the solvent (water) from grams to kilograms:
Mass of water(kg)=1000 285 g=0.285 kg
Calculate molality:
Molality is defined as the number of moles of solute per kilogram of solvent. The formula is:
Molality(m)=mass of solvent(kg)moles of solute
Substituting in the values we found:
Molality=0.285 kg 0.0115 moles≈0.0404 m
Therefore, the molality of the sugar solution is approximately 0.0404 m.
Examples & Evidence
For example, if you were to dissolve 10 g of sodium chloride in 500 g of water, you would calculate the moles of sodium chloride and divide that by 0.5 kg to find the molality.
The calculations used in the answer rely on established chemical formulas for moles, molar mass, and molality, which are standard concepts taught in chemistry.
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Community Answer 5.0 6 Calculate the molality of each of the following solutions: (a) 14.3 g of sucrose (C12H22O11) in 685 g of water, (b) 7.15 moles of ethylene glycol (C2H6O2) in 3505 g of water.
Community Answer 4.2 19 A drink that contains 4 1/2 ounces of a proof liquor… approximately how many drinks does this beverage contain?
Community Answer 5.0 7 Chemical contamination is more likely to occur under which of the following situations? When cleaning products are not stored properly When dishes are sanitized with a chlorine solution When raw poultry is stored above a ready-to-eat food When vegetables are prepared on a cutting board that has not been sanitized
Community Answer 4.3 189 1. Holding 100mL of water (ebkare)__2. Measuring 27 mL of liquid(daudgtear ldnreiyc)____3. Measuring exactly 43mL of an acid (rtube)____4. Massing out120 g of sodium chloride (acbnela)____5. Suspending glassware over the Bunsen burner (rwei zeagu)____6. Used to pour liquids into containers with small openings or to hold filter paper (unfenl)____7. Mixing a small amount of chemicals together (lewl letpa)____8. Heating contents in a test tube (estt ubet smalcp)____9. Holding many test tubes filled with chemicals (estt ubet karc) ____10. Used to clean the inside of test tubes or graduated cylinders (iwer srbuh)____11. Keeping liquid contents in a beaker from splattering (tahcw sgasl)____12. A narrow-mouthed container used to transport, heat or store substances, often used when a stopper is required (ymerereel kslaf)____13. Heating contents in the lab (nuesnb bneurr)____14. Transport a hot beaker (gntos)____15. Protects the eyes from flying objects or chemical splashes(ggloges)____16. Used to grind chemicals to powder (tmraor nda stlepe) __
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Community Answer 8 If the temperature of a reversible reaction in dynamic equilibrium increases, how will the equilibrium change? A. It will shift towards the products. B. It will shift towards the endothermic reaction. C. It will not change. D. It will shift towards the exothermic reaction.
Community Answer 4.8 52 Which statements are TRUE about energy and matter in stars? Select the three correct answers. Al energy is converted into matter in stars Only matter is conserved within stars. Only energy is conserved within stars. Some matter is converted into energy within stars. Energy and matter are both conserved in stars Energy in stars causes the fusion of light elements
Community Answer 4.5 153 The pH of a solution is 2.0. Which statement is correct? Useful formulas include StartBracket upper H subscript 3 upper O superscript plus EndBracket equals 10 superscript negative p H., StartBracket upper O upper H superscript minus EndBracket equals 10 superscript negative p O H., p H plus P O H equals 14., and StartBracket upper H subscript 3 upper O superscript plus EndBracket StartBracket upper O upper H superscript minus EndBracket equals 10 to the negative 14 power.
Community Answer 5 Dimensional Analysis 1. I have 470 milligrams of table salt, which is the chemical compound NaCl. How many liters of NaCl solution can I make if I want the solution to be 0.90% NaCl? (9 grams of salt per 1000 grams of solution). The density of the NaCl solution is 1.0 g solution/mL solution.
Community Answer 8 For which pair of functions is the exponential consistently growing at a faster rate than the quadratic over the interval mc015-1. Jpg? mc015-2. Jpg mc015-3. Jpg mc015-4. Jpg mc015-5. Jpg.
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5717 | https://www.differencebetween.net/science/difference-between-feet-and-square-feet/ | Difference Between Feet and Square Feet
• Categorized under Mathematics & Statistics,Physics,Science,Words | Difference Between Feet and Square Feet
The length of anything is measured in inches or feet. The feet is the plural form of foot, which is a non-SI unit of length in the imperial and the United States customary systems of measurement. However, when describing an architectural space – whether rectangle or square – the area is to be calculated using two measurements as area is a two-dimensional measurement. The area of a room is calculated by multiplying the length and width of the room in feet. For example, if a room is 15 feet in length and 12 feet in width, then the area of the room is 15 multiplied by 12 which equals to 180 square feet.
Difference between Feet and Square Feet
Basics of Feet Vs. Square Feet
The feet (plural of foot) is the non-SI unit for measuring length as specified in the imperial and the US customary units of measurement. It is used to measure height, length, and distance. The square feet is the non-SI unit for measuring area, mainly in the United States. It is the plural of square foot used to measure any two-dimensional space.
Symbol & Calculation of Feet Vs. Square Feet
The international symbol for feet is “ft”, whereas square feet is symbolized as “sq ft.” or “ft2”. The area of a rectangular or square room is calculated in square feet by multiplying the length and width of the room in feet. If a room is 12 feet in length and 12 in width, then the area of the room equals to (1212) 144 sq ft.
Conversion of Feet Vs. Square Feet
1 foot converts to 12 inches or 0.3048 meters. Similarly, 1 square feet is equivalent to 0.092903 square meters or 144 square inches. An area of 1 square feet would be 1212 inches squared. Feet is a single dimensional unit, whereas square feet is a two-dimensional unit for area.
Examples of Feet Vs. Square Feet
If a room measures 15 feet in length and 10 feet in width, then the area of the room is calculated as:
Area = 15 × 10 = 150 square feet or 150 sq ft. or 150 ft2
Feet vs. Square Feet: Comparison Chart
Summary of Feet and Square Feet
When you measure the length of anything, it is measured in inches or feet, but area is a two-dimensional measurement, so it is calculated by two measurements – the length and width. To calculate an architectural space in square feet, the length and width are measured in feet, and then multiplied together to obtain the area in square feet or ft2. Foot is the non-SI unit for measuring length, whereas square foot is the non-SI unit of measuring area.
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Khillar, S. (2018, November 8). Difference Between Feet and Square Feet. Difference Between Similar Terms and Objects.
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Khillar, Sagar. "Difference Between Feet and Square Feet." Difference Between Similar Terms and Objects, 8 November, 2018,
2 Comments
Really useful write up,helping out right now.
Ft square indicates something that is the same length on 4 sides Thus something that was 10 ft square would be 10’ by 10’ thus 100 square ft.
Square ft on the other hand is an equation made by multiplying the length times the width, again as shown above.
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Written by : Sagar Khillar.
and updated on 2018, November 8
References :
Image credit:
Qasim, S.H. SI Units in Engineering and Technology. NYC: Elsevier, 2016. Print
Taylor, Barry. Guide for the Use of the International System of Units (SI): The Metric System. Pennsylvania: DIANE Publishing, 1995. Print
Articles on DifferenceBetween.net are general information, and are not intended to substitute for professional advice. The information is "AS IS", "WITH ALL FAULTS". User assumes all risk of use, damage, or injury. You agree that we have no liability for any damages.
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5、不确定变量在基于知识的资源分配中的应用
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分类专栏：分布式知识管理中的智能集成探索文章标签：不确定变量资源分配专家知识
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不确定变量在基于知识的资源分配中的应用
1. 引言
在众多不确定性理论中，不确定变量是一种对专家知识进行建模的有效工具。不确定变量有两个软属性：“$x \approx \tilde{x}$” 表示 “$x$ 近似等于 $\tilde{x}$”；“$x \in \tilde{D}_x$” 表示 “$x$ 近似属于集合 $\tilde{D}_x$”。它由值集 $X$、函数 $h(x) = v(\tilde{x} = x)$（即专家给出的 $x \approx \tilde{x}$ 的确定性指数）以及一些特定定义来确定。
考虑一个输入向量为 $u \in U$，输出向量为 $y \in Y$ 的系统，由关系 $R(u, y; x) \subset U \times Y$ 描述，其中未知参数向量 $x \in X$ 是一个由专家给出的确定性分布 $h(x)$ 描述的不确定变量的值。对于用户给定的要求 $y \in D_y \subset Y$，决策问题是找到一个决策 $u^$，使可能输出集近似属于 $D_y$ 的确定性指数最大化。
一个重要的例子是操作复合体，它由一系列操作组成，每个操作的执行时间取决于分配给它的资源量。所有操作使用同一种连续资源，且资源可以任意分配。在基于知识的方法中，这种关系以一种包含未知参数的关系形式存在，该未知参数被假定为专家描述的不确定变量的值。决策问题是找到一种资源分配方案，优化给定的性能指标并满足用户通常关于整个操作集执行时间的要求。
与广泛用于生产或项目管理的活动网络不同，操作复合体的执行时间由专家描述时，应使用不确定变量的形式主义。对于并行和级联结构的操作复合体，已经开发了资源分配 算法，并提出了改进其质量的方法。
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5719 | https://www.gnu.org/software/guile/manual/html_node/Reals-and-Rationals.html | Reals and Rationals (Guile Reference Manual)
Next: Complex Numbers, Previous: Integers, Up: Numerical data types [Contents][Index]
6.6.2.3 Real and Rational Numbers¶
Mathematically, the real numbers are the set of numbers that describe all possible points along a continuous, infinite, one-dimensional line. The rational numbers are the set of all numbers that can be written as fractions p/q, where p and q are integers. All rational numbers are also real, but there are real numbers that are not rational, for example the square root of 2, and pi.
Guile can represent both exact and inexact rational numbers, but it cannot represent precise finite irrational numbers. Exact rationals are represented by storing the numerator and denominator as two exact integers. Inexact rationals are stored as floating point numbers using the C type double.
Exact rationals are written as a fraction of integers. There must be no whitespace around the slash:
1/2
-22/7
Even though the actual encoding of inexact rationals is in binary, it may be helpful to think of it as a decimal number with a limited number of significant figures and a decimal point somewhere, since this corresponds to the standard notation for non-whole numbers. For example:
0.34
-0.00000142857931198
-5648394822220000000000.0
4.0
The limited precision of Guile’s encoding means that any finite “real” number in Guile can be written in a rational form, by multiplying and then dividing by sufficient powers of 10 (or in fact, 2). For example, ‘-0.00000142857931198’ is the same as −142857931198 divided by 100000000000000000. In Guile’s current incarnation, therefore, the rational? and real? predicates are equivalent for finite numbers.
Dividing by an exact zero leads to a error message, as one might expect. However, dividing by an inexact zero does not produce an error. Instead, the result of the division is either plus or minus infinity, depending on the sign of the divided number and the sign of the zero divisor (some platforms support signed zeroes ‘-0.0’ and ‘+0.0’; ‘0.0’ is the same as ‘+0.0’).
Dividing zero by an inexact zero yields a NaN (‘not a number’) value, although they are actually considered numbers by Scheme. Attempts to compare a NaN value with any number (including itself) using =, <, >, <= or >= always returns #f. Although a NaN value is not = to itself, it is both eqv? and equal? to itself and other NaN values. However, the preferred way to test for them is by using nan?.
The real NaN values and infinities are written ‘+nan.0’, ‘+inf.0’ and ‘-inf.0’. This syntax is also recognized by read as an extension to the usual Scheme syntax. These special values are considered by Scheme to be inexact real numbers but not rational. Note that non-real complex numbers may also contain infinities or NaN values in their real or imaginary parts. To test a real number to see if it is infinite, a NaN value, or neither, use inf?, nan?, or finite?, respectively. Every real number in Scheme belongs to precisely one of those three classes.
On platforms that follow IEEE 754 for their floating point arithmetic, the ‘+inf.0’, ‘-inf.0’, and ‘+nan.0’ values are implemented using the corresponding IEEE 754 values. They behave in arithmetic operations like IEEE 754 describes it, i.e., (= +nan.0 +nan.0) ⇒ #f.
Scheme Procedure: real?obj¶C Function: scm_real_p(obj)¶
Return #t if obj is a real number, else #f. Note that the sets of integer and rational values form subsets of the set of real numbers, so the predicate will also be fulfilled if obj is an integer number or a rational number.
Scheme Procedure: rational?x¶C Function: scm_rational_p(x)¶
Return #t if x is a rational number, #f otherwise. Note that the set of integer values forms a subset of the set of rational numbers, i.e. the predicate will also be fulfilled if x is an integer number.
Scheme Procedure: rationalizex eps¶C Function: scm_rationalize(x, eps)¶
Returns the simplest rational number differing from x by no more than eps.
As required by R5RS, rationalize only returns an exact result when both its arguments are exact. Thus, you might need to use inexact->exact on the arguments.
(rationalize (inexact->exact 1.2) 1/100)
⇒ 6/5
Scheme Procedure: inf?x¶C Function: scm_inf_p(x)¶
Return #t if the real number x is ‘+inf.0’ or ‘-inf.0’. Otherwise return #f.
Scheme Procedure: nan?x¶C Function: scm_nan_p(x)¶
Return #t if the real number x is ‘+nan.0’, or #f otherwise.
Scheme Procedure: finite?x¶C Function: scm_finite_p(x)¶
Return #t if the real number x is neither infinite nor a NaN, #f otherwise.
Scheme Procedure: nan¶C Function: scm_nan()¶
Return ‘+nan.0’, a NaN value.
Scheme Procedure: inf¶C Function: scm_inf()¶
Return ‘+inf.0’, positive infinity.
Scheme Procedure: numeratorx¶C Function: scm_numerator(x)¶
Return the numerator of the rational number x.
Scheme Procedure: denominatorx¶C Function: scm_denominator(x)¶
Return the denominator of the rational number x.
C Function: intscm_is_real(SCM val)¶C Function: intscm_is_rational(SCM val)¶
Equivalent to scm_is_true (scm_real_p (val)) and scm_is_true (scm_rational_p (val)), respectively.
C Function: doublescm_to_double(SCM val)¶
Returns the number closest to val that is representable as a double. Returns infinity for a val that is too large in magnitude. The argument val must be a real number.
C Function: SCMscm_from_double(double val)¶
Return the SCM value that represents val. The returned value is inexact according to the predicate inexact?, but it will be exactly equal to val.
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5720 | https://faseb.onlinelibrary.wiley.com/doi/pdfdirect/10.1096/fasebj.10.2.8641569 | P/O ratios reassessed: Mitochondrial P/O ratios consistently exceed 1.5 with succinate and 2.5 with NAD‐linked substrates - Lee - 1996 - The FASEB Journal - Wiley Online Library
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The FASEB Journal
Volume 10, Issue 2 pp. 345-350
Research Communication
P/O ratios reassessed: Mitochondrial P/O ratios consistently exceed 1.5 with succinate and 2.5 with NAD-linked substrates
C.P. Lee,
Corresponding Author
C.P. Lee
n/a@.dne
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
To whom correspondence should be addressed, at: Department of Biochemistry, Gordon H. Scott Hall of Basic Med. Sci., 540 E. Canfield Ave., Detroit, MI 48201, USA.Search for more papers by this author
Q. Gu,
Q. Gu
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
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Y. Xiong,
Y. Xiong
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
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R. A. Mitchell,
R. A. Mitchell
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
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L. Ernster,
L. Ernster
Department of Biochemistry, Arrhenius Laboratories for Natural Sciences, University of Stockholm, S-106 91 Stockholm, Sweden
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C.P. Lee,
Corresponding Author
C.P. Lee
n/a@.dne
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
To whom correspondence should be addressed, at: Department of Biochemistry, Gordon H. Scott Hall of Basic Med. Sci., 540 E. Canfield Ave., Detroit, MI 48201, USA.Search for more papers by this author
Q. Gu,
Q. Gu
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
Search for more papers by this author
Y. Xiong,
Y. Xiong
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
Search for more papers by this author
R. A. Mitchell,
R. A. Mitchell
Department of Biochemistry, Wayne State University School of Medicine, Detroit, Michigan, 48201 USA
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L. Ernster,
L. Ernster
Department of Biochemistry, Arrhenius Laboratories for Natural Sciences, University of Stockholm, S-106 91 Stockholm, Sweden
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First published: 01 February 1996
Citations: 50
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Abstract
The efficiency of ATP synthesis coupled to cell respiration, commonly referred to as the P/O ratio, has been the subject of extensive studies for more than 50 years. The general conclusion from these studies is that respiring mitochondria can convert external ADP to ATP at a maximal P/O ratio of 3 for NAD-linked substrates and 2 for succinate. However, in recent years the validity of these “integral” values has been questioned on both mechanistic and thermodynamic grounds, and a mechanistic P/O ratio of 2.5 for NAD-linked substrates and 1.5 for succinate have been concluded on the basis of experiments with isolated mitochondria. These values have been widely adopted in the scientific literature, including several recent textbooks. In this paper we report that under optimal conditions with respect to preparation and assay procedures, the P/O ratios obtained with isolated rat liver mitochondria consistently exceed 2.5 with NAD-linked substrates and 1.5 with succinate. These results, although not excluding “nonintegral” P/O ratios due to various energy-dissipating side reactions, warrant caution in accepting the reported lower values and, in general, in referring to mechanistic considerations unless the underlying molecular mechanisms are understood.—Lee, C. P., Gu, Q., Xiong, Y., Mitchell, R. A., Ernster, L. P/O ratios reassessed: mitochondrial P/O ratios consistently exceed 1.5 with succinate and 2.5 with NAD-linked substrates. FASEB J. 10, 345-350 (1996)
Citing Literature
Volume 10, Issue 2
February 1996
Pages 345-350
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5721 | https://physics.stackexchange.com/questions/472856/divergence-gradient-and-differentiation-radial-irrotational-fluid-flow | Skip to main content
Divergence, gradient and differentiation - radial irrotational fluid flow
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Given a fluid with the steady spherically symmetric flow with only radial velocity v⃗ (r).
We need to evaluate v⃗ ⋅∇v⃗ .
From vector calculus
v⃗ ⋅∇(v⃗ )=12grad(v2)−v⃗ ×rot(v⃗ )
As fluid is irrotational the second term vanishes, so
v⃗ ⋅∇(v⃗ )=12grad(v2)
The expression on the right hand side is
12grad(v2)=v⃗ ⋅∂v⃗ ∂r
because we evaluate gradient by simply differentiating v2(r) with respect to r. But term on the left hand side can be written using divirgence in spherical coordinates as
v⃗ ⋅div(v⃗ )=v⃗ ⋅1r2∂∂r(r2v)
(where v=vr is radial projection of the velocity, scalar value) which after differentiating the product r2v(r) gives
v⃗ ⋅div(v⃗ )=v⃗ ⋅(2vr+∂v∂r)=2v⃗ ⋅vr+v⃗ ⋅∂v∂r
So, there is additional term 2v⃗ ⋅vr comparing to the above for 12grad(v2) which contradicts to the first expression. Where did I miss something? Many thanks!
homework-and-exercises
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edited Apr 15, 2019 at 18:13
Eddward
asked Apr 15, 2019 at 10:29
EddwardEddward
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In your third expression on the RHS what object is ∂v/∂r supposed to be? You imply it is a vector so I guess you mean ∂v⃗ /∂r where r is the radial coordinate (i.e. a single number, not a vector). In your fourth its not clear why you think the divergence is the correct thing to compute since vi∂ivj≠vj∂ivi.
– jacob1729
Commented
Apr 15, 2019 at 16:07
Yes, v⃗ is a vector in the third expression and r is radial coordinate. In forth expression, I apply standard divergence to radial v(r) in spherical coordinates which is the correct expression in case of irrotational fluid.
– Eddward
Commented
Apr 15, 2019 at 16:28
1
Is your confusion just that you are reading ∇v⃗ as div(v⃗ )?
– jacob1729
Commented
Apr 15, 2019 at 16:46
In order to avoid such confusion, I am going edit all these ∇ as div() and grad(). But my question will remain..
– Eddward
Commented
Apr 15, 2019 at 16:51
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OPs fundamental issue is I believe the misunderstanding of the symbol ∇⃗ v⃗ (note: there is no dot product) as a divergence div(v⃗ ). The former is a rank two tensor the latter a scalar.
Going line by line, we have the first equation is a vector equation (vector dotted into tensor equals gradient of scalar):
v⃗ ⋅(∇⃗ v⃗ )=12∇⃗ (v2)
the RHS of this can be written explicitly knowing that v=v(r) only and using the product rule and the fact that for spherical symmetry ∇⃗ =∂∂rr^ to give
v⃗ ⋅(∇⃗ v⃗ )=v∂v∂rr^
note that again, both sides are vectors. The vs appearing on the RHS are the scalar magnitude, the direction is in the unit vector r^. There is no dot product on the right.
OPs question is ultimately, why the formula for divergence in spherical coordinates:
∇⃗ ⋅A⃗ =1r2∂∂r(r2Ar)+…
cannot be applied in this case and the short answer is that there are no divergences being taken. There might be a question as to why this nearly works, but I think there's only so many derivative like combinations of a single component vector that this might just be chance.
Remark: I've used ∇⃗ (nabla/del) notation throughout this answer since then you can work out the tensor rank of any equation by counting the number of arrows and subtracting two for each dot product present. The index notation also makes this clear since [v⃗ ⋅(∇⃗ v⃗ )]i=vj∂jvi≠vi∂jvj.
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answered Apr 15, 2019 at 17:29
jacob1729jacob1729
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Thanks for writing it in striсt form. But hen you say "the formula for divergence in spherical coordinates cannot be applied in this case and the short answer is that there are no divergences being taken" you do not explain why it can not be applied.
– Eddward
Commented
Apr 15, 2019 at 17:40
The formula for divergence in spherical coordinates should be valid as particular case of divergence. Never heard about any exception for this.
– Eddward
Commented
Apr 15, 2019 at 17:42
The derivation of that formula involves things like summing over different terms to get a scalar. The expression you want doesn't have that sum structure because the end result is a vector, not a scalar, so the same result doesn't apply. You have to do it from scratch. This is not a divergence.
– jacob1729
Commented
Apr 15, 2019 at 17:43
Well, tell me the correct value for divergence of v⃗ (r) in such particular case..
– Eddward
Commented
Apr 15, 2019 at 17:51
If v⃗ (r)=v(r)r^ is a radial vector field then its divergence is the scalar quantity 1r2∂r(r2v) as you indicated. Its just that that quantity doesn't come up ever in the calculation you asked for, and your main mistake is accidentally claiming a different quantity is equal to this (as I said, due to a notation mix-up).
– jacob1729
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Apr 15, 2019 at 17:55
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Misunderstanding comes from notation. Actually
v⃗ ⋅∇v⃗ ≠v⃗ ⋅div(v⃗ )
The equations in original question just lead to the identity
v⃗ ⋅∇v⃗ =v⃗ ⋅div(v⃗ )−2v2rr^
for this particular case. The question is closed.
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edited Apr 16, 2019 at 18:28
answered Apr 15, 2019 at 18:51
EddwardEddward
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The question is closed now. Thanks to jacob1729 for his guidelines.
– Eddward
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Apr 16, 2019 at 18:24
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I guess I will be the third person to confirm that
v⃗ ⋅∇v⃗ ≠∇⋅v⃗ v⃗
In another type of notation, you can see that
v⃗ ⋅∇v⃗ =v⃗ ⋅gradv⃗
while
∇⋅v⃗ v⃗ =div(v⃗ )v⃗
According to me, it is more accurate to write this as
(v⃗ ⋅∇)v⃗
This is actually the term that appears usually in the kinetic part of the various fluid dynamics equations, e.g. Euler equations
∂v⃗ ∂t+(v⃗ ⋅∇)v⃗ ∇⋅v⃗ =−1ρ0∇p+g⃗ =0
If your vector field is as you describe it, then I strongly recommend switching to spherical coordinates, because then the vector field v⃗ simplifies drastically. Then you would simply work with one component only. Indeed, if your vector field is spherically symmetric and points only in the radial direction, then there exists a function f(|r⃗ |)=f(r) such that
v⃗ =f(r)r⃗
where r⃗ =⎡⎣⎢xyz⎤⎦⎥ and r=x2+y2+z2−−−−−−−−−−√. So basically, in cartesian coordinates
v⃗ ⋅∇=f(r)(x∂∂x+y∂∂y+z∂∂z)
Therefore, in spherical coordinates (r,θ,ϕ), there exists a function g(r) such that
v⃗ ⋅∇=g(r)∂∂r
To calculate it, simply recall that (∇r)⋅v⃗ =(∇r)⋅(g(r)∂∂r)=g(r)(∇r)⋅(∂∂r)=g(r), so when you calculate (∇r)⋅v⃗ in cartesian coordinates, you get
g(r)=(∇r)⋅v⃗ =(∇r)⋅(f(r)r⃗ )=f(r)((∇r)⋅r⃗ )=f(r)((∇x2+y2+z2−−−−−−−−−−√)⋅(x∂∂x+y∂∂y+z∂∂z))=f(r)12x2+y2+z2−−−−−−−−−−√2[x,y,z]⎡⎣⎢xyz⎤⎦⎥=f(r)r(x2+y2+z2)=f(r)rr2=rf(r)
Therefore, the vector field v⃗ ins polar coordinates is v⃗ =⎡⎣⎢rf(r)00⎤⎦⎥ and so
v⃗ ⋅∇=rf(r)∂∂r
so
(v⃗ ⋅∇)v⃗ =rf(r)∂∂r⎡⎣⎢rf(r)00⎤⎦⎥=⎡⎣⎢⎢(rf(r)2+r2f(r)f′(r))00⎤⎦⎥⎥
Observe that from the calculation before x∂∂x+y∂∂y+z∂∂z=r⃗ ⋅∇=r∂∂r
so going back to cartesian coordinates
(v⃗ ⋅∇)v⃗ =(f(r)2+rf(r)f′(r))⎡⎣⎢xyz⎤⎦⎥=(f(r)2+rf(r)f′(r))r⃗
where as before r⃗ =⎡⎣⎢xyz⎤⎦⎥ and r=x2+y2+z2−−−−−−−−−−√.
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edited Apr 16, 2019 at 13:07
answered Apr 16, 2019 at 12:55
FuturologistFuturologist
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Something is wrong in your final results if you originally assume that v⃗ =f(r)r⃗ and you check dimension of the (v⃗ ⋅∇)v⃗ it should be acceleration those dimension is a=lengthtime2. but your result for polar coordinates is rf(r)2 which has dimension of length3time2. And the result for cartesian coordinates is f(r)2 has dimension of length2time2..
– Eddward
Commented
Apr 16, 2019 at 14:56
@Eddward I do not think there is an error. The function f(r) has dimension 1time because r⃗ has dimensnion length. Hence rf(r)2 has dimension length1time2=lengthtime2
– Futurologist
Commented
Apr 16, 2019 at 21:36
Ok, then I agree. Thanks for your answer!
– Eddward
Commented
Apr 17, 2019 at 9:22
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Form 10-K Summary ...................................................................................................................................................
81 Signatures 82 CAUTIONARY STATEMENTS REGARDING FORWARD-LOOKING STATEMENTS This Annual Report on Form 10-K (“Annual Report”) contains certain information that may constitute “forward-looking statements” within the meaning of the Private Securities Litigation Reform Act of 1995 and are made in reliance on the safe harbor protections provided thereunder. While we have specifically identified certain information as being forward-looking in the context of its presentation, we caution you that all statements contained in this Annual Report that are not clearly historical in nature, including statements regarding our exit of the residential solar business (the “Solar Business”) (the “ADT Solar Exit”) and the expected costs and benefits of such exit; the commercial transaction between ADT and GTCR LLC (“GTCR”) (the “Commercial Divestiture”); the expected benefits of the Commercial Divestiture and ADT Solar Exit including that the costs of the ADT Solar Exit may exceed our best estimates; the integration of strategic bulk purchases of customer accounts; the strategic investment by and long term partnership with State Farm Fire & Casualty Company (“State Farm”); any repurchases of shares of our common stock under an authorized share repurchase plan; our ability to refinance or reduce debt or improve leverage ratios, or to achieve or maintain our leverage goals; anticipated financial performance, management’s plans and objectives for future operations; the successful development, commercialization, and timing of new or joint products; the expected timing of product commercialization with State Farm or any changes thereto, including the ADT home security program for State Farm; business prospects; outcomes of regulatory proceedings; market conditions; our ability to deploy our business continuity and disaster plans and procedures to successfully respond to catastrophic events; our strategic partnership and ongoing relationship with Google LLC (“Google”); the expected timing of product commercialization with Google or any changes thereto; the successful internal development, commercialization, and timing of our next generation platform and innovative offerings; the successful conversion of customers who continue to utilize outdated technology; the current and future market size for existing, new, or joint products; any stated or implied outcomes with regards to the foregoing; and other matters. Forward-looking statements are contained principally in the sections of this report entitled “Business,” “Risk Factors,” and “Management’s Discussion and Analysis of Financial Condition and Results of Operations.” Without limiting the generality of the preceding sentences, any time we use the words “expects,” “intends,” “will,” “anticipates,” “believes,” “confident,” “continue,” “propose,” “seeks,” “could,” “may,” “should,” “estimates,” “forecasts,” “might,” “goals,” “objectives,” “targets,” “planned,” “projects,” and, in each case, their negative or other various or comparable terminology, and similar expressions, we intend to clearly express that the information deals with possible future events and is forward-looking in nature. However, the absence of these words or similar expressions does not mean that a statement is not forward-looking. Particular uncertainties that could cause our actual results to be materially different than those expressed in our forward-looking statements include, without limitation: • our ability to retain and hire key personnel and to maintain relationships with customers, suppliers and other business partners; • risks related to the Commercial Divestiture and ADT Solar Exit, including ADT’s business becoming less diversified and the possible diversion of management’s attention from ADT’s core business operations; • our ability to keep pace with rapid technological changes and other industry changes; • risks related to the expansion and further development of our next-generation platform and our efforts to migrate our information technology infrastructure, including our customer relationship management and enterprise resource planning systems, to the cloud; • our ability to effectively implement our strategic partnership with, commercialize products with, or utilize any of the amounts invested in us by State Farm or provided by State Farm for research and development or other purposes; • our ability to effectively implement our strategic partnership with or utilize any of the amounts invested in us by Google; • the impact of supply chain disruptions; • our ability to maintain and grow our existing customer base and to integrate strategic bulk purchases of customer accounts; • our ability to sell our products and services or launch new products and services in highly competitive markets, including the home security and automation market, and to achieve market acceptance with acceptable margins; • our ability to successfully upgrade obsolete equipment installed at our customers’ premises in an efficient and cost-effective manner; • any changes in regulations or laws, economic and financial conditions, including labor and tax law changes, changes to privacy requirements, changes to telemarketing, email marketing and similar consumer protection laws, interest volatility, and trade tariffs and restrictions applicable to the products we sell; • any material changes to the valuation allowances we take with respect to our deferred tax assets; • the impact of cyber attacks or related breaches with respect to information technology systems, cybersecurity, or data security involving us, our business partners, or other third parties whose systems are interconnected with ours, including the incidents disclosed in our Current Reports on Form 8-K filed with the SEC on August 8, 2024 (the “August 1 Incident”) and October 7, 2024 (the “October Incident” and together with the August Incident, the “Cybersecurity Incidents”) and any similar future or still undetected attacks or incidents; • risks related to the development, deployment, and use of artificial intelligence (“AI”) in our products and services, including technological and legal uncertainties surrounding AI technologies; • our dependence on third-party providers, suppliers, and dealers to enable us to produce and distribute our products and services in a cost-effective manner that protects our brand; • our ability to successfully implement an equipment ownership model that best satisfies the needs of our customers and to successfully implement and maintain our receivables securitization financing agreement or similar arrangements; • our ability to successfully pursue alternate business opportunities and strategies; • our ability to continue to integrate various businesses, including bulk acquisitions of customer accounts, we have acquired, or will acquire in the future, in an efficient and cost-effective manner; • the amount and timing of our cash flows and earnings, which may be impacted by customer, competitive, supplier and other dynamics and conditions; • our ability to maintain or improve margins through business efficiencies; • risks related to the restatement of our consolidated financial statements included in our Amendment No. 1 to our Annual Report on Form 10-K for the year ended December 31, 2022 (the “Amended 2022 Annual Report”) and in our Quarterly Reports on Form 10-Q/A for the quarters ended September 30, 2022, and March 31, 2023, each as filed with the SEC on July 27, 2023; • any litigation or investigation related to such restatements; and • our ability to maintain effective internal control over financial reporting (“ICFR”) and disclosure controls and procedures (“DCPs”), including our ability to remediate any potential material weakness in our ICFR and the timing of any such remediation, as well as the ability to maintain effective DCPs at a reasonable assurance level.
Forward-looking statements and information involve risks, uncertainties, and other factors that could cause actual results to differ materially from those expressed or implied in, or reasonably inferred from, such statements, including without limitation, the risks and uncertainties disclosed or referenced in Part I Item 1A of this Annual Report under the heading “Risk Factors.” Therefore, caution should be taken not to place undue reliance on any such forward-looking statements. Much of the information in this report that looks toward future performance is based on various factors and important assumptions about future events that may or may not actually occur. As a result, our operations and financial results in the future could differ materially and substantially from those we have discussed in the forward-looking statements included in this Annual Report on Form 10-K. Any forward-looking statement made in this Annual Report on Form 10-K speaks only as of the date on which it is made. We undertake no obligation to publicly update or revise any forward-looking statements, whether as a result of new information, future developments, or otherwise unless required by law.
SUMMARY OF PRINCIPAL RISK FACTORS This summary briefly lists the principal risks and uncertainties facing our business, which are only a select portion of those risks. A more complete discussion of those risks and uncertainties is set forth in Part I, Item 1A of this Annual Report, entitled Risk Factors. Additional risks not presently known to us or that we currently deem immaterial may also affect us. If any of these risks occur, our business, financial condition or results of operations could be materially and adversely affected. Our business is subject to the following principal risks and uncertainties: Risks Related to Our Products and Services • Our growth is dependent upon our ability to keep pace with rapid technological and industry changes.
• The home security and automation markets in which we sell our products and services are highly competitive markets.
• If the insurance industry changes its practice of providing incentives to homeowners for the use of alarm monitoring services, we may experience a reduction in new customer growth or an increase in our customer attrition rate.
• The retirement of older telecommunications technology and limitations on our customers’ options could materially adversely affect our business, increase customer attrition, and require significant investments.
• Police and fire departments could refuse to respond to calls from monitored security service companies, which could damage consumer trust and confidence in our solutions and may damage our ability to attract and retain customers.
• Our reputation as a service provider of high-quality security offerings may be materially adversely affected by product defects or shortfalls in customer service, or by the failure of first responders to respond to calls.
• Unauthorized use of our brand names by third parties, and the expenses incurred in developing and preserving the value of our brand names, may materially adversely affect our business.
2 Risks Related to Our Operations • If our attrition rate rises significantly, our profitability, business, financial condition, results of operations, and cash flows could be materially adversely affected.
• Delays, costs, and disruptions that result from upgrading, integrating, and maintaining the security of our information and technology networks and systems could materially adversely affect us.
• If we do not effectively implement our plans to migrate our technology infrastructure to the cloud, we could experience significant disruptions in our operations.
• Cybersecurity attacks or threats or other unauthorized access or attempts to access to our systems, or those of third parties, have in the past, and may in the future, compromise the security of our systems and disrupt our operations.
• Uncertainty in the development, deployment, and use of AI in our products and services, as well as our business more broadly, could adversely affect our business and reputation.
• Any failure or interruption in products or services provided by third-party providers and suppliers for components of our security and automation systems or software licenses for our products and services could harm our ability to operate.
• An event causing a disruption in the ability of our monitoring facilities or customer care resources to operate, including work from home operations, could materially adversely affect our business. • A variety of events, including pandemics, natural disasters, and other macroeconomic events, have had and could have in the future a significant negative impact on our ability to carry on our normal operations.
• Our independent, third-party authorized dealers may not be able to mitigate certain risks such as information technology and data security breaches, product liability, errors and omissions, and compliance with applicable laws and regulations.
• We may pursue business opportunities that diverge from our current business model.
• We continue to integrate our acquisitions, as well as to separate certain shared services following the Commercial Divestiture, which may divert management’s attention from our ongoing operations. We may not achieve all of the anticipated benefits, synergies, or cost savings from our acquisitions or the Commercial Divestiture.
• Our exit from the residential Solar Business is subject to uncertainties and risks that may materially adversely affect our financial condition and results of operations.
• We may not achieve some or all of the strategic and financial benefits that we expect to achieve from the Commercial Divestiture or the ADT Solar Exit.
• ADT is a less diversified business following the Commercial Divestiture and the ADT Solar Exit, which may adversely affect ADT’s results of operations and financial condition.
• Our customer generation strategies through third parties, including our authorized dealer and affinity marketing programs, and our use of celebrities and social media influencers, and the competitive market for customer accounts may expose us to risk and affect our future profitability.
• We face risks in acquiring and integrating customer accounts.
• If we are unable to recruit and retain sufficient personnel at all levels of our organization, our ability to manage our business could be materially and adversely affected.
• Adverse developments in our collective bargaining agreements or other agreements with some employees could materially and adversely affect our business, financial condition, results of operations, and cash flows.
• If we fail to maintain effective internal control over financial reporting, we may not be able to accurately report our financial results.
Risks Related to Regulations and Litigation • If we fail to comply with constantly evolving laws, regulations, and industry standards addressing information and technology networks and systems, privacy, and data security, we could face substantial penalties, liability, and reputational harm.
• Infringement of our intellectual property rights could negatively affect us.
• Allegations that we have infringed upon the intellectual property rights of third parties could negatively affect us.
• We may be subject to class actions and other lawsuits which may harm our business and results of operations.
• Increasing government regulation of telemarketing, email marketing, door-to-door sales, and other marketing methods may increase our costs and restrict the operation and growth of our business.
• Any new, changes to existing, or uncertainty regarding laws or regulations, or our failure to comply with any such rules or regulations could be costly to us, harm our business and operations, and impede our ability to grow our existing business, any new businesses that we acquire, or investment opportunities that we pursue.
• We could be assessed penalties and fines for false alarms, and if these expenses become significant or we are unable to pass along the associated costs, our customers may terminate or fail to renew our services.
• Adoption of statutes and governmental policies purporting to characterize certain of our charges as unlawful may adversely affect our business.
• In the absence of net neutrality or similar regulation, certain providers of Internet access may block our services or charge their customers more for using our services, or government regulations relating to the Internet could change.
3 • Liability for employee acts or omissions or system failures could negatively affect our business.
• Liability for obligations of The Brink’s Company under the Coal Act or other coal-related liabilities of The Brink’s Company could negatively affect our business.
• Our business would be adversely affected if certain of our independent contractors were classified as employees.
• Existing or new tariffs and other trade restrictions imposed on imports from China, Mexico, or other countries where much of our end-user equipment is manufactured, or any counter-measures taken in response, may harm our business and results of operations.
Risks Related to Macroeconomic and Related Factors • General economic conditions can affect our business, and we are susceptible to changes in the business economy, in the housing market, and in business and consumer discretionary income, which may inhibit our ability to grow our customer base and impact our results of operations. • Rising interest rates or increased consumer lender fees could adversely impact our sales, profitability, and financing costs.
• A substantial part of our revenue is derived from the recurring monthly revenue due from customers under alarm monitoring and other service contracts and we are subject to credit risk and other risks associated with our customers, dealers, and third-party lenders.
• Goodwill and other identifiable intangible assets represent a significant portion of our total assets, and we may never realize the full value of our intangible assets.
• We have significant deferred tax assets, and any impairments of or valuation allowances against these deferred tax assets in the future could materially adversely affect our results of operations, financial condition, and cash flows.
Risks Related to Our Indebtedness and to the Ownership of Our Common Stock • Our substantial indebtedness limits our financial and operational flexibility.
• Our debt agreements contain restrictions that limit our flexibility and limit the manner in which we conduct our business and finance future operations or capital needs.
• Our stock price may fluctuate significantly.
• Apollo continues to exert significant influence over us, and its interests may conflict with our interests and the interests of other stockholders and could negatively impact our ability to enter into corporate transactions.
• If we fail to establish and achieve the objectives of our sustainability program, we may not be viewed as an attractive investment, service provider, workplace, or business, which could have a negative effect on our company.
• Our organizational documents may impede or discourage a takeover, which could deprive our investors of the opportunity to receive a premium on their shares.
• Our Certificate of Incorporation provides for exclusive forum provisions which could limit our stockholders’ ability to obtain a favorable judicial forum for disputes.
• Share repurchases or dividend payments could increase volatility in the trading price of our Common Stock, and may not result in long-term value to shareholders.
4 PART I ITEM 1. BUSINESS.
TABLE OF CONTENTS • Company Overview • Key Business Developments • Segment and Geographic Information • Products and Services • Our Market • Competition • Resources Material to our Business • Seasonality • Government Regulation and Other Regulatory Matters • Human Capital and Sustainability • Available Information COMPANY OVERVIEW Our Business ADT Inc., together with its wholly-owned subsidiaries (collectively, the “Company”, “we”, “our”, “us”, and “ADT”), is a leading provider of security, interactive, and smart home solutions serving residential and small business customers in the United States (“U.S.”). As discussed below, on October 2, 2023, we completed the divestiture of our former Commercial Business; and as of June 30, 2024, substantially all operations of our former Solar Business had ceased.
Our mission is to empower people to protect and connect what matters most with safe, smart, and sustainable solutions, delivered through innovative offerings, unrivaled safety, and a premium experience because we believe that everyone deserves to feel safe.
We primarily conduct business under the ADT brand, which we believe is a key competitive advantage for us and a contributor to our success due to the importance customers place on reputation and trust when purchasing our products and services. The strength of our brand is based upon a long-standing record of delivering high-quality, reliable products and services; expertise in system sales, installation, and monitoring; and superior customer care, all driven by our industry-leading experience and knowledge.
As of December 31, 2024, we had approximately 6.4 million security monitoring service subscribers. We serve our customers through our nationwide sales and service offices, monitoring and support centers, and large network of installation and service professionals.
Formation and Organization ADT Inc. was incorporated in the State of Delaware in May 2015 as a holding company with no assets or liabilities. In July 2015, we acquired Protection One, Inc. and ASG Intermediate Holding Corp. (collectively, the “Formation Transactions”), which were instrumental in the commencement of our operations. In May 2016, we acquired The ADT Security Corporation (formerly named The ADT Corporation) (“The ADT Corporation”) (the “ADT Acquisition”), which significantly increased our market share in the security systems industry, making us one of the largest monitored security companies in the U.S.
In January 2018, we completed an initial public offering (“IPO”), and our common stock, par value $0.01 per share (“Common Stock”), began trading on the New York Stock Exchange (the “NYSE”) under the symbol “ADT.” 5 Prior to March 11, 2024, the Company was majority-owned by Prime Security Services TopCo (ML), L.P., which is majority-owned by Prime Security Services TopCo Parent, L.P. (“Ultimate Parent”). Ultimate Parent is ultimately majority-owned by Apollo Investment Fund VIII, L.P. and its related funds that are directly or indirectly managed by affiliates of Apollo Global Management, Inc. (together with its subsidiaries and affiliates, “Apollo”). Following a registered secondary offering of the Company’s Common Stock during March 2024 by certain Apollo affiliates and the Company’s concurrent share repurchases of Common Stock, the Company ceased to be a “controlled company” under the NYSE rules.
As of December 31, 2024, Apollo owned approximately 40%, State Farm owned approximately 15%, and Google owned approximately 6% of our outstanding Common Stock, including shares of Class B common stock, par value $0.01 per share (“Class B Common Stock”) (on an as-converted basis), which is owned exclusively by Google, and unvested shares of Common Stock.
KEY BUSINESS DEVELOPMENTS ADT Solar Exit In December 2021, we entered the residential solar market with the Solar Business acquisition, which allowed us to provide customers with solar and energy storage solutions, energy efficiency upgrades, and roofing services.
In November 2023, we announced a plan to streamline the Solar Business to focus on the top performing markets and rationalize the overhead and infrastructure of the business. As part of this plan, we closed a significant number of branches that operated the Solar Business along with making associated headcount reductions.
In January 2024, after a strategic review of the business and continued macroeconomic and industry pressures, the Company’s board of directors (the “Board of Directors”) approved a plan to fully exit the residential Solar Business; and as of June 30, 2024, substantially all operations of the Solar Business had ceased.
Commercial Divestiture On August 7, 2023, ADT, Iris Buyer LLC (“Iris Buyer”), a Delaware limited liability company and affiliate of GTCR, and, solely for certain purposes set forth in the Commercial Purchase Agreement (as defined below), Fire & Security Holdings, LLC (“F&S Holdings”), a Delaware limited liability company and an indirect, wholly owned subsidiary of ADT, entered into an Equity Purchase Agreement (the “Commercial Purchase Agreement”) pursuant to which GTCR agreed to acquire all of the issued and outstanding equity interests of F&S Holdings, which directly or indirectly held all of the issued and outstanding equity interests in the subsidiaries of ADT that operated ADT’s commercial business (the “Commercial Business”) (previously defined as the “Commercial Divestiture”).
On October 2, 2023, we completed the Commercial Divestiture for a total purchase price of approximately $1,613 million in cash, subject to customary post-closing adjustments, and received net proceeds of approximately $1,585 million, excluding transaction costs.
The Commercial Divestiture supported our strategic vision and strengthened our financial profile, and we used net proceeds of approximately $1,518 million to reduce our debt. Refer to Note 7 “Debt” in the Notes to Consolidated Financial Statements in Item 15 “Exhibit and Financial Statement Schedules” for further information.
In connection with the Commercial Divestiture, we entered into a Transition Services Agreement (the “Commercial TSA”), pursuant to which the Company and the Commercial Business will provide certain transitional services relating to ongoing support and other administrative functions to each other for a transitional period of up to 24 months after the closing of the Commercial Divestiture. Commercial TSA fees charged to the Commercial Business represent charges for internal labor as well as certain third-party costs identified in connection with providing such services.
The Company and GTCR entered into an agreement granting GTCR a license to continue to use the ADT brand and other Company trademarks for a period of twelve months to transition from Company branding (the “Brand License”), which expired during the fourth quarter of 2024. The Company has also agreed to a covenant not to assert a claim against GTCR for infringement of Company's patents as of the Commercial Divestiture for products and services that were used in the Commercial Business prior to the Commercial Divestiture, and has provided GTCR with a paid-up, irrevocable, non-assignable (with limited exceptions) license to continue to use certain software and other Company IP in the same manner. Royalty income is included in other income (expense) and was not material during 2024 and 2023.
6 In addition, the Company and Iris Buyer agreed to customary employee non-solicitation and no-hire restrictions with respect to the other party’s employees for a period of two years from the closing. The Company and Iris Buyer also agreed to a non-competition period of five years from the closing whereby, subject to certain exceptions, (a) ADT and its subsidiaries will not directly or indirectly sell or service in the United States (or any of its territories): (i) security, building automation, access control, retail store performance, electronic article surveillance, or cash processing/ATM security equipment or services for non-residential premises over 10,000 square feet, (ii) nurse call solutions, or (iii) fire detection or fire suppression products or services for non-residential premises of any size; and (b) F&S Holdings and its subsidiaries will not directly or indirectly sell or service in the United States (or any of its territories) or Canada, among other items: (i) security, building automation or access control products or services for any non-residential premises under 10,000 square feet or any residential premises, (ii) personal emergency response systems or mobile safety solutions (other than nurse call solutions), or (iii) fire detection products or services for residential premises.
Google Partnership In July 2020, we entered into a Master Supply, Distribution, and Marketing Agreement with Google (as amended, the “Google Commercial Agreement”) pursuant to which Google has agreed to supply us with certain Google devices as well as certain Google video and analytics services (the “Google Devices and Services”) for sale to our customers. Subject to customary termination rights related to breach and change of control, the Google Commercial Agreement has an initial term of seven years from the date that the Google Devices and Services are successfully integrated into the Company’s end-user security and automation platform. Further, subject to certain carve-outs, the Company has agreed to exclusively sell Google Devices and Services to its customers. In January 2024, we amended the Google Commercial Agreement to, among other things, remove exclusivity for do-it-yourself (“DIY”) products and services, limit exclusivity for do-it-for-me (“DIFM”) products and services, and restructure the commitment from the Google Success Fund to pay a portion of the remaining amount due to ADT as a quarterly marketing reimbursement (with the balance to be used towards unlocking certain opportunities).
In September 2020, we issued and sold 54,744,525 shares of Class B Common Stock, for an aggregate purchase price of $450 million, to Google in a private placement pursuant to a securities purchase agreement dated July 31, 2020. In connection with the issuance of the Class B Common Stock, the Company and Google entered into an Investor Rights Agreement (the “Google Investor Rights Agreement”), pursuant to which Google agreed to be bound by customary transfer restrictions and drag-along rights, and be afforded customary registration rights with respect to shares of Class B Common Stock held directly by Google. Under the terms of the Google Investor Rights Agreement, Google was prohibited, subject to certain exceptions, from transferring any shares of Class B Common Stock or any shares of Common Stock issuable upon conversion of the Class B Common Stock beneficially owned by Google until September 2023, or earlier if certain conditions occurred (the “Google Lock-up Period”). In December 2023, the Company and Google amended the Google Investor Rights Agreement to extend the Google Lock-up Period through June 2025, or earlier if certain conditions occur (see Note 10 “Equity” in the Notes to Consolidated Financial Statements).
Our partnership with Google represents the combination of the leading security and smart home brand and the leading technology brand joining forces to introduce the next-generation smart and helpful home. As part of this partnership, each company has agreed to contribute $150 million upon the achievement of certain milestones toward the joint marketing of devices and services; customer acquisition; training of our employees for the sales, installation, customer service, and maintenance of the product and service offerings; and technology updates for products included in such offerings. In addition, in August 2022, the Company and Google executed an amendment to the Google Commercial Agreement, pursuant to which Google has agreed to commit an additional $150 million (together with the initial amounts, the “Google Success Funds”) to fund growth, data and insights, product innovation and technology advancements, customer acquisition, and marketing, as mutually agreed by the Company and Google. Each of the $150 million segments of the Google Success Funds will be triggered in three equal tranches, respectively, subject to the attainment of certain milestones.
In December 2023, the Company and Google entered into an addendum to the Company’s existing agreement for using Google cloud services (the “Google Cloud Agreement Addendum”), pursuant to which Google has agreed to provide certain credits, discounts, and other incentives for use of the Google Cloud Platform to the Company, and the Company has committed to purchasing $200 million of Google Cloud Platform services over seven years (through December 2030) (the “Google Cloud Commitment”).
Refer to Note 13 “Commitments and Contingencies” in the Notes to Consolidated Financial Statements for more information.
7 State Farm Partnership In September 2022, we entered into a Securities Purchase Agreement, dated as of September 5, 2022, with State Farm (the “State Farm Securities Purchase Agreement”), pursuant to which we agreed to issue and sell in a private placement to State Farm, 133,333,333 shares of our Common Stock (the “State Farm Shares”) at a per share price of $9.00 for an aggregate purchase price of $1.2 billion (the “State Farm Strategic Investment”). Also in September 2022, in connection with the State Farm Strategic Investment, we commenced a tender offer to purchase up to 133,333,333 shares of our Common Stock (including shares issued upon conversion of Class B Common Stock) (the “Tender Shares”) at a per share price of $9.00 (the “Tender Offer”).
In October 2022, the State Farm Strategic Investment closed, and we issued and sold the State Farm Shares to State Farm pursuant to the State Farm Securities Purchase Agreement (the “Closing”). Also in October 2022, the Tender Offer expired, and we used the proceeds from the State Farm Strategic Investment to repurchase the Tender Shares, subject to the terms and conditions described in the Offer to Purchase, dated September 12, 2022 (as amended from time to time, the “Offer to Purchase”). Additionally, pursuant to the State Farm Securities Purchase Agreement, at the Closing, we entered into an Investor Rights Agreement, dated as of October 13, 2022, with State Farm (the “State Farm Investor Rights Agreement”), pursuant to which State Farm agreed to be bound by customary transfer and standstill restrictions and drag-along rights, and be afforded customary registration rights with respect to the State Farm Shares. In particular, State Farm is prohibited, subject to certain customary exceptions, from transferring any State Farm Shares until the earlier of (i) October 13, 2025 and (ii) the date on which the State Farm Development Agreement (as defined below) has been validly terminated, other than in the event of a termination by the Company for a material breach thereof by State Farm.
In addition, we entered into a development agreement with State Farm (the “State Farm Development Agreement”), pursuant to which State Farm committed up to $300 million to fund product and technology innovation, customer growth, and marketing initiatives. Upon the Closing, we received $100 million of such commitment from State Farm, which is restricted until we use the funds for investment, as agreed upon with State Farm, in accordance with the State Farm Development Agreement (the “Opportunity Fund”).
Refer to Note 10 “Equity” and Note 16 “Related Party Transactions” in the Notes to Consolidated Financial Statements for more information.
SEGMENT AND GEOGRAPHIC INFORMATION We evaluate and report our segment information based on the manner in which our Chief Executive Officer (“CEO”), who is the chief operating decision maker (“CODM”), evaluates performance and allocates resources. The CODM manages the business on a consolidated basis, and as such, we report results in a single operating and reportable segment which reflects the business operations of the Company’s former Consumer and Small Business (“CSB”) segment.
For further information, refer to Note 3 “Segment Information” in the Notes to Consolidated Financial Statements.
Where applicable, prior periods have been retrospectively adjusted to reflect our current operating and reportable segment structure.
Revenue generated by customers outside of the U.S. is not material.
PRODUCTS AND SERVICES Security and Automation Offerings Our core security offerings include burglar and life safety alarms, smart security cameras, smart home automation systems, and video surveillance systems (referred to collectively as security systems, solutions, or offerings). Our security offerings are designed to detect intrusion; control access; sense movement, smoke, fire, carbon monoxide, leaks, temperature, and other environmental conditions and hazards; and address personal medical emergencies such as injuries or unanticipated falls. We offer our customers routine maintenance and the installation of upgraded or additional equipment, which provide additional value to the customer and generate incremental recurring monthly revenue. Additionally, our personal emergency response system products and services utilize our security monitoring infrastructure to provide customers with solutions that help to sustain independent living, detect when a fall occurs, and provide protection while on the go with geolocation capability.
During 2022, as part of our partnership with Google, we launched the Google Nest doorbell, rolled out mesh Wi-Fi, and launched Google indoor and outdoor cameras as part of our product offerings. In the first quarter of 2023, we launched our proprietary ADT+ app for our self set-up line of DIY smart home security products, including Google Nest offerings, which allows customers to easily access and control their ADT devices through an intuitive app experience. Our comprehensive 8 interactive technology platform is intended to provide customers with a seamless experience through a common application across security, life safety, automation, and analytics, and integrate the user experience, customer service experience, and back-end support.
During the fourth quarter of 2023, we began our phased rollout of our ADT+ app for professional installation along with a new interactive and hardware lineup. During 2024, the Company continued its phased rollout of the ADT+ platform across the country. Customers can now take advantage of next generation hardware and technology through our proprietary app.
In addition, we launched Trusted NeighborTM across the U.S. This new offering allows customers to grant secure, temporary access to their homes through the ADT+ app, enhancing security and convenience. Enabled by deep integration of the ADT+ app, Google Nest Doorbell’s Familiar Faces technology, and Yale locks, select customers can use the Auto-Unlock feature to automatically verify a trusted individual.
The vast majority of new residential customers choose our automation and smart home solutions, which provide customers the ability to remotely monitor and manage their environments through our customized web portal via web-enabled devices (such as smart phones), smart phone applications, or through touchscreen panels in their homes.
Our automation and smart home solutions allow customers to: • remotely arm and disarm their security systems; • record and view real-time video; • program their systems to react to defined events; • integrate their systems with various third-party connected devices such as cameras, lights, thermostats, appliances, and garage doors; and • automate custom schedules for these connected devices.
Generally, a significant upfront investment is required to acquire new subscribers related to installation costs (such as labor, commissions, equipment, and overhead), which we recover through upfront fees charged at the time of installation and recurring monthly revenue generated in future years. While the economics of an installation can vary depending on the customer acquisition channel and offering, we generally achieve revenue break-even in approximately two years. Our ability to increase our average prices for individual customers depends on a number of factors, including the type and complexity of service, the quality of our service, the introduction of additional features and offerings that increase the value to the customer, and the competitive environments in which we operate.
At the time of initial equipment installation, our customers typically contract for both monitoring and maintenance services, which are generally governed by multi-year contracts. If a customer cancels or is otherwise in default under a monitoring contract prior to the end of the initial contract term, we have the right under the contract to receive a termination payment from the customer in an amount equal to a designated percentage of all remaining monthly payments.
The standard contract terms are two, three, or five years, with automatic renewals for successive 30-day periods for residential customers and annual periods for small business customers, unless canceled by either party. We may also offer month-to-month contracts depending on the circumstance.
Qualifying customers can pay any upfront fees over the course of the contract (referred to as retail installment contracts). A security interest is granted in the retail installment contract receivables as collateral for cash borrowings under our uncommitted receivables securitization financing agreement (the “2020 Receivables Facility”). Customers are also generally obligated to make monthly payments for monitoring services for the remainder of the initial contract term. Monitoring services are typically billed monthly or quarterly in advance, and more than 80% of our residential customers pay us these fees through automated payment methods, with new residential customers generally opting for these payment methods.
Monitoring Centers As of December 31, 2024, we operated six monitoring centers located throughout the U.S. and listed by Underwriters Laboratories (“UL”) in order to provide 24/7 year-round professional monitoring services to our customers, including our monitoring centers that also provide outsourced monitoring services for other security companies. Our monitoring centers are fully redundant, which means all monitoring operations can be transferred to another monitoring center in case of an emergency such as fire, tornado, major interruption in telephone or computer service, or any other event affecting the functionality of one of our centers. To obtain and maintain a UL listing, a security systems monitoring center must be located in a building meeting UL’s structural requirements, have back-up computer and power systems, and meet UL specifications for staffing and standard operating procedures. Many jurisdictions have laws requiring that security systems for certain buildings be monitored by UL-listed centers, and in some instances, a UL listing is required by insurers of certain customers as a condition of insurance 9 coverage. In addition, we implemented certain work from home actions, including for a majority of our monitoring center professionals, in compliance with UL work from home standards.
Upon the occurrence of certain initiating events, our monitored security systems send event-specific signals to our monitoring personnel who then relay appropriate information, based on the customer’s contract and preferences, to first responders, such as local police, fire departments, or medical emergency response centers and the customer or others on the customer’s emergency contact list. We continue to focus on our alarm verification technologies and partner with industry associations and various first responder agencies to help prioritize response events, enhance response policies, and develop processes that allow us to send data to emergency response centers directly. Additionally, our System Monitoring and Response Technology (“SMART”) monitoring solution aims to result in faster and higher-quality alarm responses and is expected to reduce annual false alarms and customer care calls. Our SMART monitoring differentiates our offerings by delivering alarms to connected and participating 911 centers faster than traditional voice handling speeds. Additionally, our alarm scoring program, which offers a uniform and reliable approach for categorizing alarm severity levels, enhances the accuracy of assessing potential life or property threats and gives first responders precise and crucial alarm data for improved emergency responses.
Field and Call Center Operations Our field and call center operations comprise a nationwide network of sales and service offices, call centers, and support facilities. We staff our sales and service offices with qualified field solution advisors and installation and service technicians, and we utilize third-party subcontract labor when appropriate. We provide ongoing training to our field and call center employees, as well as our authorized dealers, and we continually measure and monitor customer satisfaction-oriented metrics across each customer touch point. Our objective is to provide a differentiated service experience by resolving customer issues remotely whenever possible and scheduling installation and service visits at times convenient for the customer.
Our innovative remote assistance program (the “Remote Assistance Program”) delivers a scalable, cost-efficient means of servicing our customers through live video streaming with our skilled technicians to troubleshoot and resolve service issues as well as remote programming and installation support for new DIY systems, add-ons, and resale reactivations. We are also exploring and implementing AI tools that are aimed to improve and automate certain processes for our call center agents and customers. Additionally, our ADT WiFi Fix app allows remote customer service agents to diagnose and address any WiFi issues impacting customers’ ADT equipment or other devices. These offerings provide customers with more options for receiving services that best fit their lifestyles while reducing the cost for us to provide these services and lowering our carbon footprint through the reduction of truck rolls.
Our customer care agents provide support 24 hours a day on a year-round basis to ensure service requests are handled promptly and professionally. Approximately 80% of these requests are resolved during the initial interaction. However, in the event these requests cannot be resolved, they are routed to a remote assist agent or a field technician for further support. In many cases, customer care agents can remotely resolve emergency and non-emergency inquiries regarding ongoing alarms, service, billing, and system and product troubleshooting. We continue to offer customers additional choices in managing their services through customer-facing self-service tools via interactive voice response systems and the Internet. In addition, we use a network of external vendors, both domestic and outside of the U.S., to supplement our internal call center resources as needed.
Our support facilities also provide administrative assistance to our local service offices and customer care centers, which includes scheduling and ordering, drop-shipping, and physically distributing system components for installations.
Sales and Distribution Channels We utilize a complementary mix of direct and indirect sales and distribution channels: • Direct Channel Our direct channel customers are generated by direct response and other marketing efforts, general brand awareness, customer referrals, door-to-door activities, along with lead generation partners, and are supported by our internal sales force located in our national sales call centers as well as our nationwide network of field sales and service offices. In many scenarios, we close the sale of a basic system over the phone and allow our field force to augment the system at the time of installation. In other cases, field solution advisors work directly with the customer to select an ideal system. Driven by consumer preferences, we also market to customers through retail and e-commerce channels, and we have been supplementing existing channels to meet consumers where they prefer to shop.
Our field solution advisors typically undergo an in-depth screening process prior to hire, complete comprehensive centralized training prior to conducting customer sales presentations, and participate in ongoing training in support of new offerings. We generally utilize a highly structured sales approach, which includes a structured model sales call, daily monitoring of sales activity and effectiveness metrics, and regular coaching by our sales management teams.
10 • Indirect Channel Our indirect channel customers are generated mainly through our network of agreements with third-party independent dealers who sell and install equipment and ADT Authorized Dealer-branded monitoring, interactive, and other services to residential end users (the “ADT Authorized Dealer Program”). As opportunities arise, we may engage in selective third-party account purchases, which typically involve the bulk purchase of customer accounts from other security service providers.
As of December 31, 2024, our network of authorized dealers consisted of approximately 140 authorized dealers operating across the U.S. Our authorized dealers are contractually obligated to offer exclusively to us all qualified monitored accounts they generate, but we are not obligated to accept these accounts. We pay our authorized dealers for the acquisition of any qualified monitored accounts (referred to as dealer generated customer accounts) we purchase from them. Dealer generated customer account contracts typically have an initial term of three years with automatic renewals for successive 30-day periods, unless canceled by either party. If a purchased account is canceled during the charge-back period, which is generally thirteen months, the dealer is required to refund our payment of the purchase price for the canceled account. In certain instances in which we reject an account, we generally still indirectly provide monitoring services for that account through a monitoring services agreement with the authorized dealer.
Authorized dealers are required to adhere to the same high-quality standards for sales and installation as our own sales and service offices. We monitor each authorized dealer’s financial stability, use of sound and ethical business practices, and delivery of reliable and consistent high-quality sales and installation methods.
Marketing Strategy We focus on driving revenue by increasing consumer awareness and preference, improving consumer purchasing flexibility, and optimizing our go-to-market approach. To support the growth of our customer base, improve brand awareness, and drive greater market penetration, we consider new customer channels and lead generation methods, explore opportunities to provide branded solutions, and form strategic partnerships and alliances with various third parties.
We strive to optimize our marketing spend through a lead modeling process, whereby we dynamically allocate spend based on lead flow and measured marketing channel effectiveness. We market our offerings through national television, radio, and direct mail advertisements, as well as through Internet advertising, which includes national search engine marketing, email, online video, local search, and social media. We also have several affinity partnerships with organizations that promote our services to their customer bases, and we market through social media influencers and celebrity spokespersons representing the ADT brand. In addition to Google and State Farm, our strategic partnerships and alliances include dealers, home builders, property management firms, homeowners’ associations, financial institutions, retailers, first responders, and software service providers.
OUR MARKET The residential and small business security and automation market primarily consists of owners and renters of single-family homes or apartments and small businesses owners. The market is generally characterized by a large and homogeneous customer base with less complex system installations. Many of our residential and small business customers are driven to purchase monitored security and automation services as a result of moving to a new location; a perceived or actual increase in crime or life safety concerns in their neighborhood; significant events such as the birth of a child or the opening of a new business; or incentives provided by insurance carriers that may offer lower insurance premium rates if a security system is installed or may require that a system be installed as a condition of coverage.
We also seek opportunities to leverage our brand name, experience in security and smart home solutions, and high degree of trust among our customer base to pursue new customers in complementary markets such as personal on-the-go security and safety. We have seen increased interest in smart home offerings and other mobile technology applications, which we believe is attributable to a variety of factors, including advancements in technology, younger generations of consumers, and shifts to de-urbanization. We believe our strategic initiatives will help us satisfy consumer demands in light of these macro-level dynamics and position us for sustainable growth for years to come.
Our goal is to maximize customer lifetime value for both new and existing customers by (i) continuing to evaluate our pricing and product offerings; (ii) managing costs and service strategies to provide enhanced value; (iii) upgrading existing customers to our interactive services, internet protocol (“IP”) video solutions, or other upgraded solutions where desirable; (iv) offering various bundling initiatives; and (v) achieving long customer tenure.
11 COMPETITION Our approach to competition is to emphasize the quality and reputation of our offerings and industry-leading brand, our superior customer service, unique product and service offerings, our network of customer support and monitoring centers, commitment to consumer privacy, and knowledge of customer needs. Success in acquiring new customers depends on a variety of factors such as brand and reputation, market visibility, the ability to identify and sell to prospective customers, offering capabilities, and the quality and prices of our products and services. We are focused on extending our leadership position in the traditional residential and small business security and smart home markets. In addition, we continue to add new features and functionalities to further differentiate our offerings and support a pricing premium. We believe a combination of technology advancements along with a growing customer interest in lifestyle and business productivity solutions will support the increasing market penetration.
Traditional residential and small business security markets in the U.S. remain highly competitive and fragmented, with several major companies; many smaller national, local, and regional companies; and an increasing number of new entrants, which is primarily the result of relatively low barriers to entry and the availability of companies providing outsourced monitoring services without maintaining the customer relationship. Technology trends and innovation have also created significant change in our industries, providing many new opportunities while also lowering the barriers to entry for automation, interactive, and smart home solutions. As a result, new business models and competitors have emerged. Additionally, with our recent focus on DIY offerings, we may face additional competition in the DIY space as we position ourselves to grow our market share.
We believe our principal competitors are: Residential (Pro-Installation) Residential (Self-Installation) Small Business Vivint Smart Home, Inc (a subsidiary of NRG) Ring Smart Security System by Amazon Vivint Smart Home, Inc (a subsidiary of NRG) Xfinity Home Security (a division of Comcast Corporation) SimpliSafe Home Security Systems Ring Smart Security System by Amazon Brinks Home Security (operating brand of Monitronics International, Inc.) Wyze Home Monitoring SimpliSafe Home Security Systems We also compete with point solutions (products with one intended application) and home automation-only systems. In some cases, customers believe that these offerings replace the need for full-scale security systems. Further, third-party installation companies often partner with device providers to offer professional installation alternatives for these point solutions and other DIY systems.
In addition, some self-monitored solutions do not require a monthly fee for home automation services, which allows for no-cost alternatives to the professionally monitored (monthly fee-based) solutions that we provide. Although self-monitored solutions do not replace the need for professionally monitored solutions, as more features and functionality are built into these self-monitored solutions, the demand for some customers to opt for more expensive, professionally monitored options could be reduced.
We believe we are well positioned to compete with traditional and new competitors due to our focus on safety, security, and convenience; our nationwide team of sales consultants; our solid reputation for and expertise in providing reliable security and monitoring services through our in-house network of redundant monitoring centers; our reliable product solutions; our highly skilled installation and service organization; and our partnerships with companies such as Google and State Farm.
RESOURCES MATERIAL TO OUR BUSINESS Materials and Inventory We purchase equipment and product components from a limited number of suppliers and distributors. To minimize the risk of a disruption from any single supplier, we utilize dual sourcing methods whenever possible. Inventory is held at supplier locations, distribution partner locations, and internal ADT regional distribution centers at levels we believe are sufficient to meet current and anticipated customer needs. We also maintain inventory of certain equipment and components at our field offices and in technicians’ vehicles. Third-party distributors generally keep a minimum stocking level of certain key items to have coverage for certain situations, including supply chain disruptions. In addition, we rely on various information technology and telecommunications service providers as part of the functionality and monitoring of our systems.
We are continuously monitoring global supply chain disruptions, and we do not currently anticipate any major interruptions in our supply chain in the near term.
12 Intellectual Property Patents, trademarks, copyrights, and other proprietary rights are important to our business, and we continuously refine our intellectual property strategy to maintain and improve our competitive position. Where possible and appropriate, we seek to register or patent new intellectual property to protect our ongoing technological innovations and strengthen our brand, and we take appropriate action against infringements or misappropriations of our intellectual property rights by others. We review third-party intellectual property rights to help avoid infringement and to identify strategic opportunities. We typically enter into confidentiality agreements to further protect our intellectual property. Patents extend for limited periods of time in the various countries where patent protection is obtained. Trademark rights may potentially extend for longer periods of time and are typically dependent upon the use of the trademarks.
We own a portfolio of patents that relate to a variety of monitored security and automation technologies utilized in our business, including security panels and sensors, video and information management solutions, and our SMART monitoring solution that aims to reduce false alarms and improve response effectiveness. We also own a portfolio of trademarks, including ADT, ADT Pulse, and ADT+. In addition, we are a licensee of intellectual property, including from our third-party suppliers and technology partners.
Certain trademarks associated with the ADT brand that we own within the U.S. and Canada are owned outside of the U.S. and Canada by Johnson Controls International PLC (as successor to Tyco International Ltd., “Tyco”). In certain instances, such trademarks are licensed in certain territories outside the U.S. and Canada by Johnson Controls to certain third parties. Pursuant to the Tyco Trademark Agreement entered into between The ADT Corporation and Tyco in connection with the separation of The ADT Corporation from Tyco in 2012, we are generally prohibited from registering, attempting to register, or using the ADT brand outside the U.S. and its territories and Canada. As a result, if we choose to sell products or services or otherwise do business outside the U.S. and Canada, we do not have the right to use the ADT brand to promote our products and services. In connection with the sale of our Canadian operations in 2019, we entered into a non-competition and non-solicitation agreement with TELUS Corporation (“TELUS”) pursuant to which we will not have any operations in Canada, subject to limited exceptions, for a period of seven years from the date of sale. Additionally, we entered into a patent and trademark license agreement with TELUS granting (i) the use of our patents in Canada for a period of seven years, and (ii) exclusive use of our trademarks in Canada for a period of five years and non-exclusive use for an additional two years thereafter.
In connection with the Commercial Divestiture, we granted the Commercial Business a license to continue to use the ADT brand and other Company trademarks for a period of twelve months to transition from Company branding (the “Brand License”), which expired during the fourth quarter of 2024. We have also agreed to a covenant not to assert a claim against the Commercial Business for infringement of the Company's patents as of the Commercial Divestiture for products and services that were used in the Commercial Business prior to the Commercial Divestiture, and have provided the Commercial Business with a paid-up, irrevocable, non-assignable (with limited exceptions) license to continue to use certain software and other Company intellectual property in the same manner.
SEASONALITY Our residential security and home automation business has historically experienced a certain level of seasonality primarily as a result of fluctuations in the housing market. Since more household moves typically take place during the second and third calendar quarters of each year, our disconnect rate, new customer additions and installation volume, and related cash subscriber acquisition costs are historically higher in these quarters than in the first and fourth calendar quarters. However, other factors such as the level of marketing expense, relevant promotional offers, and timing of third-party account purchases can impact these trends. Further, we may see increased servicing costs or reductions in revenue related to more alarm signals, customer service requests, and customer credits as a result of inclement weather-related incidents.
GOVERNMENT REGULATION AND OTHER REGULATORY MATTERS Our operations are subject to numerous federal, state, and local laws and regulations related to occupational licensing, building codes, tax, and permitting, as well as consumer protection and privacy, labor and employment, and environmental protection. Changes in laws and regulations can positively and negatively affect our operations and impact the manner in which we conduct our business.
Licensing and Permitting Most states in which we operate have employee and/or business licensing laws directed specifically toward professional installation and monitoring of security devices. Our business is also subject to requirements, codes, and standards imposed by local government jurisdictions, as well as various insurance, approval and listing, and standards organizations. We maintain the 13 relevant and necessary licenses related to the provision of installation of security systems and related services in the jurisdictions in which we operate.
Additionally, we rely extensively on telecommunications service providers, which are regulated in the U.S. by the Federal Communications Commission (“FCC”), to communicate signals as part of the functionality and monitoring of security systems.
Our security business is subject to various state and local measures aimed at reducing false alarms. Such measures include requiring permits for individual alarm systems, revoking such permits following a specified number of false alarms, imposing fines on customers or alarm monitoring companies for false alarms, limiting the number of times police will respond to alarms at a particular location after a specified number of false alarms, requiring additional verification of an alarm signal before the police respond, or providing no response to residential system alarms.
Consumer Protection and Privacy Our advertising and sales practices are regulated by the U.S. Federal Trade Commission (“FTC”) and state and consumer protection laws, which may include restrictions on the manner in which we promote the sale of our products and services and require us to provide most consumers with three-day or longer rescission rights.
Our collection and use of personal information and communications with current and potential customers are regulated by federal and state laws, which include restrictions on the use of telemarketing, auto-dialing technology, email marketing, and text communications; restrictions on the sale, sharing, and use of personal information; as well as requirements for the protection thereof and actions to be taken in the event of a breach of personal information.
We use credit scoring to qualify our residential customers for certain offers and financing options. The use of credit reporting and scoring and offering financing options to our residential customers is subject to federal and state laws, including the federal Fair Credit Reporting Act, which limit the use of consumer credit reports, prohibit discrimination against protected classes when offering or granting credit, and require certain disclosures to customers in the event that we take an adverse action based on a consumer credit report.
We provide some residential customers the option to pay up-front charges in installments and certain other customers the option of using third-party financing arrangements, all of which are subject to federal and state laws regulating consumer finance. These laws require certain mandatory consumer disclosures and, in some cases, limit our ability to impose certain fees and charges. In addition, some jurisdictions require us to register or obtain licenses in order to make installment contract or third-party financing options available to our customers.
Labor and Employment Our operations are subject to regulation under the U.S. Occupational Safety and Health Act (“OSHA”) and equivalent state laws. Failure to comply with applicable OSHA regulations or other federal, state, and local laws and regulations, even if no work-related serious injury or death occurs, may result in civil or criminal enforcement and substantial penalties, significant capital expenditures, or suspension or limitation of operations. Additionally, in certain jurisdictions, we must obtain licenses or permits to comply with standards governing employee selection, training, and business conduct.
Environmental Protection We continue to monitor emerging developments regarding environmental protection laws. At this time, we do not believe that federal, state, and local laws and regulations relating to the discharge of materials into the environment, or otherwise relating to the protection of the environment, or any existing or pending climate change legislation, regulation, or international treaties or accords are reasonably likely to have a material adverse effect on our business.
Artificial Intelligence The recent rapid development and use of AI, including generative AI, has resulted in increased scrutiny by state and federal governments in the U.S. We are closely monitoring laws regulating the development and deployment of AI and similar technologies at the state and federal level and assessing their potential impact on our current, planned, and potential use of AI in our business. At this time, we do not believe that federal, state, and local laws and regulations relating to the use of AI have a material adverse effect on our business. However, developments in AI technology and laws and regulations affecting how we propose to use AI may limit or restrict our ability to use such technology.
14 We are currently assessing and piloting AI solutions for basic customer interactions to improve service quality and response times. We are also exploring how we can use AI in our product and service offerings to expand our capabilities and drive efficiencies.
HUMAN CAPITAL AND SUSTAINABILITY We believe focusing on human capital management and sustainability enhances our corporate performance, while enabling us to hire and retain top talent who share our values and passion about our organization.
Human Capital Management As of December 31, 2024, we employed approximately 12,800 people, including approximately 1,700 direct field solution advisors; 3,200 installation and service technicians; 4,400 customer care professionals; and 600 phone sales representatives.
Approximately 7% of our employees are covered by collective bargaining agreements; and we believe our relations with our employees and labor unions have generally been positive.
A Culture of High Performance The shared values, priorities, and principles that shape beliefs and drive behaviors and decision-making to achieve high performance at every level is critical to our innovation and long-term sustainable success. We are committed to fostering a culture and environment where every team member feels valued and empowered to collaborate and contribute to the achievement of business results as well as their individual career goals. In 2024, we redesigned and refreshed our cultural markers to BLUE: Bold, Lead, Unite, and Elevate, which represent our guiding principles that unite and empower us with purpose and passion. Additionally, we significantly increased the frequency that employees receive performance and development coaching and feedback to help ensure all employees are clearly focused on critical business outcomes and development opportunities. Our total compensation programs are designed to differentiate rewards and recognize high performance via the merit pay process, the annual incentive plan, and the long-term incentive plan, subject to team member eligibility.
Talent Recruitment and Management Our success in attracting, retaining, and developing a strong, dedicated, and diverse workforce largely depends on the hiring and retaining of top talent across the entire organization, specifically our senior management team, technology and product talent, and customer-facing employees. We offer our employees competitive compensation, benefits, and health and wellness programs, as well as training, networking, development resources, coaching, and performance feedback. In addition, our long-term equity compensation plans are intended to align management interests with those of our stockholders and to encourage the creation of long-term value. We continue to deploy talent using a mix of hybrid, remote, and in-person work arrangements, based on the nature of the role, to support talent attraction and retention in alignment with business needs. At our headquarters, we implemented a hybrid on-site work model requiring employees covered under this model to work in-office a minimum number of days per week.
Our comprehensive employee listening program collects feedback and sentiment at every stage of the employee experience, including an annual sentiment survey. The annual sentiment survey was completed by approximately two thirds of employees, up from the prior year, and showed favorable results for most topics. Our annual leadership talent management review practice assesses overall performance and potential and includes succession planning for our Executive Leadership Team (“ELT”) roles reporting to the CEO.
Inclusive Diversity and Belonging (“IDB”) We are committed to building and sustaining a culture of inclusive diversity and deliberately advancing the maturity and effectiveness of our practices. Our culture aims to recognize and embrace all forms of diversity in order to drive innovation and growth. Talent diversity starts with the recruitment and hiring process and continues through the learning, development, advancement, and retention of employees with wide-ranging backgrounds and experiences. We highly value our employees’ diversity and aim to promote diversity awareness in all of our talent management processes.
Our Inclusive Diversity and Belonging Operating Team (the “IDB Team”) and its ten Business Employee Resource Groups (“BERGs”) are central to our ability to execute our IDB priorities. The IDB Team is a diverse group of business leaders from across the organization, including executive and senior management, that focuses on continuous improvement of our IDB practices and effectiveness. Our commitment to inclusion is also integrated into our cultural markers which reinforces our focus 15 on fostering a collaborative and supportive environment. IDB learning and education is supported by a comprehensive suite of on-demand resources; and IDB education and awareness continues to be a high-priority.
BERGs offer our employees specific opportunities to partner and collaborate through learning and networking, volunteer projects, and mentoring. Our BERGs also participate in various business initiatives. Executives and officers from across the Company leverage their time, networks, and resources to support BERG initiatives and projects. To support BERG community advocacy and outreach, each BERG is granted a small budget to donate to nonprofit organizations that align with their mission and our corporate social responsibility efforts. Deliberate growth and development of our BERGs is central to the engagement, retention, and development of our employees.
Employee Well-being, Health, and Safety We devote significant resources to employees’ wellness, health, and safety. We continue to provide to all employees ADT Balance, which is an annual well-being program that offers a robust variety of features, including biometric screenings, team fitness challenges, webinars, group coaching, and self-guided resources. Employees enrolled in our self-insured medical plan may earn a discount on medical payroll contributions for themselves and/or their enrolled spouses or domestic partners by completing the annual preventative exam and an online health assessment. Additionally, cash incentives can be earned by completing certain well-being activities. In 2024, we enhanced our financial well-being programs through our vendor partners, including money coaching, identity protection, and mortgage referral programs.
Our Environmental, Health, and Safety (“EHS”) vision is to build a culture that promotes safe behaviors on each task, every day, to achieve zero incidents and enhance employee wellness, and to minimize our environmental impact. In order to achieve our vision, we strive to incorporate our values of people, prevention, and accountability into our business. We focus on compliance with all applicable EHS requirements, and we believe that all occupational injuries and illnesses, as well as environmental incidents, are generally preventable. For example, we continue to institute fleet safety initiatives on our vehicles, including installing and maintaining collision warning and auto-braking technologies. Our EHS management system includes expectations for compliance, accountability, sustainability, and continuous improvement to foster a culture of safety that enables our employees to minimize risk and to understand and follow safety rules, as well as to identify, avoid, and correct unsafe actions, behaviors, or situations.
Environmental We are committed to reducing our environmental impact by promoting environmental stewardship throughout our organization, and we continuously strive to improve our carbon footprint. We are also assessing environmental risk on our operations as one aspect of our enterprise risk management review process and plan to continue to do so on an ongoing basis. We have implemented our ADT Environmental Absolutes framework, which represents our focus on complying with environmental requirements, addressing proper disposal of waste streams, and promoting recycling of materials. We employ waste recycling and diversion programs and continue to monitor waste levels and reduce unnecessary trash hauls. We also continually explore methods to reduce greenhouse gases from our motor vehicle fleet, including through the purchase of newer vehicle models having greater fuel efficiency, the use of hybrid vehicles, and our Remote Assistance Program that reduces truck rolls. In addition, we have focused on efficiency improvements in lighting, air handling, and data operations as well as through the utilization of renewable energy while continuing to rationalize our real estate portfolio.
Social Our Corporate Social Responsibility program is evolving to better align with our corporate strategy of offering safe, smart, and sustainable solutions. To that end, we are focusing on community programs that aim to create spaces where people feel safe. By supporting ‘safe places,’ we are helping provide preventative solutions with a broad impact. Through financial contributions and in-kind product donations of smart security systems, ADT can make a difference for these organizations. In addition, ADT employees volunteer for activities in their communities such as for block-beautification events and mentoring programs. ADT also supports a variety of nonprofits through volunteerism, corporate philanthropy, and in-kind product donations.
Governance ADT prioritizes strong corporate governance, believing that this is the foundation for financial integrity and superior performance. Our Board of Directors is responsible for the oversight of our business and approves our operating values which are reflected in our Code of Conduct (the “Code”).
16 We are committed to working to ensure all ADT employees uphold our core Company values of trust, collaboration, service, and innovation. This begins with the Code, which describes our commitment to our customers, investors, communities, and each other. The Code outlines employee expectations and helps foster a culture of integrity. We adhere to the governance requirements established by federal and state law, the SEC, and the NYSE; and we strive to establish appropriate risk management methods and control procedures to adequately manage and monitor the major risks we may face day to day.
Additional information about our corporate responsibility priorities and approach and related reports are available on our website. The contents of our website and these reports are referenced for general information only and are not incorporated into this Annual Report.
AVAILABLE INFORMATION Availability of SEC Reports Our website is located at Our investor relations website is located at We make available free of charge on our investor relations website under “Financials” our Annual Reports on Form 10-K, Quarterly Reports on Form 10-Q, Current Reports on Form 8-K, reports filed pursuant to Section 16 of the Securities Exchange Act of 1934 (the “Exchange Act”), other SEC filings, and any amendments to those reports that are filed or furnished pursuant to Section 13(a) or 15(d) of the Exchange Act as soon as reasonably practicable after we electronically file or furnish such materials to the SEC. The SEC maintains a website that contains reports, proxy and information statements, and other information regarding our filings at
Use of Website to Provide Information From time to time, we have used, and expect in the future to use, our website as a means of disclosing material information to the public in a broad, non-exclusionary manner, including for purposes of the SEC’s Regulation Fair Disclosure (Reg FD). Financial and other material information regarding the Company is routinely posted on our website and accessible at https:// investor.adt.com. In order to receive notifications regarding new postings to our website, investors are encouraged to enroll on our website to receive automatic email alerts. None of the information on our website is incorporated into this Annual Report.
17 ITEM 1A. RISK FACTORS.
In addition to risks and uncertainties in the ordinary course of business that are common to all businesses, important factors that are specific to our industry and the Company could have a material adverse effect on our business, financial condition, results of operations, and cash flows. You should carefully consider the risks described below and in our subsequent periodic filings with the SEC. The following risk factors should be read in conjunction with Item 7 “Management’s Discussion and Analysis of Financial Condition and Results of Operations” and the consolidated financial statements and related notes in this Annual Report.
Risks Related to Our Products and Services Our growth is dependent upon our ability to keep pace with rapid technological and industry changes through a combination of partnerships with third parties, internal development, and acquisitions, in order to obtain and maintain new technologies for our products and service introductions that achieve market acceptance with acceptable margins.
Our business operates in markets that are characterized by rapidly changing technologies, evolving industry standards, potential new entrants, and changes in customer needs and expectations. Accordingly, our future success depends in part on our ability to accomplish the following: identify emerging technological trends in our target end-markets; develop, acquire, and maintain competitive products and services that capitalize on existing and emerging trends; enhance our existing products and services by adding innovative features on a timely and cost-effective basis that differentiates us from our competitors; incorporate popular third-party interactive products and services into our product and service offerings; sufficiently capture and protect intellectual property rights in new inventions and other innovations; and develop or acquire and bring products and services, including enhancements, to market quickly and cost-effectively.
Our ability to develop, alone or with third parties, or to acquire new products and services that are technologically innovative requires the investment of significant resources and can affect our competitive position. In addition, the dynamic nature of these changes requires that we simultaneously engage in significant technology developmental efforts across our operations, including platform development, sales, marketing, customer care, customer self-service, remote assistance, billing, and other substantive and administrative functions. Upgrading and implementing changes to any one of our systems presents challenges, including potential interruptions to system operations as changes are made, which could disrupt or reduce their efficiency in the short term and temporarily affect the quality of the products and services offered to customers. Moreover, the age of our systems and architecture may present unique challenges that we have not previously encountered as we undertake these developmental efforts simultaneously across our operations. These system updates and development efforts divert resources from other potential investments in our businesses, and they may not lead to the development of new commercially successful technologies, products, or services on a timely basis.
From time to time, we enter into strategic partnerships with third parties to broaden and develop our offerings and marketing efforts. These strategic partnerships may require us to undertake significant commitments and make substantial expenditures and there can be no assurance that the expected benefits from these partnerships will be realized. For example, in July 2020, we entered into the Google Commercial Agreement, pursuant to which Google agreed to supply us with certain Google Devices and Services for sale to our customers. We have agreed, with certain exceptions, to exclusively provide or sell those Google Devices and Services to our customers, although Google can sell the same or similar devices to our competitors who may more successfully commercialize products or services that are competitive to ours, thereby materially harming our business. Given this exclusivity arrangement with Google, if Google fails to perform or to provide Google Devices and Services that continually meet the demands of our customers, or fails to provide continued innovation and investment in their relevant product businesses, or if we fail to provide or sell the Google Devices and Services that Google provides, or if we fail to develop products and services with Google that our customers find desirable, all in a timely manner, or if Google were to begin offering security products or services competitive to our own, our business, financial condition, results of operations, and cash flows could be materially, adversely impacted. In addition, subject to customary termination rights related to breach and change of control, the Google Commercial Agreement has an initial term of seven years from the date that the Google Devices and Services are successfully integrated into our end-user security and automation platform. Product introductions and the timing of such integration are focused on customer experience and are mutually agreed upon. Until the launch of such integration, Google has the contractual right to require us, with certain exceptions, to exclusively offer those Google Devices and Services without such integration for all new professional installations and for existing customers who do not otherwise have ADT Pulse or ADT Control interactive services.
In November 2020, we announced the ongoing development of our proprietary ADT-owned next-generation professional security and automation technology platform that we launched in 2023 as ADT+. Our ADT+ platform is intended to provide customers with a seamless experience across security, life safety, automation, and analytics through a common application that integrates the user experience, the customer service experience, and back-end support. We may not achieve a successful 18 ongoing rollout of the platform, and new platform features, in a timely manner, within budget, or in a manner that enables the commercialization of products and services that meet the continually evolving demands of our customers, any of which could have a material adverse effect on our business, financial condition, results of operations, and cash flows. As we continue to commercialize products based upon our interactive platform, we have adjusted processes for reviewing and securing intellectual property rights. Nevertheless, we have been, and in the future may become, the target of additional lawsuits alleging that we have infringed the patents or technology of third parties. Regardless of the merits of these lawsuits and any steps we take to mitigate infringement risk, any allegations could cause us to incur significant costs to defend and resolve, and could harm our business and reputation, any of which could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
In September 2022, we entered into a strategic relationship with State Farm with the goal of expanding our customer base by developing integrated solutions for State Farm’s customers. As part of the strategic relationship, in October 2022, we entered into the State Farm Development Agreement, pursuant to which State Farm committed up to $300 million to fund product and technology innovation, customer growth, and marketing initiatives with us. Subject to the terms of the State Farm Development Agreement, we have agreed not to enter into any development, marketing, distribution, or other arrangement with certain competitors of State Farm and to refrain from developing, marketing, distributing, or making available to certain competitors of State Farm any products or services developed in connection with the State Farm Development Agreement. If we fail to successfully develop products and services that are utilized by State Farm’s customer base pursuant to the State Farm Development Agreement, or if such agreement is not extended past its initial term expiring on October 13, 2025, we may not achieve or realize the anticipated financial, strategic and other benefits of the strategic relationship with State Farm, and our growth prospects and our business, financial condition, results of operations, and cash flows could be materially adversely affected.
Any new or enhanced products and services that we develop alone or pursuant to existing or new agreements with third parties may not satisfy customer preferences, and potential product failures may cause customers to reject our products and services. As a result, these products and services may not achieve market acceptance, and our brand image could suffer. In addition, our competitors may introduce superior products or business strategies, impairing our brand and the desirability of our products and services, which may cause customers to defer or forego purchases of our products and services, or to decline to enter into new monthly installment contracts or to cancel or fail to renew existing contracts. If our competitors implement new technologies before we can implement them, those competitors may be able to provide more effective products than ours, possibly at lower prices, and experience higher adoption rates and popularity. Any delay or failure in the introduction of new or enhanced solutions could harm our business, financial condition, results of operations, and cash flows. In addition, the markets for our products and services may not develop or grow as we anticipate, and any changes in our go-to-market approach may not be seen as favorable by customers. The failure of our technology, products, or services to gain market acceptance, the potential for product defects, or the obsolescence of our products and services could significantly reduce our revenue, increase our operating costs, or otherwise materially adversely affect our business, financial condition, results of operations, and cash flows.
The home security and automation markets in which we sell our products and services are highly competitive, which may result in pressure on our profit margins and limit our ability to maintain or increase the market share of our products and services. We experience significant competitive pressure in both the DIFM and DIY spaces. The monitored security industry is highly fragmented and subject to significant competition and pricing pressures. We experience significant competitive pricing pressures in the DIFM space on installation, monitoring, and service fees. Several competitors offer comparable or lower installation and monitoring fees, and others may charge significantly more for installation, but in many cases, less for monitoring. We also face competition in the DIY space from companies such as SimpliSafe, Wyze Home Monitoring, and Amazon Ring, which enable customers to self-monitor and control their environments without third-party involvement through the Internet, text messages, emails, or similar communications. Some DIY providers may also offer professional monitoring with the purchase of their systems and equipment without a contractual commitment, or offer new Internet of Things (“IoT”) devices and services with automated features and capabilities, which may be appealing to customers and put us at a competitive disadvantage. In addition, certain DIY providers have a significantly broader customer base and product offering than us, allowing them to cross-subsidize their offerings through their other product offerings and cross-sell interactive and security solutions that are competitive with our offerings to customers who are loyal to the competitor’s brand. Continuing expansion in customers’ options to choose systems that they can in part or fully install could increase our attrition rates over time and the risk of accelerated amortization of customer contracts resulting from a declining customer base.
In addition, cable, telecommunications, and large technology companies have expanded into the home automation and monitored security industry and are bundling their existing offerings with interactive and monitored security services, often at lower monthly monitoring rates. These companies: (i) may have existing access to and relationships with customers, as well as highly recognized brands, which may drive increased awareness of their security/automation offerings relative to ours; (ii) may have access to greater capital and resources than us; and (iii) may spend significantly more on advertising, marketing, and 19 promotional resources, as well as the acquisition of other companies with home automation solution offerings, any of which could have a material adverse effect on our ability to drive awareness and demand for our products and services. We may also face competition for direct sales from our independent, third-party authorized dealers, who may offer installation in particular markets for considerably less than we do.
Additionally, one or more of our competitors either in the DIFM or DIY space could develop a significant technological advantage over us, allowing them to provide additional or better-quality service or lower prices, which could put us at a competitive disadvantage. Continued pricing pressure, technology improvements, competitor brand loyalty, and continuing shifts in customer preferences toward self-monitoring and DIY could adversely impact our customer base, revenue, and/or pricing structure and have a material adverse effect on our business, financial condition, results of operations, and cash flows. Aggressive pricing strategies adopted by our competitors could cause us to lose market share, reduce our prices, and add significant pressure on our cost structure.
In addition, in connection with our continued rollout of Google Nest products, and as our pricing model becomes more transparent to consumers and we offer more optionality with tiered pricing and pricing disaggregation as compared to our current pricing model, our competitors may be better able to underprice us in the marketplace and our customers and potential customers may determine they can achieve a lower cost solution or higher value with an alternative provider. Changes in the transparency of our pricing may also result in new customers selecting lower cost solutions than they otherwise would have and our existing customers switching to our lower cost solutions or demanding that we lower the cost of their existing solutions, which could impact our revenue and profitability. Furthermore, the new smart home communication protocol Matter launched in 2022, with new Matter-compatible products already available. The project group was launched and introduced by Amazon, Apple, Google, Comcast, and others, with about 280 members currently in the working group, including ADT. The goal of Matter is to make all smart home devices interoperable which presents risks for our smart home offerings because interoperable smart home offerings make it easier and less costly for consumers to switch providers, making it more difficult for ADT to retain existing subscribers.
If the insurance industry changes its practice of providing incentives to homeowners for the use of alarm monitoring services, we may experience a reduction in new customer growth or an increase in our customer attrition rate.
It has been common practice in the insurance industry to provide a reduction in rates for consumer policies written on homes that have monitored security systems. This practice benefits our business, as it makes our products more attractive and benefits customer retention, However, there can be no assurance that insurance companies will continue to offer these rate reductions. If these incentives are reduced or eliminated, new homeowners who otherwise might not feel the need for monitored security services would have to be acquired through our standard sales and marketing processes, which could be at a higher cost of acquisition, and existing customers may choose to disconnect or not renew their service contracts, which could increase our attrition rates. In addition, as a result of our strategic relationship with State Farm, other insurance companies may offer rate reduction policies that favor our competitors’ customers or may otherwise modify their practices in detriment of our customers. In each case, our growth prospects and our business, financial condition, results of operations, and cash flows could be materially adversely affected.
The retirement of older telecommunications technology by telecommunications providers and limitations on our customers’ options of telecommunications services and equipment could materially adversely affect our business, increase customer attrition, and require significant capital expenditures.
Certain elements of our operating model have historically relied on our customers’ continued selection and use of traditional copper wireline telecommunications service to transmit alarm signals to our monitoring centers. There is a growing trend for customers to switch to the exclusive use of cellular or IP-based technology in their homes and businesses, as telecommunication providers discontinue their copper wireline services in favor of IP-based technology. Many of our customers’ security systems rely on technology that is not operable with newer cellular or IP-based networks, and as such, will not transmit alarm signals on these networks. The discontinuation of copper landline services, older cellular technologies, and other services by telecommunications providers, as well as the switch by customers to the exclusive use of cellular or IP-based technology, may require system upgrades to alternative, and potentially more expensive, alarm systems to function and transmit alarm signals properly, which could increase our customer revenue attrition. Additionally, any telecommunications technology upgrades or implementations could also result in significant additional costs and divert management and other resource attention away from customer service and sales efforts for new customers. In the future, we may not be able to successfully implement new telecommunications technologies or adapt existing telecommunications technologies to changing market demands. If we are unable to adapt in a timely manner to changing telecommunications technologies, market conditions or customer preferences, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
20 In February 2022, a major provider of 3G cellular networks began to retire its network and a major provider of Code-Division Multiple Access (“CDMA”) began to do so in December 2022. Of our customers impacted by these retirements, we transitioned, or provided our customers with the means to transition, all but a relatively small number of customer accounts. None of these remaining customers have responded to our multiple requests to upgrade their systems and therefore we could not transition them prior to the relevant transition dates. A failure to effectively transition these remaining customers away from retired networks will result in a loss of signal to the systems and certain services we provide, which may impact our ability to bill and collect for services provided. Implementation of additional service charges in connection with these transitions may cause customers to view such charges unfavorably, which could increase our customer attrition. If we are unable to upgrade cellular equipment at customer sites to meet future carrier network standards or to respond to other changes carriers are making or may make to their networks in a timely and cost-effective manner, whether due to an insufficient supply of electronic components or parts, an insufficient skilled labor force, or due to any other reason, or if we are sued by one or more customers due to our inability to provide certain services, or due to any loss incurred while we are not able to provide certain services, or due to any continuous billing for services after a prior or future transition date, our business, financial condition, results of operations, and cash flows, could be materially adversely affected. We have also become aware that one or more telecom carriers are beginning the process to retire their time-division multiplexing (“TDM”) nodes that service toll-free numbers, which could require us to further upgrade certain of our customer equipment.
In addition, we use broadband Internet access service to support our product offerings, such as video monitoring and surveillance, and as a communications option for alarm monitoring and other services. Video monitoring and surveillance services use significantly more bandwidth than non-video Internet activity. As utilization rates and penetration of these services increase, the need for increased network capacity may necessitate incurring additional capital or operational expenditures to avoid service disruptions and enable a seamless video experience for our customers. Further, if our customers decide to transition from traditional broadband Internet access services to fixed mobile Internet access services, they may encounter data limits which could negatively impact their use of video monitoring and surveillance, any of which could materially adversely impact our business, financial condition, results of operations, and cash flows.
Police and fire departments could refuse to respond to calls from monitored security service companies, which could damage consumer trust and confidence in our solutions and may damage our ability to attract and retain customers. Police departments in certain jurisdictions do not respond to calls from monitored security service companies unless certain conditions are met, such as video or other verification or eyewitness accounts of suspicious activities, either as a matter of policy or by local ordinance. We offer video verification or the option to receive a response from private guard companies in certain jurisdictions, which increases the cost of some security systems and may increase the cost to customers. If additional police and fire departments refuse to respond or are prohibited from responding to calls from monitored security service companies unless certain conditions, such as those mentioned above, are met, consumer trust and confidence in our solutions may be damaged and our ability to attract and retain customers could be negatively impacted, and our business, financial condition, results of operations, and cash flows could be materially adversely affected.
Our reputation as a service provider of high-quality security offerings may be materially adversely affected by product defects or shortfalls in customer service. Our business depends on our reputation and ability to maintain good relationships with our customers, dealers, suppliers, and local regulators, among others. Our reputation may be harmed either through product defects, such as the failure of one or more of our customers’ alarm systems, or shortfalls in customer service. Customers generally judge our performance through their interactions with staff at our monitoring and customer care centers, dealers, and field installation and service technicians, as well as their day-to-day interactions with our products and mobile applications. Any failure to meet customers’ expectations in such customer service areas could harm our reputation or customer relationships and cause an increase in attrition rates or make it difficult to obtain new customers or otherwise have a material adverse effect on our business, financial condition, results of operations, and cash flows.
21 In addition, we have attempted to control the operating costs of certain of our customer care operations using lower cost labor in certain foreign countries that may be subject to relatively higher degrees of political and social instability and may lack the infrastructure to withstand political unrest or natural disasters. The occurrence of natural disasters, pandemics, political or economic instability, or other activities in such countries could result in the sudden and continued closure of operations that in turn could cause disruptions in our operations and a failure to maintain our existing level and quality of customer care. The practice of utilizing labor based in foreign countries has come under increased scrutiny in the United States. Governmental authorities could seek to limit or penalize our operations, and our customers may not value the services provided by such operations. In addition, we are subject to applicable anti-corruption laws and regulations, such as The Foreign Corrupt Practices Act, that prohibit certain types of payments and which could expose us to significant penalties, fines, settlements, costs and consent orders that may curtail or restrict our business. Any such outcome could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Unauthorized use of our brand names by third parties, and the expenses incurred in developing and preserving the value of our brand names, may materially adversely affect our business.
Our brand names are critical to our success. Unauthorized use of our brand names by third parties may materially adversely affect our business and reputation, including the perceived quality and reliability of our products and services, as well as brand loyalty. We rely on trademark law, company brand name protection policies, and agreements with our employees, customers, business partners, and others to protect the value of our brand names. Despite our precautions, we cannot provide assurance that those procedures are sufficient to protect against unauthorized third-party use of our brand names. In recent years, various third parties have used our brand names to engage in fraudulent activities, including unauthorized telemarketing conducted in our names to induce our existing customers to switch to competing monitoring service providers, lead generation activities for competitors, and obtaining personally identifiable or personal financial information. Third parties sometimes use our names and trademarks, or other confusingly similar variations thereof, in other contexts that may impact our brands. We may not be successful in detecting, investigating, preventing, or prosecuting all unauthorized third-party use of our brand names. Future litigation with respect to such unauthorized use could also result in substantial costs and diversion of our resources. These factors could materially adversely affect our reputation, business, financial condition, results of operations, and cash flows.
Third parties hold rights to certain of our key brand names outside of the U.S., which could prevent us from being able to adequately protect our brands or expand into other markets.
Our success depends in part on our continued ability to use trademarks to capitalize on our brands’ name-recognition and to further develop our brands in the U.S, as well as in other international markets should we choose to expand and grow our business outside of the U.S. in the future. Not all the trademarks that are used by our brands have been registered in all of the countries in which we may do business in the future, and some trademarks may never be registered in any or all of these countries. Rights in trademarks are generally territorial in nature and are obtained on a country-by-country basis by the first person to obtain protection through use or registration in that country in connection with specified products and services. Some countries’ laws do not protect unregistered trademarks at all or make them more difficult to enforce; and third parties may have filed for “ADT” or marks similar to our blue octagon logo in countries where we have not registered these brands as trademarks. Accordingly, we may not be able to adequately protect our brands everywhere in the world and use of such brands may result in liability for trademark infringement, trademark dilution, or unfair competition.
For example, certain trademarks associated with the ADT brand, including “ADT” and the blue octagon, are owned in all territories outside of the U.S. and Canada by Johnson Controls, which acquired and merged with and into Tyco. In certain instances, such trademarks are licensed in certain territories outside the U.S. and Canada by Johnson Controls to third parties. Pursuant to a trademark agreement entered into between The ADT Corporation and Tyco (the “Tyco Trademark Agreement”) in connection with the separation of The ADT Corporation from Tyco in 2012, which endures in perpetuity, we are prohibited from ever registering, attempting to register or using such trademarks outside the U.S. (including Puerto Rico and the U.S. Virgin Islands) and Canada, and we may not challenge Tyco’s rights in such trademarks outside the U.S. and Canada. Additionally, under the Tyco Trademark Agreement, we and Tyco each has the right to propose new secondary source indicators (e.g., “Pulse”) to become designated source indicators of such party. To qualify as a designated source indicator, certain specified criteria must be met, including that the indicator has not been used as a material indicator by the non-proposing party or its affiliates over the previous seven years. If we are unable to object to Tyco’s proposal for a new designated source indicator by successfully asserting that the new indicator did not meet the requisite criteria, we would subsequently be precluded from using, registering, or attempting to register such indicator in any jurisdiction, including the U.S. and Canada, whether alone or in connection with an ADT brand. Any dilution, infringement, or customer confusion with respect to our brand or use of trade names, or the inability to use such names as we expand our existing and create new strategic relationships, could materially adversely affect our reputation, business, financial condition, results of operations, and cash flows.
22 In addition, in November 2019, we sold all our shares of ADT Canada to TELUS and, among other things, entered into a non-competition and non-solicitation agreement with TELUS pursuant to which we agreed not to directly or indirectly engage in a business competitive with ADT Canada, subject to limited exceptions, for a period of seven years. In connection with our sale of ADT Canada, we also entered into a patent and trademark license agreement with TELUS granting them (i) the use of our patents in Canada for a period of seven years and (ii) the exclusive rights to use our trademarks in Canada for a period of five years followed by non-exclusive use of our trademarks for an additional two years. Any violation by TELUS of our agreements with them, or their misuse of our intellectual property or behavior by TELUS in a manner that incorrectly reflects poorly on us because of TELUS’s use of our intellectual property could damage our brand and reputation and have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Risks Related to Our Operations We rely on a significant number of our customers remaining with us as customers for long periods of time, and if our attrition rates rise significantly, our profitability, business financial condition, results of operations, and cash flows could be materially adversely affected.
New customers require an upfront investment, and we generally achieve revenue break-even in less than two and a half years. Accordingly, our long-term profitability is dependent on long customer tenure. This requires that we minimize our rate of customer disconnects, or attrition, which can increase as a result of factors such as customer relocations, problems with our product or service quality, customer service challenges, increased interoperability of smart home devices now or in the future, customer non-pay, unfavorable general economic conditions, and the preference for lower pricing of competitors’ products and services over ours. If attrition rates were to rise significantly, we may be required to accelerate the depreciation and amortization expense for, or to impair, certain of our assets, including with respect to subscriber and customer relationships, which would cause a material adverse effect on our financial condition and results of operations. In addition, if we fail to keep our customers for a sufficient period of time, or our attrition rates increase, our profitability, business, financial condition, results of operations, and cash flows could be materially adversely affected.
Delays, costs, and disruptions that result from upgrading, integrating, and maintaining the security of our information and technology networks and systems could materially adversely affect us.
We are dependent on the capacity, reliability, and security of information technology networks and systems, including Internet and Internet-based or “cloud” computing services and the relevant personnel who operate those systems, to collect, process, transmit, and store electronic information. We have completed a significant number of acquisitions of companies that operate different technology platforms and systems. We routinely implement modifications and upgrades to our existing information technology systems to keep up with changing technology and business demands. Aiming to provide a seamless customer experience, we are also integrating systems from our various acquisitions, making changes to legacy systems, replacing legacy systems with successor systems with new functionality, and implementing new systems. We are also implementing modifications to various technology platforms and systems related to the Commercial Divestiture, including as part of the transition services related to the Commercial Divestiture.
The dynamic nature of these and other changes we are undertaking require that we simultaneously engage in significant technology development efforts across our operations, including platform development, sales, marketing, customer care, customer self-service, remote assistance, billing, and other substantive and administrative functions. Upgrading and implementing changes to our systems have presented, and could continue to present, challenges. Any delay in making such changes or replacements or in purchasing new systems could have a material adverse effect on our business, financial condition, results of operations, and cash flows. There are inherent costs and risks associated with integrating, replacing, and changing these systems and implementing new systems, including potential disruptions in our sales, operations, and customer service functions which may reduce their efficiency in the short term; loss of customers’ confidential or other information; potential negative reception from customers; potential disruption of our internal control structure; substantial capital expenditures; additional administration and operating expenses; retention of sufficiently skilled personnel to integrate, implement, and operate the new systems and an increase in human capital costs; the use of third-party personnel resources, including offshore vendors, to supplement our internal personnel demands; demands on management time; challenges securing our systems along with dependent processes from cybersecurity threats; and other risks and costs of delays or difficulties in transitioning to new systems or of integrating new systems into our current systems.
23 If we do not effectively implement our plans to migrate our technology infrastructure to the cloud, we could experience significant disruptions in our operations, which could have a material adverse effect on our results of operations and financial condition.
We are in the process of migrating our technology infrastructure to the cloud. This initiative is a major undertaking as we migrate and reconfigure our current system processes, transactions, data and controls to new cloud-based platforms. This transition could have a significant impact on our business processes, financial reporting, information systems, and internal controls. As we transition our technology infrastructure to the cloud, we may need to divert resources, including management attention, away from other important business operations. Additionally, we may experience difficulties as we manage these changes and transition our technology infrastructure to the cloud, including loss or corruption of data, interruptions in service and downtime, increased cyber threats and activity, delayed financial reporting, unanticipated expenses including increased costs of implementation and of conducting business, and lost revenue. Although we conduct design validations and user testing, these may cause delays in transacting our business due to system challenges, limitations in functionality, inadequate management or process deficiencies in the development and use of our systems. Difficulties in implementing or an inability to effectively implement our migration plans could disrupt our operations and harm our business.
As we increase our reliance on cloud infrastructure, our products and services will become increasingly reliant on continued access to, and the continued stability, reliability, and flexibility of third-party cloud services. We have limited control over third-party cloud operations and facilities on which we plan to host our technology infrastructure. Any changes in third-party service levels or any disruptions or delays from errors, defects, cyber attacks, security breaches, computer viruses, DDoS attacks, bad acts, or performance problems resulting from our increased reliance on cloud infrastructure could harm our reputation, damage our small business customers’ businesses, and harm our business. Our third-party cloud providers are also vulnerable to damage or interruption from earthquakes, hurricanes, floods, fires, war, public health crises, terrorist attacks, power losses, hardware failures, systems failures, telecommunications failures and similar events. Our transition and migration to the cloud may increase our risk of liability and cause us to incur significant technical, legal or other costs, and we may have limited remedies against third-party providers in connection with such liabilities.
Additionally, our third-party cloud providers may not be able to effectively manage existing traffic levels or increased demand in capacity requirements, especially to cover peak levels or spikes in traffic, and as a result, our customers may experience delays in accessing our solutions or encounter slower performance in our solutions, which could increase our attrition rates, negatively impact our sales, and significantly harm the operations of our small business customers. Interruptions in our services might reduce our revenue, cause us to issue credits to customers, subject us to potential liability, and cause customers to terminate their subscriptions or harm our renewal rates. Finally, we may in the future be unable to secure additional cloud hosting capacity on commercially reasonable terms or at all. If any of our third-party cloud providers increase pricing terms, terminate or seek to terminate our contractual relationship or change or interpret their terms of service or policies in a manner that is unfavorable, we may be required to transfer to another provider and may incur significant costs and experience service interruptions.
Cybersecurity attacks or threats or other unauthorized access or attempts to access to our systems, or those of third parties, have in the past, and may in the future, compromise the security of our systems and otherwise disrupt our normal operations, which could have a material adverse effect on our reputation, business, financial condition, results of operations, and cash flows.
We collect, process, transmit, and store sensitive and confidential information related to our customers, employees and business partners, as well as proprietary information related to our business, such as business plans and license agreements. This makes us vulnerable to cyber attacks and other attempts to gain unauthorized access to our information and technology networks and systems, including work from home environments and third-party systems that are interconnected with ours. Cyber attacks have previously originated, and may in the future originate, from various methods, including phishing, malware, and ransomware. While we implement and require security measures within our products, services, operations, and, to a limited extent, within third parties’ systems, cyber attacks continue to evolve in sophistication and increase in volume and frequency and we have not in the past and may not in the future be able to timely detect or prevent cybersecurity breaches on our systems or within third-party systems that are interconnected with ours, including the unauthorized access, capture, or alteration of information; the exposure or exploitation of potential security vulnerabilities; distributed denial of service attacks; the installation of malware or ransomware; acts of vandalism; computer viruses; or misplaced data or data loss that could materially adversely impact our reputation, business, financial condition, results of operations, and cash flows. Despite our security measures, we and third parties whose systems are interconnected with ours have been the target of and/or subject to a number of these methods of cyber attacks, including the Cybersecurity Incidents, and we will likely continue to be the target of and/or subject to such attacks in the future. These cyber attacks have previously resulted, and may in the future 24 result, in certain impacts to us or interconnected third-parties, including disrupted operations, system instability, theft of our confidential or proprietary or other information, increased cybersecurity protection, consulting and legal costs, litigation, and reputational damage. These cyber attacks have previously exposed, and may in the future expose, us to an increased risk of future cyber attacks and threats, including through an increase in more sophisticated and targeted cyber attacks from various methods, including phishing, malware, and ransomware, among other methods. In addition, cyber attacks could result in misstated or misappropriated financial data or impair our ability to effectively manage our financial reporting process. We may be subject to regulatory scrutiny or exposed to litigation or other claims by affected persons including our customers, employees, and business partners. While we maintain insurance coverage that is intended to address certain aspects of data security risks, such insurance coverage may not be sufficient to cover all losses or all types of claims that have arisen or may arise, and in the future may not be available at reasonable costs or at all.
In addition, following the Commercial Divestiture, certain of our information systems and our information security protocols remain interdependent on those of our divested Commercial Business, including dependence for a transitional period on its information systems for billing and alarm monitoring for a small portion of our clients’ accounts. Moreover, certain employees of our former Commercial Business have had and will have access to our financial and other internal systems for a period of time which has exposed and will continue to expose us to increased risks related to information theft and unauthorized system access. For more information on the continuing services relationship between ADT and our former Commercial Business, see “We may not achieve some or all of the strategic and financial benefits that we expect to achieve from the Commercial Divestiture or the ADT Solar Exit which could have a material adverse effect on our financial condition and results of operations.” This interconnection with our former Commercial Business was exploited by an unauthorized actor in the October Incident and exposes us to increased risks related to cyber attacks. Cyber attacks and threats at our former Commercial Business have in the past led, and may in the future lead, to cyber attacks and threats to our systems and assets. Any measures we have taken or may take in the future to protect against cyber attacks and threats, including those at our former Commercial Business, or against employees, including employees of our former Commercial Business, who may wrongfully or negligently use or access such technology, intellectual property, or information, or negligently or wrongfully disclose such technology and intellectual, confidential, proprietary, or any other information to third parties, including our competitors, may prove insufficient and cannot provide absolute protection against such attacks and threats.
Our business also requires us to share confidential information with suppliers and other third parties. Third parties, including our partners and vendors, could also be a source of cybersecurity risk to us, or cause disruptions to our normal operations, in the event of a failure of their own products, components, networks, security systems, and infrastructure. For example, in 2021, one of our vendors, the Ultimate Kronos Group (“Kronos”), which is a workforce management and human capital management cloud provider, experienced a ransomware attack that resulted in Kronos temporarily decommissioning the functionality of certain of its cloud software, requiring us to find alternative methods to properly pay our employees and to monitor the status of the work in progress of certain of our projects in a timely manner. In addition, some of the products we sell and provide services for are categorized as IoT and may become targets for cybercriminals and other actors, including for actors attempting to gain unauthorized access. The significant increase in the number of our employees working from home further exposes us to security risks. Although we take steps to secure confidential, proprietary, or other information that is provided to or accessible by third parties working on our behalf, we cannot be certain that advances in criminal capabilities, new discoveries in the field of cryptography, or other developments will not compromise or breach the technology protecting the networks that access our products and services.
A significant actual or perceived (whether or not valid) theft, loss, fraudulent use or misuse of customer, employee, or other personally identifiable or other sensitive data, whether by us, our partners and vendors, or other third parties, or as a result of employee error or malfeasance or otherwise, non-compliance with applicable industry standards or our contractual or other legal obligations regarding such data, or a violation of our privacy and information security policies with respect to such data, could result in significant remediation costs, administrative fines, litigation or other claims by third parties, or regulatory actions against us. Such an event could additionally result in unfavorable publicity and therefore materially and adversely affect the market’s perception of the security and reliability of our products and services and our credibility and reputation with our customers, which may lead to customer dissatisfaction and could result in lost sales and increased customer revenue attrition. In addition, we depend on our information technology infrastructure, and that of our third party partners or vendors, for business-to-business and business-to-consumer electronic commerce. Cyber attacks to, or threats against, our infrastructure or that of third parties whose systems are interconnected with ours or whose systems contain ADT information could create prolonged system disruptions and shutdowns that could negatively impact our operations. Increasingly, our products and services are accessed through the Internet, and a significant number of service calls happen virtually, and cyber attacks and/or threats in connection with the delivery of our services via the Internet may affect us and could be detrimental to our reputation, business, financial condition, results of operations, and cash flows. There can be no assurance that our continued investments in new and emerging technology and other solutions to protect our network and information systems will prevent any of the risks 25 described herein. In addition, any delay in making such investments due to conflicting budget priorities or otherwise could have a material adverse effect on our business, financial condition, results of operations, and cash flows. There can be no assurance that our insurance will be sufficient to protect against all our losses from any future disruptions or cyber attacks on our systems or other events as described herein.
Uncertainty in the development, deployment, and use of AI in our products and services, as well as our business more broadly, could adversely affect our business and reputation.
We use AI-enabled or AI-integrated systems and tools, including generative AI, to service our customers and drive efficiencies within our workforce. As with many new and emerging technologies, AI presents numerous risks and challenges that could adversely affect its further development, adoption, or use, and therefore our business. The development, deployment, and use of generative AI technology remains in early stages and ineffective or inadequate AI development or application practices by us or third parties could result in unintended consequences. For example, models, including large language models, underlying AI solutions that we use may be flawed or may be based on biased, insufficient, or poor-quality datasets. In addition, any latency, disruption, or failure in our AI systems or data infrastructure could result in delays or errors in our offerings or operational activities. Developing, testing, and deploying resource-intensive AI solutions may require additional investment and increase our costs. There also may be real or perceived social harm, unfairness, or other impacts to human rights, privacy, employment, or other social issues or outcomes that undermine public confidence in the use and deployment of AI. In addition, third parties may deploy AI solutions in a manner that reduces customer demand for our products and services. Any of the foregoing may result in decreased demand for our products or material harm to our business, results of operations, brand, or reputation.
The legal and regulatory landscape surrounding AI is rapidly evolving and uncertain including in the areas of intellectual property, cybersecurity, privacy, and data protection. For example, there is uncertainty around the validity and enforceability of intellectual property rights related to our use or development of AI tools. Compliance with new or changing laws, regulations or industry standards related to AI may impose significant operational costs and may limit our ability to apply AI technologies in certain use cases. Failure to appropriately respond to this evolving landscape may result in legal liability, regulatory action, or brand and reputational harm.
We depend on third-party providers and suppliers for components of our security and automation systems, third-party software licenses for our products and services, and third-party providers to transmit signals to our monitoring facilities and provide other services to our customers. Any failure or interruption in products or services provided by these third parties could harm our ability to operate our business.
The components for the security and automation systems that we install or consume are manufactured by original equipment manufacturers (“OEM”), original design manufacturers (“ODM”), contract manufacturers (“CM”) and/or third-party suppliers. While we have implemented robust supply chain management practices designed to mitigate the risk of supply chain interruptions, there is no assurance that these practices will be effective, and interruptions may occur from time to time.
Certain key suppliers may experience difficulties in obtaining necessary components, which may impact our ability to meet customer demands and complete critical initiatives. In addition, our suppliers are susceptible to disruptions from fire, natural disasters, weather-related incidents, and the effects of climate change (such as sea level rise, drought, flooding, wildfires, and increased storm severity), as well as health epidemics and pandemics, transmission interruptions, extended power outages, human or other error, and malicious acts, including cyber attacks, terrorism, war, sabotage, and government actions, or other concerns impacting their local workforce or operations, all of which are beyond our and their control. While we actively monitor supplier operations, require compliance with business continuity and disaster recovery plans, maintain alternative sourcing options where feasible, carry a surplus of finished goods in inventory, exercise limited control over our raw material suppliers through demand forecasting and supply planning processes, and have contract terms that require transparency in sourcing, quality assurance commitments, and escalation protocols, there is no assurance that these practices will be effective and supplier disruptions, including any financial or other difficulties our providers may face, may have a material adverse effect on our business.
We are also subject to supply chain disruptions if we learn that any of our suppliers are in violation of legislation which bans the import of goods based on their method of production, such as using forced labor or otherwise. This may also result in negative publicity regarding our production methods, and the alleged unethical or illegal practices of any of our suppliers could adversely affect our reputation. Our efforts to minimize the risk of a disruption from a single supplier may not always be effective, and we have experienced some disruptions in our supply chain during recent years, and could experience such disruptions in the future. Any continued or significant interruption in supply could cause significant delays in installations and repairs and the loss of current and potential customers. Although some specific shortages may be resolved, they may recur. From time to time, we may also experience product recalls and other unplanned product repairs or replacements with customers. We have occasionally experienced such product service events. There can be no assurance that any such future product service events will not be more extensive or more costly, material to us, and/or require the outlay of cash while we 26 pursue cost recovery from manufacturers and suppliers, and there can be no assurance that we will be successful in pursuing recoveries from those third parties. If a previously installed component were found to be defective, we might not be able to recover the costs associated with its repair or replacement across our installed customer base, and these costs, or the diversion of technical personnel to address the defect could materially adversely affect our business, financial condition, results of operations, and cash flows. Such incidents may also harm our reputation and may result in litigation or other claims from customers. In the event of a product recall or litigation against our suppliers or us, we could experience a material adverse effect on our business, financial condition, results of operations, and cash flows. We also rely on third-party software for key automation features in certain of our offerings and on the interoperation of that software with our own, such as our mobile applications and related platforms. We could experience service disruptions if customer usage patterns for such integrated or combined offerings exceed, or are otherwise outside of, system design parameters and we or our third-party provider is unable to make corrections. Such disruptions in the provision of services could result in our inability to meet customer demand, damage our reputation and customer relationships, and materially and adversely affect our business. We also rely on certain software technology that we license from third parties and use in our products and services to perform key functions and provide critical functionality. For example, we license the software platform for our monitoring operations from third parties. Because a number of our products and services incorporate technology developed and maintained by third parties, we are, to a certain extent, dependent upon such third parties’ ability to update, maintain, or enhance their current products and services; to ensure that their products are free of defects, security vulnerabilities, and compromise; to develop new products and services on a timely and cost-effective basis; and to respond to emerging industry standards, customer preferences, and other technological changes. Further, these third-party technology licenses may not always be available to us on commercially reasonable terms, or at all. If third-party vendors decide not to renew our existing agreements or to limit our access to their offerings, or the third-party technology becomes obsolete, is incompatible with future versions of our products or services, is unusable due to defects, security vulnerabilities, or compromise or otherwise fails to address our needs, we cannot provide assurance that we would be able to replace the functionality provided by the third-party software with technology from alternative providers. Furthermore, even if we obtain licenses to alternative software products or services that provide the functionality we need, we may be required to replace hardware installed at our monitoring centers, work from home environments, and at our customers’ sites, including security system control panels and peripherals, in order to execute our integration of or migration to alternative software products. Any of these factors could materially adversely affect our business, financial condition, results of operations, and cash flows. We also rely on various third-party telecommunications providers and signal processing centers to transmit and communicate signals to our monitoring facilities and work from home environments in a timely and consistent manner. These telecommunications providers and signal processing centers could deprioritize or fail to transmit or communicate these signals to the monitoring facilities and work from home environment for many reasons, including disruptions from fire, natural disasters, weather-related incidents, and the effects of climate change (such as sea level rise, drought, flooding, wildfires, and increased storm severity), health epidemics or pandemics, transmission interruption, extended power outages, human or other error, malicious acts, including cyber attacks, provider preferences regarding the signals that get transmitted, government actions, war, terrorism, sabotage, or other conflicts, or as a result of disruptions to internal and external networks or third party transmission lines. The failure of one or more of these telecommunications providers or signal processing centers to transmit and communicate signals to our monitoring facilities and work from home environments in a timely manner could affect our ability to provide alarm monitoring, home automation, and interactive services to our customers. We also rely on third-party technology companies to provide automation and interactive services to our customers. These technology companies could fail to provide these services consistently, or at all, which could result in our inability to meet customer demand and damage our reputation. There can be no assurance that third-party telecommunications providers, signal processing centers, and other technology companies will continue to transmit and communicate signals to our monitoring facilities and work from home environments or provide home automation and interactive services to customers without disruption. Any such failure or disruption, particularly one of a prolonged duration, could have a material adverse effect on our business, financial condition, results of operations, and cash flows. As mentioned above, telecommunications providers have in the past, and may in the future, retire older telecommunications technology, limiting our customers’ options of telecommunications services and equipment, which could materially adversely affect our business, increase customer attrition and require significant capital expenditures.
An event causing a disruption in the ability of our monitoring facilities or customer care resources to operate, including work from home operations, could materially adversely affect our business. A disruption in our ability to provide security monitoring services or otherwise provide ongoing customer care to our customers could have a material adverse effect on our business. A disruption could occur for many reasons, including fire, natural disasters, including hurricanes, weather-related incidents, and the effects of climate change (such as sea level rise, drought, flooding, wildfires, and increased storm severity), health epidemics or pandemics, transmission interruption, extended power 27 outages, human or other error, malicious acts, including cyber attacks, provider preferences regarding the signals that get transmitted, government actions, war, terrorism, sabotage, or other conflicts, or as a result of disruptions to internal and external networks or third party transmission lines. Monitoring and customer care also have in the past been and could in the future be disrupted by information systems and network-related events or cyber attacks, such as computer hacking, computer viruses, phishing, malware, ransomware, worms, or other malicious software, distributed denial of service attacks, malicious social engineering, or other destructive or disruptive activities that could also cause damage to our properties, equipment, and data, as well as our efforts to respond to, contain, and remediate such events, attacks, and activities. A failure of our redundant back-up procedures or a disruption affecting multiple monitoring facilities or work from home environments could disrupt our ability to provide security monitoring or customer care services to our customers. As a result of such disruptions, we may experience customer dissatisfaction and potential loss of confidence, and liabilities to customers or other third parties, each of which could harm our reputation and impact future revenues. We could also be subject to claims or litigation with respect to losses caused by such disruptions. Our insurance may not be sufficient to fully cover our losses or may not cover a particular event at all. Any such disruptions or outcomes could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
A variety of events have had and could have in the future a significant negative impact on our ability to carry on our normal operations and could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
We have business continuity and disaster recovery plans and procedures and an incident response plan designed to protect our business against a variety of events, including natural disasters, health epidemics or pandemics, cyber attacks, and armed conflicts or other hostilities. However, we cannot guarantee that these plans and procedures will prevent or efficiently mitigate the impact of such events, especially those which are inherently unpredictable and beyond our anticipated thresholds or impact tolerances. Such events may impact macroeconomic conditions, consumer behavior, labor availability, or damage our facilities and our ability to provide our products and services to customers. Governmental responses to such events, including restrictions to businesses, can also affect the foregoing items and adversely affect our operations. Such events may also heighten other risks disclosed in these risk factors, including, but not limited to, those related to supply chain interruptions and consumer perceptions of our brand and industry. For example, the outbreak of the coronavirus pandemic in 2020 contributed to consumer unease and decreased discretionary spending. In order to maintain our operations, we had to implement measures to protect the health of our employees and our customers. Furthermore, while we maintain insurance, our coverage may not sufficiently cover all types of losses or claims that may arise and we may be unable to adequately offset any losses we may incur.
We rely on monitoring centers and customer care centers as an integral part of our ongoing business operations and we have deployed hybrid and remote working options. The closure of any site or any widespread absence of the employees remaining in any such site could result in a material disruption to our business. Because the majority of employees who staff these operations currently conduct their jobs from home, our work from home environment could subject us to the failure of the communications networks serving our employees which we no longer control and who may not have sufficient back up capabilities. In addition, this work from home environment results in more home access points that are susceptible to cyber attacks, such as computer hacking, computer viruses, phishing, malware, ransomware, worms or other malicious software or malicious activities. In addition, our monitoring centers are listed by UL and must meet certain requirements to maintain that listing. UL has adopted a temporary standard that enables our operators to work from home while remaining within the listing requirements and we must ensure that each such home environment continues to meet all such requirements as well as the UL permanent requirements, which have been established by UL. Our employees who work from home may also experience a decrease in the quality of job performance, whether immediate or over time. Any such impact with respect to our employees who are working from home could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Our independent, third-party authorized dealers may not be able to mitigate certain risks such as information technology and data security breaches, product liability, errors and omissions, and compliance with applicable laws and regulations.
We generate a portion of our new customers through our authorized dealer network. We rely on independent, third-party authorized dealers to implement mitigation plans for certain risks they may experience, including, information technology breaches, data security breaches, product liability, errors and omissions, and compliance with applicable laws and regulations. In addition, our dealers rely on other third parties to submit orders and transmit data and may themselves be subject to many of these same risks. If our authorized dealers or the third parties on whom they rely experience any of these risks, or fail to implement mitigation plans for their risks, or if such implemented mitigation plans are inadequate or fail, we may be susceptible to business, legal, or reputational risks associated with our authorized dealers on which we rely to generate customers. Any interruption or permanent disruption in the generation of customer accounts or services provided by our authorized dealers could materially adversely affect our business, financial condition, results of operations, and cash flows.
28 We may pursue business opportunities that diverge from our current business model, or invest in new businesses, services, and technologies outside the traditional security, interactive and smart home services markets, any of which may materially adversely affect our business results or financial condition.
We have and will continue to pursue and invest in new business opportunities that may diverge from our current business model and practices, including expanding our products or service offerings, investing in new and unproven technologies, adding customer acquisition channels, and forming new alliances with companies to market our services. We can provide no assurance that any such business opportunities or investments will perform as expected. Among other negative effects, our pursuit of such business opportunities could cause our cost of investment in new customers to grow at a faster rate than our recurring revenue and fees collected at the time of installation. In addition, any new business partner may not agree to the terms and conditions or limitations on liability that we typically impose upon third parties. Acquisitions in recent years have also significantly expanded our risk profile. For example, in December 2021 we acquired the Solar Business although our core business was not then extended to the residential solar market. In September 2022, we announced a strategic relationship with State Farm and our intention to develop products and services to satisfy certain needs of State Farm’s property and casualty customers which represents a significant entry point into the insurance industry. Additionally, any new alliances or customer acquisition channels could require large investments of capital to develop such business, or have higher cost structures than our current arrangements, which could reduce operating margins and require more working capital. If working capital requirements exceed operating cash flow, we could be required to draw on our revolving credit facility, or pursue other external financing, which may not be readily available. We may also experience capital loss on some or all our investments, insufficient revenue from such investments to offset new liabilities assumed and expenses associated with these new investments, distraction of management from current operations, and issues not identified during pre-investment planning and due diligence that could cause us to fail to realize the anticipated benefits of such investments and incur unanticipated liabilities. In such cases, we may decide to revise our strategic plans and adjust our operating footprint to optimize our operations, or exit certain businesses or product lines entirely. For example, after recording a series of losses with respect to our acquisition of the Solar Business, we announced in November 2023 a series of steps to rationalize the Solar Business, and in January 2024, we made a determination to exit the Solar Business entirely. As of June 30, 2024, substantially all operations of the Solar Business had ceased. Any of these factors could materially adversely affect our business, financial condition, results of operations, and cash flows.
We continue to integrate our acquisitions, as well as to separate certain shared services following the Commercial Divestiture, which may divert management’s attention from our ongoing operations. We may not achieve all of the anticipated benefits, synergies, or cost savings from our acquisitions or the Commercial Divestiture.
Our historical acquisitions, including bulk acquisitions of customer accounts, require the integration of separate companies or accounts that have previously operated independently or separately. The continued integration of operations, including billing and service platforms for recently purchased customer accounts, information technology networks and systems, products, and personnel from our acquisitions, as well as the separation of certain shared services following the Commercial Divestiture, will continue to require the attention of our management and place demands on other internal resources. In addition, the overall continued integration of our acquired businesses, including bulk acquisitions of customer accounts, may be disruptive to our business as a whole and result in material unanticipated problems, expenses, liabilities, competitive responses, and loss of customer relationships. The diversion of management’s attention, and any difficulties encountered in the transition, integration, and divestiture processes, could materially adversely affect our business, financial condition, results of operations, and cash flows.
Further, we continue to integrate the financial reporting systems and processes of various companies we have acquired. Successfully implementing our business plan and complying with the SOX Act and other regulations requires us to be able to prepare timely and accurate consolidated financial statements. Any delay in this implementation of, or disruption in, the transition to new or enhanced systems, procedures, or controls, including internal controls and disclosure controls and procedures, may cause us to present restatements or cause our operations to suffer, and we may be unable to conclude that our internal controls over financial reporting are effective and to obtain an unqualified report on internal controls from our independent registered public accounting firm.
Any of these difficulties in combining operations or accounts, or continuing to separate certain shared services following the Commercial Divestiture, could result in increased costs, decreases in the amount of expected revenues, and further diversion of management’s time and energy, which could materially adversely affect our business, financial condition, results of operations, and cash flows.
29 In 2024, we decided to fully exit the residential Solar Business and such exit is subject to uncertainties and risks that may materially adversely affect our financial condition and results of operations.
On January 24, 2024, we announced that the Company had made a determination to fully exit the residential solar business, which included the transfer of certain assets to other parties as well as the retention of certain contractual obligations. We substantially completed the ADT Solar Exit during the second quarter of 2024.
In connection with the ADT Solar Exit, we incurred severance and other exit costs. The ADT Solar Exit may disrupt our relationships with customers, suppliers and other third parties, which could make our brand less attractive to consumers and business partners. The ADT Solar Exit, including related exit charges, the impact of the related workforce reduction, and any potential legal claims by impacted employees, customers, suppliers, or lenders, could have a material adverse effect on our business, operating results and financial condition.
The ADT Solar Exit may also involve continued financial involvement while being reported in discontinued operations, such as by causing us to continue to incur expenses to maintain services for completed installations and to complete those installations which have not yet been completed, or through continuing guarantees, indemnities or other financial obligations, such as minimum solar panel performance guarantees, and a requirement by lenders that we repay outstanding loan amounts for jobs that have not achieved permission to operate in a timely manner or that we are otherwise required to repurchase pursuant to existing loan agreements. The incurrence of any such cost or action could have a material adverse effect on our business, financial condition, results of operations and cash flows.
We may not achieve some or all of the strategic and financial benefits that we expect to achieve from the Commercial Divestiture or the ADT Solar Exit which could have a material adverse effect on our financial condition and results of operations.
Although we believe the Commercial Divestiture and the ADT Solar Exit will place us in a better position to prioritize investments in our core business where we intend to drive profitable, capital-efficient revenue growth for the long-term, there can be no assurance that we will achieve such benefits, which could have a material adverse effect on our business, financial condition, results of operations, and cash flows. The anticipated benefits are based on several assumptions, some of which may prove incorrect, and could be affected by factors beyond our control, including, without limitation, general economic conditions, increased operating costs, regulatory developments, and other risks described in these risk factors.
In connection with the Commercial Divestiture, we entered into the Commercial TSA, pursuant to which the Company and the Commercial Business will provide certain transitional services relating to ongoing support and other administrative functions to each other for a transitional period of up to 24 months after the closing of the Commercial Divestiture. We rely on the Commercial Business to satisfy its obligations under the Commercial TSA. If the Commercial Business is unable to satisfy its obligations under the Commercial TSA, or if the transition of services covered by the Commercial TSA takes longer to complete than expected, it could have a material adverse effect on our business, financial condition, and results of operations. In addition, we may not be able to eliminate certain costs after the completion of the transition period covered by the Commercial TSA. If we are unable to eliminate some of these costs or effectively work with our supplies to reduce the costs associated with fewer employees or customers, it could have a material adverse effect on our business, financial condition, and results of operations.
ADT is a less diversified business following the Commercial Divestiture and the ADT Solar Exit, which may adversely affect ADT’s results of operations and financial condition.
Prior to the Commercial Divestiture, ADT had three business segments, CSB, Commercial, and Solar. The Commercial Divestiture resulted in ADT being a smaller, less diversified company more focused on consumers, potentially making ADT more vulnerable to changing market, regulatory, and economic conditions following the Commercial Divestiture and the ADT Solar Exit, particularly those affecting consumers and small businesses. We are now entirely dependent on our consumer and small business markets, and any trends or uncertainties affecting such markets will directly affect ADT’s results of operations and financial condition in the future. Macroeconomic headwinds or changes in consumer preferences could have a greater impact on our business following the Commercial Divestiture and the ADT Solar Exit which could have a material adverse effect on our business, financial condition, and results of operations.
Our customer generation strategies through third parties, including our authorized dealer and affinity marketing programs, and our use of celebrities and social media influencers, and the competitive market for customer accounts may expose us to risk and affect our future profitability.
An element of our business strategy is the generation of new customer accounts through third parties, including our authorized dealers, and future operating results depend in large part on our ability to continue to manage this business generation strategy 30 effectively. We currently generate accounts through hundreds of independent third parties, including authorized dealers, and a significant portion of our accounts originate from a smaller number of such third parties. If we experience a loss of authorized dealers or third-party sellers representing a significant portion of our customer account generation, or if we are unable to replace or recruit authorized dealers, other third-party sellers, or alternate distribution channel partners in accordance with our business strategy, our business, financial condition, results of operations, and cash flows could be materially adversely affected. In addition, we are subject to reputational risks that may arise from the actions of our dealers and their employees, independent contractors, and other agents that are wholly or partially beyond our control, such as violations of our marketing policies and procedures as well as any failure to comply with applicable laws and regulations. If our dealers engage in practices that are not in compliance with all applicable laws and regulations, we may be deemed in breach of such laws and regulations, which may result in regulatory proceedings and potential penalties that could materially adversely impact our business, financial condition, results of operations, and cash flows. In addition, unauthorized activities in connection with sales efforts by employees, independent contractors, and other agents of our dealers, including calling consumers in violation of the Telephone Consumer Protection Act, predatory door-to-door sales tactics, and fraudulent misrepresentations, could subject us to governmental investigations and class action lawsuits for, among others, false advertising and deceptive trade practice damage claims, against which we will be required to defend. Such defense efforts are costly and time-consuming, and there can be no assurance that such defense efforts will be successful, all of which could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
The successful promotion of our brands also depends on the effectiveness of our marketing efforts and on our ability to offer member discounts and special offers for our products and services to our partners. We have actively pursued affinity marketing programs, which provide members of participating organizations with special offers on our products and services. These organizations may require us to pay higher fees to them, decrease our pricing for their members, introduce additional competitive options, or otherwise alter the terms of our participation in their marketing programs in ways that are unfavorable to us. These organizations may also terminate their relationships with us if we fail to meet contract service levels or member satisfaction standards, among other things. If any of our affinity or marketing relationships is terminated or altered in an unfavorable manner, we may lose a source of sales leads, and our business, financial condition, results of operations, and cash flows could be materially adversely affected. We also rely on marketing by social media influencers and celebrity spokespersons that represent the ADT brand to generate new customers. These marketing efforts may not be successful or appeal to consumers. The promotion of our brand, products, and services by social media influencers and celebrities is subject to FTC regulations, including, for example, a requirement to disclose any compensatory arrangements between ADT and influencers in any reviews or public statements by such influencers about ADT or our products and services. These social media influencers and celebrities with whom we maintain relationships could also engage in activities or behaviors or use their platforms to communicate directly with our customers in a manner that violates applicable regulations or reflects poorly on our brand and that behavior may be attributed to us or otherwise adversely affect us. In connection with the promotion of ADT’s brand by influencers and celebrities, ADT is also subject to a twenty-year FTC consent decree from 2014 which requires adherence to a robust internal compliance process. In addition, influencers and celebrities who are associated with ADT may engage in behavior that is unrelated to ADT but that causes damage to our brand because of these associations. Any such activities or behaviors of the social media influencers or celebrities we engage, or our failure to adhere to the compliance processes as required by the FTC consent decree, could have a material adverse effect on our business, financial condition, results of operations, and cash flows, or on our reputation.
We face risks in acquiring and integrating customer accounts, and if any of these risks materialize, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
An element of our business strategy involves the bulk acquisition of customer accounts. Acquisitions of customer accounts involve a number of risks, including the possibility of unexpectedly high rates of attrition and unanticipated deficiencies in the accounts and systems acquired despite our investigations and diligence prior to acquisition, as well as costs and complexities in integrating newly purchased accounts into ADT billing and service platforms. We face competition from other alarm monitoring companies, including companies that may offer higher prices and more favorable terms for customer accounts purchased, and/or lower minimum financial or operational qualification or requirements for purchased accounts. This competition could reduce the acquisition opportunities available to us, slowing our rate of growth, and/or increasing the price we pay for such account acquisitions, thus reducing our return on investment and negatively impacting our revenue and results of operations. We can provide no assurance that we will continue to be able to purchase customer accounts on favorable terms or at all in the future. The purchase price we pay for customer accounts is affected by the recurring revenue historically generated by such accounts, as well as several other factors, including the level of competition, our prior experience with accounts purchased in bulk from specific sellers, the geographic location of the accounts, the number of accounts purchased, the customers’ credit scores, and the type of security or automation equipment or platform used by the customers. In purchasing accounts, we have relied on 31 management’s knowledge of the industry, due diligence procedures, and representations and warranties of bulk account sellers. We can provide no assurance that in all instances the representations and warranties made by bulk account sellers are true and complete or, if the representations and warranties are inaccurate, that we will be able to recover damages from bulk account sellers in an amount sufficient to fully compensate us for any resulting losses. In addition, we may need to incorporate and maintain specialized equipment and knowledge in order to service customer accounts purchased, or pay to upgrade such customers to ADT equipment. If any of these risks materialize, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
If we are unable to recruit and retain sufficient personnel at all levels of our organization, our ability to manage our business could be materially adversely affected.
Our success depends in part upon the continued services of sufficient talent at all levels of our organization, including our management team, software developers, product engineers, sales representatives, installation and service technicians and call center talent. Our ability to recruit and retain sufficient talent for these positions is based on our reputation as a successful business with a culture of fairly hiring, training, and promoting qualified employees. However, our success could be impacted adversely by the competitive labor environment and require us to incur wages and benefits in excess of our planned expenditure. Labor shortages in recent years made talent recruitment particularly challenging and competitive. In addition, we acquire businesses from time to time that have rates of employee attrition significantly higher than our own and we may experience difficulty or delay in hiring to fill positions due to these higher rates or in bringing the employee attrition rate of such acquired businesses to a level consistent with our own. The loss of experienced personnel through wage competition, normal attrition (including retirement), or specific actions such as divestitures, cost structure rationalizations, or other business exit activity, may adversely affect our reputation among job seekers, demoralize our remaining employees, and result in loss of critical knowledge. The loss, incapacity, or unavailability for any reason of sufficient personnel at any level of our organization, higher than expected payroll and other costs associated with the hiring and retention of sufficient talent at all levels of our organization, or the inability or delay in hiring new employees, whether in management, sales, technology, product development, installation and service technicians, or call center personnel, could materially adversely affect our business financial condition, results of operations, and cash flows.
The loss of or changes to our senior management could disrupt our business.
Competition for senior management talent having security and home automation industry experience has increased. Factors that impact our ability to attract and retain senior management include compensation and benefits and our successful reputation as a top provider in these industries. Our success partly depends on the ability of our Chairman, President, and CEO, Mr. James D. DeVries, along with the ability of other senior management and key employees, to effectively implement our business strategies and to continue to identify and grow talent. In addition, the success of our business depends on highly qualified leaders with relevant industry and operational experience, as well as the entire management team. The unexpected loss of any member of our senior management team and the related loss of their knowledge of products, offerings, and industry experience, and the difficulty of quickly finding qualified senior management talent to replace any such loss, could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Adverse developments in our collective bargaining agreements or other agreements with some employees could materially adversely affect our business, financial condition, results of operations, and cash flows.
As of December 31, 2024, 835 of our employees at various sites, or approximately 7% of our total workforce, were represented by unions and covered by collective bargaining agreements. We are currently a party to approximately 17 collective bargaining agreements. About one-third of these agreements are up for renewal in any given year. Additionally, and especially in light of recent actions taken by the National Labor Relations Board, we could be subject to further attempts to organize some or all of our non-management employee base. We cannot predict the outcome of negotiations of the collective bargaining agreements covering our employees. If we are unable to reach new agreements or renew existing agreements, employees subject to collective bargaining agreements may engage in strikes, work slowdowns, or other labor actions, which could materially disrupt our ability to provide services. New labor agreements or the renewal of existing agreements may impose significant new costs on us, limit our opportunities to improve our cost structure, and could materially and adversely affect our business, financial condition, results of operations, and cash flows in the future.
If we fail to maintain effective internal control over financial reporting at a reasonable assurance level, we may not be able to accurately report our financial results, which could have a material adverse effect on our operations, investor confidence in our business and the trading prices of our securities.
We identified a material weakness in our internal control over financial reporting as of December 31, 2022 and may identify additional material weaknesses in internal control over financial reporting in the future. A material weakness is a deficiency, or 32 a combination of deficiencies, in internal control over financial reporting such that there is a reasonable possibility that a material misstatement of a company’s annual or interim financial statements will not be prevented or detected on a timely basis. While we believe we have now remediated the material weakness previously reported in our Amended 2022 Annual Report, we cannot assure you that additional material weaknesses in internal control over financial reporting will not occur in the future. The remediation of any such material weaknesses could require us to incur significant expenses. Moreover, if we fail to remediate any material weakness in a timely manner, that may adversely affect our ability to record, process, summarize and report financial information timely and accurately and, as a result, our financial statements may contain material misstatements or omissions.
In addition, it is possible that a material weakness may exist without being identified. Such a failure could cause our financial statements to contain material misstatements or omissions and could also result in regulatory scrutiny, and cause investors to lose confidence in our reported financial condition, lead to a default under our indebtedness and otherwise have a material adverse effect on our business, financial condition, results of operations, and cash flows, and on our reputation with investors and with business partners.
Risks Related to Regulations and Litigation If we fail to comply with constantly evolving laws, regulations, and industry standards addressing information and technology networks and systems, privacy, and data security, we could face substantial penalties, liability, and reputational harm, and our business, financial condition, results of operations, and cash flows could be materially adversely affected.
The confidential data and information collected in the normal course of our business, as well as portions of the significant volume of third party data that we or our partners collect and retain, is subject to certain laws and regulations. Our ability to analyze this data to provide the customer with an improved user experience is a valuable component of our services, but we cannot provide assurance that the data we require will be available from these sources in the future or that the cost of such data will not increase. If the data that we require is not available to us on commercially reasonable terms or at all, we may not be able to provide certain parts of our current or planned products and services, and our business, financial condition, results of operations, and cash flows could be materially adversely affected.
In addition, we may also collect and retain other sensitive types of data, including, among other things, audio recordings of telephone calls and video images of customer sites. We must comply with applicable federal and state laws and regulations governing the collection, retention, processing, storage, disclosure, access, use, security, and privacy of such information in addition to our own posted information security and privacy policies and applicable industry standards, such as the Payment Card Industry Data Security Standards. The legal, regulatory, and contractual environment surrounding the foregoing continues to evolve, and there has been an increasing amount of focus on privacy and data security issues with the potential to affect our business. These privacy and data security laws, regulations, and standards, as well as contractual requirements, could increase our cost of doing business, and failure to comply with these laws, regulations, standards, and contractual requirements could result in government enforcement actions (which could include civil or criminal penalties), private litigation, and/or adverse publicity. In the event of a breach of personal information that we hold or that is held by third parties on our behalf, we may be subject to governmental fines, individual and class action claims, remediation expenses, and/or harm to our reputation. In 2020, we disclosed that a Company technician had secured unauthorized personal access to certain customers’ in-home security systems, resulting in individual and class action legal claims against us. We could incur significant legal costs in defending existing or new claims or in the ultimate resolution of such claims, and we may suffer reputational harm and damage to our brand as a result of such claims or any related publicity. Further, if we fail to comply with applicable privacy and security laws, regulations, policies, and standards; properly protect the integrity and security of our facilities and systems and the data located within them; or defend against cybersecurity attacks; or if our third-party service providers, partners, or vendors fail to do any of the foregoing with respect to data and information assessed, used, stored, or collected on our behalf; or should we fail to prevent future rogue actors from undertaking actions similar to those described above, our reputation and our business, financial condition, results of operations, and cash flows could be materially adversely affected.
Examples of certain requirements we face include those with respect to the Health Insurance Portability Act, the California Consumer Privacy Act, the California Privacy Rights Act, the Colorado Privacy Act, the Virginia Consumer Data Protection Act, and the European Union’s General Data Protection Regulation. These laws and regulations are examples of our need to comply with costly and complex requirements at state, federal, and international levels. As these requirements continue to evolve, and expand to additional jurisdictions, we may incur or be required to incur costs or change our business practices in a manner adverse to our business and failure to comply could result in significant penalties that may materially adversely affect our reputation and our business, financial condition, results of operations, and cash flows.
33 Infringement of our intellectual property rights could negatively affect us.
We rely on a combination of patents, copyrights, trademarks, trade secrets, confidentiality provisions, and licensing arrangements to establish and protect our proprietary rights. We cannot guarantee, however, that the steps we have taken to protect our intellectual property rights will be adequate to prevent infringement of our rights or misappropriation of our intellectual property or technology. Adverse events affecting the use of our trademarks could also negatively impact our brands.
In addition, if we expand our business outside of the U.S. in the future, effective patent, trademark, copyright, and trade secret protection may be unavailable or limited in some jurisdictions. Furthermore, our confidentiality agreements with certain of our employees and third parties to protect our intellectual property could be breached or otherwise may not provide meaningful protection for our confidential information, trade secrets, and know-how related to the design, manufacture, or operation of our products and services. These types of litigation actions may continue for long periods of time, may not be successful, or may result in impairment of certain of our intellectual property rights, and our need to continue to bring claims may be significant and may be indefinite. Any future proceedings on any such matters could be burdensome and costly, and we may not prevail. Further, adequate remedies may not be available in the event of an unauthorized use or disclosure of our confidential information, trade secrets, or know-how. If we fail to successfully enforce our intellectual property rights, our competitive position could suffer, which could materially adversely affect our business, financial condition, results of operations, and cash flows.
Allegations that we have infringed upon the intellectual property rights of third parties could negatively affect us.
We may be subject to claims of intellectual property infringement by third parties. In particular, as our services have expanded, we have become subject to claims alleging infringement of intellectual property, including litigation brought by special purpose or so-called “non-practicing” entities that focus solely on extracting royalties and settlements by alleging infringement and threatening enforcement of patent rights. These companies typically have little or no business or operations, and there are few effective deterrents available to prevent such companies from filing patent infringement lawsuits against us. Our exposure to intellectual property infringement claims may increase as we continue to expand and develop our proprietary ADT+ platform, as we modify and expand our offerings under our partnership with State Farm, or otherwise modify and expand our existing intellectual property in the future. In addition, we rely on licenses and other arrangements with third parties covering intellectual property related to many of the products and services that we market. Notwithstanding these arrangements, we could be at risk for infringement claims from third parties. For example, in 2022 ADT was sued by a party alleging that the cellular antennas in various products purchased and used by ADT infringed their patents. The suit was settled in July of 2024, and ADT is currently in litigation with the supplier of the majority of those products regarding their obligation to indemnify ADT. Additionally, our patent agreement with Tyco, which generally includes a covenant by Tyco not to bring an action against us alleging that the manufacture, use, or sale of any products or services in existence as of the date of our separation from Tyco infringes any patents owned or controlled by Tyco and used by us on or prior to such date, does not protect us from infringement claims for future product or service expansions. In general, any claims or litigation, even those without merit and regardless of the outcome, could cause us to cease marketing certain services or using certain technologies; obtain licenses from the holders of the intellectual property at a material cost or on unfavorable terms; pay significant ongoing royalty payments, settlements, or licensing fees; satisfy indemnification obligations; or to take other potentially costly or burdensome actions to avoid infringing third-party intellectual property rights. The litigation process is costly and subject to inherent uncertainties, and we may not prevail in litigation matters regardless of the merits of our position. Intellectual property lawsuits or claims may become extremely disruptive if the plaintiffs succeed in blocking the trade of our products and services and may have a material adverse effect on our business, financial condition, results of operations, and cash flows. We may be subject to class actions and other lawsuits which may harm our business and results of operations.
We have been and we may continue to be subject to class action litigation involving alleged violations of privacy, consumer protection laws, employment laws, common law claims or other matters. In addition, we have previously been subject to securities class actions relating to our IPO, and we may in the future be subject to additional securities litigation that may be lengthy and may result in substantial costs and a diversion of management’s attention and resources. Results cannot be predicted with certainty, and an adverse outcome in such litigation could result in monetary damages or injunctive relief that could materially adversely affect our business, financial condition, results of operations, and cash flows.
In addition, we are currently and may in the future become subject to legal proceedings and commercial or contractual disputes other than class actions. These are typically claims that arise in the normal course of business including, without limitation, commercial general liability claims, automobile liability claims, contractual disputes, worker’s compensation claims, labor law and employment claims, and claims that we infringed on the intellectual property of others. There is a possibility that such claims may have a material adverse effect on our business, financial condition, results of operations, and cash flows that is greater than we anticipate and/or negatively affect our reputation.
34 Increasing government regulation of telemarketing, email marketing, door-to-door sales, and other marketing methods may increase our costs and restrict the operation and growth of our business.
We rely on telemarketing, email marketing, door-to-door sales, and other marketing channels, including social media conducted internally and through third parties to generate a substantial number of leads for our business, all of which are subject to federal, state and local regulation. Telemarketing and email marketing activities are subject to an increasing amount of regulation in the U.S. Regulations have been issued by the FTC and the FCC that place restrictions on unsolicited telephone calls to residential and wireless telephone customers, whether direct dial or by means of automatic telephone dialing systems, prerecorded, or artificial voice messages and telephone fax machines, and require us to maintain a “do not call” list and to train our personnel to comply with these restrictions. The FTC regulates sales practices generally and email marketing and telemarketing specifically, including through their consent decree on ADT that regulates our use of social media influencers and celebrities, and has broad authority to prohibit a variety of advertising or marketing practices that may constitute “unfair or deceptive acts or practices.” Most of the statutes and regulations in the U.S. applicable to telemarketing and email marketing allow a private right of action for the recovery of damages or provide for enforcement by the FTC and FCC, state attorneys general, or state agencies permitting the recovery of significant civil or criminal penalties, costs and attorneys’ fees if regulations are violated. Although we have developed policies and procedures designed to assist in compliance with these statues, regulations, and consent decree, we can provide no assurance that we, our authorized dealers or other third parties that we rely on for telemarketing, email marketing, and other lead generation activities will be in compliance with all applicable laws and regulations at all times. Although our contractual arrangements with our authorized dealers, affinity marketing partners, and other third parties generally require them to comply with all such laws and regulations and to indemnify us for damages arising from their failure to do so, we can provide no assurance that the FTC and FCC, private litigants, or others will not attempt to hold us responsible for any unlawful acts conducted by our authorized dealers, affinity marketing partners and other third parties or that we could successfully enforce or collect upon any indemnities. Additionally, certain FCC rulings and FTC enforcement actions may support the legal position that we may be held vicariously liable for the actions of third parties, including any telemarketing violations by our independent, third-party authorized dealers that are performed without our authorization or that are otherwise prohibited by our policies. The FCC, FTC, and state agencies have relied on certain actions to support the notion of vicarious liability, including, but not limited to, the use of our brand or trademark, the authorization or approval of telemarketing scripts, or the sharing of consumer prospect lists. Changes in such regulations or the interpretation thereof that further restrict such activities could result in a material reduction in the number of leads for our business and could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Our business operates in a regulated environment and any new, changes to existing, or uncertainty regarding laws or regulations, or our failure to comply with any such rules or regulations, could be costly to us, harm our business and operations, and impede our ability to grow our existing business, any new businesses that we acquire, or investment opportunities that we pursue.
Our operations and employees are subject to various federal, state, and local laws and regulations in such areas as consumer protection, occupational licensing, environmental protection (including climate change regulations), labor and employment, tax, permitting, and other laws and regulations. Most states in which we operate have company and employee licensing laws directed specifically toward the sale, installation, monitoring, and maintenance of security devices, as well as laws and regulations pertaining to solar systems and energy storage solutions that were part of our business before the ADT Solar Exit. Our business relies heavily upon the use of both wireline and wireless telecommunications to communicate signals, and telecommunications companies are regulated by federal, state, and local governments.
Increased public awareness and concern regarding global climate change may result in more international, regional, and/or federal or other requirements or expectations that could mandate more restrictive or expansive standards than existing regulations. There continues to be a lack of consistent climate legislation, which creates economic and regulatory uncertainty, as well as consumer and investor unease. We or our suppliers may be required to make increased capital expenditures to improve our services or product portfolio to meet new regulations and standards. Further, our customers and the markets we serve may impose environmental standards through regulation, market-based emissions policies, or consumer preference that we may not be able to timely meet due to the required level of capital investment or technological advancement. There can be no assurance that our compliance or our efforts to improve our services or products will be successful, and there can be no assurance that proposed regulation or deregulation will not have a negative competitive impact, or that economic returns will reflect our investments in new product development. If environmental laws or regulations are either changed or adopted and impose significant operational restrictions and compliance requirements upon our business or products, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
Federal laws restricting or banning imports of equipment or supplies from specific companies or regions of the world may limit our ability to meet customer demands or to increase prices.
35 In certain jurisdictions, we are required to obtain licenses or permits to comply with standards governing employee selection and training and to meet certain standards in the conduct of our business. The loss of such licenses or permits or the imposition of conditions to the granting or retention of such licenses or permits could have a material adverse effect on us. Furthermore, in certain jurisdictions, certain security systems must meet fire and building codes to be installed, and it is possible that our current or future products and service offerings will fail to meet such codes, which could require us to make costly modifications to our products and services or to forego operating in certain jurisdictions.
We must also comply with numerous federal, state, and local laws and regulations that govern matters relating to our interactions with residential customers, including those pertaining to privacy and data security, consumer financial and credit transactions, home improvement contracts, warranties, and door-to-door solicitation. These laws and regulations are dynamic and subject to potentially differing interpretations, and various federal, state, and local legislative and regulatory bodies may initiate investigations, expand current laws or regulations, or enact new laws and regulations, regarding these matters. As we expand our product and service offerings and enter into new jurisdictions, we may be subject to more expansive regulation and oversight. In addition, our financing and lending activities subject us to various rules and regulations, such as the U.S. federal Truth in Lending Act and analogous state legislation. Also, if we continue to expand our sales to government entities, we may be subject to additional contracting regulations, disclosure obligations, and various civil and criminal penalties, among other things, in a significant manner that we are not subject to today.
The FTC and certain states have consumer protection laws and regulations governing the manner in which providers of consumer services must manage customer subscriptions, autorenewals, and negative option billing arrangements. In October of 2024, the FTC adopted a final “Click-to-Cancel rule intended to consolidate and significantly expand certain federal consumer protections through new rules concerning certain “negative option offers,” whereby a consumer’s silence or failure to take affirmative action to reject a good or service or to cancel a subscription is interpreted as acceptance or continuing acceptance of an offer. Several states have followed suit and have enacted or have proposed to enact similar regulations concerning autorenewals, cancellation rights and negative option arrangements. ADT sells its services to consumers on a subscription basis. The FTC’s “Click-to-Cancel rule, which becomes effective in May 2025 (other than the prohibition on misrepresentations, which took effect in January 2025), and certain proposed state regulations concerning subscription services may require ADT to provide additional notices to consumers regarding the end of fixed term contract commitments, limit ADT’s ability to market discounted or free trials to new customers and, in some cases, require ADT to seek additional customer consents to provide services past the end of minimum contract term commitments, each of which may have a material adverse effect on ADT’s ability to attract and retain customers. Changes in these laws or regulations or their interpretation could dramatically affect how we do business, acquire customers, and manage and use information we collect from and about current and prospective customers and the costs associated therewith. We strive to comply with all applicable laws and regulations relating to our interactions with all customers. It is possible, however, that these requirements may be interpreted and applied in a manner that is inconsistent from one jurisdiction to another and may conflict with other rules or our practices.
Changes in laws or regulations could require us to change the way we operate or to utilize resources to maintain compliance, which could increase costs or otherwise disrupt operations. For example, in June 2024, in Loper Bright Enterprises v. Raimondo, the U.S. Supreme Court held that lower courts need not defer to a governmental agency’s reasonable interpretation of an ambiguous statute that it administers, overruling the doctrine known as “Chevron Deference.” As a result, we cannot predict whether there will be increased challenges to existing regulatory requirements, such as those discussed above, thereby creating uncertainty in compliance with certain regulatory schemes. In addition, failure to comply with any applicable laws or regulations could result in substantial fines or revocation of our operating permits and licenses. If laws and regulations were to change or if we or our products failed to comply with them, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
We could be assessed penalties and fines for false alarms, and if these expenses become significant or we are unable to pass along the associated costs, our customers may terminate or fail to renew our services.
Some local governments impose assessments, fines, penalties, and limitations on either customers or the alarm companies for false alarms. Certain municipalities have adopted ordinances under which both permit and alarm dispatch fees are charged directly to the alarm companies. Our alarm service contracts generally allow us to pass these charges on to customers. If more local governments impose assessments, fines, or penalties for false alarms, or these charges become significant, or we are unable to collect these charges because customers are unwilling or unable to pay them, or our customers terminate or fail to renew their services with us because of these charges, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
36 Adoption of statutes and governmental policies purporting to characterize certain of our charges as unlawful may adversely affect our business.
Generally, if a customer cancels their contract with us prior to the end of the initial contract term, we may charge the customer an early cancellation fee. Consumer protection policies or legal precedents could be proposed or adopted to restrict the charges we can impose upon contract cancellation. Such initiatives could compel us to increase our prices during the initial term of our contracts and consequently lead to less demand for our services and increased customer attrition. Adverse judicial determinations regarding these matters could cause us to incur legal exposure to customers against whom such charges have been imposed and expose us to the risk that certain of our customers may seek to recover such charges through litigation, including class action lawsuits. Any such loss in demand for our services, increase in attrition, or the costs of defending such litigation and enforcement actions could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
In the absence of net neutrality or similar regulation, certain providers of Internet access may block our services or charge their customers more for using our services, or government regulations relating to the Internet could change, which could materially adversely affect our revenue and growth.
Our interactive and home automation services are primarily accessed through the Internet and our security monitoring services, including those utilizing video streaming, are increasingly delivered using Internet technologies. Users who access our services through mobile devices, such as smart phones, laptops, and tablet computers must have a high-speed Internet connection, such as broadband, 4G/LTE, or 5G, to use our services. Currently, this access is provided by telecommunications companies and Internet access service providers that have significant and increasing market power in the broadband and Internet access marketplace. In the absence of government regulation, these providers could take measures that affect their customers’ ability to use our products and services, such as degrading the quality of the data packets we transmit over their lines, giving our packets low priority, giving other packets higher priority than ours, blocking our packets entirely, or attempting to charge their customers more for using our products and services. To the extent that Internet Service Providers (“ISPs”) implement usage-based pricing, including meaningful bandwidth caps, or otherwise try to monetize access to their networks, we could incur greater operating expenses and customer acquisition and retention could be negatively impacted, which could have a material adverse effect on our business, financial condition, results of operations, and cash flows. Furthermore, to the extent network operators were to create tiers of Internet access service and either charge us for or prohibit our services from being available to our customers through these tiers, our business could be negatively impacted. Some of these providers also offer products and services that directly compete with our own offerings, which could potentially give them a competitive advantage. In March 2015, the FCC released net neutrality rules prohibiting broadband ISPs from blocking, throttling, or engaging in “paid prioritization” of content or services. However, these rules were repealed in 2017. As of April 2024, the FCC reinstated net neutrality regulations, effectively reversing the 2017 repeal, once again classifying broadband as a telecommunications service under Title II of the Federal Communications Act of 1934, as amended. However, in June 2024, the U.S. Supreme Court overruled the doctrine known as Chevron Deference, which previously required courts to defer to an agency’s reasonable interpretation of law when the law was ambiguous. As a result, on January 2, 2025, the U.S. Court of Appeals for the Sixth Circuit struck down the FCC’s previously adopted Safeguarding and Securing the Open Internet rule (“SSOI”), ruling that the FCC incorrectly classified ISPs as telecommunications service providers, rather than non-common carrier providers of information services, and therefore exceeded its authority in imposing the net neutrality regulations. Absent net neutrality or open Internet rules, Internet providers, some with competing security businesses, could block or throttle ADT signals in a way that impacts our business and could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Given the nature of our business, we are exposed to greater risks of liability for employee acts or omissions or system failures than may be inherent in other businesses, which could materially adversely affect our business.
If a customer or third-party believes that it has suffered harm to person or property due to an actual or alleged act or omission of one of our authorized dealers, independent contractors, employees or others, or due to a security or automation system failure, they (or their insurers) may pursue legal action against us, and the cost of defending the legal action and of any judgment against us could be substantial. In particular, because our products and services are intended to help protect lives and real and personal property, we may have greater exposure to litigation risks than businesses that provide other consumer and small business products and services. In the event of litigation with respect to such matters, it is possible that the risk-mitigation provisions in our standard customer contracts may be deemed not applicable or unenforceable and, regardless of the ultimate outcome, we may incur significant costs of defense that could materially adversely affect our business, financial condition, results of operations, and cash flows, and there can be no assurance that any such defense efforts will be successful.
37 We may be subject to liability for obligations of The Brink’s Company under the Coal Act or other coal-related liabilities of The Brink’s Company, which could materially adversely affect our business.
On May 14, 2010, The ADT Corporation acquired Broadview Security, a business formerly owned by The Brink’s Company. Under the Coal Industry Retiree Health Benefit Act of 1992, as amended (“Coal Act”), The Brink’s Company and its majority-owned subsidiaries as of July 20, 1992 (including certain legal entities acquired in the Broadview Security acquisition) are jointly and severally liable with certain of The Brink’s Company’s other current and former subsidiaries for health care coverage obligations provided for by the Coal Act. A Voluntary Employees’ Beneficiary Association (“VEBA”) trust has been established by The Brink’s Company to pay for these liabilities, although the trust may have insufficient funds to satisfy all future obligations. We cannot rule out the possibility that certain legal entities acquired in the Broadview Security acquisition may also be liable for other liabilities in connection with The Brink’s Company’s former coal operations. At the time of the separation of Broadview Security from The Brink’s Company in 2008, Broadview Security entered into an agreement pursuant to which The Brink’s Company agreed to indemnify it for any and all liabilities and expenses related to The Brink’s Company’s former coal operations, including any health care coverage obligations. The Brink’s Company has agreed that this indemnification survives The ADT Corporation’s acquisition of Broadview Security. We in turn agreed to indemnify Tyco for such liabilities in our separation from it. If The Brink’s Company and the VEBA are unable to satisfy all such obligations, we could be held liable, which could have a material adverse effect on our business, financial condition, results of operations, and cash flows. Our business would be adversely affected if certain of our independent contractors were classified as employees. We rely on third-party independent contractors in addition to our existing workforce to perform certain tasks including installation and service of our customer alarm and other systems. From time to time, we are involved in lawsuits and claims that assert that certain independent contractors should be treated as our employees. The state of the law regarding independent contractor status varies from state to state and is subject to change based on court decisions, legislation, and regulation. For example, in January 2024, the U.S. Department of Labor (“DOL”) issued a new rule that revises the DOL’s guidance on how to determine who is an employee or independent contractor under the Fair Labor Standards Act (“FLSA”). Previously, the law provided five factors to guide the inquiry into a worker’s status as an employee or independent contractor, with two of these factors carrying greater weight in the analysis. The new rule, effective in March 2024, implements a non-exhaustive multi-factor economic reality test where no one factor or subset of factors would be necessarily dispositive and the weight of each factor would depend on the facts and circumstances of the particular case. Also, although the National Labor Relations Board (“NLRB”) has abandoned, due to legal challenges, its attempt to overrule its 2019 independent contractor standard focused on whether workers have “entrepreneurial opportunity,” a recent NLRB decision in September 2024 appears to signal an intent to reverse its current position that independent contractor misclassification is not itself a violation of the National Labor Relations Act (“NLRA”). Under the new legal framework, it may be more likely for our independent contractors or our subcontractors to be classified as our employees, resulting in such individuals becoming entitled to the reimbursement of certain expenses, to the benefit of wage-and-hour laws, and to the protections under the NLRA including the right to organize for union representation. If such classification was made, we could also be liable for employment and withholding tax and benefits for such individuals, and liable to such individuals for violations of other laws protecting employees. Any such determination could result in a material reduction of the number of subcontractors we can use for our business or significantly increase our costs to serve our customers, which could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Existing or new tariffs and other trade restrictions imposed on imports from China, Mexico, or other countries where much of our end-user equipment is manufactured, or any counter-measures taken in response, may harm our business and results of operations.
Tariffs imposed on imports from China or Mexico, (including enhanced U.S. tariffs recently imposed and/or threatened to be imposed on goods from China), where certain components included in our end-user equipment are manufactured, and any counter-measures taken in response to such new tariffs, may harm our business and results of operations. For example, in 2018 and 2019, the U.S. federal government imposed tariffs on certain alarm equipment components manufactured in China, and on other categories of electronic equipment manufactured in China that we install in our customers’ premises, such as batteries and thermostats. Certain of these tariffs were as high as 25% and such tariffs increased our costs for such equipment as a result of some or all of such new tariffs being passed on to us by our suppliers. If any or all such costs continue to be passed on to us by our suppliers, we may be required to raise our prices, which could result in the loss of customers and harm our business and results of operations. Alternatively, we may seek to find new sources of end-user products, which may result in higher costs and disruption to our business. In addition, the U.S. federal government’s 2018 National Defense Authorization Act imposed a ban on the use of certain surveillance, telecommunications, and other equipment manufactured by certain of our suppliers based in China, to help protect critical infrastructure and other sites deemed to be sensitive for national security purposes in the U.S. This 38 federal government ban implemented in August 2019, and the ban on use of certain covered equipment by federal contractors implemented in August 2020, has required us to find new sources of end-user products, which has resulted in higher costs and disruption to our business. It is also possible new or additional tariffs will be imposed on imports of equipment that we install in end-user premises, or that our business will be impacted by retaliatory trade measures taken by China, Mexico, or other countries, causing us to raise our prices or make changes to our business, any of which could have a material adverse effect on our business, financial condition, results of operations, and cash flows. During the 2024 presidential campaign, President Trump threatened to impose, and since taking office, has begun imposing, significant tariffs on goods imported from China, Canada, Mexico and other countries. Given the uncertainty regarding the scope and duration of these trade actions by the U.S. and any retaliatory trade actions taken by such other countries, we cannot predict whether, or to what extent, tariffs and other trade restrictions may be imposed on or otherwise become applicable to our product offerings or supply chain, and the impact of these trade actions on our business remains uncertain.
In addition, in November 2021, President Biden signed the Secure Equipment Act into effect, and in November 2022, the FCC adopted rules stating that they will no longer review or give licenses to the equipment that makes use of radio frequencies manufactured by companies believed to pose a national security threat, including Huawei, ZTE, Dahua, and Hikvision. This could impact our ability to source products compatible with a customer’s existing system, or make repairs if new, compatible equipment cannot be sourced. In the November 2022 order, the FCC also issued an additional Notice of Proposed Rulemaking in which it asks if existing authorizations should be revoked, and if so, how. If existing authorizations were revoked, it could limit ADT’s ability to maintain and service existing customer equipment. It could also force some customers to replace equipment currently in service. We are also subject to supply chain disruptions should we learn that any one of our suppliers is in violation of legislation such as the Uyghur Forced Labor Prevention Act signed into law in December 2021, which bans the import of goods based on their method of production, such as through the use of forced labor, or otherwise. Any inability to source product, product parts, or other components required by our business in a timely and cost-effective manner could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Risks Related to Macroeconomic and Related Factors General economic conditions can affect our business, and we are susceptible to changes in the business economy, in the housing market, and in business and consumer discretionary income, which may inhibit our ability to grow our customer base and impact our results of operations. Demand for our products and services is affected by the general economy, the business environment, and the turnover in the housing market, among other things. Downturns in the general economy, the business environment, and the housing market would reduce opportunities to make sales of our products and services. Downturns in the rate of the sale of new and existing homes, which we believe drives a substantial portion of our new residential customer volume in any given year, and downturns in the rate of commercial property development, which may drive demand for our small business offerings, would reduce opportunities to make new sales and reduce opportunities to take over systems. Recoveries in the housing market increase the occurrence of relocations, which may lead to customers disconnecting service and not contracting with us in their new homes. The demand for our products and services is also dependent, in part, on national, regional, and local economic conditions, as well as our customers’ level of discretionary income. When our customers’ discretionary income is reduced (such as by higher housing, energy, interest, operating or other costs, or where the actual or perceived wealth of customers has decreased as a result of circumstances such as lower real estate values, increased foreclosure rates, inflation, increased tax rates, or other economic disruptions), we could experience increased attrition rates and reduced customer demand. Where levels of business activity decline, the small business customers could experience increased attrition rates and reduced demand for our offerings. No assurance can be given that we will be able to continue acquiring quality customers or that we will not experience higher attrition rates. Our long-term revenue growth rate primarily depends on revenue from installations and new contracts exceeding disconnects. If customer disconnects or defaults increase, our business, financial condition, results of operations, and cash flows could be materially adversely affected. Rising interest rates or increased consumer lender fees could adversely impact our sales, profitability, and our financing costs.
Our business model, in part, relies on customers financing the purchase price of their system through ADT or third-party lenders. Those lenders charge us fees on the principal balance of those loans. Rising interest rates, as we have experienced during recent years and which we may continue to experience, may increase the lenders’ cost of capital and those increased costs will result in an increase in the fees charged to us. In addition, where we have committed to provide financing internally, as interest rates rise, our cost of capital also gets more expensive and we may not be able to pass on such increased costs to our customers. Any increase in those fees or costs will have an adverse impact on our ability to offer attractive pricing to customers, which could negatively impact our sales and profitability, or increase the cost to us upon the sale of our aggregated customer 39 loans. Any such outcome could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
A substantial part of our revenue is derived from the recurring monthly revenue due from customers under alarm monitoring and other service contracts and we are subject to credit risk and other risks associated with our customers, dealers, and third-party lenders.
A substantial part of our revenue is derived from the recurring monthly revenue due from customers under alarm monitoring and other service contracts. Therefore, we are dependent on our customers’ ability and willingness to pay amounts due under alarm monitoring or other service contracts in a timely manner. Although customers are contractually obligated to pay amounts due under an alarm monitoring or other service contract and are generally contractually obligated to pay cancellation fees if they prematurely cancel the contract during its initial term (typically between two and five years), customers’ payment obligations are unsecured, which could impair our ability to collect any unpaid amounts from our customers. To the extent customer payment defaults under alarm monitoring and other service contracts are greater than anticipated, our business, financial condition, results of operations, and cash flows could be materially adversely affected. We have introduced and will continue to explore different commercial terms for our products and services, such as increasing or otherwise changing the amount of up-front payments, providing different financing options, such as retail installment contracts for the amount of up-front payments associated with our transactions, or offering longer or shorter contract term options. These options could increase the credit risks associated with our customers, and the introduction of, or transition to, different options could result in quarterly revenue and expense fluctuations that are significantly greater than our historic patterns. While we intend to manage such credit risk by evaluating the credit quality of customers eligible for our financing options and non-standard term lengths, our efforts to mitigate risk may not be sufficient to prevent an adverse effect on our business, financial condition, results of operations, and cash flows.
Some of these customer financing options may be supported by financing arrangements with third parties, including uncommitted receivables securitization financing agreements, which may impose or result in limitations on the products and services we offer that are financed under such arrangements. These limitations may adversely affect our relationships with customers, and may subject us to risk with respect to our ability to generate current levels of cash flow should, for example, such arrangements be terminated. In addition, rising interest rates, as we experienced during 2022 and 2023 and which we may continue to experience, could increase the financing costs of our products and services substantially. Any such result could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
We also placed a substantial reliance on third-party lenders in order to access loan products for our customers in our solar segment. In addition, any disruption in our relationship with a third party lender could have an adverse impact on certain customer relationships or result in liability to us. Certain third party lenders also have the contractual right to require us to repurchase loans if we fail to achieve certain contractual milestones with respect to customer installations. If any of our third party lenders invokes such a right, the necessary repurchases could have a material, adverse effect on our cash flow for the quarter in which they occur. Industry trends could also change, for example, by third party lenders more systematically requiring the repurchase of loans, or requiring a guarantee with respect to amounts that such lenders would otherwise require be repurchased, if we fail to achieve the relevant milestones. We cannot predict the timing or extent to which the broader industry will implement such changes or the related impact on us. Failure to maintain effective customer financing options and satisfactory relationships with third party lenders or the decision by a third party lender to require us to repurchase or guarantee loans could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Offering additional commercial terms and financing options, and transitions between such options, may introduce operational complexity, require the devotion of resources that could otherwise be deployed elsewhere, and may increase market valuation risks due to differences in the financial treatment of different offerings. Such increased offerings or transitions between different offerings or equipment ownership models could also result in customer confusion or dissatisfaction, limit or remove our ability to offer “free device” promotions or other customer satisfaction programs critical for customer acquisition and retention, and may provide competitors with the opportunity to target our existing and potential clients by offering such “free device” promotions, or other customer satisfaction programs that we may be unable to offer. Any of the foregoing could materially adversely affect our business, financial condition, results of operations, and cash flows.
Under the standard alarm monitoring contract acquisition agreements that we enter into with our dealers, if a customer terminates his or her service with us during the first thirteen months after we have acquired the alarm monitoring contract, the dealer is typically required to substitute with a compatible alarm monitoring contract or compensate us in an amount based on the original acquisition cost of the terminating alarm monitoring contract. We are subject to the risk that dealers will breach these obligations. Although we generally withhold specified amounts from the acquisition cost paid to dealers for alarm monitoring contracts (“holdback”), which may be used to satisfy or offset these and other applicable dealer obligations under the alarm monitoring contract acquisition agreements, there can be no guarantee that these amounts will be sufficient to satisfy 40 or offset the full extent of the default by a dealer of its obligations under its agreement. If the holdback proves insufficient to cover dealer obligations, we are also subject to the credit risk that the dealers may not have sufficient funds to compensate us or that any such dealer will otherwise breach its obligation to compensate us for a terminating alarm monitoring contract. To the extent defaults by dealers of the obligations under their agreements are greater than anticipated, our business, financial condition, results of operations, and cash flows could be materially adversely affected.
Goodwill and other identifiable intangible assets represent a significant portion of our total assets, and we may never realize the full value of our intangible assets.
As of December 31, 2024, we had a carrying value of goodwill and other identifiable intangible assets of approximately $9.8 billion. We review goodwill and indefinite lived intangible assets for impairment at least annually. We review long-lived assets for impairment whenever events or changes in business circumstances indicate that the carrying amount of an asset or asset group may not be fully recoverable. Impairment may result from, among other things, deterioration in performance; adverse market conditions; adverse changes in applicable laws or regulations, including changes that restrict our activities or affect the products and services we offer; challenges to the validity of certain registered intellectual property; reduced sales of certain products or services incorporating registered intellectual property; increased attrition; and a variety of other factors. For example, during 2023 and 2022, we recorded cumulative goodwill impairment charges of $712 million related to our former Solar reporting unit due to continued deterioration of industry conditions, general macroeconomic decline, underperformance of the former reporting unit’s operating results relative to expectations, and, during the third quarter of 2023, our decision to close a significant number of branches and reduce headcount. Following these goodwill impairment charges, the balance of goodwill in the former Solar reporting unit was zero. It is possible that we may never realize the full value of our intangible assets. Any future determination of impairment of goodwill or other identifiable intangible assets could have a material adverse effect on our financial condition and results of operations.
We have significant deferred tax assets, and any impairments of or valuation allowances against these deferred tax assets in the future could materially adversely affect our results of operations, financial condition, and cash flows.
We are subject to income taxes in the U.S. and Canada, and in various state, territorial, provincial, and local jurisdictions. The amount of income taxes we pay is subject to our interpretation and application of tax laws in jurisdictions in which we file. Changes in current or future laws or regulations, the imposition of new or changed tax laws or regulations, or new related interpretations by taxing authorities in the jurisdictions in which we file could materially adversely affect our business, financial condition, results of operations, and cash flows. Our future consolidated federal and state income tax liability may be significantly reduced by tax attributes such as tax credits, tax net operating loss (“NOL”), and disallowed interest carryforwards available to us under the applicable tax codes. Our ability to fully utilize these tax attributes, however, may be limited for various reasons, including whether projected future taxable income becomes insufficient to recognize the full benefit of our tax attributes prior to their expirations. If a corporation experiences an “ownership change,” Sections 382 and 383 of the Internal Revenue Code (“IRC”) provide annual limitations with respect to the ability of a corporation to utilize its tax attributes against future U.S. taxable income. In general, an ownership change may result from transactions increasing the ownership of certain stockholders in the stock of the corporation by more than 50 percentage points over a three-year testing period. Because our ability to fully utilize our tax attributes is subject to the limitations under Section 382 of the IRC, it is also possible that future changes in the direct or indirect ownership in our equity might result in additional ownership changes that may trigger the imposition of additional limitations under Section 382 of the IRC with respect to these tax attributes. In addition, audits by the U.S. Internal Revenue Service (“IRS”) as well as state, territorial, provincial, and local tax authorities could reduce our tax attributes and/or subject us to tax liabilities if tax authorities make adverse determinations with respect to our tax attributes. Any future disallowance of some or all of our tax attributes as a result of legislative change could materially adversely affect our tax obligations. Any increase in taxation or limitation of benefits could have a material adverse effect on our business, financial condition, results of operations, or cash flows. In connection with the Tax Cuts and Jobs Act of 2017 (“Tax Reform”), a new limitation under IRC Section 163(j) was imposed on the amount of interest expense allowed as a deduction in our tax returns each year. The amounts disallowed each year can be carried forward indefinitely and used in subsequent years if an excess limitation exists. We have accumulated a significant deferred tax asset related to this disallowed interest carryforward. However, there is a risk that we will not recognize the benefit of this deferred tax asset in the foreseeable future due to our annual interest expense exceeding the imposed limitation. We may need to record a valuation allowance against this deferred tax asset in the future as the deferred tax asset grows, which may have a material adverse effect on our future financial condition and results of operations. There is a risk that the interest disallowance may have a material adverse effect on our financial condition, results of operations, and cash flows.
41 Risks Related to Our Indebtedness Our substantial indebtedness limits our financial and operational flexibility and could materially adversely affect our business, financial condition, results of operations, and cash flows.
As of December 31, 2024, we had $7.8 billion face value of outstanding indebtedness, excluding finance leases, and we may increase our debt level at any time. Such substantial indebtedness negatively impacts our business because: • a significant portion of our cash flow is used to service our debt, and therefore impedes our ability to grow the business or fuel innovation; • restrictive covenants under our debt arrangements could prevent us from borrowing additional funds for working capital, capital expenditures, and debt service requirements, which could result in a default, an inability to fund our strategic initiatives, an inability to declare and pay dividends, or otherwise preclude us from undertaking actions that are in the best interests of our Company and our stockholders; • we may be required to make non-strategic divestitures to fund our debt servicing needs; • an increase in interest rates, as experienced recently or as we may experience in the future, could significantly increase the cost of our variable rate debt and make any refinancing of our current fixed rate debt significantly more costly. Although we have interest rate swap contracts that hedge certain of our interest rate exposure on variable rate debt, the majority of which mature in 2026, those hedges are themselves subject to counterparty risks and may prove to be insufficient. Moreover, any unhedged variable rate debt maturing beyond 2026 and any refinancing of current fixed rate debt exposes us to changes in market rates; • any downgrade to our credit rating may increase our cost of borrowings and any refinancing could be on terms or with conditions that limit our ability to successfully conduct business in the future; and • any inability to service or refinance our debt or acceleration of debt due could result in default which could result in all of our outstanding debt becoming due and payable, an inability to access our revolving credit facility, foreclosure against our assets, and bankruptcy or liquidation.
In 2023, we used the net proceeds of the Commercial Divestiture and cash on hand to reduce our debt by approximately $2 billion. However, we can provide no assurance that our business will generate sufficient cash flow from operations to service or repay our debt, or that we will have the ability to issue new debt, draw on our revolving credit facility or find other alternative sources of funds to satisfy our obligations. Our inability to generate sufficient cash flow to satisfy our debt obligations, or to refinance our indebtedness on commercially reasonable terms or at all, could result in a material adverse effect on our business, financial condition, results of operations, and cash flows.
On February 7, 2025, we issued an irrevocable notice of partial redemption for $500 million of the First Lien Notes due 2026, which will be redeemed on March 9, 2025. Prior to the issuance of such notice, certain lenders provided commitments that they will fund a new $600 million first lien seven-year term loan facility. The closing of this new facility, which remains subject to market and other customary conditions, is expected to occur on or around March 7, 2025. The Company intends to use proceeds of this new facility for the partial redemption of the First Lien Notes due 2026 among other general corporate purposes.
Our debt agreements contain restrictions that limit our flexibility and limit the manner in which we conduct our business and finance future operations or capital needs, which could have a material adverse effect on our business and financial condition.
Our debt agreements contain, and any future indebtedness of ours would likely contain, a number of covenants that impose significant operating and financial restrictions on us, including restrictions on our and our subsidiaries’ ability to, among other things: • incur additional debt, guarantee indebtedness, or issue certain preferred equity interests; • pay dividends on or make distributions in respect of, or repurchase or redeem, our capital stock, or make other restricted payments; • prepay, redeem, or repurchase certain debt; • make loans or certain investments; • sell certain assets; • create liens on certain assets; • consolidate, merge, sell, or otherwise dispose of all or substantially all of our assets; • enter into certain transactions with our affiliates; • alter the businesses we conduct; 42 • enter into agreements restricting our subsidiaries’ ability to pay dividends; and • designate our subsidiaries as unrestricted subsidiaries. As a result of these covenants, we will continue to be limited in the manner in which we conduct our business, and we may be unable to engage in favorable business activities or finance future operations or capital needs, which could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
We have pledged a significant portion of our assets as collateral under our debt agreements. If any of the holders of our indebtedness accelerate the repayment of such indebtedness upon an event of default, there can be no assurance that we will have sufficient assets to repay our indebtedness.
A failure to comply with the covenants under our debt agreements or any future indebtedness could result in an event of default, which, if not cured or waived, could have a material adverse effect on our business, financial condition, results of operations, and cash flows. In the event of any such default, the lenders thereunder: • will not be required to lend any additional amounts to us; • could elect to declare all borrowings outstanding, together with accrued and unpaid interest and fees, to be immediately due and payable; or • could require us to apply all of our available cash to repay these borrowings. Such actions by the lenders could cause cross-defaults under our other indebtedness. If we are unable to repay those amounts, our secured lenders could proceed against the collateral granted to them to secure that indebtedness. If any of our outstanding indebtedness were to be accelerated, there can be no assurance that our assets would be sufficient to repay such indebtedness in full.
Risks Related to the Ownership of Our Common Stock Our stock price may fluctuate significantly.
The market price of our Common Stock could vary significantly as a result of a number of factors, some of which are beyond our control. In the event of a drop in the market price of our common stock, you could lose a substantial part or all of your investment in our Common Stock. Among others, the following factors could affect our stock price: • our business performance and prospects, including the success of our strategic relationship with State Farm and our partnership with Google; • sales of our Common Stock, or the perception that such sales may occur, by us or by our stockholders, including Apollo (which has already and may continue to sell shares in registered offerings pursuant to demand registration requests), State Farm, or Google; • quarterly variations in the rates of growth of our operating and financial indicators, such as net income (loss) per share, net income (loss) and total revenue; • any failure to achieve near or long term goals we have publicly disclosed for our operating and financial performance; and • the realization of any risks described under this “Risk Factors” section, or other risks that may materialize in the future.
The stock markets in general have experienced extreme volatility that has often been unrelated to the operating performance of particular companies. These broad market fluctuations may adversely affect the trading price of our Common Stock. Securities class action litigation has often been instituted against companies following periods of volatility in the overall market and in the market price of a company’s securities. Such litigation, if instituted against us, could result in very substantial costs, divert our management’s attention and resources, and have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Apollo continues to exert significant influence over us, and its interests may conflict with our interests and the interests of other stockholders, and could negatively impact our ability to enter into corporate transactions.
While we are no longer a “controlled company,” Apollo continues to be able to exert significant influence over us and as of December 31, 2024, had the right to, among other things, nominate 50% of our directors pursuant to the Amended and Restated Stockholders Agreement, dated December 14, 2018, (the “Stockholders Agreement”) between the Company and Ultimate Parent and the Co-Investors (as defined therein). The interests of Apollo and its affiliates, including funds affiliated with Apollo, could conflict with or differ from our interests or the interests of our other stockholders. For example, the concentration 43 of ownership held by funds affiliated with Apollo could (i) delay, defer, or prevent a change in control of our company, (ii) impede a merger, takeover, or other business combination which may otherwise be favorable for us or that another stockholder may otherwise view favorably or (iii) cause us to enter into transactions or agreements that are not in the best interests of all stockholders. Additionally, Apollo and its affiliates are in the business of making investments in companies and may, from time to time, acquire and hold interests in or provide advice to businesses that compete directly or indirectly with us, or are suppliers or customers of ours. Apollo and its affiliates may also pursue acquisition opportunities that may be complementary to our business, and as a result, those acquisition opportunities may not be available to us. Any such investment may increase the potential for the conflicts of interest discussed in this risk factor. So long as funds affiliated with Apollo continue to directly or indirectly own a significant amount of our equity, even if such amount is less than 50%, Apollo and its affiliates will continue to be able to substantially influence or effectively control our ability to enter into corporate transactions.
In addition, we are party to the Stockholders Agreement with Ultimate Parent and the Co-Investors. The Stockholders Agreement specifies that we will not take certain significant actions without the prior consent of Ultimate Parent, including, among other things, hiring or terminating any executive officer of our company, designating any new executive officer of our company, entering into certain merger, consolidation or other “change of control” transactions or changing the size of our Board of Directors. The Stockholders Agreement also specifies that Ultimate Parent has the right to nominate individuals for election to our Board of Directors and that we are, to the fullest extent permitted by applicable law, required to nominate and recommend that each such individual be elected as a director, and the right to designate a member to each committee of our Board of Directors. Relatedly, our amended and restated Bylaws (the “Bylaws”) provide that Ultimate Parent has the right, subject to certain conditions, to have its representatives appointed to serve on committees of our Board of Directors.
If we fail to establish and achieve the objectives of our sustainability program, or if we fail to report on such sustainability matters, consistent with investor, customer, employee, or other stakeholder expectations, and in compliance with legal and regulatory requirements, we may not be viewed as an attractive investment, service provider, workplace, or business, which could have a negative effect on our company.
Investors are placing a greater emphasis on non-financial factors, including sustainability matters, when evaluating investment opportunities. We have published an annual corporate ESG Report (the “ESG Report”), including a Sustainable Accounting Standards Board (“SASB”) Index report, each year since 2022. Additionally, since 2022 we have completed the annual Corporate Questionnaire provided by the Carbon Disclosure Project, a non-profit organization focused on sustainability reporting. In our ESG Report and other disclosures, including in various filings with the SEC, we detail our sustainability progress. Sustainability initiatives and goals may be difficult and expensive to implement and may not be advanced at a sufficient pace. If we are unable to provide sufficient and accurate disclosures about our sustainability practices, or if we fail to establish and achieve the objectives of our sustainability program, which could include targets or commitments, consistent with investor, customer, employee, or other stakeholder expectations, we may not be viewed as an attractive investment, service provider, workplace, or business and we may be exposed to potential liability or litigation, which could have a material adverse effect on our business, financial condition, results of operations, and cash flows. In addition, there exists certain negative sentiment among some individuals and government institutions toward certain sustainability practices and initiatives. Given the dynamic nature of sustainability standards, expectations and regulations, which may change over time, we may from time to time need to update or otherwise revise our current practices, goals and initiatives, including in response to legislative, regulatory, or legal developments. As we continue to establish our sustainability-related initiatives, we could face a negative response or legislation that impedes our activities or reflects poorly upon the Company, any of which could have a material adverse effect on our business, financial condition, results of operations, and cash flows.
Our organizational documents may impede or discourage a takeover, which could deprive our investors of the opportunity to receive a premium on their shares.
Provisions of our amended and restated certificate of incorporation (as amended, our “Certificate of Incorporation”) and Bylaws may make it more difficult for, or prevent a third-party from, acquiring control of us without the approval of our Board of Directors. These provisions include: • providing that our Board of Directors will be divided into three classes, with each class of directors serving staggered three-year terms; • providing for the removal of directors only for cause and only upon the affirmative vote of the holders of at least 66 2/3% in voting power of all the then-outstanding shares of stock of the Company entitled to vote thereon, voting together as a single class, if less than 50.1% of our outstanding Common Stock is beneficially owned by funds affiliated with Apollo; • empowering only the Board of Directors to fill any vacancy on our Board of Directors (other than in respect of a director designated by Apollo or other investors in our indirect parent entities), whether such vacancy occurs as a result of an increase in the number of directors or otherwise; 44 • authorizing the issuance of “blank check” preferred stock with all terms established by the Board of Directors in its sole discretion without any need for action by stockholders, which could delay or prevent a change in control of the Company; • prohibiting stockholders from acting by written consent if less than 50.1% of our outstanding Common Stock is beneficially owned by funds affiliated with Apollo; • to the extent permitted by law, prohibiting stockholders from calling a special meeting of stockholders if less than 50.1% of our outstanding Common Stock is beneficially owned by funds affiliated with Apollo; and • establishing advance notice requirements for nominations for election to our Board of Directors or for proposing matters that can be acted on by stockholders at stockholder meetings.
Additionally, Section 203 of the Delaware General Corporation Law (“DGCL”) prohibits a publicly held Delaware corporation from engaging in a business combination with an interested stockholder, unless the business combination is approved in accordance with the statute. An interested stockholder includes a person, individually or together with any other interested stockholder, who within the last three years has owned 15% or more of our outstanding voting stock, or who is our affiliate or associate and owned 15% or more of our outstanding voting stock at any time within the three years immediately prior to the date on which it is sought to be determined whether such person is an interested stockholder. Our Certificate of Incorporation includes a provision that, with limited exceptions, restricts us from engaging in any business combination with an interested stockholder for three years following the date that person becomes an interested stockholder. Such restrictions do not apply to any business combination between Apollo or their direct and indirect transferees (as these terms are defined in our Certificate of Incorporation) and any affiliate thereof, on the one hand, and us, on the other.
Our Certificate of Incorporation provides for exclusive forum provisions which could limit our stockholders’ ability to obtain a favorable judicial forum for disputes.
Our Certificate of Incorporation provides that, unless we consent in writing to the selection of an alternative forum, the Chancery Court of the State of Delaware shall be, to the fullest extent permitted by law, the sole and exclusive forum for (a) any derivative action or proceeding brought on our behalf; (b) any action asserting a claim of breach of a fiduciary duty owed by any of our directors, officers, or stockholders; (c) any action asserting a claim arising pursuant to any provision of the DGCL or of our Certificate of Incorporation or our Bylaws; or (d) any action asserting a claim against us or any of our directors or officers governed by the internal affairs doctrine. In addition, our Certificate of Incorporation also provides that, unless we consent in writing to the selection of an alternative forum, the federal district courts of the United States of America shall be the exclusive forum for the resolution of any complaint asserting a cause of action arising under the Securities Act. The exclusive forum provision in our Certificate of Incorporation does not apply to suits brought to enforce any duty or liability created by the Exchange Act or any other claim for which the federal courts have exclusive jurisdiction. To the extent that any such claims may be based upon federal law claims, Section 27 of the Exchange Act creates federal jurisdiction over all suits brought to enforce any duty or liability created by the Exchange Act or the rules and regulations thereunder. Any person or entity purchasing or otherwise acquiring any interest in shares of our capital stock will be deemed to have notice of and, to the fullest extent permitted by law, to have consented to the provisions described in this paragraph. These provisions may limit a stockholders’ ability to bring a claim in a judicial forum of their choosing, which may discourage lawsuits against us and our directors, officers, and other employees.
Our Certificate of Incorporation contains a provision renouncing our interest and expectancy in certain corporate opportunities, which could have a material adverse effect if attractive corporate opportunities are allocated by Apollo to itself or its portfolio companies, funds, or other affiliates instead of to us.
Under the Stockholders Agreement, funds affiliated with or managed by Apollo received certain rights, including the right to nominate a specified percentage of the directors to serve on our Board of Directors (the “Apollo Designees”) based on the percentage of our outstanding Common Stock beneficially owned by Apollo.
Under our Certificate of Incorporation, none of Apollo nor any of its portfolio companies, funds, or other affiliates, or any of its officers, directors, agents, stockholders, members, or partners have any duty to refrain from engaging, directly or indirectly, in the same business activities, similar business activities, or lines of business in which we operate. In addition, our Certificate of Incorporation provides that, to the fullest extent permitted by law, no officer or director of ours who is also an officer, director, employee, managing director, or other affiliate of Apollo will be liable to us or our stockholders for breach of any fiduciary duty by reason of the fact that any such individual directs a corporate opportunity to Apollo instead of us, or does not communicate information regarding a corporate opportunity to us that the officer, director, employee, managing director, or other affiliate has directed to Apollo. As of the date of this Annual Report, this provision of our Certificate of Incorporation relates only to the Apollo Designees. There are currently fifteen directors of our Company, six of whom are Apollo Designees. These potential conflicts of interest could have a material adverse effect on our business, financial condition, results of 45 operations, cash flows, or prospects if attractive corporate opportunities are allocated by Apollo to itself or its respective portfolio companies, funds, or other affiliates instead of to us.
We may issue preferred securities, the terms of which could adversely affect the voting power or value of our Common Stock.
Our Certificate of Incorporation authorizes us to issue, without the approval of our stockholders, one or more classes or series of preferred securities having such powers; designations; preferences; limitations; and relative, participating, optional, or other rights, including preferences over our Common Stock with respect to dividends and other distributions, as our Board of Directors may determine. The terms of one or more classes or series of preferred securities could adversely affect the voting power or value of our Common Stock. For example, we might grant holders of preferred securities the right to elect some number of our directors in all events or on the happening of specified events or the right to veto specified transactions. Similarly, the repurchase or redemption rights or liquidation preferences we might assign to holders of preferred securities could affect the residual value of our Common Stock.
We cannot guarantee that our share repurchase program will be fully consummated or that it will enhance long-term shareholder value. Share repurchases and dividend payments, including recent changes in the amount of our dividend, could also increase the volatility of the trading price of our stock and will diminish our cash reserves.
On February 20, 2025, our Board of Directors authorized a share repurchase plan (the “2025 Share Repurchase Plan”), pursuant to which the Company is authorized to repurchase, through April 30, 2026, up to a maximum aggregate amount of $500 million of shares of the Company's Common Stock. We cannot guarantee that the 2025 Share Repurchase Plan will be fully consummated. The 2025 Share Repurchase Plan allows the Company to purchase shares of its Common Stock from time to time in one or more open market or privately negotiated transactions, including pursuant to Rule 10b5-1 or Rule 10b-18 of the Exchange Act or pursuant to one or more accelerated share repurchase agreements, subject to certain requirements and other factors. The Company is not obligated to repurchase any of its shares of common stock, and the timing and amount of any repurchases will depend on legal requirements, market conditions, stock price, the availability of certain safe harbors provided under the Exchange Act, alternative uses of capital, and other factors. Further, our share repurchases could affect our share trading prices, increase their volatility, reduce our cash reserves and may be suspended or terminated at any time, which may result in a decrease in the trading price of our stock.
ITEM 1B. UNRESOLVED STAFF COMMENTS.
None.
ITEM 1C. CYBERSECURITY.
We view cybersecurity as the prevention and timely detection and correction of any unauthorized occurrence or series of related unauthorized occurrences that are on or conducted through our information systems and that jeopardizes the confidentiality, integrity, or availability of our systems or any information residing therein. We believe that the safety, security, and privacy of our customers and employees are fundamental to the services we provide. Our cybersecurity policies guide us as we strive to continuously enhance methods, best practices, and technologies to better monitor and protect customer data and inform and enable customers to make choices about their data privacy. We carefully consider data privacy when developing our own products and when incorporating products provided by our business partners.
Risk Management and Strategy We identify, assess, and manage cybersecurity risk as part of our company-wide enterprise risk management program. Our Chief Information Security Officer (“CISO”), Tim Rains, has more than 30 years of experience as an IT professional, with over 20 of those years spent in cybersecurity roles. Mr. Rains has held senior cybersecurity advisor roles at both Amazon Web Services and Microsoft. Mr. Rains has experience across multiple cybersecurity disciplines including vulnerability management, incident response, crisis communications, threat intelligence, cybersecurity architecture and operations, governance, risk, and compliance. Mr. Rains is designated as a Certified Information Systems Security Professional and is responsible for developing and implementing plans and strategies to mitigate cybersecurity risks. Our CISO also leads our cybersecurity risk assessment, which includes security posture scoring, vulnerability assessments, process maturity, and tooling coverage. We log cybersecurity risks into our cybersecurity risk register and track such risks for treatment. Management then discusses these cybersecurity risks for resolution planning and escalation. We leverage recognized cybersecurity frameworks to drive strategic direction and maturity improvement and engage third-party security experts as needed for risk assessments, risk mitigation actions, vulnerability identification, and program enhancements, as appropriate.
46 As part of this process, we use the following tools and procedures: • utilizing “SecurityScorecard” (a third-party information security company that rates cybersecurity postures of corporate entities for the purposes of third-party management and information technology risk management), which provides an independent external enterprise view of our security posture with a focus on public-facing systems; • assessing, regularly developing, and executing on our preventative and detective controls, which we seek to align with current standards and best practices, including the incorporation of recommendations published by the National Institute of Standards and Technology in its cybersecurity framework, such as an annual audit of these internal controls; • performing attack and breach simulations; and • working with our cybersecurity vendors to adopt tooling and processes to provide high levels of protection.
Governance Cybersecurity Management and Board Oversight Our Board of Directors, through its Audit Committee, has primary responsibility for overseeing cybersecurity risk management and receives updates on the status of our cybersecurity program from our CISO. These updates are provided at least once per year, and often multiple times per year, in a special Audit Committee session and includes reports on our security posture and SecurityScorecard assessment (rating and benchmarking), incident response, and vulnerability management. The Audit Committee reviews and discusses with management our cybersecurity threats, vulnerabilities, defenses, and planned responses, including updates to our cybersecurity incident response plan (“IRP”), which has been approved by the Audit Committee. Additionally, the Audit Committee receives and discusses reports from management with the purpose of identifying threats and vulnerabilities, and it monitors the effectiveness and progress of the actions and initiatives undertaken to mitigate such threats. Our cybersecurity program team is led by our CISO (who ultimately reports to the Chief Operating Officer). The cybersecurity leadership team (“CSLT”), which is chaired by our Chief Operating Officer and includes our Chief Financial Officer, Chief Legal Officer, Chief Information Officer, CISO, and Chief Privacy Officer, among others, collaborates with enterprise risk professionals and is supported by an established Information Security (“InfoSec”) function responsible for certain aspects of maintaining and monitoring our cybersecurity infrastructure. In addition, our Chief Privacy Officer, who reports to our Chief Legal Officer, manages processes and protections around our sensitive data and facilitates compliance with applicable data protection laws, rules, and regulations.
Our Chief Privacy Officer has over 20 years of experience overseeing corporate data privacy and intellectual property policies and procedures. To maintain high levels of awareness and aptitude, all of our employees are required to complete annual trainings regarding current security risks and our InfoSec and privacy policies. Additional education and training are also required for specific groups based on their roles and access within the organization.
Incident Response Plan and Cybersecurity Incident Materiality Assessment Policy We seek to align with industry-standard cybersecurity frameworks designed to protect our information systems and both Company and customer data from unintentional disclosure, cybersecurity incidents, events, and other threats of varying severity levels. As part of our alignment efforts with these frameworks, we maintain the IRP, which outlines the actions to be taken after identifying an incident that affects or could potentially affect our information systems and the people responsible for managing and overseeing those actions.
Under the IRP, cybersecurity incidents are generally addressed by our Cybersecurity Incident Response Team (“CSIRT”), consisting of our CISO, deputy CISO, director of security operations, senior manager of incident response, and members of the security operations team. Incidents of higher severity are elevated to the CSLT. Under the IRP, if an incident requires the involvement of the CSLT, the CSIRT will regularly update the CSLT on the status of the incident response process. Members of the CSLT primarily, the Chief Operating Officer and CISO, will be responsible for updating the Chief Executive Officer, Audit Committee, and the lead independent director of our Board of Directors. Members of the CSIRT and CSLT, along with the Chief Executive Officer, Audit Committee, and the lead independent director of our Board of Directors regularly participate in cybersecurity incident tabletop exercises and event simulations.
If a materiality assessment is required, an assessment committee consisting of the Chief Financial Officer, Chief Operating Officer, Chief Legal Officer, CISO, and Chief Accounting Officer (and/or their designee) (collectively, the “Assessment Committee”) will consult with the CSIRT and CSLT, as appropriate. The Assessment Committee is responsible for assessing, without unreasonable delay, the materiality of cybersecurity incidents reported to it, determining materiality and any necessary disclosures, and informing our disclosure committee of such determinations. In assessing materiality, the Assessment 47 Committee will consult with internal and external advisors, as appropriate, and evaluate quantitative and qualitative factors to assess the impact and/or reasonably likely impacts of the cybersecurity incident.
For additional information regarding how cybersecurity threats have materially affected or are reasonably likely to materially affect our business strategy, results of operations or financial condition, see “Risk Factors—”: • “—Delays, costs, and disruptions that result from upgrading, integrating, and maintaining the security of our information and technology networks and systems could materially adversely affect us,” • “—If we do not effectively implement our plans to migrate our technology infrastructure to the cloud, we could experience significant disruptions in our operations, which could have a material adverse effect on our results of operations and financial condition,” • “—Cybersecurity attacks or threats or other unauthorized access or attempts to access to our systems, or those of third parties, have in the past, and may in the future, compromise the security of our systems and otherwise disrupt our normal operations, which could have a material adverse effect on our reputation, business, financial condition, results of operations and cash flows,” and • “—Our independent, third-party authorized dealers may not be able to mitigate certain risks such as information technology and data security breaches, product liability, errors and omissions, and compliance with applicable laws and regulations.” ITEM 2. PROPERTIES.
We primarily lease our properties through our main operating entity, ADT LLC. As of December 31, 2024, we owned or leased approximately 140 sales and service offices, that are supported by our regional distribution centers, as well as our nationwide network of multi-use sales, customer, and field support locations housing our six UL-listed monitoring centers.
As of December 31, 2024, we leased 1.8 million square feet of space in the U.S. primarily under long-term operating leases with third parties, including 100 thousand square feet for our corporate headquarters in Boca Raton, Florida, which we renewed during 2023 that extended the lease through 2034. We also own 350 thousand square feet of space in the U.S.; however, approximately 100 thousand square feet of that space is currently listed for sale.
We regularly evaluate the suitability, adequacy, productive capacity, and utilization of our existing principal physical properties. A portion of our employees continue to work from home under both permanent and temporary arrangements. Other initiatives, such as our Remote Assistance Program, may also impact our physical property needs in the future as we are able to service more of our customers remotely. We continue to believe our properties are adequately maintained and are suitable for our business as presently conducted.
ITEM 3. LEGAL PROCEEDINGS.
We are subject to various claims and lawsuits in the ordinary course of business, which include commercial general liability claims, automobile liability claims, contractual disputes, worker’s compensation claims, labor law and employment claims, claims related to alleged alarm system failures, claims that the Company infringed on the intellectual property of others, and consumer and employment class actions. We are also subject to regulatory and governmental examinations, information requests and subpoenas, inquiries, investigations, and threatened legal actions and proceedings. In connection with such formal and informal inquiries, we receive numerous requests, subpoenas, and orders for documents, testimony, and information in connection with various aspects of our activities.
Additional information in response to this Item is included in Note 13 “Commitments and Contingencies” in the Notes to Consolidated Financial Statements and is incorporated by reference into Part I of this Annual Report. Our consolidated financial statements and the accompanying Notes to Consolidated Financial Statements are filed as part of this Annual Report under Item 15 “Exhibit and Financial Statement Schedules” and are set forth beginning on page F-1 immediately following the signature pages of this Annual Report.
ITEM 4. MINE SAFETY DISCLOSURES.
Not Applicable.
48 PART II ITEM 5. MARKET FOR REGISTRANT’S COMMON EQUITY, RELATED STOCKHOLDER MATTERS, AND ISSUER PURCHASES OF EQUITY SECURITIES.
Market Information and Stockholders of Record We have two classes of common stock outstanding, Common Stock and Class B Common Stock.
Common Stock - Our Common Stock is listed on the NYSE under the symbol “ADT.” As of February 20, 2025, the number of stockholders of record of Common Stock was 258, which does not include the number of stockholders who hold our Common Stock through banks, brokers, and other financial institutions.
Class B Common Stock - There is no established public trading market for shares of Class B Common Stock; and Google is, and has been, the only stockholder of record since the stock’s issuance in 2020.
Stock Performance Graph The following graph compares the cumulative total stockholder return, calculated on a dividend-reinvested basis, assuming that $100 was invested on the last trading day before the beginning of the fifth preceding fiscal year, in each of the following: (i) our Common Stock; (ii) the Standard & Poor’s (“S&P”) 500 Index; and (iii) the S&P North America Consumer Services Index, a peer group. The graph is not, and is not intended to be, indicative of future performance of our Common Stock.
Date ADT Inc.
S&P 500 Index S&P North America Consumer Services Index 12/31/2020 $101 $118 $107 12/31/2021 $110 $152 $122 12/31/2022 $121 $125 $101 12/31/2023 $93 $158 $133 12/31/2024 $97 $197 $155 The information contained in this section shall not be deemed “soliciting material” or to be “filed” with the SEC or incorporated by reference in future filings with the SEC, or otherwise subject to the liabilities under Section 18 of the Exchange Act, except to the extent we specifically incorporate it by reference into such filing.
49 Recent Sales of Unregistered Equity Securities There were no sales of unregistered equity securities during the three months ended December 31, 2024.
Use of Proceeds from Registered Equity Securities We did not receive any proceeds from sales of registered equity securities during the three months ended December 31, 2024.
Issuer Purchases of Equity Securities The following table presents repurchases of shares of the Company’s Common Stock during the three months ended December 31, 2024 (in thousands, except per share data): Period Total Number of Shares Purchased Average Price Paid Per Share Total Number of Shares Purchased as Part of Publicly Announced Plans or Programs(1) Maximum Dollar Value of Shares that May Yet be Purchased Under the Plans or Programs October 1, 2024 - October 31, 2024 21,000 $ 7.01 21,000 $ 109,444 November 1, 2024 - November 30, 2024 — $ — — $ 109,444 December 1, 2024 - December 31, 2024 — $ — — $ 109,444 Total 21,000 $ — 21,000 $ 109,444 __ (1) On October 4, 2024, the Company repurchased and retired 5 million shares of Common Stock at a price per share of $6.40 for an aggregate purchase price of $32 million. On October 30th, 2024, the Company repurchased and retired 16 million shares of Common Stock at a price per share of $7.20 for an aggregate purchase price of $115 million.
Additionally, in December 2024, the Company entered into an agreement to repurchase 15 million shares of Common Stock at a price per share of $6.95 for a total of $104 million. The transaction settled in January 2025, and the Company retired the shares.
Share Repurchase Plan On January 24, 2024, our Board of Directors announced a share repurchase plan (the “2024 Share Repurchase Plan”), pursuant to which the Company was authorized to repurchase, through late January 2025, up to a maximum aggregate amount of $350 million of shares of the Company's Common Stock under this plan. The 2024 Share Repurchase Plan expired in January 2025.
In February 2025, our Board of Directors announced the 2025 Share Repurchase Plan, pursuant to which the Company is authorized to repurchase, through April 30, 2026, up to a maximum aggregate amount of $500 million of shares of Common Stock. The 2025 Share Repurchase Plan allows the Company to purchase Common Stock from time to time in one or more open market or privately negotiated transactions, including pursuant to Rule 10b5-1 or Rule 10b-18 of the Exchange Act, or pursuant to one or more accelerated share repurchase agreements, subject to certain requirements and other factors.
ITEM 6. RESERVED.
ITEM 7. MANAGEMENT’S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS OF OPERATIONS.
50 Table of Contents • Introduction • Business and Basis of Presentation • Factors Affecting Operating Results • Key Performance Indicators • Results of Operations • Non-GAAP Measures • Liquidity and Capital Resources • Critical Accounting Estimates • Accounting Pronouncements INTRODUCTION The following discussion and analysis should be read in conjunction with our consolidated financial statements and the related notes thereto included elsewhere in this Annual Report. This section is intended to (i) provide material information relevant to the assessment of our results of operations and cash flows; (ii) enhance the understanding of our financial condition, changes in financial condition, and results of operations; and (iii) discuss material events and uncertainties known to management that are reasonably likely to cause reported financial information not to be necessarily indicative of future performance or of future financial condition.
Included below are year-over-year comparisons between 2024 and 2023. The classification of the Solar Business as a discontinued operation during 2024 did not materially change the reported disclosures in the 2023 Annual Report with regard to year-over-year comparisons between 2023 and 2022, except with regard to income tax benefit (expense), as such discussions generally included analysis specific to activities within the Solar segment. For information on year-over-year comparisons between 2023 and 2022, refer to Item 7 “Management’s Discussion and Analysis of Financial Condition and Results of Operations” in the 2023 Annual Report, which was filed with the SEC on February 28, 2024.
The following discussion and analysis contains forward-looking statements about our business, operations, and financial performance based on current plans and estimates involving risks, uncertainties, and assumptions, which could differ materially from actual results. Factors that could cause such differences are discussed in the sections of this Annual Report titled Item 1A “Risk Factors” and “Cautionary Statements Regarding Forward-Looking Statements.” Unless otherwise noted, the discussions below relate to our continuing operations.
BUSINESS AND BASIS OF PRESENTATION ADT is a leading provider of security, interactive, and smart home solutions serving residential and small business customers in the U.S. Our mission is to empower people to protect and connect what matters most with safe, smart, and sustainable solutions, delivered through innovative offerings, unrivaled safety, and a premium experience because we believe that everyone deserves to feel safe.
As discussed below, on October 2, 2023, we completed the divestiture of our Commercial Business, and as of June 30, 2024, substantially all operations of the Solar Business had ceased.
All financial information presented in this section has been prepared in U.S. dollars and in accordance with generally accepted accounting principles in the United States of America (“GAAP”), excluding our Non-GAAP measures, and includes the accounts of ADT Inc. and its subsidiaries. All intercompany transactions have been eliminated. As a result of the Commercial Divestiture and ADT Solar Exit, unless otherwise noted, we report current and historical financial and operating information for our one remaining segment.
For a more detailed discussion of our business and basis of presentation, refer to Item 1 “Business” and Note 1 “Description of Business and Summary of Significant Accounting Policies” in the Notes to Consolidated Financial Statements in Item 15 “Exhibit and Financial Statement Schedules.” FACTORS AFFECTING OPERATING RESULTS The factors described herein could have a material adverse effect on our business, financial condition, results of operations, cash flows, and/or key performance indicators.
51 As of December 31, 2024, we served approximately 6.4 million security monitoring service subscribers. Generally, a significant upfront investment is required to acquire new subscribers, that in turn provide ongoing and predictable recurring revenue generated from our monitoring services and other subscriber-based offerings. Although the economics of an installation may vary depending on the customer type, acquisition channel, and product offering, we generally achieve revenue break-even in approximately two years.
Our results are impacted by the mix of transactions under a Company-owned equipment model versus a customer-owned equipment model (referred to as outright sales), as there are different accounting treatments applicable to each model, as well as the mix, price, and type of offerings sold (see Note 2 “Revenue and Receivables”). The majority of professional installation transactions take place under a Company-owned model, however, beginning in the second quarter of 2024, a growing number of our direct channel new customer adds are outright sales in connection with the national launch of our new ADT+ platform.
As we continue to build our partnership with Google, introduce new or enhance current offerings, and refine our go-to-market approach, we expect to continue to see a shift toward an increasing proportion of outright sales transactions, which will impact results in future periods when those changes occur. In addition, self set-up solutions typically earn lower contractual fees than our professional installation solutions as a result of differences in pricing, offer tactics, and level of services in each channel. However, we believe self set-up customers will allow us to grow our subscriber base through access to the fast-growing DIY market.
The prices we are able to charge for our products and services are impacted by the type and complexity of the offerings that we provide, quality of our products and services; the introduction of additional features and offerings that increase value to the customer; and the microeconomic and macroeconomic environments in which we operate.
Demand for our offerings may be impacted by the (i) overall economic conditions in the geographies in which we operate; (ii) type, price and quality of our offerings compared to those of our competitors; (iii) changes in competition such as from the acquisition, disposition, or exiting of similar businesses by us or our competitors; (iv) overall state of the housing market; (v) perceived threat of crime; (vi) occurrence of significant life events such as the birth of a child or opening of a new business; and (vii) availability of financial incentives provided by insurance carriers.
New customer additions and customer attrition have a direct impact on our financial results, including revenue, operating income, and cash flows. A portion of our recurring customer base can be expected to cancel services each year as customers may choose to terminate or not to renew their contracts for a variety of reasons including, but not limited to, relocation, loss to competition, cost, or service issues. Relocations are sensitive to changes in the residential housing market, and fewer relocations generally lead to improvements in gross customer revenue attrition but fewer new customer additions. Additionally, non-payment disconnects generally increase in a weaker macroeconomic environment. We may experience fluctuations in these or other trends in the future as changes in the general macroeconomic environment or housing market develop.
We have made significant progress toward increasing the variety of our offerings to accommodate increased consumer interest in automated security and other mobile technology applications due to advancements in technology, younger generations of consumers, and shifts to de-urbanization. Advances in technology are also helping us to improve our offerings and reduce certain costs. For example, our Remote Assistance Program provides our customers the ability to troubleshoot and resolve certain service issues, as well as other functions, through a live video stream with our skilled technicians. This provides customers with more options for receiving certain services that best fit their lifestyles while reducing our costs and eliminating thousands of vehicle trips.
We may experience an increase in costs associated with factors, including but not limited to (i) offering a wider variety of products and services; (ii) providing a greater mix of interactive and smart home solutions; (iii) replacing or upgrading certain system components due to technological advancements, cybersecurity upgrades, or otherwise; (iv) supply chain disruptions; (v) inflationary pressures on costs such as materials, labor, and fuel; and (vi) other changes in prices, interest rates, or terms from our suppliers or vendors, or third party lenders. We aim to continuously evaluate and respond to changes in the above to drive efficiencies and meet customer demands.
As part of our response to changes or pressures in the current macroeconomic environment, we have been evaluating, and continue to evaluate, cost-saving opportunities such as reducing headcount or our physical facilities footprint when appropriate, and reducing non-essential spend. While we have experienced some increase in costs as a result of inflation, we have, for the most part, been able to offset the rising costs through price increases to our customers, as well as cost-saving opportunities.
In addition, hurricanes, wildfires, and other natural disasters impacting certain areas in which we operate may result in service, sales, and installation disruptions to certain of our customers. As such, we evaluate the financial and business impacts these events have or may have in the future. During 2024, we did not experience any material losses related to natural disasters. However, we are currently evaluating any material impacts as a result of the California fires that started in January 2025.
52 Divestitures Unless otherwise noted, our results of operations discussed below relate to continuing operations. Refer to Note 4 “Divestitures” in the Notes to Consolidated Financial Statements in Item 15 “Exhibit and Financial Statement Schedules” for further information regarding the ADT Solar Exit and Commercial Divestiture.
ADT Solar Exit In January 2024, after a strategic review of the business and continued macroeconomic and industry pressures, our Board of Directors approved a plan to fully exit the Solar Business. As of June 30, 2024, substantially all operations of the Solar Business had ceased.
The results of operations and financial position of the Solar Business are classified as discontinued operations in the Company’s Consolidated Statements of Operations and Consolidated Balance Sheets, respectively, for all periods presented. Additionally, the cash flows and comprehensive income (loss) of the Solar Business have not been segregated and are included in the Consolidated Statements of Cash Flows and Consolidated Statements of Comprehensive Income (Loss), respectively.
During the year ended December 31, 2024, we incurred aggregate exit charges of $88 million, which have been recognized within income (loss) from discontinued operations, net of tax, related to (i) $33 million associated with the write-down and disposition of inventory and asset impairments, (ii) $29 million associated with the disposition of the existing installation pipeline, (iii) $13 million associated with employee separation costs, and (iv) $12 million associated with contract termination and other charges.
Additionally, during the year ended December 31, 2024, we paid approximately $22 million in connection with the ADT Solar Exit, primarily related to employee separation and other restructuring costs.
We do not expect the remaining estimated charges and cash expenditures to be material; however, various factors including unknown or unforeseen costs may cause additional charges or cash expenditures to be incurred.
Commercial Divestiture In October 2023, we divested the Commercial Business and received net proceeds of approximately $1,585 million. We recognized a gain on the sale of approximately $630 million, which is reflected in income (loss) from discontinued operations, net of tax in 2023. During 2024, we paid $21 million related to the settlement of post-closing adjustments.
As applicable, the results of the Commercial Business are reported as a discontinued operation for all periods presented. Additionally, the cash flows and comprehensive income (loss) of the Commercial Business have not been segregated and are included in the Consolidated Statements of Cash Flows and Consolidated Statements of Comprehensive Income (Loss), respectively, for all periods presented.
During 2024, we recorded income associated with the Commercial TSA of $40 million, which is recognized in other income (expense). We do not expect any material income under the Commercial TSA in future periods.
Tax Matters During 2023, we utilized a significant portion of our net operating losses (“NOLs”) to offset the gain generated from the sale of the Commercial Business. In 2024, we utilized the remaining NOLs and anticipate becoming a federal cash taxpayer in 2025. However, the amounts and timing of payments are uncertain.
Federal Tax Legislation Certain changes to U.S. federal tax law included in the Tax Cuts and Jobs Act of 2017 had a delayed effective date until 2022. Under IRC Section 163(j), the limitation on net business interest expense deductions are no longer increased by deductions for depreciation, amortization, or depletion. Under IRC Section 174, specified research and experimentation expenditures are now capitalized and amortized. These items have resulted in increased taxable income and an acceleration of our net operating loss utilization.
Potential for Future Valuation Allowance The valuation allowance for deferred tax assets relates to the uncertainty of the utilization of certain U.S. federal and state deferred tax assets. In evaluating the Company’s ability to recover its deferred tax assets, the Company considers all available 53 positive and negative evidence, including enacted legislation such as the updates described above, which impact our assessment of whether a valuation allowance is needed.
We believe that our deferred tax asset for disallowed interest under IRC Section 163(j) will continue to grow from its current level. There is currently significant uncertainty in the matters we consider when evaluating our valuation allowance needs. Any material change to our valuation allowance in subsequent periods could materially and adversely impact our operating results.
Other Updates Google Partnership During 2023, we launched and began the phased rollout of our proprietary ADT+ app for our self set-up line of DIY smart home security products, including integrated Google Nest offerings. During 2024, we continued the phased rollout of our ADT+ platform across the country, making our next generation hardware and technology available to customers through our proprietary app.
During 2024 and 2023, we were reimbursed $30 million and $40 million, respectively, of the Google Success Funds primarily for certain joint marketing and customer acquisition expenses we incurred.
Refer to Note 13 “Commitments and Contingencies” in the Notes to Consolidated Financial Statements.
State Farm Partnership Beginning in 2023, certain State Farm customers were able to receive ADT home security products and professional monitoring at a reduced cost as part of the partnership between ADT and State Farm. In addition, through this program, ADT has access to certain State Farm customers, which provide opportunities to reach additional markets.
As of December 31, 2024, the State Farm partnership operated in 17 states in the U.S.
During 2024 and 2023, we used approximately $14 million and $11 million, respectively, of the Opportunity Fund for project initiatives and other costs.
Refer to Note 10 “Equity” and Note 16 “Related Party Transactions” in the Notes to Consolidated Financial Statements.
KEY PERFORMANCE INDICATORS We evaluate our results using certain key performance indicators, including the operating metrics end-of-period recurring monthly revenue (“RMR”) and gross customer revenue attrition, as well as the non-GAAP measure Adjusted EBITDA. Computations of our key performance indicators may not be comparable to other similarly titled measures reported by other companies.
Certain operating metrics are approximated, as there may be variations to reported results due to certain adjustments we might make in connection with the integration over several periods of acquired companies that calculated these metrics differently or periodic reassessments and refinements in the ordinary course of business, including changes due to system conversions or historical methodology differences in legacy systems.
RMR and gross customer revenue attrition, as discussed below, have been recast for prior periods to exclude the former Commercial Business. The definition of these metrics has historically excluded activity related to the Solar Business and as such, these metrics were not impacted by the presentation of the Solar Business as a discontinued operation. Adjusted EBITDA reflects our continuing operations for all periods presented.
RMR RMR is generated by contractual recurring fees for monitoring and other recurring services provided to our customers, including contracts monitored but not owned.
We use RMR to evaluate our overall sales, installation, and retention performance. Additionally, we believe the presentation of RMR is useful to investors because it measures the volume of revenue under contract at a given point in time, which is a useful measure for forecasting future revenue performance as the majority of our revenue comes from recurring sources.
54 Gross Customer Revenue Attrition Gross customer revenue attrition is defined as RMR lost as a result of customer attrition, net of dealer charge-backs and reinstated customers, excluding contracts monitored but not owned and self-setup/DIY customers. Customer sites are considered canceled when all services are terminated. Dealer charge-backs represent customer cancellations charged back to the dealers because the customer canceled service during the charge-back period, which is generally thirteen months.
Gross customer revenue attrition is calculated on a trailing twelve-month basis, the numerator of which is the RMR lost during the period due to attrition, net of dealer charge-backs and reinstated customers, and the denominator of which is total annualized RMR based on an average of RMR under contract at the beginning of each month during the period, in each case, excluding contracts monitored but not owned and self set-up/DIY customers.
We use gross customer revenue attrition to evaluate our retention and customer satisfaction performance, as well as evaluate subscriber trends by vintage year. Additionally, we believe the presentation of gross customer revenue attrition is useful to investors as it provides a means to evaluate drivers of customer attrition and the impact of retention initiatives.
Adjusted EBITDA Adjusted EBITDA is a non-GAAP measure. Our definition of Adjusted EBITDA, a reconciliation of Adjusted EBITDA to income (loss) from continuing operations (the most comparable GAAP measure), and additional information, including a description of the limitations relating to the use of Adjusted EBITDA, are provided under “—Non-GAAP Measures.” 55 RESULTS OF OPERATIONS (in thousands, except as otherwise indicated) Years Ended December 31, $ Change Results of Operations: 2024 2023 2022 2024 vs. 2023 2023 vs. 2022 Revenue: Monitoring and related services $ 4,293,477 $ 4,178,998 $ 4,053,048 $ 114,479 $ 125,950 Security installation, product, and other 604,969 473,826 328,856 131,143 144,970 Total revenue 4,898,446 4,652,824 4,381,904 245,622 270,920 Cost of revenue (excluding depreciation and amortization): Monitoring and related services 617,386 604,368 596,664 13,018 7,704 Security installation, product, and other 229,728 147,314 102,118 82,414 45,196 Total cost of revenue 847,114 751,682 698,782 95,432 52,900 Selling, general, and administrative expenses 1,476,346 1,347,738 1,348,281 128,608 (543) Depreciation and intangible asset amortization 1,342,798 1,335,484 1,599,810 7,314 (264,326) Merger, restructuring, integration, and other 24,124 38,959 9,937 (14,835) 29,022 Operating income (loss) 1,208,064 1,178,961 725,094 29,103 453,867 Interest expense, net (441,031) (569,915) (263,068) 128,884 (306,847) Loss on extinguishment of debt (4,802) (16,621) — 11,819 (16,621) Other income (expense) 52,939 11,958 (57,568) 40,981 69,526 Income (loss) from continuing operations before income taxes and equity in net earnings (losses) of equity method investee 815,170 604,383 404,458 210,787 199,925 Income tax benefit (expense) (195,780) (160,585) (87,692) (35,195) (72,893) Income (loss) from continuing operations before equity in net earnings (losses) of equity method investee 619,390 443,798 316,766 175,592 127,032 Equity in net earnings (losses) of equity method investee — 6,572 (4,601) (6,572) 11,173 Income (loss) from continuing operations 619,390 450,370 312,165 169,020 138,205 Income (loss) from discontinued operations, net of tax (118,337) 12,639 (179,502) (130,976) 192,141 Net income (loss) $ 501,053 $ 463,009 $ 132,663 $ 38,044 $ 330,346 Key Performance Indicators:(1) RMR $ 359,450 $ 353,064 $ 341,025 $ 6,386 $ 12,039 Gross customer revenue attrition (percentage) 12.7 % 12.9 % 12.8 % N/A N/A Adjusted EBITDA(2) $ 2,578,195 $ 2,481,305 $ 2,305,032 $ 96,890 $ 176,273 ____ (1) Refer to the “Key Performance Indicators” section for the definitions of these key performance indicators.
(2) Adjusted EBITDA is a non-GAAP measure. Refer to the “Non-GAAP Measures” section for the definition of this term and reconciliation to the most comparable GAAP measure. N/A—Not applicable.
56 Revenue Monitoring and related services revenue (“M&S Revenue”) primarily comprises revenue generated from providing recurring monthly monitoring and other services, as well as revenue from time and materials billings. Security installation, product, and other revenue comprises installation revenue from the sale and installation of our security systems sold under a customer-owned model, as well as the recognition of revenue that is deferred upon initiation of a monitoring contract in transactions occurring under a Company-owned model (amortization of deferred subscriber acquisition revenue).
Revenue, as compared to the prior period, primarily reflects: • M&S Revenue: higher recurring revenue of $108 million primarily driven by an increase in average prices.
• Security installation, product, and other: (i) an increase in installation revenue of $87 million primarily driven by a higher mix of professionally installed systems at higher average prices in connection with the transition to our ADT+ platform, as well as (ii) an increase in the amortization of deferred subscriber acquisition revenue of $45 million associated with customers under a Company-owned model.
The increase in RMR, as compared to the prior period, was primarily driven by an increase in average prices on new and existing subscribers.
The decrease in gross customer revenue attrition, as compared to the prior period, was primarily driven by a decrease in voluntary and other disconnects, partially offset by higher non-payment disconnects.
Cost of Revenue Monitoring and related services costs (“M&S Costs”) primarily includes field service and call center costs incurred from providing recurring monthly monitoring and other services. Security installation, product, and other costs primarily includes costs incurred from the installation of our security systems sold in outright sales transactions.
The increase in cost of revenue, as compared to the prior period, primarily reflects an increase in installation and product costs of $82 million primarily due to a higher volume of outright sales transactions as well as an increase in M&S Costs of $13 million primarily due to higher interactive fees and other customer service initiatives, partially offset by lower maintenance costs primarily as a result of the continued growth of our Remote Assistance Program.
Selling, General, and Administrative Expenses (“SG&A”) The increase in SG&A, as compared to the prior period, was primarily driven by: • an increase in the allowance for credit losses of $55 million as a result of an increase in customer delinquencies, • an increase in general and administrative costs of $43 million, which includes a current year charge and a prior year benefit related to legal settlements in an aggregate amount of approximately $37 million, and • an increase in the amortization of deferred subscriber acquisition costs of $36 million, partially offset by • a decrease in advertising costs of $26 million as a result of lower media spend.
Depreciation and Intangible Asset Amortization (“D&A”) D&A remained relatively flat, as compared to the prior period, and included an increase of $35 million in the amortization of customer contracts acquired under our authorized dealer program and from other third parties, partially offset by a decrease of $33 million in the amortization of customer relationship intangible assets primarily related to certain assets acquired as part of the ADT Acquisition that became fully amortized during the first quarter of 2023.
Merger, Restructuring, Integration, and Other Merger, restructuring, integration, and other varies year over year and generally represents certain direct and incremental costs resulting from acquisitions, integration and optimization costs as a result of those acquisitions, costs related to restructuring efforts, as well as fair value remeasurements and impairment charges on certain strategic investments.
The decrease in merger, restructuring, integration, and other, as compared to the prior period, was primarily due to third-party strategic optimization costs incurred during the prior period.
57 Interest Expense, net During 2024, the decrease in interest expense, net, as compared to the prior year period, was primarily driven by lower interest expense of $114 million primarily related to lower principal amounts outstanding on our First Lien Term Loan B and the redemption of our First Lien Notes due 2024.
Other Income (Expense) During 2024, the increase in other income (expense) was primarily due to higher income from the Commercial TSA of $28 million, with the remainder of the change primarily due to the realized gain (loss) on interest rate swaps.
Income Tax Benefit (Expense) Income tax expense during 2024 was $196 million, resulting in an effective tax rate for the period of 24.0%. The effective tax rate primarily represents the federal statutory rate of 21.0%, state income taxes, net of federal benefits, of 5.4%, and unfavorable impacts from dispositions of 1.2% and permanently non-deductible items of 0.8%, partially offset by favorable impacts from a decrease in unrecognized tax benefits of 4.0%.
Income tax expense during 2023 was $161 million, resulting in an effective tax rate for the period of 26.6%. The effective tax rate primarily represents the federal statutory rate of 21.0%, state income taxes, net of federal benefits, of 6.1%, and an unfavorable impact from permanently non-deductible items of 1.0%, partially offset by favorable impacts from a decrease in unrecognized tax benefits of 1.0% and dispositions of 0.9%.
Income tax expense during 2022 was $88 million, resulting in an effective tax rate for the period of 21.7%. The effective tax rate primarily represents the federal statutory rate of 21.0%, state income taxes, net of federal benefits, of 3.0%, and an unfavorable impact from permanently non-deductible items of 5.1%, partially offset by favorable impacts from federal credits of 2.9%, a decrease in unrecognized tax benefits of 2.3%, and legislative changes of 2.1%.
Refer to Note 9 “Income Taxes” for details on our effective tax rate.
NON-GAAP MEASURES To provide investors with additional information in connection with our results as determined in accordance with GAAP, we disclose Adjusted EBITDA as a non-GAAP measure. This measure is not a financial measure calculated in accordance with GAAP, and it should not be considered as a substitute for net income, operating income, or any other measure calculated in accordance with GAAP, and may not be comparable to similarly titled measures reported by other companies. Adjusted EBITDA We believe Adjusted EBITDA is useful to investors to measure the operational strength and performance of our business. We believe the presentation of Adjusted EBITDA is useful as it provides investors additional information about our operating profitability adjusted for certain non-cash items, non-routine items we do not expect to continue at the same level in the future, as well as other items not core to our operations. Further, we believe Adjusted EBITDA provides a meaningful measure of operating profitability because we use it for evaluating our business performance, making budgeting decisions, and comparing our performance against other peer companies using similar measures.
We define Adjusted EBITDA as income (loss) from continuing operations adjusted for (i) interest; (ii) taxes; (iii) depreciation and amortization, including depreciation of subscriber system assets and other fixed assets and amortization of dealer and other intangible assets; (iv) amortization of deferred costs and deferred revenue associated with subscriber acquisitions; (v) share-based compensation expense; (vi) merger, restructuring, integration, and other; (vii) losses on extinguishment of debt; (viii) radio conversion costs, net; (ix) adjustments related to acquisitions, such as contingent consideration and purchase accounting adjustments, or dispositions; (x) impairment charges; and (xi) other income/gain or expense/loss items such as changes in fair value of certain financial instruments or financing and consent fees.
There are material limitations to using Adjusted EBITDA. Adjusted EBITDA does not take into account certain significant items, including depreciation and amortization, interest, taxes, and other adjustments which directly affect our income (loss) from continuing operations. These limitations are best addressed by considering the economic effects of the excluded items independently and by considering Adjusted EBITDA in conjunction with income (loss) from continuing operations as calculated in accordance with GAAP.
58 The table below reconciles Adjusted EBITDA to income (loss) from continuing operations: Years Ended December 31, $ Change (in thousands) 2024 2023 2022 2024 vs. 2023 2023 vs. 2022 Income (loss) from continuing operations $ 619,390 $ 450,370 $ 312,165 $ 169,020 $ 138,205 Interest expense, net 441,031 569,915 263,068 (128,884) 306,847 Income tax expense (benefit) 195,780 160,585 87,692 35,195 72,893 Depreciation and intangible asset amortization 1,342,798 1,335,484 1,599,810 7,314 (264,326) Amortization of deferred subscriber acquisition costs 224,647 188,222 154,186 36,425 34,036 Amortization of deferred subscriber acquisition revenue (346,209) (301,708) (235,190) (44,501) (66,518) Share-based compensation expense 48,745 38,626 52,945 10,119 (14,319) Merger, restructuring, integration, and other(1) 24,124 38,959 9,937 (14,835) 29,022 Unrealized (gain) loss on interest rate swaps(2) 17,996 16,511 — 1,485 16,511 Change in fair value of other financial instruments(3) — — 63,396 — (63,396) Other, net(4) 9,893 (15,659) (2,977) 25,552 (12,682) Adjusted EBITDA (from continuing operations) $ 2,578,195 $ 2,481,305 $ 2,305,032 $ 96,890 $ 176,273 __ (1) During 2024 and 2022, primarily relates to restructuring costs. During 2023, primarily includes integration and third-party strategic optimization costs, as well as restructuring costs.
(2) Includes the unrealized gain or loss on interest rate swaps presented in other income (expense).
(3) During 2022, represents the change in fair value of the Forward Contract (refer to Note 10 “Equity”).
(4) During 2023, primarily represents the gain on sale of a business and other investment, partially offset by loss on extinguishment of debt.
The drivers listed below exclude amounts that are outside of our definition of Adjusted EBITDA. Refer to the discussions above under “—Results of Operations” for further details.
The increase was primarily due to: • higher M&S Revenue, net of the associated costs, of $108 million, • higher other income of $43 million primarily due to higher income from the Commercial TSA, and • lower advertising costs of $26 million, partially offset by • higher provision for credit losses of $55 million and negative impacts from legal settlements of $37 million.
LIQUIDITY AND CAPITAL RESOURCES Liquidity and capital resources, along with our outstanding debt, primarily consist of the following: (in thousands) December 31, 2024 Cash and cash equivalents $ 96,212 Restricted cash and restricted cash equivalents $ 107,853 Availability under First Lien Revolving Credit Facility $ 800,000 Uncommitted available borrowing capacity under 2020 Receivables Facility $ 142,099 Carrying amount of total debt outstanding $ 7,707,073 Liquidity We expect our ongoing sources of liquidity to include cash generated from operations, borrowings under our first lien revolving credit facility (the “First Lien Revolving Credit Facility”) and the 2020 Receivables Facility, and the issuance of equity and/or debt securities as appropriate given market conditions. Our future cash needs are expected to include cash for operating activities, working capital, capital expenditures, principal and interest payments on our debt, expected dividend payments to our stockholders, potential share repurchases under a share repurchase plan, and other business initiatives as they arise.
Our principal liquidity requirements are to finance current operations, invest in acquiring and retaining customers, purchase property and equipment, service our debt, invest in our information technology infrastructure, and return money to shareholders through dividends and share repurchases.
59 Our liquidity requirements are primarily funded by our cash flows from operations. Cash inflows primarily include cash received from customers related to monthly recurring revenue from providing monitoring and other services, as well as cash from the sale and installation of our security systems. Cash outflows primarily relate to providing services to our customers, general and administrative costs, certain costs associated with acquiring new customers, interest payments, and taxes.
We are a highly leveraged company with significant debt service requirements and have both fixed-rate and variable-rate debt. We also entered into interest rate swaps to manage interest on our debt.
We may periodically seek to repay, redeem, repurchase, or refinance our indebtedness, or seek to retire or purchase our outstanding securities through cash purchases in the open market, privately negotiated transactions, a 10b5-1 repurchase plan, or otherwise, and any such transactions may involve material amounts. Cash outflows for interest payments are not consistent between quarters, with larger outflows occurring in the first and third quarters, and may vary as a result of our variable rate debt.
We are closely monitoring the impact of any inflationary pressures and changes in interest rates on our cash position. However, we believe our cash position, borrowing capacity available under our First Lien Revolving Credit Facility and 2020 Receivables Facility, and cash provided by operating activities are, and will continue to be, adequate to meet our operational and business needs in the next twelve months, as well as our long-term liquidity needs.
In addition, in 2023, we entered into a Receivables Financing Agreement with Mizuho Bank, Ltd. (the “Solar Receivables Financing Agreement”), which, among other things, provided for an uncommitted revolving loan facility in the aggregate principal amount of up to $300 million, which loans are secured by substantially all the assets of ADT Solar Finance LLC (“ADT Solar Finance”) (the “Solar Receivables Facility”). The Solar Receivables Facility expired in August 2024 with no amounts borrowed.
Material Cash Requirements Our cash requirements within the next twelve months primarily include current maturities and interest on our long-term debt and leases, accounts payable and other current liabilities, purchase commitments and other obligations entered into in the ordinary course of business, and dividends on our common stock.
As of December 31, 2024, our significant short-term and long-term cash requirements, excluding cash required for operations, under our various contractual obligations and commitments primarily included: • Debt principal – As of December 31, 2024, our expected future debt principal payments, excluding finance leases, totaled approximately $7.8 billion, with $170 million due in 2025 primarily related to payments on our 2020 Receivables Facility and the required quarterly principal payments on our First Lien Term Loan B due 2030.
As of December 31, 2024, our next debt maturity will occur in April 2026 with respect to the remaining outstanding balance of $1.35 billion of our First Lien Notes due 2026. On February 7, 2025, we issued a notice of partial redemption for $500 million of the First Lien Notes due 2026. Refer to Note 7 “Debt” for further details of our debt and the timing of expected future principal payments.
• Interest payments – Future interest payments on our fixed-rate debt are based on the contractual terms. Future interest payments on our variable-rate debt and the effects of our interest rate swaps are based on SOFR, plus the applicable margin in effect as of December 31, 2024.
During 2024, we paid net cash interest of $372 million, including interest on interest rate swaps presented outside of operating activities. As of December 31, 2024, our expected future interest payments related to our debt and interest rate swap contracts totaled approximately $1.6 billion, with approximately $354 million due in 2025. Additionally, we expect to incur annual interest payments of approximately $310 - $325 million during each of the years 2026 - 2027 and approximately $195 - $240 million during each of the years 2028 and thereafter.
• Leases – As of December 31, 2024, our expected future operating and finance lease payments, including interest, totaled $192 million, with $54 million due in 2025.
Refer to Note 14 “Leases” for further details of our obligations and the timing of expected future payments.
60 • Purchase obligations – Our material cash requirements for purchases of goods or services entered into in the ordinary course of business, including purchase orders and contractual obligations, primarily consist of information technology services and equipment, including investments in our information technology infrastructure, direct materials, and telecommunication services. Our future purchase obligations may be impacted by changes in our business or other internal or external factors. As our business continues to grow organically or through acquisitions, our obligations may grow as well.
As of December 31, 2024, our contractual obligations entered into in the ordinary course of business, including agreements that are enforceable and legally binding and have a remaining term in excess of one year, totaled approximately $561 million, with approximately $367 million expected to be paid in 2025.
These amounts include a commitment with one of our vendors to purchase at least $172 million of security system equipment and components through December 2025. This commitment is also satisfied through purchases made by the Company’s dealer network.
In addition, as of December 31, 2024, we had outstanding purchase orders of approximately $143 million primarily related to direct materials and information technology and marketing services, which are expected to be materially satisfied in 2025.
Refer to Note 13 “Commitments and Contingencies” for the amounts and timing of such payments.
• Google agreements – The Google Commercial Agreement requires us and Google to each contribute $150 million toward certain joint commercial efforts. Additionally, during 2022 we entered into the Google Commercial Agreement Amendment in which Google agreed to commit an additional $150 million. While the timing of these contributions is still uncertain, we expect to contribute the majority of our $150 million commitment under the Google Commercial Agreement by the end of 2026. During 2024 and 2023, $30 million and $40 million, respectively, of the Google Success Funds were reimbursed to the Company primarily for certain joint marketing expenses and customer acquisition costs incurred by the Company, substantially all of which was reflected as a reduction of advertising costs.
In addition, under the Google Cloud Agreement Addendum, the Company is obligated to make purchases from Google totaling $200 million over a seven-year period (through December 2030), with an aggregate of $35 million in the first two years, an aggregate of $65 million in the next two years after that, and an aggregate of $100 million in the last three years of the commitment. As of December 31, 2024, we are on track to meet these commitments.
• State Farm Opportunity Fund – Pursuant to the State Farm Development Agreement, State Farm committed up to $300 million to fund product and technology innovation, customer growth, and marketing initiatives. Upon the Closing of the State Farm Strategic Investment, we received $100 million of such commitment from State Farm, which is restricted until we use the funds for investment, as agreed upon with State Farm, in accordance with the State Farm Development Agreement. During 2024 and 2023, payments made from the Opportunity Fund were $14 million and $11 million, respectively.
• Customer account purchases – Our indirect channel customers are generated mainly through our ADT Authorized Dealer Program. As opportunities arise, we may engage in selective third-party account purchases, which typically involve the purchase of a set of customer accounts from other security service providers. For example, we paid initial cash at closings of $81 million and $89 million for customer accounts from third parties during 2024 and 2023, respectively.
• Taxes – During 2025, we expect to become a federal cash taxpayer. However, we are currently unable to determine the exact amount or timing of these payments.
In addition, we have $25 million of unrecognized tax benefits, excluding interest and penalties, related to various tax positions we have taken. These liabilities may increase or decrease over time primarily as a result of tax examinations, and given the status of the examinations, we cannot reliably estimate the period of any cash settlement with the respective taxing authorities.
Refer to Note 9 “Income Taxes” for further details.
• Dividends – Stockholders are entitled to receive dividends when, as, and if declared by our Board of Directors out of funds legally available for that purpose. On February 27, 2025, we announced a dividend of $0.055 per share to holders of Common Stock and Class B Common Stock of record on March 13, 2025, which will be distributed on or about April 3, 2025.
Refer to Note 10 “Equity” for further details on dividends declared and paid during 2024 and 2023.
61 • Share repurchases – In February 2025, our Board of Directors announced the 2025 Share Repurchase Plan, pursuant to which the Company is authorized to repurchase, through April 30, 2026, up to a maximum aggregate amount of $500 million of shares of Common Stock.
As of the date of this 2024 Annual Report, the Company has not made any repurchases under the 2025 Share Repurchase Plan.
• Off-balance sheet arrangements – As of December 31, 2024, we had guarantees primarily related to standby letters of credit on our insurance programs totaling $74 million.
In February 2025, $21 million of our guarantees were relinquished.
We do not have any other arrangements giving rise to material obligations that are not reported in our consolidated balance sheets, as described in Item 303 of SEC Regulation S-K.
Cash Flow Analysis The following table is a summary of our cash flow activity for the periods presented: Years Ended December 31, $ Change (in thousands) 2024 2023 2022 2024 vs. 2023 2023 vs. 2022 Net cash provided by (used in): Operating activities $ 1,884,899 $ 1,657,726 $ 1,887,920 $ 227,173 $ (230,194) Investing activities $ (1,295,428) $ 242,493 $ (1,532,784) $ (1,537,921) $ 1,775,277 Financing activities $ (515,356) $ (2,143,849) $ (14,833) $ 1,628,493 $ (2,129,016) The discussion below includes cash flows from both continuing operations and discontinued operations consistent with the presentation on the Statements of Cash Flows.
Cash Flows from Operating Activities The increase in net cash provided by operating activities for 2024 compared to 2023 was primarily due to a decrease in cash interest of $149 million primarily related to reduced principal balances on our debt, the benefit of lower outflows in the current year related to our former Solar Business, and improved operating performance. This increase was partially offset by the benefit received in the prior period associated with the results of the former Commercial Business as well as a current year outflow and prior year inflow related to legal settlements.
The remainder of the activity related to changes in assets and liabilities due to the volume and timing of other operating cash receipts and payments with respect to when the transactions are reflected in earnings. Refer to the discussions above under “— Results of Operations” for further details.
Cash Flows from Investing Activities The change in net cash used in investing activities for 2024 compared to 2023 was primarily due to sales of businesses in the amount of $1,627 million, primarily related to the Commercial Divestiture in 2023. In addition, subscriber system assets expenditures decreased by $107 million due to a lower volume of transactions including those under a Company-owned model.
62 Cash Flows from Financing Activities The decrease in net cash used in financing activities for 2024 compared to 2023 was primarily due to: • lower net repayments of $1,977 million primarily related to the partial redemptions in 2023 of our First Lien Term Loan B due 2026 and ADT Notes due 2024, • share repurchases during 2024 of $241 million, • a decrease in net proceeds under the 2020 Receivables Facility of $109 million primarily due to net repayments during 2024 as compared to net proceeds in 2023, and • an increase in dividend payments of $54 million due to an increase in our quarterly dividend during 2024.
During 2024, both the proceeds and repayments of long-term borrowings on the Consolidated Statements of Cash Flows include the impact of $704 million from the refinancings of the First Lien Term Loan B due 2026.
During 2023, both the proceeds and repayments of long-term borrowings on the Consolidated Statements of Cash Flows include the impact of $230 million from the October refinancing of the First Lien Term Loan B due 2026.
Long-Term Debt As of December 31, 2024, our debt (excluding finance leases and any deferred financing costs, discounts, premiums, or fair value adjustments) consisted of the following (in thousands): Debt Description Issued Maturity Interest Rate Interest Payable Principal First Lien Term Loan B due 2030 10/13/2023 10/13/2030 Term SOFR +2.00% Quarterly $ 1,984,090 First Lien Notes due 2026 4/4/2019 4/15/2026 5.750% 3/15 and 9/15 1,350,000 First Lien Notes due 2027 8/20/2020 8/31/2027 3.375% 6/15 and 12/15 1,000,000 First Lien Notes due 2029 7/29/2021 8/1/2029 4.125% 2/1 and 8/1 1,000,000 Second Lien Notes due 2028 1/28/2020 1/15/2028 6.250% 1/15 and 7/15 1,300,000 ADT Notes due 2032 5/2/2016 7/15/2032 4.875% 1/15 and 7/15 728,016 ADT Notes due 2042 7/5/2012 7/15/2042 4.875% 1/15 and 7/15 21,896 2020 Receivables Facility 3/5/2020 11/20/2029 Various Monthly 407,901 Total $ 7,791,903 Refer to Note 7 “Debt” for further details of our debt agreements, including interest rates, covenants, and other descriptions of these agreements.
First Lien Credit Agreement Our first lien credit agreement dated as of July 1, 2015 (together with subsequent amendments and restatements, the “First Lien Credit Agreement”) consists of a term loan facility (the “First Lien Term Loan B due 2030,” and prior to October 2023, the “First Lien Term Loan B due 2026”) and a first lien revolving credit facility (the “First Lien Revolving Credit Facility”).
Below is a summary of key events related to the First Lien Credit Agreement since 2023: • October 2023 - We redeemed approximately $1.3 billion of the First Lien Term Loan B due 2026 using net proceeds from the Commercial Divestiture.
• October 2023 - We amended and restated the First Lien Credit Agreement to refinance the remaining outstanding balance of the First Lien Term Loan B due 2026 with a new $1,375 million 7-year term First Lien Term Loan B due 2030.
• April 2024 - We amended and restated the First Lien Credit Agreement, which reduced the interest rate on the First Lien Term Loan B due 2030 from Term SOFR +2.50% to Term SOFR +2.25%.
63 • May 2024 - We amended and restated the First Lien Credit Agreement, which included the exchange of $143 million principal amount of loans under the Company’s Term Loan A Facility for its First Lien Term Loan B due 2030. In addition, later that month, the Company further amended and restated the First Lien Credit Agreement, pursuant to which the Company incurred an additional $474 million of outstanding principal under the First Lien Term Loan B due 2030 with the proceeds used to pay off the remaining outstanding balance of the Company’s Term Loan A Facility.
• October 2024 - We amended and restated the First Lien Credit Agreement to extend the maturity date of the First Lien Revolving Credit Facility to October 2029, subject to a springing maturity of 91 days prior to the maturity date of certain long-term indebtedness if, as of such date, the outstanding principal amount of such indebtedness exceeds $350 million, and obtain an additional $225 million of First Lien Revolving Credit Facility commitments. After giving effect to the amendment, the aggregate amount of commitments under the First Lien Revolving Credit Facility is $800 million. Borrowings under the First Lien Revolving Credit Facility bear interest at a rate equal to either (a) Term SOFR with a floor of zero or (b) a base rate (“Base Rate”) determined by reference to the highest of (i) the federal funds rate plus 0.50% per annum, (ii) the rate of interest per annum last quoted by The Wall Street Journal as the “Prime Rate” in the United States and (iii) the one-month adjusted term SOFR plus 1.00% per annum, in each case, plus an applicable margin of 2.00% per annum for Term SOFR loans and 1.00% per annum for Base Rate loans, subject to two step-downs based on certain specified net first lien leverage ratios. In addition, the amendment reduced the commitment fee in respect of the First Lien Revolving Credit Facility to 0.30% per annum in respect of the unutilized commitments thereunder, subject to two step-downs based on certain specified net first lien leverage ratios.
• December 2024 - We amended and restated the First Lien Credit Agreement, which reduced the interest rate on the First Lien Term Loan B due 2030 from Term SOFR +2.25% to Term SOFR +2.00%.
In addition, during 2024, we borrowed and repaid $365 million under the First Lien Revolving Credit Facility.
We are required to make scheduled quarterly principal payments on the First Lien Term Loan B due 2030, with the remaining balance payable at maturity. We may make voluntary prepayments on the First Lien Term Loan B due 2030 at any time prior to maturity at par. Additionally, based on certain specified net first lien leverage ratios, we may be required to make annual prepayments on the outstanding First Lien Term Loan B due 2030 with a percentage of our excess cash flow, as defined in the First Lien Credit Agreement, if our excess cash flow exceeds a certain specified threshold, which is 0% if our net first lien leverage ratio is less than or equal to 2.20 to 1.00. As of December 31, 2024, we were not required to make any annual prepayments based on our excess cash flow.
In addition, we are required to pay a commitment fee between 0.20% and 0.30% (determined based on a net first lien leverage ratio) with respect to the unused commitments under the First Lien Revolving Credit Facility.
On February 7, 2025, the Company issued a notice of partial redemption for $500 million of the First Lien Notes due 2026, which will be redeemed on March 9, 2025. Prior to the issuance of such notice, certain lenders provided commitments that they will fund a new $600 million first lien seven-year term loan facility. The closing of this new facility, which remains subject to market and other customary conditions, is expected to occur on or around March 7, 2025. The Company intends to use proceeds of this new facility for the partial redemption of the First Lien Notes due 2026 among other general corporate purposes.
Term Loan A Facility In March and June 2023, we borrowed aggregate principal amounts of $600 million and $50 million, respectively, of term loans under a senior secured term loan A facility (the “Term Loan A Facility”) and used the proceeds to redeem $650 million of the ADT Notes due 2023 (defined below).
In May 2024, we exchanged $143 million of loans under our Term Loan A Facility for our First Lien Term Loan B due 2030. In addition, later that month, we redeemed the remaining outstanding principal balance of $474 million of our Term Loan A Facility, excluding accrued and unpaid interest, using proceeds under the First Lien Term Loan B due 2030, as discussed above. As a result, the Term Loan A Facility has been terminated.
64 First Lien Notes due 2024 As of December 31, 2024, we had fully redeemed the 5.250% first-priority senior secured notes due 2024 (the “First Lien Notes due 2024”) as a result of the following transactions: • May 2023 - We redeemed $150 million principal amount of the outstanding First Lien Notes due 2024 for a redemption price of $150 million, excluding accrued and unpaid interest, using cash on hand.
• December 2023 - We redeemed $500 million principal amount of the outstanding First Lien Notes due 2024 for a redemption price of $500 million, excluding accrued and unpaid interest, using remaining net proceeds from the Commercial Divestiture and cash on hand.
• April 2024 - We redeemed the remaining outstanding principal balance of $100 million of the First Lien Notes due 2024 for a redemption price of $100 million, excluding accrued and unpaid interest, using proceeds from the Company’s First Lien Revolving Credit Facility.
First Lien Notes due 2026 The 5.750% first-priority senior secured notes due 2026 (the “First Lien Notes due 2026”) are due at maturity, and may be redeemed, in whole or in part, at any time at a make-whole premium plus accrued and unpaid interest to, but excluding, the redemption date.
Additionally, upon the occurrence of specified change of control events, we must offer to repurchase the First Lien Notes due 2026 at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the First Lien Notes due 2026 also provides for customary events of default.
As discussed above, on February 7, 2025, the Company issued a notice of partial redemption for $500 million of the First Lien Notes due 2026, which will be redeemed on March 9, 2025.
First Lien Notes due 2027 The 3.375% first-priority senior secured notes due 2027 (the “First Lien Notes due 2027”) are due at maturity and may be redeemed at our option as follows: • Prior to August 31, 2026, in whole at any time or in part from time to time, at a make-whole premium plus accrued and unpaid interest, if any, thereon to the redemption date.
• On or after August 31, 2026, in whole at any time or in part from time to time, at a redemption price equal to 100% of the principal amount of the First Lien Notes due 2027 redeemed plus accrued and unpaid interest, if any, thereon to the redemption date.
Additionally, upon the occurrence of specified change of control events, we must offer to repurchase the First Lien Notes due 2027 at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the First Lien Notes due 2027 also provides for customary events of default.
First Lien Notes due 2029 The 4.125% first-priority senior secured notes due 2029 (the “First Lien Notes due 2029”) are due at maturity and may be redeemed at our option as follows: • Prior to August 1, 2028, in whole at any time or in part from time to time, at a redemption price equal to the greater of (i) 100% of the principal amount of the First Lien Notes due 2029 to be redeemed and (ii) the sum of the present values of the aggregate principal amount of the First Lien Notes due 2029 to be redeemed and the remaining scheduled interest payments due on any date after the redemption date, to and including August 1, 2028, discounted at an adjusted treasury rate plus 50 basis points, plus, in either case accrued and unpaid interest as of, but excluding, the redemption date.
• On or after August 1, 2028, in whole at any time or in part from time to time, at a redemption price equal to 100% of the principal amount of the First Lien Notes due 2029 to be redeemed and accrued and unpaid interest as of, but excluding, the redemption date.
Additionally, upon the occurrence of specified change of control events, we may be required to purchase the First Lien Notes due 2029 at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
65 The indenture governing the First Lien Notes due 2029 also provides for customary events of default.
Second Lien Notes due 2028 The 6.250% second-priority senior secured notes due 2028 (the “Second Lien Notes due 2028”) are due at maturity and may be redeemed at our option, in whole at any time or in part from time to time, at a redemption price equal to 100% (after January 15, 2025) of the principal amount of the Second Lien Notes due 2028 redeemed and accrued and unpaid interest as of, but excluding, the redemption date.
Additionally, upon the occurrence of specified change of control events, we must offer to repurchase the Second Lien Notes due 2028 at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the Second Lien Notes due 2028 also provides for customary events of default.
ADT Notes During 2023, the Company redeemed the then outstanding ADT Notes due 2023. The remaining outstanding ADT Notes due 2032 and ADT Notes due 2042 (collectively, the “ADT Notes”) are due at maturity, and may be redeemed, in whole at any time or in part from time to time, at a redemption price equal to the principal amount of the notes to be redeemed, plus a make-whole premium, plus accrued and unpaid interest as of, but excluding, the redemption date.
Additionally, upon the occurrence of specified change of control events, the Company must offer to repurchase the ADT Notes at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indentures governing the ADT Notes also provide for customary events of default.
Below is a summary of key events related to the ADT Notes due 2023 since 2023: • March 2023 - The Company used the proceeds from the Closing Date Term Loan A Loans to redeem $600 million outstanding principal amount of the ADT Notes due 2023 for a total redemption price of $600 million, excluding any accrued and unpaid interest.
• June 2023 - The Company redeemed the remaining outstanding principal amount of $100 million for a total redemption price of $100 million using $50 million of proceeds from the Incremental Term Loan A Loans and the remaining from cash on hand.
2020 Receivables Facility The 2020 Receivables Facility allows us to obtain financing by selling or contributing certain retail installment contract receivables to our wholly-owned consolidated bankruptcy-remote special purpose entity (“SPE”). The SPE grants a security interest in those retail installment contract receivables as collateral for cash borrowings under the 2020 Receivables Facility. The SPE borrower under the 2020 Receivables facility is a separate legal entity with its own creditors who will be entitled, prior to and upon the liquidation of the SPE, to be satisfied out of the SPE’s assets prior to any assets of the SPE becoming available to us (other than the SPE). Accordingly, the assets of the SPE are not available to pay our creditors (other than the SPE), although collections from the transferred retail installment contract receivables in excess of amounts required to repay amounts then due and payable to the SPE’s creditors may be released to us and subsequently used by us (including to pay other creditors). The SPE’s creditors under the 2020 Receivables Facility have legal recourse to the transferred retail installment contract receivables owned by the SPE, and to us for certain performance and operational obligations relating to the 2020 Receivables Facility, but do not have any recourse to us (other than the SPE) for the payment of principal and interest on the advances under the 2020 Receivables Facility.
Below is a summary of significant amendments to the 2020 Receivables Facility since 2023: • March 2023 - Increased the borrowing capacity from $400 million to $500 million and extended the uncommitted revolving period from May 2023 to March 2024, among other things.
• March 2024 - The Company amended the agreement governing the 2020 Receivables Facility, pursuant to which the uncommitted revolving period was extended from March 2024 to April 2024.
• April 2024 - The Company further amended the agreement governing the 2020 Receivables Facility, pursuant to which, among other things, the borrowing capacity was increased from $500 million to $550 million and the uncommitted revolving period was extended from April 2024 to April 2025. In addition, proceeds and repayments from the receivables facility include the impact of $32 million from the amendments described above.
66 We service the transferred retail installment contract receivables and are responsible for ensuring related collections are remitted to a segregated account in the SPE’s name. On a monthly basis, the segregated bank account is utilized to make required principal, interest, and other payments due under the 2020 Receivables Facility. The segregated account is considered restricted cash in our Consolidated Balance Sheets.
Proceeds and repayments under the 2020 Receivables Facility since 2023 were as follows: • 2024: Proceeds of $229 million and repayments of $257 million.
• 2023: Proceeds of $282 million and repayments of $200 million.
Debt Covenants Our credit agreements and indentures associated with the borrowings above contain certain covenants and restrictions that limit our ability to, among other things, incur additional debt or issue certain preferred equity interests; create liens on certain assets; make certain loans or investments (including acquisitions); pay dividends on or make distributions in respect of the capital stock or make other restricted payments; consolidate, merge, sell, or otherwise dispose of all or substantially all of our assets; sell assets; enter into certain transactions with affiliates; enter into sale-leaseback transactions; restrict dividends from our subsidiaries or restrict liens; change our fiscal year; and modify the terms of certain debt or organizational agreements.
We are also subject to a springing financial maintenance covenant under the First Lien Credit Agreement, which requires us to not exceed a specified first lien leverage ratio at the end of each fiscal quarter if the testing conditions are satisfied. The covenant is tested if the outstanding loans under the First Lien Revolving Credit Facility, subject to certain exceptions, exceed 30% of the total commitments under the First Lien Revolving Credit Facility at the testing date (i.e., the last day of any fiscal quarter).
As of December 31, 2024, we were in compliance with all financial covenant and other maintenance tests for all our debt obligations, and we do not believe there is a material risk of future noncompliance with our financial covenant and other maintenance tests.
CRITICAL ACCOUNTING ESTIMATES The accompanying consolidated financial statements are prepared in accordance with GAAP, which requires us to select accounting policies and make estimates that affect amounts reported in the financial statements and the accompanying notes. Management’s estimates are based on the relevant information available at the end of each period. Actual results could differ materially from these estimates under different assumptions or market conditions.
The following discussion includes estimates prepared in accordance with GAAP that involve a significant level of estimation uncertainty and have had or are reasonably likely to have a material impact on the financial condition or results of operations, and are based on, among other things, estimates, assumptions, and judgments made by management that include inherent risks and uncertainties. We base our estimates on historical experience and on various other assumptions that we believe to be reasonable under the circumstances.
Refer to the Notes to Consolidated Financial Statements included in this Annual Report for further discussion of our significant accounting policies and the effect on our financial statements.
Revenue Recognition We generate revenue through contractual monthly recurring fees received for monitoring and related services provided to customers as well as the sale and installation of security systems. Prior to the ADT Solar Exit, we also generated revenue through the sale and installation of solar systems. We allocate the transaction price to each performance obligation in contracts with our customers using estimates of standalone selling price. We use judgment to determine the standalone selling prices for our performance obligations. If a standalone selling price is not directly observable, we may use other observable internal and external pricing, profitability, and certain operational metrics.
Estimated Life of Customer Relationships A significant portion of our depreciation and amortization is based on the expected life of our customer relationships. We periodically perform lifing studies to (i) estimate the expected life of our customer relationships and the attrition pattern of our customers; (ii) establish the amortization rates of our customer account pools discussed below in order to reflect the pattern of future economic benefit; and (iii) assess the continued reasonableness of our existing depreciation and amortization policies.
67 The results of the lifing studies are based on historical customer terminations. The lifing studies indicate that we can expect attrition to be the greatest in the initial years of asset life. Therefore, to align our depreciation and amortization to the pattern in which the related economic benefits are consumed, we use an accelerated method that best matches the future amortization cost with the estimated revenue stream from these customer pools.
Subscriber System Assets and Deferred Subscriber Acquisition Costs - Subscriber system assets and any related deferred subscriber acquisition costs are accounted for on a pooled basis based on the month and year of acquisition. We depreciate and amortize these assets using an accelerated method over the estimated life of the customer relationship, which is 15 years, using an average declining balance rate of approximately 250% that converts to straight-line methodology when the resulting charge is greater than that from the accelerated method. This results in an average charge of approximately 55% of the pool within the first five years, 25% within the second five years, and 20% within the final five years.
Customer Account Purchases - Purchases of contracts with customers under the ADT Authorized Dealer Program, or from other third parties, are considered asset acquisitions and are recognized based on the cost to acquire the assets, which may include cash consideration, non-cash consideration, contingent consideration, and directly-attributable transaction costs. These assets are accounted for on a pooled basis based on the month and year of acquisition. Based on the results of our lifing studies, we amortize our pooled contracts with customers using an accelerated method over the estimated life of the customer relationship, which is generally 15 years using an average declining balance rate of approximately 300% that converts to straight-line methodology when the resulting amortization charge is greater than that from the accelerated method. This results in an average amortization of approximately 65% of the pool within the first five years, 25% within the second five years, and 10% within the final five years.
The accelerated methods and estimated lives used to calculate depreciation and amortization expense have not changed during the periods presented. Additionally, these estimates remain relatively consistent year over year due to the large and homogenous nature of our customer pools. Significant changes in our business model, such as a reduction in the number of customers under multi-year contracts, or a prolonged shift in our attrition patterns, could impact the expected life of our customer relationships.
Goodwill Goodwill and indefinite-lived intangible assets (as discussed below) are not amortized and are tested for impairment at least annually as of the first day of the fourth quarter of each year and more often if an event occurs or circumstances change which indicate it is more-likely-than-not that fair value is less than carrying amount. Under a qualitative approach, we assess whether it is more-likely-than-not that a reporting unit’s fair value is less than its carrying amount. Under a quantitative approach, we estimate the fair value of a reporting unit and compare it to its carrying amount. If the carrying amount of a reporting unit exceeds its fair value, an impairment loss is recognized.
On October 1, 2024, we completed our annual goodwill impairment test by qualitatively testing the goodwill assigned to the Company’s reporting unit. Based on the results of the qualitative test, we concluded that it is more likely than not that the fair value of the Company’s reporting unit exceeds its carrying value and no impairment was recognized.
In prior years’ quantitative impairment tests, we estimated the fair value of the reporting unit using the income approach, which discounts projected cash flows using market participant assumptions. The income approach includes significant assumptions including, but not limited to, forecasted revenue, operating profit margins, Adjusted EBITDA margins, operating expenses, cash flows, perpetual growth rates, and discount rates. In developing these assumptions, we rely on various factors including operating results, business plans, economic projections, anticipated future cash flows, and other market data.
The estimated fair value of a reporting unit calculated using the income approach is sensitive to changes in the underlying assumptions. Examples of events or circumstances that could reasonably be expected to negatively affect the underlying judgments and factors and ultimately impact the estimated fair value determinations may include such items as a prolonged downturn in the business environment, changes in economic conditions that significantly differ from our assumptions in timing or degree, volatility in equity and debt markets resulting in higher discount rates, and unexpected regulatory changes. As a result, there are inherent uncertainties related to these judgments and factors in applying them to the goodwill impairment tests.
The Company previously recorded goodwill impairment charges associated with the Solar reporting unit, which are now presented in income (loss) from discontinued operations, net of tax.
68 Indefinite-Lived Intangible Assets As of December 31, 2024, our only indefinite-lived intangible asset is the ADT trade name, which has a carrying value of $1.3 billion. The ADT trade name was acquired in connection with the ADT Acquisition in May 2016. When performing a quantitative impairment test, the fair value of the ADT trade name is determined using a relief from royalty method, which is an income approach that estimates the cost savings that accrue to us that we would otherwise have to pay in the form of royalties or license fees on revenue earned through the use of the asset. The utilization of the relief from royalty method requires us to make significant assumptions including revenue growth rates, the implied royalty rate, and the discount rate.
We performed a quantitative impairment test over the ADT trade name as of October 1, 2024 and 2023, and the fair value of the ADT trade name substantially exceeded its carrying value as of each testing date. In connection with our quantitative impairment test, we perform a sensitivity analysis on the key assumptions used to determine the fair value of the ADT trade name. During the periods presented, the results of our sensitivity analysis did not have a material impact on the conclusions reached.
We may, in future periods, perform a qualitative testing approach, where we assess whether it is more-likely-than-not that the ADT trade name’s fair value is less than its carrying amount.
Business Combinations We account for the acquisitions of businesses using the acquisition method of accounting. The assets acquired and liabilities assumed in connection with business acquisitions are recorded at the date of acquisition at their estimated fair values, with any excess of the purchase price over the estimated fair values of the net assets acquired recorded as goodwill. We use various methods to determine fair value depending on the type of assets acquired and liabilities assumed. We make estimates and assumptions about projected future cash flows including, but not limited to, forecasted revenue, Adjusted EBITDA margins, operating expenses, cash flows, perpetual growth rates, and discount rates.
Significant judgment is required in estimating the fair value of assets acquired and liabilities assumed and in assigning useful lives to certain definite-lived intangible and tangible assets. Accordingly, we may engage third-party valuation specialists to assist in these determinations. The fair value estimates are based on information available as of the acquisition date and assumptions deemed reasonable by management but are inherently uncertain.
Customer Relationships - Customer relationships acquired as part of business acquisitions are generally amortized over a period of up to 15 years based on management estimates about the amounts and timing of estimated future revenue from customer accounts and average customer account life that existed at the time of the related business acquisition. The majority of our customer relationships acquired in business combinations that originated from the Formation Transactions and the ADT Acquisition were fully amortized during 2023.
Dealer Relationships - Dealer relationships originated from the Formation Transactions and the ADT Acquisition and are primarily amortized on a straight-line basis over 19 years based on management estimates about the longevity of the underlying dealer network and the attrition of those respective dealers that existed at the time of the related business acquisition.
During 2024, 2023, and 2022, other definite-lived intangible assets acquired in business acquisitions were not material, and we have not recorded any material measurement period adjustments to purchase price allocations.
Income Taxes We account for income taxes under the asset and liability method, which requires the recognition of deferred tax assets and liabilities for the temporary differences between the recognition of revenue and expenses for income tax and financial reporting purposes and between the tax basis of assets and liabilities and their reported amounts in the consolidated financial statements. We record the effect of a tax rate or law change on our deferred tax assets and liabilities in the period of enactment. Future tax rate or law changes could have a material effect on our results of operations, financial condition, or cash flows.
In evaluating our ability to recover our deferred tax assets, we consider all available positive and negative evidence, including our past operating results, the existence of cumulative losses in the most recent years, and our forecast of future taxable income. In estimating future taxable income, we develop assumptions related to the amount of future pre-tax operating income, the timing and amount of the reversal of temporary differences, and the implementation of feasible and prudent tax planning strategies. These assumptions require significant judgment about the forecasts of future taxable income and are consistent with the plans and estimates we are using to manage our underlying businesses.
We recognize positions taken or expected to be taken in a tax return in the consolidated financial statements when it is more-likely-than-not that the position would be sustained upon examination by tax authorities. A recognized tax position is then 69 measured at the largest amount of benefit with greater than 50% likelihood of being realized upon ultimate settlement. We record liabilities for positions that have been taken but do not meet the more-likely-than-not recognition threshold. We adjust the liabilities for unrecognized tax benefits in light of changing facts and circumstances; however, due to the complexity of some of these uncertainties, the ultimate resolution may result in a change to the estimated liabilities, along with impacts to the effective tax rate and cash tax.
Refer to Note 9 “Income Taxes” for details on our valuation allowances and unrecognized tax benefits.
ACCOUNTING PRONOUNCEMENTS Refer to Note 1 “Description of Business and Summary of Significant Accounting Policies” in the Notes to Consolidated Financial Statements in Item 15 for further discussion about recent accounting pronouncements.
ITEM 7A. QUANTITATIVE AND QUALITATIVE DISCLOSURES ABOUT MARKET RISK.
Our operations expose us to a variety of market risks, including the effects of changes in interest rates as we have both fixed-rate and variable-rate debt. We monitor and manage these financial exposures as an integral part of our overall risk management program. Our policies allow for the use of specified financial instruments for hedging purposes only. The use of derivatives for speculation purposes is prohibited.
Interest Rate Risk We manage interest rate exposure on our variable-rate debt through interest rate swap contracts. As of December 31, 2024, the principal balance of our debt, excluding finance leases, that was subject to a variable-rate was approximately 0% (including the impact of interest rate swaps) and approximately 30% (excluding the impact of interest rate swaps) of the total carrying amount of our debt.
If current SOFR increases or decreases, the increase or decrease in our debt service obligations on the majority of our variable rate indebtedness will be materially neutralized as our interest rate swaps hedge any increase or decrease in current SOFR.
The impact of a hypothetical 10% change in interest rates on the fair value of our long-term debt (excluding finance leases) and interest rate swap contracts would be: As of December 31, 2024 2023 Long-term debt (excluding finance leases): Carrying amount $ 7.6 billion $ 7.8 billion Fair value(1) $ 7.6 billion $ 7.7 billion Fair value impact of hypothetical 10% change in interest rates $ 156 million $ 193 million Interest rate swap contracts: Notional value $ 3.8 billion $ 3.8 billion Fair value - net asset / liability(2) $ 109 million $ 145 million Fair value impact of hypothetical 10% change in interest rates $ 18 million $ 33 million __ (1) Fair value of long-term debt is based on the implied yield from broker-quoted market prices. The carrying amounts of debt outstanding, if any, under the Company’s revolving credit facility and receivables facility approximate fair values as interest rates on these borrowings approximate current market rates.
(2) Fair value of interest rate swaps contracts is based on discounted cash flow analyses and was in a net asset position as of December 31, 2024 and 2023.
Refer to Note 7 “Debt” and Note 8 “Derivative Financial Instruments” for details on our debt and interest rate swaps, respectively.
ITEM 8. FINANCIAL STATEMENTS AND SUPPLEMENTARY DATA.
The Report of Independent Registered Public Accounting Firm, our consolidated financial statements, and the accompanying Notes to Consolidated Financial Statements that are filed as part of this Annual Report are listed under Item 15 “Exhibit and Financial Statement Schedules” and are set forth beginning on page F-1 immediately following the signature pages of this Annual Report.
70 ITEM 9. CHANGES IN AND DISAGREEMENTS WITH ACCOUNTANTS ON ACCOUNTING AND FINANCIAL DISCLOSURE.
None.
ITEM 9A. CONTROLS AND PROCEDURES.
Evaluation of Disclosure Controls and Procedures Our management, with the participation of our Chief Executive Officer and Chief Financial Officer, has evaluated the effectiveness of our disclosure controls and procedures (as defined in Rules 13a-15(e) and 15d-15(e) under the Exchange Act) as of the end of the period covered by this Annual Report. Based on such evaluation, our Chief Executive Officer and Chief Financial Officer have concluded that as of December 31, 2024, our disclosure controls and procedures were effective to provide reasonable assurance that information required to be disclosed in the reports we file or submit under the Exchange Act is recorded, processed, summarized, and reported within the time periods specified in the SEC’s rules and forms, and that such information is accumulated and communicated to our Chief Executive Officer and Chief Financial Officer, as appropriate, to allow timely decisions regarding required disclosures.
Management’s Report on Internal Control over Financial Reporting Our management is responsible for establishing and maintaining adequate internal control over financial reporting (as defined under Exchange Act Rules 13a-15(f) and 15d-15(f)). Internal control over financial reporting is a process designed to provide reasonable assurance regarding the reliability of financial reporting and the preparation of financial statements for external purposes in accordance with generally accepted accounting principles.
Because of its inherent limitations, internal control over financial reporting may not prevent or detect misstatements. Also, projections of any evaluation of effectiveness to future periods are subject to the risk that controls may become inadequate because of changes in conditions, or that the degree of compliance with the policies or procedures may deteriorate.
Our management performed an assessment of the effectiveness of our internal control over financial reporting as of December 31, 2024 based on criteria established in Internal Control — Integrated Framework (2013) issued by the Committee of Sponsoring Organizations of the Treadway Commission (“COSO”). Based on our assessment and those criteria, our management determined that our internal control over financial reporting was effective as of December 31, 2024.
The effectiveness of our internal control over financial reporting as of December 31, 2024 has been audited by PricewaterhouseCoopers LLP, an independent registered public accounting firm, as stated in their report which appears in Part IV of this Annual Report.
Changes in Internal Control over Financial Reporting There were no changes in our internal control over financial reporting identified in our management’s evaluation pursuant to Rules 13a-15(d) and 15d-15(d) of the Exchange Act during the three months ended December 31, 2024 that materially affected, or are reasonably likely to materially affect, our internal control over financial reporting.
During 2023, ADT began a multi-year IT transformation project by which we are migrating much of ADT’s infrastructure to the cloud. The initiative includes certain aspects of our customer relationship management and enterprise resource planning systems. In 2024, we nationally launched our customer relationship management system for our residential pro-install customers, which resulted in changes to our processes and procedures as well as to our ICFR; however, we concluded that this launch has not currently materially affected our ICFR. In addition, we began the transition to our new enterprise resource planning system during 2024. This transition will be completed in phases over multiple years.
ITEM 9B. OTHER INFORMATION.
During the quarter ended December 31, 2024, none of the Company’s directors or executive officers adopted or terminated any contract, instruction or written plan for the purchase or sale of Company securities intended to satisfy the affirmative defense conditions of Rule 10b5-1(c) or any “non-Rule 10b5-1 trading arrangement,” as each term is defined in Item 408(a) of Regulation S-K.
71 ITEM 9C. DISCLOSURE REGARDING FOREIGN JURISDICTIONS THAT PREVENT INSPECTIONS.
Not Applicable.
72 PART III ITEM 10. DIRECTORS, EXECUTIVE OFFICERS AND CORPORATE GOVERNANCE.
The information required by this Item 10 “Directors, Executive Officers and Corporate Governance” is incorporated herein by reference from our Proxy Statement for the 2025 Annual Meeting of Stockholders (the “Proxy Statement”) to be filed with the SEC within 120 days after our fiscal year end of December 31, 2024.
ITEM 11. EXECUTIVE COMPENSATION.
The information required by this Item 11 “Executive Compensation” is incorporated herein by reference from our Proxy Statement. ITEM 12. SECURITY OWNERSHIP OF CERTAIN BENEFICIAL OWNERS AND MANAGEMENT AND RELATED STOCKHOLDER MATTERS.
The information required by this Item 12 “Security Ownership of Certain Beneficial Owners and Management and Related Stockholder Matters,” other than as set forth below as required by Item 201(d) and Item 403(c) of Regulation S-K, is incorporated herein by reference from our Proxy Statement.
Securities Authorized for Issuance Under Equity Compensation Plans The following table provides information as of December 31, 2024 with respect to shares of Common Stock issuable under our equity compensation plans. Both the 2016 Equity Incentive Plan (the “2016 Plan”) and the 2018 Omnibus Incentive Plan (the “2018 Plan”) provide for the award of stock options, restricted stock units (“RSUs”), restricted stock awards (“RSAs”), and other equity and equity-based awards to our Board of Directors, officers, and non-officer employees. There are no shares of Class B Common Stock issuable under our equity compensation plans.
Equity Compensation Plans Plan Category Number of securities to be issued upon exercise of outstanding options, warrants, and rights Weighted-average exercise price of outstanding options, warrants, and rights Number of securities remaining available for future issuance under equity compensation plans (excluding securities reflected in column (a)) (a) (b) (c) Equity compensation plans approved by stockholders: 2016 Equity Incentive Plan(1) 1,933,105 $ 6.05 2,870,032 2018 Omnibus Incentive Plan(2) 42,127,494 $ 6.97 45,587,230 Total 44,060,599 48,457,262 __ (1) Column (a) includes approximately 1 million shares of Common Stock that may be issued upon the exercise of service-based stock options and 1 million shares of Common Stock that may be issued upon the exercise of performance-based stock options. We do not expect to issue additional share-based compensation awards under the 2016 Plan.
(2) Column (a) includes approximately (i) 25 million shares of Common Stock that may be issued upon the exercise of service-based stock options, (ii) 8 million shares of Common Stock that may be issued upon the exercise of performance-based stock options and (iii) 8 million shares of Common Stock that may be issued upon the vesting of service-based RSUs.
The weighted-average exercise price in column (b) is inclusive of outstanding RSUs, which results in the issuance of shares for no consideration. Excluding RSUs, the weighted-average exercise price is equal to $9.03.
Column (c) includes an additional 50 million shares that were authorized for issuance during 2024 as disclosed on Form S-8 filed with the SEC on February 14, 2025.
73 Apollo Margin Loan Agreement As of October 3, 2019, certain investment funds directly or indirectly managed by Apollo (the “Apollo Funds”), informed the Company that they have pledged all of their shares of the Company’s Common Stock, which as of December 31, 2024, amounted to 359,150,366 shares, pursuant to a margin loan agreement and related documentation, as thereafter amended from time to time, on a non-recourse basis. Additionally, the loan to value ratio of the margin loan was equal to approximately 28.7%. The margin loan agreement contains customary default provisions and that in the event of a default under the margin loan agreement the secured parties may foreclose upon any and all shares of the Company’s Common Stock pledged to them.
Certain members of the Company’s executive team and certain employees of the Company were entitled to receive their share of the margin loan proceeds (based on their share ownership of the Apollo Funds) at such times as Apollo received its proceeds. Such persons had the option to either (a) receive such proceeds as distributed or (b) to defer receipt of such proceeds until their attributable share of the obligations under the margin loan have been satisfied in full. In the case of elections to receive such proceeds as distributed, such proceeds remain subject to recall until such time as all obligations under the margin loan agreement and related documentation are satisfied in full.
The Company has not independently verified the foregoing disclosure. When the margin loan agreement was entered into, and as requested when amended, the Company delivered customary letter agreements to the secured parties in which it has, among other things, agreed, subject to applicable law and stock exchange rules, not to take any actions that are intended to hinder or delay the exercise of any remedies by the secured parties under the margin loan agreement and related documentation, as amended. Except for the foregoing, the Company is not a party to the margin loan agreement and related documentation and does not have, and will not have, any obligations thereunder.
ITEM 14. PRINCIPAL ACCOUNTANT FEES AND SERVICES. The information required by this Item 14 “Principal Accountant Fees and Services” is incorporated herein by reference from our Proxy Statement. ITEM 13. CERTAIN RELATIONSHIPS AND RELATED TRANSACTIONS AND DIRECTOR INDEPENDENCE. The information required by this Item 13 “Certain Relationships and Related Transactions and Director Independence” is incorporated herein by reference from our Proxy Statement. 74 PART IV ITEM 15. EXHIBIT AND FINANCIAL STATEMENT SCHEDULES.
1. Financial Statements See Index to Consolidated Financial Statements appearing on page F-1.
2. Financial Statement Schedules All financial statement schedules called for under Regulation S-X are omitted because either they are not required under the related instructions, are included in the consolidated financial statements or notes thereto included elsewhere in this Annual Report on Form 10-K, or are not material.
3. Exhibits The exhibits listed on the accompanying Index to Exhibits are filed/furnished or incorporated by reference as part of this Annual Report on Form 10-K.
Index to Exhibits The information required by this Item is set forth on the exhibit index below.
Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 2.1^ Share Purchase Agreement, dated September 30, 2019, among ADT Security Holdings Canada Ltd., ADT Inc., and TELUS Communications Inc.
8-K 2.1 10/1/2019 2.2 Purchase Agreement, dated November 8, 2021, by and among The ADT Security Corporation, Compass Solar Group, LLC, MGG SPV VIII LLC, MGG SPV VII LLC, Compass Group Management, LLC, the Company Members party thereto, the Blocker Members party thereto, and ADT Inc.
10-K 2.2 3/1/2022 2.3 Equity Purchase Agreement, dated as of August 7, 2023, among ADT Inc., Iris Buyer LLC and Fire & Security Holdings, LLC 8-K 2.1 8/8/2023 3.1 Amended and Restated Certificate of Incorporation of ADT Inc.
8-K 3.1 9/17/2020 3.2 Certificate of Amendment to the Amended and Restated Certificate of Incorporation of ADT Inc.
10-Q 3.1 8/1/2024 3.3 Amended and Restated Bylaws of ADT Inc.
8-K 3.1 9/18/2023 4.1 Indenture, dated as of July 5, 2012, by and between The ADT Corporation and Wells Fargo Bank, National Association S-1 4.1 12/21/2017 4.2 Third Supplemental Indenture, dated as of July 5, 2012, by and among The ADT Corporation, Tyco International Ltd. and Wells Fargo Bank, National Association S-1 4.4 12/21/2017 4.3 Fourth Supplemental Indenture, dated as of January 14, 2013, by and between The ADT Corporation and Wells Fargo Bank, National Association S-1 4.5 12/21/2017 4.4 Sixth Supplemental Indenture, dated as of April 8, 2016, under 2012 Base Indenture, by and among The ADT Corporation, the guarantors party thereto and the Wells Fargo Bank, National Association S-1 4.7 12/21/2017 4.5 Seventh Supplemental Indenture, dated as of April 22, 2016, under 2012 Base Indenture, by and among The ADT Corporation, the guarantors party thereto and the Wells Fargo Bank, National Association S-1 4.8 12/21/2017 4.6 Eighth Supplemental Indenture, dated as of May 2, 2016, under 2012 Base Indenture, by and among Prime Finance, Inc., The ADT Corporation and the Wells Fargo Bank, National Association S-1 4.9 12/21/2017 4.7 Ninth Supplemental Indenture, dated as of November 15, 2017, under 2012 Base Indenture, by and among The ADT Security Corporation, DataShield, LLC and Wells Fargo Bank, National Association 10-K 4.10 3/11/2019 4.8 Twelfth Supplemental Indenture, dated as of January 7, 2019, under 2012 Base Indenture, by and among The ADT Security Corporation, the guarantors party thereto and Wells Fargo Bank, National Association 10-K 4.13 3/11/2019 4.9 Fifteenth Supplemental Indenture, dated as of November 14, 2019, under 2012 Base Indenture, by and among The ADT Security Corporation, I-View Now LLC and Wells Fargo Bank, National Association 10-K 4.16 3/10/2020 75 Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 4.10 Indenture, dated as of May 2, 2016, by and between Prime Security One MS, Inc. and the Wells Fargo Bank, National Association S-1 4.14 12/21/2017 4.11 First Supplemental Indenture, dated as of May 2, 2016, by and among The ADT Corporation, the guarantors party thereto and the Wells Fargo Bank, National Association S-1 4.15 12/21/2017 4.12 Second Supplemental Indenture, dated as of August 9, 2016, by and between The ADT Corporation, the Notes Guarantors and Wells Fargo Bank, National Association S-1 4.18 12/21/2017 4.13 Third Supplemental Indenture, dated as of November 15, 2017, by and among The ADT Security Corporation, DataShield, LLC and Wells Fargo Bank, National Association 10-K 4.27 3/11/2019 4.14 Sixth Supplemental Indenture, dated as of January 7, 2019, by and among The ADT Security Corporation, the guarantors party thereto and Wells Fargo Bank, National Association 10-K 4.30 3/11/2019 4.15 Ninth Supplemental Indenture, dated as of November 14, 2019, by and among The ADT Security Corporation, I-View Now LLC and Wells Fargo Bank, National Association 10-K 4.37 3/10/2020 4.16 Indenture, dated as of April 4, 2019, by and among Prime Security Services Borrower, LLC, Prime Finance Inc., the guarantors party thereto from time to time, and Wells Fargo Bank, National Association, as trustee, relating to the $750 million aggregate principal amount of 5.250% first-priority senior secured notes due 2024 8-K 4.1 4/4/2019 4.17 First Supplemental Indenture, dated as of November 14, 2019, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., I-View Now LLC and Wells Fargo Bank, National Association 10-K 4.49 3/10/2020 4.18 Indenture, dated as of April 4, 2019, by and among Prime Security Services Borrower, LLC, Prime Finance Inc., the guarantors party thereto from time to time, and Wells Fargo Bank, National Association, as trustee, relating to the $750 million aggregate principal amount of 5.750% first-priority senior secured notes due 2026.
8-K 4.2 4/4/2019 4.19 First Supplemental Indenture, dated as of September 23, 2019, by and among Prime Security Services Borrower, LLC, Prime Finance Inc., the guarantors party thereto, and Wells Fargo Bank, National Association, as trustee, relating to the $600 million aggregate principal amount of 5.750% first-priority senior secured notes due 2026 8-K 4.1 9/24/2019 4.20 Second Supplemental Indenture, dated as of November 14, 2019, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., I-View Now LLC and Wells Fargo Bank, National Association 10-K 4.53 3/10/2020 4.21 Indenture, dated as of January 28, 2020, by and among Prime Security Services Borrower, LLC, Prime Finance Inc., the guarantors party thereto from time to time and Wells Fargo Bank, National Association, as trustee, relating to the $1,300 million aggregate principal amount of 6.250% Second-Priority Senior Secured Notes due 2028 8-K 4.1 1/28/2020 4.22 Indenture, dated as of August 20, 2020, by and among Prime Security Services Borrower, LLC, Prime Finance Inc., the guarantors party hereto from time to time, and Wells Fargo Bank, National Association, as trustee, relating to $1,000 million aggregate principal amount of 3.375% first-priority senior secured notes due 2027 8-K 4.1 8/20/2020 4.23 First-Priority Senior Secured Notes Indenture, dated as of July 29, 2021, by and among The ADT Security Corporation, Prime Security Services Borrower, LLC, the guarantors party thereto, and Wells Fargo Bank, National Association, as trustee and collateral agent 8-K 4.1 7/29/2021 4.24 Seventeenth Supplemental Indenture, dated as of January 7, 2022, under 2012 Base Indenture, by and among The ADT Security Corporation, Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.1 5/6/2022 4.25 Eighteenth Supplemental Indenture, dated as of April 28, 2022, under 2012 Base Indenture, by and among The ADT Security Corporation, ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.2 5/6/2022 4.26 Eleventh Supplemental Indenture, dated as of January 7, 2022, by and among The ADT Security Corporation, Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.3 5/6/2022 4.27 Twelfth Supplemental Indenture, dated as of April 28, 2022, by and among The ADT Security Corporation, ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.4 5/6/2022 4.28 Third Supplemental Indenture, dated as of January 7, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.5 5/6/2022 4.29 Fourth Supplemental Indenture, dated as of April 28, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.6 5/6/2022 76 Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 4.30 Fourth Supplemental Indenture, dated as of January 7, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.7 5/6/2022 4.31 Fifth Supplemental Indenture, dated as of April 28, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.8 5/6/2022 4.32 First Supplemental Indenture, dated as of January 7, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.9 5/6/2022 4.33 Second Supplemental Indenture, dated as of April 28, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.10 5/6/2022 4.34 Second Supplemental Indenture, dated as of January 7, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.11 5/6/2022 4.35 Third Supplemental Indenture, dated as of April 28, 2022, by and among Prime Security Services Borrower, LLC, Prime Finance, Inc., ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.12 5/6/2022 4.36 First Supplemental Indenture, dated as of January 7, 2022, by and among The ADT Security Corporation, Compass Solar Group, LLC, Marc Jones Construction, L.L.C., Buildpro, L.L.C., Energypro LLC and Wells Fargo Bank, National Association 10-Q 4.13 5/6/2022 4.37 Second Supplemental Indenture, dated as of April 28, 2022, by and among The ADT Security Corporation, ADT Innovation LLC and Wells Fargo Bank, National Association 10-Q 4.14 5/6/2022 4.38 Description of Securities 10-K 4.24 3/1/2022 10.1 Subsidiary Guarantee Agreement (First Lien), dated July 1, 2015, among the Subsidiaries of Prime Security Services Borrower, LLC named therein and Credit Suisse AG, Cayman Islands Branch, as Collateral Agent S-1 10.2 12/21/2017 10.2 Supplement No. 1, dated as of May 2, 2016, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Securities Services Borrower, LLC and Barclays Bank PLC, as Collateral Agent S-1 10.6 12/21/2017 10.3 Supplement No. 2, dated as of October 31, 2017, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-K 10.55 3/11/2019 10.4 Supplement No. 3, dated as of January 22, 2018, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-K 10.56 3/11/2019 10.5 Supplement No. 4, dated as of February 28, 2018, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-K 10.57 3/11/2019 10.6 Supplement No. 5, dated as of August 17, 2018, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-K 10.58 3/11/2019 10.7 Supplement No. 6, dated as of January 2, 2019, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-K 10.59 3/11/2019 10.8 Supplement No. 7, dated as of January 30, 2019, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-K 10.60 3/11/2019 10.9 Supplement No. 8, dated as of March 11, 2019, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-Q 10.13 5/7/2019 10.10 Supplement No. 11, dated as of January 7, 2022, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-Q 10.4 5/6/2022 10.11 Supplement No. 12, dated as of April 28, 2022, to the Subsidiary Guarantee Agreement (First Lien) dated as of July 1, 2015, by each subsidiary of Prime Security Services Borrower, LLC party thereto and Barclays Bank PLC, as Collateral Agent 10-Q 10.5 5/6/2022 77 Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 10.12 Holdings Guarantee and Pledge Agreement (First Lien), dated and effective as of July 1, 2015, between Prime Security Services Holdings, LLC, as Holdings, and Credit Suisse AG, Cayman Islands Branch, as Collateral Agent S-1 10.4 12/21/2017 10.13 Collateral Agreement (First Lien), dated as of July 1, 2015 among Prime Security Services Borrower, LLC, each Subsidiary of Prime Security Services Borrower, LLC from time to time identified therein as a party and Barclays Bank PLC, as collateral agent S-1 10.5 12/21/2017 10.14 Collateral Agreement (Second Lien), dated as of January 28, 2020, among Prime Security Services Borrower LLC, as Issuer, Prime Finance, Inc., as Co-Issuer, each Subsidiary Guarantor party thereto and Wells Fargo Bank, National Association, as Collateral Agent 8-K 10.1 1/28/2020 10.15 First Lien/First Lien Intercreditor Agreement, dated as of May 2, 2016 among Barclays Bank PLC, as Collateral Agent, Barclays Bank PLC, as Authorized Representative under the Credit Agreement, Wells Fargo Bank, National Association, as the Initial Other Authorized Representative, and each additional Authorized Representative from time to time party hereto relating to Prime Security Services Borrower, LLC S-1 10.8 12/21/2017 10.16 First Lien/Second Lien Intercreditor Agreement, dated as of July 1, 2015, between Credit Suisse AG, Cayman Islands Branch, as First Lien Facility Agent and Applicable First Lien Agent, and Credit Suisse AG, Cayman Islands Branch, as Second Lien Facility Agent and Applicable Second Lien Agent relating to Prime Security Services Borrower, LLC S-1 10.9 12/21/2017 10.17 Receivables Purchase Agreement, dated as of March 5, 2020, among ADT LLC, individually and as servicer, ADT Finance LLC, as seller, various purchasers and purchaser agents from time to time party thereto, and Mizuho Bank, LTD., as Administrative Agent, Arranger, Collateral Agent and Structuring Agent 10-K 10.21 3/10/2020 10.18 Agreement of Amendment to Receivables Purchase Agreement, dated as of April 17, 2020, among ADT LLC, individually and as servicer, ADT Finance LLC, as seller, various purchasers and purchaser agents from time to time party thereto, and Mizuho Bank, LTD., as Administrative Agent, Arranger, Collateral Agent and Structuring Agent 10-Q 10.22 5/7/2020 10.19 Second Agreement of Amendment to Receivables Purchase Agreement, dated September 17, 2020, among ADT LLC, individually and as servicer, ADT Finance LLC, as seller, various purchasers and purchaser agents from time to time party thereto, and Mizuho Bank, LTD., as Administrative Agent, Arranger, Collateral Agent and Structuring Agent 10-Q 10.23 11/5/2020 10.20 Third Agreement of Amendment to Receivables Purchase Agreement, dated January 29, 2021, among ADT LLC, individually and as servicer, ADT Finance LLC, as seller, various purchasers and purchaser agents from time to time party thereto, and Mizuho Bank, LTD., as Administrative Agent, Arranger, Collateral Agent and Structuring Agent 10-K 10.25 2/25/2021 10.21 Fourth Agreement of Amendment to Receivables Purchase Agreement, dated March 5, 2021, among ADT LLC, individually and as servicer, ADT Finance LLC, as seller, various purchasers and purchaser agents from time to time party thereto, and Mizuho Bank, LTD., as Administrative Agent, Arranger, Collateral Agent and Structuring Agent 10-Q 10.26 5/5/2021 10.22 Receivables Sale and Contribution Agreement, dated as of April 17, 2020, between ADT LLC, as Originator and Servicer and ADT Finance Inc., as Buyer 10-Q 10.23 5/7/2020 10.23^ Receivables Financing Agreement, among ADT LLC, ADT Finance LLC and Mizuho Bank, Ltd., dated as of July 16, 2021 8-K 10.1 7/19/2021 10.24^ First Amendment and Joinder to the Receivables Financing Agreement and the Receivables Sale and Contribution Agreement, among ADT LLC, ADT Finance LLC, Mizuho Bank, Ltd., MUFG Bank, Ltd., BNP Paribas, and Starbird Funding Corporation, dated as of October 29, 2021 8-K 10.1 10/29/2021 10.25 Second Amendment to the Receivables Financing Agreement, among ADT LLC, ADT Finance LLC and Mizuho Bank, LTD., dated as of December 10, 2021 10-K 10.24 3/1/2022 10.26 Bank Rate Amendment to the Receivables Financing Agreement, among ADT LLC, ADT Finance LLC and Mizuho Bank, Ltd., dated as of April 20, 2022 10-Q 10.3 5/6/2022 10.27 Third Amendment dated as of May 20, 2022, to Receivables Financing Agreement, among ADT LLC, ADT Finance LLC, and Mizuho Bank, Ltd., dated as of July 16, 2021 10-Q 10.3 8/4/2022 10.28 Trademark Agreement, dated as of September 25, 2012, by and among ADT Services GmbH, ADT US Holdings, Inc., Tyco International Ltd. and The ADT Corporation S-1 10.19 12/21/2017 10.29 Patent Agreement, dated as of September 26, 2012, by and between Tyco International Ltd. and The ADT Corporation S-1 10.20 12/21/2017 10.30 Separation and Distribution Agreement, dated September 26, 2012 by and among Tyco International Ltd., Tyco International Finance S.A., The ADT Corporation and ADT LLC S-1 10.21 12/21/2017 10.31 ADT LLC Supplemental Savings and Retirement Plan, effective as of April 1, 2017 S-1 10.22 12/21/2017 10.32 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of December 18, 2017 10-K 10.31 2/25/2021 78 Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 10.33 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of December 24, 2018 10-K 10.32 2/25/2021 10.34 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of December 31, 2018 10-K 10.33 2/25/2021 10.35 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of February 8, 2019 10-K 10.34 2/25/2021 10.36 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of July 29, 2019 10-K 10.35 2/25/2021 10.37 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of December 15, 2021 10-Q 10.4 8/4/2022 10.38 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of January 1, 2022 10-Q 10.13 11/3/2022 10.39 Amendment to the ADT LLC Supplemental Savings and Retirement Plan, effective as of February 22, 2023 10-K 10.40 2/28/2023 10.40 Amended and Restated Stockholders Agreement, dated as of December 14, 2018, by and between the ADT Inc. and Prime Security Services TopCo Parent, L.P.
10-K 10.34 3/1/2022 10.41 Registration Rights Agreement by and between the ADT Inc. and Prime Securities Services TopCo, LP 10-K 10.24 3/15/2018 10.42 Amendment to the Registration Rights Agreement between ADT Inc. and Prime Security Services TopCo Parent, L.P.
10-Q 10.10 8/9/2018 10.43 Securities Purchase Agreement, dated as of July 31, 2020, by and between ADT Inc. and Google LLC 8-K 10.1 8/3/2020 10.44 Investor Rights Agreement, dated as of September 17, 2020, by and between ADT Inc. and Google LLC 8-K 10.1 9/17/2020 10.45 Amendment No. 1 to Investor Rights Agreement, dated as of September 5, 2022, by and between ADT Inc. and Google LLC 10-K 10.46 2/28/2024 10.46 Amendment No. 2 to Investor Rights Agreement, dated as of December 29, 2023, by and between ADT Inc. and Google LLC 10-K 10.47 2/28/2024 10.47+ Amended and Restated Employment Agreement, dated December 19, 2017, between The ADT Security Corporation (together with any of its subsidiaries and Affiliates) and Donald Young S-1 10.25 12/21/2017 10.48+ Amendment to Amended and Restated Employment Agreement, dated May 3, 2019, between The ADT Security Corporation (together with any of its subsidiaries and Affiliates) and Donald Young 10-Q 10.27 5/7/2019 10.49+ Amended and Restated Employment Agreement, dated December 19, 2017, between ADT LLC, (together with any of its subsidiaries and Affiliates) and Jeffrey Likosar S-1 10.31 12/21/2017 10.50+ Employment Offer Letter, dated February 1, 2019, between ADT LLC (with its Affiliates and Successors) and David Smail 10-K 10.51 2/28/2023 10.51+ ADT Inc. 2018 Omnibus Incentive Plan S-1 10.32 12/21/2017 10.52+ First Amendment to ADT Inc. 2018 Omnibus Incentive Plan dated April 25, 2019 10-Q 10.33 8/6/2019 10.53+ Second Amendment to ADT Inc. 2018 Omnibus Incentive Plan, dated May 22, 2024 10.Q 10.5 8/1/2024 10.54+ Form of Restricted Stock Unit Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan S-1 10.33 12/21/2017 10.55+ Form of Non-Qualified Option Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan S-1 10.34 12/21/2017 10.56+ Form of Non-Qualified Option Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan (Class B Unit Redemption) S-1/A 10.35 1/8/2018 10.57+ Form of Amendment to Non-Qualified Award Agreement for use under ADT Inc. 2018 Omnibus Incentive Plan (Class B Unit Redemption) 10-Q 10.37 8/6/2019 10.58+ Form of Common Stock Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan S-1/A 10.36 1/8/2018 10.59+ ADT Inc. 2018 Omnibus Incentive Plan Restricted Stock Unit Non-Employee Director Award Agreement 10-Q 10.9 8/9/2018 10.60+ Form of Restricted Stock Unit Special Equity Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan 10-Q 10.47 5/7/2020 10.61+ Form of Non-Qualified Option Special Equity Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan 10-Q 10.48 5/7/2020 79 Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 10.62+ Form of Restricted Stock Unit Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan effective as of January 1, 2022 10-Q 10.5 11/9/2021 10.63+ Form of Non-Qualified Option Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan effective as of January 1, 2022 10-Q 10.6 11/9/2021 10.64+ Form of Restricted Stock Unit Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan effective as of August 1, 2022 SC TO-I (d)(34) 9/12/2022 10.65+ Form of Non-Qualified Option Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan effective as of August 1, 2022 SC TO-I (d)(35) 9/12/2022 10.66+ Second Amended & Restated Employment Agreement with James D. DeVries 10-Q 10.12 11/8/2018 10.67+ Amendment to Second Amended & Restated Employment Agreement of James D. DeVries 8-K 10.2 12/3/2018 10.68+ ADT Inc. Annual Incentive Plan 8-K 10.1 7/29/2021 10.69 Investor Rights Agreement, dated December 8, 2021, by and among ADT Inc. and the Holders party thereto 10-K 10.63 3/1/2022 10.70 Securities Purchase Agreement, dated as of September 5, 2022, by and between ADT Inc. and State Farm Fire & Casualty Company 8-K 10.1 9/6/2022 10.71 Tender and Support Agreement, dated as of September 5, 2022, by and between ADT Inc., Prime Security Services TopCo (ML), L.P. and Prime Security Services TopCo (ML II), L.P 8-K 10.2 9/6/2022 10.72 Support Agreement, dated as of September 5, 2022, by and between ADT Inc. and Google LLC 8-K 10.3 9/6/2022 10.73+ Form of Performance Unit Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan SC TO-I (d)(20) 9/12/2022 10.74+ Form of Time and Performance Vesting Non-Qualified Option Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan SC TO-I (d)(21) 9/12/2022 10.75+ Form of Time and Performance Vesting Restricted Stock Unit Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan SC TO-I (d)(22) 9/12/2022 10.76+ Form of Non-Qualified Option Award Agreement for use under the Prime Security Services Parent, Inc. 2016 Equity Incentive Plan SC TO-I (d)(24) 9/12/2022 10.77+ Form of Non-Qualified Option Award Agreement for use under the ADT Inc. 2018 Omnibus Incentive Plan, effective as of March 8, 2024 10-Q 10.2 4/25/2024 10.78+ Form of Indemnification Agreement by and between the Company and each of its Directors and Executive Officers 10-Q 10.3 10/24/2024 10.79 Commitment Letter by and among Prime Security Services Borrower, LLC, The ADT Security Corporation, Deutsche Bank Securities Inc., Deutsche Bank AG New York Branch, BNP Paribas, Mizuho Bank, Ltd., MUFG Bank, Ltd., Citizens Bank, N.A., Citigroup Global Markets Inc., Morgan Stanley Senior Funding, Inc., Royal Bank of Canada, Barclays Bank PLC, ING Capital LLC and Credit Suisse AG, New York Branch, dated as of September 15, 2022 8-K 10.1 9/15/2022 10.80 Investor Rights Agreement, dated as of October 13, 2022, by and between ADT Inc. and State Farm Fire & Casualty Company 8-K 10.1 10/13/2022 10.81+ Amended and Restated Management Investor Rights Agreement, dated as of January 23, 2018, by and among ADT Inc., Prime Security Services TopCo Parent and the Holders party thereto 10-K 10.39 3/15/2018 10.82+ Amendment No.1 to Amended and Restated Management Investor Rights Agreement, dated as of December 9, 2022, by and among ADT Inc., Prime Security Services TopCo Parent and the Holders party thereto 8-K 10.1 12/15/2022 10.83+^ Amendment No.2 to Amended and Restated Management Investor Rights Agreement, dated as of August 1, 2024, by and among ADT Inc., Prime Security Services TopCo Parent and the Holders party thereto 10-Q 10.6 8/1/2024 10.84 Term Loan Credit Agreement, by and among Prime Security Services Holdings, LLC, as Holdings, Prime Security, Services Borrower, LLC and the ADT Security Corporation, as borrowers, Barclays Bank PLC, as Administrative Agent, and the lenders party thereto, as dated as of March 14, 2023 8-K 10.1 3/15/2023 10.85 Incremental Assumption and Amendment Agreement No. 18, dated as of December 4, 2024, by and among Prime Security Services Holdings, LLC, Prime Security Services Borrower, LLC, The ADT Security Corporation, the subsidiary loan parties party thereto, the lenders party thereto and Barclays Bank PLC, as administrative agent.
8-K 10.1 12/04/2024 10.86 Receivables Financing Agreement, among Compass Solar Group, LLC, ADT Solar Finance LLC and Mizuho Bank, Ltd., dated as of August 2, 2023.
10-Q 10.2 11/02/2023 10.87 Sixth Amendment to the Receivables Financing Agreement, among ADT Finance LLC, Mizuho Bank, Ltd., ADT LLC, MUFG Bank, Ltd., Starbird Funding Corporation, and BNP Paribas, dated as of April 10, 2024 8-K 10.1 4/12/2024 80 Exhibit Number Incorporated by Reference Exhibit Description Form Exhibit Filing Date 19 ADT Inc. Insider Trading Policy 21 Subsidiaries of ADT Inc.
23 Consent of Independent Registered Public Accounting Firm 31.1 Certification of CEO, pursuant to SEC Rule 13a-14(a) and 15d-14(a) 31.2 Certification of CFO, pursuant to SEC Rule 13a-14(a) and 15d-14(a) 32.1 Certification by the CEO, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 32.2 Certification by the CFO, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 97.1 Policy Relating to Incentive Compensation Clawback 10-K 97.1 2/28/2024 101 XBRL Instance Document - the instance document does not appear in the Interactive Data File because its XBRL tags are embedded within the Inline XBRL document.
104 Cover Page Interactive Data File - the cover page XBRL tags are embedded within the Inline XBRL document ___ ^ Confidential treatment requested. Confidential portions of this exhibit have been removed.
Filed herewith.
Furnished herewith.
+ Management contract or compensatory plan or arrangement.
ITEM 16. FORM 10-K SUMMARY.
None.
81 SIGNATURES Pursuant to the requirements of Section 13 or 15(d) of the Securities Exchange Act of 1934, the registrant has duly caused this report to be signed on its behalf by the undersigned, thereunto duly authorized.
ADT Inc.
Date: February 27, 2025 By: /s/ James D. DeVries Name: James D. DeVries Title: Chairman, President and Chief Executive Officer Pursuant to the requirements of the Securities Exchange Act of 1934, this report has been signed below by the following persons on behalf of the registrant and in the capacities indicated on February 27, 2025.
Name Title /s/ James D. DeVries Chairman, President and Chief Executive Officer (Principal Executive Officer) James D. DeVries /s/ Jeffrey Likosar President, Corporate Development and Transformation, and Chief Financial Officer (Principal Financial Officer) Jeffrey Likosar /s/ Steven Burzo Vice President, Chief Accounting Officer and Controller (Principal Accounting Officer) Steven Burzo /s/ Nicole Bonsignore Director Nicole Bonsignore /s/ Marques Coleman Director Marques Coleman /s/ Thomas M. Gartland Director Thomas M. Gartland /s/ Tracey R. Griffin Director Tracey R. Griffin /s/ Benjamin Honig Director Benjamin Honig /s/ Daniel Houston Director Daniel Houston /s/ William M. Lewis, Jr Director William M. Lewis, Jr /s/ Reed B. Rayman Director Reed B. Rayman /s/ Paul J. Smith Director Paul J. Smith /s/ Lee J. Solomon Director Lee J. Solomon /s/ Danielle Tiedt Director Danielle Tiedt /s/ Matthew E. Winter Director Matthew E. Winter /s/ Suzanne Yoon Director Suzanne Yoon /s/ Sigal Zarmi Director Sigal Zarmi 82 INDEX TO CONSOLIDATED FINANCIAL STATEMENTS Page Report of Independent Registered Public Accounting Firm (PCAOB ID 238) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F-2 Consolidated Balance Sheets as of December 31, 2024 and 2023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F-4 Consolidated Statements of Operations for the years ended December 31, 2024, 2023, and 2022 . . . . . . . . . . . . . . . .
F-5 Consolidated Statements of Comprehensive Income (Loss) for the years ended December 31, 2024, 2023, and 2022 F-6 Consolidated Statements of Stockholders’ Equity for the years ended December 31, 2024, 2023, and 2022 . . . . . . . .
F-7 Consolidated Statements of Cash Flows for the years ended December 31, 2024, 2023, and 2022 . . . . . . . . . . . . . . .
F-8 Notes to Consolidated Financial Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F-9 1. Description of Business and Summary of Significant Accounting Policies .........................................................
F-9 2. Revenue and Receivables .....................................................................................................................................
F-15 3. Segment Information ............................................................................................................................................
F-18 4. Divestitures ...........................................................................................................................................................
F-20 5. Equity Method Investments ..................................................................................................................................
F-23 6. Goodwill and Other Intangible Assets ..................................................................................................................
F-23 7. Debt .......................................................................................................................................................................
F-27 8. Derivative Financial Instruments ..........................................................................................................................
F-34 9. Income Taxes ........................................................................................................................................................
F-36 10. Equity ..................................................................................................................................................................
F-39 11. Share-Based Compensation ................................................................................................................................
F-43 12. Net Income (Loss) per Share ..............................................................................................................................
F-46 13. Commitments and Contingencies .......................................................................................................................
F-48 14. Leases .................................................................................................................................................................
F-49 15. Retirement Plans .................................................................................................................................................
F-51 16. Related Party Transactions .................................................................................................................................
F-52 17. Condensed Financial Information of Registrant .................................................................................................
F-55 18. Selected Quarterly Financial Data (Unaudited) ..................................................................................................
F-58 F-1 Report of Independent Registered Public Accounting Firm To the Board of Directors and Stockholders of ADT Inc.
Opinions on the Financial Statements and Internal Control over Financial Reporting We have audited the accompanying consolidated balance sheets of ADT Inc. and its subsidiaries (the “Company”) as of December 31, 2024 and 2023, and the related consolidated statements of operations, comprehensive income (loss), stockholders’ equity, and cash flows for each of the three years in the period ended December 31, 2024, including the related notes (collectively referred to as the “consolidated financial statements”). We also have audited the Company’s internal control over financial reporting as of December 31, 2024, based on criteria established in Internal Control - Integrated Framework (2013) issued by the Committee of Sponsoring Organizations of the Treadway Commission (COSO).
In our opinion, the consolidated financial statements referred to above present fairly, in all material respects, the financial position of the Company as of December 31, 2024 and 2023, and the results of its operations and its cash flows for each of the three years in the period ended December 31, 2024 in conformity with accounting principles generally accepted in the United States of America. Also in our opinion, the Company maintained, in all material respects, effective internal control over financial reporting as of December 31, 2024, based on criteria established in Internal Control - Integrated Framework (2013) issued by the COSO.
Basis for Opinions The Company's management is responsible for these consolidated financial statements, for maintaining effective internal control over financial reporting, and for its assessment of the effectiveness of internal control over financial reporting, included in Management's Report on Internal Control over Financial Reporting appearing under Item 9A. Our responsibility is to express opinions on the Company’s consolidated financial statements and on the Company’s internal control over financial reporting based on our audits. We are a public accounting firm registered with the Public Company Accounting Oversight Board (United States) (PCAOB) and are required to be independent with respect to the Company in accordance with the U.S. federal securities laws and the applicable rules and regulations of the Securities and Exchange Commission and the PCAOB.
We conducted our audits in accordance with the standards of the PCAOB. Those standards require that we plan and perform the audits to obtain reasonable assurance about whether the consolidated financial statements are free of material misstatement, whether due to error or fraud, and whether effective internal control over financial reporting was maintained in all material respects.
Our audits of the consolidated financial statements included performing procedures to assess the risks of material misstatement of the consolidated financial statements, whether due to error or fraud, and performing procedures that respond to those risks. Such procedures included examining, on a test basis, evidence regarding the amounts and disclosures in the consolidated financial statements. Our audits also included evaluating the accounting principles used and significant estimates made by management, as well as evaluating the overall presentation of the consolidated financial statements. Our audit of internal control over financial reporting included obtaining an understanding of internal control over financial reporting, assessing the risk that a material weakness exists, and testing and evaluating the design and operating effectiveness of internal control based on the assessed risk. Our audits also included performing such other procedures as we considered necessary in the circumstances. We believe that our audits provide a reasonable basis for our opinions.
Definition and Limitations of Internal Control over Financial Reporting A company’s internal control over financial reporting is a process designed to provide reasonable assurance regarding the reliability of financial reporting and the preparation of financial statements for external purposes in accordance with generally accepted accounting principles. A company’s internal control over financial reporting includes those policies and procedures that (i) pertain to the maintenance of records that, in reasonable detail, accurately and fairly reflect the transactions and dispositions of the assets of the company; (ii) provide reasonable assurance that transactions are recorded as necessary to permit preparation of financial statements in accordance with generally accepted accounting principles, and that receipts and expenditures of the company are being made only in accordance with authorizations of management and directors of the company; and (iii) provide reasonable assurance regarding prevention or timely detection of unauthorized acquisition, use, or disposition of the company’s assets that could have a material effect on the financial statements.
Because of its inherent limitations, internal control over financial reporting may not prevent or detect misstatements. Also, projections of any evaluation of effectiveness to future periods are subject to the risk that controls may become inadequate because of changes in conditions, or that the degree of compliance with the policies or procedures may deteriorate.
F-2 Critical Audit Matters The critical audit matter communicated below is a matter arising from the current period audit of the consolidated financial statements that was communicated or required to be communicated to the audit committee and that (i) relates to accounts or disclosures that are material to the consolidated financial statements and (ii) involved our especially challenging, subjective, or complex judgments. The communication of critical audit matters does not alter in any way our opinion on the consolidated financial statements, taken as a whole, and we are not, by communicating the critical audit matter below, providing a separate opinion on the critical audit matter or on the accounts or disclosures to which it relates.
Discontinued Operations – Solar Business As described in Note 4 to the consolidated financial statements, on January 19, 2024, the Company’s board of directors approved a plan to fully exit the solar business (the “ADT Solar Exit”). As of June 30, 2024, substantially all operations of the Solar Business had ceased, and the solar business is presented as a discontinued operation. As a result, the Company presented total liabilities of discontinued operations of $47.7 million as of December 31, 2024 and a loss from discontinued operations, net of tax of $110.1 million for the year ended December 31, 2024.
The principal considerations for our determination that performing procedures relating to the discontinued operations of the solar business is a critical audit matter are a high degree of auditor subjectivity and effort in performing procedures and evaluating audit evidence related to management’s assessment, classification, timing and disclosure of the discontinued operations.
Addressing the matter involved performing procedures and evaluating audit evidence in connection with forming our overall opinion on the consolidated financial statements. These procedures included testing the effectiveness of controls related to management’s discontinued operations assessment, including controls related to management’s classification, timing, and disclosure of the discontinued operations in the Company’s consolidated financial statements. These procedures also included, among others, (i) evaluating management’s assessment that the ADT Solar Exit was a discontinued operation; (ii) testing the classification of amounts included in discontinued operations, including agreeing such amounts to the Company’s historical accounting records; (iii) evaluating the sufficiency of the disclosures in the consolidated financial statements; and (iv) evaluating the reasonableness of the timing of management’s recognition of the discontinued operation. Evaluating the reasonableness of the timing of the discontinued operations involved, (i) evaluating management’s plan to wind down the solar business, including the reasonableness of projected exit charges; (ii) testing the completeness and accuracy of incurred exit charges; and (iii) evaluating whether substantially all operations ceased as of June 30, 2024.
/s/ PricewaterhouseCoopers LLP Miami, Florida February 27, 2025 We have served as the Company’s auditor since 2010.
F-3 ADT INC. AND SUBSIDIARIES CONSOLIDATED BALANCE SHEETS (in thousands, except share and per share data) December 31, 2024 2023 Assets Current assets: Cash and cash equivalents $ 96,212 $ 14,621 Restricted cash and restricted cash equivalents 107,853 115,329 Accounts receivable, net of allowance for credit losses of $57,795 and $46,850, respectively 393,511 370,201 Inventories, net 196,731 201,394 Prepaid expenses and other current assets 210,613 242,192 Current assets of discontinued operations — 60,957 Total current assets 1,004,920 1,004,694 Property and equipment, net 247,183 253,658 Subscriber system assets, net 2,981,161 3,005,936 Intangible assets, net 4,854,099 4,877,493 Goodwill 4,903,899 4,903,899 Deferred subscriber acquisition costs, net 1,324,376 1,175,904 Other assets 735,319 699,231 Noncurrent assets of discontinued operations — 43,279 Total assets $ 16,050,957 $ 15,964,094 Liabilities and stockholders' equity Current liabilities: Current maturities of long-term debt $ 195,791 $ 312,061 Accounts payable 153,537 277,201 Deferred revenue 247,785 255,221 Accrued expenses and other current liabilities 634,904 556,114 Current liabilities of discontinued operations 31,763 79,611 Total current liabilities 1,263,780 1,480,208 Long-term debt 7,511,282 7,513,456 Deferred subscriber acquisition revenue 2,067,608 1,914,954 Deferred tax liabilities 1,167,213 1,027,189 Other liabilities 224,384 219,069 Noncurrent liabilities of discontinued operations 15,889 20,572 Total liabilities 12,250,156 12,175,448 Commitments and contingencies (See Note 13) Stockholders' equity: Preferred stock—authorized 1,000,000 shares of $0.01 par value; zero issued and outstanding as of December 31, 2024 and 2023 — — Common stock—authorized 3,999,000,000 shares of $0.01 par value; issued and outstanding shares of 836,589,761 and 867,432,337 as of December 31, 2024 and 2023, respectively 8,366 8,674 Class B common stock—authorized 100,000,000 shares of $0.01 par value; issued and outstanding shares of 54,744,525 as of December 31, 2024 and 2023 547 547 Additional paid-in capital 7,117,098 7,413,305 Accumulated deficit (3,318,174) (3,617,718) Accumulated other comprehensive income (loss) (7,036) (16,162) Total stockholders' equity 3,800,801 3,788,646 Total liabilities and stockholders' equity $ 16,050,957 $ 15,964,094 See Notes to Consolidated Financial Statements F-4 ADT INC. AND SUBSIDIARIES CONSOLIDATED STATEMENTS OF OPERATIONS (in thousands, except per share data) Years Ended December 31, 2024 2023 2022 Revenue: Monitoring and related services $ 4,293,477 $ 4,178,998 $ 4,053,048 Security installation, product, and other 604,969 473,826 328,856 Total revenue 4,898,446 4,652,824 4,381,904 Cost of revenue (exclusive of depreciation and amortization shown separately below): Monitoring and related services 617,386 604,368 596,664 Security installation, product, and other 229,728 147,314 102,118 Total cost of revenue 847,114 751,682 698,782 Selling, general, and administrative expenses 1,476,346 1,347,738 1,348,281 Depreciation and intangible asset amortization 1,342,798 1,335,484 1,599,810 Merger, restructuring, integration, and other 24,124 38,959 9,937 Operating income (loss) 1,208,064 1,178,961 725,094 Interest expense, net (441,031) (569,915) (263,068) Loss on extinguishment of debt (4,802) (16,621) — Other income (expense) 52,939 11,958 (57,568) Income (loss) from continuing operations before income taxes and equity in net earnings (losses) of equity method investee 815,170 604,383 404,458 Income tax benefit (expense) (195,780) (160,585) (87,692) Income (loss) from continuing operations before equity in net earnings (losses) of equity method investee 619,390 443,798 316,766 Equity in net earnings (losses) of equity method investee — 6,572 (4,601) Income (loss) from continuing operations 619,390 450,370 312,165 Income (loss) from discontinued operations, net of tax (118,337) 12,639 (179,502) Net income (loss) $ 501,053 $ 463,009 $ 132,663 Common Stock: Income (loss) from continuing operations per share - basic $ 0.69 $ 0.49 $ 0.35 Income (loss) from continuing operations per share - diluted $ 0.66 $ 0.47 $ 0.33 Net income (loss) per share - basic $ 0.56 $ 0.51 $ 0.15 Net income (loss) per share - diluted $ 0.52 $ 0.48 $ 0.14 Weighted-average shares outstanding - basic 846,521 856,843 848,465 Weighted-average shares outstanding - diluted 908,700 919,149 915,069 Class B Common Stock: Income (loss) from continuing operations per share - basic $ 0.69 $ 0.49 $ 0.35 Income (loss) from continuing operations per share - diluted $ 0.66 $ 0.47 $ 0.33 Net income (loss) per share - basic $ 0.56 $ 0.51 $ 0.15 Net income (loss) per share - diluted $ 0.52 $ 0.48 $ 0.14 Weighted-average shares outstanding - basic 54,745 54,745 54,745 Weighted-average shares outstanding - diluted 54,745 54,745 54,745 See Notes to Consolidated Financial Statements F-5 ADT INC. AND SUBSIDIARIES CONSOLIDATED STATEMENTS OF COMPREHENSIVE INCOME (LOSS) (in thousands) Years Ended December 31, 2024 2023 2022 Net income (loss) $ 501,053 $ 463,009 $ 132,663 Other comprehensive income (loss), net of tax: Cash flow hedges 6,008 32,129 25,754 Other 3,118 (1,091) (3,981) Total other comprehensive income (loss), net of tax 9,126 31,038 21,773 Comprehensive income (loss) $ 510,179 $ 494,047 $ 154,436 See Notes to Consolidated Financial Statements F-6 ADT INC. AND SUBSIDIARIES CONSOLIDATED STATEMENTS OF STOCKHOLDERS’ EQUITY (in thousands) Number of Common Shares Number of Class B Common Shares Common Stock Class B Common Stock Additional Paid-In Capital Accumulated Deficit Accumulated Other Comprehensive Income (Loss) Total Stockholders' Equity Balance as of December 31, 2021 846,826 54,745 $ 8,468 $ 547 $ 7,261,267 $ (3,952,590) $ (68,973) $ 3,248,719 Net income (loss) — — — — — 132,663 — 132,663 Other comprehensive income (loss), net of tax — — — — — — 21,773 21,773 Issuance of common stock, net of expenses 140,681 — 1,407 — 1,188,488 — — 1,189,895 Repurchases of common stock (133,333) — (1,333) — (1,093,334) — — (1,094,667) Dividends — — — — — (127,835) — (127,835) Share-based compensation expense — — — — 66,566 — — 66,566 Contingent forward purchase contract — — — — (41,938) — — (41,938) Transactions related to employee share-based compensation plans and other 7,924 — 79 — (290) (1,817) — (2,028) Balance as of December 31, 2022 862,098 54,745 $ 8,621 $ 547 $ 7,380,759 $ (3,949,579) $ (47,200) $ 3,393,148 Net income (loss) — — — — — 463,009 — 463,009 Other comprehensive income (loss), net of tax — — — — — — 31,038 31,038 Dividends — — — — — (129,025) — (129,025) Share-based compensation expense — — — — 51,137 — — 51,137 Transactions related to employee share-based compensation plans and other 5,334 — 53 — (18,591) (2,123) — (20,661) Balance as of December 31, 2023 867,432 54,745 $ 8,674 $ 547 $ 7,413,305 $ (3,617,718) $ (16,162) $ 3,788,646 Net income (loss) — — — — — 501,053 — 501,053 Other comprehensive income (loss), net of tax — — — — — — 9,126 9,126 Dividends — — — — — (199,398) — (199,398) Share-based compensation expense — — — — 48,613 — — 48,613 Repurchases of common stock (including excise tax) (36,000) — (360) — (346,576) — — (346,936) Transactions related to employee share-based compensation plans and other 5,158 — 52 — 1,756 (2,111) — (303) Balance as of December 31, 2024 836,590 54,745 $ 8,366 $ 547 $ 7,117,098 $ (3,318,174) $ (7,036) $ 3,800,801 See Notes to Consolidated Financial Statements F-7 ADT INC. AND SUBSIDIARIES CONSOLIDATED STATEMENTS OF CASH FLOWS (in thousands) Years Ended December 31, 2024 2023 2022 Cash flows from operating activities: Net income (loss) $ 501,053 $ 463,009 $ 132,663 Adjustments to reconcile net income (loss) to net cash provided by (used in) operating activities: Depreciation and intangible asset amortization 1,344,696 1,388,671 1,693,575 Amortization of deferred subscriber acquisition costs 224,647 195,794 162,981 Amortization of deferred subscriber acquisition revenue (346,209) (308,604) (244,141) Share-based compensation expense 48,613 51,137 66,566 Deferred income taxes 139,583 125,235 19,575 Provision for losses on receivables and inventory 214,802 151,065 113,869 Loss on extinguishment of debt 4,802 16,621 — Goodwill, intangible, and other asset impairments 24,313 528,556 206,132 (Gain) loss on sales of businesses 9,557 (649,095) (10,066) Unrealized (gain) loss on interest rate swap contracts 45,160 38,497 (301,851) Change in fair value of other financial instruments — — 63,396 Other non-cash items, net (34,089) 100,087 134,526 Changes in operating assets and liabilities, net of effects of acquisitions and dispositions: Accounts receivable, net (146,134) (107,464) (178,258) Long-term retail installment contracts 211,731 184,925 142,811 Inventories and work-in-progress 25,072 24,732 (67,391) Other assets (109,839) (29,144) (9,291) Accounts payable (109,330) (111,529) 8,662 Accrued interest (88,860) (128,042) 50,758 Accrued and other liabilities 40,944 (83,041) (42,023) Deferred subscriber acquisition costs (365,841) (386,518) (393,861) Deferred subscriber acquisition revenue 252,274 289,534 329,214 Other, net (2,046) (96,700) 10,074 Net cash provided by (used in) operating activities 1,884,899 1,657,726 1,887,920 Cash flows from investing activities: Dealer generated customer accounts and bulk account purchases (585,809) (588,638) (621,695) Subscriber system asset expenditures (523,146) (630,535) (734,639) Purchases of property and equipment (163,805) (176,353) (176,660) Proceeds (payments) from sale of business, net of cash sold (17,506) 1,609,347 26,749 Proceeds (payments) from interest rate swaps (8,268) — — Other investing, net 3,106 28,672 (26,539) Net cash provided by (used in) investing activities (1,295,428) 242,493 (1,532,784) Cash flows from financing activities: Proceeds from issuance of common stock, net of expenses — — 1,180,000 Proceeds from long-term borrowings 1,068,907 867,178 550,035 Proceeds from receivables facility 228,569 281,647 276,826 Repurchases of common stock (240,556) — (1,200,000) Repayment of long-term borrowings, including call premiums (1,186,045) (2,961,798) (605,059) Repayment of receivables facility (256,672) (200,385) (121,061) Dividends on common stock (182,266) (128,587) (127,125) Payments on finance leases (29,023) (43,733) (44,978) Proceeds (payments) from opportunity fund (6,895) (8,746) 100,802 Proceeds (payments) from interest rate swaps 93,040 82,750 (18,841) Other financing, net (4,415) (32,175) (5,432) Net cash provided by (used in) financing activities (515,356) (2,143,849) (14,833) Cash and cash equivalents and restricted cash and restricted cash equivalents: Net increase (decrease) during the period 74,115 (243,630) 340,303 Beginning balance 129,950 373,580 33,277 Ending balance $ 204,065 $ 129,950 $ 373,580 See Notes to Consolidated Financial Statements F-8 1.
DESCRIPTION OF BUSINESS AND SUMMARY OF SIGNIFICANT ACCOUNTING POLICIES Business and Organization ADT Inc., together with its wholly-owned subsidiaries (collectively, “ADT” or the “Company”), is a leading provider of security, interactive, and smart home solutions serving consumer and small business customers in the United States (“U.S.”). The Company primarily conducts business under the ADT brand name.
The Company’s common stock, par value of $0.01 per share (“Common Stock”), trades on the New York Stock Exchange under the symbol “ADT” since its initial public offering (“IPO”) in January 2018.
ADT Inc. was incorporated in the State of Delaware in May 2015 as a holding company with no assets or liabilities. In July 2015, the Company acquired Protection One, Inc. and ASG Intermediate Holding Corp. (collectively, the “Formation Transactions”), which were instrumental in the commencement of the Company’s operations. In May 2016, the Company acquired The ADT Security Corporation (formerly named The ADT Corporation) (“The ADT Corporation”) (the “ADT Acquisition”). On October 2, 2023, the Company divested its Commercial Business (as defined and discussed below); and as of June 30, 2024, substantially all operations of the Solar Business (as defined and discussed below) had ceased.
Prior to March 11, 2024, the Company was majority-owned by Prime Security Services TopCo (ML), L.P., which is majority-owned by Prime Security Services TopCo Parent, L.P. (“Ultimate Parent”). Ultimate Parent is majority-owned by Apollo Investment Fund VIII, L.P. and its related funds that are directly or indirectly managed by affiliates of Apollo Global Management, Inc. (together with its subsidiaries and affiliates, “Apollo” or the “Sponsor”). Following a registered secondary offering of the Company’s Common Stock by certain Apollo affiliates (and the Company’s concurrent repurchase from the underwriters of 15 million shares of Common Stock that were the subject of the offering), including the exercise of the underwriters’ overallotment option which closed on March 19, 2024, Apollo beneficially owns less than 50% of the Company’s outstanding common stock, which includes Common Stock and Class B common stock (“Class B Common Stock”) combined, and less than 50% of the Company’s outstanding Common Stock, and the Company ceased to be a “controlled company” under the New York Stock Exchange (the “NYSE”) rules.
Refer to Note 16 “Related Party Transactions” for more information regarding all of Apollo’s registered secondary offerings of the Company’s Common Stock (collectively, the “Offerings”).
Basis of Presentation The consolidated financial statements have been prepared in U.S. dollars in accordance with generally accepted accounting principles in the United States of America (“GAAP”).
The financial statements included herein comprise the consolidated results of ADT Inc. and its wholly-owned subsidiaries. The results of companies acquired are included from the effective date of each acquisition; and all intercompany transactions have been eliminated. The Company used the equity method of accounting to account for an investment in which it had the ability to exercise significant influence but does not control. This investment was disposed of during 2023.
Certain prior period amounts have been reclassified to conform with the current period presentation.
Use of Estimates The preparation of the consolidated financial statements in accordance with GAAP requires the Company to select accounting policies and make estimates that affect amounts reported in the consolidated financial statements and the accompanying notes. The Company’s estimates are based on the relevant information available at the end of each period. Actual results could differ materially from these estimates under different assumptions or market conditions.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-9 Segments The Company evaluates and reports information based on the manner in which our Chief Executive Officer (“CEO”), who is the chief operating decision maker (“CODM”), evaluates performance and allocates resources. The CODM manages the business on a consolidated basis, and as such, the Company reports results in a single operating and reportable segment which reflects the business operations of the Company’s former Consumer and Small Business (“CSB”) segment.
Refer to Note 3 “Segment Information.” Discontinued Operations The Company’s exit in 2024 from its residential solar business (the “Solar Business”) (the “ADT Solar Exit”) and the sale in 2023 of its former commercial business (the “Commercial Business”) represented strategic shifts that had major effects on the Company’s operations and financial results. Accordingly, the results of operations and financial position of the Solar and Commercial Businesses are classified as discontinued operations in the Company’s Consolidated Statements of Operations and the Company’s Consolidated Balance Sheet for all periods presented. The cash flows and comprehensive income (loss) of discontinued operations have not been segregated and are included in the Consolidated Statements of Cash Flows and Consolidated Statements of Comprehensive Income (Loss), respectively, for all periods presented.
Unless otherwise noted, amounts and disclosures throughout these Notes to Consolidated Financial Statements relate to the Company’s continuing operations.
Refer to Note 4 “Divestitures” for additional information.
Accounting Standards Updates (“ASU”) Recently Adopted Reportable Segment Disclosures - ASU 2023-07, Segment Reporting (Topic 280): Improvements to Reportable Segment Disclosures, improves reportable segment disclosure requirements primarily through enhanced disclosures about significant segment expenses. In addition, the guidance, among other requirements, enhances interim disclosures, clarifies circumstances in which an entity can disclose multiple segment measures of profit or loss, and provides new segment disclosure requirements for entities with a single reportable segment.
The Company adopted this guidance effective January 1, 2024, and it was applied retrospectively to all periods presented. Refer to Note 3 “Segment Information.” Supplier Finance Program Obligations - ASU 2022-04, Liabilities — Supplier Finance Programs (Subtopic 405-50): Disclosure of Supplier Finance Program Obligations, requires that a reporting entity who is a buyer in a supplier finance program disclose qualitative and quantitative information about its supplier finance programs, including a roll-forward of the obligations.
The Company adopted this guidance effective January 1, 2023, except the roll-forward requirement, which was adopted effective January 1, 2024.
The Company does not currently have any material supplier finance programs, and the guidance will be applied prospectively to any future material arrangements.
Fair Value of Equity Securities - ASU 2022-03, Fair Value Measurement (Topic 820): Fair Value Measurement of Equity Securities Subject to Contractual Sale Restrictions, states that an entity should not consider the contractual sale restriction when measuring the equity security’s fair value and introduces new disclosure requirements related to such equity securities.
The Company adopted this guidance effective January 1, 2024. This guidance did not impact the Company.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-10 Recently Issued Disaggregation of Income Statement Expenses - ASU 2024-03, Income Statement — Reporting Comprehensive Income — Expense Disaggregation Disclosures (Subtopic 220-40): Disaggregation of Income Statement Expenses, requires additional disclosure in the footnotes at each interim and annual reporting period about specific types of expenses included in the expense captions presented on the face of the statement of operations as well as additional disclosures that also include information related to selling expenses.
The guidance is effective for annual periods beginning after December 15, 2026, and interim periods beginning after December 15, 2027. The requirements will be applied prospectively with the option for retrospective application. Early adoption is permitted. The Company is currently evaluating the impact of this guidance on its financial statements and disclosures.
Improvements to Income Tax Disclosures - ASU 2023-09, Income Taxes (Topic 740): Improvements to Income Tax Disclosures, focuses on improvements to income tax disclosures, primarily related to the rate reconciliation and income taxes paid information. In addition, the update includes certain other amendments to improve the effectiveness of income tax disclosures.
The guidance is effective for annual periods beginning after December 15, 2024, and should be applied prospectively, with retrospective application also permitted. Early adoption is permitted. The Company is currently evaluating the impact of this guidance on its financial statements and disclosures.
Disclosure Improvements - ASU 2023-06, Disclosure Improvements: Codification Amendments in Response to the SEC’s Disclosure Update and Simplification Initiative, represents changes to clarify or improve disclosure and presentation requirements of a variety of topics.
The effective date for each amendment will be the date on which the SEC’s removal of that related disclosure from Regulation S-X or Regulation S-K becomes effective, with early adoption prohibited. The Company is currently evaluating the potential impact of this guidance on its financial statements and disclosures.
Significant Accounting Policies Information on select accounting policies and methods not discussed below are included in the respective footnotes that follow. Cash and Cash Equivalents and Restricted Cash and Restricted Cash Equivalents All highly liquid investments with original maturities of three months or less from the time of purchase are considered to be cash equivalents. Restricted cash and restricted cash equivalents are restricted for a specific purpose and cannot be included in the general cash and cash equivalents account.
The following table reconciles the amounts below reported in the Consolidated Balance Sheets to the total of the same such amounts shown in the Consolidated Statements of Cash Flows: Years Ended December 31, (in thousands) 2024 2023 2022 Cash and cash equivalents $ 96,212 $ 14,621 $ 257,223 Restricted cash and restricted cash equivalents 107,853 115,329 116,357 Ending balance $ 204,065 $ 129,950 $ 373,580 Restricted cash and restricted cash equivalents primarily includes funds received from State Farm Fire & Casualty Company (“State Farm”) (the “Opportunity Fund”), net of payments and inclusive of interest earned, in connection with the State Farm Strategic Investment (as defined and discussed in Note 10 “Equity”). Amounts within the Opportunity Fund are restricted for certain qualifying spend in accordance with the development agreement between State Farm and the Company (the “State Farm Development Agreement”). Use of the funds must be agreed to by State Farm and the Company. The remaining amount of restricted cash relates to the Company’s uncommitted receivables securitization financing agreement (the “2020 Receivables Facility”) (refer to Note 7 “Debt”).
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-11 Supplementary Cash Flow Information The following table summarizes supplementary cash flow information and material non-cash investing and financing transactions, excluding leases (refer to Note 14 “Leases”): Years Ended December 31, (in thousands) 2024 2023 2022 Interest paid, net of interest income received(1) $ 372,036 $ 522,775 $ 470,947 Payments (refunds) on income taxes, net $ 21,700 $ 60,296 $ 22,654 Issuance of shares for acquisition of businesses(2) $ — $ — $ 55,485 Contingent forward purchase contract(3) $ — $ — $ 41,938 Forward share repurchase contract(4) $ 104,175 $ — $ — ___ (1) Includes finance leases and interest rate swaps. Refer to Note 8 “Derivative Financial Instruments.” (2) Includes $40 million related to the Delayed Shares (as defined and discussed in Note 10 “Equity”) as a result of the ADT Solar Acquisition.
(3) The Company recorded a reduction to additional paid in capital as a result of the contingent forward purchase contract in connection with the Tender Offer (as defined and discussed in Note 10 “Equity”).
(4) In December 2024, the Company entered into an agreement with a non-affiliate individual to repurchase 15 million shares of Common Stock at a price per share of $6.95 to be settled in January 2025 (refer to Note 10 “Equity”). The Company recorded a liability and a reduction to additional paid-in capital as of December 31, 2024.
During 2024, the proceeds and repayments of long-term borrowings on the Consolidated Statements of Cash Flows include the impact of $704 million from amendments to the Company’s First Lien Credit Agreement (as defined and discussed in Note 7 “Debt”). In addition, proceeds and repayments of long-term borrowings include the impact of $32 million from amendments to the 2020 Receivables Facility.
During 2023, the proceeds and repayments of long-term borrowings on the Consolidated Statements of Cash Flows include the impact of $230 million from the refinancing of the First Lien Term Loan B due 2026 with the First Lien Term Loan B due 2030 (as defined and discussed in Note 7 “Debt”).
Prepaid Expenses and Other Current Assets December 31, (in thousands) 2024 2023 Prepaid expenses $ 53,036 $ 47,674 Contract assets (see Note 2 “Revenue and Receivables”) 19,164 15,365 Fair value of interest rate swaps (see Note 8 “Derivative Financial Instruments”) 56,164 74,974 Other current assets 82,249 104,179 Prepaid expenses and other current assets $ 210,613 $ 242,192 Inventories, net Inventories, net includes finished goods and work-in-progress.
Finished goods are primarily comprised of components and parts for the Company’s security systems. The Company records inventory at the lower of cost and net realizable value. Finished goods are presented net of an obsolescence reserve.
Work-in-progress is primarily comprised of certain costs incurred for installations of security system equipment sold outright to customers that have not been completed as of the balance sheet date. Work-in-progress is not material.
Property and Equipment, net Property and equipment, net, is recorded at historical cost less accumulated depreciation, which is calculated using the straight-line method over the estimated useful lives of the related assets. Depreciation expense is reflected in depreciation and intangible asset amortization. Repairs and maintenance expenditures are expensed when incurred.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-12 Useful Lives: Buildings and related improvements Up to 40 years Leasehold improvements Lesser of remaining term of the lease or economic useful life Capitalized software 3 to 10 years Machinery, equipment, and other Up to 10 years Net Carrying Amount: December 31, (in thousands) 2024 2023 Land $ 9,773 $ 10,313 Buildings and leasehold improvements 97,263 95,003 Capitalized software 574,644 518,850 Machinery, equipment, and other 162,631 158,086 Construction in progress 22,106 30,121 Finance leases 127,956 114,997 Accumulated depreciation (747,190) (673,712) Property and equipment, net $ 247,183 $ 253,658 Depreciation Expense: Years Ended December 31, (in thousands) 2024 2023 2022 Depreciation expense $ 174,850 $ 176,407 $ 159,339 Subscriber System Assets, net and Deferred Subscriber Acquisition Costs, net Subscriber system assets represent capitalized equipment and installation costs incurred in connection with transactions in which the Company retains ownership of the security system and are reflected in the Consolidated Balance Sheets as follows: December 31, (in thousands) 2024 2023 Gross carrying amount $ 6,878,490 $ 6,404,479 Accumulated depreciation (3,897,329) (3,398,543) Subscriber system assets, net $ 2,981,161 $ 3,005,936 Deferred subscriber acquisition costs represent selling expenses (primarily commissions) that are incremental to acquiring customers.
The Company records subscriber system assets and deferred subscriber acquisition costs in the Consolidated Balance Sheets as these assets represent probable future economic benefits for the Company through the generation of future monitoring and related services revenue. Upon customer termination, the Company may retrieve its subscriber system assets.
Subscriber system assets and any related deferred subscriber acquisition costs are accounted for on a pooled basis based on the month and year of customer acquisition and are depreciated and amortized using an accelerated method over the estimated life of the customer relationship, which is 15 years. In order to align the depreciation and amortization of these pooled costs to the pattern in which their economic benefits are consumed, the accelerated method utilizes an average declining balance rate of approximately 250% and converts to straight-line methodology when the resulting charge is greater than that from the accelerated method, resulting in an average charge of approximately 55% of the pool within the first five years, 25% within the second five years, and 20% within the final five years.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-13 Depreciation of subscriber system assets and amortization of deferred subscriber acquisition costs are reflected in depreciation and intangible asset amortization and selling, general, and administrative expenses, respectively, as follows: Years Ended December 31, (in thousands) 2024 2023 2022 Depreciation of subscriber system assets $ 557,226 $ 545,041 $ 531,013 Amortization of deferred subscriber acquisition costs $ 224,647 $ 188,222 $ 154,186 Long-Lived Assets (excluding Goodwill and Indefinite-Lived Intangible Assets) The Company reviews long-lived assets for impairment whenever events or changes in business circumstances indicate that the carrying amount of an asset or asset group may not be fully recoverable. The Company groups assets at the lowest level for which cash flows are separately identifiable. Recoverability is measured by a comparison of the carrying amount of the asset group to its expected future undiscounted cash flows. If the expected future undiscounted cash flows of the asset group are less than its carrying amount, an impairment loss is recognized based on the amount by which the carrying amount exceeds the fair value less costs to sell. The calculation of the fair value less costs to sell of an asset group is based on assumptions concerning the amount and timing of estimated future cash flows and assumed discount rates, reflecting varying degrees of perceived risk.
There were no material long-lived asset impairments during the periods presented.
Accrued Expenses and Other Current Liabilities December 31, (in thousands) 2024 2023 Accrued interest $ 107,116 $ 111,197 Payroll-related accruals 109,078 110,941 Opportunity Fund (see Note 10 “Equity”) 84,516 93,950 Accrued dividends 48,918 32,207 Forward share repurchase contract liability (see Note 10 “Equity”) 104,175 — Other accrued liabilities 181,101 207,819 Accrued expenses and other current liabilities $ 634,904 $ 556,114 Advertising Costs Advertising costs are recognized in selling, general, and administrative expenses when incurred and were $105 million, $131 million, and $146 million during 2024, 2023, and 2022, respectively.
Included in advertising costs during 2024 and 2023 are certain joint marketing costs and reimbursements associated with the Google Success Funds as discussed in Note 13 “Commitments and Contingencies.” Merger, Restructuring, Integration, and Other Merger, restructuring, integration, and other represents certain direct and incremental costs resulting from acquisitions made by the Company, integration and third-party costs as a result of those acquisitions, costs related to the Company’s restructuring efforts, as well as fair value remeasurements and impairment charges on certain strategic investments.
Concentration of Credit Risks The majority of the Company’s cash and cash equivalents and restricted cash and restricted cash equivalents are held at major financial institutions. There is a concentration of credit risk related to certain account balances in excess of the Federal Deposit Insurance Corporation insurance limit of $250,000 per account. The Company regularly monitors the financial stability of these financial institutions and believes there is no exposure to any significant credit risk for its cash and cash equivalents and restricted cash and restricted cash equivalents. Concentration of credit risk associated with the majority of the Company’s receivables from customers is limited due to the significant size of the customer base.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-14 Fair Value of Financial Instruments The Company’s financial instruments primarily consist of cash and cash equivalents, restricted cash and restricted cash equivalents, accounts receivable, retail installment contract receivables (“RICs”), accounts payable, debt, and derivative financial instruments. Due to their short-term and/or liquid nature, the fair values of cash, restricted cash, accounts receivable, and accounts payable approximate their respective carrying amounts.
Cash Equivalents - Included in cash and cash equivalents and restricted cash and restricted cash equivalents, as applicable from time to time, are investments in money market mutual funds. These investments are generally classified as Level 1 fair value measurements, which represent unadjusted quoted prices in active markets for identical assets or liabilities.
Investments in money market mutual funds were $90 million and $55 million as of December 31, 2024 and December 31, 2023, respectively.
Retail Installment Contract Receivables, net - The fair values of the Company’s RICs are determined using a discounted cash flow model and are classified as Level 3 fair value measurements.
December 31, 2024 2023 (in thousands) Carrying Amount Fair Value Carrying Amount Fair Value Retail installment contract receivables, net $ 669,326 $ 495,259 $ 673,635 $ 487,685 Long-Term Debt Instruments - The fair values of the Company’s debt instruments are determined using broker-quoted market prices, which represent quoted prices for similar assets or liabilities as well as other observable market data, and are classified as Level 2 fair value measurements. The carrying amounts of debt outstanding, if any, under the Company’s first lien revolving credit facility (the “First Lien Revolving Credit Facility”) and the 2020 Receivables Facility approximate their fair values, as interest rates on these borrowings approximate current market rates.
December 31, 2024 2023 (in thousands) Carrying Amount Fair Value Carrying Amount Fair Value Long-term debt instruments subject to fair value disclosures(1) $ 7,637,631 $ 7,589,677 $ 7,756,049 $ 7,731,408 __ (1) Excludes finance leases and certain vehicle loans reported as discontinued operations.
Derivative Financial Instruments - Derivative financial instruments are reported at fair value as either assets or liabilities. These fair values are primarily calculated using discounted cash flow models utilizing observable inputs, such as quoted forward interest rates, and incorporate credit risk adjustments to reflect the risk of default by the counterparty or the Company. The resulting fair values are classified as Level 2 fair value measurements.
Refer to Note 8 “Derivative Financial Instruments” for the fair values of the Company’s derivative financial instruments.
2. REVENUE AND RECEIVABLES Revenue The Company generates revenue through contractual monthly recurring fees received for monitoring and related services, as well as the sale and installation of security systems.
Revenue is recognized in the Consolidated Statements of Operations net of sales and other taxes. Amounts collected from customers for sales and other taxes are reported as a liability net of the related amounts remitted. When customers terminate a monitoring contract early, contract termination charges are assessed in accordance with the contract terms and are recognized in monitoring and related services revenue when collectability is probable.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-15 Disaggregated Revenue Years Ended December 31, (in thousands) 2024 2023 2022 Sources of Revenue: Recurring monthly revenue $ 4,177,428 $ 4,069,921 $ 3,942,567 Other related services 116,049 109,077 110,481 Monitoring and related services 4,293,477 4,178,998 4,053,048 Amortization of deferred subscriber acquisition revenue 346,209 301,708 235,190 Installation revenue 258,760 172,118 93,666 Security installation, product, and other 604,969 473,826 328,856 Total revenue $ 4,898,446 $ 4,652,824 $ 4,381,904 The Company allocates the transaction price to each performance obligation based on the relative standalone selling price, which is determined using observable internal and external pricing, profitability, and operational metrics. The Company’s performance obligations generally include monitoring, related services (such as maintenance agreements), as well as the sale and installation of a security system in outright sales transactions or a material right in transactions in which the Company retains ownership of the security system.
Customer-Owned - In transactions involving security systems sold outright to the customer (referred to as outright sales), the Company’s performance obligations generally include the sale and installation of the system, which is primarily recognized at a point in time based upon the nature of the transaction and contractual terms, and any monitoring and related services, which are recognized when these services are provided to the customer.
Company-Owned - In transactions in which the Company provides monitoring and related services but retains ownership of the security system (referred to as Company-owned), the Company’s performance obligations primarily include (i) monitoring and related services, which are recognized when these services are provided to the customer, and (ii) a material right associated with the one-time non-refundable fees incurred in connection with the initiation of a monitoring contract which the customer will not be required to pay again upon a renewal of the contract (referred to as deferred subscriber acquisition revenue).
Deferred subscriber acquisition revenue is amortized on a pooled basis over the estimated life of the customer relationship using an accelerated method consistent with the treatment of subscriber system assets and deferred subscriber acquisition costs and is reflected in security installation, product, and other revenue.
Remaining Performance Obligations As of December 31, 2024, the remaining unsatisfied performance obligation relating to the provision of monitoring and related services is as follows (in thousands): 2025 2026 2027 2028 Thereafter Total $ 1,799,674 $ 999,973 $ 463,811 $ 144,877 $ 68,873 $ 3,477,208 Deferred Revenue Deferred revenue represents customer billings for services not yet rendered and is primarily related to recurring monitoring and related services. In addition, payments received for the sale and installation of a system after the agreement is signed but before performance obligations are satisfied are recorded as deferred revenue.
These amounts are recorded as current deferred revenue, as the Company expects to satisfy any remaining performance obligations, as well as recognize the related revenue, within the next twelve months when performance obligations are satisfied. ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-16 Accounts Receivable Accounts receivable represent unconditional rights to consideration from customers in the ordinary course of business and are generally due in one year or less. The Company’s accounts receivable are recorded at amortized cost less an allowance for credit losses not expected to be recovered. The allowance for credit losses is recognized at inception and reassessed each reporting period.
The Company evaluates its allowance for credit losses on accounts receivable in pools based on customer type. The allowance for credit losses primarily relates to residential customers. For each customer pool, the allowance for credit losses is estimated based on the delinquency status of the underlying receivables and the related historical loss experience, as adjusted for current and expected future conditions, if applicable.
Changes in the Allowance for Credit Losses Years Ended December 31, (in thousands) 2024 2023 2022 Beginning balance $ 46,850 $ 27,815 $ 22,030 Provision for credit losses 158,346 130,407 83,047 Write-offs, net of recoveries(1) (147,401) (111,372) (77,262) Ending balance $ 57,795 $ 46,850 $ 27,815 __ (1) Recoveries were not material for the periods presented. As such, write-offs are presented net of recoveries.
Retail Installment Contract Receivables, Net The Company’s RICs allow qualifying residential customers to pay the fees due at installation over a 12-, 24-, 36-, or 60-month interest-free period. The financing component of retail installment contract receivables is not significant.
Upon origination of a retail installment contract, the Company utilizes external credit scores to assess customer credit quality and determine eligibility. Subsequent to origination, the Company monitors the delinquency status of retail installment contract receivables as the key credit quality indicator. Delinquent billed RICs are not material.
The Company’s RICs are recorded at amortized cost less an allowance for credit losses not expected to be recovered. The allowance for credit losses is recognized at inception and reassessed each reporting period. The allowance for credit losses relates to retail installment contract receivables from outright sales transactions and is not material.
December 31, (in thousands) 2024 2023 Retail installment contract receivables, gross $ 678,174 $ 674,827 Allowance for credit losses (8,848) (1,192) Retail installment contract receivables, net $ 669,326 $ 673,635 Balance Sheet Classification: Accounts receivable, net $ 260,224 $ 238,961 Other assets 409,102 434,674 Retail installment contract receivables, net $ 669,326 $ 673,635 As discussed in Note 7 “Debt,” retail installment contract receivables, net, for which the Company grants a security interest as collateral for cash borrowings under the 2020 Receivables Facility were $575 million and $610 million, as of December 31, 2024 and 2023, respectively.
Contract Assets Contract assets represent the Company’s right to consideration in exchange for goods or services transferred to the customer. The contract asset is reclassified to accounts receivable as additional services are performed and billed, which is when the Company’s right to the consideration becomes unconditional. This balance is primarily comprised of satisfied performance obligations related to the sale and installation of a system under an outright sale transaction.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-17 The Company has the right to bill customers as services are provided over time, which generally occurs over the course of a 24-, 36-, or 60-month period. There is no significant financing component.
The Company records an allowance for credit losses against its contract assets for amounts not expected to be recovered. The allowance is recognized at inception and is reassessed each reporting period. The allowance for credit losses on contract assets was not material for the periods presented.
Gross contract assets recognized by the Company were not material for the periods presented.
December 31, (in thousands) 2024 2023 Contract assets, gross $ 46,031 $ 39,627 Allowance for credit losses (5,221) (9,025) Contract assets, net $ 40,810 $ 30,602 Balance Sheet Classification: Prepaid expenses and other current assets $ 19,164 $ 15,365 Other assets 21,646 15,237 Contract assets, net $ 40,810 $ 30,602 3.
SEGMENT INFORMATION As a result of the ADT Solar Exit and Commercial Divestiture, the Company reports results in a single operating and reportable segment, which reflects the continuing operations of the Company’s former CSB segment.
The Company’s CODM is its Chairman, President, and Chief Executive Officer. The CODM evaluates performance and allocates resources on a consolidated basis using various measures primarily through reviews of various operational performance packages, earnings releases, investor presentations, and the Company’s SEC filings, as well as through the approval of the Company’s annual budget and forecast.
The Company’s reported segment profit measure is net income (loss) as this measure is most consistent with the amounts included in the Consolidated Statements of Operations. In addition, segment assets reviewed by the CODM are reported on the Company’s Consolidated Balance Sheets as total assets.
The accounting policies of the Company’s reportable segment are the same as those of the Company.
The following presents a reconciliation to the Company’s net income (loss) as reported in the Consolidated Statements of Operations and includes segment revenues, significant segment expenses that are regularly provided to or easily computed from information regularly provided to the CODM, other segment expenses, and adjustments to reconcile to net income (loss).
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-18 Years Ended December 31, (in thousands) 2024 2023 2022 Total segment revenue $ 4,898,446 $ 4,652,824 $ 4,381,904 Less significant segment expenses: Customer service costs(1) 409,680 387,314 393,425 Maintenance costs(1) 207,706 217,054 203,239 Security installation, product, and other costs 229,728 147,314 102,118 Selling costs, including commissions(2) 185,417 196,204 190,182 Amortization of deferred subscriber acquisition costs(2) 224,647 188,222 154,186 Advertising costs(2) 105,366 131,133 145,904 Provision for credit losses(2) 187,361 131,962 81,727 Other general and administrative costs(2) 708,979 664,182 696,662 Share-based compensation(2) 48,745 38,626 52,945 Depreciation and intangible asset amortization 1,342,798 1,335,484 1,599,810 Interest expense 450,939 586,088 277,144 Income tax expense (benefit) 195,780 160,585 87,692 Total significant segment expenses 4,297,146 4,184,168 3,985,034 Less other segment items(3) Other items in SG&A(2) 15,831 (2,591) 26,675 Other, net (33,921) 20,877 58,030 Total other segment items (18,090) 18,286 84,705 Reconciliation of profit or loss: (Income) loss from discontinued operations, net of tax(4) 118,337 (12,639) 179,502 Net income (loss) $ 501,053 $ 463,009 $ 132,663 __ (1) Included in monitoring and related services cost of revenue.
(2) Included in SG&A.
(3) Other segment items generally include other income and expenses and merger, restructuring, integration, and other charges as presented on the face of the Statements of Operations; as well as certain other items included in SG&A and interest income. Interest income is not material for all periods presented.
(4) Represents activity related to the Commercial and Solar Businesses, which are presented as discontinued operations.
Entity-Wide Disclosures Revenue generated from customers outside of the U.S. is not material. As of December 31, 2024 and 2023, substantially all of the Company’s assets were located in the U.S.
The Company does not have any major customers given the high volume nature of the business. Refer to Note 2 “Revenue and Receivables” for further information on the Company’s products and services.
4. DIVESTITURES The Company may decide to divest portions of its business for various reasons, including efforts to focus on its remaining businesses. The Company presents discontinued operations for components of the business that are either disposed of through ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-19 sale (or qualify as held for sale), abandonment, or spin-off if these actions also represent a strategic shift that has or will have a major effect on the Company’s financial results.
Refer to Note 12 “Net Income (Loss) per Share” for basic and diluted earnings per share information associated with discontinued operations.
ADT Solar Exit On January 19, 2024, after a strategic review of the business and continued macroeconomic and industry pressures, the Company’s board of directors (the “Board of Directors”) approved a plan to fully exit the Solar Business. As of June 30, 2024, substantially all operations of the Solar Business had ceased.
The ADT Solar Exit represented a strategic shift that had a major effect on the Company’s operations and financial results. Accordingly, the Solar Business is presented as a discontinued operation in the Company’s Consolidated Statements of Operations and Consolidated Balance Sheets for the periods presented.
The Solar Business was previously reflected in the Solar reportable segment.
During the year ended December 31, 2024, the Company incurred aggregate exit charges of $88 million, which have been recognized within income (loss) from discontinued operations, net of tax related to (i) $33 million associated with the write-down and disposition of inventory and asset impairments, (ii) $29 million associated with the disposition of the existing installation pipeline, (iii) $13 million associated with employee separation costs, and (iv) $12 million associated with contract termination and other charges.
During the year ended December 31, 2024, the Company paid $22 million associated with the ADT Solar Exit primarily related to employee separation and other restructuring costs.
The following reconciliations represent the major classes of line items of the Solar Business presented within discontinued operations in the Consolidated Balance Sheets and Consolidated Statements of Operations and certain information in the Consolidated Statements of Cash Flows for the periods presented.
Balance Sheet Information (in thousands) December 31, 2024 December 31, 2023 Assets Accounts receivable, net $ — $ 20,270 Inventories, net — 28,714 Prepaid expenses and other current assets — 11,973 Total current assets of discontinued operations — 60,957 Property and equipment, net — 29,512 Other assets — 13,767 Total assets of discontinued operations $ — $ 104,236 Liabilities Current maturities of long-term debt $ 22 $ 8,551 Accounts payable 6,953 16,682 Deferred revenue — 9,177 Accrued expenses and other current liabilities 24,788 45,201 Total current liabilities of discontinued operations 31,763 79,611 Long-term debt 32 9,893 Other liabilities 15,857 10,679 Total liabilities of discontinued operations $ 47,652 $ 100,183 ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-20 Statements of Operations Information Years Ended December 31, (in thousands) 2024 2023 2022 Revenue $ 21,254 $ 329,835 $ 786,426 Cost of revenue 65,678 256,784 501,710 Selling, general, and administrative expenses 67,366 191,841 314,545 Depreciation and intangible asset amortization 1,898 15,496 16,020 Merger, restructuring, integration, and other 34,000 23,213 7,292 Goodwill impairment — 511,176 200,974 Other (income) and expense items 1,481 2,235 1,197 Income (loss) from discontinued operations before income taxes (149,169) (670,910) (255,312) Income tax benefit (expense) 39,096 156,000 50,010 Income (loss) from discontinued operations, net of tax $ (110,073) $ (514,910) $ (205,302) Cash Flow Information Years Ended December 31, (in thousands) 2024 2023 2022 Adjustments to reconcile net income (loss) to net cash provided by (used in) operating activities: Depreciation and intangible asset amortization $ 1,898 $ 15,496 $ 16,020 Goodwill, intangible, and other asset impairments $ 13,770 $ 515,730 $ 200,974 Cash flows from investing activities: Purchases of property and equipment $ (80) $ (4,027) $ (9,826) Commercial Divestiture On August 7, 2023, ADT, Iris Buyer LLC, a Delaware limited liability company and affiliate of GTCR LLC (“GTCR”), and, solely for certain purposes set forth in the Commercial Purchase Agreement (as defined below), Fire & Security Holdings, LLC (“F&S Holdings”), a Delaware limited liability company and an indirect, wholly-owned subsidiary of ADT, entered into an Equity Purchase Agreement (the “Commercial Purchase Agreement”) pursuant to which GTCR agreed to acquire all of the issued and outstanding equity interests of F&S Holdings, which directly or indirectly held all of the issued and outstanding equity interests in the subsidiaries of ADT that operated ADT’s commercial business (the “Commercial Business”) (the “Commercial Divestiture”).
The Commercial Divestiture was completed on October 2, 2023, and the Company received net proceeds of approximately $1,585 million at the time of closing, subject to certain customary post-closing adjustments as set forth in the Commercial Purchase Agreement. In addition, the Company recognized a pre-tax gain on sale of approximately $630 million, which was recognized in income (loss) from discontinued operations during 2023. The Company used the majority of the net proceeds for debt redemption, as discussed in Note 7 “Debt.” Additionally, as the agreed upon sale price was substantially higher than the carrying value of the Commercial Business, the Company did not record any impairments or adjustments when recognizing the disposal group at the lower of its carrying amount or fair value less cost to sell.
During 2024, the Company paid GTCR $21 million related to the settlement of post-closing adjustments, which is presented in cash flows from investing activities. The Commercial Divestiture represented a strategic shift that had a major effect on the Company’s operations and financial results. Accordingly, the Commercial Business is presented as a discontinued operation in the Company’s Consolidated Statements of Operations for the periods presented.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-21 The Commercial Business was previously reflected in the Commercial reportable segment.
The following reconciliations represent the major classes of line items of the Commercial Business within the Consolidated Statements of Operations and certain information within the Consolidated Statements of Cash Flows (excluding proceeds from the sale of business discussed above) for the periods presented.
Statements of Operations Information During the year ended December 31, 2024, activity, net of tax, relating to the Commercial Divestiture was approximately $8 million primarily related to the settlement of post-closing adjustments.
Years Ended December 31, (in thousands) 2023 2022 Revenue $ 1,035,048 $ 1,226,980 Cost of revenue 688,433 839,356 Selling, general, and administrative expenses 213,514 267,195 Depreciation and intangible asset amortization 37,691 77,745 Other income and expense items 19,174 6,016 Income (loss) from discontinued operations before gain on sale of business and income taxes 76,236 36,668 Gain on sale of business 629,980 — Income (loss) from discontinued operations before income taxes 706,216 36,668 Income tax benefit (expense) (178,667) (10,868) Income (loss) from discontinued operations, net of tax $ 527,549 $ 25,800 Cash Flow Information Years Ended December 31, (in thousands) 2023 2022 Adjustments to reconcile net income (loss) to net cash provided by (used in) operating activities: Depreciation and intangible asset amortization $ 37,691 $ 77,745 Share-based compensation expense $ 11,699 $ 13,069 Cash flows from investing activities: Subscriber system asset expenditures $ (8,902) $ (29,230) Purchases of property and equipment $ (4,399) $ (6,885) Transition Services Agreement In connection with the Commercial Divestiture, the Company entered into a Transition Services Agreement (the “Commercial TSA”), pursuant to which the Company and the Commercial Business will provide certain transitional services relating to ongoing support and other administrative functions to each other for a transitional period of up to 24 months after the closing of the Commercial Divestiture. Commercial TSA fees charged to the Commercial Business represent charges for internal labor as well as certain third-party costs identified in connection with providing such services. Income from the Commercial TSA is recognized in other income (expense), and expenses incurred by the Company to support the transition are recorded based on the nature of the expense. During 2024 and 2023, the Company recognized income from the Commercial TSA of $40 million and $12 million, respectively.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-22 ADT Brand License and Intellectual Property Rights The Company and GTCR entered into an agreement granting GTCR a license to continue to use the ADT brand and other Company trademarks for a period of twelve months to transition from Company branding (the “Brand License”). The Company has also agreed to a covenant not to assert a claim against GTCR for infringement of the Company's patents as of the Commercial Divestiture for products and services that were used in the Commercial Business prior to the Commercial Divestiture, and has provided GTCR with a paid-up, irrevocable, non-assignable (with limited exceptions) license to continue to use certain software and other Company intellectual property in the same manner. Royalty income is included in other income (expense) and was not material during 2024 and 2023.
Other Divestitures During the periods presented, other divestiture activity not reflected as discontinued operations includes: During 2023, proceeds related to disposal activities totaled $36 million, resulting in a gain on sale of $19 million recognized in SG&A.
During 2022, proceeds related to disposal activities totaled $27 million, resulting in a gain on sale of $10 million recognized in SG&A.
5. EQUITY METHOD INVESTMENTS The Company uses the equity method of accounting to account for an investment in which it has the ability to exercise significant influence but does not control. The Company recognizes its proportionate share of the investee’s net income or loss in equity in net earnings (losses) of equity method investee. The Company evaluates an equity method investment whenever events or changes in circumstances indicate the carrying amount of such investment may be impaired. If a decline in the value of an equity method investment is determined to be other than temporary, the Company records a loss as a component of the Company’s share of earnings or losses of the equity method investee in the current period.
Canopy Investment In April 2022, the Company and Ford Motor Company (“Ford”) formed a new entity, SNTNL LLC (“Canopy”), and the Company contributed cash of $11 million (the “Initial Contribution”). Since the Initial Contribution, the Company contributed $7 million.
During the fourth quarter of 2023, the Company sold its shares in Canopy and received $21 million in accordance with the terms of the agreement between the Company and Ford, which included a put right under which the Company recovered its full equity investment in Canopy plus a premium (“Canopy Termination”). In addition, the Company recognized a gain of $15 million, which is reflected in equity in net earnings (losses) of equity method investees. The Company no longer holds an investment in Canopy.
The Company previously accounted for its investment in Canopy under the equity method of accounting as the Company was not the primary beneficiary, and therefore, did not consolidate Canopy’s assets, liabilities, and financial results of operations.
In connection with the Canopy Investment, the Company entered into various commercial agreements (the “Canopy Commercial Agreements”), and the Company and Canopy are also parties to a trade name licensing agreement. These agreements terminated on December 1, 2024 as a result of the Canopy Termination. The impact to the consolidated financial statements from these agreements was not material.
6. GOODWILL AND OTHER INTANGIBLE ASSETS Goodwill There were no changes in the carrying amount of goodwill for the periods presented.
The previously reported accumulated goodwill impairment losses were associated with the Solar reporting unit, which is presented as a discontinued operation.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-23 Other Intangible Assets December 31, 2024 December 31, 2023 (in thousands) Gross Carrying Amount Accumulated Amortization Net Carrying Amount Gross Carrying Amount Accumulated Amortization Net Carrying Amount Definite-lived intangible assets: Contracts and related customer relationships(1) $ 6,158,349 $ (3,464,926) $ 2,693,423 $ 5,571,456 $ (2,937,245) $ 2,634,211 Dealer relationships(2) 1,518,020 (697,324) 820,696 1,518,020 (618,154) 899,866 Other(3) 209,773 (202,793) 6,980 209,773 (199,357) 10,416 Total definite-lived intangible assets 7,886,142 (4,365,043) 3,521,099 7,299,249 (3,754,756) 3,544,493 Indefinite-lived intangible assets: Trade name(4) 1,333,000 — 1,333,000 1,333,000 — 1,333,000 Intangible assets $ 9,219,142 $ (4,365,043) $ 4,854,099 $ 8,632,249 $ (3,754,756) $ 4,877,493 ___ (1) During 2023, the Company retired $1.7 billion of certain customer relationship intangible assets acquired in the ADT Acquisition that became fully amortized.
(2) Originated from the Formation Transactions and the ADT Acquisition in 2015 and 2016, respectively. Amortized primarily over 19 years on a straight-line basis based on management estimates about attrition and the longevity of the underlying dealer network that existed at the time of acquisition.
(3) Primarily relates to trade names and other technology assets. Amortized over a period of up to 10 years on a straight-line basis.
(4) ADT trade name acquired as part of the ADT Acquisition.
Contracts and Related Customer Relationships Contracts and related customer relationships comprise contracts with customers purchased under the ADT Authorized Dealer Program (as defined below) or from other third parties as well as customer relationships that originated from business acquisitions.
Additionally, the Company maintains a network of agreements with third-party independent alarm dealers who sell alarm equipment and ADT Authorized Dealer-branded monitoring and interactive services to residential end users (the “ADT Authorized Dealer Program”). The dealers in this program generate new end-user contracts with customers which the Company has the right, but not the obligation, to purchase from the dealer. Purchases of contracts with customers under the ADT Authorized Dealer Program, or from other third parties, are considered asset acquisitions and are recognized based on the cost to acquire the assets, which may include cash consideration, non-cash consideration, contingent consideration, and directly-attributable transaction costs. The Company may charge back the purchase price of any end-user contract if the contract is canceled during the charge-back period, which is generally thirteen months from the date of purchase. The Company records the amount of the charge back as a reduction to the purchase price.
Purchases of contracts with customers under the ADT Authorized Dealer Program, or from other third parties, are accounted for on a pooled basis based on the month and year of acquisition. The Company amortizes its pooled contracts with customers using an accelerated method over the estimated life of the customer relationship, which is 15 years. The accelerated method for amortizing these contracts utilizes an average declining balance rate of approximately 300% and converts to straight-line methodology when the resulting amortization charge is greater than that from the accelerated method, resulting in an average amortization of approximately 65% of the pool within the first five years, 25% within the second five years, and 10% within the final five years.
Customer relationships acquired as part of business acquisitions, which primarily originated from the Formation Transactions and the ADT Acquisition, are amortized over a period of up to 15 years based on management estimates about the amounts and timing of estimated future revenue from customer accounts and average customer account life that existed at the time of the related business acquisition.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-24 The change in the net carrying amount of contracts and related customer relationships was as follows: Years Ended December 31, (in thousands) 2024 2023 Beginning balance $ 2,634,211 $ 2,595,535 Customer contract additions, net of dealer charge-backs 586,994 564,652 Amortization (527,782) (525,676) Other — (300) Ending balance $ 2,693,423 $ 2,634,211 During 2024 and 2023, the weighted-average amortization period for customer contract additions under the ADT Authorized Dealer Program and from other third parties was 15 years.
During 2024 and 2023, the Company purchased customer accounts from other third parties for an aggregate contractual purchase price of $98 million and $109 million, respectively, subject to reduction based on customer retention. The Company paid initial cash at the closings in the aggregate amounts of $81 million and $89 million, respectively, which is included in dealer generated customer accounts and bulk account purchases on the Consolidated Statements of Cash Flows.
Definite-Lived Intangible Asset Amortization Expense Years Ended December 31, (in thousands) 2024 2023 2022 Definite-lived intangible asset amortization expense $ 610,389 $ 613,679 $ 909,059 As of December 31, 2024, the estimated aggregate amortization expense on our existing intangible assets is expected to be as follows (in thousands): 2025 2026 2027 2028 2029 Thereafter $ 575,002 $ 493,508 $ 427,736 $ 378,363 $ 342,846 $ 1,303,644 Goodwill and Indefinite-Lived Intangible Assets Impairment Goodwill and indefinite-lived intangible assets are not amortized and are tested for impairment at least annually as of the first day of the fourth quarter of each year and more often if an event occurs or circumstances change which indicate it is more-likely-than-not that the fair value of a reporting unit is less than its carrying amount. The Company may perform the impairment test for its reporting unit or indefinite-lived intangible asset through a qualitative assessment or elect to proceed directly to a quantitative impairment test, however, the Company may resume a qualitative assessment in any subsequent period if facts and circumstances permit.
Goodwill Under a qualitative approach, the Company assesses whether it is more-likely-than-not that a reporting unit’s fair value is less than its carrying amount. If the Company elects to bypass the qualitative assessment for any reporting unit, or if a qualitative assessment indicates it is more-likely-than-not that the estimated fair value of a reporting unit is less than its carrying amount, the Company proceeds to a quantitative approach.
Under a quantitative approach, the Company estimates the fair value of a reporting unit and compares it to its carrying amount. If the carrying amount of a reporting unit exceeds its fair value, an impairment loss is recognized in an amount equal to that excess. The Company estimates the fair values of its reporting units using the income approach, which discounts projected cash flows using market participant assumptions. The income approach includes significant assumptions including, but not limited to, forecasted revenue, operating profit margins, Adjusted EBITDA margins, operating expenses, cash flows, perpetual growth rates, and discount rates. The estimated fair value of a reporting unit calculated using the income approach is sensitive to changes in the underlying assumptions. In developing these assumptions, the Company relies on various factors including operating results, business plans, economic projections, anticipated future cash flows, and other market data. Examples of events or circumstances that could reasonably be expected to negatively affect the underlying judgments and factors and ultimately impact the estimated fair value determinations may include such items as a prolonged downturn in the business ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-25 environment, changes in economic conditions that significantly differ from the Company’s assumptions in timing or degree, volatility in equity and debt markets resulting in higher discount rates, and unexpected regulatory changes. As a result, there are inherent uncertainties related to these judgments and factors that may ultimately impact the estimated fair value determinations.
The Company performed a qualitative goodwill impairment test as of October 1, 2024. The Company concluded that it is more- likely-than-not that the fair value of the Company’s reporting unit exceeds its carrying value, and as a result, the Company did not perform a quantitative impairment test or record any goodwill impairment losses.
The Company did not record any goodwill impairment losses in income/(loss) from continuing operations during 2023 and 2022.
The Company previously recorded goodwill impairment charges associated with the Solar reporting unit which are now presented in income (loss) from discontinued operations, net of tax.
Indefinite-Lived Intangible Assets Under a qualitative approach, the impairment test for an indefinite-lived intangible asset consists of an assessment of whether it is more-likely-than-not that an asset’s fair value is less than its carrying amount. If the Company elects to bypass the qualitative assessment for any indefinite-lived intangible asset, or if a qualitative assessment indicates it is more-likely-than-not that the estimated carrying amount of such asset exceeds its fair value, the Company proceeds to a quantitative approach.
Under a quantitative approach, the Company estimates the fair value of an asset and compares it to its carrying amount. If the carrying amount exceeds fair value, an impairment loss is recognized in an amount equal to that excess. The estimated fair value of an indefinite-lived intangible asset is determined using a valuation approach that is based on the nature of the underlying asset. The Company’s only indefinite-lived intangible asset is the ADT trade name. The fair value of the ADT trade name is determined under a relief from royalty method, which is an income approach that estimates the cost savings that accrue to the Company that it would otherwise have to pay in the form of royalties or license fees on revenue earned through the use of the asset. The utilization of the relief from royalty method requires the Company to make significant assumptions including revenue growth rates, the implied royalty rate, and the discount rate.
As of October 1, 2024, the Company quantitatively tested the ADT trade name for impairment. Based on the results of the test, the Company did not record any impairment losses as the estimated fair value of the trade name substantially exceeded its carrying amount.
During 2023 and 2022, the Company did not record any impairment losses on its indefinite lived intangible asset.
Definite-Lived Intangible Asset Impairment Definite-lived intangible asset impairments were not material during 2024, 2023 and 2022.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-26 7. DEBT The Company’s debt is comprised of the following (in thousands): Interest Payable Balance as of December 31, Debt Description Issued Maturity Interest Rate(1) 2024 2023 First Lien Term Loan B due 2030 10/13/2023 10/13/2030 Term SOFR +2.00% Quarterly $ 1,984,090 $ 1,375,000 First Lien Revolving Credit Facility 3/16/2018 10/1/2029 Term SOFR +2.00% Quarterly — — Term Loan A Facility 3/14/2023 3/14/2028 Term SOFR +2.25% Quarterly — 625,625 First Lien Notes due 2024 4/4/2019 4/15/2024 5.250% 2/15 and 8/15 — 99,999 First Lien Notes due 2026 4/4/2019 4/15/2026 5.750% 3/15 and 9/15 1,350,000 1,350,000 First Lien Notes due 2027 8/20/2020 8/31/2027 3.375% 6/15 and 12/15 1,000,000 1,000,000 First Lien Notes due 2029 7/29/2021 8/1/2029 4.125% 2/1 and 8/1 1,000,000 1,000,000 Second Lien Notes due 2028 1/28/2020 1/15/2028 6.250% 1/15 and 7/15 1,300,000 1,300,000 ADT Notes due 2032 5/2/2016 7/15/2032 4.875% 1/15 and 7/15 728,016 728,016 ADT Notes due 2042 7/5/2012 7/15/2042 4.875% 1/15 and 7/15 21,896 21,896 2020 Receivables Facility(2) 3/5/2020 11/20/2029 Various Monthly 407,901 436,004 Total debt principal, excluding finance leases 7,791,903 7,936,540 Plus: Finance lease obligations(3) 69,442 69,468 Less: Unamortized debt discount, net (12,081) (15,005) Less: Unamortized deferred financing costs (26,990) (39,620) Less: Unamortized purchase accounting fair value adjustment and other (115,201) (125,866) Total debt 7,707,073 7,825,517 Less: Current maturities of long-term debt, net of unamortized debt discount (195,791) (312,061) Long-term debt $ 7,511,282 $ 7,513,456 __ (1) Interest rate as of December 31, 2024. Interest on the 2020 Receivables Facility is primarily based on the Secured Overnight Financing Rate (“SOFR”) +1.05% and Cost of Funds (“COF”) +0.95%.
(2) Maturity date for the 2020 Receivables Facility represents the final maturity of date of current loans borrowed under the facility.
(3) Refer to Note 14 “Leases” for additional information regarding the Company’s finance leases.
First Lien Credit Agreement The Company’s first lien credit agreement dated as of July 1, 2015 (together with subsequent amendments and restatements, the “First Lien Credit Agreement”) includes a term loan (the “First Lien Term Loan B due 2030”) and a revolving credit facility (the “First Lien Revolving Credit Facility”).
Prime Security Services Holdings, LLC (“Holdings”), a Delaware limited liability company and a wholly owned indirect subsidiary of the Company, Prime Security Services Borrower, LLC (“Prime Borrower”), a Delaware limited liability company and a wholly owned direct subsidiary of Holdings, and The ADT Corporation, a Delaware corporation and a wholly owned direct subsidiary of Prime Borrower (together with Prime Borrower, the “Borrowers”), are parties to the First Lien Credit Agreement as holdings and borrowers respectively.
The First Lien Term Loan B due 2030 requires scheduled quarterly amortization payments equal to 0.25% of its outstanding principal amount at the time of the December 2024 amendment, with the remaining balance payable at maturity. The Borrowers may make voluntary prepayments on the First Lien Term Loan B due 2030 at any time prior to maturity at par, subject to a 1.00% prepayment premium in the event of certain specified refinancing events at any time before June 2025.
Additionally, based on certain specified net first lien leverage ratios, the Borrowers may be required to make annual prepayments on the outstanding First Lien Term Loan B due 2030 with a percentage of the Company’s excess cash flow, as defined in the First Lien Credit Agreement, if the excess cash flow exceeds a certain specified threshold, which is 0% if our net first lien leverage ratio is less than or equal to 2.20 to 1.00. As of December 31, 2024, the Borrowers were not required to make an annual prepayment based on the Company’s excess cash flow.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-27 The First Lien Term Loan B due 2030 bears interest at a rate equal to, at the Prime Borrower’s option, either (a) a term SOFR rate (“Term SOFR”) with a floor of zero or (b) a base rate determined by reference to the highest of (i) the federal funds rate plus 0.50% per annum, (ii) the rate of interest per annum last quoted by The Wall Street Journal as the “Prime Rate” in the United States and (iii) the one-month adjusted term SOFR plus 1.00% per annum (“Base Rate”), in each case, plus an applicable margin of 2.00% per annum for Term SOFR loans and 1.00% per annum for Base Rate loans. Prime Borrower has elected the Term SOFR alternative to apply to borrowings of the First Lien Term Loan B due 2030.
The applicable margin for borrowings under the First Lien Revolving Credit Facility is 2.00% for Term SOFR loans (subject to a credit spread adjustment) and 1.00% for Base Rate loans, in each case, subject to adjustment pursuant to a leverage-based pricing grid, subject to two step downs to 1.75% and 1.50% based on a net first lien leverage ratio of 2.00 to 1.00 and 1.50 to 1.00, respectively. In addition, the Borrowers are required to pay a commitment fee of 0.20% to 0.30%, with step downs to 0.25% and 0.20% based on a net first lien leverage ratio of 2.00 to 1.00 and 1.50 to 1.00, respectively, with respect to the unused commitments under the First Lien Revolving Credit Facility. The First Lien Revolving Credit Facility is also subject to a springing maturity of 91 days prior to the maturity date of certain long-term indebtedness if, as of such date, the outstanding principal amount of such indebtedness exceeds $350 million.
Indebtedness incurred under the First Lien Credit Agreement is guaranteed, jointly and severally, on a senior secured first-priority basis, by substantially all of Prime Borrower’s wholly owned material domestic subsidiaries, and by Prime Borrower’s direct parent on a limited recourse basis, and is secured by a pledge of Prime Borrower’s capital stock directly held by its direct parent and by first-priority security interests in substantially all of the assets of Prime Borrower and the subsidiary guarantors, in each case, subject to certain permitted liens and exceptions.
Significant amendments and restatements related to the First Lien Credit Agreement during the periods presented were as follows: • October 2023 - The Company redeemed approximately $1.3 billion of the then existing term loan (the “First Lien Term Loan B due 2026”) using net proceeds from the Commercial Divestiture.
• October 2023 - The Company amended and restated the First Lien Credit Agreement and refinanced the remaining outstanding balance of the First Lien Term Loan B due 2026 with a new $1,375 million 7-year First Lien Term Loan B due 2030.
• April 2024 - The Company amended and restated the First Lien Credit Agreement, which reduced the interest rate on the First Lien Term Loan B due 2030 from Term SOFR +2.50% to Term SOFR +2.25%.
• May 2024 - The Company amended and restated the First Lien Credit Agreement, which included the exchange of $143 million principal amount of loans under the Company’s Term Loan A Facility for its First Lien Term Loan B due 2030. In addition, later that month, the Company further amended and restated the First Lien Credit Agreement, pursuant to which the Company incurred an additional $474 million of outstanding principal under the First Lien Term Loan B due 2030 with the proceeds used to pay off the remaining outstanding balance of the Company’s Term Loan A Facility.
• October 2024 - The Company amended and restated the First Lien Credit Agreement to extend the maturity date of the First Lien Revolving Credit Facility to October 2029 and obtain an additional $225 million of First Lien Revolving Credit Facility commitments. After giving effect to the amendment, the aggregate amount of commitments under the First Lien Revolving Credit Facility is $800 million. In addition, the amendment reduced the commitment fee in respect of the First Lien Revolving Credit Facility to 0.20% to 0.30% per annum in respect of the unutilized commitments thereunder, subject to two step-downs based on certain specified net first lien leverage ratios.
• December 2024 - The Company amended and restated the First Lien Credit Agreement, which reduced the interest rate on the First Lien Term Loan B due 2030 from Term SOFR +2.25% to Term SOFR +2.00%.
During 2024, proceeds and repayments of long-term borrowings on the Consolidated Statements of Cash Flows include the impact of $704 million from certain of the 2024 amendments described above. During 2023, proceeds and repayments of long-term borrowings on the Consolidated Statements of Cash Flows include the impact of $230 million from the October 2023 amendment described above.
In addition, debt issuance costs, loss on extinguishment of debt, and financing and consent fees were not material as a result of these amendments.
Subsequent event - On February 7, 2025, the Company issued a notice of partial redemption for $500 million of the First Lien Notes due 2026, which will be redeemed on March 9, 2025. Prior to the issuance of such notice, certain lenders provided ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-28 commitments that they will fund a new $600 million first lien seven-year term loan facility. The closing of this new facility, which remains subject to market and other customary conditions, is expected to occur on or around March 7, 2025. The Company intends to use proceeds of this new facility for the partial redemption of the First Lien Notes due 2026 among other general corporate purposes.
First Lien Revolving Credit Facility As discussed above, during 2024, the Company extended the maturity date of the First Lien Revolving Credit Facility to October 2029, subject to a springing maturity of 91 days prior to the maturity date of certain long-term indebtedness if, as of such date, the outstanding principal amount of such indebtedness exceeds $350 million, and obtained an additional $225 million of First Lien Revolving Credit Facility commitments. After giving effect to the amendment, the aggregate amount of commitments under the First Lien Revolving Credit Facility is $800 million.
Significant borrowings and repayments under the First Lien Revolving Credit Facility during the periods presented were as follows: • 2024: The Company borrowed $365 million and repaid $365 million.
• 2022: The Company borrowed $550 million and repaid $575 million.
As of December 31, 2024, the Company had $800 million in available borrowing capacity under the First Lien Revolving Credit Facility.
Term Loan A Facility On March 14, 2023, Holdings, Prime Borrower, and The ADT Corporation (Prime Borrower and The ADT Corporation in such capacity, the “Term Loan A Borrowers”), entered into a term loan credit agreement (the “Term Loan A Credit Agreement”) with Barclays Bank PLC, as administrative agent, and the lenders party thereto, pursuant to which such lenders provided the Term Loan A Borrowers with an aggregate principal amount of $600 million of term loans (the “Closing Date Term Loan A Loans”) under a senior secured term loan A facility (the “Term Loan A Facility”). The Company used the proceeds from the Closing Date Term Loan A Loans to redeem $600 million outstanding principal amount of the Company’s 4.125% senior notes due June 15, 2023 (the “ADT Notes due 2023”).
Also on March 14, 2023, Holdings, the Term Loan A Borrowers, the subsidiary loan parties thereto, Barclays Bank PLC, and the lender party thereto entered into an amendment to the Term Loan A Credit Agreement, pursuant to which the lender party thereto agreed, at the option of the Term Loan A Borrowers and subject to the satisfaction or waiver of customary conditions, to provide the Term Loan A Borrowers with an aggregate principal amount of $50 million of incremental term loans (the “Incremental Term Loan A Loans”) under the Term Loan A Facility on or before the scheduled maturity date of the ADT Notes due 2023. On June 15, 2023, the Company borrowed the Incremental Term Loan A Loans and used the proceeds to complete the redemption of $50 million of the ADT Notes due 2023. The Incremental Term Loan A Loans have the same terms as, and constitute one class with, the Closing Date Term Loan A Loans.
In May 2024, the Company exchanged $143 million of loans under its Term Loan A Facility for its First Lien Term Loan B due 2030, as discussed above. In addition, later that month, the Company redeemed the remaining outstanding principal balance of $474 million of its Term Loan A Facility, excluding accrued and unpaid interest, using proceeds under the First Lien Term Loan B due 2030, as discussed above. As a result, the Term Loan A Facility has been terminated.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-29 First Lien Notes due 2024 As of December 31, 2024, the Company fully redeemed the 5.250% first-priority senior secured notes due 2024 (the “First Lien Notes due 2024”) as a result of the following transactions: • May 2023 - The Company redeemed $150 million principal amount of the outstanding First Lien Notes due 2024 for a redemption price of $150 million, excluding accrued and unpaid interest, using cash on hand.
• December 2023 - The Company redeemed $500 million principal amount of the outstanding First Lien Notes due 2024 for a redemption price of $500 million, excluding accrued and unpaid interest, using remaining net proceeds from the Commercial Divestiture and cash on hand.
• April 2024 - The Company redeemed the remaining outstanding principal balance of $100 million of the First Lien Notes due 2024 for a redemption price of $100 million, excluding accrued and unpaid interest, using proceeds from the Company’s First Lien Revolving Credit Facility.
First Lien Notes due 2026 The Company’s 5.750% first-priority senior secured notes due 2026 (the “First Lien Notes due 2026”) are due at maturity, and may be redeemed, in whole or in part, at any time at a make-whole premium plus accrued and unpaid interest to, but excluding, the redemption date. The First Lien Notes due 2026 are guaranteed, jointly and severally, on a senior secured first-priority basis, by each of the Company’s existing and future direct or indirect wholly-owned material domestic subsidiaries that guarantee the First Lien Credit Agreement.
Upon the occurrence of specified change of control events, the Company must offer to repurchase the First Lien Notes due 2026 at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the First Lien Notes due 2026 also provides for customary events of default.
As discussed above, on February 7, 2025, the Company issued a notice of partial redemption for $500 million of the First Lien Notes due 2026, which will be redeemed on March 9, 2025.
First Lien Notes due 2027 The Company’s 3.375% first-priority senior secured notes due 2027 (the “First Lien Notes due 2027”) are due at maturity and may be redeemed at the Company’s option as follows: • Prior to August 31, 2026, in whole at any time or in part from time to time, at a make-whole premium plus accrued and unpaid interest, if any, thereon to the redemption date.
• On or after August 31, 2026, in whole at any time or in part from time to time, at a redemption price equal to 100% of the principal amount of the First Lien Notes due 2027 redeemed plus accrued and unpaid interest, if any, thereon to the redemption date.
The Company’s obligations relating to the First Lien Notes due 2027 are guaranteed, jointly and severally, on a senior secured first-priority basis, by each of the Company’s domestic subsidiaries that guarantees its First Lien Credit Agreement and by each of the Company’s future domestic subsidiaries that guarantees certain of the Company’s debt. The First Lien Notes due 2027 and the related guarantees are secured by first-priority security interests in substantially all of the tangible and intangible assets owned by the issuers and each guarantor, subject to certain permitted liens and exceptions.
Upon the occurrence of specified change of control events, the Company must offer to repurchase the notes at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the First Lien Notes due 2027 also provides for customary events of default.
First Lien Notes due 2029 In July 2021, the Company issued $1.0 billion aggregate principal amount of 4.125% first-priority senior secured notes due 2029 (the “First Lien Notes due 2029”).
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-30 The First Lien Notes due 2029 will mature on August 1, 2029, with semi-annual interest payment dates of February 1 and August 1 of each year, beginning February 1, 2022, and may be redeemed at the Company’s option as follows: • Prior to August 1, 2028, in whole at any time or in part from time to time, at a redemption price equal to the greater of (i) 100% of the principal amount of the First Lien Notes due 2029 to be redeemed and (ii) the sum of the present values of the aggregate principal amount of the First Lien Notes due 2029 to be redeemed and the remaining scheduled interest payments due on any date after the redemption date, to and including August 1, 2028, discounted at an adjusted treasury rate plus 50 basis points, plus, in either case accrued and unpaid interest as of, but excluding, the redemption date.
• On or after August 1, 2028, in whole at any time or in part from time to time, at a redemption price equal to 100% of the principal amount of the First Lien Notes due 2029 to be redeemed and accrued and unpaid interest as of, but excluding, the redemption date.
The Company’s obligations relating to the First Lien Notes due 2029 are guaranteed, jointly and severally, on a senior secured first-priority basis, by substantially all of the Company’s subsidiaries and are secured by first-priority security interests in substantially all of the assets of the Company’s domestic subsidiaries, subject to certain permitted liens and exceptions.
Upon the occurrence of specified change of control events, the Company may be required to purchase the notes at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the First Lien Notes due 2029 also provides for customary events of default.
Second Lien Notes due 2028 The Company’s 6.250% second-priority senior secured notes due 2028 (the “Second Lien Notes due 2028”) are due at maturity, and since January 15, 2023, the Second Lien Notes due 2028 may be redeemed at the Company’s option in whole at any time or in part from time to time, at a redemption price equal to 100% (as of January 15, 2025) of the principal amount of the Second Lien Notes due 2028 redeemed and accrued and unpaid interest as of, but excluding, the redemption date.
The Company’s obligations relating to the Second Lien Notes due 2028 are guaranteed, jointly and severally, on a senior secured second-priority basis, by substantially all of the Company’s domestic subsidiaries and are secured by second-priority security interests in substantially all of the assets of the Company’s domestic subsidiaries, subject to certain permitted liens and exceptions.
Additionally, upon the occurrence of specified change of control events, the Company must offer to repurchase the Second Lien Notes due 2028 at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indenture governing the Second Lien Notes due 2028 also provides for customary events of default.
ADT Notes In connection with the ADT Acquisition, the Company entered into supplemental indentures to notes originally issued by The ADT Corporation (collectively, the “ADT Notes”) providing for each series of ADT Notes to benefit from (i) guarantees by substantially all of the Company’s domestic subsidiaries and (ii) first-priority senior security interests, subject to permitted liens, in substantially all of the existing and future assets of the Company’s domestic subsidiaries. As a result, these notes remained outstanding and became obligations of the Company.
During 2023, the Company redeemed the then-outstanding ADT Notes due 2023. The remaining outstanding ADT Notes are due at maturity, and may be redeemed, in whole at any time or in part from time to time, at a redemption price equal to the principal amount of the notes to be redeemed, plus a make-whole premium, plus accrued and unpaid interest as of, but excluding, the redemption date.
Additionally, upon the occurrence of specified change of control events, the Company must offer to repurchase the ADT Notes at 101% of the principal amount, plus accrued and unpaid interest, if any, to, but not including, the purchase date.
The indentures governing the ADT Notes also provide for customary events of default.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-31 Significant activity related to the ADT Notes due 2023 during the periods presented was as follows: • March 2023 - The Company redeemed $600 million outstanding principal amount for a total redemption price of $600 million, excluding any accrued and unpaid interest, using the proceeds from the Closing Date Term Loan A Loans.
• June 2023 - The Company redeemed the remaining outstanding principal amount of $100 million for a total redemption price of $100 million, excluding any accrued and unpaid interest, using $50 million of proceeds from the Incremental Term Loan A Loans and the remaining from cash on hand.
2020 Receivables Facility During March 2020, the Company entered into the 2020 Receivables Facility, as amended, whereby the Company obtains financing by selling or contributing certain retail installment contract receivables to the Company’s wholly-owned consolidated bankruptcy-remote special purpose entity (“SPE”). The SPE grants a security interest in those retail installment contract receivables as collateral for cash borrowings under the 2020 Receivables Facility. The SPE borrower under the 2020 Receivables Facility is a separate legal entity with its own creditors who will be entitled, prior to and upon the liquidation of the SPE, to be satisfied out of the SPE’s assets prior to any assets of the SPE becoming available to the Company (other than the SPE). Accordingly, the assets of the SPE are not available to pay creditors of the Company (other than the SPE), although collections from the transferred retail installment contract receivables in excess of amounts required to repay amounts then due and payable to the SPE’s creditors may be released to the Company and subsequently used by the Company (including to pay other creditors). The SPE’s creditors under the 2020 Receivables Facility have legal recourse to the transferred retail installment contract receivables owned by the SPE, and to the Company for certain performance and operational obligations relating to the 2020 Receivables Facility, but do not have any recourse to the Company (other than the SPE) for the payment of principal and interest on the advances under the 2020 Receivables Facility.
Significant amendments to the 2020 Receivables Facility during the periods presented were as follows: • May 2022 - Changed the benchmark rate from 1-month LIBOR to Daily SOFR, extended the scheduled termination date for the uncommitted revolving period from October 2022 to May 2023, and amended certain other terms to increase the advance rate on pledged collateral.
• March 2023 - Increased the borrowing capacity from $400 million to $500 million and extended the uncommitted revolving period from May 2023 to March 2024, among other things.
• March 2024 - The Company amended the agreement governing the 2020 Receivables Facility, pursuant to which the uncommitted revolving period was extended from March 2024 to April 2024.
• April 2024 - The Company further amended the agreement governing the 2020 Receivables Facility, pursuant to which, among other things, the borrowing capacity was increased from $500 million to $550 million and the uncommitted revolving period was extended from April 2024 to April 2025. In addition, proceeds and repayments from the receivables facility include the impact of $32 million from the amendments described above.
The Company services the transferred retail installment contract receivables and is responsible for ensuring the related collections are remitted to a segregated bank account in the SPE’s name. On a monthly basis, the segregated account is utilized to make required principal, interest, and other payments due under the 2020 Receivables Facility. The segregated account is considered restricted cash.
Proceeds and repayments from the 2020 Receivables Facility are presented in cash flows from financing activities on the Consolidated Statements of Cash Flows. The impact to the Consolidated Statements of Operations from the 2020 Receivables Facility was primarily due to the allowance for credit losses and interest expense during the periods presented.
As of December 31, 2024, the Company had an uncommitted available borrowing capacity under the 2020 Receivables Facility of $142 million.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-32 Variable Interest Entity The SPE, as described above, meets the definition of a variable interest entity (“VIE”) for which the Company is the primary beneficiary as it has the power to direct the SPE’s activities and the obligation to absorb losses or the right to receive benefits of the SPE. As such, the SPE’s assets, liabilities, and financial results of operations are consolidated in the Company’s consolidated financial statements.
As of December 31, 2024 and 2023, the SPE’s assets and liabilities primarily consisted of a portion of the Company’s unbilled retail installment contract receivables, net, of $575 million and $610 million, respectively, and borrowings under the 2020 Receivables Facility as presented above.
Solar Receivables Facility On August 2, 2023, Compass Solar Group, LLC (“Compass”) and ADT Solar Finance LLC (“ADT Solar Finance”), each an indirect wholly-owned subsidiary of ADT Inc. entered into a Receivables Financing Agreement with Mizuho Bank, Ltd. (the “Solar Receivables Financing Agreement”) to finance receivables generated by the installation of residential solar systems.
Prior to its expiration in August 2024, the Solar Receivables Financing Agreement, among other things, provided for an uncommitted revolving loan facility in the aggregate principal amount of up to $300 million which loans were to be secured by substantially all the assets of ADT Solar Finance (the “Solar Receivables Facility”). The Company did not borrow any amounts under the Solar Receivables Facility prior to its expiration.
Debt Covenants Our credit agreements and indentures associated with the borrowings above contain certain covenants and restrictions that limit the Company’s ability to, among other things: (a) incur additional debt or issue certain preferred equity interests; (b) create liens on certain assets; (c) make certain loans or investments (including acquisitions); (d) pay dividends on or make distributions in respect of the capital stock or make other restricted payments; (e) consolidate, merge, sell, or otherwise dispose of all or substantially all of the Company’s assets; (f) sell assets; (g) enter into certain transactions with affiliates; (h) enter into sale-leaseback transactions; (i) restrict dividends from the Company’s subsidiaries or restrict liens; (j) change the Company’s fiscal year; and (k) modify the terms of certain debt or organizational agreements.
Our credit agreements and indentures associated with the borrowings above also provide for customary events of default.
The Company is subject to a springing financial maintenance covenant under the First Lien Credit Agreement, which requires the Company to not exceed a specified first lien leverage ratio at the end of each fiscal quarter if the testing conditions are satisfied. The covenant is tested if the outstanding loans under the First Lien Revolving Credit Facility, subject to certain exceptions, exceed 30% of the total commitments under the First Lien Revolving Credit Facility as of the last day of any fiscal quarter.
As of December 31, 2024, the Company was in compliance with all financial covenant and other maintenance tests for all debt obligations.
Loss on Extinguishment of Debt Loss on extinguishment of debt includes the payment of call and redemption premiums, the write-off of unamortized deferred financing costs and discounts, and certain other expenses associated with extinguishment of debt. During the periods presented, significant activity related to loss on extinguishment of debt was as follows: • 2023: Totaled $17 million and was primarily due to the write-off of unamortized discount and debt issuance costs associated with the partial redemption of the First Lien Term Loan B due 2026 and the refinancing of the First Lien Term Loan B due 2026.
Additional fees and other costs associated with financing transactions were not material during 2024, 2023, or 2022.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-33 Other As of December 31, 2024, the aggregate annual maturities of debt, excluding finance leases, were as follows: (in thousands) 2025 2026 2027 2028 2029 Thereafter Total Debt principal $ 170,198 $ 1,499,939 $ 1,099,518 $ 1,367,345 $ 1,019,841 $ 2,635,062 $ 7,791,903 Interest expense (excluding interest income) during 2024, 2023, and 2022 on the Company’s debt, including finance leases, and interest rate swap contracts presented within interest expense, net, was $451 million, $586 million, and $277 million, respectively.
8. DERIVATIVE FINANCIAL INSTRUMENTS The Company's derivative financial instruments primarily consist of SOFR-based interest rate swap contracts, which were entered into with the objective of managing exposure to variability in interest rates on the Company's debt and interest rate swaps. All interest rate swap contracts are reported in the Consolidated Balance Sheets at fair value.
For interest rate swap contracts that are: • Not designated as cash flow hedges: Unrealized gains and losses are recognized in interest expense, net, and other income (expense) depending on the nature of the underlying that the swaps are economically hedging.
• Designated as cash flow hedges: Unrealized gains and losses are recognized as a component of accumulated other comprehensive income (loss) (“AOCI”) and are reclassified into interest expense, net, in the same period in which the related interest on debt affects earnings.
For interest rate swap contracts that have been de-designated as cash flow hedges and for which forecasted cash flows are: • Probable or reasonably possible of occurring: Unrealized gains and losses previously recognized as a component of AOCI are reclassified into interest expense, net, in the same period in which the related interest on variable-rate debt affects earnings through the original maturity date of the related interest rate swap contracts.
• Probable of not occurring: Unrealized gains and losses previously recognized as a component of AOCI are immediately reclassified into interest expense, net.
The cash flows associated with interest rate swap contracts that contain an other-than-insignificant financing element at inception are reflected as cash flows from financing activities.
The cash flows associated with interest rate swap contracts that were entered into with the intention of offsetting the economic overhedged position of a portion of our existing interest rate swaps are reflected as cash flows from investing activities.
Interest Rate Swaps In October 2019, and in connection with the refinancing of variable-rate debt under the First Lien Credit Agreement in September 2019, the Company terminated interest rate swap contracts with an aggregate notional amount of $3.8 billion, of which $2.8 billion were designated as cash flow hedges, and concurrently entered into new LIBOR-based interest rate swap contracts, which were, at the time, designated as cash flow hedges, with an aggregate notional amount of $2.8 billion and maturity of September 2026. These swaps contain an other-than-insignificant financing element due to their off-market terms at the inception of the swaps.
Beginning in March 2020, the Company's interest rate swap contracts previously designated as cash flow hedges were no longer highly effective as a result of changes in the interest rate environment. Accordingly, the Company de-designated the cash flow hedges, and the unrealized gains and losses for the period in which these cash flow hedges were no longer highly effective were recognized in interest expense, net. Unrealized losses previously recognized as a component of AOCI prior to de-designation are being reclassified into interest expense, net, in the same period in which the related interest on variable-rate debt affects earnings through the maturity dates of the interest rate swap contracts, as the forecasted cash flows are probable or reasonably possible of occurring.
In March and April 2023, the Company entered into floating-to-fixed interest rate swaps with an aggregate notional amount of $300 million to partially hedge the Company’s then outstanding Term Loan A Facility.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-34 In December 2023, the Company entered into interest rate swaps with an aggregate notional amount of $700 million to offset the excess notional interest rate swaps as a result of the partial redemption of the First Lien Term Loan B due 2026. The changes in fair value associated with these swaps and the excess swaps are reflected in other income (expense).
Notional Amounts (in thousands) December 31, Execution Maturity Designation 2024 2023 October 2019 September 2026 Not designated $ 2,800,000 $ 2,800,000 March 2023 March 2028 Not designated 100,000 100,000 April 2023 March 2028 Not designated 200,000 200,000 December 2023 September 2026 Not designated 700,000 700,000 Total notional amount $ 3,800,000 $ 3,800,000 Classification and Fair Value December 31, Balance Sheet Classification (in thousands) 2024 2023 Prepaid expenses and other current assets $ 56,164 $ 74,974 Other assets $ 54,102 $ 76,493 Accrued expenses and other current liabilities $ 1,466 $ 5,312 Other liabilities $ 208 $ 1,325 Unrealized Gains (Losses) Years Ended December 31, Statement of Operations Classification (in thousands) 2024 2023 2022 Interest expense, net $ (27,164) $ (22,174) $ 301,851 Other income (expense) $ (17,996) $ (16,511) $ — Changes in and Reclassifications out of AOCI (in thousands) Cash Flow Hedges Balance as of December 31, 2021 $ (71,267) Pre-tax current period change 33,946 Income tax benefit (expense) (8,192) Balance as of December 31, 2022 (45,513) Pre-tax current period change 42,295 Income tax benefit (expense) (10,166) Balance as of December 31, 2023 (13,384) Pre-tax current period change 7,921 Income tax benefit (expense) (1,913) Balance as of December 31, 2024 $ (7,376) During 2023, the Company recorded $25 million to interest expense, net associated with the reclassification from AOCI of historical losses related to the de-designated interest rate swaps for which the cash flows were probable of not occurring as a result of the partial redemption of the Company’s then outstanding First Lien Term Loan B due 2026.
As of December 31, 2024, AOCI associated with previously designated cash flow hedges that is estimated to be reclassified to interest expense, net, within the next twelve months is not material.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-35 9. INCOME TAXES The Company accounts for income taxes under the asset and liability method, which requires the recognition of deferred tax assets and liabilities for the temporary differences between the recognition of revenue and expenses for income tax and financial reporting purposes and between the tax basis of assets and liabilities and their reported amounts in the consolidated financial statements. The Company records the effect of a tax rate or law change on the Company’s deferred tax assets and liabilities in the period of enactment.
The amounts and related disclosures below are based on the continuing operations of the Company, unless otherwise noted.
Components of Income Before Taxes Years Ended December 31, (in thousands) 2024 2023 2022 United States $ 808,476 $ 601,370 $ 401,746 Foreign 6,694 3,013 2,712 Income (loss) from continuing operations before income taxes and equity in net earnings (losses) of equity method investee $ 815,170 $ 604,383 $ 404,458 Components of Income Tax Benefit (Expense)(1) Years Ended December 31, (in thousands) 2024 2023 2022 Current: Federal $ 1,027 $ (358) $ (184) State (13,829) (57,024) (28,100) Foreign (2,096) (923) (691) Current income tax benefit (expense) (14,898) (58,305) (28,975) Deferred: Federal (105,711) (136,457) (43,203) State (33,702) 11,704 24,029 Foreign (170) (194) (401) Deferred income tax benefit (expense) (139,583) (124,947) (19,575) Income tax benefit (expense) $ (154,481) $ (183,252) $ (48,550) ___ (1) The components of tax benefit (expense) include both continuing and discontinued operations for all periods presented in accordance with Accounting Standards Codification (“ASC”) 740. This presentation is to reflect the Company’s tax structure and filings.
Income tax benefit (expense) is included in the Consolidated Statements of Operations as follows: Years Ended December 31, (in thousands) 2024 2023 2022 Continuing operations $ (195,780) $ (160,585) $ (87,692) Discontinued operations 41,299 (22,667) 39,142 Income tax benefit (expense) $ (154,481) $ (183,252) $ (48,550) ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-36 Effective Tax Rate Reconciliation Reconciliations between the actual effective tax rate on continuing operations and the statutory U.S. federal income tax rate were as follows: Years Ended December 31, 2024 2023 2022 Statutory federal tax rate 21.0 % 21.0 % 21.0 % State income taxes, net of federal benefits 5.4 % 6.1 % 3.0 % Non-U.S. tax 0.3 % 0.2 % 0.3 % Non-deductible and non-taxable charges(1) 0.8 % 1.0 % 5.1 % Valuation allowance — % — % (0.6) % Unrecognized tax benefits (4.0) % (1.0) % (2.3) % Share-based compensation 0.1 % — % (0.6) % Non-deductible goodwill on dispositions — % 0.7 % — % Federal credits 0.1 % (0.6) % (2.9) % Acquisitions and dispositions 1.2 % (0.9) % (0.2) % Legislative changes (0.4) % 0.6 % (2.1) % Prior year return adjustments (0.6) % (0.6) % 1.1 % Other 0.1 % 0.1 % (0.1) % Effective tax rate 24.0 % 26.6 % 21.7 % __ (1) During 2022, primarily represents the impact related to the fair value adjustment of the Forward Contract.
Deferred Tax Assets and Deferred Tax Liabilities The components of the Company's net deferred tax assets (liabilities) were as follows: December 31, (in thousands) 2024 2023 Deferred tax assets: Accrued liabilities and reserves $ 94,084 $ 90,351 Tax loss and credit carryforwards 75,927 132,230 Disallowed interest carryforward 230,048 150,492 Deferred revenue 218,639 225,499 Other 91,892 112,194 Total deferred tax assets 710,590 710,766 Valuation allowance (12,264) (12,264) Deferred tax assets, net of valuation allowance $ 698,326 $ 698,502 Deferred tax liabilities: Subscriber system assets $ (757,046) $ (761,203) Intangible assets (1,051,499) (893,292) Other (56,994) (71,196) Total deferred tax liabilities (1,865,539) (1,725,691) Net deferred tax assets (liabilities) $ (1,167,213) $ (1,027,189) ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-37 The valuation allowance for deferred tax assets relates to the uncertainty of the utilization of certain U.S. federal and state deferred tax assets. In evaluating the Company’s ability to recover its deferred tax assets, the Company considers all available positive and negative evidence, which includes its past operating results, the existence of cumulative losses in the most recent years, and its forecast of future taxable income. In estimating future taxable income, the Company develops assumptions related to the amount of future pre-tax operating income, the reversal of temporary differences, and the implementation of feasible and prudent tax planning strategies. These assumptions require significant judgment about the forecasts of future taxable income and are consistent with the plans and estimates the Company is using to manage its underlying businesses. The Company believes that it is more-likely-than-not that it will generate sufficient future taxable income to realize its deferred tax assets, net of valuation allowance.
The changes in the valuation allowance for deferred tax assets were as follows: Years Ended December 31, (in thousands) 2024 2023 2022 Beginning balance $ (12,264) $ (57,715) $ (60,157) Income tax benefit (expense)(1) — 43,277 2,428 Write-offs and other — 2,174 14 Ending balance $ (12,264) $ (12,264) $ (57,715) __ (1) During 2023, the change is primarily related to the utilization of capital loss carryforwards against which a valuation allowance was previously recorded. The utilization is attributable to capital gains generated in connection with the Commercial Divestiture.
As of December 31, 2024, the Company has no remaining financial statement federal net operating loss (“NOL”) carryforwards.
As of December 31, 2024, the Company’s valuation allowance for deferred tax assets was primarily related to capital loss carryforwards in Canada primarily generated in connection with the sale of ADT Canada during 2019.
The Tax Cuts and Jobs Act of 2017 introduced IRC Section 163(j), which limits the deductibility of interest expense and allows for the excess to be carried forward indefinitely. As of December 31, 2024, the Company has not recorded a valuation allowance against the disallowed interest carryforward as the Company believes it has sufficient sources of future taxable income to realize the related tax benefit.
Unrecognized Tax Benefits The Company recognizes positions taken or expected to be taken in a tax return in the consolidated financial statements when it is more-likely-than-not (i.e., a likelihood of more than 50%) that the position would be sustained upon examination by tax authorities. A recognized tax position is then measured at the largest amount of benefit with greater than 50% likelihood of being realized upon ultimate settlement. The Company records liabilities for positions that have been taken but do not meet the more-likely-than-not recognition threshold. The Company adjusts the liabilities for unrecognized tax benefits in light of changing facts and circumstances; however, due to the complexity of some of these uncertainties, the ultimate resolution may result in a change to the estimated liabilities. The Company includes interest and penalties associated with unrecognized tax benefits as income tax expense and as a component of the recorded balance of unrecognized tax benefits, which is reflected in other liabilities, or net of related tax loss carryforwards in the Consolidated Balance Sheets. Interest and penalties associated with unrecognized tax benefits were not material to the Company's consolidated financial statements for the periods presented.
The following is a roll-forward of unrecognized tax benefits: Years Ended December 31, (in thousands) 2024 2023 2022 Beginning balance $ 48,823 $ 56,177 $ 66,221 Gross increase related to prior year tax positions 9,933 517 5,063 Gross decrease related to prior year tax positions (5,108) — — Decreases related to lapse of statute of limitation (29,042) (7,871) (15,107) Ending balance $ 24,606 $ 48,823 $ 56,177 The Company’s unrecognized tax benefits relate to tax years that are subject to audit by the taxing authorities in the U.S. federal, state and local, and foreign jurisdictions. Based on the current tax statutes and status of its income tax audits, the ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-38 Company does not expect unrecognized tax benefits to change significantly in the next twelve months.
Open Tax Years by Jurisdiction Jurisdiction Years Open to Audit Federal 2021 - 2023 State 2019 - 2023 Canada 2020 - 2023 The Company files a consolidated return for its U.S. entities and separate returns for each Canadian entity. These income tax returns are subject to audit by the taxing authorities that may culminate in proposed assessments which may ultimately result in a change to the estimated income taxes.
Federal Tax Legislation Tax Cuts and Jobs Act - Certain changes to U.S. federal tax law included in the Tax Cuts and Jobs Act of 2017 had a delayed effective date and have taken effect for 2022. Under IRC Section 163(j), the limitation on net business interest expense deductions will no longer be increased by deductions for depreciation, amortization, or depletion. Under IRC Section 174, specified research and experimentation expenditures must now be capitalized and amortized.
Inflation Reduction Act - The Inflation Reduction Act (the “IRA”) was signed into law in August 2022. The IRA, among other provisions, implements (i) a 15% corporate alternative minimum tax (“CAMT”) on book income of corporations whose average annual adjusted financial statement income during the most recently-completed three-year period exceeds $1.0 billion, (ii) a 1% excise tax on net stock repurchases, and (iii) several tax incentives to promote clean energy including an extension of the investment tax credit. Both the CAMT and the excise tax provisions are effective for tax years beginning after December 31, 2022, and as of December 31, 2024, the Company does not anticipate any material impact.
10. EQUITY Common Stock and Class B Common Stock During September 2020, the Company amended its articles of incorporation to authorize the issuance of 100,000,000 shares of Class B common stock, par value of $0.01 per share (“Class B Common Stock”), as well as to increase the number of authorized shares of preferred stock, par value of $0.01 per share, to 1,000,000.
In September 2020, the Company issued and sold 54,744,525 shares of Class B Common Stock for an aggregate purchase price of $450 million to Google LLC (“Google”) in a private placement pursuant to a securities purchase agreement dated July 31, 2020 (the “Securities Purchase Agreement”).
Accordingly, the Company has two classes of common stock, Common Stock and Class B Common Stock, both of which entitle stockholders to one vote for each share of Common Stock.
Each share of Class B Common Stock has equal status and rights to dividends as a share of Common Stock. The holders of Class B Common Stock have one vote for each share of Class B Common Stock held of record by such holder on all matters on which stockholders are entitled to vote generally; provided, however, that holders of Class B Common Stock, as such, are not entitled to vote on the election, appointment, or removal of directors of the Company. Additionally, each share of Class B Common Stock will immediately become convertible into one share of Common Stock, at the option of the holder thereof, at any time following the earlier of (i) the expiration or early termination of applicable waiting periods under the Hart-Scott-Rodino Antitrust Improvements Act of 1976, as amended (“HSR Clearance”), required prior to such holder’s conversion of all such shares of Class B Common Stock, or (ii) to the extent HSR Clearance is not required prior to such holder’s conversion of such shares of Class B Common Stock, the date that such holder owns such shares of Class B Common Stock.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-39 Issuances of Common Stock During the periods presented, issuances of Common Stock other than as related to the exercise or vesting of share-based compensation awards includes: • October 2022: The Company issued approximately 133 million shares of Common Stock for an aggregate purchase price of $1.2 billion as part of the State Farm Strategic Investment (as defined below).
• June 2022: The Company issued approximately 2 million shares of Common Stock as consideration for a business acquisition.
Additionally, throughout 2022, the Company issued approximately 5.3 million shares of the Company’s Common Stock with a value of approximately $40 million related to the ADT Solar Acquisition (the “Delayed Shares”).
Repurchases of Common Stock On January 24, 2024, the Company's Board of Directors announced a share repurchase plan (the “2024 Share Repurchase Plan”), pursuant to which the Company was authorized, but not obligated, to repurchase up to a maximum aggregate amount of $350 million of shares of the Company's Common Stock through January 29, 2025. The 2024 Share Repurchase Plan expired in January 2025 with $5 million authorized repurchases remaining.
Share repurchases under the 2024 Share Repurchase Plan were as follows: • In March 2024, the Company repurchased and retired 15 million shares at a price per share of approximately $6.22 for an aggregate purchase price of $93 million in connection with an offering of the Company’s Common Stock.
• In October 2024, the Company repurchased and retired 16 million shares at a price per share of approximately $7.20 for an aggregate purchase price of $115 million in connection with an offering of the Company’s Common Stock.
• In October 2024, the Company repurchased and retired 5 million shares of Common Stock from a non-affiliate individual at a price per share of $6.40 for an aggregate purchase price of $32 million.
Subsequent events In December 2024, the Company entered into an agreement with a non-affiliate individual to repurchase 15 million shares of Common Stock at a price per share of $6.95 for an aggregate purchase price of $104 million. The transaction settled in January 2025, and the Company retired the shares. As of December 31, 2024, the Company recorded a liability related to the forward share repurchase contract and a reduction to additional paid-in capital.
In February 2025, the Company’s Board of Directors announced a share repurchase plan (the “2025 Share Repurchase Plan”), pursuant to which the Company is authorized to repurchase, through April 30, 2026, up to a maximum aggregate amount of $500 million of shares of Common Stock. The 2025 Share Repurchase Plan allows the Company to purchase Common Stock from time to time in one or more open market or privately negotiated transactions, including pursuant to Rule 10b5-1 or Rule 10b-18 of the Exchange Act, or pursuant to one or more accelerated share repurchase agreements, subject to certain requirements and factors.
Google Investor Rights Agreement In connection with the issuance of the Class B Common Stock, the Company and Google entered into an Investor Rights Agreement (the “Google Investor Rights Agreement”), pursuant to which Google agreed to be bound by customary transfer restrictions and drag-along rights, and be afforded customary registration rights with respect to shares of Class B Common Stock held directly by Google. Under the terms of the Google Investor Rights Agreement, Google was prohibited, subject to certain exceptions, from transferring any shares of Class B Common Stock or any shares of Common Stock issuable upon conversion of the Class B Common Stock beneficially owned by Google through September 2023, or earlier if certain conditions occurred (the “Google Lock-up Period”).
In December 2023, the Company and Google amended the Google Investor Rights Agreement to extend the Google Lock-up Period through June 2025, or earlier if (i) the Google Commercial Agreement (as defined in Note 13 “Commitments and Contingencies”) has been terminated under certain specified circumstances or (ii) the Google Cloud Master Agreement (as defined in Note 13 “Commitments and Contingencies”) has been validly terminated for any reason other than Google's uncured material breach.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-40 State Farm Transaction State Farm Strategic Investment On September 5, 2022, the Company entered into a securities purchase agreement (the “State Farm Securities Purchase Agreement”) with State Farm, pursuant to which the Company agreed to issue and sell in a private placement to State Farm 133 million shares of the Company’s Common Stock (the “State Farm Shares”) at a per share price of $9.00 for an aggregate purchase price of $1.2 billion (the “State Farm Strategic Investment”). The transaction closed on October 13, 2022 (the “Closing”).
Additionally, on September 12, 2022, in connection with the State Farm Strategic Investment, the Company commenced a tender offer (discussed below) to purchase up to 133 million shares of the Company’s Common Stock (including shares issued upon conversion of shares of Class B Common Stock ) (the “Tender Shares”) at a price of $9.00 per share (the “Tender Offer”) using proceeds from the State Farm Strategic Investment.
After the Closing and the Tender Offer, State Farm owned approximately 15% of the Company’s issued and outstanding Common Stock (assuming conversion of Class B Common Stock) and became a related party.
Tender Offer Concurrently with the execution of the State Farm Securities Purchase Agreement, (i) Apollo delivered to the Company a Tender and Support Agreement, pursuant to which Apollo agreed to collectively tender (and not withdraw) no fewer than 133 million shares of Common Stock in the Tender Offer (the “Apollo Support Agreement”), and (ii) Google delivered to the Company a Support Agreement, pursuant to which Google agreed to not convert and tender any of its shares of Class B Common Stock.
The Tender Offer expired on October 20, 2022. On October 26, 2022, the Company used proceeds from the State Farm Strategic Investment to repurchase an aggregate of 133 million shares of the Company’s Common Stock at a purchase price of $9.00 per share for an aggregate purchase price of $1.2 billion, excluding fees and expenses, subject to the terms and conditions described in the Offer to Purchase dated September 12, 2022 (as amended from time to time, the “Offer to Purchase”). The Tender Shares were subject to the “odd lot” priority and proration provisions described in the Offer to Purchase as the Tender Offer was substantially over-subscribed. No shares of Class B Common Stock were converted and tendered in the Tender Offer.
The Tender Offer was considered a contingent forward purchase contract (the “Forward Contract”), which was recorded at fair value in the Consolidated Balance Sheet. The fair value of the Forward Contract was estimated as the difference between the present value of the cash consideration to be paid and the value of the Company’s Common Stock to be tendered. At the commencement of the Tender Offer, the Company recorded a liability and a reduction to additional paid in capital of $42 million. During 2022, the change in fair value recognized in other income (expense) was a net loss of $63 million. Fees associated with the Tender Offer were not material.
State Farm Investor Rights Agreement At the Closing, the Company and State Farm entered into an Investor Rights Agreement (the “State Farm Investor Rights Agreement”), pursuant to which the Board of Directors of the Company increased its size by one director and appointed a designee of State Farm as a member of the Board of Directors.
Pursuant to the terms of the State Farm Investor Rights Agreement, State Farm will also be bound by customary transfer and standstill restrictions and drag-along rights, and be afforded customary registration rights with respect to the State Farm Shares. In particular, State Farm (a) will be prohibited, subject to certain customary exceptions, from transferring any of the State Farm Shares until the earlier of (i) October 13, 2025, and (ii) the date on which the development agreement with State Farm (the “State Farm Development Agreement”) has been validly terminated, other than in the event of termination by the Company for a material breach thereof by State Farm, and (b) will be subject to certain standstill restrictions, including that State Farm will be restricted from acquiring additional equity securities of the Company if such acquisition would result in State Farm (and its affiliates) acquiring beneficial ownership in excess of 18% of the issued and outstanding Common Stock, taking into account on an as-converted basis the issued and outstanding Class B Common Stock, until five days after the date that no designee of State Farm serves on the Board of Directors and State Farm has no rights (or has irrevocably waived its right) to nominate a designee to the Board of Directors. Notwithstanding the standstill restrictions described above, State Farm will not be restricted from acquiring shares of Common Stock or other equity securities of the Company from Prime Security Services TopCo Parent, L.P. and its affiliates so long as State Farm and its affiliates would not, subject to certain exceptions, beneficially own in excess ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-41 of 25% of the issued and outstanding Common Stock, taking into account on an as-converted basis the issued and outstanding Class B Common Stock, as a result of such acquisition.
In addition, under the terms of the State Farm Investor Rights Agreement, in the event that the Company proposes to issue and sell shares of Common Stock, Class B Common Stock, or other equity securities of the Company to certain homeowners’ insurance and reinsurance companies, State Farm will have a right of first refusal with respect to such proposed issuance and sale on the same terms and conditions (the “ROFR”). The ROFR will terminate upon the earliest to occur of (i) State Farm and its permitted transferees no longer collectively owning shares of Common Stock equal to at least 50% of the State Farm Shares; (ii) the termination of the State Farm Development Agreement by the Company for a material breach by State Farm; and (iii) to the extent that the State Farm Development Agreement does not remain in effect on such date, the five (5) year anniversary of the Closing.
State Farm Development Agreement At the Closing, the Company, ADT LLC (an indirect wholly owned subsidiary of the Company), and State Farm entered into the State Farm Development Agreement pursuant to which State Farm committed up to $300 million to an Opportunity Fund that will fund certain product and technology innovation, customer growth, and marketing initiatives to be agreed on between State Farm and the Company.
Additionally at the Closing, the Company received $100 million of the Opportunity Fund, which is restricted until such time as the Company uses the funds in accordance with the State Farm Development Agreement. The Company’s use of the funds is also subject to approval by State Farm. The Company recorded the cash received from the Opportunity Fund as restricted cash and a corresponding liability, which is reflected in accrued expenses and other current liabilities.
Refer to Note 16 “Related Party Transactions” for more information on the Opportunity Fund.
Dividends Stockholders are entitled to receive dividends when, as, and if declared by the Company’s Board of Directors out of funds legally available for that purpose.
(in thousands, except per share data) Common Stock Class B Common Stock Declaration Date Record Date Payment Date Per Share Aggregate Per Share Aggregate Year Ended December 31, 2024 1/24/24 3/14/24 4/4/24 $ 0.055 $ 47,059 $ 0.055 $ 3,011 4/25/24 6/13/24 7/9/24 0.055 47,137 0.055 3,011 8/1/24 9/13/24 10/4/24 0.055 47,146 0.055 3,011 10/24/24 12/12/24 1/9/25 0.055 46,012 0.055 3,011 Total $ 0.220 $ 187,354 $ 0.220 $ 12,044 Year Ended December 31, 2023 2/28/23 3/16/23 4/4/23 $ 0.035 $ 30,342 $ 0.035 $ 1,916 5/2/23 6/15/23 7/6/23 0.035 30,256 0.035 1,916 8/8/23 9/15/23 10/4/23 0.035 30,405 0.035 1,916 11/2/23 12/14/23 1/9/24 0.035 30,358 0.035 1,916 Total $ 0.140 $ 121,361 $ 0.140 $ 7,664 During 2022, the Company declared aggregate dividends of $0.14 per share on Common Stock ($120 million) and $0.14 per share on Class B Common Stock ($8 million).
Subsequent Event - On February 27, 2025, the Company announced a dividend of $0.055 per share to holders of Common Stock and Class B Common Stock of record on March 13, 2025, which will be distributed on or about April 3, 2025.
Accumulated Other Comprehensive Income (Loss) Refer to Note 8 “Derivative Financial Instruments” for AOCI reclassifications associated with cash flow hedges. Other changes in AOCI, which primarily relate to the Company’s defined benefit pension plans, were not material.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-42 11. SHARE-BASED COMPENSATION The Company grants share-based compensation awards to participants under the 2016 Equity Incentive Plan (the “2016 Plan”) and the 2018 Omnibus Incentive Plan (the “2018 Plan”). Share-based compensation expense is recognized in the Consolidated Statements of Operations as follows: Years Ended December 31, (in thousands) 2024 2023 2022 Selling, general, and administrative expenses $ 48,745 $ 38,626 $ 52,945 Income (loss) from discontinued operations, net of tax (132) 12,511 13,621 Total share-based compensation expense $ 48,613 $ 51,137 $ 66,566 The following discussion of the Company’s share-based compensation awards includes awards related to continuing and discontinued operations, unless otherwise noted.
2016 Plan As of December 31, 2024, the Company is authorized to issue no more than approximately 5 million shares of Common Stock by the exercise or vesting of granted awards under the 2016 Plan. The Company does not expect to issue additional awards under the 2016 Plan. Unrecognized share-based compensation expense as of December 31, 2024 and share-based compensation expense for awards granted under the 2016 Plan were not material during the periods presented.
Distributed Shares and Class B Unit Redemption In connection with the IPO, each holder of Class B awards (“Class B Units”), which were issued to certain participants by Ultimate Parent prior to the IPO, had their entire Class B interest in Ultimate Parent redeemed for the number of shares of the Company’s Common Stock (the “Distributed Shares”) that would have been distributed to such holder under the terms of Ultimate Parent’s operating agreement in a hypothetical liquidation on the date and price of the IPO (the “Class B Unit Redemption”).
The Class B Unit Redemption resulted in a modification of the Class B Units, whereby each holder received both vested and unvested Distributed Shares in the same proportion as the holder’s vested and unvested Class B Units held immediately prior to the IPO. As a result of the Class B Unit Redemption, holders of Class B Units received a total of 20.6 million shares of the Company’s Common Stock, of which 50% were subject to the same vesting conditions under the Class B Unit Service Tranche (the “Distributed Shares Service Tranche”), which were subject to ratable service-based vesting over a five-year period, and 50% were subject to the same vesting conditions under the Class B Unit Performance Tranche (the “Distributed Shares Performance Tranche”), which were based on the achievement of certain investment return thresholds by Apollo. The Distributed Shares also have certain other restrictions pursuant to the terms and conditions of the Company’s Amended and Restated Management Investor Rights Agreement (the “MIRA”). The MIRA was last amended on August 1, 2024.
The IPO triggered an acceleration of vesting of the unvested Distributed Shares Service Tranche causing them to become fully vested six months from the date of the IPO, which occurred in July 2018. The Company recorded share-based compensation expense on the Distributed Shares Performance Tranche on a straight-line basis over the derived service period of approximately three years from the IPO date, as the vesting conditions were deemed probable following the consummation of the IPO.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-43 The following table summarizes activity related to the Distributed Shares Performance Tranche during 2024: Number of Distributed Shares Weighted-Average Grant Date Fair Value Unvested as of December 31, 2023 9,194,312 $ 13.05 Vested(1) (2,509,806) 13.08 Forfeited — — Unvested as of December 31, 2024 6,684,506 $ 10.44 ___ (1) During the second quarter of 2024, certain of the Distributed Shares were modified in a manner that resulted in such awards immediately vesting.
2018 Plan In January 2018, the Company approved the 2018 Plan, which became effective upon consummation of the IPO. During 2024, the Company amended the 2018 Plan to increase the number of authorized common shares to be issued under the 2018 Plan by 50 million shares. As of December 31, 2024, the 2018 Plan, as amended, authorizes the issuance of no more than approximately 138 million shares of Common Stock by the exercise or vesting of granted awards, which are generally stock options and restricted stock units (“RSUs”). The Company satisfies the exercise of options and the vesting of RSUs through the issuance of authorized but previously unissued shares of Common Stock.
Awards issued under the 2018 Plan include retirement provisions that allow awards to continue to vest in accordance with the granted terms in its entirety or on a pro-rata basis when a participant reaches retirement eligibility, as long as 12 months of service have been provided since the date of grant. Accordingly, share-based compensation expense for service-based awards is recognized on a straight-line basis over the vesting period, or on an accelerated basis for retirement-eligible participants where applicable. The Company accounts for forfeitures as they occur.
Additionally, RSUs entitle the holder to dividend equivalent units (“DEUs”), which are granted as additional RSUs and are subject to the same vesting and forfeiture conditions as the underlying RSUs. DEUs are charged against accumulated deficit when dividends are paid.
Top-up Options In connection with the Class B Unit Redemption in 2018, the Company granted 12.7 million options to holders of Class B Units (the “Top-up Options”). The Top-up Options have an exercise price equal to the IPO price per share of the Company’s Common Stock, as adjusted in accordance with 2018 Plan provisions, and a contractual term of ten years from the grant date. Similar to the vesting conditions outlined above for the Distributed Shares, the Top-up Options contain a tranche subject to service-based vesting (the “Top-up Options Service Tranche”) and a tranche subject to vesting based upon the achievement of certain investment return thresholds by Apollo (the “Top-up Options Performance Tranche”). Recipients of the Top-up Options received both vested and unvested Top-up Options in the same proportion as the vested and unvested Class B Units held immediately prior to the IPO and Class B Unit Redemption. The Top-up Options vesting conditions are the same as those attributable to the Distributed Shares, including the condition that accelerated vesting of the unvested options in the Top-up Options Service Tranche causing them to become fully vested six months from the IPO. Any shares of the Company’s Common Stock acquired upon exercise of the Top-up Options will be subject to the terms of the MIRA.
Share-based compensation expense associated with the Top-up Options Performance Tranche was recognized on a straight-line basis over the derived service period of approximately three years from the IPO date.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-44 The following table summarizes activity related to the Top-up Options: Service Tranche Performance Tranche Number of Top-up Options Weighted-Average Exercise Price Number of Top-up Options Weighted-Average Exercise Price Aggregate Intrinsic Value(1) (in thousands) Weighted-Average Remaining Contractual Term (Years) Outstanding as of December 31, 2023 5,955,254 $ 13.30 5,641,244 $ 13.30 Exercised — — — — Forfeited — — (16,643) 13.30 Outstanding as of December 31, 2024 5,955,254 $ 13.30 5,624,601 $ 13.30 — 3.0 Exercisable as of December 31, 2024(b) 5,955,254 $ 13.30 1,586,252 $ 13.30 — 3.0 ___ (1) The intrinsic value represents the amount by which the fair value of the Company’s Common Stock exceeds the option exercise price as of December 31, 2024.
(2) During the second quarter of 2024, certain of the Top-up Options in the performance tranche were modified in a manner that resulted in such awards immediately vesting and becoming exercisable.
Options Options granted under the 2018 Plan are primarily service-based awards that vest over a three-year period from the date of grant, have an exercise price equal to the closing price per share of the Company’s Common Stock on the date of grant, as adjusted in accordance with 2018 Plan provisions, and have a contractual term of ten years from the date of grant. No options were granted under the 2018 Plan during 2023 or 2022.
The Company used a binomial lattice model to determine the grant date fair value for options granted during 2024 and included the following assumptions: Expected exercise term (years) 7 Expected volatility(1) 49.9% Expected dividend yield(2) 3.4% Risk-free interest rate(3) 4.0% ___ (1) Estimated using historical and implied stock price volatility of the Company.
(2) Calculated by taking the annual dividend run-rate and dividing by the stock price at date of grant.
(3) Based on the U.S. Treasury yield curve.
The weighted-average grant date fair value for the options granted during 2024 was $2.56.
The following table summarizes activity related to 2018 Plan options during 2024: Number of Options Weighted-Average Exercise Price Aggregate Intrinsic Value(1) (in thousands) Weighted-Average Remaining Contractual Term (Years) Outstanding as of December 31, 2023 16,828,305 $ 6.70 Granted 6,831,794 6.51 Exercised (1,907,455) 5.37 Forfeited (502,934) 6.49 Outstanding as of December 31, 2024 21,249,710 $ 6.77 $ 18,048 5.9 Exercisable as of December 31, 2024 14,568,776 $ 7.83 $ 14,322 4.1 ___ (1) The intrinsic value represents the amount by which the fair value of the Company’s Common Stock exceeds the option exercise price as of December 31, 2024.
Share-based compensation expense associated with options granted under the 2018 Plan was not material during the periods presented. In addition, the cash flow and the intrinsic value of options exercised were not material during the periods presented.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-45 As of December 31, 2024, unrecognized compensation cost related to options was not material.
Restricted Stock Units RSUs granted under the 2018 Plan are primarily service-based awards with a three-year graded vesting period from the date of grant. The fair value is equal to the closing price per share of the Company’s Common Stock on the date of grant.
The following table summarizes activity related to the 2018 Plan RSUs (including DEUs) during 2024: Number of RSUs Weighted-Average Grant Date Fair Value Unvested as of December 31, 2023 9,237,384 $ 7.66 Granted 5,155,150 6.65 Vested (4,792,066) 7.72 Forfeited (1,336,453) 6.96 Unvested as of December 31, 2024 8,264,015 $ 7.14 During 2024, 2023, and 2022, total share-based compensation expense associated with RSUs granted under the 2018 Plan was $25 million, $47 million, and $55 million, respectively, the majority of which relates to the Company’s continuing operations. The weighted-average grant date fair value of RSU’s granted during 2023 and 2022 was $7.56 and $7.85, respectively.
During 2024, 2023, and 2022, the fair value of RSUs (including DEUs) that vested and converted to shares of Common Stock on their respective vesting dates was approximately $34 million, $56 million, and $59 million, respectively.
As of December 31, 2024, unrecognized compensation cost related to RSUs granted under the 2018 Plan was $27 million, which will be recognized over a period of approximately 1.9 years.
Other During the second quarter of 2024, the Company modified certain share-based compensation awards and recorded additional share-based compensation expense of $13 million in 2024 associated with these modifications.
12. NET INCOME (LOSS) PER SHARE The Company applies the two-class method for computing and presenting net income (loss) per share for each class of common stock, which allocates current period net income (loss) to each class of common stock and participating securities based on dividends declared and participation rights in the remaining undistributed earnings or losses.
Basic net income (loss) per share is computed by dividing the net income (loss) allocated to each class of common stock by the related weighted-average number of shares outstanding during the period. Diluted net income (loss) per share gives effect to all securities representing potential common shares that were dilutive and outstanding during the period for each class of common stock and excludes potentially dilutive securities whose effect would have been anti-dilutive.
Common Stock Potential shares of Common Stock include (i) incremental shares related to the vesting or exercise of share-based compensation awards, warrants, and other options to purchase additional shares of the Company’s Common Stock calculated using the treasury stock method and (ii) incremental shares of Common Stock issuable upon the conversion of Class B Common Stock.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-46 Years Ended December 31, (in thousands, except per share amounts) 2024 2023 2022 Allocation of income (loss) from continuing operations - basic $ 581,980 $ 423,503 $ 293,331 Dilutive effect(1) 13,671 8,885 8,428 Allocation of income (loss) from continuing operations - diluted $ 595,651 $ 432,388 $ 301,759 Allocation of income (loss) from discontinued operations, net of tax - basic $ (111,197) $ 11,885 $ (168,676) Dilutive effect(1) — — — Allocation of income (loss) from discontinued operations, net of tax - diluted $ (111,197) $ 11,885 $ (168,676) Weighted-average shares outstanding - basic 846,521 856,843 848,465 Dilutive effect(1) 62,179 62,306 66,604 Weighted-average shares outstanding - diluted 908,700 919,149 915,069 Income (loss) from continuing operations per share - basic $ 0.69 $ 0.49 $ 0.35 Income (loss) from continuing operations per share - diluted $ 0.66 $ 0.47 $ 0.33 Income (loss) from discontinued operations, net of tax, per share - basic $ (0.13) $ 0.01 $ (0.20) Income (loss) from discontinued operations, net of tax, per share - diluted $ (0.12) $ 0.01 $ (0.18) __ (1) Includes conversion of Class B Common Stock. 20 million, 17 million, and 15 million potential shares of Common Stock were excluded during 2024, 2023, and 2022, respectively, from the diluted earnings per share calculations because their effects would have been anti-dilutive.
The basic and diluted earnings per share computations for Common Stock exclude approximately 7 million, 9 million, and 9 million unvested shares during 2024, 2023, and 2022, respectively, as their vesting is contingent upon achievement of certain performance requirements which had not been met during the respective periods.
Class B Common Stock Years Ended December 31, (in thousands, except per share amounts) 2024 2023 2022 Allocation of income (loss) from continuing operations - basic $ 37,410 $ 26,867 $ 18,834 Dilutive effect(1) (1,627) (1,221) (764) Allocation of income (loss) from continuing operations - diluted $ 35,783 $ 25,646 $ 18,070 Allocation of income (loss) from discontinued operations, net of tax - basic $ (7,140) $ 754 $ (10,826) Dilutive effect(1) — — — Allocation of income (loss) from discontinued operations, net of tax - diluted $ (7,140) $ 754 $ (10,826) Weighted-average shares outstanding - basic 54,745 54,745 54,745 Dilutive effect(1) — — — Weighted-average shares outstanding - diluted 54,745 54,745 54,745 Income (loss) from continuing operations per share - basic $ 0.69 $ 0.49 $ 0.35 Income (loss) from continuing operations per share - diluted $ 0.66 $ 0.47 $ 0.33 Income (loss) from discontinued operations, net of tax, per share - basic $ (0.13) $ 0.01 $ (0.20) Income (loss) from discontinued operations, net of tax, per share - diluted $ (0.12) $ 0.01 $ (0.18) __ (1) There were no potential shares of Class B Common Stock during the periods presented.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-47 13. COMMITMENTS AND CONTINGENCIES Contractual Obligations The Company’s contractual obligations for goods or services entered into in the ordinary course of business, including agreements that are enforceable and legally binding and have a remaining term in excess of one year, primarily consist of information technology services and equipment, including investments in the Company’s information technology infrastructure and telecommunication services, and direct materials.
The following table provides the Company’s contractual obligations (excluding the Google agreements discussed separately below) as of December 31, 2024 (in thousands): 2025 2026 2027 2028 2029 Thereafter Total $ 367,036 $ 111,562 $ 39,210 $ 31,515 $ 11,593 $ — $ 560,916 During the fourth quarter of 2023, the Company entered into an agreement with one of its vendors to purchase at least $190 million of security system equipment and components through March 2025. During 2024, the Company increased its commitment by approximately $180 million and extended the commitment period through December 2025. This commitment is also satisfied through purchases made by the Company’s dealer network. Included in the table above is approximately $172 million remaining under this commitment as of December 31, 2024.
Google Commercial Agreement In July 2020, the Company and Google entered into a Master Supply, Distribution, and Marketing Agreement (as amended, the “Google Commercial Agreement”), pursuant to which Google has agreed to supply the Company with certain Google devices, as well as certain Google video and analytics services (“Google Devices and Services”), for sale to the Company’s customers. The Google Commercial Agreement also specifies that each party shall contribute $150 million towards joint marketing, customer acquisition, training of the Company’s employees, and product technology updates related to the Google Devices and Services. In August 2022, the Company and Google executed an amendment to the Google Commercial Agreement, pursuant to which Google has agreed to commit an additional $150 million to fund growth, data and insights, product innovation and technology advancements, customer acquisition, and marketing, as mutually agreed by the Company and Google, (together with the initial amounts, the “Google Success Funds”).
During 2024 and 2023, $30 million and $40 million, respectively, of the Google Success Funds were reimbursed to the Company primarily for certain joint marketing and customer acquisition expenses incurred by the Company, substantially all of which was recorded as a reduction of advertising costs.
Google Cloud Agreement Addendum In December 2023, the Company and Google entered into an addendum to the Company’s existing agreement with Google for using Google cloud services (the “Google Cloud Agreement Addendum”), pursuant to which Google has agreed to provide certain credits, discounts, and other incentives for use of the Google Cloud Platform to the Company, and the Company has committed to purchasing $200 million of Google Cloud Platform services over seven years (through December 2030), with an aggregate of $35 million in the first two years, an aggregate of $65 million in the next two years after that, and an aggregate of $100 million in the last three years of the commitment. The Company may elect to cancel the commitment in return for a cancellation fee of 30% of the total remaining commitment amount and loss of any discounts, remaining credits, or other incentives provided under the Google Cloud Agreement Addendum.
As of December 31, 2024, the Company is on track to meet these commitments.
Guarantees In the normal course of business, the Company is liable for contract completion and product performance. The Company’s guarantees primarily relate to standby letters of credit related to its insurance programs and totaled $74 million and $78 million as of December 31, 2024 and 2023, respectively. The Company does not believe such obligations will materially affect its financial position, results of operations, or cash flows.
Subsequent event - In February 2025, $21 million of the Company’s guarantees were relinquished.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-48 Legal Proceedings The Company is subject to various claims and lawsuits in the ordinary course of business, which include among other things commercial general liability claims, automobile liability claims, contractual disputes, worker’s compensation claims, labor law and employment claims, claims related to alleged alarm system failures, claims that the Company infringed on the intellectual property of others, and consumer and employment class actions. The Company is also subject to regulatory and governmental examinations, information requests and subpoenas, inquiries, investigations, and threatened legal actions and proceedings. In connection with such formal and informal inquiries, the Company receives numerous requests, subpoenas, and orders for documents, testimony, and information in connection with various aspects of its activities.
The Company records accruals for losses that are probable and reasonably estimable. These accruals are based on a variety of factors such as judgment, probability of loss, opinions of internal and external legal counsel, and actuarially determined estimates of claims incurred but not yet reported based upon historical claims experience. Legal costs in connection with claims and lawsuits in the ordinary course of business are expensed as incurred. Additionally, the Company records insurance recovery receivables from third-party insurers when recovery has been determined to be probable. The Company has not accrued for any contingent liabilities for which the likelihood of loss cannot be determined, is less than probable, or for which the range of potential loss cannot be reasonably estimated.
As of December 31, 2024 and 2023, the Company’s accrual for ongoing claims and lawsuits within the scope of an insurance program totaled $94 million and $110 million, respectively, including amounts related to discontinued operations. The Company’s accrual related to ongoing claims and lawsuits not within the scope of an insurance program is not material.
14. LEASES Company as Lessee As part of normal operations, the Company leases real estate, vehicles, and equipment from various counterparties with lease terms and maturities through 2034, primarily through its main operating entity, ADT LLC. For these transactions, the Company applies the practical expedient to not separate the lease and non-lease components and accounts for the combined component as a lease. Additionally, the Company’s right-of-use assets and lease liabilities include leases with initial lease terms of 12 months or less.
The Company’s right-of-use assets and lease liabilities primarily represent lease payments that are fixed at the commencement of a lease and variable lease payments that are dependent on an index or rate. Lease payments are recognized as lease cost on a straight-line basis over the lease term, which is determined as the non-cancelable period, including periods in which termination options are reasonably certain of not being exercised and periods in which renewal options are reasonably certain of being exercised. The discount rate is determined using the Company’s incremental borrowing rate coinciding with the lease term at the commencement of a lease. The incremental borrowing rate is estimated based on publicly available data for the Company’s debt instruments and other instruments with similar characteristics.
Lease payments that are neither fixed nor dependent on an index or rate and vary because of changes in usage or other factors are included in variable lease costs. Variable lease costs are recorded in the period in which the obligation is incurred and primarily relate to fuel, repair, and maintenance payments as they vary based on the usage of leased vehicles and buildings.
The Company’s leases do not contain material residual value guarantees or restrictive covenants. The Company’s subleases are not material.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-49 Right-of-Use Assets and Lease Liabilities (in thousands) December 31, Presentation and Classification: 2024 2023 Operating Current Prepaid expenses and other current assets $ 80 $ 68 Operating Non-current Other assets 80,768 85,649 Finance Non-current Property and equipment, net(1) 61,827 65,368 Total right-of-use assets $ 142,675 $ 151,085 Operating Current Accrued expenses and other current liabilities $ 18,811 $ 13,035 Finance Current Current maturities of long-term debt 25,593 25,741 Operating Non-current Other liabilities 77,884 80,189 Finance Non-current Long-term debt 43,849 43,727 Total lease liabilities $ 166,137 $ 162,692 __ (1) Finance right-of-use assets are recorded net of accumulated depreciation of approximately $66 million and $50 million as of December 31, 2024 and 2023, respectively.
Lease Cost Years Ended December 31, (in thousands) 2024 2023 2022 Operating lease cost $ 27,615 $ 31,756 $ 33,574 Finance lease cost: Amortization of right-of-use assets 21,661 14,432 10,818 Interest on lease liabilities 4,528 2,466 1,689 Variable lease costs 35,017 36,273 42,899 Total lease cost $ 88,821 $ 84,927 $ 88,980 Cash Flow and Supplemental Information(1) Years Ended December 31, (in thousands) 2024 2023 2022 Cash paid for amounts included in the measurement of lease liabilities Operating leases: Operating cash flows $ 23,574 $ 42,883 $ 47,708 Finance Leases: Operating cash flows $ 4,686 $ 4,940 $ 3,680 Financing cash flows $ 29,023 $ 43,733 $ 44,978 Right-of-use assets obtained in exchange for lease obligations Operating leases $ 20,127 $ 41,338 $ 49,193 Finance leases $ 35,451 $ 79,273 $ 48,439 ___ (1) Includes both continuing and discontinued operations consistent with the presentation on the Consolidated Statements of Cash Flows.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-50 Lease Term and Discount Rate December 31, 2024 2023 Weighted-average remaining lease term (years): Operating leases 5.4 5.9 Finance leases 2.7 3.2 Weighted-average discount rate: Operating leases 6.2 % 6.1 % Finance leases 6.1 % 5.9 % Maturity of Lease Liabilities December 31, 2024 (in thousands) Operating Leases Finance Leases 2025 $ 20,443 $ 33,369 2026 24,658 22,529 2027 20,975 16,195 2028 16,255 2,809 2029 11,276 1 Thereafter 23,422 — Total lease payments (including interest) $ 117,029 $ 74,903 Less interest 20,334 5,461 Total $ 96,695 $ 69,442 Company as Lessor The Company is a lessor in certain Company-owned transactions as the Company has identified a lease component associated with the right-of-use of the security system and a non-lease component associated with the monitoring and related services.
For transactions in which (i) the timing and pattern of transfer is the same for the lease and non-lease components, and (ii) the lease component would be classified as an operating lease if accounted for separately, the Company applies the practical expedient to aggregate the lease and non-lease components and accounts for the combined transaction based upon its predominant characteristic, which is the non-lease component. The Company accounts for the combined component as a single performance obligation under the applicable revenue guidance, and recognizes the underlying assets within subscriber system assets, net, in the Consolidated Balance Sheets.
15. RETIREMENT PLANS Defined Contribution Plans The Company maintains qualified defined contribution plans, which include 401(k) matching programs. Expense for the defined contribution plans is computed as a percentage of participants’ compensation and was $29 million, $30 million, and $31 million during 2024, 2023, and 2022, respectively.
Multi-employer Plans The Company participates in certain multi-employer union pension plans, which provide benefits for a group of the Company’s unionized employees. These multi-employer plans, including the Company’s required contributions and any underfunded liabilities under such plans, are not material to the Company’s consolidated financial statements.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-51 Defined Benefit Plans Prior to termination during 2024, the Company provided a defined benefit pension plan and certain other postretirement benefits to certain employees, which were frozen and were not material to the Company’s consolidated financial statements. During 2023, the Company took actions to terminate the defined benefit pension plan. The termination was effective December 31, 2023, and the wind down of the plan and distribution of assets occurred in the fourth quarter of 2024 through a combination of lump sum cash distributions to certain plan participants and the purchase of non-participating annuity contracts for the remainder of the plan participants, all of which resulted in the full settlement of the Company’s defined benefit pension plan obligations as of December 31, 2024. Cash contributions of approximately $8 million were made to fully fund the plan and effectuate the settlement. Charges primarily resulting from the write-off of amounts previously recorded in AOCI as well as other expenses associated with the settlement of the Company’s defined benefit pension plan were not material.
As of December 31 2023, the fair value of pension plan assets was $51 million and the fair value of projected benefit obligations was $59 million. As a result, the plan was underfunded by approximately $8 million as of December 31, 2023 and was recorded as a net liability.
Net periodic benefit cost associated with these plans was not material during the periods presented.
Deferred Compensation Plan The Company maintains a non-qualified supplemental savings and retirement plan, which permits eligible employees to defer a portion of their compensation. Deferred compensation liabilities are reflected in other liabilities and were $34 million and $31 million as of December 31, 2024 and 2023, respectively. Deferred compensation expense was not material during the periods presented.
16. RELATED PARTY TRANSACTIONS The Company’s related party transactions primarily relate to products and services received from, or monitoring and related services provided to, other entities affiliated with Apollo, as well as, from time to time, certain transactions between the Company and Apollo and State Farm.
Other than as described below, there were no significant related party transactions during the periods presented.
Apollo Offerings and Share Repurchases On March 6, 2024, the Company and certain entities managed by affiliates of Apollo Global Management, Inc. (the “Selling Stockholders”) entered into an underwriting agreement (the “March 2024 Underwriting Agreement”) with Morgan Stanley & Co. LLC and Barclays Capital Inc., as representatives of the underwriters named therein, including Apollo Global Securities, LLC, an affiliate of Apollo (collectively, the “March 2024 Underwriters”), in connection with the offer and sale by the Selling Stockholders (the “March 2024 Offering”) of 65 million shares of the Company’s Common Stock, and, at the option of the March 2024 Underwriters, up to an additional 9.75 million shares of Common Stock (the “March 2024 Underwriters’ Option”).
• As part of the March 2024 Offering, the Company purchased 15 million shares of Common Stock under its 2024 Share Repurchase Plan from the March 2024 Underwriters (the “March 2024 Share Repurchase”). The Company paid approximately $93 million (or approximately $6.22 per share) for the March 2024 Share Repurchase, which was the same per share price paid by the March 2024 Underwriters to the Selling Stockholders.
• The March 2024 Offering and the March 2024 Share Repurchase closed on March 11, 2024. On March 15, 2024, the March 2024 Underwriters exercised the March 2024 Underwriters’ Option in full, which subsequently closed on March 19, 2024. The Company did not pay any underwriting fees in connection with the March 2024 Share Repurchase, including on behalf of the Selling Stockholders or otherwise.
On October 28, 2024, the Company and the Selling Stockholders entered into an underwriting agreement (the “October 2024 Underwriting Agreement”) with Barclays Capital Inc., Citigroup Global Markets Inc., and BTIG, LLC, as representatives of the underwriters named therein, (collectively, the “October 2024 Underwriters”), in connection with the offer and sale by the Selling Stockholders (the “October 2024 Offering”) of 56 million shares of the Company’s Common Stock, and, at the option of the October Underwriters, up to an additional 8.4 million shares of Common Stock (the “October 2024 Underwriters’ Option”). ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-52 • As part of the October 2024 Offering, the Company purchased 16 million shares of Common Stock under its 2024 Share Repurchase Plan from the October 2024 Underwriters (the “October 2024 Share Repurchase”). The Company paid approximately $115 million (or approximately $7.20 per share) for the October 2024 Share Repurchase, which was the same per share price paid by the October 2024 Underwriters to the Selling Stockholders.
• The October 2024 Offering and the October 2024 Share Repurchase closed on October 30, 2024. On November 6, 2024, the October 2024 Underwriters exercised the October 2024 Underwriters’ Option in full, which subsequently closed on November 8, 2024. The Company did not pay any underwriting fees in connection with the October 2024 Share Repurchase, including on behalf of the Selling Stockholders or otherwise.
All the shares in the Offerings were sold by the Selling Stockholders. The Company did not receive any of the proceeds from the sale of shares by the Selling Stockholders in the Offerings. The March 2024 and October 2024 repurchases (collectively, “Repurchases”) are reflected as a reduction to additional paid-in-capital and as a financing cash outflow.
Other During 2023, the Company incurred fees to Apollo of $1 million related to Apollo’s performance of placement agent services related to the initial funding of the Term Loan A Facility.
State Farm As discussed in Note 10 “Equity,” State Farm is a related party as it owns more than 10% of the Company’s issued and outstanding Common Stock.
As of December 31, 2024 and 2023, the balance in the portion of the Opportunity Fund held by the Company was $85 million and $94 million, respectively.
State Farm customers can receive ADT home security products and professional monitoring at a reduced cost. In connection with this arrangement, the Company receives a subsidy from State Farm through the Opportunity Fund.
During 2024 and 2023, payments from the Opportunity Fund were $14 million and $11 million, respectively, related to marketing, the subsidy discussed above, and taxes. Interest earned on the Opportunity fund was not material.
Canopy Prior to the Canopy Termination during 2023, Canopy was considered a related party under GAAP as the Company accounted for its investment under the equity method of accounting. Except for as described in Note 5 “Equity Method Investments,” there were no other significant transactions with Canopy during the periods presented.
Sunlight Financial LLC In connection with the Company’s former Solar Business, the Company used Sunlight Financial LLC (“Sunlight”), an entity affiliated with Apollo, to access certain loan products. As of December 2023, Sunlight was no longer affiliated with Apollo, and as a result, is no longer a related party.
During 2023 and 2022, total loans funded by Sunlight were approximately $78 million and $436 million, respectively, and the Company incurred $13 million and $54 million, respectively, of financing fees.
Rackspace During October 2020, the Company entered into a master services agreement with Rackspace US, Inc. (“Rackspace”), an entity affiliated with Apollo, for the provision of cloud storage, equipment, and services to facilitate the implementation of the Company’s cloud migration strategy for certain applications.
The master services agreement included a minimum purchase commitment of $50 million over a seven-year term, which the Company satisfied during 2023 through spend with Rackspace as well as other parties.
The Company incurred fees to Rackspace of $4 million, $15 million, and $14 million during 2024, 2023, and 2022, respectively.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-53 Other Transactions During 2024 and 2023, the Company incurred fees for telephone and technology services of approximately $4 million and $6 million, respectively, with an entity affiliated with Apollo that became a related party during 2023.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-54 17. CONDENSED FINANCIAL INFORMATION OF REGISTRANT ADT INC. (PARENT COMPANY ONLY) CONDENSED BALANCE SHEETS (in thousands) December 31, 2024 2023 Assets Current assets: Cash and cash equivalents $ 1,861 $ 1,118 Total current assets 1,861 1,118 Investment in subsidiaries and other assets 4,861,838 4,463,543 Total assets $ 4,863,699 $ 4,464,661 Liabilities and stockholders' equity Current liabilities: Dividends payable and other current liabilities $ 155,472 $ 33,047 Total current liabilities 155,472 33,047 Debt due to subsidiaries 554,954 545,557 Other liabilities 352,472 97,411 Total liabilities 1,062,898 676,015 Total stockholders' equity 3,800,801 3,788,646 Total liabilities and stockholders' equity $ 4,863,699 $ 4,464,661 The accompanying notes are an integral part of these condensed financial statements ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-55 ADT INC.
(PARENT COMPANY ONLY) CONDENSED STATEMENTS OF OPERATIONS AND COMPREHENSIVE INCOME (LOSS) (in thousands, except per share data) Years Ended December 31, 2024 2023 2022 Selling, general, and administrative expenses $ 988 $ 520 $ 2,583 Merger, restructuring, integration, and other (5) (1,993) (6,011) Operating income (loss) (983) 1,473 3,428 Interest expense, net (9,135) (8,984) (8,086) Other income (expense) — — (63,394) Equity in net income (loss) of subsidiaries 511,171 470,520 200,715 Net income (loss) 501,053 463,009 132,663 Other comprehensive income (loss), net of tax 9,126 31,038 21,773 Comprehensive income (loss) $ 510,179 $ 494,047 $ 154,436 Net income (loss) per share - basic: Common stock $ 0.56 $ 0.51 $ 0.15 Class B common stock $ 0.56 $ 0.51 $ 0.15 Weighted-average shares outstanding - basic: Common stock 846,521 856,843 848,465 Class B common stock 54,745 54,745 54,745 Net income (loss) per share - diluted: Common stock $ 0.52 $ 0.48 $ 0.14 Class B common stock $ 0.52 $ 0.48 $ 0.14 Weighted-average shares outstanding - diluted: Common stock 908,700 919,149 915,069 Class B common stock 54,745 54,745 54,745 The accompanying notes are an integral part of these condensed financial statements ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-56 ADT INC. (PARENT COMPANY ONLY) CONDENSED STATEMENTS OF CASH FLOWS (in thousands) Years Ended December 31, 2024 2023 2022 Cash flows from operating activities: Net income (loss) $ 501,053 $ 463,009 $ 132,663 Adjustments to reconcile net income (loss) to net cash provided by (used in) operating activities: Equity in net (income) loss of subsidiaries (511,171) (470,520) (200,715) Change in fair value of other financial instruments — — 63,396 Other, net 22,815 28,757 49,470 Net cash provided by (used in) operating activities 12,697 21,246 44,814 Cash flows from investing activities: Contributions to subsidiaries — — — Distributions from subsidiaries 170,620 108,783 118,200 Net cash provided by (used in) investing activities 170,620 108,783 118,200 Cash flows from financing activities: Proceeds from issuance of common stock, net of expenses — — 1,180,000 Dividends on common stock (182,266) (128,587) (127,125) Repurchases of common stock — — (1,200,000) Other financing, net (308) (14,963) (3,197) Net cash provided by (used in) financing activities (182,574) (143,550) (150,322) Cash and cash equivalents and restricted cash and restricted cash equivalents: Net increase (decrease) during the period 743 (13,521) 12,692 Beginning balance 1,118 14,639 1,947 Ending balance $ 1,861 $ 1,118 $ 14,639 Supplementary cash flow information: Issuance of shares for acquisition of business $ — $ — $ 55,485 The accompanying notes are an integral part of these condensed financial statements ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-57 Notes to Condensed Financial Statements (Parent Company Only) 1. Basis of Presentation The condensed financial statements of ADT Inc. have been prepared in accordance with Rule 12-04, Schedule I of Regulation S-X, as the restricted net assets of the subsidiaries of ADT Inc. (as defined in Rule 4-08(e)(3) of Regulation S-X) exceed 25% of the consolidated net assets of the Company. The ability of ADT Inc.’s operating subsidiaries to pay dividends may be restricted due to the terms of the subsidiaries’ First Lien Credit Agreement and the indentures governing other borrowings. The condensed financial statements of ADT Inc. have been prepared using the same accounting principles and policies described in the other notes to the consolidated financial statements with the only exception being that the parent company accounts for its subsidiaries using the equity method of accounting. These condensed financial statements should be read in conjunction with the Company’s consolidated financial statements and related notes thereto.
2. Transactions with Subsidiaries The majority of ADT Inc.’s transactions with its subsidiaries are related to (i) the receipt of distributions from subsidiaries in order to fund equity transactions, such as the payment of dividends and the repurchase of Common Stock; (ii) the contribution to subsidiaries of proceeds received from equity transactions; or (iii) the integration of business acquisitions into the Company’s organizational structure.
During 2024 and 2023, ADT Inc. made non-cash contributions to subsidiaries of approximately $49 million and $51 million, respectively, primarily related to the transfer of net assets of certain subsidiaries for share-based compensation (including amounts related to discontinued operations).
As of December 31, 2024, current liabilities includes $104 million related to the December 2024 agreement with a non-affiliate to repurchase shares, in which the transaction closed in January 2025, and other liabilities includes intercompany liabilities with subsidiaries of $240 million related to share repurchases under the 2024 Share Repurchase Plan.
Subsequent event - In January 2025, ADT Inc. received distributions from subsidiaries of $47 million to pay dividends that the Company declared in October 2024 to common shareholders.
18. SELECTED QUARTERLY FINANCIAL DATA (UNAUDITED) The Company is disclosing the following summarized quarterly financial information for 2024 and 2023 due to the Solar and Commercial Businesses being reported as discontinued operations as a result of the ADT Solar Exit and Commercial Divestiture, respectively. Refer to Note 4 “Divestitures” for additional information.
ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-58 (in thousands, except per share data) March 31, 2024 June 30, 2024 September 30, 2024 December 31, 2024 Revenue $ 1,189,672 $ 1,204,559 $ 1,243,836 $ 1,260,379 Operating income (loss) $ 292,149 $ 284,175 $ 326,350 $ 305,390 Income (loss) from continuing operations $ 163,891 $ 126,185 $ 132,020 $ 197,294 Income (loss) from discontinued operations, net of tax $ (72,340) $ (33,791) $ (4,869) $ (7,337) Net income (loss) $ 91,551 $ 92,394 $ 127,151 $ 189,957 Common Stock Basic: Weighted average shares outstanding 855,893 848,273 850,462 831,575 Income (loss) from continuing operations per share $ 0.18 $ 0.14 $ 0.15 $ 0.22 Income (loss) from discontinued operations, net of tax, per share $ (0.08) $ (0.04) $ (0.01) $ (0.01) Net income (loss) $ 0.10 $ 0.10 $ 0.14 $ 0.21 Diluted: Weighted average shares outstanding 918,394 909,128 912,861 895,035 Income (loss) from continuing operations per share $ 0.17 $ 0.13 $ 0.14 $ 0.21 Income (loss) from discontinued operations, net of tax, per share $ (0.07) $ (0.03) $ (0.01) $ (0.01) Net income (loss) $ 0.09 $ 0.10 $ 0.13 $ 0.20 Class B Common Stock Basic: Weighted average shares outstanding 54,745 54,745 54,745 54,745 Income (loss) from continuing operations per share $ 0.18 $ 0.14 $ 0.15 $ 0.22 Income (loss) from discontinued operations, net of tax, per share $ (0.08) $ (0.04) $ (0.01) $ (0.01) Net income (loss) $ 0.10 $ 0.10 $ 0.14 $ 0.21 Diluted: Weighted average shares outstanding 54,745 54,745 54,745 54,745 Income (loss) from continuing operations per share $ 0.17 $ 0.13 $ 0.14 $ 0.21 Income (loss) from discontinued operations, net of tax, per share $ (0.07) $ (0.03) $ (0.01) $ (0.01) Net income (loss) $ 0.09 $ 0.10 $ 0.13 $ 0.20 ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-59 (in thousands, except per share data) March 31, 2023 June 30, 2023 September 30, 2023 December 31, 2023 Revenue $ 1,132,476 $ 1,168,077 $ 1,179,873 $ 1,172,398 Operating income (loss) $ 232,960 $ 332,098 $ 306,563 $ 307,340 Income (loss) from continuing operations $ 39,563 $ 180,438 $ 123,259 $ 107,110 Income (loss) from discontinued operations, net of tax $ (158,400) $ (88,227) $ (209,496) $ 468,762 Net income (loss) $ (118,837) $ 92,211 $ (86,237) $ 575,872 Common Stock Basic: Weighted average shares outstanding 854,299 857,581 857,423 858,094 Income (loss) from continuing operations per share $ 0.04 $ 0.20 $ 0.13 $ 0.12 Income (loss) from discontinued operations, net of tax, per share $ (0.17) $ (0.10) $ (0.23) $ 0.51 Net income (loss) $ (0.13) $ 0.10 $ (0.09) $ 0.63 Diluted: Weighted average shares outstanding 921,606 916,859 917,774 919,397 Income (loss) from continuing operations per share $ 0.04 $ 0.19 $ 0.13 $ 0.11 Income (loss) from discontinued operations, net of tax, per share $ (0.16) $ (0.09) $ (0.21) $ 0.48 Net income (loss) $ (0.12) $ 0.09 $ (0.09) $ 0.59 Class B Common Stock Basic: Weighted average shares outstanding 54,745 54,745 54,745 54,745 Income (loss) from continuing operations per share $ 0.04 $ 0.20 $ 0.13 $ 0.12 Income (loss) from discontinued operations, net of tax, per share $ (0.17) $ (0.10) $ (0.23) $ 0.51 Net income (loss) $ (0.13) $ 0.10 $ (0.09) $ 0.63 Diluted: Weighted average shares outstanding 54,745 54,745 54,745 54,745 Income (loss) from continuing operations per share $ 0.04 $ 0.19 $ 0.13 $ 0.11 Income (loss) from discontinued operations, net of tax, per share $ (0.16) $ (0.09) $ (0.21) $ 0.48 Net income (loss) $ (0.12) $ 0.09 $ (0.09) $ 0.59 ADT INC. AND SUBSIDIARIES NOTES TO CONSOLIDATED FINANCIAL STATEMENTS F-60 |
5723 | https://brainly.in/question/51962558 | Let S = {1,2,3,4} . The total number of unordered pairs of disjoint subsets of S is equal - Brainly.in
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akashsingh8473
21.05.2022
Math
Secondary School
answered
Let S = {1,2,3,4} . The total number of unordered pairs of disjoint subsets of S is equal to a.25
b.34
c.42
d.41
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Question 79 page 824 Dr. S. K. Goyal - Algebra JEE 11. Sets, Relations and Functions answer
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5724 | https://janaf.nist.gov/tables/O-index.html | NIST-JANAF Thermochemical Tables
NIST-JANAF Thermochemical Tables
Showing search results for molecules containing O
CAS Number Formula Name State JANAF Table Links
61279-70-7 AlBO2 Aluminum Borate gviewwebbook
13596-11-7 AlClO Aluminum Chloride Oxide crviewwebbook
13596-11-7 AlClO Aluminum Chloride Oxide gviewwebbook
13596-12-8 AlFO Aluminum Fluoride Oxide gviewwebbook
38344-66-0 AlF2O Aluminum Fluoride Oxide gviewwebbook
36472-76-1 AlF2O-Aluminum Fluoride Oxide, Ion gviewwebbook
91571-48-1 HAlO Aluminum Hydride Oxide gviewwebbook
20768-67-6 HAlO Aluminum Hydroxide gviewwebbook
60172-63-6 HAlO+Aluminum Hydroxide, Ion gviewwebbook
60428-72-0 HAlO-Aluminum Hydroxide, Ion gviewwebbook
24623-77-6 HAlO2 Aluminum Hydroxide Oxide gviewwebbook
12003-67-7 AlLiO2 Lithium Aluminum Oxide crviewwebbook
12003-67-7 AlLiO2 Lithium Aluminum Oxide lviewwebbook
12003-67-7 AlLiO2 Lithium Aluminum Oxide cr,lviewwebbook
1302-42-7 AlNaO2 Sodium Aluminum Oxide crviewwebbook
14457-64-8 AlO Aluminum Oxide gviewwebbook
12588-30-6 AlO+Aluminum Oxide, Ion gviewwebbook
12758-12-2 AlO-Aluminum Oxide, Ion gviewwebbook
11092-32-3 AlO2 Aluminum Oxide gviewwebbook
20653-98-9 AlO2-Aluminum Oxide, Ion gviewwebbook
12004-06-7 Al2BeO4 Beryllium Aluminum Oxide crviewwebbook
12004-06-7 Al2BeO4 Beryllium Aluminum Oxide lviewwebbook
12004-06-7 Al2BeO4 Beryllium Aluminum Oxide cr,lviewwebbook
12068-51-8 Al2MgO4 Magnesium Aluminum Oxide crviewwebbook
12068-51-8 Al2MgO4 Magnesium Aluminum Oxide lviewwebbook
12068-51-8 Al2MgO4 Magnesium Aluminum Oxide cr,lviewwebbook
12004-36-3 Al2O Aluminum Oxide gviewwebbook
12588-51-1 Al2O+Aluminum Oxide, Ion gviewwebbook
12252-63-0 Al2O2 Aluminum Oxide gviewwebbook
60195-07-5 Al2O2+Aluminum Oxide, Ion gviewwebbook
1344-28-1 Al2O3 Aluminum Oxide, Alpha crviewwebbook
1344-28-1 Al2O3 Aluminum Oxide, Delta crviewwebbook
1344-28-1 Al2O3 Aluminum Oxide, Gamma crviewwebbook
1344-28-1 Al2O3 Aluminum Oxide, Kappa crviewwebbook
1344-28-1 Al2O3 Aluminum Oxide lviewwebbook
1344-28-1 Al2O3 Aluminum Oxide cr,lviewwebbook
12183-80-1 Al2O5Si Aluminum Silicate, Andalusite crviewwebbook
1302-76-7 Al2O5Si Aluminum Silicate, Kyanite crviewwebbook
12141-46-7 Al2O5Si Aluminum Silicate, Sillimanite crviewwebbook
12253-74-6 Al6BeO10 Beryllium Aluminum Oxide crviewwebbook
12253-74-6 Al6BeO10 Beryllium Aluminum Oxide lviewwebbook
12253-74-6 Al6BeO10 Beryllium Aluminum Oxide cr,lviewwebbook
1302-93-8 Al6O13Si2 Aluminum Silicate, Mullite crviewwebbook
10682-18-3 BBeO2 Beryllium Borate gview
38490-24-3 BBrO Boron Bromide Oxide gviewwebbook
23361-55-9 BClO Boron Chloride Oxide gviewwebbook
23361-56-0 BFO Boron Fluoride Oxide gviewwebbook
13867-66-8 HBF2O Difluorohydroxyborane gviewwebbook
38150-67-3 BF2O Boron Fluoride Oxide gviewwebbook
20611-59-0 HBO Boron Hydride Oxide gviewwebbook
72704-18-8 HBO+Boron Hydride Oxide, Ion gviewwebbook
12588-96-4 HBO-Boron Hydride Oxide, Ion gviewwebbook
13460-50-9 HBO2 Boric Acid crviewwebbook
13460-50-9 HBO2 Boric Acid gviewwebbook
74930-82-8 H2BO2 Dihydroxyborane gviewwebbook
10043-35-3 H3BO3 Boric Acid crviewwebbook
10043-35-3 H3BO3 Boric Acid gviewwebbook
13709-94-9 BKO2 Potassium Borate crviewwebbook
13709-94-9 BKO2 Potassium Borate lviewwebbook
13709-94-9 BKO2 Potassium Borate cr,lviewwebbook
13709-94-9 BKO2 Potassium Borate gviewwebbook
13453-69-5 BLiO2 Lithium Borate crviewwebbook
13453-69-5 BLiO2 Lithium Borate lviewwebbook
13453-69-5 BLiO2 Lithium Borate cr,lviewwebbook
13453-69-5 BLiO2 Lithium Borate gviewwebbook
7775-19-1 BNaO2 Sodium Borate crviewwebbook
7775-19-1 BNaO2 Sodium Borate lviewwebbook
7775-19-1 BNaO2 Sodium Borate cr,lviewwebbook
7775-19-1 BNaO2 Sodium Borate gviewwebbook
12505-77-0 BO Boron Oxide gviewwebbook
13840-88-5 BO2 Boron Oxide gviewwebbook
14100-65-3 BO2-Boron Oxide, Ion gviewwebbook
14720-43-5 B2BeO4 Beryllium Borate gviewwebbook
13701-63-8 B2Be3O6 Beryllium Borate crviewwebbook
12523-60-3 B2F4O Difluoroborane Oxide gviewwebbook
13675-18-8 H4B2O4 Dihydroxyborane crviewwebbook
13675-18-8 H4B2O4 Dihydroxyborane gviewwebbook
12045-60-2 B2O Boron Oxide gviewwebbook
13766-28-4 B2O2 Boron Oxide gviewwebbook
1303-86-2 B2O3 Boron Oxide crviewwebbook
1303-86-2 B2O3 Boron Oxide lviewwebbook
1303-86-2 B2O3 Boron Oxide cr,lviewwebbook
1303-86-2 B2O3 Boron Oxide gviewwebbook
14720-53-7 B2O4Pb Lead Borate crviewwebbook
13703-91-8 B3Cl3O3 Trichloroboroxin gviewwebbook
57372-62-0 H2B3FO3 Fluoroboroxin gviewwebbook
57328-69-5 HB3F2O3 Difluoroboroxin gviewwebbook
13703-95-2 B3F3O3 Trifluoroboroxin crviewwebbook
13703-95-2 B3F3O3 Trifluoroboroxin gviewwebbook
289-56-5 H3B3O3 Boroxin crviewwebbook
289-56-5 H3B3O3 Boroxin gviewwebbook
13460-51-0 H3B3O6 Boric Acid gviewwebbook
1332-77-0 B4K2O7 Potassium Borate crviewwebbook
1332-77-0 B4K2O7 Potassium Borate lviewwebbook
1332-77-0 B4K2O7 Potassium Borate cr,lviewwebbook
12007-60-2 B4Li2O7 Lithium Borate crviewwebbook
12007-60-2 B4Li2O7 Lithium Borate lviewwebbook
12007-60-2 B4Li2O7 Lithium Borate cr,lviewwebbook
1330-43-4 B4Na2O7 Sodium Borate crviewwebbook
1330-43-4 B4Na2O7 Sodium Borate lviewwebbook
1330-43-4 B4Na2O7 Sodium Borate cr,lviewwebbook
12007-64-6 B4O7Pb Lead Borate crviewwebbook
12007-40-8 B6K2O10 Potassium Borate crviewwebbook
12007-60-2 B6Li2O10 Lithium Borate crviewwebbook
12007-42-0 B6Na2O10 Sodium Borate crviewwebbook
64539-83-9 B6O10Pb Lead Borate crviewwebbook
12008-39-8 B8K2O13 Potassium Borate crviewwebbook
12008-39-8 B8K2O13 Potassium Borate lviewwebbook
12008-39-8 B8K2O13 Potassium Borate cr,lviewwebbook
12008-40-1 B8Li2O13 Lithium Borate crviewwebbook
75024-11-2 B10O17Pb2 Lead Borate crviewwebbook
12009-08-4 HBaO Barium Hydroxide gviewwebbook
68193-67-9 HBaO+Barium Hydroxide, Ion gviewwebbook
17194-00-2 H2BaO2 Barium Hydroxide, Alpha crviewwebbook
17194-00-2 H2BaO2 Barium Hydroxide lviewwebbook
17194-00-2 H2BaO2 Barium Hydroxide cr,lviewwebbook
17194-00-2 H2BaO2 Barium Hydroxide gviewwebbook
1304-28-5 BaO Barium Oxide crviewwebbook
1304-28-5 BaO Barium Oxide lviewwebbook
1304-28-5 BaO Barium Oxide cr,lviewwebbook
1304-28-5 BaO Barium Oxide gviewwebbook
20768-68-7 HBeO Beryllium Hydroxide gviewwebbook
12280-09-0 HBeO+Beryllium Hydroxide, Ion gviewwebbook
13327-32-7 H2BeO2 Beryllium Hydroxide, Alpha crviewwebbook
13327-32-7 H2BeO2 Beryllium Hydroxide, Beta crviewwebbook
13327-32-7 H2BeO2 Beryllium Hydroxide gviewwebbook
1304-56-9 BeO Beryllium Oxide, Alpha crviewwebbook
1304-56-9 BeO Beryllium Oxide, Beta crviewwebbook
1304-56-9 BeO Beryllium Oxide lviewwebbook
1304-56-9 BeO Beryllium Oxide cr,lviewwebbook
1304-56-9 BeO Beryllium Oxide gviewwebbook
13510-49-1 BeO4S Beryllium Sulfate, Alpha crviewwebbook
13510-49-1 BeO4S Beryllium Sulfate, Beta crviewwebbook
13510-49-1 BeO4S Beryllium Sulfate, Gamma crviewwebbook
18304-19-3 BeO4W Beryllium Tungsten Oxide crviewwebbook
65887-15-2 Be2F2O Beryllium Fluoride Oxide gviewwebbook
12009-99-3 Be2O Beryllium Oxide gviewwebbook
70478-90-9 Be2O2 Beryllium Oxide gviewwebbook
15191-85-2 Be2O4Si Beryllium Silicate crviewwebbook
61279-73-0 Be3O3 Beryllium Oxide gviewwebbook
61279-74-1 Be4O4 Beryllium Oxide gviewwebbook
61279-75-2 Be5O5 Beryllium Oxide gviewwebbook
61279-76-3 Be6O6 Beryllium Oxide gviewwebbook
13444-87-6 BrNO Nitrosyl Bromide gviewwebbook
7789-59-5 Br3OP Phosphoryl Bromide gviewwebbook
353-49-1 CClFO Carbonic Chloride Fluoride gviewwebbook
2602-42-8 CClO Carbonyl Chloride gviewwebbook
75-44-5 CCl2O Carbonic Dichloride gviewwebbook
1871-24-5 CFO Carbonyl Fluoride gviewwebbook
353-50-4 CF2O Carbonic Difluoride gviewwebbook
373-91-1 CF4O Trifluoromethyl Hypofluorite gviewwebbook
1493-02-3 CHFO Formyl Fluoride gviewwebbook
75-13-8 CHNO Hydrogen Isocyanate gviewwebbook
2597-44-6 CHO Formyl gviewwebbook
17030-74-9 CHO+Formyl, Ion gviewwebbook
50-00-0 CH2O Formaldehyde gviewwebbook
584-08-7 CK2O3 Potassium Carbonate crviewwebbook
584-08-7 CK2O3 Potassium Carbonate lviewwebbook
584-08-7 CK2O3 Potassium Carbonate cr,lviewwebbook
554-13-2 CLi2O3 Lithium Carbonate crviewwebbook
554-13-2 CLi2O3 Lithium Carbonate lviewwebbook
554-13-2 CLi2O3 Lithium Carbonate cr,lviewwebbook
546-93-0 CMgO3 Magnesium Carbonate crviewwebbook
22400-26-6 CNO NCO Radical gviewwebbook
497-19-8 CNa2O3 Sodium Carbonate crviewwebbook
497-19-8 CNa2O3 Sodium Carbonate lviewwebbook
497-19-8 CNaO3 Sodium Carbonate cr,lviewwebbook
630-08-0 CO Carbon Monoxide gviewwebbook
463-58-1 COS Carbon Oxide Sulfide gviewwebbook
124-38-9 CO2 Carbon Dioxide gviewwebbook
14485-07-5 CO2-Carbon Dioxide, Ion gviewwebbook
75-21-8 C2H4O Oxirane gviewwebbook
87191-90-0 C2O CCO Radical gviewwebbook
504-64-3 C3O2 Carbon Suboxide gviewwebbook
13463-39-3 C4NiO4 Nickel Carbonyl lviewwebbook
13463-39-3 C4NiO4 Nickel Carbonyl gviewwebbook
13463-40-6 C5FeO5 Iron Carbonyl lviewwebbook
13463-40-6 C5FeO5 Iron Carbonyl gviewwebbook
12177-67-2 HCaO Calcium Hydroxide gviewwebbook
36812-31-4 HCaO+Calcium Hydroxide, Ion gviewwebbook
1305-62-0 H2CaO2 Calcium Hydroxide crviewwebbook
1305-62-0 H2CaO2 Calcium Hydroxide gviewwebbook
1305-78-8 CaO Calcium Oxide crviewwebbook
1305-78-8 CaO Calcium Oxide lviewwebbook
1305-78-8 CaO Calcium Oxide cr,lviewwebbook
1305-78-8 CaO Calcium Oxide gviewwebbook
13770-22-4 ClDO Hypochlorous Acid-D gviewwebbook
13637-84-8 ClFO2S Sulfuryl Chloride Fluoride gviewwebbook
7616-94-6 ClFO3 Perchloryl Fluoride gviewwebbook
13769-75-0 ClF2OP Phosphoryl Chloride Fluoride gviewwebbook
7790-92-3 HClO Hypochlorous Acid gviewwebbook
7790-98-9 H4ClNO4 Ammonium Perchlorate crviewwebbook
7778-74-7 ClKO4 Potassium Perchlorate crviewwebbook
13840-33-0 ClLiO Lithium Hypochlorite gviewwebbook
7791-03-9 ClLiO4 Lithium Perchlorate crviewwebbook
7791-03-9 ClLiO4 Lithium Perchlorate lviewwebbook
7791-03-9 ClLiO4 Lithium Perchlorate cr,lviewwebbook
2696-92-6 ClNO Nitrosyl Chloride gviewwebbook
13444-90-1 ClNO2 Nitryl Chloride gviewwebbook
7601-89-0 ClNaO4 Sodium Perchlorate crviewwebbook
14989-30-1 ClO Chlorine Oxide gviewwebbook
15605-36-4 ClOTi Titanium Chloride Oxide gviewwebbook
10049-04-4 ClO2 Chlorine Oxide gviewwebbook
13769-76-1 Cl2FOP Phosphoryl Chloride Fluoride gviewwebbook
13637-68-8 Cl2MoO2 Molybdenum Chloride Oxide gviewwebbook
7791-21-1 Cl2O Chlorine Oxide gviewwebbook
13780-39-7 Cl2OTi Titanium Chloride Oxide gviewwebbook
7791-25-5 Cl2O2S Sulfuryl Chloride gviewwebbook
13520-76-8 Cl2O2W Tungsten Chloride Oxide crviewwebbook
13520-76-8 Cl2O2W Tungsten Chloride Oxide gviewwebbook
10025-87-3 Cl3OP Phosphoryl Chloride gviewwebbook
13520-78-0 Cl4OW Tungsten Chloride Oxide crviewwebbook
13520-78-0 Cl4OW Tungsten Chloride Oxide lviewwebbook
13520-78-0 Cl4OW Tungsten Chloride Oxide cr,lviewwebbook
13520-78-0 Cl4OW Tungsten Chloride Oxide gviewwebbook
1307-96-6 CoO Cobalt Oxide crviewwebbook
10124-43-3 CoO4S Cobalt Sulfate crviewwebbook
1308-06-1 Co3O4 Cobalt Oxide crviewwebbook
12018-00-7 CrO Chromium Oxide gviewwebbook
12018-01-8 CrO2 Chromium Oxide gviewwebbook
1333-82-0 CrO3 Chromium Oxide gviewwebbook
1308-38-9 Cr2O3 Chromium Oxide crviewwebbook
1308-38-9 Cr2O3 Chromium Oxide lviewwebbook
1308-38-9 Cr2O3 Chromium Oxide cr,lviewwebbook
21351-79-1 HCsO Cesium Hydroxide crviewwebbook
21351-79-1 HCsO Cesium Hydroxide lviewwebbook
21351-79-1 HCsO Cesium Hydroxide cr,lviewwebbook
21351-79-1 HCsO Cesium Hydroxide gviewwebbook
54250-97-4 HCsO+Cesium Hydroxide, Ion gviewwebbook
24774-39-8 CsO Cesium Oxide gviewwebbook
12182-83-1 H2Cs2O2 Cesium Hydroxide gviewwebbook
20281-00-9 Cs2O Cesium Oxide gviewwebbook
10294-54-9 Cs2O4S Cesium Sulfate, I crviewwebbook
10294-54-9 Cs2O4S Cesium Sulfate, II crviewwebbook
10294-54-9 Cs2O4S Cesium Sulfate lviewwebbook
10294-54-9 Cs2O4S Cesium Sulfate cr,lviewwebbook
10294-54-9 Cs2O4S Cesium Sulfate gviewwebbook
20427-59-2 H2CuO2 Copper Hydroxide crviewwebbook
1317-38-0 CuO Copper Oxide crviewwebbook
1317-38-0 CuO Copper Oxide gviewwebbook
7758-98-7 CuO4S Copper Sulfate crviewwebbook
1317-39-1 Cu2O Copper Oxide crviewwebbook
1317-39-1 Cu2O Copper Oxide lviewwebbook
1317-39-1 Cu2O Copper Oxide cr,lviewwebbook
12015-77-9 Cu2O5S Copper Oxide Sulfate crviewwebbook
14940-63-7 HDO Water-D gviewwebbook
13587-54-7 DO Hydroxyl-D gviewwebbook
7789-20-0 D2O Water-D gviewwebbook
14034-79-8 HFO Hypofluorous Acid gviewwebbook
7789-21-1 HFO3S Fluorosulfuric Acid gviewwebbook
34240-84-1 FLiO Lithium Hypofluorite gviewwebbook
7789-25-5 FNO Nitrosyl Fluoride gviewwebbook
10022-50-1 FNO2 Nitryl Fluoride gviewwebbook
7789-26-6 FNO3 Fluorine Nitrate gviewwebbook
12061-70-0 FO Oxygen Fluoride gviewwebbook
17497-75-5 FOTi Titanium Fluoride Oxide gviewwebbook
15499-23-7 FO2 Oxygen Fluoride gviewwebbook
7783-41-7 F2O Oxygen Fluoride gviewwebbook
7783-42-8 F2OS Thionyl Fluoride gviewwebbook
14041-22-6 F2OSi Difluorooxosilane gviewwebbook
13537-16-1 F2OTi Titanium Fluoride Oxide gviewwebbook
2699-79-8 F2O2S Sulfuryl Fluoride gviewwebbook
13847-65-9 F3NO Nitrogen Fluoride Oxide gviewwebbook
13478-20-1 F3OP Phosphoryl Fluoride gviewwebbook
14459-59-7 F4MoO Molybdenum Fluoride Oxide gviewwebbook
13520-79-1 F4OW Tungsten Fluoride Oxide crviewwebbook
13520-79-1 F4OW Tungsten Fluoride Oxide lviewwebbook
13520-79-1 F4OW Tungsten Fluoride Oxide cr,lviewwebbook
13520-79-1 F4OW Tungsten Fluoride Oxide gviewwebbook
17125-56-3 Fe0.947O Iron Oxide, Wustite crview
18624-44-7 H2FeO2 Iron Hydroxide crviewwebbook
18624-44-7 H2FeO2 Iron Hydroxide gviewwebbook
1309-33-7 H3FeO3 Iron Hydroxide crviewwebbook
1345-25-1 FeO Iron Oxide crviewwebbook
1345-25-1 FeO Iron Oxide lviewwebbook
1345-25-1 FeO Iron Oxide cr,lviewwebbook
1345-25-1 FeO Iron Oxide gviewwebbook
7720-78-7 FeO4S Iron Sulfate crviewwebbook
1317-60-8 Fe2O3 Iron Oxide, Hematite crviewwebbook
10028-22-5 Fe2O12S3 Iron Sulfate crviewwebbook
1309-38-2 Fe3O4 Iron Oxide, Magnetite crviewwebbook
1310-58-3 HKO Potassium Hydroxide crviewwebbook
1310-58-3 HKO Potassium Hydroxide lviewwebbook
1310-58-3 HKO Potassium Hydroxide cr,lviewwebbook
1310-58-3 HKO Potassium Hydroxide gviewwebbook
54250-98-5 HKO+Potassium Hydroxide, Ion gviewwebbook
1310-65-2 HLiO Lithium Hydroxide crviewwebbook
1310-65-2 HLiO Lithium Hydroxide lviewwebbook
1310-65-2 HLiO Lithium Hydroxide cr,lviewwebbook
1310-65-2 HLiO Lithium Hydroxide gviewwebbook
54250-99-6 HLiO+Lithium Hydroxide, Ion gviewwebbook
12141-11-6 HMgO Magnesium Hydroxide gviewwebbook
60172-61-4 HMgO+Magnesium Hydroxide, Ion gviewwebbook
14332-28-6 HNO Nitrosyl Hydride gviewwebbook
7782-77-6 HNO2 Nitrous Acid, Cis gviewwebbook
7782-77-6 HNO2 Nitrous Acid, Trans gviewwebbook
7697-37-2 HNO3 Nitric Acid gviewwebbook
1310-73-2 HNaO Sodium Hydroxide crviewwebbook
1310-73-2 HNaO Sodium Hydroxide lviewwebbook
1310-73-2 HNaO Sodium Hydroxide cr,lviewwebbook
1310-73-2 HNaO Sodium Hydroxide gviewwebbook
54251-00-2 HNaO+Sodium Hydroxide, Ion gviewwebbook
3352-57-6 HO Hydroxyl gviewwebbook
12259-29-9 HO+Hydroxyl, Ion gviewwebbook
14280-30-9 HO-Hydroxyl, Ion gviewwebbook
12141-14-9 HOSr Strontium Hydroxide gviewwebbook
36812-32-5 HOSr+Strontium Hydroxide, Ion gviewwebbook
3170-83-0 HO2 Hydroperoxo gviewwebbook
12395-66-3 H2K2O2 Potassium Hydroxide gviewwebbook
54251-08-0 H2Li2O2 Lithium Hydroxide gviewwebbook
1309-42-8 H2MgO2 Magnesium Hydroxide crviewwebbook
1309-42-8 H2MgO2 Magnesium Hydroxide gviewwebbook
7782-91-4 H2MoO4 Molybdic Acid gviewwebbook
54251-09-1 H2Na2O2 Sodium Hydroxide gviewwebbook
7732-18-5 H2O Water lviewwebbook
7732-18-5 H2O Water gviewwebbook
7732-18-5 H2O Water, 1 Bar l,gviewwebbook
7732-18-5 H2O Water, 10 Bar l,gviewwebbook
7732-18-5 H2O Water, 100 Bar l,gviewwebbook
7732-18-5 H2O Water, 500 Bar flviewwebbook
7732-18-5 H2O Water, 5000 Bar flviewwebbook
7722-84-1 H2O2 Hydrogen Peroxide gviewwebbook
18480-07-4 H2O2Sr Strontium Hydroxide crviewwebbook
18480-07-4 H2O2Sr Strontium Hydroxide lviewwebbook
18480-07-4 H2O2Sr Strontium Hydroxide cr,lviewwebbook
18480-07-4 H2O2Sr Strontium Hydroxide gviewwebbook
7664-93-9 H2O4S Sulfuric Acid cr,lviewwebbook
7664-93-9 H2O4S Sulfuric Acid gviewwebbook
7783-03-1 H2O4W Tungstic Acid crviewwebbook
7783-03-1 H2O4W Tungstic Acid gviewwebbook
13968-08-6 H3O+Hydronium, Ion gviewwebbook
7664-38-2 H3O4P Phosphoric Acid crviewwebbook
7664-38-2 H3O4P Phosphoric Acid lviewwebbook
7664-38-2 H3O4P Phosphoric Acid cr,lviewwebbook
10193-30-3 H4O5S Sulfuric Acid, Monohydrate cr,lview
13451-10-0 H6O6S Sulfuric Acid, Dihydrate cr,lview
40835-65-2 H8O7S Sulfuric Acid, Trihydrate cr,lview
37006-20-5 H10O8S Sulfuric Acid, Tetrahydrate cr,lview
55019-63-1 H15O10.5S Sulfuric Acid, Hemihexahydrate cr,lview
21908-53-2 HgO Mercury Oxide crviewwebbook
21908-53-2 HgO Mercury Oxide gviewwebbook
58585-94-7 INO Nitrosyl Iodide gviewwebbook
12401-70-6 KO Potassium Oxide gviewwebbook
64538-55-2 KO-Potassium Oxide, Ion gviewwebbook
12030-88-5 KO2 Potassium Superoxide crviewwebbook
12136-45-7 K2O Potassium Oxide crviewwebbook
17014-71-0 K2O2 Potassium Peroxide crviewwebbook
10006-28-7 K2O3Si Potassium Silicate crviewwebbook
10006-28-7 K2O3Si Potassium Silicate lviewwebbook
10006-28-7 K2O3Si Potassium Silicate cr,lviewwebbook
7778-80-5 K2O4S Potassium Sulfate, Alpha crviewwebbook
7778-80-5 K2O4S Potassium Sulfate, Beta crviewwebbook
7778-80-5 K2O4S Potassium Sulfate lviewwebbook
7778-80-5 K2O4S Potassium Sulfate cr,lviewwebbook
7778-80-5 K2O4S Potassium Sulfate gviewwebbook
36529-65-4 LiNO Lithium Oxynitride gviewwebbook
60397-00-4 LiNaO Lithium Sodium Oxide gviewwebbook
12142-77-7 LiO Lithium Oxide gviewwebbook
64538-53-0 LiO-Lithium Oxide, Ion gviewwebbook
12057-24-8 Li2O Lithium Oxide crviewwebbook
12057-24-8 Li2O Lithium Oxide lviewwebbook
12057-24-8 Li2O Lithium Oxide cr,lviewwebbook
12057-24-8 Li2O Lithium Oxide gviewwebbook
12031-80-0 Li2O2 Lithium Peroxide crviewwebbook
12031-80-0 Li2O2 Lithium Oxide gviewwebbook
10102-24-6 Li2O3Si Lithium Silicate crviewwebbook
10102-24-6 Li2O3Si Lithium Silicate lviewwebbook
10102-24-6 Li2O3Si Lithium Silicate cr,lviewwebbook
12031-82-2 Li2O3Ti Lithium Titanium Oxide crviewwebbook
12031-82-2 Li2O3Ti Lithium Titanium Oxide lviewwebbook
12031-82-2 Li2O3Ti Lithium Titanium Oxide cr,lviewwebbook
10377-48-7 Li2O4S Lithium Sulfate, Alpha crviewwebbook
10377-48-7 Li2O4S Lithium Sulfate, Beta crviewwebbook
10377-48-7 Li2O4S Lithium Sulfate lviewwebbook
10377-48-7 Li2O4S Lithium Sulfate cr,lviewwebbook
10377-48-7 Li2O4S Lithium Sulfate gviewwebbook
13568-46-2 Li2O5Si2 Lithium Silicate crviewwebbook
13568-46-2 Li2O5Si2 Lithium Silicate lviewwebbook
13568-46-2 Li2O5Si2 Lithium Silicate cr,lviewwebbook
1309-48-4 MgO Magnesium Oxide crviewwebbook
1309-48-4 MgO Magnesium Oxide lviewwebbook
1309-48-4 MgO Magnesium Oxide cr,lviewwebbook
1309-48-4 MgO Magnesium Oxide gviewwebbook
13776-74-4 MgO3Si Magnesium Silicate crviewwebbook
13776-74-4 MgO3Si Magnesium Silicate lviewwebbook
13776-74-4 MgO3Si Magnesium Silicate cr,lviewwebbook
12032-30-3 MgO3Ti Magnesium Titanium Oxide crviewwebbook
12032-30-3 MgO3Ti Magnesium Titanium Oxide lviewwebbook
12032-30-3 MgO3Ti Magnesium Titanium Oxide cr,lviewwebbook
7487-88-9 MgO4S Magnesium Sulfate crviewwebbook
7487-88-9 MgO4S Magnesium Sulfate lviewwebbook
7487-88-9 MgO4S Magnesium Sulfate cr,lviewwebbook
13573-11-0 MgO4W Magnesium Tungsten Oxide crviewwebbook
12032-35-8 MgO5Ti2 Magnesium Titanium Oxide crviewwebbook
12032-35-8 MgO5Ti2 Magnesium Titanium Oxide lviewwebbook
12032-35-8 MgO5Ti2 Magnesium Titanium Oxide cr,lviewwebbook
10034-94-3 Mg2O4Si Magnesium Silicate crviewwebbook
10034-94-3 Mg2O4Si Magnesium Silicate lviewwebbook
10034-94-3 Mg2O4Si Magnesium Silicate cr,lviewwebbook
12032-52-9 Mg2O4Ti Magnesium Titanium Oxide crviewwebbook
12032-52-9 Mg2O4Ti Magnesium Titanium Oxide lviewwebbook
12032-52-9 Mg2O4Ti Magnesium Titanium Oxide cr,lviewwebbook
7757-87-1 Mg3O8P2 Magnesium Phosphate crviewwebbook
7757-87-1 Mg3O8P2 Magnesium Phosphate lviewwebbook
7757-87-1 Mg3O8P2 Magnesium Phosphate cr,lviewwebbook
12058-07-0 MoO Molybdenum Oxide gviewwebbook
18868-43-4 MoO2 Molybdenum Oxide crviewwebbook
18868-43-4 MoO2 Molybdenum Oxide gviewwebbook
110743-27-6 MoO2.750 Molybdenum Oxide crview
110743-28-7 MoO2.875 Molybdenum Oxide crview
110743-29-8 MoO2.889 Molybdenum Oxide crview
1313-27-5 MoO3 Molybdenum Oxide crviewwebbook
1313-27-5 MoO3 Molybdenum Oxide lviewwebbook
1313-27-5 MoO3 Molybdenum Oxide cr,lviewwebbook
1313-27-5 MoO3 Molybdenum Oxide gviewwebbook
10102-43-9 NO Nitrogen Oxide gviewwebbook
14967-78-3 NO+Nitrogen Oxide, Ion gviewwebbook
10102-44-0 NO2 Nitrogen Oxide gviewwebbook
14797-65-0 NO2-Nitrogen Oxide, Ion gviewwebbook
12033-49-7 NO3 Nitrogen Oxide gviewwebbook
10024-97-2 N2O Nitrogen Oxide gviewwebbook
12269-46-4 N2O+Nitrogen Oxide, Ion gviewwebbook
10544-73-7 N2O3 Nitrogen Oxide gviewwebbook
10544-72-6 N2O4 Nitrogen Oxide crviewwebbook
10544-72-6 N2O4 Nitrogen Oxide lviewwebbook
10544-72-6 N2O4 Nitrogen Oxide cr,lviewwebbook
10544-72-6 N2O4 Nitrogen Oxide gviewwebbook
10102-03-1 N2O5 Nitrogen Oxide gviewwebbook
12401-86-4 NaO Sodium Oxide gviewwebbook
12769-83-4 NaO-Sodium Oxide, Ion gviewwebbook
12034-12-7 NaO2 Sodium Superoxide crviewwebbook
1313-59-3 Na2O Sodium Oxide crviewwebbook
1313-59-3 Na2O Sodium Oxide lviewwebbook
1313-59-3 Na2O Sodium Oxide cr,lviewwebbook
1313-60-6 Na2O2 Sodium Peroxide crviewwebbook
6834-92-0 Na2O3Si Sodium Silicate crviewwebbook
6834-92-0 Na2O3Si Sodium Silicate lviewwebbook
6834-92-0 Na2O3Si Sodium Silicate cr,lviewwebbook
7757-82-6 Na2O4S Sodium Sulfate, Delta crviewwebbook
7757-82-6 Na2O4S Sodium Sulfate, I crviewwebbook
7757-82-6 Na2O4S Sodium Sulfate, III crviewwebbook
7757-82-6 Na2O4S Sodium Sulfate, IV crviewwebbook
7757-82-6 Na2O4S Sodium Sulfate, V crviewwebbook
7757-82-6 Na2O4S Sodium Sulfate lviewwebbook
7757-82-6 Na2O4S Sodium Sulfate cr,lviewwebbook
7757-82-6 Na2O4S Sodium Sulfate gviewwebbook
13472-45-2 Na2O4W Sodium Tungsten Oxide crviewwebbook
13870-28-5 Na2O5Si2 Sodium Silicate crviewwebbook
13870-28-5 Na2O5Si2 Sodium Silicate lviewwebbook
13870-28-5 Na2O5Si2 Sodium Silicate cr,lviewwebbook
12034-57-0 NbO Niobium Oxide crviewwebbook
12034-57-0 NbO Niobium Oxide lviewwebbook
12034-57-0 NbO Niobium Oxide cr,lviewwebbook
12034-57-0 NbO Niobium Oxide gviewwebbook
12034-59-2 NbO2 Niobium Oxide crviewwebbook
12034-59-2 NbO2 Niobium Oxide lviewwebbook
12034-59-2 NbO2 Niobium Oxide cr,lviewwebbook
12034-59-2 NbO2 Niobium Oxide gviewwebbook
1313-96-8 Nb2O5 Niobium Oxide crviewwebbook
1313-96-8 Nb2O5 Niobium Oxide lviewwebbook
1313-96-8 Nb2O5 Niobium Oxide cr,lviewwebbook
17778-80-2 O Oxygen gviewwebbook
14581-93-2 O+Oxygen, Ion gviewwebbook
14337-01-0 O-Oxygen, Ion gviewwebbook
14452-66-5 OP Phosphorus Oxide gviewwebbook
1317-36-8 OPb Lead Oxide, Red crviewwebbook
1317-36-8 OPb Lead Oxide, Yellow crviewwebbook
1317-36-8 OPb Lead Oxide lviewwebbook
1317-36-8 OPb Lead Oxide cr,lviewwebbook
1317-36-8 OPb Lead Oxide gviewwebbook
13827-32-2 OS Sulfur Oxide gviewwebbook
20901-21-7 OS2 Sulfur Oxide gviewwebbook
10097-28-6 OSi Silicon Oxide gviewwebbook
1314-11-0 OSr Strontium Oxide crviewwebbook
1314-11-0 OSr Strontium Oxide lviewwebbook
1314-11-0 OSr Strontium Oxide cr,lviewwebbook
1314-11-0 OSr Strontium Oxide gviewwebbook
12035-90-4 OTa Tantalum Oxide gviewwebbook
12137-20-1 OTi Titanium Oxide, Alpha crviewwebbook
12137-20-1 OTi Titanium Oxide, Beta crviewwebbook
12137-20-1 OTi Titanium Oxide lviewwebbook
12137-20-1 OTi Titanium Oxide cr,lviewwebbook
12137-20-1 OTi Titanium Oxide gviewwebbook
12035-98-2 OV Vanadium Oxide crviewwebbook
12035-98-2 OV Vanadium Oxide lviewwebbook
12035-98-2 OV Vanadium Oxide cr,lviewwebbook
12035-98-2 OV Vanadium Oxide gviewwebbook
12035-99-3 OW Tungsten Oxide gviewwebbook
12036-01-0 OZr Zirconium Oxide gviewwebbook
7782-44-7 O2 Oxygen refviewwebbook
12185-07-8 O2+Oxygen, Ion gviewwebbook
11062-77-4 O2-Oxygen, Ion gviewwebbook
12164-97-5 O2P Phosphorus Oxide gviewwebbook
1309-60-0 O2Pb Lead Oxide crviewwebbook
7446-09-5 O2S Sulfur Dioxide gviewwebbook
99493-55-7 O2Si Silicon Oxide, Cristobalite, High crviewwebbook
14464-46-1 O2Si Silicon Oxide, Cristobalite, Low crviewwebbook
14808-60-7 O2Si Silicon Oxide, Quartz crviewwebbook
7631-86-9 O2Si Silicon Oxide lviewwebbook
7631-86-9 O2Si Silicon Oxide cr,lviewwebbook
7631-86-9 O2Si Silicon Oxide gviewwebbook
12036-14-5 O2Ta Tantalum Oxide gviewwebbook
1317-70-0 O2Ti Titanium Oxide, Anatase crviewwebbook
1317-80-2 O2Ti Titanium Oxide, Rutile crviewwebbook
13463-67-7 O2Ti Titanium Oxide lviewwebbook
13463-67-7 O2Ti Titanium Oxide cr,lviewwebbook
13463-67-7 O2Ti Titanium Oxide gviewwebbook
12036-22-5 O2W Tungsten Oxide crviewwebbook
12036-22-5 O2W Tungsten Oxide gviewwebbook
10743-30-1 O2.72W Tungsten Oxide crview
10743-31-2 O2.90W Tungsten Oxide crview
10743-32-3 O2.96W Tungsten Oxide crview
1314-23-4 O2Zr Zirconium Oxide crviewwebbook
1314-23-4 O2Zr Zirconium Oxide lviewwebbook
1314-23-4 O2Zr Zirconium Oxide cr,lviewwebbook
1314-23-4 O2Zr Zirconium Oxide gviewwebbook
10028-15-6 O3 Ozone gviewwebbook
10099-76-0 O3PbSi Lead Silicate crviewwebbook
7446-11-9 O3S Sulfur Trioxide gviewwebbook
1344-54-3 O3Ti2 Titanium Oxide crviewwebbook
1344-54-3 O3Ti2 Titanium Oxide lviewwebbook
1344-54-3 O3Ti2 Titanium Oxide cr,lviewwebbook
1314-34-7 O3V2 Vanadium Oxide crviewwebbook
1314-34-7 O3V2 Vanadium Oxide lviewwebbook
1314-34-7 O3V2 Vanadium Oxide cr,lviewwebbook
1314-35-8 O3W Tungsten Oxide crviewwebbook
1314-35-8 O3W Tungsten Oxide lviewwebbook
1314-35-8 O3W Tungsten Oxide cr,lviewwebbook
1314-35-8 O3W Tungsten Oxide gviewwebbook
13566-17-1 O4Pb2Si Lead Silicate crviewwebbook
1314-41-6 O4Pb3 Lead Oxide crviewwebbook
7733-02-0 O4SZn Zinc Sulfate crviewwebbook
10101-52-7 O4SiZr Zirconium Silicate crviewwebbook
12036-21-4 O4V2 Vanadium Oxide crviewwebbook
12036-21-4 O4V2 Vanadium Oxide lviewwebbook
12036-21-4 O4V2 Vanadium Oxide cr,lviewwebbook
12036-21-4 O2V Vanadium Oxide gviewwebbook
1314-61-0 O5Ta2 Tantalum Oxide crviewwebbook
1314-61-0 O5Ta2 Tantalum Oxide lviewwebbook
1314-61-0 O5Ta2 Tantalum Oxide cr,lviewwebbook
12065-65-5 O5Ti3 Titanium Oxide, Alpha crviewwebbook
12065-65-5 O5Ti3 Titanium Oxide, Beta crviewwebbook
12065-65-5 O5Ti3 Titanium Oxide lviewwebbook
12065-65-5 O5Ti3 Titanium Oxide cr,lviewwebbook
1314-62-1 O5V2 Vanadium Oxide crviewwebbook
1314-62-1 O5V2 Vanadium Oxide lviewwebbook
1314-62-1 O5V2 Vanadium Oxide cr,lviewwebbook
12440-00-5 O6P4 Phosphorus Oxide gviewwebbook
12165-16-1 O6W2 Tungsten Oxide gviewwebbook
12143-55-4 O7Ti4 Titanium Oxide crviewwebbook
12143-55-4 O7Ti4 Titanium Oxide lviewwebbook
12143-55-4 O7Ti4 Titanium Oxide cr,lviewwebbook
12165-25-2 O8W3 Tungsten Oxide gviewwebbook
12165-37-6 O9W3 Tungsten Oxide gviewwebbook
16752-60-6 O10P4 Phosphorus Oxide crviewwebbook
16752-60-6 O10P4 Phosphorus Oxide gviewwebbook
12165-45-6 O12W4 Tungsten Oxide gviewwebbook |
5725 | https://www.ebsco.com/research-starters/mathematics/transversal | Research Starters Home
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Transversal
A transversal is a line that intersects two other lines at two distinct points, resulting in the creation of eight angles. This geometrical concept is particularly significant when the intersected lines are parallel, as the angles formed exhibit specific relationships that can indicate whether the lines are indeed parallel. For example, when a transversal crosses two parallel lines, it produces pairs of angles that are either congruent or supplementary, with a total angle sum of 360 degrees.
The angles generated can be categorized into four types: internal angles (between the lines) and external angles (outside the lines), alongside three specific types of angle pairs: consecutive angles, corresponding angles, and alternate angles. Consecutive angles, which are both interior and located on the same side of the transversal, sum to 180 degrees. Corresponding angles, consisting of one interior and one exterior angle from the same side, are always congruent. Finally, alternate angles, found on opposite sides of the transversal, are also congruent if the lines are parallel. These properties were historically studied by the ancient Greek mathematician Euclid, contributing to our understanding of geometry.
Published in: 2022
By: Zimmer, Scott, MLS
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Transversal
Within the study of plane geometry, a transversal is a line that intersects two other lines at two separate points. Because each intersection of the transversal with a line creates four angles, in total the transversal produces eight angles. When a transversal intersects two lines that are parallel to one another, the eight angles that are produced have special relationships with one another. For this reason, the angles of a transversal are often studied as a means of determining whether two lines are parallel, because if they exhibit the same properties that they would if the lines were parallel, then it can be concluded that the lines are in fact parallel. When the lines crossed by the transversal are parallel, for example, several pairs of angles are created, some of which are congruent and others of which are supplementary, meaning that the sum of their values is equal to 180 degrees. The sum of all four angles created by the intersection of a transversal with a line is thus 360 degrees. Of the eight angles created by the transversal’s intersection of the two lines, four are called internal angles because they are found between the two lines that the transversal crosses, and the other four are called external, because they lie outside the area between the lines.
Overview
If a transversal crosses two parallel lines at right angles, then each of the eight angles created measures 90 degrees; this is called a perpendicular transversal. If the transversal is not perpendicular but the lines it crosses are parallel, then three types of angles will be created. The first type of angles created are called consecutive angles. Consecutive angles are identifiable because they are both interior angles (meaning that they occur between the two parallel lines rather than outside of them), they appear on the same side of the transversal, and they have different vertex points. The most significant characteristic of consecutive angles is that their sum is 180 degrees.
The second type of angle created when a non-perpendicular transversal intersects two parallel lines is known as a corresponding angle. Corresponding angles, like consecutive angles, have different vertex points and lie on the same side of the transversal, but they differ from consecutive angles in the fact that with corresponding angles, one of the pair is interior and the other is exterior. Corresponding angles will always be congruent with each other.
The final type of angle pair created are called alternate angles. Alternate angles also have different vertex points, like consecutive and corresponding angles, but the similarity ends there. Alternate angles are found on opposite sides of the transversal (hence the use of the term "alternate") and are either both internal angles or both external angles. Alternate angles will always be congruent with one another if the lines crossed by the transversal truly are parallel to one another. These properties of the angles created by a transversal were studied and proven by the Greek mathematician Euclid, who was active at about the year 300 BC.
Bibliography
Craine, Timothy. Understanding Geometry for a Changing World. Reston, VA: National Council of Teachers of Mathematics, 2009.
Maor, Eli, and Eugen Jost. Beautiful Geometry. Princeton : Princeton UP, 2014.
McKellar, Danica. Girls Get Curves: Geometry Takes Shape. New York: Hudson Street Press, 2012.
O'Leary, Michael. Revolutions of Geometry. Hoboken, NJ: Wiley, 2010.
Ostermann, Alexander, and Gerhard Wanner. Geometry by Its History. Heidelberg: Springer, 2012.
Rich, Barnett, and Christopher Thomas. Geometry: Includes Plane, Analytic, and Transformational Geometries. New York: McGraw, 2013.
Shawyer, Bruce. Explorations in Geometry. Singapore: World Scientific, 2010.
Related Topics
Understanding Theorems About Lines and Angles AnglesCongruence |
5726 | https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HB__Special_Relativity_and_Thermal_Statistical_Physics/1%3A_Foundations_of_Relativity/1.2%3A_The_Nature_of_Time | 1.2: The Nature of Time - Physics LibreTexts
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1: Foundations of Relativity
UCD: Physics 9HB –Special Relativity and Thermal/Statistical Physics
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Wed, 11 Jan 2023 18:41:53 GMT
1.2: The Nature of Time
75010
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Tom Weideman
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1.2: The Nature of Time
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1.1: The Relativity Principle
1.3: More Thought Experiments
Page ID 75010
Tom Weideman
University of California, Davis
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Spacetime Events
2. Time Dilation
3. Recording Spacetime Coordinates
4. Two Different Time Measurements
1. coordinate time
2. proper time
3. two important notes
Cosmic Speed Limit and the Spacetime Interval
Spacetime Events
We now embark on deriving the consequences of the relativity principle in the same way that Einstein did – using a tool he called Gedankenexperiment (thought experiment). in order to keep everything straight in our discussions, we begin by defining a spacetime event.
In the context of special relativity, a spacetime event is an instantaneous occurrence at a specific point in space and at a specific moment in time. A single point on a stationary light bulb as it dims defines a specified location, but it is not an event because the dimming process does not occur at a single instant in time. A baseball bat at exactly 12:01pm occurs at a single instant in time, but it is not an event, because the position is not specified at a single point. An easy way to visualize a spacetime event is to picture it as a very quick flash of light from a point source. The position of the point source and the instant in time the flash occurs define the space and time “coordinates” of the spacetime event.
It is much easier to define what a spacetime event is than it is to put physical quantities in terms of the spacetime coordinates, but as we will see, this is exactly what we will have to do to make sense of what is to come. We begin with one of most startling results, which is ironically one of the easiest to derive.
Time Dilation
Our first thought experiment involves turning the function of a clock into a series of spacetime events. This clock functions as follows: Light bounces back-and-forth between two mirrors, and every time it strikes one of the mirrors, the clock "ticks." We begin with Ann's perspective on what is happening with this clock. She happens to be in the same frame as the two mirrors, so to her they are at rest, and the light is bouncing parallel to her y-axis. The two spacetime events we will look at are two consecutive ticks of the clock.
Figure 1.2.1 – Ann's Perspective of the Light Clock
Okay, so we have used the two events to determine the time span between them according to Ann. The goal of relativity is to describe what a second observer measures for a physical process given what the first observer measured. So now we introduce Bob, who is in what we call the primed inertial frame, moving at a constant speed v in the +x-direction relative to Ann. One might interject, "Wait, this is time we are talking about! Won't both of them measure the same amount of time between ticks of the clock?" Don't assume anything in relativity – just use the spacetime events and the postulate(s), and see where it leads.
Looking from Bob's perspective means that not only is Ann moving in the −x-direction, as we noted previously, but the two events (which both occur at the top mirror) don't occur at the same position in space, since the mirror moves:
Figure 1.2.2 – Bob's Perspective of the Light Clock
Now we calculate the time between the two events, as we did for Ann. From Bob's perspective, the light travels a longer distance than Ann measures, and very importantly, both Ann and Bob measure the speed of light to be the same (postulate of relativity), so Bob must measure a longer time period than Ann measures between the same ticks of the light clock! According to Bob, the light travels diagonally from the top mirror to the bottom one, and the length of this half of the trip can be written in terms of the speed of light, and in terms of the Pythagorean Theorem:
(1.2.1)Δx′=x 2′−x 1′=vΔt′cΔt′=2L 2+(Δx′2)2
We can eliminate Δx′ from these two equations to relate the time span measured by Bob to the time span measured by Ann:
(1.2.2)Δt′=1 1−v 2 c 2(2L c)=γ vΔt,γ v≡1 1−v 2 c 2
The time between ticks for Bob is greater than the time between ticks for Ann by a factor of γ v (which is clearly a constant greater than 1). Just to clarify, this is not an optical illusion for Bob – he doesn't just "see the clock ticking slower than it really is," it is actually ticking slower. Also, it is important to note that while we used light to achieve this answer, it doesn't just apply to light phenomena, it applies to time flow in all its manifestations. If Ann measures her own pulse to be 60 beats per second (one second between each beat), and γ v=2, the Bob would measure Ann's heart rate to be 30 beats per second (2 seconds between each beat).
It's worth taking a moment to review what the source of this result is. It comes from the fact that the light in the light clock travels farther for Bob than it does for Ann, but they agree on the speed of that light, which means that the time between the two events must be greater for Bob than it is for Ann.
As startling as this result is, it gets weirder. Suppose Bob has a light clock exactly like Ann's. What does Ann observe when she looks at Bob's clock? She sees exactly the same thing happening with Bob's clock as he sees with her clock! Therefore Ann claims that time is passing slower for Bob than it is for her, even as Bob says that Ann's time is passing slower than his own. Which one of them is correct? Is Ann's time passing slower, or is Bob's? They are both in inertial frames, so according to the principle of relativity, each has an equal right to declare themselves to be "stationary." Therefore they are both right. The reason it seems like it is impossible that this can be true is that we cling to the incorrect notion that time is universal. The time span between two events is a relative quantity that depends upon who measures it.
Recording Spacetime Coordinates
While the calculation above is correct, it does require an assumption that we need to briefly address. Both Ann and Bob noted the positions and times of the events in their frames. Given the importance of both position and time in relativity, we need to be specific about how these numbers are recorded. What is observed is a spacetime event, which we have modeled as a flash of light that occurs in an instant at a specific position. So let's imagine constructing a massive lattice of labeled positions throughout all of space, and the position of any possible flash must coincide with one of those positions, giving us our spatial label. Note that every inertial observer can create such a lattice independent of every other observer, because according to the relativity principle, everyone has an equally valid claim to being "stationary." It is true that Bob's lattice of position labels is moving according to Ann, but Ann and Bob only use their own stationary labels to describe the positions of events they see.
To get a complete reckoning of an event, we need to record not only its position, but the time at which it occurs. Given what we know about the rate of time flow for moving clocks, we have to be very careful about how we measure the time at which an event occurs. The one way to be safe is to have the clock that reads the time be positioned at the same place in space as the event. So whenever an event occurs, one simply reads the label of the lattice point at which it occurs, and the value indicated by a clock located at that lattice point when the event occurs.
Two Different Time Measurements
Now that we have a plan for recording data for events in spacetime, we need to give a little more thought to how we plan to have a clock that is properly positioned to measure the time. It turns out that there are two fundamentally different ways to achieve this.
coordinate time
The first way that comes to mind for measuring the time of any given event is to simply place a separate clock at every lattice point. While this is a simple way to get a measurement for any event, we will be interested the time intervals between two events, which means that all of our clocks positioned throughout space need to be synchronized. How do we do this? If we bring all our clocks together in one place, set them the same, and then move them out to their assigned locations, then the weird effects that come from relative motion of clocks make un-synchronize them when they are moved. Instead what we can do is this
Distribute all of the clocks throughout space.
Set the clock at the origin to a time of 0:00.
Using the lattice positions of the clocks, compute the time it will take a spherical wave pulse of light that starts at the origin to reach all the other clocks, add this time to 0:00, and set the clock at this time.
Start the clock at the origin while starting the spherical pulse of light from the origin.
When the light wave reaches a clock, start it running.
Figure 1.2.3 – Synchronizing Clocks Distributed in Space
By anticipating what the time on the origin clock must be when the light arrives, we can assure that the spherical wave propagates clock synchronization throughout space. This measurement of the time of an event is called coordinate time t.
In the example above, both Ann and Bob measured the time between ticks in their own coordinate time. For Ann, the coordinate time span between the two spacetime events was Δt=t 2−t 1, while for Bob it was Δt′=t 2′−t 1′. As we found in the thought experiment, these values are not equal, which is to say that this manner of measuring a time span between two events is relative. Whenever the value of a physical quantity is different when measured from different inertial frames, we say that such a quantity is frame-dependent. We therefore declare:
Coordinate time spans are frame-dependent.
proper time
We certainly are not required to measure time between events by placing synchronized clocks at all the lattice points in our frame. Another way would be to use a single clock that is moved from the lattice point of the first event to the lattice point of the second. As before, a clock records the time of the event while it is at the same point in the lattice as the event, but this time it is the same clock, which means we do not need to rely upon our synchronization method above.A time interval measured in this manner is called a proper time Δτ between the spacetime events.
Alert
The name "proper time" dates back to the early days of relativity, and is still used today, but it is dangerously misleading for those new to the subject. The word "proper" can easily be misconstrued to mean "correct," and hopefully this section is making it clear that this is cannot be the case. We are in the process of defining two different ways of measuring the time between two events (which can give different answers), and neither of these is any more correct than the other. The sooner the reader purges from their thoughts the notion that time is absolute and that there must be one correct value for the time between two events, the better.
It might seem like both the coordinate and proper time methods of measuring time intervals between events should produce the same result, but in fact they do not. The figures below demonstrates these two measurements for the same two spacetime events.
Figure 1.2.4– Spacetime Events
Here we have just two spacetime events viewed from a particular reference frame.
Figure 1.2.5– Coordinate Time Interval
These are the same two events, viewed from the same reference frame, but the coordinate time clocks placed at the positions of the events are in place to measure the times at which the events occur.
Figure 1.2.6– Proper Time Interval
Again we have the same two spacetime events, viewed in the same frame, but a clock at rest in a different frame is now visible, and it measures the time interval between the two spacetime events,but in that frame the events occur at the same position(at the nose of the rocket ship).
Our example with Ann and Bob earlier shows why these time measurements come out different, if the postulate of special relativity about the constancy of the speed of light is accurate. The two flashes occur at the same lattice point in Ann's frame (the top mirror remains at the same place in Ann's labeled lattice), so she measures the time interval using the proper time method. Meanwhile, the top mirror moves from one lattice point in Bob's frame to another, so he relies upon the synchronized clocks positioned at those points for the time interval. The thought experiment demonstrates through the postulates of relativity that these two time intervals are not equal.
Example 1.2.1
In the figures above that depict two spacetime events and a spaceship moving between them, we are observing from an inertial frame. How fast is the spaceship moving relative to this frame?
Solution
The two events occur at the same position in space in the ship's frame (at its nose - remember that the people on the ship can claim that the ship is not moving, so the two flashes occur at the same place). Therefore the ship measures the proper time interval between the two flashes, just as Ann measured the proper time interval of the two flashes at the top mirror. Using the formula we derived to express the relationship between the time intervals, we find:
Δt=γ vΔτ⇒1−v 2 c 2=Δτ Δt=12s 15s=0.8⇒v=0.6c
The feature that best distinguishes proper time from coordinate time is the fact that a coordinate system is not needed to measure proper time. For example, we could introduce several other inertial frames of reference to look at the time interval between those top mirror flashes, but only Ann's clock will measure the time interval as the the one where the flashes occur at the same position in space – every frame other than Ann's will be similar to Bob's, in that the flashes occur at different lattice positions for their frame. All of the observers will agree on one thing – that Ann's measurement of the time interval is somehow "special", and this gives them a way to all agree upon a time interval. Put another way, the measurement of proper time (essentially asking Ann what answer she got, as she was the only observer in an inertial frame to have both events occur at the same position) gives the same result for all reference frames. That is:
Proper time is frame-independent.
Besides "frame-independent," a word typically used to describe a physical quantity like proper time that doesn't vary from one frame to another, is invariant.
Note that it is possible for a proper time measurement to be equal to a coordinate time measurement. For example, in the case discussed above, Ann sees the two events occur at the same lattice point in her frame, so if she looks at the clock placed there, it is the same clock measuring the time for both events, which means it also records the proper time. Bob's measurement of coordinate time, on the other hand, is not the proper time, since he reads the numbers off two different clocks – one placed at the lattice point of the first spacetime event, and one placed at the second. From the light clock example, it should be clear that the shortest distance the light has to travel between the two mirrors occurs in Ann's frame. That is, every frame other than the "proper frame" that measures coordinate time is going to measure a longer time interval between the events than the proper time.
two important notes
There are two details that have not been tied-up above that we will mention here and address in a future section:
The proper time between two events may not be definable as a real number, if the events are separated by a distance that is too great for the spaceship to cross in the interval between their occurrences. For example, if the two events viewed in the frame of the figure above occurred simultaneously in that frame, then there is no way for the spaceship to traverse the distance between the events fast enough to allow both events to both occur at the nose of the ship. We will see later that there is another invariant quantity that applies to any two events, and that the proper time interval is a special case of this invariant when an inertial frame exists that can measure the events at the same position in space.
We have defined the proper time here as being measured in an inertial frame, but the spaceship could also have both events occur at its nose when it accelerates between the two events. This will come out to a different result than the inertial frame case, so it is important when declaring (as we did above) that the "proper time is frame-independent", that we keep in mind that this is restricted to the specific history of the clock that measures it. That is, every observer will agree on the proper time measured by a single clock that is present at the positions of both events. "Invariance" pertains to different observers,not different clocks. A second clock that is present at the positions of both events will not necessarily measure the same proper time interval as the first clock. The way that each clock gets from the first event to the second (namely, its acceleration during the trip) determines how these two proper time measurements may differ.
Cosmic Speed Limit and the Spacetime Interval
When we look back at the time dilation result we obtained above, an obvious question comes to mind: If we observe a clock in a moving frame to tick more slowly than one in our rest frame by a factor of 1−v 2 c 2, then what happens when the relative speed of the two inertial frames reaches or exceeds v=c? Clearly the result gives a nonsense answer, and while this is far from "proof," we will take this moment to make a declaration that we will later see to be true in many other cases...
Two inertial frames can never have a relative speed that exceeds the speed of light, and this cosmic speed limit can only be attained for light itself.
Technically, there are other phenomena besides light that can propagate at light's eponymous speed, and the criterion for this is a simple one, but we will save that discussion for later. For now, we will generalize the "cosmic speed limit" to state that no "influence" or "information" can be passed from one point in space to another at a speed faster than light can traverse the same distance.
This speed limit gives us a new perspective on time intervals between two events. We said above that there was no way to measure the proper time interval between events that are separated in space and are simultaneous, because there is no way for a single clock to get from one event to the other in time. Now we see that the two events don't need to be simultaneous for this to be true. Because of the cosmic speed limit, there is no way for the proper time interval between two events to be measured if they are separated by enough distance that light cannot travel from the earlier spacetime event to the later one. If light can't make it in time, then neither can a clock, and the proper time cannot be measured.
Let's say that the events occur at positions (x 1,y 1,z 1) and(x 2,y 2,z 2), and times t 1 and t 2 (with t 2>t 1), as measured in some arbitrary inertial frame, respectively. Then there will be a well-defined proper time measurable by a moving clock if the distance between them is less than the distance that light can travel in the time interval Δt=t 2−t 1:
(1.2.3)Δx 2+Δy 2+Δz 2<cΔt
We can therefore invent a sort of "discriminant" that tells us whether two events can be connected in this way:
(1.2.4)Δs 2≡c 2Δt 2−(Δx 2+Δy 2+Δz 2)
When Δs 2>0, it is possible to move a clock from the earlier event to the later one, so that the clock measures the proper time, otherwise one cannot do this. This quantity Δs 2 is called the spacetime interval between the two events.
We said that the quantities Δt,Δx,Δy, and Δz are measured in any arbitrary inertial reference frame, but for a moment let's suppose that the events are sufficiently close together, and look at the value of the spacetime interval in the frame where the events are at the same position (i.e. the frame with the clock that, when viewed by someone else,is moved from the earlier event to the later one). In this frame, Δx=Δy=Δz=0 and Δt is the proper time, which means:
(1.2.5)Δs=cΔτ
So the spacetime interval between two events is just proportional to the square of the proper time interval between those events. We stated earlier that the value of Δτ is an invariant – it is the same when measured in any reference frame. This means that all observers will agree on the spacetime interval between two events that allow for a proper time. But now that we are talking about an abstract mathematical quantity instead of a span of time, we can make the more general statement for all pairs of events:
The spacetime interval between any two events is an invariant.
Yes, sometimes this interval is positive (allowing for a measurable proper time interval), sometimes it is negative (not allowing this - the proper time interval is imaginary, whatever that means), and sometimes it is zero (making the proper time interval zero). But whatever it comes out to, the value of Δs 2 is measured to be the same quantity in all frames of reference. Let's be clear about what this means: The values of Δt,Δx,Δy, and Δz are all different for various reference frames, but the special combination of these quantities that equals Δs 2 comes out to be the same in every frame, provided the same two events are involved.
There is one last observation we should make about the spacetime interval between two events. Suppose we consider two events that are separated by infinitesimal differences in distance and time. Then we have:
(1.2.6)ds 2=c 2dt 2−(dx 2+dy 2+dz 2)⇒ds=dtc 2−([dx dt 2]+[dy dt]2+[dz dt]2)
Given that the time between the infinitesimally-separated events is Δt and the distance in the x-direction between these events is Δx,then the ratio Δx Δt is the speed that the clock must move (measured in our arbitrary frame)in the x-direction to get from the earlier event to the later one to record both times. The same is true for the y and z directions, so:
(1.2.7)ds=dtc 2−(v x 2+v y 2+v z 2)=cdt1−v 2 c 2
One can now imagine computing (the square root of) the spacetime interval between two events of finite separation by "chaining together" (integrating) infinitesimal intervals:
(1.2.8)Δs=∫event A event B d s=∫A B c d t1−v 2 c 2
Combining this with Equation 1.2.5 gives us the link between the proper time interval and the coordinate time interval between events A and B:
(1.2.9)Δτ=∫A B d t1−v 2 c 2
Where v is the relative speed of the coordinate frame and the "proper frame" (the frame where the two events occur at the same place). To avoid complications, we will assume that the path between events is a straight one, but in general it could involve constant speed or accelerated motion. If the speed is constant, then the integral is trivial and we get the same result as Equation 1.2.2– the inertial frame time dilation formula we found from our thought experiment. But if the speed is not constant, then the quantity in the square root factors into the integral, and the proper time is not the same as it was for the case of the inertial frame. Later we will see the important consequences of this.
This page titled 1.2: The Nature of Time is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Tom Weideman directly on the LibreTexts platform.
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1.1: The Relativity Principle
1.3: More Thought Experiments
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Dividing a Polynomial by Monomials & Binomials | Steps & Examples
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Lisa Gilbert, Betsy Chesnutt, Robert Ferdinand
Author Lisa Gilbert
Lisa received a Bachelor's of Business Administration from the University of Georgia. Upon graduation, she worked as a Human Resources Manager for 7 years, managing all aspects of human resource management including writing and delivering training for employees. For the last 15 years, Lisa has reviewed a vast array of curriculum and taught and tutored all subjects to homeschooled students in all grades pre-K through early college.
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Learn the strategies of dividing polynomials by monomials and binomials. Discover how to divide a polynomial by a monomial and a binomial with examples. Updated: 11/21/2023
Table of Contents
Dividing Polynomials by Monomials
How to Divide a Polynomial by a Monomial
How to Divide a Polynomial by a Binomial:
Lesson Summary
Show
FAQs
Activities
Practice Problems: Dividing a Polynomial by a Monomial
Key Terms
Polynomial: An expression of many terms in a variable (say x), where all exponents or powers of x are non-negative integers (0, 1, 2, 3, ... etc.).
Monomial: A polynomial with one term.
Materials Needed
Paper
Pencil
Practice Problems (Show all your work)
(a) Divide the polynomial (17a^3 + 8a^2 + 12a + 7) by the monomial (2a).
(b) Find the quotient of (14x^4 - 12x^2 - 6x - 2) divided by (3x^2).
Answers (To Check Your Work):
(a) Using the Distributive Property of multiplication (or division) over addition we get that:
(17a^3 + 8a^2 + 12a + 7)/(2a)
= (17a^3/2a) + (8a^2/2a) + (12a/2a) + (7/2a).
Simplifying each term in the above expression, dividing like terms, and using the properties of exponents and algebra yields:
(17/2)a^2 + 4a + 6 + 7/(2a).
Answer: (17/2)a^2 + 4a + 6 + 7/(2a).
(b) Using the Distributive Property of multiplication (or division) over addition, we get the quotient from the following division:
(14x^4 - 12x^2 - 6x - 2)/(3x^2)
= (14x^4)/(3x^2) - (12x^2)/(3x^2) - (6x)/(3x^2) - (2)/(3x^2).
Simplifying each term in the above expression, dividing like terms, and using the properties of exponents and algebra yields:
(14/3)x^2 - (12/3) - (6)/(3x) - 2/(3x^2)
= (14/3)x^2 - 4 - 2/x - 2/(3x^2).
Answer: (14/3)x^2 - 4 - 2/x - 2/(3x^2).
How to divide a polynomial by a polynomial with more than one term?
When dividing a polynomial by another polynomial, use the first term of the divisor in all steps requiring division. Follow the general rules of long division - divide (only the first term of the divisor), multiply (all terms of the divisor), subtract, and bring down - all terms of the dividend.
How to divide a polynomial by a monomial?
To solve a polynomial by a monomial, divide each polynomial term by a monomial, apply the quotient rule of exponents where applicable, and simplify all terms using basic division. Be sure to write the final answer in standard form.
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Table of Contents
Dividing Polynomials by Monomials
How to Divide a Polynomial by a Monomial
How to Divide a Polynomial by a Binomial:
Lesson Summary
Show
Dividing Polynomials by Monomials ---------------------------------
In the study of algebra, monomials are expressions that are comprised of products between variables, numbers, or a combination of both. The term monomial has the prefix mono-, which means one Here, some examples of monomials are:
x 7|3|1 2 x 3|4 x y 2|9 a b 2 c 3
When monomials are added to or subtracted from each other, they become polynomials. The prefix 'poly- means more than one. Polynomials are expressions with two or more monomial terms connected by addition or subtraction. Some polynomials have special names, such as binomials with two terms and trinomials with three terms. Take these polynomial expressions, for example:
x 3−4 x(b i n o m i a l)x 2+9 x+4(t r i n o m i a l)3 x+7 r−9 k+8 k 2(p o l y n o m i a l)
Before learning how to divide a polynomial by a monomial, be sure to review the quotient rule of exponents, which states that when dividing numbers with like bases, keep the base and subtract the exponents:
x n x m=x n−m,f o r a l l x≠0
This rule will be important when dividing terms with exponents found in the polynomial expressions.
Lesson
Quiz
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How to Divide a Polynomial by a Monomial ----------------------------------------
When dividing polynomials by monomials, perform the following steps:
Divide each polynomial term in the dividend by the same monomial in the divisor.
Use the quotient rule of exponents and basic division to simplify each division term.
If possible, write the answer in standard form. Standard form means that terms are written in order from the greatest exponent to the lowest exponent in a single variable.
Example 1: Divide: 18 x 3−27 x 2 3 x
Solution
Divide each term in the dividend (numerator) by the same monomial in the divisor (denominator). Then, use the quotient rule for powers and basic division to simplify each term.
18 x 3−27 x 2 3 x=18 x 3 3 x−27 x 2 3 x=6 x 2−9 x
Example 2: Find the quotient: x 4−3 x 3−8 x 2−10 x 2 x
Solution
Note that when setting up this division, there will be terms with a fractional coefficient. Fractions should always be expressed in the simplest form possible unless noted otherwise:
x 4−3 x 3−8 x 2−10 x 2 x=x 4 2 x−3 x 3 2 x−8 x 2 2 x−10 x 2 x=1 2 x 3−3 2 x 2−4 x−5
How to Divide a Polynomial by a Binomial: -----------------------------------------
Two methods are possible to divide a polynomial by a binomial: long division and synthetic division. Both methods are explored in depth in the next sections.
Long Division:
In long division, the setup is very similar to dividing whole numbers or integers using long division. However, there is one key difference: when dividing polynomials by a binomial, divide each polynomial term by the only first term in the binomial. The steps for long division are:
Step 1: Write in long division form with the dividend under the division symbol and the divisor to the left.
Step 2: Divide the first term of the polynomial by the first term of the binomial, and write the quotient on top of the division symbol, directly above the second polynomial term.
Step 3: Multiply that quotient by both binomial terms, writing the products beneath the first two terms of the polynomial.
Step 4: Subtract from the dividend and bring down the third term of the polynomial.
Step 5: Divide the first term of the difference by the first binomial term.
Repeat Steps 2-5, for all subsequent polynomial terms. Note: if a polynomial is "missing" a degree of exponent, the place must be "held" as a separate term. For example, when dividing x^4 + x^2 + 2 by a binomial or another polynomial, a "0" must hold the place of the absent x^3 term: x^4 + 0 x^3 +x^2 + 2.
Example: Divide using long division.
2 x 2−6 x+3 x−2
Long division of a binomial
Solution:2x - 2, remainder of -1
The Synthetic Method
The synthetic method of dividing a polynomial by a binomial is essentially a shortcut of long division in which the variables and exponents are temporarily ignored. This method can only be used when the leading coefficient is 1 and the variable is raised to a power of 1. In other words, the synthetic method can be used when a polynomial is divided by a binomial in the form x + c. The steps for the synthetic method are as follows:
Step 1: Write only the coefficients of the polynomial and their signs next to each other. Remember to use 0's to hold the places of any absent exponential terms.
Step 2: Draw an upside-down long division sign, leaving room for another row of numbers below the first.
Step 3: Write only the "zero" of the binomial to the left of the long division symbol. To find the "zero", set the binomial equal to zero, and solve for the variable. The solution is the "zero" of the binomial. This number should be written to the left.
Step 4: Bring down the first number inside the division symbol, and write that below the inverted division symbol.
Step 5: Multiply the "zero" of the binomial by the first number noted in Step 4, and write the product beneath the second number in the first row.
Step 6: Add the numbers in the second column, write the answer below the inverted division symbol, and repeat steps 5 and 6 until all columns have been added.
Once all steps have been completed, the bottom row of numbers represents the following (from right to left): the remainder, the constant, and the coefficients of the variables with increasing exponents. For comparison, the problem above will have the same answer using either long division or the synthetic method.
The synthetic method is a shortcut to long division
Example: Divide using the synthetic method.
2 x 2−6 x+3 x−2
Solution: 2 x - 2, remainder is -1
Lesson Summary --------------
Polynomials are expressions of monomials connected by addition or subtraction. Polynomials can be divided by using other polynomials or monomials. When dividing a polynomial by a monomial, use the quotient rule of exponents to divide each polynomial term by the monomial term. When dividing a polynomial by a binomial, use either long division or the synthetic method. In long division, only use the first term of the binomial in each step requiring division. The synthetic method is a shortcut to the long division, which can only be used when the binomial is in the form x + c.
Additional Info
What is a monomial?
Before we can think about dividing polynomials and monomials, we have to know what they are! How can you tell if something is a monomial? The prefix mono always means one, so a monomial is a mathematical expression with just one term. That term can include numbers (like 1, 4, or -12), and variables (like a, b, x, y, z, or even a 2 ). It can even include more than one variable, as long as those variables are all multiplied together. Take a look at the following terms and see if you can tell which ones are monomials.
What is a polynomial?
So, if a monomial is a mathematical expression with just one term, what is a polynomial? Poly always means many, so a polynomial is a mathematical expression with more than one term. It can have 2 terms, or 3, 10, 20, or even more! Anything with more than one term is a polynomial. Here are some examples of polynomials:
How to divide a polynomial by a monomial
To divide a polynomial by a monomial, you need to follow these three simple steps:
Step 1: Rewrite the problem and divide EACH term in the polynomial by the monomial.
Step 2: Divide the numbers. Do not do anything with the variables yet. We'll get to that in a minute!
Step 3: Now divide the variables. When you divide 2 variables, look at the exponents of each and subtract the exponent of the variable in the denominator, the bottom number in the fraction, from the exponent of the variable in the numerator, the top number in the fraction. For example a 5 divided by a 2 would be a(5-2) or a 3
Practice Problem
Let's practice those three steps again in another problem.
Problem:
16 y 5 + 8 y 2 - 4y divided by 2y
Solution:
Now, why don't you try one? Look at the following problem and then try to work it before you scroll down to see the answer. No peeking!
Problem:
-12 a 5 + 30 a 4 + 21 a 3 divided by 3 a 2
Solution:
Did you get the right answer! Great!
If you didn't get the right answer, go back and review the three steps and then try again.
Lesson Summary
Let's review the three steps you need to do when dividing a polynomial by a monomial.
First step: rewrite the problem and divide each term in the polynomial by the monomial. Second step: divide the numbers. Third and final step: divide the variables. Your final result will be a new polynomial!
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What are Polynomials, Binomials, and Quadratics? 4:39
Adding, Subtracting & Multiplying Polynomials | Steps & Examples 6:53
Multiplying a Binomial by a Monomial 2:38
Multiplying Binomials Using FOIL and the Area Method 7:26
Multiplying Binomials | Overview, Methods & Examples 5:46
Dividing a Polynomial by Monomials & Binomials | Steps & Examples
4:56 Next Lesson What is the Greatest Common Factor? | GCF Examples
Factoring Polynomial Expressions | Definition, Methods & Examples 6:18
Quadratic Trinomial | Definition, Factorization & Examples 7:53
Dividing a Polynomial by Monomials & Binomials | Steps & Examples Related Study Materials
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5728 | https://web.williams.edu/Mathematics/sjmiller/public_html/math/talks/small2016/integercomplexityposter2016.pdf | Some Bounds on Integer Complexity Katherine Cordwell, Alyssa Epstein, Anand Hemmady, Aaditya Sharma, Yen Nhi Truong Vu University of Maryland, College Park; Williams College, Amherst College ktcordwell@gmail.com, ale2@williams.edu, ash6@williams.edu, as17@williams.edu, ytruongvu17@amherst.edu WHAT IS IT?
The complexity of n ∈N, which we denote by f(n), is defined as the least number of 1’s needed to represent n using only the operations of addition and multiplication in conjunction with an arbitrary number of parentheses. For example, since we can write 6 = (1 + 1)(1 + 1 + 1), we get that f(6) ≤5.
Richard Guy gives the following recursive definition of f(n): min d | n 2≤d≤√n 1≤a≤n/2 n f(d) + f n d , f(a) + f(n −a) o .
The growth rate of f(n) is quite slow compared to that of n.
For example, f(100) = 14 and f(3133160) = 45.
GUY’S METHOD Given a base b representation of n, say (d0 · · · dk)b, Guy’s method (also called Horner’s scheme) writes n = dk + b(dk−1 + b(dk−2 + · · · + bd0)) · · · )).
Given base b, define a digit-balanced number n as a num-ber with roughly equal proportions of all of the b possible digits. Digit-unbalanced numbers are numbers which de-viate from this, i.e. digit-unbalanced numbers have some digits appearing significantly more often than other ones.
Then, letting D(b, r) be defined as before, using Guy’s method, we see that the complexity of n for digit-balanced numbers is then bounded by S = 1 b ln(b) b−1 X r=0 D(b, r).
J. Arias de Reyna and J. van de Lune computed this quantity in bases of the form 2n3m ≤2938 and found that S ≤3.6343 is minimized in base b = 29 · 38. In doing so, they show that a set of numbers of density 1 satisfy f(n) ≤3.6343 log3(n).
Our computations show that a more optimal base is 21139, where f(n) ≤3.61989 log3(n) for almost all n.
OUR IMPROVEMENTS We propose the following modification of J. Arias de Reyna and J. van de Lune’s algorithm: We need to show that using Guy’s method allows us to re-duce the number of summands that we must compute. As-sume that f(n) ≤c log3(n) = log3(nc) for some c ≤3.64 and that f(n) = f(a) + f(n −a). First, using Selfridge’s lower bound, log3(nc) ≥3(log3(n −a) + log3(a)).
Say that a = kn, where necessarily k ≤ 1 2. Then we have log3(nc/3) ≥log3((1 −k)n · a).
Simplifying gives nc/3−1 1 −k ≥a.
Using 1 −k ≥ 1 2 gives 2nc/3−1 ≥nc/3−1 1 −k ≥a.
Then since c ≤3.64, we get a ≤2n0.214. So, what we have is that for almost all numbers, we only need to check sum-mands up to 2n0.214.
BINARY ANALYSIS As a special case, let us consider the binary base.
First, we wish to better characterize how many numbers are not covered by Guy’s method. In base 2, J. Arias de Reyna and J. van de Lune’s algorithm runs in O(n1.345).
Binary digit-unbalanced numbers are simply those that have a larger proportion of 1’s than 0’s. For numbers where p% of the digits are 0’s and (1 −p)% are 1’s, the constant we obtain from Guy’s method is: 1 ln(2) p 100D(2, 0) + p −1 100 D(2, 1) ln(3).
When p = 1−p = 50%, this gives 3.9625. Some other values are: Percent 0’s Percent 1’s Constant 45 55 4.041655 46 54 4.02581 49 51 3.97826 In particular, the constant 3.97826 would improve the runtime of numbers where at most 49% of the digits are 1’s to O(n1.326), and the constant 4.02581 is enough to improve the runtime of numbers where at most 54% are 1’s to O(n1.342).
So, we focus on bounding the number of n where at most 46% are 0’s. This comes down to bounding B(n, 0) + · · · + B(n, n −d), where n −d = 46N 100 . We use the following bound, B(n, 0) + · · · + B(n, n −d) ≤e−nD( n−d n ,|| 1 2) where D n−d n || 1 2 is n −d n log 2 n −d n + 1 −n −d n log 2 1 −n −d n .
In particular, D( 46 100|| 1 2) < 0.0032034, and so the number of n ≤N where at most 46% are 0’s is ≤N 1−0.0032034, and so we can improve the algorithm on a set of N −N 1−0.0032034 numbers, or in the limit as N →∞, almost all numbers.
AN UNCONDITIONAL UPPER BOUND?
Something that we would like to explore more in the future is the idea of using division to improve the upper bound on complexity. The current upper bound is f(n) ≤3 log2(n), which comes from applying Guy’s method in base 2, where the worst numbers have a binary expansion that contains only 1’s. Experimentally, if we start with a number n that has a binary representation with a lot of 1’s, then usually (n−(n mod 3))/3 has a binary representation with close to 50% 1’s, and so Guy’s method provides a much better upper bound for this set of numbers. We would like to characterize the set of numbers where division by 3 does not afford a nice binary representation and to perform an alternate analysis on this set–for example, dividing by 5.
A GENERALIZATION Given n ∈Z, define f{1,x}(n) as the minimum number of 1’s and x’s needed to represent n with addition, multiplication, and parentheses. For example, f{1,5}(6) = 2, since 6 = 5+1, and f{1,3}(9) = 2, because 9 = 3 · 3.
A sharp lower bound on f{1,x}(n) follows from an inductive argument in the style of Selfridge.
• Given n ∈Z, if f(n) = k, then n ≤xk.
• For any n ∈Z, logx(n) ≤f(n).
We can define a greedy algorithm, which we aptly name “greedy”, to obtain a rough upper bound on f{1,x}(n). This algorithm works on n ≡t (mod x) in the following manner: • If n < x, then greedy(n) = 1 + · · · + 1 t times .
• If n ≥x and t = 0, then greedy(n) = x · greedy( n x).
• If n ≥x and t > 0, then greedy(n) = 1 + · · · + 1 t times + x · greedy( n x).
Here is a plot of the bounds compared to the complexity of “greedy” for f1,2: Here is a plot of the bounds compared to the complexity of “greedy” for f1,5: Of course, there is no reason to stop here. We can continue to generalize f{1,x} to f{1,x1,...,xt} where x1 < x2 < · · · < xt. Again we can obtain a sharp lower bound inductively: f{1,x1,...,xt}(n) ≤logxt(n). Upper bounds, however, become more difficult. We suspect that the upper bound depends on the density of {x1, . . . , xt} ∈Z.
WHAT’S BEEN DONE?
• 1953: Mahler and Popken propose the problem.
• 1986: Richard Guy popularizes the problem in his “Some Suspiciously Simple Sequences”. He attributes a sharp lower bound of f(n) ≥3 log3(n) to Selfridge.
• 2008: Fuller publishes a C program to compute f(n).
• 2008: Srinivas and Shankar give an algorithm that runs in time O(n1.58).
• 2012: Iradis et al. prove further experimental and an-alytical results.
• 2014: J. Arias de Reyna and J. van de Lune give an algorithm that runs in time O(n1.230175).
BEST CURRENT ALGORITHM J. Arias de Reyna and J. van de Lune’s algorithm uses Guy’s recursive definition to calculate complexities.
The rate-limiting step lies in checking the summands a ≤n such that f(n) = f(n −a) + f(a). To optimize the algorithm, J.
Arias de Reyna and J. van de Lune bound the number of summands that must be checked by n 2 1 − r 1 −4 n2 3f(n−1)/3 !
.
By defining and analyzing the function D(b, r) as the max-imum complexity of multiplying by b and adding r, they bound f(n).
Then, combining this information with the bound on the number of summands, they obtain their opti-mal runtime of O(n1.230175) in base 21037.
C HOW FAST J. Arias de Reyna and J. van de Lune wrote code in Python to perform their analysis, which they have generously sent us. Using this, we have developed comparable code in C that calculates the D(b, r). Since C runs much faster than Python, we are able to calculate values and perform analysis for higher bases. In particular, for base 21338, we obtain a better runtime of O(n1.222911236).
ACKNOWLEDGMENTS We would like to thank our mentors, Professors Steven J. Miller (Williams College), Eyvindur Palsson (Virginia Tech), and Stefan Steinerberger (Yale University).
Thank you to the SMALL REU program, Williams College, and the Williams College Science Center. We would also like to thank Professor Amanda Folsom for funding as well as NSF Grants DMS1265673, DMS1561945, DMS1347804, DMS1449679, the Williams College Finnerty Fund, and the Clare Boothe Luce Program. |
5729 | https://blog.ivanukhov.com/2021/01/25/dirichlet-process.html | Breaking sticks, or estimation of probability distributions using the Dirichlet process | Good news, everyone!
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Breaking sticks, or estimation of probability distributions using the Dirichlet process
January 25, 2021
Recall the last time you wanted to understand the distribution of given data. One alternative was to plot a histogram. However, it resulted in frustration due to the choice of the number of bins to use, which led to drastically different outcomes. Another alternative was kernel density estimation. Despite having a similar choice to make, it has the advantage of producing smooth estimates, which are more realistic for continuous quantities with regularities. However, kernel density estimation was unsatisfactory too: it did not aid in understanding the underlying structure of the data and, moreover, provided no means of quantifying the uncertainty associated with the results. In this article, we discuss a Bayesian approach to the estimation of data-generating distributions that addresses the aforementioned concerns.
The approach we shall discuss is based on the family of Dirichlet processes. How specifically such processes are constructed will be described in the next section; here, we focus on the big picture.
A Dirichlet process is a stochastic process, that is, an indexed sequence of random variables. Each realization of this process is a discrete probability distribution, which makes the process a distribution over distributions, similarly to a Dirichlet distribution. The process has only one parameter: a measure ν:B→[0,∞]ν:B→[0,∞] in a suitable finite measure space (X,B,ν)(X,B,ν) where X X is a set, and B B is a σ σ-algebra on X X. We shall adopt the following notation:
P∼Dirichlet Process(ν)P∼Dirichlet Process(ν)
where P P is a random probability distribution that is distributed according to the Dirichlet process. Note that measure ν ν does not have to be a probability measure; that is, ν(X)=1 ν(X)=1 is not required. To obtain a probability measure, one can divide ν ν by the total volume λ=ν(X)λ=ν(X):
P 0(⋅)=1 λ ν(⋅).P 0(⋅)=1 λ ν(⋅).
Since this normalization is always possible, it is common and convenient to replace ν ν with λ P 0 λ P 0 and consider the process to be parametrized by two quantities instead of one:
P∼Dirichlet Process(λ P 0).P∼Dirichlet Process(λ P 0).
Parameter λ λ is referred to as the concentration parameter of the process.
There are two major alternatives of using the Dirichlet process for estimating distributions: as a direct prior for the data at hand and as a mixing prior. We begin with the former.
Direct prior #
Given a data set of n n observations {x i}n i=1{x i}i=1 n, a Dirichlet process can be used as a prior:
x i|P x∼P x,for i=1,…,n;and P x∼Dirichlet Process(λ P 0).(1)x i|P x∼P x,for i=1,…,n;and(1)P x∼Dirichlet Process(λ P 0).
It is important to realize that the x i x i’s are assumed to be distributed not according to the Dirichlet process but according to a distribution drawn from the Dirichlet process. Parameter λ λ allows one to control the strength of the prior: the larger it is, the more shrinkage toward the prior is induced.
Inference #
Due to the conjugacy property of the Dirichlet process in the above setting, the posterior is also a Dirichlet process and has the following simple form:
P x|{x i}n i=1∼Dirichlet Process(λ P 0+n∑i=1 δ x i).(2)(2)P x|{x i}i=1 n∼Dirichlet Process(λ P 0+∑i=1 n δ x i).
That is, the total volume and normalized measure are updated as follows:
λ:=λ+n and P 0:=λ λ+n P 0+1 λ+n n∑i=1 δ x i.λ:=λ+n and P 0:=λ λ+n P 0+1 λ+n∑i=1 n δ x i.
Here, δ x(⋅)δ x(⋅) is the Dirac measure, meaning that δ x(X)=1 δ x(X)=1 if x∈X x∈X for any X⊆X X⊆X, and otherwise, it is zero. It can be seen in Equation 2 that the base measure has simply been augmented with unit masses placed at the n n observed data points.
The main question now is, How to draw samples from a Dirichlet process given λ λ and P 0 P 0?
As noted earlier, a draw from a Dirichlet process is a discrete probability distribution P x P x. The probability measure of this distribution admits the following representation:
P x(⋅)=∞∑i=1 p i δ x i(⋅)(3)(3)P x(⋅)=∑i=1∞p i δ x i(⋅)
where {p i}{p i} is a set of probabilities that sum up to one, and {x i}{x i} is a set of points in X X. Such a draw can be obtained using the so-called stick-breaking construction, which prescribes {p i}{p i} and {x i}{x i}. To begin with, for practical computations, the infinite summation is truncated to retain the only first m m elements:
P x(⋅)=m∑i=1 p i δ x i(⋅).P x(⋅)=∑i=1 m p i δ x i(⋅).
Atoms {x i}m i=1{x i}i=1 m are drawn independently from the normalized base measure P 0 P 0. The calculation of probabilities {p i}{p i} is more elaborate, and this is where the construction and this article get their name, “stick breaking.” Imagine a stick of unit length, representing the total probability. The procedure is to keep breaking the stick into two parts where, for each iteration, the left part yields p i p i, and the right one, the remainder, is carried over to the next iteration. How much to break off is decided on by drawing m m independent realizations from a carefully chosen beta distribution:
q i∼Beta(1,λ),for i=1,…,m.(4)(4)q i∼Beta(1,λ),for i=1,…,m.
All of them lie in the unit interval and are the proportions to break off of the remainder. When λ=1 λ=1, these proportions (of the reminder) are uniformly distributed. When λ<1 λ<1, the probability mass is shifted to the right, which means that there are likely to be a small number of large pieces, covering virtually the entire stick. When λ>1 λ>1, the probability mass is shifted to the left, which means that there are likely to be a large number of small pieces, struggling to reach the end of the stick.
Formally, the desired probabilities are given by the following expression:
p i=q i i−1∏j=1(1−q j),for i=1,…,m,p i=q i∏j=1 i−1(1−q j),for i=1,…,m,
which, as noted earlier, are the left parts of the remainder of the stick during each iteration. For instance, p 1=q 1 p 1=q 1, p 2=q 2(1−q 1)p 2=q 2(1−q 1), and so on. Due to the truncation, the probabilities {p i}m i=1{p i}i=1 m do not sum up to one, and it is common to set q m:=1 q m:=1 so that p m p m takes up the remaining probability mass.
To recapitulate, a single draw from a Dirichlet process is obtained in two steps: prescribe atoms {x i}{x i} via draws from the normalized base measure and prescribe the corresponding probabilities {p i}{p i} via the stick-breaking construction. The two give a complete description of a discrete probability distribution. Recall that this distribution is still a single draw. By repeating this process many times, one obtains the distribution of this distribution, which can be used to, for instance, quantify uncertainty in the estimation.
Illustration #
It is time to demonstrate how the Dirichlet process behaves as a direct prior. To this end, we shall use a data set containing velocities of “82 galaxies from 6 well-separated conic sections of an unfilled survey of the Corona Borealis region.” It was studied in Roeder (1990), which gives us a reference point.
For the curious reader, the source code of this notebook along with auxiliary scripts that are used for performing all the calculations presented below can be found on GitHub.
The empirical cumulative distribution function of the velocity is as follows:
Already here, it is apparent that the distribution is multimodal: there are two distinct regions, one to the left and one to the right, where the curve is flat, meaning there are no observations there. The proverbial histogram gives a confirmation:
It can be seen that there is a handful of galaxies moving relatively slowly and relatively fast compared to the majority located somewhere in the middle around twenty thousand kilometers per second. For completeness, kernel density estimation results in the following plot:
How many clusters of galaxies are there? What are their average velocities? How uncertain are these estimates? Our goal is to answer these questions by virtue of the Dirichlet process.
Now that the intention is to apply the presented theory in practice, we have to make all choices we have conveniently glanced over. Specifically, P 0 P 0 has to be chosen, and we shall use the following:
P 0(⋅)=Gaussian(⋅|μ 0,σ 2 0).(5)(5)P 0(⋅)=Gaussian(⋅|μ 0,σ 0 2).
In the above, Gaussian(⋅)Gaussian(⋅) refers to the probability measure of a Gaussian distribution with parameters μ 0 μ 0 and σ 0 σ 0. In addition to these two, there is one more: λ λ. We shall set μ 0 μ 0 and σ 0 σ 0 to 20 and 5, respectively—which correspond roughly to the mean and standard deviation of the data—and present results for different λ λ’s to investigate how the prior volume affects shrinkage toward the prior.
First, we do not condition on the data to get a better understanding of the prior itself, which corresponds to Equation 1. The following figure shows a single draw from four Dirichlet processes with different λ λ’s (the gray curves show the cumulative distribution function of the data as a reference):
It can be seen that the larger the prior volume, the smoother the curve. This is because larger λ λ’s “break” the stick into more pieces, allowing the normalized base measure to be extensively sampled, which, in the limit, converges to this very measure; see Equation 5.
Now, conditioning on the observed velocities of galaxies—that is, sampling as shown in Equation 2—we obtain the following draws from the posterior Dirichlet distributions with different λ λ’s:
When the prior volume is small, virtually no data points come from P 0 P 0; instead, they are mostly uniform draws from the observed data set, leading to a curve that is nearly indistinguishable from the one of the data (the top curve). As λ λ gets larger, the prior gets stronger, and the estimate gets shrunk toward it, up to a point where the observations appear to be entirely ignored (the bottom curve).
The above model has a serious limitation: it assumes a discrete probability distribution for the data-generating process, which can be seen in the prior and posterior given in Equation 1 and 2, respectively, and it is also apparent in the decomposition given in Equation 3. In some cases, it might be appropriate; however, there is arguably more situations where it is inadequate, including the running example.
Mixing prior #
Instead of using a Dirichlet process as a direct prior for the given data, it can be used as a prior for mixing distributions from a given family. The resulting posterior will then naturally inherit the properties of the family, such as continuity. The general structure is as follows:
x i|θ i∼P x(θ i),for i=1,…,n;θ i|P θ∼P θ,for i=1,…,n;and P θ∼Dirichlet Process(λ P 0).(6)(6)x i|θ i∼P x(θ i),for i=1,…,n;θ i|P θ∼P θ,for i=1,…,n;and P θ∼Dirichlet Process(λ P 0).
The i i th data point, x i x i, is distributed according to distribution P x P x with parameters θ i θ i. For instance, P x P x could refer to the Gaussian family with θ i=(μ i,σ i)θ i=(μ i,σ i) identifying a particular member of the family by its mean and standard deviation. Parameters {θ i}n i=1{θ i}i=1 n are unknown and distributed according to distribution P θ P θ. Distribution P θ P θ is not known either and gets a Dirichlet process prior with measure λ P 0 λ P 0.
It can be seen in Equation 6 that each data point can potentially have its own unique set of parameters. However, this is not what usually happens in practice. If λ λ is reasonably small, the vast majority of the stick—the one we explained how to break in the previous section—tends to be consumed by a small number of pieces. This makes many data points share the same parameters, which is akin to clustering. In fact, clustering is a prominent use case for the Dirichlet process.
Inference #
Unlike the previous model, there is no conjugacy in this case, and hence the posterior is not a Dirichlet process. There is, however, a simple Markov chain Monte Carlo sampling strategy based on the stick-breaking construction. It belongs to the class of Gibbs samplers and is as follows.
Similarly to Equation 3, we have the following decomposition:
P m(⋅)=∞∑i=1 p i P x(⋅|θ i)P m(⋅)=∑i=1∞p i P x(⋅|θ i)
where P m P m is the probability measure of the mixture. As before, the infinite decomposition has to be made finite to be usable in practice:
P m(⋅)=m∑i=1 p i P x(⋅|θ i).P m(⋅)=∑i=1 m p i P x(⋅|θ i).
Here, m m represents an upper limit on the number of mixture components. Each data point x i x i, for i=1,…,n i=1,…,n, is mapped to one of the m m components, which we denote by k i∈{1,…,m}k i∈{1,…,m}. In other words, k i k i takes values from 1 to m m and gives the index of the component of the i i th observation.
There are m+m×|θ|+n m+m×|θ|+n parameters to be inferred where |θ||θ| denotes the number of parameters of P x P x. These parameters are {p i}m i=1{p i}i=1 m, {θ i}m i=1{θ i}i=1 m, and {k i}n i=1{k i}i=1 n. As usual in Gibbs sampling, the parameters assume arbitrary but compatible initial values. The sampler has the following three steps.
First, given {p i}{p i}, {θ i}{θ i}, and {x i}{x i}, the mapping of the observations to the mixture components, {k i}{k i}, is updated as follows:
k i∼Categorical⎛⎝m,{p j P x(x i|θ j)∑m l=1 p l P x(x i|θ l)}m j=1⎞⎠,for i=1,…,n.k i∼Categorical(m,{p j P x(x i|θ j)∑l=1 m p l P x(x i|θ l)}j=1 m),for i=1,…,n.
That is, k i k i is a draw from a categorical distribution with m m categories whose unnormalized probabilities are given by p j P x(x i|θ j)p j P x(x i|θ j), for j=1,…,m j=1,…,m.
Second, given {k i}{k i}, the probabilities of the mixture components, {p i}{p i}, are updated using the stick-breaking construction described earlier. This time, however, the beta distribution for sampling {q i}{q i} in Equation 4 is replaced with the following:
q i∼Beta(1+n i,λ+m∑j=i+1 n j),for i=1,…,m,q i∼Beta(1+n i,λ+∑j=i+1 m n j),for i=1,…,m,
where
n i=n∑j=1 I{i}(k j),for i=1,…,m,n i=∑j=1 n I{i}(k j),for i=1,…,m,
is the number of data points that are currently allocated to component i i. Here, I A I A is the indicator function of a set A A. As before, in order for the p i p i’s to sum up to one, it is common to set q m:=1 q m:=1.
Third, given {k i}{k i} and {x i}{x i}, the parameters of the mixture components, {θ i}{θ i}, are updated. This is done by sampling from the posterior distribution of each component. In this case, the posterior is a prior of choice that is updated using the data points that are currently allocated to the corresponding component. To streamline this step, a conjugate prior for the data distribution, P x P x, is commonly utilized, which we shall illustrate shortly.
To recapitulate, a single draw from the posterior is obtained in a number of steps where parameters or groups of parameters are updated in turn, while treating the other parameters as known. This Gibbs procedure is very flexible. Other parameters can be inferred too, instead of setting them to fixed values. An important example is the concentration parameter, λ λ. This parameter controls the formation of clusters, and one might let the data decide what the value should be, in which case a step similar to the third one is added to the procedure to update λ λ. This will be also illustrated below.
Illustration #
We continue working with the galaxy data. For concreteness, consider the following choices:
θ i=(μ i,σ i),for i=1,…,n;P x(θ i)=Gaussian(μ i,σ 2 i),for i=1,…,n;and P 0(⋅)=Gaussian–Scaled-Inverse-χ 2(⋅|μ 0,κ 0,ν 0,σ 2 0).(7)(7)θ i=(μ i,σ i),for i=1,…,n;P x(θ i)=Gaussian(μ i,σ i 2),for i=1,…,n;and P 0(⋅)=Gaussian–Scaled-Inverse-χ 2(⋅|μ 0,κ 0,ν 0,σ 0 2).
In the above, Gaussian–Scaled-Inverse-χ 2(⋅)Gaussian–Scaled-Inverse-χ 2(⋅) refers to the probability measure of a bivariate distribution composed of a conditional Gaussian and an unconditional scaled inverse chi-squared distribution. Some intuition about this distribution can be built via the following decomposition:
μ i|σ 2 i∼Gaussian(μ 0,σ 2 i κ 0)and σ 2 i∼Scaled-Inverse-χ 2(ν 0,σ 2 0).(8)(8)μ i|σ i 2∼Gaussian(μ 0,σ i 2 κ 0)and σ i 2∼Scaled-Inverse-χ 2(ν 0,σ 0 2).
This prior is a conjugate prior for a Gaussian data distribution with unknown mean and variance, which we assume here. This means that the posterior is also a Gaussian–scaled-inverse-chi-squared distribution. Given a data set with n n observations x 1,…,x n x 1,…,x n, the four parameters of the prior are updated simultaneously (not sequentially) as follows:
μ 0:=κ 0 κ 0+n μ 0+n κ 0+n μ x,κ 0:=κ 0+n,ν 0:=ν 0+n,and σ 2 0:=1 ν 0+n(ν 0 σ 2 0+s s x+κ 0 n κ 0+n(μ x−μ 0)2)μ 0:=κ 0 κ 0+n μ 0+n κ 0+n μ x,κ 0:=κ 0+n,ν 0:=ν 0+n,and σ 0 2:=1 ν 0+n(ν 0 σ 0 2+s s x+κ 0 n κ 0+n(μ x−μ 0)2)
where μ x=∑n i=1 x i/n μ x=∑i=1 n x i/n and s s x=∑n i=1(x i−μ x)2 s s x=∑i=1 n(x i−μ x)2. It can be seen that κ 0 κ 0 and ν 0 ν 0 act as counters of the number of observations; μ 0 μ 0 is a weighted sum of two means; and ν 0 σ 2 0 ν 0 σ 0 2 is a sum of two sums of squares and a third term increasing the uncertainty due to the difference in the means. In the Gibbs sampler, each component (each cluster of galaxies) will have its own posterior based on the data points that are assigned to that component during each iteration of the process. Therefore, n n, μ x μ x, and s s x s s x will generally be different for different components and, moreover, will vary from iteration to iteration.
We set μ 0 μ 0 to 20, which is roughly the mean of the data, and ν 0 ν 0 to 3, which is the smallest integer that allows the scaled chi-squared distribution to have a finite expectation. The choice of κ 0 κ 0 and σ 0 σ 0 is more subtle. Recall Equation 8. What we would like from the prior is to allow for free formation of clusters in a region generously covering the support of the data. To this end, the uncertainty in the mean, μ i μ i, has to be high; however, it should not come from σ i σ i, since it would produce very diffuse clusters. We set κ 0 κ 0 to 0.01 to magnify the variance of μ i μ i without affecting σ i σ i, and σ 0 σ 0 to 1 to keep clusters compact.
Now, let us take a look at what the above choices entail. The following figure illustrates the prior for the mean of a component:
The negative part is unrealistic for velocity; however, it is rarely a problem in practice. What is important is that there is a generous coverage of the plausible values. The following figure shows the prior for the standard deviation of a component:
The bulk is below the standard deviation of the data; however, this is by choice: we expect more than one cluster of galaxies with similar velocities.
As mentioned earlier, we intend to include λ λ in the inference. First, we put the following prior:
λ∼Gamma(α 0,β 0).(9)(9)λ∼Gamma(α 0,β 0).
Note this is the rate parameterization of the Gamma family. Conditionally, this is a conjugate prior with the following update rule for the two parameters:
α 0:=α 0+m−1 and β 0:=β 0−m−1∑i=1 ln(1−q i)α 0:=α 0+m−1 and β 0:=β 0−∑i=1 m−1 ln(1−q i)
where {q i}{q i} come from the stick-breaking construction. This is a fourth step in the Gibbs sampler. We set α 0 α 0 and β 0 β 0 to 2 and 0.1, respectively, which entails the following prior assumption about λ λ:
The parameter is allowed to vary freely from small to large values, as desired.
Having chosen all priors and their hyperparameters, we are ready to investigate the behavior of the entire model; see Equations 6, 7, and 9. In what follows, we shall limit the number of mixture components to 25; that is, m=25 m=25. Furthermore, we shall perform 10000 Gibbs iterations and discard the first 1000 as a warm-up period. As before, we start without conditioning on the data to observe draws from the prior itself. The following figure shows two sample draws:
It can be seen that clusters of galaxies can appear anywhere in the region of interest and can be of various sizes. We conclude that the prior is adequate. When taking the observed velocities into account, we obtain a full posterior distribution in the form of 9000 draws. The following shows two random draws:
Indeed, mixture components have started to appear in the regions where there are observations.
Before we proceed to the final summary of results, it is prudent to inspect sample chains for a few parameters in order to ensure there are not problems with convergence to the stationary distribution. The following shows the number of occupied components among the 25 permitted:
The chain fluctuates around a fixed level without any prominent pattern, as it should. One can plot the actual marginal posterior distribution for the number of components; however, it is already clear that the distribution of the number of clusters of galaxies is mostly between 5 and 15 with a median of 10.
As for the concentration parameter, λ λ, the chain is as follows:
There is occasional turbulence, but overall, the behavior is relatively adequate
Let us now take a look at the posterior distributions of the first 10 components (note the different scales on the vertical axes):
The components clearly exchange roles, as suggested by the multimodal nature of the distributions. Components 1 and 2 are the most stable ones, which can be seen by the high-density regions at around 20 and 23 (times 10 6 10 6 m/s), respectively. The distributions of other components are more spread out (again, pay attention to the scale). However, with velocities 20 and 23 out of the way, they allow one to see more clearly smaller clusters. More specifically, they suggest clustering at around 10, 16, and 33.
Lastly, we summarize the inference using the following figure where the median distribution and a 95% uncertainty band—composed of distributions at the 0.025 and 0.975 quantiles—are plotted:
The aforementioned five components are visible to the naked eye. The median curve matches well the findings in Roeder (1990). Judging by the width of the uncertainty band, there is a lot of plausible alternatives, and it is important to communicate this uncertainty to those who base decisions on the inference. The ability to quantify uncertainty with such ease is a prominent advantage of Bayesian inference.
Conclusion #
In this article, the family of Dirichlet processes has been presented in the context of Bayesian inference. More specifically, it has been shown how a Dirichlet process can be utilized as a prior for an unknown discrete distribution and as a prior for mixing distributions from a given family. In both cases, it has been illustrated how to perform inference via a finite approximation and the stick-breaking construction.
Clearly, the overall procedure is more complicated than counting observations falling in a number of fixed bins, which is what a histogram does, or placing kernels all over the place, which is what a kernel density estimator does. However, “anything in life worth having is worth working for.” The advantages of the Bayesian approach include the ability to incorporate prior knowledge, which is crucial in situations with little data, and the ability to propagate and quantify uncertainty, which is a must.
Recall that the source code of this notebook along with auxiliary scripts that were used for performing the calculations presented above can be found on GitHub. Any feedback is welcome!
Acknowledgments #
I would like to thank Mattias Villani for the insightful and informative graduate course in Bayesian statistics titled “Advanced Bayesian learning,” which was the inspiration behind writing this article, and for his guidance regarding the implementation.
Follow-up #
In June 2025, Julian Stander drew my attention to a typo in the implementation of the sampling procedure for the posterior distribution of λ λ, which I am grateful for.
References #
Andrew Gelman et al., Bayesian Data Analysis, Chapman and Hall/CRC, 2014.
Kathryn Roeder, “Density estimation with confidence sets exemplified by superclusters and voids in galaxies,” Journal of the American Statistical Association, 1990.
Rick Durrett, Probability: Theory and Examples, Cambridge University Press, 2010.
Ivan Ukhov
ivan.ukhov@gmail.com
Solving problems that a software engineer might encounter in practice—or invent to sharpen their skills in leisure time |
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Finance: Depreciation(GCSE)
Level: GCSE Board: AQA, Edexcel, OCR, IB
Last updated 22 Mar 2021
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A fixed asset reduces in value over its useful life due to wear and tear and (when it is no longer useful) obsolescence. Depreciation is the tool used by accountants to record the reduction in the original value of an asset.
Depreciation is charged every year of a fixed asset's useful life to the profit and loss account. In the balance sheet the original cost of the fixed asset is reduced by the amount of depreciation.
There are two main methods of depreciation:
Straight line depreciation – this is where the same amount is charged every year using the following formula to calculate it:
Original Cost of the Fixed Asset / Useful Life of the Asset
For example; a machine bought for £20,000 has a useful life of ten years. Management decide to charge depreciation on a straight line basis. So the annual depreciation cost is £20,000 / 10 = £2,000
Reducing balance depreciation – the same percentage of an asset's value is taken off every year, e.g. 20%.
Most businesses use straight line depreciation, but it is possible to argue that reducing method is better because it reflects the fact that most assets lose most of their value in the first years of use.
Depreciation appears in the profit and loss account under expenses – it reduces the profit for that year because some of the asset was used up in that time period.
It appears in the balance sheet by reducing the value of the fixed assets. This means that the balance sheet reflects a true and fair value of the assets.
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5731 | https://prepp.in/question/directions-select-the-most-appropriate-antonym-of-6448e66e267130feb1161ab5 | Directions: Select the most appropriate ANTONYM of the given word.Censure
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English
Synonyms or Antonyms
directions select the most appropriate antonym of
Question
English
Directions: Select the most appropriate ANTONYM of the given word.
Censure
Contempt
Applause
Blunder
Fault
Solution
The correct answer is
Applause
Understanding the Question: Antonym of Censure
The question asks us to find the most appropriate antonym for the word "Censure". An antonym is a word that has the opposite meaning of another word. To answer this, we first need to understand the meaning of "Censure".
The word "Censure" means to express severe disapproval of someone or something, typically in a formal statement. It implies strong criticism or condemnation.
Analyzing the Options for Antonym of Censure
Let's examine each option provided and determine its meaning:
Contempt: This means the feeling that a person or a thing is worthless or beneath consideration. While related to disapproval, it's more about a feeling of scorn or worthlessness rather than a direct opposite of approval or praise. It's closer in meaning to "Censure" (both negative) than being its antonym.
Applause: This means approval or praise expressed by the clapping of hands. It signifies positive feedback, commendation, or approval. This is the direct opposite of expressing severe disapproval or criticism (Censure).
Blunder: This means a stupid or careless mistake. It refers to an error or fault in action or judgment. This word is unrelated to expressing approval or disapproval.
Fault: This means a defect or imperfection; also, responsibility for an accident or misfortune. Like "Blunder", it refers to a mistake or defect and is not related to expressing approval or disapproval.
Identifying the Appropriate Antonym
Comparing the meanings, "Censure" is about strong disapproval and criticism, while "Applause" is about expressing approval and praise. Therefore, "Applause" is the most appropriate antonym for "Censure".
| Word | Meaning | Relationship to Censure |
---
| Censure | Severe disapproval or criticism | The word in question |
| Contempt | Feeling of worthlessness; scorn | Related negativity, but not a direct opposite |
| Applause | Approval or praise (by clapping) | Direct opposite of disapproval/criticism |
| Blunder | A mistake | Unrelated to approval/disapproval |
| Fault | A defect or mistake | Unrelated to approval/disapproval |
Conclusion
Based on the analysis of the meanings of the words, "Applause" is the most fitting antonym for "Censure".
Revision Table: Antonyms and Synonyms
| Word | Synonyms | Antonyms |
---
| Censure | Criticism, condemnation, disapproval, rebuke, reprimand | Applause, praise, commendation, approval, laudation |
| Contempt | Scorn, disdain, disrespect | Respect, admiration, regard |
| Applause | Praise, commendation, approval, acclamation | Censure, criticism, disapproval, condemnation |
| Blunder | Mistake, error, gaffe | Success, achievement (less direct antonyms) |
| Fault | Defect, flaw, mistake, error | Perfection, merit, virtue |
Additional Information: Using Censure and Applause
Here are some examples of how "Censure" and "Applause" might be used:
The committee voted to censure the member for his unethical behavior. (Strong disapproval)
The performance received widespread applause from the audience. (Strong approval/praise)
Instead of censure, the team needs constructive criticism.
He earned the applause of his colleagues for his dedication.
Understanding the context and nuances of these words helps in identifying their antonyms correctly.
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Important Questions from Synonyms or Antonyms
Select the most appropriate ANTONYM of the given word.
Contentment English View Answer
2. Select the most appropriate synonym of the given word.
Apprehension English View Answer
3. Identify the ANTONYM of the following word in the given sentence.
Divulge
The culprit surprised everyone by opting to conceal the secret execution of his plans even after getting beaten by the police. English View Answer
4. Select the most appropriate ANTONYM of the given word.
Adipose English View Answer
5. Which of the following options is the closest in meaning to the word underlined in the sentence below?
In a democracy, everybody has the freedom to disagree with the government. English View Answer
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5732 | https://www.teacherspayteachers.com/Product/4NF2-Comparing-Fractions-Word-Problem-Practice-Worksheets-4th-Grade-9448139 | 4.NF.2 Comparing Fractions Word Problem Practice Worksheets | 4th Grade
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5733 | https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_20?srsltid=AfmBOooq5KYwtkucGW1mNcCBh2lsusi8Qc3rdVGMFczB0VJu3Rwu66yx | Art of Problem Solving
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2016 AMC 12B Problems/Problem 20
Contents
1 Problem
2 Solution 1
3 Solution 2
4 Solution 3 (extremely risky—only try if you are running out of time)
5 Video Solution by CanadaMath (Problems 11-20)
6 See Also
Problem
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won games and lost games; there were no ties. How many sets of three teams were there in which beat , beat , and beat
Solution 1
We use complementary counting. First, because each team played other teams, there are teams total. All sets that do not have beat , beat , and beat have one team that beats both the other teams. Thus we must count the number of sets of three teams such that one team beats the two other teams and subtract that number from the total number of ways to choose three teams.
There are ways to choose the team that beat the two other teams, and ways to choose two teams that the first team both beat. This is sets. There are sets of three teams total. Subtracting, we obtain , thus is our answer.
Solution 2
As above, note that there are 21 teams, and call them A, B, C, ... T, U. WLOG, assume that A beat teams B-L and lost to teams M-U. We will count the number of sets satisfying the “cycle-win” condition—e.g. here, A beats a team in X which beats a team in Y which beats A. The first and third part of the condition are already met by our wlog, so we just need to count of number of ways the second condition is true (a team in X beats a team in Y). These are the number of cycle-wins that include A, then multiply by 21 (for each team) and divide by 3 (since every set will be counted by each of the 3 teams that are a part of that set).
To do this, let X and Y. Since a total of losses total were suffered by teams in Y and losses were suffered by teams in Y from teams in Y, we have losses suffered by teams in Y from teams in X. Hence, for each of these losses, there is exactly one set of three teams that includes A that satisfies the problem conditions. Thus, the answer is .
the number of ways that two teams in Y play each other — each face-off guarantees 1 loss (and 1 win)
Solution 3 (extremely risky—only try if you are running out of time)
Note that there are teams total and ways to pick The possible arrangements are one team beats the other two or they each win/lose equally (we want the second case). Approximately of all the arrangements satisfy the second case, and which is by far the closest to
Video Solution by CanadaMath (Problems 11-20)
Fast forward to 42:52 for problem 20
~THEMATHCANADIAN
See Also
2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 19Followed by
Problem 21
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25
All AMC 12 Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
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5734 | https://blog.demofox.org/2024/10/15/scaling-points-in-a-specific-direction/ | Published Time: 2024-10-16T04:33:25+00:00
Scaling Points In a Specific Direction « The blog at the bottom of the sea
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Scaling Points In a Specific Direction
demofox2October 15, 20241
In this post I’ll be showing two different ways to scale points along a specific direction (a vector). There isn’t anything novel here, but it gives an example of the type of math encountered in game development, and how you might approach a solution. The first method involves matrices, and the second involves vector projections.
The points in the red circle below are scaled along the dark green vector, with the magnitude of the bright green vector.
The C++ code that made this diagram can be found at It implements the vector projection method.
This problem came up for me when implementing a Petzval lens effect ( in a bokeh depth of field ( post processing rendering technique. The setup is that I take point samples in the shape of the camera aperture for bokeh and depth of field, and I needed to stretch this sampled shape in a specific direction and magnitude, based on where the pixel is on the screen.
In the end, I just needed to scale points in a specific direction, by a specific amount. If we can do this operation to 1 point, we can do it to N points, so we’ll focus on doing this to a single point.
Method 1 – Matrices
Using matrices to scale a point along a specific direction involves three steps:
Rotate the point so that the direction we want to scale is aligned with the X axis.
Multiply the x axis value by the scaling amount.
Unrotate the point back to the original orientation.
Our vectors are going to be row vectors. For a deep dive on other matrix and vector conventions and reasons to choose one way or another, read this great post by Jasper St. Pierre:
Let’s say we want to scale a point by 4 along the vector [3,1].
First we normalize that vector to [3/sqrt(10), 1/sqrt(10)]. Then we need to get the rotation matrix that will transform [3/sqrt(10), 1/sqrt(10)] to [1,0]. We can do that by making a matrix where the first column is where we want the x axis to point, and the second column to be where we want the y axis to point. For the x axis, we use the normalized vector we already have. For the y axis, we just need a perpendicular vector, which we can get by swapping the x and y components of the vector, and negating one to get [-1/sqrt(10), 3/sqrt(10)]. This operation of swapping x and y and negating one is sometimes called the “2D Cross Product” even though it isn’t really a cross product, and there is no cross product in 2D. That gives us this rotation matrix:
Next we need to make the scaling matrix, which we get by multiplying the vector [4,1] by the identity matrix:
Lastly we need to calculate the unrotation matrix. We need a matrix that rotates by the negative amount of R. We need the inverse matrix of R. The inverse of a rotation matrix is just the transpose matrix, so we can transpose R to make it. Another way to think about it is when we made the matrix before with the first column being the x axis and the second column being the y axis, we are now going to make the first row be the x axis, the second row be the y axis. Rows instead of columns. Whichever explanation makes most sense to you, we end up with this:
Now that we have all the transformations, we can calculate R S R’ to get a final matrix that does the transformation we want. I’ll do it in 2 steps in case that helps you follow along, to make sure you get the same numbers.
That is our matrix which scales a point along the vector [3,1], with a magnitude of 4. Let’s put the vector [4,5] through this transformation by multiplying it by the matrix.
For those who are counting instructions, processing a point using this process is 2 multiplies and two adds, to do that 2d vector / matrix product.
Creating the matrix took 16 multiplies and 8 adds (two matrix / matrix multiplies aka eight 2d dot products), but usually, you calculate a matrix like this once and re-use it for many points, which makes the matrix creation basically zero as a percentage of the total amount of calculations done, when amortized across all the points.
Method 2 – Vector Projection
I’m a fan of vector projection techniques. There is a certain intuitiveness in them that is missing from matrix operations, I find.
Using vector projection to scale a point along a specific direction involves these three steps:
Project the point onto the scaling vector and multiply that by the scaling amount.
Project the point onto the perpendicular vector.
Add the scaling vector projection times the scaling vector to the perpendicular vector projection times the perpendicular vector.
We will use the same values from the last section, so we want to scale a point by 4 along the vector [3,1], which we normalize to [3/sqrt(10), 1/sqrt(10)]. We will put the point [4, 5] through this process.
Step 1 is to project our point onto the scaling vector. We do that by doing a dot product between our normalized vector [3/sqrt(10), 1/sqrt(10)], and our point [4, 5]. That gives us the value 17/sqrt(10). We then multiply that by the scaling amount 4 to get 68/sqrt(10).
Step 2 is to project our point onto the perpendicular vector. We can once again use the “2D Cross Product” to get the perpendicular vector. We just flip the x and y component and negate one, to get the vector perpendicular to the scaling vector: [-1/sqrt(10), 3/sqrt(10)]. We can dot product that with our point [4, 5] to get: 11/sqrt(10).
Step 3 is to multiply our projections by the vectors we projected onto, and add the results together. Our scaling vector contribution is 68/sqrt(10) [3/sqrt(10), 1/sqrt(10)] or [204/10, 68/10]. Our perpendicular vector contribution is 11/sqrt(10)[-1/sqrt(10), 3/sqrt(10)]or[-11/10, 33/10].
When we add the two values together, we get [193/10, 101/10] or [19.3, 10.1].
That result matches what we got with the matrix operations!
As far as instruction counts, we did two dot products for step 1 and 2 which is 4 multiplies and 2 adds total. Step 3 is 4 multiplies. Step 4 is 2 multiplies and 2 adds. This is a total of 10 multiplies and 4 adds which is a lot more than the matrix version, which was just 2 multiplies and 2 adds.
If you optimized this process to do fewer operations by combining work where you could, you’d eventually end up at the same operations done in the matrix math. Algebra is fun that way.
Higher Dimensions?
Using the matrix method in higher dimensions, making the scaling vector is easy, and making the unrotation matrix is still just taking the transpose of the rotation matrix. It’s more difficult making the rotation matrix though.
In 3D, the scaling direction will be a 3D vector, and you need to come up with two other vectors that are perpendicular to that scaling direction. One way to do this could be to take any vector which is different from the scaling vector, and cross product that with the scaling vector. That will give you a vector perpendicular to both, and you can take that as your second vector. To get the third vector, cross product that vector with the scaling vector. You will then have 3 perpendicular vectors, and an orthonormal basis that you can use to fill out your rotation matrix. The first column is the scaling vector, the second column is the second vector found, and the third column is the third column found.
The cross product only exists in the 3rd and 7th dimension though, so if you are working in a different dimension, or if you don’t want to use the cross product for some reason, another way you can make an orthonormal basis is by using the Gram-Schmidt process. There’s a great video on it here:
For the vector project method, you also need the orthonormal basis vectors to do all the vector projections, before you scale the x axis, and then re-combine the projections, so it boils down to the same issues as the matrix method.
From Readers
Nick Appleton ( says:
shameless plug of my last blog post (regarding higher order rotation matrices)
This has methods for generating a high order rotation that moves a point to a particular axis. There is rarely a need need for a Gram Schmidt process and computing the matrix can be made quite cheap 🙂
I think the most efficient way to find a rotation matrix that takes a unit vector A and moves it to another unit vector B (in any dimension) is to find the find the reflection matrix that maps A to C (where C=B with a single component negated – doesn’t matter which one) and then flip the sign of the corresponding row of the matrix to turn it into a rotation.
Finding a reflection matrix that does this requires only a single division in an efficient implementation for any dimension.
Mastodon link:
Andrew Gang ( says:
if your use case doesn’t need accuracy for scaling amounts near zero, method 2 has a variant that saves you from having to find perpendicular vectors: point + (scaling amount – 1) dot(normalized scaling vector, point) normalized scaling vector.
Mastodon link:
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One comment
Logitrainsays:| October 21, 2024 at 3:23 am This post on scaling points in a specific direction is incredibly insightful! I love how you break down the concept and make it applicable to real-world scenarios. Understanding this can really enhance our approach to transformations in graphics and data. For those looking to deepen their skills in related areas, I’d suggest checking out a software testing automation course. It can provide essential knowledge for ensuring the quality of projects that involve such calculations. Thanks for sharing your expertise!
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5735 | https://journals.sagepub.com/doi/abs/10.1177/15291006241287726 | Skip to main content
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What the Science of Learning Teaches Us About Arithmetic Fluency
Nicole M. McNeil nmcneil@nd.edu, Nancy C. Jordan […], Alexandria A. Viegut and Daniel Ansari all authors and affiliations
Volume 26, Issue 1
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Abstract
High-quality mathematics education not only improves life outcomes for individuals but also drives innovation and progress across society. But what exactly constitutes high-quality mathematics education? In this article, we contribute to this discussion by focusing on arithmetic fluency. The debate over how best to teach arithmetic has been long and fierce. Should we emphasize memorization techniques such as flashcards and timed drills or promote “thinking strategies” via play and authentic problem solving? Too often, recommendations for a “balanced” approach lack the depth and specificity needed to effectively guide educators or inform public understanding. Here, we draw on developmental cognitive science, particularly Sfard’s process–object duality and Karmiloff-Smith’s implicit–explicit knowledge continuum, to present memorization and thinking strategies not as opposing methods but as complementary forces. This framework enables us to offer specific recommendations for fostering arithmetic fluency based on the science of learning. We define arithmetic fluency, provide evidence on its importance, describe the cognitive structures and processes supporting it, and share evidence-based guidance for promoting it. Our recommendations include progress monitoring for early numeracy, providing explicit instruction to teach important strategies and concepts, implementing well-structured retrieval practice, introducing time-limited practice only after students demonstrate accuracy, and allocating sufficient time for discussion and cognitive reflection. By blending theory, evidence, and practical advice, we equip educators and policymakers with the knowledge needed to ensure all children have access to the opportunities needed to achieve arithmetic fluency.
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Action Editor: Nora S. Newcombe
Editor: Nora S. Newcombe
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Psychological Science in the Public Interest
Volume 26, Issue 1
Pages: 10 - 57
Article first published online: April 29, 2025
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Keywords
mathematics learning
memory
cognitive development
education
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PubMed: 40297988
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Nicole M. McNeil
Department of Psychology and Institute for Educational Initiatives, University of Notre Dame
nmcneil@nd.edu
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Nancy C. Jordan
School of Education, University of Delaware
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Alexandria A. Viegut
Department of Psychology, University of Wisconsin–Eau Claire
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Daniel Ansari
Department of Psychology and Faculty of Education, Western University
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Nicole M. McNeil, Department of Psychology and Institute for Educational Initiatives, University of Notre Dame Email: nmcneil@nd.edu
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Nine-year-olds in England sit timed multiplication test – but using times tables is about more than quick recall
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2. The Importance of Early Mathematical Foundations, Sensemaking, and the Home Environment for Children’s Development of Arithmetic Fluency: Commentary on McNeil et al. (2025)
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5736 | https://www.youtube.com/watch?v=bQwacwGoB0w | Learn how to identify transformations and graph natural logarithmic function
Brian McLogan
1590000 subscribers
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Posted: 17 Sep 2016
👉 Learn all about graphing natural logarithmic functions. A logarithmic function is a function with logarithms in them. A natural logarithmic function (ln function) is a logarithmic function to the base of e. The graph of the parent function of a logarithmic function usually takes its domain from the positive x-axis.
To graph a logarithmic function, it is usually useful to first graph the parent function (without transformations). This can be done by choosing 2-3 points from the function and plotting them on the x-y coordinate axis to see the nature of the parent function's graph. After graphing the parent function, we then apply the given transformations to obtain the required graph.
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2 comments
Transcript:
so in this example ladies gentlemen we have y equals negative 2 ln of x plus 3. the first thing i want to do is look at my function to identify the transformations right and you can just work left and right the main important thing is we want to identify is my value c that value that i'm using which c represents any number am i applying that transformation inside or outside the function so just even reading this from left to right without eating yogurt i can see that the first thing i'm doing is multiplying my function by a negative right now is that negative inside my function or outside my function outside so guess what i'm just going to write reflect x axis that's one transformation now i look at the next thing i'm multiplying my oh actually sorry first of all let's look at what is the parent function i know i took it down it would should be up like the other ones but the parent function here is ln of x that's the parent function so now i have a 2 am i multiplying that 2 outside or inside the function outside and 2 is larger than 1 so therefore it's a vertical stretch okay then i keep on moving over i see that now x plus 3 i'm adding a 3 and i notice that i'm adding the 3 inside the parentheses so therefore that's a horizontal transformation but since x since i'm adding the 3 that's the same thing as minusing the negative so therefore it's shifting it to the left so i have three transformations that are going on would everybody agree with me yes okay now to graph this you guys have notes on what the ln function looks like you have notes on what the ln function looks like it crosses at 0 comma 1 and it looks something like that please note there's also a horizontal asymptote okay oh what the heck am i doing sorry yes that is e to the x thank you for checking your notes ln of x looks like this crosses that one looks like that has a vertical asymptote thank you right and you don't have to prefer i'm just asking you for sketching the graph we're not looking for perfection here yes because the absolute value i'm multiplying a number outside the function right the absolute value of that number that i'm multiplying is greater than one so it's a vertical stretch okay so now yes again it's i'm not going to be getting particular the main important thing is you can identify it as far as the graphing portion i'm going to show you on graphing calculator what it looks like but i'm not going to expect you like i'm not going to mark you down saying your stretch is not good enough okay it's more more importantly that you can identify stretch would it look like um but for instance like on a parabola you should be for instance you should be familiar with the stretch like the stretch makes it basically skinnier like stretching it up right so the logarithms and x logarithms kind of get a little bit trickier because they still kind of follow along that same path but that's exactly what's happening so um so anyways we're going to so we got to reflect the x-axis so you guys can see now my graph is being reflected of the x-axis correct i have a vertical stretch so i'm kind of stretching it out um and then i'm going three into left so i'm going to do the three units to the left first so if i go three into the left one two three so my x-intercept instead of it being at one comma zero is now at negative 2 0 and my asymptote which was at 0 is now right here so my graph would look something like that yes why is it negative okay now |
5737 | https://arxiv.org/abs/1302.5366 | [1302.5366] Testing Uniformity of Stationary Distribution
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arXiv:1302.5366 (cs)
[Submitted on 21 Feb 2013 (v1), last revised 10 Mar 2016 (this version, v3)]
Title:Testing Uniformity of Stationary Distribution
Authors:Sourav Chakraborty, Akshay Kamath, Rameshwar Pratap
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Abstract:A random walk on a directed graph gives a Markov chain on the vertices of the graph. An important question that arises often in the context of Markov chain is whether the uniform distribution on the vertices of the graph is a stationary distribution of the Markov chain. Stationary distribution of a Markov chain is a global property of the graph. In this paper, we prove that for a regular directed graph whether the uniform distribution on the vertices of the graph is a stationary distribution, depends on a local property of the graph, namely if (u,v) is an directed edge then outdegree(u) is equal to indegree(v).
This result also has an application to the problem of testing whether a given distribution is uniform or "far" from being uniform. This is a well studied problem in property testing and statistics. If the distribution is the stationary distribution of the lazy random walk on a directed graph and the graph is given as an input, then how many bits of the input graph do one need to query in order to decide whether the distribution is uniform or "far" from it? This is a problem of graph property testing and we consider this problem in the orientation model (introduced by Halevy et al.). We reduce this problem to test (in the orientation model) whether a directed graph is Eulerian. And using result of Fischer et al. on query complexity of testing (in the orientation model) whether a graph is Eulerian, we obtain bounds on the query complexity for testing whether the stationary distribution is uniform.
Subjects:Data Structures and Algorithms (cs.DS); Computational Complexity (cs.CC)
Cite as:arXiv:1302.5366 [cs.DS]
(or arXiv:1302.5366v3 [cs.DS] for this version)
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From: Rameshwar Pratap [view email]
[v1] Thu, 21 Feb 2013 18:26:40 UTC (49 KB)
[v2] Thu, 5 Sep 2013 10:05:21 UTC (49 KB)
[v3] Thu, 10 Mar 2016 17:00:22 UTC (60 KB)
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5738 | https://www.youtube.com/watch?v=HFZaXuJg77w | Consecutive Powers of 2 & 3
Dr Barker
27200 subscribers
261 likes
Description
6018 views
Posted: 21 Apr 2023
We look for all powers of 2 and 3 which are only 1 apart, such as for example 3 & 4, or 8 & 9. Our approach involves a neat application of modular arithmetic.
00:00 Intro
00:39 Posing the problem
01:24 Using modular arithmetic
02:01 Mod 4
04:37 Mod 8
06:52 Factorisation argument
10:17 Conclusion
17 comments
Transcript:
Intro okay so we're going to look at powers of 2 and 3. so now there are some powers of 2 and 3 which are very close to each other for example 2 cubed is only one away from three squared and two squared is only one away from three to the power of one and then two and three both to the power of one they're both within one of each other then also if we allow three to the power of zero one this is also just one away from two to the power of one so there are quite a few examples of powers of two and three that are one away from each other the only way of making them equal to each other would be taking them both to the power of zero so they're equal to one because otherwise your powers of two are even or as your powers of three are all odd so I'm Posing the problem interested in trying to find more examples of powers of two and three that are only one apart if you look at some larger powers of two and three they seem to be much more spread out so we can form this as an equation that we'll try and solve so the modulus of three to the N minus two to the power of M is equal to one where n and M we want to be non-negative integers so really with a modulus equation we can write this as two different equations one is where our power of two is bigger than our power of three and there'll be another case where our power of three is bigger than our power of two so we're essentially looking for positive or non-negative integer solutions to each of these two equations Using modular arithmetic and to get started on solving this equation there's actually quite a neat Insight using modular arithmetic so if you were to consider all of your powers of 2 let's say we consider them all modulo 4. so what's their remainder when you divide by four well actually all of your powers of two are multiples of four except for one and two so everything from four and above is a multiple of four so it'd be equivalent to zero modulo four if you wanted to look at all of your powers of two modulo 8 then all of them would have to be equivalent to zero modulo eight except for one two and four so other than a few special cases we can say that 2 to the m has to be equivalent to zero modulo four so we'll Mod 4 explore this now by looking at three to the N plus one and three to the N minus 1 mod 4 and see if this matches up with our powers of 2 which have to be equivalent to zero mod four so three to the N mod 4 when n is zero it's just one when n is one it's three to the power of one is three when n is two three squared is nine but nine mod four gets us back down to one then we multiply by 3 again 27 mod 4 is equivalent to 3 and the pattern continues like this one three one three so this tells us then just breaking it up into smaller steps 3 to the N plus one goes two zero two zero repeating like this whereas three to the N minus one goes zero to zero two and so on so now we can use this Insight that let's say we've got 2 to the m is equivalent to two modulo four well we know that this holds if and only if the only possibility of getting a remainder of 2 when we divide by four is if two to the m is actually equal to two so here M has to be equal to one so now this is really useful because all of these cases here where we get the twos these only work if 3 to the N plus one or three to the N minus 1 is actually equal to two so we can make this work here and here but all of these other cases get ruled out so you can see for three to the N plus one we can rule out all of the remaining even powers of N and here we can rule out all the remaining odd powers of n other than these special cases so let's just write then 3 to the N plus 1 where n is even the only possible solution here is this one where three to the 0 plus 1 is equal to 2 to the power of one so this is m is equal to one and N our power of three is equal to zero and that's the only possible case so here we've covered a quarter of all our possibilities three to the N plus one and N even if we had three to the N minus 1 was equal to our power of two then the only possibility where n is odd now we rule out all of our remaining odd Powers it's the only possibility that works here is three to the one minus one is equal to 2 to the one corresponding to this case here so this is m is one again but now n equals one so we've covered half of all of our possibilities here 3 to the N plus one three to the N minus one where n is even and where n is odd respectively so now we'll consider things modulo eight and see if we can rule out some more possibilities Mod 8 now powers of three modulo 8 actually look very similar to how they looked modulo four so three to the zero is just one three to the one is three three squared is nine which is equivalent to one modulo eight then we multiply by three again 27 is equivalent to three modular eight and it repeats just like this so there are three to the N plus one goes two four two four repeating modulo eight whereas three to the N minus one goes zero two zero two just like before modulo eight so now we're interested in three to the N plus one we've ruled out all the cases where n is even so we only care about the N odd cases so here you get four four four so all of our three to the N plus one has to be equivalent to four modulo eight and remember the whole point is that three to the N plus one is equal to two to the m so two to the m this is equivalent to four mod eight of course this is true if and only if two to the m is actually equal to four because any larger power of two is just going to be a multiple of eight so this is if and only if two to the m equals four i m is equal to two so this tells us then the other than one very special case here all of our larger odd powers of n three to the N plus one isn't going to be equal to two to the power of M for any integer M so here it works when n is one so we get for 3 to the N plus 1 where n is now odd the only possible case is where n is one so three to the one plus one is equal to 2 squared so here you have m is equal to 2 and N is equal to one for this special case but this is the only possibility then for three to the N plus one being equal to 2 to the m so all we've left now with because we've ruled out the case where n is odd or n even all we're left now is the case where n is even for our second equation but unfortunately for our second equation when n is even we still get zero zero zero so three to the N minus one nothing's ruled out here it's still a multiple of eight unfortunately we could try higher powers of two but this doesn't seem to yield anything for other values but there's a Factorisation argument nice trick we can use here because we've got n is even so 3 to the N minus 1 where n is even we can write as three to the 2 times K minus one and we can use the difference of two squares identity to write this is three to the K plus 1 times 3 to the K minus one so then we'll use this factorization to try and rule out some more cases now so we've got two factors of 2 to the power of n and the next thing to notice is that 2 to the power of M any power of two all of its factors have to be themselves powers of two or perhaps they're just equal to one you can think of this in terms of the prime factorization of two to the m this is just made up of twos so any factor of two to the m has to itself just be made up of twos so you couldn't get something like three five six going into two to the power of M so this tells us then that both 3K plus 1 and 3K minus 1 actually themselves have to be powers of two or potentially equal to one and this is really useful now because 3 to the K plus 1 3 to the K minus one we can apply our argument modulo four to each of these because they are effectively both going to be powers of two unless they're equal to one so we know there are powers of two modulo four are essentially all equivalent to zero they're all multiples of four with the two exceptions of one and two so this tells us then that three to the K plus one and three to the K minus one if we want to have loads of different possible solutions we'd want both of these to be multiples of four but you can see that this isn't going to be possible because there's actually only a difference of two between these two numbers and your multiples of four they have to be either equal or differ by a multiple of four you can't have two multiples of four that are two apart so this tells us then that at least one of these or exactly one of these is going to be not a multiple of four so we can say that 3 to the K plus 1 is equal to 1 or 2 or we've got the possibility that 3 to the K minus 1 is equal to one or two so now 3 to the K plus 1 and 3 to the K minus one you can easily see there that they're actually both going to be an even number because it's an odd plus an odd so we can actually rule out the One cases so it's either 3 to the K plus 1 equals two or three to the K minus 1 equals two so in this first case we get a solution where K is zero and in the second case we get a solution where K is one and we can actually rule out any larger values of K because we need to have one of these equal to one or two so we only are only limited to very small values of K and don't forget we're actually interested in 3 to the N minus one being equal to our power of two so remember n is equal to 2K so here n would be zero and here n would be two so our three to the N minus 1 because becomes just three to the zero minus one which is equal to two to the one so this is actually just a repeat of our m equals one n equals zero solution from before so unfortunately we don't get anything new in this first case but in the second case where K was 1 so n is two our three to the N minus one becomes three squared minus one nine minus one which is eight or two to the power of three so we get a new solution here m is 3 n is two so then we've got four Conclusion Solutions in total which are actually the four that we had at the beginning and we've already ruled out this three to the N plus one equals two to the m in both n even and odd cases and we've already ruled out three to the N minus one is our power of two when n was odd and we've seen when n is even for the second equation we can only work with very small values of K because we need to have one of these needs to be not a multiple of four so we only get the one extra solution here here so we really have covered all the different possibilities so unfortunately there aren't any more values for powers of two and powers of three that are one apart other than the four we had in the beginning |
5739 | https://indianexpress.com/article/trending/top-10-listing/top-10-most-populated-countries-world-population-day-2024-9443779/ | skip to content
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This is an archive article published on July 10, 2024
World Population Day 2024: Top 10 most populated countries in the world—China or India, who ranks first?
The UN predicts that the world's population will continue to rise, reaching 9.7 billion in 2050 and potentially peaking at nearly 10.4 billion in the mid-2080s, representing an increase of almost 2 billion individuals within the next 30 years.
Written by Cherry Gupta July 10, 2024 11:24 AM IST
4 min read
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the world’s population will continue to rise, reaching 9.7 billion in 2050
On World Population Day 2024, the top 10 most populated countries will be: The global population has experienced remarkable growth in recent centuries. It was expected to take hundreds of thousands of years to reach 1 billion people; however, within just 200 years, the population grew to seven times that size.
In 2011, the world’s population reached 7 billion, and projections by the UN suggest that the global population will grow to approximately 8.5 billion by 2030, 9.7 billion by 2050, and 10.9 billion by 2100.
Primarily driven by increased life expectancy, changes in fertility rates, urbanisation, and migration, these factors are likely to have widespread implications for generations to come, impacting economic development, employment, income distribution, poverty, and social welfare.
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Notably, with its population estimated to touch 142.86 crores, marginally ahead of China, India overtook China as the world’s most populous country, according to the UNFPA’s State of the World Population Report in 2023. This was a result of 11.1 million deaths and 9 million births in China, marking the second year in which China’s total population has been reduced.
Read | World’s top 10 largest cities by population in 2024: Indian cities claim two spots
The UNFPA report indicates that if India’s population continues to grow at the current rate of just under one per cent annually, it will double from its current value in the next 75 years. Experts say that India’s large population is a result of the “population momentum” from earlier decades and that the country’s population is likely to start its decline closer to 2050.
Graph showing an increase in India’s population between 2011 and 2035.
This trend would also apply to the global population, which is currently slightly above 8 billion. However, it is expected that both India’s and the world’s populations will stabilise long before that.
Recognising the growth trends in population, the United Nations Development Programme (UNDP) emphasised the urgency of global population issues by establishing the Day of the Five Billion in 1987, which they have been observing as World Population Day on July 11th annually since 1990.
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World Population Day seeks to raise awareness of global population issues. (Source: United Nations)
World Population Day serves as a platform to raise awareness and address crucial population-related challenges, in addition to highlighting the significance of family planning, gender equality, maternal health, and human rights.
This year, in 2024, it will fall on a Thursday and will emphasise the theme Investingin data collection is important to understanding problems, tailoring solutions, and driving progress.”
On World Population Day, we aim to raise awareness and understanding of global population issues, including demographic trends, growth projections, and their impact on sustainable development. Discover and take a look at the most populous countries in the world as of 2024.
Here are the top 10 most populated countries in the world, as of 2024:
| | | | | | |
--- --- --- |
| Rank | Country Name | 2024 Population | 2023 Population | Growth Rate | World % |
| 1 | India | 1,441,719,852 | 1,428,627,663 | 0.92% | 18.01% |
| 2 | China | 1,425,178,782 | 1,425,671,352 | -0.03% | 17.80% |
| 3 | United States of America | 341,814,420 | 339,996,563 | 0.53% | 4.27% |
| 4 | Indonesia | 279,798,049 | 277,534,122 | 0.82% | 3.50% |
| 5 | Pakistan | 245,209,815 | 240,485,658 | 1.96% | 3.06% |
| 6 | Nigeria | 229,152,217 | 223,804,632 | 2.39% | 2.86% |
| 7 | Brazil | 217,637,297 | 216,422,446 | 0.56% | 2.72% |
| 8 | Bangladesh | 174,701,211 | 172,954,319 | 1.01% | 2.18% |
| 9 | Russia | 143,957,079 | 144,444,359 | -0.34% | 1.80% |
| 10 | Ethiopia | 129,719,719 | 126,527,060 | 2.52% | 1.62% |
Source: World Population Review
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The global human population surpassed 8.0 billion in mid-November 2022, marking a significant increase from an estimated 2.5 billion people in 1950. This growth includes an additional 1 billion people since 2010 and 2 billion since 1998, according to the UN. Notably, China and India are the most populated countries, each with over 1 billion people, constituting nearly 18% of the world’s population.
Hence, the UN predicts that the world’s population will continue to rise, reaching 9.7 billion in 2050 and potentially peaking at nearly 10.4 billion in the mid-2080s, representing an increase of almost 2 billion individuals within the next 30 years.
Cherry Gupta
Cherry Gupta is an Assistant Manager – Content at The Indian Express. She leads the Top 10 section, curating list-based features on key national and international developments, and manages daily news content. She also produces SEO-driven articles and collaborates with the Lifestyle team to conduct interviews with notable artists and write workplace culture features. ... Read More
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5740 | https://uomustansiriyah.edu.iq/media/lectures/5/5_2020_06_12!08_23_23_PM.pdf | Power Plant Steam Turbine 1 Chapter Four Steam Turbine 4.1 Introduction : Steam turbine is used to produce power. Choice of steam turbine The choice of steam turbine depends on the following factors : (i) Capacity of plant (ii) Plant load factor and capacity factor (iii) Thermal efficiency (iv) Reliability (v) Location of plant with reference to availability of water for condensate. Power Plant Steam Turbine 2 The high pressure steam is expanded in the turbine. During expansion, the rotor (blades) of the turbine rotates, thus giving work output. Flow through Nozzle: A nozzle is a duct which the velocity of fluid through it increases at the expense of pressure drop. At the entrance to the nozzle, the velocity of steam is very low but its pressure is very high. In flowing through the nozzle the steam expands, its pressure drops but its velocity increases. Figure 4.1: Flow through Steam Nozzle Power Plant Steam Turbine 3 4. 2 Classification of Steam Turbines: Depending upon the types of blades used and the method of energy transfer from the fluid to the rotor wheel, the turbine may be into two types: 1. Impulse turbine (De-Laval , Curtis and Rateau): There is no change in the pressure of the steam as it passes through the moving blades. There is change only in the velocity of the steam flow. Steam at high pressure passes through nozzle where the velocity of steam increases. The high velocity jet of steam strikes on the blades of impulse turbine. The blades change the direction of steam flow without changing its pressure. The force due to change of momentum causes the rotation of the turbine shaft. Figure 4.2: Impulse Turbine 2. Reaction turbine: There is change in both pressure and velocity as the steam flows through the moving blades.The steam leaving from a fixed blade (acting as a nozzle) enters into the curved blade at and glides over the inside surface of the blades and leaves from the other edge. Power Plant Steam Turbine 4 Figure 4.3: Reaction Turbine 4. 3 Principle Operation of Simple Impulse Turbine The single-stage impulse turbine is also called the de Laval turbine after its inventor. The turbine consists of a single rotor to which impulse blades are attached. The steam is fed through one or several convergent-divergent nozzles which do not extend completely around the circumference of the rotor, so that only part of the blades is impinged upon by the steam at any one time. The nozzles also allow governing of the turbine by shutting off one or more them. The single-stage impulse turbine has been shown in Fig. 4.4. Power Plant Steam Turbine 5 Figure 4.4: Single– Stage Impulse Turbine (De-Laval) Power Plant Steam Turbine 6 4. 3 .1 Velocity Diagram for Impulse Turbine The velocity diagram for a single-stage impulse has been shown in Fig. 4.5. Figure 4.5 shows the velocity diagram indicating the flow through the turbine blades. Figure 4.5: Single– Stage velocity diagram Impulse Turbine (De-Laval) Power Plant Steam Turbine 7 A velocity triangle paves the way for a better understanding of the relationship between the various velocities. In the adjacent figure we have: Tangential force on a blade or, Blade efficiency It is ratio of power developed by the turbine to the energy entering the blade per second. Maximum blade efficiency Power Plant Steam Turbine 8 Blade speed ratio Stage efficiency: It is the ratio of work done/sec in one stage to the isentropic heat (enthalpy) drop in one stage. Power Plant Steam Turbine 9 If blade friction coefficient (K) is given then Unless otherwise stated, we can take 4.4 Compounding in Impulse Turbine If high velocity of steam is allowed to flow through one row of moving blades, it produces a rotor speed of about 30000 rpm which is too high for practical use. It is therefore essential to incorporate some improvements for practical use and also to achieve high performance. This is possible by making use of more than one set of nozzles, and rotors, in a series, keyed to the shaft so that either the steam pressure or the jet velocity is absorbed by the turbine in stages. This is called compounding. In an Impulse steam turbine compounding can be achieved in the following three ways: 1. Velocity compounding 2. Pressure compounding 3. Pressure-Velocity Compounding Power Plant Steam Turbine 10 1. Velocity compounding The Curtis stage turbine, as it came to be called, is composed of one stage of nozzles as the single-stage turbine, followed by two rows of moving blades instead of one. These two rows are separated by one row of fixed blades attached to the turbine stator, which has the function of redirecting the steam leaving the first row of moving blades to the second row of moving blades. A Curtis stage impulse turbine is shown in Fig. 4.6 with schematic pressure and absolute steam-velocity changes through the stage. In the Curtis stage, the total enthalpy drop and hence pressure drop occur in the nozzles so that the pressure remains constant in all three rows of blades. Power Plant Steam Turbine 11 Figure 4.6: Schematic Diagram of Curtis Stage Impulse Turbine Power Plant Steam Turbine 12 where, Pi = pressure of steam at inlet Vi = velocity of steam at inlet Po = pressure of steam at outlet Vo = velocity of steam at outlet In the above figure there are two rings of moving blades separated by a single of ring of fixed blades. As discussed earlier the entire pressure drop occurs in the nozzle, and there are no subsequent pressure losses in any of the following stages. Velocity drop occurs in the moving blades and not in fixed blades. Disadvantages of Velocity Compounding Due to the high steam velocity there are high friction losses Work produced in the low-pressure stages is much less. The designing and fabrication of bsexyes which can withstand such high velocities is difficult. 2. Pressure Compounding of Impulse Turbine The pressure compounded Impulse turbine is also called as Rateau turbine, after its inventor. This is used to solve the problem of high blade velocity in the single-stage impulse turbine. It consists of alternate rings of nozzles and turbine blades. The nozzles are fitted to the casing and the blades are keyed to the turbine shaft. Power Plant Steam Turbine 13 In this type of compounding the steam is expanded in a number of stages, instead of just one (nozzle) in the velocity compounding. It is done by the fixed blades which act as nozzles. The steam expands equally in all rows of fixed blade. The steam coming from the boiler is fed to the first set of fixed blades i.e. the nozzle ring. The steam is partially expanded in the nozzle ring. Hence, there is a partial decrease in pressure of the incoming steam. This leads to an increase in the velocity of the steam. Therefore the pressure decreases and velocity increases partially in the nozzle. Figure 4.7: Schematic Diagram of Pressure compounded Impulse Turbine Power Plant Steam Turbine 14 Figure 4.8: Velocity Diagram of Pressure compounded Impulse Turbine 3. Pressure-Velocity compounded Impulse Turbine It is a combination of the above two types of compounding. The total pressure drop of the steam is divided into a number of stages. Each stage consists of rings of fixed and moving blades. Each set of rings of moving blades is separated by a single ring of fixed blades. In each stage there is one ring of fixed blades and 3-4 rings of moving blades. Each stage acts as a velocity compounded impulse turbine. Power Plant Steam Turbine 15 Figure 4.9: Schematic Diagram of Pressure-Velocity compounded Impulse Turbine Power Plant Steam Turbine 16 4.5 Reaction turbine: (Also called Impulse-Reaction turbine) In the reaction turbine, the rotor blades themselves are arranged to form convergent nozzles. This type of turbine makes use of the reaction force produced as the steam accelerates through the nozzles formed by the rotor. Steam is directed onto the rotor by the fixed vanes of the stator. It leaves the stator as a jet that fills the entire circumference of the rotor. The steam then changes direction and increases its speed relative to the speed of the blades. A pressure drop occurs across both the stator and the rotor, with steam accelerating through the stator and decelerating through the rotor, with no net change in steam velocity across the stage but with a decrease in both pressure and temperature, reflecting the work performed in the driving of the rotor. In reaction turbine, steam expands both in fixed and moving blades continuously as the steam passes over them. The pressure drop and heat drop occurs continuously over both moving and fixed blades. The example for reaction turbine is Parson’s turbine. The steam expands while flowing over the moving blades and thus gives reaction to the moving blades. Hence this turbine is known as reaction turbine Number of stages, each stage consisting of set of fixed and moving blades. Power Plant Steam Turbine 17 4. 5 .1 Velocity Diagram for Reaction Turbine Figure 4.10: Schematic Diagram of Reaction Turbine Power Plant Steam Turbine 18 4. 5 .2 Degree of Reaction The degree of reaction is defined as the ratio of isentropic heat drop in the moving blades to isentropic heat drop in the entire stage of reaction turbine. A very widely used design has half degree of reaction or 50% reaction and this is known as Parson’s turbine. This consists of symmetrical rotor and stator blades. For this turbine the velocity triangle is similar and we have: 4. 5 .3 Comparing Efficiencies of Impulse and Reaction turbines Power Plant Steam Turbine 19 Power Plant Steam Turbine 20 4. 6 Losses in steam turbines 1. Admission losses: The decrease in kinetic energy is due to the following reasons Viscous forces between steam particles Heat loss from steam before entering the nozzle Deflection of flow in the nozzle Boundary layer development in the nozzle Turbulence in the nozzle The friction in the nozzle 2. Leakage losses 3. Friction losses: Frictional resistance is offered during flow of steam through nozzles on moving and stationary blades 4. Exhaust loss: The energy content of steam is not fully utilized in the turbine. Despite of being at very low pressure the exhaust coming out of the turbine and entering the condenser carries some of kinetic energy and useful enthalpy, which is direct energy loss. 5. Radiation and convection losses: The steam turbine operates at a relatively high temperature; therefore some of the heat energy of steam is radiated and convected from the body of the turbine to its surrounding. 6. Losses due to moisture: The steam passing through the last stage of turbine has high velocity and large moisture content. Power Plant Steam Turbine 21 EXAMPLE 1 A superheated steam leaves a jet nozzle at 800m/s and with inclined 20º and enters a single stage impulse turbine. Determine the work output rate of the rotor, the angle of rotor blade and the blade efficiency when the blade rotates at optimum velocity. SOLUTION: 2 20 800 2 1 1 , Cos Cos V V V opt B B s m VB / 87 .
375 ) ( 2 1 1 .
.
B B V Cos V V m W ) ( 2 1 1 .
.
B B V Cos V V m W W Power Plant Steam Turbine 22 ) 87 .
375 20 800 ( 87 .
375 2 Cos W Kg KJ W / 55 .
282 B r V Cos V Cos V 1 1 1 1 87 .
375 20 800 Cos 88 .
375 1 1 Cos Vr ……….. 1 1 1 1 1 Sin V Sin Vr 20 800 1 1 Sin Sin Vr 61 .
273 1 1 Sin Vr ………….. 2 Dividing 2 to 1 7279 .
0 88 .
375 61 .
273 tan 1 05 .
36 1 = 2 (impulse) 88 .
375 05 .
36 1 Cos Vr s m Vr / 8 .
464 1 2 1 2 1V W b 2 ) 800 ( 2 1 1000 55 .
282 b % 3 .
88 8829 .
0 b Power Plant Steam Turbine 23 EXAMPLE 2 A steam enter and leave nozzle to an impulse turbine with single stage at 2.5 MPa and 300Cº with pressure leave stage at 1.2MPa with 50 kg/s steam flow rate. The steam leave nozzle with angle 20º and speed 546.8m/s. The blade rotates with optimum velocity and has a velocity coefficient of 0.97. Determine the blade angle, stage power and the efficiencies of blade and stage. SOLUTION: 2 20 8 .
546 2 1 1 , Cos Cos V V V opt B B s m VB / 9 .
256 B r V Cos V Cos V 1 1 1 1 9 .
256 20 8 .
546 Cos 9 .
256 1 1 Cos Vr ……… 1 Power Plant Steam Turbine 24 1 1 1 1 Sin V Sin Vr 20 8 .
546 1 1 Sin Sin Vr 01 .
187 1 1 Sin Vr ………….. 2 Dividing 2 to 1 7279 .
0 9 .
256 01 .
187 tan 1 05 .
36 1 = 2 (impulse) 9 .
256 05 .
36 1 Cos Vr s m Vr / 7 .
317 1 7 .
317 97 .
0 1 2 r r V K V s m Vr / 1 .
308 2 2 2 2 2 Sin V Sin Vr 2 2 05 .
36 1 .
308 Sin V Sin 3 .
181 2 2 Sin V ……….. 3 B r V Cos V Cos V 2 2 2 2 9 .
256 05 .
36 1 .
308 2 2 Cos V Cos 8 .
7 2 2 Cos V …….. 4 From 3, 4 clockwise 53 .
87 53 .
87 2 s m V / 4 .
181 2 Power Plant Steam Turbine 25 ) ( 2 2 1 1 .
.
Cos V Cos V V m W B ) 53 .
87 4 .
181 20 8 .
546 ( 9 .
256 50 .
Cos Cos W MW W 5 .
6 .
2 1 .
.
2 1 V m W b 2 6 ) 8 .
546 ( 50 2 1 10 5 .
6 b % 9 .
86 869 .
0 b s s h m W .
.
Nozzle Inlet pN = 2.5 MPa hN = 3008.81 kJ/kg TN = 300ºC sN = 6.6437 kJ/kg. K Nozzle Outlet pB = 1.2 MPa hB = 2842.7 kJ/kg (Blade inlet) sN = sB = 6.6437 B N s h h h 28427 81 .
3008 Power Plant Steam Turbine 26 11 .
166 s h kJ/kg 1000 11 .
166 50 10 5 .
6 6 .
.
s s h m W % 2 .
78 782 .
0 s Power Plant Steam Turbine 27 EXAMPLE 3 A Superheated steam leave nozzle to a reaction turbine at 400m/s with inclined 25º. Steam strikes the rotor blade of 2m, 1m at tip and root diameter respectively. The reaction turbine operates with maximum velocity at tip blade. Determine : 1. The No. of revaluation per minutes of the rotor blade 2.The angle of blade rotor at tip , mean and root 3. The work done per unit mass at the mean diameter. SOLUTION: SOLUTION.1 25 400 1 1 , Cos Cos V V V opt B B s m VB / 5 .
362 (at tip) 60 , DN V opt B 60 2 5 .
362 N m p r N .
.
6 .
3461 Power Plant Steam Turbine 28 SOLUTION.2 At 50% reaction turbine 1 2 2 1 V Vr 1 2 V Vr at tip s m V Vr / 400 1 2 1 1 1 1 Sin V Sin Vr 25 400 90 1 Sin Sin Vr 2 1 / 09 .
169 V s m Vr 90 1 At optimum Vb Power Plant Steam Turbine 29 at mean point Dm=1.5m 60 DN VB 60 6 .
3461 5 .
1 B V s m VB / 8 .
271 1 1 1 1 Cos V V Cos V r B 1 1 8 .
271 25 400 Cos V Cos r 1 1 723 .
90 Cos Vr 1 1 1 1 Sin V Sin V r 2 1 7 .
61 at mean point 1 1 25 400 Sin V Sin r 1 1 04 .
169 Sin Vr 7 .
61 04 .
169 1Sin Vr 2 1 / 192 V s m Vr at root point Dr=1 m 60 DN VB 60 6 .
3461 1 B V s m VB / 2 .
181 Power Plant Steam Turbine 30 1 1 1 1 Cos V V Cos V r B 1 1 2 .
181 25 400 Cos V Cos r 1 1 323 .
181 Cos Vr 1 1 1 1 Sin V Sin V r 43 1 at mean point 1 1 25 400 Sin V Sin r 1 1 04 .
169 Sin Vr 43 04 .
169 1Sin Vr 2 1 / 86 .
247 V s m Vr 25 1 2 at tip , mean and root SOLUTION.3 At mean point ) 2 ( 1 1 .
.
B B V Cos V V m W ) 8 .
271 5 .
362 2 ( 8 .
271 .
.
m W Kg KJ m W / 18 .
123 .
.
Power Plant Steam Turbine 31 EXAMPLE 4 A steam enters and leaves a 50% reaction turbine with four stages. Steam operated with 10 MPa and 600Cº at inlet and 100 KPa at outlet. Nozzle inclined with 20º. Determine the work output in KJ per unit mass and the rotor angles. SOLUTION: Turbine Inlet pTi = 10 MPa hi = 3625.34 kJ/kg TTi = 600ºC si = 6.9028 kJ/kg. K Turbine Outlet pTo = 1.2 MPa Sf = 1.3025 kJ/kg. K So = Si = 6.9028 Sfg =6.0568 kJ/kg. K S = Sf + X Sfg Power Plant Steam Turbine 32 6.9028 = 1.3025 + X 6.0568 X = 0.924 ho = hf + X hfg ho = 417.44+ 0.924(2258.02) ho = 2503.8 kJ/kg (∆h)tot = hi -ho = 3625.34 - 2503.8 (∆h)tot = 1121.54 kJ/kg Work output n h)tot ( h)stage ( 4 121.54 1 h)stage ( 38 .
280 h)stage ( kJ/kg 2 h)stage ( h)rotor ( h)stator ( 19 .
140 h)rotor ( h)stator ( kJ/kg 2 h)stator ( 2 1 V 2 40190 1 2 1 V s m V / 5 .
529 1 Power Plant Steam Turbine 33 20 5 .
529 1 1 , Cos Cos V V V opt B B s m V opt B / 5 .
497 , At 50% reaction turbine 20 1 2 90 1 Power Plant Steam Turbine 34 Power Plant Engineering TUTORIAL SHEET ‐ 4 Steam Turbine 1- A superheated steam leaves nozzle to an impulse turbine with single stage at 500m/s. The rotor angle is with angle 45º that operated with optimum speed. Determine the work output in KJ per unit mass and the nozzle angle. Answers: [ 100 kJ/kg , 26.56º ] 2‐ A single row impulse turbine develops 150 kW at a blade speed of 175 m/s, using 2.27 kg of steam per second. Steam leaves the nozzle at 400 m/s. Velocity coefficient of blades is 0.9. Steam leaves the turbine blades axially. Determine nozzle angle, blade angles at entry and exit. 3- A steam expands with ideal operation in a 50% reaction turbine. Steam operated with 15 MPa and 600Cº at inlet to 20 KPa at outlet. The turbine operates at an optimum speed of 500m/s knowing that the nozzle inclination angle is 20º. How many stages should be used?. Also determine the work output per unit mass of steam for one stage. Answers: 5 stages, 276.588kJ/kg 4- A steam with 100 kg/s expands with ideal operation in a 50% reaction turbine. Steam operated with 5 MPa and 400Cº at inlet to 50 KPa at outlet. The nozzle inclination angle is 25º. The turbine operates at an optimum speed of 450m/s. Determines the number of stages should be used and the output power. Answers: 4 stages, 88.636 MW الجامعه المستنصريه– كليه الهندسه قسم الهندسه الميكانيكيه Power Plant Steam Turbine 35 5 -In one stage of a reaction steam turbine, both the fixed and moving blades have inlet and outlet blade tip angles of 35 º and 20 º respectively. The mean blade speed is 80 m/s and the steam consumption is 22500 kg/hr. Determine the power developed and stage efficiency if the isentropic heat drops in both fixed and moving rows is 23.5 kJ/kg in the pair. Answers: 125 kW, 85.1% |
5741 | https://ruggeri.store/blogs/come-arredare-casa/set-asciugamani-da-cosa-e-composto?srsltid=AfmBOoromKF9J5wOpaHr1qu8STcC7am4svRzkiSyJG_lHYoNFF-_5_FN | Set asciugamani: da cosa è composto?
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Articolo:Set asciugamani: da cosa è composto
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Set asciugamani: da cosa è composto
Set 3 pezzi, set 5 pezzi, 2 teli: quali sono le salviette che compongono il set di asciugamani perfetto? Scopriamolo insieme.
Avere un set di asciugamani completo è fondamentale per prenderci cura di noi in modo pratico e confortevole. Quali sono le salviette che compongono il set da bagno perfetto? Quali le dimensioni degli asciugamani da avere sempre in casa? Facciamo un po’ di chiarezza su cosa sono gli asciugamani viso, gli asciugamani ospite e i teli da bagno, e quali sono le salviette che non possono mancare nel tuo set di biancheria per il bagno.
Asciugamani viso
Gliasciugamani viso svolgono un ruolo fondamentale nella nostra routine quotidiana di cura personale e sono il tipo di salviette che sta alla base del perfetto set da bagno.
La loro dimensione standard è di 50x100 cm, ma si possono trovare anche set con asciugamani viso da 60x100 cm che, essendo leggermente più grandi, sono più pratici e versatili. Gli asciugamani viso vengono usati per asciugarsi le mani e il volto, ma sono perfetti anche per legare i capelli in un turbante o per fare una pedicure. Inoltre, sono gli asciugamani che possiamo portare con noi in palestra per restare sempre freschi e asciutti durante gli allenamenti.
Asciugamani ospite
Con il termine asciugamano ospite si intende una salvietta più piccola, in genere 40x50 cm, che accompagna l’asciugamano viso nel set da bagno. Viene usato solo per asciugarsi le mani, per questo è di dimensioni ridotte, e spesso è lasciato a disposizione di chi frequenta la casa saltuariamente e non necessita di un asciugamano viso per la sua cura quotidiana. Nel bagno patronale, può essere posizionato vicino al bidet e, se abbinato agli altri asciugamani, trasforma il bagno in un ambiente ordinato e ben curato.
Telo bagno
Il terzo elemento che compone un set standard di asciugamani bagno è il telo. Questa salvietta è molto ampia, con una dimensione media di 100x150 cm, e viene usata per avvolgere tutto il corpo dopo una doccia veloce o un rilassante bagno caldo. In genere il telo bagno viene posizionato vicino alla vasca o vicino alla doccia, meglio se ben piegato sopra uno sgabello, e se è scelto in coordinato con l’asciugamano viso e l’asciugamano ospite, rende l’ambiente elegante e ricercato.
I set di asciugamani disponibili
Sebbene il set di asciugamani 3 pezzi sia quello standard che si può trovare nella maggior parte dei negozi di biancheria per il bagno, può capitare di trovare anche dei set da 5 pezzi. In questo caso, gli asciugamani che compongono il set sono sempre viso, ospite e telo bagno, ma in quantità differenti. Alcuni set bagno possono contenere 2 asciugamani viso, 2 asciugamani ospite e un telo bagno, mentre altri sono composti da 2 teli bagno, 2 asciugamani viso e 1 solo asciugamano ospite. Inoltre, ci sono anche set da bagno che prevedono salviette tutte della stessa dimensione, come ad esempio 4 asciugamani ospite o 3 asciugamani viso. La differenza tra un set con salviette diversificate e un set di asciugamani con salviette della stessa dimensione è nella sua praticità di utilizzo. Non sempre hai bisogno di un set completo per arredare il tuo bagno. Pensa ad esempio ai bagni di servizio, dove non è prevista una doccia o una vasca da bagno. Acquistare un telo per il set da destinare a questo bagno sarebbe uno spreco! Inoltre, tra di noi c’è chi preferisce usare solo asciugamani viso, più grandi e avvolgenti, rinunciando così all’asciugamano ospite, oppure c’è chi preferisce usare solo salviette di piccole dimensioni per asciugarsi le mani.
Insomma, anche se quello più comune è il set con 3 asciugamani, quando devi scegliere un nuovo set da bagno devi valutare il tipo di utilizzo che farai delle diverse salviette, il bagno dove vorrai posizionarle e le tue abitudini quotidiane. Solo così potrai acquistare un set di asciugamani composto da salviette che per te sono davvero indispensabili!
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Bitwise operations equivalent of greater than operator
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I am working on a function that will essentially see which of two ints is larger. The parameters that are passed are 2 32-bit ints. The trick is the only operators allowed are ! ~ | & << >> ^ (no casting, other data types besides signed int, , /, -, etc..).
My idea so far is to ^ the two binaries together to see all the positions of the 1 values that they don't share. What I want to do is then take that value and isolate the 1 farthest to the left. Then see of which of them has that value in it. That value then will be the larger. (Say we use 8-bit ints instead of 32-bit). If the two values passed were 01011011 and 01101001 I used ^ on them to get 00100010. I then want to make it 00100000 in other words 01xxxxxx -> 01000000 Then & it with the first number !! the result and return it. If it is 1, then the first # is larger.
Any thoughts on how to 01xxxxxx -> 01000000 or anything else to help?
Forgot to note: no ifs, whiles, fors etc...
c
binary
bit-manipulation
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edited Aug 26, 2018 at 13:21
Shubham
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asked Apr 10, 2012 at 21:17
GekctekGekctek
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What do you mean by "I then want to make it 00100000 in other words 01xxxxxx-> 01000000"? How do you get 01xxxxxx from 00100000?Manlio –Manlio 2012-04-10 21:21:32 +00:00 Commented Apr 10, 2012 at 21:21
Oh come on... what can you use? Can you use the ternary conditional?Luchian Grigore –Luchian Grigore 2012-04-10 21:55:56 +00:00 Commented Apr 10, 2012 at 21:55
nope, no ternary, its to teach us how to essentially create the building blocks Gekctek –Gekctek 2012-04-10 22:14:12 +00:00 Commented Apr 10, 2012 at 22:14
I mean you get 01000000 from 01xxxxxx where x could be 0 or 1 Gekctek –Gekctek 2012-04-10 22:14:48 +00:00 Commented Apr 10, 2012 at 22:14
1 Doing bitwise stuff in signed integers is dubious. The C standard doesn't specify any particular binary representation for negative numbers. The real-world representation is almost always twos complement, but some compilers (e.g. GCC) exploit the undefinedness to do optimisations. As the sign can clearly be significant, this is at least potentially a problem.user180247 –user180247 2012-04-11 09:48:22 +00:00 Commented Apr 11, 2012 at 9:48
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Here's a loop-free version which compares unsigned integers in O(lg b) operations where b is the word size of the machine. Note the OP states no other data types than signed int, so it seems likely the top part of this answer does not meet the OP's specifications. (Spoiler version as at the bottom.)
Note that the behavior we want to capture is when the most significant bit mismatch is 1 for a and 0 for b. Another way of thinking about this is any bit in a being larger than the corresponding bit in b means a is greater than b, so long as there wasn't an earlier bit in a that was less than the corresponding bit in b.
To that end, we compute all the bits in a greater than the corresponding bits in b, and likewise compute all the bits in a less than the corresponding bits in b. We now want to mask out all the 'greater than' bits that are below any 'less than' bits, so we take all the 'less than' bits and smear them all to the right making a mask: the most significant bit set all the way down to the least significant bit are now 1.
Now all we have to do is remove the 'greater than' bits set by using simple bit masking logic.
The resulting value is 0 if a <= b and nonzero if a > b. If we want it to be 1 in the latter case we can do a similar smearing trick and just take a look at the least significant bit.
```c
include
// Works for unsigned ints.
// Scroll down to the "actual algorithm" to see the interesting code.
// Utility function for displaying binary representation of an unsigned integer
void printBin(unsigned int x) {
for (int i = 31; i >= 0; i--) printf("%i", (x >> i) & 1);
printf("\n");
}
// Utility function to print out a separator
void printSep() {
for (int i = 31; i>= 0; i--) printf("-");
printf("\n");
}
int main()
{
while (1)
{
unsigned int a, b;
printf("Enter two unsigned integers separated by spaces: ");
scanf("%u %u", &a, &b);
getchar();
printBin(a);
printBin(b);
printSep();
/ The actual algorithm starts here /
// These are all the bits in a that are less than their corresponding bits in b.
unsigned int ltb = ~a & b;
// These are all the bits in a that are greater than their corresponding bits in b.
unsigned int gtb = a & ~b;
ltb |= ltb >> 1;
ltb |= ltb >> 2;
ltb |= ltb >> 4;
ltb |= ltb >> 8;
ltb |= ltb >> 16;
// Nonzero if a > b
// Zero if a <= b
unsigned int isGt = gtb & ~ltb;
// If you want to make this exactly '1' when nonzero do this part:
isGt |= isGt >> 1;
isGt |= isGt >> 2;
isGt |= isGt >> 4;
isGt |= isGt >> 8;
isGt |= isGt >> 16;
isGt &= 1;
/ The actual algorithm ends here /
// Print out the results.
printBin(ltb); // Debug info
printBin(gtb); // Debug info
printSep();
printBin(isGt); // The actual result
}
}
```
Note: This should work for signed integers as well if you flip the top bit on both of the inputs, e.g. a ^= 0x80000000.
Spoiler
If you want an answer that meets all of the requirements (including 25 operators or less):
```c
int isGt(int a, int b)
{
int diff = a ^ b;
diff |= diff >> 1;
diff |= diff >> 2;
diff |= diff >> 4;
diff |= diff >> 8;
diff |= diff >> 16;
diff &= ~(diff >> 1) | 0x80000000;
diff &= (a ^ 0x80000000) & (b ^ 0x7fffffff);
return !!diff;
}
```
I'll leave explaining why it works up to you.
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edited Apr 11, 2012 at 14:27
answered Apr 10, 2012 at 22:07
KaganarKaganar
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To convert 001xxxxx to 00100000, you first execute:
c
x |= x >> 4;
x |= x >> 2;
x |= x >> 1;
(this is for 8 bits; to extend it to 32, add shifts by 8 and 16 at the start of the sequence).
This leaves us with 00111111 (this technique is sometimes called "bit-smearing"). We can then chop off all but the first 1 bit:
c
x ^= x >> 1;
leaving us with 00100000.
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answered Apr 11, 2012 at 7:30
cafcaf
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Kaganar
KaganarOver a year ago
Well, sometimes called "bit-smearing" by me. I can't find any solid references to various versions of the term via some fast Googling.
2012-04-11T13:34:05.113Z+00:00
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caf
cafOver a year ago
@Kaganar: That's the name by which I've known it too. I can't recall when I first heard it, though.
2012-04-12T02:10:49.063Z+00:00
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An unsigned variant given that one can use logical (&&, ||) and comparison (!=, ==).
c
int u_isgt(unsigned int a, unsigned int b)
{
return a != b && ( / If a == b then a !> b and a !< b. /
b == 0 || / Else if b == 0 a has to be > b (as a != 0). /
(a / b) / Else divide; integer division always truncate /
); / towards zero. Giving 0 if a < b. /
}
!= and == can easily be eliminated., i.e.:
c
int u_isgt(unsigned int a, unsigned int b)
{
return a ^ b && (
!(b ^ 0) ||
(a / b)
);
}
For signed one could then expand to something like:
c
int isgt(int a, int b)
{
return
(a != b) &&
(
(!(0x80000000 & a) && 0x80000000 & b) || / if a >= 0 && b < 0 /
(!(0x80000000 & a) && b == 0) ||
/ Two more lines, can add them if you like, but as it is homework
I'll leave it up to you to decide.
Hint: check on "both negative" and "both not negative". /
)
;
}
Can be more compact / eliminate ops. (at least one) but put it like this for clarity.
Instead of 0x80000000 one could say ie:
```c
include
static const int INT_NEG = (1 << ((sizeof(int) CHAR_BIT) - 1));
```
Using this to test:
c
void test_isgt(int a, int b)
{
fprintf(stdout,
"%11d > %11d = %d : %d %s\n",
a, b,
isgt(a, b), (a > b),
isgt(a, b) != (a>b) ? "BAD!" : "OK!");
}
Result:
c
33 > 0 = 1 : 1 OK!
-33 > 0 = 0 : 0 OK!
0 > 33 = 0 : 0 OK!
0 > -33 = 1 : 1 OK!
0 > 0 = 0 : 0 OK!
33 > 33 = 0 : 0 OK!
-33 > -33 = 0 : 0 OK!
-5 > -33 = 1 : 1 OK!
-33 > -5 = 0 : 0 OK!
-2147483647 > 2147483647 = 0 : 0 OK!
2147483647 > -2147483647 = 1 : 1 OK!
2147483647 > 2147483647 = 0 : 0 OK!
2147483647 > 0 = 1 : 1 OK!
0 > 2147483647 = 0 : 0 OK!
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edited Apr 11, 2012 at 9:29
answered Apr 10, 2012 at 23:49
MorpfhMorpfh
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Greg Price
Greg PriceOver a year ago
Logical operators like && and || are equivalent to if/else -- in particular, they get compiled to exactly the same machine code. So this doesn't meet the constraints. (Also the actual statement of the constraints lists operators and these aren't on the list.)
2013-11-08T08:03:20.2Z+00:00
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A fully branchless version of Kaganar's smaller isGt function might look like so:
```c
int isGt(int a, int b)
{
int diff = a ^ b;
diff |= diff >> 1;
diff |= diff >> 2;
diff |= diff >> 4;
diff |= diff >> 8;
diff |= diff >> 16;
//1+ on GT, 0 otherwise.
diff &= ~(diff >> 1) | 0x80000000;
diff &= (a ^ 0x80000000) & (b ^ 0x7fffffff);
//flatten back to range of 0 or 1.
diff |= diff >> 1;
diff |= diff >> 2;
diff |= diff >> 4;
diff |= diff >> 8;
diff |= diff >> 16;
diff &= 1;
return diff;
}
```
This clocks in at around 60 instructions for the actual computation (MSVC 2010 compiler, on an x86 arch), plus an extra 10 stack ops or so for the function's prolog/epilog.
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edited May 23, 2017 at 10:30
CommunityBot
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answered Jan 6, 2013 at 7:19
Philip ConradPhilip Conrad
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EDIT:
Okay, there were some issues with the code, but I revised it and the following works.
This auxiliary function compares the numbers' n'th significant digit:
```c
int compare ( int a, int b, int n )
{
int digit = (0x1 << n-1);
if ( (a & digit) && (b & digit) )
return 0; //the digit is the same
if ( (a & digit) && !(b & digit) )
return 1; //a is greater than b
if ( !(a & digit) && (b & digit) )
return -1; //b is greater than a
}
```
The following should recursively return the larger number:
c
int larger ( int a, int b )
{
for ( int i = 8sizeof(a) - 1 ; i >= 0 ; i-- )
{
if ( int k = compare ( a, b, i ) )
{
return (k == 1) ? a : b;
}
}
return 0; //equal
}
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edited Jun 20, 2020 at 9:12
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answered Apr 10, 2012 at 21:39
Luchian GrigoreLuchian Grigore
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Luchian Grigore
Luchian GrigoreOver a year ago
@Daniel I made a typo, it should be digit not n.
2012-04-10T21:43:58.197Z+00:00
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Daniel Lubarov
Daniel LubarovOver a year ago
Ah I see, don't you just want int digit = 1 << n though? This looks like it'd shift a quadratic amount.
2012-04-10T21:46:12.187Z+00:00
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Luchian Grigore
Luchian GrigoreOver a year ago
@Daniel there were several issues, now it's ok :)
2012-04-10T21:52:05.617Z+00:00
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Daniel Lubarov
Daniel LubarovOver a year ago
Looks good :-) except are you sure about the n-1? I guess that'd be right if you treat the last significant bit as bit 1, but your example code seems to go down to 0.
2012-04-10T22:05:04.58Z+00:00
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As much as I don't want to do someone else's homework I couldn't resist this one.. :) I am sure others can think of a more compact one..but here is mine..works well, including negative numbers..
Edit: there are couple of bugs though. I will leave it to the OP to find it and fix it.
```c
include
#include<stdio.h>
int a, b, i, ma, mb, a_neg, b_neg, stop;
int flipnum(int num, int is_neg) {
num = ~(num) + 1;
is_neg = 1;
return 0;
}
int print_num1() {
return ((a_neg && printf("bigger number %d\n", mb)) ||
printf("bigger number %d\n", ma));
}
int print_num2() {
return ((b_neg && printf("bigger number %d\n", ma)) ||
printf("bigger number %d\n", mb));
}
int check_num1(int j) {
return ((a & j) && print_num1());
}
int check_num2(int j) {
return ((b & j) && print_num2());
}
int recursive_check (int j) {
((a & j) ^ (b & j)) && (check_num1(j) || check_num2(j)) && (stop = 1, j = 0);
return(!stop && (j = j >> 1) && recursive_check(j));
}
int main() {
int j;
scanf("%d%d", &a, &b);
ma = a; mb = b;
i = (sizeof (int) 8) - 1;
j = 1 << i;
((a & j) && flipnum(&a, &a_neg));
((b & j) && flipnum(&b, &b_neg));
j = 1 << (i - 1);
recursive_check(j);
(!stop && printf("numbers are same..\n"));
}
```
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edited Apr 11, 2012 at 7:00
answered Apr 11, 2012 at 6:55
ManoharManohar
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I think I have a solution with 3 operations:
Add one to the first number, the subtract it from the largest possible number you can represent (all 1's). Add that number to the second number. If it it overflows, then the first number is less than the second.
I'm not 100% sure if this is correct. That is you might not need to add 1, and I don't know if it's possible to check for overflow (if not then just reserve the last bit and test if it's 1 at the end.)
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answered May 11, 2015 at 5:23
HoushalterHoushalter
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EDIT: The constraints make the simple approach at the bottom invalid. I am adding the binary search function and the final comparison to detect the greater value:
c
unsigned long greater(unsigned long a, unsigned long b) {
unsigned long x = a;
unsigned long y = b;
unsigned long t = a ^ b;
if (t & 0xFFFF0000) {
x >>= 16;
y >>= 16;
t >>= 16;
}
if (t & 0xFF00) {
x >>= 8;
y >>= 8;
t >>= 8;
}
if (t & 0xf0) {
x >>= 4;
y >>= 4;
t >>= 4;
}
if ( t & 0xc) {
x >>= 2;
y >>= 2;
t >>= 2;
}
if ( t & 0x2) {
x >>= 1;
y >>= 1;
t >>= 1;
}
return (x & 1) ? a : b;
}
The idea is to start off with the most significant half of the word we are interested in and see if there are any set bits in there. If there are, then we don't need the least significant half, so we shift the unwanted bits away. If not, we do nothing (the half is zero anyway, so it won't get in the way). Since we cannot keep track of the shifted amount (it would require addition), we also shift the original values so that we can do the final and to determine the larger number. We repeat this process with half the size of the previous mask until we collapse the interesting bits into bit position 0.
I didn't add the equal case in here on purpose.
Old answer:
The simplest method is probably the best for a homework. Once you've got the mismatching bit value, you start off with another mask at 0x80000000 (or whatever suitable max bit position for your word size), and keep right shifting this until you hit a bit that is set in your mismatch value. If your right shift ends up with 0, then the mismatch value is 0.
I assume you already know the final step required to determine the larger number.
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edited Apr 10, 2012 at 22:40
answered Apr 10, 2012 at 21:41
vhallacvhallac
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Gekctek
GekctekOver a year ago
limited to 25 operators and no loops
2012-04-10T22:20:35.31Z+00:00
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vhallac
vhallacOver a year ago
In that case you need a hand rolled binary search (with a twist). Updating the answer with source - it would take too long to describe. :)
2012-04-10T22:32:35.893Z+00:00
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vhallac
vhallacOver a year ago
No if would make things a lot harder. I'll update answer if I can get an answer.
2012-04-10T22:48:16.463Z+00:00
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5744 | https://math-gpt.org/problems/question/find-the-number-of-triples-a-b-c-of-decimal-digits-a-b-and-c-wit | To solve the problem, we start with the given condition invo | MathGPT
To solve the problem, we start with the given condition involving the three-digit integers a b c and c b a . The thre...
Question:
To solve the problem, we start with the given condition involving the three-digit integers a b c and c b a . The three-digit integer a b c can be expressed as:100a + 10b + cSimilarly, the integer c b a can be expressed as:100c + 10b + aAccording to the problem, we need to solve the equation:100a + 10b + c100c + 10b + a = 2 - caCross-multiplying gives:(100a + 10b + c) = (2 - ca)(100c + 10b + a)Expanding the right side:(100a + 10b + c) = 200c + 20b + 2a - 100c^2a - 10bca - c^2aRearranging all terms leads to a non-linear equation in terms of a, b, c . However, as we are looking for integer solutions with a, b, c being decimal digits and a 0 , we can explore a more straightforward approach by substituting possible values for a , b , and c .Let’s analyze the condition a 0 . This means a can take any value from 1 to 9 . The digits b and c can take values between 0 to 9 .Given the complexity of the equations, we can iterate through all possible values of a, b, c and check when the equation holds, keeping in mind the digits' constraints. For each valid a from 1 to 9 , and for each pair (b, c) from 0 to 9 , we can check the validity of the equation. This can be efficiently computed via programming as the total number of checks would not exceed 9 10 10 = 900 .After implementing a simple script or manual checking, we keep a count of the valid (a, b, c) that satisfy the original condition.Finally, through checking or by running a systematic approach, we conclude the total number of valid triples (a, b, c) . Hence, the answer to the problem is:The number of triples (a, b, c) is Xwhere X is the final count obtained from the checks.
Answer:
Given that divided by equals , we can write this as:
Cross-multiplying gives:
Expanding both sides:
[
100a^2 + 10ab + ac =...
Unlock with MathGPT
200ac + 20ab + 2a^2 - 100c^2 - 10bc - ac.
]
Collecting like terms and simplifying:
Since , we can further simplify and try to find a relationship between , , and that satisfies the equation. However, we notice that the equation involves variables in a way that suggests we should look for specific properties or patterns rather than directly solving it algebraically.
Given the condition , we note that for the division to result in a proper fraction, . This is because must be positive, implying .
Since , , and are decimal digits, ranges from 1 to 9, and and range from 0 to 9.
However, the key insight comes from realizing that must be a rational number that can be expressed as a fraction of two integers where the denominator is not zero. Given this, and the initial setup, it's clear that the problem simplifies into finding valid combinations of , , and rather than solving the equation directly for each variable.
Given and the decimal digit constraint, let's consider the possible values of and that would satisfy being a rational number that results from the division of two three-digit numbers.
For , can range from 0 to 1 (to satisfy ).
For , can range from 0 to 3.
For , can range from 0 to 5.
For , can range from 0 to 7.
For , can range from 0 to 9.
For , can range from 0 to 9.
For , can range from 0 to 9.
For , can range from 0 to 9.
For , can range from 0 to 9.
Since is independent of the condition and can range from 0
MathGPTNext Problem
; |
5745 | https://www.vedantu.com/chemistry/cannizzaro-reaction-mechanism | Cannizzaro Reaction Mechanism: Steps, Examples & Class 12 Notes
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Chemistry
Cannizzaro Reaction Mechanism: Stepwise Explanation, Types & Examples
Cannizzaro Reaction Mechanism: Stepwise Explanation, Types & Examples
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Ritika Singla
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Which compounds undergo the Cannizzaro reaction and why?
Cannizzaro reaction is a reaction of aldehydes with no ɑ-H atom in the presence of a concentrated base/alkali to give carboxylic acid and alcohol. In this reaction, 2 equivalents of aldehydes are used of which one equivalent gets converted to carboxylic acid and the other one gets converted to alcohol. It is a therefore a type of self-oxidation and reduction reaction (redox) i.e. disproportionation reaction in which the same molecule undergoes oxidation as well as reduction. Cannizzaro reaction can occur if and only if the aldehyde is devoid of ɑ-H atom and the base/alkali used is concentrated. The reason for the same can be better understood after the proper understanding of the mechanism of the reaction. So, here it goes.
Cannizzaro Reaction Mechanism is essential in chemistry and helps students understand various practical and theoretical applications related to this topic. It explains how certain aldehydes react under basic conditions, offering a foundation for mastering organic redox and name reactions in the Class 12 curriculum.
What is Cannizzaro Reaction Mechanism in Chemistry?
A Cannizzaro reaction mechanism refers to the base-induced self-oxidation and reduction (disproportionation) of non-enolizable aldehydes to yield a carboxylic acid salt and an alcohol. This concept appears in chapters related to Redox Reactions, Aldehydes and Ketones, and Organic Reaction Mechanisms, making it a foundational part of your chemistry syllabus.
Molecular Formula and Composition
There is no single molecular formula for the Cannizzaro reaction mechanism because it is a general reaction type. For example, using benzaldehyde (C 6 H 5 CHO) with strong base (NaOH), the products are benzyl alcohol (C 6 H 5 CH 2 OH) and sodium benzoate (C 6 H 5 COONa). The process is categorized under organic disproportionation reactions.
Preparation and Synthesis Methods
The Cannizzaro reaction is typically carried out in the laboratory by mixing a non-enolizable aldehyde (like benzaldehyde or formaldehyde) with a concentrated aqueous solution of a strong base such as NaOH or KOH. The reaction is not industrially significant due to low yields but is a must-know method for labs and academic study. The base must be strong and concentrated to drive the reaction to completion.
Physical Properties of Cannizzaro Reaction Mechanism
The reaction itself is not a substance but involves aldehyde substrates and their products. For example, benzyl alcohol (a Cannizzaro product) is a colorless liquid, soluble in water, and sodium benzoate is a white crystalline salt. Reaction conditions usually require room temperature or mild warming and concentrated base solutions.
Chemical Properties and Reactions
The Cannizzaro reaction mechanism highlights base-induced disproportionation. In the classic version, two molecules of an aldehyde without an alpha-hydrogen react—one is oxidized to a carboxylate ion (salt), and the other is reduced to an alcohol. Key steps include:
Nucleophilic attack of hydroxide ion on the carbonyl carbon
Formation of a tetrahedral intermediate and hydride ion transfer
One aldehyde molecule oxidized (to acid salt), the other reduced (to alcohol)
This mechanism is different from Aldol condensation, which operates only for aldehydes with alpha-hydrogen atoms.
Frequent Related Errors
Confusing the Cannizzaro reaction with Aldol reaction, especially regarding the need for alpha-hydrogen.
Thinking all aldehydes show Cannizzaro; only those without alpha hydrogen do.
Forgetting that the base must be concentrated, not dilute, for this reaction to proceed efficiently.
Misidentifying the oxidation and reduction half in cross Cannizzaro reactions.
Uses of Cannizzaro Reaction Mechanism in Real Life
The Cannizzaro reaction mechanism is used in organic synthesis and research, particularly for preparing alcohols from aromatic aldehydes and for demonstrating classic redox processes in education. Although rare in industrial settings, it helps illustrate how organic compounds can undergo simultaneous oxidation and reduction—an important concept for students and scientists. Vedantu educators often use this reaction to clarify the difference between Aldol and Cannizzaro mechanisms for competitive exams.
Relevance in Competitive Exams
Students preparing for NEET, JEE, and Olympiads should be familiar with the Cannizzaro reaction mechanism, as it often features in reaction-based problems, mechanism drawing, and concept-testing questions related to aldehydes and name reactions. Understanding this reaction also helps in quickly identifying which compounds do NOT give Aldol but do give Cannizzaro reactions.
Relation with Other Chemistry Concepts
Cannizzaro reaction mechanism is closely related to topics such as Aldol Condensation and Redox Reactions. Comparing Cannizzaro with Aldol helps students distinguish conditions (alpha-hydrogen presence/absence) and types of products formed. This also builds bridges to the understanding of Benzoin Condensation and other organic reaction mechanisms.
Step-by-Step Reaction Example
Start with two molecules of benzaldehyde and add concentrated NaOH.
2C 6 H 5 CHO + NaOH → C 6 H 5 COONa + C 6 H 5 CH 2 OH
Hydroxide ion attacks the carbonyl carbon of first benzaldehyde.
Forms a tetrahedral intermediate.
Hydride shifts from this intermediate to the carbonyl carbon of the second benzaldehyde molecule.
First molecule oxidizes to benzoate ion, second molecule reduces to benzyl alcohol.
Carboxylate ion (C 6 H 5 COO-) picks up the sodium ion; the alcohol is liberated.
Lab or Experimental Tips
Remember the Cannizzaro reaction mechanism by the rule: "No alpha-hydrogen, Cannizzaro can go!" Always use concentrated base and avoid aldehydes with CH 2 groups next to the carbonyl. Vedantu educators recommend drawing arrows for each mechanistic step to avoid confusion with Aldol reaction on exam day.
Try This Yourself
Write the IUPAC name of benzyl alcohol and sodium benzoate.
Identify if formaldehyde will undergo Cannizzaro reaction.
Give two real-life examples of using the Cannizzaro reaction mechanism in organic synthesis lab.
Final Wrap-Up
We explored Cannizzaro reaction mechanism—its concept, stepwise mechanism, related errors, and exam significance. Mastery of this reaction helps students tackle competitive questions and build clear organic chemistry fundamentals. For more detailed notes and LIVE explanations, explore Vedantu’s organic chemistry resources and interactive classes.
Explore related concepts:
Aldol Condensation | Benzoin Condensation | Haloform Reaction | Redox Reactions
FAQs on Cannizzaro Reaction Mechanism: Stepwise Explanation, Types & Examples
What is the Cannizzaro reaction mechanism?
The Cannizzaro reaction is a base-induced disproportionation of non-enolizable aldehydes. It involves a nucleophilic attack by hydroxide ion on the carbonyl carbon, followed by a hydride ion transfer to another aldehyde molecule. This results in the formation of a carboxylate ion and an alcohol. The reaction requires a strong base and aldehydes lacking α-hydrogens.
Which aldehydes undergo the Cannizzaro reaction?
Only aldehydes lacking α-hydrogens undergo the Cannizzaro reaction. These aldehydes cannot undergo aldol condensation because they lack the acidic α-hydrogen needed for enolate formation. Examples include benzaldehyde and formaldehyde.
What is the role of the base in the Cannizzaro reaction?
The strong base, typically a concentrated alkali like NaOH or KOH, acts as a nucleophile, initiating the reaction by attacking the carbonyl carbon of the aldehyde. It also deprotonates the resulting intermediate, leading to the formation of the carboxylate ion.
What is the difference between a simple and a crossed Cannizzaro reaction?
In a simple Cannizzaro reaction, two equivalents of the same aldehyde react to produce a carboxylate salt and an alcohol. In a crossed Cannizzaro reaction, two different aldehydes lacking α-hydrogens react. One aldehyde is oxidized to the carboxylate, while the other is reduced to the alcohol. The outcome is determined by the relative ease of oxidation of the two aldehydes.
Why don't all aldehydes undergo the Cannizzaro reaction?
Aldehydes possessing α-hydrogens preferentially undergo aldol condensation in the presence of a base. The α-hydrogen is more acidic than the carbonyl carbon, thus reacting with the base to form an enolate before nucleophilic attack at the carbonyl carbon can occur. The Cannizzaro reaction is only favored when α-hydrogens are absent.
What are some examples of aldehydes that undergo the Cannizzaro reaction?
Common examples include benzaldehyde (yielding benzyl alcohol and benzoic acid) and formaldehyde (yielding methanol and formic acid). Cross-Cannizzaro reactions can involve combinations of these, or other aldehydes lacking α-hydrogens.
What is the rate-determining step in the Cannizzaro reaction?
The rate-determining step is the hydride ion transfer between the two aldehyde molecules. This step is slower than the initial nucleophilic attack by the hydroxide ion.
How is the Cannizzaro reaction applied in organic synthesis?
The Cannizzaro reaction is a valuable tool for synthesizing specific alcohols and carboxylic acids from aldehydes lacking α-hydrogens. It's particularly useful when other methods are less efficient or selective.
What are the limitations of the Cannizzaro reaction?
Limitations include the requirement for aldehydes lacking α-hydrogens and the use of concentrated base, which can lead to side reactions. The yields can also be moderate.
How does the Cannizzaro reaction compare to the aldol condensation?
The Cannizzaro reaction and aldol condensation are both base-catalyzed reactions of aldehydes, but they differ significantly. The Cannizzaro reaction occurs with aldehydes lacking α-hydrogens, resulting in a redox reaction (disproportionation). The aldol condensation requires α-hydrogens, leading to the formation of β-hydroxy aldehydes or α,β-unsaturated aldehydes.
Can ketones undergo a Cannizzaro reaction?
No, ketones generally do not undergo the Cannizzaro reaction. The reaction mechanism requires a relatively electrophilic carbonyl carbon susceptible to nucleophilic attack, which is less pronounced in ketones due to steric hindrance and the electron-donating alkyl groups.
What is the historical significance of the Cannizzaro reaction?
The Cannizzaro reaction, discovered by Stanislao Cannizzaro in 1853, is historically significant because it provided early insights into the reactivity of aldehydes and redox reactions in organic chemistry. It helped establish fundamental principles of organic reaction mechanisms and continues to be a core concept in organic chemistry education.
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5746 | https://opencw.aprende.org/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2015/ | Home
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Mathematics for Computer Science
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Unit 1: Proofs
1.1 Intro to Proofs
1.2 Proof Methods
1.3 Well Ordering Principle
1.4 Logic & Propositions
1.5 Quantifiers & Predicate Logic
1.6 Sets
1.7 Binary Relations
1.8 Induction
1.9 State Machines - Invariants
1.10 Recursive Definition
1.11 Infinite Sets
Unit 2: Structures
2.1 GCDs
2.2 Congruences
2.3 Euler's Theorem
2.4 RSA Encryption
2.5 Digraphs: Walks & Paths
2.6 Directed Acyclic Graphs (DAGs) & Scheduling
2.7 Partial Orders and Equivalence
2.8 Degrees & Isomorphism
2.9 Coloring & Connectivity
2.10 Trees
2.11 Stable Matching
Unit 3: Counting
3.1 Sums & Products
3.2 Asymptotics
3.3 Counting with Bijections
3.4 Repetitions & Binomial Theorem
3.5 Pigeonhole Principle, Inclusion-Exclusion
Unit 4: Probability
4.1 Intro to Discrete Probability
4.2 Conditional Probability
4.3 Independence & Causality
4.4 Random Variables, Density Functions
4.5 Expectation
4.6 Deviation: Markov & Chebyshev Bounds
4.7 Sampling & Confidence
4.8 Random Walks & Pagerank
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6.042 serves as an introduction to discrete mathematics, probability, and mathematical thinking for computer scientists. (Image by OpenCourseWare, based on an image by Nick Matsakis.)
Instructor(s)
Prof. Albert R. Meyer
Prof. Adam Chlipala
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Spring 2015
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Ocean Wave Interaction with Ships and Offshore Energy Systems (13.022)
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| As Taught In | Spring 2002 |
| Course Number | 2.24 |
| Level | Undergraduate/Graduate |
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This subject offers an interactive introduction to discrete mathematics oriented toward computer science and engineering. The subject coverage divides roughly into thirds:
Fundamental concepts of mathematics: Definitions, proofs, sets, functions, relations.
Discrete structures: graphs, state machines, modular arithmetic, counting.
Discrete probability theory.
On completion of 6.042J, students will be able to explain and apply the basic methods of discrete (noncontinuous) mathematics in computer science. They will be able to use these methods in subsequent courses in the design and analysis of algorithms, computability theory, software engineering, and computer systems.
Interactive site components can be found on the Unit pages in the left-hand navigational bar, starting with Unit 1: Proofs.
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6.042J Mathematics for Computer Science (SMA 5512) (Fall 2002)
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Albert Meyer, and Adam Chlipala. 6.042J Mathematics for Computer Science. Spring 2015. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.
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5747 | https://www.math.uci.edu/~ndonalds/math2b/notes/11-5.pdf | 11.5 Alternating Series An alternating sequence is a sequence whose terms alternate between positive and negative. Often such sequences are written in the form an = (−1)nbn or an = (−1)n+1bn where (bn) is a sequence of positive terms, although sometimes they are somewhat disguised. An alternating series is the sum of an alternating sequence. For example, ∞ ∑ n=1 (−1)n+1 n = 1 −1 2 + 1 3 + 1 4 −1 5 + · · · is the alternating harmonic series.
The alternating series test is a convergence test which may be applied to alternating series. It is very easy to use.
Theorem (Alternating Series Test). Suppose that (bn) is a decreasing sequence of positive values with limit zero. Then the alternating series ∑(−1)nbn converges.
Like the other series tests, it does not matter which value of n denotes the initial term. As long as a series is alternating and decreasing, then it will converge. Just make sure that you observe all these facts when using the alternating series test.
Examples 1. The alternating harmonic series ∞ ∑ n=1 (−1)n+1 n is certainly alternating, and the sequence ( 1 n) decre-ases with limit zero. The test applies and so the series converges.
2. Consider the series ∞ ∑ n=1 (−1)n−1e−(n2+7n−2) Since the exponential term is always positive, this is certainly an alternating series. We should check that the exponential term is decreasing. For this, compute the derivative d dxe−(x2+7x−2) = (−2x −7)e−(x2+7x−2) < 0 whenever x ≥1 It follows that the alternating series test applies, and so the series converges.
3. Similarly, the series ∞ ∑ n=3 (−1)n(n2 + n) en is alternating and, since d dx (x2 + x) ex = (2x + 1)ex −(x2 + x)ex e2x = 2x + 1 −x2 + x ex = −x(x −3) −1 ex < 0 if x ≥3, the alternating series test applies.
1 4. Be careful! Not all alternating series converge!
∞ ∑ n=1 (−1)n−1 r 1 + 2 n is alternating, and bn = q 1 + 2 n is decreasing. The series does not converge, since bn does not converge to zero (nth term/divergence test).
Advanced: estimates of alternating series If you read the proof of the alternating series test (below) you may be able to convince yourself of the following: Theorem. If s = ∑(−1)n−1bn is a convergent alternating series, then the nth partial sum sn is at most a distance bn+1 from the value s of the series. That is |s −sn| ≤bn+1 This result is mostly of academic interest, for alternating series typically converge to their limits very slowly. . .
Example It can be shown that the infinite series ∞ ∑ n=0 (−1)n 1 + 2n = 1 −1 3 + 1 5 −1 7 + 1 9 −· · · converges to s = π 4 . How many terms would we need to sum in order to be sure that sn is an approximation to s which is correct to 2 decimal places? To guarantee this, we solve |s −sn| ≤bn+1 < 0.01 = ⇒ 1 1 + 2n < 1 100 = ⇒n > 49.5 Advanced: proving the alternating series test Like many similar proofs, this one relies on the monotone convergence theorem. We consider the sequence (sn) of partial sums of a (decreasing) alternating series and show that half of this sequence (the even terms (s2m)) is decreasing and bounded below, while the other half (s2m+1) is increasing and bounded above. Both halves converge. It remains to see that both halves converge to the same value. At all stages we need the fact that an = (−1)nbn where bn is a decreasing sequence.
Sketch Proof. For clarity, we assume that the series has the form ∑∞ n=0(−1)nbn where (bn) is a sequence which decreases to zero.
Consider the sequence of partial sums (sn). Depending on whether n is even or odd, we have diffe-rent expressions, whose terms may be grouped differently: n = 2m even s2m = 2m ∑ i=0 (−1)ibi = b0 −(b1 −b2) −(b3 −b4) −· · · −(b2m−1 −b2m) 2 n = 2m + 1 odd s2m+1 = 2m+1 ∑ i=0 (−1)ibi = (b0 −b1) + (b2 −b3) + · · · + (b2m −b2m+1) Since (bn) is decreasing, it follows that each of the bracketed terms above is positive. It follows that the subsequence (s2m) is decreasing and that (s2m+1) is increasing.
Moreover, s2m = (b0 −b1) + (b2 −b3) + · · · + (b2m−2 −b2m−1) + b2m > 0 s2m+1 = b0 −(b1 −b2) −(b3 −b4) −· · · −(b2m −b2m+1) < b0 (s2m) is decreasing and bounded below, while (s2m+1) is increasing and bounded above. The mono-tone convergence theorem says that both subsequences converge.
Finally, s2m+1 −s2m = −b2m+1 →0 so that both subsequences converge to the same limit.
Suggested problems 1.
(a) Show that ∞ ∑ k=2 (−1)k k + √ k converges.
(b) Why doesn’t the alternating series test apply to the series ∑aj, where (aj) = 1, −3 2, 1 3, −3 4, 1 5, −3 6, 1 7, −3 8, 1 9, . . .
?
2. Determine whether the following series converge.
(a) ∞ ∑ k=3 (−1)k(k −1) k2 + 2 (b) ∞ ∑ n=1 (−1)n+1n1/n 3. You are given that π2 = ∞ ∑ n=1 12(−1)n−1 n2 . How many terms of the series is it necessary to sum in order to approximate π2 to within 0.03? Use a calculator to do so, and check your answer with the calculator’s value for π2.
3 |
5748 | https://divisible.info/long-division/20/how-to-calculate-9800-divided-by-20-using-long-division.html | How to calculate 9800 divided by 20 using long division
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How to calculate 9800 divided by 20
Here we will show you an image that illustrates how to calculate 9800 divided by 20 using long division. We will also give you the answer to 9800 divided by 20 (9800 ÷ 20) as a fraction and as a decimal.
Note that in the problem 9800 divided by 20, the numbers are defined as follows:
9800 = dividend
20 = divisor
The illustration below shows you how to calculate 9800 divided by 20 using long division. As you can see, the dividend value of 9800 is located to the right of the "⟌" and the divisor value of 20 is located to the left of the "⟌".
The image displays 490 at the top, which is the quotient, and 0 at the bottom, which is the remainder. Therefore, the answer to 9800 divided by 20 (9800 ÷ 20) using long division is 490 remainder 0, or simply 490 R 0.
490 = quotient
0 = remainder
Now we will show you the answer to 9800 divided by 20 as a fraction, or more specifically, as a whole number.
9800 divided by 20 as a whole number
= 490
To find the answer to 9800 divided by 20 as a decimal, we entered 9800 divided by 20 into our pocket calculator and got this result:
9800 divided by 20 as a decimal
= 490
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Variance of sample variance?
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What is the variance of the sample variance? In other words I am looking for V a r(S 2).
I have started by expanding out V a r(S 2) into E(S 4)−[E(S 2)]2
I know that [E(S 2)]2 is σ to the power of 4. And that is as far as I got.
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edited Oct 16, 2011 at 14:33
cardinal
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Your expressions are very difficult to read. You need to edit and present your question in a better way.smanoos –smanoos 2011-10-16 04:44:13 +00:00 Commented Oct 16, 2011 at 4:44
2 One way of expressing V a r(S 2) is given on the Wikipedia page for variance.Mike Spivey –Mike Spivey 2011-10-16 04:50:15 +00:00 Commented Oct 16, 2011 at 4:50
It doesn't show how they derived it.MathMan –MathMan 2011-10-16 04:51:51 +00:00 Commented Oct 16, 2011 at 4:51
1 The solution to the question is in many books. You can easily find it.smanoos –smanoos 2011-10-16 05:17:18 +00:00 Commented Oct 16, 2011 at 5:17
2 There is a derivation on MathWorld's Sample Variance Distribution page. They use the "divide by N" convention rather than the "divide by N−1" convention, though, so you might have to adjust for that.Mike Spivey –Mike Spivey 2011-10-16 05:39:53 +00:00 Commented Oct 16, 2011 at 5:39
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Here's a general derivation that does not assume normality.
Let's rewrite the sample variance S 2 as an average over all pairs of indices: S 2=1(n 2)∑{i,j}1 2(X i−X j)2. Since E[(X i−X j)2/2]=σ 2, we see that S 2 is an unbiased estimator for σ 2.
The variance of S 2 is the expected value of (1(n 2)∑{i,j}[1 2(X i−X j)2−σ 2])2.
When you expand the outer square, there are 3 types of cross product terms [1 2(X i−X j)2−σ 2][1 2(X k−X ℓ)2−σ 2] depending on the size of the intersection {i,j}∩{k,ℓ}.
When this intersection is empty, the factors are independent and the expected cross product is zero.
There are n(n−1)(n−2) terms where |{i,j}∩{k,ℓ}|=1 and each has an expected cross product of (μ 4−σ 4)/4.
There are (n 2) terms where |{i,j}∩{k,ℓ}|=2 and each has an expected cross product of (μ 4+σ 4)/2.
Putting it all together shows that Var(S 2)=μ 4 n−σ 4(n−3)n(n−1). Here μ 4=E[(X−μ)4] is the fourth central moment of X.
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edited Oct 23, 2011 at 22:04
answered Oct 16, 2011 at 16:49
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1 a related question on stats.SE asks provides a different solution, and asks for a reference, your input would be appreciated: stats.stackexchange.com/q/29905/2750Abe –Abe 2012-06-06 16:56:05 +00:00 Commented Jun 6, 2012 at 16:56
2 @Abe Sorry, I don't have any references or worthwhile input. The above is a solution that I made up to teach my students.user940 –user940 2012-06-06 19:07:39 +00:00 Commented Jun 6, 2012 at 19:07
2 thanks, an answer to the stats.SE question solved my confusion: the discrepancy was use of kurtosis (μ 4, the fourth central moment) vs excess kurtosis (κ=μ 4 σ 4−3); one reference is Mood Graybill and Boes, 1974, Introduction to the Theory of StatisticsAbe –Abe 2012-06-06 22:39:58 +00:00 Commented Jun 6, 2012 at 22:39
2 @ByronSchmuland It's probably too basic, but I have problems with the first expression of variance as a pair of indices. Is there any way you can send a reference for this equation? Ty Antoni Parellada –Antoni Parellada 2015-08-03 14:21:38 +00:00 Commented Aug 3, 2015 at 14:21
9 In the derivation, how do we see claims 2 and 3, i.e. that the expected value of [1 2(X−Y)2−σ 2][1 2(X−Y)2−σ 2] is (μ 4+σ 4)/2, for X,Y i.i.d?Emolga –Emolga 2017-07-28 11:35:13 +00:00 Commented Jul 28, 2017 at 11:35
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Maybe, this will help. Let's suppose the samples are taking from a normal distribution. Then using the fact that (n−1)S 2 σ 2 is a chi squared random variable with (n−1) degrees of freedom, we get Var(n−1)S 2 σ 2=Var χ 2 n−1(n−1)2 σ 4 Var S 2=2(n−1)Var S 2=2(n−1)σ 4(n−1)2=2 σ 4(n−1),
where we have used that fact that Var χ 2 n−1=2(n−1).
Hope this helps.
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edited Oct 16, 2011 at 14:26
answered Oct 16, 2011 at 5:52
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25 Remember that (n−1)S 2/σ 2 is only guaranteed to be χ 2 when the sample is taken from a normal distribution, though.Mike Spivey –Mike Spivey 2011-10-16 06:09:29 +00:00 Commented Oct 16, 2011 at 6:09
14 The question posed is a general one, whereas the answer is distribution-specific. Not appropriate, I am afraid.wolfies –wolfies 2013-04-26 07:49:18 +00:00 Commented Apr 26, 2013 at 7:49
1 @afsdfdfsaf Perhaps, you should ask that as a separate question.Nana –Nana 2014-04-12 18:48:08 +00:00 Commented Apr 12, 2014 at 18:48
9 The answer is extremely useful, but would have been even more useful if someone could reference why (n−1)S2/σ2 is a Chi squared moldovean –moldovean 2015-03-09 07:06:42 +00:00 Commented Mar 9, 2015 at 7:06
1 @moldovean About as to why (n−1)S 2/σ 2 is a Ki2 distribution, I see it this way : ∑(x i−¯x)2 is the sum of the square value of N variables following normal distribution with expected value 0 and variance σ 2. Then, since all the (x i−¯x)/σ 2 follow a normal standard distribution, Y=∑N((x i−¯x)/σ)2=1 σ 2∑N(x i−¯x)2=(n−1)S 2 σ 2 follows a ki2 with N degrees of freedom, and not with N-1 degrees of freedom. I don't know what I am missing...mocquin –mocquin 2019-09-27 10:01:17 +00:00 Commented Sep 27, 2019 at 10:01
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This is quite a well-known result in statistics, and it can be found in a number of books and papers on sampling theory. You can find a range of useful moment results of this kind in O'Neill (2014) (this one is given in Result 3, p. 284). Consider a distribution with mean μ, variance σ 2, skewness γ and kurtosis κ (where all these moments are finite).† Taking n IID draws from this distribution and taking the variance of the sample variance S 2 n gives:
V(S 2 n)=(κ−n−3 n−1)σ 4 n
In the special case where the underlying distribution is mesokurtic (e.g., for a normal distribution) we have κ=3 and this expression then reduces to:
V(S 2 n)=2 n n−1⋅σ 4 n=2 σ 4 n−1.
You might also be interested to note that, in general, the sample variance and sample mean are correlated. Their covariance is C o v(ˉ X n,S 2 n)=γ σ 3/n and their corresponding correlation coefficient is:
C o r r(ˉ X n,S 2 n)=C o v(ˉ X n,S 2 n)S(ˉ X n)⋅S(S 2 n)=γ σ 3 n/σ√n⋅√(κ−n−3 n−1)σ 4 n=γ σ 3 n/σ 3 n⋅√κ−n−3 n−1=γ√κ−(n−3)/(n−1),
and as n→∞ you get:
C o r r(ˉ X n,S 2 n)→γ√κ−1,
which is the adjusted skewness of the underlying distribution. You can find further discussion of moments of the sample moments (including correlation between them) in O'Neill (2014).
† Actually, you can just assume that the kurtosis is finite, and this implies that all the lower-order moments are also finite.
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edited Oct 6, 2022 at 16:53
answered Sep 4, 2021 at 21:51
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1 Thanks for clarifying and bringing the reference O´Neill (2014). Very useful. It is a common mistake taking the particular formula for V(S 2 n), from a Normal sample, as the general formula. And thanks again for the bonus formula for the correlation between ˉ X n and S 2 n. I knew they were not independent in general but have never seen this formula before.bluemaster –bluemaster 2022-08-30 03:02:41 +00:00 Commented Aug 30, 2022 at 3:02
@bluemaster: Yes, that is a common mistake, not just in this particular case but in many other contexts too. There are a number of general moment formulae in statistics that reduce down to special cases when you use a normal distribution (taking γ=0 and κ=3). It is fairly common that people unfamiliar with the field will use the formulae for the special cases without being aware that the general formulae depend on skewness and kurtosis.Ben –Ben 2022-08-30 23:13:17 +00:00 Commented Aug 30, 2022 at 23:13
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There can be some confusion in defining the sample variance ... 1/n vs 1/(n-1). The OP here is, I take it, using the sample variance with 1/(n-1) ... namely the unbiased estimator of the population variance, otherwise known as the second h-statistic:
h2 = HStatistic
These sorts of problems can now be solved by computer. Here is the solution using the mathStatica add-on to Mathematica. In particular, we seek the Var[h2], where the variance is just the 2nd central moment, and express the answer in terms of central moments of the population:
CentralMomentToCentral[2, h2]
We could just as easily find, say, the 4th central moment of the sample variance, as:
CentralMomentToCentral[4, h2]
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edited May 30, 2017 at 8:19
answered Apr 26, 2013 at 8:02
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Does your program also let you handle dependent random variables? Presumably, then the result would be in terms of higher-order covariances.Thomas Ahle –Thomas Ahle 2019-08-15 15:17:45 +00:00 Commented Aug 15, 2019 at 15:17
Yes - it works for dependent random variables too. There is a multivariate example at: math.stackexchange.com/questions/589865/…wolfies –wolfies 2019-08-16 21:54:56 +00:00 Commented Aug 16, 2019 at 21:54
+1 Please, take a look at my new question: math.stackexchange.com/questions/4640597/…user1420303 –user1420303 2023-02-16 19:43:47 +00:00 Commented Feb 16, 2023 at 19:43
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Showing the derivation of E([1 2(X−Y)2−σ 2]2)=(μ 4+σ 4)/2 of user940:
LHS:
E([1 2(X−Y)2−σ 2]2)=E(1 4(X−Y)4−(X−Y)2 σ 2+σ 4)=E(1 4(X−Y)4)−2 σ 2 σ 2+σ 4=E(1 4(X−Y)4)−σ 4=1 4 E(X 4−4 X 3 Y+6 X 2 Y 2−4 X Y 3+Y 4)−σ 4=1 4(2 E(X 4)−8 E(X)E(X 3)+6 E(X 2)(X 2))−σ 4=1 2(E(X 4)−4 E(X)E(X 3)+3 E(X 2)(X 2)−2 σ 4)
I use the fact that E((x−y)2)=2 σ 2 here.
RHS:
(μ 4+σ 4)/2=1 2(E((X−μ)4)+σ 4)=1 2(E((X−E(X))4)+σ 4)=1 2(E(X 4−4 X 3 E(X)+6 X 2 E(X)2−4 X E(X)3+E(X)4)+σ 4)=1 2(E(X 4−4 X 3 E(X)+6 X 2 E(X 2)−6 X 2 σ 2−4 X E(X)(E(X 2)−σ 2)+(E(X 2)−σ 2)2)+σ 4)=1 2(E(X 4)−4 E(X)3 E(X)+6 E(X)2 E(X 2)−6 E(X)2 σ 2−4 E(X)2(E(X 2)−σ 2)+(E(X 2)−σ 2)2+σ 4)=1 2(E(X 4)−4 E(X)3 E(X)+6 E(X)2 E(X 2)−6 E(X)2 σ 2−4 E(X 2)E(X 2)+4 E(X 2)σ 2+4 E(X 2)σ 2−4 σ 4+E(X 2)2−2 E(X 2)σ 2+σ 4+σ 4)=1 2(E(X 4)−4 E(X)3 E(X)+3 E(X)2 E(X 2)−2 σ 4)
I use the fact that E(x)=μ and that E(x)2=E(x 2)−σ 2
Now LHS = RHS.
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answered Feb 10, 2020 at 14:56
Maverick MeerkatMaverick Meerkat
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I know that this question is very old but wanted to contribute nonetheless as I found it very hard to find a proof online that was satisfying enough, yet easy enough to follow. Proofs by induction are not constructive and do not tell you where the expression comes from in the first place.
While user940's answer is helpful, some parts are not very obvious, especially the first line and the later lines involving counting.
I have come up with a solution that is easier to follow and intuit, and I hope it helps others.
Let μ:=E(X i) denote the population mean and μ k:=E[(X i−μ)k] denote the k th centered population moment. Notice that with this notation, μ 2 is the population variance (what you are used to notating with σ 2, so μ 2 2 would be the square of this (or σ 4).
Then, if the kurtosis exists, V a r(S 2)=1 n(μ 4−n−3 n−1 μ 2 2).
Beginning from the definition of sample variance: S 2:=1 n−1 n∑i=1(X i−ˉ X)2, let us derive the following useful lemma:
Lemma (reformulation of S 2 as the average distance between two datapoints). Let X be a sample of size n and S 2 be the sample variance. Then S 2≡1 2 n(n−1)n∑i=1 n∑j=1(X i−X j)2.
Proof. Pick some X j and note that: S 2≡1 n−1 n∑i=1(X i−X j+X j−ˉ X)2⟹(n−1)S 2=n∑i=1(X i−X j)2+2 n∑i=1(X i−X j)(X j−ˉ X)+n∑i=1(X j−ˉ X)2. Now sum this over j: n(n−1)S 2=n∑j=1 n∑i=1(X i−X j)2+2 n∑j=1 n∑i=1(X i−X j)(X j−ˉ X)+n∑j=1 n∑i=1(X j−ˉ X)2. The final term is simply n(n−1)S 2 again, so we have: n∑i=1 n∑j=1(X i−X j)2=−2 n∑i=1 n∑j=1(X i−X j)(X j−ˉ X)=2 n∑i=1 n∑j=1(X j−ˉ X+ˉ X−X i)(X j−ˉ X)=2 n∑i=1 n∑j=1(X j−ˉ X)2+2 n∑i=1(ˉ X−X i)n∑j=1(X j−ˉ X)⏟0=2 n(n−1)S 2, giving the result.
Using this Lemma, we can now find the sample variance. Let's begin by using a small trick: Let Z i:=X i−μ. This means that E(Z k i)=μ k gives you the k th central moment. This makes the algebra much easier.
Proceed as so: S 2≡1 2 n(n−1)n∑i=1 n∑j=1(Z i⏞(X i−μ)−Z j⏞(X j−μ))2⟹V a r(S 2)≡V a r[1 2 n(n−1)(n∑i=1 n∑j=1(Z 2 i+Z 2 j−2 Z i Z j))]=V a r[1 2 n(n−1)(2 n n∑i=1 Z 2 i−2 n∑i=1 n∑j=1 Z i Z j)]=V a r[1 n(n−1)(n n∑i=1 Z 2 i−n∑i=1 n∑j=1 Z i Z j)]=1 n 2(n−1)2[n 2 V a r(n∑i=1 Z 2 i)+V a r(n∑i=1 n∑j=1 Z i Z j)=−2 n C o v(n∑i=1 Z 2 i,n∑i=1 n∑j=1 Z i Z j)].
Now we must calculate each of these variances and covariances. For the first one, V a r(n∑i=1 Z 2 i)=n V a r(Z 2 1)=n(E(Z 4 1)−E(Z 2 1)2)=n(μ 4−μ 2 2). (Note that μ 2 2 is the variance squared.)
For the second variance, V a r(n∑i=1 n∑j=1 Z i Z j)=E((n∑i=1 n∑j=1 Z i Z j)2)−E(n∑i=1 n∑j=1 Z i Z j)2.
To find the first expectation, we write: E((n∑i=1 n∑j=1 Z i Z j)2)=n∑i=1 n∑j=1 n∑k=1 n∑l=1 E(Z i Z j Z k Z l).
To compute this, we must consider all the possible ways that the indices (i,j,k,l) could be different. If there is any index that is different from all the other indices, then the expectation is zero. Thus, the only way for \mathop{\mathbb{E}}\left(Z_{i}Z_{j}Z_{k}Z_{l}\right) to be non-zero is if all indices are the same, or if there are two distinct pairs of equal indices (e.g. i = j \neq k = l).
In the first case, the expectation becomes \mathop{\mathbb{E}}\left(Z_{i}^{4}\right) = \mu_{4}, and there are n ways that this happens.
In the second case, suppose i = j and k = l and k > i (We take one to be strictly larger to avoid double counting in what follows). Then the expectation becomes \mathop{\mathbb{E}}\left(Z_{i}^{2} Z_{k}^{2}\right) = \mathop{\mathbb{E}}\left(Z_{i}^{2}\right) \mathop{\mathbb{E}}\left(Z_{k}^{2}\right) = \mu_{2}^{2}. The number of ways for i = j > k = l is {n \choose 2} = \frac{1}{2}n(n - 1). But there are other ways of choosing two pairs. In particular, there are {4 \choose 2} = 6 ways of choosing two pairs of indices from 4 indices, giving a total of 3n(n - 1) ways that the expectation equals \mu_{2}^{2}. Thus, \mathop{\mathbb{E}}\left(\left(\sum_{i=1}^{n}\sum_{j=1}^{n} Z_{i}Z_{j}\right)^{2}\right) = n \mu_{4} + 3n(n - 1)\mu_{2}^{2}.
Now we must calculate the expectation \mathop{\mathbb{E}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right) = \sum_{i=1}^{n}\sum_{j=1}^{n}\mathop{\mathbb{E}}\left(Z_{i}Z_{j}\right). Since Z_{i} and Z_{j} are independent, \mathop{\mathbb{E}}\left(Z_{i}Z_{j}\right) = 0 whenever i \neq j. Thus, the expectation becomes \mathop{\mathbb{E}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right) = \mathop{\mathbb{E}}\left(\sum_{i=1}^{n}Z_{i}^{2}\right) = n \mu_{2}.
Subtracting \mathop{\mathbb{E}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right)^{2} = n^{2}\mu_{2}^{2} from the previous expectation, we have: \begin{align} \mathop{\mathrm{Var}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right) &= n \mu_{4} + 2 n^{2} \mu_{2}^{2} - 3n \mu_{2}^{2}. \tag{(\spadesuit)} \end{align}
Finally, we calculate the covariance: \begin{align} \mathop{\mathrm{Cov}}\left(\sum_{i=1}^{n}Z_{i}^{2}, \sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right) &= \mathop{\mathbb{E}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}Z_{i}^{2}Z_{j}Z_{k}\right) - \mathop{\mathbb{E}}\left(\sum_{i=1}^{n}Z_{i}^{2}\right)\mathop{\mathbb{E}}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right). \end{align}
We have already found that the final two expectations are equal to n \mu_{2}, so the second term becomes n^{2} \mu_{2}^{2}.
For \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n} \mathop{\mathbb{E}}\left(Z_{i}^{2}Z_{j}Z_{k}\right), notice that the expectation is 0 whenever j \neq k. When i = j = k, the expectation is \mathop{\mathbb{E}}\left(Z_{1}^{4}\right) = \mu_{4} and there are n ways of achieving this. When i \neq j = k, the expectation is \mu_{2}^{2} and there are n(n - 1) ways of achieving this (now the order of i, j matters, so we do double count).
We therefore get: \begin{align} \mathop{\mathrm{Cov}}\left(\sum_{i=1}^{n}Z_{i}^{2}, \sum_{i=1}^{n}\sum_{j=1}^{n}Z_{i}Z_{j}\right) &= n(\mu_{4} - \mu_{2}^{2}). \tag{(\clubsuit)} \end{align}
Substituting \clubsuit, \spadesuit, \blacktriangle the values we just found into \bigstar, we get \begin{align} \mathop{\mathrm{Var}}\left(S^{2}\right) &= \frac{1}{n^{2}(n - 1^{2})} \left[(n^{3} - 2n^{2} + n)\mu_{4} - (n^{3} - 4n^{2} + 3n) \mu_{2}^{2}\right] \ &= \frac{1}{n^{2}(n - 1)^{2}} \left[n(n - 1)^{2} \mu_{4} - n(n - 1)(n - 3)\mu_{2}^{2}\right] \ &= \frac{1}{n} \left(\mu_{4} - \frac{n - 3}{n - 1} \mu_{2}^{2}\right) \end{align}
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edited Oct 22, 2023 at 12:40
Arto
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answered Jan 28, 2023 at 23:16
itoscholesitoscholes
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I will use this as an example of this theorem(from Seber, G.A. and Lee, A.J. (2012))
Let X_1, X_2, ... , X_n be independent rvs with means (\theta_1, \theta_2, ... ,\theta_n),common \mu_2,\mu_3,\mu_4. If A is any n x n symmetric matrix and a is a column vector of the diagonal elements of A, then
var[X'AX]=(\mu_4-3\mu^2_2)a'a+2\mu^2_2tr(A^2)+4\mu_2\theta'A^2\theta+4\mu_3\theta'Aa denote 1_n as n-dim column vector that all elements are 1, notice that for sample variance S^2=\frac{1}{n-1}X'AX, where A=I_n-\frac{1}{n}1_n1_n' and we have A^2=A, a=(1-\frac{1}{n})1_n
since X_i in our case are iid, let's say their mean is \mu, then \theta=\mu1_n
so the third and fourth term is 0,since A^2\theta=A\theta=\mu(1_n-\frac{1}{n}1_n(1_n'1_n))=0\ Aa=(1-\frac{1}{n})(1_n-\frac{1}{n}1_n(1_n'1_n))=0 then var[S^2]=\frac{1}{(n-1)^2}[(\mu_4-3\mu_2^2)(1-\frac{1}{n})^2n+2\mu_2^2(n-1)]=\frac{\mu_4}{n}-\frac{n-3}{n(n-1)}\mu_2^2
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answered Jan 9, 2021 at 16:08
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5750 | https://www.masterorganicchemistry.com/reaction-guide/oxidation-of-aldehydes-to-carboxylic-acids-with-ag2o/ | Skip to content
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Master Organic Chemistry Reaction Guide
Oxidation of aldehydes to carboxylic acids with Ag2O
Description:When Ag2O is added to aldehydes, carboxylic acids are formed. This results in the formation of silver metal (“silver mirror”) a reaction known as the Tollens test.
The rest of this page is available to MOC Members only. To get access to this page, plus over 2500 quizzes, the Reaction Encyclopedia, Org 1 / Org 2 summary sheets, and flashcards, sign up here for only 30 cents/ day!
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Org. Synth. 1953, 33, 94
DOI Link: 10.15227/orgsyn.033.0094
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Org. Synth. 1950, 30, 101
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7 thoughts on “Oxidation of aldehydes to carboxylic acids with Ag2O”
Will ketones undergo oxidation with Ag2O?
Reply
No, only aldehydes. The only way ketones will undergo oxidation is with a reagent like mCPBA (doing the Baeyer-Villiger) or with something like KMnO4
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2. Could this also be done with other oxidation agents?
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Sure you can use any other mild or strong oxidising agents like Fehling Solution, HNO3,acidic/basic KMnO4/K2Cr2O7 and many more..
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2. pretty sure acidified Potassium DiChromate does the same thing :)
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3. Is this reagent the same as Ag(NH3)2+?
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The oxidation state of Ag+ is the same, and there will be some similar properties, but they are not the same thing. Ag(NH3)2+ is much better at forming complexes with alkenes, for example.
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5751 | https://artofproblemsolving.com/wiki/index.php/Isogonal_conjugate?srsltid=AfmBOoq7Z_cjMQncduthVE7mIGEVcPOirG9vB262-iIBTs8jXie9lb_S | Art of Problem Solving
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Isogonal conjugate
Isogonal conjugates are pairs of points in the plane with respect to a certain triangle.
Contents
1 The isogonal theorem
1.1 Isogonal lines definition
1.2 Projective transformation
1.3 The isogonal theorem
1.3.1 Proof
1.4 The isogonal theorem in the case of parallel lines
1.4.1 Proof
1.5 Converse theorem
1.5.1 Proof
2 Parallel segments
3 Perpendicularity
4 Fixed point
5 Bisector
6 Isogonal of the diagonal of a quadrilateral
7 Isogonals in trapezium
8 Isogonals in complete quadrilateral
9 Isogonal of the bisector of the triangle
10 Points on isogonals
11 Trapezoid
12 Definition of isogonal conjugate of a point
13 Three points
14 Second definition
15 Distance to the sides of the triangle
16 Sign of isogonally conjugate points
17 Circumcircle of pedal triangles
18 Common circumcircle of the pedal triangles as the sign of isogonally conjugate points
19 Two pares of isogonally conjugate points
20 Circles
21 Equidistant isogonal conjugate points
22 Simplified distance formula for isogonal points
23 Point on circumcircle
24 Fixed point on circumcircle
25 Distance formula for isogonal points
26 Miquel point for isogonal conjugate points
27 Point on circumcircle
27.1 Simplified problem
28 Isogonal of line BC with respect to angle BAC
29 Isogonal bijection lines and points
30 Miquel point for two pare isogonal points
31 Isogonic center’s conjugate point
32 Three pairs isogonal points
33 Ratio for three pairs of isogonal points
34 Problems
The isogonal theorem
Isogonal lines definition
Let a line and a point lying on be given. A pair of lines symmetric with respect to and containing the point be called isogonals with respect to the pair
Sometimes it is convenient to take one pair of isogonals as the base one, for example, and are the base pair. Then we call the remaining pairs as isogonals with respect to the angle
Projective transformation
It is known that the transformation that maps a point with coordinates into a point with coordinates is projective.
If the abscissa axis coincides with the line and the origin coincides with the point then the isogonals define the equations and the lines symmetrical with respect to the line become their images.
It is clear that, under the converse transformation (also projective), such pairs of lines become isogonals, and the points equidistant from lie on the isogonals.
The isogonal theorem
Let two pairs of isogonals and with respect to the pair be given. Denote
Prove that and are the isogonals with respect to the pair
Proof
Let us perform a projective transformation of the plane that maps the point into a point at infinity and the line maps to itself. In this case, the isogonals turn into a pair of straight lines parallel to and equidistant from
The converse (also projective) transformation maps the points equidistant from onto isogonals. We denote the image and the preimage with the same symbols.
Let the images of isogonals are vertical lines. Let coordinates of images of points be Equation of a straight line is
Equation of a straight line is
The abscissa of the point is
Equation of a straight line is
Equation of a straight line is
The abscissa of the point is
Preimages of the points and lie on the isogonals.
The isogonal theorem in the case of parallel lines
Let and are isogonals with respect
Let lines and intersect at point
Prove that and line through parallel to are the isogonals with respect
Proof
The preimage of is located at infinity on the line
The equality implies the equality the slopes modulo of and to the bisector of
Converse theorem
Let lines and intersect at point
Let and be the isogonals with respect
Prove that and are isogonals with respect
Proof
The preimage of is located at infinity on the line so the slope of is known.
Suppose that
The segment and the lines are fixed
intersects at
but there is the only point where line intersect Сontradiction.
vladimir.shelomovskii@gmail.com, vvsss
Parallel segments
Let triangle be given. Let and be the isogonals with respect Let
Prove that lies on bisector of and
Proof
Both assertions follow from The isogonal theorem in the case of parallel lines
vladimir.shelomovskii@gmail.com, vvsss
Perpendicularity
Let triangle be given. Right triangles and with hypotenuses and are constructed on sides and to the outer (inner) side of Let Prove that
Proof
Let be the bisector of
and are isogonals with respect to the pair
and are isogonals with respect to the pair
and are isogonals with respect to the pair in accordance with The isogonal theorem.
is the diameter of circumcircle of
Circumradius and altitude are isogonals with respect bisector and vertex of triangle, so
Circumradius
vvsss
Fixed point
Let fixed triangle be given. Let points and on sidelines and respectively be the arbitrary points.
Let be the point on sideline such that
Prove that line pass through the fixed point.
Proof
We will prove that point symmetric with respect lies on .
and are isogonals with respect to
points and lie on isogonals with respect to in accordance with The isogonal theorem.
Point symmetric with respect lies on isogonal with respect to that is
vvsss
Bisector
Let a convex quadrilateral be given. Let and be the incenters of triangles and respectively.
Let and be the A-excenters of triangles and respectively.
Prove that is the bisector of
Proof
and are isogonals with respect to the angle
and are isogonals with respect to the angle in accordance with The isogonal theorem.
Denote
WLOG,
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Isogonal of the diagonal of a quadrilateral
Given a quadrilateral and a point on its diagonal such that
Let
Prove that
Proof
Let us perform a projective transformation of the plane that maps the point to a point at infinity and the line into itself.
In this case, the images of points and are equidistant from the image of
the point (midpoint of lies on
contains the midpoints of and
is the Gauss line of the complete quadrilateral bisects
the preimages of the points and lie on the isogonals and
vvsss
Isogonals in trapezium
Let the trapezoid be given. Denote
The point on the smaller base is such that
Prove that
Proof
Therefore and are isogonals with respect
Let us perform a projective transformation of the plane that maps the point to a point at infinity and the line into itself.
In this case, the images of points and are equidistant from the image of contains the midpoints of and , that is, is the Gauss line of the complete quadrilateral
bisects
The preimages of the points and lie on the isogonals and
vvsss
Isogonals in complete quadrilateral
Let complete quadrilateral be given. Let be the Miquel point of
Prove that is isogonal to and is isogonal to with respect
Proof
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Isogonal of the bisector of the triangle
The triangle be given. The point chosen on the bisector
Denote Prove that
Proof
Let us perform a projective transformation of the plane that maps the point to a point at infinity and the line into itself.
In this case, the images of segments and are equidistant from the image of
Image of point is midpoint of image and midpoint image
Image is parallelogramm
distances from and to are equal
Preimages and are isogonals with respect
vladimir.shelomovskii@gmail.com, vvsss
Points on isogonals
The triangle be given. The point chosen on The point chosen on such that and are isogonals with respect
Prove that
Proof
Denote
We use the Law of Sines and get:
vladimir.shelomovskii@gmail.com, vvsss
Trapezoid
The lateral side of the trapezoid is perpendicular to the bases, point is the intersection point of the diagonals .
Point is taken on the circumcircle of triangle diametrically opposite to point
Prove that
Proof
WLOG, is not the diameter of Let sidelines and intersect at points and respectively.
is rectangle
is isogonal to with respect
is isogonal to with respect
In accordance with The isogonal theorem in case parallel lines
is isogonal to with respect
in accordance with Converse theorem for The isogonal theorem in case parallel lines.
vladimir.shelomovskii@gmail.com, vvsss
Definition of isogonal conjugate of a point
Let triangle be given. Let be the circumcircle of Let point be in the plane of Denote by the lines respectively. Denote by the lines , , , respectively. Denote by , , the reflections of , , over the angle bisectors of angles , , , respectively.
Prove that lines , , concur at a point This point is called the isogonal conjugate of with respect to triangle .
Proof
By our constructions of the lines , , and this statement remains true after permuting . Therefore by the trigonometric form of Ceva's Theorem so again by the trigonometric form of Ceva, the lines concur, as was to be proven.
Corollary
Let points P and Q lie on the isogonals with respect angles and of triangle
Then these points lie on isogonals with respect angle
Corollary 2
Let point be in the sideline of
Then the isogonal conjugate of a point is a point
Points and do not have an isogonally conjugate point.
vladimir.shelomovskii@gmail.com, vvsss
Three points
Let fixed triangle be given. Let the arbitrary point not be on sidelines of Let be the point on isogonal of with respect angle Let be the crosspoint of isogonal of with respect angle and isogonal of with respect angle
Prove that lines and are concurrent.
Proof
Denote
and are isogonals with respect
and S lie on isogonals of
is isogonal conjugated of with respect
and lie on isogonals of
Therefore points and lie on the same line which is isogonal to with respect
vladimir.shelomovskii@gmail.com, vvsss
Second definition
Let triangle be given. Let point lies in the plane of Let the reflections of in the sidelines be
Then the circumcenter of the is the isogonal conjugate of
Points and have not isogonal conjugate points.
Another points of sidelines have points respectively as isogonal conjugate points.
Proof is common therefore Similarly is the circumcenter of the
From definition 1 we get that is the isogonal conjugate of
It is clear that each point has the unique isogonal conjugate point.
Let point be the point with barycentric coordinates Then has barycentric coordinates
vladimir.shelomovskii@gmail.com, vvsss
Distance to the sides of the triangle
Let be the isogonal conjugate of a point with respect to a triangle
Let and be the projection on sides and respectively.
Let and be the projection on sides and respectively.
Then
Proof
Let vladimir.shelomovskii@gmail.com, vvsss
Sign of isogonally conjugate points
Let triangle and points and inside it be given.
Let be the projections on sides respectively.
Let be the projections on sides respectively.
Let Prove that point is the isogonal conjugate of a point with respect to a triangle
One can prove a similar theorem in the case outside
Proof
Denote Similarly Hence point is the isogonal conjugate of a point with respect to a triangle
vladimir.shelomovskii@gmail.com, vvsss
Circumcircle of pedal triangles
Let be the isogonal conjugate of a point with respect to a triangle
Let be the projection on sides respectively.
Let be the projection on sides respectively.
Prove that points are concyclic.
The midpoint is circumcenter of
Proof
Let
Hence points are concyclic.
is trapezoid,
the midpoint is circumcenter of
Similarly points are concyclic and points are concyclic.
Therefore points are concyclic, so the midpoint is circumcenter of
vladimir.shelomovskii@gmail.com, vvsss
Common circumcircle of the pedal triangles as the sign of isogonally conjugate points
Let triangle and points and inside it be given. Let be the projections on sides respectively. Let be the projections on sides respectively.
Let points be concyclic and none of them lies on the sidelines of
Then point is the isogonal conjugate of a point with respect to a triangle
This follows from the uniqueness of the conjugate point and the fact that the line intersects the circle in at most two points.
vladimir.shelomovskii@gmail.com, vvsss
Two pares of isogonally conjugate points
Let triangle and points and be given. Let points and be the isogonal conjugate of a points and with respect to a triangle respectively.
Let cross at and cross at
Prove that point is the isogonal conjugate of a point with respect to
Proof
There are two pairs of isogonals and with respect to the angle are isogonals with respect to the in accordance with The isogonal theorem.
Similarly are the isogonals with respect to the
Therefore the point is the isogonal conjugate of a point with respect to
vladimir.shelomovskii@gmail.com, vvsss
Circles
Let be the isogonal conjugate of a point with respect to a triangle
Let be the circumcenter of
Let be the circumcenter of
Prove that points and are inverses with respect to the circumcircle of
Proof
The circumcenter of point and points and lies on the perpendicular bisector of Similarly
vladimir.shelomovskii@gmail.com, vvsss
Equidistant isogonal conjugate points
Let triangle with incenter be given. Denote
Let point be the isogonal conjugate of the point with respect to
Prove that iff
Proof
Let WLOG, Point
Point is the isogonal conjugate of the point with respect to So points and are concyclic.
Let Then is the center of
Let
Points and are symmetric with respect
Suppose that
Let be the center of be the center of
It is known that points and are inverted with respect to the circumcircle of
Points and belong to bisector
Therefore divide and
WLOG (see diagram) contradiction.
vladimir.shelomovskii@gmail.com, vvsss
Simplified distance formula for isogonal points
Let triangle points and and be given. Let point be the isogonal conjugate of a point with respect to a triangle Prove that
Proof
and are both subtended by arc Similarly Product of isogonal segments
vladimir.shelomovskii@gmail.com, vvsss
Point on circumcircle
Let triangle points and be given.
Denote Prove that
Proof
WLOG, the order of the points is as shown on diagram.
The spiral symilarity centered at maps to and point to point
is the external angle of
Corollary
is the isogonal conjugate to with respect
vladimir.shelomovskii@gmail.com, vvsss
Fixed point on circumcircle
Let triangle point on circumcircle and point be given.
Point lies on point be the isogonal conjugate of a point with respect to a triangle
Prove that is fixed point and not depends from position of
Proof
WLOG, the order of points on sideline is point is closer to than to
Denote
Spiral similarity centered at which maps into transform point into point Points and are collinear.
It is known (Ratio of isogonal segments) that
We use the ratio of the areas and get: Denote Therefore which means (Problems | Simple) that is the radical axes of and
and not depends from position of
Fixed point on circumcircle
vladimir.shelomovskii@gmail.com, vvsss
Distance formula for isogonal points
Let triangle and point be given.
Let point be the isogonal conjugate of a point with respect to a triangle
Let lines and cross sideline at and and circumcircle of at and respectively.
We apply the Isogonal’s property and get
We apply the Ptolemy's theorem to and get
We apply the barycentric coordinates and get
Barycentric coordinates
vladimir.shelomovskii@gmail.com, vvsss
Miquel point for isogonal conjugate points
Let triangle points and be given. Let point be the isogonal conjugate of a point with respect to a triangle Let be the Miquel point of a complete quadrilateral
Prove that lies on the circumcircle of
Proof
Point is the isogonal conjugate of a point with respect to a triangle so point is the isogonal conjugate of a point with respect to a triangle
Points and lies on the same line, therefore
Point lies on circles and spiral similarity centered at transform triangle to
vladimir.shelomovskii@gmail.com, vvsss
Point on circumcircle
Let triangle and points and be given.
Let
Let lines and be the isogonals with respect to the angle
Let be an arbitrary point on
Prove that lies on
Simplified problem
Let and points and be given,
Let lines and be the isogonals with respect to
Prove that
Proof, Simplified problem
points are concyclic on
Proof
Let points and be the isogonal conjugate of a points and with respect to a triangle
It is known that
points are concyclic on
vladimir.shelomovskii@gmail.com, vvsss
Isogonal of line BC with respect to angle BAC
Let triangle be given,
Let lines and be the isogonals with respect to
Prove that is tangent to
Proof
Let and be the circumcenter and the orthocenter of respectively. is isogonal to with respect to is tangent to
vladimir.shelomovskii@gmail.com, vvsss
Isogonal bijection lines and points
Let triangle and line be given,
Define the point with property
Prove that is equal the angle between and
Proof
WLOG, the configuration is the same as shown on diagram, is the tangent to
is isogonal to is isogonal to with respect to A bijection has been established between the set of lines parallel to a given one and the set of points of the circumcircle.
vladimir.shelomovskii@gmail.com, vvsss
Miquel point for two pare isogonal points
Let triangle and points and be given.
Let points and be the isogonal conjugate of the points and with respect to is the Miquel point of quadrilateral
Prove that
Proof
Denote
Then is the Miquel point of quadrilateral
Denote
Let be the point with property
WLOG, configuration is similar as shown in diagram.
(Isogonal_bijection_lines_and_points).
vladimir.shelomovskii@gmail.com, vvsss
Isogonic center’s conjugate point
Let triangle with isogonic center or be given. Denote
Let line be the axial symmetry of line according to the sideline
Define lines and similarly.
Prove that the lines and are concurrent.
Proof
Let be the incenter of Let is simmetric to with respect
The diameter of lies on
Therefore is the isogonal conjugate of with respect to
Similarly and are the isogonal conjugate of and so point is the isogonal conjugate of point with respect to
The second diagram show construction in the case The proof is similar.
vladimir.shelomovskii@gmail.com, vvsss
Three pairs isogonal points
Let a triangle points and be given, Points and are the isogonal conjugate of the points and respectively, with respect to
Prove that
Proof
Denote We use isogonal properties and get By applying the Law of Sines, we get Symilarly, We multiply these equations and get vladimir.shelomovskii@gmail.com, vvsss
Ratio for three pairs of isogonal points
Let a triangle points and be given,
Points and are the isogonal conjugate of the points and respectively, with respect to
Denote and the circumradii of triangles and respectively.
Prove that
Proof
Denote
where is the area of the figure Similarly, Similarly, It is known that (Three pairs isogonal points), therefore Comment: The main idea of the proof was found by Leonid Shatunov.
vladimir.shelomovskii@gmail.com, vvsss
Problems
Given a nonisosceles, nonright triangle let denote the center of its circumscribed circle, and let and be the midpoints of sides and respectively. Point is located on the ray so that is similar to . Points and on rays and respectively, are defined similarly. Prove that lines and are concurrent. (Source)
Let be a given point inside quadrilateral . Points and are located within such that , , , . Prove that if and only if . (Source)
Let and be the bisectors of a triangle
The segments and meet at point Let be the projection of to Points and on the sides and respectively, are such that Prove that (Source)
IMO 2007 Short list/G3
The diagonals of a trapezoid intersect at point
Point lies between the parallel lines and such that and line separates points and
Prove that
Proof
and are isogonals with respect
is isogonal to with respect
From the converse of The isogonal theorem we get
vladimir.shelomovskii@gmail.com, vvsss
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5752 | https://ektalks.blogspot.com/2018/10/additive-and-multiplicative-3x3-magic.html | skip to main | skip to sidebar
A science blog designed for community education. Climate Change, Sustainable Development, Lifestyle and Health are recurring themes. Biographies of famous scientists like Curie, Watt, Fleming, Einstein, Galileo etc. are available. Biomedical Imaging, Nanotechnology,Theory of Relativity, Physics of the Nucleus, Making of the Atomic Bomb, Plate Tectonics and many other subjects are discussed in detail without the use of higher maths. For further information - Contact ektalks@yahoo.co.uk
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: Science communication is important in today's technologically advanced society. A good part of the adult community is not science savvy and lacks the background to make sense of rapidly changing technology. My blog attempts to help by publishing articles of general interest in an easy to read and understand format without using mathematics. You can contact me at ektalks@yahoo.co.uk
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Tuesday, 16 October 2018
Additive and Multiplicative 3X3 Magic Squares - Construction and Some Not So Well Known Properties
I had discussed a couple of party games with 4x4 magic squares that have proved really popular among friends and in gatherings. Magic squares have fascinated people for thousands of years - they have an aura of mysticism and intrigue that is irresistible.
Wiki's article on magic squares comes with a lot of historic background etc. (also see 1, 2). In this blog, I wish to concentrate on the additive 3x3 magic squares - particularly on their construction and its less well known relative - the multiplicative magic square (MMS), and describe some other interesting variations. I shall end this blog with a discussion of the majestic 16x16 magic square constructed by Benjamin Franklin more than 200 years ago. Let us look at magic squares with sequential numbers.
A general way to find the number in the central cell is described in the next slide
I shall now discuss two variations of the magic squares that are not well known, but have really interesting properties. The Multiplicative Magic Square: In the magic squares that we have considered so far, the numbers in each row, column and diagonal add up to the same value. In a multiplicative magic square, theproduct of the numbers in each row, column and diagonal has the same value.
This is from Wiki:
Sum of Products of Rows and Columns in a Magic Square: This is a property of magic squares that is relatively unknown. It was analysed by Professor Hahn in 1975. Essentially, in an additive magic square, the sum the products of numbers in each row is equal to the sum of products of numbers in each column. Hahn shows, in a rather formal looking paper, that this property is always true for a 3x3 magic square but only holds for some (balanced) 4x4 and higher order magic squares. I refer you to the paper that is available online to read. A second important point here is that the sum of products of numbers in rows or columns is not equal to the sum of products of numbers in the diagonals. I could algebraically prove these results for a general 3x3 magic square but the calculation is too long and tedious to present here. I give some examples in the following:
The Majestic 16x16 Magic Square: Professor Bill Richardson has described this magic square that was constructed by Franklin more than 200 years ago - without the advantage of computers!! I refer you to the 1991 publication for all the details. In the following is a brief summary: The 16x16 magic square is shown in the slide:
Posted by Ravi Singhal at Tuesday, October 16, 2018
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5754 | https://acf.gov/sites/default/files/documents/opre/bloom.pdf | _ _ Paper 2: How should we calculate effect sizes? What are common mistakes in the calculation of effect sizes? How does research design influence the calculation of effect sizes? Dr. Howard Bloom, Senior methodologist at MDRC There are two considerations when computing or interpreting Effect Sizes: (a) different definitions of effect size convey different impressions and (b) interpretations of the magnitudes of effect sizes depend on their context. The Standardized Mean Difference Effect Size (ES) ES = (Y_s - Y_c)/g ES = (410 - 40 0)/50 = .20g The standardized mean difference effect size is a relative concept that depends upon the standard deviation used. Dividing the difference by a large standard deviation, results in a relatively small standardized mean difference (top curve) whereas dividing the difference by a relatively small standard deviation, results in a larger effect size (bottom curve). Figure 1 The top of Figure One is a depiction of two normal curve distributions, one for treatment and one for control. There is a line connecting the mean of the two distributions marked with a delta as the change score. The two distributions are normal and have some overlap between the distributions. The figure depicts if the standardized mean difference is calculated using a large standard deviation, the mean difference will be small. The bottom of Figure One is a depiction of two normal curve distributions that do not overlap, one for treatment and one for control but the normal curves are much taller and narrow (more kurtosis). The calculation of the standardized mean difference between the two means in this diagram with a smaller standard deviation will result in a larger effect size, though the means are actually the same between the two figures. Variance Components Framework A central tenet of effect sizes is understanding that the choice of standard deviation used to calculate an effect size influences its magnitude and thus its meaning. This point can be illustrated using a variance components framework in terms of test scores. By definition, there is some overall standard national variance in individual test scores. The total national variance can be decomposed into the variations across state, and within state across various districts, within districts across schools, subgroups of students, students within groups and then each one of these test scores is a measure of some latent achievement variable. With all of these variances, different researchers will choose one or a combination of variances for their study and provide justification for their choice. Then, perhaps another researcher will perform a meta-analysis, but the effect sizes are not comparable unless the same metric is applied. g^ 2 _ ( U S ) = g ^ 2_ ( s t ate) + g ^2_(di st rict) + g ^2 (schoo l) + g^2(s ub group) + g ^2_(student) + g^2_(error) g_ (U S) = sqrt(g^ 2_(sta te ) + g ^ 2 (distric t) + g^2(s ch oo l) + g^2_ (s ub group) + g ^2 (student) + g^2(error)) Here is an example of student-level versus school-level standard deviations: τ 2 = between − school − variance σ 2 = within − school − variance g_ s tu dent l = sqrt(t^2+g^2) g school = - sqr t( t^2 _ 2 + g ^ 2/ n p = t^2/(t^2+g^2) = ICC The table below depicts the ratio of student-level to school-level standard deviations. As intra-class correlations increase, schools are more different from each another on average. For example, if you have 50 third graders per school and the intra-class correlation is .20, the ratio of the student-level standard deviation to the school-level standard deviation is 2.15. In other words, the student-level standard deviation is a little over twice the size of the school-level standard deviation. Depending on the standard deviation used, the magnitude of the effect size varies greatly. Students in a grade per school (n) Intra-class correlation (ρ) .05 .10 .20 50 3.81 2.91 2.15 100 4.10 3.03 2.19 200 4.27 3.09 2.21 400 4.37 3.13 2.22 Adjusted versus Unadjusted Standard Deviation Another factor to consider is whether to use an unadjusted standard deviation or a regression-adjusted standard deviation. Once again, the choice in standard deviation can result in effect sizes that look very different. g^ 2 (unplan ned) = (1-R^2)g^2_ (total) R2 Ratio of unadjusted to adjusted standard deviations 0.1 1.05 0.3 1.20 0.5 1.41 0.7 1.83 0.9 3.16 The following table provides a sense of the calculations and the implications of using an unadjusted or a reliability-adjusted standard deviation. g^ 2_ (true ) = lg^ 2_(observed) Ratio of unadjusted to adjusted standard deviations Reliability 0.9 1.05 0.7 1.20 0.5 1.41 0.3 1.83 0.1 3.16 Assessing and Interpreting an Effect Size The research community should develop better conventions and contingencies for understanding effect sizes. This effort will assist the field in proceeding in a more orderly manner, require researchers to report their methodology and justification, and give the reader the ability to review the same findings. The context of "when" and the term "how" can assist in the calculation and discussion of effect sizes. For example, effect size calculations should be considered when designing an intervention study and determining the level of precision needed; when interpreting the results of an intervention study; when assigning adjectives in the final report; and when synthesizing intervention studies. Another way to discuss effect sizes can be discussed in terms of “how.” For example, comparing the external criterion or standard to either a related outcome construct or a related context. Cohen and Lipsey offer the prevailing guidelines for interpreting an effect size Cohen Small = 0.20 s Lipsey Small = 0.15 s Medium = 0.50 s Medium = 0.45 s Large = 0.80 s Large = 0.90 s Cohen, Jacob (1988) Statistical Power Analysis for the Behavioral Lipsey, Mark W. (1990) Design Sensitivity: Statistical Power for Sciences 2nd edition (Hillsdale, NJ: Lawrence Erlbaum). Experimental Research (Newbury Park, CA: Sage Publications). A common guideline for gauging achievement effects in education is an effect size equal to, or greater than, .25. This level has been defined as “educationally significant.” In Tallmadge’s The Joint Dissemination Review Panel IDEABOOK (1977) he stated, “One widely applied rule is that the effect must equal or exceed some proportion of a standard deviation– usually one-third, but at times as small as one-fourth– to be considered educationally significant” (p. 34). Conclusion When interpreting the magnitudes of effect sizes, “one size” does not fit all. Instead, researchers should interpret magnitudes of effects in context of the interventions being studied, of the outcomes being measured, and of the samples/subsamples being examined. Consider different frames of reference in context, instead of a universal standard such as external criteria, normative change, policy-relevant gaps, observed effect size distributions, or intervention costs. As a parting word of caution, standardized effect size measures should only be used when necessary. For example, when the original outcome measure does not have a meaningful metric, or when comparing intervention effects that are measured in different metrics. |
5755 | https://faculty.wharton.upenn.edu/wp-content/uploads/2018/09/DeterministicBoundsPart2.pdf | Deterministic Inequalities for Smooth M-estimators Arun Kumar Kuchibhotla University of Pennsylvania e-mail: arunku@wharton.upenn.edu Abstract: Ever since the proof of asymptotic normality of maximum likeli-hood estimator by Cram´ er (1946), it has been understood that a basic tech-nique of the Taylor series expansion suffices for asymptotics of M-estimators with smooth/differentiable loss function. Although the Taylor series expansion is a purely deterministic tool, the realization that the asymptotic normality results can also be made deterministic (and so finite sample) received far less attention. With the advent of big data and high-dimensional statistics, the need for finite sample results has increased. In this paper, we use the (well-known) Banach fixed point theorem to derive various deterministic in-equalities that lead to the classical results when studied under randomness. In addition, we provide applications of these deterministic inequalities for cross-validation/subsampling, marginal screening and uniform-in-submodel results that are very useful for post-selection inference and in the study of post-regularization estimators. Our results apply to many classical estimators, in particular, generalized linear models, non-linear regression and cox propor-tional hazards model. Extensions to non-smooth and constrained problems are also discussed.
1. Introduction One of the basic problems of statistics concerns estimation of parameters or func-tionals of a population based on a sample of observations. A large class of estimators in statistics are obtained as minimizers of some function of the observations, that is, estimators ˆ θn are obtained as ˆ θn := arg min θ∈Θn Mn(θ; Z1, . . . , Zn), (1) for some parameter space Θn (possibly depending on the sample size n) and a func-tion Mn(θ; Z1, . . . , Zn) written explicitly as a function of the parameter argument θ and observations Z1, . . . , Zn. Here the random variables Z1, . . . , Zn take values in a measurable space (not necessarily Euclidean) and are not required to be either in-dependent or to be identically distributed. For instance, the ordinary least squares 1 Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 2 (OLS) linear regression estimator ˆ βn based on regression data Z1 = (X1, Y1), . . ., Zn = (Xn, Yn) ∈Rp+1 is given by ˆ βn := arg min θ∈Rp 1 n n X i=1 Yi −X⊤ i θ 2 .
(2) Estimators of the type (1) are referred to as M-estimators in van der Vaart and Wellner (1996).
The study of the (asymptotic) properties of M-estimators is an ever-evolving research area in statistics. One of the most general and widely used frameworks for this study can be found in Section 3.2 of van der Vaart and Wellner (1996).
The results in this section require a stochastic equicontinuity assumption which in turn requires controlling the supremum of a stochastic process. Under certain differentiability assumptions on Mn, this equicontinuity assumption can be easily validated. Although there are numerous empirical process techniques to verify the equicontinuity assumption under independence, we do not know of such general techniques in case of dependent observations. In this paper, we provide a general way of proving deterministic inequalities for understanding the estimator ˆ θn which only requires randomness in verifying convergence of remainder terms to zero. This approach sacrifices the level of generality of van der Vaart and Wellner (1996) with certain smoothness assumptions but provides optimal rates as well as tail bounds.
Deterministic inequalities for M-estimators to be discussed were shown in Kuchib-hotla et al. (2018a) for the OLS estimator (2). The proofs there are much easier because of the explicit/closed-form OLS solution. It should be mentioned here that the idea of deriving results for M-estimators without assuming a particular depen-dence structure is not new and can be found in the works of van der Vaart and Wellner (1996), Yuan and Jennrich (1998), Hjort and Pollard (2011), Geyer (2013) and Kuchibhotla and Basu (2017). This list is by no means exhaustive.
1.1. The need for Deterministic Inequalities A starting point for this paper is Hjort and Pollard (2011); this paper was avail-able since 1993. One of the main conclusions of Hjort and Pollard (2011) is the following: pointwise convergence of a random convex objective function to a fixed function implies the consistency, rate of convergence and asymptotic distribution of the (global) minimizer. Some natural follow-up questions are “what happens if the parameter space changes with n? What if the dimension grows? What if the dependence between the observations changes with n?”. These questions are also Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 3 hard to answer from classical asymptotic normality results. However, the study of remainders in a single deterministic inequality can answer all these questions in a simple way. In this respect, deterministic inequalities unify the study of properties of the solutions of an estimating equation.
Another source of motivation for deterministic inequalities is the need for un-derstanding a collection of many estimators in some statistical applications. For instance, model-selection plays a pivotal role in data analysis where it is of interest to choose a “good parsimonious” model out of a (fixed) collection of models. In this case, to understand how a model-selection procedure works, it is necessary to study simultaneously the properties of all the estimators in the collection of models.
Depending on the number of models in the collection and the dependence between the observations, classical asymptotic results do not provide a clear understanding while deterministic inequalities (if they exist) provide detailed knowledge without any difficulty. Some of these examples will be presented later.
1.2. Can we expect Deterministic Inequalities?
As hinted in the abstract, classical asymptotic normality results are essentially based on the Taylor series expansion which is a deterministic tool. So, a first order Taylor series expansion of the estimating function (with explicit bounds on the remainder) and inversion implies explicit bounds for the solutions of the estimating equation.
A clearer picture can be seen through functionals. Let ˆ θn := θ(Pn) and θ0 := θ(P) be defined, respectively, as solutions of the equations Z ψ(θ; w)dPn(w) = 0 and Z ψ(θ; w)dP(w) = 0, where ψ(·; ·) is a fixed function and Pn is the empirical probability measure based on the observations W1, W2, . . ., Wn. The functional here is θ(·) which is a function on the space of all probability measures. Under most independence and dependence settings, it is known that Pn is “close” to P (in a suitable metric). So, if the functional θ(·) is continuous, then we get θ(Pn) −θ(P) ≈0.
Further if the functional θ(·) is Fr´ echet differentiable, then θ(Pn) −θ(P) −θ′(P; Pn −P) = o(d(Pn, P)), Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 4 for some metric d(·, ·). Here θ′(P; Pn −P) denotes the directional derivative of θ(·) at P in the direction of Pn −P. This is essentially the framework of von Mises calculus; see Clarke (2018) for details. It is clear that if the continuity assumption is replaced by H¨ older continuity, that is, ∥θ(Pn) −θ(P)∥≤dα(Pn, P) for some metric d(·, ·) and α > 0, then we got a deterministic inequality for estimation error. Similarly, if the differentiability assumption is made precise in terms of some explicit bounds, then we can write ∥θ(Pn) −θ(P) −θ′(P; Pn −P)∥≤Cd1+α(Pn, P), for some constant C > 0 and α > 0. This again provides a deterministic inequal-ity for an expansion of the estimator. In the following sections, we provide these explicit bounds for various M-estimation problems.
There is a also a rich literature in the field of mathematical programming where the problem Z ψ(θ; w)dPn(w) = 0, is seen as a “perturbation” of the problem R ψ(θ; w)dP(w) = 0 (since Pn ≈P).
There are a large number of sensitivity and stability results that show how much different θ(Pn) is from θ(P); see, for example, R¨ omisch and Wets (2007), Rock-afellar and Wets (2009). In the current paper, we restrict mostly to smooth M-estimators, meaning differentiable ψ(·, w) with a H¨ older continuous derivative.
There are numerous results in mathematical programming literature that work for non-smooth functions and also, M-estimators with constraints. We hope to tackle these additional problems in the future.
1.3. Organization The remainder of the paper is organized as follows. In Section 2, we state a Banach fixed point theorem in a form suitable for our purposes. This result appeared in a similar form in Yuan and Jennrich (1998) and Jacod and Sørensen (2018). Also, a result similar to Newton-Kantorovich theorem is provided. All our subsequent results follow from these results. Applications to M-estimators based on convex loss functions including generalized linear models (GLMs) and Cox proportional hazards model are given in Sections 3 and 4, respectively. In Section 5, we provide deterministic inequalities for least squares non-linear regression. In Section 6, we provide a deterministic inequality for an equality constrained minimization prob-lem. In Section 7, we present three applications of our deterministic inequalities Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 5 for cross-validation/subsampling, marginal screening and post-selection inference.
We conclude with some remarks and future directions in Section 8.
2. The Basic Result In this section, we state a basic inversion theorem that implies existence of solutions to an equation in a certain neighborhood. This result is of primary importance to us since it gives explicit bounds on the radius of the neighborhood and so can provide finite sample results in statistical applications. The following result is stated in a similar form in Yuan and Jennrich (1998). Define for any θ0 ∈Rq and r > 0, the closed ball as B(θ0, r) := {θ ∈Rq : ∥θ −θ0∥2 ≤r} .
For any matrix A, let ∥A∥op denote the operator norm of A. Also, for any function f(·), ∇f(·) and ∇2f(·) denote the gradient and Hessian of f(·).
Theorem 2.1. Let f(·) be an everywhere differentiable mapping from an open subset of Rq into Rq. Let A be a non-singular matrix. If for some θ0 ∈Rq, r > 0 and ε ∈[0, 1], A−1 (A −∇f(θ)) op ≤ε for all θ ∈B(θ0, r), (3) and A−1f(θ0) 2 ≤r(1 −ε), (4) then there exists a unique vector θ∗∈B(θ0, r) satisfying f(θ∗) = 0 and 1 1 + ε A−1f(θ0) 2 ≤∥θ⋆−θ0∥2 ≤ 1 1 −ε A−1f(θ0) 2 .
The proof of this theorem is given in Appendix A. Also, see Jacod and Sørensen (2018) for related results and applications of this result for the asymptotics of statistical estimating functions.
Interestingly (from the statistical viewpoint), Theorem 2.1 does not require any special properties for θ0. This particular fact becomes important in applications related to subsampling/cross-validation in Section 7.1. In the following sections, we apply this result with f(θ) = ∇Mn(θ), A = ∇2Mn(θ0) (which requires the objective function Mn(·) to be twice differentiable) and deterministically, the result shows that the estimation error ∥ˆ θ −θ0∥2 is up to a constant factor same as ∥∇Mn(θ0)∥2.
The classical results mostly use the choice θ0 that satisfies E [∇Mn(θ0)] = 0. If the objective function Mn(·) is an average, then the control of ∥∇Mn(θ0)∥2 is the same Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 6 as controlling a mean zero average which is studied in probability and statistics for almost all practically useful dependence structures.
For simplicity, the result is stated with Rq as the domain but it is not difficult to extend the proof to more general Banach spaces under Fr´ echet differentiabil-ity. This extension is useful in deriving deterministic inequalities for smoothing spline estimators for non-parametric regression/density estimation; see Shang et al.
(2010) and Shang et al. (2013). Another interesting extension can be obtained by replacing Euclidean norm-∥·∥2 by a general norm ∥·∥N on Rq. In this case, the assumptions (3) and (4) need to be rewritten as A−1(A −∇f(θ)) N→N ≤ε, for all θ ∈Br,N(θ0), and A−1f(θ0) N ≤r(1 −ε), where ∥·∥N represents a norm on Rq and for any matrix K, ∥K∥N→N := sup{x⊤Kx : ∥x∥N ≤1}, and Br,N(θ0) := {θ ∈Rq : ∥θ −θ0∥N ≤r}.
In many statistical applications of interest, it is also of interest to prove a first order (influence function) approximation for the estimator. For these applications, we present the following extended result under a strengthening of the assumptions of Theorem 2.1. The following theorem is closely related to the well-known Newton-Kantorovich Theorem about Newton’s method of root finding.
Theorem 2.2. Let f(·) be an everywhere differentiable mapping from an open subset of Rq into Rq. If for some θ0 ∈Rq, L ≥0, and α ∈(0, 1], [∇f(θ0)]−1 (∇f(θ0) −∇f(θ)) op ≤L ∥θ −θ0∥α 2 , (5) whenever (3L)1/α ∥θ −θ0∥2 ≤1 and [∇f(θ0)]−1f(θ0) 2 ≤ 2 3(3L)1/α, (6) then there exists a unique solution θ⋆of f(θ) = 0 in B(θ0, 1.5∥[∇f(θ0)]−1f(θ0)∥2) and that solution θ⋆satisfies θ⋆−θ0 + [∇f(θ0)]−1f(θ0) 2 ≤(1.5)1+αL [∇f(θ0)]−1f(θ0) 1+α 2 .
(7) Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 7 The expansion (7) of Theorem 2.2 is essentially proving that the first iteration of Newton’s scheme is “close” to the true solution θ⋆and is a very special case of the general superlinear convergence statement1. The classical Newton-Kantorovich theorem (Theorem 1.1 of Yamamoto (1985)) usually requires a slightly stronger condition on the derivative of f with α = 1 and proves explicit bounds for all iterations of the Newton’s scheme. Also, see Clarke and Futschik (2007) for some applications in statistical problems.
An important message from Theorem 2.2 is that any iterative algorithm that re-quires conditions only on the initial point2 (θ0) and proves superlinear convergence can be used to prove expansion results like (7). Apart from the classical Newton’s method (that requires differentiable f), there are numerous extensions allowing for non-smooth f including B-differentiable functions (Qi and Sun (1993)) and nor-mal mappings (Robinson (1994)). For a general treatment of Newton’s method, see Argyros (2008).
From the proof, it is easy to replace the right hand side of assumption (5) by any non-decreasing function ω(·) of ∥θ −θ0∥2; this extension is useful for nonlinear regression as in Section 5. As before, an extension of Theorem 2.2 to Banach spaces is possible. In fact, Theorem 1.1 of Yamamoto (1985) holds for Banach spaces.
Most commonly used M-estimators in statistics or machine learning are based on objective functions that are averages. In this case, f(·) and ∇f(·) are also averages. Averages (under independence as well as dependence) have been the subject of investigation for decades in statistics and probability literature. Thus, our results imply that randomness plays the role only in controlling the averages appearing as the remainders.
Remark 2.1 (Implications for the Landscape of Non-convex Losses) Determin-istic inequalities of the type obtained in Theorems 2.1 and 2.2 have implications for local minimizers in statistical applications. Suppose there are n identically dis-tributed observations W1, . . . , Wn and the parameter of interest is θ0 ∈Rp that is defined as a global minimizer of E[ℓ(θ, W1)]. Then a natural estimator of θ0 is ˆ θn := arg min θ∈Rp 1 n n X i=1 ℓ(θ, Wi).
1A sequence of iterates {θn}n≥0 is said to converge superlinearly if ∥θn+1 −θn∥2 = o(∥θn −θn−1∥2) as n →∞. In our case θ0 is the initial point and θ1 = θ0 −(∇f(θ0))−1f(θ0) is the first iterate.
2Convergence analysis that require conditions only on the initial point is usually referred to as semilocal analysis.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 8 If the loss function ℓ(·, w) is a non-convex function, then it is in general very hard to obtain a global minimizer ˆ θn. For this reason, it is of significant interest to understand the behavior of local minimizers or critical points of the sample loss function. Recently Mei et al. (2016) proved that the landscape of the sample loss function is similar to that the population loss function under certain assumptions including independent and identically distributed (iid) observations.
Using a deterministic inequality, this fact becomes clear. Suppose, for some K ≥ 1, θ(1) 0 , θ(2) 0 , . . . , θ(K) 0 represent the critical values of E[ℓ(θ, W1)], that is, ∇E[ℓ(θ, W1)] θ=θ(j) 0 = 0, for all 1 ≤j ≤K.
Then under various dependence settings, it is expected that for any 1 ≤j ≤K, 1 n n X i=1 ∇ℓ(θ(j) 0 , Wi) 2 = op(1).
This implies that assumption (6) of Theorem 2.2 is satisfied in probability. Thus, by Theorem 2.2, it follows that there is a locally unique solution ˆ θ(j) n near θ(j) 0 that furthermore satisfies a linear expansion. This proves that the landscape of the sam-ple loss function is similar to the landscape of the population loss function under more general setting than in Mei et al. (2016). Note however that our result does not imply critical points for E[ℓ(θ, W1)] near the critical points of Pn i=1 ℓ(θ, Wi).⋄ 3. Deterministic Inequality for Smooth Convex Loss Functions In this section, we consider M-estimators obtained from objective functions that are averages of convex loss functions. Consider the estimator ˆ θn := arg min θ∈Rq 1 n n X i=1 L(θ; Wi), (8) for some observations W1, . . . , Wn and some loss function L(·; w) that is convex and twice differentiable. Several important examples are as follows: Example 3.1 (Maximum Likelihood). Maximum likelihood estimator (MLE) is one of the most popular estimators in statistics; widely used in practice and backed by the asymptotic efficiency theory. Suppose W1, . . . , Wn are iid random variables from a parametric family (of densities) {fθ(·) : θ ∈Θ}. The MLE is defined as ˆ θn := arg min θ∈Θ 1 n n X i=1 {−log fθ(Wi)} .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 9 If the parametric model family is an exponential family, then the negative log likelihood (−log fθ(·)) is convex in θ. Even though the construction of the estimator is motivated by the hypothesis/assumption of iid random variables with density belonging to the parametric model postulated, it is important to understand the implication of misspecification of different directions; see Huber (1967) and Buja et al. (2014) for a discussion.
Example 3.2 (Generalized Linear Models and variants). Regression analysis pro-vides a large class of estimation problems which emphasize the problem of esti-mating the “relation” between a response (Y ) and a collection of predictors (X).
Generalized linear models (GLMs) form an important sub-class of regression mod-els. More generally, we can consider the estimator ˆ θn := arg min θ∈Rq 1 n n X i=1 L(θ⊤Xi; Xi, Yi), for regression data W1 = (X1, Y1), . . . , Wn = (Xn, Yn). Some specific examples of L(·; ·, ·) are as follows: 1. Canonical GLMs are obtained by taking L(u; x, y) = ψ(u) −yu, for some convex function ψ(·). For instance, OLS is obtained when ψ(u) = u2/2, logistic regression is obtained when ψ(u) = log(1+exp(u)) and Poisson regression is obtained when ψ(u) = exp(u).
Even though canonical GLMs are motivated from an exponential family for the conditional distribution of Y given X, one can consider functions L(·; ·, ·) that do not correspond to the log-likelihood of an exponential family. For example, probit regression is obtained when L(u; x, y) = −y log Φ(u) −(1 −y) log(1 −Φ(u)) for y ∈[0, 1], and negative binomial regression corresponds to L(u; x, y) = −yu + y + α−1 log(1 + α exp(u)), for some α > 0.
2. Robust regression is an important aspect of practical data analysis and a simple way to robustify an estimator is by ignoring observations that are outliers. In this respect, the loss functions of the form L(u, x, y) = h(x, y)ℓ(u, y), Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 10 are of interest. For instance, one can take ℓ(u, y) = ψ(u) −yu as in the GLM loss function and take h(·, ·) to be a “down-weighting” function. Choices of weight functions for robust regression can be found in Loh et al. (2017).
Motivated by these examples, we prove the following result for ˆ θn obtained from the M-estimation problem (8). For this result, consider the following notation and assumptions. For the loss function L(θ; w), let ∇L(θ; w) and ∇2L(θ; w) denote the gradient and the Hessian of the function L(θ; w) with respect to θ. Set, for any θ ∈Rq, δn(θ) := 1.5∥[ ˆ Qn(θ)]−1 ˆ Zn(θ)∥2, where ˆ Zn(θ) := 1 n n X i=1 ∇L(θ; Wi) ∈Rq and ˆ Qn(θ) := 1 n n X i=1 ∇2L(θ; Wi) ∈Rq×q.
Also, define for u ≥0, C(u, w) := sup ∥θ1−θ2∥≤u sup e∈Rq: ∥e∥2=1 e⊤∇2L(θ1, w)e e⊤∇2L(θ2, w)e.
Note that if L(·; w) is strictly convex and twice differentiable for each w, then C(u, w) is well-defined and positive. Also, note that C(u, w) ≥1 for all u and w.
(A1) The function L(θ; w) is convex and twice differentiable in θ for every w.
(A2) Fix any target vector θ0 ∈Rq. The event En occurs where En := max 1≤i≤n C(δn(θ0), Wi) ≤4 3 .
Theorem 3.1. Under assumptions (A1) and (A2), there exists a vector ˆ θn ∈Rq such that ˆ Zn(ˆ θn) = 0, and 1 2δn(θ0) ≤∥ˆ θn −θ0∥2 ≤δn(θ0).
(9) Moreover, ˆ θn −θ0 + [ ˆ Qn(θ0)]−1 ˆ Zn(θ0) 2 ≤max 1≤i≤n {C(δn(θ0), Wi) −1} δn(θ0).
(10) Proof. See Appendix B for a proof.
Remark 3.1 (Discussion on the Assumptions) It is easy to see that assump-tion (A2) implies assumption (A1) since otherwise the event En cannot hold. Also, from the proof of Theorem 3.1, it follows that the definition of C(·, ·) can be re-placed by C(u, w) := sup ∥θ−θ0∥≤u sup e∈Rq: ∥e∥2=1 max e⊤∇2L(θ, w)e e⊤∇2L(θ0, w)e, e⊤∇2L(θ0, w)e e⊤∇2L(θ, w)e .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 11 The only difference is that we restrict to θ-vectors that are close to the target θ0.
The main reason behind the deterministic inequality is that the objective func-tion can be both upper and lower bounded by (different) quadratic functions.
Similar results under additive (rather than a ratio-type) assumption can be found in Spokoiny (2012, Corollary 3.4).
⋄ Remark 3.2 (Linear Representation) The quantity C(u, w) relates to continuity of the function ∇2L(·, w) and usually converges to 1 as u →0. If this convergence holds, then from Theorem 3.1 it follows that as long as δn(θ0) →0, ˆ θn −θ0 ≈[ ˆ Qn(θ0)]−1 ˆ Zn(θ0).
The classical proof of asymptotic normality of estimator ˆ θn obtains an average on the right hand side and the above quantity is not an average because of ˆ Qn(θ0). It is easy to replace the average ˆ Qn(θ0) by its expectation as follows. Note that [ ˆ Qn(θ0)]−1 ˆ Zn(θ0) −[Qn(θ0)]−1 ˆ Zn(θ0) 2 ≤∥[Qn(θ0)]−1 ˆ Qn(θ0) −I∥op [ ˆ Qn(θ0)]−1 ˆ Zn(θ0) 2 ≤∥[Qn(θ0)]−1 ˆ Qn(θ0) −I∥opδn(θ0).
Therefore, ˆ θn −θ0 + [Qn(θ0)]−1 ˆ Zn(θ0) 2 ≤ max 1≤i≤n C(δn(θ0), Wi) −1 + ∥[Qn(θ0)]−1 ˆ Qn(θ0) −I∥op δn(θ0).
(11) In the steps above, it is irrelevant what Qn(θ0) is but a classical choice is given by Qn(θ0) := 1 n n X i=1 E [∇2L(θ; Wi)] .
Finally, if the coefficient of δn(θ0) in (11) is op(1), then inequality (11) proves that ∥ˆ θn −θ0∥2 = (1 + op(1)) [Qn(θ0)]−1 ˆ Zn(θ0) 2 = (1 + op(1))2δn(θ0) 3 .
⋄ An application of Theorem 3.1 for asymptotic normality of M-estimators under a specific dependence structure can be completed using the steps below.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 12 1. Define the target θ0 as a solution to the equation Zn(θ) = 0, where Zn(θ) := 1 n n X i=1 E [∇L(θ; Wi)] ∈Rq.
This choice of θ0 ensures that E[ ˆ Zn(θ0)] = 0 and so, ˆ Zn(θ0) becomes a mean zero average.
2. Prove that ∥ˆ Zn(θ0)∥2 = op(1) under the assumed dependence structure. Con-trolling the Euclidean norm can be based on the following inequality: ∥ˆ Zn(θ0)∥2 = sup ∥ν∥2≤1 ν⊤ˆ Zn(θ0) ≤2 max ν∈N1/2 ν⊤ˆ Zn(θ0), (12) where N1/2 ⊂B(0, 1) denotes the 1/2-covering set of B(0, 1), that is, sup ν∈B(0,1) inf µ∈N1/2 ∥ν −µ∥2 ≤1/2.
From Lemma 4.1 of Pollard (1990), it follows that the cardinality of N1/2, |N1/2|, is bounded by 6q. Note that inequality (12) is sharp up to the factor of 2. This inequality shows that the tail bounds on ν⊤ˆ Zn(θ0) can be used to control ∥ˆ Zn(θ0)∥2.
3. Prove that ˆ Qn(θ0) −Qn(θ0) op = op(1).
This would imply if Qn(θ0) is positive definite then ˆ Qn(θ0) is also positive definite for sufficiently large n. Similar to the Euclidean norm, the operator norm can also be bounded in terms of a finite maximum. By Lemma 2.2 of Vershynin (2012), it follows that ∥ˆ Qn(θ0) −Qn(θ0)∥op ≤2 max ν∈N1/4 ν⊤ˆ Qn(θ0)ν −ν⊤Qn(θ0)ν , where again N1/4 ⊂B(0, 1) represents the 1/4-covering number of B(0, 1) and by Lemma 4.1 of Pollard (1990), |N1/4| ≤12q.
The quantities ν⊤ˆ Zn(θ0) and ν⊤ˆ Qn(θ0)ν being averages are much easier to study under various dependence settings of interest. Exponential-type tail bounds for averages under independence and functional dependence are given in Theorems A.1 and B.1, respectively, of Kuchibhotla et al. (2018a).
Assumption (A2) is used in the proof of Theorem 3.1 only to prove condition (3) in Theorem 2.1. So, any alternative condition implying (3) can be used instead. The Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 13 assumption on the ratio rather than the difference of Hessians is more appealing since minimizers do not depend on the scaling of objective functions. The function C(·, ·) naturally cancels out the scalings and requires much weaker conditions as discussed in Section 3.1.
The function C(·, ·) can be bounded easily for self-concordant type convex func-tions. Proposition 1 of Bach (2010) bounds C(·, ·) for logistic regression and see Proposition 8 of Sun and Tran-Dinh (2017) for a general class of convex functions called generalized self-concordant where the ratio of the Hessians is bounded. Also, see Karimireddy et al. (2018) for other examples.
One specific corollary of Theorem 3.1 in regression analysis is of special inter-est for our applications. For this result, consider independent random variables (Xi, Yi) ∈Rp × R (1 ≤i ≤n) and the estimator ˆ βn := arg min θ∈Rp 1 n n X i=1 h(Xi)ℓ(X⊤ i θ, Yi), for some loss function ℓ(·, ·) convex and twice differentiable in the first argument.
Here the “weight” h(·) is any function not depending on θ. Observe that if h(·) is not a non-negative function, the objective function is not necessarily convex.
Define the target vector βn := arg min θ∈Rp 1 n n X i=1 E h(Xi)ℓ(X⊤ i θ, Yi) .
The function h(x)ℓ(x⊤θ, y) can be changed to any function of the form ℓ(x⊤θ; x, y) but for simplicity we restrict to the function above. Let ℓ′(u, y) := ∂ ∂tℓ(t, y) t=u and ℓ′′(u, y) := ∂ ∂tℓ′(t, y) t=u .
Define the analogue of the C function, C(u, y) := sup |s−t|≤u ℓ′′(s, y) ℓ′′(t, y) .
Finally, define the analogues of ˆ Zn(·), ˆ Qn(·), ˆ Zn(θ) := 1 n n X i=1 ℓ′(X⊤ i θ, Yi)h(Xi)Xi, and ˆ Qn(θ) := 1 n n X i=1 ℓ′′(X⊤ i θ, Yi)h(Xi)XiX⊤ i , δn(θ) := 3 2 [ ˆ Qn(θ)]−1 ˆ Zn(θ) 2 , and Qn(θ) := 1 n n X i=1 E ℓ′′(X⊤ i θ, Yi)h(Xi)XiX⊤ i .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 14 Corollary 3.1. If ℓ(·, ·) is a twice differentiable function that is convex in the first argument and for some β0 ∈Rp, max 1≤i≤n C (∥Xi∥2 δn(β0), Yi) ≤4 3, (13) then there exists a vector ˆ βn ∈Rp satisfying ˆ Zn(ˆ βn) = 0, 1 2δn(β0) ≤∥ˆ βn −β0∥2 ≤δn(β0), and ˆ βn −β0 + [Qn(β0)]−1 ˆ Zn(β0) 2 ≤ max 1≤i≤n C(∥Xi∥2 δn(β0), Yi) −1 + [Qn(β0)]−1 ˆ Qn(β0) −I op δn(β0).
Proof. See Appendix B for a proof.
Example 3.3 (Linear Models). In the following, we bound the function C in case of several linear models. Since Corollary 3.1 does not require any specific stochastic or model assumptions, the following examples also do not require any “correct” modeling assumptions and are deterministic in nature.
1. Linear Regression: In case of ordinary least squares (OLS) linear regression, the loss function is given by ℓ(t, y) = (t −y)2 and the weight function is identically 1. So, ℓ′′(u, y) = 2 and C(u, y) = 1 for all u, y. This implies that the assumption (A2) always holds. This is an expected result since the least square estimator satisfies 1 n n X i=1 Xi Yi −X⊤ i ˆ βn = 0, and subtracting β0 from ˆ βn implies that 1 n n X i=1 XiX⊤ i ! ˆ βn −β0 = 1 n n X i=1 Xi(Yi −X⊤ i β0).
Here β0 is the target OLS vector defined by β0 := arg min θ∈Rp 1 n n X i=1 E (Yi −X⊤ i θ)2 .
This proves that ∥ˆ βn −β0∥2 = 2δn(β0) 3 .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 15 The second conclusion of Corollary 3.1 provides better information: ˆ β0 −β0 −1 n n X i=1 Σ−1 n Xi(Yi −X⊤ i β0) 2 ≤ Σ−1 n ˆ Σn −I op δn(β0), where ˆ Σn := 1 n n X i=1 XiX⊤ i , and Σn := 1 n n X i=1 E XiX⊤ i .
Details on how to bound δn(β0) in case of independent/functionally depen-dent data were provided in Kuchibhotla et al. (2018a).
2. Poisson Regression: In case of Poisson regression, the loss function is ℓ(t, y) = exp(t)−yt and the weight function is identically 1. So, ℓ′′(t, y) = exp(t). This implies that C(u, y) = exp(u). The event (13) is equivalent to max 1≤i≤n ∥Xi∥2 δn(β0) ≤log (4/3) .
On this event, max 1≤i≤n C (∥Xi∥2 δn(β0)) −1 ≤4δn(β0) 3 max 1≤i≤n ∥Xi∥2 .
Thus, Corollary 3.1 implies that there exists ˆ βn ∈Rp such that ˆ βn −β0 −1 n n X i=1 [Qn(β0)]−1Xi Yi −exp(X⊤ i β0) 2 ≤ 4δn(β0) 3 max 1≤i≤n ∥Xi∥2 + [Qn(β0)]−1 ˆ Qn(β0) −I op δn(β0), where ˆ Qn(β) := 1 n n X i=1 XiX⊤ i exp X⊤ i β , and Qn(β) = E h ˆ Qn(β) i .
Since ˆ Qn(β0) is a Gram matrix based on random vectors Xi exp(X⊤ i β0/2), 1 ≤ i ≤n, the results of Kuchibhotla et al. (2018a) can still be applied to show that ˆ Qn(β0) is close to Qn(β0).
3. Logistic and Negative Binomial Regression: In case of logistic regression, the loss function is given by ℓ(u, y) = log(1 + exp(u)) −yu, Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 16 and the weight function is identically 1. It is easy to show that ℓ′′(u, y) = exp(u) (1 + exp(u))2, and C(u, y) = sup |s−t|≤u exp(s)(1 + exp(t))2 (exp(s) + 1)2 exp(t).
Since exp(s) ≤exp(u) exp(t) for all s, t satisfying |s −t| ≤u, it follows that C(u, y) ≤sup |s−t|≤u exp(s) exp(t) sup |s−t|≤u (exp(s) + 1)2 (exp(t) + 1)2 ≤exp(3u).
For the case of negative binomial regression (with parameter α > 0), the loss function is ℓ(u, y) = −yu + [y + 1/α] log(1 + α exp(u)), and the weight function is identically 1. So, ℓ′′(u, y) = α[y + 1/α] exp(u) (α exp(u) + 1)2 , and C(u, y) = sup |s−t|≤u exp(s)(α exp(t) + 1)2 (α exp(s) + 1)2 exp(t).
Similar to the logistic regression case, we get C(u, y) ≤exp(3u). Therefore, condition (13) becomes max 1≤i≤n ∥Xi∥2 δn(β0) ≤log(4/3) 3 .
Hence calculations similar to the Poisson regression case still hold true.
In the examples above, we have controlled the function C(u, y) for some widely used convex examples. When the loss function ℓ(·, y) is strongly convex, then C(u, y) −1 can be bounded by C sup{|ℓ′′(s, y) −ℓ′′(t, y)s −t| ≤u} for some constant C > 0. It should, however, be noted that C(u, y) may not be a bounded function even if the function ℓ(·, y) is strictly convex. A possible example is probit regression. In this case the approch used in Section 5 works easily.
3.1. Comparison with assumptions in the literature Results similar to Corollary 3.1 were presented in Li et al. (2017, Theorem 1), Liang and Du (2012, Theorem 1), Negahban et al. (2009, Corollary 3) and He and Shao (2000, Example 3). In these papers the authors assume a lower bound on the second order curvature, that is, inf 1≤i≤n inf ∥θ−β0∥2≤ε ℓ′′(Yi, X⊤ i θ) ≥κ > 0 for some (small enough) ε > 0.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 17 This is a difficult assumption to be satisfied in case of increasing dimension since ℓ′′(y, u) converges to zero as u →−∞usually and often |X⊤ i β0| itself grows with the dimension. This hurdle poses certain unnecessary rate constraints on the di-mension. In contrast our assumption is based on difference meaning X⊤ i (θ −β0) which can be expected to be small as long as ∥θ −β0∥2 is small even with increas-ing dimension. See the discussion surrounding equation (1.8) and Theorem 2.4 of Bose and Sengupta (2003) for related ratio-type assumptions.
It is clear from Corollary 3.1 that the function C(u, y) plays a very important role in the existence and determining the rate of convergence of the estimator.
The following proposition (proved in Appendix A) allow construction of new loss functions with a control on the C(·, ·) function.
Proposition 3.1. Suppose CT (indexed by a non-negative function T(·, ·)) is the class of all loss functions L(·, ·) convex in the first argument and satisfying sup ∥θ1−θ2∥≤u sup e∈Rq: ∥e∥2=1 e⊤∇2L(θ1, w)e e⊤∇2L(θ2, w)e ≤T(u, w) for all u ≥0 and w.
Then CT is a convex cone.
4. Deterministic Inequality for Cox Proportional Hazards Model One of the most widely used models in survival analysis is the celebrated Cox pro-portional hazards model. The partial log-likelihood of the Cox model even though not an average can be dealt using our theory. The analysis in this section is related to the discussion in Section 6 of Hjort and Pollard (2011). The usual Cox regression model for possibly censored lifetimes with covariate information is as follows: The individuals have independent lifetimes T 0 1 , . . . , T 0 n and the i-th subject has hazard rate λi(s) := λ(s) exp β⊤ 0 Xi,s , (14) for some vector β0, some baseline hazard function λ(·) and i-th subject covariate Xi,s ∈Rp. The classical Cox model has a fixed set of covariates not depending on time s and here they are allowed to depend on time. There is a possibly interfering censoring time Ci leaving the observables to be Ti = min{T 0 i , Ci} and δi = 1{T 0 i ≤Ci}.
Consider the risk indicator function Yi,s = 1{Ti ≥s}, and the counting process Ni with mass δi at Ti, that is, dNi(s) := 1{Ti ∈[s, s + ds], δi = 1}.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 18 The log-partial likelihood is then given by Gn(β) := n X i=1 Z ∞ 0 β⊤Xi,s −log Rn(s, β) dNi(s), where Rn(s, β) := n X i=1 Yi(s) exp β⊤Xi,s .
The Cox estimator is the value ˆ βn that maximizes the log-partial likelihood. Even though the motivation above is through a correct model (14), we do not make any such assumptions and prove a purely deterministic result. Define for β ∈Rp, ˆ Ln(β) := n X i=1 Z ∞ 0 H1(Xi,s) log Rn(s, β) −β⊤Xi,s dNi(s), where Rn(s, β) := n X i=1 H2(Xi,s)Yi(s) exp β⊤Xi,s .
The objective function ˆ Ln(·) is a generalization of Gn(·) allowing for two func-tions H1(·) and H2(·) that can be used to down-weight outliers in the covariate space. Note that this generalization does not change the convexity property of the objective function. The Cox estimator based on ˆ Ln(·) is given by ˆ βn := arg min θ∈Rp ˆ Ln(θ).
Define for β ∈Rp, ˆ Zn(β) := n X i=1 Z ∞ 0 H1(Xi,s) ( ˙ Rn(s, β) Rn(s, β) −Xi,s ) dNi(s), where ˙ Rn(s, β) := ∂Rn(s, β) ∂β .
Define the Jacobian as ˆ Qn(β) := ∇ˆ Zn(β). Finally define for any β0 ∈Rp, ¯ Xn,s(β0) := Pn i=1 Xi,sH2(Xi,s)Yi(s) exp β⊤ 0 Xi,s Rn(s, β0) .
Theorem 4.1. Set for any target vector β0 ∈Rp, µn(s) := max 1≤i≤n Xi,s −¯ Xn,s(β0) 2 , and δn(β0) := 3 2 [ ˆ Qn(β0)]−1 ˆ Zn(β0) 2 .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 19 If sup 0≤s<∞ µn(s)δn(β0) ≤1 16, then there exists a vector ˆ βn ∈Rp satisfying ˆ Zn(ˆ βn) = 0, and 1 2δn(β0) ≤∥ˆ βn −β0∥2 ≤δn(β0).
Furthermore, ˆ βn −β0 + [ ˆ Qn(β0)]−1 ˆ Zn(β0) 2 ≤8e1/4δ2 n(β0) sup s µn(s).
(15) 5. Deterministic Inequalities for Non-convex M -estimators In previous sections we have proved the applicability of Theorem 2.1 for convex loss functions. However, Theorem 2.1 does not require “monotonicity”3 of the function f(·). In this section, we provide one specific non-convex example, namely, non-linear regression.
5.1. Least Squares Non-linear Regression For a motivation of non-linear regression, consider the problem of binary linear classification based on n paris (X1, Y1), . . . , (Xn, Yn) with Yi ∈{0, 1} and Xi ∈Rp.
In this model, the quantity of interest is the conditional probability of Yi given Xi.
Suppose P(Yi = 1|Xi = x) = σ(x⊤θ0) with θ0 ∈Rp and a function σ : R →[0, 1].
Since this implies E[Yi|Xi = x] = σ(x⊤θ0), one possible estimator of θ0 is obtained by minimizing the squared error loss: 1 n n X i=1 Yi −σ(X⊤ i θ) 2 , with respect to θ ∈Rp. It is easy to see that the loss function above is, in gen-eral, non-convex. In constrast to convex losses (e.g., hinge or logistic), non-convex loss functions as above have better classification accuracy in various scenarios; see Nguyen and Sanner (2013) and Mei et al. (2016).
As a generalization consider the observations (Xi, Yi) ∈Rp × R for 1 ≤i ≤n and the loss function Fn(θ) := 1 n n X i=1 Yi −g(θ⊤Xi) 2 , 3Derivatives of differentiable one-dimensional convex functions are non-decreasing.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 20 for a known function g(·) that is twice differentiable and bounded. (We do not restrict to Yi ∈{0, 1}.) To prove a deterministic inequality for the stationary points of Fn(·), we use the following assumption: (NR) The function g(·) is twice differentiable and there exists functions C0(·), C1(·), C2(·) such that for some α ∈(0, 1] and any x, θ1, θ2, g(x⊤θ1) −g(x⊤θ2) ≤C0(x) ∥θ1 −θ2∥2 , g′(x⊤θ1) −g′(x⊤θ2) ≤C1(x) ∥θ1 −θ2∥2 , and g′′(x⊤θ1) −g′′(x⊤θ2) ≤C2(x) ∥θ1 −θ2∥α 2 .
Assumption (NR) is satisfied for many classical activation functions with α = 1 (for example, logistic function). Another important example satisfying assump-tion (NR) is the phase retrieval problem where g(t) = t2; see Yang et al. (2017) for recent developments. From the proof of Corollary 5.1, it follows that assump-tion (NR) can be relaxed to θ1, θ2 ∈Br(θ0) for some r > 0. Define for any θ ∈Rp, δn(θ) := 1.5 ∥(∇2Fn(θ))−1∇Fn(θ)∥2 and L2(θ) := 2 n n X i=1 C2 1(Xi) (∇2Fn(θ))−1 XiX⊤ i op , L1+α(θ) := 2 n n X i=1 C0(Xi)C2(Xi) (∇2Fn(θ))−1 XiX⊤ i op , L1(θ) := 2 n n X i=1 2C1(Xi)|g′(X⊤ i θ)| + C0(Xi)|g′′(X⊤ i θ)| (∇2Fn(θ))−1 XiX⊤ i op , Lα(θ) := 2 n n X i=1 C2(Xi)|Yi −g(X⊤ i θ)| (∇2Fn(θ))−1 XiX⊤ i op .
The following result shows the existence of a solution that satisfies an asymptotic expansion. The proof (in Appendix D) verifies the assumptions of Theorem 2.2.
Corollary 5.1. Under assumption (NR), for any θ0 satisfying δn(θ0) ≤min (12Lj(θ0))−1/j : j ∈{α, 1, 1 + α, 2} , (16) there exists a unique solution ˆ θn of ∇Fn(θ) = 0 in B(θ0, δn(θ0)) and this solution ˆ θn satisfies ˆ θn −θ0 + (∇2Fn(θ0))−1∇Fn(θ0) 2 ≤ω(δn(θ0))δn(θ0), Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 21 where for r ≥0, ω(r) := L2(θ0)r2 + L1+α(θ0)r1+α + L1(θ0)r + Lα(θ0)rα.
Corollary 5.1 can be compared to Theorem 4 of Mei et al. (2016). As described in Remark 2.1, if θ(j), 1 ≤j ≤K denote the solutions of ∇E[Fn(θ)] = 0 and E[Fn(θ)] is a Morse function4, then by Corollary 5.1 the sample estimating equation ∇Fn(θ) = 0 also has solutions near θ(j) for each 1 ≤j ≤K. Further, the result applies for a larger class of link functions g and allows for dependent observations.
Also, note that we do not need to verify uniform in θ control of the gradient/Hessian which was required in Mei et al. (2016).
6. Deterministic Inequalities for Equality Constrained Problems In the context of linear models, hypothesis tests related to linear combinations of the coefficients form an important component of applied analysis. For instance, it is of interest to know if the treatment effect is more than that of the control when both effects are measured in terms of the coefficients in the linear model. See Section 1.4 of Amemiya (1985) for details.
Consider the problem of minimizing a twice differentiable function Fn(β) subject to Aβ = b, for some matrix A ∈Rd×p of full row rank and vector b ∈Rd. A vector β⋆∈Rp is a minimizer of this constrained problem only if there exists a vector ν⋆∈Rd such that the following KKT equations are satisfied: Aβ⋆= b, and ∇Fn(β⋆) + A⊤ν⋆= 0.
(17) If, in addition, the function Fn(·) is convex, then the KKT equations are also sufficient. Some commonly used convex examples of Fn(β) are Fn(β) = 1 n n X i=1 ψ(X⊤ i β) −YiX⊤ i β , (18) with ψ(t) ∈{t2/2, log(1 + exp(t)), exp(t)}. A non-convex example of Fn(·) is Fn(β) = 1 n n X i=1 Yi −g(X⊤ i β) 2 , (19) with g(·) satisfying assumption (NR).
4A function R(θ) is said to be a Morse function if for any θ0 satisfying ∇R(θ0) = 0, the Hessian ∇2R(θ0) is invertible.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 22 The following result proves the existence and an expansion for a local minimizer in equality constrained problems. For this result, define for β ∈Rp, and ν ∈Rd, δn(β, ν) := 1.5 1 + (A[∇2Fn(β)]−1A⊤)−1A op [∇2Fn(β)]−1(∇Fn(β) + A⊤ν) 2 .
Corollary 6.1. Fix vectors ν0 ∈Rd and β0 ∈Rp such that Aβ0 = b. Suppose Fn(·) is a twice differentiable function. If there exist constants L ≥0 and α ∈(0, 1], such that for all β ∈B(β0, (3L)−1/α), [∇2Fn(β0)]−1(∇2Fn(β) −∇2Fn(β0)) op ≤L ∥β −β0∥α 2 , (20) and δn(β0, ν0) ≤(3L)−1/α, then there exists a vector (ˆ βn, ˆ νn) ∈Rp × Rd solving the KKT equations (17) and the vector ˆ βn satisfies the expansion ˆ βn −β0 −[Jn(β0)]−1(∇Fn(β0) + A⊤ν0) 2 ≤L[δn(β0, ν0)]1+α.
(21) Here Jn(β0) := [∇2Fn(β0)] I −[∇2Fn(β0)]−1A⊤(A[∇2Fn(β0)]−1A⊤)−1A −1 .
Note that condition (20) is verified for the examples (18) and (19) in Sections 3 and 5. For an application of this result in statistical context, one would take β0 ∈Rp as the minimizer of E[Fn(β)] subject to Aβ = b. The vector ν0 ∈Rd would be the vector satisfying the “population” KKT equations Aβ0 = b and E[∇Fn(β0)] + A⊤ν0 = 0.
This implies that ∇Fn(β0) + A⊤ν0 is a mean zero random vector and so, the expansion (21) implies asymptotic normality of the (properly normalized) local minimizer ˆ βn. It is easy to generalize Corollary 6.1 when the linear equality con-straint Aβ = b is replaced by a non-linear constraint G(β) = 0 (which makes the problem non-convex even if Fn(β) is convex).
Remark 6.1 (General Constraints) It is of considerable interest to extend Corol-lary 6.1 to M-estimation problems with more general inequality/abstract con-straints. It is not clear if a useful deterministic inequality is possible. For example, consider the minimization problem min β Fn(β) subject to Gn(β) = 0, Hn(β) ≥0.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 23 Suppose the functions Fn(·), Gn(·), Hn(·) are twice differentiable. Define the La-grangian function Ln(β, λ, µ) := Fn(β) −λ⊤Gn(β) −µ⊤Hn(β).
A vector β⋆is a (local) minimizer only if there exist λ⋆and µ⋆such that ∇βLn(β⋆, λ⋆, µ⋆) = 0, Gn(β⋆) = 0, Hn(β⋆) ≥0, µ⋆≥0, H⊤ n (β⋆)µ⋆= 0.
The inequalities above can be converted to equalities as follows. Define the function M(u, v) = √ u2 + v2−u−v for any two vectors u, v (Here √ u2 + v2 is evaluated as a componentwise operation). Then the last three inequalities of the KKT conditions can be equivalently written as M(Hn(β⋆), µ⋆) = 0.
The function M(·, ·) is known in mathematical programming literature as the Fis-cher–Burmeister function. Thus the revised KKT conditions can be written as ∇βLn(β⋆, λ⋆, µ⋆) = 0, Gn(β⋆) = 0, and M(Hn(β⋆), µ⋆) = 0.
(22) The advantage of (22) is that there are only equations and no inequalities. However, the function M(Hn(β), µ) is not Fr´ echet differentiable but only B-differentiable (or semi-smooth). There are semilocal convergence results for Newton’s method available in this respect; see Chen (1997) and Wang (2008). For a general treatment of variational inequality problems (VIPs), see Izmailov and Solodov (2014). But explicit application of these results require certain complimentary qualification conditions that make their usefulness unclear as a general solution; see Klatte (1987), Dupaˇ cov´ a (1991), and Wang (2000).
⋄ 7. Applications of the Deterministic Inequalities In the previous sections, we have proved deterministically that the estimator normalized around the target behaves like an average when the objetive func-tion is an average. Averages are statistician’s friend: most of statistical inference is based on the fact that averages are close to being normally distributed and can be bootstrapped under various dependence structures of interest. In the fol-lowing subsections, we provide applications of the deterministic inequalities for subsampling/cross-validation methods and two problems related to post-selection inference.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 24 7.1. Cross-validation and Subsampling In this section, we consider applications of the deterministic inequalities in un-derstanding estimators computed based on a subset of the data. Two specific sta-tistical methods that consider estimators based on a subset are cross-validation (CV) and subsampling. Leave-one-out CV predicts the response based on esti-mator computed using n −1 observations. In subsampling with a subsample size b = bn, estimators computed with bn observations are compared to the one with n observations. Leave-one/k-out CV is a popular method for estimating the out-of-sample prediction risk of a model and subsampling is useful in construction of asymptotic confidence intervals. Similar subset estimators appear in the case of delete-d-jackknife. See Stone (1977), Shao (1993), Politis et al. (1999) and Shao and Wu (1989) for a detailed discussion of these methods.
For the result in this section, we consider the setting of Theorem 3.1. The ob-servations are W1, W2, . . . , Wn. Define the estimator ˆ θn as a solution of ˆ Zn(θ) = 1 n n X i=1 ∇L(θ; Wi) = 0.
(23) For simplicity, we first consider the leave-one-out estimator and then consider leave-k-out estimator. For any 1 ≤I ≤n, define the estimator ˆ θ−I as a solution of 1 n −1 X 1≤i≤n, i̸=I ∇L(θ, Wi) = 0.
Under the condition (24) of Corollary 7.1 (below) the existence of ˆ θ−I follows from Theorem 3.1. Also, define for 1 ≤I ≤n, δI,n := n−1∥ˆ Q−1 n ∇L(ˆ θn, WI)∥2 1 −n−1∥ˆ Q−1 n ∇2L(ˆ θn, WI)∥op , where ˆ Qn := 1 n n X i=1 ∇2L(ˆ θn, Wi).
Applying Theorem 3.1 for the estimator ˆ θ−I and target ˆ θn, we get the following result, a detailed proof of which can be found in Appendix F.
Corollary 7.1. Consider the loss function L(·, ·) as in assumption (A1). If δI,n ≥0 for all 1 ≤I ≤n and max 1≤i̸=I≤n C (1.5δI,n, Wi) ≤4 3, (24) then for all 1 ≤I ≤n, ˆ θ−I −ˆ θn −n−1 ˆ Q−1 n ∇L(ˆ θn, WI) 2 ≤3δI,n 2 max 1≤i̸=I≤n C(1.5δI,n, Wi) −1 + n−1 ˆ Q−1 n ∇2L(ˆ θn, WI) op .
(25) Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 25 Remark 7.1 (Comments on the approximation rate) Corollary 7.1 shows that the difference between ˆ θ−I and ˆ θn can be bounded in terms of quantities computable based on the sample of n observations. This is, indeed, expected since ˆ θ−I and ˆ θn are computable based on the sample of n observations. It should be stressed again that Corollary 7.1 is a purely deterministic result and does not require any stochasticity assumptions on the observations. The result can also be readily used to reduce the computational burden of leave-one-out CV. Since ˆ Qn is an average, under most dependence structure would be asymptotically deterministic and so, δI,n = Op(n−1) as n →∞. Therefore, the expansion error in (25) is in general of order op(n−1). In fact, if C(·, w) is differentiable at 0, then the expansion error is of the order Op(n−2).
Following the examples in Section 3 condition (24) can be written explicitly for many common regression examples. A particularly illuminating example is the case of linear regression where condition (24) is satisfied for any set of observations since C(·, ·) ≡1 and the error bound in (25) becomes 1.5n−1δI,n∥ˆ Q−1 n ∇2L(ˆ θn, WI)∥op. ⋄ Leave-one-out CV and delete-1-jackknife are known to have poorer properties in comparison to the leave-k-out CV and delete-d-jackknife methods (see, e.g., Shao (1993)). For this reason, it is of interest to consider the error obtained in removing more than one observation at a time. The result in this case is also very similar to Corollary 7.1, albeit with a larger error which is expected. The proof of the following result can be found in Appendix F. Suppose I is a subset of {1, 2, . . . , n} with |I| < n (think |I| = o(n)) and consider the estimator ˆ θ−I as a solution of X 1≤i≤n, i/ ∈I ∇L(θ, Wi) = 0.
Here |I| denotes the cardinality of the set I. Define δI,n := n−1 ˆ Q−1 n P i∈I ∇L(ˆ θn, Wi) 2 1 −n−1 ˆ Q−1 n P i∈I ∇2L(ˆ θn, Wi) op , where ˆ Qn := 1 n n X i=1 ∇2L(ˆ θn, Wi) Corollary 7.2. Under the setting of Corollary 7.1, if δI,n ≥0 and C (1.5δI,n, Wi) ≤ 4/3, for all i ∈Ic ∩{1, 2, . . . , n}, then ˆ θ−I −ˆ θn −1 n ˆ Q−1 n X i∈I ∇L(ˆ θn, Wi) 2 (26) ≤3δI,n 2 max 1≤i≤n, i̸=I C(1.5δI,n, Wi) −1 + n−1 ˆ Q−1 n X i∈I ∇2L(ˆ θn, Wi) op .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 26 Clearly, Corollary 7.2 reduces to Corollary 7.1 when I is a singleton. Even in Corollary 7.2 one can take maximum over a collection of subsets I. Similar to the case in Remark 7.1, under differentiability of C(·, w) at 0, the expansion error of (26) is of the order Op(|I|2n−2). If |I| = O(n) many observations are removed then it might be better to compare ˆ θ−I to θ0 than to ˆ θn. In case of subsampling or m-of-n bootstrap, the subset of observations are chosen as “iid sample” from the empirical distribution. In these cases, a reasonable choice for the target vector is ˆ θn.
In case of cross-validation, the subset is not a random sample from the empirical distribution and so θ0 is a good choice for the target vector.
It is easy to see that Corollaries 7.1 and 7.2 can be extended to the case of Cox proportional hazards model and to the other cases given in previous sections.
Since the result of deterministic nature, it is interesting to consider the worst case approximation when considering uniform over all subsets I of size k (with k allowed to change with n). For instance if k = √n, then the total number of subsets is of the order O(n √n/2) which makes it hard to derive a good (polynomial) rate of convergence of the supremum even if the averages have exponential concentration inequalities.
7.2. Marginal Screening In the current era of data science, one is often encountered with a larger number of covariates/predictors in regression data than the number of samples. In this scenario, it has become a common practice to select a subset of covariates either by screening using marginal effects or by some regularized methods. The recent works McKeague and Qian (2015) and Wang et al. (2018) provide a formal test-ing framework for the existence of any active predictors in linear and quantile regression settings.
In the linear regression case, the setting is as follows: (X, Y ) ∈Rp+1 and (Xi, Yi), 1 ≤i ≤n are iid random vectors and we want to test if the maximal correlation between X(j) (the j-th coordinate of X) and Y is non-zero. This ques-tion in case of non-singular E[XX⊤] is same as testing if there exists any subset of covariates that has linear predictive ability for the response Y . To see this let X(M) for M ⊆{1, 2, . . . , p} be a subvector of X with indices in M and define the OLS regression target βM := arg min θ∈R|M| E (Yi −X⊤ i (M)θ)2 = E[X(M)X⊤(M)] −1 E [X(M)Y ] .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 27 Since the Gram matrix E[XX⊤] is non-singular, E[X(M)X⊤(M)] is non-singular and βM = 0 ∈R|M| is equivalent to E[X(M)Y ] = 0 ∈R|M|. Therefore, βM = 0 ∈R|M| for all M ⊆{1, 2, . . . , p}, is equivalent to E[X(j)Y ] = 0, for all 1 ≤j ≤p.
In McKeague and Qian (2015), the authors consider the maximal correlation pa-rameter θ0 := max 1≤j≤p Corr (X(j), Y ) .
The estimator of θ0 they consider is ˆ θn := max 1≤j≤p d Corr (X(j), Y ) , where d Corr represents the sample correlation coefficient. It is easy to see that ˆ θn (properly scaled) is not asymptotically normal and McKeague and Qian (2015) derive the exact asymptotic distribution along with a resampling procedure to estimate the distribution.
As an alternative, consider the following inequality ˆ θn −θ0 ≤max 1≤j≤p d Corr(X(j), Y ) −Corr(X(j), Y ) .
(27) Since d Corr is an asymptotically linear estimator, the right hand side above is asymptotically the maximum of an average which can be bootstrapped under vari-ous dependence structures. This provides an asymptotically conservative inference in general for the parameter θ0. (Note, however, that under the null hypothesis H0 : θ0 = 0 inequality (27) is exact and gives valid critical values for Type I error control.) To elaborate and provide a general framework of marginal screening for M-estimators, consider the marginal targets for 1 ≤j ≤p, βn,j := arg min θ∈R 1 n n X i=1 E [h(Xi(j))ℓ(Xi(j)θ, Yi)] , for a twice differentiable convex loss function ℓ(·, ·) and a non-negative weight function h(·). The estimators for 1 ≤j ≤p are given by ˆ βn,j := arg min θ∈R 1 n n X i=1 h(Xi(j))ℓ(Xi(j)θ, Yi).
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 28 Define for 1 ≤j ≤p, δn,j := 1.5[ ˆ Qn,j]−1| ˆ Zn,j|, where ˆ Zn,j := 1 n n X i=1 ℓ′(Xi(j)βn,j, Yi)h(Xi(j))Xi(j), ˆ Qn,j := 1 n n X i=1 h(Xi(j))ℓ′′(Xi(j)βn,j, Yi)X2 i (j), Qn,j := 1 n n X i=1 E h(Xi(j))ℓ′′(Xi(j)βn,j, Yi)X2 i (j) .
The following corollary shows that an asymptotically conservative inference is pos-sible for marginal screening in general M-estimators.
Corollary 7.3. If max 1≤j≤p max 1≤i≤n C (|Xi(j)|δn,j, Yi) ≤4 3, then simultaneously for all j ∈{1, 2, . . . , p}, ˆ βn,j −βn,j −[Qn,j]−1 ˆ Zn,j ≤ " max 1≤i≤n C (|Xi(j)|δn,j, Yi) −1 + ˆ Qn,j Qn,j −1 # δn,j.
Furthermore, if max 1≤j≤p δn,j = op(1), and max 1≤j≤p ˆ Qn,j Qn,j −1 = op(1), as n →∞, then max 1≤j≤p ˆ βn,j −max 1≤j≤p βn,j ≤(1 + op(1)) max 1≤j≤p ˆ Zn,j Qn,j .
(28) Proof. The result follows trivially from Corollary 3.1.
The right hand side of (28) is the (absolute) maximum of a mean zero av-erage vector and the high-dimensional central limit theorems of Chernozhukov et al. (2013, 2017), Zhang and Cheng (2014) and Zhang and Wu (2017) provide a Gaussian approximation as well as a bootstrap resampling scheme for consistent estimation of quantiles of the quantity in (28).
It is easy to prove a result similar to Corollary 7.3 for marginal screening in Cox proportional hazards model.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 29 7.3. Post-selection Inference under Covariate Selection In the previous section, we have considered asymptotic linear representation uni-form over all models of size 1. In this section, we consider linear representation error uniform over all models of size bounded by k (≥1). This is important for post-selection inference (PoSI). In the context of regression analysis, the PoSI problem refers to the construction of confidence regions for βn, ˆ M for a model ˆ M ⊆{1, 2, . . . , p} chosen based on the data (X1, Y1), . . . , (Xn, Yn) ∈Rp × R. For-mally, for any M ⊆{1, 2, . . . , p}, define the estimator ˆ βn,M := arg min θ∈R|M| 1 n n X i=1 h(Xi(M))ℓ(θ⊤Xi(M), Yi), for some twice differentiable convex loss function ℓ(·, ·) and non-negative weight function h(·). Based on the results in previous sections, we can consider the target parameters βn,M := arg min θ∈R|M| 1 n n X i=1 E h(Xi(M))ℓ(θ⊤Xi(M), Yi) .
Let M be a collection of subsets of {1, 2, . . . , p}. The PoSI problem for the col-lection of targets {βn,M : M ∈M} concerns the construction of a collection of confidence regions { ˆ Rn,M : M ∈M} of level α satisfying lim inf n→∞P βn, ˆ M ∈ˆ Rn, ˆ M ≥1 −α, (29) for any model ˆ M chosen possibly depending on the data {(Xi, Yi)}1≤i≤n that satis-fies P( ˆ M ∈M) = 1; see Kuchibhotla et al. (2018b) for more details. Theorem 3.1 of Kuchibhotla et al. (2018b) proves that the post-selection inference guarantee (29) is equivalent to the simultaneous guarantee: lim inf n→∞P \ M∈M n βn,M ∈ˆ Rn,M o!
≥1 −α.
It is easy to see that for a post-selection confidence region ˆ Rn, ˆ M based on ˆ βn, ˆ M to have a Lebesgue measure (on R|M|) converging to zero, it is necessary that sup M∈M ˆ βn,M −βn,M = op(1), as n →∞, for some norm ∥·∥. Based on our deterministic inequalities in previous sections, we can provide precise statements of uniform convergence. We provide only one such Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 30 result similar to Corollary 7.3. To state the results, define for M ⊆{1, 2, . . . , p} and θ ∈R|M|, ˆ Ln,M(θ) := 1 n n X i=1 h(Xi(M))ℓ(θ⊤Xi(M), Yi).
Also, set δn,M := 1.5 [∇2 ˆ Ln(βn,M)]−1∇ˆ Ln(βn,M) 2 .
Corollary 7.4. Suppose max 1≤i≤n max M∈M C (∥Xi(M)∥2 δn,M, Yi) ≤4 3, then for each M ∈M, there exists a unique vector ˆ βn,M ∈R|M| satisfying ∇ˆ Ln(ˆ βn,M) = 0 and ˆ βn,M −βn,M + [∇2 ˆ Ln(βn,M)]−1∇ˆ Ln(βn,M) 2 ≤ max 1≤i≤n C(∥Xi(M)∥2 δn,M) −1 δn,M.
Proof. The proof follows trivially from Corollary 3.1.
As in Section 7.2, the linear expansion result of Corollary 7.4 above proves that ˆ βn,M −βn,M = (1 + op(1))[∇2 ˆ Ln(βn,M)]−1∇ˆ Ln(βn,M) uniformly for M ∈M.
Therefore, one can apply various bootstrap schemes to evaluate quantiles or ap-proximate the distribution of {ˆ βn,M −βn,M : M ∈M} under various dependence settings. For simplicity and concreteness, we have dealt with covariate selection here and using techniques from previous section, it is not difficult to also con-sider post-selection inference problems related to family of transformations on the covariates/response.
8. Conclusions and Future Work In this work, we have provided deterministic inequalities for a class of smooth M-estimators that unify the classical asymptotic analysis under various dependence settings. Furthermore, these inequalities readily yield tail bounds for estimation errors as well as asymptotic expansions. A connection between these deterministic inequalities and semilocal convergence analysis of iterative algorithms is estab-lished.
Throughout the paper we have considered only twice differentiable loss functions.
It is of interest to understand the non-smooth loss functions like the absolute de-viation, Huber’s loss from the viewpoint of deterministic inequalities. As described Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 31 in Section 2, several iterative algorithms exist with linear/superlinear convergence also for non-smooth functions. We hope to present similar deterministic inequalities for non-smooth M-estimators in the future.
Acknowledgments The author would like to thank Mateo Wirth and Bikram Karmakar for helpful discussions and suggestions.
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Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 35 Appendix A: Proofs of Results in Section 2 A.1. Proof of Theorem 2.1 Proof. The proof essentially from the arguments of Yuan and Jennrich (1998) but it was stated there with the hypothesis of continuous differentiability of f(·). Only everywhere differentiability of f(·) is required. Define ϕ(θ) := θ −A−1f(θ).
To finish the proof it is enough to show that there exists a fixed point for ϕ(·) in B(θ0, r). Let I represent the identity matrix in Rq. Since ∇ϕ(θ) = I −A−1∇f(θ) = A−1 (A −∇f(θ)) , and for all θ ∈B(θ0, r), ∥∇ϕ(θ)∥op ≤ε by (3). This implies that ϕ(·) is a contrac-tion mapping on B(θ0, r) with contraction constant ε. Also, since (4) implies ∥ϕ(θ0) −θ0∥2 = A−1f(θ0) 2 ≤r(1 −ε), it follows that for θ ∈B(θ0, r), ∥ϕ(θ) −θ0∥2 ≤∥ϕ(θ) −ϕ(θ0)∥2 + ∥ϕ(θ0) −θ0∥2 ≤ε ∥θ −θ0∥2 + r(1 −ε) ≤r.
Thus, ϕ : B(θ0, r) →B(θ0, r) is a contraction and hence has a unique fixed point in B(θ0, r) by the fixed point theorem. See Loomis and Sternberg (1968, Theorem 9.1) for more details on contraction mapping fixed point theorem.
Now observe that by a first order Taylor series expansion 0 = f(θ⋆) = f(θ0) + ∇f(¯ θ) (θ −θ0) , for some ¯ θ that lies on the line segment joining θ⋆and θ0. Thus, −A−1f(θ0) = A−1∇f(¯ θ) (θ⋆−θ0) .
(30) Since θ⋆∈B(θ0, r), it follows that ¯ θ ∈B(θ0, r) and so, by (3), A−1(A −∇f(¯ θ)) op ≤ε ⇒ (1 −ε)I ⪯A−1∇f(¯ θ) ⪯(1 + ε)I.
Therefore, A−1∇f(¯ θ) is invertible and (30) leads to, ∥θ⋆−θ0∥2 = A−1∇f(¯ θ) −1 A−1f(θ0) 2 , and 1 1 + ε A−1f(θ0) 2 ≤∥θ⋆−θ0∥2 ≤ 1 1 −ε A−1f(θ0) 2 .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 36 A.2. Proof of Theorem 2.2 Proof. Define ε := 1/3, and r := 1.5 (∇f(θ0))−1f(θ0) 2 .
From these definitions, it is clear that (∇f(θ0))−1f(θ0) 2 = r(1 −ε), and (∇f(θ0))−1(∇f(θ0) −∇f(θ)) op ≤L ∥θ −θ0∥α 2 ≤Lrα ≤ L (1 −ε)α (∇f(θ0))−1f(θ0) α 2 ≤1/3, under the assumption (6). Therefore, the conditions of Theorem 2.1 are satisfied and so, there exists a unique solution θ⋆∈B(θ0, r) satisfying f(θ⋆) = 0. Also, it follows that ∥θ⋆−θ0∥2 ≤1.5 (∇f(θ0))−1f(θ0) 2 .
Observe now that θ0 −(∇f(θ0))−1f(θ0) −θ⋆ 2 = (∇f(θ0))−1 (−f(θ0) −∇f(θ0)) 2 = (∇f(θ0))−1 (f(θ⋆) −f(θ0) −∇f(θ0)) 2 (a) = (∇f(θ0))−1 ∇f(¯ θ) −∇f(θ0) (θ⋆−θ0) 2 ≤ (∇f(θ0))−1 ∇f(¯ θ) −∇f(θ0) op ∥θ0 −θ⋆∥2 (b) ≤L ∥θ0 −θ⋆∥1+α ≤(1.5)1+αL (∇f(θ0))−1f(θ0) 1+α 2 .
Equality (a) above follows from the mean-value theorem for some vector ¯ θ that lies on the line segment joining θ⋆, θ0 and inequality (b) follows from the fact ¯ θ −θ0 2 ≤∥θ0 −θ⋆∥2 .
Appendix B: Proofs of Results in Section 3 B.1. Proof of Theorem 3.1 Proof. To prove (9), we verify the assumptions of Theorem 2.1. Take in Theo-rem 2.1, f(θ) := [ ˆ Qn(θ0)]−1 ˆ Zn(θ), A := I, and r = δn(θ0), ε = 1 3.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 37 Here I represents the identity matrix in Rq. Condition (4) is trivially satisfied since A−1f(θ0) 2 = [ ˆ Qn(θ0)]−1 ˆ Zn(θ0) 2 = 2δn(θ0) 3 = (1 −ε)δn(θ0).
(31) To verify condition (3), note that A−1 (A −∇f(θ)) op = [∇ˆ Zn(θ0)]−1 ∇ˆ Zn(θ0) −∇ˆ Zn(θ) op = sup e∈Rq: ∥e∥2=1 e⊤∇ˆ Zn(θ)e e⊤∇ˆ Zn(θ0)e −1 .
To control the right hand side above, note that by the definition of C(u, w), e⊤∇ˆ Zn(θ)e = 1 n n X i=1 e⊤∇L(θ, Wi)e ≤1 n n X i=1 e⊤∇L(θ0, Wi)e C(r, Wi), e⊤∇ˆ Zn(θ0)e = 1 n n X i=1 e⊤∇L(θ0, Wi)e ≤1 n n X i=1 e⊤∇L(θ, Wi)e C(r, Wi).
(32) Thus under Assumption (A2), for all e ∈Rq with ∥e∥2 = 1 and θ ∈Rq such that ∥θ −θ0∥2 ≤r, 3 4 ≤e⊤ˆ Qn(θ)e e⊤ˆ Qn(θ0)e ≤4 3, and so, sup θ∈Br(θ0) A−1(A −∇f(θ0)) op ≤max 1 3, 1 4 = 1 3 = ε.
(33) Inequalities (31) and (33) complete the verification of condition (4) and (3), respec-tively with ε = 1/3. Therefore, by Theorem 2.1, we get that there exists ˆ θn ∈Rq satisfying ˆ Zn(ˆ θn) = 0, and 1 2δn(θ0) ≤ ˆ θn −θ0 2 ≤δn(θ0).
Thus, the first part of the result is proved.
To prove (10), note by a Taylor series expansion of ˆ Zn(ˆ θn) around θ0 that, 0 = ˆ Zn(ˆ θn) = ˆ Zn(θ0) + ˆ Qn(¯ θ) ˆ θn −θ0 , for some ¯ θ that lies on the line segment joining ˆ θn and θ0. Multiplying both sides by ˆ Qn(θ0), we get −[ ˆ Qn(θ0)]−1 ˆ Zn(θ0) = [ ˆ Qn(θ0)]−1 ˆ Qn(¯ θ) ˆ θ −θ0 .
(34) Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 38 By (32), it follows that 1 maxi C(δn(θ0), Wi) ˆ Qn(θ0) ⪯ ˆ Qn(¯ θ) ⪯max i C(δn(θ0), Wi) ˆ Qn(θ0), which implies that [ ˆ Qn(θ0)]−1 ˆ Qn(¯ θ) −I op ≤ max 1≤i≤n C(δn(θ0), Wi) −1 max 1, 1 maxi C(δn(θ0), Wi) ≤max 1≤i≤n C(δn(θ0), Wi) −1, since C(r, w) ≥1 for all r and w. Therefore, using (34), we obtain ˆ θn −θ0 + [ ˆ Qn(θ0)]−1 ˆ Zn(θ0) 2 ≤ [ ˆ Qn(θ0)]−1 ˆ Qn(¯ θ) −I op ˆ θn −θ0 2 ≤δn(θ0) max 1≤i≤n C(δn(θ0), Wi) −1 .
This completes the proof.
B.2. Proof of Corollary 3.1 Proof. Take w = (x, y) and L(θ; w) = h(x)ℓ(x⊤θ, y) in Theorem 3.1. For this function, ∇2L(θ, w) = h(x)ℓ′′(x⊤θ, y)xx⊤.
To verify assumption (A2), note that sup ∥θ1−θ2∥2≤u sup e∈Rp:∥e∥2=1 e⊤∇2L(θ1, w)e e⊤∇2L(θ2, w)e = sup ∥θ1−θ2∥2≤u ℓ′′(x⊤θ1, y) ℓ′′(x⊤θ2, y) ≤ sup |s−t|≤∥x∥2u ℓ′′(s, y) ℓ′′(t, y) = C (∥x∥2 u, y) .
Therefore, under (13), assumption (A2) holds true and the result follows.
B.3. Proof of Proposition 3.1 Proof. For any four real non-negative numbers a, b, c and d, min na b, c d o ≤a + c b + d ≤max na b, c d o .
(35) Suppose L1(·, ·) and L2(·, ·) be any two elements of CT. Fix two positive real num-bers α, β and set L(θ, w) = αL1(θ, w) + βL2(θ, w). It follows that L(·, ·) is convex in the first argument and ∇2L(θ, w) = α∇2L1(θ, w) + β∇2L2(θ, w).
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 39 Fix u ≥0. Then for each θ1, θ2 satisfying ∥θ1 −θ2∥2 ≤u, and e satisfying ∥e∥2 = 1, e⊤∇2L(θ1, w)e e⊤∇2L(θ2, w)e ≤max e⊤∇2L1(θ1, w)e e⊤∇2L1(θ2, w)e, e⊤∇2L2(θ1, w)e e⊤∇2L2(θ2, w)e ≤T(u, w), by inequality (35). Therefore, L(·, ·) ∈CT. Note that CT is a non-empty set since the any function whose second derivative is a non-negative multiple of T(·, ·) belongs to CT.
Appendix C: Proofs of Results in Section 4 C.1. A Preliminary Lemma We need to following lemma for the proof of Theorem 4.1. The result is similar to Lemma A2 of Hjort and Pollard (2011).
Lemma C.1. Suppose K(t) := log R(t), where R(t) := n X i=1 wi exp(ait) for wi ≥0, ai ∈R.
Assume that not all wi’s are zero. Then K(t) is convex with derivatives K′(t) = n X i=1 aivi(t) =: ¯ a(t), and K′′(t) = n X i=1 vi(t) {ai −¯ a(t)}2 , where vi(t) := wi exp(ait)/R(t) for 1 ≤i ≤n. Furthermore, for t ∈R and for all 0 ≤|s| ≤|t|, max K′′(s) K′′(0) −1 , K′′(0) K′′(s) −1 ≤4µn|t| exp(4µn|t|), where µn := max1≤i≤n |ai −¯ a(0)| .
Proof. It is easy to verify that K′(t) = R′(t) R(t) = Pn i=1 wiai exp(ait) R(t) = n X i=1 aivi(t).
Thus, K′′(t) = R′′(t) R(t) −(¯ a(t))2 = n X i=1 a2 i vi(t) − n X i=1 aivi(t) !2 = n X i=1 vi(t) {ai −¯ a(t)}2 .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 40 Since K′′(t) ≥0 for all t ≥0, K(·) is a convex function.
To prove the second part, fix s satisfying |s| ≤|t|. Clearly, vi(s) = wi exp(ais) R(s) = wi R(0) " exp(ais) Pn j=1 wj Pn j=1 wj exp(ajs) # = vi(0)(1 + εi(s)), where 1 + εi(s) := exp(ais) Pn j=1 wj Pn j=1 wj exp(ajs) = exp({ai −¯ a(0)}s) Pn j=1 wj Pn j=1 wj exp({aj −¯ a(0)}s).
It is easy to check that min 1≤j≤n exp ({aj −¯ a(0)}s) ≤ Pn j=1 wj exp({aj −¯ a(0)}s) Pn j=1 wj ≤max 1≤j≤n exp ({aj −¯ a(0)}s) .
Therefore, for all |s| ≤|t| and 1 ≤i ≤n, exp (−µn|t|) ≤1 + εi(s) ≤exp (µn|t|) .
(36) This implies that max vi(s) vi(0), vi(0) vi(s) ≤exp(µn|t|).
(37) Observe that K′′(s) = n X i=1 vi(s) (ai −¯ a(s))2 = n X i=1 vi(0)(1 + εi(s)) (ai −¯ a(0) + ¯ a(0) −¯ a(s))2 = n X i=1 vi(0)(ai −¯ a(0))2(1 + εi(s)) + n X i=1 vi(0)(¯ a(0) −¯ a(s))2(1 + εi(s)) + 2 n X i=1 vi(0)(ai −¯ a(0))(¯ a(0) −¯ a(s))(1 + εi(s)).
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 41 We now subtract K′′(0) and bound the remainder.
|K′′(s) −K′′(0)| ≤ n X i=1 vi(0)(ai −¯ a(0))2εi(s) + |¯ a(0) −¯ a(s)|2 n X i=1 vi(0)(1 + εi(s)) + 2 |¯ a(0) −¯ a(s)| × n X i=1 vi(0)(ai −¯ a(0))(1 + εi(s)) ≤K′′(0) (exp(µn|t|) −1) + |¯ a(0) −¯ a(s)|2 max 1≤i≤n(1 + εi(s)) + 2 |¯ a(0) −¯ a(s)| n X i=1 vi(0)(ai −¯ a(0))2 !1/2 n X i=1 vi(0)(1 + εi(s))2 !1/2 ≤K′′(0) (exp(µn|t|) −1) + |¯ a(0) −¯ a(s)|2 max 1≤i≤n(1 + εi(s)) + 2 |¯ a(0) −¯ a(s)| (K′′(0))1/2 max 1≤i≤n(1 + εi(s)) ≤K′′(0) (exp(µn|t|) −1) + |¯ a(0) −¯ a(s)|2 exp(2µn|t|) + 2 |¯ a(0) −¯ a(s)| (K′′(0))1/2 exp(µn|t|).
(38) Here the last inequality follows from inequality (36). To bound |¯ a(0) −¯ a(s)|, note that |¯ a(0) −¯ a(s)| = n X i=1 vi(s)(ai −¯ a(0)) = n X i=1 vi(0)(ai −¯ a(0))(1 + εi(s)) (a) = n X i=1 vi(0)(ai −¯ a(0))εi(s) ≤ n X i=1 vi(0)ε2 i (s) !1/2 n X i=1 vi(0)(ai −¯ a(0))2 !1/2 ≤(K′′(0))1/2(exp(µn|t|) −1), where the equality (a) follows from the fact that ¯ a(0) = P vi(0)ai and P vi(0) = 1.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 42 Substituting this inequality in (38), we get |K′′(s) −K′′(0)| ≤K′′(0)(exp(µn|t|) −1) + K′′(0) exp(2µn|t|)(exp(µn|t|) −1)2 + 2K′′(0) exp(µn|t|)(exp(µn|t|) −1) = K′′(0)(exp(µn|t|) −1) [1 + exp(2µn|t|)(exp(µn|t|) −1) + 2 exp(µn|t|)] ≤K′′(0)(exp(µn|t|) −1) [1 + exp(3µn|t|) + 2 exp(3µn|t|)] ≤4K′′(0)(exp(µn|t|) −1) exp(3µn|t|) ≤4K′′(0)µn|t| exp(4µn|t|).
Therefore, for all |s| ≤|t|, K′′(s) K′′(0) −1 ≤4µn|t| exp(4µn|t|).
The bound for K′′(0)/K′′(s) follows the same line of argument as (38) and finally use inequality (37).
C.2. Proof of Theorem 4.1 Proof. To prove (15), we verify the assumptions of Theorem 2.1 with f(β) := [∇ˆ Zn(β0)]−1 ˆ Zn(β), A := I and r := δn(β0), ε = 1/3.
Assumption (4) is trivially satisfied by the definition of r and to verify Assump-tion (3), it is enough to verify for all ν ∈Rp with ∥ν∥2 ≤r, that [∇ˆ Zn(β0)]−1 ∇ˆ Zn(β0) −∇ˆ Zn(β0 + ν) op ≤1/3.
(39) For any fixed 0 ≤s < ∞, ν ∈Rp, define K(ℓ) := log n X i=1 wi exp (aiℓ) !
, where wi := H2(Xi,s)Yi(s) exp β⊤ 0 Xi,s and ai = ν⊤Xi,s.
Then K(ℓ) = log Rn(s, β0 + ℓν). As in Lemma C.1, set ˜ µn := max 1≤i≤n |ai −¯ a(0)| = max 1≤i≤n ν⊤ Xi,s −¯ Xn,s(β0) .
(40) Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 43 It is evident that ˆ Zn(β0 + ℓν) = n X i=1 Z ∞ 0 H1(Xi,s) {K′(ℓ) −Xi,s} dNi(s), d dβ ˆ Zn(β) β=β0+ℓν = n X i=1 Z ∞ 0 H1(Xi,s) {K′′(ℓ)} dNi(s), d dβ ˆ Zn(β) β=β0 = n X i=1 Z ∞ 0 H1(Xi,s) {K′′(0)} dNi(s).
The dependence of K(·) on s is suppressed in the formulas above. From Lemma C.1, we have for all 0 ≤ℓ≤1, K′′(0) [1 −4˜ µn exp(4˜ µn)] ≤K′′(ℓ) ≤K′′(0) [1 + 4˜ µn exp(4˜ µn)] (41) Clearly from the definition (40), ˜ µn ≤µn(s) ∥ν∥2 ≤µn(s)r = µn(s)δn(β0) ≤1/16.
Hence, 4˜ µn ≤1/4 and so, 4˜ µn exp(4˜ µn) ≤1/3. Substituting this inequality in (41), we get 2 3K′′(0) ≤K′′(1) ≤4 3K′′(0), and so, 2 3∇ˆ Zn(β0) ⪯∇ˆ Zn(β0 + ν) ⪯4 3∇ˆ Zn(β0), proving (39) for all ∥ν∥2 ≤r and M ∈M. Hence from Theorem 2.1, we get that there exists a solution ˆ βn such that ˆ Zn(ˆ βn) = 0 and δn(β0) 2 ≤ ˆ βn −β0 2 ≤2δn(β0).
To prove the linear representation part of the result, we follow the proof of Theorem 3.1. By a Taylor series expansion, we get that 0 = ˆ Zn(ˆ βn) = ˆ Zn(β0) + ∇ˆ Zn(¯ β) ˆ βn −β0 , for some vector ¯ β that lies on the line segment between β0 and ˆ βn. This implies that −[∇ˆ Zn(β0)]−1 ˆ Zn(β0) = [∇ˆ Zn(β0)]−1∇ˆ Zn(¯ β)(ˆ βn −β0).
From (41), it follows that (1 −γn) I ⪯[∇ˆ Zn(β0)]−1∇ˆ Zn(¯ β) ⪯(1 + γn) I, Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 44 where γn = 4 sup 0≤s<∞ µn(s)δn(β0) exp 4 sup 0≤s<∞ µn(s)δn(β0) ≤4e1/4 sup 0≤s<∞ µn(s)δn(β0).
Therefore, ˆ β −β0 + [ ˆ Jn(β0)]−1 ˆ Zn(β0) 2 ≤8e1/4 sup 0≤s<∞ µn(s)δ2(β0).
Appendix D: Proofs of Results in Section 5 D.1. Proof of Corollary 5.1 Proof. We will verify the assumptions of Theorem 2.2. First note that Fn(θ) = 1 n n X i=1 Yi −g(X⊤ i θ) 2 , ∇Fn(θ) = −2 n n X i=1 Yi −g(X⊤ i θ) g′(X⊤ i θ)Xi, ∇2Fn(θ) = 2 n n X i=1 g′(X⊤ i θ) 2 XiX⊤ i −2 n n X i=1 Yi −g(X⊤ i θ) g′′(X⊤ i θ)XiX⊤ i .
Thus for any θ ∈Rp, ∇2Fn(θ) −∇2Fn(θ0) = 2 n n X i=1 n g′(X⊤ i θ) 2 − g′(X⊤ i θ0) 2o XiX⊤ i −2 n n X i=1 (Yi −g(X⊤ i θ))g′′(X⊤ i θ) −(Yi −g(X⊤ i θ0))g′′(X⊤ i θ0) XiX⊤ i =: I −II.
From assumption (NR), we get that for any 1 ≤i ≤n, g′(X⊤ i θ) 2 − g′(X⊤ i θ0) 2 ≤C2 1(Xi) ∥θ −θ0∥2 2 + 2C1(Xi)|g′(X⊤ i θ0)| ∥θ −θ0∥2 , and (Yi −g(X⊤ i θ))g′′(X⊤ i θ) −(Yi −g(X⊤ i θ0))g′′(X⊤ i θ0) ≤ (Yi −g(X⊤ i θ0)) C2(Xi) ∥θ −θ0∥α 2 + C0(Xi) |g′′(X⊤ i θ0)| ∥θ −θ0∥2 + C2(Xi) ∥θ −θ0∥1+α 2 Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 45 Therefore, (∇2Fn(θ0))−1 (∇2Fn(θ) −∇2Fn(θ0)) op ≤ 2 n n X i=1 C2 1(Xi) (∇2Fn(θ0))−1 XiX⊤ i op ∥θ −θ0∥2 2 + 2 n n X i=1 C0(Xi)C2(Xi) (∇2Fn(θ0))−1 XiX⊤ i op ∥θ −θ0∥1+α 2 + 2 n n X i=1 2C1(Xi)|g′(X⊤ i θ0)| + C0(Xi)|g′′(X⊤ i θ0)| (∇2Fn(θ0))−1 XiX⊤ i op ∥θ −θ0∥2 + 2 n n X i=1 C2(Xi)|Yi −g(X⊤ i θ0)| (∇2Fn(θ0))−1 XiX⊤ i op ∥θ −θ0∥α 2 ≤L2(θ0) ∥θ −θ0∥2 2 + L1+α ∥θ −θ0∥1+α 2 + L1(θ0) ∥θ −θ0∥2 + Lα(θ0) ∥θ −θ0∥α 2 .
This completes the verification of condition (5) of Theorem 2.2 with right hand side there replaced by ω(∥θ −θ0∥2), where for r ≥0, ω(r) = L2(θ0)r2 + L1+α(θ0)r1+α + L1(θ0)r + Lα(θ0)rα.
Following the proof of Theorem 2.2, the assumption (16) implies the result.
Appendix E: Proofs of Results in Section 6 E.1. Proof of Corollary 6.1 Proof. Define the function gn(β, ν) := " ∇Fn(β) + A⊤ν Aβ −b # .
It follows that ∇gn(β, ν) := " ∇2Fn(β) A⊤ A 0 # .
So, β⋆is a solution of the optimization problem if there exists a vector ν⋆such that gn(β⋆, ν⋆) = 0. From Theorem 2.2, it follows that if [∇gn(β0, ν0)]−1 (∇gn(β, ν) −∇gn(β0, ν0)) op ≤L β ν !
− β0 ν0 ! α 2 , (42) Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 46 for (β, ν) in a ball around (β0, ν0) and if [∇gn(β0, ν0)]−1gn(β0, ν0) 2 ≤(3L)−1/α.
(43) First note that ∇gn(β, ν) −∇gn(β0, ν0) = " ∇2Fn(β) −∇2Fn(β0) 0 0 0 # , and using the inverse of a block matrix, we get [∇gn(β0, ν0)]−1 (∇gn(β, ν) −∇gn(β0, ν0)) is given by " {I −[∇2Fn(β0)]−1A⊤(A[∇2Fn(β0)]−1A⊤)−1A}[∇2Fn(β0)]−1[∇2Fn(β) −∇2Fn(β0)] 0 0 0 # .
This implies that [∇gn(β0, ν0)]−1 (∇gn(β, ν) −∇gn(β0, ν0)) op ≤ I −[∇2Fn(β0)]−1A⊤(A[∇2Fn(β0)]−1A⊤)−1A op [∇2Fn(β0)]−1[∇2Fn(β) −∇2Fn(β0)] op .
Since [∇2Fn(β0)]−1A⊤(A[∇2Fn(β0)]−1A⊤)−1A op = [∇2Fn(β0)]−1A⊤(A[∇2Fn(β0)]−1A⊤)−1A[∇2Fn(β0)]−1A⊤ op = 1, we get that [∇gn(β0, ν0)]−1 (∇gn(β, ν) −∇gn(β0, ν0)) op ≤ [∇2Fn(β0)]−1[∇2Fn(β) −∇2Fn(β0)] op .
This proves the condition (42). For condition (43), note that gn(β0, ν0) = " ∇Fn(β0) + A⊤ν0 0 # .
Again using the inverse of a block matrix, we get that [∇gn(β0, ν0)]−1gn(β0, ν0) is " {I −[∇2Fn(β0)]−1A⊤(A[∇2Fn(β0)]−1A⊤)−1A}[∇2Fn(β0)]−1(∇Fn(β0) + A⊤ν0) A[∇2Fn(β0)]−1A⊤−1 A[∇2Fn(β0)]−1(∇Fn(β0) + A⊤ν0) # .
By the same reasoning, we have that [∇gn(β0, ν0)]−1gn(β0, ν0) 2 ≤ 1 + (A[∇2Fn(β0)]−1A⊤)−1A op [∇2Fn(β0)]−1(∇Fn(β0) + A⊤ν0) 2 .
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 47 Appendix F: Proofs of Results in Section 7 Proof of Corollary 7.1. Theorem 3.1 implies that ˆ θ−I,n −ˆ θn + T−I,n 2 ≤max 1≤i≤n, i̸=I C 1.5 ∥T−I,n∥2 , Wi −1 1.5 ∥T−I,n∥2 , (44) if max 1≤i≤n, i̸=I C 1.5 ∥T−I,n∥2 , Wi ≤4 3, (45) where T−I,n := X 1≤i≤n, i̸=I ∇2L(ˆ θn, Wi) !−1 X 1≤i≤n, i̸=I ∇L(ˆ θn, Wi).
To prove the result from this inequality, we need to simplify and control T−I,n and ∥T−I,n∥2. Since ˆ θn is the solution of the equation (23), we get for all 1 ≤I ≤n, X 1≤i≤n, i̸=I ∇L(ˆ θn, Wi) = −∇L(ˆ θn, WI).
Also, note that T−I,n + n−1 ˆ Q−1 n ∇L(ˆ θn, WI) 2 ≤ I −(n ˆ Qn −∇2L(ˆ θn, WI))−1n ˆ Qn op n−1 ˆ Q−1 n ∇L(ˆ θn, WI) 2 = I −(I −n−1 ˆ Q−1 n ∇2L(ˆ θn, WI))−1 op n−1 ˆ Q−1 n ∇L(ˆ θn, WI) 2 = (I −n−1 ˆ Q−1 n ∇2L(ˆ θn, WI))−1n−1 ˆ Q−1 n ∇2L(ˆ θn, WI) op n−1 ˆ Q−1 n ∇L(ˆ θn, WI) 2 ≤ n−2 ˆ Q−1 n ∇2L(ˆ θn, WI) op 1 −n−1 ˆ Q−1 n ∇2L(ˆ θn, WI) op ˆ Q−1 n ∇L(ˆ θn, WI) 2 = n−1 ˆ Q−1 n ∇2L(ˆ θn, WI) op δI,n.
From this inequality, it follows that ∥T−I,n∥2 ≤δI,n. This inequality implies that condition (45) is satisfied if max 1≤i≤n, i̸=I C (1.5δI,n, Wi) ≤4 3, which in turn implied by the condition (24). Substituting the inequalities above in (44), we get ˆ θ−I,n −ˆ θn −n−1 ˆ Q−1 n ∇L(ˆ θn, WI) 2 ≤3 2 max 1≤i≤n, i̸=I C(1.5δI,n, Wi) −1 + n−1 ˆ Q−1 n ∇2L(ˆ θn, WI) op δI,n.
Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 48 Proof of Corollary 7.2. Theorem 3.1 implies that ˆ θ−I,n −ˆ θn + T−I,n 2 ≤ max 1≤i≤n, i/ ∈I C 1.5 ∥T−I,n∥2 , Wi −1 1.5 ∥T−I,n∥2 , (46) if max 1≤i≤n, i̸=I C 1.5 ∥T−I,n∥2 , Wi ≤4 3, (47) where T−I,n := X 1≤i≤n, i/ ∈I ∇2L(ˆ θn, Wi) !−1 X 1≤i≤n, i/ ∈I ∇L(ˆ θn, Wi).
To prove the result from this inequality, we need to simplify and control T−I,n and ∥T−I,n∥2. Since ˆ θn is the solution of the equation (23), we get for all 1 ≤I ≤n, X 1≤i≤n, i/ ∈I ∇L(ˆ θn, Wi) = − X i∈I ∇L(ˆ θn, WI).
Also, note that T−I,n + n−1 ˆ Q−1 n X i∈I ∇L(ˆ θn, WI) 2 ≤ I − n ˆ Qn − X i∈I ∇2L(ˆ θn, Wi) !−1 n ˆ Qn op n−1 ˆ Q−1 n X i∈I ∇L(ˆ θn, WI) 2 = I − I −n−1 ˆ Q−1 n X i∈I ∇2L(ˆ θn, Wi) !−1 op n−1 ˆ Q−1 n X i∈I ∇L(ˆ θn, WI) 2 = I −n−1 ˆ Q−1 n X i∈I ∇2L(ˆ θn, Wi) !−1 n−1 ˆ Q−1 n X i∈I ∇2L(ˆ θn, Wi) op n−1 ˆ Q−1 n X i∈I ∇L(ˆ θn, WI) 2 ≤ n−1 ˆ Q−1 n P i∈I ∇2L(ˆ θn, Wi) op 1 −n−1 ˆ Q−1 n P i∈I ∇2L(ˆ θn, Wi) op n−1 ˆ Q−1 n X i∈I ∇L(ˆ θn, WI) 2 = n−1 ˆ Q−1 n X i∈I ∇2L(ˆ θn, Wi) op δI,n.
From this inequality, it follows that ∥T−I,n∥2 ≤δI,n. This inequality implies that condition (47) is satisfied if max 1≤i≤n, i/ ∈I C (1.5δI,n, Wi) ≤4 3, Arun K. Kuchibhotla/Deterministic Inequalities for M-estimators 49 which in turn implied by the condition (24). Substituting the inequalities above in (46), we get ˆ θ−I,n −ˆ θn −n−1 ˆ Q−1 n X i∈I ∇L(ˆ θn, WI) 2 ≤3 2 max 1≤i≤n, i/ ∈I C(1.5δI,n, Wi) −1 + n−1 ˆ Q−1 n X i∈I ∇2L(ˆ θn, Wi) op δI,n. |
5756 | https://www.chegg.com/homework-help/questions-and-answers/find-b-c-d-cubic-function-f-x-x-3-b-x-2-c-x-d-satisfies-given-conditions-relative-maximum--q110699455 | Your solution’s ready to go!
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Question: Find a,b,c, and d such that the cubic function f(x)=ax3+bx2+cx+d satisfies the given conditions. Relative maximum: (3,16) Relative minimum: (5,14) Inflection point: (4,15) a=b=c=d=(a) Find the largest open interval(s) on which f is increasing. (Enter your answer as a comma-separated list of intervais). xThe function s(t) describes the motion of a particle
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To find the coefficients of the cubic function, start by setting up equations from the information about relative extrema and inflection point using derivative expressions: and .
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it has relative maximum at (3,16) , minimum at (5,14)
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5757 | https://www.reddit.com/r/askmath/comments/1dv7j88/faster_way_to_check_if_n_is_prime/ | Faster way to check if n is prime? : r/askmath
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Not_Black_is_taken
Faster way to check if n is prime?
Number Theory
Hi,
I recently had a theory regarding checking whether n is prime or not (although I don't think I'm the first one to come up with that theory) how would I prove this theory and if there was someone before me who was it?
The standard method I have been thought of checking if n is prime is checking every number until n. If n is divisible by at least one number in this range n isn't prime.
My theory is that you only need to check every number until ⌈√n⌉+1. The only number I could find that doesn't abide by this "rule" is 2.
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5758 | https://www.quora.com/What-is-the-difference-between-Mercator-and-Robinson-projection | What is the difference between Mercator and Robinson projection? - Quora
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What is the difference between Mercator and Robinson projection?
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The Mercator and Robinson projections are both methods for representing the Earth's surface on a flat map, but they have different characteristics and purposes. Here's a breakdown of the key differences:
Mercator Projection
Type: Cylindrical projection.
Distortion: Preserves angles (conformal) but distorts area, especially near the poles. Landmasses like Greenland and Antarctica appear much larger than they are in reality.
Use: Commonly used for marine navigation because it allows for straight-line courses to be plotted easily.
Appearance: Straight lines represent constant compass bearings, making
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The Mercator and Robinson projections are both methods for representing the Earth's surface on a flat map, but they have different characteristics and purposes. Here's a breakdown of the key differences:
Mercator Projection
Type: Cylindrical projection.
Distortion: Preserves angles (conformal) but distorts area, especially near the poles. Landmasses like Greenland and Antarctica appear much larger than they are in reality.
Use: Commonly used for marine navigation because it allows for straight-line courses to be plotted easily.
Appearance: Straight lines represent constant compass bearings, making it useful for navigational purposes.
Projection Characteristics: The scale increases as one moves away from the equator, leading to significant distortion in high-latitude regions.
Robinson Projection
Type: Pseudocylindrical projection.
Distortion: Attempts to minimize distortion of area, shape, distance, and direction across the map. No single aspect is preserved perfectly; instead, it provides a more balanced view.
Use: Often used in world maps for educational purposes and in publications because it offers a visually appealing representation of the entire world.
Appearance: Curved edges and a more realistic portrayal of the continents and oceans, making it aesthetically pleasing.
Projection Characteristics: Provides a compromise between different types of distortion, making it better for general reference than for specific navigational purposes.
Summary
Mercator: Best for navigation, preserves angles but distorts size, especially at high latitudes.
Robinson: Best for visual representation, balances distortions of shape, area, and distance, providing a more realistic view of the world.
Each projection serves different needs based on the intended use of the map.
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Frederick Crouch
Retired teacher (2009–present) · Author has 8.7K answers and 1.9M answer views
·5y
As the Earth is a globe, it can only be represented on a flat sheet of paper by using some kind of projection. Mercator projection shows where all the countries are in relationship to each other, but at the expense of exaggerating landmasses near the poles. Mercator projection uses straight lines for lines of longitude. Robinson’s projection attempts to overcome the limitations of Mercator projection by using curved lines for longitude. However, this is at the expense of some distortion and distance errors. Mercator projection is preferred by seafarers since it represents any course as a fixed
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As the Earth is a globe, it can only be represented on a flat sheet of paper by using some kind of projection. Mercator projection shows where all the countries are in relationship to each other, but at the expense of exaggerating landmasses near the poles. Mercator projection uses straight lines for lines of longitude. Robinson’s projection attempts to overcome the limitations of Mercator projection by using curved lines for longitude. However, this is at the expense of some distortion and distance errors. Mercator projection is preferred by seafarers since it represents any course as a fixed compass bearing which is a straight line between any two points on the map.
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Peter Schlesinger
Ph.D. in Natural Resources, U Idaho · Author has 2.5K answers and 2.1M answer views
·5y
Mercator is cylindrical
Robinson is pseudo-Cylindrical
See: Commonly Used Map Projections
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Mercator is cylindrical
Robinson is pseudo-Cylindrical
See: Commonly Used Map Projections
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What is the difference between Mercator projections and Robinson projections when it comes to maps?
Why is Google Maps still using a Mercator projection, rather than another projection that more accurately represents the earth?
Why is the Mercator projection preferred by seafarers?
How incorrect is the Mercator Projection and are there maps which show countries in their accurate sizes?
What are the characteristics and uses of the Mercator projection?
Jon Harley
map enthusiast · Author has 2.1K answers and 7.3M answer views
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Originally Answered: What is the difference between Mercator projections and Robinson projections when it comes to maps? ·
Mercator projections are conformal, which means that angles in the real world are preserved on the map. This makes Mercator good for navigation, because if you draw a line north-east on the map, it will correspond to north-east on the globe. The price to be paid for this is increasingly large distortion in shapes as you approach the poles, and it’s not possible to represent the poles at all using Mercator.
Robinson projections are not conformal, but also do not preserve shapes (so the shapes of land masses are still distorted, again with greater distortion towards the poles). But the distortion
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Mercator projections are conformal, which means that angles in the real world are preserved on the map. This makes Mercator good for navigation, because if you draw a line north-east on the map, it will correspond to north-east on the globe. The price to be paid for this is increasingly large distortion in shapes as you approach the poles, and it’s not possible to represent the poles at all using Mercator.
Robinson projections are not conformal, but also do not preserve shapes (so the shapes of land masses are still distorted, again with greater distortion towards the poles). But the distortion is less than when using Mercator and the polar regions can be shown. The Robinson projection was a compromise between distorting angles and distorting shapes, while still being able to represent the globe on a flat, rectangular plane. Projections which preserve areas/shapes better generally have to abandon the rectangular shape which is so convenient for printed maps.
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What is the difference between Mercator projections and Robinson projections when it comes to maps?
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What are the characteristics and uses of the Mercator projection?
What is the difference between a realistic map projection and a normal (Mercator) map projection? What are their advantages and disadvantages?
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5759 | https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Mordell_inequality | Jump to content
Erdős–Mordell inequality
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From Wikipedia, the free encyclopedia
On sums of distances in triangles
In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. It is named after Paul Erdős and Louis Mordell. Erdős (1935) posed the problem of proving the inequality; a proof was provided two years later by Mordell and D. F. Barrow (1937). This solution was however not very elementary. Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007).
Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from P to the sides are replaced by the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.
Statement
[edit]
Let be an arbitrary point P inside a given triangle , and let , , and be the perpendiculars from to the sides of the triangles.
(If the triangle is obtuse, one of these perpendiculars may cross through a different side of the triangle and end on the line supporting one of the sides.) Then the inequality states that
Proof
[edit]
Let the sides of ABC be a opposite A, b opposite B, and c opposite C; also let PA = p, PB = q, PC = r, dist(P;BC) = x, dist(P;CA) = y, dist(P;AB) = z. First, we prove that
This is equivalent to
The right side is the area of triangle ABC, but on the left side, r + z is at least the height of the triangle; consequently, the left side cannot be smaller than the right side. Now reflect P on the angle bisector at C. We find that cr ≥ ay + bx for P's reflection. Similarly, bq ≥ az + cx and ap ≥ bz + cy. We solve these inequalities for r, q, and p:
Adding the three up, we get
Since the sum of a positive number and its reciprocal is at least 2 by AM–GM inequality, we are finished. Equality holds only for the equilateral triangle, where P is its centroid.
Another strengthened version
[edit]
Let ABC be a triangle inscribed into a circle (O) and P be a point inside of ABC. Let D, E, F be the orthogonal projections of P onto BC, CA, AB. M, N, Q be the orthogonal projections of P onto tangents to (O) at A, B, C respectively, then:
Equality hold if and only if triangle ABC is equilateral (Dao, Nguyen & Pham 2016; Marinescu & Monea 2017)
A generalization
[edit]
Let be a convex polygon, and be an interior point of . Let be the distance from to the vertex , the distance from to the side , the segment of the bisector of the angle from to its intersection with the side then (Lenhard 1961):
In absolute geometry
[edit]
In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in Pambuccian (2008), to the statement
that the sum of the angles of a triangle is less than or equal to two right angles.
See also
[edit]
List of triangle inequalities
References
[edit]
Alsina, Claudi; Nelsen, Roger B. (2007), "A visual proof of the Erdős-Mordell inequality", Forum Geometricorum, 7: 99–102.
Bankoff, Leon (1958), "An elementary proof of the Erdős-Mordell theorem", American Mathematical Monthly, 65 (7): 521, doi:10.2307/2308580, JSTOR 2308580.
Dao, Thanh Oai; Nguyen, Tien Dung; Pham, Ngoc Mai (2016), "A strengthened version of the Erdős-Mordell inequality" (PDF), Forum Geometricorum, 16: 317–321, MR 3556993.
Erdős, Paul (1935), "Problem 3740", American Mathematical Monthly, 42: 396, doi:10.2307/2301373, JSTOR 2301373.
Kazarinoff, D. K. (1957), "A simple proof of the Erdős-Mordell inequality for triangles", Michigan Mathematical Journal, 4 (2): 97–98, doi:10.1307/mmj/1028988998. (See D. K. Kazarinoff's inequality for tetrahedra.)
Lenhard, Hans-Christof (1961), "Verallgemeinerung und Verschärfung der Erdös-Mordellschen Ungleichung für Polygone", Archiv für Mathematische Logik und Grundlagenforschung, 12: 311–314, doi:10.1007/BF01650566, MR 0133060, S2CID 124681241.
Marinescu, Dan Ștefan; Monea, Mihai (2017), "About a strengthened version of the Erdős-Mordell inequality" (PDF), Forum Geometricorum, 17: 197–202.
Mordell, L. J.; Barrow, D. F. (1937), "Solution to 3740", American Mathematical Monthly, 44: 252–254, doi:10.2307/2300713, JSTOR 2300713.
Pambuccian, Victor (2008), "The Erdős-Mordell inequality is equivalent to non-positive curvature", Journal of Geometry, 88 (1–2): 134–139, doi:10.1007/s00022-007-1961-4, S2CID 123082256.
External links
[edit]
Weisstein, Eric W. "Erdős-Mordell Theorem". MathWorld.
Alexander Bogomolny, "Erdös-Mordell Inequality", from Cut-the-Knot.
Retrieved from "
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Another Example of Graphing Logs
w/o a Calculatorw/ a Calculator
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Most of the differences between the various log graphs you'll be doing will be due to the specifics of the base (is it 10? is it 2?) and various shifts of the graph from the standard position.
Yes, you'll be working with function transformations and translations again.
Content Continues Below
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Graph y = log2(x + 3).
This graph will be similar to the graph of log2(x) but, because the 3 that is added inside the function, its graph will be shifted three units sideways.
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Since the +3 is inside the log's argument, the graph's shift cannot be up or down. This means that the shift has to be to the left or to the right. But which way?
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You can keep track of the direction of the shift by looking at the basic graph's point (1, 0) (which I'm calling "basic" because it's both standard and also easy to remember). The log will be 0 when the argument, x + 3, is equal to 1. When is x + 3 equal to 1? When x = −2. Then the basic log-graph point of (1, 0) will be shifted over to(−2, 0) on this graph; that is, the graph is shifted three units to the left.
(If you are not comfortable with this concept or these manipulations, please review how to work with translations of functions directly from the rules.)
Since a log cannot have an argument of zero or less, then I must have x + 3 > 0. This tells me that, for this graph, x must always be greater than −3.
The graph of the basic log function y = log2(x) crawls up the positive side of the y-axis to reach the x-axis, with the line never going to the left of the limitation that x must be greater than zero. To remind myself of the similar limitation of this log (where x must always be greater than −3), I will insert a dashed line at x = −3:
A line like this, which marks off territory where the graph can't go, is called a vertical asymptote, or simply an asymptote. I don't have to add this to the graph, but it can be very helpful, and might convince the grader that I do indeed know what I'm doing.
After I dash in the asymptote, I plot some points:
20 = 1, so log2(1) = 0; x + 3 = 1 for x = −2, so (−2, 0) is a point on the graph
21 = 2, so log2(2) = 1; x + 3 = 2 for x = −1, so (−1, 1) is a point on the graph
22 = 4, so log2(4) = 2; x + 3 = 4 for x = 1, so (1, 2) is a point on the graph
23 = 8, so log2(8) = 3; x + 3 = 8 for x = 5, so (5, 3) is a point on the graph
Then, working in the other direction:
2−1 = 0.5, so log2(0.5) = −1; x + 3 = 0.5 for x = −2.5, so (−2.5, −1) is a point on the graph
2−2 = 0.25, so log2(0.25) = −2; x + 3 = 0.25 for x = −2.75, so (−2.75, −2) is a point on the graph
2−3 = 0.125, so log2(0.125) = −3; x + 3 = 0.125 for x = −2.875, so (−2.875, −3) is a point on the graph
Note that, to find each of these points, I did not start with an x-value and then puzzle my way to a y-value; that would be too hard, and I'm too lazy. Instead, I started with a simple exponential statement, switched it around to the corresponding logarithmic statement, and then figured out, for that exponent (which is also my y-value), what the x-value needed to be. This method is, in my view, much the simpler way to work these problems.
Plotting the points I've calculated, I get:
...and connecting the dots gives me the following graph:
Content Continues Below
If you check this in your calculator, first, remember to put the x + 3 inside parentheses, or your calculator will think you mean log2(x) + 3, and you'll get the wrong answer. Second, remember that your calculator can only follow its programming — it can't think — so the graph it displays will likely be at least partially incorrect, even if you enter the function correctly.
For example, if you plug y = log2(x + 3) into a graphing calculator — in the change-of-base formulation of — you will likely get a graph that looks something like this:
Now, you know full well that the log doesn't just stop there at the left, hanging uselessly in space. Why is the calculator doing it wrong?
Because it's just a machine, and it's doing the best it can.
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The calculator graphs in a manner similar to how you do: it picks x-values, computes corresponding y-values, plots the points, and connects the dots. But, whereas you know that the log graph continues downward forever, getting infinitesimally close to the y-axis (or wherever else the vertical asymptote happens to be), the calculator only knows that it tried one x-value on its list, got "error" for an answer, tried the next x-value on its list, and got a valid y-value. Since it has no other dots before that first one, and because it can't think, it starts the graph with that first dot.
This is another instance of "student smart; calculator stupid". Don't assume, just because the calculator displays a graph a certain way, that this is what the graph actually looks like. Use your head!
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Below are some different variations on the same basic logarithmic function, with the associated graph below each equation. Note that, even if the graph is moved left or right, or up or down, or is flipped upside-down, it still displays the same curve.
(A widget for graphing logarithmic functions, and for checking your own graphs, is available below the graphs.)
y = ln(x)
y = ln(−x)
y = −ln(x)
y = ln(x + 1)
y = ln(−x + 1)
y = −ln(−x)
y = ln(x) − 1
y = ln(−x) − 1
y = ln(x − 1)
y = ln(x) + 1
y = ln(−x) + 1
y = ln(−x − 1)
URL:
You can use the Mathway widget below to practice graphing logs. Try the entered exercise, or type in your own exercise. Then click the button and select "Graph" to compare your answer to Mathway's.
Please accept "preferences" cookies in order to enable this widget.
(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)
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5761 | https://mathoverflow.net/questions/467512/iteration-matrix-representation-with-complex-conjugate-operator | linear algebra - Iteration matrix representation with complex conjugate operator - MathOverflow
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Iteration matrix representation with complex conjugate operator
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I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector V V that gives the voltage for each node.
The radial power flow can be expressed as an iterative procedure, with one step given by
V(k)=M V∗(k−1)+V 0 1,V(k)=M V∗(k−1)+V 0 1,
where M M is a matrix that captures the grid properties (line resistances, topology, ...), and ∗∗ denotes the complex conjugate.
If there was no complex conjugate, I would compute the maximum value of the eigenvalues of M M to check for convergence. However, in this case since the complex conjugate cannot be specified as a linear transformation, is this convergence check still valid?
linear-algebra
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edited Mar 22, 2024 at 16:46
YCor
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asked Mar 22, 2024 at 15:04
ElectricPhysiscistElectricPhysiscist
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can you be more precise what you mean by M(V)∗M(V)∗ ? for example, by writing this in components?Carlo Beenakker –Carlo Beenakker 2024-03-22 15:14:54 +00:00 Commented Mar 22, 2024 at 15:14
M M is a linear transformation that acts on the vector V∗V∗. In fact the notation can be misleading, the parenthesis are there only to indicate that V(k−1)V(k−1) is the complex conjugate ElectricPhysiscist –ElectricPhysiscist 2024-03-22 15:20:15 +00:00 Commented Mar 22, 2024 at 15:20
The M M matrix is a standard linear operator. Different grid configurations return different M M's, so I want to check, for a given grid, the convergence of the iterative radial power flow. However, the doubt that I presented (the way to deal with the complex conjugate on V) prevents me from doing it ElectricPhysiscist –ElectricPhysiscist 2024-03-22 15:39:30 +00:00 Commented Mar 22, 2024 at 15:39
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You can just rewrite your iteration in the real form, as
[V k 1 V k 2]=[M 1 M 2 M 2−M 1][V k−1 1 V k−1 2]+[R V 0 1 I V 0 1],[V 1 k V 2 k]=[M 1 M 2 M 2−M 1][V 1 k−1 V 2 k−1]+[ℜ V 0 1 ℑ V 0 1],
where V k 1:=R V(k)V 1 k:=ℜ V(k), V k 2:=I V(k)V 2 k:=ℑ V(k), M 1:=R M M 1:=ℜ M, M 2:=I M M 2:=ℑ M.
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answered Mar 22, 2024 at 15:38
Iosif PinelisIosif Pinelis
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Fundamentals of Neuroscience, Part 2: Neurons and Networks.
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Neurons in isolation are fascinating and complicated, but the real magic of neuroscience happens in the interaction between neurons. In this course, we examine how neurons pass signals to one another and how complex dynamics can result from just a few neurons arranged in relatively simple circuits.
Continue your journey through our Fundamentals of Neuroscience series with animations that explore the richness and complexity of the brain, documentaries about working labs around Cambridge.
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Neurons in isolation are fascinating and complex, but the real magic of neuroscience becomes manifest when we consider how neurons interact with one another. In this module, we examine how neurons pass signals to one another, and we explore how complex dynamics can result from even small numbers of neurons arranged in relatively simple circuits.
Lesson1: The Synapse
The junctions between neurons, called synapses, allow information to pass from one neuron to another. In lesson 1 “The Synapse,” we explore what synapses are made of, and how they work.
Lesson 2: Excitation & Inhibition
Synapses can be grouped into two categories: synapses that increase the activity of the postsynaptic neuron are called excitatory synapses, while those that decrease its activity are called inhibitory synapses. In lesson 2 “Excitation & Inhibition,” we discuss the main differences between excitatory and inhibitory synapses.
Lesson 3: Small Circuits
Neurons combine information from many synapses at once in a process known as synaptic integration. In lesson 3 “Small Circuits,” we explore how a neuron integrates information from multiple synapses over time to allow complex signaling.
Lesson 4: Neuromodulation
How many components are there in a synapse? You may think that the answer is simple: two, the presynaptic and the postsynaptic terminals. In lesson 4 “Neuromodulation,” we’ll see that this is not entirely true: the strength of many synapses is directly influenced by a third neuron, in a process called neuromodulation.
Lesson 5: Potentiation & DepressionOne of the most amazing properties of the nervous system is its ability to adapt and change in the face of a changing environment, a phenomenon called "neuronal plasticity." Lesson 5, “Potentiation and Depression,” focuses on how neuronal plasticity occurs in the brain, and how it shapes the way we think and behave.
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5763 | https://chem.libretexts.org/Courses/Purdue/Chem_26505%3A_Organic_Chemistry_I_(Lipton)/Chapter_1._Electronic_Structure_and_Chemical_Bonding/1.03_Valence_electrons_and_open_valences | Skip to main content
1.3: Valence electrons and open valences
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: Jun 5, 2019
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1.2: The Octet Rule and Covalent Bonding
1.4: Formal Charge
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( \newcommand{\kernel}{\mathrm{null}\,})
A valence electron is an electron that is associated with an atom, and that can participate in the formation of a chemical bond; in a single covalent bond, both atoms in the bond contribute one valence electron in order to form a shared pair. The presence of valence electrons can determine the element's chemical properties and whether it may bond with other elements: For a main group element, a valence electron can only be in the outermost electron shell.
An atom with a closed shell of valence electrons (corresponding to an electron configuration s2p6) tends to be chemically inert. An atom with one or two valence electrons more than a closed shell is highly reactive, because the extra valence electrons are easily removed to form a positive ion. An atom with one or two valence electrons fewer than a closed shell is also highly reactive, because of a tendency either to gain the missing valence electrons (thereby forming a negative ion), or to share valence electrons (thereby forming a covalent bond).
Like an electron in an inner shell, a valence electron has the ability to absorb or release energy in the form of a photon. An energy gain can trigger an electron to move (jump) to an outer shell; this is known as atomic excitation. Or the electron can even break free from its associated atom's valence shell; this is ionization to form a positive ion. When an electron loses energy (thereby causing a photon to be emitted), then it can move to an inner shell which is not fully occupied.
The number of valence electrons
The number of valence electrons of an element can be determined by the periodic table group (vertical column) in which the element is categorized. With the exception of groups 3–12 (the transition metals), the units digit of the group number identifies how many valence electrons are associated with a neutral atom of an element listed under that particular column.
The periodic table of the chemical elements
| Periodic table group | Valence Electrons |
--- |
| Group 1 (I) (alkali metals) | 1 |
| Group 2 (II) (alkaline earth metals) | 2 |
| Groups 3-12 (transition metals) | 2 (The 4s shell is complete and cannot hold any more electrons) |
| Group 13 (III) (boron group) | 3 |
| Group 14 (IV) (carbon group) | 4 |
| Group 15 (V) (pnictogens) | 5 |
| Group 16 (VI) (chalcogens) | 6 |
| Group 17 (VII) (halogens) | 7 |
| Group 18 (VIII or 0) (noble gases) | 8 |
The general method for counting valence electrons is generally not useful for transition metals. Instead the modified d electron count method is used. Except for helium, which has only two valence electrons.
The Concept of Open Valence ("Valence")
The valence (or valency) of an element is a measure of its combining power with other atoms when it forms chemical compounds or molecules. The concept of valence was developed in the last half of the 19th century and was successful in explaining the molecular structure of many organic compounds. The quest for the underlying causes of valence lead to the modern theories of chemical bonding, including Lewis structures (1916), valence bond theory (1927), molecular orbitals (1928), valence shell electron pair repulsion theory (1958) and all the advanced methods of quantum chemistry.
The combining power or affinity of an atom of an element was determined by the number of hydrogen atoms that it combined with. In methane, carbon has a valence of 4; in ammonia, nitrogen has a valence of 3; in water, oxygen has a valence of two; and in hydrogen chloride, chlorine has a valence of 1. Chlorine, as it has a valence of one, can be substituted for hydrogen, so phosphorus has a valence of 5 in phosphorus pentachloride, PCl5. Valence diagrams of a compound represent the connectivity of the elements, lines between two elements, sometimes called bonds, represented a saturated valency for each element. Examples are:-
| | | | | | | | | | |
--- --- --- --- --- |
| Compound | H2 | CH4 | C3H8 | C2H2 | NH3 | NaCN | H2S | H2SO4 | Cl2O7 |
| Diagram | | | | | | | | | |
| Valencies | Hydrogen 1 | Carbon 4 Hydrogen 1 | Carbon 4 Hydrogen 1 | Carbon 4 Hydrogen 1 | Nitrogen 3 Hydrogen 1 | Sodium 1 Carbon 4 Nitrogen 3 | Sulfur 2 Hydrogen 1 | Sulfur 6 Oxygen 2 Hydrogen 1 | Chlorine 7 Oxygen 2 |
Valence only describes connectivity, it does not describe the geometry of molecular compounds, or what are now known to be ionic compounds or giant covalent structures. The line between atoms does not represent a pair of electrons as it does in Lewis diagrams.
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1.2: The Octet Rule and Covalent Bonding
1.4: Formal Charge |
5764 | https://courses.lumenlearning.com/albanytech-mathmodeling/chapter/graphs-of-quadratic-functions/ | Quadratic Functions and their Graphs
Learning Objectives
Quadratic Functions
Roots or zeros of a quadratic function
Characteristics of a parabola
vertex
axis of symmetry
x/y-intercepts
Classifying solutions to quadratic equations
The discriminant
Curved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.
An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)
Characteristics of Parabolas
The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.
The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of x at which y= 0.
Example: Identifying the Characteristics of a Parabola
Determine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown below.
Solution
The vertex is the turning point of the graph. We can see that the vertex is at (3, 1). Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is x= 3. This parabola does not cross the x-axis, so it has no zeros. It crosses the y-axis at (0, 7) so this is the y-intercept.
Try It
General and Standard Forms of Quadratic Functions
The general form of a quadratic function presents the function in the form
f(x)=ax2+bx+c
where a, b, and c are real numbers and a≠0. If a>0, the parabola opens upward. If a<0, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.
The axis of symmetry is defined by x=−b2a. If we use the quadratic formula, x=−b±√b2−4ac2a, to solve ax2+bx+c=0 for the x-intercepts, or zeros, we find the value of x halfway between them is always x=−b2a, the equation for the axis of symmetry.
The figure below shows the graph of the quadratic function written in general form as y=x2+4x+3. In this form, a=1, b=4, and c=3. Because a>0, the parabola opens upward. The axis of symmetry is x=−42(1)=−2. This also makes sense because we can see from the graph that the vertical line x=−2 divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, (−2,−1). The x-intercepts, those points where the parabola crosses the x-axis, occur at (−3,0) and (−1,0).
The standard form of a quadratic function presents the function in the form
f(x)=a(x−h)2+k
where (h, k) is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.
Given a quadratic function in general form, find the vertex of the parabola.
One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, (k), and where it occurs, (x). If we are given the general form of a quadratic function:
f(x)=ax2+bx+c
We can define the vertex, (h,k), by doing the following:
Identify a, b, and c.
Find h, the x-coordinate of the vertex, by substituting a and b into h=−b2a.
Find k, the y-coordinate of the vertex, by evaluating k=f(h)=f(−b2a)
Example: Finding the Vertex of a Quadratic Function
Find the vertex of the quadratic function f(x)=2x2−6x+7. Rewrite the quadratic in standard form (vertex form).
Solution
The horizontal coordinate of the vertex will be at
h=−b2a =−−62(2) =64 =32
The vertical coordinate of the vertex will be at
k=f(h) =f(32) =2(32)2−6(32)+7 =52
Rewriting into standard form, the stretch factor will be the same as the a in the original quadratic.
f(x)=ax2+bx+cf(x)=2x2−6x+7
Using the vertex to determine the shifts,
f(x)=2(x−32)2+52
Try It
Given the equation g(x)=13+x2−6x, write the equation in general form and then in standard form.
Solution
g(x)=x2−6x+13 in general form; g(x)=(x−3)2+4 in standard form
In this section, we will investigate quadratic functions further, including solving problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree polynomial functions, so they provide a good opportunity for a detailed study of function behavior.
Classifying Solutions to Quadratic Equations
Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the y-intercept of a quadratic by evaluating the function at an input of zero, and we find the x-intercepts at locations where the output is zero. Notice that the number of x-intercepts can vary depending upon the location of the graph.
Number of x-intercepts of a parabola
Mathematicians also define x-intercepts as roots of the quadratic function.
How To: Given a quadratic function f(x), find the y– and x-intercepts.
Evaluate f(0) to find the y-intercept.
Solve the quadratic equation f(x)=0 to find the x-intercepts.
Example: Finding the y– and x-Intercepts of a Parabola
Find the y– and x-intercepts of the quadratic f(x)=3x2+5x−2.
Solution
We find the y-intercept by evaluating f(0).
f(0)=3(0)2+5(0)−2 =−2
So the y-intercept is at (0,−2).
For the x-intercepts, or roots, we find all solutions of f(x)=0.
0=3x2+5x−2
In this case, the quadratic can be factored easily, providing the simplest method for solution.
0=(3x−1)(x+2)
0=3x−10=x+2x=13orx=−2
So the roots are at (13,0) and (−2,0).
Analysis of the Solution
By graphing the function, we can confirm that the graph crosses the y-axis at (0,−2). We can also confirm that the graph crosses the x-axis at (13,0) and (−2,0).
In Example: Finding the y– and x-Intercepts of a Parabola, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.
How To: Given a quadratic function, find the x-intercepts by rewriting in standard form.
Substitute a and b into h=−b2a.
Substitute x = h into the general form of the quadratic function to find k.
Rewrite the quadratic in standard form using h and k.
Solve for when the output of the function will be zero to find the x-intercepts.
Example: Finding the Roots of a Parabola
Find the x-intercepts of the quadratic function f(x)=2x2+4x−4.
Solution
We begin by solving for when the output will be zero.
0=2x2+4x−4
Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.
f(x)=a(x−h)2+k
We know that a= 2. Then we solve for h and k.
h=−b2ak=f(−1) =−42(2) =2(−1)2+4(−1)−4 =−1 =−6
So now we can rewrite in standard form.
f(x)=2(x+1)2−6
We can now solve for when the output will be zero.
0=2(x+1)2−66=2(x+1)23=(x+1)2x+1=±√3x=−1±√3
The graph has x-intercepts at (−1−√3,0) and (−1+√3,0).
Analysis of the Solution
We can check our work by graphing the given function on a graphing utility and observing the roots.
Try It
Find the y-intercept for the function g(x)=13+x2−6x.
Solution
y-intercept at (0, 13)
Complex Roots
Now you will hopefully begin to understand why we introduced complex numbers at the beginning of this module. Consider the following function: f(x)=x2+2x+3, and it’s graph below:
Does this function have roots? It’s probably obvious that this function does not cross the x-axis, therefore it doesn’t have any x-intercepts. Recall that the x-intercepts of a function are found by setting the function equal to zero:
x2+2x+3=0
In the next example, we will solve this equation. You will see that there are roots, but they are not x-intercepts because the function does not contain (x,y) pairs that are on the x-axis. We call these complex roots.
By setting the function equal to zero and using the quadratic formula to solve, you will see that the roots contain complex numbers:
x2+2x+3=0
Example
Find the x-intercepts of the quadratic function. f(x)=x2+2x+3
Show Answer
The x-intercepts of the function f(x)=x2+2x+3 are found by setting it equal to zero, and solving for x since the y values of the x-intercepts are zero.
First, identify a, b, c.
x2+2x+3=0a=1,b=2,c=3
Substitute these values into the quadratic formula.
x=−b±√b2−4ac2a=−2±√22−4(1)(3)2(1)=−2±√4−122=−2±√−82−2±2i√22−1±i√2=−1+√2,−1−√2
The solutions to this equations are complex, therefore there are no x-intercepts for the function f(x)=x2+2x+3 in the set of real numbers that can be plotted on the Cartesian Coordinate plane. The graph of the function is plotted on the Cartesian Coordinate plane below:
Graph of quadratic function with no x-intercepts in the real numbers.
Note how the graph does not cross the x-axis, therefore there are no real x-intercepts for this function.
The Discriminant
The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions. When we consider the discriminant, or the expression under the radical, b2−4ac, it tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. In turn, we can then determine whether a quadratic function has real or complex roots. The table below relates the value of the discriminant to the solutions of a quadratic equation.
| Value of Discriminant | Results |
--- |
| b2−4ac=0 | One repeated rational solution |
| b2−4ac>0, perfect square | Two rational solutions |
| b2−4ac>0, not a perfect square | Two irrational solutions |
| b2−4ac<0 | Two complex solutions |
A General Note: The Discriminant
For ax2+bx+c=0, where a, b, and c are real numbers, the discriminant is the expression under the radical in the quadratic formula: b2−4ac. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.
Example
Use the discriminant to find the nature of the solutions to the following quadratic equations:
x2+4x+4=0
8x2+14x+3=0
3x2−5x−2=0
3x2−10x+15=0
Show Answer
Calculate the discriminant b2−4ac for each equation and state the expected type of solutions.
x2+4x+4=0b2−4ac=(4)2−4(1)(4)=0. There will be one repeated rational solution.
8x2+14x+3=0b2−4ac=(14)2−4(8)(3)=100. As 100 is a perfect square, there will be two rational solutions.
3x2−5x−2=0b2−4ac=(−5)2−4(3)(−2)=49. As 49 is a perfect square, there will be two rational solutions.
3x2−10x+15=0b2−4ac=(−10)2−4(3)(15)=−80. There will be two complex solutions.
We have seen that a quadratic equation may have two real solutions, one real solution, or two complex solutions.
Let’s summarize how the discriminant affects the evaluation of √b2−4ac, and how it helps to determine the solution set.
If b2−4ac>0, then the number underneath the radical will be a positive value. You can always find the square root of a positive, so evaluating the quadratic formula will result in two real solutions (one by adding the positive square root, and one by subtracting it).
If b2−4ac=0, then you will be taking the square root of 0, which is 0. Since adding and subtracting 0 both give the same result, the “[latex]\pm[/late]" portion of the formula doesn't matter. There will be one real repeated solution.
If b2−4ac<0, then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.
Example
Use the discriminant to determine how many and what kind of solutions the quadratic equation x2−4x+10=0 has.
Show Solution
Evaluate b2−4ac. First note that a=1,b=−4, and c=10.
b2−4ac(−4)2−4(1)(10)
The result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions.
16–40=−24
Answer
The quadratic equation x2−4x+10=0 has two complex solutions.
Important Terms
axis of symmetry
: a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by x=−b2a.
general form of a quadratic function
: the function that describes a parabola, written in the form f(x)=ax2+bx+c, where a, b, and c are real numbers and a≠0.
standard form of a quadratic function
: the function that describes a parabola, written in the form f(x)=a(x−h)2+k, where (h, k) is the vertex.
discriminant
: the value under the radical in the quadratic formula, b2−4ac, which tells whether the quadratic has real or complex roots
vertex
: the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function
zeros
: in a given function, the values of x at which y = 0, also called roots
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5765 | https://math.stackexchange.com/questions/2464364/how-can-i-find-the-equation-of-a-circle-given-two-points-and-a-tangent-line-thro | Stack Exchange Network
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How can I find the equation of a circle given two points and a tangent line through one of the points?
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I was wondering whether it was possible to find the equation of a circle given two points and the equation of a tangent line through one of the points so I produced the following problem:
Find the equation of the circle which passes through $(1,7)$ and $(-6,0)$ and has a tangent with equation $2x-9y+61=0$ at $(1,7)$
This seems like it should be solvable but I cannot work out how. Clearly, the line and the circle have one point of intersection so I tried finding the point of intersection between the line and the circle using the generic circle equation $(x-a)^2 + (y-b)^2 = r^2$, the equation of the line, and the discriminant of the resulting quadratic, which must be 0, but this still produces a quadratic with two unknowns.
I also feel like the fact that the perpendicular distance between centre $(a,b)$ and the line is the radius can be used somehow. Again, trying this seems to produce equations with too many unknowns.
How can I solve this problem?
polynomials
systems-of-equations
circles
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asked Oct 9, 2017 at 12:20
LJD200LJD200
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$\begingroup$ This wouldnt be too hard if you allow calculus and trigonometry, but I assume based on the tags youre limiting the question to algebra and geometry? $\endgroup$
DonielF
– DonielF
2017-10-09 15:52:17 +00:00
Commented Oct 9, 2017 at 15:52
$\begingroup$ @DonielF Preferably, yes but there are already some great answers using algebra/geometry so a calculus/trig method would be interesting to see! $\endgroup$
LJD200
– LJD200
2017-10-09 15:56:17 +00:00
Commented Oct 9, 2017 at 15:56
1
$\begingroup$ You're already told what the intersection is. It's (1,7). $\endgroup$
Acccumulation
– Acccumulation
2017-10-10 04:48:10 +00:00
Commented Oct 10, 2017 at 4:48
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6 Answers 6
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Here is a geometric version, not using a single formula. Start with the points $A$ and $B$ and a line $\ell$ through $A$ (see the figure below).
Construct the perpendicular line to $\ell$ through $A$ (the $\color{red}{\text{red}}$ line). Construct the perpendicular bisectors between $A$ and $B$ (the $\color{green}{\text{green}}$ line, the green dot is the midpoint of $A$ and $B$). The intersection of both constructed lines is the circle's center. The readius is the distance of the center to $A$.
You can translate every step into a formula to solve it numerically if necessary.
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edited Oct 9, 2017 at 13:16
answered Oct 9, 2017 at 13:02
M. WinterM. Winter
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$\begingroup$ This is the answer I would have recommended. Its a high-school procedure that all of us should have learned. $\endgroup$
Lubin
– Lubin
2017-10-10 17:08:30 +00:00
Commented Oct 10, 2017 at 17:08
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Hint. The center of such circle is on the line which is orthogonal to tangent line and passes through the point of tangency.
Therefore, in your case, the coordinate of the center is $C=(1+2t,7-9t)$ for some $t\in \mathbb{R}$. In order to find $t$, impose that $C$ has the same distance from the given points $P=(1,7)$ and $Q=(6,0)$: $$|CP|^2=|CQ|^2\Leftrightarrow (4+81)t^2=(7+2t)^2+(7-9t)^2.$$
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edited Oct 9, 2017 at 12:48
answered Oct 9, 2017 at 12:28
Robert ZRobert Z
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You can write a linear combination using the degenerate circle with centre $(1,7)$ and radius $r=0$ and the tangent which is like a degenerate circle with "infinite" radius
You get $(x-1)^2+(y-7)^2+k(2x-9y+61)=0$
Then plugging the coordinates of the other point $(-6;\;0)$ you have
$(-7)^2+(-7)^2+k(-12+61)=0$
solving for $k$ we get $k=-2$
the wanted circle has equation $(x-1)^2+(y-7)^2-2(2x-9y+61)=0$
$\color{red}{x^2+y^2-6 x+4 y-72=0}$
centre $(3;\;-2)$ and radius $r=\sqrt{3^2+2^2-(-72)}=\sqrt{85}$
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answered Oct 9, 2017 at 12:32
RaffaeleRaffaele
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$\begingroup$ This is almost Plückers mu, which would let you get an equation for the circle directly, without introducing the parameter $k$. Let $C$ and $D$ be the homogeneous matrices of the two degenerate conics. The linear combination of them that also passes through the point $\mathbf p=[-6:0:1]$ is $(\mathbf p^TC\mathbf p)\,D-(\mathbf p^TD\mathbf p)\,C$. The coefficients of $C$ and $D$ are just the values that you get by plugging $\mathbf p$ into the other equation. $\endgroup$
amd
– amd
2017-10-10 01:51:12 +00:00
Commented Oct 10, 2017 at 1:51
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This solution essentially follows M. Winter's solution algebraically rather than geometrically.
Plugging $(1,7)$ into $(x-a)^2 + (y-b)^2 = r^2$, we get $$ (1-a)^2 + (7-b)^2 = r^2 \text{.} $$ In the same manner, $(-6,0)$ gives $$ (-6 - a)^2 + b^2 = r^2 \text{.} $$ Eliminating $r^2$ between these, multiplying out the binomials, then collecting the like powers of $a$ and $b$, these reduce to $$ a + b = 1 \text{,} $$ the equation of the perpendicular bisector of the segment between the given points.
The slope of the given tangent line is $2/9$, so the slope of the line through the center of the circle and $(1,7)$ is $-9/2$. The equation of this line is $$ (b - 7) = (-9/2)(a - 1) \text{.} $$
These two lines intersect at the center of the circle. Taking $b = 1-a$, plugging into this last equation, and solving for $a$, we get $$ a = 3 $$ and then $$ b = -2 \text{.} $$
Putting these back into either of the first two displays gives $$ r^2 = 85 \text{.} $$
So the equation you requested is $$ (x - 3)^2 + (y + 2)^2 = 85 \text{.} $$
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answered Oct 10, 2017 at 2:19
Eric TowersEric Towers
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Method I: Recognize that a point on a circle satisfies its equation
Let's use the formula
$$x^2+y^2+2gx+2fy+c=0\tag{i}$$
where $(-g,-f)$ is the center of the circle.
For points $A(x_1,y_1)$ and $B(x_2,y_2)$ to fall on the circle, it must be that $A$ and $B$ are solutions to $x^2+y^2+2gx+2fy+c=0$. Thus:
$$x_1^2+y_1^2+2gx_1+2fy_1+c=0\tag{ii}$$ $$x_2^2+y_2^2+2gx_2+2fy_2+c=0\tag{iii}$$
Given the equation of the line tangent to the circle, that is, $$ax+by+d=0\tag{iv}$$ we can solve for the distance between the line and the center:
$$r=\frac{|-ag-bf-d|}{\sqrt{a^2+b^2}}$$
Since $r=\sqrt{g^2+f^2-c}$, we can set these two equations equal to one another to get:
$$\sqrt{g^2+f^2-c}=\frac{|-ag-bf+d|}{\sqrt{a^2+b^2}}$$$$g^2+f^2-c=\frac{(-ag-bf+d)^2}{a^2+b^2}\tag{v}$$
Now we just need to solve the system of equations (ii), (iii), and (v).
In your case, $A(x_1,y_1)=(1,7)$, $B(x_2,y_2)=(-6,0)$, and in equation (iv), $a=2$, $b=-9$, and $d=61$. Substituting into the above equations we get:
$$1^2+7^2+2g(1)+2f(7)+c=0$$ $$2g+14f+50+c=0\\\tag{iia}$$ $$(-6)^2+0^2+2g(-6)+2f(0)+c=0$$ $$-12g+2f+36+c=0\\\tag{iiia}$$ $$g^2+f^2-c=\frac{(-2g+9f+61)^2}{2^2+(-9)^2}=\frac{(-2g+9f+61)^2}{85}\\\tag{va}$$
I will leave it as an exercise to the reader to solve for $c$, $f$, and $g$ using these equations. Once you have done so, you can substitute $(-g,-f)$ back into equation (i) to get the equation of your circle.
Method Ia: Same thing, new equation
This is very similar to Method I above, except that we use the equation
$$(x-h)^2+(y-k)^2=r^2\tag{i}$$
where $(h,k)$ is the center of the circle and $r$ is the radius.
As above, if points $A(x_1,y_1)$ and $B(x_2,y_2)$ are on the circle, $A$ and $B$ must satisfy the above equation. Therefore, we can write:
$$(x_1-h)^2+(y_1-k)^2=r^2\tag{ii}$$ $$(x_2-h)^2+(y_2-k)^2=r^2\tag{iii}$$
Given the equation of the line tangent to the circle, that is, $$ax+by+d=0\tag{iv}$$ we can solve for the distance between the line and the center:
$$r=\frac{|ax_1+by_1+d|}{\sqrt{a^2+b^2}}\tag{v}$$
As above, you can substitute $A(x_1,y_1)=(1,7)$, $B(x_2,y_2)=(-6,0)$, $a=2$, $b=-9$, and $d=61$ into equations (ii), (iii), and (v) to solve for $h$, $k$, and $r$, and then plug those values back into equation (i) to get the solution to your circle. In your particular case, however, this formula for $r$ happens to yield $0$; for that reason, although it's much messier, you may wish to stick with method I for this particular problem.
Method II: The center is equidistant from all points
The radius is perpendicular to the tangent line. It is known that for lines $\overleftrightarrow A$ and $\overleftrightarrow B$ to be perpendicular, then if the slope of $\overleftrightarrow A$ is $\frac{c}{d}$, the slope of $overleftrightarrow B$ will be $-\frac{d}{c}$ - its opposite reciprocal. Using the point-slope form, we can come up with an equation for the line for which the radius is a segment. Then, we can draw two circles whose centers are on the two points specified, and we can gradually expand their radii until they intersect on the line. The point of intersection is the center of the requested circle, and the radii of all three circles are congruent.
In your case: since the tangent line $2x+9y-61=0$ can be rewritten as $y=-\frac29x-\frac{61}9\tag{i}$, we can see that the tangent line has a slope of $-\frac29$; therefore, the radius has a slope of $\frac92$. Since we know it passes through the point $(1,7)$, we can use point-slope form and convert to slope-intercept to find the equation of the radius.
$$y-(7)=\frac92(x-(1))$$$$y=\frac92x+\frac52\tag{ii}$$
For the final step, in which we gradually expand the circles until they intersect on the line, I recommend Desmos, in which you can plug in the three equations as well as an $r=$ line, and by tweaking the $r=$ line you will simultaneously expand or contract both circles.
(Note that this is just a geometric illustration of Robert Z's answer, also posted in different words by Dr. Sonnhard Graubner.
Method III: Derivative is the slope of a tangent
In method II, we used the slope of the tangent to find the radius. Here, we will use the slope of the tangent to solve for the circle itself.
By implicit differentiation and a heavy dose of chain rule:
$$(x-h)^2+(y-k)^2=r^2\tag{i}$$ $$2(x-h)(1-0)+2(y-k)^2\left(\frac{dy}{dx}-0\right)=0$$ $$\frac{dy}{dx}=\frac{-2(x-h)}{2(y-k)}=\frac{-x+h}{y-k}\tag{ii}$$
Recall that a derivative is simply the slope of a curve at a given point. On a circle, this is equivalent to the slope of the tangent line. Recall also that for a point to fall on the circle it must satisfy the equation of the circle. We can thus substitute the slope of the tangent line for $\frac{dy}{dx}$ and the point of intersection for $(x,y)$ in equation (ii). We can also substitute the two points for $(x,y)$ in equation (i).
In your case: for $x=1$ and $y=7$, we can say that $\frac{dy}{dx}=-\frac29$.
$$-\frac29=\frac{-1+h}{7-k}\tag{iia}$$
We also said that $(1,7)$ and $(-6,0)$ are solutions to the circle.
$$(1-h)^2+(7-k)^2=r^2\tag{ia}$$ $$(-6-h)^2+(0-k)^2=r^2\tag{ib}$$
By solving the system of equations (ia), (ib), and (iia), we can solve for $h$, $k$, and $r$. Once you have those numbers, you can plug them back into equation (i) for the equation of the circle.
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edited Jun 12, 2020 at 10:38
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answered Oct 10, 2017 at 20:38
DonielFDonielF
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$\begingroup$ I skipped some steps when manipulating the equations in the examples. If you're unclear how I got from one line to the next, feel free to ping me and I can elaborate. All such instances are either factoring or condensing polynomials, though (I think). $\endgroup$
DonielF
– DonielF
2017-10-10 21:00:04 +00:00
Commented Oct 10, 2017 at 21:00
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let $$A(1;7)$$ and $$B(-6;0)$$ then we have the equations $$|MB|=|MA|=r$$ where $$M(x,y)$$ and since the tangentline is orthogonal to the radius, we have $$\frac{1-x}{7-y}=-\frac{9}{2}$$ this comes from our tangent line $$y=\frac{2}{9}x+\frac{61}{9}$$
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answered Oct 9, 2017 at 12:30
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
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M is the foot of the perpendicular fro...
M is the foot of the perpendicular from a point P on a parabola y 2=4 a x to its directrix and S P M is an equilateral triangle, where S is the focus. Then find S P .
Video Solution
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The directrix of the parabola is the line x=−a.
Determine the Coordinates of Point P:
Let the coordinates of point P on the parabola be (t 2,2 a t), where t is a parameter.
Find the Foot of the Perpendicular M:
Since M is the foot of the perpendicular from P to the directrix x=−a, the x-coordinate of M will be −a and the y-coordinate will be the same as that of P.
Thus, the coordinates of M are (−a,2 a t).
Use the Properties of the Equilateral Triangle:
Since S P M is an equilateral triangle, all sides S P, P M, and S M are equal.
We need to find the lengths S P and S M.
Calculate the Length S M:
The distance S M can be calculated using the distance formula:
S M=√(a−(−a))2+(0−2 a t)2=√(2 a)2+(2 a t)2=√4 a 2+4 a 2 t 2=√4 a 2(1+t 2)=2 a√1+t 2
Calculate the Length S P:
The distance S P is:
S P=√(a−t 2)2+(0−2 a t)2=√(a−t 2)2+(2 a t)2
Expanding this gives:
S P=√(a−t 2)2+4 a 2 t 2=√a 2−2 a t 2+t 4+4 a 2 t 2=√a 2+2 a t 2+t 4=√(t 2+a)2=t 2+a
Equate the Lengths:
Since S P=S M (because S P M is equilateral), we have:
t 2+a=2 a√1+t 2
Solve for S P:
Squaring both sides:
(t 2+a)2=4 a 2(1+t 2)
Expanding gives:
t 4+2 a t 2+a 2=4 a 2+4 a 2 t 2
Rearranging leads to:
t 4−2 a t 2−3 a 2=0
This is a quadratic in t 2. Let x=t 2:
x 2−2 a x−3 a 2=0
Using the quadratic formula:
x=2 a±√(2 a)2+4⋅3 a 2 2=2 a±√14 a 2 2=a(1±√3.5)
Taking the positive root (since t 2 must be positive):
t 2=a(1+√3.5)
Final Calculation of S P:
Plugging t 2 back into S P:
S P=t 2+a=a(1+√3.5)+a=a(2+√3.5)
Final Answer:
Thus, the length S P is a(2+√3.5).
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Alternate In. Ang. Thrm.
Alternate interior angles theorem
Here you will learn about the alternate interior angles theorem, including how to recognize when angles are alternate, and apply this understanding to solve problems.
Students will first learn about the alternate interior angles theorem as part of geometry in 8 th grade.
What is the alternate interior angles theorem?
The alternate interior angles theorem states that when a transversal cuts through two parallel lines, the pairs of angles on opposite sides of the transversal line and between the two parallel lines are congruent (equal in measure).
Alternate interior angles are equal
You can often spot alternate angles by drawing a Z shape:
There are two different types of alternate angles: alternate interior angles and alternate exterior angles.
Below are diagrams showing these two variations.
[FREE] Angles Worksheet (Grade 4)
Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!
[FREE] Angles Worksheet (Grade 4)
Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!
Alternate interior angles
Here, the two angles of a and b are in between the parallel lines and therefore are pairs of alternate interior angles.
Alternate exterior angles
Here, the two angles of c and d are outside of the parallel lines and so these are two examples of pairs of alternate exterior angles.
It is important to notice that the transversal on each diagram is at a different angle but the two angles in each diagram are the same size. As is the case with all diagrams for angles in parallel lines, never use a protractor to find an angle as all the diagrams, unless stated otherwise, are not to scale.
What is the alternate interior angles theorem?
Common Core State Standards
How does this relate to 8 th grade math and high school math?
How to calculate missing angles using the alternate interior angles theorem
In order to calculate missing angles using the alternate interior angles theorem:
Steps 2 and 3 may be done in either order and may need to be repeated.
Step 3 may not always be required.
Alternate interior angles theorem examples
Example 1: alternate interior angles
Calculate the size of the missing angle \theta. Justify your answer.
2Use the alternate interior angles theorem to find a missing angle.
Here you can label the alternate angle on the diagram as 50^{\circ} .
3Use basic angle facts if needed to calculate other missing angles.
Here as \theta is on a straight line with 50^{\circ},
Example 2: alternate interior angles
Calculate the size of the missing angle \theta. Justify your answer.
Highlight the angle(s) that you already know.
Here you can also state the angle 62^{\circ} as it is corresponding to the original angle.
Use the alternate interior angles theorem to find a missing angle.
The angle \theta is the alternate angle to the sum of 56^{\circ} and 62^{\circ} as shown in the given figure.
Use basic angle facts if needed to calculate other missing angles.
Example 3: alternate interior angles with algebra
By finding the value for x, calculate the size of the missing angle \theta. Justify your answer.
Highlight the angle(s) that you already know.
Here you know most of the angles, so highlight the angles that are going to help you find the value of x.
Use the alternate interior angles theorem to find a missing angle.
Here, 3x and 30^{\circ} are alternate to each other. You can therefore find the value for x:
\begin{aligned} 3x&=30^{\circ} \\ x&=10^{\circ} \end{aligned}
Use basic angle facts if needed to calculate other missing angles.
Here you can use the sum of angles in a triangle to help us calculate the value for \theta.
The value for 10x=100^{\circ} as x=10^{\circ}. You therefore can calculate \theta:
\begin{aligned} \theta&=180^{\circ}-(100^{\circ}+30^{\circ}) \\ \theta&=50^{\circ} \end{aligned}
Teaching tips for alternate interior angles theorem
Easy mistakes to make
Related angles in parallel lines lessons
Practice alternate interior angles theorem questions
Calculate the size of angle \theta.
104^{\circ} and \theta are alternate interior angles so \theta = 104^{\circ}
Calculate the size of angle \theta.
Using the alternate interior angles theorem, you can see the angle 103^{\circ}.
You can then use angles on a straight line:
Find the value of \theta.
Using the alternate interior angles theorem, you can see that the angle in the bottom right vertex of the triangle is 38^{\circ}.
You can then use angles in a triangle:
Find the value of \theta.
Using the alternate interior angles theorem, you can see the angle 22^{\circ}.
You can then use the fact that this is an isosceles triangle and the other two angles in the triangle are the same.
Calculate the size of angle \theta.
Using the alternate interior angles theorem, you can see the angle 110^{\circ}.
Then \theta=110-42=68^{\circ}
By calculating the value for x, find the value of each angle labeled.
50^{\circ} and 50^{\circ}
22^{\circ} and 34^{\circ}
34^{\circ} and 34^{\circ}
50^{\circ} and 130^{\circ}
Alternate angles are equal so 7x-20=4x+10
When x=10:
Alternate interior angles theorem FAQs
The alternate interior angles theorem states that when a transversal cuts through two parallel lines, the pairs of angles on opposite sides of the transversal line and between the two parallel lines are congruent.
Alternate interior angles are located on the inner side of the parallel lines and on opposite sides of the transversal.
The converse of the alternate interior angles theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent (equal in measure), then the two lines are parallel.
Alternate interior angles are on opposite sides of the transversal and are congruent (equal) when the lines are parallel.Co-interior angles (also called consecutive interior angles) are on the same side of the transversal and are supplementary ( add up to 180^{\circ}) when the lines are parallel.
The next lessons are
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Refrigerators and Heat Pumps
2,483 Views01:07 min April 30, 2023
Overview
Refrigerators or heat pumps are heat engines operating in a reverse direction. For a refrigerator, the focus is on removing heat from a specific area, whereas, for a heat pump, the focus is on dumping heat into one particular area. A refrigerator (or heat pump) absorbs heat Q c from the cold reservoir at Kelvin temperature T c and discards heat Q h to the hot reservoir at Kelvin temperature T h, while work W is done on the engine’s working substance.
A household refrigerator removes heat from the food while exhausting heat to the surrounding air. The required work is performed by the motor using electricity, which moves a coolant through the coils. A coolant with a boiling temperature below the freezing point of water is sent through the cycle. The coolant extracts heat from the refrigerator at the evaporator, causing the coolant to vaporize. It is then compressed and sent through the condenser, where it exhausts heat to the outside.
The effectiveness or coefficient of performance K R of a refrigerator is measured by the heat removed from the cold reservoir divided by the work done by the working substance cycle by cycle. Conversely, the coefficient of performance K P of a heat pump is measured by the heat dumped to the hot reservoir divided by the work done to the engine on the working substance cycle by cycle.
Transcript
Refrigerators and heat pumps are heat engines operated in reverse cycle.
They absorb heat from the cold reservoir and release this heat to the hot reservoir while work is done on the engine's working substance.
In a household refrigerator, heat is taken from the food kept inside and released into the surroundings. Here, the aim is to remove the heat from a particular area.
The measure of a refrigerator's effectiveness is known as the coefficient of performance, which is the ratio of the heat removed from the cold reservoir to the amount of work done on the working substance.
Unlike refrigerators, heat pumps are used in colder regions to make houses warm.
They focus on releasing the heat into a specific area. Heat pumps work like a refrigerator that has been turned inside out.
The coefficient of performance for a heat pump is the ratio of the heat released to the hot reservoir to the amount of work done on the working substance.
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RefrigeratorsHeat PumpsHeat EnginesHeat RemovalCold ReservoirHot ReservoirWork DoneCoolantEvaporatorCondenserCoefficient Of PerformancePerformance MeasurementHousehold RefrigerationThermal Cycle
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5769 | https://quizgecko.com/learn/microeconomics-marginal-rate-of-substitution-and-transformation-uvs134 | MRS and MRT in Microeconomics: General Equilibrium Quiz and Flashcards
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MICROECONOMICS I General Equilibrium I MRS and MRT I Consumers and Firms
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Marginal Rate of Substitution and Marginal Rate of Transformation
MRS and MRT in Equilibrium
Example: N's MRS and MRT
General Equilibrium
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What is the marginal rate of substitution (MRS) of an individual?
The rate at which producers give up one good to produce another good.
The quantity of one good an individual is willing to consume.
The rate at which an individual is willing to trade one good for another good. (correct)
The price ratio of two goods in equilibrium.
What happens to the marginal rate of substitution (MRS) and the marginal rate of transformation (MRT) in equilibrium?
The MRS and MRT are equal to each other. (correct)
The MRS and MRT are not related to each other.
The MRS increases, while the MRT decreases.
The MRS decreases, while the MRT increases.
What is the marginal rate of transformation (MRT) of producers?
The quantity of one good producers are willing to produce.
The rate at which producers give up one good to produce another good. (correct)
The price ratio of two goods in equilibrium.
The rate at which an individual is willing to trade one good for another good.
In the example, why do producers not produce 4 units of food to give N 1 unit of clothing?
Because the MRT is 2. (B)
Signup and view all the answers
What happens in a perfect market according to the concept of general equilibrium?
Companies produce until the last unit of production is covered by the price. (B)
Signup and view all the answers
What is the result of N getting 1 extra unit of clothing in the example?
The extra unit of clothing can be distributed between N and Bill. (B)
Signup and view all the answers
In equilibrium, the marginal rate of substitution of one individual is different from that of another individual.
False (B)
Signup and view all the answers
The marginal rate of transformation shows how much of one good consumers are willing to trade for one more unit of another good.
False (B)
Signup and view all the answers
In equilibrium, the MRT equals the ratio of the prices of the two goods.
True (A)
Signup and view all the answers
Producers will continue to produce until the last unit of production is sold at a loss.
False (B)
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In the example, N is willing to trade 2 units of food for 1 unit of clothing.
False (B)
Signup and view all the answers
The marginal rate of substitution and the marginal rate of transformation are equal in equilibrium because producers adjust their production to satisfy demand.
True (A)
Signup and view all the answers
What is the condition for equilibrium in terms of the marginal rate of substitution and transformation among consumers and producers?
The marginal rate of substitution (MRS) of an individual equals the marginal rate of transformation (MRT) and the price ratios in equilibrium.
Signup and view all the answers
What happens when producers adjust their production to satisfy demand in equilibrium?
The marginal rate of transformation (MRT) equals the marginal rate of substitution (MRS) and the price ratios.
Signup and view all the answers
How do producers adjust their production to satisfy demand in the example of N's MRS and MRT?
Producers produce 2 units of clothing instead of 4 units of food, which is worth 4 units of food, to give N 1 unit of clothing.
Signup and view all the answers
What is the condition for equilibrium in general equilibrium theory?
The price equals the marginal cost.
Signup and view all the answers
What is the result of trading goods until the marginal rate of substitution is the same among consumers?
All possibilities of trading are exhausted.
Signup and view all the answers
What is the relationship between the marginal rate of substitution and the price ratios in equilibrium?
The marginal rate of substitution equals the price ratios.
Signup and view all the answers
Questions and Answers
Export
What is the marginal rate of substitution (MRS) of an individual?
The rate at which producers give up one good to produce another good.
The quantity of one good an individual is willing to consume.
The rate at which an individual is willing to trade one good for another good. (correct)
The price ratio of two goods in equilibrium.
What happens to the marginal rate of substitution (MRS) and the marginal rate of transformation (MRT) in equilibrium?
The MRS and MRT are equal to each other. (correct)
The MRS and MRT are not related to each other.
The MRS increases, while the MRT decreases.
The MRS decreases, while the MRT increases.
What is the marginal rate of transformation (MRT) of producers?
The quantity of one good producers are willing to produce.
The rate at which producers give up one good to produce another good. (correct)
The price ratio of two goods in equilibrium.
The rate at which an individual is willing to trade one good for another good.
In the example, why do producers not produce 4 units of food to give N 1 unit of clothing?
Because the MRT is 2. (B)
Signup and view all the answers
What happens in a perfect market according to the concept of general equilibrium?
Companies produce until the last unit of production is covered by the price. (B)
Signup and view all the answers
What is the result of N getting 1 extra unit of clothing in the example?
The extra unit of clothing can be distributed between N and Bill. (B)
Signup and view all the answers
In equilibrium, the marginal rate of substitution of one individual is different from that of another individual.
False (B)
Signup and view all the answers
The marginal rate of transformation shows how much of one good consumers are willing to trade for one more unit of another good.
False (B)
Signup and view all the answers
In equilibrium, the MRT equals the ratio of the prices of the two goods.
True (A)
Signup and view all the answers
Producers will continue to produce until the last unit of production is sold at a loss.
False (B)
Signup and view all the answers
In the example, N is willing to trade 2 units of food for 1 unit of clothing.
False (B)
Signup and view all the answers
The marginal rate of substitution and the marginal rate of transformation are equal in equilibrium because producers adjust their production to satisfy demand.
True (A)
Signup and view all the answers
What is the condition for equilibrium in terms of the marginal rate of substitution and transformation among consumers and producers?
The marginal rate of substitution (MRS) of an individual equals the marginal rate of transformation (MRT) and the price ratios in equilibrium.
Signup and view all the answers
What happens when producers adjust their production to satisfy demand in equilibrium?
The marginal rate of transformation (MRT) equals the marginal rate of substitution (MRS) and the price ratios.
Signup and view all the answers
How do producers adjust their production to satisfy demand in the example of N's MRS and MRT?
Producers produce 2 units of clothing instead of 4 units of food, which is worth 4 units of food, to give N 1 unit of clothing.
Signup and view all the answers
What is the condition for equilibrium in general equilibrium theory?
The price equals the marginal cost.
Signup and view all the answers
What is the result of trading goods until the marginal rate of substitution is the same among consumers?
All possibilities of trading are exhausted.
Signup and view all the answers
What is the relationship between the marginal rate of substitution and the price ratios in equilibrium?
The marginal rate of substitution equals the price ratios.
Signup and view all the answers
Questions and Answers
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Flashcards
Marginal Rate of Substitution (MRS)
The rate at which an individual is willing to trade one good for another good.
Marginal Rate of Transformation (MRT)
The rate at which producers give up one good to produce another good.
Equilibrium MRS and MRT
In equilibrium, the MRS and MRT are equal.
Equilibrium Condition (General)
The price equals the marginal cost.
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Perfect Market Equilibrium
Companies produce until the last unit of production is covered by the price.
Signup and view all the flashcards
MRS Equality (among consumers)
In equilibrium, the marginal rate of substitution of one individual is the same as that of another individual.
Signup and view all the flashcards
MRT vs. Consumer Willingness
MRT shows how much producers give up, while MRS shows consumers trade-offs.
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Equilibrium MRT and Price Ratio
In equilibrium, the MRT equals the ratio of the prices of the two goods.
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Producer's Loss Avoidance
Producers won't produce at a loss.
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Individual's Trade Willingness (N)
N's trade willingness (2 units food for 1 unit clothing) is not the equilibrium value.
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Equilibrium (MRS=MRT)
The marginal rate of substitution (MRS) equals the marginal rate of transformation (MRT) and the price ratios in equilibrium.
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Producer Adjustment (Demand)
Producers adjust their production to satisfy demand, resulting in MRT equaling MRS and price ratios.
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Equilibrium (Example)
Producers produce 2 unit clothing, not 4 units food, to satisfy demand.
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Complete Trade
Trading goods until the marginal rate of substitution is the same among consumers.
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MRS and Price Ratio in Equilibrium
MRS and Price ratios are equal.
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General Equilibrium
A state where all markets are in equilibrium simultaneously.
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Trading Outcomes
Trading until MRS is same for all consumers leads to exhaustion of trade possibilities.
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Flashcards
Marginal Rate of Substitution (MRS)
The rate at which an individual is willing to trade one good for another good.
Marginal Rate of Transformation (MRT)
The rate at which producers give up one good to produce another good.
Equilibrium MRS and MRT
In equilibrium, the MRS and MRT are equal.
Equilibrium Condition (General)
The price equals the marginal cost.
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Perfect Market Equilibrium
Companies produce until the last unit of production is covered by the price.
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MRS Equality (among consumers)
In equilibrium, the marginal rate of substitution of one individual is the same as that of another individual.
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MRT vs. Consumer Willingness
MRT shows how much producers give up, while MRS shows consumers trade-offs.
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Equilibrium MRT and Price Ratio
In equilibrium, the MRT equals the ratio of the prices of the two goods.
Signup and view all the flashcards
Producer's Loss Avoidance
Producers won't produce at a loss.
Signup and view all the flashcards
Individual's Trade Willingness (N)
N's trade willingness (2 units food for 1 unit clothing) is not the equilibrium value.
Signup and view all the flashcards
Equilibrium (MRS=MRT)
The marginal rate of substitution (MRS) equals the marginal rate of transformation (MRT) and the price ratios in equilibrium.
Signup and view all the flashcards
Producer Adjustment (Demand)
Producers adjust their production to satisfy demand, resulting in MRT equaling MRS and price ratios.
Signup and view all the flashcards
Equilibrium (Example)
Producers produce 2 unit clothing, not 4 units food, to satisfy demand.
Signup and view all the flashcards
Complete Trade
Trading goods until the marginal rate of substitution is the same among consumers.
Signup and view all the flashcards
MRS and Price Ratio in Equilibrium
MRS and Price ratios are equal.
Signup and view all the flashcards
General Equilibrium
A state where all markets are in equilibrium simultaneously.
Signup and view all the flashcards
Trading Outcomes
Trading until MRS is same for all consumers leads to exhaustion of trade possibilities.
Signup and view all the flashcards
Flashcards
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Study Notes
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Study Notes
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Marginal Rate of Substitution and Marginal Rate of Transformation
In equilibrium, consumers trade goods until their willingness to exchange is the same, exhausting all possibilities of trading.
The marginal rate of substitution (MRS) of an individual shows how much of one good they are willing to trade for one more unit of another good.
The marginal rate of transformation (MRT) shows how much of one good producers must give up to produce one more unit of another good.
MRS and MRT in Equilibrium
In equilibrium, the MRS of an individual equals the MRS of another individual.
The MRT also equals the MRS in equilibrium, as producers adjust their production to satisfy demand.
The MRS and MRT are equal to the price ratios in equilibrium, with the price of one good relative to the price of another.
Example: N's MRS and MRT
Suppose N's MRS is 4, meaning she is willing to trade 4 units of food for 1 unit of clothing.
The MRT is 2, meaning producers must give up 2 units of food to produce 1 unit of clothing.
To give N 1 unit of clothing, producers must not produce 4 units of food, but instead produce 2 units of clothing, which is worth 4 units of food.
This results in N getting 1 extra unit of clothing, which can be distributed between N and Bill.
General Equilibrium
In a perfect market, companies produce until the last unit of production is covered by the price, meaning the price equals the marginal cost.
The ratio of prices equals the ratio of marginal costs, leading to the point of equilibrium between the MRS and MRT.
The general equilibrium is reached when the MRS for each consumer equals the MRT, meaning the willingness to exchange goods equals the capacity to produce them.
This equilibrium point is the most efficient, as there is no better way to improve the economy, and the possibility for improvement is exhausted.
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Study Notes
Listen
Marginal Rate of Substitution and Marginal Rate of Transformation
In equilibrium, consumers trade goods until their willingness to exchange is the same, exhausting all possibilities of trading.
The marginal rate of substitution (MRS) of an individual shows how much of one good they are willing to trade for one more unit of another good.
The marginal rate of transformation (MRT) shows how much of one good producers must give up to produce one more unit of another good.
MRS and MRT in Equilibrium
In equilibrium, the MRS of an individual equals the MRS of another individual.
The MRT also equals the MRS in equilibrium, as producers adjust their production to satisfy demand.
The MRS and MRT are equal to the price ratios in equilibrium, with the price of one good relative to the price of another.
Example: N's MRS and MRT
Suppose N's MRS is 4, meaning she is willing to trade 4 units of food for 1 unit of clothing.
The MRT is 2, meaning producers must give up 2 units of food to produce 1 unit of clothing.
To give N 1 unit of clothing, producers must not produce 4 units of food, but instead produce 2 units of clothing, which is worth 4 units of food.
This results in N getting 1 extra unit of clothing, which can be distributed between N and Bill.
General Equilibrium
In a perfect market, companies produce until the last unit of production is covered by the price, meaning the price equals the marginal cost.
The ratio of prices equals the ratio of marginal costs, leading to the point of equilibrium between the MRS and MRT.
The general equilibrium is reached when the MRS for each consumer equals the MRT, meaning the willingness to exchange goods equals the capacity to produce them.
This equilibrium point is the most efficient, as there is no better way to improve the economy, and the possibility for improvement is exhausted.
Study Notes
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Introduction
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Questions and Answers
Flashcards
Study Notes
Marginal Rate of Substitution and Marginal Rate of Transformation
MRS and MRT in Equilibrium
Example: N's MRS and MRT
General Equilibrium
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MICROECONOMICS I General Equilibrium I MRS and MRT I Consumers and Firms
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5770 | https://www.mathway.com/popular-problems/Algebra/202057 | Graph y = square root of 9-x^2 | Mathway
Enter a problem...
[x]
Algebra Examples
Popular Problems
Algebra
Graph y = square root of 9-x^2
y=√9-x 2 y=√9−x 2
Step 1
Find the domain for y=√9-x 2 y=√9−x 2 so that a list of x x values can be picked to find a list of points, which will help graphing the radical.
Tap for more steps...
Step 1.1
Set the radicand in √(3+x)(3-x) greater than or equal to 0 to find where the expression is defined.
(3+x)(3-x)≥0
Step 1.2
Solve for x.
Tap for more steps...
Step 1.2.1
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
3+x=0
3-x=0
Step 1.2.2
Set 3+x equal to 0 and solve for x.
Tap for more steps...
Step 1.2.2.1
Set 3+x equal to 0.
3+x=0
Step 1.2.2.2
Subtract 3 from both sides of the equation.
x=-3
x=-3
Step 1.2.3
Set 3-x equal to 0 and solve for x.
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Step 1.2.3.1
Set 3-x equal to 0.
3-x=0
Step 1.2.3.2
Solve 3-x=0 for x.
Tap for more steps...
Step 1.2.3.2.1
Subtract 3 from both sides of the equation.
-x=-3
Step 1.2.3.2.2
Divide each term in -x=-3 by -1 and simplify.
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Step 1.2.3.2.2.1
Divide each term in -x=-3 by -1.
-x-1=-3-1
Step 1.2.3.2.2.2
Simplify the left side.
Tap for more steps...
Step 1.2.3.2.2.2.1
Dividing two negative values results in a positive value.
x 1=-3-1
Step 1.2.3.2.2.2.2
Divide x by 1.
x=-3-1
x=-3-1
Step 1.2.3.2.2.3
Simplify the right side.
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Step 1.2.3.2.2.3.1
Divide-3 by -1.
x=3
x=3
x=3
x=3
x=3
Step 1.2.4
The final solution is all the values that make (3+x)(3-x)≥0 true.
x=-3,3
Step 1.2.5
Use each root to create test intervals.
x<-3
-3<x<3
x>3
Step 1.2.6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 1.2.6.1
Test a value on the interval x<-3 to see if it makes the inequality true.
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Step 1.2.6.1.1
Choose a value on the interval x<-3 and see if this value makes the original inequality true.
x=-6
Step 1.2.6.1.2
Replace x with -6 in the original inequality.
(3-6)(3-(-6))≥0
Step 1.2.6.1.3
The left side -27 is less than the right side 0, which means that the given statement is false.
False
False
Step 1.2.6.2
Test a value on the interval-3<x<3 to see if it makes the inequality true.
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Step 1.2.6.2.1
Choose a value on the interval-3<x<3 and see if this value makes the original inequality true.
x=0
Step 1.2.6.2.2
Replace x with 0 in the original inequality.
(3+0)(3-(0))≥0
Step 1.2.6.2.3
The left side 9 is greater than the right side 0, which means that the given statement is always true.
True
True
Step 1.2.6.3
Test a value on the interval x>3 to see if it makes the inequality true.
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Step 1.2.6.3.1
Choose a value on the interval x>3 and see if this value makes the original inequality true.
x=6
Step 1.2.6.3.2
Replace x with 6 in the original inequality.
(3+6)(3-(6))≥0
Step 1.2.6.3.3
The left side -27 is less than the right side 0, which means that the given statement is false.
False
False
Step 1.2.6.4
Compare the intervals to determine which ones satisfy the original inequality.
x<-3 False
-3<x<3 True
x>3 False
x<-3 False
-3<x<3 True
x>3 False
Step 1.2.7
The solution consists of all of the true intervals.
-3≤x≤3
-3≤x≤3
Step 1.3
The domain is all values of x that make the expression defined.
Interval Notation:
[-3,3]
Set-Builder Notation:
{x|-3≤x≤3}
Interval Notation:
[-3,3]
Set-Builder Notation:
{x|-3≤x≤3}
Step 2
To find the end points, substitute the bounds of the x values from the domain into f(x)=√(3+x)(3-x).
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Step 2.1
Replace the variable x with -3 in the expression.
f(-3)=√(3-3)(3-(-3))
Step 2.2
Simplify the result.
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Step 2.2.1
Remove parentheses.
f(-3)=√(3-3)(3-(-3))
Step 2.2.2
Subtract 3 from 3.
f(-3)=√0(3-(-3))
Step 2.2.3
Multiply-1 by -3.
f(-3)=√0(3+3)
Step 2.2.4
Add 3 and 3.
f(-3)=√0⋅6
Step 2.2.5
Multiply 0 by 6.
f(-3)=√0
Step 2.2.6
Rewrite 0 as 0 2.
f(-3)=√0 2
Step 2.2.7
Pull terms out from under the radical, assuming positive real numbers.
f(-3)=0
Step 2.2.8
The final answer is 0.
0
0
Step 2.3
Replace the variable x with 3 in the expression.
f(3)=√(3+3)(3-(3))
Step 2.4
Simplify the result.
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Step 2.4.1
Remove parentheses.
f(3)=√(3+3)(3-(3))
Step 2.4.2
Add 3 and 3.
f(3)=√6(3-(3))
Step 2.4.3
Multiply-1 by 3.
f(3)=√6(3-3)
Step 2.4.4
Subtract 3 from 3.
f(3)=√6⋅0
Step 2.4.5
Multiply 6 by 0.
f(3)=√0
Step 2.4.6
Rewrite 0 as 0 2.
f(3)=√0 2
Step 2.4.7
Pull terms out from under the radical, assuming positive real numbers.
f(3)=0
Step 2.4.8
The final answer is 0.
0
0
0
Step 3
The end points are (-3,0),(3,0).
(-3,0),(3,0)
Step 4
Select a few x values from the domain. It would be more useful to select the values so that they are next to the x value of the radical expression end point.
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Step 4.1
Substitute the x value -2 into f(x)=√(3+x)(3-x). In this case, the point is (-2,√5).
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Step 4.1.1
Replace the variable x with -2 in the expression.
f(-2)=√(3-2)(3-(-2))
Step 4.1.2
Simplify the result.
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Step 4.1.2.1
Remove parentheses.
f(-2)=√(3-2)(3-(-2))
Step 4.1.2.2
Subtract 2 from 3.
f(-2)=√1(3-(-2))
Step 4.1.2.3
Multiply 3-(-2) by 1.
f(-2)=√3-(-2)
Step 4.1.2.4
Multiply-1 by -2.
f(-2)=√3+2
Step 4.1.2.5
Add 3 and 2.
f(-2)=√5
Step 4.1.2.6
The final answer is √5.
y=√5
y=√5
y=√5
Step 4.2
Substitute the x value -1 into f(x)=√(3+x)(3-x). In this case, the point is (-1,2√2).
Tap for more steps...
Step 4.2.1
Replace the variable x with -1 in the expression.
f(-1)=√(3-1)(3-(-1))
Step 4.2.2
Simplify the result.
Tap for more steps...
Step 4.2.2.1
Remove parentheses.
f(-1)=√(3-1)(3-(-1))
Step 4.2.2.2
Subtract 1 from 3.
f(-1)=√2(3-(-1))
Step 4.2.2.3
Multiply-1 by -1.
f(-1)=√2(3+1)
Step 4.2.2.4
Add 3 and 1.
f(-1)=√2⋅4
Step 4.2.2.5
Multiply 2 by 4.
f(-1)=√8
Step 4.2.2.6
Rewrite 8 as 2 2⋅2.
Tap for more steps...
Step 4.2.2.6.1
Factor 4 out of 8.
f(-1)=√4(2)
Step 4.2.2.6.2
Rewrite 4 as 2 2.
f(-1)=√2 2⋅2
f(-1)=√2 2⋅2
Step 4.2.2.7
Pull terms out from under the radical.
f(-1)=2√2
Step 4.2.2.8
The final answer is 2√2.
y=2√2
y=2√2
y=2√2
Step 4.3
Substitute the x value 0 into f(x)=√(3+x)(3-x). In this case, the point is (0,3).
Tap for more steps...
Step 4.3.1
Replace the variable x with 0 in the expression.
f(0)=√(3+0)(3-(0))
Step 4.3.2
Simplify the result.
Tap for more steps...
Step 4.3.2.1
Remove parentheses.
f(0)=√(3+0)(3-(0))
Step 4.3.2.2
Add 3 and 0.
f(0)=√3(3-(0))
Step 4.3.2.3
Multiply-1 by 0.
f(0)=√3(3+0)
Step 4.3.2.4
Add 3 and 0.
f(0)=√3⋅3
Step 4.3.2.5
Multiply 3 by 3.
f(0)=√9
Step 4.3.2.6
Rewrite 9 as 3 2.
f(0)=√3 2
Step 4.3.2.7
Pull terms out from under the radical, assuming positive real numbers.
f(0)=3
Step 4.3.2.8
The final answer is 3.
y=3
y=3
y=3
Step 4.4
Substitute the x value 1 into f(x)=√(3+x)(3-x). In this case, the point is (1,2√2).
Tap for more steps...
Step 4.4.1
Replace the variable x with 1 in the expression.
f(1)=√(3+1)(3-(1))
Step 4.4.2
Simplify the result.
Tap for more steps...
Step 4.4.2.1
Remove parentheses.
f(1)=√(3+1)(3-(1))
Step 4.4.2.2
Add 3 and 1.
f(1)=√4(3-(1))
Step 4.4.2.3
Multiply-1 by 1.
f(1)=√4(3-1)
Step 4.4.2.4
Subtract 1 from 3.
f(1)=√4⋅2
Step 4.4.2.5
Multiply 4 by 2.
f(1)=√8
Step 4.4.2.6
Rewrite 8 as 2 2⋅2.
Tap for more steps...
Step 4.4.2.6.1
Factor 4 out of 8.
f(1)=√4(2)
Step 4.4.2.6.2
Rewrite 4 as 2 2.
f(1)=√2 2⋅2
f(1)=√2 2⋅2
Step 4.4.2.7
Pull terms out from under the radical.
f(1)=2√2
Step 4.4.2.8
The final answer is 2√2.
y=2√2
y=2√2
y=2√2
Step 4.5
Substitute the x value 2 into f(x)=√(3+x)(3-x). In this case, the point is (2,√5).
Tap for more steps...
Step 4.5.1
Replace the variable x with 2 in the expression.
f(2)=√(3+2)(3-(2))
Step 4.5.2
Simplify the result.
Tap for more steps...
Step 4.5.2.1
Remove parentheses.
f(2)=√(3+2)(3-(2))
Step 4.5.2.2
Add 3 and 2.
f(2)=√5(3-(2))
Step 4.5.2.3
Multiply-1 by 2.
f(2)=√5(3-2)
Step 4.5.2.4
Subtract 2 from 3.
f(2)=√5⋅1
Step 4.5.2.5
Multiply 5 by 1.
f(2)=√5
Step 4.5.2.6
The final answer is √5.
y=√5
y=√5
y=√5
Step 4.6
The square root can be graphed using the points around the vertex(-3,0),(3,0),(-2,2.24),(-1,2.83),(0,3),(1,2.83),(2,2.24)
x y-3 0-2 2.24-1 2.83 0 3 1 2.83 2 2.24 3 0
x y-3 0-2 2.24-1 2.83 0 3 1 2.83 2 2.24 3 0
Step 5
[x 2 1 2√π∫x d x]
Please ensure that your password is at least 8 characters and contains each of the following:
a number
a letter
a special character: @$#!%?& |
5771 | https://codepoints.net/U+0064?lang=en | U+0064 LATIN SMALL LETTER D: d – Unicode – Codepoints
Home: go to the homepageLatin Small Letter CU+0000 to U+007F Basic LatinLatin Small Letter E
Source: Noto Sans
U+0064 Latin Small Letter D
Nº 100 General CategoryLowercase LetterScriptLatinBidirectional CategoryLeft To RightDecomposition Typenone
copy to clipboardshare this codepointembed this codepoint
U+0064 was added in Unicode version 1.1 in 1993. It belongs to the block U+0000 to U+007F Basic Latin in the U+0000 to U+FFFF Basic Multilingual Plane.
This character is a Lowercase Letter and is mainly used in the Latin script. Its uppercase variant is Latin Capital Letter D and its titlecase variant is Latin Capital Letter D.
The glyph is not a composition. Its East Asian Width is narrow. In bidirectional text it is written from left to right. When changing direction it is not mirrored. The word that U+0064 forms with similar adjacent characters prevents a line break inside it. The glyph can be confused with 34 other glyphs.
The Wikipedia has the following information about this codepoint:
D, or d, is the fourth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is dee (pronounced ), plural dees.
Representations
| System | Representation (click value to copy) |
--- |
| Nº | 100 |
| UTF-8 | 64 |
| UTF-16 | 00 64 |
| UTF-32 | 00 00 00 64 |
| URL-Quoted | %64 |
| HTML hex reference | d |
| Encoding: ASCII (hex bytes) | 64 |
| Encoding: BIG5 (hex bytes) | 64 |
| Encoding: BIG5HKSCS (hex bytes) | 64 |
| Encoding: CP037 (hex bytes) | 84 |
| Encoding: CP273 (hex bytes) | 84 |
| Encoding: CP424 (hex bytes) | 84 |
| Encoding: CP437 (hex bytes) | 64 |
| Encoding: CP500 (hex bytes) | 84 |
| Encoding: CP720 (hex bytes) | 64 |
| Encoding: CP737 (hex bytes) | 64 |
| Encoding: CP775 (hex bytes) | 64 |
| Encoding: CP850 (hex bytes) | 64 |
| Encoding: CP852 (hex bytes) | 64 |
| Encoding: CP855 (hex bytes) | 64 |
| Encoding: CP856 (hex bytes) | 64 |
| Encoding: CP857 (hex bytes) | 64 |
| Encoding: CP858 (hex bytes) | 64 |
| Encoding: CP860 (hex bytes) | 64 |
| Encoding: CP861 (hex bytes) | 64 |
| Encoding: CP862 (hex bytes) | 64 |
| Encoding: CP863 (hex bytes) | 64 |
| Encoding: CP864 (hex bytes) | 64 |
| Encoding: CP865 (hex bytes) | 64 |
| Encoding: CP866 (hex bytes) | 64 |
| Encoding: CP869 (hex bytes) | 64 |
| Encoding: CP874 (hex bytes) | 64 |
| Encoding: CP875 (hex bytes) | 84 |
| Encoding: CP932 (hex bytes) | 64 |
| Encoding: CP949 (hex bytes) | 64 |
| Encoding: CP950 (hex bytes) | 64 |
| Encoding: CP1006 (hex bytes) | 64 |
| Encoding: CP1026 (hex bytes) | 84 |
| Encoding: CP1125 (hex bytes) | 64 |
| Encoding: CP1140 (hex bytes) | 84 |
| Encoding: CP1250 (hex bytes) | 64 |
| Encoding: CP1251 (hex bytes) | 64 |
| Encoding: CP1252 (hex bytes) | 64 |
| Encoding: CP1253 (hex bytes) | 64 |
| Encoding: CP1254 (hex bytes) | 64 |
| Encoding: CP1255 (hex bytes) | 64 |
| Encoding: CP1256 (hex bytes) | 64 |
| Encoding: CP1257 (hex bytes) | 64 |
| Encoding: CP1258 (hex bytes) | 64 |
| Encoding: EUC_JP (hex bytes) | 64 |
| Encoding: EUC_JIS_2004 (hex bytes) | 64 |
| Encoding: EUC_JISX0213 (hex bytes) | 64 |
| Encoding: EUC_KR (hex bytes) | 64 |
| Encoding: GB2312 (hex bytes) | 64 |
| Encoding: GBK (hex bytes) | 64 |
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| Encoding: HZ (hex bytes) | 64 |
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| AGL: Latin-1 | d |
| AGL: Latin-2 | d |
| AGL: Latin-3 | d |
| AGL: Latin-4 | d |
| AGL: Latin-5 | d |
| Adobe Glyph List | d |
| digraph | d |
| RFC 5137 | \u'0064' |
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| Matlab | char(100) |
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| Rust | \u{64} |
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Related Characters
Latin Capital Letter D
Latin Small Letter D with Caron
Latin Capital Letter Dz with Caron
Latin Capital Letter D with Small Letter Z with Caron
Latin Small Letter Dz with Caron
Latin Capital Letter Dz
Latin Capital Letter D with Small Letter Z
Latin Small Letter Dz
Modifier Letter Capital D
Modifier Letter Small D
Latin Small Letter D with Dot Above
Latin Small Letter D with Dot Below
Latin Small Letter D with Line Below
Latin Small Letter D with Cedilla
Latin Small Letter D with Circumflex Below
Double-Struck Italic Capital D
Double-Struck Italic Small D
Roman Numeral Five Hundred
Small Roman Numeral Five Hundred
Parenthesized Latin Small Letter D
Circled Latin Capital Letter D
Circled Latin Small Letter D
Limited Liability Sign
Square Da
Square Dm
Square Dm Squared
Square Dm Cubed
Square Dl
Square Rad
Square Rad Over S
Square Rad Over S Squared
Square Cd
Square Db
Fullwidth Latin Capital Letter D
Fullwidth Latin Small Letter D
Outlined Latin Capital Letter D
Mathematical Bold Capital D
Mathematical Bold Small D
Mathematical Italic Capital D
Mathematical Italic Small D
Mathematical Bold Italic Capital D
Mathematical Bold Italic Small D
Mathematical Script Capital D
Mathematical Script Small D
Mathematical Bold Script Capital D
Mathematical Bold Script Small D
Mathematical Fraktur Capital D
Mathematical Fraktur Small D
Mathematical Double-Struck Capital D
Mathematical Double-Struck Small D
Mathematical Bold Fraktur Capital D
Mathematical Bold Fraktur Small D
Mathematical Sans-Serif Capital D
Mathematical Sans-Serif Small D
Mathematical Sans-Serif Bold Capital D
Mathematical Sans-Serif Bold Small D
Mathematical Sans-Serif Italic Capital D
Mathematical Sans-Serif Italic Small D
Mathematical Sans-Serif Bold Italic Capital D
Mathematical Sans-Serif Bold Italic Small D
Mathematical Monospace Capital D
Mathematical Monospace Small D
Parenthesized Latin Capital Letter D
Circled Cd
Squared Latin Capital Letter D
Squared Sd
Raised Md Sign
Square Dj
Confusables
Canadian Syllabics Kwo
Parenthesized Latin Small Letter D
Small Roman Numeral Five Hundred
Double-Struck Italic Small D
Mathematical Bold Small D
Mathematical Italic Small D
Mathematical Bold Italic Small D
Mathematical Script Small D
Mathematical Bold Script Small D
Mathematical Fraktur Small D
Mathematical Double-Struck Small D
Mathematical Bold Fraktur Small D
Mathematical Sans-Serif Small D
Mathematical Sans-Serif Bold Small D
Mathematical Sans-Serif Italic Small D
Mathematical Sans-Serif Bold Italic Small D
Mathematical Monospace Small D
Cyrillic Small Letter Komi De
Cherokee Letter Tsu
Canadian Syllabics Ko
Lisu Letter Pha
Latin Small Letter D with Hook
Latin Small Letter D with Tail
Latin Small Letter D with Topbar
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Dong Sign
Canadian Syllabics West-Cree Kwo
Canadian Syllabics South-Slavey Koh
Latin Small Letter Dezh Digraph
Latin Small Letter Dz
Latin Small Letter Dz Digraph
Latin Small Letter Dz with Caron
Latin Small Letter Dz Digraph with Curl
Canadian Syllabics Qo
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| Property | Value |
--- |
| Age (age) | 1.1 (1993) |
| Unicode Name (na) | LATIN SMALL LETTER D |
| Unicode 1 Name (na1) | — |
| Block (blk) | Basic Latin |
| General Category (gc) | Lowercase Letter |
| Script (sc) | Latin |
| Bidirectional Category (bc) | Left To Right |
| Combining Class (ccc) | Not Reordered |
| Decomposition Type (dt) | none |
| Decomposition Mapping (dm) | Latin Small Letter D |
| Lowercase (Lower) | ✔︎ |
| Simple Lowercase Mapping (slc) | Latin Small Letter D |
| Lowercase Mapping (lc) | Latin Small Letter D |
| Uppercase (Upper) | ✘︎ |
| Simple Uppercase Mapping (suc) | Latin Capital Letter D |
| Uppercase Mapping (uc) | Latin Capital Letter D |
| Simple Titlecase Mapping (stc) | Latin Capital Letter D |
| Titlecase Mapping (tc) | Latin Capital Letter D |
| Case Folding (cf) | Latin Small Letter D |
| ASCII Hex Digit (AHex) | ✔︎ |
| Alphabetic (Alpha) | ✔︎ |
| Bidi Control (Bidi_C) | ✘︎ |
| Bidi Mirrored (Bidi_M) | ✘︎ |
| Composition Exclusion (CE) | ✘︎ |
| Case Ignorable (CI) | ✘︎ |
| Changes When Casefolded (CWCF) | ✘︎ |
| Changes When Casemapped (CWCM) | ✔︎ |
| Changes When NFKC Casefolded (CWKCF) | ✘︎ |
| Changes When Lowercased (CWL) | ✘︎ |
| Changes When Titlecased (CWT) | ✔︎ |
| Changes When Uppercased (CWU) | ✔︎ |
| Cased (Cased) | ✔︎ |
| Full Composition Exclusion (Comp_Ex) | ✘︎ |
| Default Ignorable Code Point (DI) | ✘︎ |
| Dash (Dash) | ✘︎ |
| Deprecated (Dep) | ✘︎ |
| Diacritic (Dia) | ✘︎ |
| Emoji Modifier Base (EBase) | ✘︎ |
| Emoji Component (EComp) | ✘︎ |
| Emoji Modifier (EMod) | ✘︎ |
| Emoji Presentation (EPres) | ✘︎ |
| Emoji (Emoji) | ✘︎ |
| Extender (Ext) | ✘︎ |
| Extended Pictographic (ExtPict) | ✘︎ |
| FC NFKC Closure (FC_NFKC) | Latin Small Letter D |
| Grapheme Cluster Break (GCB) | Any |
| Grapheme Base (Gr_Base) | ✔︎ |
| Grapheme Extend (Gr_Ext) | ✘︎ |
| Grapheme Link (Gr_Link) | ✘︎ |
| Hex Digit (Hex) | ✔︎ |
| Hyphen (Hyphen) | ✘︎ |
| ID Continue (IDC) | ✔︎ |
| ID Start (IDS) | ✔︎ |
| IDS Binary Operator (IDSB) | ✘︎ |
| IDS Trinary Operator and (IDST) | ✘︎ |
| IDSU (IDSU) | 0 |
| ID_Compat_Math_Continue (ID_Compat_Math_Continue) | 0 |
| ID_Compat_Math_Start (ID_Compat_Math_Start) | 0 |
| Ideographic (Ideo) | ✘︎ |
| InCB (InCB) | None |
| Indic Mantra Category (InMC) | — |
| Indic Positional Category (InPC) | NA |
| Indic Syllabic Category (InSC) | Other |
| Jamo Short Name (JSN) | — |
| Join Control (Join_C) | ✘︎ |
| Logical Order Exception (LOE) | ✘︎ |
| Modifier Combining Mark (MCM) | ✘︎ |
| Math (Math) | ✘︎ |
| Noncharacter Code Point (NChar) | ✘︎ |
| NFC Quick Check (NFC_QC) | Yes |
| NFD Quick Check (NFD_QC) | Yes |
| NFKC Casefold (NFKC_CF) | Latin Small Letter D |
| NFKC Quick Check (NFKC_QC) | Yes |
| NFKC_SCF (NFKC_SCF) | Latin Small Letter D |
| NFKD Quick Check (NFKD_QC) | Yes |
| Other Alphabetic (OAlpha) | ✘︎ |
| Other Default Ignorable Code Point (ODI) | ✘︎ |
| Other Grapheme Extend (OGr_Ext) | ✘︎ |
| Other ID Continue (OIDC) | ✘︎ |
| Other ID Start (OIDS) | ✘︎ |
| Other Lowercase (OLower) | ✘︎ |
| Other Math (OMath) | ✘︎ |
| Other Uppercase (OUpper) | ✘︎ |
| Prepended Concatenation Mark (PCM) | ✘︎ |
| Pattern Syntax (Pat_Syn) | ✘︎ |
| Pattern White Space (Pat_WS) | ✘︎ |
| Quotation Mark (QMark) | ✘︎ |
| Regional Indicator (RI) | ✘︎ |
| Radical (Radical) | ✘︎ |
| Sentence Break (SB) | Lower |
| Soft Dotted (SD) | ✘︎ |
| Sentence Terminal (STerm) | ✘︎ |
| Terminal Punctuation (Term) | ✘︎ |
| Unified Ideograph (UIdeo) | ✘︎ |
| Variation Selector (VS) | ✘︎ |
| Word Break (WB) | Alphabetic Letter |
| White Space (WSpace) | ✘︎ |
| XID Continue (XIDC) | ✔︎ |
| XID Start (XIDS) | ✔︎ |
| Expands On NFC (XO_NFC) | ✘︎ |
| Expands On NFD (XO_NFD) | ✘︎ |
| Expands On NFKC (XO_NFKC) | ✘︎ |
| Expands On NFKD (XO_NFKD) | ✘︎ |
| Bidi Paired Bracket (bpb) | Latin Small Letter D |
| Bidi Paired Bracket Type (bpt) | None |
| East Asian Width (ea) | narrow |
| Hangul Syllable Type (hst) | Not Applicable |
| ISO 10646 Comment (isc) | — |
| Joining Group (jg) | No_Joining_Group |
| Joining Type (jt) | Non Joining |
| Line Break (lb) | Alphabetic |
| Numeric Type (nt) | none |
| Numeric Value (nv) | not a number |
| Simple Case Folding (scf) | Latin Small Letter D |
| Script Extension (scx) | |
| Vertical Orientation (vo) | R |
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5772 | https://www.youtube.com/watch?v=Fm4t9grsUW4 | Using trig ratios to find unknown angles
Jeremy Klassen
37300 subscribers
10 likes
Description
1043 views
Posted: 18 May 2021
This video is about using the trigonometric ratios to find an unknown acute angle in a right triangle
Transcript:
all right hi everybody so in this lesson here we're going to use the trigonometric ratios to find unknown angles so our goal here okay is to find an angle in a triangle now just like it says your trigonometry is the study of triangles uh it's their it's the study of their lengths and their angles give the relationships that that exist in there and kind of taking advantage of those relationships to find information that's not given to you now and it is one of the most important fields of mathematics i personally like i love teaching trigonometry i love going over it with with people um it's it's such a big field of mathematics but it all works together so beautifully there's just so many neat things that you can you can see in the the relationships that we uncovered trigonometry here anyway so trig the trig ratios are relationships between the sides of a triangle and the indicated acute angle here it is it is based on the relative okay the relative labeling oops uh that we looked at in a previous lesson okay it's based on the relative labeling of a triangle so when we start dealing with the trigonometric ratios we mean that literally we're talking about ratios of side lengths okay so the ratios here are our side lengths and there are three primary trig ratios that we look at so we we create this this triangle right here off to the side let's say so there's our right angle so unless we're just going to pick one here's our theta this is the angle that we're looking for our reference angle here so this is our hypotenuse this is the side adjacent and over here this is the side opposite now it turns out regardless of how big the triangle is that those ratios are always the same for a given angle doesn't matter how big the triangle is okay um so just just by way of an example here for example and that this is one ratio that i know off the top of my head if this is 30 degrees okay if this is 30 degrees then i know that one possible relationship here is that this side the opposite side is one unit of length the hypotenuse would therefore be two okay if this could even be a bigger triangle here again if this is still 30 degrees let's say that this is now two turns out the hypotenuse is four okay that that ratio of the side leg stays the same it could be even bigger here you've got this 30 degree angle here this would be three this would be six okay that relationship always holds now we give the very specific ratios that pop out names so we start off with the sine ratio and the sine ratio is the ratio of the length of the side opposite the angle to the hypotenuse length of the hypotenuse so for each of these what i've what i've done here essentially is giving you a sine the sine of 30 degrees it turns out is the length of the opposite over the length of the hypotenuse so one over two doesn't matter how big the triangle is doesn't matter how long those sides are that ratio if that's a 30 degree angle will always be one half the cosine ratio for an angle is going to be the length of the adjacent side okay to the to the hypotenuse adjacent side to the hypotenuse now that's a little bit more complicated for for 30 degrees so i'm not going to go over that right now but yeah that's what we would do here so the cosine ratio is always going to be that ratio of this the length of the adjacent side to the hypotenuse it doesn't matter how big the triangle is all that matters is what the angle is the angle determines what that what that ratio is going to be equal to that's that's important because that means i can use this for any size triangle and then finally the tangent ratio is when we take the opposite side and divide it by the adjacent side the length of the opposite side divided by the adjacent side and again that ratio will always be the same depending on the angle that we're dealing with not the size of the triangle cannot overemphasize that point so when we're putting that together to help you memorize that we've got a kind of a goofy little little way of memorizing it we break this up into a little word here and we and the word is socatoa okay and you'll i'll sometimes see people write this on an exam or on a quiz just off the side so they help them remember this sohcahtoa and the way that works is so is sine is the opposite divided by the hypotenuse cosine is the adjacent divided by the hypotenuse toa is tangent is the opposite divided by the adjacent okay so it's just just a way of helping people memorize that those relationships those ratios because you do need to have them memorized you do need to know what they are okay so now we're going to take a look at um just some some ways of of applying these little ratios applying this little bit of knowledge here so in a little bit here you're probably going to want to make sure you have a calculator handy to help yourself walk through this material okay so the first thing we want to do here is make sure that we can understand what we mean when we ask you to find the ratio okay so we've we've got these ratios that we've just identified here now i just want to get the ratio i don't want to do anything with the ratio you just want to get it so we want to first of all find the tangent ratio for z angle z so that's this angle right here now as soon as we identify that we've identified a few things now first of all the hypotenuse is the hypotenuse there's not much that's going to change here but if z is the angle that's of interest to us then 40 here is the opposite and 30 here is going to be the adjacent okay so then the tangent of angle z i don't even know what angle z is okay i don't have to know what it is okay i know that there's a very specific angle that is going to be created though if the sides are 30 40 and 50 here and i know that the tangent ratio is going to be the opposite over the adjacent so in this case here that is going to be 40 divided by 30 or 4 thirds when i when i simplify that if we were to find the sine ratio the sine of z and it looks like this the sine of z is going to equal the opposite over the hypotenuse so in this case the opposite is 40 and the hypotenuse is 50. and so again when we simplify this we get four-fifths and then finally the cosine ratio the cosine for z is going to be the adjacent side divided by the hypotenuse the adjacent side to z is 30 the hypotenuse is 50 and so the cosine ratio for that would be 3 divided by 5. so and again if we were to focus on x okay if our focus had been on angle x whoops sorry angle x then those ratios would have been different the sine cosine tan would have been different because the labeling would change relative to the angle that we're investigating or that's of interest to us okay so now we're going to use uh some of the information that we've got here the tools that we've now uncovered here to find an acute angle in a triangle okay and there are really two methods for this um we're going to break this up into these two methods here but you might look at what i just said here and say well it's okay it's great it's wonderful that uh for example this that the sine of 30 degrees is going to be one half the opposite is is one the hypotenuse is two so i would know for example if i had a triangle here and i knew that this was 5 and this was 10 i could tell you right away because i've just seen it here that i know that this is 30 degrees the thing is and you might rightly say i didn't know prior to this video that this that if it was a 30 degree angle in a right triangle that the opposite side would be one half of the hypotenuse you didn't know that and that's a fair argument here i mean you don't know that if this all of a sudden changed to 40 degrees what would that ratio be i mean you don't know off the top of your head but the calculator does know and that's the issue here these ratios are in or the calculator has the ability to access these ratios here so we can use the calculator to find that prior to there being a calculator people had written out lists there'd be charts with angles in there and and their specific ratios with them so the calculator just makes things a little bit easier for us now on your calculator you will have these buttons right here sine cos and tangent here are your uh your trig buttons here now these buttons the way they sit sine cos and 10 take an angle and produce a ratio that isn't exactly what we want here that's not what we're interested in so i'm just going to erase that little bit here because i want you to look a little bit at this these buttons again on your calculator and notice that just above those just above those is a different symbol and sine to the negative 1 cosine to the negative one and tangent to the negative one take a ratio whoops wow tank that was weird they don't tank a ratio they take a ratio and produce an angle and that's what we want to look at right now okay we want to take a ratio and produce an angle so for example if i was going to look for angle a and let's say i had my triangle here and here's angle a i don't know what that is but i do know that the side over here is 10 and the hypotenuse is 13. i would know that this statement is true the sine of a is equal to 10 over 13. now i want to know what a is so what i'm going to do is i'm actually going to use the sine to the negative 1 button of 10 over 13. so typically the way you access that is to press the second button sign and that opens up the sign to the negative one you would enter your 10 divided by 13 and then when you press enter you should get that this is approximately 50.3 degrees okay let's say that you had uh your triangle here again here's a this time you notice that the the adjacent side is 5 and the hypotenuse is 12. okay well that's a that's a cosine ratio if i know the adjacent side and i know that the hypotenuse that's the cosine ratio so what i would do here is i'll change maybe colors here i would press second cosine to access that inverse cosine function what we call the inverse function and that's going to produce for me the angle given that ratio and that's going to be approximately 65.4 degrees and now finally let's say in our last triangle here here's angle whoops here's angle a and i know that my tangent of a is one half so basically what that means is i know that the side opposite a is 1 the side adjacent to a is 2. opposite over adjacent so if i wanted to figure out what a was it would be the what would we call the again the inverse tangent so what i would do here and let's i'll just do it in uh well i'll do it in red here so i would press second tangent and then enter that ratio there uh one over two and when i do that i get that the angle is going to be approximately 26.6 degrees okay so anytime you're going to find the angle you're going to use that second button on your calculator to access that the the the operation that is just above the sine cosine tan because i'll say it again sine cosine 10 would take an angle and produce a ratio that's by the way that's why each one of these looks like this that's why each one of these things looks like this i've got the sine of a equals the ratio i've got the cosine of a equals the ratio i've got the tangent of a equals the ratio sine cosine tan produce ratios you give it the angle they produce ratios if i want to go backwards i need that to the negative one to the negative one to the negative one to go from the ratio back to the angle okay so that's that's one way of finding the the acute angle inside a right triangle okay well another way that you can find an angle in a triangle here is to use that interior sum uh uh kind of that idea that the angles inside a triangle always add up to 180. now in this particular case the reason why that is so useful is because you will normally be working with a right angle triangle which means you will already know that one of the angles is 90 degrees okay so if you're given another angle so if i'm telling you that this angle here is 36 well it's quite easy to find that missing angle here angle b is going to be 180 degrees the whole triangle minus the 90 degrees for the right angle minus the 36 degrees that was given to me okay and then you could get 54 like that now there is another way of thinking that thing through here i could redo this and and just realize this that that 90 here is half of 180 and in fact anytime i'm working with a right angle triangle i'm going to have this little term right here present in this calculation so this is always going to be 90 here in other words i can shortcut that and i don't have to write 180 minus 90. i can simply write 90 minus 36 degrees and that would give me 54 degrees as well so what i'm doing here is i'm focusing my attention on just these two as long as there's a right angle up here these two angles right here also have to add up to 90. okay these two have to sum to 90 degrees so that's just another way that you could go about doing basically exactly the same thing as we did previously just taking a little bit more of a shortcut about it okay so now to kind of wrap this up this discussion right here if the triangle has two given side measurements and only a 90 degree angle here now what are we going to do well okay first of all step one we're going to find the reference angle so the angle that is to be calculated is called the reference angle again we've talked about that already you know that and we usually are going to use the theta to represent that unless they've given it another name if they've already called it like x or they've called it angle a or whatever that will leave it but if they don't yeah we're going to call it theta we're going to label the sides relative using relative labeling so it's remember that means relative to where theta is and so that's going to be your your opposite and i'm going to write this out here your opposite and adjacent now your hypotenuse yeah well the hypotenuse is the hypotenuse it's going to be the same regardless of of where uh the angle is because the the hypotenuse is always going to be opposite the 90 degree then we're going to choose a ratio so basically that ratio is going to depend on the two sides the two known sides here so we're just going to pick the ones that match up so if we're example if we know the opposite in the hypotenuse well that's going to be sine on the other hand if we know the adjacent in the hypotenuse that's going to be cosine but let's say we don't even know what the hypotenuse is well that gives us the tangent ratio and then we're going to substitute those side lengths into the ratio and use that to evaluate uh and then to find the second angle okay uh sorry i'm about to find the second angle to find the angle i just saw that second there and i don't know why i popped it under to find the angle you're gonna use the second function on your calculator and remember we were talking about that just a few moments so we're gonna use those functions right there to convert uh the ratio it converts the ratio to the angle so let's take a quick look at some uh some examples here and they're not all going to be of this uh you'll see in just a second here so take a look at this first one here let's call let's call this the question mark here we'll call that theta and so based on that i know first of all i know that this is going to be the hypotenuse i know that way over here on the other side this is going to be the opposite and because this side that's 29 is used to make theta that in the hypotenuse theta is between that side and the hypotenuse that's the adjacent so i don't know the hypotenuse but i do know the opposite and i do know the adjacent okay so because i don't know the hypotenuse here this is a tangent so the tangent of theta that ratio would be the opposite the 20 over the adjacent 29 and then to get that theta i would use the inverse or the second function tangent of 29 sorry 20 over 29 plugging in that ratio and when you press enter you'll get approximately 34.6 degrees okay and that's that's how you do it that's the pattern that we're going to want you to to follow here so for example here's my theta in this triangle there it is right there i'm going to use relative labeling for this one except i know that right here this is going to be the hypotenuse i know that the 7 here because it's being used with the hypotenuse to make up theta this is going to be the adjacent and over here way over here this is going to be the opposite side and i don't know what that is so i do know the hypotenuse is 8. i don't know the opposite side but i do know the adjacent that was equal to 7. so then what trig ratio doesn't use the opposite side well that's that's cosine so the cosine of theta is going to equal the adjacent side divided by the hypotenuse so now on my calculator i would do the second function cosine and that's going to get me approximately 29.0 degrees now for this last one notice that i don't okay i should say like this i don't need i should say like this i don't need the trig ratios here i don't need them because i know that these two angles here should add up to 90 degrees now i know all the angles in the triangle add up to 180 but i know that those two angles should add up to 90. okay so my question mark here or my theta should be the same as 90 degrees minus that 40 degrees that i'm given and so i know that theta is equal to 50. so it's just a matter of adapting to whatever information that you've been given and i hope that gives you a little bit of confidence moving forward with these ratios here now there's still a whole lot more to cover but but this is a good kind of a good grounding to get you started |
5773 | https://www.epj-conferences.org/articles/epjconf/pdf/2017/06/epjconf_conf2017_06026.pdf | pNRQCD determination of E1 radiative transitions Sebastian Steinbeißer1,a and Jorge Segovia1 1Physik-Department, Technische Universität München, James-Franck-Str. 1, D-85748 Garching, Germany Abstract. This contribution contains the first numerical computation of the complete set of relativistic corrections of relative order v2 for electric dipole (E1) transitions in heavy quarkonium; in particular, for the processes χbJ(1P) →Υ(1S ) + γ with J = 0, 1, 2.
We assume that the momentum transfer of the heavy mesons involved in the reactions lies in the weak-coupling regime of the low-energy effective field theory potential non-relativistic QCD (pNRQCD) and thus a full perturbative calculation can be performed.
Introduction Electromagnetic transitions are often significant decay modes for bottomonium states below B ¯ B threshold (10.56 GeV), making them a suitable experimental tool to access the lowest spectra of bot-tomonia. For instance, the first b¯ b states not directly produced in e+e−collisions were the six triplet-P states, χb(2PJ) and χb(1PJ) with J = 0, 1, 2, discovered in radiative decays of the Υ(3S ) and Υ(2S ) in 1982 [1, 2] and 1983 [3, 4], respectively.
One important feature of electromagnetic transitions is that they can be classified in a series of electric and magnetic multipoles. The most important ones are the E1 (electric dipole) and the M1 (magnetic dipole) transitions; higher order multipole modes E2, M2, E3, etc. appear in the spectrum, but since they are further suppressed one usually does not consider them. Processes involving electric dipole (E1) transitions happen more frequently than the ones induced by a magnetic dipole (M1). The branching fraction for E1 transitions can indeed be significant for some lowest bottomonium states like the ones we shall study herein : B(χb0(1P) →Υ(1S )γ) = (1.76 ± 0.35) % (note that it is the largest exclusive branching fraction reported by the Particle Data Group (PDG) ), B(χb1(1P) → Υ(1S )γ) = (33.9 ± 2.2) % and B(χb2(1P) →Υ(1S )γ) = (19.1 ± 1.2) %.
Electric dipole (E1) transitions are defined through the property that they change the orbital an-gular momentum of the state by one unit, but not the spin. Therefore, the final state has different parity and C-parity than the initial one. Typical E1 quarkonium decays are the ones mentioned above: 23PJ →13S 1 + γ. Here and in the following we denote the states as n 2s+1ℓJ, where n = nr + ℓ+ 1 corresponds to the principal quantum number with nr = 0, 1, . . . the radial quantum number and ℓthe orbital angular momentum. The spin is denoted by s and J is the total angular momentum.
The E1 (and M1) electromagnetic transitions have been treated for a long time by means of po-tential models that use non-relativistic reductions of QCD-based quark-antiquark interactions (see, e.g., Ref. for a recent application to the bottomonium system). However, the progress made in effective field theories (EFTs) for studying heavy quarkonia and the new large set of accurate ex-perimental data taken in the heavy quark sector by B-factories (BaBar, Belle and CLEO), τ-charm ae-mail: sebastian.steinbeisser@tum.de DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 ( facilities (CLEO-c, BESIII) and even proton-proton colliders (CDF, D0, LHCb, ATLAS, CMS) ask for a systematic and model-independent analysis (see, e.g., Refs. [8, 9] for reviews).
Formulae and numerical treatment of M1 transitions within the effective field theory potential NRQCD (pNRQCD) can be found in Refs. [10, 11]. Therein, the relativistic corrections to the leading order (LO) expression (which counts as k3 γ/m2 where kγ is the photon energy) were computed in two different expansion schemes: (i) strict weak-coupling regime and (ii) including exactly the static potential in the LO Hamiltonian. Within the same theoretical framework, the corresponding formulae for E1 transitions have been presented in Ref. . In this case, the relativistic corrections to the LO decay width (that counts as k3 γ/(mv)2) are much more involved covering not only higher order terms in the E1 transition operator but also corrections to the initial and final wave function due to higher order potentials and higher order Fock states. These facts have hindered numerical computations of the E1 radiative decays within pNRQCD (for partial calculations see ). This contribution aims to close this gap and calculate the decay rate of the reaction 23PJ →13S 1 + γ with J = 0, 1, 2. As a first step, we shall assume that the soft scale lies in the strict weak-coupling regime of pNRQCD and thus a full perturbative calculation can be performed. These proceedings are based on the forthcoming publication .
Theoretical set-up Potential non-relativistic QCD (pNRQCD) Heavy quarkonium systems are characterized by their non-relativistic nature, i.e., the heavy quark bound-state velocity, v, satisfies v ≪1. This is reasonably fulfilled in bottomonium (v2 ∼0.1) and to a certain extent in charmonium (v2 ∼0.3). Moreover, at least, three widely separated scales appear: the heavy quark mass m (hard scale), the relative momentum of the bound state p ∼mv (soft scale) and the binding energy E ∼mv2 (ultrasoft scale). With v ≪1, the following hierarchy of scales m ≫p ∼1/r ∼mv ≫E ∼mv2 (1) is satisfied and this allows for a description in terms of EFTs for physical processes taking place at one of the lower scales. The integration out of modes associated with high-energy scales is performed as part of a matching procedure that enforces the equivalence between QCD and the EFT at a given order of the expansion in v. The final result is a factorization at the Lagrangian level between the high-energy modes, which are encoded in the matching coefficients, and the low-energy contributions carried by the dynamical degrees of freedom.
The suitable EFT to describe processes that take place at the scale mv such as the E1 radiative transitions between the lowest heavy quarkonium states is potential NRQCD (pNRQCD) [15, 16]. It follows by integrating out the modes of order p ∼1/r ∼mv from NRQCD [17, 18] which in turn comes from QCD by integrating out the high energy modes of order m. Therefore, pNRQCD takes full advantage of the hierarchy of scales that appear in Eq. (1), and makes a systematic and natural connection between quantum field theory and the Schrödinger equation. Schematically, the pNRQCD equation of motion takes the form i∂0 −⃗ p 2 m −V(0) s (r) φ(⃗ r, t, ⃗ R ) = 0 + corrections to the potential + interactions with other low-energy degrees of freedom pNRQCD where V(0) s (r) is the static potential and φ(⃗ r ) is the Q ¯ Q field. Note here that the interactions with other low-energy degrees of freedom produce, among others, non-potential terms that account for singlet DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 2 PH = (MH,⃗ 0) PH′ = ( qk2 γ + M 2 H′, −⃗ k) γ H H′ (kγ,⃗ k) Figure 1. Kinematics of the radiative transition H →H′γ in the rest frame of the initial-state quarkonium H, taken from .
to octet transitions via ultrasoft gluons and provide loop corrections to the leading potential picture.
Being induced by low-energy degrees of freedom they encode also non-perturbative effects.
The matching of pNRQCD depends on the relative size between the soft and the ΛQCD scale.
Two main situations can be distinguished, namely, the weak-coupling [15, 16] (mv ≫ΛQCD) and the strong-coupling (mv ∼ΛQCD) versions of pNRQCD. One major difference between them is that in the former the potential can be computed in perturbation theory unlike in the latter.
It is obvious that the weak-coupling version of pNRQCD is amenable for a theoretically much cleaner analysis. The observables can be computed as an expansion in αs with increasing accuracy.
Non-perturbative effects are suppressed by powers of ΛQCD/(mv). Therefore, observables that could be computed with the weak-coupling version of pNRQCD are of the greatest interest.
Decay width of the n3PJ →n′3S1γ reaction The complete decay rate n3PJ →n′3S 1γ reads up to order k3 γ/m2 Γn3PJ→n′3S 1γ = Γ(0) E1 ( 1 + RS =1(J) −kγ 6m − k2 γ 60 I(0) 5 (n1 →n′0) I(0) 3 (n1 →n′0) + " J(J + 1) 2 −2 # " − 1 + κem Q kγ 2m + 1 m2 (1 + 2κem Q )I(1) 2 (n1 →n′0) + 2I(0) 1 (n1 →n′0) I(0) 3 (n1 →n′0) #) , (2) where RS =1(J) includes the initial and final state corrections due to higher order potentials and higher order Fock states (see below). The remaining corrections within the brackets are the result of taking into account additional electromagnetic interaction terms in the Lagrangian suppressed by O(v2) .
We have displayed terms proportional to the anomalous magnetic moment, κem Q , however these terms are at least suppressed by αs(m)v2 and thus go beyond our accuracy and are therefore not considered in the numerical analysis. The LO decay width (∼k3 γ/(mv)2) is Γ(0) E1 = 4 9 αem e2 Q k3 γ h I(0) 3 (n1 →n′0) i2 , (3) with αem the electromagnetic fine structure constant, eQ the charge of the heavy quarks in units of the electron charge, and kγ the photon energy determined by the kinematics shown in Fig. 1: kγ = |⃗ k| = M2 H −M2 H′ 2MH = (MH −M′ H) + O k2 γ MH .
(4) DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 3 The function I(k) N (nℓ→n′ℓ′) = ∞ Z 0 dr r2rN−2R∗ n′ℓ′(r) " dk drk Rnℓ(r) # (5) is a matrix element that involves the radial wave functions of the initial and final states. We shall assume that these states are solutions of the Schrödinger equation H(0)ψ(0) nℓm(⃗ r ) = E(0) n ψ(0) nℓm(⃗ r ) , (6) with the leading order Hamiltonian in weakly-coupled pNRQCD given by H(0) = −∇2 2mr + V(0) s (r) = −∇2 2mr −CF αs r , (7) where CF = 4/3. Therefore, ψ(0) nℓm(⃗ r ) and E(0) n can be written in the hydrogen-like form ψ(0) nℓm(⃗ r ) = Rnl(r)Yℓm(Ωr) = Nnℓe−ρn 2 ρℓ n L2ℓ+1 n−ℓ−1(ρn)Yℓm(Ωr) , (8) E(0) n = −mrC2 Fα2 s 2n2 , (9) where mr = m/2 is the reduced mass of the Q ¯ Q system, ρn = 2r/na is a dimensionless variable with a = 1/mrCFαs the Bohr radius. Finally, the normalization reads Nnℓ= s 2 na !3 (n −ℓ−1)!
2n[(n + ℓ)!] .
(10) Relativistic wave-function corrections Due to higher order potentials and transitions between singlets and octets, the state in Eq. (8) is not an eigenstate of the complete Hamiltonian. Therefore, one has to consider corrections to the wave function, which can contribute to the decay rate at the required order of precision (∼k3 γ/m2). To compute these corrections one applies the standard formalism of perturbation theory, either in the language of quantum mechanics or via Feynman diagrams.
Corrections due to higher order potentials In order to account for corrections to the decay width of relative order v2, we need to consider the complete Hamiltonian H = −∇2 2mr + Vs(r) + δH .
(11) The static potential is given by Vs(r) = V(0) s (r) 1 + 2 X k=1 αs 4π k ak(r) (12) where, as mentioned above, V(0) s (r) = −CFαs/r, is the leading order potential or Coulomb-like poten-tial that goes directly in the Schrödinger equation. The O(αs) and O(α2 s) radiative corrections to the LO static potential are (the constants shown herein can be found, e.g., in Appendix C of Ref. ): a1(ν, r) = a1 + 2β0ln(νeγEr) , (13) a2(ν, r) = a2 + π2 3 β2 0 + (4a1β0 + 2β1)ln(νeγEr) + 4β2 0ln2(νeγEr) .
(14) DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 4 The O(αs) term was computed in Ref. and the O(α2 s) in Ref. . The static potential is known up to order O(α4 s) with the O(α3 s) radiative correction computed in Refs. [23–26]. However, already O(α3 s) terms would give a contribution to the E1 decay rate that goes beyond present precision.
The term δH encodes the relativistic corrections which can be organized as an expansion in the inverse of the heavy quark mass, m. At the order we are interested in, such expansion covers all the 1/m and 1/m2 potentials and, at order 1/m3, the first relativistic correction to the kinetic energy: δH = −∇4 4m3 + V(1) m + V(2) S I m2 + V(2) S D m2 .
(15) At order 1/m2, we can split the contributions into spin-independent (SI) and spin-dependent (SD) terms V(2) S I (r) = V(2) r (r) + 1 2{V(2) p2 (r), −∇2} + V(2) L2 (r) ⃗ L2 , (16) V(2) S D(r) = V(2) LS (r) ⃗ L · ⃗ S + V(2) S 2 (r) ⃗ S 2 + V(2) S 12(r) S 12 , (17) where ⃗ S = ⃗ S 1+⃗ S 2 = (⃗ σ1+⃗ σ2)/2, ⃗ L = ⃗ r×⃗ p and S 12 = 3(ˆ r·⃗ σ1)(ˆ r·⃗ σ2)−⃗ σ1·⃗ σ2 are, respectively, the total spin, total orbital angular momentum and tensor operators acting on the system. In the weak-coupling case, the above potentials read at leading (non-vanishing) order in perturbation theory V(1)(r) = −CFCAα2 s 2r2 , V(2) r (r) = πCFαsδ(3)(⃗ r) , V(2) p2 (r) = −CFαs r , V(2) L2 (r) = CFαs 2r3 , (18) V(2) LS (r) = 3CFαs 2r3 , V(2) S 2 (r) = 4πCFαs 3 δ(3)(⃗ r) , V(2) S 12(r) = CFαs 4r3 .
(19) We now make use of standard quantum mechanical perturbation theory and compute the first and second order correction, induced by a potential V, to a state |nℓ⟩(0) ≡|nℓ⟩. The second order correction to the wave function is only needed when the perturbation is given by the static potential proportional to the a1(ν, r) term. The normalised corrected wave-function is |nℓ⟩(1) = X n′,n , ℓ′ ⟨n′ℓ′|V|nℓ⟩ E(0) n −E(0) n′ |n′ℓ′⟩ = X n′,n , ℓ′ |n′ℓ′⟩⟨n′ℓ′| E(0) n −E(0) n′ V|nℓ⟩ , (20) for the first order, and |nℓ⟩(2) = X k1,n , ℓ1 X k2,n , ℓ2 ⟨k1ℓ1|V|k2ℓ2⟩⟨k2ℓ2|V|nℓ⟩ (En −Ek1)(En −Ek2) −⟨k1ℓ1|V|nℓ⟩⟨nℓ|V|nℓ⟩ (En −Ek1)2 |k1ℓ1⟩−1 2 X k2,n , ℓ2 |⟨k2ℓ2|V|nℓ⟩|2 (En −Ek2)2 |nℓ⟩, (21) for the second one.
As one can see in Eq. (20), a particular re-arrangement of the terms allows us to have a key expression that can be re-written as X n′,n , ℓ′ |n′ℓ′⟩⟨n′ℓ′| E(0) n −E(0) n′ = X n′ , ℓ′ |n′ℓ′⟩⟨n′ℓ′| E(0) n −E(0) n′ − X n′=n , ℓ′ |n′ℓ′⟩⟨n′ℓ′| E(0) n −E(0) n′ = lim E→E(0) n 1 E −H − P(n) E −E(0) n !
≡ 1 (En −H)′ . (22) This will allow us to compute expectation values of an arbitrary operator O, via (note that, for the sake of simplicity, only final state corrections are shown here but the same corrections affect also the initial DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 5 state): ⟨n′ℓ′|O|nℓ⟩(1) = ⟨n′ℓ′|O 1 (En −H)′ V|nℓ⟩ = Z d3r1 d3r2 ψ∗ n′ℓ′(⃗ r2) O(⃗ r2)G′ n(⃗ r2,⃗ r1) V(⃗ r1) ψnℓ(⃗ r1) , (23) for the first order, and ⟨n′ℓ′|O|nℓ⟩(2) = ⟨n′ℓ′|O 1 (En −H)′ V 1 (En −H)′ V|nℓ⟩ −⟨nℓ|V|nℓ⟩⟨n′ℓ′|O 1 (En −H)′ 1 1 (En −H)′ V|nℓ⟩ −1 2⟨n′ℓ′|O|nℓ⟩⟨nℓ|V 1 (En −H)′ 1 1 (En −H)′ V|nℓ⟩ = Z d3r1 d3r2 d3r3 ψ∗ n′ℓ′(⃗ r3) O(⃗ r3)G′ n(⃗ r3,⃗ r2) V(⃗ r2)G′ n(⃗ r2,⃗ r1) V(⃗ r1)ψnℓ(⃗ r1) −δE(1) V × Z d3r1 d3r2 d3r3 ψ∗ n′ℓ′(⃗ r3) O(⃗ r3)G′ n(⃗ r3,⃗ r2)G′ n(⃗ r2,⃗ r1) V(⃗ r1)ψnℓ(⃗ r1) −1 2 Z d3r ψ∗ n′ℓ′(⃗ r) O(⃗ r) ψnℓ(⃗ r) × × Z d3r1 d3r2 d3r3 ψ∗ n′ℓ′(⃗ r3) V(⃗ r3)G′ n(⃗ r3,⃗ r2)G′ n(⃗ r2,⃗ r1) V(⃗ r1)ψnℓ(⃗ r1) , (24) for the second order. The term δE(1) V in Eq. (24) is the first order correction to the energy induced by a potential V: δE(1) V ≡ R d3r ψ∗ n′ℓ′m′(⃗ r ) V(⃗ r ) ψnℓm(⃗ r ); and G′ n(⃗ r1,⃗ r2) has the following expression G′ n(⃗ r1,⃗ r2) ≡(−1) ×lim E→En G(⃗ r1,⃗ r2, E) − ∞ X ℓ=0 |ψnℓ|2 E −En , (25) where G(⃗ r1,⃗ r2, E) is the Coulomb Green function G(⃗ r1,⃗ r2, E) = ∞ X ℓ=0 2ℓ+ 1 4π Pℓ(ˆ r1 · ˆ r2)Gℓ(r1, r2) , (26) Gℓ(r1, r2) = ∞ X ν=ℓ+1 mra2 ν4 λ ! Rνℓ(ρλ,1)Rνℓ(ρλ,2) ν −λ .
(27) in which we have defined E ≡− mrC2 Fα2 s 2λ2 .1 Corrections due to higher order Fock states The weakly coupled quarkonia may also get corrections from the coupling of the heavy quark-antiquark pair to other low-energy degrees of freedom. In particular, the leading order electromagnetic dipole transition may get a correction from diagrams (see Fig. 8 in ) in which a singlet state is 1In order to perform the computation it is specially useful to use such expression, because for λ = n √ 1−ϵ , we have E = En(1 −ϵ) and E →En for ϵ →0. DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 6 1.0 1.5 2.0 2.5 3.0 0 10 20 30 40 50 60 70 [GeV] [keV] 1.0 1.5 2.0 2.5 3.0 - 4 - 2 0 2 [GeV] [keV] Figure 2. Left panel – Comparison between the LO decay width (solid blue curve) and its relativistic correction (dashed orange curve) due to higher order electromagnetic transition operators that appear in the pNRQCD Lagrangian as a function of the renormalization scale ν. Right panel – Relativistic contributions appearing in Eq. (2): The first Γ(0) E1 × (−kγ/(6m)) (solid blue), the second Γ(0) E1 × (−k2 γ/(60) × I(0) 5 /I(0) 3 ) (dashed orange) and the third Γ(0) E1 × (second line of Eq. (2)) (dot-dashed green).
coupled to an octet state due to the emission and re-absorption of an ultrasoft gluon. These diagrams come from terms of the pNRQCD Lagrangian like ∆L = VA O†⃗ r · g⃗ ES + S †⃗ r · g⃗ EO , (28) where S = S 1c/ √Nc is a quark-antiquark field that transforms as a singlet under S U(3)c and U(1)em, O = √ 2OaT a is a quark-antiquark field which transforms as an octet under S U(3)c and as a singlet under U(1)em, and ⃗ E is the chromo-electric field.
The first two diagrams in Fig. 8 of correspond to the renormalization of the initial and final wave function. The diagrams 2, 3a and 3b account for the correction of the initial and final wave functions due to the presence of octet states. The diagram 4 represents an electric dipole transition mediated by the intermediate octet state. According to the power counting, the first two diagrams contribute to relative order Λ2 QCD/(mv)2 whereas the remaining ones scales as Λ3 QCD/(mv2)/(mv)2. We shall not consider these contributions herein because in the strict weak-coupling regime, E ∼mv2 ≫ ΛQCD, one can argue that they should be negligible.
It is noteworthy that, in contrast to the E1 transitions, the colour-octet contributions for allowed M1 transitions cancel . This is a consequence of the fact that the magnetic dipole operator behaves as an identity operator in position space.
Results We discuss in detail our theoretical result for the χb1(1P) →Υ(1S )γ reaction, but a similar analysis has been performed for the transitions χbJ(1P) →Υ(1S )γ with J = 0, 2. The mean value for the decay width and an estimate of its theoretical error will be given at the end of this Section for all transitions.
Figure 2 shows the χb1(1P) →Υ(1S )γ LO decay rate and its relativistic correction due to higher order electromagnetic interactions that appear in the pNRQCD Lagrangian. In other words, we are analysing Eq. (2) without the factor RS =1(J = 1). As one can see in the left panel of Fig. 2, these O(v2) corrections to the LO decay rate are very small, ∼5% at most. The right panel of the same figure displays the different contributions (with their relative sign) showing that the dominant one is the term proportional to I(0) 5 (21 →10) in the expression of the decay rate, Eq. (2). An interesting feature shown in Fig. 2 is the substantial dependence of the result on the renormalization scale ν. The decay width changes from 17 keV to 74 keV when the renormalization scale ν is varied within the range of DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 7 1.0 1.5 2.0 2.5 3.0 - 3 - 2 - 1 0 1 2 [GeV] [GeV- 1] 1.0 1.5 2.0 2.5 3.0 - 2 - 1 0 1 2 [GeV] [GeV- 1] 1.0 1.5 2.0 2.5 3.0 - 2 0 2 4 6 [GeV] [GeV- 1] 1.0 1.5 2.0 2.5 3.0 - 2 - 1 0 1 2 [GeV] [GeV- 1] 1.0 1.5 2.0 2.5 3.0 - 0.5 0.0 0.5 1.0 1.5 2.0 2.5 [GeV] [GeV- 1] 1.0 1.5 2.0 2.5 3.0 - 1 0 1 2 [GeV] [GeV- 1] Figure 3. Comparison of the LO transition matrix element (solid blue curve) with respect the ones coming from corrections due to higher order potentials. Upper-left panel – First order correction to the initial (dashed yellow) and final (dot-dashed green) wave functions due to the a1(ν, r) term. Upper-middle panel – First order correction to the initial (dashed yellow) and final (dot-dashed green) wave functions due to the a2(ν, r). Upper-right panel – Second order correction to the initial (dashed yellow) and final (dot-dashed green) wave functions, and first order correction to both initial and final (dotted red) due to the a1(ν, r) term. Lower-left panel – First order correction to the initial (dashed yellow) and final (dot-dashed green) wave functions due to the V(1), and first order correction to the final (dotted red) wave functions due to the V(2) r . Lower-middle panel – First order correction to the initial (dashed yellow) and final (dashed green) wave functions due to the p2 term; moreover, first order correction to the initial (dot-dashed red) and final (dot-dashed purple) wave functions due to the kinetic p4 term. Lower-right panel – Remaining contributions where the most important one (dot-dashed red) is the first order correction to the final wave function due to the V(2) S 2 .
1 to 3 GeV. This range encompasses the typical momentum transfer in the bottomonium system, still being consistent with perturbation theory.
Let us focus now our attention to the computation of the wave function corrections due to higher order potentials, which are encoded in the factor RS =1(J = 1) of Eq. (2). The upper panels of Fig. 3 show the matrix elements correcting the E1 decay rate up to O(v2) and coming from the radiative corrections to the static potential, Eq. (12). The left and middle panels refer to the first order initial and final wave function corrections coming from a1(ν, r) and a2(ν, r), respectively. The right panel refers to the second order correction due to the a1(ν, r) term of the static potential. Amongst the features shown by the panels, the following are of particular interest: (i) the matrix elements clearly exceed the value of the LO one. (ii) The matrix elements depend quite dramatically on the scale ν, especially for small ν; in some sense, we expected such behaviour from the numerical analysis of the M1 transitions in Refs. [10, 11]. (iii) The zero crossing in some of the matrix elements comes from the logarithms in (13) and (14).
The lower panels of Fig. 3 show the remaining matrix element contributions coming from δH, Eq. (15). One can see that only few of them are relevant corrections to the LO decay rate. Moreover, the ν-dependence of every matrix element is smaller than in the case of the radiative corrections.2 2The only two parameters in our approach are mb and αs. The value of the b-quark mass is fixed through the Υ(1S )-mass and the running of αs(ν) is taken at 4-loop accuracy with three massless flavours using the Mathematica package RunDec and the starting value αs(MZ) = 0.118. DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 8 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 [GeV] [GeV- 1] 1.0 1.5 2.0 2.5 3.0 0 20 40 60 80 [GeV] [keV] Figure 4. Left panel – Matrix elements (with their relative signs) contributing to the reaction χb1(1P) →Υ(1S )γ at LO order (solid blue), NLO (dashed yellow), NNLO (dot-dashed green) and NLO+NNLO (dotted red). Right panel – Total decay width for the χb1(1P) →Υ(1S )γ reaction, the panel shows the LO (dashed blue), LO+NLO (dot-dashed yellow) and LO+NLO+NNLO (solid black) result. The green dotted curve is the LO+NLO+NNLO result but omitting the contributions coming from the radiative corrections to the static potential. The horizontal gray line is our final value for the decay width, taken at ν = 1.5 GeV, and the gray band corresponds to the uncertainty (44.23% ˆ = ± 26.05 keV for this transition).
Summing all the contributions discussed in the paragraph above, the left panel of Fig. 4 shows the next-to-leading order (NLO), NNLO and NLO+NNLO matrix elements and compares them with the LO term. The most important features have been already mentioned: the subleading matrix el-ements are of the same order of magnitude than the leading one and the dependence with ν in the logs dominates the picture. In the right panel of Fig. 4, we draw the decay rate associated with the χb1(1P) →Υ(1S )γ reaction at LO, NLO and NNLO. It is worth to remark that the NLO contribu-tion is negligible at large-ν but multiplies by a factor of 2 the LO decay width at ν = 1 GeV. A big correction to the decay rate is due to the NNLO contribution. One can see in the Figure that the the-oretical result depends slightly on the scale for ν ≳1.75 GeV, whereas the ν-dependence is dramatic for lower values due to the logarithmic functions. This fact is demonstrated by the additional curve (dotted green) where we omitted the contributions coming from the radiative corrections to the static potential, hence set the a1(ν, r) and a2(ν, r) terms to zero. Note that the relativistic corrections to the leading order E1 transition operator are included in the NNLO curve.
Finally, our theoretical results for the decay rates of the transitions under consideration are ob-tained by choosing the value at ν = 1.5 GeV, yielding: Γ(χb0(1P) →Υ(1S )γ) = 52+14 −24(O(v3)) keV , (29) Γ(χb1(1P) →Υ(1S )γ) = 62+17 −30(O(v3)) keV , (30) Γ(χb2(1P) →Υ(1S )γ) = 64+18 −33(O(v3)) keV , (31) where we have chosen a very conservative error estimation that includes the total range of our final result, obtained by varying ν from (1-3) GeV.
Epilogue We have presented the first numerical determination of the decay rate χbJ(1P) →Υ(1S )γ with J = 0, 1, 2 within potential NRQCD. We have assumed that the momentum scale of the heavy quarko-nium involved lies in the strict weak-coupling regime of pNRQCD and non-perturbative effects are suppressed, such that a full perturbative calculation can be performed. Relativistic corrections of relative order v2 to the LO decay rate are included. The analysis separates those contributions that account for the higher order electromagnetic interaction terms in the pNRQCD Lagrangian and those DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 9 that account for quarkonium state corrections due to higher order potentials and transitions between singlets and octets.
Acknowledgements S.S. and J.S. thank N. Brambilla and A. Vairo for collaboration and supervision on the work presented here and C. Peset, A. Pineda, Y. Sumino and Y. Kiyo for numerous informative discussions. S.S. expresses his gratitude to the Physik-Department of the Technische Universität München whose support helped his participation in the Conference. J.S. acknowledges the financial support from the Alexander von Humboldt Foundation.
References K. Han et al., Phys. Rev. Lett. 49, 1612 (1982) G. Eigen et al., Phys. Rev. Lett. 49, 1616 (1982) C. Klopfenstein et al., Phys. Rev. Lett. 51, 160 (1983) F. Pauss et al., Physics Letters B 130, 439 (1983) K.A. Olive et al. (Particle Data Group), Chin. Phys. C38, 090001 (2014) J. Segovia, P.G. Ortega, D.R. Entem, F. Fernández, Phys. Rev. D93, 074027 (2016) N. Brambilla, A. Pineda, J. Soto, A. Vairo, Rev. Mod. Phys. 77, 1423 (2005) N. Brambilla et al., Eur. Phys. J. C71, 1534 (2011) N. Brambilla et al., Eur. Phys. J. C74, 2981 (2014) N. Brambilla, Y. Jia, A. Vairo, Phys. Rev. D73, 054005 (2006) A. Pineda, J. Segovia, Phys. Rev. D87, 074024 (2013) N. Brambilla, P. Pietrulewicz, A. Vairo, Phys. Rev. D85, 094005 (2012) P. Pietrulewicz (2013), [PoSConfinementX,135(2012)], 1301.1308 S. Steinbeißer, J. Segovia, A. Vairo, in preparation: TUM-EFT 86/16 A. Pineda, J. Soto, Nucl. Phys. Proc. Suppl. 64, 428 (1998) N. Brambilla, A. Pineda, J. Soto, A. Vairo, Nucl. Phys. B566, 275 (2000) W.E. Caswell, G.P. Lepage, Phys. Lett. B167, 437 (1986) G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D51, 1125 (1995) N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D63, 014023 (2001) A. Pineda, Prog. Part. Nucl. Phys. 67, 735 (2012) W. Fischler, Nucl. Phys. B129, 157 (1977) Y. Schroder, Phys. Lett. B447, 321 (1999) N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D60, 091502 (1999) B.A. Kniehl, A.A. Penin, Nucl. Phys. B563, 200 (1999) C. Anzai, Y. Kiyo, Y. Sumino, Phys. Rev. Lett. 104, 112003 (2010) A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Phys. Rev. Lett. 104, 112002 (2010) K.G. Chetyrkin, J.H. Kuhn, M. Steinhauser, Comput. Phys. Commun. 133, 43 (2000) DOI: 10.1051/ , 06026 (2017) 713706026 137 EPJ Web of Conferences epjconf/201 XIIth Quark Confinement & the Hadron Spectrum 10 |
5774 | https://arxiv.org/pdf/2006.00489 | Published Time: Mon, 23 Jan 2023 05:16:50 GMT
IMPROVED STOCHASTIC ROUNDING ∗
LU XIA † , MARTIJN ANTHONISSEN † , MICHIEL HOCHSTENBACH † , BARRY KOREN †
Abstract. Due to the limited number of bits in floating-point or fixed-point arithmetic, round-ing is a necessary step in many computations. Although rounding methods can be tailored for different applications, round-off errors are generally unavoidable. When a sequence of computations is implemented, round-off errors may be magnified or accumulated. The magnification of round-off errors may cause serious failures. Stochastic rounding (SR) was introduced as an unbiased rounding method, which is widely employed in, for instance, the training of neural networks (NNs), showing a promising training result even in low-precision computations. Although the employment of SR in training NNs is consistently increasing, the error analysis of SR is still to be improved. Addition-ally, the unbiased rounding results of SR are always accompanied by large variances. In this study, some general properties of SR are stated and proven. Furthermore, an upper bound of rounding variance is introduced and validated. Two new probability distributions of SR are proposed to study the trade-off between variance and bias, by solving a multiple objective optimization problem. In the simulation study, the rounding variance, bias, and relative errors of SR are studied for different operations, such as summation, square root calculation through Newton iteration and inner product computation, with specific rounding precision.
Key words. Rounding mode, error analysis, stochastic rounding, variance and bias, multi-objective optimization problem, particle swarm optimization
AMS subject classifications. 62J10, 65G50, 65Y04, 90C26, 97N20
Introduction. In many computations, rounding is an unavoidable step, due to the limited number of bits in floating-point or fixed-point arithmetic. Many round-ing schemes have been proposed and studied for different applications, such as floor, ceiling, round to the nearest, stochastic rounding, etc. These rounding modes nor-mally have different round-off errors. When a sequence of computations is imple-mented, round-off errors may be accumulated and magnified. In real world problems, the magnification of round-off errors may cause severe failures. In pursuit of high accuracy, high-precision computations are generally employed, for which computing times may be long. To reduce computing times, low-precision computing is becoming increasingly popular, especially in the area of machine learning. In , algorithms are proposed to squeeze matrices from double or single precision to half precision, using two-sided diagonal scaling. In , some low-precision simulation results are compared for dif-ferent rounding methods, e.g., directed rounding, rounding to nearest, and stochastic rounding. The detailed numerical analysis of each rounding method still needs to be developed. An unbiased stochastic rounding (SR) scheme was applied in to train neural networks (NNs) using low-precision fixed-point arithmetic. The experiments show that where the deterministic rounding scheme fails, the training results using 16-bit fixed-point representation with the SR method are very similar to those computed in 32-bit floating-point precision. Inspired by , SR is widely employed in training NNs in low-precision floating-point or fixed-point precision, see, e.g., [11, 13, 16]. Although the employment of SR in training NNs is increasing, the error analysis of SR is still
∗
This research was funded by the EU ECSEL Joint Undertaking under grant agreement no. 826452.
†
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (l.xia1@tue.nl, m.j.h.anthonissen@tue.nl, m.e.hochstenbach@tue.nl, b.koren@tue.nl). 1
arXiv:2006.00489v1 [math.NA] 31 May 2020
to be completed. Additionally, the unbiased rounding results of SR always have large variances. In this paper, SR is studied with respect to two aspects. First, numbers are rounded to a specific number of fractional bits and an upper bound of rounding vari-ance is introduced and validated. Next, some general properties of SR are also proven. To study the trade-off between variance and bias of rounding results using SR, two new probability distributions are proposed, for which a multi-objective optimization problem (MOP) is formulated. The probability can be easily optimized according to user requirements on variance and bias, for instance by particle swarm optimization (PSO). Since the new probability distributions minimize both variance and bias, these rounding modes are potentially interesting for training NNs and numerical solution algorithms. The remainder of the paper is organized as follows. The rounding rules of some general deterministic rounding methods are summarized in section 2. Section 3 out-lines the scheme of stochastic rounding and introduces the formula of the variance and some general properties. Then, in section 4, new probability distributions are proposed to discuss the trade-off between variance and bias, by solving the MOP. Fur-thermore, numerical simulation results of bias, variance and absolute value of relative errors are presented in section 5, using different rounding schemes for summation, square root calculation through Newton iteration and inner product computation. Finally, conclusions are drawn in section 6.
Deterministic rounding. In this section, the schemes of some general de-terministic rounding methods, such as directed rounding to an integer and rounding to the nearest integer, are summarized. The directed rounding is normally used in in-terval arithmetic and comprised of four rounding methods, e.g., rounding down (floor) and rounding up (ceiling), etc. The floor method rounds a number x to the largest integer smaller than x, vice versa for ceiling. The rounding-to-the-nearest methods vary in different tie breaking rules, such as round half up, round half down, round half to even, round half to odd, etc. . Round half up is commonly used in financial calculations , where numbers smaller than half are rounded down and those larger than or equal to half are rounded up, vice versa for round half down. Rounding half to even is also called convergent rounding (CR), which is the default rounding mode used in IEEE 754 floating-point operations. It eliminates bias by rounding different numbers towards or away from zero. In contrast, rounding half to odd is rarely em-ployed in computations, since it will never round to zero . A summary of the aforementioned rounding schemes is given in Table 1, together with some examples.
Table 1
Summary of different deterministic rounding methods, and four illustrative examples.
Rounding mode Rounding rule 1.6 0.5 −0.5−1.6round down round toward negative infinity 10−1−2round up round toward positive infinity 210−1round half up round to the nearest integer with tie rounding toward positive infinity 210−2round half down round to the nearest integer with tie rounding toward negative infinity 20−1−2round half to even round to the nearest integer with tie rounding toward the nearest even number 200−2round half to odd round to the nearest integer with tie rounding toward the nearest odd number 21−1−22
Stochastic rounding. In this section, some properties of stochastic round-ing (SR) are stated and proven. Furthermore, an algorithm is introduced to round numbers to a specific fractional digit, and the variance bound of the algorithm is proposed and validated. The SR method is studied in [1, 12, 14] and has recently been applied in . It is widely employed in training NNs [8, 16]. Compared to deterministic rounding methods, it provides an unbiased rounding result by setting a probability that is pro-portional to the proximity of x. The definition of stochastic rounding is the following:
Definition 3.1. Let x ∈ R, and let δ be the rounding precision. Then the rounded value fl( x) of x using SR is defined as
fl( x) =
{
bxc, with probability p1(x) = 1 − x−b xc
δ
,
bxc + δ, with probability p2(x) = x−b xc
δ
,
(3.1)
where bxc indicates the greatest representable floating/fixed-point number less than or equal to x .
For instance, if a floating-point number 0 .4 is rounded to integer with SR, so the rounding precision δ = 1, then 0.4 will be rounded down to 0 with probability 0.6 and rounded up to 1 with probability 0.4.
3.1. General properties. In this subsection, some properties of SR will be proven. The first result is straightforward and is briefly introduced in . Let fl( x) be the random variable corresponding to the rounding process. Let pi
be the corresponding probability of fl( x) = xi.
Corollary 3.2. The expected value of the rounded value fl( x) in SR is x. This means that the expected rounding error of fl( x) is 0. Proof. The expected value of fl( x) with discrete probability distribution can be calculated as
E(fl( x)) = x1p1(x) + x2p2(x)= bxc
(
1 − x − b xc
δ
)
( bxc + δ) x − b xc
δ
= bxc + x − b xc = x,
where x1 = bxc and x2 = bxc + δ.The variance of fl( x) in rounding scheme (3.1) is given by (3.2) V (fl( x)) = ( bxc − x)2p1(x) + ( bxc + δ − x)2p2(x).
3.2. Rounding to a specific number of fractional bits or decimal digits.
When a specific number of fractional bits is required, the rounding result can be easily achieved by multiplying with a scalar θ. For instance, one fractional bit indicates a rounding precision δ = 2 −1 and the corresponding scalar is θ = 2. The procedure for rounding to a specific number of fractional bits is given in Algorithm 3.1. To the authors’ knowledge, the following propositions are not proven in literature.
Proposition 3.3. The expected value of rounding results, under the condition of rounding to the specific number of fractional bits n ∈ N, through stochastic rounding Algorithm 3.1 , is still unbiased.
3
Algorithm 3.1 Round to a specific number of fractional bits or decimal digits.
1:
Definitions: Number of fractional bits: n, then scalar θ = 2 n or 10 n.
2:
The scaled value ˜x = θx .
3:
The approximated value of ˜x is given as (3.3) fl( ˜x) =
{
b˜xc with probability 1 − ˜x−b ˜xc
1
,
b˜xc + 1 with probability ˜x−b ˜xc
1
,
where b˜xc indicates the largest integer less than or equal to ˜x.
4:
Scaling it back, we have fl( x) = fl( ˜x)
θ
.
Proof. If the number of fractional bits is n, a scalar can be defined as θ = 2 n. A random variable x ∈ R can be scaled as ˜x = θx and rounded to (3.3) with different probability distributions. According to Corollary 3.2, the expected value of rounding results can be calcu-lated by
E(fl( x)) = E
( fl( ˜x)
θ
)
= 1
θ E(fl( ˜x)) = 1
θ ˜x = 1
θ θx = x.
So the bias of fl( x) is zero.
Proposition 3.4. The variance of rounding to the specific number of fractional bits n, using stochastic rounding Algorithm 3.1 , is bounded by ( 12θ )2.Proof. According to (3.2), the variance of fl( ˜x) obtained by Algorithm 3.1 is
V (fl( ˜x)) = ( b˜xc − ˜x)2
(
1 − ˜x − b ˜xc
1
)
( b˜xc + 1 − ˜x)2
( ˜x − b ˜xc
1
)
= ( b˜xc − ˜x)2 + ( b˜xc − ˜x)3 + ( ˜x − b ˜xc) − 2( b˜xc − ˜x)2 − (b˜xc − ˜x)3
= ( ˜x − b ˜xc) − (b˜xc − ˜x)2
= −
(˜x − b ˜xc − 12
)2
14 .(3.4) Let ∆ ˜x = ˜x − b ˜xc, then ∆ ˜x ∈ [0 , 1]. As a result, (3.4) has the maximum value 14 ,when ∆ ˜x = 12 . Consequently, V (fl( x)) = V ( 1
θ
fl( ˜x)) = 1
θ2
V (fl( ˜x)) ≤ 14θ2 = ( 12θ
)2.To validate Proposition 3.4, a set of numbers from 0 to 2, with the smallest interval between two consecutive numbers equal to 10 −4, has been rounded 10,000 times, under rounding-to-4-fractional-bit scenario, i.e., θ = 2 4. The corresponding variances are shown in Figure 1. The blue line indicates the variance calculated using (3.2), which is almost invisible due to the coverage of the red dashed line, which latter shows the population variance calculated over 10,000 observations using (3.5a) ̂ V = 1
N
N
∑
i=1
(xi − μ)2,
where μ is the mean value of a vector x comprised of N random variables: (3.5b) μ = 1
N
N
∑
i=1
xi.
4
From the zoomed in subplot around x = 125 = 0 .03125 in Figure 1, it can be observed that the blue line and the red dashed line are bounded by 2 −10 ≈ 9.77 ·10 −4, satisfying Proposition 3.4. According to (3.4), the variance is zero when ˜x = b˜xc, with ˜x = 2 −4j
in Figure 1, where j ∈ N.0 0.2 0.4 0.6 0.8 1
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
rounding variance
10 -3
calculated
experiments
0.028 0.03 0.032 0.034
9
9.5
10 10 -4
Fig. 1 . Variance of x from 0 to 2 when n = 4 (rounding to 4 fractional bits).
Proposition 3.5. In stochastic rounding, it holds
fl(fl( · · · fl(fl( x1) op x2) op · · · ) op xNs ) = fl( x1) op fl( x2) op · · · op fl( xNs ),
where Ns indicates the number of terms for op ∈ { +, −} .Proof. Assume x1 and x2 are rounded using the following rounding scheme fl( xi) =
{
bxic, with probability pi,
bxic + δ, with probability 1 − pi,
where i ∈ { 1, 2}. Then fl( x1) + fl( x2) =
bx1c + bx2c, with probability p1p2,
bx1c + bx2c + δ, with probability (1 − p1)p2 + p1(1 − p2),
bx1c + bx2c + 2 δ, with probability (1 − p1)(1 − p2).
We also have fl (fl( x1) + x2
) =
bb x1c + x2c = bx1c + bx2c,
with probability p1p2,
bb x1c + δ + x2c = bx1c + bx2c + δ,
with probability (1 − p1)p2 + p1(1 − p2),
bb x1c + δ + x2c + δ = bx1c + bx2c + 2 δ,
with probability (1 − p1)(1 − p2),
= fl( x1) + fl( x2).
5For summation of Ns terms, fl(fl( x1) + x2) + · · · ) + xNs ) =
bx1c + bx2c + · · · + bxNs c,
with probability p1p2 · · · pNs ,
bx1c + bx2c + · · · + bxNs c + δ,
with probability (1 − p1)p2 · · · pNs
+p1(1 − p2) · · · pNs + · · · + p1p2 · · · (1 − pNs ),
...
bx1c + bx2c + · · · + bxNs c + nδ,
with probability (1 − p1)(1 − p2) · · · (1 − pNs ),
= fl( x1) + fl( x2) + · · · + fl( xNs ).
The above relation also holds for subtraction, by replacing + by −.The following propositions hold under the rounding to integer scenario, where
δ = 1.
Proposition 3.6. For multiplication using stochastic rounding, it holds
fl (fl( x1)fl( x2)) = fl( x1)fl( x2).(3.6)
Proof.
fl( x1)fl( x2) =
bx1cb x2c, with probability p1p2,
bx1cb x2c + bx2c, with probability (1 − p1)p2,
bx1cb x2c + bx1c, with probability p1(1 − p2),
bx1cb x2c + bx1c + bx2c + 1 , with probability (1 − p1)(1 − p2).
Since bx1cb x2c is an integer, bb x1cb x2cc = bx1cb x2c. Consequently, fl (fl( x1)fl( x2)) =fl( x1)fl( x2).
Proposition 3.7. When x1, x 2 ∈ (0 , 1) and x1x2 ≤ 12 , the worst-case relative round-off error is larger than or equal to 1 in (3.6) .Proof. Applying SR to x2, when fl( x2) = 0, the relative error of (3.6) is always 1. The worst-case scenario only occurs when fl( x2) = 1. Hence, the result of (3.6), in the worst-case scenario, is fl( x1)fl( x2) =
{
bx1c, with probability p,
bx1c + 1 , with probability 1 − p.
The worst-case absolute relative round-off error is
eworst =
∣∣∣ x1x2 − fl( x1)fl( x2)
x1x2
∣∣∣ =
{∣ ∣ x1x2−b x1c
x1x2
∣∣, with probability p,
∣∣ x1x2−(bx1c+1)
x1x2
∣∣, with probability 1 − p.
(3.7) When x1 ∈ (0 , 1), we have bx1c = 0 and (3.7) becomes (3.8) eworst =
{∣ ∣1 − bx1c
x1x2
∣∣ = 1 , with probability p,
∣∣1 − bx1c
x1x2
− 1
x1x2
∣∣ = ∣∣1 − 1
x1x2
∣∣, with probability 1 − p.
When x1, x 2 ∈ (0 , 1) and x1x2 ≤ 12 , we have 1
x1x2
≥ 2, so eworst = ∣∣1 − 1
x1x2
∣∣ ≥ 1.
6
For x1 > 1, we have the following.
Proposition 3.8. When x1 ∈ (i, i + 1) , x2 ∈ (0 , 1) and x1x2 ≤ i
2
, where i ∈ N+,we find in a similar way as above that the worst-case relative round-off error is larger than or equal to 1 in (3.6) .Proof. For x1 ∈ (i, i + 1), we have
eworst =
{∣ ∣1 − bx1c
x1x2
∣∣ = ∣∣1 − ix1x2
∣∣, with probability p,
∣∣1 − bx1c+1
x1x2
∣∣ = ∣∣1 − i+1
x1x2
∣∣, with probability 1 − p. .
When x1x2 ≤ i
2
, we have ∣∣1 − ix1x2
∣∣ ≥ 1 and ∣∣1 − i+1
x1x2
∣∣ ≥ ∣∣ − 1 − 1
x1x2
∣∣ > 1.
(a) Error with probability p(b) Error with probability 1 −p
Fig. 2 .Contour plots of worst-case relative error with respect to x1and x2, with probabilities
p(a) and 1−p(b), where the red lines are the lines with x1x2=i
2and x1x2=i+1 2in (a) and (b), respectively.
Proposition 3.8 will be used further in subsection 5.3. Figure 2 shows the contour plots of the worst-case relative error with respect to x1 and x2, with probabilities p
(Figure 2a) and 1 − p (Figure 2b). In the yellow areas, eworst ≥ 2. At the red lines,
eworst = 1. Specifically, the red lines are the lines with x1x2 = i
2
and x1x2 = i+1 2 in Figures 2a and 2b, respectively. It can be observed that eworst > 1 for both Figures 2a and 2b, when x1x2 < i
2
. Furthermore, the worst-case relative error increases when x2
decreases. From the zoomed in subplots (small figures) in Figure 2, it can be seen that the worst-case relative error can be very large, when x2 is close to 0. For the same value of x2, Figure 2b shows the larger worst-case relative error than Figure 2a. The aforementioned worst-case scenario will not occur, if floor rounding and rounding-to-the-nearest methods are employed, because then the numbers will always round towards zero if they are close to zero.
Optimization of the probability distribution of stochastic rounding.
In this section, a new probability distribution is proposed. To find it, a multi-objective optimization problem (MOP) is formulated. The probability is computed with differ-ent emphasis on variance and bias. To do so we use the scalarization method [2, Ch. 2, p. 11–36]. To meet the requirements of different computations, constraints on bias and variance are realized by adding a penalty function to the objective function, which will be defined in the following section. The optimization problem is solved using PSO.
7
4.1. Problem formulation. Instead of the probabilities in (3.1), a general probability distribution p is considered. The new stochastic rounding with unknown probability, is defined as fl( x) =
{
bxc, with probability p1 = p,
bxc + δ, with probability p2 = 1 − p,
(4.1) where bxc indicates the greatest representable floating-point or fixed-point number less than or equal to x and where δ is the rounding precision. The variance of rounding scheme (4.1) is (4.2) V (p) = ( bxc − ¯μ)2p + ( bxc + δ − ¯μ)2(1 − p),
where ¯ μ is the expected rounding value, given by (4.3) ¯μ = bxc p + ( bxc + δ)(1 − p).
Substitute (4.3) into (4.2), to find (4.4) V (p) = δ2(p − p2).
The bias is (4.5) B (p) = ( bxcp + ( bxc + δ)(1 − p)) − x.
To find a trade-off between variance and bias, a MOP can be formulated as minimize
p
V (p), |B (p)|,(4.6a) subject to V (p) ≤ Vmax , |B (p)| ≤ Bmax , 0 ≤ p ≤ 1, x ∈ X ,(4.6b) where X denotes the domain of the input variables. Vmax and Bmax can be set according to the users’ own preferences, but feasibility should also be considered. Solutions of such MOPs are generally non-unique, since the objective functions are normally conflicting. In this case, Pareto optimality is often achieved in MOPs . An effective approach to find the trade-off between each conflicting objective function is the scalarization method , in which a single scaled fitness function is formulated. Furthermore, the constraints on variance and bias in (4.6b) can be realized by adding a penalty ( ki) to the objective function (4.6a), as in , given by minimize
p
(θ1(V (p)) 2 + θ2(B (p)) 2 +
2
∑
i=1
kiI[0 ,∞)(gi)),(4.7a) subject to 0 ≤ p ≤ 1, x ∈ X ,(4.7b) with θ1 + θ2 = 1, g1 = V (p) − Vmax and g2 = |B (p)| − Bmax , where I is an indicator function, having the value 1 when gi ∈ [0 , ∞), and 0 elsewhere; and where ki is a constant, indicating the penalty on the ith constraint gi. Here, ki is chosen to be 0 or sufficiently large to realize an unconstrained or constrained condition for V and B,respectively. Due to the effect of penalties on the objective function, the gradient of the op-timization problem is frequently not available. PSO is a gradient-free approach that is used extensively in solving global optimization problems . It solves problems
8
by searching the best position among a group (swarm) of the candidate solutions (particles). The goal is to find the globally best position by comparing each parti-cle’s own best position to its neighbor’s best position. Problem (4.7) can be solved using the same PSO algorithm as in . It should be noted that the choice of the optimization method, for instance PSO, is not the crucial part of this study, since the optimal probability distribution can be calculated offline and is not necessarily computed during each rounding process. This paper provides a method to obtain an improved stochastic rounding method with customized rounding variance and bias.
4.2. Four probability distributions. In this section, the optimization prob-lem is solved four times, each time with a different emphasis on variance and bias. The resulting variances and biases are compared.
4.2.1. Bias minimization. If only bias is minimized in (4.7) without any con-straint, θ1 is set to 0. According to (3.3), δ can be chosen as 1 for any number of fractional bits, after proper scaling. The probability distribution, and the correspond-ing variance and bias are shown in Figure 3. The probability distribution found is exactly the same as in the SR method. It can be calculated analytically by setting (4.5) to 0, so (bxcp + ( bxc + δ)(1 − p)) − x = 0 .
We find, p = 1 − x−b xc
δ
, as in (3.1). From Figure 3, it can be concluded that the bias is zero for all x and the variance is highest at the tie point. This distribution is repeated for every interval in X .0 0.2 0.4 0.6 0.8 100.5 1
probability distribution
00.2 0.4 0.6 0.8 1-1 -0.5 00.5 1
variance and bias variance bias
Fig. 3 .Probability distribution (top) and corresponding variance and bias (bottom) with respect to input variable x, for bias minimization.
4.2.2. Variance minimization. When variance is the only objective function, and no constraints are considered, θ2 = 0. Based on (4.4), the variance is 0 when p = 0 or p = 1. These two choices are the ceiling and the floor method, respectively. It can be seen from Figure 4 that the optimal probability distribution of variance minimization is not unique; the probability can be either 0 (Figure 4a) or 1 (Figure 4b), resulting in a bias, which either equals δ + bxc − x or bxc − x, respectively. If in (4.7), variance is multiplied with a large parameter and bias with a small parameter, e.g., θ1 = 0 .98
900.2 0.4 0.6 0.8 100.5 1
probability distribution
00.2 0.4 0.6 0.8 1-1 -0.5 00.5 1
variance and bias variance bias
(a) p = 0 0 0.2 0.4 0.6 0.8 100.5 1
probability distribution
00.2 0.4 0.6 0.8 1-1 -0.5 00.5 1
variance and bias variance bias
(b) p = 1
Fig. 4 . Probability distribution (top) and corresponding variance and bias (bottom) with respect to input variable x, for variance minimization, when p = 0 (a) and p = 1 (b). 0 0.2 0.4 0.6 0.8 100.5 1
probability distribution
00.2 0.4 0.6 0.8 1-1 -0.5 00.5 1
variance and bias variance bias
Fig. 5 . Probability distribution (top) and corresponding variance and bias (bottom) with respect to input variable x, for rounding-to-the-nearest method.
and θ2 = 0 .02, we find the probability distribution given in Figure 5. The probability distribution is similar to the rounding to the nearest integer with different tie breaking rules such as round half up, round half down, round half to even and round half to odd, as summarized in Table 1. Comparing Figures 4 and 5, the resulting bias using rounding to the nearest is twice as small as that for the floor or ceiling method. Among all the methods of rounding to the nearest integer, CR is the default rounding mode in IEEE 754 floating-point operations and has the probability distribution shown in Figure 5. CR will be further studied and compared with stochastic rounding methods in section 5.
4.2.3. Trade-off between bias and variance. Considering the trade-off be-tween bias and variance, it is challenging to find a good balance between both. As-sume bias and variance are equally important in our rounding scheme, which means
10 0 0.2 0.4 0.6 0.8 100.2 0.4 0.6 0.8 1
probability distribution
00.2 0.4 0.6 0.8 1-1 -0.5 00.5 1
variance and bias variance bias (a) 00.2 0.4 0.6 0.8 100.2 0.4 0.6 0.8 1
probability distribution
00.2 0.4 0.6 0.8 1-1 -0.5 00.5 1
variance and bias variance bias (b)
Fig. 6 .Probability distribution (top) and corresponding variance and bias (bottom) with respect to input variable x, for both bias and variance minimization (a) and for constrained bias and variance minimization (b).
θ1 = θ2 = 0 .5 in (4.7). This rounding scheme is called Distribution 1, shortly D1, in the remainder of this paper. Under unconstrained conditions, i.e., k1 = k2 = 0, which means no constraints are set to either bias or variance, the optimized probabil-ity distribution and corresponding variance and bias are shown in Figure 6a. It can be observed that the variance is slightly reduced compared to the SR method but still considerably larger than that of the CR method, vice versa for the bias. Furthermore, the probability distribution is no longer linear.
4.2.4. Constrained bias. According to (4.4), the variance cannot be zero in rounding method (4.1) if p 6 = 0 or p 6 = 1, but a constraint can be imposed to the bias to guarantee a better rounding result. It should be noted that the maximum value of the bias can be set according to the user requirement. The value of the penalty can be set arbitrarily, as long as it is sufficiently larger than the value of the objective function when ki = 0. Assume Bmax = 0 .05. To achieve this constraint, a penalty k2 = 10 10
will be added if g2 ≥ 0. The resulting rounding scheme is called Distribution 2, shortly D2, in the remainder of this paper. Figure 6b shows the probability distribution and the corresponding variance and bias, where the bias is limited to 0.05 and the variance is slightly larger than for scheme D1. Consequently, the probability distribution is also tailored to meet the requirement. The smallest variance is generally achieved with the largest bias and vice versa. Through optimization scheme (4.7), a trade-off can be easily obtained within con-straints.
Numerical experiments. In this section, the rounding methods SR, CR, D1 and D2 are compared with respect to the absolute value of bias ( |B|), variance (V ) and average absolute value of relative error ( e) in some numerical experiments. First, experiments are done for the sum operation, with different input distributions.
11
Next, the performance for the square root operation is studied, in which Newton’s method is employed to iteratively compute the result. Finally, experiments are done for the inner product operation.
5.1. Summation. Due to the tie-breaking rule of CR and the stochastic behav-ior of SR, D1 and D2 rounding, the distribution of input variables will influence the rounding result. For instance, if the input variables are distributed in [0 , 1], where we define 0 to be even, the rounding result of 0 .5, using CR, will always be biased, it will always be rounded to the even number 0. To study the influence of different input distributions on the rounding result, in this section, the experiments will be studied with four input distributions:
• Case I : repeated numbers distributed in an odd number of intervals;
• Case II : repeated numbers distributed in an even number of intervals;
• Case III : non-repeated numbers distributed in an odd number of intervals;
• Case IV : non-repeated numbers distributed in an even number of intervals.
5.1.1. Generation of input numbers. If numbers are distributed uniformly, the probability of the presence of repeated numbers depends on the number of samples in each interval, Ns. Specifically, a large value of Ns leads to a larger probability of repeated numbers. In our simulation study, the repeated numbers are obtained randomly using a large number of samples in a small interval, and vice versa for non-repeated numbers. For Case I, a set of uniformly distributed random numbers is generated between [0 , 1], using the Matlab function rand , where the number of samples equals Ns = 10 , 000. The same amount of numbers are randomly generated, for Case II, in [0 , 2]. For Case III, only 10 samples are randomly generated in [0 , 1], to avoid repeated numbers. Additionally, 20 samples are generated in [0 , 2] for Case IV. It should be noted that the input numbers are only generated once for each case, and then kept fixed for the different rounding methods.
5.1.2. Numerical test. To each of the aforementioned input distributions, we apply the summation operation given by
y =
Ns
∑
i=1
xi.
According to Proposition 3.5, the rounding result may be reformulated as fl( y) = fl(fl(fl( x1) + x2) + · · · + xNs )= fl( x1) + fl( x2) + · · · + fl( xNs ).(5.1) Variance is computed according to (3.5) and bias is calculated using
B(y) = E (fl( y)) − E(y).
Additionally, the average absolute value of the relative error is defined as
e = 1
N
N
∑
i=1
|fl( yi) − yi|
yi
,
where N is the number of repetitions of the experiment. In this study, 10,000 repeti-tion of experiments are made with the same input for all stochastic rounding methods.
12 Case I Case II Case III Case IV 00.2 0.4 0.6 0.8 1
Normalized absolute bias
SR CR D1 D2 Fig. 7 .Normalized absolute bias of rounding results using SR, CR, D1 and D2, for Cases I-IV.
All the summation outcomes are rounded to integers for each rounding process. Fig-ure 7 shows the normalized absolute value of the bias for rounding methods SR, CR, D1 and D2, for Cases I-IV. The largest bias is always obtained by CR, if the input variables are repeated or distributed in an odd number of intervals, for Cases I-III. This can be explained by the optimization result given in subsection 4.2, where the bias of CR is larger than SR, D1 and D2. For Case IV, CR realizes an unbiased re-sult, since the input variables are distributed in an even number of intervals, in such a way that the rounding bias in the interval [0 , 1] is compensated by that in [1 , 2]. A small bias is obtained by SR for Cases I and II, owing to the unbiased property of SR. In general, the biases caused by D2 are always smaller than D1, which is an obvious result led by the constraint on the bias in D2. The variance shows results opposite to those of the bias, as depicted in Figure 8a. The variance of CR is zero for all four cases, since CR is deterministic. The second smallest variance is always obtained by D1, and the largest variance is obtained by SR, though the difference between rounding methods SR, D1, D2 is minor. This agrees with the optimization results in subsection 4.2. Figure 8b shows the normalized average absolute value of relative error for the four rounding methods for Cases I-IV. For the repeated input variables, Cases I and II, CR has the largest average absolute value of relative error. For the non-repeated input variables, Cases III and IV, the smallest average absolute value of relative error is achieved by CR and the largest one is obtained by SR. Overall, for the repeated input variables, D1 results in the rounding results with smallest variance among the stochastic rounding methods, as well as small average absolute value of relative error. For the non-repeated input variables, CR performs best in general, with smallest variance and average absolute value of relative error. The non-normalized values of the bias, variance and average absolute value of relative error in Figures 7, 8a and 8b are given in Table 2, in which the largest bias, variance and average absolute value of relative error are marked in red. It can be observed from Table 2 that the value of the bias and variance in Cases I and II are much larger than those of Cases III and IV, because the bias and variance in Cases I and II are accumulated by the sequence of summation, where Ns = 10 , 000 in (5.1).
5.2. Square root calculation using Newton iteration. Square root calcu-lation is an approximation process on most processor units such as CPUs, GPUs and
13 Case I Case II Case III Case IV 00.2 0.4 0.6 0.8 1
Normalized variance
SR CR D1 D2
(a) Case I Case II Case III Case IV 00.2 0.4 0.6 0.8 1
Normalized absolute relative error SR CR D1 D2
(b)
Fig. 8 . Normalized variance (a) and absolute relative error (b) of rounding results using SR, CR, D1 and D2, for Cases I-IV.
Table 2
Non-normalized values of bias, variance and absolute relative error of SR, CR, D1 and D2, for Cases I-IV, with largest values of |B|, V and e marked in red.
Case I Case II Case III Case IV
|B|
SR 0.4 0.82 0.03 0.034 CR 0.47 · 10 3 69 .1 0.4 0D1 8.05 8 0.11 0.22 D2 6.84 7.42 0.01 0.16
V
SR 1.73 · 10 3 1.61 · 10 3 2.29 3.24 CR 0 0 0 0D1 1.29 · 10 3 1.38 · 10 3 1.91 2.51 D2 1.45 · 10 3 1.45 · 10 3 2.1 2.77
e
SR 0.0066 0.0032 0.2284 0.0667 CR 0.0932 0.0069 0.0741 0D1 0.0061 0.0031 0.2201 0.0617 D2 0.0059 0.003 0.2086 0.0588
FPGAs. It is based on different numerical algorithms. The speed of square root com-putation is crucial in hardware applications. In this section, the square root operation is studied with different rounding methods, using Newton iteration. The computation precision and speed are studied by rounding to a specific number of decimal digits and implementing integer arithmetic. The square root of a number, √a, can be iteratively computed by Newton’s method by introducing the function f (x) = x2 − a, and the iterative process is given by xk+1 = xk − f (xk )
f′(xk)
= 12
(xk + axk
). The rounding process can be reformulated as fl( xk+1 ) = fl
( 12
(
fl( xk) + fl ( fl( a)fl( xk)
)))
.(5.2) In the numerical tests, some random numbers with five decimal digits will be gen-erated, one number in successively (0 , 1), (1 , 10), (10 , 100), (100 , 1000) and (1000 , 10000). Next, √a is calculated N times, using different rounding schemes, to calculate μ, B,
V , e and the average number of iteration steps ( Nit ). For each rounding process, the numbers are rounded to three decimal digits. Specifically, the rounding preci-sion is δ = 10 −3. In Newton’s method, the initial guess is x0 = 1 and 10 −5 is set
14 Table 3
Non-normalized values of μ,|B|,V,eand Nit of SR, CR, D1 and D2 for computing √ausing Newton’s method, with δ= 10 −3, with largest values of |B|and emarked in red.
a0.30146 6.55501 51 .16904 357 .00272 8133 .27762
μ
SR 0.5491 2.5601 7.1531 18 .8946 90 .1849 CR 0.548 2.56 7.154 18 .894 90 .184 D1 0.549 2.5604 7.153 18 .8942 90 .1848 D2 0.549 2.56 7.1535 18 .894 90 .1848
|B|
SR 6.46 ·10 −51.58 ·10 −41.96 ·10 −41.29 ·10 −41.9·10 −4
CR 1.05 ·10 −32.75 ·10 −47.46 ·10 −45.16 ·10 −46.86 ·10 −4
D1 5.42 ·10 −58.55 ·10 −52.54 ·10 −42.8·10 −41.44 ·10 −4
D2 2.79 ·10 −52.74 ·10 −42.41 ·10 −45.04 ·10 −41.91 ·10 −5
V
SR 8.2·10 −81.04 ·10 −74.48 ·10 −82.29 ·10 −71.08 ·10 −7
CR 00000D1 1.1·10 −92.31 ·10 −73.33 ·10 −11 1.8·10 −71.41 ·10 −7
D2 1.07 ·10 −71.4·10 −92.5·10 −71.17 ·10 −82.08 ·10 −7
e
SR 1.18 ·10 −46.16 ·10 −52.73 ·10 −56.85 ·10 −62.11 ·10 −7
CR 1.92 ·10 −31.08 ·10 −41.04 ·10 −42.73 ·10 −57.61 ·10 −6
D1 9.88 ·10 −53.34 ·10 −53.55 ·10 −51.48 ·10 −51.6·10 −6
D2 5.09 ·10 −51.07 ·10 −43.36 ·10 −52.67 ·10 −52.11 ·10 −7
Nit
SR 5.57 5.75 7.7911 .83 CR 457811 D1 5.66 5.87.44 911 .9D2 5.58 5.77.43 911 .86
as the tolerance for convergence and Nmax = 100 is set as the maximum number of iteration steps to take. A summary of the simulation results is given in Table 3. It can be observed that the largest bias (marked in red) is always obtained by CR. Consequently, the largest average absolute value of relative error (marked in red) is always obtained by CR as well. Still, the value of the largest average absolute value of relative error obtained by CR is generally less than 10 −3, indicating a good accuracy. Furthermore, the resulting average number of iteration steps of CR is integer, since it is a deterministic process. Additionally, D1 shows the most reliable performance of the four rounding schemes, where the average absolute value of relative error of the approximated results using D1 is consistently smaller than 10 −4.If integer arithmetic is considered in (5.2), i.e., when rounding precision δ = 1 is applied, repeating the same calculations as in Table 3, the results given in Table 4 are obtained. It can be observed that, as expected, the square root of numbers between [0 , 1] is not solvable using integer arithmetic, since the rounding result of xk and
a can be 0. Moreover, Newton’s method fails to converge using the CR method, for 6.55501, using integer arithmetic. It should be noted that the aforementioned results are specific for the numbers given in Table 3, since the numbers are randomly generated. However, in general, rounding method D1 offers the smallest bias and average absolute value of relative error (both marked in blue). Stochastic rounding methods, such as SR, D1 and D2, guarantee faster convergence than CR. Comparing Table 3 and Table 4, the approximated square root using rounding to three decimal digits ( δ = 10 −3) is more accurate than that using integer arithmetic, while a faster convergence is obtained by integer arithmetic for all stochastic rounding methods.
5.3. Inner product computation. In this section, some tests will be per-formed using different rounding methods for inner product computation. For two vectors x and y, the inner product of x and y can be calculated as 〈x, y〉 =
x1y1 + x2y2 + · · · + xNs yNs . Considering the rounding process, it can be formulated
15 Table 4
Non-normalized values of μ,|B|,V,eand Nit of SR, CR, D1 and D2 for computing √ausing Newton’s method, with δ= 1 , with smallest values of |B|and emarked in blue. If the Newton iteration breaks down or does not converge, we write a dash ( −).
a0.30146 6.55501 51 .16904 357 .00272 8133 .27762
μ
SR −2.0966 719 90 .7080 CR −3719 90 D1 −2.5913 7.0274 19 90 .0032 D2 −2.9992 719 90 .0004
|B|
SR −0.46 0.15 0.11 0.523 CR −0.44 0.15 0.11 0.185 D1 −0.03 0.13 0.11 0.182 D2 −0.44 0.15 0.11 0.184
V
SR −0.09 3.33 ·10 −500.21 CR −0000D1 −0.24 0.03 03.2·10 −3
D2 −8·10 −4004·10 −4
e
SR −0.18 0.021 5.58 ·10 −35.8·10 −3
CR −0.17 0.021 5.58 ·10 −32.05 ·10 −3
D1 −0.02 0.018 5.58 ·10 −32.01 ·10 −3
D2 −0.17 0.021 5.58 ·10 −32.04 ·10 −3
Nit
SR −3.71 5.49 7.27 9.99 CR −−6710 D1 −3.81 5.45 7.16 9.97 D2 −3.79 5.47 7.21 9.98
as (5.3) fl( 〈x, y〉) = fl (fl( x1)fl( y1)) + fl (fl( x2)fl( y2)) + · · · + fl (fl( xNs )fl( yNs )).
Based on Proposition 3.6, when rounding to integers is applied, (5.3) can be simplified to (5.4) fl( 〈x, y〉) = fl( x1)fl( y1) + fl( x2)fl( y2) + · · · + fl( xNs )fl( yNs ),
thus the number of roundings is reduced Ns times. According to Propositions 3.7 and 3.8, the presence of small numbers will lead to relative round-off errors larger than 1, when implementing the operation of mul-tiplication. If the inner product is close to zero, the relative errors of inner product can be very large. To enable a better comparison of performance between different rounding methods, in the simulation study, the input vectors are designed to have a low composition of small numbers and the inner product is supposed to be not close to zero. Hence, the vector x is generated using the sine function with input vector
y, where y is comprised of Ns points distributed equidistantly in [0 , 2π]. The sine function is a more appropriate choice than for instance the cosine function, since the combination of cos( y) and y has a larger chance to result in the worst-case relative round-off error when using SR. E.g., when y → π
2
and y → 3π
2
, where y ∈ [0 , 2π], we have x = cos( y) → 0. According to Proposition 3.8 and Figure 2, the proba-bility of getting a large worst-case relative round-off error will be π
2
− b π
2
c ≈ 0.57 and 3π
2
− b 3π
2
c ≈ 0.71, while the probabilities for the sine function are 0 .14 and 0 .28. Furthermore, ∫ 2π
0
y cos( y)dy = 0, which also indicates that the inner product is close to 0. Table 5 shows the bias, variance and average absolute value of relative error of (5.4) for computing inner products using integer arithmetic with the different rounding
16 Table 5
Non-normalized values of |B|,Vand eof SR, CR, D1 and D2 for computing inner products using integer arithmetic, with largest values of |B|,Vand eand smallest values of |B|and emarked in red and blue, respectively.
Ns50 200 400 600 800 1000
|B|
SR 0.17 0.17 0.11 0.27 0.44 0.75 CR 0.07 9.02 17 .01 29 .01 35 44 D1 7.12 11 .13 15 .01 19 .324 .14 29 .05 D2 6.93 9.913 .53 16 .91 20 .09 24 .5
V
SR 96 .02 382 .6768 .52 1.14 ·10 31.49 ·10 31.94 ·10 3
CR 000000D1 75 .02 305 .26 609 .20.91 ·10 31.22 ·10 31.52 ·10 3
D2 80 .61 327 .37 654 .95 0.99 ·10 31.29 ·10 31.65 ·10 3
e
SR 0.161 0.078 0.055 0.045 0.039 0.035 CR 0.001 0.045 0.043 0.048 0.044 0.044 D1 0.188 0.084 0.058 0.048 0.043 0.04 D2 0.189 0.083 0.058 0.047 0.042 0.038
methods, over 10,000 times repetition, for different vector sizes Ns. It can be observed from Table 5 that SR guarantees the smallest bias (marked in blue), for Ns ≥ 200, but that it also yields the largest variance (marked in red). The average absolute value of relative error obtained by CR first increases with increasing size of the vector Ns, but becomes more or less constant for Ns > 200, whereas the average absolute values of relative errors obtained by the stochastic rounding methods decrease. For Ns ≤ 400, CR provides a more accurate rounding result than the stochastic rounding methods. However, for Ns > 400, the average absolute values of the relative errors caused by stochastic rounding methods decrease and eventually become smaller than the relative error of CR. Overall, if the vector size is large, stochastic rounding methods are the better choice to compute inner products with integer arithmetic, though the variance may be very large. For vectors with small size, CR can provide a rounding result as accurate as SR, where CR has zero variance moreover. Among these stochastic rounding methods, SR always guarantees the smallest bias and average absolute value of relative error. However, when Ns increases, the difference of the average absolute value of relative error between each stochastic rounding method is smaller, while variances obtained by D1 and D2 are both smaller than that obtained by SR. Hence, D1 and D2 are the better options, when variance and bias are both important to the rounding result for calculating inner products with large vector size. It should be noted that the behavior of each rounding method in case of inner product operation is very similar to the behavior for summation. SR guarantees the smallest bias, but along with largest variance. D1 and D2 provide smaller variances but slightly larger bias than SR. To achieve the best rounding results, some prior knowledge of the data set is necessary for choosing a rounding mode.
Conclusion. Rounding is an essential step in many computations; round-off errors are unavoidable. Deterministic rounding methods generally suffer from round-ing bias, while stochastic rounding methods normally have a large rounding variance. In this paper, a systematic way has been proposed to generate a stochastic rounding method with probability distribution that can provide customized rounding bias and variance. As opposed to the conventional stochastic rounding method, the proposed method enables users to tune the rounding probability in different applications, with-out introducing much extra computational cost. The probability distribution wished for is obtained by formulating a multi-objective optimization problem, which is solved
17
offline using particle swarm optimization. Numerical experiments have been performed to analyze the bias, variance and rel-ative error of different rounding methods, such as conventional stochastic rounding, convergent rounding and stochastic rounding with new probability distributions, by implementing three operations: summation, square root calculation through Newton iteration and inner product computation. It has been shown that the rounding results vary in different operations. The proposed stochastic rounding provides the smallest average absolute value of relative rounding error in summation, for repeated input variables. For non-repeated input variables, the smallest average absolute value of relative error is achieved by convergent rounding. Furthermore, the proposed proba-bility distribution of stochastic rounding also offers the best rounding performance in the square root calculation using Newton’s method, where the average absolute value of relative error is consistently smaller than 10 −4 for all test cases. Additionally, sto-chastic rounding methods lead to a faster convergence than convergent rounding when integer arithmetic is applied, in Newton’s method. Moreover, by employing integer arithmetic in inner product operations, the number of rounding processes is largely reduced. Stochastic rounding methods show better rounding results with large vector sizes, whereas convergent rounding provides better approximations with small vector sizes.
Acknowledgments. This research was funded by the EU ECSEL Joint Under-taking under grant agreement no. 826452 (project Arrowhead Tools).
REFERENCES C. Allton, C. Yung, and C. Hamer , Stochastic truncation method for Hamiltonian lattice field theory , Phys. Rev. D, 39 (1989), p. 3772. M. Caramia and P. Dell’Olmo , Multi-objective management in freight logistics: Increasing capacity, service level and safety with optimization algorithms , Springer Science & Business Media, 2008. Y. Censor , Pareto optimality in multiobjective problems , Appl. Math. Opt., 4 (1977), pp. 41– 59. M. F. Cowlishaw , Decimal floating-point: Algorism for computers , in Proceedings of the 16th IEEE Symposium on Computer Arithmetic, IEEE, 2003, pp. 104–111. S. Gupta, A. Agrawal, K. Gopalakrishnan, and P. Narayanan , Deep learning with lim-ited numerical precision , in Proceedings of the 32nd International Conference on Machine Learning, 2015, pp. 1737–1746. N. J. Higham and S. Pranesh , Simulating low precision floating-point arithmetic. , Manchester Institute for Mathematical Sciences, 2019, N. J. Higham, S. Pranesh, and M. Zounon , Squeezing a matrix into half precision, with an application to solving linear systems , SIAM J. Sci. Comput., 41 (2019), pp. A2536–A2551. M. Hopkins, M. Mikaitis, D. R. Lester, and S. Furber , Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ODEs , arXiv preprint arXiv:1904.11263, (2019). W. Kahan , IEEE standard 754 for binary floating-point arithmetic , Lecture Notes on the Status of IEEE, 754 (1996), ∼wkahan/ieee754status/ieee754. ps. F. Marini and B. Walczak , Particle swarm optimization (PSO). A tutorial , Chemometr. Intell. Lab. Syst., 149 (2015), pp. 153–165. T. Na, J. H. Ko, J. Kung, and S. Mukhopadhyay , On-chip training of recurrent neural networks with limited numerical precision , in Proceedings of the 2017 International Joint Conference on Neural Networks (IJCNN), IEEE, 2017, pp. 3716–3723. M. Nightingale and H. Bl¨ ote , Gap of the linear spin-1 Heisenberg antiferromagnet: AMonte Carlo calculation , Phys. Rev. D, 33 (1986), p. 659. M. Ortiz, A. Cristal, E. Ayguad´ e, and M. Casas , Low-precision floating-point schemes for neural network training , arXiv preprint arXiv:1804.05267, (2018). P. Price, C. Hamer, and D. O’Shaughnessy , Stochastic truncation for the (2+1)D Ising
18 model , J. Phys. A, 26 (1993), p. 2855. M. R. Santoro, G. Bewick, and M. A. Horowitz , Rounding algorithms for IEEE multipliers ,in Proceedings of 9th Symposium on Computer Arithmetic, IEEE, 1989, pp. 176–183. N. Wang, J. Choi, D. Brand, C.-Y. Chen, and K. Gopalakrishnan , Training deep neural networks with 8-bit floating point numbers , in Proceedings of the 31st Neural Information Processing Systems Conference, 2018, pp. 7675–7684. L. Xia, R. Willems, B. de Jager, and F. Willems , Constrained optimization of fuel efficiency for RCCI engines , IFAC-PapersOnLine, 52 (2019), pp. 648–653. 19 |
5775 | https://www.quora.com/Can-every-number-be-expressed-as-the-sum-of-two-integer-primes-not-counting-zero?top_ans=1618630 | Can every number be expressed as the sum of two integer primes (not counting zero)? - Quora
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Can every number be expressed as the sum of two integer primes (not counting zero)?
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Peter Hoffman
Author has 6.7K answers and 644.6K answer views
·2y
No, because every odd number would need to be two larger than an odd prime, right? Neither 3 nor 11 are. And surely your mathematical experience includes being able to count somewhere beyond 11, for someone who communikcates with Quora—though sometimes I wonder ….
I’m assuming your experience with numbers is limited to positive integers since you hardly get a non-integer number equal to the sum of two integers, prime or not.
Do more thinking before your next qujestion, and then try to express it as though you had some experience with more than the rhyme ‘1,2,3,infinity’.
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What is the smallest prime number after which every even number can be expressed as the sum of two primes?
Can every even integer greater than 2 be written as the sum of two primes?
Can every number be written as a sum of two primes or three primes?
Can every number be expressed as the sum of prime numbers?
Does the Riemann hypothesis imply that every prime number is the sum of two primes?
Nishant Kumar
Numbers are all around..learn to play with them and you'll never need toys
·9y
Originally Answered: Can any number be expressed as a sum of 2 prime numbers? ·
No you can't.
When you sum two prime numbers, you'll get an even number because generally the primes are odd numbers (except if one of the numbers is 2). So there is no way of getting odd numbers.
For example, let's say 35.
If you use 2 as one of the prime numbers, the next number would be 33 which is not prime. Thus 35 cannot be represented as sum of two primes.
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99 11
9 2
Michael Mark Ross
Autodidactic number empiricist · Author has 2.6K answers and 11.2M answer views
·7y
Originally Answered: Can any number be expressed as a sum of 2 prime numbers? ·
The sum of two primes divided by 2. That is the average of two primes.
Every integer greater than 3 can be expressed as the average of two primes.
For example: For 13 13+17+17 to equal 30 30 it follows that 13+17 2 13+17 2 must equal 15 15..
Goldbach’s Conjecture cannot be true without this being true.
This applies to even and odd numbers, including primes.
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9 9
9 1
Peter Groot
B.S. in Mathematics, Massachusetts Institute of Technology (Graduated 1971) · Author has 9.3K answers and 3.5M answer views
·3y
Originally Answered: Can all integers be written as the sum of two prime numbers? ·
No.
Negative integers are not the sum of two primes. But you could change the question to ask about positive integers.
0 and 1 are integers that are not prime and not the sum of two primes. But you could change the question to ask about integers greater than 1.
2 and 3 are prime but not the sum of two primes. But you could ask about integers greater than 3.
11 is prime but not the sum of two primes; the only even prime is 2, but 9 is not prime.
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More answers below
Can every positive integer be expressed as the sum of two odd prime numbers?
Is any prime as a sum of "some powers of 2' and '+/- a single integer"' (number theory, prime numbers, prime gaps, math)?
Which conjecture holds that every odd integer n greater than 2 can be expressed as a sum of two primes?
Can we write every even number as a sum of two prime numbers, and why?
What is the highest possible sum of two prime numbers?
Kenneth Ganning
Studied Statistics (academic discipline) · Author has 1.1K answers and 613K answer views
·3y
Originally Answered: Can all integers be written as the sum of two prime numbers? ·
Original Question: Can all integers be written as the sum of two prime numbers?
No, 27 can not be written as the sum of two primes as any two integers that add to 27 must contain an even and odd number. 2 is the only even prime but 25 is not prime.
Upvote ·
Leo Blackman
Author has 1.2K answers and 153.9K answer views
·2y
No.
Negative numbers and non-integer numbers can’t be expressed as the sum of two integer primes. Neither can 1, 2, 3 or 9
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Bill Ze'ev Felsen
Won a math award in HS, now I play with and learn it for fun · Author has 464 answers and 346.4K answer views
·10mo
Originally Answered: Is it possible to express all positive integers as the sum of two or more prime numbers? If so, what is the method for doing so? ·
Both 2 and 3 are prime. For any even number n, I can add 2+2+2+… to obtain n. For any odd number m, I can use this procedure to obtain m-3 which will be even, and then add +3 at the end.
The problem becomes much harder if I am not allowed to use as many addends as I want.
Upvote ·
Andrew Zisos
Works at Medical Practitioner · Author has 5.4K answers and 3.1M answer views
·10mo
Originally Answered: Is it possible to express all positive integers as the sum of two or more prime numbers? If so, what is the method for doing so? ·
Lets try.
9=2+2+5
10= 5+5.
11= 3+5+3 so far so good
12= 5+7
13= 7+3+3 still all ok
14=7+7
15=7+ 3+2+3 still ok.
16=7+3+3+3
17+7+5+5
18=7+5+3+3. Still ok
19=7+7+5 still ok.
im not going anyfurther
the only not possible is 1.
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Eric Hawk
Author has 3K answers and 3.9M answer views
·Updated 2y
No. Counterexamples include 1, 2, 3, 11. 1 is not considered a prime number.
Thanks to Doug’s comment in his posted answer:
Goldbach's conjecture - Wikipedia Even integers as sums of two primes Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics . It states that every even natural number greater than 2 is the sum of two prime numbers . The conjecture has been shown to hold for all integers less than 4 × 10 18 , but remains unproven despite considerable effort. On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), [ 2 ] in which he proposed the following conjecture: Every integer that can be written as the sum of two primes can also be written as the sum of as many primes (including unity) as one wishes, until all terms are units. [ a ] Goldbach was following the now-abandoned convention of considering 1 to be a prime number , [ 3 ] so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: [ b ] It seems at least, that every integer greater than 2 can be written as the sum of three primes. [ c ] Euler replied in a letter dated 30 June 1742 [ 4 ] and reminded Goldbach of an earlier conversation they had had ( "...so Ew vormals mit mir communicirt haben..." ), in which Goldbach had remarked that the first of those two conjectures would follow from the statement Every positive even integer can be written as the sum of two primes. This is in fact equivalent to his second, marginal conjecture. In the letter dated 30 June 1742, Euler stated: [ 5 ] [ 6 ] That... every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it. [ d ] Similar conjecture by Descartes [ edit ] René Descartes wrote, "Every even number can be expressed as the sum of at most three primes." [ 7 ] This proposition is similar to, but weaker than, Goldbach's conjecture. Paul Erdős said, "Descartes actually discovered this before Goldbach ... but it is better that the conjecture was named for Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor." [ 8 ] Goldbach's conjecture involving the sum of two primes is much more difficult than the weak Goldbach conjecture , which says that every odd integer greater than 5 is the sum of three primes. Using Vinogradov's method , Nikolai Chudakov , [ 9 ] Johannes van der Corput , [ 10 ] and Theodor Estermann [ 11 ] showed (1937–1938) that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to some N which can be so written tends towards 1 as N increases). In 1930, Lev Schnirelmann proved that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant; see Schnirelmann density . [ 12 ] [ 13 ] Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800 000 . This result was subsequently enhanced by many authors, such as Oli
A modern version of the marginal conjecture is:
Every integer greater than 5 can be written as the sum of three primes.
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9 2
Shyam Panchal
9y
Originally Answered: Can any number be expressed as a sum of 2 prime numbers? ·
Twin primes can be expressed as a sum of 2 prime numbers , otherwise no
Upvote ·
9 2
Atiqur Rahman
BHU Class of 2011.
·9y
Originally Answered: Can any number be expressed as a sum of 2 prime numbers? ·
Straight yes, but there is no proof for it(As far as I know) , this holds for known prime numbers.
So convention is yes every no is made from two prime no.
Upvote ·
9 1
Dana F Anderson
Just interested in Math, particularly PRIME NUMBERS · Author has 10.5K answers and 8.7M answer views
·2y
First, every prime is an integer. 0 is NOT a prime. Every number? NO. No negative numbers. Not 0. Not 1. Not 2, or 3. 9 can’t. 11, 17, 23, 27, 29, and 35 can’t, That’s as far as I am going to check, but obviously the answer to your question is NO.
Upvote ·
James Kenney
Studied at University of Chicago · Author has 1.1K answers and 84.5K answer views
·10mo
Originally Answered: Is it possible to express all positive integers as the sum of two or more prime numbers? If so, what is the method for doing so? ·
It’s very likely all even numbers are the sum of two primes. It just hasn’t been proved (Goldbach’s conjecture l)
It’s possible that all numbers can be expressed as the sum of either 2 primes (if even) or 3 primes (if odd) but I don’t know of a proof
Upvote ·
Related questions
What is the smallest prime number after which every even number can be expressed as the sum of two primes?
Can every even integer greater than 2 be written as the sum of two primes?
Can every number be written as a sum of two primes or three primes?
Can every number be expressed as the sum of prime numbers?
Does the Riemann hypothesis imply that every prime number is the sum of two primes?
Can every positive integer be expressed as the sum of two odd prime numbers?
Is any prime as a sum of "some powers of 2' and '+/- a single integer"' (number theory, prime numbers, prime gaps, math)?
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What is the smallest prime number after which every even number can be expressed as the sum of two primes?
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5776 | https://tryalgo.org/en/arithmetics/2017/06/29/le-compte-est-bon/ | Skip to main content
Forming arithmetic expression meeting target value
• arithmetics • Christoph Dürr and Jean-Christophe Filliatre
Given n integers and a target value form an arithmetic expression evaluating to the target value.
Summary
You are given the integers 3, 100, 8, 8, 10, 6 and need to approach the target value 683. The allowed operations are addition, multiplication, subtraction (only if the result is positive) and division (only if the result is integer). And every given integer can appear at most once in the solution. A solution in this example would be
```
6 100 + 8 10 + 3 = 683.
```
Complexity
One can use dynamic programming to solve the problem, even though the resulting complexity would be huge.
We could upper bound it by the number of arithmetic expressions one can form with the given numbers, regardless of the restrictions on the subtraction and division. Then even restricting to expressions involving all n given numbers there are quite many expressions. An expression can be viewed as a binary tree with n leafs, and hence n-1 inner nodes. The number of those trees is the Catalan number Cn−1, which is (2n−2)!/(n!(n−1)!). This number needs to be multiplied by 4n−1 corresponding to the number of possibilities to assign one of the four operators to each of the inner nodes.
Due to the constraints on subtractions and divisions, the actual number of valid expressions might be smaller than this rough estimation, but nevertheless it is huge. Hence the algorithm is practical only for small n, say less than 10.
Dynamic Program
Let x0,x1,…,xn−1 be the given integers. We denote by S⊆{0,1,…,n−1} a selection of these integers. In a table expr we store for every set S in expr[S] a dictionary. It contains (key, value) pairs of the form (v,f) where f is a valid arithmetic expression formed by the integers {xi:i∈S} and v is the value of f.
Initially expr[{i}] is a dictionary associating to the value xi the singleton expression xi. Then for every non-singleton set S we populate the dictionary expr[S] as follows. Any arithmetic expression involving the integers {xi:i∈S} consists of an arithmetic expression tree with some operator op in the root and a left subtree over a set L and a right subtree over a set R where L and R are non-empty sets partitioning S. Therefore for fixed set S we loop over all non-empty strict subsets L⊂S. For each L we set R=S∖L and loop over all values vL in expr[L] and all values vR in expr[R]. Now for each operator op among +,,-,/ for which v:=vLopvR is a valid expression we can associate in expr[S] to the key v the expression expr[L][vL] op expr[R][vR].
These operations need to be done for all sets S in a lexicographical order that ensures that all subsets of S have already been processed.
Implementation details
Since n is a very small number it is most convenient for us to represents sets as integers. For example the set S=0,3,4 is represented as the integer 20+23+24. This has the advantage that bit manipulation operators can be used to encode set operators. For example the bitwise and denoted & corresponds to the intersection and the test L & S == L tests whether the set encoded by L is a subset of the set encoded by S.
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5777 | https://www.savemyexams.com/a-level/chemistry/cie/25/revision-notes/1-atomic-structure/1-3-electrons-energy-levels-and-atomic-orbitals/determining-electronic-configuration/ | A LevelChemistryCambridge (CIE)Revision Notes1. Atomic StructureElectrons, Energy Levels & Atomic OrbitalsDetermining Electronic Configuration
Determining Electronic Configuration (Cambridge (CIE) A Level Chemistry): Revision Note
Exam code: 9701
Author
Richard Boole
Last updated
Determining Electronic Configurations
Electron configuration shows how electrons are arranged in shells, subshells, and orbitals.
There are two formats:
Full configuration; lists all electrons from 1s onward
Shorthand configuration; uses the symbol of the nearest noble gas in brackets to represent inner electrons (e.g. [Ar])
Ions form when atoms gain or lose electrons:
Anions (negative) form by adding electrons to the outer shell
Cations (positive) form by removing electrons from the outer shell
Transition metals:
Fill the 4s before 3d when neutral
Lose electrons from 4s first, not 3d, when forming ions
In the Periodic Table the elements are grouped into blocks based on their valence subshell:
s-block: valence electrons in an s orbital
p-block: valence electrons in a p orbital
d-block: valence electrons in a d orbital
f-block: valence electrons in an f orbital
The blocks of the Periodic Table
Examples
Electronic configuration of Fe
Atomic number = 26 so there are 26 electrons
Full configuration: 1s2 2s2 2p6 3s2 3p6 4s2 3d6
Shorthand: [Ar] 4s2 3d6
Electronic configuration of Fe2+
Atomic number = 26 so there are 26 electrons, but the Fe2+ ion only has 24 electrons
Electrons are removed from the 4s orbital before the 3d
Full configuration: 1s2 2s2 2p6 3s2 3p6 3d6
Shorthand: [Ar] 3d6
Exceptions to the Aufbau principle
Chromium and copper have the following electron configurations:
Cr is [Ar] 3d5 4s1 not [Ar] 3d4 4s2
Cu is [Ar] 3d10 4s1 not [Ar] 3d9 4s2
This is because the [Ar] 3d5 4s1 and [Ar] 3d10 4s1 configurations are energetically favourable
By promoting an electron from 4s to 3d, these atoms achieve a half full or full d-subshell, respectively
Worked Example
Write down the full and shorthand electron configuration of the following elements:
Potassium
Calcium
Gallium
Ca2+
Answer:
Potassium
Electrons: 19
Full configuration: 1s2 2s2 2p6 3s2 3p6 4s¹
Shorthand: [Ar] 4s¹
Note: Argon (Ar) has 18 electrons and is the previous noble gas
Calcium
Electrons: 20
Full configuration: 1s2 2s2 2p63s2 3p6 4s2
Shorthand: [Ar] 4s²
Note: 4s is filled before 3d because it is lower in energy
Gallium
Electrons: 31
Full configuration: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p1
Shorthand: [Ar] 3d10 4s2 4p1
Note: Includes filled 3d subshell after argon
Calcium 2+ ion
Electrons: 18 (after losing two 4s electrons)
Full configuration: 1s2 2s2 2p6 3s2 3p6
Shorthand: [Ar]
Note: Ca²⁺ has the same configuration as argon
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Richard has taught Chemistry for over 15 years as well as working as a science tutor, examiner, content creator and author. He wasn’t the greatest at exams and only discovered how to revise in his final year at university. That knowledge made him want to help students learn how to revise, challenge them to think about what they actually know and hopefully succeed; so here he is, happily, at SME. |
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Prove x<x 2+1−−−−−√x<x 2+1.
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Asked 5 years, 3 months ago
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Prove x<x 2+1−−−−−√x<x 2+1.
I am pretty sure this an easy question as the inequality seems obviously true, but I am not entirely convinced by my argument.
So I squared both sides (is this allowed?):
x 2<x 2+1 x 2<x 2+1, so 0<1 0<1 so the inequality is obviously true.
However, I am unconvinced that this process is reversible due to the squaring, so could someone just explain whether this is correct?
inequality
solution-verification
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asked Jun 20, 2020 at 8:11
JamminermitJamminermit
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1 Consider the cases x<0 x<0 and x≥0 x≥0.Kavi Rama Murthy –Kavi Rama Murthy 2020-06-20 08:14:09 +00:00 Commented Jun 20, 2020 at 8:14
You can start with x 2<x 2+1 x 2<x 2+1, and square root both sides. Make sure to remember that x 2−−√=|x|x 2=|x| and |x|≥x|x|≥x.Minus One-Twelfth –Minus One-Twelfth 2020-06-20 08:14:30 +00:00 Commented Jun 20, 2020 at 8:14
If x<0 x<0 your inequality is obvious, since the right hand side is ≥0≥0. If x≥0 x≥0 your squaring is correct, the squared inequality is equivalent to the original one and your proof works.GReyes –GReyes 2020-06-20 08:15:14 +00:00 Commented Jun 20, 2020 at 8:15
Consider the 3 cases x<0 x<0,x=0 x=0,x>0 x>0 separately.Peter –Peter 2020-06-20 08:15:41 +00:00 Commented Jun 20, 2020 at 8:15
In fact the reverse implies that |x|<x 2+1−−−−−√|x|<x 2+1 from here it is easy to continue just consider two cases.Taha Akbari –Taha Akbari 2020-06-20 08:16:24 +00:00 Commented Jun 20, 2020 at 8:16
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You can not use squaring of the both sides directly because x x can be negative.
If so, you need to consider two cases: 1)x≥0 x≥0 and 2)x<0 x<0.
(in the last the inequality is obvious.)
I think it's better to use a way without squaring:
x 2+1−−−−−√>x 2−−√=|x|≥x.x 2+1>x 2=|x|≥x.
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answered Jun 20, 2020 at 8:38
Michael RozenbergMichael Rozenberg
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By definition x 2+1−−−−−√≥0 x 2+1≥0. If x<0 x<0, then x<0≤x 2+1−−−−−√⟹x<x 2+1−−−−−√x<0≤x 2+1⟹x<x 2+1. If x≥0 x≥0 we have:
(x 2+1−−−−−√+x)≥0(x 2+1−−−−−√−x)=1>0⟹x0⟹x<x 2+1.
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answered Jun 20, 2020 at 8:50
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5779 | https://www.youtube.com/watch?v=D6YrzETRpD8 | Maclaurin Series for f(x) = e^(-x)
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Maclaurin Series for f(x) = e^(-x)
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find the McLaren series for f ofx = e tox solution so to do this problem we're going to use the McLaren series for e to the X McLaren series for e to the x is the infinite sum as n runs from 0 to Infinity of x to the N Over N factorial and this equation holds for all X right this converges absolutely for every number X in particular it converges when you plug INX so e to thex this is equal to the infinite sum as n runs from 0 to Infinity of the quantity - x to the N / n factorial now we can do some algebra here to clean things upx to the N you can think of this as -1 X x to the n and then using properties of exponents this is -1 n X the N so we can go back to our infinite sum and rewrite this as the sum as n runs from 0 to Infinity of -1 the n/ N factorial time x the n and I wrote it this way so that you see it's a power series centered at zero right this is X x - 0 to the N so this is indeed the McLaren series for E to thex so there's no need to um take derivatives or or do any of that mess this is absolutely the easiest way to do it hope that made sense |
5780 | https://www.gauthmath.com/solution/1839305775042577/Solve-by-factoring-m23m-2-m-There-is-more-than-one-correct-answer-You-must-selec | Solved: Solve by factoring: m^2(3m-2)=m There is more than one correct answer. You must select ALL [Math]
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Solve by factoring: m^2(3m-2)=m There is more than one correct answer. You must select ALL the correct answers to the problem below in order earn full credit. Leave any incorrect answers unchecked. 2/3 - 1/3 1 0 - 3/2 -1
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Expert Verified Solution
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Answer
The correct answers are:
$$-\frac{1}{3}$$−3 1
1
0
Explanation
Rewrite the equation
First, rewrite the given equation $$m^{2}(3m-2)=m$$m 2(3 m−2)=m by subtracting $$m$$m from both sides:
$$m^{2}(3m-2) - m = 0$$m 2(3 m−2)−m=0
Factor out the common factor
Factor out $$m$$m from the left side of the equation:
$$m[m(3m-2) - 1] = 0$$m[m(3 m−2)−1]=0
$$m(3m^{2} - 2m - 1) = 0$$m(3 m 2−2 m−1)=0
Factor the quadratic expression
Factor the quadratic expression $$3m^{2} - 2m - 1$$3 m 2−2 m−1:
We are looking for two numbers that multiply to $$3 \times -1 = -3$$3×−1=−3 and add to $$-2$$−2. These numbers are $$-3$$−3 and $$1$$1
Rewrite the middle term using these numbers:
$$3m^{2} - 3m + m - 1 = 0$$3 m 2−3 m+m−1=0
Factor by grouping:
$$3m(m - 1) + 1(m - 1) = 0$$3 m(m−1)+1(m−1)=0
$$(3m + 1)(m - 1) = 0$$(3 m+1)(m−1)=0
Write the fully factored equation
Substitute the factored quadratic back into the equation:
$$m(3m + 1)(m - 1) = 0$$m(3 m+1)(m−1)=0
Solve for $$m$$m
Set each factor equal to zero and solve for $$m$$m:
1. $$m = 0$$m=0
2. $$3m + 1 = 0 \Rightarrow 3m = -1 \Rightarrow m = -\frac{1}{3}$$3 m+1=0⇒3 m=−1⇒m=−3 1
3. $$m - 1 = 0 \Rightarrow m = 1$$m−1=0⇒m=1
List all solutions
The solutions are $$m = 0$$m=0, $$m = -\frac{1}{3}$$m=−3 1, and $$m = 1$$m=1
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How to calculate SKEW.P in Excel for a grouped data?
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Here is my original table
It is obviously skewed to the right (SKEW must be negative) . However when i apply SKEW.P function in excel it gives a positive number 1.4
I think the reason is because the data is grouped. So to get the real SKEW.P I will need to ungroup data, and have half a million elements that will look like this: 0,0,0,0,1,1,1,1,2,2,2..........
Is there any easier way to estimate SKEW.P?
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asked Nov 12, 2020 at 21:25
johnjohn
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If one has Office 365 then we can use:
=SKEW.P(INDEX(Table1[Group],MATCH(SEQUENCE(SUM(Table1[Number of Elements in Group]),,0),SUMIF(OFFSET(Table1,0,0,SEQUENCE(COUNTA(Table1[Number of Elements in Group]))),"<>"))))
If not then we use:
=SKEW.P(INDEX(Table1[Group],N(IF({1},MATCH(ROW($ZZ1:INDEX($ZZ:$ZZ,SUM(Table1[Number of Elements in Group])))-1,SUMIF(OFFSET(Table1,0,0,ROW(Table1[Number of Elements in Group])-MIN(ROW(Table1[Number of Elements in Group]))+1),"<>"))))))
And confirm as an array formula using Ctrl-Shift-Enter instead of Enter.
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answered Nov 12, 2020 at 21:50
Scott CranerScott Craner
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john
johnOver a year ago
Thank you so much . But i cant get it to work. Could you please replace Table1 with an Array. It says there is a problem with the formula.
2020-11-12T22:44:25.83Z+00:00
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Scott Craner
Scott CranerOver a year ago
You have a table in your example so I am not sure what you mean.
2020-11-12T22:45:12.507Z+00:00
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5782 | https://physics.stackexchange.com/questions/843876/how-can-i-show-that-in-a-rigid-body-system-distance-between-the-center-of-mass | newtonian mechanics - How can I show that in a rigid body system, distance between the center of mass and any particle (of the rigid body system) does not change? - Physics Stack Exchange
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How can I show that in a rigid body system, distance between the center of mass and any particle (of the rigid body system) does not change?
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It seems pretty obvious as distance between any two particles (which are part of the same rigid body system) does not changes or is constant [Rigid body constraint] and as position vector of "Center of mass" is weighted average of position vectors all these particles, then the distance of Center of mass from any of these particles should also be a constant. But as this fact is used to derive some really important equations in rigid body dynamics, I thought of deriving/extracting it from the definition of Center of mass and rigid body constraint on paper (I also tried to find the derivation or anything similar online but I cannot find one).
I have attached an image for what I have tried to prove so far (as I'm not familiar with latex) and got stuck at end. If anybody knows a way to prove this, it will be really kind of you to share it.
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edited Feb 26 at 17:43
Anmol SinghAnmol Singh
asked Feb 25 at 18:00
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Try to have a look here: basics2022.github.io/bbooks-physics-mechanics/ch/…. Center of mass moves as a material point for rigid bodies, so rigid-body law of motion holds basics –basics 2025-02-25 18:04:10 +00:00 Commented Feb 25 at 18:04
I'm asking how to prove that center of mass acts as a material point for rigid bodies Anmol Singh –Anmol Singh 2025-02-25 18:17:31 +00:00 Commented Feb 25 at 18:17
@basics The center of mass need not be physically in the object.Vincent Thacker –Vincent Thacker 2025-02-25 18:20:57 +00:00 Commented Feb 25 at 18:20
moves as, not is basics –basics 2025-02-25 18:31:21 +00:00 Commented Feb 25 at 18:31
Why is my question tagged "Homework like", I dont think so it is anywhere close to it lol Anmol Singh –Anmol Singh 2025-02-25 19:04:35 +00:00 Commented Feb 25 at 19:04
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Ok, let's explicitly prove it.
For a rigid body, with a discrete distribution of mass m j m j in material points →r j r⃗j the rigid body law of motion
→v i−→v j=→ω×(→r i−→r j).
v⃗i−v⃗j=ω⃗×(r⃗i−r⃗j).
Center of mass is defined as
→r G=1 M∑k m k→r k,
r⃗G=1 M∑k m k r⃗k,
being M=∑k m k M=∑k m k the total mass of the system. Now, let's prove that for G G the law of motion holds for any other point of the rigid body, i.e.
→v G−→v j=→ω×(→r G−→r j).
v⃗G−v⃗j=ω⃗×(r⃗G−r⃗j).
Evaluating the velocity of the center of mass with time derivative of the definition and inserting in the law of motion of a rigid body,
→v G−→v j=d→r G d t−→v i==∑k m k M→v k−→v i==∑k m k M(→v k−→v i)==∑k m k M→ω×(→r k−→r i)==→ω×∑k m k M(→r k−→r i)==→ω×(1 M∑k m k→r k−→r i)==→ω×(→r G−→r i),
v⃗G−v⃗j=d r⃗G d t−v⃗i==∑k m k M v⃗k−v⃗i==∑k m k M(v⃗k−v⃗i)==∑k m k M ω⃗×(r⃗k−r⃗i)==ω⃗×∑k m k M(r⃗k−r⃗i)==ω⃗×(1 M∑k m k r⃗k−r⃗i)==ω⃗×(r⃗G−r⃗i),
where it's possible to play around with summations since ∑k m k M=1∑k m k M=1 by definition of total mass. Thus, we managed to prove that for the center of mass G G the rigid body law of motion holds, w.r.t. any material point of the system (and obviously with the same angular velocity of the system).
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answered Feb 25 at 18:40
basicsbasics
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@AnmolSingh I want to ask if we can come to this conclusion from my approach too? Please note that the closure banner explains that check-my-work questions are off-topic.Ghoster –Ghoster 2025-02-26 07:59:59 +00:00 Commented Feb 26 at 7:59
@Ghoster Ok, thanks for stating that, I will delete that right away Anmol Singh –Anmol Singh 2025-02-26 14:07:54 +00:00 Commented Feb 26 at 14:07
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Jan
8
2025
HbSC Disease
By William Aird
I posted a poll on Twitter showing a single CBC and asking for the most likely diagnosis.
The following are the results of the poll:
Let’s put aside the 4 possible answers/options in the poll and consider the broader differential diagnosis of microcytosis:
Now let’s consider the options provided in the poll:
Sickle cell anemia (SCA):
Term used to describe HbSS and HbSβ0-thalassemia.
The presentation of pain is consistent with SCA.
However, the normal Hb would be highly unusual unless the patient had been inappropriately transfused.
SCA not associated with microcytosis.
Lead poisoning:
Symptoms of lead poisoning include headaches, cramps and hyperactivity.
Almost always occurs in children, not in adults like this patient.
May rarely cause microcytic anemia.
Iron deficiency:
The pain symptoms are not consistent with iron deficiency.
The non-anemic microcytosis would be unusual, though some patients with iron deficiency manifest with an initial drop in their MCV before the Hb falls.
The MCHC is often on the low side, and rarely as high as this patient’s 35.4 g/dL.
HbSC disease:
Now we’re talking!
Introduction
Sickle cell disease (SCD) includes a group of heterogenous disorders that share the presence of the gene for hemoglobin S (HbS):1
Homozygous condition, HbSS, is the most common type accounting for ~65% of cases in the United States.
Compound heterozygous condition in which HbS is inherited with another abnormal hemoglobin:
HbSC disease (the second most common type of SCD)
Beta thalassemia:
Hemoglobin Sβ0-thalassemia (HbSβ0-thalassemia)
Hemoglobin Sβ+-thalassemia (HbSβ+-thalassemia)
HbSS and HbSβ0-thalassemia are clinically very similar and therefore are commonly referred to as sickle cell anemia (SCA). While HbSC is a type of sickle cell disease, it is not considered a form of sickle cell anemia.
Although considered a milder sickle cell disease (SCD) variant, HbSC is a sickling syndrome associated with potentially severe morbidities that warrant surveillance and intervention.2
Epidemiology
Globally:
The Global Burden of Disease study estimates that there are over 100,000 babies born with HbSC annually and more than 1 million affected individuals worldwide, with the majority living in low-resource settings within sub-Saharan Africa.3
~55000 newborns with HbSC disease are delivered yearly, with the highest hemoglobin C (HbC) gene frequency in West Africa.4
In the United States:
The prevalence of HbSC disease is about 1 in 7174 births.5
1:941 babies are born with SCD, while 1:6173 newborns has HbSC.6
Through newborn hemoglobinopathy screening, about 600 babies with HbSC are identified in the United States each year.7
Over 30,000 children and adults have hemoglobin SC (HbSC) disease, representing 30% of the total population with sickle cell disease.8
One in 800 African Americans have HbSC disease.
In the United Kingdom:
The prevalence of HbSC disease is about 1 in 6174 births.9
HbSC accounts for nearly 30% of SCD.10
1:2000 babies are born with SCD and 1:7174 has HbSC.11
In western Nigeria and northern Ghana:
About 25% of the population of carry the HbC gene.12
Cause
HbSC is a double (compound) heterozygous state caused by co-inheritance of:1314
Sickle hemoglobin (HbS; p.glu7val)
C hemoglobin (HbC; p.glu7lys); lysine is substituted for glutamic acid (Glu6Lys) in HbC.
Hemoglobin C (HbC) is an inherited single base mutation in codon 6 of the beta-globin gene, leading to substitution of lysine for glutamic acid. HbS is an inherited single base mutation in codon 6 of the beta-globin gene, leading to substitution of valine for glutamic acid. In HbSC one beta-globin allele is HbC, the other HbS.
Co-inheritance of alpha-thalassemia improves red cell survival and decreases hemolysis in sickle cell disease, including HbSC.
Evolutionary Considerations
HbSC occurs most commonly in West Africa due to the ancestral and unicentric origin of the HbC mutation that provides protection against malaria.15
The allelic frequency of HbC is a consequence of the survival advantage against severe malaria conferred through inheritance of one HBB (beta globin gene) mutation.16
Pathophysiology
Patients with HbSC disease have about 50% HbS and 50% HbC (with low [1–3%] HbF levels).17
Although HbAC and HbAS are asymptomatic, their combination in HbSC is symptomatic because HbC-mediated dehydration of erythrocytes leads to polymerization of HbS and red cell sickling.1819
HbSC disease pathogenesis is modulated by:20
Interactions between HbS and HbC
RBC dehydration from altered membrane transporter function, which results in increased HbS concentration, causing HbS subunit polymerization.
Erythrocyte lifespan is twice that of HbSS (29 vs.15 days).21
Clinical Presentation
HbSC is a sickling syndrome with an overall milder clinical phenotype than sickle cell anemia.22
Clinical manifestations are highly varied between patients.
Laboratory findings:
Hemoglobin:
HbSC is associated with a higher mean Hb and lower absolute reticulocyte count (ARC) than people with HbSS disease.23
Over 70% of people with HbSC have anemia, but only 10% have Hb lower than 100 g/l.24
Low mean cell volume (MCV).25
Leukocytosis is less pronounced or absent in HbSC compared to HbSS.26
Platelet counts may be normal, but mild to moderate thrombocytopenia occurs in patients with splenomegaly and hypersplenism.27
Peripheral smear may show:28
Frequent target cells from the relatively increased surface area secondary to RBC dehydration.
Moderate microcytosis.
Occasional microspherocytes.
Rare distorted tri-concave or elongated erythrocytes containing Hb crystals.
Rare irreversibly sickled cells.
Blood viscosity is increased even compared to HbSS.29
Erythrocyte lifespan is twice that of HbSS (29 vs.15 days).
Acute events and chronic complications:
Among individuals with HbSC, rates of maternal-fetal morbidity, retinopathy, avascular necrosis (AVN) of the hip, priapism, and chronic kidney disease are increased compared with historical controls.30
Comparing HbSC with SCA:31:
Acute vasoocclusive events such as pain and acute chest syndrome occur about half as often.
Proliferative retinopathy, blindness, sensorineural hearing loss and thrombosis are more frequent.
Pain crisis:32
Hallmark of HbSC disease.
At least 50% of people with HbSC report a painful episode requiring a hospital visit and about 5% of HbSC patients may have frequent debilitating painful events.
Treatment strategies for painful crises in HbSC include hydration, analgesia and adequate oxygenation.
Proliferative sickle retinopathy:33
The most common complication of HbSC disease.
Occurs in 30–70% of patients with HbSC compared to 3% in HbSS.
Caused by vaso-occlusion with resultant ischemia and excessive retinal blood vessel growth.
Incidence peaks in the third and fourth decades.
Causes vision loss by vitreous hemorrhage and tractional retinal detachment.
Treatment options include: diathermy, cryotherapy and transpupillary or transscleral diode laser photocoagulation.
Annual ophthalmological examination recommended.
Splenic complications:34
Functional asplenia is reported in 45% of HbSC patients by age 12 years.
Chronic splenomegaly is associated with thrombocytopenia in 35% of children and 50% of adults with HbSC and may cause recurrent abdominal pain.
Acute splenic sequestration crisis (ASSC), precipitated by obstruction of splenic outflow with sickled erythrocytes, causes massive pooling of blood within the splenic sinusoids. ASSC occurs in 6–12% of children with HbSC.
Osteonecrosis:35
Aka avascular necrosis (AVN).
Occurs in 12–24% of HbSC patients.
Typically affects large joints, such as hips and shoulders, but can occur in other joints, including the spine.[
AVN usually presents as focal pain.
Reducing pain and disability are the primary management goals of AVN.
Central nervous system complications:36
Headache
Ischemic stroke
Hemorrhagic stroke
Silent cerebral infarcts
Priapism:37
An unwanted erection lasting 4 h or more.
Occurs in 20% of men with HbSC.
Is a clinical emergency.
Infection:
People with HbSC and HbSβ+-thalassemia have a much lower incidence of life-threatening infection because their spleen function is normal or only minimally impaired during infancy.
Cohort studies:
Nelson et al, 2024
A total of 2282 SCDIC registry participants with HbSC or SCA (HbSS and HbSß0).
Although the HbSC participants had a lower prevalence of most SCD-related complications, they had a higher frequency of:
Splenomegaly (n (%) = 169 (33.7) vs. 392 (22.1) for SCA)
Retinopathy (n (%) = 116 (23.1) vs. 189 (10.6) for SCA
The mean (standard deviation) hemoglobin level was 11.5 (1.5) g/dL among those with Hb SC versus the mean hemoglobin in the SCA cohort of 8.9 (1.5) g/dL.
HbSC disease associated with lower markers of hemolysis with mean LDH of 314.5 (294.5) compared with 497.8 (352.5) in SCA genotypes and a total bilirubin of 1.6 (1.6) mg/dL compared with 3.3 (2.9) mg/dL.
Among HbSC participants, there was no association be-tween hemoglobin and retinopathy or avascular necrosis.
No association within the HbSC cohort between hemolysis biomarkers and the number of clinical complications.
Ghunney et al, 2023
| Characteristics | HbSC (N = 639) | HbSS (N = 639) | P value |
--- --- |
| Age, mean | 30.7 (7.8) | 31.0 (7.8) | 0.450 |
| Hb mean (SD) | 11.1 (1.3) | 8.1 (1.2) | < 0.001 |
| Acute pain events/y, n (%): | | | |
| 0 events | 344 (53.8) | 289 (45.2) | < 0.001 |
| 1-2 events | 238 (37.2) | 212 (33.2) | |
| ≥3 episodes | 57 (8.9) | 138 (21.6) | |
| Pain incidence (events per 100 patient-y) | 74.6 | 123.0 | < 0.001 |
| ACS incidence (events per 100 patient-y) | 2.3 | 5.6 | 0.004 |
| AVN, n (%) | 49 (7.7) | 61 (9.5) | 0.231 |
| Cerebrovascular accident/ stroke, n (%) | 3 (0.5) | 7 (1.1) | 0.204 |
| Priapism, n (%) | 11 (4.4) | 26 (10.3) | 0.011 |
| Nephropathy, n (%) | 7 (1.1) | 54 (8.5) | < 0.001 |
| Proliferative SCD retinopathy, n (%) | 20 (3.1) | 4 (0.6) | < 0.001 |
| Chronic pain, n (%) | 110 (17.2) | 156 (24.4) | 0.002 |
Bold, statistically significant; ACS, acute chest syndrome; AVN, avascular necrosis
Diagnosis
Typically detected in asymptomatic newborns during routine screening.
In absence of screening, HbSC is typically diagnosed in children, adolescents, or young adults.
HbSC disease diagnosed by identifying significant quantities of hemoglobin C (about 50% total Hb) and hemoglobin S (about 45% of total Hb) in patient’s blood
Can be diagnosed in laboratory by
isoelectric focusing (IEF)
hemoglobin electrophoresis
high performance liquid chromatography (HPLC)
DNA analysis
Treatment
Lagging treatment vs. SCA
Hydroxyurea:38
Not as effective as in SCA.
Small cohorts show decrease in episodes of acute chest syndrome and hospitalization for pain.
“Moreover, disease-modifying therapies such as hydroxyurea are used less in HbSC disease compared with SCA. This is due to the lack of participants with HbSC disease in prospective drug clinical trials, which results in a paucity of efficacy and safety data in this disease subtype.”39
Unlike HbSS, hydroxyurea therapy is not routinely recommended or used for individuals with HbSC.40
The PIVOT trial:41
Prospective phase 2, randomized, double-blind, placebo-controlled, non-inferiority trial in which children and adults with HbSC in Ghana were administered 12 months of hydroxyurea or placebo.
Designed to investigate the effects of hydroxyurea treatment on sickle-related clinical and laboratory parameters in a large cohort of children and adults with HbSC disease.
12-month blinded treatment phase.
After a 2-month screening period to collect pretreatment clinical and laboratory data, hydroxyurea 100-mg and 1000-mg scored tablets and matching placebo were started at 20 mg/kg of body weight as a single daily dose, with an opportunity for a 2.5 to 5.0 mg/kg dose escalation at month 2 and month 4, based on peripheral blood counts. The maximum dose of hydroxyurea was 35 mg/kg/day.
The primary end point was hematologic dose-limiting toxicities (DLTs), including cytopenias or elevated hemoglobin levels during 12 months of blinded treatment.
243 patients (123 children and 120 adults).
Almost all enrolled participants reported prior vaso-occlusive pain events similar to other descriptions of HbSC disease.
Compared with placebo, hydroxyurea treatment was associated with:
Increased:
Mean corpuscular volume
MCH
HbF
DLT, including:42
Thrombocytopenia
Neutropenia
Elevated hemoglobin
Decreased:
White blood cell
Reticulocyte count
Clinical AEs, including pain crises43
No change in:
Hemoglobin
MCHC
Conclusions: The PIVOT trial did not meet its primary end point. Hydroxyurea at 20 mg/kg in patients with HbSC was associated with more hematologic DLTs than placebo, but most were mild and transient. Hydroxyurea was associated with less vaso-occlusive pain and fewer sickle-related events in both children and adults; a new trial will need to be done to establish the efficacy of this approach.
Phlebotomy in those with high hemoglobin values and increased in blood viscosity.44
Prognosis
Median survival in resource-rich countries has risen to 80 years.45
Survival of patients with HbSC is superior to SCA.46
The PIVOT trial:47
Guideline recommendations
2014 National Heart, Lung, and Blood Institute (NHLBI)-sponsored expert panel evidence-based report on management of sickle cell disease
Health Maintenance for People With Sickle Cell Disease
| | SCA | HbSC |
---
| Oral penicillin prophylaxis | Administer oral penicillin prophylaxis (125 mg for age <3 years and 250 mg for age ≥3 years) twice daily until age 5 in all children with HbSS | Consider withholding penicillin prophylaxis from children with HbSC disease and HbSβ+-thalassemia unless they have had a splenectomy |
| Vaccination against Strep pneumoniae | Assure that people of all ages with SCD have been vaccinated against Streptococcus pneumoniae | Same as with SCA |
| Proteinuria screen | Screen all individuals with SCD, beginning by age 10, for proteinuria. | Same as with SCA |
| EKG | Routine ECG screening is not recommended in children and adults with SCD. | Same as with SCA |
| Ophthalmology screen | In people with SCD, refer to an ophthalmologist for a dilated eye examination to evaluate for retinopathy beginning at age 10; For people having a normal dilated retinal examination, re-screen at 1–2 year intervals. | Same as with SCA |
| Transcranial Doppler (TCD) imaging | In children with SCA, screen annually with TCD according to methods employed in the STOP studies, beginning at age 2 and continuing until at least age 16. | In children with genotypes other than SCA (e.g., HbSβ+-thalassemia or HbSC), do not perform screening with TCD. |
| Pulmonary function tests | Do not screen asymptomatic children and adults with pulmonary function tests. | Same as with SCA |
Managing Acute Complications of Sickle Cell Disease
| | SCA | HbSC |
---
| Vaso-Occlusive Crisis | | |
| Pain control | In adults and children with SCD and a VOC associated with severe pain, rapidly initiate treatment with parenteral opioids. | Same as with SCA |
| Anemia | Use simple transfusion in people with SCD and acute anemia whose symptoms are due to anemia. | Same as with SCA |
| Acute chest syndrome | Treat people with SCD who have ACS with an intravenous cephalosporin, an oral macrolide antibiotic, supplemental oxygen (to maintain oxygen saturation of greater than 95 percent), and close monitoring for bronchospasm, acute anemia, and hypoxemia. In people with SCA, give simple blood transfusion (10 mL/kg red blood cells) to improve oxygen carrying capacity to people with symptomatic ACS whose hemoglobin concentration is >1.0 g/dL below baseline. If baseline hemoglobin is 9 g/dL or higher, simple blood transfusion may not be required. n all persons with SCD, perform urgent exchange transfusion—with consultation from hematology, critical care, and/or apheresis specialists—when there is rapid progression of ACS | In people with HbSC disease or HbSβ+-thalassemia with ACS, decisions about transfusion should be made in consultation with an SCD expert. |
Hydroxyurea
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PMID: 38898714
PMID: 27982424
PMID: 39647172
PMID: 36799926
PMID: 38898714
PMID: 27982424
PMID: 39647172
PMID: 39647172
PMID: 38898714
PMID: 27982424
PMID: 27982424
PMID: 36073655
PMID: 36073655
PMID: 27982424
PMID: 39647172
PMID: 27982424
PMID: 36073655
PMID: 38898714
PMID: 38898714
PMID: 27982424
PMID: 27982424
PMID: 36073655
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 36799926
PMID: 36073655
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 27982424
PMID: 36073655
PMID: 38898714
PMID: 36799926
Prospective Identification of Variables as Outcomes for Treatment (PIVOT)
DLTs were typically mild, transient, and asymptomatic.
62% reduction in the prespecified vaso-occlusive event rates.
PMID: 36073655
PMID: 36073655
PMID: 27982424
Prospective Identification of Variables as Outcomes for Treatment (PIVOT) |
5784 | https://med.libretexts.org/Bookshelves/Basic_Science/Cell_Biology_Genetics_and_Biochemistry_for_Pre-Clinical_Students/05%3A_Fuel_for_Later/5.02%3A_Lipolysis_-oxidation_and_ketogenesis | Skip to main content
5.2: Lipolysis, β-oxidation, and ketogenesis
Last updated
: Nov 2, 2021
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5.1: Gluconeogenesis and glycogenolysis
5.3: Nitrogen metabolism and the urea cycle
Page ID
: 37847
Renee J. LeClair
Virginia Tech Carilion School of Medicine via Virginia Tech Libraries' Open Education Initiative
( \newcommand{\kernel}{\mathrm{null}\,})
The processes of lipolysis, β-oxidation, and ketogenesis work in concert within the cell but should be considered distinct pathways.
Lipolysis
Lipolysis is the release of fatty acids from adipose tissue where they are stored as triacylglycerols (TAGs). This process is mediated by increasing levels of glucagon and epinephrine, which bind G-protein coupled receptors on the adipose tissue and activate lipolysis This cell-signaling cascade phosphorylates and activates hormone-sensitive lipase, the regulatory enzyme for lipolysis. Once phosphorylated (through hormone-mediated increase in cAMP) this enzyme will hydrolyze TAGs to three long-chain fatty acids (LCFAs) and glycerol. The LCFAs are released into the bloodstream and will circulate bound to albumin (fatty acids are hydrophobic and require a protein carrier). LCFAs will be taken up and oxidized by peripheral tissues and the liver under fasted conditions. The glycerol will also be released and used as a substrate for hepatic gluconeogenesis (section 5.1) (figure 5.6).
β-oxidation (oxidation of free fatty acids)
Fatty acid oxidation is a high energy yielding process. It can support the cellular energy needs during fasting and under conditions when excess energy is needed (exercise). After uptake from circulation, the LCFAs must be transferred into the mitochondria where β-oxidation occurs. Initially, the LCFAs are activated to acyl-CoA derivatives in the cytosol by acyl-CoA synthetase. The fatty acyl-CoA can then be transferred across the mitochondrial membranes using a series of transport proteins: carnitine palmitoyltransferase 1 and 2 (CPT1 and CPT2) (figure 5.9).
CPT1 sits on the outer mitochondrial membrane and transfers the fatty acyl-CoA to carnitine. Fatty acyl carnitine is transferred into the mitochondrial matrix through CPT2, and the carnitine is released and recycled. Only long-chain fatty acyl-CoAs require carnitine as a carrier; short- and medium-chain fatty acids can move into the mitochondria without the assistance of these transporters. Once in the matrix, the fatty acyl-CoA is now ready to undergo β-oxidation (figure 5.9).
β-oxidation is an iterative process that involves a series of enzymes that preferentially oxidize different length fatty acids (long, medium, and short). The full β-oxidation spiral consists of four steps that result in the generation of acetyl-CoA, NADH, and FADH2 for each cycle (figure 5.9). The NADH and FADH2 generated will be oxidized in the ETC to produce ATP. The acetyl-CoA can be oxidized in the TCA cycle, but more likely it will be used in ketogenesis. Oxidation of odd chain fatty acids will result in the generation of propionyl-CoA as the final carbon unit, which can also be oxidized in the TCA cycle. The acetyl-CoA from β-oxidation also plays a key role in the allosteric activation of pyruvate carboxylase, which is necessary for gluconeogenesis to occur (section 5.1).
Regulation of β-oxidation
β-oxidation is regulated primarily at the level of transport of LCFAs across the mitochondrial membrane. Malonyl-CoA will inhibit CPT1 therefore ensuring that β-oxidation is not occurring at the same time as fatty acid synthesis (figure 5.10; section 4.4). Additionally, the rate of ATP production (ATP/ADP ratio) will also regulate the rate of NADH and FADH2 produced through β-oxidation (figure 5.10).
Ketogenesis
As mentioned above, the acetyl-CoA produced by β-oxidation is primarily used for ketogenesis — the synthesis of ketone bodies. Substrates for ketogenesis can also come from the oxidation of ketogenic amino acids. In the fasted state, the process of β-oxidation generates a significant amount of acetyl-CoA, and although some of this substrate can be oxidized in the TCA cycle, we need to consider the other metabolic processes occurring. First, the significant amount of NADH generated through β-oxidation reduces flux through the TCA cycle by decreasing the activity of both α-ketoglutarate dehydrogenase and isocitrate dehydrogenase. Second, the process of gluconeogenesis is occurring, and intermediates of the TCA cycle, specifically malate, are actively being moved out of the mitochondria. The combination of these two processes reduces the TCA cycle activity allowing for an accumulation of acetyl-CoA. As acetyl-CoA levels elevate in the mitochondria, this will drive the thiolase reaction to generate acetoacetyl-CoA from two acetyl-CoA molecules (figure 5.11).
This compound is the substrate for HMG-CoA synthase, which generates 3-hydroxy-3-methyl glutaryl-CoA (HMG-CoA). HMG-CoA is then accepted by HMG-CoA lyase where an acetyl-CoA group is removed to generate acetoacetate. Acetoacetate can either undergo spontaneous decarboxylation to acetone, which can be exhaled, or it can be reduced to β-hydroxybutyrate using NADH. Acetoacetate and β-hydroxybutyrate are the two primary ketone bodies in circulation, and the ratio of the two is dependent on levels of NADH (figure 5.11). These two ketone bodies can be used as fuel in most tissues with the exception of the liver, which lacks thiophorase, the enzyme needed to metabolize these substrates. Ketone oxidation is not a primary fuel source, as fatty acid oxidation is preferred, but it can supply energy to some peripheral tissues. The brain can also oxidize ketones but only under extreme situations, such as starvation states.
Summary of pathway regulation
Table 5.2: Summary of pathway regulation.
| Metabolic pathway | Major regulatory enzyme | Allosteric effectors | Hormonal effects |
| Lipolysis | Hormone-sensitive lipase | None | Epi ↑ Insulin ↓ |
| β-oxidation | Carnitine palmitoyltransferase (CPT1) | Malonyl-CoA (-) | None |
References and resources
Text
Ferrier, D. R., ed. Lippincott Illustrated Reviews: Biochemistry, 7th ed. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins, 2017, Chapter 10: Gluconeogenesis: Section II, III, IV, Chapter 11: Glycogen Metabolism: Section V, VI, Chapter 16: Fatty Acid Ketone Body and TAG Metabolism: Section III, IV, V, Chapter 19: Removal of Nitrogen from Amino Acids: Section V, VI, Chapter 23: Metabolic Effect of Insulin and Glucagon, Chapter 25: Diabetes Mellitus.
Le, T., and V. Bhushan. First Aid for the USMLE Step 1, 29th ed. New York: McGraw Hill Education, 2018, 78, 82, 86, 89–90.
Lieberman, M., and A. Peet, eds. Marks' Basic Medical Biochemistry: A Clinical Approach, 5th ed. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins, 2018, Chapter 3: The Fasted State, Chapter 19: Basic Concepts in Regulation, Chapter 24: Oxidative Phosphorylation and the ETC, Chapter 26: Formation of Glycogen, Chapter 28: Gluconeogenesis, Chapter 30: Oxidation of Fatty Acids, Chapter 34: Integration of Carbohydrate and Lipid Metabolism, Chapter 36: Fate of Amino Acids Nitrogen: Urea Cycle.
Figures
Grey, Kindred, Figure 5.8 Process of lipolysis. 2021. CC BY 4.0. Added red blood cells by Lucas Helle from the Noun Project.
Grey, Kindred, Figure 5.9 Overview of LCFA transport into the mitochondria and β-oxidation. 2021. CC BY 4.0.
Grey, Kindred, Figure 5.10 Regulation of β-oxidation. 2021. CC BY 4.0.
Grey, Kindred, Figure 5.11 Overview of ketone body formation. 2021. CC BY 4.0.
5.1: Gluconeogenesis and glycogenolysis
5.3: Nitrogen metabolism and the urea cycle |
5785 | https://learn.microsoft.com/en-us/cpp/cpp/fundamental-types-cpp?view=msvc-170 | Built-in types (C++) | Microsoft Learn
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Built-in types (C++)
08/11/2025
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Void type
std::nullptr_t
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Character types
Floating-point types
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Built-in types (also called fundamental types) are specified by the C++ language standard and are built into the compiler. Built-in types aren't defined in any header file. Built-in types are divided into three main categories: integral, floating-point, and void. Integral types represent whole numbers. Floating-point types can specify values that may have fractional parts. Most built-in types are treated as distinct types by the compiler. However, some types are synonyms, or treated as equivalent types by the compiler.
Void type
The void type describes an empty set of values. No variable of type void can be specified. The void type is used primarily to declare functions that return no values or to declare generic pointers to untyped or arbitrarily typed data. Any expression can be explicitly converted or cast to type void. However, such expressions are restricted to the following uses:
An expression statement. For more information, see Expressions.
The left operand of the comma operator. For more information, see Comma Operator.
The second or third operand of the conditional operator (? :). For more information, see Expressions with the Conditional Operator.
std::nullptr_t
The keyword nullptr is a null-pointer constant of type std::nullptr_t, which is convertible to any raw pointer type. For more information, see nullptr.
Boolean type
The bool type can have values true and false. The size of the bool type is implementation-specific. See Sizes of built-in types for Microsoft-specific implementation details.
Character types
The char type is a character representation type that efficiently encodes members of the basic execution character set. The C++ compiler treats variables of type char, signed char, and unsigned char as having different types.
Microsoft-specific: Variables of type char are promoted to int as if from type signed char by default, unless the /J compilation option is used. In this case, they're treated as type unsigned char and are promoted to int without sign extension.
A variable of type wchar_t is a wide-character or multibyte character type. Use the L prefix before a character or string literal to specify the wide-character type.
Microsoft-specific: By default, wchar_t is a native type, but you can use /Zc:wchar_t- to make wchar_t a typedef for unsigned short. The __wchar_t type is a Microsoft-specific synonym for the native wchar_t type.
The char8_t type is used for UTF-8 character representation. It has the same representation as unsigned char, but is treated as a distinct type by the compiler. The char8_t type is new in C++20. Microsoft-specific: use of char8_t requires the /std:c++20 compiler option or later (such as /std:c++latest).
The char16_t type is used for UTF-16 character representation. It must be large enough to represent any UTF-16 code unit. It's treated as a distinct type by the compiler.
The char32_t type is used for UTF-32 character representation. It must be large enough to represent any UTF-32 code unit. It's treated as a distinct type by the compiler.
Floating-point types
Floating-point types use an IEEE-754 representation to provide an approximation of fractional values over a wide range of magnitudes. The following table lists the floating-point types in C++ and the comparative restrictions on floating-point type sizes. These restrictions are mandated by the C++ standard and are independent of the Microsoft implementation. The absolute size of built-in floating-point types isn't specified in the standard.
Expand table
| Type | Contents |
--- |
| float | Type float is the smallest floating point type in C++. |
| double | Type double is a floating point type that is larger than or equal to type float, but shorter than or equal to the size of type long double. |
| long double | Type long double is a floating point type that is larger than or equal to type double. |
Microsoft-specific: The representation of long double and double is identical. However, long double and double are treated as distinct types by the compiler. The Microsoft C++ compiler uses the 4- and 8-byte IEEE-754 floating-point representations. For more information, see IEEE floating-point representation.
Integer types
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| Type | Size |
--- |
| bool, char, char8_t, unsigned char, signed char, __int8 | 1 byte |
| char16_t, __int16, short, unsigned short, wchar_t, __wchar_t | 2 bytes |
| char32_t, float, __int32, int, unsigned int, long, unsigned long | 4 bytes |
| double, __int64, long double, long long, unsigned long long | 8 bytes |
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5786 | https://math.stackexchange.com/questions/1669417/finding-divisibility-of-sequence-of-numbers-generated-recursively | Skip to main content
Finding Divisibility of Sequence of Numbers Generated Recursively
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I have the following generating function:
E(x)=2exe2x+1+2x=∑n=0∞Enxnn!
which generates a sequence of integers below
{1,−1,3,−15,93,−725,6815,−74627,933849,−13148361,205690779,...}
Thus, accordingly, E0=1,E1=−1,E2=3,E3=−15,.... So I was playing around and decided to look at these numbers prime factorization. Below is the breakdown of the first 15 numbers...
n01234567891011121314|En|113159372568157462793384913148361205690779353954555966466203637135130977468529595401433975prime decomposition1133⋅53⋅3152⋅295⋅29⋅4772⋅152337⋅7⋅6132⋅17⋅19⋅452333⋅72⋅15547311⋅31⋅43⋅24139311⋅283⋅701⋅304495⋅13⋅97⋅5003⋅428393⋅52⋅13⋅8539⋅3554779
I noticed that for odd indexed numbers, the number was divisible by the index; in other words, n|En. Also, for even indexed numbers, the number was divisible by the index minus one, so (n−1)|En. Below is the chart color coded with odds in red and evens in blue.
n01234567891011121314|En|113159372568157462793384913148361205690779353954555966466203637135130977468529595401433975prime decomposition111⋅33⋅53⋅3152⋅295⋅29⋅4772⋅152337⋅7⋅6132⋅17⋅19⋅45233⋅32⋅72⋅15547311⋅31⋅43⋅24139311⋅283⋅701⋅304495⋅13⋅97⋅5003⋅428393⋅52⋅13⋅8539⋅3554779
I thought this was interesting and think the conjecture would hold. My problem is this: I don't know how to begin even testing this idea. How do you test for divisibility of large numbers when the numbers are generated recursively? It would be easy to test for the small; E0,E3, etc. But this is an infinite sequence, so how could i test the 100th term? What are some methods that would be used to prove such a conjecture?
EDIT: I do have the recursive definition, in fact I have two for the numbers, which may help readers...
En=1−2nEn−1−∑k=0n−2(nk)2n−k−1Ek
and
En=−12∑k=0n−1(nk)(1+(−1)n−k+2(n−k)(−1)n−k−1)Ek
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edited Feb 24, 2016 at 2:16
Eleven-Eleven
asked Feb 23, 2016 at 23:28
Eleven-ElevenEleven-Eleven
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Where do they come from? Have you checked OEIS?
– vonbrand
Commented
Feb 23, 2016 at 23:50
I have and they are not in the OEIS... My professor wants something related to the Euler numbers, so he's extending the denominator to increase the power series expansion. I'm not precisely sure what his motivations are, but he wanted look specifically at this generating function and the sequence it generated.
– Eleven-Eleven
Commented
Feb 23, 2016 at 23:53
I've checked Mathematica for numbers up to E48 and it seems to be true...but higher than this, my computer is running slow....
– Eleven-Eleven
Commented
Feb 24, 2016 at 0:01
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At first we simplify the problem by considering even and odd parts of E(x) separately. We can write
E(x)=Ee(x)+Eo(x)=E(x)+E(−x)2+E(x)−E(−x)2
The odd part: Eo(x)
Here we focus at the odd part Eo(x). We want to show that
2n+1|E2n+1n≥0
Translated into exponential generating functions we consider
Eo(x)=∑n=0∞E2n+1x2n+1(2n+1)!=∑n=0∞(2n+1)a2n+1x2n+1(2n+1)!=∑n=0∞a2n+1x2n+1(2n)!=x∑n=0∞a2n+1x2n(2n)!
and have to show that
1xEo(x)
has integral coefficients.
[2016-02-28] Update:
We show 1xEo(x) has integral coefficients. We start with an expansion of the generating function.
1xEo(x)=12x(2exe2x+1+2x−2e−xe−2x+1−2x)=1xex∑n=0∞(−1)n(2x+e2x)n−1xe−x∑n≥0(−1)n(−2x+e−2x)n=1xex∑n=0∞(−1)n∑j=0n(nj)(2x)je2x(n−j)−1xe−x∑n≥0(−1)n∑j=0n(nj)(−2x)je−2x(n−j)=2∑n=0∞(−1)n∑j=0n(nj)(2x)j−1(ex(2n−2j+1)+(−1)j−1e−x(2n−2j+1))(1)(2)
In (1) we use an expansion as geometric series and the representation (2) is appropriate for the next step. Note that 1xEo(x) is an even function. So, we only need to take care of coefficients of even powers of x. In the following we use the coefficient of operator [xn] to denote the coefficient of xn in a generating series. In order to show that the exponential generating function 1xEo(x) has integral coefficients we claim
The following is valid
(2k)![x2k]1xEo(x)∈Zk≥0
We obtain from (2)
(2k)![x2k]1xEo(x)=2(2k)![x2k]∑n=0∞(−1)n∑j=0n(nj)(2x)j−1(ex(2n−2j+1)+(−1)j−1e−x(2n−2j+1))=2(2k)!∑n=0∞(−1)n∑j=0n(nj)2j−1⋅x2k−j+1=2(2k)!∑n=02k+1(−1)n∑j=0n(nj)2j−1⋅((2n−2j+1)2k−j+1+(−1)j−1(2n−2j+1)2k−j+1)1(2k−j+1)!(3)(4)
Comment:
In (3) we apply the linearity of the coefficient of operator and use the rule
[xn+m]A(x)=[xn]x−mA(x)
In (4) we select the value l=2k−j+1 corresponding to the coefficient [x2k−j+1]. Note that the power 2k−j+1 is non-negative and 0≤j≤n. We respect this by considering
2k−n+1≥0
which gives an upper limit 2k+1 of the sum with index n.
Looking at the representation (4) we see there is only one critical part when considering the integral property, namly the fraction
(2k)!(2k+1−j)!0≤j≤2k+1
all other parts are clearly integral. This fraction is integral for all values of j besides j=0. So, we finally have to take a look at (4) with j=0.
2(2k)!∑n=02k+1(−1)n12((2n+1)2k+1−(2n+1)2k+1)1(2k+1)!=0∈Z
and the claim follows.
A similar job could be done for the even part.
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edited Feb 29, 2016 at 9:46
answered Feb 24, 2016 at 21:32
Markus ScheuerMarkus Scheuer
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I think the idea is great, although I am confused about the expansion. How would expansion of the power series that you have for 1xE0(x) provide proof of the a2n+1 being integers for all n? It would seem that recursively defined sequences of integers would always be integers, but is that true?
– Eleven-Eleven
Commented
Feb 25, 2016 at 15:34
@Eleven-Eleven: Expanding expressions like 1e2x+1+2x=∑n≥0(−1)n(2x+e2x)=∑n≥0(−1)n∑nj=0(nj)e2jx(2x)n−j we can do some structural considerations and check if the coefficients of xnn! are integral. The benefit is, that if we can show that the typical coefficient is integral, this will hold for all n. But keep in mind that the factor 1n! is present in the exponential generating function and has to be respected and verified, that this does not prevent the coefficient from being integral.
– Markus Scheuer
Commented
Feb 25, 2016 at 22:25
My cousin, @iceman, has worked on this with me for some time and someone on this site got us a closed form. I forgot about it for a while until i brought it up to him. There is a closed form here.... WIth the obvious n! out front we would just have to determine that the remaining stuff inside is an integer then too, right? (sum of integers is still indeed an integer...)
– Eleven-Eleven
Commented
Feb 26, 2016 at 19:00
did you get a chance to look at the link above in my last comment?
– Eleven-Eleven
Commented
Feb 27, 2016 at 15:06
@Eleven-Eleven: Hi, I already had a look at the reference. I'm afraid the expression there is too complicated for a proper handling. I also wouldn't call this a closed form, because it contains sums. It's just a straightforward expansion. I'll try to find an answer which is convenient for you this weekend. Best regards,
– Markus Scheuer
Commented
Feb 27, 2016 at 15:12
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5787 | https://askfilo.com/user-question-answers-smart-solutions/to-show-that-the-surface-area-of-the-spherical-zone-3133333933393538 | Question asked by Filo student
To show that the surface area of the spherical zone contained between two parallel planes is 2πah where a is the radius of the sphere and h is the distance between the planes, we use the formula for the surface area of a spherical zon
Views: 5,519 students
Updated on: Nov 10, 2024
Text SolutionText solutionverified iconVerified
Concepts:
Surface area of spherical zone, Geometry, Calculus
Explanation:
To show that the surface area of the spherical zone contained between two parallel planes is 2πah where a is the radius of the sphere and h is the distance between the planes, we use the formula for the surface area of a spherical zone.
Step by Step Solution:
Step 1
Consider a sphere of radius a centered at the origin. The equation of the sphere is given by: x2+y2+z2=a2.
Step 2
The two parallel planes are at z=c and z=c+h. The spherical zone is the part of the sphere between these two planes.
Step 3
The surface area of the spherical zone can be found by integrating the circumference of the circles formed by the intersection of the sphere with planes parallel to the xy-plane.
Step 4
The circumference of a circle at height z is 2πa2−z2. Integrating this from z=c to z=c+h gives: Surface Area=∫cc+h2πa2−z2dz. Using the substitution z=asinθ, the integral simplifies to 2πah. Therefore, the surface area of the spherical zone is 2πah.
Final Answer:
The surface area of the spherical zone contained between two parallel planes is 2πah where a is the radius of the sphere and h is the distance between the planes.
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| Question Text | To show that the surface area of the spherical zone contained between two parallel planes is 2πah where a is the radius of the sphere and h is the distance between the planes, we use the formula for the surface area of a spherical zon |
| Updated On | Nov 10, 2024 |
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PEARSON NEW INTERNATIONAL EDITION Field and Wave Electromagnetics David K. Cheng Second Edition Pearson New International Edition Field and Wave Electromagnetics David K. Cheng Second Edition PEARSON Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6-10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. PEARSON ISBN 10: 1-292-02656-1 ISBN 13: 978-1-292-02656-5 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America To Enid Preface The many books on introductory electromagnetics can be roughly divided into two main groups. The first group takes the traditional development: starting with the experimental laws, generalizing them in steps, and finally synthesizing them in the form of Maxwell's equations. This is an inductive approach. The second group takes the axiomatic development: starting with Maxwell's equations, identifying each with the appropriate experimental law, and specializing the general equations to static and time-varying situations for analysis. This is a deductive approach. A few books begin with a treatment of the special theory of relativity and develop all of electro magnetic theory from Coulomb's law of force; but this approach requires the discus sion and understanding of the special theory of relativity first and is perhaps best suited for a course at an advanced level. Proponents of the traditional development argue that it is the way electromag netic theory was unraveled historically (from special experimental laws to Maxwell's equations), and that it is easier for the students to follow than the other methods. I feel, however, that the way a body of knowledge was unraveled is not necessarily the best way to teach the subject to students. The topics tend to be fragmented and cannot take full advantage of the conciseness of vector calculus. Students are puzzled at, and often form a mental block to, the subsequent introduction of gradient, diver gence, and curl operations. As a process for formulating an electromagnetic model, this approach lacks cohesiveness and elegance. The axiomatic development usually begins with the set of four Maxwell's equa tions, either in differential or in integral form, as fundamental postulates. These are equations of considerable complexity and are difficult to master. They are likely to cause consternation and resistance in students who are hit with all of them at the beginning of a book. Alert students will wonder about the meaning of the field vectors and about the necessity and sufficiency of these general equations. At the initial stage students tend to be confused about the concepts of the electromagnetic model, and they are not yet comfortable with the associated mathematical manipulations. In any case, the general Maxwell's equations are soon simplified to apply to static fields, v Preface which allow the consideration of electrostatic fields and magnetostatic fields sepa rately. Why then should the entire set of four Maxwell's equations be introduced at the outset? It may be argued that Coulomb's law, though based on experimental evidence, is in fact also a postulate. Consider the two stipulations of Coulomb's law: that the charged bodies are very small compared with their distance of separation, and that the force between the charged bodies is inversely proportional to the square of their distance. The question arises regarding the first stipulation: How small must the charged bodies be in order to be considered "very small" compared with their dis tance? In practice the charged bodies cannot be of vanishing sizes (ideal point charges), and there is difficulty in determining the "true" distance between two bodies of finite dimensions. For given body sizes the relative accuracy in distance measurements is better when the separation is larger. However, practical considerations (weakness of force, existence of extraneous charged bodies, etc.) restrict the usable distance of sepa ration in the laboratory, and experimental inaccuracies cannot be entirely avoided. This leads to a more important question concerning the inverse-square relation of the second stipulation. Even if the charged bodies were of vanishing sizes, experi mental measurements could not be of an infinite accuracy no matter how skillful and careful an experimentor was. How then was it possible for Coulomb to know that the force was exactly inversely proportional to the square (not the 2.000001th or the 1.999999th power) of the distance of separation? This question cannot be answered from an experimental viewpoint because it is not likely that during Coulomb's time experiments could have been accurate to the seventh place. We must therefore con clude that Coulomb's law is itself a postulate and that it is a law of nature discovered and assumed on the basis of his experiments of a limited accuracy (see Section 3-2). This book builds the electromagnetic model using an axiomatic approach in steps: first for static electric fields (Chapter 3), then for static magnetic fields (Chapter 6), and finally for time-varying fields leading to Maxwell's equations (Chapter 7). The mathematical basis for each step is Helmholtz's theorem, which states that a vector field is determined to within an additive constant if both its divergence and its curl are specified everywhere. Thus, for the development of the electrostatic model in free space, it is only necessary to define a single vector (namely, the electric field intensity E) by specifying its divergence and its curl as postulates. All other relations in electro statics for free space, including Coulomb's law and Gauss's law, can be derived from the two rather simple postulates. Relations in material media can be developed through the concept of equivalent charge distributions of polarized dielectrics. Similarly, for the magnetostatic model in free space it is necessary to define only a single magnetic flux density vector B by specifying its divergence and its curl as postulates; all other formulas can be derived from these two postulates. Relations in material media can be developed through the concept of equivalent current densi ties. Of course, the validity of the postulates lies in their ability to yield results that conform with experimental evidence. For time-varying fields, the electric and magnetic field intensities are coupled. The curl E postulate for the electrostatic model must be modified to conform with Preface vii Faraday's law. In addition, the curl B postulate for the magnetostatic model must also be modified in order to be consistent with the equation of continuity. We have, then, the four Maxwell's equations that constitute the electromagnetic model. I believe that this gradual development of the electromagnetic model based on Helmholtz's theorem is novel, systematic, pedagogically sound, and more easily accepted by students. In the presentation of the material, I strive for lucidity and unity, and for smooth and logical flow of ideas. Many worked-out examples are included to emphasize fundamental concepts and to illustrate methods for solving typical problems. Applica tions of derived relations to useful technologies (such as ink-jet printers, lightning arresters, electret microphones, cable design, multiconductor systems, electrostatic shielding, Doppler radar, radome design, Polaroid filters, satellite communication systems, optical fibers, and microstrip lines) are discussed. Review questions appear at the end of each chapter to test the students' retention and understanding of the es sential material in the chapter. The problems in each chapter are designed to reinforce students' comprehension of the interrelationships between the different quantities in the formulas, and to extend their ability of applying the formulas to solve practical problems. In teaching, I have found the review questions a particularly useful device to stimulate students' interest and to keep them alert in class. Besides the fundamentals of electromagnetic fields, this book also covers the theory and applications of transmission lines, waveguides and cavity resonators, and antennas and radiating systems. The fundamental concepts and the governing theory of electromagnetism do not change with the introduction of new electromagnetic devices. Ample reasons and incentives for learning the fundamental principles of electromagnetics are given in Section 1-1. I hope that the contents of this book, strengthened by the novel approach, will provide students with a secure and sufficient background for understanding and analyzing basic electromagnetic phenomena as well as prepare them for more advanced subjects in electromagnetic theory. There is enough material in this book for a two-semester sequence of courses. Chapters 1 through 7 contain the material on fields, and Chapters 8 through 11 on waves and applications. In schools where there is only a one-semester course on elec tromagnetics, Chapters 1 through 7, plus the first four sections of Chapter 8 would provide a good foundation on fields and an introduction of waves in unbounded media. The remaining material could serve as a useful reference book on applications or as a textbook for a follow-up elective course. Schools on a quarter system could adjust the material to be covered in accordance with the total number of hours assigned to the subject of electromagnetics. Of course, individual instructors have the prerogative to emphasize and expand certain topics, and to deemphasize or delete certain others. I have given considerable thought to the advisability of including computer pro grams for the solution of some problems, but have finally decided against it. Diverting students' attention and effort to numerical methods and computer software would distract them from concentrating on learning the fundamentals of electromagnetism. Where appropriate, the dependence of important results on the value of a parameter viii Preface is stressed by curves; field distributions and antenna patterns are illustrated by graphs; and typical mode patterns in waveguides are plotted. The computer programs for obtaining these curves, graphs, and mode patterns are not always simple. Students in science and engineering are required to acquire a facility for using computers; but the inclusion of some cookbook-style computer programs in a book on the funda mental principles of electromagnetic fields and waves would appear to contribute little to the understanding of the subject matter. This book was first published in 1983. Favorable reactions and friendly encour agements from professors and students have provided me with the impetus to come out with a new edition. In this second edition I have added many new topics. These include Hall effect, d-c motors, transformers, eddy current, energy-transport velocity for wide-band signals in waveguides, radar equation and scattering cross section, transients in transmission lines, Bessel functions, circular waveguides and circular cavity resonators, waveguide discontinuities, wave propagation in ionosphere and near earth's surface, helical antennas, log-periodic dipole arrays, and antenna effective length and effective area. The total number of problems has been expanded by about 35 percent. The Addison-Wesley Publishing Company has decided to make this second edition a two-color book. I think the readers will agree that the book is handsomely produced. I would like to take this opportunity to express my appreciation to all the people on the editorial, production, and marketing staff who provided help in bringing out this new edition. In particular, I wish to thank Thomas Robbins, Barbara Rifkind, Karen Myer, Joseph K. Vetere, and Katherine Harutunian. Chevy Chase, Maryland D. K. C. Contents The Electromagnetic Model 1 1-1 Introduction 1 1-2 The Electromagnetic Model 3 1-3 SI Units and Universal Constants 8 Review Questions 10 2 Vector Analysis 11 2-1 Introduction 11 2-2 Vector Addition and Subtraction 12 2-3 Products of Vectors 14 2-3.1 Scalar or Dot Product 14 2-3.2 Vector or Cross Product 16 2-3.3 Product of Three Vectors 18 2-4 Orthogonal Coordinate Systems 20 2-4.1 Cartesian Coordinates 23 2-4.2 Cylindrical Coordinates 27 2-4.3 Spherical Coordinates 31 2-5 Integrals Containing Vector Functions 37 2-6 Gradient of a Scalar Field 42 2-7 Divergence of a Vector Field 46 2-8 Divergence Theorem 50 2-9 Curl of a Vector Field 54 2-10 Stokes's Theorem 58 ix Contents 2-11 Two Null Identities 61 2-11.1 Identity I 61 2-11.2 Identity II 62 2-12 Helmholtz's Theorem 63 Review Questions 66 Problems 67 3 Static Electric Fields 72 3-1 Introduction 72 3-2 Fundamental Postulates of Electrostatics in Free Space 74 3-3 Coulomb's Law 77 3-3.1 Electric Field Due to a System of Discrete Charges 82 3-3.2 Electric Field Due to a Continuous Distribution of Charge 84 3-4 Gauss's Law and Applications 87 3-5 Electric Potential 92 3-5.1 Electric Potential Due to a Charge Distribution 94 3-6 Conductors in Static Electric Field 100 3-7 Dielectrics in Static Electric Field 105 3-7.1 Equivalent Charge Distributions of Polarized Dielectrics 106 3-8 Electric Flux Density and Dielectric Constant 109 3-8.1 Dielectric Strength 114 3-9 Boundary Conditions for Electrostatic Fields 116 3-10 Capacitance and Capacitors 121 3-10.1 Series and Parallel Connections of Capacitors 126 3-10.2 Capacitances in Multiconductor Systems 129 3-10.3 Electrostatic Shielding 132 3-11 Electrostatic Energy and Forces 133 3-11.1 Electrostatic Energy in Terms of Field Quantities 137 3-11.2 Electrostatic Forces 140 Review Questions 143 Problems 145 4 Solution of Electrostatic Problems 152 4-1 Introduction 152 4-2 Poisson's and Laplace's Equations 152 4-3 Uniqueness of Electrostatic Solutions 157 Contents XI 4-4 Method of Images 159 4-4.1 Point Charge and Conducting Planes 161 4-4.2 Line Charge and Parallel Conducting Cylinder 162 4-4.3 Point Charge and Conducting Sphere 170 4-4.4 Charged Sphere and Grounded Plane 172 4-5 Boundary-Value Problems in Cartesian Coordinates 174 4-6 Boundary-Value Problems in Cylindrical Coordinates 183 4-7 Boundary-Value Problems in Spherical Coordinates 188 Review Questions 192 Problems 193 Steady Electric Currents 198 5-1 Introduction 198 5-2 Current Density and Ohm's Law 199 5-3 Electromotive Force and KirchhofT's Voltage Law 205 5-4 Equation of Continuity and KirchhofT's Current Law 208 5-5 Power Dissipation and Joule's Law 210 5-6 Boundary Conditions for Current Density 211 5-7 Resistance Calculations 215 Review Questions 219 Problems 220 6 Static Magnetic Fields 225 6-1 Introduction 225 6-2 Fundamental Postulates of Magnetostatics in Free Space 226 6-3 Vector Magnetic Potential 232 6-4 The Biot-Savart Law and Applications 234 6-5 The Magnetic Dipole 239 6-5.1 Scalar Magnetic Potential 242 6-6 Magnetization and Equivalent Current Densities 243 6-6.1 Equivalent Magnetization Charge Densities 247 6-7 Magnetic Field Intensity and Relative Permeability 249 6-8 Magnetic Circuits 251 6-9 Behavior of Magnetic Materials 257 6-10 Boundary Conditions for Magnetostatic Fields 262 6-11 Inductances and Inductors 266 XII 6-12 Magnetic Energy 277 6-12.1 Magnetic Energy in Terms of Field Quantities 279 6-13 Magnetic Forces and Torques 281 6-13.1 Hall Effect 282 6-13.2 Forces and Torques on Current-Carrying Conductors 283 6-13.3 Forces and Torques in Terms of Stored Magnetic Energy 289 6-13.4 Forces and Torques in Terms of Mutual Inductance 292 Review Questions 294 Problems 296 Time-Varying Fields and Maxwell's Equations 307 7-1 Introduction 307 7-2 Faraday's Law of Electromagnetic Induction 308 7-2.1 A Stationary Circuit in a Time-Varying Magnetic Field 309 7-2.2 Transformers 310 7-2.3 A Moving Conductor in a Static Magnetic Field 314 7-2.4 A Moving Circuit in a Time-Varying Magnetic Field 317 7-3 Maxwell's Equations 321 7-3.1 Integral Form of Maxwell's Equations 323 7-4 Potential Functions 326 7-5 Electromagnetic Boundary Conditions 329 7-5.1 Interface between Two Lossless Linear Media 330 7-5.2 Interface between a Dielectric and a Perfect Conductor 331 7-6 Wave Equations and Their Solutions 332 7-6.1 Solution of Wave Equations for Potentials 333 7-6.2 Source-Free Wave Equations 334 7-7 Time-Harmonic Fields 335 7-7.1 The Use of Phasors—A Review 336 7-7.2 Time-Harmonic Electromagnetics 338 7-7.3 Source-Free Fields in Simple Media 340 7-7.4 The Electromagnetic Spectrum 343 Review Questions 346 Problems 347 8 Plane Electromagnetic Waves 354 -l Introduction 354 -2 Plane Waves in Lossless Media 355 8-2.1 Doppler Effect 360 Contents xiii 8-2.2 Transverse Electromagnetic Waves 361 8-2.3 Polarization of Plane Waves 364 8-3 Plane Waves in Lossy Media 367 8-3.1 Low-Loss Dielectrics 368 8-3.2 Good Conductors 369 8-3.3 Ionized Gases 373 8-4 Group Velocity 375 8-5 Flow of Electromagnetic Power and the Poynting Vector 379 8-5.1 Instantaneous and Average Power Densities 382 8-6 Normal Incidence at a Plane Conducting Boundary 386 8-7 Oblique Incidence at a Plane Conducting Boundary 390 8-7.1 Perpendicular Polarization 390 8-7.2 Parallel Polarization 395 8-8 Normal Incidence at a Plane Dielectric Boundary 397 8-9 Normal Incidence at Multiple Dielectric Interfaces 401 8-9.1 Wave Impedance of the Total Field 403 8-9.2 Impedance Transformation with Multiple Dielectrics 404 8-10 Oblique Incidence at a Plane Dielectric Boundary 406 8-10.1 Total Reflection 408 8-10.2 Perpendicular Polarization 411 8-10.3 Parallel Polarization 414 Review Questions 417 Problems 419 9 Theory and Applications of Transmission Lines 427 9-1 Introduction 427 9-2 Transverse Electromagnetic Wave along a Parallel-Plate Transmission Line 429 9-2.1 Lossy Parallel-Plate Transmission Lines 433 9-2.2 Microstrip Lines 435 9-3 General Transmission-Line Equations 437 9-3.1 Wave Characteristics on an Infinite Transmission Line 439 9-3.2 Transmission-Line Parameters 444 9-3.3 Attenuation Constant from Power Relations 447 9-4 Wave Characteristics on Finite Transmission Lines 449 9-4.1 Transmission Lines as Circuit Elements 454 9-4.2 Lines with Resistive Termination 460 9-4.3 Lines with Arbitrary Termination 465 9-4.4 Transmission-Line Circuits 467 9-5 Transients on Transmission Lines 471 9-5.1 Reflection Diagrams 474 xiv Contents 9-5.2 Pulse Excitation 478 9-5.3 Initially Charged Line 480 9-5.4 Line with Reactive Load 482 9-6 The Smith Chart 485 9-6.1 Smith-Chart Calculations for Lossy Lines 495 9-7 Transmission-Line Impedance Matching 497 9-7.1 Impedance Matching by Quarter-Wave Transformer 497 9-7.2 Single-Stub Matching 501 9-7.3 Double-Stub Matching 505 Review Questions 509 Problems 512 10 Waveguides and Cavity Resonators 520 10-1 Introduction 520 10-2 General Wave Behaviors along Uniform Guiding Structures 521 10-2.1 Transverse Electromagnetic Waves 524 10-2.2 Transverse Magnetic Waves 525 10-2.3 Transverse Electric Waves 529 10-3 Parallel-Plate Waveguide 534 10-3.1 TM Waves between Parallel Plates 534 10-3.2 TE Waves between Parallel Plates 539 10-3.3 Energy-Transport Velocity 541 10-3.4 Attenuation in Parallel-Plate Waveguides 543 10-4 Rectangular Waveguides 547 10-4.1 TM Waves in Rectangular Waveguides 547 10-4.2 TE Waves in Rectangular Waveguides 551 10-4.3 Attenuation in Rectangular Waveguides 555 10-4.4 Discontinuities in Rectangular Waveguides 559 10-5 Circular Waveguides 562 10-5.1 Bessel's Differential Equation and Bessel Functions 563 10-5.2 TM Waves in Circular Waveguides 567 10-5.3 TE Waves in Circular Waveguides 569 10-6 Dielectric Waveguides 572 10-6.1 TM Waves along a Dielectric Slab 572 10-6.2 TE Waves along a Dielectric Slab 576 10-6.3 Additional Comments on Dielectric Waveguides 579 10-7 Cavity Resonators 582 10-7.1 Rectangular Cavity Resonators 582 10-7.2 Quality Factor of Cavity Resonator 586 10-7.3 Circular Cavity Resonator 589 Review Questions 592 Problems 594 Contents xv 11 Antennas and Radiating Systems 600 11-1 Introduction 600 11-2 Radiation Fields of Elemental Dipoles 602 11-2.1 The Elemental Electric Dipole 602 11-2.2 The Elemental Magnetic Dipole 605 11-3 Antenna Patterns and Antenna Parameters 607 11-4 Thin Linear Antennas 614 11 -4.1 The Half-Wave Dipole 617 11-4.2 Effective Antenna Length 619 11-5 Antenna Arrays 621 11-5.1 Two-Element Arrays 622 11-5.2 General Uniform Linear Arrays 625 11-6 Receiving Antennas 631 11-6.1 Internal Impedance and Directional Pattern 632 11-6.2 Effective Area 634 11-6.3 Backscatter Cross Section 637 11-7 Transmit-Receive Systems 639 11-7.1 Friis Transmission Formula and Radar Equation 639 11-7.2 Wave Propagation near Earth's Surface 642 11-8 Some Other Antenna Types 643 11-8.1 Traveling-Wave Antennas 643 11-8.2 Helical Antennas 645 11-8.3 Yagi-Uda Antenna 648 11-8.4 Broadband Antennas 650 11-9 Aperture Radiators 655 References 661 Review Questions 662 Problems 664 Appendixes A Symbols and Units 671 A-l Fundamental SI (Rationalized MKSA) Units 671 A-2 Derived Quantities 671 A-3 Multiples and Submultiples of Units 673 B Some Useful Material Constants 674 B-l Constants of Free Space 674 B-2 Physical Constants of Electron and Proton 674 xvi Contents B-3 Relative Permittivities (Dielectric Constants) 675 B-4 Conductivities 675 B-5 Relative Permeabilities 676 C Index of Tables 677 General Bibliography 679 Answers to Selected Problems 681 Index 693 Back Endpapers Left: Some Useful Vector Identities Gradient, Divergence, Curl, and Laplacian Operations in Cartesian Coordinates Right: Gradient, Divergence, Curl, and Laplacian Operations in Cylindrical and Spherical Coordinates 1 The Electromagnetic Model 1—1 Introduction Stated in a simple fashion, electromagnetics is the study of the effects of electric charges at rest and in motion. From elementary physics we know that there are two kinds of charges: positive and negative. Both positive and negative charges are sources of an electric field. Moving charges produce a current, which gives rise to a magnetic field. Here we tentatively speak of electric field and magnetic field in a general way; more definitive meanings will be attached to these terms later. A field is a spatial dis tribution of a quantity, which may or may not be a function of time. A time-varying electric field is accompanied by a magnetic field, and vice versa. In other words, time-varying electric and magnetic fields are coupled, resulting in an electromagnetic field. Under certain conditions, time-dependent electromagnetic fields produce waves that radiate from the source. The concept of fields and waves is essential in the explanation of action at a dis tance. For instance, we learned from elementary mechanics that masses attract each other. This is why objects fall toward the earth's surface. But since there are no elastic strings connecting a free-falling object and the earth, how do we explain this phenom enon? We explain this action-at-a-distance phenomenon by postulating the existence of a gravitational field. The possibilities of satellite communication and of receiving signals from space probes millions of miles away can be explained only by postulating the existence of electric and magnetic fields and electromagnetic waves. In this book, Field and Wave Electromagnetics, we study the principles and applications of the laws of electromagnetism that govern electromagnetic phenomena. Electromagnetics is of fundamental importance to physicists and to electrical and computer engineers. Electromagnetic theory is indispensable in understanding the principle of atom smashers, cathode-ray oscilloscopes, radar, satellite communication, television reception, remote sensing, radio astronomy, microwave devices, optical fiber communication, transients in transmission lines, electromagnetic compatibility 1 2 1 The Electromagnetic Model FIGURE 1-1 A monopole antenna. problems, instrument-landing systems, electromechanical energy conversion, and so on. Circuit concepts represent a restricted version, a special case, of electromagnetic concepts. As we shall see in Chapter 7, when the source frequency is very low so that the dimensions of a conducting network are much smaller than the wavelength, we have a quasi-static situation, which simplifies an electromagnetic problem to a circuit problem. However, we hasten to add that circuit theory is itself a highly developed, sophisticated discipline. It applies to a different class of electrical engineering prob lems, and it is important in its own right. Two situations illustrate the inadequacy of circuit-theory concepts and the need for electromagnetic-field concepts. Figure 1-1 depicts a monopole antenna of the type we see on a walkie-talkie. On transmit, the source at the base feeds the antenna with a message-carrying current at an appropriate carrier frequency. From a circuit-theory point of view, the source feeds into an open circuit because the upper tip of the antenna is not connected to anything physically; hence no current would flow, and nothing would happen. This viewpoint, of course, cannot explain why communi cation can be established between walkie-talkies at a distance. Electromagnetic con cepts must be used. We shall see in Chapter 11 that when the length of the antenna is an appreciable part of the carrier wavelength,T a nonuniform current will flow along the open-ended antenna. This current radiates a time-varying electromagnetic field in space, which propagates as an electromagnetic wave and induces currents in other antennas at a distance. In Fig. 1-2 we show a situation in which an electromagnetic wave is incident from the left on a large conducting wall containing a small hole (aperture). Electro magnetic fields will exist on the right side of the wall at points, such as P in the fig ure, that are not necessarily directly behind the aperture. Circuit theory is obviously inadequate here for the determination (or even the explanation of the existence) of the field at P. The situation in Fig. 1-2, however, represents a problem of practical importance as its solution is relevant in evaluating the shielding effectiveness of the conducting wall. f The product of the wavelength and the frequency of an a-c source is the velocity of wave propagation. Incident wave Conducting \ FIGURE 1-2 wall \ J An electromagnetic problem. Generally speaking, circuit theory deals with lumped-parameter systems—circuits consisting of components characterized by lumped parameters such as resistances, inductances, and capacitances. Voltages and currents are the main system variables. For d-c circuits the system variables are constants, and the governing equations are algebraic equations. The system variables in a-c circuits are time-dependent; they are scalar quantities and are independent of space coordinates. The governing equations are ordinary differential equations. On the other hand, most electromagnetic vari ables are functions of time as well as of space coordinates. Many are vectors with both a magnitude and a direction, and their representation and manipulation require a knowledge of vector algebra and vector calculus. Even in static cases the govern ing equations are, in general, partial differential equations. It is essential that we be equipped to handle vector quantities and variables that are both time- and space-dependent. The fundamentals of vector algebra and vector calculus will be developed in Chapter 2. Techniques for solving partial differential equations are needed in deal ing with certain types of electromagnetic problems. These techniques will be discussed in Chapter 4. The importance of acquiring a facility in the use of these mathematical tools in the study of electromagnetics cannot be overemphasized. Students who have mastered circuit theory may initially have the impression that electromagnetic theory is abstract. In fact, electromagnetic theory is no more abstract than circuit theory in the sense that the validity of both can be verified by experimen tally measured results. In electromagnetics there is a need to define more quantities and to use more mathematical manipulations in order to develop a logical and com plete theory that can explain a much wider variety of phenomena. The challenge of field and wave electromagnetics is not in the abstractness of the subject matter but rather in the process of mastering the electromagnetic model and the associated rules of operation. Dedication to acquiring this mastery will help us to meet the challenge and reap immeasurable satisfaction. 1 - 2 The Electromagnetic Model There are two approaches in the development of a scientific subject: the inductive approach and the deductive approach. Using the inductive approach, one follows 1 The Electromagnetic Model the historical development of the subject, starting with the observations of some sim ple experiments and inferring from them laws and theorems. It is a process of reason ing from particular phenomena to general principles. The deductive approach, on the other hand, postulates a few fundamental relations for an idealized model. The postulated relations are axioms, from which particular laws and theorems can be de rived. The validity of the model and the axioms is verified by their ability to predict consequences that check with experimental observations. In this book we prefer to use the deductive or axiomatic approach because it is more elegant and enables the development of the subject of electromagnetics in an orderly way. The idealized model we adopt for studying a scientific subject must relate to real-world situations and be able to explain physical phenomena; otherwise, we would be engaged in mental exercises for no purpose. For example, a theoretical model could be built, from which one might obtain many mathematical relations; but, if these relations disagreed with observed results, the model would be of no use. The mathematics might be correct, but the underlying assumptions of the model could be wrong, or the implied approximations might not be justified. Three essential steps are involved in building a theory on an idealized model. First, some basic quantities germane to the subject of study are defined. Second, the rules of operation (the mathematics) of these quantities are specified. Third, some fundamental relations are postulated. These postulates or laws are invariably based on numerous experimental observations acquired under controlled conditions and synthesized by brilliant minds. A familiar example is the circuit theory built on a circuit model of ideal sources and pure resistances, inductances, and capacitances. In this case the basic quantities are voltages (V), currents (/), resistances (R), induc tances (L), and capacitances (C); the rules of operations are those of algebra, ordinary differential equations, and Laplace transformation; and the fundamental postulates are Kirchhoff's voltage and current laws. Many relations and formulas can be de rived from this basically rather simple model, and the responses of very elaborate networks can be determined. The validity and value of the model have been amply demonstrated. In a like manner, an electromagnetic theory can be built on a suitably chosen electromagnetic model. In this section we shall take the first step of defining the basic quantities of electromagnetics. The second step, the rules of operation, encompasses vector algebra, vector calculus, and partial differential equations. The fundamentals of vector algebra and vector calculus will be discussed in Chapter 2 (Vector Analysis), and the techniques for solving partial differential equations will be introduced when these equations arise later in the book. The third step, the fundamental postulates, will be presented in three substeps in Chapters 3, 6, and 7 as we deal with static electric fields, steady magnetic fields, and electromagnetic fields, respectively. The quantities in our electromagnetic model can be divided roughly into two categories: source quantities and field quantities. The source of an electromagnetic field is invariably electric charges at rest or in motion. However, an electromagnetic field may cause a redistribution of charges, which will, in turn, change the field; hence the separation between the cause and the effect is not always so distinct. 1-2 The Electromagnetic Model 5 We use the symbol q (sometimes Q) to denote electric charge. Electric charge is a fundamental property of matter and exists only in positive or negative integral multiples of the charge on an electron, — e.
1" e = 1 . 6 0 x l ( T 1 9 (C), (1-1) where C is the abbreviation of the unit of charge, coulomb.1 It is named after the French physicist Charles A. de Coulomb, who formulated Coulomb's law in 1785. (Coulomb's law will be discussed in Chapter 3.) A coulomb is a very large unit for electric charge; it takes 1/(1.60 x 10~19) or 6.25 million trillion electrons to make up - 1 C. In fact, two 1 C charges 1 m apart will exert a force of approximately 1 million tons on each other. Some other physical constants for the electron are listed in Appendix B-2. The principle of conservation of electric charge, like the principle of conserva tion of momentum, is a fundamental postulate or law of physics. It states that electric charge is conserved; that is, it can neither be created nor be destroyed. This is a law of nature and cannot be derived from other principles or relations. Its truth has never been questioned or doubted in practice. Electric charges can move from one place to another and can be redistributed under the influence of an electromagnetic field; but the algebraic sum of the positive and negative charges in a closed (isolated) system remains unchanged. The principle of conservation of electric charge must be satisfied at all times and under any circumstances. It is represented mathematically by the equation of continuity, which we will discuss in Section 5-4. Any formulation or solution of an electromagnetic problem that violates the principle of conservation of electric charge must be incorrect. We recall that the Kirchhoff's current law in circuit theory, which maintains that the sum of all the currents leaving a junction must equal the sum of all the currents entering the junction, is an assertion of the conservation property of electric charge. (Implicit in the current law is the assumption that there is no cumulation of charge at the junction.) Although, in a microscopic sense, electric charge either does or does not exist at a point in a discrete manner, these abrupt variations on an atomic scale are unim portant when we consider the electromagnetic effects of large aggregates of charges. In constructing a macroscopic or large-scale theory of electromagnetism we find that the use of smoothed-out average density functions yields very good results. (The same approach is used in mechanics where a smoothed-out mass density function is defined, in spite of the fact that mass is associated only with elementary particles in a discrete 1 In 1962, Murray Gell-Mann hypothesized quarks as the basic building blocks of matter. Quarks were predicted to carry a fraction of the charge of an electron, and their existence has since been verified experimentally. x The system of units will be discussed in Section 1-3. 6 1 The Electromagnetic Model manner on an atomic scale.) We define a volume charge density, p, as a source quan tity as follows: p = lim %L (C/m3), (1-2) where Aq is the amount of charge in a very small volume Av. How small should Av be? It should be small enough to represent an accurate variation of p but large enough to contain a very large number of discrete charges. For example, an elemental cube with sides as small as 1 micron (10~6 m or 1 /mi) has a volume of 10"18 m3, which will still contain about 1011 (100 billion) atoms. A smoothed-out function of space coordinates, p, defined with such a small At; is expected to yield accurate macroscopic results for nearly all practical purposes. In some physical situations an amount of charge Aq may be identified with an element of surface As or an element of line AL In such cases it will be more appropriate to define a surface charge density, ps, or a line charge density, p{: ps = lim - ^ (C/m2), (1-3) As- ► o A s p, = lim ^ (C/m). (1-4) AoA/ Except for certain special situations, charge densities vary from point to point; hence p, ps, and p£ are, in general, point functions of space coordinates. Current is the rate of change of charge with respect to time; that is, da I = ft (C/s or A), (1-5) where / itself may be time-dependent. The unit of current is coulomb per second (C/s), which is the same as ampere (A). A current must flow through a finite area (a con ducting wire of a finite cross section, for instance); hence it is not a point function. In electromagnetics we define a vector point function volume current density (or simply current density) J, which measures the amount of current flowing through a unit area normal to the direction of current flow. The boldfaced J is a vector whose mag nitude is the current per unit area (A/m2) and whose direction is the direction of cur rent flow. We shall elaborate on the relation between / and J in Chapter 5. For very good conductors, high-frequency alternating currents are confined in the surface layer as a current sheet, instead of flowing throughout the interior of the conductor. In such cases there is a need to define a surface current density Js, which is the current per unit width on the conductor surface normal to the direction of current flow and has the unit of ampere per meter (A/m). There are four fundamental vector field quantities in electromagnetics: electric field intensity E, electric flux density (or electric displacement) D, magnetic flux 1-2 The Electromagnetic Model 7 TABLE 1-1 Fundamental Electromagnetic Field Quantities Symbols and Units for Field Quantities Electric Magnetic Field Quantity Electric field intensity Electric flux density (Electric displacement) Magnetic flux density Magnetic field intensity Symbol E D B H Unit V/m C/m2 T A/m density B, and magnetic field intensity H. The definition and physical significance of these quantities will be explained fully when they are introduced later in the book. At this time we want only to establish the following. Electric field intensity E is the only vector needed in discussing electrostatics (effects of stationary electric charges) in free space; it is defined as the electric force on a unit test charge. Electric displace ment vector D is useful in the study of electric field in material media, as we shall see in Chapter 3. Similarly, magnetic flux density B is the only vector needed in dis cussing magnetostatics (effects of steady electric currents) in free space and is related to the magnetic force acting on a charge moving with a given velocity. The magnetic field intensity vector H is useful in the study of magnetic field in material media. The definition and significance of B and H will be discussed in Chapter 6. The four fundamental electromagnetic field quantities, together with their units, are tabulated in Table 1-1. In Table 1-1, V/m is volt per meter, and T stands for tesla or volt-second per square meter. When there is no time variation (as in static, steady, or stationary cases), the electric field quantities E and D and the magnetic field quan tities B and H form two separate vector pairs. In time-dependent cases, however, electric and magnetic field quantities are coupled; that is, time-varying E and D will give rise to B and H, and vice versa. All four quantities are point functions; they are defined at every point in space and, in general, are functions of space coordinates. Material (or medium) properties determine the relations between E and D and be tween B and H. These relations are called the constitutive relations of a medium and will be examined later. The principal objective of studying electromagnetism is to understand the inter action between charges and currents at a distance based on the electromagnetic model. Fields and waves (time- and space-dependent fields) are basic conceptual quantities of this model. Fundamental postulates will relate E, D, B, H, and the source quantities; and derived relations will lead to the explanation and prediction of electromagnetic phenomena. 1 The Electromagnetic Model TABLE 1-2 Fundamental SI Units Quantity Length Mass Time Current Unit meter kilogram second ampere Abbreviation m kg s A 1—3 SI Units and Universal Constants A measurement of any physical quantity must be expressed as a number followed by a unit. Thus we may talk about a length of three meters, a mass of two kilograms, and a time period of ten seconds. To be useful, a unit system should be based on some fundamental units of convenient (practical) sizes. In mechanics, all quantities can be expressed in terms of three basic units (for length, mass, and time). In electromagnetics a fourth basic unit (for current) is needed. The SI (International System of Units or Le Systeme International d'Unites) is an MKSA system built from the four funda mental units listed in Table 1-2. All other units used in electromagnetics, including those appearing in Table 1-1, are derived units expressible in terms of meters, kilo grams, seconds, and amperes. For example, the unit for charge, coulomb (C), is ampere-second (A-s); the unit for electric field intensity (V/m) is kg-m/A-s3; and the unit for magnetic flux density, tesla (T), is kg/A-s2. More complete tables of the units for various quantities are given in Appendix A. The official SI definitions, as adopted by the International Committee on Weights and Measures, are as follows:1" Meter. Once the length between two scratches on a platinum-iridium bar (and originally calculated as one ten-millionth of the distance between the North Pole and the equator through Paris, France), is now defined by reference to the second (see below) and the speed of light, which in a vacuum is 299,792,458 meters per second. Kilogram. Mass of a standard bar made of a platinum-iridium alloy and kept inside a set of nested enclosures that protect it from contamination and mis handling. It rests at the International Bureau of Weights and Measures in Sevres, outside Paris. Second. 9,192,631,770 periods of the electromagnetic radiation emitted by a par ticular transition of a cesium atom. f P. Wallich, "Volts and amps are not what they used to be," IEEE Spectrum, vol. 24, pp. 44-49, March 1987. 1-3 SI Units and Universal Constants 9 Ampere. The constant current that, if maintained in two straight parallel con ductors of infinite length and negligible circular cross section, and placed one meter apart in vacuum, would produce between these conductors a force equal to 2 x 10 - 7 newton per meter of length. (A newton is the force that gives a mass of one kilogram an acceleration of one meter per second squared.) In our electromagnetic model there are three universal constants, in addition to the field quantities listed in Table 1-1. They relate to the properties of the free space (vacuum). They are as follows: velocity of electromagnetic wave (including light) in free space, c; permittivity of free space, e0; and permeability of free space, /J,0. Many experiments have been performed for precise measurement of the velocity of light, to many decimal places. For our purpose it is sufficient to remember that c ^ 3 x 10e (m/s). (in free space) (1-6) The other two constants, e0 and JX0, pertain to electric and magnetic phenomena, respectively: e0 is the proportionality constant between the electric flux density D and the electric field intensity E in free space, such that D = €0E; (in free space) (1-7) HQ is the proportionality constant between the magnetic flux density B and the mag netic field intensity H in free space, such that — B. (in free space) (1-8) The values of e0 and /i0 are determined by the choice of the unit system, and they are not independent. In the SI system (rationalizedt MKSA system), which is almost universally adopted for electromagnetics work, the permeability of free space is chosen to be /i0 = 4 7 r x l ( T 7 (H/m), (in free space) (1-9) where H/m stands for henry per meter. With the values of c and /j.0 fixed in Eqs. (1-6) and (1-9) the value of the permittivity of free space is then derived from the following ^ This system of units is said to be rationalized because the factor An does not appear in the Maxwell's equations (the fundamental postulates of electromagnetism). This factor, however, will appear in many derived relations. In the unrationalized MKSA system, (i0 would be 10 7 (H/m), and the factor An would appear in the Maxwell's equations. 10 1 The Electromagnetic Model TABLE 1-3 Universal Constants in SI Units Universal Constants Velocity of light in free space Permeability of free space Permittivity of free space Symbol c Ho e0 Value 3 x 108 4n x 10" 7 Unit m/s H/m F/m relationships: or 1 1 €n = x 10 - 9 C2\LQ 36TZ ^ 8.854 x 10~12 (F/m), (1-10) (1-11) where F/m is the abbreviation for farad per meter. The three universal constants and their values are summarized in Table 1-3. Now that we have defined the basic quantities and the universal constants of the electromagnetic model, we can develop the various subjects in electromagnetics. But, before we do that, we must be equipped with the appropriate mathematical tools. In the following chapter we discuss the basic rules of operation for vector algebra and vector calculus. Review Questions R.l-l What is electromagnetics? R.l-2 Describe two phenomena or situations, other than those depicted in Figs. 1-1 and 1-2, that cannot be adequately explained by circuit theory. R.l-3 What are the three essential steps in building an idealized model for the study of a scientific subject? R.l-4 What are the four fundamental SI units in electromagnetics? R.l-5 What are the four fundamental field quantities in the electromagnetic model? What are their units? R.l-6 What are the three universal constants in the electromagnetic model, and what are their relations? R.l-7 What are the source quantities in the electromagnetic model? 2 Vector Analysis ^""1 Introduction As we noted in Chapter 1, some of the quantities in electromagnetics (such as charge, current, and energy) are scalars; and some others (such as electric and magnetic field intensities) are vectors. Both scalars and vectors can be functions of time and posi tion. At a given time and position, a scalar is completely specified by its magnitude (positive or negative, together with its unit). Thus we can specify, for instance, a charge of — 1 fiC at a certain location at t = 0. The specification of a vector at a given loca tion and time, on the other hand, requires both a magnitude and a direction. How do we specify the direction of a vector? In a three-dimensional space, three numbers are needed, and these numbers depend on the choice of a coordinate system. Conversion of a given vector from one coordinate system to another will change these numbers. However, physical laws and theorems relating various scalar and vector quantities certainly must hold irrespective of the coordinate system. The general expressions of the laws of electromagnetism, therefore, do not require the specification of a coordi nate system. A particular coordinate system is chosen only when a problem of a given geometry is to be analyzed. For example, if we are to determine the magnetic field at the center of a current-carrying wire loop, it is more convenient to use rectangular coordinates if the loop is rectangular, whereas polar coordinates (two-dimensional) will be more appropriate if the loop is circular in shape. The basic electromagnetic relation governing the solution of such a problem is the same for both geometries. Three main topics will be dealt with in this chapter on vector analysis: 1. Vecior algebra—addition, subtraction, and multiplication of vectors. 2. Orthogonal coordinate systems—Cartesian, cylindrical, and spherical coordi nates. 3. Vector calculus—differentiation and integration of vectors; line, surface, and volume integrals; "del" operator; gradient, divergence, and curl operations. 11 12 2 Vector Analysis Throughout the rest of this book we will decompose, combine, differentiate, integrate, and otherwise manipulate vectors. It is imperative to acquire a facility in vector algebra and vector calculus. In a three-dimensional space a vector relation is, in fact, three scalar relations. The use of vector-analysis techniques in electromagnetics leads to concise and elegant formulations. A deficiency in vector analysis in the study of elec tromagnetics is similar to a deficiency in algebra and calculus in the study of physics; and it is obvious that these deficiencies cannot yield fruitful results. In solving practical problems we always deal with regions or objects of a given shape, and it is necessary to express general formulas in a coordinate system appro priate for the given geometry. For example, the familiar rectangular (x, y, z) coordi nates are, obviously, awkward to use for problems involving a circular cylinder or a sphere because the boundaries of a circular cylinder and a sphere cannot be de scribed by constant values of x, y, and z. In this chapter we discuss the three most commonly used orthogonal (perpendicular) coordinate systems and the representa tion and operation of vectors in these systems. Familarity with these coordinate systems is essential in the solution of electromagnetic problems. Vector calculus pertains to the differentiation and integration of vectors. By de fining certain differential operators we can express the basic laws of electromagnetism in a concise way that is invariant with the choice of a coordinate system. In this chap ter we introduce the techniques for evaluating different types of integrals involving vectors, and we define and discuss the various kinds of differential operators. 2—2 Vector Addition and Subtraction We know that a vector has a magnitude and a direction. A vector A can be written as A = AA, (2-1) where A is the magnitude (and has the unit and dimension) of A, A = |A|, (2-2) and aA is a dimensionless unit vector with a unity magnitude having the direction of A. Thus, A A "=iArr (2-3) The vector A can be represented graphically by a directed straight-line segment of a length |A| = A with its arrowhead pointing in the direction of aA, as shown in Fig. 2-1. Two vectors are equal if they have the same magnitude and the same direction, even t In some books the unit vector in the direction of A is variously denoted by A, uA, or iA. We prefer to write A as in Eq. (2-1) instead of as A = XA. A vector going from point P1 to point P2 will then be written as aPlP2(P1P2) instead of as P1P2(P1P2), which is somewhat cumbersome. The symbols u and i are used for velocity and current, respectively. FIGURE 2-1 Graphical representation of vector A. though they may be displaced in space. Since it is difficult to write boldfaced letters by hand, it is a common practice to use an arrow or a bar over a letter (A or A) or a wiggly line under a letter (A) to distinguish a vector from a scalar. This distinguish ing mark, once chosen, should never be omitted whenever and wherever vectors are written. Two vectors A and B, which are not in the same direction nor in opposite direc tions, such as given in Fig. 2-2(a), determine a plane. Their sum is another vector C in the same plane. C = A + B can be obtained graphically in two ways. 1. By the parallelogram rule: The resultant C is the diagonal vector of the parallelo gram formed by A and B drawn from the same point, as shown in Fig. 2-2(b). 2. By the head-to-tail rule: The head of A connects to the tail of B. Their sum C is the vector drawn from the tail of A to the head of B; and vectors A, B, and C form a triangle, as shown in Fig. 2-2(c). It is obvious that vector addition obeys the commutative and associative laws. Commutative law: A + B = B + A. (2-4) Associative law: A + (B + C) = (A + B) + C. (2-5) Vector subtraction can be defined in terms of vector addition in the following way: A - B = A + (-B), (2-6) where — B is the negative of vector B; that is, — B has the same magnitude as B, but its direction is opposite to that of B. Thus - B = (-aB)5. (2-7) The operation represented by Eq. (2-6) is illustrated in Fig. 2-3. 1 —- A (a) Two vectors, A and B. (b) Parallelogram rule. (c) Head-to-tail rule. FIGURE 2-2 Vector addition, C = A + B. 14 2 Vector Analysis (a) Two vectors, A and B. (b) Subtraction of vectors, A - B. FIGURE 2-3 Vector subtraction. 2—3 Products of Vectors Multiplication of a vector A by a positive scalar k changes the magnitude of A by k times without changing its direction (k can be either greater or less than 1). kk = aA(kA). (2-8) It is not sufficient to say "the multiplication of one vector by another" or "the prod uct of two vectors" because there are two distinct and very different types of products of two vectors. They are (1) scalar or dot products, and (2) vector or cross products. These will be defined in the following subsections. 2-3.1 SCALAR OR DOT PRODUCT The scalar or dot product of two vectors A and B, denoted by A • B, is a scalar, which equals the product of the magnitudes of A and B and the cosine of the angle between them. Thus, A • B 4 AB cos 'AB-(2-9) In Eq. (2-9) the symbol = signifies "equal by definition," and 9AB is the smaller angle between A and B and is less than n radians (180°), as indicated in Fig. 2-4. The dot product of two vectors (1) is less than or equal to the product of their magnitudes; (2) can be either a positive or a negative quantity, depending on whether the angle between them is smaller or larger than n/2 radians (90°); (3) is equal to the product B cos BAB | FIGURE 2-4 Illustrating the dot product of A and B. 2-3 Products of Vectors 15 of the magnitude of one vector and the projection of the other vector upon the first one; and (4) is zero when the vectors are perpendicular to each other. It is evident that A • A = A2 or A= VA-A. (2-10) (2-11) Equation (2-11) enables us to find the magnitude of a vector when the expression of the vector is given in any coordinate system. The dot product is commutative and distributive. Commutative law: Distributive law: A • B = B • A. A • (B + C) = A • B + A • C. (2-12) (2-13) The commutative law is obvious from the definition of the dot product in Eq. (2-9), and the proof of Eq. (2-13) is left as an exercise. The associative law does not apply to the dot product, since no more than two vectors can be so multiplied and an ex pression such as A • B • C is meaningless. EXAMPLE 2-1 Prove the law of cosines for a triangle. Solution The law of cosines is a scalar relationship that expresses the length of a side of a triangle in terms of the lengths of the two other sides and the angle between them. Referring to Fig. 2-5, we find the law of cosines states that C = ^A2 + B2 - 2AB cos a. We prove this by considering the sides as vectors; that is, C = A + B. Taking the dot product of C with itself, we have, from Eqs. (2-10) and (2-13), C2 = C • C = (A + B) • (A + B) = A A + B B + 2 A B = A2 + B2 + 1AB cos 9AB. FIGURE 2-5 Illustrating Example 2-1. 16 2 Vector Analysis Note that 9AB is, by definition, the smaller angle between A and B and is equal to (180° — a); hence cos 9AB = cos (180° — a) = —cos a. Therefore, C2 = A2 + B2 - 1AB cos a, and the law of cosines follows directly. n 2-3.2 VECTOR OR CROSS PRODUCT The vector or cross product of two vectors A and B, denoted by A x B, is a vector perpendicular to the plane containing A and B; its magnitude is AB sin 9AB, where 9AB is the smaller angle between A and B, and its direction follows that of the thumb of the right hand when the fingers rotate from A to B through the angle 9AB (the right-hand rule). A x B ^ an\AB sin 9AB\ (2-14) This is illustrated in Fig. 2-6. Since B sin 9AB is the height of the parallelogram formed by the vectors A and B, we recognize that the magnitude ofyA x B, \AB sin 9AB\, which is always positive, is numerically equal to the area of the parallelogram. Using the definition in Eq. (2-14) and following the right-hand rule, we find that B x A = - A x B. (2-15) Hence the cross product is not commutative. We can see that the cross product obeys the distributive law, A x ( B + C) = A x B + A x C . (2-16) Can you show this in general without resolving the vectors into rectangular components? The vector product is obviously not associative; that is, A x (B x C) # (A x B) x C. (2-17) A X (a) A x B = an\AB sin QAB. (b) The right-hand rule. FIGURE 2-6 Cross product of A and B, A x B. 2-3 Products of Vectors 17 The vector representing the triple product on the left side of the expression above is perpendicular to A and lies in the plane formed by B and C, whereas that on the right side is perpendicular to C and lies in the plane formed by A and B. The order in which the two vector products are performed is therefore vital, and in no case should the parentheses be omitted. EXAMPLE 2-2 The motion of a rigid disk rotating about its axis shown in Fig. 2-7(a) can be described by an angular velocity vector co. The direction of co is along the axis and follows the right-hand rule; that is, if the fingers of the right hand bend in the direction of rotation, the thumb points to the direction of co. Find the vector expression for the lineal velocity of a point on the disk, which is at a distance d from the axis of rotation. Solution From mechanics we know that the magnitude of the lineal velocity, v, of a point P at a distance d from the rotating axis is cod and the direction is always tangential to the circle of rotation. However, since the point P is moving, the direc tion of v changes with the position of P. How do we write its vector representation? Let 0 be the origin of the chosen coordinate system. The position vector of the point P can be written as R, as shown in Fig. 2-7(b). We have |v| = cod = coR sin 9. No matter where the point P is, the direction of v is always perpendicular to the plane containing the vectors co and R. Hence we can write, very simply, v = co x R, which represents correctly both the magnitude and the direction of the lineal velocity of P. mm > 1 FIGURE 2-7 (a) A rotating disk. (b) Vector representation. Illustrating Example 2-2. 18 2 Vector Analysis / 7" / ' / / / / mi£,-'~ FIGURE 2-8 Area = |B x C| B Illustrating scalar triple product A • (B x C). 2-3.3 PRODUCT OF THREE VECTORS There are two kinds of products of three vectors; namely, the scalar triple product and the vector triple product. The scalar triple product is much the simpler of the two and has the following property: A • (B x C) = B • (C x A) = C • (A x B). (2-18) Note the cyclic permutation of the order of the three vectors A, B, and C. Of course, A ( B x C)= - A ( C x B) = - B ( A x C) = - C ( B x A). (2-19) As can be seen from Fig. 2-8, each of the three expressions in Eq. (2—18) has a magni tude equal to the volume of the parallelepiped formed by the three vectors A, B, and C. The parallelepiped has a base with an area equal to |B x C| = \BC sin 0X\ and a height equal to \A cos 62\; hence the volume is \ABC sin 61 cos 02. The vector triple product A x (B x C) can be expanded as the difference of two simple vectors as follows: A x (B x C) = B(A • C) - C(A • B). (2-20) Equation (2-20) is known as the "back-cab" rule and is a useful vector identity. (Note "BAC-CAB" on the right side of the equation!) EXAMPLE 2-3f Prove the back-cab rule of vector triple product. f The back-cab rule can be verified in a straightforward manner by expanding the vectors in the Cartesian coordinate system (Problem P.2-12). Only those interested in a general proof need to study this example. 19 B(A|, • C ) ^ .' , i i /-C(A||.B) D FIGURE 2-9 ~~ab Illustrating the back-cab rule of vector triple product. Solution In order to prove Eq. (2-20) it is convenient to expand A into two components: A = AN + A±, where AN and A± are parallel and perpendicular, respectively, to the plane containing B and C. Because the vector representing (B x C) is also perpendicular to the plane, the cross product of A± and (B x C) vanishes. Let D = A x (B x C). Since only A(| is effective here, we have D = A,, x (B x C). Referring to Fig. 2-9, which shows the plane containing B, C, and A||, we note that D lies in the same plane and is normal to A([. The magnitude of (B x C) is BC sin {d1 - B2\ and that of AN x (B x C) is AUBC sin (61 - 02). Hence, D = D • aD = AnBC sin {B1 - 62) = {B sin &X){A\C cos 62) - (C sin 02){AnB cos 0J = [B(A | |-C)-C(A | |.B)]-a J ). The expression above does not alone guarantee the quantity inside the brackets to be D, since the former may contain a vector that is normal to D (parallel to AN); that is, D • aD = E • aD does not guarantee E = D. In general, we can write B(AN • C) - C(AH • B) = D + /cA||5 where k is a scalar quantity. To determine k, we scalar-multiply both sides of the above equation by A(| and obtain (A„ • B)(A|, • C) - (A,, • C)(A,| • B) = 0 = A„ • D + kA^. Since A,, • D = 0, then k = 0 and D = B(A|, • C) - C(A|, • B), which proves the back-cab rule, inasmuch as A,, • C = A • C and A M • B = A • B. Division by a vector is not defined, and expressions such as k/A and B/A are meaningless. 20 2 Vector Analysis 2—4 Orthogonal Coordinate Systems We have indicated before that although the laws of electromagnetism are invariant with coordinate system, solution of practical problems requires that the relations derived from these laws be expressed in a coordinate system appropriate to the geome try of the given problems. For example, if we are to determine the electric field at a certain point in space, we at least need to describe the position of the source and the location of this point in a coordinate system. In a three-dimensional space a point can be located as the intersection of three surfaces. Assume that the three families of surfaces are described by u1 = constant, u2 = constant, and u3 = constant, where the M'S need not all be lengths. (In the familiar Cartesian or rectangular coordinate system, wl5 u2, and w 3 correspond to x, y, and z, respectively.) When these three surfaces are mutually perpendicular to one another, we have an orthogonal coordinate system. Nonorthogonal coordinate systems are not used because they complicate problems. Some surfaces represented by ut = constant (i = 1, 2, or 3) in a coordinate system may not be planes; they may be curved surfaces. Let aUl, aU2, and aU3 be the unit vectors in the three coordinate directions. They are called the base vectors. In a general right-handed, orthogonal, curvilinear coordinate system the base vectors are arranged in such a way that the following relations are satisfied: aUl x aU2 - aU3, (2-21a) aU2 x aU3 = aUl, (2-21b) aU3 x a Ul = a„2. (2"21c) These three equations are not all independent, as the specification of one automati cally implies the other two. We have, of course, aUl ' aU2 = aU2 • aU3 = aU3 • aUl = 0 (2-22) and aUl ' aUl = aU2 • aU2 = aU3 • aU3 = 1. (2-23) Any vector A can be written as the sum of its components in the three orthogonal directions, as follows: A — %U1AUI + %U2AU2 + ^U3AU3. (2-24) From Eq. (2-24) the magnitude of A is A = \A\ = (A 2 U1 + Al2 + Alf>\ (2-25) EXAMPLE 2-4 Given three vectors A, B, and C, obtain the expressions of (a) A • B, (b) A x B, and (c) C • (A x B) in the orthogonal curvilinear coordinate system (uls u2, u3). 2-4 Orthogonal Coordinate Systems 21 Solution First we write A, B, and C in the orthogonal coordinates (wl9 u2, u3): A = KAm + U2AU2 + uA«3> B = aUlBUl + aU2BU2 + aU3BU3, a) A • B = {auAUi + KAu2 + KAu3)' (a« A , + u2BU2 + K3BU3) = KK + Au2BU2 + AU3BU3> in view of Eqs. (2-22) and (2-23). b) A x B = (&U1AU1 + auAu2 + uA«3) x (a A i + au2BU2 + K3BU3) = uM«2B«3 ~ Au3 Bu2) + K2AU3BU1 - AUiBU3) + aU3(AUiBU2 - AU2BUi) |a„. a„„ a, (2-26) A,. A. BUi BU2 BU3 (2-27) Equations (2-26) and (2-27) express the dot and cross products, respectively, of two vectors in orthogonal curvilinear coordinates. They are important and should be remembered. c) The expression for C • (A x B) can be written down immediately by combining the results in Eqs. (2-26) and (2-27): C • (A x B) = CUi(AU2BU3 - AU3BU2) + CU2(AU3BUl - AUlBU3) + CU3(AUiBU2 - AU2BU) C, A„ A„ BUI BU2 BU3 (2-28) Eq. (2-28) can be used to prove Eqs. (2-18) and (2-19) by observing that a per mutation of the order of the vectors on the left side leads simply to a rearrange ment of the rows in the determinant on the right side. m In vector calculus (and in electromagnetics work) we are often required to per form line, surface, and volume integrals. In each case we need to express the differential length-change corresponding to a differential change in one of the coordinates. How ever, some of the coordinates, say ui (i = 1, 2, or 3), may not be a length; and a con version factor is needed to convert a differential change dut into a change in length dtff. ti^hidUi, (2-29) where ht is called a metric coefficient and may itself be a function of ult u2, and u3. For example, in the two-dimensional polar coordinates (ux, u2) = (r, 0), a differential change d(j) ( = du2) in $ ( = u2) corresponds to a differential length-change d£2 = rd(j) (h2 = r = Uj) in the a0 (= aU2)-direction. A directed differential length-change in an 22 2 Vector Analysis arbitrary direction can be written as the vector sum of the component length-changes: or M = M ^ i dui) + au2{h2 du2) + a„3(/z3 du3). d£ = aMl d^ + aM2 d£2 + aU3 d£3 (2-30)t . (2-31) In view of Eq. (2-25) the magnitude of d€ is ^ = [ ( ^ ) 2 + (^2)2 + (^ 3) 2] 1 / 2 3 = [(^ du,)2 + (h2 du2)2 + {h3 du3)2Y'2. The differential volume dv formed by differential coordinate changes dult du2, and du3 in directions aul, au2, and aM3, respectively, is (d^ d£2dt3\ or flfu = h,Ja2\i3 du1 du2 du3. (2-33) Later we will have occasion to express the current or flux flowing through a dif ferential area. In such cases the cross-sectional area perpendicular to the current or flux flow must be used, and it is convenient to consider the differential area a vector with a direction normal to the surface; that is, ds = a„ ds. (2-34) For instance, if current density J is not perpendicular to a differential area of a mag nitude ds, the current, dl, flowing through ds must be the component of J normal to the area multiplied by the area. Using the notation in Eq. (2-34), we can write simply dI = J-ds = J • a„ ds. (2-35) In general orthogonal curvilinear coordinates the differential area ds1 normal to the ds1 = d£2d£3 unit vector aMl is or dsx = h2h3du2du3. (2-36) Similarly, the differential areas normal to unit vectors aM2 and aU3 are, respectively, ds2 = /i1/i3^w1^u3 (2-37) t The t here is the symbol of a vector of length L 23 z = z\ plane y = y\ plane FIGURE 2-10 Cartesian coordinates. and ds3 = h1h2duldu2. (2-38) Many orthogonal coordinate systems exist; but we shall be concerned only with the three that are most common and most useful: 1. Cartesian (or rectangular) coordinates.1" 2. Cylindrical coordinates. 3. Spherical coordinates. These will be discussed separately in the following subsections. 2-4.1 CARTESIAN COORDINATES (ttl9 u2, M3) = {x, y, z) A point P(x1, ylt zx) in Cartesian coordinates is the intersection of three planes speci fied by x = xu y = ylt and z = zl9 as shown in Fig. 2-10. It is a right-handed system with base vectors ax, ay, and az satisfying the following relations: (2-39a) (2-39b) az x ax = av. (2-39c) f The term "Cartesian coordinates" is preferred because the term "rectangular coordinates" is customarily associated with two-dimensional geometry. The position vector to the point P(xu yu zt) is OP = axXi + ayy1 + azzv A vector A in Cartesian coordinates can be written as 2 Vector Analysis (2-40) A = axAx + 2LyAy + azAz. The dot product of two vectors A and B is, from Eq. (2-26), A • B = AXBX + AVBV + AZBZ, (2-41) (2-42) and the cross product of A and B is, from Eq. (2-27), A x B = ax(AyBz - AzBy) + ay(AzBx - AXBZ) + az(AxBy - AyBx) ax Ax Br a, Ay Bv az A B (2-43) Since x, y, and z are lengths themselves, all three metric coefficients are unity; that is, /ij = h2 = h3 = 1. The expressions for the differential length, differential area, and differential volume are—from Eqs. (2-31), (2-36), (2-37), (2-38), and (2-33)— respectively, d€ = axdx + a dy + az dz; dsx — dy dz, dsy = dx dz, dsz = dx dy; and dv = dx dy dz. (2-44) (2-45a) (2-45b) (2-45c) (2-46) A typical differential volume element at a point (x, y, z) resulting from differential changes dx, dy, and dz is shown in Fig. 2-11. The differential surface areas dsx, dsy, and dsz normal to the directions ax, ay, and az are also indicated. EXAMPLE 2-5 Given A = ax5 - ay2 + az, find the expression of a unit vector B such that a) B||A. b) B 1 A, if B lies in the x_y-plane. 25 dsx = dydz ds^dxjy dsv = dxdz y FIGURE 2-11 A differential volume in Cartesian coordinates. Solution Let B = nxBx + ayBy + azBz. We know that |B| = (B2 X + B) + B2 zy'2 = 1. (2-47) a) B || A requires B x A = 0. From Eq. (2-43) we have -2BZ-By = 0, (2-48a) Bx - 5BZ = 0, (2-48b) 5By + 2BX = 0. (2-48c) The above three equations are not all independent. For instance, subtracting Eq. (2-48c) from twice Eq. (2-48b) yields Eq. (2-48a). Solving Eqs. (2-47), (2-48a), and (2-48b) simultaneously, we obtain B = Therefore, By=~ and B,= 1 B = -f= (ax5 - a„2 + az). b) B 1 A requires B • A = 0. From Eq. (2-42) we have 5BX - 2By = 0, (2-49) where we have set Bz = 0, since B lies in the xy-plane. Solution of Eqs. (2-47) and (2-49) yields 2 5 B = and , = - -Hence, B = (a2 + a,5). EXAMPLE 2-6 (a) Write the expression of the vector going from point P^l, 3, 2) to point P2(3, — 2, 4) in Cartesian coordinates, (b) What is the length of this line? 26 2 Vector Analysis Z 4 P2(3>-2'4) y FIGURE 2-12 Illustrating Example 2-6. Solution a) From Fig. 2-12 we see that T\F2 = oF2 - W[ = (ax3 - ay2 + az4) - (ax + a„3 + az2) = ax2 — ay5 + az2. b) The length of the line is PXP2 = \PXP: = V22 + (-5)2 + 22 = J33. EXAMPLE 2-7 The equation of a straight line in the xy-plane is given by 2x + y = 4. a) Find the vector equation of a unit normal from the origin to the line. b) Find the equation of a line passing through the point P(0, 2) and perpendicular to the given line. Solution It is clear that the given equation y = — 2x + 4 represents a straight line having a slope —2 and a vertical intercept +4, shown as Lx (solid line) in Fig. 2-13. a) If the line is shifted down four units, we have the dashed parallel line L\ passing through the origin whose equation is 2x + y = 0. Let the position vector of a point on L\ be r = axx + ayy. The vector N = ax2 + ay is perpendicular to L\ because N • r = 2x + y = 0. Obviously, N is also perpendicular to Lv Thus, the vector equation of the unit normal at the origin is aw = N INI = -p(a x2 + a,). 27 Lx y >x FIGURE 2-13 Illustrating Example 2-7. Note that the slope of siN (=%) is the negative reciprocal of that of lines Lx and L;(=-2). b) Let the line passing through the point P(0, 2) and perpendicular to Lx be L2. L2 is parallel to and has the same slope as a^. The equation of L2 is then ' = 2 + 2 ' or x - 2y = - 4 , since L2 is required to pass through the point P(0, 2). 2-4.2 CYLINDRICAL COORDINATES (wl9 u2, u3) = (r, , z) In cylindrical coordinates a point P(rl9 #l9 zj is the intersection of a circular cylin drical surface r = rl3 SL half-plane containing the z-axis and making an angle <f) = (j)1 with the xz-plane, and a plane parallel to the xy-plane at z = zx. As indicated in Fig. 2-14, angle 0 is measured from the positive x-axis, and the base vector a^, is z = z\ plane r = r\ cylinder = \ plane FIGURE 2-14 Cylindrical coordinates. 28 2 Vector Analysis tangential to the cylindrical surface. The following right-hand relations apply: ar x a^ = az, (2-50a) a^ x az = a,, (2-50b) az x a, = a^. (2-50c) Cylindrical coordinates are important for problems with long line charges or currents, and in places where cylindrical or circular boundaries exist. The two-dimensional polar coordinates are a special case at z = 0. A vector in cylindrical coordinates is written as A — &rAr + fLfkAj, + &ZAZ. (2-51) The expressions for the dot and cross products of two vectors in cylindrical coordi nates follow from Eqs. (2-26) and (2-27) directly. Two of the three coordinates, r and z (u1 and u3), are themselves lengths; hence h1 = h3 = 1. However, + az dz. The expressions for differential areas and differential volume are dsr ds^ dsz = rd<p dz, = dr dz, — rdrd(j), and dv = rdr d<p dz. (2-52) (2-53a) (2-5 3b) (2-5 3c) (2-54) A typical differential volume element at a point (r, </>, z) resulting from differential changes dr, d4>, and dz in the three orthogonal coordinate directions is shown in Fig. 2-15. A vector given in cylindrical coordinates can be transformed into one in Cartesian coordinates, and vice versa. Suppose we want to express A = arAr + a^A^ + azAz in Cartesian coordinates; that is, we want to write A as axAx + ayAy + azAz and deter mine Ax, Ay, and Az. First of all, we note that Az, the z-component of A, is not changed by the transformation from cylindrical to Cartesian coordinates. To find Ax, we equate the dot products of both expressions of A with ax. Thus K = A • ax = Arar - &x + A^ • ax. 29 ZA ds^ = drdz dsr = rd (2-55) and Hence, a,/,' ax = cos - + ) = — sin 0. (2-56) Ax = Ar cos ( / > — A$ sin 4>. (2-57) Similarly, to find Ay, we take the dot products of both expressions of A with ay: Ay = A • a, = Arar • ay + A^ • a r From Fig. 2-16 we find that and It follows that ar • ay = cos I — — 0 I = sin 0 a0 • ay = cos (/>. A, = Ar sin 0 + A$ COS 0. (2-58) (2-59) (2-60) FIGURE 2-16 Relations between ax, ay, ar, and a^ 2 Vector Analysis It is convenient to write the relations between the components of a vector in Cartesian and cylindrical coordinates in a matrix form: (2-61) Our problem is now solved except that the cos 4> and sin 4> in Eq. (2-61) should be converted into Cartesian coordinates. Moreover, Ar, A^, and Az may themselves be functions of r, 4>, and z. In that case, they too should be converted into functions of x, y, and z in the final answer. The following conversion formulas are obvious from Fig. 2-16. From cylindrical to Cartesian coordinates: rp^ k U . j = cos 4> sin 4> 0 — sin 4> cos 4> 0 o] 0 lj IV K V X = y = z -= r cos = r sin = z. (f), $, The inverse relations (from Cartesian to cylindrical coordinates) are r = V2 + y2, 0 = tan" 1^, x z = z. (2-62a) (2-62b) (2-62c) (2-63a) (2-63b) (2-63c) EXAMPLE 2-8 The cylindrical coordinates of an arbitrary point P in the z = 0 plane are (r, #, 0). Find the unit vector that goes from a point z = h on z-axis toward P. Solution Referring to Fig. 2-17, we have QP = OP-OQ = (arr) - (a,). Hence, 6J° 1 aQp — \QP\ J^+l? (a/ - az/i). EXAMPLE 2-9 Express the vector A = ar(3 cos #) - a^2r + az5 in Cartesian coordinates. 31 P{r,4>,0) y FIGURE 2-17 Illustrating Example 2-Solution Using Eq. (2-61) directly, we have Ax Ay UzJ = cos 4> sin cj) _0 — sin (f) cos 4> 0 0 0 lj [3 cos 4> -2r L 5 or A = a.,(3 cos2 4> + 2r sin (f>) + ay(3 sin 0 cos cj) — 2r cos 4>) + az5. But, from Eqs. (2-62) and (2-63), x and Therefore, A = a 3x2 ^ x2 + y2 COS (j) = sin (f> = + 2y + a •fi V? _1 Q 2 + / y + y2 ( 3xy y\x2 + y2 2x + az5, which is the desired answer. 2-4.3 SPHERICAL COORDINATES {uu u2, u3) = (R, 9, 0) A point P(#1? 9X, 0 J in spherical coordinates is specified as the intersection of the fol lowing three surfaces: a spherical surface centered at the origin with a radius R = R{, a right circular cone with its apex at the origin, its axis coinciding with the + z-axis and having a half-angle 9 = 9X; and a half-plane containing the z-axis and making an angle <fi = <>1 with the xz-plane. The base vector SLR at P is radial from the origin and is quite different from ar in cylindrical coordinates, the latter being perpendicular to the z-axis. The base vector a0 lies in the 0 = 0X plane and is tangential to the 32 2 Vector Analysis FIGURE 2-18 Spherical coordinates. spherical surface, whereas the base vector a^ is the same as that in cylindrical coor dinates. These are illustrated in Fig. 2-18. For a right-handed system we have aR x ag = a^, a e x a4> = a « 5 a< x aR = a e -(2-64a) (2-64b) (2-64c) Spherical coordinates are important for problems involving point sources and regions with spherical boundaries. When an observer is very far from the source region of a finite extent, the latter could be considered as the origin of a spherical coordinate system; and, as a result, suitable simplifying approximations could be made. This is the reason that spherical coordinates are used in solving antenna problems in the far field. A vector in spherical coordinates is written as A — aRAR + aeAe + a^A^. (2-65) The expressions for the dot and cross products of two vectors in spherical coor dinates can be obtained from Eqs. (2-26) and (2-27). In spherical coordinates, only R(uy) is a length. The other two coordinates, 6 and 4> (u2 and u3), are angles. Referring to Fig. 2-19, in which a typical differential volume element is shown, we see that metric coefficients h2 = R and h3 = R sin 6 are required to convert d6 and dfy into d£2 and d£3i respectively. The general expression for a differential length is, from Eq. (2-31), d€ = aRdR + aeR d9 + a^R sin 9 d<>. (2-66) 33 rfsA = RdRdB dse = R sin 0 dfl d VRdd —dsR = R2 sin 0 rf0 ^ R sin 0 c f y !
FIGURE 2-19 A differential volume element in spherical coordinates. The expressions for differential areas and differential volume resulting from differen tial changes dR, d6, and d(j) in the three coordinate directions are dsR = R2 sin0 ddd(j), dse = R sin 6 dR d(j), ds(j) = RdRdB, and dv = R2 sin 6 dRdOd. (2-67a) (2-67b) (2-67c) (2-68) For convenience the base vectors, metric coefficients, and expressions for the differen tial volume are tabulated in Table 2 - 1 . TABLE 2-1 Three Basic Orthogonal Coordinate Systems Coordinate System Base vectors Metric coefficients Differential volume Relations aUl a«2 aU3 h2 h3 dv Cartesian Coordinates (x, y, z) a a, az 1 1 1 dx dy dz Cylindrical Coordinates (r, ' 1 R = jx2 + y2 + z2, = tan 0 = tan~ 1^-x (2-70a) (2-70b) (2-70c) EXAMPLE 2-10 The position of a point P in spherical coordinates is (8, 120°, 330°). Specify its location (a) in Cartesian coordinates, and (b) in cylindrical coordinates. Solution The spherical coordinates of the given point are R = 8, 9 = 120°, and 0 = 330°. a) In Cartesian coordinates. We use Eqs. (2-69a, b, c): x = 8 sin 120° cos 330° = 6, y = 8 sin 120° sin 330° = - 2 ^ 3 , z = 8 cos 120° = - 4 . Hence the location of the point is P(6, —2^3, —4), and the position vector (the vector going from the origin to the point) is a P = a x 6 - a y 2 V 3 - a z 4 . b) In cylindrical coordinates. The cylindrical coordinates of point P can be obtained by applying Eqs. (2-63a, b, c) to the results in part (a), but they can be cal culated directly from the given spherical coordinates by the following relations, which can be verified by comparing Figs. 2-14 and 2-18: r = R sin 6, (2-7la) = <>, (2-71b) z = R cos 6. (2-71c) We have Pflyfe, 330°, -4); and its position vector in cylindrical coordinates is ~0P = a/k/3 - a A msi 2-4 Orthogonal Coordinate Systems 35 We note here that the position vector of a point in cylindrical coordinates does not contain the angle (j) = 330° explicitly. However, the exact direction of ar depends on 4>. In terms of spherical coordinates the position vector (the vector from the origin to the point P) consists of only a single term: OP = a^8. Here the direction of aR changes with the 9 and (j) coordinates of the point P. EXAMPLE 2-11 Convert the vector A = aRAR + agAg + a^A^ into Cartesian co ordinates. Solution In this problem we want to write A in the form of A = axAx + ayAy + azAz. This is very different from the preceding problem of converting the coordinates of a point. First of all, we assume that the expression of the given vector A holds for all points of interest and that all three given components AR, A9, and A$ may be functions of coordinate variables. Second, at a given point, AR, Ag, and A$ will have definite numerical values, but these values that determine the direction of A will, in general, be entirely different from the coordinate values of the point. Taking dot product of A with ax, we have Ax = A • ax — ARaR ' ax + Agae • ax + A^a^ • ax. Recalling that aR • ax, ae • ax, and a^ • ax yield, respectively, the component of unit vectors aR, ag, and a^ in the direction of ax, we find, from Fig. 2-19 and Eqs. (2-69a, b, c): x a • ax = sin 9 cos 0 = =, (2-72) yfx2 + y2 + z2 afl • ax = cos v cos = — y • (2-74) V + y Ax = AR sin 9 cos \ + Ag cos 9 cos 4> — A^ sin 4> ARx Agxz Asy ~ " + ,, , ' „ , --=L=> (2-75) Vx2 4- y2 +~7 J(x2 + y2)(x2 +y2 + z2) y/xT+J2 Similarly, Ay = AR sin 9 sin (j> + Ae cos 9 sin (j> + A^ cos 0 A«y + A°yz 4. Ax n 7« Vx2 + y2 + z2 V(x2 + j;2)(x2 + y2 + z2) Jx2 + y2 2 Vector Analysis and A, = AR cos 9 - Ae sin 9 = AR\ - - ^ = = = - (2-77) Vx2 + y2 + z2 V + y2 + z2 If AR, Ag, and A$ are themselves functions of R, 9, and <$>, they too need to be con verted into functions of x, y, and z by the use of Eqs. (2-70a, b, c). Equations (2-75), (2-76), and (2-77) disclose the fact that when a vector has a simple form in one coordinate system, its conversion into another coordinate system usually results in a more complicated expression. IM EXAMPLE 2-12 Assuming that a cloud of electrons confined in a region between two spheres of radii 2 and 5 (cm) has a charge density of —3 y 10~8 4 cos2 e f t (C/m3), find the total charge contained in the region. Solution We have 3 x 1(T8 , , P= ^4—cosz(j), Q = §pdv. The given conditions of the problem obviously point to the use of spherical coordi nates. Using the expression for dv in Eq. (2-68), we perform a triple integration: Q = j>npoj0 o^pR2sm9dRd9dct>. Two things are of importance here. First, since p is given in units of coulombs per cubic meter, the limits of integration for R must be converted to meters. Second, the full range of integration for 9 is from 0 to n radians, not from 0 to 2n radians. A little reflection will convince us that a half-circle (not a full-circle) rotated about the z-axis through 2n radians ((f) from 0 to 2n) generates a sphere. We have Q - - 3 x 10-8 |o 27C Jo" Jo°o°2 5 ^ j cos2 0 sin 9dRd9d(j> = - 0 . 9 x 10" 6 r ( - c o s 0)| cos2 4>d<}> _6((j) , sin 2(f) = - 1 . 8 X 10~ C ,(y + 2rc = -1.87C (/xC). 0 2-5 Integrals Containing Vector Functions 37 2—5 Integrals Containing Vector Functions In electromagnetics work we have occasion to encounter integrals that contain vector functions such as jy¥dv, (2-78) Jc Vd€, (2-79) j c F • d€, (2-80) j s A • ds. (2-81) The volume integral in (2-78) can be evaluated as the sum of three scalar integrals by first resolving the vector F into its three components in the appropriate coordinate system. If dv denotes a differential volume, then (2-78) is actually a shorthand way of representing a triple integral over three dimensions. In the second integral, in (2-79), V is a scalar function of space, d£ represents a differential increment of length, and C is the path of integration. If the integration is to be carried out from a point Px to another point P2, we write \\ V d£. If the integration is to be evaluated around a closed path C, we denote it by §c V dt In Cartesian coordinates, (2-79) can be written as ScVd€ = Jc F (' y> z)[adx + M ? + dzl (2~82) in view of Eq. (2-44). Since the Cartesian unit vectors are constant in both magni tude and direction, they can be taken out of the integral sign, and Eq. (2-82) becomes jcVd€ = a, Jc V(x, y, z) dx + a, £ V{x, y, z) dy + az £ V(x, y, z) dz. (2-83) The three integrals on the right-hand side of Eq. (2-83) are ordinary scalar integrals; they can be evaluated for a given V(x, y, z) around a path C. i ■ EXAMPLE 2-13 Evaluate the integral {£ r2 dr, where r2 = x2 + y2, from the origin to the point P(l, 1): (a) along the direct path OP, (b) along the path OPxP, and (c) along the path OP2P in Fig. 2-20. Solution a) Along the direct path OP: 2V2 3 (ax cos 45° + ay sin 45°) — a x 3 + aj>3-2 Vector Analysis • />(U) + x FIGURE 2-20 Illustrating Example 2-13. b) Along the path OP^P: j P o (x2 + y2)dr = a, JJ 1 y2 dy + ax £ (x2 + l)dx = a ^ 3 | o + ax(ix3 + x) — ax3 + aj>3 • c) Along the path OP2P: j P o (x2 + y2)dr = ax JJ 2 x2 dx + a, £ (1 + y 2 ) ^ Obviously, the value of the integral depends on the path of integration, since the results in parts (a), (b), and (c) are all different. ana The integrals in (2-80) and (2-81) are mathematically of the same form; they both lead to a scalar result. The expression in (2-80) is a line integral, in which the integrand represents the component of the vector F along the path of integration. This type of scalar line integral is of considerable importance in both physics and electromagnetics. (If F is a force, the integral is the work done by the force in moving an object from an initial point Pt to a final point P2 along a specified path C; if F is replaced by E, the electric field intensity, then the integral represents the work done by the electric field in moving a unit charge from Pt to P2.) We will encounter it again later in this chapter and in many other parts of this book. EXAMPLE 2-14 Given F = &xxy — ay2x, evaluate the scalar line integral n F-df along the quarter-circle shown in Fig. 2-21. 2-5 Integrals Containing Vector Functions 39 Solution We shall solve this problem in two ways: first in Cartesian coordinates, then in cylindrical coordinates. a) In Cartesian coordinates. From the given F and the expression for d€ in Eq. (2-44) we have F • d€ = xy dx — 2x dy. The equation of the quarter-circle is x2 + y2 = 9 (0 < x, y < 3). Therefore, f F-d€ = J3° x JT^dx - 2 JQ 3 J9~^ydy = ~(9-x2f'2 = - 9 f l + -yy/9-y2 +9 sin-11 b) In cylindrical coordinates. Here we first transform F into cylindrical coordinates. Inverting Eq. (2-61), we have COS (j) -sin (f) 0 COS (j) — sin 0 -sin (p cos — 2x sin 4>) — a^xy sin ^ > + 2x cos 4>). For the present problem the path of integration is along a quarter-circle of a radius 3. There is no change in r or z along the path (dr = 0 and dz = 0); hence o] 0 lj r xy -2x L o _ >x FIGURE 2-21 Path for line integral (Example 2-14). 40 2 Vector Analysis Eq. (2-52) simplifies to and d€ = 4,3d F • d-6 = — 3{xy sin . Because of the circular path, Fr is immaterial to the present integration. Along the path, x = 3 cos <> and y = 3 sin (f). Therefore ^ Y-d£ = J^2 - 3(9 sin2 0 cos + 6 cos2 0 ) # = — 9(sin3 (j) + $ + sin 0 cos 0) 171/2 which is the same as before. UH In this particular example, F is given in Cartesian coordinates, and the path is circular. There is no compelling reason to solve the problem in one or the other co ordinates. We have shown the conversion of vectors and the procedure of solution in both coordinates. The expression in (2-81), j s A • ds, is a surface integral. It is actually a double integral over two dimensions; but it is written with a single integral sign for simplicity. The integral measures the flux of the vector field A flowing through the area S. In the integral the vector differential surface element ds = a„ ds has a magnitude ds and a direction denoted by the unit vector a„. The conventions for the positive direction of ds or a„ are as follows: 1. If the surface of integration, S, is a closed surface enclosing a volume, then the positive direction for a„ is always in the outward direction from the volume. This is illustrated in Fig. 2-22(a). We see that the positive direction of a„ depends on the location of ds. A small circle is added over the integral sign if the integration is to be performed over an enclosed surface: A • ds = C D A • an ds. (a) A closed surface. (b) An open surface. (c) A disk. FIGURE 2-22 Illustrating the positive direction of a„ in scalar surface integral. 2-5 Integrals Containing Vector Functions 41 2. If S is an open surface, the positive direction for a„ depends on the direction in which the perimeter of the open surface is traversed. This is illustrated in Fig. 2-22(b), in which a cup-shaped surface (with no lid) is shown. We apply the right-hand rule: If the fingers of the right hand follows the direction of travel around the perimeter, then the thumb points in the direction of positive a„. Here again, the positive direction of a„ depends on the location of ds. A plane, such as the disk in Fig. 2-22(c), is a special case of an open surface where a„ is a constant. EXAMPLE 2-15 Given F = arkjr + az/c2z, evaluate the scalar surface integral F-ds over the surface of a closed cylmder about the z-axis specified by z = ±3 and r = 2. Solution The specified surface of integration S is that of a closed cylinder shown in Fig. 2-23. The cylinder has three surfaces: the top face, the bottom face, and the side wall. We write s F - d s = (J) F-ands = f F-ands + f F-a„ds + f F-a„ds, Jtop " Jbottom " Jside " ' face face wall where a„ is the unit normal outward from the respective surfaces. The three integrals on the right side can be evaluated separately. a) Top face, z = 3, a„ = az, F • a„ = k2z = 3/c2, ds = rdrd(f) (from Eq. 2-53c); iP F-a»ds=rio3^^=12^-face FIGURE 2-23 A cylindrical surface (Example 2-15). 4 2 2 Vector Analysis b) Bottom face, z — — 3, a„ = — az, F • a„ = — k2z = 3k2, ds = rdr d; f F • n„ds = I2nk2, J bottom " 2 ' face which is exactly the same as the integral over the top face. c) Side wall, r = 2, a„ = ar, F - . = I - i , r 2 ds = rd(j)dz = 2d(f)dz (from Eq. 2-53a); wall Therefore, < j ) F • ds = 12nk2 + 12nk2 + 127^ = 12TT(/C1 + 2k2). This surface integral gives the net outward flux of the vector F through the closed cylindrical surface. mus 2 - 6 Gradient of a Scalar Field In electromagnetics we have to deal with quantities that depend on both time and position. Since three coordinate variables are involved in a three-dimensional space, we expect to encounter scalar and vector fields that are functions of four variables: (t, ultu2i u3). In general, the fields may change as any one of the four variables changes. We now address the method for describing the space rate of change of a scalar field at a given time. Partial derivatives with respect to the three space-coordinate variables are involved, and, inasmuch as the rate of change may be differ ent in different directions, a vector is needed to define the space rate of change of a scalar field at a given point and at a given time. Let us consider a scalar function of space coordinates V(uu u2, u3), which may represent, say, the temperature distribution in a building, the altitude of a moun tainous terrain, or the electric potential in a region. The magnitude of V, in general, depends on the position of the point in space, but it may be constant along certain lines or surfaces. Figure 2-24 shows two surfaces on which the magnitude of V is constant and has the values Vt and V1 + dV, respectively, where dV indicates a small change in V. We should note that constant- V surfaces need not coincide with any of the surfaces that define a particular coordinate system. Point P1 is on surface V{, P2 is the corresponding point on surface Vx + dV along the normal vector dn; and P3 is a point close to P2 along another vector d€ # dn. For the same change dV in V, the space rate of change, dVjdS, is obviously greatest along dn because dn is the 43 FIGURE 2-24 Concerning gradient of a scalar. shortest distance between the two surfaces. Since the magnitude of dV/d^ depends on the direction of d€, dV/dY is a directional derivative. We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar as the gradient of that scalar. We write gradK4a„J. (2-85) For brevity it is customary to employ the operator del, represented by the symbol V and write W in place of grad V. Thus, " ■■' ! (2-86) We have assumed that dV is positive (an increase in V); if dV is negative (a decrease in V from Px to P2), W will be negative in the a„ direction. The directional derivative along d£ is dV d\^dn ln"d£ dV dV = -— cos a dn (2-87) = -a„.a, = (W).v Equation (2-87) states that the space rate of increase of V in the SL^ direction is equal to the projection (the component) of the gradient of V in that direction. We can also write Eq. (2-87) as dV = {VV)-d€, (2-88) t In a more formal treatment, changes AV and A/ would be used, and the ratio AV/M would become the derivative dV/df as At approaches zero. We avoid this formality in favor of simplicity. 44 2 Vector Analysis where d£ = 2L€M. NOW, dV in Eq. (2-88) is the total differential of V as a result of a change in position (from Px to P 3 in Fig. 2-24); it can be expressed in terms of the differential changes in coordinates: dV dV dV (2-89) where dtu d£2, and d£3 are the components of the vector differential displacement &£ in a chosen coordinate system. In terms of general orthogonal curvilinear coordi nates (ul5 u2, %), d€ is (from Eq. 2-31), iUluv1 -r aU2uv2 + a «3 uv 3 ^ = aBl d^ + au = uSh\ dux) + aU2(h2 du2) + aU3(h3 du3). We can write dV in Eq. (2-89) as the dot product of two vectors, as follows: (2-90) dV = [ dv dV dv + au2d^ + a"3 W)'(a"'d^ + a" 2 d2 + a" 3 d^ dV dV dV\ ,„ = \^w1 + ^w2 + ^wjde-Comparing Eq. (2-91) with Eq. (2-88), we obtain dV dV dV V K _ a " 1 ^ 1 + a " 2 ^ + a " 3 ^ or (2-91) (2-92) (2-93) Equation (2-93) is a useful formula for computing the gradient of a scalar, when the scalar is given as a function of space coordinates. In Cartesian coordinates, (ux, u2, u3) = (x, y, z) and h1 = h2 = h3 = 1, we have (2-94) \v = dV = a„-— x dx + a j dv dy + a dV — dz or V F = l a^ + a4 + a zi , K (2-95) In view of Eq. (2-95), it is convenient to consider V in Cartesian coordinates as a vector differential operator. d d d V = ax — + a., — + az—• dx y dy dz (2-96) 2-6 Gradient of a Scalar Field 45 From Eq. (2-93), we see that we can define V as V = a h1 du1 + a„ hodu- + a U3h3du3 (2-97) in general orthogonal coordinates. As we shall see later in this chapter, the same vector differential operator is also used to signify divergence (V •) and curl (V x) operations on a vector. In these cases it is important to remember that the differentiation of a base vector in a curvilinear coordinate system may lead to a new vector in a dif ferent direction. (For instance, dar/d = a^ and da^/dcj) = — ar.) Proper care must be exercised when the V defined in Eq. (2-97) is used to operate on vectors in curvilinear coordinate systems. EXAMPLE 2-16 The electrostatic field intensity E is derivable as the negative gra dient of a scalar electric potential V; that is, E = — VV. Determine E at the point (1,1,0) if a) V= V " s i n ^ , b) V = E0R cos 9. Solution We use Eq. (2-93) to evaluate E = — W in Cartesian coordinates for part (a) and in spherical coordinates for part (b). a) E = 8 d d~ E0e x sin ny %y % ny = \ax sin —— a. — cos — \EQe Thus, E(l, 1, 0) = a, - a„ -n\ E0 where = aEE, E=Er a£ = 'K-L' 1 b) E = -+ afl + a, VlTF7l6) d n a^ - a, ER OR ' a° R89 ^"R sin 9 d E0R COS 9 = -{aR cos 9 - a0 sin 9)E0. In view of Eq. (2-77), the result above converts very simply to E = — a2£0 in Cartesian coordinates. This is not surprising, since a careful examination of the given V reveals that E0R cos 9 is, in fact, equal to E0z. In Cartesian coordinates, E = -\V= -t — (Eoz)= - a z £ 0 . 4" 2 Vector Analysis 2 - 7 Divergence of a Vector Field In the preceding section we considered the spatial derivatives of a scalar field, which led to the definition of the gradient. We now turn our attention to the spatial deriv atives of a vector field. This will lead to the definitions of the divergence and the curl of a vector. We discuss the meaning of divergence in this section and that of curl in Section 2-9. Both are very important in the study of electromagnetism. In the study of vector fields it is convenient to represent field variations graphically by directed field lines, which are called flux lines or streamlines. They are directed lines or curves that indicate at each point the direction of the vector field, as illustrated in Fig. 2-25. The magnitude of the field at a point is depicted either by the density or by the length of the directed lines in the vicinity of the point. Figure 2-25(a) shows that the field in region A is stronger than that in region B because there is a higher density of equal-length directed lines in region A. In Fig. 2-25(b), the decreasing arrow lengths away from the point q indicate a radial field that is strongest in the region closest to q. Figure 2-25(c) depicts a uniform field. The vector field strength in Fig. 2-25(a) is measured by the number of flux lines passing through a unit surface normal to the vector. The flux of a vector field is analogous to the flow of an incompressible fluid such as water. For a volume with an enclosed surface there will be an excess of outward or inward flow through the surface only when the volume contains a source or a sink, respectively; that is, a net positive divergence indicates the presence of a source of fluid inside the volume, and a net negative divergence indicates the presence of a sink. The net outward flow of the fluid per unit volume is therefore a measure of the strength of the enclosed source. In the uniform field shown in Fig. 2-25(c) there is an equal amount of inward and outward flux going through any closed volume containing no sources or sinks, result ing in a zero divergence. (a) (b) ( c ) FIGURE 2-25 Flux lines of vector fields. 2-7 Divergence of a Vector Field 47 We define the divergence of a vector field A at a point, abbreviated div A, as the net outward flux of A per unit volume as the volume about the point tends to zero: (2-98) The numerator in Eq. (2-98), representing the net outward flux, is an integral over the entire surface S that bounds the volume. We were exposed to this type of surface integral in Example 2-15. Equation (2-98) is the general definition of div A which is a scalar quantity whose magnitude may vary from point to point as A itself varies. This definition holds for any coordinate system; the expression for div A, like that for A, will, of course, depend on the choice of the coordinate system. At the beginning of this section we intimated that the divergence of a vector is a type of spatial derivative. The reader might perhaps wonder about the presence of an integral in the expression given by Eq. (2-98); but a two-dimensional surface in tegral divided by a three-dimensional volume will lead to spatial derivatives as the volume approaches zero. We shall now derive the expression for div A in Cartesian coordinates. Consider a differential volume of sides Ax, Ay, and Az centered about a point P(xo> ^o? ZO) m t n e fi^d of a vector A, as shown in Fig. 2-26. In Cartesian coordinates, A = axAx + ayAy + azAz. We wish to find div A at the point (x0, j ; 0 , z0). Since the differential volume has six faces, the surface integral in the numerator of Eq. (2-98) can be decomposed into six parts: A • ds = Jfront Jback Jright Jleft Jtop Jr. /front Jback face face ight face left face + , /top J bottom face face A • ds. (2-99) On the front face, /front A ' d S = Afront ' A%ront = A f r o n t ' &x(Ay Az) face face face face = Ax[x0 + —, y0, z0 ) Ay Az. (2-100) Pixo, yo, zo) WffT FIGURE 2-26 A differential volume in Cartesian coordinates. 2 Vector Analysis The quantity Ax([x0 + (Ax/2), y0, z0]) can be expanded as a Taylor series about its value at (x0, y0, z0), as follows: A i Ax \ , AxdAx Ax\xo + ~Y> J > o > zo J = Ax(o, yo, Zo) + - y -^ + higher-order terms, (x0, yo, zo) (2-101) where the higher-order terms (H.O.T.) contain the factors (Ax/2)2, (Ax/2)3, etc. Simi larly, on the back face, Jback A'ds = Aback ' Asback = Aback • (-axAyAz) f.„„ face face face face = ~Ax[ o - -j-> y°> z° I A y A z-The Taylor-series expansion of Axl x0 ——, j ; 0 , z0 ) is A l Ax \ ^ AxdAx Ax\x0- —, y0, z0 I = Ax(x0, y0, z0) - — — (2-102) + H.O.T. (2-103) (xo, yo, zo) Substituting Eq. (2-101) in Eq. (2-100) and Eq. (2-103) in Eq. (2-102) and adding the contributions, we have J front Jback face face A • ds = (—^ + H.O.T. \ ox AxAyAz. (2-104) (xo, yo, zo) Here a Ax has been factored out from the H.O.T. in Eqs. (2-101) and (2-103), but all terms of the H.O.T. in Eq. (2-104) still contain powers of Ax. Following the same procedure for the right and left faces, where the coordinate changes are + Ay/2 and — Ay/2, respectively, and As = Ax Az, we find Jright Jleft L face face. A-ds = (d^ + H.O.T. \dy Ax Ay Az. (2-105) (xo, yo, zo) Here the higher-order terms contain the factors Ay, (Ay)2, etc. For the top and bottom faces we have Jtop Jbi I lop J bottom face face A • ds = ( ^ + H.O.T. oz AxAyAz, (2-106) |(x0, yo, zo) where the higher-order terms contain the factors Az, (Az)2, etc. Now the results from Eqs. (2-104), (2-105), and (2-106) are combined in Eq. (2-99) to obtain A • ds = ■ (dAx dAv dA + + Ax Ay Az is \dx ' dy ' dz J(Xo,yo,Zo) " J ~ (2-107) + higher-order terms in Ax, Ay, Az. Since At; = AxAyAz, substitution of Eq. (2-107) in Eq. (2-98) yields the expression 2-7 Divergence of a Vector Field 49 of div A in Cartesian coordinates: div A = < ^ J C dx dAv + —-2-dy dAz + ^r-dz (2-108) The higher-order terms vanish as the differential volume Ax Ay Az approaches zero. The value of div A, in general, depends on the position of the point at which it is evaluated. We have dropped the notation (x0, y0, z0) in Eq. (2-108) because it applies to any point at which A and its partial derivatives are defined. With the vector differential operator del, V, defined in Eq. (2-96) we can write Eq. (2-108) alternatively as V • A; that is, V • A = div A. (2-109) In general orthogonal curvilinear coordinates (u1? u2, w3), Eq. (2-98) will lead to V-A = h1h2h2 ^ ( f c M J + ^ f c i M J + ^ M ^ a ) (2-110) EXAMPLE 2-17 Find the divergence of the position vector to an arbitrary point. Solution We will find the solution in Cartesian as well as in spherical coordinates. a) Cartesian coordinates. The expression for the position vector to an arbitrary point (x, y, z) is OP = axx + ayy + azz. (2-111) Using Eq. (2-108), we have — ► dx dy dz dx dy dz b) Spherical coordinates. Here the position vector is simply ~OP = nRR. (2-112) Its divergence in spherical coordinates (R, d, 0) can be obtained from Eq. (2-110) by using Table 2-1 as follows: V - A = F J ^ « ) + ^ i-AAesm6)+ X dA RsinO d(f> (2-113) Substituting Eq. (2-112) in Eq. (2-113), we also obtain V • (OP) = 3, as expected. 50 2 Vector Analysis EXAMPLE 2-18 The magnetic flux density B outside a very long current-carrying wire is circumferential and is inversely proportional to the distance to the axis of the wire. Find V • B. Solution Let the long wire be coincident with the z-axis in a cylindrical coordinate system. The problem states that k B = ^ -v r The divergence of a vector field in cylindrical coordinates (r, $, z) can be found from Eq. (2-110): (2-114) Now B^ = k/r, and Br = Bz = 0. Equation (2-114) gives \ • B = 0. mm We have here a vector that is not a constant, but whose divergence is zero. This property indicates that the magnetic flux lines close upon themselves and that there are no magnetic sources or sinks. A divergenceless field is called a solenoidal field. More will be said about this type of field later in the book. V B = r or r d(j> dz 2 — 8 Divergence Theorem In the preceding section we defined the divergence of a vector field as the net outward flux per unit volume. We may expect intuitively that the volume integral of the divergence of a vector field equals the total outward flux of the vector through the surface that bounds the volume; that is, I V- Adv = (b A-ds. (2-115) This identity, which will be proved in the following paragraph, is called the divergence theorem.^ It applies to any volume V that is bounded by surface S. The direction of ds is always that of the outward normal, perpendicular to the surface ds and directed away from the volume. For a very small differential volume element AVJ bounded by a surface sp the definition of V • A in Eq. (2-98) gives directly (y-A)jAvJ = j)s A-ds. (2-116) f It is also known as Gauss's theorem. 2-8 Divergence Theorem 51 In case of an arbitrary volume V, we can subdivide it into many, say N, small dif ferential volumes, of which Ay,- is typical. This is depicted in Fig. 2-27. Let us now combine the contributions of all these differential volumes to both sides of Eq. (2-116). We have lim yv-A)^. U=i = lim I L/=i A-ds The left side of Eq. (2-117) is, by definition, the volume integral of V • A: lim Avj-+0 L/=i £(V-A)jAi;J = JK(V-A)di;. (2-117) (2-118) The surface integrals on the right side of Eq. (2-117) are summed over all the faces of all the differential volume elements. The contributions from the internal surfaces of adjacent elements will, however, cancel each other, because at a common internal surface the outward normals of the adjacent elements point in opposite directions. Hence the net contribution of the right side of Eq. (2-117) is due only to that of the external surface S bounding the volume V; that is, N lim Avj->0 V f A • ds JSj U=i = C p A • da. (2-119) The substitution of Eqs. (2-118) and (2-119) in Eq. (2-117) yields the divergence theorem in Eq. (2-115). The validity of the limiting processes leading to the proof of the divergence the orem requires that the vector field A, as well as its first derivatives, exist and be con tinuous both in V and on S. The divergence theorem is an important identity in vector analysis. It converts a volume integral of the divergence of a vector to a closed surface integral of the vector, and vice versa. We use it frequently in establishing other theorems and relations in electromagnetics. We emphasize that, although a single integral sign is used on both sides of Eq. (2-115) for simplicity, the volume and surface integrals represent triple and double integrations, respectively. FIGURE 2-27 Subdivided volume for proof of divergence theorem. 52 2 Vector Analysis EXAMPLE 2-19 Given A = SLXX2 + ayxy + azyz, verify the divergence theorem over a cube one unit on each side. The cube is situated in the first octant of the Cartesian coordinate system with one corner at the origin. Solution Refer to Fig. 2-28. We first evaluate the surface integral over the six faces. 1. Front face: x = 1, ds = stxdydz; f A-ds = P r dydz = l. Jfront JO JO J face 2. Back face: x = 0, ds = — axdy dz; f A • ds = 0. Jback face 3. Left face: y = 0, ds = —aydxdz; f A • ds = 0. Jleft face 4. Right face: y = 1, ds = aydxdz; A • ds = xdxdz = j . Jrighl JO JO z face 5. Top face: z = 1, ds = a^xdy; lop A,dS = Jo J 0 ^ ^ ^ = i face 6. Bottom face: z = 0, ^s = — az dx dy; f A • rfs = 0. J bottom face Adding the above six values, we have (j). A- ds = 1 + 0 + 0 + ^ + ^ + 0 = 2. (2-120) Now the divergence of A is FIGURE 2-28 A unit cube (Example 2-19). 2-8 Divergence Theorem 53 Hence, j v V • A dv = j j j (3x + y) dx dy dz = 2, (2-121) which is the same as the result of the closed surface integral in (2-120). The divergence theorem is therefore verified. m EXAMPLE 2-20 Given F = aRkR, determine whether the divergence theorem holds for the shell region enclosed by spherical surfaces at R = R1 and R = R2(R2 > #i) centered at the origin, as shown in Fig. 2-29. Solution Here the specified region has two surfaces, at R = R1 and R = R2. At the outer surface: R = R2, ds = aRRj sin ddddfc /outer F ' d S = J T Jo" (kR2>R2 S h l 0 dd W = ^kR. surface At the inner surface: R = Rlt ds = —&RRj sin OdOdfc jnner F ' dS = ~ j ^ ^0 ^ R ^ S h l Q dQ d4> = ~^kR. surface Actually, since the integrand is independent of 9 or 0 in both cases, the integral of a constant over a spherical surface is simply the constant multiplied by the area of the surface {4nRj for the outer surface and 4nRj for the inner surface), and no integration is necessary. Adding the two results, we have 4 F • ds = 4nk{R\ - Rl). (2-122) To find the volume integral, we first determine V • F for an F that has only an FR component. From Eq. (2-113), we have Since V • F is a constant, its volume integral equals the product of the constant and the volume. The volume of the shell region between the two spherical surfaces with FIGURE 2-29 A spherical shell region (Example 2-20). 54 2 Vector Analysis radii Rx and R2 is 4n(Rj - Rf)/3. Therefore, j y V • F dv = (V • F)V = 47tfe(R| - Rl), (2-123) which is the same as the result in Eq. (2-122). This example shows that the divergence theorem holds even when the volume has holes inside—that is, even when the volume is enclosed by a multiply connected surface. BUB 2 - 9 Curl of a Vector Field In Section 2-7 we stated that a net outward flux of a vector A through a surface bounding a volume indicates the presence of a source. This source may be called a flow source, and div A is a measure of the strength of the flow source. There is another kind of source, called vortex source, which causes a circulation of a vector field around it. The net circulation (or simply circulation) of a vector field around a closed path is defined as the scalar line integral of the vector over the path. We have Circulation of A around contour C = ( D A • dt. (2-124) Equation (2-124) is a mathematical definition. The physical meaning of circulation depends on what kind of field the vector A represents. If A is a force acting on an object, its circulation will be the work done by the force in moving the object once around the contour; if A represents an electric field intensity, then the circulation will be an electromotive force around the closed path, as we shall see later in the book. The familiar phenomenon of water whirling down a sink drain is an example of a vortex sink causing a circulation of fluid velocity. A circulation of A may exist even when div A = 0 (when there is no flow source). Since circulation as defined in Eq. (2-124) is a line integral of a dot product, its value obviously depends on the orientation of the contour C relative to the vector A. In order to define a point function, which is a measure of the strength of a vortex source, we must make C very small and orient it in such a way that the circulation is a maximum. We define1 curl A = V x A 4 lim-^-As->0 AS n§c±'d€ max In words, Eq. (2-125) states that the curl of a vector field A, denoted by curl A or V x A, is a vector whose magnitude is the maximum net circulation of A per unit f In books published in Europe, the curl of A is often called the rotation of A and written as rot A. 55 FIGURE 2-30 Relation between a„ and d€ in denning curl. area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum. Because the normal to an area can point in two opposite directions, we adhere to the right-hand rule that when the fingers of the right hand follow the direction of d€, the thumb points to the a„ direction. This is illustrated in Fig. 2-30. Curl A is a vector point function and is conventionally written as V x A (del cross A). The component of V x A in any other direction a„ is a„-(V x A), which can be determined from the circulation per unit area normal to a„ as the area approaches zero. (2-126) where the direction of the line integration around the contour Cu bounding area Asu and the direction a„ follow the right-hand rule. We now use Eq. (2-126) to find the three components of V x A in Cartesian coordinates. Refer to Fig. 2-31, in which a differential rectangular area parallel to the yz-plane and having sides Ay and Az is drawn about a typical point P{x0, y0, z0). We have a„ = ax and Asu = Ay Az, and the contour Cu consists of the four sides 1, 2, 3, P(xo, .vo, o) FIGURE 2-31 Determining (V x A)x. 56 2 Vector Analysis and 4. Thus, (VxA),= lim -r^—[(p.A A-d€. AyAz^O AyAZ \ Jsides (2-127) 1, 2, 3,4 In Cartesian coordinates, A = axAx + ayAy + azAz. The contributions of the four sides to the line integral are as follows. Side 1: di = az Az, k-d€ = Az(x0, y0 + -y, z0JAz, I &y \ where Az\ x0, y0 + —. z0 I can be expanded as a Taylor series: A i Ay \ , Ay dAz K\ x0, y0 + -y' Zo I = Az{xQ, y0, z0) + — — + H.O.T, (2-128) (x0,yo,z0) where H.O.T. (higher-order terms) contain the factors {Ay)2, (Ay)3, etc. Thus, A^dA, 2 fy (x0,yo,z0) Ay £ d e l A - ^ = |^z(x0,j;0,z0) + + H.O.T. \ Az. (2-129) Side 3: di = -azAz,A-d€ = Ag[x0,y0-—,z0) Az, where . Ay \ Ay dA. M o> yo - y > z°) = ^ x ° ' y°> zo) - y y + H.O.T.; (2-130) Iide 3 A' ^ = \AZ(X0, y0, z0)- AydAz 2 dy (xo, y0, z0) + H.O.T. [(-Az). (2-131) (xo,yo,zo) Combining Eqs. (2-129) and (2-131), we have Jsides \ 3v I 1 & 3 v J 7 AyAz. (2-132) |(x 0, yo, z 0) The H.O.T. in Eq. (2-132) still contain powers of Ay. Similarly, it may be shown that AyAz. (2-133) f A . ^ = f - ^ + H.O.T.) Jsides \ CZ } 2 & 4 x 7 (x0, yo, zo) Substituting Eqs. (2-132) and (2-133) in Eq. (2-127) and noting that the higher-order terms tend to zero as Ay -» 0, we obtain the x-component of V x A: v ,x dy dz (2-134) |
5789 | https://brainly.com/question/30509595 | [FREE] Using digits 1 through 9, find the greatest value of GHI in the equation ABC + DEF = GHI, where each letter - brainly.com
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Using digits 1 through 9, find the greatest value of GHI in the equation ABC + DEF = GHI, where each letter represents a different digit.
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The greatest value of GHI that can be achieved using digits 1 through 9, where each letter is represented by a different digit, is 984.
The greatest value of GHI can be found using the following formula:
GHI = 987 - (ABC + DEF)
Using this formula, we can solve for GHI by substituting the values for ABC, DEF, and 987. For the highest value of GHI, we can set ABC and DEF to the lowest possible values, 1 and 2 respectively. This gives us the following equation:
GHI = 987 - (1 + 2)
GHI = 984
Therefore, the greatest value of GHI that can be achieved using digits 1 through 9, where each letter is represented by a different digit, is 984.
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The greatest value of GHI that can be achieved in the equation ABC + DEF = GHI, using the digits 1 through 9 with no repeats, is 987. This value occurs when ABC is set to 219 and DEF to 768. Therefore, GHI achieves its maximum of 987 while satisfying the conditions of the problem.
Explanation
To find the greatest value of GHI in the equation ABC + DEF = GHI using the digits 1 through 9 (with each letter representing a different digit), we can follow this step-by-step method:
Determine the Maximum Value of GHI: The maximum GHI can be formed by using the digits that yield the largest numbers. Since GHI is a 3-digit number, let's maximize the digits assigned to G, H, and I. The highest three unique digits we can use from 1 to 9 are 9, 8, and 7, which will give us GHI = 987.
Set Up the Equation: The equation we are working with is ABC + DEF = GHI. This means that to achieve the maximum GHI of 987, we need to find values for ABC and DEF that will add up to 987 and still use all different digits.
Assign Values to ABC and DEF: To maximize GHI effectively, minimize ABC and DEF. Looking at the available digits, we can effectively use:
ABC = 123 (using 1, 2, 3)
DEF = 456 (using 4, 5, 6)
Now let’s check the sum:
123 + 456 = 579.
Rearrange the Remaining Digits: The unused digits from 1 to 9 are 7, 8, and 9. These digits, if we form a number with them, give us GHI = 789 (which is less than 987). However, we want to maximize GHI.
Test Various Combinations: The maximum valid combinations from available digits can ultimately lead to:
ABC = 219 and DEF = 768 yielding:
219 + 768 = 987.
This confirmation meets our equation and successfully uses all different digits.
Final Verification: Therefore, the greatest value of GHI achieved using unique digits 1 through 9 in this equation is indeed 987.
Conclusively, we establish that the maximum value of GHI using digits from 1 to 9, with each represented by different digits in the equation ABC + DEF = GHI, is:
G H I ma x=987
Examples & Evidence
For example, if we assign ABC the digits 219 and DEF the digits 768, the addition would look like this:
219+768=987 This satisfies the equation while using different digits from the available set of numbers.
We can verify this solution as all digits from 1 to 9 (1, 2, 3, 4, 5, 6, 7, 8, 9) are used exactly once across ABC, DEF, and GHI, ensuring that they are distinct and meet the requirements of the problem.
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5790 | https://www.youtube.com/watch?v=oVkYn13RLIM | Circle Angles I (Parallel lines intercept congruent arcs)
Mr. Estrada
794 subscribers
48 likes
Description
7311 views
Posted: 4 Mar 2016
I go over how to solve problems that involve parallel chords and arcs in circles.
4 comments
Transcript:
[Music] hey it's Mr E here and in this video I'm going to show you how we could find the measurement of arcs intercepted by parallel lines so let's begin with this definition of an arc an arc is a portion of the circumference of a circle and it can be measured in degrees relative to the central angle so let me give you an example of what I mean by an arc so it's a portion of the circumference of the circle so this highlighted portion of the circumference remember circumference is all this around the circle and an arc is just a portion of that circumference so we can call this Arc ab and Arc AB would measure we could say Arc a measures 90° because the angle the central angle that's intercepting the arc is 90° and this is a central angle because we're going to go ahead and assume that this vert vertex here what's called vertex o is the center of the circle so I could write that the measure of Ark AB is 90° right um You can also think about it this way the whole Circ circle is 360° and a quarter of the circle or a quarter of that portion of the circumference should be 90° right because 90° is a quarter of 360 all right so that's what in Arc is little uh brief summary on it and let me give you a theorem that we're going to be using in this video so there's a theorem that says in a circle parallel chords intercept congruent arcs and what do we mean by that so let's suppose that segment AB is parallel to segment CD so these two lines are parallel the theorem essentially says that if you have parallel chords which which chords are these lines a chord is just a line that connects two points on the circle so it says parallel chords intercept congruent Arc so these two lines are parallel to each other therefore Arc a is congruent to Arc BD these two arcs measure the same because these two segments or these two chords are parallel to each other all right so here's our first example that we want to go over it says in the diagram of circle o below towards ab and CD are parallel and BD is the diameter of the circle what is the measure of Arc BC well we just learned that if you have parallel chords that they intercept congruent arcs so Arc a D should be equal to Arc BC therefore Arc BC must also be 60° let's try another example it says in the diagram of circle o below chord CD is parallel to diameter a o and the measure of Arc AC is equal to 40° so so the way we read this notation here that says the measure of Ark AC equals 40 deg okay so um let's label what we got so far well we know that these two chords are parallel or these two segments are parallel to each other and we are also told that the measure of Arc AC C is 40° so that's 40° then we know automatically that Arc DB must also be 40° but the question is asking for the measurement of Arc CD so we want to find the measurement of that Arc so hm I wonder how I could find that is there a clue in the question there is we are told that a o is a diameter okay now what you need to know about a diameter is that a diameter Cuts right through the middle of the circle so here's here's where I'm going with this I I want you to see that Ark a plus Arc CD plus Arc DB should be equal to 180° now why is that well remember the diameter is cutting the circle in half and what that means means is that this half over here is of course 180° and then the top half will also be 180° so that means that these three arcs added together Arc AC Arc CD and Arc DB must be equal to 18 180° so if we know Arc AC is 40 and we also know Arc DB is 40 and the three of them add up to 180 we know that Arc CD must be equal to 100° right because 40 + 100 + 40 equal 180 all right that's it for this video I hope that it helps take care |
5791 | https://www.rvrhs.com/ourpages/auto/2014/2/19/45536567/4%20%20%208_2%20_B_1b_%20Dividing%20Monomials%20-%20Negative%20Exponents%20Homework.pdf | ©m V2G021r4f 2KmuwtCay cSyosfHtpwWaOrVeV kLELYCC.B r WAplNlh xr6ihgIh0tIsA urfecsZe9rfvHebdq.R J 0M8a7dOef kw5iHtVh7 8I5nVf0ihnfiftgeq ZALlxgbeSbarlaB d1K.s Worksheet by Kuta Software LLC Algebra 1B Name_____ Period____ ©M 32h01194t hKXu2tEar JSHoufhtcwZaxr3ec BLsLoCk.9 e FAulTlz frcizgChOtvsU drCewsXeErfvLeodu.p 8.2 Dividing Monomials - Negative Exponents Homework Simplify. 1) xy y 2) x xy 3) n mn 4) mn n 5) xy xy 6) xy xy 7) mn mn 8) y xy -1-©T V2i0t114M 7K0uptDaj tScopf7tMwTa9rNet XLmLLC7.6 O 6AgltlY nr1iygEh1t4sx CrYeLsoeXrmvueUdS.E 9 TMbacd5e9 1wniMtRhI AIQnufgiWnriitYeV QASlogtesbWraak 11W.C Worksheet by Kuta Software LLC 9) mpq mq 10) yxz xyz 11) abc abc 12) xy yzx 13) xy (y) 14) (a) ba 15) ab (b) 16) (xy) xy 17) mn m 18) xy (xy) -2- |
5792 | https://www.quora.com/In-ABC-the-median-AM-is-extended-to-ray-AM-and-point-P-on-AM-is-taken-so-that-PM-AM-What-is-the-distance-from-point-P-to-vertices-C-and-B-if-AB-c-and-AC-b | In △ABC the median AM is extended to ray AM and point P on AM is taken so that PM=AM. What is the distance from point P to vertices C and B if AB=c and AC=b? - Quora
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In △ABC the median AM is extended to ray AM and point P on AM is taken so that PM=AM. What is the distance from point P to vertices C and B if AB=c and AC=b?
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John Barrow
Studied Physics,Fluid Dynamics (Graduated 1975) · Author has 100 answers and 151.9K answer views
·6y
This needs a diagram. The solution goes like this …
In the triangle ABC, the median AM bisects side BC and is extended to P such that PM = AM. Side AB = c and side AC = b.
Using the sine rule in triangle ABM gives … AM / sin(ABM) = c / sin(AMB) [equn 1]
and again in triangle MCP … PM / sin(PCM) = PC / sin(PMC) [equn 2]
Dividing equn 1 by equn 2 and noting that AM = PM results in
c sin(ABM) = PC sin(PCM)
Then, if it can be shown that angle ABM = angle PCM, distance PC = c.
In triangle ABM the three angles ABM + BAM + AMB = 180 [equn 3]
Similarly in triangle MCP, angles PCM + CPM + PMC = 180 [equn 4]
Ang
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This needs a diagram. The solution goes like this …
In the triangle ABC, the median AM bisects side BC and is extended to P such that PM = AM. Side AB = c and side AC = b.
Using the sine rule in triangle ABM gives … AM / sin(ABM) = c / sin(AMB) [equn 1]
and again in triangle MCP … PM / sin(PCM) = PC / sin(PMC) [equn 2]
Dividing equn 1 by equn 2 and noting that AM = PM results in
c sin(ABM) = PC sin(PCM)
Then, if it can be shown that angle ABM = angle PCM, distance PC = c.
In triangle ABM the three angles ABM + BAM + AMB = 180 [equn 3]
Similarly in triangle MCP, angles PCM + CPM + PMC = 180 [equn 4]
Angle AMB = angle PMC and BAM = CPM, so subtracting equn 4 from equn 3 gives ABM - PCM = 0
So, angle ABM = angle PCM as required and PC does equal c.
Repeat this procedure with the other pair of triangles (AMC and BMP) to prove distance BP = b.
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What is the distance of point (a, b, C) from y-axis?
AM is a median of a triangle ABC. Is AB+BC+CA > 2AM?
What is the length of the median from A to side BC of the triangle A (2, -2), B (-4, -4), and C (0, 4)?
N ∆ABC, AB = 26, AC = 30 and median AM = 4√37. What is the area of 4ABC?
What is the length of the shortest median if AC=4, BC=2 and angle A=60 degrees?
Steve's Mathy Stuff
Former Taxpayer Service Specialist (1986–2015)
·6y
Originally Answered: In ∆ABC, the median AM is extended to ray AM and point P on AM is taken so that PM = AM. What is the distance from point P to vertices C and B if AB = c and AC = b? ·
The angles AMB and PMC are congruent. AM = PM. BM = CM. So triangles AMB and PMC are congruent. So PC = AB = c. By a similar argument PB = AC = b.
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Ragu Rajagopalan
Passionate Maths solver ;Reviving knowledge after 3 decades · Author has 10.1K answers and 7.6M answer views
·2y
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Consider a triangle △A B C,△A B C, right at C C and a point P P at a distance 4 4 from vertex A,7 A,7 from vertex B,B, and 1 1 from vertex C.C. What are the smallest and largest lengths that side A C A C can have?
Consider the below diagram with the coordinates as :C(0,0),B(m,0),A(0,n),P(x,y)Consider the below diagram with the coordinates as :C(0,0),B(m,0),A(0,n),P(x,y)
Applying the property : Sum of 2 sides of a triangle > 3rd side Applying the property : Sum of 2 sides of a triangle > 3rd side
In△A C P:In△A C P:
A P+P C>A C⟹A C<5 A P+P C>A C⟹A C<5
P C+A C>A P⟹A C>A P=4−P C=1⟹A C>3 P C+A C>A P⟹A C>A P⏟=4−P C⏟=1⟹A C>3
⟹3<A C<5⟹3<A C<5
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Consider the below diagram with the coordinates as :C(0,0),B(m,0),A(0,n),P(x,y)Consider the below diagram with the coordinates as :C(0,0),B(m,0),A(0,n),P(x,y)
Applying the property : Sum of 2 sides of a triangle > 3rd side Applying the property : Sum of 2 sides of a triangle > 3rd side
In△A C P:In△A C P:
A P+P C>A C⟹A C<5 A P+P C>A C⟹A C<5
P C+A C>A P⟹A C>A P=4−P C=1⟹A C>3 P C+A C>A P⟹A C>A P⏟=4−P C⏟=1⟹A C>3
⟹3<A C<5⟹3<A C<5
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Enrico Gregorio
Associate professor in Algebra · Author has 18.4K answers and 16M answer views
·Apr 25
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If the sides of triangle ABC are AB = 16, BC = 24, AC = 20, what is the length of the median from vertex B to side AC?
Refer to the figure
The side A C A C has length b=2 x b=2 x and the median B M B M has length m.m. By the cosine law,
m 2=a 2+x 2−2 a x cos C m 2=a 2+x 2−2 a x cosC
We also have
c 2=a 2+b 2−2 a b cos C c 2=a 2+b 2−2 a b cosC
whereby
cos C=a 2+b 2−c 2 2 a b cosC=a 2+b 2−c 2 2 a b
and we get
m 2=a 2+b 2 4−a b a 2+b 2−c 2 2 a b m 2=a 2+b 2 4−a b a 2+b 2−c 2 2 a b
and therefore
4 m 2=4 a 2+b 2−2 a 2−2 b 2+2 c 2 4 m 2=4 a 2+b 2−2 a 2−2 b 2+2 c 2
so finally
4 m 2=2 a 2−b 2+2 c 2 4 m 2=2 a 2−b 2+2 c 2
With a=24,b=20,c=16 a=24,b=20,c=16 we obtain
4 m 2=1152−400+512=1264 4 m 2=1152−400+512=1264
hence
m 2=316=4⋅79 m 2=316=4⋅79
so
m=2√79 m=2 79
In general, the median relative to the side b b is
m=1 2√2 a 2−b 2+2 c 2 m=1 2 2 a 2−b 2+2 c 2
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Refer to the figure
The side A C A C has length b=2 x b=2 x and the median B M B M has length m.m. By the cosine law,
m 2=a 2+x 2−2 a x cos C m 2=a 2+x 2−2 a x cosC
We also have
c 2=a 2+b 2−2 a b cos C c 2=a 2+b 2−2 a b cosC
whereby
cos C=a 2+b 2−c 2 2 a b cosC=a 2+b 2−c 2 2 a b
and we get
m 2=a 2+b 2 4−a b a 2+b 2−c 2 2 a b m 2=a 2+b 2 4−a b a 2+b 2−c 2 2 a b
and therefore
4 m 2=4 a 2+b 2−2 a 2−2 b 2+2 c 2 4 m 2=4 a 2+b 2−2 a 2−2 b 2+2 c 2
so finally
4 m 2=2 a 2−b 2+2 c 2 4 m 2=2 a 2−b 2+2 c 2
With a=24,b=20,c=16 a=24,b=20,c=16 we obtain
4 m 2=1152−400+512=1264 4 m 2=1152−400+512=1264
hence
m 2=316=4⋅79 m 2=316=4⋅79
so
m=2√79 m=2 79
In general, the median relative to the side b b is
m=1 2√2 a 2−b 2+2 c 2 m=1 2 2 a 2−b 2+2 c 2
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Aside A triangle ABC has vertices A(4,3), B(6,1) and C (-2,-3). What is the equation of AM the median of A?
Point D is the midpoint of median AM of triangle ABC. Point E is the middle point of AB, and point T is the intersection of BD and ME. What is the area of triangle DMT if ABC =180?
ABCD is a //gm G is a point on AB such that AG=2GB, E is a point on DC such that CE=2DE and F is a point on BC such that BF=2FC. Can you prove that A(EBG) =A(EFC)?
What is the distance between the points A and B 0 C?
What is the distance from the origin to points A, B, and C, given their coordinates?
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Let M be midpoint of BC, so BM = CM = (a/2) where a = BC. AM is median. Point P lies on ray AM beyond M with PM = AM. Using vectors or coordinates gives distances from P to B and C as follows.
Place coordinates: A = (0,0), B = (c,0) (so AB = c), and C at (x,y) with AC = b. Solve for x,y:
From AB along x-axis, AC^2 = x^2 + y^2 = b^2,
BC^2 = (x - c)^2 + y^2 = a^2.
Midpoint M of BC: M = ((c + x)/2, y/2). Vector AM = M − A = ((c + x)/2, y/2). Point P on ray AM beyond M with PM = AM means P = M + (AM)/|AM| |AM| = M + AM = 2M − A = 2M. Since A is origin, P = 2M = (c + x, y).
Thus coordinates:
B = (c,0)
C
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Let M be midpoint of BC, so BM = CM = (a/2) where a = BC. AM is median. Point P lies on ray AM beyond M with PM = AM. Using vectors or coordinates gives distances from P to B and C as follows.
Place coordinates: A = (0,0), B = (c,0) (so AB = c), and C at (x,y) with AC = b. Solve for x,y:
From AB along x-axis, AC^2 = x^2 + y^2 = b^2,
BC^2 = (x - c)^2 + y^2 = a^2.
Midpoint M of BC: M = ((c + x)/2, y/2). Vector AM = M − A = ((c + x)/2, y/2). Point P on ray AM beyond M with PM = AM means P = M + (AM)/|AM| |AM| = M + AM = 2M − A = 2M. Since A is origin, P = 2M = (c + x, y).
Thus coordinates:
B = (c,0)
C = (x,y)
P = (c + x, y)
Distances:
PB^2 = (c + x − c)^2 + (y − 0)^2 = x^2 + y^2 = AC^2 = b^2. Hence PB = b.
PC^2 = (c + x − x)^2 + (y − y)^2 = c^2. Hence PC = c.
Conclusion: PB = AC = b and PC = AB = c.
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Avaneesh Awasthi
B.Sc(Hons.) in Electronics, University of Delhi · Author has 59 answers and 150.8K answer views
·8y
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AM is a median of a triangle ABC. Is AB+BC+CA > 2AM?
The sum of length of two sides of triangle is always greater than third side.
By following above rule we have……
In triangle ABM
AB+ BM > AM…….(1)
In rriangle ACM
AC + CM > AM…….(2)
By adding the equation 1 & 2 we get
AB + AC +( BM+CM) > 2AM
where BM +CM = BC ( acc. to figure)
Hence …. AB + BC + CA > 2 AM
PROVED………..
Here the information that AM is median is not worthy for solving the problem.
Continue Reading
The sum of length of two sides of triangle is always greater than third side.
By following above rule we have……
In triangle ABM
AB+ BM > AM…….(1)
In rriangle ACM
AC + CM > AM…….(2)
By adding the equation 1 & 2 we get
AB + AC +( BM+CM) > 2AM
where BM +CM = BC ( acc. to figure)
Hence …. AB + BC + CA > 2 AM
PROVED………..
Here the information that AM is median is not worthy for solving the problem.
Upvote ·
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Devesh Dube
5y
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In a triangle ABC, AB =AC, how do you prove that angle B=C?
Well you can prove it by RHS congruence rule
1- draw a triangle ABC in which AB = AC.
2- draw a perpendicular AD such that it cut BC at point D.
3 - now we have two triangle ABD and and triangle ACD
In which,
AD common side
AB = AC given which is hypotenuse
Hence , t r i a n g l e A B D i s c o n g r u e n t t o t r a n g l e A C D t r i a n g l e A B D i s c o n g r u e n t t o t r a n g l e A C D
By this we get , angle B = angle C (by cpct )
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Well you can prove it by RHS congruence rule
1- draw a triangle ABC in which AB = AC.
2- draw a perpendicular AD such that it cut BC at point D.
3 - now we have two triangle ABD and and triangle ACD
In which,
AD common side
AB = AC given which is hypotenuse
Hence , t r i a n g l e A B D i s c o n g r u e n t t o t r a n g l e A C D t r i a n g l e A B D i s c o n g r u e n t t o t r a n g l e A C D
By this we get , angle B = angle C (by cpct )
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Gary Ward
MaEd in Education&Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views
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Point A is at (-3,4) and point C is at (2,-6). What are the coordinates of point B on AC such that the ratio of AB to AC is 4:5?
Point A is at (-3,4) and point C is at (2,-6). What are the coordinates of point B on AC such that the ratio of AB to AC is 4:5?
C - A = <2 - (-3), -6 - 4> = <5, -10>
(4/5)<5, -10> =<4, -8>
A + <4, -8> = B = (-3, 4) + <4, -8> = (1, -4)
B is at (1, -4)
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Point A is at (-3,4) and point C is at (2,-6). What are the coordinates of point B on AC such that the ratio of AB to AC is 4:5?
C - A = <2 - (-3), -6 - 4> = <5, -10>
(4/5)<5, -10> =<4, -8>
A + <4, -8> = B = (-3, 4) + <4, -8> = (1, -4)
B is at (1, -4)
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Schlitzer
dropout · Author has 325 answers and 243.9K answer views
·2y
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In acute-angled triangle ABC, let D be the leg of the perpendicular drawn from vertex A to side [BC], and let M be the midpoint of side [AC]. A point P is taken on the line segment [BM] such that ∠PAM = ∠MBA. If ∠BAP=41 and ∠PDB=115,what is the ∠BAC?
Let k k denote the circle with diameter A C A C, its midpoint is M M, its radius is r=M A r=M A. As ∠C D A=90°∠C D A=90°, by inverse Thales one has D∈k D∈k. Let E E and F F denote the points of intersection of line B M B M with k k (so that E E is closer to B B). By secant theorem,
B D⋅B C=B E⋅B F=(B M−r)(B M+r)=B M 2−M A 2(1)(1)B D⋅B C=B E⋅B F=(B M−r)(B M+r)=B M 2−M A 2
As ∠P A M=∠P B A∠P A M=∠P B A, by inverse tangent-chord theorem the line A C A C is tangent to the circumcircle of triangle B P A B P A. Hence
M A 2=M P⋅M B(2)(2)M A 2=M P⋅M B
Therefore
B P⋅B M=(B M−M P)M B=B M 2−M P⋅M B(2)=B M 2−M A 2(1)=B D⋅B C B P⋅B M=(B M−M P)M B=B M 2−M P⋅M B=(2)B M 2−M A 2=(1)B D⋅B C
This implies that the quadrilateral D C M P D C M P is cyclic. Hence \angl\angl
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Let k k denote the circle with diameter A C A C, its midpoint is M M, its radius is r=M A r=M A. As ∠C D A=90°∠C D A=90°, by inverse Thales one has D∈k D∈k. Let E E and F F denote the points of intersection of line B M B M with k k (so that E E is closer to B B). By secant theorem,
B D⋅B C=B E⋅B F=(B M−r)(B M+r)=B M 2−M A 2(1)(1)B D⋅B C=B E⋅B F=(B M−r)(B M+r)=B M 2−M A 2
As ∠P A M=∠P B A∠P A M=∠P B A, by inverse tangent-chord theorem the line A C A C is tangent to the circumcircle of triangle B P A B P A. Hence
M A 2=M P⋅M B(2)(2)M A 2=M P⋅M B
Therefore
B P⋅B M=(B M−M P)M B=B M 2−M P⋅M B(2)=B M 2−M A 2(1)=B D⋅B C B P⋅B M=(B M−M P)M B=B M 2−M P⋅M B=(2)B M 2−M A 2=(1)B D⋅B C
This implies that the quadrilateral D C M P D C M P is cyclic. Hence ∠P M C=180°−∠C D P=∠P D B∠P M C=180°−∠C D P=∠P D B.
On the other hand, the exterior angle theorem says ∠P M C=∠M B A+∠B A C∠P M C=∠M B A+∠B A C. Recalling ∠M B A=∠P A M=∠B A C−∠B A P∠M B A=∠P A M=∠B A C−∠B A P, we get ∠P D B=2∠B A C−∠B A P∠P D B=2∠B A C−∠B A P, or
∠B A C=1/2(∠P D B+∠B A P)=1/2(115°+41°)=78°∠B A C=1/2(∠P D B+∠B A P)=1/2(115°+41°)=78°
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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.
Upvote ·
999 197
9 6
9 6
Burhaan Rasheed Zargar
Studied Mathematics&Science at Jodhamal Public School (Graduated 2022) · Author has 65 answers and 133.4K answer views
·5y
Related
ABC is an isosceles triangle with AB=AC. A circle through B is touching AC at the middle point intersect AB at P. What is AP:AB?
G i v e n:A B=A C a n d A Q=Q C G i v e n:A B=A C a n d A Q=Q C
T o f i n d:A P A B T o f i n d:A P A B
S o l u t i o n:S o l u t i o n:
I n△A B Q a n d△A Q P,I n△A B Q a n d△A Q P,
∠B A Q=∠Q A P...[C o m m o n a n g l e]∠B A Q=∠Q A P...[C o m m o n a n g l e]
∠A B Q=∠A Q P...[A n g l e b e t w e e n a c h o r d a n d t a n g e n t∠A B Q=∠A Q P...[A n g l e b e t w e e n a c h o r d a n d t a n g e n t
i s e q u a l t o a n g l e s u b t e n d e d b y t h a t c h o r d i n t h e a l t e r n a t e i s e q u a l t o a n g l e s u b t e n d e d b y t h a t c h o r d i n t h e a l t e r n a t e
s e g m e n t]s e g m e n t]
S o,b y A A s i m i l a r i t y c r i t e r i a,S o,b y A A s i m i l a r i t y c r i t e r i a,
△A B Q∼△A Q P△A B Q∼△A Q P
T h u s,T h u s,
A B A Q=A Q A P...[C P S T]A B A Q=A Q A P...[C P S T]
⟹A B⋅A P=(A Q)2⟹A B⋅A P=(A Q)2
⟹A B⋅A P=(A C 2)2⟹A B⋅A P=(A C 2)2
⟹A B⋅A P=(A C)2 4⟹A B⋅A P=(A C)2 4
B u t A C=A B B u t A C=A B
⟹A B⋅A P=(A B)2 4⟹A B⋅A P=(A B)2 4
⟹A P=(A B)2 4 A B⟹A P=(A B)2 4 A B
⟹A P=A B 4⟹A P=A B 4
⟹\dfr⟹\dfr
Continue Reading
G i v e n:A B=A C a n d A Q=Q C G i v e n:A B=A C a n d A Q=Q C
T o f i n d:A P A B T o f i n d:A P A B
S o l u t i o n:S o l u t i o n:
I n△A B Q a n d△A Q P,I n△A B Q a n d△A Q P,
∠B A Q=∠Q A P...[C o m m o n a n g l e]∠B A Q=∠Q A P...[C o m m o n a n g l e]
∠A B Q=∠A Q P...[A n g l e b e t w e e n a c h o r d a n d t a n g e n t∠A B Q=∠A Q P...[A n g l e b e t w e e n a c h o r d a n d t a n g e n t
i s e q u a l t o a n g l e s u b t e n d e d b y t h a t c h o r d i n t h e a l t e r n a t e i s e q u a l t o a n g l e s u b t e n d e d b y t h a t c h o r d i n t h e a l t e r n a t e
s e g m e n t]s e g m e n t]
S o,b y A A s i m i l a r i t y c r i t e r i a,S o,b y A A s i m i l a r i t y c r i t e r i a,
△A B Q∼△A Q P△A B Q∼△A Q P
T h u s,T h u s,
A B A Q=A Q A P...[C P S T]A B A Q=A Q A P...[C P S T]
⟹A B⋅A P=(A Q)2⟹A B⋅A P=(A Q)2
⟹A B⋅A P=(A C 2)2⟹A B⋅A P=(A C 2)2
⟹A B⋅A P=(A C)2 4⟹A B⋅A P=(A C)2 4
B u t A C=A B B u t A C=A B
⟹A B⋅A P=(A B)2 4⟹A B⋅A P=(A B)2 4
⟹A P=(A B)2 4 A B⟹A P=(A B)2 4 A B
⟹A P=A B 4⟹A P=A B 4
⟹A P A B=1 4⟹A P A B=1 4
Upvote ·
9 7
Haresh Sagar
Studied Science&Mathematics (Graduated 1988) · Author has 6.2K answers and 7M answer views
·Apr 25
Related
If the sides of triangle ABC are AB = 16, BC = 24, AC = 20, what is the length of the median from vertex B to side AC?
Get cosine of ∠A∠A or ∠C∠C. Since cosine of an angle is ratio of adjacent to hypotenuse, we can take a similar triangle (4,5,6) to make computations easy.
c o s A=16+25−36 40=1 8 c o s A=16+25−36 40=1 8
Again use cosine law to get median opp. ∠A∠A. Here too, 16 and 10 can be taken as 8 and 5 and the result into 2.
B M 2=64+25−10=79 B M 2=64+25−10=79
B M=2√79≈17.78 B M=2 79≈17.78
Continue Reading
Get cosine of ∠A∠A or ∠C∠C. Since cosine of an angle is ratio of adjacent to hypotenuse, we can take a similar triangle (4,5,6) to make computations easy.
c o s A=16+25−36 40=1 8 c o s A=16+25−36 40=1 8
Again use cosine law to get median opp. ∠A∠A. Here too, 16 and 10 can be taken as 8 and 5 and the result into 2.
B M 2=64+25−10=79 B M 2=64+25−10=79
B M=2√79≈17.78 B M=2 79≈17.78
Upvote ·
99 12
Pradeep Hebbar
Many years of Structural Engineering & Math enthusiasm · Author has 9.3K answers and 6.2M answer views
·2y
Related
A and B are the points (12, 0) and (0, -5) respectively. What is the length AB and the length of the median through the origin O of the triangle OAB?
Given 3 3 points O(0,0)O(0,0), A(12,0)A(12,0) and B(0,−5)B(0,−5)
In △O A B△O A B we observe that,
O A⊥O B O A⊥O B
Hence △O A B△O A B is a right triangle.
O A=12−0=12 O A=12−0=12
O B=0−(−5)=5 O B=0−(−5)=5
A B 2=O A 2+O B 2=12 2+5 2=169 A B 2=O A 2+O B 2=12 2+5 2=169
A B=13 A B=13 cm
Let O M O M be the median through O O
A M=B M=13 2 A M=B M=13 2
O M O M can be found by more than one approach
Method 1
Using Apollonius's theorem,
O A 2+O B 2=2(O M 2+A M 2)O A 2+O B 2=2(O M 2+A M 2)
A B 2=2(O M 2+A M 2)A B 2=2(O M 2+A M 2)
13 2=2(O M 2+(13 2)2)13 2=2(O M 2+(13 2)2)
O M=13 2 O M=13 2
Method 2
In right triangle, midpoint of hypotenuse is equidistant from all the vertices, as it is the circumcenter.
O M=B M=A M O M=B M=A M
Therefore,
O M=13 2 O M=13 2
Method 3
Let (p,q)(p,q) be the
Continue Reading
Given 3 3 points O(0,0)O(0,0), A(12,0)A(12,0) and B(0,−5)B(0,−5)
In △O A B△O A B we observe that,
O A⊥O B O A⊥O B
Hence △O A B△O A B is a right triangle.
O A=12−0=12 O A=12−0=12
O B=0−(−5)=5 O B=0−(−5)=5
A B 2=O A 2+O B 2=12 2+5 2=169 A B 2=O A 2+O B 2=12 2+5 2=169
A B=13 A B=13 cm
Let O M O M be the median through O O
A M=B M=13 2 A M=B M=13 2
O M O M can be found by more than one approach
Method 1
Using Apollonius's theorem,
O A 2+O B 2=2(O M 2+A M 2)O A 2+O B 2=2(O M 2+A M 2)
A B 2=2(O M 2+A M 2)A B 2=2(O M 2+A M 2)
13 2=2(O M 2+(13 2)2)13 2=2(O M 2+(13 2)2)
O M=13 2 O M=13 2
Method 2
In right triangle, midpoint of hypotenuse is equidistant from all the vertices, as it is the circumcenter.
O M=B M=A M O M=B M=A M
Therefore,
O M=13 2 O M=13 2
Method 3
Let (p,q)(p,q) be the coordinates of M M
p=12+0 2=6 p=12+0 2=6
q=−5+0 2=−5 2 q=−5+0 2=−5 2
O M 2=(6−0)2+(−5 2−0)2 O M 2=(6−0)2+(−5 2−0)2
O M=13 2 O M=13 2
Ans: OM=6.5 cm
Upvote ·
9 3
9 1
9 1
Gary Ward
MaEd in Education&Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views
·3y
Related
In triangle ABC, angle A is 40 degrees; angle C is 65 degrees, AB = 45m, what is the length of the median draw from vertex A to side BC?
In triangle ABC, angle A is 40 degrees; angle C is 75 degrees, AB = 45 m. What is the length of the median drawn from vertex A to side BC?
Draw a sketch of the problem.
Use the Law of Sines to find BC
45/sin65° = BC/sin40° --> BC = 45× sin40°/sin65°
BC = 31.915; BM = 15.958
Use the Law of Cosines to find AM
AM = sqrt(45^2+15.958^2-2×45×15.958×cos75°)
AM = 43.6799 m
The length of median AM is about 43.68 meters.
Continue Reading
In triangle ABC, angle A is 40 degrees; angle C is 75 degrees, AB = 45 m. What is the length of the median drawn from vertex A to side BC?
Draw a sketch of the problem.
Use the Law of Sines to find BC
45/sin65° = BC/sin40° --> BC = 45× sin40°/sin65°
BC = 31.915; BM = 15.958
Use the Law of Cosines to find AM
AM = sqrt(45^2+15.958^2-2×45×15.958×cos75°)
AM = 43.6799 m
The length of median AM is about 43.68 meters.
Upvote ·
9 5
Related questions
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Related questions
What is the distance of point (a, b, C) from y-axis?
AM is a median of a triangle ABC. Is AB+BC+CA > 2AM?
What is the length of the median from A to side BC of the triangle A (2, -2), B (-4, -4), and C (0, 4)?
N ∆ABC, AB = 26, AC = 30 and median AM = 4√37. What is the area of 4ABC?
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Aside A triangle ABC has vertices A(4,3), B(6,1) and C (-2,-3). What is the equation of AM the median of A?
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5793 | https://ideas.repec.org/a/elg/rokejn/v9y2021i3p368-393.html | Omitted-variable bias in demand-regime estimations: the role of household credit and wage inequality in Brazil
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Omitted-variable bias in demand-regime estimations: the role of household credit and wage inequality in Brazil
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Julia Burle
(University of São Paulo (FEA-USP), Brazil)
Laura Carvalho
(University of São Paulo (FEA-USP), Brazil)
Registered:
Laura Barbosa de Carvalho
Abstract
In the Kaleckian theoretical framework, an economy's demand regime is characterized as either wage-led or profit-led depending on the relative effect of an increase in the wage share on consumption, investment, and net exports. Based on this framework, a vast empirical literature has focused on estimating demand regimes in numerous countries. Although they contribute to a better understanding of the relationship between distribution and demand in different economies and time periods, they also face various critiques on theoretical and methodological grounds. This paper aims to address one dimension of these critiques by investigating a potential omitted-variable bias in the estimated relationship between distribution and demand in the Brazilian economy between 1997 and 2014. Our results suggest that when controlling for some of the relevant factors in Brazil's inclusive growth experience of the early twenty-first century, namely wage inequality, commodity prices, and household credit, the empirical characterization of the Brazilian demand regime as profit-led loses its statistical significance. Also, the demand-regime definition was found to be most sensitive to intra-wage distribution, confirming previous findings in the Kaleckian empirical literature for the Brazilian case.
Suggested Citation
Julia Burle & Laura Carvalho, 2021. "Omitted-variable bias in demand-regime estimations: the role of household credit and wage inequality in Brazil," Review of Keynesian Economics, Edward Elgar Publishing, vol. 9(3), pages 368-393, July.
Handle: RePEc:elg:rokejn:v:9:y:2021:i:3:p368-393
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Paul Carrillo‐Maldonado, 2023. "Partial identification for growth regimes: The case of Latin American countries," Metroeconomica, Wiley Blackwell, vol. 74(3), pages 557-583, July.
More about this item
Keywords
demand regimes; functional distribution of income; Brazilian inclusive growth; omitted-variable bias; distribution of wage; All these keywords
JEL classification:
E12 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Keynes; Keynesian; Post-Keynesian; Modern Monetary Theory
E25 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Aggregate Factor Income Distribution
D33 - Microeconomics - - Distribution - - - Factor Income Distribution
C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
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5794 | https://mathworld.wolfram.com/Curvature.html | Curvature -- from Wolfram MathWorld
TOPICS
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Differential Geometry
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Curvature
In general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.
After the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the shape operator. Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature.
Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface, whereas mean curvature and the shape operator are not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides. The importance of Gaussian curvature to an inhabitant is that it controls the surface area of spheres around the inhabitant.
Riemann and many others generalized the concept of curvature to sectional curvature, scalar curvature, the Riemann tensor, Ricci curvature tensor, and a host of other intrinsic and extrinsic curvatures. General curvatures no longer need to be numbers, and can take the form of a map, group, groupoid, tensor field, etc.
The simplest form of curvature and that usually first encountered in calculus is an extrinsic curvature. In two dimensions, let a plane curve be given by Cartesianparametric equations and . Then the curvature , sometimes also called the "first curvature" (Kreyszig 1991, p.47), is defined by
(1)
(2)
(3)
(4)
where is the tangential angle and is the arc length. As can readily be seen from the definition, curvature therefore has units of inverse distance. The derivative in the above equation can be found using the identity
(5)
(6)
(7)
so
(8)
and
(9)
(10)
(11)
(12)
Combining equations (◇), 3), (10), and (12) then gives
(13)
For a two-dimensional curve written in the form , the equation of curvature becomes
(14)
If the two-dimensional curve is instead parameterized in polar coordinates, then
(15)
where (Gray 1997, p.89). In pedal coordinates, the curvature is given by
(16)
The curvature for a two-dimensional curve given implicitly by is given by
(17)
(Gray 1997).
Now consider a parameterized space curve in three dimensions for which the tangent vector is defined as
(18)
Therefore,
(19)
(20)
(21)
where is the normal vector. But
(22)
(23)
so taking norms of both sides gives
(24)
Solving for then gives
(25)
(26)
(27)
(Gray 1997, p.192).
The curvature of a two-dimensional curve is related to the radius of curvature of the curve's osculating circle. Consider a circle specified parametrically by
(28)
(29)
which is tangent to the curve at a given point. The curvature is then
(30)
or one over the radius of curvature. The curvature of a circle can also be repeated in vector notation. For the circle with , the arc length is
(31)
(32)
(33)
so and the equations of the circle can be rewritten as
(34)
(35)
The radius vector is then given by
(36)
and the tangent vector is
(37)
(38)
so the curvature is related to the radius of curvature by
(39)
(40)
(41)
(42)
as expected.
Four very important derivative relations in differential geometry related to the Frenet formulas are
(43)
(44)
(45)
(46)
where is the tangent vector, is the normal vector, is the binormal vector, and is the torsion (Coxeter 1969, p.322).
The curvature at a point on a surface takes on a variety of values as the plane through the normal varies. As varies, it achieves a minimum and a maximum (which are in perpendicular directions) known as the principal curvatures. As shown in Coxeter (1969, pp.352-353),
(47)
(48)
where is the Gaussian curvature, is the mean curvature, and det denotes the determinant.
The curvature is sometimes called the first curvature and the torsion the second curvature. In addition, a third curvature (sometimes called total curvature)
(49)
is also defined. A signed version of the curvature of a circle appearing in the Descartes circle theorem for the radius of the fourth of four mutually tangent circles is called the bend.
See also
Bend, Binormal Vector, Curvature Center, Extrinsic Curvature, Four-Vertex Theorem, Gaussian Curvature, Intrinsic Curvature, Lancret Equation, Line of Curvature, Mean Curvature, Multivariable Calculus, Normal Curvature, Normal Vector, Osculating Circle, Principal Curvatures, Radius of Curvature, Ricci Curvature Tensor, Riemann Tensor, Scalar Curvature, Sectional Curvature, Shape Operator, Special Affine Curvature, Soddy Circles, Tangent Vector, Third Curvature, Torsion, Total CurvatureExplore this topic in the MathWorld classroom
Explore with Wolfram|Alpha
More things to try:
curvature of epicycloid
curvature of a circle
curvature of a sphere
References
Casey, J. Exploring Curvature. Wiesbaden, Germany: Vieweg, 1996.Coxeter, H.S.M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Fischer, G. (Ed.). Plates 79-85 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp.74-81, 1986.Gray, A. "Curvature of Curves in the Plane," "Drawing Plane Curves with Assigned Curvature," and "Drawing Space Curves with Assigned Curvature." §1.5, 6.4, and 10.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp.14-17, 140-146, and 222-224, 1997.Kreyszig, E. "Principal Normal, Curvature, Osculating Circle." §12 in Differential Geometry. New York: Dover, pp.34-36, 1991.Yates, R.C. "Curvature." A Handbook on Curves and Their Properties. Ann Arbor, MI: J.W.Edwards, pp.60-64, 1952.
Referenced on Wolfram|Alpha
Curvature
Cite this as:
Weisstein, Eric W. "Curvature." From MathWorld--A Wolfram Resource.
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5795 | https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-8/section/11.2/primary/lesson/area-of-a-trapezium/ | Skip to content
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11.2
Area of Trapezium (Trapezoid)
Written by:Neha Khandelwal
Fact-checked by:The CK-12 Editorial Team
Last Modified: Aug 02, 2025
Lesson
What is a Trapezium (Trapezoid)?
A trapezium is a quadrilateral with exactly one pair of parallel sides.
Formula of Area of Trapezium (Trapezoid)
Trapezium or trapezoid can be thought of as half a parallelogram by rotating it 180∘about the midpoint of one of its non-parallel sides.
The base of this parallelogram is b1+b2 and the altitude is h. The area of the parallelogram is (b1+b2)h. Therefore, the area of the trapezium is (b1+b2)h2.In other words, the area of a trapezium equals half the sum of parallel sides multiplied by the altitude (height).
The following examples will illustrate the use of the above formula.
CK-12 Interactive: Area of Trapezium (Trapezoid)
Area of Trapezium (Trapezoid) - Examples
Example 1
The figure shows a trapezium PQRS where PQ=16.5 cm, QR=6 cm, RS=8.5 cm and SP=7 cm. If ST=5 cm, calculate the area and the perimeter of the trapezium.
Area of the trapezium=12× (sum of parallel sides) × altitude
=12×(16.5+8.5)×5=12×25×5=62.5 cm2
Perimeter of the trapezium=PQ+QR+RS+SP
=16.5+8+8.5+7=40 cm
Example 2
The figure shows a trapezium ABCD where AB=12 cm and AD=10 cm. If the area and the perimeter of the trapezium are 145 cm2 and 50.18 cm respectively, calculate the length of
i. CD
ii. BC
The height of the trapezium is given by the length AD=10 cm.
Area of the trapezium=12× (sum of lengths of parallel sides) × height=145 cm2
12×(12+CD)×10=14512+CD=145×21012+CD=29CD=29−12CD=17
Length of CD=17 cm.
ii. Perimeter of the trapezium=AB+BC+CD+DA=50.18 cm
12+BC+17+10=50.18BC+39=50.18BC=11.18
Length of BC=11.18 cm.
Example 3
In the figure, a semicircle is removed from a trapezium PQRS. XOY is the diameter of the semicircle with centre O. If PQ=35 cm, RS=SP=18 cm, XY=12 cm, calculate the area of the figure.
Area of the figure = Area of the trapezium − Area of the semicircle
={12× (sum of parallel sides) × height }−12πr2
={12×(35+18)×18}−12×227×(122)2=477−56.57=420.43 cm2
Example 4
The figure shows a picture frame with outer dimensions 24 cm × 28 cm and the width of the picture frame as 4 cm. Find the area of each section of the frame.
Area of section (1) = Area of section (3)=12(24+16)×4
=80 cm2
Area of section (2) = Area of section (4)=12(28+20)×4
=96 cm2
Area of section (5) =20×16
=320 cm2
Summary
Formula of area of a trapezium=(b1+b2)h2.
Area of Trapezium (Trapezoid) - Review Questions
The parallel sides of a trapezium are 20 cm and 35.5 cm and the distance between them is 15 cm. Find the area of the trapezium.
The figure shows a trapezium UVWX where WX=8 cm, XU=UV=5 cmand VW=4 cm. Find the area of the trapezium.
The figure shows a trapezium ABCD where AB=24 cm and CD=18 cm. If the area of the trapezium is 273 cm2, calculate the altitude of the trapezium.
The lengths of the parallel sides of a trapezium are in the ratio 3 : 1 and the distance between them is 25 cm. If the area of the trapezium is 450 cm2, find the length of the parallel sides.
The figure shows a trapezium PQRS where RS⊥QR, PQ=14 cm, QR=12 cm and RS=10 cm. Find the area of the shaded region.
Find the area of the shaded region.
Find the area of the shaded region.
Find the area of the figure.
A line connects the midpoint of PQ i.e., point T with vertex S in the square PQRS. Calculate the area of the required trapezium if the square has a side of 8 m.
A grassy park in the shape of a trapezium whose parallel sides measure 64 m and 46 m and are at a distance of 20 m from each other. A 2 m wide concrete walkway is constructed which runs perpendicular to the parallel sides. Calculate the area of the grassy land.
Vocabulary
Quadrilateral
Parallel
Trapezoid
midpoint
parallelogram
Area
altitude
perimeter
semicircle
diameter
Dimensions
Test Your Knowledge
Question 1
Find the area of the following trapezoid.
Trapezoid with bases 9 in and 4 in and height of 1 in.
a
@$\begin{align}380 \ in^2\end{align}@$
b
@$\begin{align}15 \ in^2\end{align}@$
c
@$\begin{align}160 \ in^2\end{align}@$
d
@$\begin{align}6.5 \ in^2\end{align}@$
Let:
@$\begin{align}b_1 = 9 \text{ in}\end{align}@$
@$\begin{align}b_2 = 4 \text{ in}\end{align}@$
@$\begin{align}h = 1 \text{ in}\end{align}@$
The formula for the area of a trapezoid is @$\begin{align}A = \frac{1}{2}h(b_1 + b_2)\end{align}@$.
Substitute the given values into the below formula:
@$$\begin{align}\eqalign{
A &=\frac{1}{2}h(b_1 + b_2) \
A &=\frac{1}{2}(1)(9 + 4) \
A &=\frac{1}{2}(1)(13) \
A &=\frac{1}{2}(13) \
A &=\frac{13}{2} \
A &=6.5
}\end{align}@$$
The area of the trapezoid is @$\begin{align}6.5 \text{ in}^2\end{align}@$.
Question 2
A trapezium has bases of @$\begin{align}1.8 \text{ m}\end{align}@$ and @$\begin{align}2.2 \text{ m}\end{align}@$, and an altitude of @$\begin{align}0.5 \text{ m}\end{align}@$. What is its area?
a@$\begin{align}2.0 \text{ m}^2\end{align}@$
b@$\begin{align}2.5 \text{ m}^2\end{align}@$
c@$\begin{align}1.5 \text{ m}^2\end{align}@$
d@$\begin{align}1.0 \text{ m}^2\end{align}@$
The area of the trapezium is calculated using the formula: @$\begin{align}\frac{1}{2} \times (1.8 + 2.2) \times 0.5 = \frac{1}{2} \times 4.0 \times 0.5 = 1.0 \text{ m}^2\end{align}@$.
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Area and Perimeter of Trapezoids Examples
Area and Perimeter of Trapezoids
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5796 | https://rheckathorn.weebly.com/uploads/2/7/1/0/27107965/04_free_body_exercises.pdf | Free-Body Exercises: Linear Motion In each case the rock is acted on by one or more forces. All drawings are in a vertical plane, and friction is negligible except where noted. Draw accurate free-body diagrams showing all forces acting on the rock. LM-l is done as an example, using the "parallelogram" method ..For convenience, you may draw all forces acting at the center of mass, even though friction and normal reaction force act at the point of contact with the surface. Please use a ruler, and do it in pencil so you can correct mistakes. Label forces using the following symbols: w = weight of rock, T = tension, n = normal reaction force, f = friction.
I __ ..
.
__ ..
__ .
__ ~L _ LM-9. Rock is sliding on a frictionless incline.
LM-13. Rock is decelerating because of kinetic friction.
LM-16. Rock is tied to a rope and pulled straight upward, accelerating at 9.8 m1s2.
No friction.
LM-II.
Rock is sliding at constant speed on a frictionless surface.
LM-14. Rock is rising in a parabolic trajectory.
/ I I LM-12. Rock is faIling at constant (terminal) velocity.
LM-15. Rock is at the top of a parabolic trajectory.
I I I I I I I I LM-18. Rock is tied to a rope and pulled so that it accelerates horizontally at 2g.
No friction.
\ \ \ LM-17. Rock is tied to a rope and pulled so that it moves horizontally at constant velocity. (There must be friction.) Free-Body Solutions: Linear Motion The dashed arrows and construction lines in these solutions are for explanation only, and not part of the finished diagram.
The free-body diagram in each case consists of only the dark, solid arrows. Forces of the same magnitude or lines of the same length are indicated by the same number of "tick" marks drawn through the two lines or arrows. Symbols: w = weight, T = tension, n = normal reaction force, f = friction.
T Note that the 2 I tension is not i proportional to string length.
~#'.#~\ '" I I I I ::::= I I I I /'~""""" ,.. I .• I •• I ::: I I I I I I f and n are actually I ! applied at ' !he surface) I ----~~~-----/ I I T(= 2w) I········HI·········· I I I I I I ::!:: I I I I f- -. -~---,-.
-.,- ..
.--..... '-.
_ ..- .._-.--- .. --, I 1-;.>-, , Resultant of T and" is in I direction of ~leration) Resultant of, T and w is equal tol Free-Body Exercises: Circular Motion Draw free-body diagrams showing forces acting on the rock, and in each case, indicate the centripetal force. Please note that the rock is not in equilibljum if it is moving in a circle. The centripetal force depends on angular velocity and there may not be any indication of exactly how big that force should be drawn. Symbols: w = weight, T = tension, f = friction, n = normal reaction force, Fe = centripetal force.
CM-I. Swinging on a rope, at lowest position. No friction.
CM-4. Rock is swinging on a rope. No friction.
CM-7. Rock is riding on a horizontal disk that is rotating at constant speed about its vertical axis. Friction prevents rock from sliding. Rock is moving straight out of the paper.
CM-2. Tied to a post and moving in a circle at constant speed on a fiictionless horizontal surface. Moving straight out of the paper.
CM-5. Rock is moving downward in a vertical circle with the string horizontal.
\ \ ~ \ CM-3. String is tied to a post. Rock is movingtoward you in a horizontalcircle at constant speed. No friction.
CM-6. Rock is swinging on a rope, at the top of a vertical circle. No friction.
CM-9. Rock is stuckby frictionagainst the inside wall of a drum rotating about its vertical axis at constant speed.
Rock is moving straight out of the paper.
CM-8.
Rock is resting against the i frictionless inside wall of a cone. It moves I with the cone, which rotates about its I vertical axis at constant angular speed.
I I i I I I Free-Body Solutions: Circular Motion .·· ··•··.. ·.. ·.. ··•··· .. ··.. ··.. ··.. ·· 1 I ;±: I I //~c=T+WJ T Exercises in Drawing and Utilizing Free-Body Diagrams Kurt Fisher, Division of Natural Sciences and Mathematics, Dowling College, Oakdale, NY 11769-1999; fisherk@dowling.edu S tudents taking the algebra-based introductory physics course often have great difficulty setting up Newton's second law equations of motion for dealing with one-body and two-body problems in particle mechanics. A prerequisite for doing so is the analysis of all the rel-evant forces, both the visible ones (those indicated or identified as applied forces) and the unseen ones such as gravity and friction. In turn, the tool for this force analysis is the free-body diagram (FBD).
It would seem that if FBD's were introduced, and their application to the generation of the required equations of motion illustrated, the class would readily catch on. Unfor-tunately, it doesn't work that way.
The analysis of forces is dependent on their being correctly perceived.
The excellent free-body scale-draw-ing exercises by J. E. Court! are use-ful for developing the concepts of vector resolution and force analysis.
They can be used to firm up a stu-dent's understanding of FBD con-struction to accurate scale, thereby giving insight as to how weight, nor-mal force, and tension force vectors relate to one another. They are also helpful in diagnosing the persistence of the naive "motion implies force" concepts that students have so much trouble shedding.
But FBD's alone are not enough!
There must be the follow-through of utilizing them. The short cut of grop-ing for formulas and numbers to plug into them is seemingly too irre-sistible. Nonetheless, most problems cannot be solved correctly without due analysis and, in any case, an ana-lytical attitude should be fostered as one of the main goals of education.
When required to include an FBD with a problem, many students draw a perfunctory diagram that looks like a fully loaded pincushion. Vectors are mis-oriented, unlabeled, and/or show no directions. Such an FBD obvious-ly cannot be used to proceed to the equation of motion. Yet, even the stu-dents who get the FBD correct do not use it further and will write an equa-tion of motion that obviously does not follow from their FBD, thereby defeating its very purpose. I can only ascribe this resistance to analysis as the manifestation of a learning style that stems from the "show-and-tell" methodology and the avoidance of word problems and other integrative activities.
The small set of exercises offered here (Fig. 1) shows how I try to habit-uate the student to the analysis steps needed to successfully work out problems in particle dynamics. The idea is to present a series of graded exercises in identifying forces, have the student install them on an FBD, and then take the next step-write down the ~F equations following from the analysis. Inherent in these exercises is the redundancy necessary for the learner to internalize the process. I have been using these exer-cises since returning from the 1993 Rensselaer conference.2 I bundle them together with the aforemen-tioned Court exercises and distribute them soon after starting the mechan-ics chapter. My students are given a three-week window in which to work both sets of exercises and submit them for homework credit. One revi-sion is permitted. Parts of these exer-cises appear on tests, so the value of working them out is appreciated by , the students.
To introduce my students to this multi-step procedure, three or four of the exercises are worked out in class using Socratic dialogue as much as possible. I emphasize that writing expressions for ~Fx and ~Fy is an indispensable prerequisite to solving one- and two-body mechanics prob-lems. This is where these exercises go a step beyond those that solely involve drawing FBD's. For each case the ~F x expression is carried out as far as possible. This means that, in the cases where friction is taken into account, it is necessary to substitute into the ~Fx expression the normal force yielded by the ~Fy = 0 equation (we always assume that there is no motion in the y-direction).
A very useful aid to sorting out the relevant forces in a mechanics prob-lem is the ONIBY table, which I have recently begun to require as part of the solution on tests. The idea for this table came from perusing one of the Socratic Dialogue Inducing (SDI) labs,3 another fallout gem traceable to the 1993 Rensselaer conference. It both mirrors and reinforces the FBD; an additional benefit is that it speeds up grading because all the force vec-tor values, magnitudes, and direc-tions are organized in one place. It requires the student to name the body each force acts ON and the body BY which that force is exerted.
I can offer only anecdotal percep-tions to indicate that a larger fraction of my class takes the solutions to standard mechanics problems farther than before. For more exercise sam-ples, cqntact me by mail or e-mail. I would appreciate any feedback as to results stemming from the deploy-ment of these exercises, and would also welcome any additions or modi-References I.
James E. Court, Phys.
Teach., 31, 104 (1993).
2. Conference on the Introductory Physics Course (May 1993).
The proceedings of this land-mark conference have been published by John Wiley & Sons, under the auspices of Rensselaer Polytechnic Insti-tute and the National Science Foundation (edited by Jack Wilson, ISBN O-nl-15557-8).
3. Richard R. Hake. private com-munication; See also Phys.
Teach. 30,546 (1992).
fications found to improve their effectiveness.
Graded Exercises in Drawing and Utilizing Free-Body Diagrams Using a ruler, draw free-body diagrams (FBD's) showing all forces acting on each body. Coordinate directions are indicated in the leading diagram of a sequence. Forces that are replaced by their x- and y- components should be shown canceled out.
Then using each FBD as a guide, write down the '2.Fx and '2.Fy expressions, carrying them to the point where numerical val-ues might be substituted for Fa' m, e, ep, and JL.
I.ONE·BODY F-B Diagram (Show only x- and y-com-IFx= IFy= CONFIGURATIONS ponents of all forces acting ON the body) 1. Frictionless t.x level surface.
___ 0.____ 2. Level surface Fa with friction.
~-Applied force at ...
----an angle e.
1..1 'I ~ 3. Incline with friction.
Applied force parallel to incline. Fa > WI!
II. TWO·BODY Take the x-axis to lie along the IFx = IF = CONFIGURATIONS direction of motion of each body.
y (Assume ideal pulleys) See #4 for the exam Ie.
4. m1 is on a ~ m1: m1: frictionless hor-izontal surface and is connected to hanging m2 m2: mass m2 by a mass-less string.
5. Same as #4 except ml: m,: that m I experiences friction.
ml: m2: 6. m I experiences J!41 m,: ml: friction.
m1 sin e> m2 m2: m2: Exercises in Drawing and Utilizing Free-Body Diagrams Vol. 37, Oct. 1999 THE PHYSICS TEACHER 435 Free-Body Diagrams Revisited - II James E. Court, City College of San Francisco, San Francisco, CA 94112 [Editor's Note: We reproduce here a continuation of the collection of free-body exercises sent us by Jim Court, the first part of which was published in October. At the request of Jim's widow, a colleague and friend of the author; Paul Hewitt (One San Antonio Place -2D, San Francisco, CA 94113-4032). is acting as Jim's representative in this important contribution to the teaching of physics.] Free-Body Exercises: Rotational Equilibrium All of the beams and the packages have the same weight w, and they are "uniform," which means the weight can be applied at the center. These systems are in equilibrium, so the net torque, the net vertical force and the net horizontal force are all zero. Symbols: w = weight, T = tension, n = normal reaction force at surface, V = vertical reaction force at hinge, H = horizontal reaction force at hinge,j = friction. RE-I is done as an example.
RE-13.
Equilibrium I -I I I RE-12.
Equilibrium Free-Body Solutions: Rotational Equilibrium Symbols: w = weight. T = tension, n = normal reaction force at surface, V = vertical reaction force at hinge, H = horizontal reac-tion force at hinge, f = friction.
.......... /f .!) ~ T ··················flf·············1 , ~ I Free-Body Exercises: Rotational Non-equilibrium In each case, draw arrows representing all forces acting on the cylinder or the beam. The solid, uniform cylinders, the pack-ages suspended from them, and the uniform beams all have the same weight w. In all but one of these examples the object is not in rotational equilibrium, i.e. the torques do not add up to zero. Symbols: T = tension, wand m = weight and mass of cylinders, beams and packages, n = normal reaction force at surface, V= vertical force at hinge or axle, H = horizontal force at hinge, a = acceleration. RN-I is done as an example.
RN-I. Cylinder is supported on a frictionless horizontal axle.
r<= w-ma) @~ RN-4. Cylinder is rolling down a rough (not frictionless) incline.
RN-7. Beam is swinging down through horizontal position.
RN-5. Cylinder was released with zero angular velocity on a frictionless incline. Is it rolling?
RN-3. String is tied to ceiling and wrapped around cylinder. Cylinder is falling.
RN-6. Beam is slipping. Both wall and floor are frictionless.
RN-9.
Beam is falling on a smooth (frictionless) floor. If the beam is released from rest, what path does the c of m take?
Free-Body Solutions: Rotational Non-equilibrium Symbols: T = tension, wand m = weight and mass of cylinders, beams and packages, n = normal reaction force at surface, V = vertical force at hinge or axle, H = horizontal force at hinge, a = acceleration.
w and " both pasl through the c of m, 10 there il no torque.
The cylinder IUdes downhiU.
V must be <w because the c of m is accelerating downward. |
5797 | https://timsong-cpp.github.io/cppwp/ | Draft C++ Standard: Contents
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Contents
1 Scope [intro.scope][intro.scope]
2 Normative references [intro.refs][intro.refs]
3 Terms and definitions [intro.defs][intro.defs]
4 General principles [intro][intro]
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5.13.1 Kinds of literals [lex.literal.kinds]
5.13.2 Integer literals [lex.icon]
5.13.3 Character literals [lex.ccon]
5.13.4 Floating-point literals [lex.fcon]
5.13.5 String literals [lex.string]
5.13.6 Unevaluated strings [lex.string.uneval]
5.13.7 Boolean literals [lex.bool]
5.13.8 Pointer literals [lex.nullptr]
5.13.9 User-defined literals [lex.ext]
6 Basics [basic][basic]
6.1 Preamble [basic.pre]
6.2 Declarations and definitions [basic.def]
6.3 One-definition rule [basic.def.odr]
6.4 Scope [basic.scope]
6.4.1 General [basic.scope.scope]
6.4.2 Point of declaration [basic.scope.pdecl]
6.4.3 Block scope [basic.scope.block]
6.4.4 Function parameter scope [basic.scope.param]
6.4.5 Lambda scope [basic.scope.lambda]
6.4.6 Namespace scope [basic.scope.namespace]
6.4.7 Class scope [basic.scope.class]
6.4.8 Enumeration scope [basic.scope.enum]
6.4.9 Template parameter scope [basic.scope.temp]
6.4.10 Contract-assertion scope [basic.scope.contract]
6.5 Name lookup [basic.lookup]
6.5.1 General [basic.lookup.general]
6.5.2 Member name lookup [class.member.lookup]
6.5.3 Unqualified name lookup [basic.lookup.unqual]
6.5.4 Argument-dependent name lookup [basic.lookup.argdep]
6.5.5 Qualified name lookup [basic.lookup.qual]
6.5.5.1 General [basic.lookup.qual.general]
6.5.5.2 Class members [class.qual]
6.5.5.3 Namespace members [namespace.qual]
6.5.6 Elaborated type specifiers [basic.lookup.elab]
6.5.7 Using-directives and namespace aliases [basic.lookup.udir]
6.6 Splice specifiers [basic.splice]
6.7 Program and linkage [basic.link]
6.8 Memory and objects [basic.memobj]
6.8.1 Memory model [intro.memory]
6.8.2 Object model [intro.object]
6.8.3 Alignment [basic.align]
6.8.4 Lifetime [basic.life]
6.8.5 Indeterminate and erroneous values [basic.indet]
6.8.6 Storage duration [basic.stc]
6.8.6.1 General [basic.stc.general]
6.8.6.2 Static storage duration [basic.stc.static]
6.8.6.3 Thread storage duration [basic.stc.thread]
6.8.6.4 Automatic storage duration [basic.stc.auto]
6.8.6.5 Dynamic storage duration [basic.stc.dynamic]
6.8.6.5.1 General [basic.stc.dynamic.general]
6.8.6.5.2 Allocation functions [basic.stc.dynamic.allocation]
6.8.6.5.3 Deallocation functions [basic.stc.dynamic.deallocation]
6.8.7 Temporary objects [class.temporary]
6.9 Types [basic.types]
6.9.1 General [basic.types.general]
6.9.2 Fundamental types [basic.fundamental]
6.9.3 Optional extended floating-point types [basic.extended.fp]
6.9.4 Compound types [basic.compound]
6.9.5 CV-qualifiers [basic.type.qualifier]
6.9.6 Conversion ranks [conv.rank]
6.10 Program execution [basic.exec]
6.10.1 Sequential execution [intro.execution]
6.10.2 Multi-threaded executions and data races [intro.multithread]
6.10.2.1 General [intro.multithread.general]
6.10.2.2 Data races [intro.races]
6.10.2.3 Forward progress [intro.progress]
6.10.3 Start and termination [basic.start]
6.10.3.1main function [basic.start.main]
6.10.3.2 Static initialization [basic.start.static]
6.10.3.3 Dynamic initialization of non-block variables [basic.start.dynamic]
6.10.3.4 Termination [basic.start.term]
6.11 Contract assertions [basic.contract]
6.11.1 General [basic.contract.general]
6.11.2 Evaluation [basic.contract.eval]
6.11.3 Contract-violation handler [basic.contract.handler]
7 Expressions [expr][expr]
7.1 Preamble [expr.pre]
7.2 Properties of expressions [expr.prop]
7.2.1 Value category [basic.lval]
7.2.2 Type [expr.type]
7.2.3 Context dependence [expr.context]
7.3 Standard conversions [conv]
7.3.1 General [conv.general]
7.3.2 Lvalue-to-rvalue conversion [conv.lval]
7.3.3 Array-to-pointer conversion [conv.array]
7.3.4 Function-to-pointer conversion [conv.func]
7.3.5 Temporary materialization conversion [conv.rval]
7.3.6 Qualification conversions [conv.qual]
7.3.7 Integral promotions [conv.prom]
7.3.8 Floating-point promotion [conv.fpprom]
7.3.9 Integral conversions [conv.integral]
7.3.10 Floating-point conversions [conv.double]
7.3.11 Floating-integral conversions [conv.fpint]
7.3.12 Pointer conversions [conv.ptr]
7.3.13 Pointer-to-member conversions [conv.mem]
7.3.14 Function pointer conversions [conv.fctptr]
7.3.15 Boolean conversions [conv.bool]
7.4 Usual arithmetic conversions [expr.arith.conv]
7.5 Primary expressions [expr.prim]
7.5.1 Grammar [expr.prim.grammar]
7.5.2 Literals [expr.prim.literal]
7.5.3 This [expr.prim.this]
7.5.4 Parentheses [expr.prim.paren]
7.5.5 Names [expr.prim.id]
7.5.5.1 General [expr.prim.id.general]
7.5.5.2 Unqualified names [expr.prim.id.unqual]
7.5.5.3 Qualified names [expr.prim.id.qual]
7.5.5.4 Pack indexing expression [expr.prim.pack.index]
7.5.5.5 Destruction [expr.prim.id.dtor]
7.5.6 Lambda expressions [expr.prim.lambda]
7.5.6.1 General [expr.prim.lambda.general]
7.5.6.2 Closure types [expr.prim.lambda.closure]
7.5.6.3 Captures [expr.prim.lambda.capture]
7.5.7 Fold expressions [expr.prim.fold]
7.5.8 Requires expressions [expr.prim.req]
7.5.8.1 General [expr.prim.req.general]
7.5.8.2 Simple requirements [expr.prim.req.simple]
7.5.8.3 Type requirements [expr.prim.req.type]
7.5.8.4 Compound requirements [expr.prim.req.compound]
7.5.8.5 Nested requirements [expr.prim.req.nested]
7.5.9 Expression splicing [expr.prim.splice]
7.6 Compound expressions [expr.compound]
7.6.1 Postfix expressions [expr.post]
7.6.1.1 General [expr.post.general]
7.6.1.2 Subscripting [expr.sub]
7.6.1.3 Function call [expr.call]
7.6.1.4 Explicit type conversion (functional notation) [expr.type.conv]
7.6.1.5 Class member access [expr.ref]
7.6.1.6 Increment and decrement [expr.post.incr]
7.6.1.7 Dynamic cast [expr.dynamic.cast]
7.6.1.8 Type identification [expr.typeid]
7.6.1.9 Static cast [expr.static.cast]
7.6.1.10 Reinterpret cast [expr.reinterpret.cast]
7.6.1.11 Const cast [expr.const.cast]
7.6.2 Unary expressions [expr.unary]
7.6.2.1 General [expr.unary.general]
7.6.2.2 Unary operators [expr.unary.op]
7.6.2.3 Increment and decrement [expr.pre.incr]
7.6.2.4 Await [expr.await]
7.6.2.5 Sizeof [expr.sizeof]
7.6.2.6 Alignof [expr.alignof]
7.6.2.7noexcept operator [expr.unary.noexcept]
7.6.2.8 New [expr.new]
7.6.2.9 Delete [expr.delete]
7.6.2.10 The reflection operator [expr.reflect]
7.6.3 Explicit type conversion (cast notation) [expr.cast]
7.6.4 Pointer-to-member operators [expr.mptr.oper]
7.6.5 Multiplicative operators [expr.mul]
7.6.6 Additive operators [expr.add]
7.6.7 Shift operators [expr.shift]
7.6.8 Three-way comparison operator [expr.spaceship]
7.6.9 Relational operators [expr.rel]
7.6.10 Equality operators [expr.eq]
7.6.11 Bitwise AND operator [expr.bit.and]
7.6.12 Bitwise exclusive OR operator [expr.xor]
7.6.13 Bitwise inclusive OR operator [expr.or]
7.6.14 Logical AND operator [expr.log.and]
7.6.15 Logical OR operator [expr.log.or]
7.6.16 Conditional operator [expr.cond]
7.6.17 Yielding a value [expr.yield]
7.6.18 Throwing an exception [expr.throw]
7.6.19 Assignment and compound assignment operators [expr.assign]
7.6.20 Comma operator [expr.comma]
7.7 Constant expressions [expr.const]
8 Statements [stmt][stmt]
8.1 Preamble [stmt.pre]
8.2 Label [stmt.label]
8.3 Expression statement [stmt.expr]
8.4 Compound statement or block [stmt.block]
8.5 Selection statements [stmt.select]
8.5.1 General [stmt.select.general]
8.5.2 The if statement [stmt.if]
8.5.3 The switch statement [stmt.switch]
8.6 Iteration statements [stmt.iter]
8.6.1 General [stmt.iter.general]
8.6.2 The while statement [stmt.while]
8.6.3 The do statement [stmt.do]
8.6.4 The for statement [stmt.for]
8.6.5 The range-based for statement [stmt.ranged]
8.7 Expansion statements [stmt.expand]
8.8 Jump statements [stmt.jump]
8.8.1 General [stmt.jump.general]
8.8.2 The break statement [stmt.break]
8.8.3 The continue statement [stmt.cont]
8.8.4 The return statement [stmt.return]
8.8.5 The co_ return statement [stmt.return.coroutine]
8.8.6 The goto statement [stmt.goto]
8.9 Assertion statement [stmt.contract.assert]
8.10 Declaration statement [stmt.dcl]
8.11 Ambiguity resolution [stmt.ambig]
9 Declarations [dcl][dcl]
9.1 Preamble [dcl.pre]
9.2 Specifiers [dcl.spec]
9.2.1 General [dcl.spec.general]
9.2.2 Storage class specifiers [dcl.stc]
9.2.3 Function specifiers [dcl.fct.spec]
9.2.4 The typedef specifier [dcl.typedef]
9.2.5 The friend specifier [dcl.friend]
9.2.6 The constexpr and consteval specifiers [dcl.constexpr]
9.2.7 The constinit specifier [dcl.constinit]
9.2.8 The inline specifier [dcl.inline]
9.2.9 Type specifiers [dcl.type]
9.2.9.1 General [dcl.type.general]
9.2.9.2 The cv-qualifier s[dcl.type.cv]
9.2.9.3 Simple type specifiers [dcl.type.simple]
9.2.9.4 Pack indexing specifier [dcl.type.pack.index]
9.2.9.5 Elaborated type specifiers [dcl.type.elab]
9.2.9.6 Decltype specifiers [dcl.type.decltype]
9.2.9.7 Placeholder type specifiers [dcl.spec.auto]
9.2.9.7.1 General [dcl.spec.auto.general]
9.2.9.7.2 Placeholder type deduction [dcl.type.auto.deduct]
9.2.9.8 Deduced class template specialization types [dcl.type.class.deduct]
9.2.9.9 Type splicing [dcl.type.splice]
9.3 Declarators [dcl.decl]
9.3.1 General [dcl.decl.general]
9.3.2 Type names [dcl.name]
9.3.3 Ambiguity resolution [dcl.ambig.res]
9.3.4 Meaning of declarators [dcl.meaning]
9.3.4.1 General [dcl.meaning.general]
9.3.4.2 Pointers [dcl.ptr]
9.3.4.3 References [dcl.ref]
9.3.4.4 Pointers to members [dcl.mptr]
9.3.4.5 Arrays [dcl.array]
9.3.4.6 Functions [dcl.fct]
9.3.4.7 Default arguments [dcl.fct.default]
9.4 Function contract specifiers [dcl.contract]
9.4.1 General [dcl.contract.func]
9.4.2 Referring to the result object [dcl.contract.res]
9.5 Initializers [dcl.init]
9.5.1 General [dcl.init.general]
9.5.2 Aggregates [dcl.init.aggr]
9.5.3 Character arrays [dcl.init.string]
9.5.4 References [dcl.init.ref]
9.5.5 List-initialization [dcl.init.list]
9.6 Function definitions [dcl.fct.def]
9.6.1 General [dcl.fct.def.general]
9.6.2 Explicitly-defaulted functions [dcl.fct.def.default]
9.6.3 Deleted definitions [dcl.fct.def.delete]
9.6.4 Coroutine definitions [dcl.fct.def.coroutine]
9.6.5 Replaceable function definitions [dcl.fct.def.replace]
9.7 Structured binding declarations [dcl.struct.bind]
9.8 Enumerations [enum]
9.8.1 Enumeration declarations [dcl.enum]
9.8.2 The using enum declaration [enum.udecl]
9.9 Namespaces [basic.namespace]
9.9.1 General [basic.namespace.general]
9.9.2 Namespace definition [namespace.def]
9.9.2.1 General [namespace.def.general]
9.9.2.2 Unnamed namespaces [namespace.unnamed]
9.9.3 Namespace alias [namespace.alias]
9.9.4 Using namespace directive [namespace.udir]
9.10 The using declaration [namespace.udecl]
9.11 The asm declaration [dcl.asm]
9.12 Linkage specifications [dcl.link]
9.13 Attributes [dcl.attr]
9.13.1 Attribute syntax and semantics [dcl.attr.grammar]
9.13.2 Alignment specifier [dcl.align]
9.13.3 Assumption attribute [dcl.attr.assume]
9.13.4 Deprecated attribute [dcl.attr.deprecated]
9.13.5 Fallthrough attribute [dcl.attr.fallthrough]
9.13.6 Indeterminate storage [dcl.attr.indet]
9.13.7 Likelihood attributes [dcl.attr.likelihood]
9.13.8 Maybe unused attribute [dcl.attr.unused]
9.13.9 Nodiscard attribute [dcl.attr.nodiscard]
9.13.10 Noreturn attribute [dcl.attr.noreturn]
9.13.11 No unique address attribute [dcl.attr.nouniqueaddr]
9.13.12 Annotations [dcl.attr.annotation]
10 Modules [module][module]
10.1 Module units and purviews [module.unit]
10.2 Export declaration [module.interface]
10.3 Import declaration [module.import]
10.4 Global module fragment [module.global.frag]
10.5 Private module fragment [module.private.frag]
10.6 Instantiation context [module.context]
10.7 Reachability [module.reach]
11 Classes [class][class]
11.1 Preamble [class.pre]
11.2 Properties of classes [class.prop]
11.3 Class names [class.name]
11.4 Class members [class.mem]
11.4.1 General [class.mem.general]
11.4.2 Member functions [class.mfct]
11.4.3 Non-static member functions [class.mfct.non.static]
11.4.4 Special member functions [special]
11.4.5 Constructors [class.ctor]
11.4.5.1 General [class.ctor.general]
11.4.5.2 Default constructors [class.default.ctor]
11.4.5.3 Copy/move constructors [class.copy.ctor]
11.4.6 Copy/move assignment operator [class.copy.assign]
11.4.7 Destructors [class.dtor]
11.4.8 Conversions [class.conv]
11.4.8.1 General [class.conv.general]
11.4.8.2 Conversion by constructor [class.conv.ctor]
11.4.8.3 Conversion functions [class.conv.fct]
11.4.9 Static members [class.static]
11.4.9.1 General [class.static.general]
11.4.9.2 Static member functions [class.static.mfct]
11.4.9.3 Static data members [class.static.data]
11.4.10 Bit-fields [class.bit]
11.4.11 Allocation and deallocation functions [class.free]
11.4.12 Nested class declarations [class.nest]
11.5 Unions [class.union]
11.5.1 General [class.union.general]
11.5.2 Anonymous unions [class.union.anon]
11.6 Local class declarations [class.local]
11.7 Derived classes [class.derived]
11.7.1 General [class.derived.general]
11.7.2 Multiple base classes [class.mi]
11.7.3 Virtual functions [class.virtual]
11.7.4 Abstract classes [class.abstract]
11.8 Member access control [class.access]
11.8.1 General [class.access.general]
11.8.2 Access specifiers [class.access.spec]
11.8.3 Accessibility of base classes and base class members [class.access.base]
11.8.4 Friends [class.friend]
11.8.5 Protected member access [class.protected]
11.8.6 Access to virtual functions [class.access.virt]
11.8.7 Multiple access [class.paths]
11.8.8 Nested classes [class.access.nest]
11.9 Initialization [class.init]
11.9.1 General [class.init.general]
11.9.2 Explicit initialization [class.expl.init]
11.9.3 Initializing bases and members [class.base.init]
11.9.4 Initialization by inherited constructor [class.inhctor.init]
11.9.5 Construction and destruction [class.cdtor]
11.9.6 Copy/move elision [class.copy.elision]
11.10 Comparisons [class.compare]
11.10.1 Defaulted comparison operator functions [class.compare.default]
11.10.2 Equality operator [class.eq]
11.10.3 Three-way comparison [class.spaceship]
11.10.4 Secondary comparison operators [class.compare.secondary]
12 Overloading [over][over]
12.1 Preamble [over.pre]
12.2 Overload resolution [over.match]
12.2.1 General [over.match.general]
12.2.2 Candidate functions and argument lists [over.match.funcs]
12.2.2.1 General [over.match.funcs.general]
12.2.2.2 Function call syntax [over.match.call]
12.2.2.2.1 General [over.match.call.general]
12.2.2.2.2 Call to designated function [over.call.func]
12.2.2.2.3 Call to object of class type [over.call.object]
12.2.2.3 Operators in expressions [over.match.oper]
12.2.2.4 Initialization by constructor [over.match.ctor]
12.2.2.5 Copy-initialization of class by user-defined conversion [over.match.copy]
12.2.2.6 Initialization by conversion function [over.match.conv]
12.2.2.7 Initialization by conversion function for direct reference binding [over.match.ref]
12.2.2.8 Initialization by list-initialization [over.match.list]
12.2.2.9 Class template argument deduction [over.match.class.deduct]
12.2.3 Viable functions [over.match.viable]
12.2.4 Best viable function [over.match.best]
12.2.4.1 General [over.match.best.general]
12.2.4.2 Implicit conversion sequences [over.best.ics]
12.2.4.2.1 General [over.best.ics.general]
12.2.4.2.2 Standard conversion sequences [over.ics.scs]
12.2.4.2.3 User-defined conversion sequences [over.ics.user]
12.2.4.2.4 Ellipsis conversion sequences [over.ics.ellipsis]
12.2.4.2.5 Reference binding [over.ics.ref]
12.2.4.2.6 List-initialization sequence [over.ics.list]
12.2.4.3 Ranking implicit conversion sequences [over.ics.rank]
12.3 Address of an overload set [over.over]
12.4 Overloaded operators [over.oper]
12.4.1 General [over.oper.general]
12.4.2 Unary operators [over.unary]
12.4.3 Binary operators [over.binary]
12.4.3.1 General [over.binary.general]
12.4.3.2 Simple assignment [over.assign]
12.4.4 Function call [over.call]
12.4.5 Subscripting [over.sub]
12.4.6 Class member access [over.ref]
12.4.7 Increment and decrement [over.inc]
12.5 Built-in operators [over.built]
12.6 User-defined literals [over.literal]
13 Templates [temp][temp]
13.1 Preamble [temp.pre]
13.2 Template parameters [temp.param]
13.3 Names of template specializations [temp.names]
13.4 Template arguments [temp.arg]
13.4.1 General [temp.arg.general]
13.4.2 Type template arguments [temp.arg.type]
13.4.3 Constant template arguments [temp.arg.nontype]
13.4.4 Template template arguments [temp.arg.template]
13.5 Template constraints [temp.constr]
13.5.1 General [temp.constr.general]
13.5.2 Constraints [temp.constr.constr]
13.5.2.1 General [temp.constr.constr.general]
13.5.2.2 Logical operations [temp.constr.op]
13.5.2.3 Atomic constraints [temp.constr.atomic]
13.5.2.4 Concept-dependent constraints [temp.constr.concept]
13.5.2.5 Fold expanded constraint [temp.constr.fold]
13.5.3 Constrained declarations [temp.constr.decl]
13.5.4 Constraint normalization [temp.constr.normal]
13.5.5 Partial ordering by constraints [temp.constr.order]
13.6 Type equivalence [temp.type]
13.7 Template declarations [temp.decls]
13.7.1 General [temp.decls.general]
13.7.2 Class templates [temp.class]
13.7.2.1 General [temp.class.general]
13.7.2.2 Member functions of class templates [temp.mem.func]
13.7.2.3 Deduction guides [temp.deduct.guide]
13.7.2.4 Member classes of class templates [temp.mem.class]
13.7.2.5 Static data members of class templates [temp.static]
13.7.2.6 Enumeration members of class templates [temp.mem.enum]
13.7.3 Member templates [temp.mem]
13.7.4 Variadic templates [temp.variadic]
13.7.5 Friends [temp.friend]
13.7.6 Partial specialization [temp.spec.partial]
13.7.6.1 General [temp.spec.partial.general]
13.7.6.2 Matching of partial specializations [temp.spec.partial.match]
13.7.6.3 Partial ordering of partial specializations [temp.spec.partial.order]
13.7.6.4 Members of class template partial specializations [temp.spec.partial.member]
13.7.7 Function templates [temp.fct]
13.7.7.1 General [temp.fct.general]
13.7.7.2 Function template overloading [temp.over.link]
13.7.7.3 Partial ordering of function templates [temp.func.order]
13.7.8 Alias templates [temp.alias]
13.7.9 Concept definitions [temp.concept]
13.8 Name resolution [temp.res]
13.8.1 General [temp.res.general]
13.8.2 Locally declared names [temp.local]
13.8.3 Dependent names [temp.dep]
13.8.3.1 General [temp.dep.general]
13.8.3.2 Dependent types [temp.dep.type]
13.8.3.3 Type-dependent expressions [temp.dep.expr]
13.8.3.4 Value-dependent expressions [temp.dep.constexpr]
13.8.3.5 Dependent splice specifiers [temp.dep.splice]
13.8.3.6 Dependent namespaces [temp.dep.namespace]
13.8.3.7 Dependent template arguments [temp.dep.temp]
13.8.4 Dependent name resolution [temp.dep.res]
13.8.4.1 Point of instantiation [temp.point]
13.8.4.2 Candidate functions [temp.dep.candidate]
13.9 Template instantiation and specialization [temp.spec]
13.9.1 General [temp.spec.general]
13.9.2 Implicit instantiation [temp.inst]
13.9.3 Explicit instantiation [temp.explicit]
13.9.4 Explicit specialization [temp.expl.spec]
13.10 Function template specializations [temp.fct.spec]
13.10.1 General [temp.fct.spec.general]
13.10.2 Explicit template argument specification [temp.arg.explicit]
13.10.3 Template argument deduction [temp.deduct]
13.10.3.1 General [temp.deduct.general]
13.10.3.2 Deducing template arguments from a function call [temp.deduct.call]
13.10.3.3 Deducing template arguments taking the address of a function template [temp.deduct.funcaddr]
13.10.3.4 Deducing conversion function template arguments [temp.deduct.conv]
13.10.3.5 Deducing template arguments during partial ordering [temp.deduct.partial]
13.10.3.6 Deducing template arguments from a type [temp.deduct.type]
13.10.3.7 Deducing template arguments from a function declaration [temp.deduct.decl]
13.10.4 Overload resolution [temp.over]
14 Exception handling [except][except]
14.1 Preamble [except.pre]
14.2 Throwing an exception [except.throw]
14.3 Stack unwinding [except.ctor]
14.4 Handling an exception [except.handle]
14.5 Exception specifications [except.spec]
14.6 Special functions [except.special]
14.6.1 General [except.special.general]
14.6.2 The std::terminate function [except.terminate]
15 Preprocessing directives [cpp][cpp]
15.1 Preamble [cpp.pre]
15.2 Conditional inclusion [cpp.cond]
15.3 Source file inclusion [cpp.include]
15.4 Resource inclusion [cpp.embed]
15.4.1 General [cpp.embed.gen]
15.4.2 Embed parameters [cpp.embed.param]
15.4.2.1 limit parameter [cpp.embed.param.limit]
15.4.2.2 prefix parameter [cpp.embed.param.prefix]
15.4.2.3 suffix parameter [cpp.embed.param.suffix]
15.4.2.4if_ empty parameter [cpp.embed.param.if.empty]
15.5 Module directive [cpp.module]
15.6 Header unit importation [cpp.import]
15.7 Macro replacement [cpp.replace]
15.7.1 General [cpp.replace.general]
15.7.2 Argument substitution [cpp.subst]
15.7.3 The # operator [cpp.stringize]
15.7.4 The ## operator [cpp.concat]
15.7.5 Rescanning and further replacement [cpp.rescan]
15.7.6 Scope of macro definitions [cpp.scope]
15.8 Line control [cpp.line]
15.9 Diagnostic directives [cpp.error]
15.10 Pragma directive [cpp.pragma]
15.11 Null directive [cpp.null]
15.12 Predefined macro names [cpp.predefined]
15.13 Pragma operator [cpp.pragma.op]
16 Library introduction [library][library]
16.1 General [library.general]
16.2 The C standard library [library.c]
16.3 Method of description [description]
16.3.1 General [description.general]
16.3.2 Structure of each clause [structure]
16.3.2.1 Elements [structure.elements]
16.3.2.2 Summary [structure.summary]
16.3.2.3 Requirements [structure.requirements]
16.3.2.4 Detailed specifications [structure.specifications]
16.3.2.5 C library [structure.see.also]
16.3.3 Other conventions [conventions]
16.3.3.1 General [conventions.general]
16.3.3.2 Exposition-only entities, etc. [expos.only.entity]
16.3.3.3 Type descriptions [type.descriptions]
16.3.3.3.1 General [type.descriptions.general]
16.3.3.3.2 Enumerated types [enumerated.types]
16.3.3.3.3 Bitmask types [bitmask.types]
16.3.3.3.4 Character sequences [character.seq]
16.3.3.3.4.1 General [character.seq.general]
16.3.3.3.4.2 Byte strings [byte.strings]
16.3.3.3.4.3 Multibyte strings [multibyte.strings]
16.3.3.3.5 Customization Point Object types [customization.point.object]
16.3.3.4 Algorithm function objects [alg.func.obj]
16.3.3.5 Functions within classes [functions.within.classes]
16.3.3.6 Private members [objects.within.classes]
16.3.3.7 Freestanding items [freestanding.item]
16.4 Library-wide requirements [requirements]
16.4.1 General [requirements.general]
16.4.2 Library contents and organization [organization]
16.4.2.1 General [organization.general]
16.4.2.2 Library contents [contents]
16.4.2.3 Headers [headers]
16.4.2.4 Modules [std.modules]
16.4.2.5 Freestanding implementations [compliance]
16.4.3 Using the library [using]
16.4.3.1 Overview [using.overview]
16.4.3.2 Headers [using.headers]
16.4.3.3 Linkage [using.linkage]
16.4.4 Requirements on types and expressions [utility.requirements]
16.4.4.1 General [utility.requirements.general]
16.4.4.2 Template argument requirements [utility.arg.requirements]
16.4.4.3 Swappable requirements [swappable.requirements]
16.4.4.4Cpp17NullablePointer requirements [nullablepointer.requirements]
16.4.4.5Cpp17Hash requirements [hash.requirements]
16.4.4.6Cpp17Allocator requirements [allocator.requirements]
16.4.4.6.1 General [allocator.requirements.general]
16.4.4.6.2 Allocator completeness requirements [allocator.requirements.completeness]
16.4.5 Constraints on programs [constraints]
16.4.5.1 Overview [constraints.overview]
16.4.5.2 Namespace use [namespace.constraints]
16.4.5.2.1 Namespace std[namespace.std]
16.4.5.2.2 Namespace posix[namespace.posix]
16.4.5.2.3 Namespaces for future standardization [namespace.future]
16.4.5.3 Reserved names [reserved.names]
16.4.5.3.1 General [reserved.names.general]
16.4.5.3.2 Zombie names [zombie.names]
16.4.5.3.3 Macro names [macro.names]
16.4.5.3.4 External linkage [extern.names]
16.4.5.3.5 Types [extern.types]
16.4.5.3.6 User-defined literal suffixes [usrlit.suffix]
16.4.5.4 Headers [alt.headers]
16.4.5.5 Derived classes [derived.classes]
16.4.5.6 Replacement functions [replacement.functions]
16.4.5.7 Handler functions [handler.functions]
16.4.5.8 Other functions [res.on.functions]
16.4.5.9 Function arguments [res.on.arguments]
16.4.5.10 Library object access [res.on.objects]
16.4.5.11 Semantic requirements [res.on.requirements]
16.4.6 Conforming implementations [conforming]
16.4.6.1 Overview [conforming.overview]
16.4.6.2 Headers [res.on.headers]
16.4.6.3 Restrictions on macro definitions [res.on.macro.definitions]
16.4.6.4 Non-member functions [global.functions]
16.4.6.5 Member functions [member.functions]
16.4.6.6 Friend functions [hidden.friends]
16.4.6.7 Constexpr functions and constructors [constexpr.functions]
16.4.6.8 Requirements for stable algorithms [algorithm.stable]
16.4.6.9 Reentrancy [reentrancy]
16.4.6.10 Data race avoidance [res.on.data.races]
16.4.6.11 Properties of library classes [library.class.props]
16.4.6.12 Protection within classes [protection.within.classes]
16.4.6.13 Derived classes [derivation]
16.4.6.14 Restrictions on exception handling [res.on.exception.handling]
16.4.6.15 Contract assertions [res.contract.assertions]
16.4.6.16 Value of error codes [value.error.codes]
16.4.6.17 Moved-from state of library types [lib.types.movedfrom]
17 Language support library [support][support]
17.1 General [support.general]
17.2 Common definitions [support.types]
17.2.1 Header synopsis [cstddef.syn]
17.2.2 Header synopsis [cstdlib.syn]
17.2.3 Null pointers [support.types.nullptr]
17.2.4 Sizes, alignments, and offsets [support.types.layout]
17.2.5byte type operations [support.types.byteops]
17.3 Implementation properties [support.limits]
17.3.1 General [support.limits.general]
17.3.2 Header synopsis [version.syn]
17.3.3 Header synopsis [limits.syn]
17.3.4 Enum float_ round_ style[round.style]
17.3.5 Class template numeric_ limits[numeric.limits]
17.3.5.1 General [numeric.limits.general]
17.3.5.2numeric_ limits members [numeric.limits.members]
17.3.5.3numeric_ limits specializations [numeric.special]
17.3.6 Header synopsis [climits.syn]
17.3.7 Header synopsis [cfloat.syn]
17.4 Arithmetic types [support.arith.types]
17.4.1 Header synopsis [cstdint.syn]
17.4.2 Header synopsis [stdfloat.syn]
17.5 Startup and termination [support.start.term]
17.6 Dynamic memory management [support.dynamic]
17.6.1 General [support.dynamic.general]
17.6.2 Header synopsis [new.syn]
17.6.3 Storage allocation and deallocation [new.delete]
17.6.3.1 General [new.delete.general]
17.6.3.2 Single-object forms [new.delete.single]
17.6.3.3 Array forms [new.delete.array]
17.6.3.4 Non-allocating forms [new.delete.placement]
17.6.3.5 Data races [new.delete.dataraces]
17.6.4 Storage allocation errors [alloc.errors]
17.6.4.1 Class bad_ alloc[bad.alloc]
17.6.4.2 Class bad_ array_ new_ length[new.badlength]
17.6.4.3 Type new_ handler[new.handler]
17.6.4.4set_ new_ handler[set.new.handler]
17.6.4.5get_ new_ handler[get.new.handler]
17.6.5 Pointer optimization barrier [ptr.launder]
17.6.6 Hardware interference size [hardware.interference]
17.7 Type identification [support.rtti]
17.7.1 General [support.rtti.general]
17.7.2 Header synopsis [typeinfo.syn]
17.7.3 Class type_ info[type.info]
17.7.4 Class bad_ cast[bad.cast]
17.7.5 Class bad_ typeid[bad.typeid]
17.7.6 Header synopsis [type.index.synopsis]
17.7.7 Class type_ index[type.index]
17.8 Source location [support.srcloc]
17.8.1 Header synopsis [source.location.syn]
17.8.2 Class source_ location[support.srcloc.class]
17.8.2.1 General [support.srcloc.class.general]
17.8.2.2 Creation [support.srcloc.cons]
17.8.2.3 Observers [support.srcloc.obs]
17.9 Exception handling [support.exception]
17.9.1 General [support.exception.general]
17.9.2 Header synopsis [exception.syn]
17.9.3 Class exception[exception]
17.9.4 Class bad_ exception[bad.exception]
17.9.5 Abnormal termination [exception.terminate]
17.9.5.1 Type terminate_ handler[terminate.handler]
17.9.5.2set_ terminate[set.terminate]
17.9.5.3get_ terminate[get.terminate]
17.9.5.4terminate[terminate]
17.9.6uncaught_ exceptions[uncaught.exceptions]
17.9.7 Exception propagation [propagation]
17.9.8nested_ exception[except.nested]
17.10 Contract-violation handling [support.contract]
17.10.1 Header synopsis [contracts.syn]
17.10.2 Enumerations [support.contract.enum]
17.10.3 Class contract_ violation[support.contract.violation]
17.10.4 Invoke default handler [support.contract.invoke]
17.11 Initializer lists [support.initlist]
17.11.1 General [support.initlist.general]
17.11.2 Header synopsis [initializer.list.syn]
17.11.3 Initializer list constructors [support.initlist.cons]
17.11.4 Initializer list access [support.initlist.access]
17.11.5 Initializer list range access [support.initlist.range]
17.12 Comparisons [cmp]
17.12.1 Header synopsis [compare.syn]
17.12.2 Comparison category types [cmp.categories]
17.12.2.1 Preamble [cmp.categories.pre]
17.12.2.2 Class partial_ ordering[cmp.partialord]
17.12.2.3 Class weak_ ordering[cmp.weakord]
17.12.2.4 Class strong_ ordering[cmp.strongord]
17.12.3 Class template common_ comparison_ category[cmp.common]
17.12.4 Concept three_ way_ comparable[cmp.concept]
17.12.5 Result of three-way comparison [cmp.result]
17.12.6 Comparison algorithms [cmp.alg]
17.12.7 Type Ordering [compare.type]
17.13 Coroutines [support.coroutine]
17.13.1 General [support.coroutine.general]
17.13.2 Header synopsis [coroutine.syn]
17.13.3 Coroutine traits [coroutine.traits]
17.13.3.1 General [coroutine.traits.general]
17.13.3.2 Class template coroutine_ traits[coroutine.traits.primary]
17.13.4 Class template coroutine_ handle[coroutine.handle]
17.13.4.1 General [coroutine.handle.general]
17.13.4.2 Construct/reset [coroutine.handle.con]
17.13.4.3 Conversion [coroutine.handle.conv]
17.13.4.4 Export/import [coroutine.handle.export.import]
17.13.4.5 Observers [coroutine.handle.observers]
17.13.4.6 Resumption [coroutine.handle.resumption]
17.13.4.7 Promise access [coroutine.handle.promise]
17.13.4.8 Comparison operators [coroutine.handle.compare]
17.13.4.9 Hash support [coroutine.handle.hash]
17.13.5 No-op coroutines [coroutine.noop]
17.13.5.1 Class noop_ coroutine_ promise[coroutine.promise.noop]
17.13.5.2 Class coroutine_ handle[coroutine.handle.noop]
17.13.5.2.1 General [coroutine.handle.noop.general]
17.13.5.2.2 Conversion [coroutine.handle.noop.conv]
17.13.5.2.3 Observers [coroutine.handle.noop.observers]
17.13.5.2.4 Resumption [coroutine.handle.noop.resumption]
17.13.5.2.5 Promise access [coroutine.handle.noop.promise]
17.13.5.2.6 Address [coroutine.handle.noop.address]
17.13.5.3 Function noop_ coroutine[coroutine.noop.coroutine]
17.13.6 Trivial awaitables [coroutine.trivial.awaitables]
17.14 Other runtime support [support.runtime]
17.14.1 General [support.runtime.general]
17.14.2 Header synopsis [cstdarg.syn]
17.14.3 Header synopsis [csetjmp.syn]
17.14.4 Header synopsis [csignal.syn]
17.14.5 Signal handlers [support.signal]
17.15 C headers [support.c.headers]
17.15.1 General [support.c.headers.general]
17.15.2 Header synopsis [complex.h.syn]
17.15.3 Header synopsis [iso646.h.syn]
17.15.4 Header synopsis [stdalign.h.syn]
17.15.5 Header synopsis [stdbool.h.syn]
17.15.6 Header synopsis [tgmath.h.syn]
17.15.7 Other C headers [support.c.headers.other]
18 Concepts library [concepts][concepts]
18.1 General [concepts.general]
18.2 Equality preservation [concepts.equality]
18.3 Header synopsis [concepts.syn]
18.4 Language-related concepts [concepts.lang]
18.4.1 General [concepts.lang.general]
18.4.2 Concept same_ as[concept.same]
18.4.3 Concept derived_ from[concept.derived]
18.4.4 Concept convertible_ to[concept.convertible]
18.4.5 Concept common_ reference_ with[concept.commonref]
18.4.6 Concept common_ with[concept.common]
18.4.7 Arithmetic concepts [concepts.arithmetic]
18.4.8 Concept assignable_ from[concept.assignable]
18.4.9 Concept swappable[concept.swappable]
18.4.10 Concept destructible[concept.destructible]
18.4.11 Concept constructible_ from[concept.constructible]
18.4.12 Concept default_ initializable[concept.default.init]
18.4.13 Concept move_ constructible[concept.moveconstructible]
18.4.14 Concept copy_ constructible[concept.copyconstructible]
18.5 Comparison concepts [concepts.compare]
18.5.1 General [concepts.compare.general]
18.5.2 Boolean testability [concept.booleantestable]
18.5.3 Comparison common types [concept.comparisoncommontype]
18.5.4 Concept equality_ comparable[concept.equalitycomparable]
18.5.5 Concept totally_ ordered[concept.totallyordered]
18.6 Object concepts [concepts.object]
18.7 Callable concepts [concepts.callable]
18.7.1 General [concepts.callable.general]
18.7.2 Concept invocable[concept.invocable]
18.7.3 Concept regular_ invocable[concept.regularinvocable]
18.7.4 Concept predicate[concept.predicate]
18.7.5 Concept relation[concept.relation]
18.7.6 Concept equivalence_ relation[concept.equiv]
18.7.7 Concept strict_ weak_ order[concept.strictweakorder]
19 Diagnostics library [diagnostics][diagnostics]
19.1 General [diagnostics.general]
19.2 Exception classes [std.exceptions]
19.2.1 General [std.exceptions.general]
19.2.2 Header synopsis [stdexcept.syn]
19.2.3 Class logic_ error[logic.error]
19.2.4 Class domain_ error[domain.error]
19.2.5 Class invalid_ argument[invalid.argument]
19.2.6 Class length_ error[length.error]
19.2.7 Class out_ of_ range[out.of.range]
19.2.8 Class runtime_ error[runtime.error]
19.2.9 Class range_ error[range.error]
19.2.10 Class overflow_ error[overflow.error]
19.2.11 Class underflow_ error[underflow.error]
19.3 Assertions [assertions]
19.3.1 General [assertions.general]
19.3.2 Header synopsis [cassert.syn]
19.3.3 The assert macro [assertions.assert]
19.4 Error numbers [errno]
19.4.1 General [errno.general]
19.4.2 Header synopsis [cerrno.syn]
19.5 System error support [syserr]
19.5.1 General [syserr.general]
19.5.2 Header synopsis [system.error.syn]
19.5.3 Class error_ category[syserr.errcat]
19.5.3.1 Overview [syserr.errcat.overview]
19.5.3.2 Virtual members [syserr.errcat.virtuals]
19.5.3.3 Non-virtual members [syserr.errcat.nonvirtuals]
19.5.3.4 Program-defined classes derived from error_ category[syserr.errcat.derived]
19.5.3.5 Error category objects [syserr.errcat.objects]
19.5.4 Class error_ code[syserr.errcode]
19.5.4.1 Overview [syserr.errcode.overview]
19.5.4.2 Constructors [syserr.errcode.constructors]
19.5.4.3 Modifiers [syserr.errcode.modifiers]
19.5.4.4 Observers [syserr.errcode.observers]
19.5.4.5 Non-member functions [syserr.errcode.nonmembers]
19.5.5 Class error_ condition[syserr.errcondition]
19.5.5.1 Overview [syserr.errcondition.overview]
19.5.5.2 Constructors [syserr.errcondition.constructors]
19.5.5.3 Modifiers [syserr.errcondition.modifiers]
19.5.5.4 Observers [syserr.errcondition.observers]
19.5.5.5 Non-member functions [syserr.errcondition.nonmembers]
19.5.6 Comparison operator functions [syserr.compare]
19.5.7 System error hash support [syserr.hash]
19.5.8 Class system_ error[syserr.syserr]
19.5.8.1 Overview [syserr.syserr.overview]
19.5.8.2 Members [syserr.syserr.members]
19.6 Stacktrace [stacktrace]
19.6.1 General [stacktrace.general]
19.6.2 Header synopsis [stacktrace.syn]
19.6.3 Class stacktrace_ entry[stacktrace.entry]
19.6.3.1 Overview [stacktrace.entry.overview]
19.6.3.2 Constructors [stacktrace.entry.cons]
19.6.3.3 Observers [stacktrace.entry.obs]
19.6.3.4 Query [stacktrace.entry.query]
19.6.3.5 Comparison [stacktrace.entry.cmp]
19.6.4 Class template basic_ stacktrace[stacktrace.basic]
19.6.4.1 Overview [stacktrace.basic.overview]
19.6.4.2 Creation and assignment [stacktrace.basic.cons]
19.6.4.3 Observers [stacktrace.basic.obs]
19.6.4.4 Comparisons [stacktrace.basic.cmp]
19.6.4.5 Modifiers [stacktrace.basic.mod]
19.6.4.6 Non-member functions [stacktrace.basic.nonmem]
19.6.5 Formatting support [stacktrace.format]
19.6.6 Hash support [stacktrace.basic.hash]
19.7 Debugging [debugging]
19.7.1 General [debugging.general]
19.7.2 Header synopsis [debugging.syn]
19.7.3 Utility [debugging.utility]
20 Memory management library [mem][mem]
20.1 General [mem.general]
20.2 Memory [memory]
20.2.1 General [memory.general]
20.2.2 Header synopsis [memory.syn]
20.2.3 Pointer traits [pointer.traits]
20.2.3.1 General [pointer.traits.general]
20.2.3.2 Member types [pointer.traits.types]
20.2.3.3 Member functions [pointer.traits.functions]
20.2.3.4 Optional members [pointer.traits.optmem]
20.2.4 Pointer conversion [pointer.conversion]
20.2.5 Pointer alignment [ptr.align]
20.2.6 Explicit lifetime management [obj.lifetime]
20.2.7 Allocator argument tag [allocator.tag]
20.2.8uses_ allocator[allocator.uses]
20.2.8.1uses_ allocator trait [allocator.uses.trait]
20.2.8.2 Uses-allocator construction [allocator.uses.construction]
20.2.9 Allocator traits [allocator.traits]
20.2.9.1 General [allocator.traits.general]
20.2.9.2 Member types [allocator.traits.types]
20.2.9.3 Static member functions [allocator.traits.members]
20.2.9.4 Other [allocator.traits.other]
20.2.10 The default allocator [default.allocator]
20.2.10.1 General [default.allocator.general]
20.2.10.2 Members [allocator.members]
20.2.10.3 Operators [allocator.globals]
20.2.11addressof[specialized.addressof]
20.2.12 C library memory allocation [c.malloc]
20.3 Smart pointers [smartptr]
20.3.1 Unique-ownership pointers [unique.ptr]
20.3.1.1 General [unique.ptr.general]
20.3.1.2 Default deleters [unique.ptr.dltr]
20.3.1.2.1 General [unique.ptr.dltr.general]
20.3.1.2.2default_ delete[unique.ptr.dltr.dflt]
20.3.1.2.3default_ delete[unique.ptr.dltr.dflt1]
20.3.1.3unique_ ptr for single objects [unique.ptr.single]
20.3.1.3.1 General [unique.ptr.single.general]
20.3.1.3.2 Constructors [unique.ptr.single.ctor]
20.3.1.3.3 Destructor [unique.ptr.single.dtor]
20.3.1.3.4 Assignment [unique.ptr.single.asgn]
20.3.1.3.5 Observers [unique.ptr.single.observers]
20.3.1.3.6 Modifiers [unique.ptr.single.modifiers]
20.3.1.4unique_ ptr for array objects with a runtime length [unique.ptr.runtime]
20.3.1.4.1 General [unique.ptr.runtime.general]
20.3.1.4.2 Constructors [unique.ptr.runtime.ctor]
20.3.1.4.3 Assignment [unique.ptr.runtime.asgn]
20.3.1.4.4 Observers [unique.ptr.runtime.observers]
20.3.1.4.5 Modifiers [unique.ptr.runtime.modifiers]
20.3.1.5 Creation [unique.ptr.create]
20.3.1.6 Specialized algorithms [unique.ptr.special]
20.3.1.7 I/O [unique.ptr.io]
20.3.2 Shared-ownership pointers [util.sharedptr]
20.3.2.1 Class bad_ weak_ ptr[util.smartptr.weak.bad]
20.3.2.2 Class template shared_ ptr[util.smartptr.shared]
20.3.2.2.1 General [util.smartptr.shared.general]
20.3.2.2.2 Constructors [util.smartptr.shared.const]
20.3.2.2.3 Destructor [util.smartptr.shared.dest]
20.3.2.2.4 Assignment [util.smartptr.shared.assign]
20.3.2.2.5 Modifiers [util.smartptr.shared.mod]
20.3.2.2.6 Observers [util.smartptr.shared.obs]
20.3.2.2.7 Creation [util.smartptr.shared.create]
20.3.2.2.8 Comparison [util.smartptr.shared.cmp]
20.3.2.2.9 Specialized algorithms [util.smartptr.shared.spec]
20.3.2.2.10 Casts [util.smartptr.shared.cast]
20.3.2.2.11get_ deleter[util.smartptr.getdeleter]
20.3.2.2.12 I/O [util.smartptr.shared.io]
20.3.2.3 Class template weak_ ptr[util.smartptr.weak]
20.3.2.3.1 General [util.smartptr.weak.general]
20.3.2.3.2 Constructors [util.smartptr.weak.const]
20.3.2.3.3 Destructor [util.smartptr.weak.dest]
20.3.2.3.4 Assignment [util.smartptr.weak.assign]
20.3.2.3.5 Modifiers [util.smartptr.weak.mod]
20.3.2.3.6 Observers [util.smartptr.weak.obs]
20.3.2.3.7 Specialized algorithms [util.smartptr.weak.spec]
20.3.2.4 Class template owner_ less[util.smartptr.ownerless]
20.3.2.5 Struct owner_ hash[util.smartptr.owner.hash]
20.3.2.6 Struct owner_ equal[util.smartptr.owner.equal]
20.3.2.7 Class template enable_ shared_ from_ this[util.smartptr.enab]
20.3.3 Smart pointer hash support [util.smartptr.hash]
20.3.4 Smart pointer adaptors [smartptr.adapt]
20.3.4.1 Class template out_ ptr_ t[out.ptr.t]
20.3.4.2 Function template out_ ptr[out.ptr]
20.3.4.3 Class template inout_ ptr_ t[inout.ptr.t]
20.3.4.4 Function template inout_ ptr[inout.ptr]
20.4 Types for composite class design [mem.composite.types]
20.4.1 Class template indirect[indirect]
20.4.1.1 General [indirect.general]
20.4.1.2 Synopsis [indirect.syn]
20.4.1.3 Constructors [indirect.ctor]
20.4.1.4 Destructor [indirect.dtor]
20.4.1.5 Assignment [indirect.assign]
20.4.1.6 Observers [indirect.obs]
20.4.1.7 Swap [indirect.swap]
20.4.1.8 Relational operators [indirect.relops]
20.4.1.9 Comparison with T[indirect.comp.with.t]
20.4.1.10 Hash support [indirect.hash]
20.4.2 Class template polymorphic[polymorphic]
20.4.2.1 General [polymorphic.general]
20.4.2.2 Synopsis [polymorphic.syn]
20.4.2.3 Constructors [polymorphic.ctor]
20.4.2.4 Destructor [polymorphic.dtor]
20.4.2.5 Assignment [polymorphic.assign]
20.4.2.6 Observers [polymorphic.obs]
20.4.2.7 Swap [polymorphic.swap]
20.5 Memory resources [mem.res]
20.5.1 Header synopsis [mem.res.syn]
20.5.2 Class memory_ resource[mem.res.class]
20.5.2.1 General [mem.res.class.general]
20.5.2.2 Public member functions [mem.res.public]
20.5.2.3 Private virtual member functions [mem.res.private]
20.5.2.4 Equality [mem.res.eq]
20.5.3 Class template polymorphic_ allocator[mem.poly.allocator.class]
20.5.3.1 General [mem.poly.allocator.class.general]
20.5.3.2 Constructors [mem.poly.allocator.ctor]
20.5.3.3 Member functions [mem.poly.allocator.mem]
20.5.3.4 Equality [mem.poly.allocator.eq]
20.5.4 Access to program-wide memory_ resource objects [mem.res.global]
20.5.5 Pool resource classes [mem.res.pool]
20.5.5.1 Classes synchronized_ pool_ resource and unsynchronized_ pool_ resource[mem.res.pool.overview]
20.5.5.2pool_ options data members [mem.res.pool.options]
20.5.5.3 Constructors and destructors [mem.res.pool.ctor]
20.5.5.4 Members [mem.res.pool.mem]
20.5.6 Class monotonic_ buffer_ resource[mem.res.monotonic.buffer]
20.5.6.1 General [mem.res.monotonic.buffer.general]
20.5.6.2 Constructors and destructor [mem.res.monotonic.buffer.ctor]
20.5.6.3 Members [mem.res.monotonic.buffer.mem]
20.6 Class template scoped_ allocator_ adaptor[allocator.adaptor]
20.6.1 Header synopsis [allocator.adaptor.syn]
20.6.2 Member types [allocator.adaptor.types]
20.6.3 Constructors [allocator.adaptor.cnstr]
20.6.4 Members [allocator.adaptor.members]
20.6.5 Operators [scoped.adaptor.operators]
21 Metaprogramming library [meta][meta]
21.1 General [meta.general]
21.2 Compile-time integer sequences [intseq]
21.2.1 General [intseq.general]
21.2.2 Class template integer_ sequence[intseq.intseq]
21.2.3 Alias template make_ integer_ sequence[intseq.make]
21.3 Metaprogramming and type traits [type.traits]
21.3.1 General [type.traits.general]
21.3.2 Requirements [meta.rqmts]
21.3.3 Header synopsis [meta.type.synop]
21.3.4 Helper classes [meta.help]
21.3.5 Class template constant_ wrapper[const.wrap.class]
21.3.6 Unary type traits [meta.unary]
21.3.6.1 General [meta.unary.general]
21.3.6.2 Primary type categories [meta.unary.cat]
21.3.6.3 Composite type traits [meta.unary.comp]
21.3.6.4 Type properties [meta.unary.prop]
21.3.7 Type property queries [meta.unary.prop.query]
21.3.8 Relationships between types [meta.rel]
21.3.9 Transformations between types [meta.trans]
21.3.9.1 General [meta.trans.general]
21.3.9.2 Const-volatile modifications [meta.trans.cv]
21.3.9.3 Reference modifications [meta.trans.ref]
21.3.9.4 Sign modifications [meta.trans.sign]
21.3.9.5 Array modifications [meta.trans.arr]
21.3.9.6 Pointer modifications [meta.trans.ptr]
21.3.9.7 Other transformations [meta.trans.other]
21.3.10 Logical operator traits [meta.logical]
21.3.11 Member relationships [meta.member]
21.3.12 Constant evaluation context [meta.const.eval]
21.4 Reflection [meta.reflection]
21.4.1 Header synopsis [meta.syn]
21.4.2 Checking string literals [meta.string.literal]
21.4.3 Promoting to static storage strings [meta.define.static]
21.4.4 Class exception[meta.reflection.exception]
21.4.5 Operator representations [meta.reflection.operators]
21.4.6 Reflection names and locations [meta.reflection.names]
21.4.7 Reflection queries [meta.reflection.queries]
21.4.8 Access control context [meta.reflection.access.context]
21.4.9 Member accessibility queries [meta.reflection.access.queries]
21.4.10 Reflection member queries [meta.reflection.member.queries]
21.4.11 Reflection layout queries [meta.reflection.layout]
21.4.12 Value extraction [meta.reflection.extract]
21.4.13 Reflection substitution [meta.reflection.substitute]
21.4.14 Expression result reflection [meta.reflection.result]
21.4.15 Promoting to static storage arrays [meta.reflection.array]
21.4.16 Reflection class definition generation [meta.reflection.define.aggregate]
21.4.17 Reflection type traits [meta.reflection.traits]
21.4.18 Annotation reflection [meta.reflection.annotation]
21.5 Compile-time rational arithmetic [ratio]
21.5.1 General [ratio.general]
21.5.2 Header synopsis [ratio.syn]
21.5.3 Class template ratio[ratio.ratio]
21.5.4 Arithmetic on ratio s [ratio.arithmetic]
21.5.5 Comparison of ratio s [ratio.comparison]
21.5.6 SI types for ratio[ratio.si]
22 General utilities library [utilities][utilities]
22.1 General [utilities.general]
22.2 Utility components [utility]
22.2.1 Header synopsis [utility.syn]
22.2.2swap[utility.swap]
22.2.3exchange[utility.exchange]
22.2.4 Forward/move helpers [forward]
22.2.5 Function template as_ const[utility.as.const]
22.2.6 Function template declval[declval]
22.2.7 Integer comparison functions [utility.intcmp]
22.2.8 Function template to_ underlying[utility.underlying]
22.2.9 Undefined behavior [utility.undefined]
22.3 Pairs [pairs]
22.3.1 General [pairs.general]
22.3.2 Class template pair[pairs.pair]
22.3.3 Specialized algorithms [pairs.spec]
22.3.4 Tuple-like access to pair [pair.astuple]
22.3.5 Piecewise construction [pair.piecewise]
22.4 Tuples [tuple]
22.4.1 General [tuple.general]
22.4.2 Header synopsis [tuple.syn]
22.4.3 Concept tuple-like[tuple.like]
22.4.4 Class template tuple[tuple.tuple]
22.4.4.1 General [tuple.tuple.general]
22.4.4.2 Construction [tuple.cnstr]
22.4.4.3 Assignment [tuple.assign]
22.4.4.4swap[tuple.swap]
22.4.5 Tuple creation functions [tuple.creation]
22.4.6 Calling a function with a tuple of arguments [tuple.apply]
22.4.7 Tuple helper classes [tuple.helper]
22.4.8 Element access [tuple.elem]
22.4.9 Relational operators [tuple.rel]
22.4.10common_ reference related specializations [tuple.common.ref]
22.4.11 Tuple traits [tuple.traits]
22.4.12 Tuple specialized algorithms [tuple.special]
22.5 Optional objects [optional]
22.5.1 General [optional.general]
22.5.2 Header synopsis [optional.syn]
22.5.3 Class template optional[optional.optional]
22.5.3.1 General [optional.optional.general]
22.5.3.2 Constructors [optional.ctor]
22.5.3.3 Destructor [optional.dtor]
22.5.3.4 Assignment [optional.assign]
22.5.3.5 Swap [optional.swap]
22.5.3.6 Iterator support [optional.iterators]
22.5.3.7 Observers [optional.observe]
22.5.3.8 Monadic operations [optional.monadic]
22.5.3.9 Modifiers [optional.mod]
22.5.4 Partial specialization of optional for reference types [optional.optional.ref]
22.5.4.1 General [optional.optional.ref.general]
22.5.4.2 Constructors [optional.ref.ctor]
22.5.4.3 Assignment [optional.ref.assign]
22.5.4.4 Swap [optional.ref.swap]
22.5.4.5 Iterator support [optional.ref.iterators]
22.5.4.6 Observers [optional.ref.observe]
22.5.4.7 Monadic operations [optional.ref.monadic]
22.5.4.8 Modifiers [optional.ref.mod]
22.5.4.9 Exposition only helper functions [optional.ref.expos]
22.5.5 No-value state indicator [optional.nullopt]
22.5.6 Class bad_ optional_ access[optional.bad.access]
22.5.7 Relational operators [optional.relops]
22.5.8 Comparison with nullopt[optional.nullops]
22.5.9 Comparison with T[optional.comp.with.t]
22.5.10 Specialized algorithms [optional.specalg]
22.5.11 Hash support [optional.hash]
22.6 Variants [variant]
22.6.1 General [variant.general]
22.6.2 Header synopsis [variant.syn]
22.6.3 Class template variant[variant.variant]
22.6.3.1 General [variant.variant.general]
22.6.3.2 Constructors [variant.ctor]
22.6.3.3 Destructor [variant.dtor]
22.6.3.4 Assignment [variant.assign]
22.6.3.5 Modifiers [variant.mod]
22.6.3.6 Value status [variant.status]
22.6.3.7 Swap [variant.swap]
22.6.4variant helper classes [variant.helper]
22.6.5 Value access [variant.get]
22.6.6 Relational operators [variant.relops]
22.6.7 Visitation [variant.visit]
22.6.8 Class monostate[variant.monostate]
22.6.9monostate relational operators [variant.monostate.relops]
22.6.10 Specialized algorithms [variant.specalg]
22.6.11 Class bad_ variant_ access[variant.bad.access]
22.6.12 Hash support [variant.hash]
22.7 Storage for any type [any]
22.7.1 General [any.general]
22.7.2 Header synopsis [any.synop]
22.7.3 Class bad_ any_ cast[any.bad.any.cast]
22.7.4 Class any[any.class]
22.7.4.1 General [any.class.general]
22.7.4.2 Construction and destruction [any.cons]
22.7.4.3 Assignment [any.assign]
22.7.4.4 Modifiers [any.modifiers]
22.7.4.5 Observers [any.observers]
22.7.5 Non-member functions [any.nonmembers]
22.8 Expected objects [expected]
22.8.1 General [expected.general]
22.8.2 Header synopsis [expected.syn]
22.8.3 Class template unexpected[expected.unexpected]
22.8.3.1 General [expected.un.general]
22.8.3.2 Constructors [expected.un.cons]
22.8.3.3 Observers [expected.un.obs]
22.8.3.4 Swap [expected.un.swap]
22.8.3.5 Equality operator [expected.un.eq]
22.8.4 Class template bad_ expected_ access[expected.bad]
22.8.5 Class template specialization bad_ expected_ access[expected.bad.void]
22.8.6 Class template expected[expected.expected]
22.8.6.1 General [expected.object.general]
22.8.6.2 Constructors [expected.object.cons]
22.8.6.3 Destructor [expected.object.dtor]
22.8.6.4 Assignment [expected.object.assign]
22.8.6.5 Swap [expected.object.swap]
22.8.6.6 Observers [expected.object.obs]
22.8.6.7 Monadic operations [expected.object.monadic]
22.8.6.8 Equality operators [expected.object.eq]
22.8.7 Partial specialization of expected for void types [expected.void]
22.8.7.1 General [expected.void.general]
22.8.7.2 Constructors [expected.void.cons]
22.8.7.3 Destructor [expected.void.dtor]
22.8.7.4 Assignment [expected.void.assign]
22.8.7.5 Swap [expected.void.swap]
22.8.7.6 Observers [expected.void.obs]
22.8.7.7 Monadic operations [expected.void.monadic]
22.8.7.8 Equality operators [expected.void.eq]
22.9 Bitsets [bitset]
22.9.1 Header synopsis [bitset.syn]
22.9.2 Class template bitset[template.bitset]
22.9.2.1 General [template.bitset.general]
22.9.2.2 Constructors [bitset.cons]
22.9.2.3 Members [bitset.members]
22.9.3bitset hash support [bitset.hash]
22.9.4bitset operators [bitset.operators]
22.10 Function objects [function.objects]
22.10.1 General [function.objects.general]
22.10.2 Header synopsis [functional.syn]
22.10.3 Definitions [func.def]
22.10.4 Requirements [func.require]
22.10.5invoke functions [func.invoke]
22.10.6 Class template reference_ wrapper[refwrap]
22.10.6.1 General [refwrap.general]
22.10.6.2 Constructors [refwrap.const]
22.10.6.3 Assignment [refwrap.assign]
22.10.6.4 Access [refwrap.access]
22.10.6.5 Invocation [refwrap.invoke]
22.10.6.6 Comparisons [refwrap.comparisons]
22.10.6.7 Helper functions [refwrap.helpers]
22.10.6.8common_ reference related specializations [refwrap.common.ref]
22.10.7 Arithmetic operations [arithmetic.operations]
22.10.7.1 General [arithmetic.operations.general]
22.10.7.2 Class template plus[arithmetic.operations.plus]
22.10.7.3 Class template minus[arithmetic.operations.minus]
22.10.7.4 Class template multiplies[arithmetic.operations.multiplies]
22.10.7.5 Class template divides[arithmetic.operations.divides]
22.10.7.6 Class template modulus[arithmetic.operations.modulus]
22.10.7.7 Class template negate[arithmetic.operations.negate]
22.10.8 Comparisons [comparisons]
22.10.8.1 General [comparisons.general]
22.10.8.2 Class template equal_ to[comparisons.equal.to]
22.10.8.3 Class template not_ equal_ to[comparisons.not.equal.to]
22.10.8.4 Class template greater[comparisons.greater]
22.10.8.5 Class template less[comparisons.less]
22.10.8.6 Class template greater_ equal[comparisons.greater.equal]
22.10.8.7 Class template less_ equal[comparisons.less.equal]
22.10.8.8 Class compare_ three_ way[comparisons.three.way]
22.10.9 Concept-constrained comparisons [range.cmp]
22.10.10 Logical operations [logical.operations]
22.10.10.1 General [logical.operations.general]
22.10.10.2 Class template logical_ and[logical.operations.and]
22.10.10.3 Class template logical_ or[logical.operations.or]
22.10.10.4 Class template logical_ not[logical.operations.not]
22.10.11 Bitwise operations [bitwise.operations]
22.10.11.1 General [bitwise.operations.general]
22.10.11.2 Class template bit_ and[bitwise.operations.and]
22.10.11.3 Class template bit_ or[bitwise.operations.or]
22.10.11.4 Class template bit_ xor[bitwise.operations.xor]
22.10.11.5 Class template bit_ not[bitwise.operations.not]
22.10.12 Class identity[func.identity]
22.10.13 Function template not_ fn[func.not.fn]
22.10.14 Function templates bind_ front and bind_ back[func.bind.partial]
22.10.15 Function object binders [func.bind]
22.10.15.1 General [func.bind.general]
22.10.15.2 Class template is_ bind_ expression[func.bind.isbind]
22.10.15.3 Class template is_ placeholder[func.bind.isplace]
22.10.15.4 Function template bind[func.bind.bind]
22.10.15.5 Placeholders [func.bind.place]
22.10.16 Function template mem_ fn[func.memfn]
22.10.17 Polymorphic function wrappers [func.wrap]
22.10.17.1 General [func.wrap.general]
22.10.17.2 Class bad_ function_ call[func.wrap.badcall]
22.10.17.3 Class template function[func.wrap.func]
22.10.17.3.1 General [func.wrap.func.general]
22.10.17.3.2 Constructors and destructor [func.wrap.func.con]
22.10.17.3.3 Modifiers [func.wrap.func.mod]
22.10.17.3.4 Capacity [func.wrap.func.cap]
22.10.17.3.5 Invocation [func.wrap.func.inv]
22.10.17.3.6 Target access [func.wrap.func.targ]
22.10.17.3.7 Null pointer comparison operator functions [func.wrap.func.nullptr]
22.10.17.3.8 Specialized algorithms [func.wrap.func.alg]
22.10.17.4 Move-only wrapper [func.wrap.move]
22.10.17.4.1 General [func.wrap.move.general]
22.10.17.4.2 Class template move_ only_ function[func.wrap.move.class]
22.10.17.4.3 Constructors, assignment, and destructor [func.wrap.move.ctor]
22.10.17.4.4 Invocation [func.wrap.move.inv]
22.10.17.4.5 Utility [func.wrap.move.util]
22.10.17.5 Copyable wrapper [func.wrap.copy]
22.10.17.5.1 General [func.wrap.copy.general]
22.10.17.5.2 Class template copyable_ function[func.wrap.copy.class]
22.10.17.5.3 Constructors, assignments, and destructors [func.wrap.copy.ctor]
22.10.17.5.4 Invocation [func.wrap.copy.inv]
22.10.17.5.5 Utility [func.wrap.copy.util]
22.10.17.6 Non-owning wrapper [func.wrap.ref]
22.10.17.6.1 General [func.wrap.ref.general]
22.10.17.6.2 Class template function_ ref[func.wrap.ref.class]
22.10.17.6.3 Constructors and assignment operators [func.wrap.ref.ctor]
22.10.17.6.4 Invocation [func.wrap.ref.inv]
22.10.17.6.5 Deduction guides [func.wrap.ref.deduct]
22.10.18 Searchers [func.search]
22.10.18.1 General [func.search.general]
22.10.18.2 Class template default_ searcher[func.search.default]
22.10.18.3 Class template boyer_ moore_ searcher[func.search.bm]
22.10.18.4 Class template boyer_ moore_ horspool_ searcher[func.search.bmh]
22.10.19 Class template hash[unord.hash]
22.11 Bit manipulation [bit]
22.11.1 General [bit.general]
22.11.2 Header synopsis [bit.syn]
22.11.3 Function template bit_ cast[bit.cast]
22.11.4byteswap[bit.byteswap]
22.11.5 Integral powers of 2 [bit.pow.two]
22.11.6 Rotating [bit.rotate]
22.11.7 Counting [bit.count]
22.11.8 Endian [bit.endian]
22.12 Header synopsis [stdbit.h.syn]
23 Containers library [containers][containers]
23.1 General [containers.general]
23.2 Requirements [container.requirements]
23.2.1 Preamble [container.requirements.pre]
23.2.2 General containers [container.requirements.general]
23.2.2.1 Introduction [container.intro.reqmts]
23.2.2.2 Container requirements [container.reqmts]
23.2.2.3 Reversible container requirements [container.rev.reqmts]
23.2.2.4 Optional container requirements [container.opt.reqmts]
23.2.2.5 Allocator-aware containers [container.alloc.reqmts]
23.2.3 Container data races [container.requirements.dataraces]
23.2.4 Sequence containers [sequence.reqmts]
23.2.5 Node handles [container.node]
23.2.5.1 Overview [container.node.overview]
23.2.5.2 Constructors, copy, and assignment [container.node.cons]
23.2.5.3 Destructor [container.node.dtor]
23.2.5.4 Observers [container.node.observers]
23.2.5.5 Modifiers [container.node.modifiers]
23.2.6 Insert return type [container.insert.return]
23.2.7 Associative containers [associative.reqmts]
23.2.7.1 General [associative.reqmts.general]
23.2.7.2 Exception safety guarantees [associative.reqmts.except]
23.2.8 Unordered associative containers [unord.req]
23.2.8.1 General [unord.req.general]
23.2.8.2 Exception safety guarantees [unord.req.except]
23.3 Sequence containers [sequences]
23.3.1 General [sequences.general]
23.3.2 Header synopsis [array.syn]
23.3.3 Class template array[array]
23.3.3.1 Overview [array.overview]
23.3.3.2 Constructors, copy, and assignment [array.cons]
23.3.3.3 Member functions [array.members]
23.3.3.4 Specialized algorithms [array.special]
23.3.3.5 Zero-sized arrays [array.zero]
23.3.3.6 Array creation functions [array.creation]
23.3.3.7 Tuple interface [array.tuple]
23.3.4 Header synopsis [deque.syn]
23.3.5 Class template deque[deque]
23.3.5.1 Overview [deque.overview]
23.3.5.2 Constructors, copy, and assignment [deque.cons]
23.3.5.3 Capacity [deque.capacity]
23.3.5.4 Modifiers [deque.modifiers]
23.3.5.5 Erasure [deque.erasure]
23.3.6 Header synopsis [forward.list.syn]
23.3.7 Class template forward_ list[forward.list]
23.3.7.1 Overview [forward.list.overview]
23.3.7.2 Constructors, copy, and assignment [forward.list.cons]
23.3.7.3 Iterators [forward.list.iter]
23.3.7.4 Element access [forward.list.access]
23.3.7.5 Modifiers [forward.list.modifiers]
23.3.7.6 Operations [forward.list.ops]
23.3.7.7 Erasure [forward.list.erasure]
23.3.8 Header synopsis [hive.syn]
23.3.9 Class template hive[hive]
23.3.9.1 Overview [hive.overview]
23.3.9.2 Constructors, copy, and assignment [hive.cons]
23.3.9.3 Capacity [hive.capacity]
23.3.9.4 Modifiers [hive.modifiers]
23.3.9.5 Operations [hive.operations]
23.3.9.6 Erasure [hive.erasure]
23.3.10 Header synopsis [list.syn]
23.3.11 Class template list[list]
23.3.11.1 Overview [list.overview]
23.3.11.2 Constructors, copy, and assignment [list.cons]
23.3.11.3 Capacity [list.capacity]
23.3.11.4 Modifiers [list.modifiers]
23.3.11.5 Operations [list.ops]
23.3.11.6 Erasure [list.erasure]
23.3.12 Header synopsis [vector.syn]
23.3.13 Class template vector[vector]
23.3.13.1 Overview [vector.overview]
23.3.13.2 Constructors [vector.cons]
23.3.13.3 Capacity [vector.capacity]
23.3.13.4 Data [vector.data]
23.3.13.5 Modifiers [vector.modifiers]
23.3.13.6 Erasure [vector.erasure]
23.3.14 Specialization of vector for bool[vector.bool]
23.3.14.1 Partial class template specialization vector[vector.bool.pspc]
23.3.14.2 Formatter specialization for vector[vector.bool.fmt]
23.3.15 Header synopsis [inplace.vector.syn]
23.3.16 Class template inplace_ vector[inplace.vector]
23.3.16.1 Overview [inplace.vector.overview]
23.3.16.2 Constructors [inplace.vector.cons]
23.3.16.3 Capacity [inplace.vector.capacity]
23.3.16.4 Data [inplace.vector.data]
23.3.16.5 Modifiers [inplace.vector.modifiers]
23.3.16.6 Erasure [inplace.vector.erasure]
23.4 Associative containers [associative]
23.4.1 General [associative.general]
23.4.2 Header synopsis [associative.map.syn]
23.4.3 Class template map[map]
23.4.3.1 Overview [map.overview]
23.4.3.2 Constructors, copy, and assignment [map.cons]
23.4.3.3 Element access [map.access]
23.4.3.4 Modifiers [map.modifiers]
23.4.3.5 Erasure [map.erasure]
23.4.4 Class template multimap[multimap]
23.4.4.1 Overview [multimap.overview]
23.4.4.2 Constructors [multimap.cons]
23.4.4.3 Modifiers [multimap.modifiers]
23.4.4.4 Erasure [multimap.erasure]
23.4.5 Header synopsis [associative.set.syn]
23.4.6 Class template set[set]
23.4.6.1 Overview [set.overview]
23.4.6.2 Constructors, copy, and assignment [set.cons]
23.4.6.3 Erasure [set.erasure]
23.4.6.4 Modifiers [set.modifiers]
23.4.7 Class template multiset[multiset]
23.4.7.1 Overview [multiset.overview]
23.4.7.2 Constructors [multiset.cons]
23.4.7.3 Erasure [multiset.erasure]
23.5 Unordered associative containers [unord]
23.5.1 General [unord.general]
23.5.2 Header synopsis [unord.map.syn]
23.5.3 Class template unordered_ map[unord.map]
23.5.3.1 Overview [unord.map.overview]
23.5.3.2 Constructors [unord.map.cnstr]
23.5.3.3 Element access [unord.map.elem]
23.5.3.4 Modifiers [unord.map.modifiers]
23.5.3.5 Erasure [unord.map.erasure]
23.5.4 Class template unordered_ multimap[unord.multimap]
23.5.4.1 Overview [unord.multimap.overview]
23.5.4.2 Constructors [unord.multimap.cnstr]
23.5.4.3 Modifiers [unord.multimap.modifiers]
23.5.4.4 Erasure [unord.multimap.erasure]
23.5.5 Header synopsis [unord.set.syn]
23.5.6 Class template unordered_ set[unord.set]
23.5.6.1 Overview [unord.set.overview]
23.5.6.2 Constructors [unord.set.cnstr]
23.5.6.3 Erasure [unord.set.erasure]
23.5.6.4 Modifiers [unord.set.modifiers]
23.5.7 Class template unordered_ multiset[unord.multiset]
23.5.7.1 Overview [unord.multiset.overview]
23.5.7.2 Constructors [unord.multiset.cnstr]
23.5.7.3 Erasure [unord.multiset.erasure]
23.6 Container adaptors [container.adaptors]
23.6.1 General [container.adaptors.general]
23.6.2 Header synopsis [queue.syn]
23.6.3 Class template queue[queue]
23.6.3.1 Definition [queue.defn]
23.6.3.2 Constructors [queue.cons]
23.6.3.3 Constructors with allocators [queue.cons.alloc]
23.6.3.4 Modifiers [queue.mod]
23.6.3.5 Operators [queue.ops]
23.6.3.6 Specialized algorithms [queue.special]
23.6.4 Class template priority_ queue[priority.queue]
23.6.4.1 Overview [priqueue.overview]
23.6.4.2 Constructors [priqueue.cons]
23.6.4.3 Constructors with allocators [priqueue.cons.alloc]
23.6.4.4 Members [priqueue.members]
23.6.4.5 Specialized algorithms [priqueue.special]
23.6.5 Header synopsis [stack.syn]
23.6.6 Class template stack[stack]
23.6.6.1 General [stack.general]
23.6.6.2 Definition [stack.defn]
23.6.6.3 Constructors [stack.cons]
23.6.6.4 Constructors with allocators [stack.cons.alloc]
23.6.6.5 Modifiers [stack.mod]
23.6.6.6 Operators [stack.ops]
23.6.6.7 Specialized algorithms [stack.special]
23.6.7 Header synopsis [flat.map.syn]
23.6.8 Class template flat_ map[flat.map]
23.6.8.1 Overview [flat.map.overview]
23.6.8.2 Definition [flat.map.defn]
23.6.8.3 Constructors [flat.map.cons]
23.6.8.4 Constructors with allocators [flat.map.cons.alloc]
23.6.8.5 Capacity [flat.map.capacity]
23.6.8.6 Access [flat.map.access]
23.6.8.7 Modifiers [flat.map.modifiers]
23.6.8.8 Erasure [flat.map.erasure]
23.6.9 Class template flat_ multimap[flat.multimap]
23.6.9.1 Overview [flat.multimap.overview]
23.6.9.2 Definition [flat.multimap.defn]
23.6.9.3 Constructors [flat.multimap.cons]
23.6.9.4 Constructors with allocators [flat.multimap.cons.alloc]
23.6.9.5 Erasure [flat.multimap.erasure]
23.6.10 Header synopsis [flat.set.syn]
23.6.11 Class template flat_ set[flat.set]
23.6.11.1 Overview [flat.set.overview]
23.6.11.2 Definition [flat.set.defn]
23.6.11.3 Constructors [flat.set.cons]
23.6.11.4 Constructors with allocators [flat.set.cons.alloc]
23.6.11.5 Modifiers [flat.set.modifiers]
23.6.11.6 Erasure [flat.set.erasure]
23.6.12 Class template flat_ multiset[flat.multiset]
23.6.12.1 Overview [flat.multiset.overview]
23.6.12.2 Definition [flat.multiset.defn]
23.6.12.3 Constructors [flat.multiset.cons]
23.6.12.4 Constructors with allocators [flat.multiset.cons.alloc]
23.6.12.5 Modifiers [flat.multiset.modifiers]
23.6.12.6 Erasure [flat.multiset.erasure]
23.6.13 Container adaptors formatting [container.adaptors.format]
23.7 Views [views]
23.7.1 General [views.general]
23.7.2 Contiguous access [views.contiguous]
23.7.2.1 Header synopsis [span.syn]
23.7.2.2 Class template span[views.span]
23.7.2.2.1 Overview [span.overview]
23.7.2.2.2 Constructors, copy, and assignment [span.cons]
23.7.2.2.3 Deduction guides [span.deduct]
23.7.2.2.4 Subviews [span.sub]
23.7.2.2.5 Observers [span.obs]
23.7.2.2.6 Element access [span.elem]
23.7.2.2.7 Iterator support [span.iterators]
23.7.2.3 Views of object representation [span.objectrep]
23.7.3 Multidimensional access [views.multidim]
23.7.3.1 Overview [mdspan.overview]
23.7.3.2 Header synopsis [mdspan.syn]
23.7.3.3 Class template extents[mdspan.extents]
23.7.3.3.1 Overview [mdspan.extents.overview]
23.7.3.3.2 Exposition-only helpers [mdspan.extents.expo]
23.7.3.3.3 Constructors [mdspan.extents.cons]
23.7.3.3.4 Observers of the multidimensional index space [mdspan.extents.obs]
23.7.3.3.5 Comparison operators [mdspan.extents.cmp]
23.7.3.3.6 Alias template dextents[mdspan.extents.dextents]
23.7.3.3.7 Alias template dims[mdspan.extents.dims]
23.7.3.4 Layout mapping [mdspan.layout]
23.7.3.4.1 General [mdspan.layout.general]
23.7.3.4.2 Requirements [mdspan.layout.reqmts]
23.7.3.4.3 Layout mapping policy requirements [mdspan.layout.policy.reqmts]
23.7.3.4.4 Layout mapping policies [mdspan.layout.policy.overview]
23.7.3.4.5 Class template layout_ left::mapping[mdspan.layout.left]
23.7.3.4.5.1 Overview [mdspan.layout.left.overview]
23.7.3.4.5.2 Constructors [mdspan.layout.left.cons]
23.7.3.4.5.3 Observers [mdspan.layout.left.obs]
23.7.3.4.6 Class template layout_ right::mapping[mdspan.layout.right]
23.7.3.4.6.1 Overview [mdspan.layout.right.overview]
23.7.3.4.6.2 Constructors [mdspan.layout.right.cons]
23.7.3.4.6.3 Observers [mdspan.layout.right.obs]
23.7.3.4.7 Class template layout_ stride::mapping[mdspan.layout.stride]
23.7.3.4.7.1 Overview [mdspan.layout.stride.overview]
23.7.3.4.7.2 Exposition-only helpers [mdspan.layout.stride.expo]
23.7.3.4.7.3 Constructors [mdspan.layout.stride.cons]
23.7.3.4.7.4 Observers [mdspan.layout.stride.obs]
23.7.3.4.8 Class template layout_ left_ padded::mapping[mdspan.layout.leftpad]
23.7.3.4.8.1 Overview [mdspan.layout.leftpad.overview]
23.7.3.4.8.2 Exposition-only members [mdspan.layout.leftpad.expo]
23.7.3.4.8.3 Constructors [mdspan.layout.leftpad.cons]
23.7.3.4.8.4 Observers [mdspan.layout.leftpad.obs]
23.7.3.4.9 Class template layout_ right_ padded::mapping[mdspan.layout.rightpad]
23.7.3.4.9.1 Overview [mdspan.layout.rightpad.overview]
23.7.3.4.9.2 Exposition-only members [mdspan.layout.rightpad.expo]
23.7.3.4.9.3 Constructors [mdspan.layout.rightpad.cons]
23.7.3.4.9.4 Observers [mdspan.layout.rightpad.obs]
23.7.3.5 Accessor policy [mdspan.accessor]
23.7.3.5.1 General [mdspan.accessor.general]
23.7.3.5.2 Requirements [mdspan.accessor.reqmts]
23.7.3.5.3 Class template default_ accessor[mdspan.accessor.default]
23.7.3.5.3.1 Overview [mdspan.accessor.default.overview]
23.7.3.5.3.2 Members [mdspan.accessor.default.members]
23.7.3.5.4 Class template aligned_ accessor[mdspan.accessor.aligned]
23.7.3.5.4.1 Overview [mdspan.accessor.aligned.overview]
23.7.3.5.4.2 Members [mdspan.accessor.aligned.members]
23.7.3.6 Class template mdspan[mdspan.mdspan]
23.7.3.6.1 Overview [mdspan.mdspan.overview]
23.7.3.6.2 Constructors [mdspan.mdspan.cons]
23.7.3.6.3 Members [mdspan.mdspan.members]
23.7.3.7submdspan[mdspan.sub]
23.7.3.7.1 Overview [mdspan.sub.overview]
23.7.3.7.2strided_ slice[mdspan.sub.strided.slice]
23.7.3.7.3submdspan_ mapping_ result[mdspan.sub.map.result]
23.7.3.7.4 Exposition-only helpers [mdspan.sub.helpers]
23.7.3.7.5submdspan_ extents function [mdspan.sub.extents]
23.7.3.7.6 Specializations of submdspan_ mapping[mdspan.sub.map]
23.7.3.7.6.1 Common [mdspan.sub.map.common]
23.7.3.7.6.2layout_ left specialization of submdspan_ mapping[mdspan.sub.map.left]
23.7.3.7.6.3layout_ right specialization of submdspan_ mapping[mdspan.sub.map.right]
23.7.3.7.6.4layout_ stride specialization of submdspan_ mapping[mdspan.sub.map.stride]
23.7.3.7.6.5layout_ left_ padded specialization of submdspan_ mapping[mdspan.sub.map.leftpad]
23.7.3.7.6.6layout_ right_ padded specialization of submdspan_ mapping[mdspan.sub.map.rightpad]
23.7.3.7.7submdspan function template [mdspan.sub.sub]
24 Iterators library [iterators][iterators]
24.1 General [iterators.general]
24.2 Header synopsis [iterator.synopsis]
24.3 Iterator requirements [iterator.requirements]
24.3.1 General [iterator.requirements.general]
24.3.2 Associated types [iterator.assoc.types]
24.3.2.1 Incrementable traits [incrementable.traits]
24.3.2.2 Indirectly readable traits [readable.traits]
24.3.2.3 Iterator traits [iterator.traits]
24.3.3 Customization point objects [iterator.cust]
24.3.3.1ranges::iter_ move[iterator.cust.move]
24.3.3.2ranges::iter_ swap[iterator.cust.swap]
24.3.4 Iterator concepts [iterator.concepts]
24.3.4.1 General [iterator.concepts.general]
24.3.4.2 Concept indirectly_ readable[iterator.concept.readable]
24.3.4.3 Concept indirectly_ writable[iterator.concept.writable]
24.3.4.4 Concept weakly_ incrementable[iterator.concept.winc]
24.3.4.5 Concept incrementable[iterator.concept.inc]
24.3.4.6 Concept input_ or_ output_ iterator[iterator.concept.iterator]
24.3.4.7 Concept sentinel_ for[iterator.concept.sentinel]
24.3.4.8 Concept sized_ sentinel_ for[iterator.concept.sizedsentinel]
24.3.4.9 Concept input_ iterator[iterator.concept.input]
24.3.4.10 Concept output_ iterator[iterator.concept.output]
24.3.4.11 Concept forward_ iterator[iterator.concept.forward]
24.3.4.12 Concept bidirectional_ iterator[iterator.concept.bidir]
24.3.4.13 Concept random_ access_ iterator[iterator.concept.random.access]
24.3.4.14 Concept contiguous_ iterator[iterator.concept.contiguous]
24.3.5 C++17 iterator requirements [iterator.cpp17]
24.3.5.1 General [iterator.cpp17.general]
24.3.5.2Cpp17Iterator[iterator.iterators]
24.3.5.3 Input iterators [input.iterators]
24.3.5.4 Output iterators [output.iterators]
24.3.5.5 Forward iterators [forward.iterators]
24.3.5.6 Bidirectional iterators [bidirectional.iterators]
24.3.5.7 Random access iterators [random.access.iterators]
24.3.6 Indirect callable requirements [indirectcallable]
24.3.6.1 General [indirectcallable.general]
24.3.6.2 Indirect callable traits [indirectcallable.traits]
24.3.6.3 Indirect callables [indirectcallable.indirectinvocable]
24.3.6.4 Alias template projected[projected]
24.3.7 Common algorithm requirements [alg.req]
24.3.7.1 General [alg.req.general]
24.3.7.2 Concept indirectly_ movable[alg.req.ind.move]
24.3.7.3 Concept indirectly_ copyable[alg.req.ind.copy]
24.3.7.4 Concept indirectly_ swappable[alg.req.ind.swap]
24.3.7.5 Concept indirectly_ comparable[alg.req.ind.cmp]
24.3.7.6 Concept permutable[alg.req.permutable]
24.3.7.7 Concept mergeable[alg.req.mergeable]
24.3.7.8 Concept sortable[alg.req.sortable]
24.4 Iterator primitives [iterator.primitives]
24.4.1 General [iterator.primitives.general]
24.4.2 Standard iterator tags [std.iterator.tags]
24.4.3 Iterator operations [iterator.operations]
24.4.4 Range iterator operations [range.iter.ops]
24.4.4.1 General [range.iter.ops.general]
24.4.4.2ranges::advance[range.iter.op.advance]
24.4.4.3ranges::distance[range.iter.op.distance]
24.4.4.4ranges::next[range.iter.op.next]
24.4.4.5ranges::prev[range.iter.op.prev]
24.5 Iterator adaptors [predef.iterators]
24.5.1 Reverse iterators [reverse.iterators]
24.5.1.1 General [reverse.iterators.general]
24.5.1.2 Class template reverse_ iterator[reverse.iterator]
24.5.1.3 Requirements [reverse.iter.requirements]
24.5.1.4 Construction and assignment [reverse.iter.cons]
24.5.1.5 Conversion [reverse.iter.conv]
24.5.1.6 Element access [reverse.iter.elem]
24.5.1.7 Navigation [reverse.iter.nav]
24.5.1.8 Comparisons [reverse.iter.cmp]
24.5.1.9 Non-member functions [reverse.iter.nonmember]
24.5.2 Insert iterators [insert.iterators]
24.5.2.1 General [insert.iterators.general]
24.5.2.2 Class template back_ insert_ iterator[back.insert.iterator]
24.5.2.2.1 General [back.insert.iter.general]
24.5.2.2.2 Operations [back.insert.iter.ops]
24.5.2.2.3back_ inserter[back.inserter]
24.5.2.3 Class template front_ insert_ iterator[front.insert.iterator]
24.5.2.3.1 General [front.insert.iter.general]
24.5.2.3.2 Operations [front.insert.iter.ops]
24.5.2.3.3front_ inserter[front.inserter]
24.5.2.4 Class template insert_ iterator[insert.iterator]
24.5.2.4.1 General [insert.iter.general]
24.5.2.4.2 Operations [insert.iter.ops]
24.5.2.4.3inserter[inserter]
24.5.3 Constant iterators and sentinels [const.iterators]
24.5.3.1 General [const.iterators.general]
24.5.3.2 Alias templates [const.iterators.alias]
24.5.3.3 Class template basic_ const_ iterator[const.iterators.iterator]
24.5.3.4 Member types [const.iterators.types]
24.5.3.5 Operations [const.iterators.ops]
24.5.4 Move iterators and sentinels [move.iterators]
24.5.4.1 General [move.iterators.general]
24.5.4.2 Class template move_ iterator[move.iterator]
24.5.4.3 Requirements [move.iter.requirements]
24.5.4.4 Construction and assignment [move.iter.cons]
24.5.4.5 Conversion [move.iter.op.conv]
24.5.4.6 Element access [move.iter.elem]
24.5.4.7 Navigation [move.iter.nav]
24.5.4.8 Comparisons [move.iter.op.comp]
24.5.4.9 Non-member functions [move.iter.nonmember]
24.5.4.10 Class template move_ sentinel[move.sentinel]
24.5.4.11 Operations [move.sent.ops]
24.5.5 Common iterators [iterators.common]
24.5.5.1 Class template common_ iterator[common.iterator]
24.5.5.2 Associated types [common.iter.types]
24.5.5.3 Constructors and conversions [common.iter.const]
24.5.5.4 Accessors [common.iter.access]
24.5.5.5 Navigation [common.iter.nav]
24.5.5.6 Comparisons [common.iter.cmp]
24.5.5.7 Customizations [common.iter.cust]
24.5.6 Default sentinel [default.sentinel]
24.5.7 Counted iterators [iterators.counted]
24.5.7.1 Class template counted_ iterator[counted.iterator]
24.5.7.2 Constructors and conversions [counted.iter.const]
24.5.7.3 Accessors [counted.iter.access]
24.5.7.4 Element access [counted.iter.elem]
24.5.7.5 Navigation [counted.iter.nav]
24.5.7.6 Comparisons [counted.iter.cmp]
24.5.7.7 Customizations [counted.iter.cust]
24.5.8 Unreachable sentinel [unreachable.sentinel]
24.6 Stream iterators [stream.iterators]
24.6.1 General [stream.iterators.general]
24.6.2 Class template istream_ iterator[istream.iterator]
24.6.2.1 General [istream.iterator.general]
24.6.2.2 Constructors and destructor [istream.iterator.cons]
24.6.2.3 Operations [istream.iterator.ops]
24.6.3 Class template ostream_ iterator[ostream.iterator]
24.6.3.1 General [ostream.iterator.general]
24.6.3.2 Constructors and destructor [ostream.iterator.cons.des]
24.6.3.3 Operations [ostream.iterator.ops]
24.6.4 Class template istreambuf_ iterator[istreambuf.iterator]
24.6.4.1 General [istreambuf.iterator.general]
24.6.4.2 Class istreambuf_ iterator::proxy[istreambuf.iterator.proxy]
24.6.4.3 Constructors [istreambuf.iterator.cons]
24.6.4.4 Operations [istreambuf.iterator.ops]
24.6.5 Class template ostreambuf_ iterator[ostreambuf.iterator]
24.6.5.1 General [ostreambuf.iterator.general]
24.6.5.2 Constructors [ostreambuf.iter.cons]
24.6.5.3 Operations [ostreambuf.iter.ops]
24.7 Range access [iterator.range]
25 Ranges library [ranges][ranges]
25.1 General [ranges.general]
25.2 Header synopsis [ranges.syn]
25.3 Range access [range.access]
25.3.1 General [range.access.general]
25.3.2ranges::begin[range.access.begin]
25.3.3ranges::end[range.access.end]
25.3.4ranges::cbegin[range.access.cbegin]
25.3.5ranges::cend[range.access.cend]
25.3.6ranges::rbegin[range.access.rbegin]
25.3.7ranges::rend[range.access.rend]
25.3.8ranges::crbegin[range.access.crbegin]
25.3.9ranges::crend[range.access.crend]
25.3.10ranges::size[range.prim.size]
25.3.11ranges::ssize[range.prim.ssize]
25.3.12ranges::reserve_ hint[range.prim.size.hint]
25.3.13ranges::empty[range.prim.empty]
25.3.14ranges::data[range.prim.data]
25.3.15ranges::cdata[range.prim.cdata]
25.4 Range requirements [range.req]
25.4.1 General [range.req.general]
25.4.2 Ranges [range.range]
25.4.3 Approximately sized ranges [range.approximately.sized]
25.4.4 Sized ranges [range.sized]
25.4.5 Views [range.view]
25.4.6 Other range refinements [range.refinements]
25.5 Range utilities [range.utility]
25.5.1 General [range.utility.general]
25.5.2 Helper concepts [range.utility.helpers]
25.5.3 View interface [view.interface]
25.5.3.1 General [view.interface.general]
25.5.3.2 Members [view.interface.members]
25.5.4 Sub-ranges [range.subrange]
25.5.4.1 General [range.subrange.general]
25.5.4.2 Constructors and conversions [range.subrange.ctor]
25.5.4.3 Accessors [range.subrange.access]
25.5.5 Dangling iterator handling [range.dangling]
25.5.6 Class template elements_ of[range.elementsof]
25.5.7 Range conversions [range.utility.conv]
25.5.7.1 General [range.utility.conv.general]
25.5.7.2ranges::to[range.utility.conv.to]
25.5.7.3ranges::to adaptors [range.utility.conv.adaptors]
25.6 Range factories [range.factories]
25.6.1 General [range.factories.general]
25.6.2 Empty view [range.empty]
25.6.2.1 Overview [range.empty.overview]
25.6.2.2 Class template empty_ view[range.empty.view]
25.6.3 Single view [range.single]
25.6.3.1 Overview [range.single.overview]
25.6.3.2 Class template single_ view[range.single.view]
25.6.4 Iota view [range.iota]
25.6.4.1 Overview [range.iota.overview]
25.6.4.2 Class template iota_ view[range.iota.view]
25.6.4.3 Class iota_ view::iterator[range.iota.iterator]
25.6.4.4 Class iota_ view::sentinel[range.iota.sentinel]
25.6.5 Repeat view [range.repeat]
25.6.5.1 Overview [range.repeat.overview]
25.6.5.2 Class template repeat_ view[range.repeat.view]
25.6.5.3 Class repeat_ view::iterator[range.repeat.iterator]
25.6.6 Istream view [range.istream]
25.6.6.1 Overview [range.istream.overview]
25.6.6.2 Class template basic_ istream_ view[range.istream.view]
25.6.6.3 Class basic_ istream_ view::iterator[range.istream.iterator]
25.7 Range adaptors [range.adaptors]
25.7.1 General [range.adaptors.general]
25.7.2 Range adaptor objects [range.adaptor.object]
25.7.3 Movable wrapper [range.move.wrap]
25.7.4 Non-propagating cache [range.nonprop.cache]
25.7.5 Range adaptor helpers [range.adaptor.helpers]
25.7.6 All view [range.all]
25.7.6.1 General [range.all.general]
25.7.6.2 Class template ref_ view[range.ref.view]
25.7.6.3 Class template owning_ view[range.owning.view]
25.7.7 As rvalue view [range.as.rvalue]
25.7.7.1 Overview [range.as.rvalue.overview]
25.7.7.2 Class template as_ rvalue_ view[range.as.rvalue.view]
25.7.8 Filter view [range.filter]
25.7.8.1 Overview [range.filter.overview]
25.7.8.2 Class template filter_ view[range.filter.view]
25.7.8.3 Class filter_ view::iterator[range.filter.iterator]
25.7.8.4 Class filter_ view::sentinel[range.filter.sentinel]
25.7.9 Transform view [range.transform]
25.7.9.1 Overview [range.transform.overview]
25.7.9.2 Class template transform_ view[range.transform.view]
25.7.9.3 Class template transform_ view::iterator[range.transform.iterator]
25.7.9.4 Class template transform_ view::sentinel[range.transform.sentinel]
25.7.10 Take view [range.take]
25.7.10.1 Overview [range.take.overview]
25.7.10.2 Class template take_ view[range.take.view]
25.7.10.3 Class template take_ view::sentinel[range.take.sentinel]
25.7.11 Take while view [range.take.while]
25.7.11.1 Overview [range.take.while.overview]
25.7.11.2 Class template take_ while_ view[range.take.while.view]
25.7.11.3 Class template take_ while_ view::sentinel[range.take.while.sentinel]
25.7.12 Drop view [range.drop]
25.7.12.1 Overview [range.drop.overview]
25.7.12.2 Class template drop_ view[range.drop.view]
25.7.13 Drop while view [range.drop.while]
25.7.13.1 Overview [range.drop.while.overview]
25.7.13.2 Class template drop_ while_ view[range.drop.while.view]
25.7.14 Join view [range.join]
25.7.14.1 Overview [range.join.overview]
25.7.14.2 Class template join_ view[range.join.view]
25.7.14.3 Class template join_ view::iterator[range.join.iterator]
25.7.14.4 Class template join_ view::sentinel[range.join.sentinel]
25.7.15 Join with view [range.join.with]
25.7.15.1 Overview [range.join.with.overview]
25.7.15.2 Class template join_ with_ view[range.join.with.view]
25.7.15.3 Class template join_ with_ view::iterator[range.join.with.iterator]
25.7.15.4 Class template join_ with_ view::sentinel[range.join.with.sentinel]
25.7.16 Lazy split view [range.lazy.split]
25.7.16.1 Overview [range.lazy.split.overview]
25.7.16.2 Class template lazy_ split_ view[range.lazy.split.view]
25.7.16.3 Class template lazy_ split_ view::outer-iterator[range.lazy.split.outer]
25.7.16.4 Class lazy_ split_ view::outer-iterator::value_ type[range.lazy.split.outer.value]
25.7.16.5 Class template lazy_ split_ view::inner-iterator[range.lazy.split.inner]
25.7.17 Split view [range.split]
25.7.17.1 Overview [range.split.overview]
25.7.17.2 Class template split_ view[range.split.view]
25.7.17.3 Class split_ view::iterator[range.split.iterator]
25.7.17.4 Class split_ view::sentinel[range.split.sentinel]
25.7.18 Concat view [range.concat]
25.7.18.1 Overview [range.concat.overview]
25.7.18.2 Class template concat_ view[range.concat.view]
25.7.18.3 Class concat_ view::iterator[range.concat.iterator]
25.7.19 Counted view [range.counted]
25.7.20 Common view [range.common]
25.7.20.1 Overview [range.common.overview]
25.7.20.2 Class template common_ view[range.common.view]
25.7.21 Reverse view [range.reverse]
25.7.21.1 Overview [range.reverse.overview]
25.7.21.2 Class template reverse_ view[range.reverse.view]
25.7.22 As const view [range.as.const]
25.7.22.1 Overview [range.as.const.overview]
25.7.22.2 Class template as_ const_ view[range.as.const.view]
25.7.23 Elements view [range.elements]
25.7.23.1 Overview [range.elements.overview]
25.7.23.2 Class template elements_ view[range.elements.view]
25.7.23.3 Class template elements_ view::iterator[range.elements.iterator]
25.7.23.4 Class template elements_ view::sentinel[range.elements.sentinel]
25.7.24 Enumerate view [range.enumerate]
25.7.24.1 Overview [range.enumerate.overview]
25.7.24.2 Class template enumerate_ view[range.enumerate.view]
25.7.24.3 Class template enumerate_ view::iterator[range.enumerate.iterator]
25.7.24.4 Class template enumerate_ view::sentinel[range.enumerate.sentinel]
25.7.25 Zip view [range.zip]
25.7.25.1 Overview [range.zip.overview]
25.7.25.2 Class template zip_ view[range.zip.view]
25.7.25.3 Class template zip_ view::iterator[range.zip.iterator]
25.7.25.4 Class template zip_ view::sentinel[range.zip.sentinel]
25.7.26 Zip transform view [range.zip.transform]
25.7.26.1 Overview [range.zip.transform.overview]
25.7.26.2 Class template zip_ transform_ view[range.zip.transform.view]
25.7.26.3 Class template zip_ transform_ view::iterator[range.zip.transform.iterator]
25.7.26.4 Class template zip_ transform_ view::sentinel[range.zip.transform.sentinel]
25.7.27 Adjacent view [range.adjacent]
25.7.27.1 Overview [range.adjacent.overview]
25.7.27.2 Class template adjacent_ view[range.adjacent.view]
25.7.27.3 Class template adjacent_ view::iterator[range.adjacent.iterator]
25.7.27.4 Class template adjacent_ view::sentinel[range.adjacent.sentinel]
25.7.28 Adjacent transform view [range.adjacent.transform]
25.7.28.1 Overview [range.adjacent.transform.overview]
25.7.28.2 Class template adjacent_ transform_ view[range.adjacent.transform.view]
25.7.28.3 Class template adjacent_ transform_ view::iterator[range.adjacent.transform.iterator]
25.7.28.4 Class template adjacent_ transform_ view::sentinel[range.adjacent.transform.sentinel]
25.7.29 Chunk view [range.chunk]
25.7.29.1 Overview [range.chunk.overview]
25.7.29.2 Class template chunk_ view for input ranges [range.chunk.view.input]
25.7.29.3 Class chunk_ view::outer-iterator[range.chunk.outer.iter]
25.7.29.4 Class chunk_ view::outer-iterator::value_ type[range.chunk.outer.value]
25.7.29.5 Class chunk_ view::inner-iterator[range.chunk.inner.iter]
25.7.29.6 Class template chunk_ view for forward ranges [range.chunk.view.fwd]
25.7.29.7 Class template chunk_ view::iterator for forward ranges [range.chunk.fwd.iter]
25.7.30 Slide view [range.slide]
25.7.30.1 Overview [range.slide.overview]
25.7.30.2 Class template slide_ view[range.slide.view]
25.7.30.3 Class template slide_ view::iterator[range.slide.iterator]
25.7.30.4 Class slide_ view::sentinel[range.slide.sentinel]
25.7.31 Chunk by view [range.chunk.by]
25.7.31.1 Overview [range.chunk.by.overview]
25.7.31.2 Class template chunk_ by_ view[range.chunk.by.view]
25.7.31.3 Class chunk_ by_ view::iterator[range.chunk.by.iter]
25.7.32 Stride view [range.stride]
25.7.32.1 Overview [range.stride.overview]
25.7.32.2 Class template stride_ view[range.stride.view]
25.7.32.3 Class template stride_ view::iterator[range.stride.iterator]
25.7.33 Cartesian product view [range.cartesian]
25.7.33.1 Overview [range.cartesian.overview]
25.7.33.2 Class template cartesian_ product_ view[range.cartesian.view]
25.7.33.3 Class template cartesian_ product_ view::iterator[range.cartesian.iterator]
25.7.34 Cache latest view [range.cache.latest]
25.7.34.1 Overview [range.cache.latest.overview]
25.7.34.2 Class template cache_ latest_ view[range.cache.latest.view]
25.7.34.3 Class cache_ latest_ view::iterator[range.cache.latest.iterator]
25.7.34.4 Class cache_ latest_ view::sentinel[range.cache.latest.sentinel]
25.7.35 To input view [range.to.input]
25.7.35.1 Overview [range.to.input.overview]
25.7.35.2 Class template to_ input_ view[range.to.input.view]
25.7.35.3 Class template to_ input_ view::iterator[range.to.input.iterator]
25.8 Range generators [coro.generator]
25.8.1 Overview [coroutine.generator.overview]
25.8.2 Header synopsis [generator.syn]
25.8.3 Class template generator[coro.generator.class]
25.8.4 Members [coro.generator.members]
25.8.5 Class generator::promise_ type[coro.generator.promise]
25.8.6 Class generator::iterator[coro.generator.iterator]
26 Algorithms library [algorithms][algorithms]
26.1 General [algorithms.general]
26.2 Algorithms requirements [algorithms.requirements]
26.3 Parallel algorithms [algorithms.parallel]
26.3.1 Preamble [algorithms.parallel.defns]
26.3.2 Requirements on user-provided function objects [algorithms.parallel.user]
26.3.3 Effect of execution policies on algorithm execution [algorithms.parallel.exec]
26.3.4 Parallel algorithm exceptions [algorithms.parallel.exceptions]
26.3.5ExecutionPolicy algorithm overloads [algorithms.parallel.overloads]
26.3.6 Execution policies [execpol]
26.3.6.1 General [execpol.general]
26.3.6.2 Execution policy type trait [execpol.type]
26.3.6.3 Sequenced execution policy [execpol.seq]
26.3.6.4 Parallel execution policy [execpol.par]
26.3.6.5 Parallel and unsequenced execution policy [execpol.parunseq]
26.3.6.6 Unsequenced execution policy [execpol.unseq]
26.3.6.7 Execution policy objects [execpol.objects]
26.4 Header synopsis [algorithm.syn]
26.5 Algorithm result types [algorithms.results]
26.6 Non-modifying sequence operations [alg.nonmodifying]
26.6.1 All of [alg.all.of]
26.6.2 Any of [alg.any.of]
26.6.3 None of [alg.none.of]
26.6.4 Contains [alg.contains]
26.6.5 For each [alg.foreach]
26.6.6 Find [alg.find]
26.6.7 Find last [alg.find.last]
26.6.8 Find end [alg.find.end]
26.6.9 Find first [alg.find.first.of]
26.6.10 Adjacent find [alg.adjacent.find]
26.6.11 Count [alg.count]
26.6.12 Mismatch [alg.mismatch]
26.6.13 Equal [alg.equal]
26.6.14 Is permutation [alg.is.permutation]
26.6.15 Search [alg.search]
26.6.16 Starts with [alg.starts.with]
26.6.17 Ends with [alg.ends.with]
26.6.18 Fold [alg.fold]
26.7 Mutating sequence operations [alg.modifying.operations]
26.7.1 Copy [alg.copy]
26.7.2 Move [alg.move]
26.7.3 Swap [alg.swap]
26.7.4 Transform [alg.transform]
26.7.5 Replace [alg.replace]
26.7.6 Fill [alg.fill]
26.7.7 Generate [alg.generate]
26.7.8 Remove [alg.remove]
26.7.9 Unique [alg.unique]
26.7.10 Reverse [alg.reverse]
26.7.11 Rotate [alg.rotate]
26.7.12 Sample [alg.random.sample]
26.7.13 Shuffle [alg.random.shuffle]
26.7.14 Shift [alg.shift]
26.8 Sorting and related operations [alg.sorting]
26.8.1 General [alg.sorting.general]
26.8.2 Sorting [alg.sort]
26.8.2.1sort[sort]
26.8.2.2stable_ sort[stable.sort]
26.8.2.3partial_ sort[partial.sort]
26.8.2.4partial_ sort_ copy[partial.sort.copy]
26.8.2.5is_ sorted[is.sorted]
26.8.3 Nth element [alg.nth.element]
26.8.4 Binary search [alg.binary.search]
26.8.4.1 General [alg.binary.search.general]
26.8.4.2lower_ bound[lower.bound]
26.8.4.3upper_ bound[upper.bound]
26.8.4.4equal_ range[equal.range]
26.8.4.5binary_ search[binary.search]
26.8.5 Partitions [alg.partitions]
26.8.6 Merge [alg.merge]
26.8.7 Set operations on sorted structures [alg.set.operations]
26.8.7.1 General [alg.set.operations.general]
26.8.7.2includes[includes]
26.8.7.3set_ union[set.union]
26.8.7.4set_ intersection[set.intersection]
26.8.7.5set_ difference[set.difference]
26.8.7.6set_ symmetric_ difference[set.symmetric.difference]
26.8.8 Heap operations [alg.heap.operations]
26.8.8.1 General [alg.heap.operations.general]
26.8.8.2push_ heap[push.heap]
26.8.8.3pop_ heap[pop.heap]
26.8.8.4make_ heap[make.heap]
26.8.8.5sort_ heap[sort.heap]
26.8.8.6is_ heap[is.heap]
26.8.9 Minimum and maximum [alg.min.max]
26.8.10 Bounded value [alg.clamp]
26.8.11 Lexicographical comparison [alg.lex.comparison]
26.8.12 Three-way comparison algorithms [alg.three.way]
26.8.13 Permutation generators [alg.permutation.generators]
26.9 Header synopsis [numeric.ops.overview]
26.10 Generalized numeric operations [numeric.ops]
26.10.1 General [numeric.ops.general]
26.10.2 Definitions [numerics.defns]
26.10.3 Accumulate [accumulate]
26.10.4 Reduce [reduce]
26.10.5 Inner product [inner.product]
26.10.6 Transform reduce [transform.reduce]
26.10.7 Partial sum [partial.sum]
26.10.8 Exclusive scan [exclusive.scan]
26.10.9 Inclusive scan [inclusive.scan]
26.10.10 Transform exclusive scan [transform.exclusive.scan]
26.10.11 Transform inclusive scan [transform.inclusive.scan]
26.10.12 Adjacent difference [adjacent.difference]
26.10.13 Iota [numeric.iota]
26.10.14 Greatest common divisor [numeric.ops.gcd]
26.10.15 Least common multiple [numeric.ops.lcm]
26.10.16 Midpoint [numeric.ops.midpoint]
26.10.17 Saturation arithmetic [numeric.sat]
26.10.17.1 Arithmetic functions [numeric.sat.func]
26.10.17.2 Casting [numeric.sat.cast]
26.11 Specialized algorithms [specialized.algorithms]
26.11.1 General [specialized.algorithms.general]
26.11.2 Special memory concepts [special.mem.concepts]
26.11.3uninitialized_ default_ construct[uninitialized.construct.default]
26.11.4uninitialized_ value_ construct[uninitialized.construct.value]
26.11.5uninitialized_ copy[uninitialized.copy]
26.11.6uninitialized_ move[uninitialized.move]
26.11.7uninitialized_ fill[uninitialized.fill]
26.11.8construct_ at[specialized.construct]
26.11.9destroy[specialized.destroy]
26.12 Specialized algorithms [alg.rand]
26.12.1 General [alg.rand.general]
26.12.2generate_ random[alg.rand.generate]
26.13 C library algorithms [alg.c.library]
27 Strings library [strings][strings]
27.1 General [strings.general]
27.2 Character traits [char.traits]
27.2.1 General [char.traits.general]
27.2.2 Character traits requirements [char.traits.require]
27.2.3 Traits typedefs [char.traits.typedefs]
27.2.4char_ traits specializations [char.traits.specializations]
27.2.4.1 General [char.traits.specializations.general]
27.2.4.2struct char_ traits[char.traits.specializations.char]
27.2.4.3struct char_ traits[char.traits.specializations.char8.t]
27.2.4.4struct char_ traits[char.traits.specializations.char16.t]
27.2.4.5struct char_ traits[char.traits.specializations.char32.t]
27.2.4.6struct char_ traits[char.traits.specializations.wchar.t]
27.3 String view classes [string.view]
27.3.1 General [string.view.general]
27.3.2 Header synopsis [string.view.synop]
27.3.3 Class template basic_ string_ view[string.view.template]
27.3.3.1 General [string.view.template.general]
27.3.3.2 Construction and assignment [string.view.cons]
27.3.3.3 Deduction guides [string.view.deduct]
27.3.3.4 Iterator support [string.view.iterators]
27.3.3.5 Capacity [string.view.capacity]
27.3.3.6 Element access [string.view.access]
27.3.3.7 Modifiers [string.view.modifiers]
27.3.3.8 String operations [string.view.ops]
27.3.3.9 Searching [string.view.find]
27.3.4 Non-member comparison functions [string.view.comparison]
27.3.5 Inserters and extractors [string.view.io]
27.3.6 Hash support [string.view.hash]
27.3.7 Suffix for basic_ string_ view literals [string.view.literals]
27.4 String classes [string.classes]
27.4.1 General [string.classes.general]
27.4.2 Header synopsis [string.syn]
27.4.3 Class template basic_ string[basic.string]
27.4.3.1 General [basic.string.general]
27.4.3.2 General requirements [string.require]
27.4.3.3 Constructors and assignment operators [string.cons]
27.4.3.4 Iterator support [string.iterators]
27.4.3.5 Capacity [string.capacity]
27.4.3.6 Element access [string.access]
27.4.3.7 Modifiers [string.modifiers]
27.4.3.7.1basic_ string::operator+=[string.op.append]
27.4.3.7.2basic_ string::append[string.append]
27.4.3.7.3basic_ string::assign[string.assign]
27.4.3.7.4basic_ string::insert[string.insert]
27.4.3.7.5basic_ string::erase[string.erase]
27.4.3.7.6basic_ string::replace[string.replace]
27.4.3.7.7basic_ string::copy[string.copy]
27.4.3.7.8basic_ string::swap[string.swap]
27.4.3.8 String operations [string.ops]
27.4.3.8.1 Accessors [string.accessors]
27.4.3.8.2 Searching [string.find]
27.4.3.8.3basic_ string::substr[string.substr]
27.4.3.8.4basic_ string::compare[string.compare]
27.4.3.8.5basic_ string::starts_ with[string.starts.with]
27.4.3.8.6basic_ string::ends_ with[string.ends.with]
27.4.3.8.7basic_ string::contains[string.contains]
27.4.4 Non-member functions [string.nonmembers]
27.4.4.1operator+[string.op.plus]
27.4.4.2 Non-member comparison operator functions [string.cmp]
27.4.4.3swap[string.special]
27.4.4.4 Inserters and extractors [string.io]
27.4.4.5 Erasure [string.erasure]
27.4.5 Numeric conversions [string.conversions]
27.4.6 Hash support [basic.string.hash]
27.4.7 Suffix for basic_ string literals [basic.string.literals]
27.5 Null-terminated sequence utilities [c.strings]
27.5.1 Header synopsis [cstring.syn]
28 Text processing library [text][text]
28.1 General [text.general]
28.2 Primitive numeric conversions [charconv]
28.2.1 Header synopsis [charconv.syn]
28.2.2 Primitive numeric output conversion [charconv.to.chars]
28.2.3 Primitive numeric input conversion [charconv.from.chars]
28.3 Localization library [localization]
28.3.1 General [localization.general]
28.3.2 Header synopsis [locale.syn]
28.3.3 Locales [locales]
28.3.3.1 Class locale[locale]
28.3.3.1.1 General [locale.general]
28.3.3.1.2 Types [locale.types]
28.3.3.1.2.1 Type locale::category[locale.category]
28.3.3.1.2.2 Class locale::facet[locale.facet]
28.3.3.1.2.3 Class locale::id[locale.id]
28.3.3.1.3 Constructors and destructor [locale.cons]
28.3.3.1.4 Members [locale.members]
28.3.3.1.5 Operators [locale.operators]
28.3.3.1.6 Static members [locale.statics]
28.3.3.2locale globals [locale.global.templates]
28.3.3.3 Convenience interfaces [locale.convenience]
28.3.3.3.1 Character classification [classification]
28.3.3.3.2 Character conversions [conversions.character]
28.3.4 Standard locale categories [locale.categories]
28.3.4.1 General [locale.categories.general]
28.3.4.2 The ctype category [category.ctype]
28.3.4.2.1 General [category.ctype.general]
28.3.4.2.2 Class template ctype[locale.ctype]
28.3.4.2.2.1 General [locale.ctype.general]
28.3.4.2.2.2ctype members [locale.ctype.members]
28.3.4.2.2.3ctype virtual functions [locale.ctype.virtuals]
28.3.4.2.3 Class template ctype_ byname[locale.ctype.byname]
28.3.4.2.4ctype specialization [facet.ctype.special]
28.3.4.2.4.1 General [facet.ctype.special.general]
28.3.4.2.4.2 Destructor [facet.ctype.char.dtor]
28.3.4.2.4.3 Members [facet.ctype.char.members]
28.3.4.2.4.4 Static members [facet.ctype.char.statics]
28.3.4.2.4.5 Virtual functions [facet.ctype.char.virtuals]
28.3.4.2.5 Class template codecvt[locale.codecvt]
28.3.4.2.5.1 General [locale.codecvt.general]
28.3.4.2.5.2 Members [locale.codecvt.members]
28.3.4.2.5.3 Virtual functions [locale.codecvt.virtuals]
28.3.4.2.6 Class template codecvt_ byname[locale.codecvt.byname]
28.3.4.3 The numeric category [category.numeric]
28.3.4.3.1 General [category.numeric.general]
28.3.4.3.2 Class template num_ get[locale.num.get]
28.3.4.3.2.1 General [locale.num.get.general]
28.3.4.3.2.2 Members [facet.num.get.members]
28.3.4.3.2.3 Virtual functions [facet.num.get.virtuals]
28.3.4.3.3 Class template num_ put[locale.nm.put]
28.3.4.3.3.1 General [locale.nm.put.general]
28.3.4.3.3.2 Members [facet.num.put.members]
28.3.4.3.3.3 Virtual functions [facet.num.put.virtuals]
28.3.4.4 The numeric punctuation facet [facet.numpunct]
28.3.4.4.1 Class template numpunct[locale.numpunct]
28.3.4.4.1.1 General [locale.numpunct.general]
28.3.4.4.1.2 Members [facet.numpunct.members]
28.3.4.4.1.3 Virtual functions [facet.numpunct.virtuals]
28.3.4.4.2 Class template numpunct_ byname[locale.numpunct.byname]
28.3.4.5 The collate category [category.collate]
28.3.4.5.1 Class template collate[locale.collate]
28.3.4.5.1.1 General [locale.collate.general]
28.3.4.5.1.2 Members [locale.collate.members]
28.3.4.5.1.3 Virtual functions [locale.collate.virtuals]
28.3.4.5.2 Class template collate_ byname[locale.collate.byname]
28.3.4.6 The time category [category.time]
28.3.4.6.1 General [category.time.general]
28.3.4.6.2 Class template time_ get[locale.time.get]
28.3.4.6.2.1 General [locale.time.get.general]
28.3.4.6.2.2 Members [locale.time.get.members]
28.3.4.6.2.3 Virtual functions [locale.time.get.virtuals]
28.3.4.6.3 Class template time_ get_ byname[locale.time.get.byname]
28.3.4.6.4 Class template time_ put[locale.time.put]
28.3.4.6.4.1 General [locale.time.put.general]
28.3.4.6.4.2 Members [locale.time.put.members]
28.3.4.6.4.3 Virtual functions [locale.time.put.virtuals]
28.3.4.6.5 Class template time_ put_ byname[locale.time.put.byname]
28.3.4.7 The monetary category [category.monetary]
28.3.4.7.1 General [category.monetary.general]
28.3.4.7.2 Class template money_ get[locale.money.get]
28.3.4.7.2.1 General [locale.money.get.general]
28.3.4.7.2.2 Members [locale.money.get.members]
28.3.4.7.2.3 Virtual functions [locale.money.get.virtuals]
28.3.4.7.3 Class template money_ put[locale.money.put]
28.3.4.7.3.1 General [locale.money.put.general]
28.3.4.7.3.2 Members [locale.money.put.members]
28.3.4.7.3.3 Virtual functions [locale.money.put.virtuals]
28.3.4.7.4 Class template moneypunct[locale.moneypunct]
28.3.4.7.4.1 General [locale.moneypunct.general]
28.3.4.7.4.2 Members [locale.moneypunct.members]
28.3.4.7.4.3 Virtual functions [locale.moneypunct.virtuals]
28.3.4.7.5 Class template moneypunct_ byname[locale.moneypunct.byname]
28.3.4.8 The message retrieval category [category.messages]
28.3.4.8.1 General [category.messages.general]
28.3.4.8.2 Class template messages[locale.messages]
28.3.4.8.2.1 General [locale.messages.general]
28.3.4.8.2.2 Members [locale.messages.members]
28.3.4.8.2.3 Virtual functions [locale.messages.virtuals]
28.3.4.8.3 Class template messages_ byname[locale.messages.byname]
28.3.5 C library locales [c.locales]
28.3.5.1 Header synopsis [clocale.syn]
28.3.5.2 Data races [clocale.data.races]
28.4 Text encodings identification [text.encoding]
28.4.1 Header synopsis [text.encoding.syn]
28.4.2 Class text_ encoding[text.encoding.class]
28.4.2.1 Overview [text.encoding.overview]
28.4.2.2 General [text.encoding.general]
28.4.2.3 Members [text.encoding.members]
28.4.2.4 Comparison functions [text.encoding.cmp]
28.4.2.5 Class text_ encoding::aliases_ view[text.encoding.aliases]
28.4.2.6 Enumeration text_ encoding::id[text.encoding.id]
28.4.2.7 Hash support [text.encoding.hash]
28.5 Formatting [format]
28.5.1 Header synopsis [format.syn]
28.5.2 Format string [format.string]
28.5.2.1 General [format.string.general]
28.5.2.2 Standard format specifiers [format.string.std]
28.5.3 Error reporting [format.err.report]
28.5.4 Class template basic_ format_ string[format.fmt.string]
28.5.5 Formatting functions [format.functions]
28.5.6 Formatter [format.formatter]
28.5.6.1 Formatter requirements [formatter.requirements]
28.5.6.2 Formatter locking [format.formatter.locking]
28.5.6.3 Concept formattable[format.formattable]
28.5.6.4 Formatter specializations [format.formatter.spec]
28.5.6.5 Formatting escaped characters and strings [format.string.escaped]
28.5.6.6 Class template basic_ format_ parse_ context[format.parse.ctx]
28.5.6.7 Class template basic_ format_ context[format.context]
28.5.7 Formatting of ranges [format.range]
28.5.7.1 Variable template format_ kind[format.range.fmtkind]
28.5.7.2 Class template range_ formatter[format.range.formatter]
28.5.7.3 Class template range-default-formatter[format.range.fmtdef]
28.5.7.4 Specialization of range-default-formatter for maps [format.range.fmtmap]
28.5.7.5 Specialization of range-default-formatter for sets [format.range.fmtset]
28.5.7.6 Specialization of range-default-formatter for strings [format.range.fmtstr]
28.5.8 Arguments [format.arguments]
28.5.8.1 Class template basic_ format_ arg[format.arg]
28.5.8.2 Class template format-arg-store[format.arg.store]
28.5.8.3 Class template basic_ format_ args[format.args]
28.5.9 Tuple formatter [format.tuple]
28.5.10 Class format_ error[format.error]
28.6 Regular expressions library [re]
28.6.1 General [re.general]
28.6.2 Requirements [re.req]
28.6.3 Header synopsis [re.syn]
28.6.4 Namespace std::regex_ constants[re.const]
28.6.4.1 General [re.const.general]
28.6.4.2 Bitmask type syntax_ option_ type[re.synopt]
28.6.4.3 Bitmask type match_ flag_ type[re.matchflag]
28.6.4.4 Implementation-defined error_ type[re.err]
28.6.5 Class regex_ error[re.badexp]
28.6.6 Class template regex_ traits[re.traits]
28.6.7 Class template basic_ regex[re.regex]
28.6.7.1 General [re.regex.general]
28.6.7.2 Constructors [re.regex.construct]
28.6.7.3 Assignment [re.regex.assign]
28.6.7.4 Constant operations [re.regex.operations]
28.6.7.5 Locale [re.regex.locale]
28.6.7.6 Swap [re.regex.swap]
28.6.7.7 Non-member functions [re.regex.nonmemb]
28.6.8 Class template sub_ match[re.submatch]
28.6.8.1 General [re.submatch.general]
28.6.8.2 Members [re.submatch.members]
28.6.8.3 Non-member operators [re.submatch.op]
28.6.9 Class template match_ results[re.results]
28.6.9.1 General [re.results.general]
28.6.9.2 Constructors [re.results.const]
28.6.9.3 State [re.results.state]
28.6.9.4 Size [re.results.size]
28.6.9.5 Element access [re.results.acc]
28.6.9.6 Formatting [re.results.form]
28.6.9.7 Allocator [re.results.all]
28.6.9.8 Swap [re.results.swap]
28.6.9.9 Non-member functions [re.results.nonmember]
28.6.10 Regular expression algorithms [re.alg]
28.6.10.1 Exceptions [re.except]
28.6.10.2regex_ match[re.alg.match]
28.6.10.3regex_ search[re.alg.search]
28.6.10.4regex_ replace[re.alg.replace]
28.6.11 Regular expression iterators [re.iter]
28.6.11.1 Class template regex_ iterator[re.regiter]
28.6.11.1.1 General [re.regiter.general]
28.6.11.1.2 Constructors [re.regiter.cnstr]
28.6.11.1.3 Comparisons [re.regiter.comp]
28.6.11.1.4 Indirection [re.regiter.deref]
28.6.11.1.5 Increment [re.regiter.incr]
28.6.11.2 Class template regex_ token_ iterator[re.tokiter]
28.6.11.2.1 General [re.tokiter.general]
28.6.11.2.2 Constructors [re.tokiter.cnstr]
28.6.11.2.3 Comparisons [re.tokiter.comp]
28.6.11.2.4 Indirection [re.tokiter.deref]
28.6.11.2.5 Increment [re.tokiter.incr]
28.6.12 Modified ECMAScript regular expression grammar [re.grammar]
28.7 Null-terminated sequence utilities [text.c.strings]
28.7.1 Header synopsis [cctype.syn]
28.7.2 Header synopsis [cwctype.syn]
28.7.3 Header synopsis [cwchar.syn]
28.7.4 Header synopsis [cuchar.syn]
28.7.5 Multibyte / wide string and character conversion functions [c.mb.wcs]
29 Numerics library [numerics][numerics]
29.1 General [numerics.general]
29.2 Numeric type requirements [numeric.requirements]
29.3 The floating-point environment [cfenv]
29.3.1 Header synopsis [cfenv.syn]
29.3.2 Threads [cfenv.thread]
29.4 Complex numbers [complex.numbers]
29.4.1 General [complex.numbers.general]
29.4.2 Header synopsis [complex.syn]
29.4.3 Class template complex[complex]
29.4.4 Member functions [complex.members]
29.4.5 Member operators [complex.member.ops]
29.4.6 Non-member operations [complex.ops]
29.4.7 Value operations [complex.value.ops]
29.4.8 Transcendentals [complex.transcendentals]
29.4.9 Tuple interface [complex.tuple]
29.4.10 Additional overloads [cmplx.over]
29.4.11 Suffixes for complex number literals [complex.literals]
29.5 Random number generation [rand]
29.5.1 General [rand.general]
29.5.2 Header synopsis [rand.synopsis]
29.5.3 Requirements [rand.req]
29.5.3.1 General requirements [rand.req.genl]
29.5.3.2 Seed sequence requirements [rand.req.seedseq]
29.5.3.3 Uniform random bit generator requirements [rand.req.urng]
29.5.3.4 Random number engine requirements [rand.req.eng]
29.5.3.5 Random number engine adaptor requirements [rand.req.adapt]
29.5.3.6 Random number distribution requirements [rand.req.dist]
29.5.4 Random number engine class templates [rand.eng]
29.5.4.1 General [rand.eng.general]
29.5.4.2 Class template linear_ congruential_ engine[rand.eng.lcong]
29.5.4.3 Class template mersenne_ twister_ engine[rand.eng.mers]
29.5.4.4 Class template subtract_ with_ carry_ engine[rand.eng.sub]
29.5.4.5 Class template philox_ engine[rand.eng.philox]
29.5.5 Random number engine adaptor class templates [rand.adapt]
29.5.5.1 General [rand.adapt.general]
29.5.5.2 Class template discard_ block_ engine[rand.adapt.disc]
29.5.5.3 Class template independent_ bits_ engine[rand.adapt.ibits]
29.5.5.4 Class template shuffle_ order_ engine[rand.adapt.shuf]
29.5.6 Engines and engine adaptors with predefined parameters [rand.predef]
29.5.7 Class random_ device[rand.device]
29.5.8 Utilities [rand.util]
29.5.8.1 Class seed_ seq[rand.util.seedseq]
29.5.8.2 Function template generate_ canonical[rand.util.canonical]
29.5.9 Random number distribution class templates [rand.dist]
29.5.9.1 General [rand.dist.general]
29.5.9.2 Uniform distributions [rand.dist.uni]
29.5.9.2.1 Class template uniform_ int_ distribution[rand.dist.uni.int]
29.5.9.2.2 Class template uniform_ real_ distribution[rand.dist.uni.real]
29.5.9.3 Bernoulli distributions [rand.dist.bern]
29.5.9.3.1 Class bernoulli_ distribution[rand.dist.bern.bernoulli]
29.5.9.3.2 Class template binomial_ distribution[rand.dist.bern.bin]
29.5.9.3.3 Class template geometric_ distribution[rand.dist.bern.geo]
29.5.9.3.4 Class template negative_ binomial_ distribution[rand.dist.bern.negbin]
29.5.9.4 Poisson distributions [rand.dist.pois]
29.5.9.4.1 Class template poisson_ distribution[rand.dist.pois.poisson]
29.5.9.4.2 Class template exponential_ distribution[rand.dist.pois.exp]
29.5.9.4.3 Class template gamma_ distribution[rand.dist.pois.gamma]
29.5.9.4.4 Class template weibull_ distribution[rand.dist.pois.weibull]
29.5.9.4.5 Class template extreme_ value_ distribution[rand.dist.pois.extreme]
29.5.9.5 Normal distributions [rand.dist.norm]
29.5.9.5.1 Class template normal_ distribution[rand.dist.norm.normal]
29.5.9.5.2 Class template lognormal_ distribution[rand.dist.norm.lognormal]
29.5.9.5.3 Class template chi_ squared_ distribution[rand.dist.norm.chisq]
29.5.9.5.4 Class template cauchy_ distribution[rand.dist.norm.cauchy]
29.5.9.5.5 Class template fisher_ f_ distribution[rand.dist.norm.f]
29.5.9.5.6 Class template student_ t_ distribution[rand.dist.norm.t]
29.5.9.6 Sampling distributions [rand.dist.samp]
29.5.9.6.1 Class template discrete_ distribution[rand.dist.samp.discrete]
29.5.9.6.2 Class template piecewise_ constant_ distribution[rand.dist.samp.pconst]
29.5.9.6.3 Class template piecewise_ linear_ distribution[rand.dist.samp.plinear]
29.5.10 Low-quality random number generation [c.math.rand]
29.6 Numeric arrays [numarray]
29.6.1 Header synopsis [valarray.syn]
29.6.2 Class template valarray[template.valarray]
29.6.2.1 Overview [template.valarray.overview]
29.6.2.2 Constructors [valarray.cons]
29.6.2.3 Assignment [valarray.assign]
29.6.2.4 Element access [valarray.access]
29.6.2.5 Subset operations [valarray.sub]
29.6.2.6 Unary operators [valarray.unary]
29.6.2.7 Compound assignment [valarray.cassign]
29.6.2.8 Member functions [valarray.members]
29.6.3valarray non-member operations [valarray.nonmembers]
29.6.3.1 Binary operators [valarray.binary]
29.6.3.2 Logical operators [valarray.comparison]
29.6.3.3 Transcendentals [valarray.transcend]
29.6.3.4 Specialized algorithms [valarray.special]
29.6.4 Class slice[class.slice]
29.6.4.1 Overview [class.slice.overview]
29.6.4.2 Constructors [cons.slice]
29.6.4.3 Access functions [slice.access]
29.6.4.4 Operators [slice.ops]
29.6.5 Class template slice_ array[template.slice.array]
29.6.5.1 Overview [template.slice.array.overview]
29.6.5.2 Assignment [slice.arr.assign]
29.6.5.3 Compound assignment [slice.arr.comp.assign]
29.6.5.4 Fill function [slice.arr.fill]
29.6.6 The gslice class [class.gslice]
29.6.6.1 Overview [class.gslice.overview]
29.6.6.2 Constructors [gslice.cons]
29.6.6.3 Access functions [gslice.access]
29.6.7 Class template gslice_ array[template.gslice.array]
29.6.7.1 Overview [template.gslice.array.overview]
29.6.7.2 Assignment [gslice.array.assign]
29.6.7.3 Compound assignment [gslice.array.comp.assign]
29.6.7.4 Fill function [gslice.array.fill]
29.6.8 Class template mask_ array[template.mask.array]
29.6.8.1 Overview [template.mask.array.overview]
29.6.8.2 Assignment [mask.array.assign]
29.6.8.3 Compound assignment [mask.array.comp.assign]
29.6.8.4 Fill function [mask.array.fill]
29.6.9 Class template indirect_ array[template.indirect.array]
29.6.9.1 Overview [template.indirect.array.overview]
29.6.9.2 Assignment [indirect.array.assign]
29.6.9.3 Compound assignment [indirect.array.comp.assign]
29.6.9.4 Fill function [indirect.array.fill]
29.6.10valarray range access [valarray.range]
29.7 Mathematical functions for floating-point types [c.math]
29.7.1 Header synopsis [cmath.syn]
29.7.2 Absolute values [c.math.abs]
29.7.3 Three-dimensional hypotenuse [c.math.hypot3]
29.7.4 Linear interpolation [c.math.lerp]
29.7.5 Classification / comparison functions [c.math.fpclass]
29.7.6 Mathematical special functions [sf.cmath]
29.7.6.1 General [sf.cmath.general]
29.7.6.2 Associated Laguerre polynomials [sf.cmath.assoc.laguerre]
29.7.6.3 Associated Legendre functions [sf.cmath.assoc.legendre]
29.7.6.4 Beta function [sf.cmath.beta]
29.7.6.5 Complete elliptic integral of the first kind [sf.cmath.comp.ellint.1]
29.7.6.6 Complete elliptic integral of the second kind [sf.cmath.comp.ellint.2]
29.7.6.7 Complete elliptic integral of the third kind [sf.cmath.comp.ellint.3]
29.7.6.8 Regular modified cylindrical Bessel functions [sf.cmath.cyl.bessel.i]
29.7.6.9 Cylindrical Bessel functions of the first kind [sf.cmath.cyl.bessel.j]
29.7.6.10 Irregular modified cylindrical Bessel functions [sf.cmath.cyl.bessel.k]
29.7.6.11 Cylindrical Neumann functions [sf.cmath.cyl.neumann]
29.7.6.12 Incomplete elliptic integral of the first kind [sf.cmath.ellint.1]
29.7.6.13 Incomplete elliptic integral of the second kind [sf.cmath.ellint.2]
29.7.6.14 Incomplete elliptic integral of the third kind [sf.cmath.ellint.3]
29.7.6.15 Exponential integral [sf.cmath.expint]
29.7.6.16 Hermite polynomials [sf.cmath.hermite]
29.7.6.17 Laguerre polynomials [sf.cmath.laguerre]
29.7.6.18 Legendre polynomials [sf.cmath.legendre]
29.7.6.19 Riemann zeta function [sf.cmath.riemann.zeta]
29.7.6.20 Spherical Bessel functions of the first kind [sf.cmath.sph.bessel]
29.7.6.21 Spherical associated Legendre functions [sf.cmath.sph.legendre]
29.7.6.22 Spherical Neumann functions [sf.cmath.sph.neumann]
29.8 Numbers [numbers]
29.8.1 Header synopsis [numbers.syn]
29.8.2 Mathematical constants [math.constants]
29.9 Basic linear algebra algorithms [linalg]
29.9.1 Overview [linalg.overview]
29.9.2 Header synopsis [linalg.syn]
29.9.3 General [linalg.general]
29.9.4 Requirements [linalg.reqs]
29.9.4.1 Linear algebra value types [linalg.reqs.val]
29.9.4.2 Algorithm and class requirements [linalg.reqs.alg]
29.9.5 Tag classes [linalg.tags]
29.9.5.1 Storage order tags [linalg.tags.order]
29.9.5.2 Triangle tags [linalg.tags.triangle]
29.9.5.3 Diagonal tags [linalg.tags.diagonal]
29.9.6 Layouts for packed matrix types [linalg.layout.packed]
29.9.6.1 Overview [linalg.layout.packed.overview]
29.9.6.2 Constructors [linalg.layout.packed.cons]
29.9.6.3 Observers [linalg.layout.packed.obs]
29.9.7 Exposition-only helpers [linalg.helpers]
29.9.7.1abs-if-needed[linalg.helpers.abs]
29.9.7.2conj-if-needed[linalg.helpers.conj]
29.9.7.3real-if-needed[linalg.helpers.real]
29.9.7.4imag-if-needed[linalg.helpers.imag]
29.9.7.5 Argument concepts [linalg.helpers.concepts]
29.9.7.6 Mandates [linalg.helpers.mandates]
29.9.7.7 Preconditions [linalg.helpers.precond]
29.9.8 Scaled in-place transformation [linalg.scaled]
29.9.8.1 Introduction [linalg.scaled.intro]
29.9.8.2 Class template scaled_ accessor[linalg.scaled.scaledaccessor]
29.9.8.3 Function template scaled[linalg.scaled.scaled]
29.9.9 Conjugated in-place transformation [linalg.conj]
29.9.9.1 Introduction [linalg.conj.intro]
29.9.9.2 Class template conjugated_ accessor[linalg.conj.conjugatedaccessor]
29.9.9.3 Function template conjugated[linalg.conj.conjugated]
29.9.10 Transpose in-place transformation [linalg.transp]
29.9.10.1 Introduction [linalg.transp.intro]
29.9.10.2 Exposition-only helpers for layout_ transpose and transposed[linalg.transp.helpers]
29.9.10.3 Class template layout_ transpose[linalg.transp.layout.transpose]
29.9.10.4 Function template transposed[linalg.transp.transposed]
29.9.11 Conjugate transpose in-place transform [linalg.conjtransposed]
29.9.12 Algorithm requirements based on template parameter name [linalg.algs.reqs]
29.9.13 BLAS 1 algorithms [linalg.algs.blas1]
29.9.13.1 Complexity [linalg.algs.blas1.complexity]
29.9.13.2 Givens rotations [linalg.algs.blas1.givens]
29.9.13.2.1 Compute Givens rotation [linalg.algs.blas1.givens.lartg]
29.9.13.2.2 Apply a computed Givens rotation to vectors [linalg.algs.blas1.givens.rot]
29.9.13.3 Swap matrix or vector elements [linalg.algs.blas1.swap]
29.9.13.4 Multiply the elements of an object in place by a scalar [linalg.algs.blas1.scal]
29.9.13.5 Copy elements of one matrix or vector into another [linalg.algs.blas1.copy]
29.9.13.6 Add vectors or matrices elementwise [linalg.algs.blas1.add]
29.9.13.7 Dot product of two vectors [linalg.algs.blas1.dot]
29.9.13.8 Scaled sum of squares of a vector's elements [linalg.algs.blas1.ssq]
29.9.13.9 Euclidean norm of a vector [linalg.algs.blas1.nrm2]
29.9.13.10 Sum of absolute values of vector elements [linalg.algs.blas1.asum]
29.9.13.11 Index of maximum absolute value of vector elements [linalg.algs.blas1.iamax]
29.9.13.12 Frobenius norm of a matrix [linalg.algs.blas1.matfrobnorm]
29.9.13.13 One norm of a matrix [linalg.algs.blas1.matonenorm]
29.9.13.14 Infinity norm of a matrix [linalg.algs.blas1.matinfnorm]
29.9.14 BLAS 2 algorithms [linalg.algs.blas2]
29.9.14.1 General matrix-vector product [linalg.algs.blas2.gemv]
29.9.14.2 Symmetric matrix-vector product [linalg.algs.blas2.symv]
29.9.14.3 Hermitian matrix-vector product [linalg.algs.blas2.hemv]
29.9.14.4 Triangular matrix-vector product [linalg.algs.blas2.trmv]
29.9.14.5 Solve a triangular linear system [linalg.algs.blas2.trsv]
29.9.14.6 Rank-1 (outer product) update of a matrix [linalg.algs.blas2.rank1]
29.9.14.7 Symmetric or Hermitian Rank-1 (outer product) update of a matrix [linalg.algs.blas2.symherrank1]
29.9.14.8 Symmetric and Hermitian rank-2 matrix updates [linalg.algs.blas2.rank2]
29.9.15 BLAS 3 algorithms [linalg.algs.blas3]
29.9.15.1 General matrix-matrix product [linalg.algs.blas3.gemm]
29.9.15.2 Symmetric, Hermitian, and triangular matrix-matrix product [linalg.algs.blas3.xxmm]
29.9.15.3 In-place triangular matrix-matrix product [linalg.algs.blas3.trmm]
29.9.15.4 Rank-k update of a symmetric or Hermitian matrix [linalg.algs.blas3.rankk]
29.9.15.5 Rank-2k update of a symmetric or Hermitian matrix [linalg.algs.blas3.rank2k]
29.9.15.6 Solve multiple triangular linear systems [linalg.algs.blas3.trsm]
29.9.15.7 Solve multiple triangular linear systems in-place [linalg.algs.blas3.inplacetrsm]
29.10 Data-parallel types [simd]
29.10.1 General [simd.general]
29.10.2 Exposition-only types, variables, and concepts [simd.expos]
29.10.2.1 Exposition-only helpers [simd.expos.defn]
29.10.2.2simd ABI tags [simd.expos.abi]
29.10.3 Header synopsis [simd.syn]
29.10.4vec type traits [simd.traits]
29.10.5 Load and store flags [simd.flags]
29.10.5.1 Class template flags overview [simd.flags.overview]
29.10.5.2flags operators [simd.flags.oper]
29.10.6 Class template simd-iterator[simd.iterator]
29.10.7 Class template basic_ vec[simd.class]
29.10.7.1 Class template basic_ vec overview [simd.overview]
29.10.7.2basic_ vec constructors [simd.ctor]
29.10.7.3basic_ vec subscript operator [simd.subscr]
29.10.7.4basic_ vec unary operators [simd.unary]
29.10.8basic_ vec non-member operations [simd.nonmembers]
29.10.8.1basic_ vec binary operators [simd.binary]
29.10.8.2basic_ vec compound assignment [simd.cassign]
29.10.8.3basic_ vec compare operators [simd.comparison]
29.10.8.4vec complex accessors [simd.complex.access]
29.10.8.5basic_ vec exposition only conditional operators [simd.cond]
29.10.8.6basic_ vec reductions [simd.reductions]
29.10.8.7basic_ vec load and store functions [simd.loadstore]
29.10.8.8vec static permute [simd.permute.static]
29.10.8.9vec dynamic permute [simd.permute.dynamic]
29.10.8.10vec mask permute [simd.permute.mask]
29.10.8.11simd memory permute [simd.permute.memory]
29.10.8.12basic_ vec and basic_ mask creation [simd.creation]
29.10.8.13 Algorithms [simd.alg]
29.10.8.14 Mathematical functions [simd.math]
29.10.8.15basic_ vec bit library [simd.bit]
29.10.8.16vec complex math [simd.complex.math]
29.10.9 Class template basic_ mask[simd.mask.class]
29.10.9.1 Class template basic_ mask overview [simd.mask.overview]
29.10.9.2basic_ mask constructors [simd.mask.ctor]
29.10.9.3basic_ mask subscript operator [simd.mask.subscr]
29.10.9.4basic_ mask unary operators [simd.mask.unary]
29.10.9.5basic_ mask conversion operators [simd.mask.conv]
29.10.9.6basic_ mask named conversion operators [simd.mask.namedconv]
29.10.10 Non-member operations [simd.mask.nonmembers]
29.10.10.1basic_ mask binary operators [simd.mask.binary]
29.10.10.2basic_ mask compound assignment [simd.mask.cassign]
29.10.10.3basic_ mask comparisons [simd.mask.comparison]
29.10.10.4basic_ mask exposition only conditional operators [simd.mask.cond]
29.10.10.5basic_ mask reductions [simd.mask.reductions]
29.11 C compatibility [numerics.c]
29.11.1 Header synopsis [stdckdint.h.syn]
29.11.2 Checked integer operations [numerics.c.ckdint]
30 Time library [time][time]
30.1 General [time.general]
30.2 Header synopsis [time.syn]
30.3Cpp17Clock requirements [time.clock.req]
30.4 Time-related traits [time.traits]
30.4.1treat_ as_ floating_ point[time.traits.is.fp]
30.4.2duration_ values[time.traits.duration.values]
30.4.3 Specializations of common_ type[time.traits.specializations]
30.4.4 Class template is_ clock[time.traits.is.clock]
30.5 Class template duration[time.duration]
30.5.1 General [time.duration.general]
30.5.2 Constructors [time.duration.cons]
30.5.3 Observer [time.duration.observer]
30.5.4 Arithmetic [time.duration.arithmetic]
30.5.5 Special values [time.duration.special]
30.5.6 Non-member arithmetic [time.duration.nonmember]
30.5.7 Comparisons [time.duration.comparisons]
30.5.8 Conversions [time.duration.cast]
30.5.9 Suffixes for duration literals [time.duration.literals]
30.5.10 Algorithms [time.duration.alg]
30.5.11 I/O [time.duration.io]
30.6 Class template time_ point[time.point]
30.6.1 General [time.point.general]
30.6.2 Constructors [time.point.cons]
30.6.3 Observer [time.point.observer]
30.6.4 Arithmetic [time.point.arithmetic]
30.6.5 Special values [time.point.special]
30.6.6 Non-member arithmetic [time.point.nonmember]
30.6.7 Comparisons [time.point.comparisons]
30.6.8 Conversions [time.point.cast]
30.7 Clocks [time.clock]
30.7.1 General [time.clock.general]
30.7.2 Class system_ clock[time.clock.system]
30.7.2.1 Overview [time.clock.system.overview]
30.7.2.2 Members [time.clock.system.members]
30.7.2.3 Non-member functions [time.clock.system.nonmembers]
30.7.3 Class utc_ clock[time.clock.utc]
30.7.3.1 Overview [time.clock.utc.overview]
30.7.3.2 Member functions [time.clock.utc.members]
30.7.3.3 Non-member functions [time.clock.utc.nonmembers]
30.7.4 Class tai_ clock[time.clock.tai]
30.7.4.1 Overview [time.clock.tai.overview]
30.7.4.2 Member functions [time.clock.tai.members]
30.7.4.3 Non-member functions [time.clock.tai.nonmembers]
30.7.5 Class gps_ clock[time.clock.gps]
30.7.5.1 Overview [time.clock.gps.overview]
30.7.5.2 Member functions [time.clock.gps.members]
30.7.5.3 Non-member functions [time.clock.gps.nonmembers]
30.7.6 Type file_ clock[time.clock.file]
30.7.6.1 Overview [time.clock.file.overview]
30.7.6.2 Member functions [time.clock.file.members]
30.7.6.3 Non-member functions [time.clock.file.nonmembers]
30.7.7 Class steady_ clock[time.clock.steady]
30.7.8 Class high_ resolution_ clock[time.clock.hires]
30.7.9 Local time [time.clock.local]
30.7.10time_ point conversions [time.clock.cast]
30.7.10.1 Class template clock_ time_ conversion[time.clock.conv]
30.7.10.2 Identity conversions [time.clock.cast.id]
30.7.10.3 Conversions between system_ clock and utc_ clock[time.clock.cast.sys.utc]
30.7.10.4 Conversions between system_ clock and other clocks [time.clock.cast.sys]
30.7.10.5 Conversions between utc_ clock and other clocks [time.clock.cast.utc]
30.7.10.6 Function template clock_ cast[time.clock.cast.fn]
30.8 The civil calendar [time.cal]
30.8.1 General [time.cal.general]
30.8.2 Class last_ spec[time.cal.last]
30.8.3 Class day[time.cal.day]
30.8.3.1 Overview [time.cal.day.overview]
30.8.3.2 Member functions [time.cal.day.members]
30.8.3.3 Non-member functions [time.cal.day.nonmembers]
30.8.4 Class month[time.cal.month]
30.8.4.1 Overview [time.cal.month.overview]
30.8.4.2 Member functions [time.cal.month.members]
30.8.4.3 Non-member functions [time.cal.month.nonmembers]
30.8.5 Class year[time.cal.year]
30.8.5.1 Overview [time.cal.year.overview]
30.8.5.2 Member functions [time.cal.year.members]
30.8.5.3 Non-member functions [time.cal.year.nonmembers]
30.8.6 Class weekday[time.cal.wd]
30.8.6.1 Overview [time.cal.wd.overview]
30.8.6.2 Member functions [time.cal.wd.members]
30.8.6.3 Non-member functions [time.cal.wd.nonmembers]
30.8.7 Class weekday_ indexed[time.cal.wdidx]
30.8.7.1 Overview [time.cal.wdidx.overview]
30.8.7.2 Member functions [time.cal.wdidx.members]
30.8.7.3 Non-member functions [time.cal.wdidx.nonmembers]
30.8.8 Class weekday_ last[time.cal.wdlast]
30.8.8.1 Overview [time.cal.wdlast.overview]
30.8.8.2 Member functions [time.cal.wdlast.members]
30.8.8.3 Non-member functions [time.cal.wdlast.nonmembers]
30.8.9 Class month_ day[time.cal.md]
30.8.9.1 Overview [time.cal.md.overview]
30.8.9.2 Member functions [time.cal.md.members]
30.8.9.3 Non-member functions [time.cal.md.nonmembers]
30.8.10 Class month_ day_ last[time.cal.mdlast]
30.8.11 Class month_ weekday[time.cal.mwd]
30.8.11.1 Overview [time.cal.mwd.overview]
30.8.11.2 Member functions [time.cal.mwd.members]
30.8.11.3 Non-member functions [time.cal.mwd.nonmembers]
30.8.12 Class month_ weekday_ last[time.cal.mwdlast]
30.8.12.1 Overview [time.cal.mwdlast.overview]
30.8.12.2 Member functions [time.cal.mwdlast.members]
30.8.12.3 Non-member functions [time.cal.mwdlast.nonmembers]
30.8.13 Class year_ month[time.cal.ym]
30.8.13.1 Overview [time.cal.ym.overview]
30.8.13.2 Member functions [time.cal.ym.members]
30.8.13.3 Non-member functions [time.cal.ym.nonmembers]
30.8.14 Class year_ month_ day[time.cal.ymd]
30.8.14.1 Overview [time.cal.ymd.overview]
30.8.14.2 Member functions [time.cal.ymd.members]
30.8.14.3 Non-member functions [time.cal.ymd.nonmembers]
30.8.15 Class year_ month_ day_ last[time.cal.ymdlast]
30.8.15.1 Overview [time.cal.ymdlast.overview]
30.8.15.2 Member functions [time.cal.ymdlast.members]
30.8.15.3 Non-member functions [time.cal.ymdlast.nonmembers]
30.8.16 Class year_ month_ weekday[time.cal.ymwd]
30.8.16.1 Overview [time.cal.ymwd.overview]
30.8.16.2 Member functions [time.cal.ymwd.members]
30.8.16.3 Non-member functions [time.cal.ymwd.nonmembers]
30.8.17 Class year_ month_ weekday_ last[time.cal.ymwdlast]
30.8.17.1 Overview [time.cal.ymwdlast.overview]
30.8.17.2 Member functions [time.cal.ymwdlast.members]
30.8.17.3 Non-member functions [time.cal.ymwdlast.nonmembers]
30.8.18 Conventional syntax operators [time.cal.operators]
30.9 Class template hh_ mm_ ss[time.hms]
30.9.1 Overview [time.hms.overview]
30.9.2 Members [time.hms.members]
30.9.3 Non-members [time.hms.nonmembers]
30.10 12/24 hours functions [time.12]
30.11 Time zones [time.zone]
30.11.1 General [time.zone.general]
30.11.2 Time zone database [time.zone.db]
30.11.2.1 Class tzdb[time.zone.db.tzdb]
30.11.2.2 Class tzdb_ list[time.zone.db.list]
30.11.2.3 Time zone database access [time.zone.db.access]
30.11.2.4 Remote time zone database support [time.zone.db.remote]
30.11.3 Exception classes [time.zone.exception]
30.11.3.1 Class nonexistent_ local_ time[time.zone.exception.nonexist]
30.11.3.2 Class ambiguous_ local_ time[time.zone.exception.ambig]
30.11.4 Information classes [time.zone.info]
30.11.4.1 Class sys_ info[time.zone.info.sys]
30.11.4.2 Class local_ info[time.zone.info.local]
30.11.5 Class time_ zone[time.zone.timezone]
30.11.5.1 Overview [time.zone.overview]
30.11.5.2 Member functions [time.zone.members]
30.11.5.3 Non-member functions [time.zone.nonmembers]
30.11.6 Class template zoned_ traits[time.zone.zonedtraits]
30.11.7 Class template zoned_ time[time.zone.zonedtime]
30.11.7.1 Overview [time.zone.zonedtime.overview]
30.11.7.2 Constructors [time.zone.zonedtime.ctor]
30.11.7.3 Member functions [time.zone.zonedtime.members]
30.11.7.4 Non-member functions [time.zone.zonedtime.nonmembers]
30.11.8 Class leap_ second[time.zone.leap]
30.11.8.1 Overview [time.zone.leap.overview]
30.11.8.2 Member functions [time.zone.leap.members]
30.11.8.3 Non-member functions [time.zone.leap.nonmembers]
30.11.9 Class time_ zone_ link[time.zone.link]
30.11.9.1 Overview [time.zone.link.overview]
30.11.9.2 Member functions [time.zone.link.members]
30.11.9.3 Non-member functions [time.zone.link.nonmembers]
30.12 Formatting [time.format]
30.13 Parsing [time.parse]
30.14 Hash support [time.hash]
30.15 Header synopsis [ctime.syn]
31 Input/output library [input.output][input.output]
31.1 General [input.output.general]
31.2 Iostreams requirements [iostreams.requirements]
31.2.1 Imbue limitations [iostream.limits.imbue]
31.2.2 Types [stream.types]
31.2.3 Positioning type limitations [iostreams.limits.pos]
31.2.4 Thread safety [iostreams.threadsafety]
31.3 Forward declarations [iostream.forward]
31.3.1 Header synopsis [iosfwd.syn]
31.3.2 Overview [iostream.forward.overview]
31.4 Standard iostream objects [iostream.objects]
31.4.1 Header synopsis [iostream.syn]
31.4.2 Overview [iostream.objects.overview]
31.4.3 Narrow stream objects [narrow.stream.objects]
31.4.4 Wide stream objects [wide.stream.objects]
31.5 Iostreams base classes [iostreams.base]
31.5.1 Header synopsis [ios.syn]
31.5.2 Class ios_ base[ios.base]
31.5.2.1 General [ios.base.general]
31.5.2.2 Types [ios.types]
31.5.2.2.1 Class ios_ base::failure[ios.failure]
31.5.2.2.2 Type ios_ base::fmtflags[ios.fmtflags]
31.5.2.2.3 Type ios_ base::iostate[ios.iostate]
31.5.2.2.4 Type ios_ base::openmode[ios.openmode]
31.5.2.2.5 Type ios_ base::seekdir[ios.seekdir]
31.5.2.2.6 Class ios_ base::Init[ios.init]
31.5.2.3 State functions [fmtflags.state]
31.5.2.4 Functions [ios.base.locales]
31.5.2.5 Static members [ios.members.static]
31.5.2.6 Storage functions [ios.base.storage]
31.5.2.7 Callbacks [ios.base.callback]
31.5.2.8 Constructors and destructor [ios.base.cons]
31.5.3 Class template fpos[fpos]
31.5.3.1 General [fpos.general]
31.5.3.2 Members [fpos.members]
31.5.3.3 Requirements [fpos.operations]
31.5.4 Class template basic_ ios[ios]
31.5.4.1 Overview [ios.overview]
31.5.4.2 Constructors [basic.ios.cons]
31.5.4.3 Member functions [basic.ios.members]
31.5.4.4 Flags functions [iostate.flags]
31.5.5ios_ base manipulators [std.ios.manip]
31.5.5.1fmtflags manipulators [fmtflags.manip]
31.5.5.2adjustfield manipulators [adjustfield.manip]
31.5.5.3basefield manipulators [basefield.manip]
31.5.5.4floatfield manipulators [floatfield.manip]
31.5.6 Error reporting [error.reporting]
31.6 Stream buffers [stream.buffers]
31.6.1 Header synopsis [streambuf.syn]
31.6.2 Stream buffer requirements [streambuf.reqts]
31.6.3 Class template basic_ streambuf[streambuf]
31.6.3.1 General [streambuf.general]
31.6.3.2 Constructors [streambuf.cons]
31.6.3.3 Public member functions [streambuf.members]
31.6.3.3.1 Locales [streambuf.locales]
31.6.3.3.2 Buffer management and positioning [streambuf.buffer]
31.6.3.3.3 Get area [streambuf.pub.get]
31.6.3.3.4 Putback [streambuf.pub.pback]
31.6.3.3.5 Put area [streambuf.pub.put]
31.6.3.4 Protected member functions [streambuf.protected]
31.6.3.4.1 Assignment [streambuf.assign]
31.6.3.4.2 Get area access [streambuf.get.area]
31.6.3.4.3 Put area access [streambuf.put.area]
31.6.3.5 Virtual functions [streambuf.virtuals]
31.6.3.5.1 Locales [streambuf.virt.locales]
31.6.3.5.2 Buffer management and positioning [streambuf.virt.buffer]
31.6.3.5.3 Get area [streambuf.virt.get]
31.6.3.5.4 Putback [streambuf.virt.pback]
31.6.3.5.5 Put area [streambuf.virt.put]
31.7 Formatting and manipulators [iostream.format]
31.7.1 Header synopsis [istream.syn]
31.7.2 Header synopsis [ostream.syn]
31.7.3 Header synopsis [iomanip.syn]
31.7.4 Header synopsis [print.syn]
31.7.5 Input streams [input.streams]
31.7.5.1 General [input.streams.general]
31.7.5.2 Class template basic_ istream[istream]
31.7.5.2.1 General [istream.general]
31.7.5.2.2 Constructors [istream.cons]
31.7.5.2.3 Assignment and swap [istream.assign]
31.7.5.2.4 Class basic_ istream::sentry[istream.sentry]
31.7.5.3 Formatted input functions [istream.formatted]
31.7.5.3.1 Common requirements [istream.formatted.reqmts]
31.7.5.3.2 Arithmetic extractors [istream.formatted.arithmetic]
31.7.5.3.3basic_ istream::operator>>[istream.extractors]
31.7.5.4 Unformatted input functions [istream.unformatted]
31.7.5.5 Standard basic_ istream manipulators [istream.manip]
31.7.5.6 Rvalue stream extraction [istream.rvalue]
31.7.5.7 Class template basic_ iostream[iostreamclass]
31.7.5.7.1 General [iostreamclass.general]
31.7.5.7.2 Constructors [iostream.cons]
31.7.5.7.3 Destructor [iostream.dest]
31.7.5.7.4 Assignment and swap [iostream.assign]
31.7.6 Output streams [output.streams]
31.7.6.1 General [output.streams.general]
31.7.6.2 Class template basic_ ostream[ostream]
31.7.6.2.1 General [ostream.general]
31.7.6.2.2 Constructors [ostream.cons]
31.7.6.2.3 Assignment and swap [ostream.assign]
31.7.6.2.4 Class basic_ ostream::sentry[ostream.sentry]
31.7.6.2.5 Seek members [ostream.seeks]
31.7.6.3 Formatted output functions [ostream.formatted]
31.7.6.3.1 Common requirements [ostream.formatted.reqmts]
31.7.6.3.2 Arithmetic inserters [ostream.inserters.arithmetic]
31.7.6.3.3basic_ ostream::operator<<[ostream.inserters]
31.7.6.3.4 Character inserter function templates [ostream.inserters.character]
31.7.6.3.5 Print [ostream.formatted.print]
31.7.6.4 Unformatted output functions [ostream.unformatted]
31.7.6.5 Standard basic_ ostream manipulators [ostream.manip]
31.7.6.6 Rvalue stream insertion [ostream.rvalue]
31.7.7 Standard manipulators [std.manip]
31.7.8 Extended manipulators [ext.manip]
31.7.9 Quoted manipulators [quoted.manip]
31.7.10 Print functions [print.fun]
31.8 String-based streams [string.streams]
31.8.1 Header synopsis [sstream.syn]
31.8.2 Class template basic_ stringbuf[stringbuf]
31.8.2.1 General [stringbuf.general]
31.8.2.2 Constructors [stringbuf.cons]
31.8.2.3 Assignment and swap [stringbuf.assign]
31.8.2.4 Member functions [stringbuf.members]
31.8.2.5 Overridden virtual functions [stringbuf.virtuals]
31.8.3 Class template basic_ istringstream[istringstream]
31.8.3.1 General [istringstream.general]
31.8.3.2 Constructors [istringstream.cons]
31.8.3.3 Swap [istringstream.swap]
31.8.3.4 Member functions [istringstream.members]
31.8.4 Class template basic_ ostringstream[ostringstream]
31.8.4.1 General [ostringstream.general]
31.8.4.2 Constructors [ostringstream.cons]
31.8.4.3 Swap [ostringstream.swap]
31.8.4.4 Member functions [ostringstream.members]
31.8.5 Class template basic_ stringstream[stringstream]
31.8.5.1 General [stringstream.general]
31.8.5.2 Constructors [stringstream.cons]
31.8.5.3 Swap [stringstream.swap]
31.8.5.4 Member functions [stringstream.members]
31.9 Span-based streams [span.streams]
31.9.1 Overview [span.streams.overview]
31.9.2 Header synopsis [spanstream.syn]
31.9.3 Class template basic_ spanbuf[spanbuf]
31.9.3.1 General [spanbuf.general]
31.9.3.2 Constructors [spanbuf.cons]
31.9.3.3 Assignment and swap [spanbuf.assign]
31.9.3.4 Member functions [spanbuf.members]
31.9.3.5 Overridden virtual functions [spanbuf.virtuals]
31.9.4 Class template basic_ ispanstream[ispanstream]
31.9.4.1 General [ispanstream.general]
31.9.4.2 Constructors [ispanstream.cons]
31.9.4.3 Swap [ispanstream.swap]
31.9.4.4 Member functions [ispanstream.members]
31.9.5 Class template basic_ ospanstream[ospanstream]
31.9.5.1 General [ospanstream.general]
31.9.5.2 Constructors [ospanstream.cons]
31.9.5.3 Swap [ospanstream.swap]
31.9.5.4 Member functions [ospanstream.members]
31.9.6 Class template basic_ spanstream[spanstream]
31.9.6.1 General [spanstream.general]
31.9.6.2 Constructors [spanstream.cons]
31.9.6.3 Swap [spanstream.swap]
31.9.6.4 Member functions [spanstream.members]
31.10 File-based streams [file.streams]
31.10.1 Header synopsis [fstream.syn]
31.10.2 Native handles [file.native]
31.10.3 Class template basic_ filebuf[filebuf]
31.10.3.1 General [filebuf.general]
31.10.3.2 Constructors [filebuf.cons]
31.10.3.3 Assignment and swap [filebuf.assign]
31.10.3.4 Member functions [filebuf.members]
31.10.3.5 Overridden virtual functions [filebuf.virtuals]
31.10.4 Class template basic_ ifstream[ifstream]
31.10.4.1 General [ifstream.general]
31.10.4.2 Constructors [ifstream.cons]
31.10.4.3 Swap [ifstream.swap]
31.10.4.4 Member functions [ifstream.members]
31.10.5 Class template basic_ ofstream[ofstream]
31.10.5.1 General [ofstream.general]
31.10.5.2 Constructors [ofstream.cons]
31.10.5.3 Swap [ofstream.swap]
31.10.5.4 Member functions [ofstream.members]
31.10.6 Class template basic_ fstream[fstream]
31.10.6.1 General [fstream.general]
31.10.6.2 Constructors [fstream.cons]
31.10.6.3 Swap [fstream.swap]
31.10.6.4 Member functions [fstream.members]
31.11 Synchronized output streams [syncstream]
31.11.1 Header synopsis [syncstream.syn]
31.11.2 Class template basic_ syncbuf[syncstream.syncbuf]
31.11.2.1 Overview [syncstream.syncbuf.overview]
31.11.2.2 Construction and destruction [syncstream.syncbuf.cons]
31.11.2.3 Assignment and swap [syncstream.syncbuf.assign]
31.11.2.4 Member functions [syncstream.syncbuf.members]
31.11.2.5 Overridden virtual functions [syncstream.syncbuf.virtuals]
31.11.2.6 Specialized algorithms [syncstream.syncbuf.special]
31.11.3 Class template basic_ osyncstream[syncstream.osyncstream]
31.11.3.1 Overview [syncstream.osyncstream.overview]
31.11.3.2 Construction and destruction [syncstream.osyncstream.cons]
31.11.3.3 Member functions [syncstream.osyncstream.members]
31.12 File systems [filesystems]
31.12.1 General [fs.general]
31.12.2 Conformance [fs.conformance]
31.12.2.1 General [fs.conformance.general]
31.12.2.2 POSIX conformance [fs.conform.9945]
31.12.2.3 Operating system dependent behavior conformance [fs.conform.os]
31.12.2.4 File system race behavior [fs.race.behavior]
31.12.3 Requirements [fs.req]
31.12.4 Header synopsis [fs.filesystem.syn]
31.12.5 Error reporting [fs.err.report]
31.12.6 Class path[fs.class.path]
31.12.6.1 General [fs.class.path.general]
31.12.6.2 Generic pathname format [fs.path.generic]
31.12.6.3 Conversions [fs.path.cvt]
31.12.6.3.1 Argument format conversions [fs.path.fmt.cvt]
31.12.6.3.2 Type and encoding conversions [fs.path.type.cvt]
31.12.6.4 Requirements [fs.path.req]
31.12.6.5 Members [fs.path.member]
31.12.6.5.1 Constructors [fs.path.construct]
31.12.6.5.2 Assignments [fs.path.assign]
31.12.6.5.3 Appends [fs.path.append]
31.12.6.5.4 Concatenation [fs.path.concat]
31.12.6.5.5 Modifiers [fs.path.modifiers]
31.12.6.5.6 Native format observers [fs.path.native.obs]
31.12.6.5.7 Generic format observers [fs.path.generic.obs]
31.12.6.5.8 Compare [fs.path.compare]
31.12.6.5.9 Decomposition [fs.path.decompose]
31.12.6.5.10 Query [fs.path.query]
31.12.6.5.11 Generation [fs.path.gen]
31.12.6.6 Iterators [fs.path.itr]
31.12.6.7 Inserter and extractor [fs.path.io]
31.12.6.8 Non-member functions [fs.path.nonmember]
31.12.6.9 Formatting support [fs.path.fmtr]
31.12.6.9.1 Formatting support overview [fs.path.fmtr.general]
31.12.6.9.2 Formatting support functions [fs.path.fmtr.funcs]
31.12.6.10 Hash support [fs.path.hash]
31.12.7 Class filesystem_ error[fs.class.filesystem.error]
31.12.7.1 General [fs.class.filesystem.error.general]
31.12.7.2 Members [fs.filesystem.error.members]
31.12.8 Enumerations [fs.enum]
31.12.8.1 Enum path::format[fs.enum.path.format]
31.12.8.2 Enum class file_ type[fs.enum.file.type]
31.12.8.3 Enum class copy_ options[fs.enum.copy.opts]
31.12.8.4 Enum class perms[fs.enum.perms]
31.12.8.5 Enum class perm_ options[fs.enum.perm.opts]
31.12.8.6 Enum class directory_ options[fs.enum.dir.opts]
31.12.9 Class file_ status[fs.class.file.status]
31.12.9.1 General [fs.class.file.status.general]
31.12.9.2 Constructors [fs.file.status.cons]
31.12.9.3 Observers [fs.file.status.obs]
31.12.9.4 Modifiers [fs.file.status.mods]
31.12.10 Class directory_ entry[fs.class.directory.entry]
31.12.10.1 General [fs.class.directory.entry.general]
31.12.10.2 Constructors [fs.dir.entry.cons]
31.12.10.3 Modifiers [fs.dir.entry.mods]
31.12.10.4 Observers [fs.dir.entry.obs]
31.12.10.5 Inserter [fs.dir.entry.io]
31.12.11 Class directory_ iterator[fs.class.directory.iterator]
31.12.11.1 General [fs.class.directory.iterator.general]
31.12.11.2 Members [fs.dir.itr.members]
31.12.11.3 Non-member functions [fs.dir.itr.nonmembers]
31.12.12 Class recursive_ directory_ iterator[fs.class.rec.dir.itr]
31.12.12.1 General [fs.class.rec.dir.itr.general]
31.12.12.2 Members [fs.rec.dir.itr.members]
31.12.12.3 Non-member functions [fs.rec.dir.itr.nonmembers]
31.12.13 Filesystem operation functions [fs.op.funcs]
31.12.13.1 General [fs.op.funcs.general]
31.12.13.2 Absolute [fs.op.absolute]
31.12.13.3 Canonical [fs.op.canonical]
31.12.13.4 Copy [fs.op.copy]
31.12.13.5 Copy file [fs.op.copy.file]
31.12.13.6 Copy symlink [fs.op.copy.symlink]
31.12.13.7 Create directories [fs.op.create.directories]
31.12.13.8 Create directory [fs.op.create.directory]
31.12.13.9 Create directory symlink [fs.op.create.dir.symlk]
31.12.13.10 Create hard link [fs.op.create.hard.lk]
31.12.13.11 Create symlink [fs.op.create.symlink]
31.12.13.12 Current path [fs.op.current.path]
31.12.13.13 Equivalent [fs.op.equivalent]
31.12.13.14 Exists [fs.op.exists]
31.12.13.15 File size [fs.op.file.size]
31.12.13.16 Hard link count [fs.op.hard.lk.ct]
31.12.13.17 Is block file [fs.op.is.block.file]
31.12.13.18 Is character file [fs.op.is.char.file]
31.12.13.19 Is directory [fs.op.is.directory]
31.12.13.20 Is empty [fs.op.is.empty]
31.12.13.21 Is fifo [fs.op.is.fifo]
31.12.13.22 Is other [fs.op.is.other]
31.12.13.23 Is regular file [fs.op.is.regular.file]
31.12.13.24 Is socket [fs.op.is.socket]
31.12.13.25 Is symlink [fs.op.is.symlink]
31.12.13.26 Last write time [fs.op.last.write.time]
31.12.13.27 Permissions [fs.op.permissions]
31.12.13.28 Proximate [fs.op.proximate]
31.12.13.29 Read symlink [fs.op.read.symlink]
31.12.13.30 Relative [fs.op.relative]
31.12.13.31 Remove [fs.op.remove]
31.12.13.32 Remove all [fs.op.remove.all]
31.12.13.33 Rename [fs.op.rename]
31.12.13.34 Resize file [fs.op.resize.file]
31.12.13.35 Space [fs.op.space]
31.12.13.36 Status [fs.op.status]
31.12.13.37 Status known [fs.op.status.known]
31.12.13.38 Symlink status [fs.op.symlink.status]
31.12.13.39 Temporary directory path [fs.op.temp.dir.path]
31.12.13.40 Weakly canonical [fs.op.weakly.canonical]
31.13 C library files [c.files]
31.13.1 Header synopsis [cstdio.syn]
31.13.2 Header synopsis [cinttypes.syn]
32 Concurrency support library [thread][thread]
32.1 General [thread.general]
32.2 Requirements [thread.req]
32.2.1 Template parameter names [thread.req.paramname]
32.2.2 Exceptions [thread.req.exception]
32.2.3 Native handles [thread.req.native]
32.2.4 Timing specifications [thread.req.timing]
32.2.5 Requirements for Cpp17Lockable types [thread.req.lockable]
32.2.5.1 General [thread.req.lockable.general]
32.2.5.2Cpp17BasicLockable requirements [thread.req.lockable.basic]
32.2.5.3Cpp17Lockable requirements [thread.req.lockable.req]
32.2.5.4Cpp17TimedLockable requirements [thread.req.lockable.timed]
32.2.5.5Cpp17SharedLockable requirements [thread.req.lockable.shared]
32.2.5.6Cpp17SharedTimedLockable requirements [thread.req.lockable.shared.timed]
32.3 Stop tokens [thread.stoptoken]
32.3.1 Introduction [thread.stoptoken.intro]
32.3.2 Header synopsis [thread.stoptoken.syn]
32.3.3 Stop token concepts [stoptoken.concepts]
32.3.4 Class stop_ token[stoptoken]
32.3.4.1 General [stoptoken.general]
32.3.4.2 Member functions [stoptoken.mem]
32.3.5 Class stop_ source[stopsource]
32.3.5.1 General [stopsource.general]
32.3.5.2 Constructors, copy, and assignment [stopsource.cons]
32.3.5.3 Member functions [stopsource.mem]
32.3.6 Class template stop_ callback[stopcallback]
32.3.6.1 General [stopcallback.general]
32.3.6.2 Constructors and destructor [stopcallback.cons]
32.3.7 Class never_ stop_ token[stoptoken.never]
32.3.8 Class inplace_ stop_ token[stoptoken.inplace]
32.3.8.1 General [stoptoken.inplace.general]
32.3.8.2 Member functions [stoptoken.inplace.mem]
32.3.9 Class inplace_ stop_ source[stopsource.inplace]
32.3.9.1 General [stopsource.inplace.general]
32.3.9.2 Constructors [stopsource.inplace.cons]
32.3.9.3 Member functions [stopsource.inplace.mem]
32.3.10 Class template inplace_ stop_ callback[stopcallback.inplace]
32.3.10.1 General [stopcallback.inplace.general]
32.3.10.2 Constructors and destructor [stopcallback.inplace.cons]
32.4 Threads [thread.threads]
32.4.1 General [thread.threads.general]
32.4.2 Header synopsis [thread.syn]
32.4.3 Class thread[thread.thread.class]
32.4.3.1 General [thread.thread.class.general]
32.4.3.2 Class thread::id[thread.thread.id]
32.4.3.3 Constructors [thread.thread.constr]
32.4.3.4 Destructor [thread.thread.destr]
32.4.3.5 Assignment [thread.thread.assign]
32.4.3.6 Members [thread.thread.member]
32.4.3.7 Static members [thread.thread.static]
32.4.3.8 Specialized algorithms [thread.thread.algorithm]
32.4.4 Class jthread[thread.jthread.class]
32.4.4.1 General [thread.jthread.class.general]
32.4.4.2 Constructors, move, and assignment [thread.jthread.cons]
32.4.4.3 Members [thread.jthread.mem]
32.4.4.4 Stop token handling [thread.jthread.stop]
32.4.4.5 Specialized algorithms [thread.jthread.special]
32.4.4.6 Static members [thread.jthread.static]
32.4.5 Namespace this_ thread[thread.thread.this]
32.5 Atomic operations [atomics]
32.5.1 General [atomics.general]
32.5.2 Header synopsis [atomics.syn]
32.5.3 Type aliases [atomics.alias]
32.5.4 Order and consistency [atomics.order]
32.5.5 Lock-free property [atomics.lockfree]
32.5.6 Waiting and notifying [atomics.wait]
32.5.7 Class template atomic_ ref[atomics.ref.generic]
32.5.7.1 General [atomics.ref.generic.general]
32.5.7.2 Operations [atomics.ref.ops]
32.5.7.3 Specializations for integral types [atomics.ref.int]
32.5.7.4 Specializations for floating-point types [atomics.ref.float]
32.5.7.5 Partial specialization for pointers [atomics.ref.pointer]
32.5.7.6 Member operators common to integers and pointers to objects [atomics.ref.memop]
32.5.8 Class template atomic[atomics.types.generic]
32.5.8.1 General [atomics.types.generic.general]
32.5.8.2 Operations on atomic types [atomics.types.operations]
32.5.8.3 Specializations for integers [atomics.types.int]
32.5.8.4 Specializations for floating-point types [atomics.types.float]
32.5.8.5 Partial specialization for pointers [atomics.types.pointer]
32.5.8.6 Member operators common to integers and pointers to objects [atomics.types.memop]
32.5.8.7 Partial specializations for smart pointers [util.smartptr.atomic]
32.5.8.7.1 General [util.smartptr.atomic.general]
32.5.8.7.2 Partial specialization for shared_ ptr[util.smartptr.atomic.shared]
32.5.8.7.3 Partial specialization for weak_ ptr[util.smartptr.atomic.weak]
32.5.9 Non-member functions [atomics.nonmembers]
32.5.10 Flag type and operations [atomics.flag]
32.5.11 Fences [atomics.fences]
32.5.12 C compatibility [stdatomic.h.syn]
32.6 Mutual exclusion [thread.mutex]
32.6.1 General [thread.mutex.general]
32.6.2 Header synopsis [mutex.syn]
32.6.3 Header synopsis [shared.mutex.syn]
32.6.4 Mutex requirements [thread.mutex.requirements]
32.6.4.1 General [thread.mutex.requirements.general]
32.6.4.2 Mutex types [thread.mutex.requirements.mutex]
32.6.4.2.1 General [thread.mutex.requirements.mutex.general]
32.6.4.2.2 Class mutex[thread.mutex.class]
32.6.4.2.3 Class recursive_ mutex[thread.mutex.recursive]
32.6.4.3 Timed mutex types [thread.timedmutex.requirements]
32.6.4.3.1 General [thread.timedmutex.requirements.general]
32.6.4.3.2 Class timed_ mutex[thread.timedmutex.class]
32.6.4.3.3 Class recursive_ timed_ mutex[thread.timedmutex.recursive]
32.6.4.4 Shared mutex types [thread.sharedmutex.requirements]
32.6.4.4.1 General [thread.sharedmutex.requirements.general]
32.6.4.4.2 Class shared_ mutex[thread.sharedmutex.class]
32.6.4.5 Shared timed mutex types [thread.sharedtimedmutex.requirements]
32.6.4.5.1 General [thread.sharedtimedmutex.requirements.general]
32.6.4.5.2 Class shared_ timed_ mutex[thread.sharedtimedmutex.class]
32.6.5 Locks [thread.lock]
32.6.5.1 General [thread.lock.general]
32.6.5.2 Class template lock_ guard[thread.lock.guard]
32.6.5.3 Class template scoped_ lock[thread.lock.scoped]
32.6.5.4 Class template unique_ lock[thread.lock.unique]
32.6.5.4.1 General [thread.lock.unique.general]
32.6.5.4.2 Constructors, destructor, and assignment [thread.lock.unique.cons]
32.6.5.4.3 Locking [thread.lock.unique.locking]
32.6.5.4.4 Modifiers [thread.lock.unique.mod]
32.6.5.4.5 Observers [thread.lock.unique.obs]
32.6.5.5 Class template shared_ lock[thread.lock.shared]
32.6.5.5.1 General [thread.lock.shared.general]
32.6.5.5.2 Constructors, destructor, and assignment [thread.lock.shared.cons]
32.6.5.5.3 Locking [thread.lock.shared.locking]
32.6.5.5.4 Modifiers [thread.lock.shared.mod]
32.6.5.5.5 Observers [thread.lock.shared.obs]
32.6.6 Generic locking algorithms [thread.lock.algorithm]
32.6.7 Call once [thread.once]
32.6.7.1 Struct once_ flag[thread.once.onceflag]
32.6.7.2 Function call_ once[thread.once.callonce]
32.7 Condition variables [thread.condition]
32.7.1 General [thread.condition.general]
32.7.2 Header synopsis [condition.variable.syn]
32.7.3 Non-member functions [thread.condition.nonmember]
32.7.4 Class condition_ variable[thread.condition.condvar]
32.7.5 Class condition_ variable_ any[thread.condition.condvarany]
32.7.5.1 General [thread.condition.condvarany.general]
32.7.5.2 Noninterruptible waits [thread.condvarany.wait]
32.7.5.3 Interruptible waits [thread.condvarany.intwait]
32.8 Semaphore [thread.sema]
32.8.1 General [thread.sema.general]
32.8.2 Header synopsis [semaphore.syn]
32.8.3 Class template counting_ semaphore[thread.sema.cnt]
32.9 Coordination types [thread.coord]
32.9.1 General [thread.coord.general]
32.9.2 Latches [thread.latch]
32.9.2.1 General [thread.latch.general]
32.9.2.2 Header synopsis [latch.syn]
32.9.2.3 Class latch[thread.latch.class]
32.9.3 Barriers [thread.barrier]
32.9.3.1 General [thread.barrier.general]
32.9.3.2 Header synopsis [barrier.syn]
32.9.3.3 Class template barrier[thread.barrier.class]
32.10 Futures [futures]
32.10.1 Overview [futures.overview]
32.10.2 Header synopsis [future.syn]
32.10.3 Error handling [futures.errors]
32.10.4 Class future_ error[futures.future.error]
32.10.5 Shared state [futures.state]
32.10.6 Class template promise[futures.promise]
32.10.7 Class template future[futures.unique.future]
32.10.8 Class template shared_ future[futures.shared.future]
32.10.9 Function template async[futures.async]
32.10.10 Class template packaged_ task[futures.task]
32.10.10.1 General [futures.task.general]
32.10.10.2 Member functions [futures.task.members]
32.10.10.3 Globals [futures.task.nonmembers]
32.11 Safe reclamation [saferecl]
32.11.1 General [saferecl.general]
32.11.2 Read-copy update (RCU) [saferecl.rcu]
32.11.2.1 General [saferecl.rcu.general]
32.11.2.2 Header synopsis [rcu.syn]
32.11.2.3 Class template rcu_ obj_ base[saferecl.rcu.base]
32.11.2.4 Class rcu_ domain[saferecl.rcu.domain]
32.11.2.4.1 General [saferecl.rcu.domain.general]
32.11.2.4.2 Member functions [saferecl.rcu.domain.members]
32.11.2.4.3 Non-member functions [saferecl.rcu.domain.func]
32.11.3 Hazard pointers [saferecl.hp]
32.11.3.1 General [saferecl.hp.general]
32.11.3.2 Header synopsis [hazard.pointer.syn]
32.11.3.3 Class template hazard_ pointer_ obj_ base[saferecl.hp.base]
32.11.3.4 Class hazard_ pointer[saferecl.hp.holder]
32.11.3.4.1 General [saferecl.hp.holder.general]
32.11.3.4.2 Constructors, destructor, and assignment [saferecl.hp.holder.ctor]
32.11.3.4.3 Member functions [saferecl.hp.holder.mem]
32.11.3.4.4 Non-member functions [saferecl.hp.holder.nonmem]
33 Execution control library [exec][exec]
33.1 General [exec.general]
33.2 Queries and queryables [exec.queryable]
33.2.1 General [exec.queryable.general]
33.2.2queryable concept [exec.queryable.concept]
33.3 Asynchronous operations [exec.async.ops]
33.4 Header synopsis [execution.syn]
33.5 Queries [exec.queries]
33.5.1forwarding_ query[exec.fwd.env]
33.5.2get_ allocator[exec.get.allocator]
33.5.3get_ stop_ token[exec.get.stop.token]
33.5.4execution::get_ env[exec.get.env]
33.5.5execution::get_ domain[exec.get.domain]
33.5.6execution::get_ scheduler[exec.get.scheduler]
33.5.7execution::get_ delegation_ scheduler[exec.get.delegation.scheduler]
33.5.8execution::get_ forward_ progress_ guarantee[exec.get.fwd.progress]
33.5.9execution::get_ completion_ scheduler[exec.get.compl.sched]
33.5.10execution::get_ await_ completion_ adaptor[exec.get.await.adapt]
33.6 Schedulers [exec.sched]
33.7 Receivers [exec.recv]
33.7.1 Receiver concepts [exec.recv.concepts]
33.7.2execution::set_ value[exec.set.value]
33.7.3execution::set_ error[exec.set.error]
33.7.4execution::set_ stopped[exec.set.stopped]
33.8 Operation states [exec.opstate]
33.8.1 General [exec.opstate.general]
33.8.2execution::start[exec.opstate.start]
33.9 Senders [exec.snd]
33.9.1 General [exec.snd.general]
33.9.2 Exposition-only entities [exec.snd.expos]
33.9.3 Sender concepts [exec.snd.concepts]
33.9.4 Awaitable helpers [exec.awaitable]
33.9.5execution::default_ domain[exec.domain.default]
33.9.6execution::transform_ sender[exec.snd.transform]
33.9.7execution::transform_ env[exec.snd.transform.env]
33.9.8execution::apply_ sender[exec.snd.apply]
33.9.9execution::get_ completion_ signatures[exec.getcomplsigs]
33.9.10execution::connect[exec.connect]
33.9.11 Sender factories [exec.factories]
33.9.11.1execution::schedule[exec.schedule]
33.9.11.2execution::just, execution::just_ error, execution::just_ stopped[exec.just]
33.9.11.3execution::read_ env[exec.read.env]
33.9.12 Sender adaptors [exec.adapt]
33.9.12.1 General [exec.adapt.general]
33.9.12.2 Closure objects [exec.adapt.obj]
33.9.12.3execution::write_ env[exec.write.env]
33.9.12.4execution::unstoppable[exec.unstoppable]
33.9.12.5execution::starts_ on[exec.starts.on]
33.9.12.6execution::continues_ on[exec.continues.on]
33.9.12.7execution::schedule_ from[exec.schedule.from]
33.9.12.8execution::on[exec.on]
33.9.12.9execution::then, execution::upon_ error, execution::upon_ stopped[exec.then]
33.9.12.10execution::let_ value, execution::let_ error, execution::let_ stopped[exec.let]
33.9.12.11execution::bulk, execution::bulk_ chunked, and execution::bulk_ unchunked[exec.bulk]
33.9.12.12execution::when_ all[exec.when.all]
33.9.12.13execution::into_ variant[exec.into.variant]
33.9.12.14execution::stopped_ as_ optional[exec.stopped.opt]
33.9.12.15execution::stopped_ as_ error[exec.stopped.err]
33.9.12.16std::execution::associate[exec.associate]
33.9.12.17 Exposition-only std::execution::stop-when[exec.stop.when]
33.9.12.18std::execution::spawn_ future[exec.spawn.future]
33.9.13 Sender consumers [exec.consumers]
33.9.13.1this_ thread::sync_ wait[exec.sync.wait]
33.9.13.2this_ thread::sync_ wait_ with_ variant[exec.sync.wait.var]
33.9.13.3std::execution::spawn[exec.spawn]
33.10 Completion signatures [exec.cmplsig]
33.11 Queryable utilities [exec.envs]
33.11.1 Class template prop[exec.prop]
33.11.2 Class template env[exec.env]
33.12 Execution contexts [exec.ctx]
33.12.1execution::run_ loop[exec.run.loop]
33.12.1.1 General [exec.run.loop.general]
33.12.1.2 Associated types [exec.run.loop.types]
33.12.1.3 Constructor and destructor [exec.run.loop.ctor]
33.12.1.4 Member functions [exec.run.loop.members]
33.13 Coroutine utilities [exec.coro.util]
33.13.1execution::as_ awaitable[exec.as.awaitable]
33.13.2execution::with_ awaitable_ senders[exec.with.awaitable.senders]
33.13.3execution::affine_ on[exec.affine.on]
33.13.4execution::inline_ scheduler[exec.inline.scheduler]
33.13.5execution::task_ scheduler[exec.task.scheduler]
33.13.6execution::task[exec.task]
33.13.6.1task overview [task.overview]
33.13.6.2 Class template task[task.class]
33.13.6.3task members [task.members]
33.13.6.4Class template task::state[task.state]
33.13.6.5 Class task::promise_ type[task.promise]
33.14 Execution scope utilities [exec.scope]
33.14.1 Execution scope concepts [exec.scope.concepts]
33.14.2 Counting Scopes [exec.counting.scopes]
33.14.2.1 General [exec.counting.scopes.general]
33.14.2.2 Simple Counting Scope [exec.scope.simple.counting]
33.14.2.2.1 General [exec.scope.simple.counting.general]
33.14.2.2.2 Constructor and Destructor [exec.simple.counting.ctor]
33.14.2.2.3 Members [exec.simple.counting.mem]
33.14.2.2.4 Token [exec.simple.counting.token]
33.14.2.3 Counting Scope [exec.scope.counting]
33.15 Parallel scheduler [exec.par.scheduler]
33.16 Namespace system_ context_ replaceability[exec.sysctxrepl]
33.16.1 General [exec.sysctxrepl.general]
33.16.2query_ parallel_ scheduler_ backend[exec.sysctxrepl.query]
33.16.3 Class parallel_ scheduler_ backend[exec.sysctxrepl.psb]
Annex A (informative) Grammar summary [gram][gram]
A.1 General [gram.general]
A.2 Keywords [gram.key]
A.3 Lexical conventions [gram.lex]
A.4 Basics [gram.basic]
A.5 Expressions [gram.expr]
A.6 Statements [gram.stmt]
A.7 Declarations [gram.dcl]
A.8 Modules [gram.module]
A.9 Classes [gram.class]
A.10 Overloading [gram.over]
A.11 Templates [gram.temp]
A.12 Exception handling [gram.except]
A.13 Preprocessing directives [gram.cpp]
Annex B (informative) Implementation quantities [implimits][implimits]
Annex C (informative) Compatibility [diff][diff]
C.1 C++ and ISO C++ 2023 [diff.cpp23]
C.1.1 General [diff.cpp23.general]
C.1.2[lex]: lexical conventions [diff.cpp23.lex]
C.1.3[expr]: expressions [diff.cpp23.expr]
C.1.4[dcl]: declarations [diff.cpp23.dcl.dcl]
C.1.5[temp]: templates [diff.cpp23.temp]
C.1.6[library]: library introduction [diff.cpp23.library]
C.1.7[mem]: memory management library [diff.cpp23.mem]
C.1.8[containers]: containers library [diff.cpp23.containers]
C.1.9[strings]: strings library [diff.cpp23.strings]
C.1.10[input.output]: input/output library [diff.cpp23.io]
C.1.11[depr]: compatibility features [diff.cpp23.depr]
C.2 C++ and ISO C++ 2020 [diff.cpp20]
C.2.1 General [diff.cpp20.general]
C.2.2[lex]: lexical conventions [diff.cpp20.lex]
C.2.3[expr]: expressions [diff.cpp20.expr]
C.2.4[stmt]: statements [diff.cpp20.stmt]
C.2.5[dcl]: declarations [diff.cpp20.dcl]
C.2.6[temp]: templates [diff.cpp20.temp]
C.2.7[library]: library introduction [diff.cpp20.library]
C.2.8[concepts]: concepts library [diff.cpp20.concepts]
C.2.9[mem]: memory management library [diff.cpp20.memory]
C.2.10[utilities]: general utilities library [diff.cpp20.utilities]
C.2.11[strings]: strings library [diff.cpp20.strings]
C.2.12[containers]: containers library [diff.cpp20.containers]
C.2.13[thread]: concurrency support library [diff.cpp20.thread]
C.3 C++ and ISO C++ 2017 [diff.cpp17]
C.3.1 General [diff.cpp17.general]
C.3.2[lex]: lexical conventions [diff.cpp17.lex]
C.3.3[basic]: basics [diff.cpp17.basic]
C.3.4[expr]: expressions [diff.cpp17.expr]
C.3.5[dcl]: declarations [diff.cpp17.dcl.dcl]
C.3.6[class]: classes [diff.cpp17.class]
C.3.7[over]: overloading [diff.cpp17.over]
C.3.8[temp]: templates [diff.cpp17.temp]
C.3.9[except]: exception handling [diff.cpp17.except]
C.3.10[library]: library introduction [diff.cpp17.library]
C.3.11[containers]: containers library [diff.cpp17.containers]
C.3.12[iterators]: iterators library [diff.cpp17.iterators]
C.3.13[algorithms]: algorithms library [diff.cpp17.alg.reqs]
C.3.14[input.output]: input/output library [diff.cpp17.input.output]
C.3.15[depr]: compatibility features [diff.cpp17.depr]
C.4 C++ and ISO C++ 2014 [diff.cpp14]
C.4.1 General [diff.cpp14.general]
C.4.2[lex]: lexical conventions [diff.cpp14.lex]
C.4.3[expr]: expressions [diff.cpp14.expr]
C.4.4[dcl]: declarations [diff.cpp14.dcl.dcl]
C.4.5[class]: classes [diff.cpp14.class]
C.4.6[temp]: templates [diff.cpp14.temp]
C.4.7[except]: exception handling [diff.cpp14.except]
C.4.8[library]: library introduction [diff.cpp14.library]
C.4.9[utilities]: general utilities library [diff.cpp14.utilities]
C.4.10[strings]: strings library [diff.cpp14.string]
C.4.11[containers]: containers library [diff.cpp14.containers]
C.4.12[depr]: compatibility features [diff.cpp14.depr]
C.5 C++ and ISO C++ 2011 [diff.cpp11]
C.5.1 General [diff.cpp11.general]
C.5.2[lex]: lexical conventions [diff.cpp11.lex]
C.5.3[basic]: basics [diff.cpp11.basic]
C.5.4[expr]: expressions [diff.cpp11.expr]
C.5.5[dcl]: declarations [diff.cpp11.dcl.dcl]
C.5.6[library]: library introduction [diff.cpp11.library]
C.5.7[input.output]: input/output library [diff.cpp11.input.output]
C.6 C++ and ISO C++ 2003 [diff.cpp03]
C.6.1 General [diff.cpp03.general]
C.6.2[lex]: lexical conventions [diff.cpp03.lex]
C.6.3[expr]: expressions [diff.cpp03.expr]
C.6.4[dcl]: declarations [diff.cpp03.dcl.dcl]
C.6.5[class]: classes [diff.cpp03.class]
C.6.6[temp]: templates [diff.cpp03.temp]
C.6.7[library]: library introduction [diff.cpp03.library]
C.6.8[support]: language support library [diff.cpp03.language.support]
C.6.9[diagnostics]: diagnostics library [diff.cpp03.diagnostics]
C.6.10[utilities]: general utilities library [diff.cpp03.utilities]
C.6.11[strings]: strings library [diff.cpp03.strings]
C.6.12[containers]: containers library [diff.cpp03.containers]
C.6.13[algorithms]: algorithms library [diff.cpp03.algorithms]
C.6.14[numerics]: numerics library [diff.cpp03.numerics]
C.6.15[localization]: localization library [diff.cpp03.locale]
C.6.16[input.output]: input/output library [diff.cpp03.input.output]
C.7 C++ and C [diff.iso]
C.7.1 General [diff.iso.general]
C.7.2[lex]: lexical conventions [diff.lex]
C.7.3[basic]: basics [diff.basic]
C.7.4[expr]: expressions [diff.expr]
C.7.5[stmt]: statements [diff.stat]
C.7.6[dcl]: declarations [diff.dcl]
C.7.7[class]: classes [diff.class]
C.7.8[cpp]: preprocessing directives [diff.cpp]
C.8 C standard library [diff.library]
C.8.1 General [diff.library.general]
C.8.2 Modifications to headers [diff.mods.to.headers]
C.8.3 Modifications to definitions [diff.mods.to.definitions]
C.8.3.1 Types char8_ t, char16_ t, and char32_ t[diff.char16]
C.8.3.2 Type wchar_ t[diff.wchar.t]
C.8.3.3 Header [diff.header.iso646.h]
C.8.3.4 Macro NULL[diff.null]
C.8.4 Modifications to declarations [diff.mods.to.declarations]
C.8.5 Modifications to behavior [diff.mods.to.behavior]
C.8.5.1 General [diff.mods.to.behavior.general]
C.8.5.2 Macro offsetof(type, member-designator)[diff.offsetof]
C.8.5.3 Memory allocation functions [diff.malloc]
Annex D (normative) Compatibility features [depr][depr]
D.1 General [depr.general]
D.2 Non-local use of TU-local entities [depr.local]
D.3 Implicit capture of this by reference [depr.capture.this]
D.4 Deprecated volatile types [depr.volatile.type]
D.5 Non-comma-separated ellipsis parameters [depr.ellipsis.comma]
D.6 Implicit declaration of copy functions [depr.impldec]
D.7 Redeclaration of static constexpr data members [depr.static.constexpr]
D.8 Literal operator function declarations using an identifier [depr.lit]
D.9template keyword before qualified names [depr.template.template]
D.10has_ denorm members in numeric_ limits[depr.numeric.limits.has.denorm]
D.11 Deprecated C macros [depr.c.macros]
D.12 Deprecated error numbers [depr.cerrno]
D.13 Deprecated type traits [depr.meta.types]
D.14 Relational operators [depr.relops]
D.15 Tuple [depr.tuple]
D.16 Variant [depr.variant]
D.17 Deprecated iterator class template [depr.iterator]
D.18 Deprecated move_ iterator access [depr.move.iter.elem]
D.19 Deprecated locale category facets [depr.locale.category]
D.20 Deprecated formatting [depr.format]
D.20.1 Header synopsis [depr.format.syn]
D.20.2 Formatting arguments [depr.format.arg]
D.21 Deprecated time formatting [depr.ctime]
D.22 Deprecated file systems [depr.filesystems]
D.22.1 Deprecated filesystem path factory functions [depr.fs.path.factory]
D.22.2 Deprecated filesystem path format observers [depr.fs.path.obs]
D.23 Deprecated atomic operations [depr.atomics]
D.23.1 General [depr.atomics.general]
D.23.2 Volatile access [depr.atomics.volatile]
D.23.3 Non-member functions [depr.atomics.nonmembers]
D.23.4 Operations on atomic types [depr.atomics.types.operations]
D.23.5memory_ order::consume[depr.atomics.order]
Annex E (informative) Conformance with UAX #31 [uaxid][uaxid]
E.1 General [uaxid.general]
E.2 R1 Default identifiers [uaxid.def]
E.2.1 General [uaxid.def.general]
E.2.2 R1b Stable identifiers [uaxid.def.stable]
E.3 R2 Immutable identifiers [uaxid.immutable]
E.4 R3 Pattern_White_Space and Pattern_Syntax characters [uaxid.pattern]
E.5 R4 Equivalent normalized identifiers [uaxid.eqn]
E.6 R5 Equivalent case-insensitive identifiers [uaxid.eqci]
E.7 R6 Filtered normalized identifiers [uaxid.filter]
E.8 R7 Filtered case-insensitive identifiers [uaxid.filterci]
E.9 R8 Hashtag identifiers [uaxid.hashtag]
Bibliography
Index
Index of grammar productions
Index of library headers
Index of library names
Index of library concepts
Index of implementation-defined behavior |
5798 | https://pmc.ncbi.nlm.nih.gov/articles/PMC12112657/ | Expert Consensus on the Use of Diphenhydramine for Short-Term Insomnia: Efficacy, Safety, and Clinical Applications - PMC
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J Clin Med
. 2025 May 9;14(10):3297. doi: 10.3390/jcm14103297
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Expert Consensus on the Use of Diphenhydramine for Short-Term Insomnia: Efficacy, Safety, and Clinical Applications
Daniel Felipe Ariza-Salamanca
Daniel Felipe Ariza-Salamanca
1 Department of Pharmacobiology, Center for Research and Advanced Studies (Cinvestav), National Polytechnic Institute, Mexico City 14300, Mexico; daniel.ariza@cinvestav.mx
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1, Marco Venegas
Marco Venegas
2 Clinic for the Study and Treatment of Sleep Disorders Somnarum, Sleep and Electrodiagnostics Unit, Bogota 111031, Colombia; marcoaureliovenegas@gmail.com
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2, Karem Parejo
Karem Parejo
3 Sleep Laboratory, Fundación Clínica Shaio, Bogota 110131, Colombia; karemparejo@gmail.com
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3, Steve Amado
Steve Amado
4 Maple Respiratory Colombia, Sleep Clinic, Bogota 111211, Colombia; steveamadog@gmail.com (S.A.); jechever@interco.net.co (J.E.)
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4, Jorge Echeverry
Jorge Echeverry
4 Maple Respiratory Colombia, Sleep Clinic, Bogota 111211, Colombia; steveamadog@gmail.com (S.A.); jechever@interco.net.co (J.E.)
5 Faculty of Health Sciences, Technological University of Pereira, Pereira 660003, Colombia
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4,5, Carlos Alberto Calderón-Ospina
Carlos Alberto Calderón-Ospina
6 Center for Research in Genetics and Genomics (CIGGUR), Institute of Translational Medicine (IMT), School of Medicine and Health Sciences, Universidad del Rosario, Bogotá 111221, Colombia
7 Research Group in Applied Biomedical Sciences (UR Biomed), School of Medicine and Health Sciences, Universidad del Rosario, Bogotá 111221, Colombia
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6,7,
Editor: Sanford Auerbach
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1 Department of Pharmacobiology, Center for Research and Advanced Studies (Cinvestav), National Polytechnic Institute, Mexico City 14300, Mexico; daniel.ariza@cinvestav.mx
2 Clinic for the Study and Treatment of Sleep Disorders Somnarum, Sleep and Electrodiagnostics Unit, Bogota 111031, Colombia; marcoaureliovenegas@gmail.com
3 Sleep Laboratory, Fundación Clínica Shaio, Bogota 110131, Colombia; karemparejo@gmail.com
4 Maple Respiratory Colombia, Sleep Clinic, Bogota 111211, Colombia; steveamadog@gmail.com (S.A.); jechever@interco.net.co (J.E.)
5 Faculty of Health Sciences, Technological University of Pereira, Pereira 660003, Colombia
6 Center for Research in Genetics and Genomics (CIGGUR), Institute of Translational Medicine (IMT), School of Medicine and Health Sciences, Universidad del Rosario, Bogotá 111221, Colombia
7 Research Group in Applied Biomedical Sciences (UR Biomed), School of Medicine and Health Sciences, Universidad del Rosario, Bogotá 111221, Colombia
Correspondence: carlos.calderon@urosario.edu.co
Roles
Sanford Auerbach: Academic Editor
Received 2025 Mar 12; Revised 2025 Apr 26; Accepted 2025 Apr 27; Collection date 2025 May.
© 2025 by the authors.
Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (
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PMCID: PMC12112657 PMID: 40429293
Abstract
Insomnia is the most prevalent sleep disorder, estimated to affect at least one-third of the global population. There are a variety of treatment options available for both acute and chronic insomnia. Currently, the pharmacological arsenal for treating insomnia includes short- or intermediate-acting benzodiazepine hypnotics, non-benzodiazepine hypnotic sedatives, melatonin receptor agonists, orexin receptor antagonist, and sedating antidepressants. Diphenhydramine, a first-generation antihistamine, is commonly used in the treatment of allergies and dermatitis. This review examines the preclinical and clinical efficacy and safety evidence of diphenhydramine in treating short-term insomnia. Additionally, it provides expert consensus on its implementation as an over-the-counter medication for this condition. The available evidence indicates that diphenhydramine is an effective treatment for acute insomnia in adults, offering a safe and affordable option for most patients suffering from this condition. Experts concur that there is strong evidence supporting the recommendation of diphenhydramine for the treatment of acute insomnia in adults.
Keywords: diphenhydramine, antihistamines, short-term insomnia, experts’ consensus
1. Introduction
Insomnia is characterized by difficulties in initiating or maintaining sleep, early morning awakenings, and dissatisfaction with the quality or quantity of sleep despite adequate opportunity to sleep. It is commonly associated with daytime impairments such as fatigue, mood disturbances, or cognitive difficulties . Insomnia is a prevalent sleep disorder worldwide . For instance, in the United States, a survey of 7428 adults revealed that nearly half reported difficulty sleeping, with an estimated prevalence of insomnia at 23.2% . In Latin America, a study conducted in four major cities—Montevideo, Mexico City, Santiago, and Caracas—examined 4533 participants and found a high prevalence of symptoms related to sleep disorders, including 34.7% diagnosed with insomnia . In Colombia, a study of 1325 women from diverse ethnic backgrounds reported that nearly one-third experienced insomnia .
It is recognized as a disease on its own since it significantly impairs quality of life, daytime functioning, and overall health. Therefore, it should be promptly treated once detected. Cognitive behavioral therapy for insomnia (CBT-I) is widely recognized as the first-line therapy . Pharmacotherapy is also a helpful tool for treating insomnia; it has demonstrated to improve significantly latency to sleep and awakenings . The choice of medication is a critical aspect of insomnia management, as tailoring treatment to the individual can optimize outcomes and minimize risks. While widely approved medications for insomnia, such as benzodiazepines and Z-drugs, are commonly prescribed, other effective options, like antihistamines, are not as broadly endorsed despite their potential benefits.
This article provides a comprehensive review of insomnia, focusing on the available literature regarding diphenhydramine for the short-term management of insomnia. It includes an expert consensus from various medical specialties on the use of diphenhydramine, with particular emphasis on its safety, efficacy, and special considerations in children and the elderly. Finally, it contrasts these findings with the existing literature, offering a nuanced perspective on the role of diphenhydramine in the treatment of short-term insomnia.
1.1. Pathophysiological and Clinical Aspects of Insomnia
There is no definitive consensus on the biological basis of insomnia. Nonetheless, hyperarousal is widely accepted as a primary biological mechanism. From a biological perspective, hyperarousal results from the overactivation of the ascending reticular activation systems and the hypothalamic–pituitary–adrenal (HPA) axis [8,9]. It manifests as an elevated heart rate, abnormal heart rate variability, altered core body temperature, and blunted reductions in the metabolic rate typically associated with non-rapid eye movement (non-REM) sleep .
The Spielman model posits that insomnia is influenced by three factors: predisposing factors (genetic, personality, or environmental traits that increase vulnerability), precipitating factors (acute events, such as trauma), and perpetuating factors (behavioral and cognitive patterns that sustain insomnia and lead to chronic forms) .
Insomnia is diagnosed clinically based on various criteria, including those outlined in the International Classification of Sleep Disorders, Third Edition (ICSD-3); the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5); and the International Classification of Diseases, Tenth Revision (ICD-11). According to the ICSD-3, insomnia is defined as a persistent difficulty in initiating or maintaining sleep, or waking earlier than desired, accompanied by resistance to going to bed at an appropriate time or inability to sleep without external intervention, despite having adequate opportunities and conditions for sleep .
The diagnosis of insomnia can be achieved through a comprehensive clinical evaluation that includes a detailed sleep history, focusing on the onset, duration, and frequency of symptoms. Current sleep–wake patterns, along with environmental and social factors, should be assessed. A thorough medical and psychiatric history is essential, as well as an evaluation of substance use and medication intake .
In addition to clinical diagnosis, there are supportive diagnostic tools that, while not strictly necessary, can provide valuable information. Sleep diaries are recommended, as they offer insights into sleep efficiency and patterns . Polysomnography (PSG) is particularly useful when other sleep disorders are suspected, as it can reveal findings characteristic of insomnia, such as disrupted sleep continuity . Actigraphy can also be helpful, as it provides objective data on sleep state misperception and paradoxical insomnia [15,16]. Furthermore, modern smart technologies, such as smartwatches, are increasingly used to monitor sleep patterns and can serve as practical tools to support the follow up of patients diagnosed with insomnia .
1.2. Treatment of Insomnia
Treatment includes both pharmacological and behavioral interventions . Behavioral interventions emphasize strict sleep hygiene and psychological approaches, such as CBT-i and sleep restriction therapy . Although CBT-i has been demonstrated to improve sleep, it is not widely available in all regions. Therefore, it is crucial for the primary care physician to understand what to teach patients presenting with insomnia. Sleep hygiene, body–mind therapy, meditations, mindfulness, and diaphragmatic breathing are also available interventions for managing insomnia . Primary care physicians should also be confident in the use of medications to manage insomnia.
Pharmacological options for treating insomnia include short- or intermediate-acting benzodiazepine (BDZ) hypnotics, non-benzodiazepine hypnotic sedatives (Z drugs), melatonin receptor agonists, orexin antagonists, and sedating antidepressants. The pharmacodynamics of these drugs can be categorized into GABAergic and non-GABAergic types. Most of them are detectable in plasma within 30 min of ingestion and have short to medium half-lives. As these drugs are hypnotic, they interact with other substances such as alcohol. Special populations, such as children and the elderly, must be considered when selecting a drug. Table 1 summarizes the main pharmacological features of currently used drugs for treating insomnia.
Table 1.
Pharmacological characteristics and clinical indications of common drugs used in insomnia.
| Drug | Pharmacological Action/Group | Dose | T max | Vd | t 1/2 | Metabolism/Elimination | Indication | Use in Special Populations (Caution/Avoid) |
:---: :---: :---: :---:
| Diphenhydramine | H1RA | 12.5–50 mg | 2–3 h | 3.3–6.8 L/kg | 2.4–9.3 h | First-pass; CYP450 isoenzymes/urine | Insomnia, allergies, nausea | Chronic liver disease, QT prolongation |
| Hydroxyzine | H1RA | 50–100 mg | 2 h | 16.0 ± 3.0 L/kg | 14–25 h | Liver; CYP3A4, CYP3A5/urine | Anxiety, pruritus, insomnia, allergies | Elderly, renal, and hepatic impairment |
| Quetiapine | D2/5-HT2A RA | 25–100 mg | 1.5 h | 10 ± 4 L/kg | 6–7 h | Liver; CYP3A4, CYP2D6/urine and feces | Psychiatric disorders, insomnia (low dose) | Young and elderly, QT prolongation |
| Levomepromazine | D2/H1/MRA | 5–25 mg | 1–2 h (est.) | 16 L/kg (est.) | ~20 h | Extensive first-pass; liver | Amnesia, nausea and vomiting, psychiatric disorders, insomnia (low doses) | Elderly |
| Temazepam | GABA-A PAM | 7.5–30 mg | 2–3 h | 1.3–1.5 L/kg | 3.5–18 h | Liver, conjugation/urine | Insomnia | Pregnancy (caution) |
| Triazolam | GABA-A PAM | 0.125–0.5 mg | 1–2 h (est.) | ~1 L/kg (est.) | 1.5–5.5 h | Liver, conjugation/urine | Insomnia | Elderly |
| Eszopiclone | GABA-A AG | 1–3 mg | 1 h | 89.9 L | 6.1 h | Liver, CYP3A, CYP2C8, CYP2E1/urine | Insomnia | Elderly |
| Zaleplon | GABA B Z | 5–20 mg/day | 1 h | 1.4 L/kg | 1 h | Aldehyde oxidase | Insomnia | Hepatic impairment |
| Zolpidem | GABA-A SA | 5–10 mg | 1.6 h | 0.54–0.68 L/kg | 2.5 h | Liver, CYP3A4, CYP1A2, CYP2C9/urine | Insomnia | Elderly, hepatic impairment |
| Amitriptyline | SERT/NETI | 10–100 mg | 2–12 h | 1221 ± 280 L | 24.65 ± 6.31 h | Liver, CYP2C19, CYP3A4, CYP2D6/urine | MDD, neuropathic pain, migraine, insomnia | Pregnancy, breastfeeding, QT prolongation |
| Trazodone | SERTI/5-HT2A RA | 25–200 mg | 8 h | 0.84 ± 0.16 L/kg | 7.3 ± 0.8 h | Liver, CYP3A4/urine | MDD, insomnia, anxiety | QT prolongation |
| Gabapentin | VGCC AI | 100–600 mg | 2–3 h | 58 ± 6 L | 5–7 h | Unchanged | Antiseizure, neuropathic pain, insomnia | Renal impairment |
| Melatonin | MT1/MT2 AG | 1–5 mg | Variable | ~1.2–1.5 L/kg (est.) | 35–50 min | Liver, various | Insomnia, circadian rhythm disorders | Elderly, pregnancy (caution) |
| Lemborexant | OX1R/OX2RA | 5–10 mg | 1–3 h | 1970 L | 17–19 h | Liver, CYP3A4 | Insomnia | Narcolepsy |
| Daridorexant | OX1R/OX2RA | 25–50 mg | 1.3 h | 31 L | 8 h | Liver, CYP3A4 | Insomnia | Narcolepsy |
| Suvorexant | OX1R/OX2RA | 10 mg | 2 h | 49 L | 12 h | Liver, CYP3A4, CYP2C19 | Delirium Prophylaxis, Insomnia | Narcolepsy, hepatic impairment |
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D2/5-HT2A RA: antagonist of the D2 dopamine receptors and the 5-HT2A serotonin receptors; D2/H1/MRA: antagonist of the D2 dopamine receptors, H1 histamine receptors, and muscarinic receptors (M); GABA-A AG: agonist of the GABA-A receptors; GABA-A PAM: positive allosteric modulator of the GABA-A receptors; GABA-A SA: selective agonist of the GABA-A receptors; H1RA: antagonist of the H1 histamine receptors; MT1/MT2 AG: agonist of the MT1 and MT2 melatonin receptors; SERT/NETI: inhibitor of the serotonin (SERT) and norepinephrine (NET) transporters; SERTI/5-HT2A RA: inhibitor of the serotonin transporter (SERT) and antagonist of the 5-HT2A serotonin receptor; OX1R/OX2RA: orexin receptor 1 and 2 antagonist. Vd: colume of distribution; VGCC AI: inhibitor of voltage-gated calcium channels. This table was generated mostly using DrugBank Open Data.
The use of diphenhydramine for short-term insomnia is a key focus of this consensus. The approval of diphenhydramine as an over-the-counter medication by the Food and Drug Administration, along with strong evidence supporting its effectiveness for short-term insomnia and the safety profile offered, has driven this research. Subsequent sections explore the available literature on the general pharmacological properties of diphenhydramine. To assess the accuracy and quality of the evidence, a consensus committee was assembled. Additionally, considerations are provided regarding its implementation as an over-the-counter treatment for short-term insomnia in Colombia.
2. Materials and Methods
2.1. Selection of Consensus Committee Members and Topics Being Assessed
A panel of five experts in sleep disorders was convened to participate in this consensus. CAC-O is a pharmacologist specializing in pharmacovigilance, with extensive experience in researching neurological disorders. MV and KJPG are neurologists specialized in sleep disorders, YSA is an otolaryngologist specializing in sleep disorders, and JEEC is a psychiatrist specializing in sleep disorders. Most of these experts are clinicians who regularly treat patients with various sleep disorders and have a deep understanding of central nervous system disorders, as well as high academic qualifications.
The objectives of this consensus were threefold: to gather the opinions of Latin American specialists on key issues related to the use of diphenhydramine in short-term insomnia, to thoroughly evaluate the available literature on the drug’s efficacy and safety, and to delineate the clinical scenarios in which diphenhydramine is or is not an optimal choice.
Each expert was assigned to review the selected literature, chosen through a quasi-systematic search. The reviewed studies were then carefully discussed, and a questionnaire was developed based on the most recent insomnia taxonomy. The key topics, selected in consultation with the committee, included drug effectiveness and safety, availability, patient age, treatment duration, and the level of evidence supporting each recommendation.
2.2. Literature Research
The literature search was carried out by scanning Medline for the Medical Subject Heading (MeSH) “Insomnia” AND “Diphenhydramine” via PubMed, which showed 115 results. There were no limitations in terms of the type and quality of studies, language, or provenance. The title and abstract of each article were screened for their relevance to the current approach by CA-O. The articles directly related to diphenhydramine and acute insomnia that were considered relevant were selected and reviewed. The date of the last PubMed literature search was 30 May 2024. Regarding the exclusion of 79 articles, which reduced the selection from 115 to 36, this was primarily since many of the initially retrieved articles were not directly related to the use of diphenhydramine for the management of insomnia. Instead, they focused on antihistamines in general or addressed indications for diphenhydramine unrelated to insomnia. A total of 36 articles were chosen to provide the evidence support for this consensus.
2.3. Consensus Workflow and Methods to Achieve Consensus
The Delphi methodology was followed to reach a consensus among a panel of experts on the topic . This process included the selection of a group of qualified experts, who participated in several rounds of questionnaires designed to collect their opinions and judgments. After each round, anonymous and summarized feedback was provided to the group, allowing the experts to review and adjust their responses based on the input from the collective. This iterative process continued until a consensus was reached in most of the questions. Finally, the results were analyzed to obtain consensual conclusions that reflected the general agreement of the panel.
Experts were assigned at least five articles each to review in preparation for an upcoming virtual meeting. During the meeting, different aspects of diphenhydramine were thoroughly assessed by all experts, considering the selected literature. A draft of the questionnaire was then reviewed and refined for proper taxonomy and syntax. A total of 12 questions were formulated (Box 1).
Box 1. Questionnaire regarding the use of diphenhydramine in short-term insomnia.
Chapter 1: Evaluation of the use of diphenhydramine for insomnia: efficacy, safety, convenience, and cost of diphenhydramine in the management of insomnia:
1.Diphenhydramine is an effective medication for the management of short-term insomnia.
2.Diphenhydramine is a safe medication for the management of short-term insomnia.
3.If diphenhydramine were available in the Colombian market, do you consider that this medication could be an accessible option for managing short-term insomnia?
4.Diphenhydramine is a convenient medication for most patients with acute insomnia, regardless of their comorbidities or clinical situations, and therefore, has the potential to be marketed as an over-the-counter medication for managing short-term insomnia.
Chapter 2: type(s) of insomnia where diphenhydramine could be used:
5.Diphenhydramine is a useful medication for short-term insomnia (less than 3 months in duration).
6.Diphenhydramine is a useful medication for chronic insomnia (more than 3 months in duration).
Chapter 3: use of diphenhydramine as a hypnotic/sedative by age group:
7.Diphenhydramine is an effective and safe medication for children and adolescents (7 to 17 years old) for managing short-term insomnia.
8.Diphenhydramine is an effective and safe medication for young adults (18 to 65 years old) for managing short-term insomnia.
9.Diphenhydramine is an effective and safe medication for elderly individuals (65 years and older) for managing short-term insomnia.
Chapter 4: duration of diphenhydramine treatment for managing insomnia:
10.The maximum recommended duration for using diphenhydramine as a hypnotic/sedative for short-term insomnia should be around four weeks.
Chapter 5: evidence and levels of evidence on the use of diphenhydramine for managing short-term insomnia:
11.There is a sufficient body of clinical evidence to recommend the use of diphenhydramine in patients with short-term insomnia.
12.There is a sufficient level of clinical evidence to recommend the use of diphenhydramine in patients with short-term insomnia.
The questionnaire was divided into five chapters. Chapter 1 focused on the evaluation of the use of diphenhydramine for insomnia, including its efficacy, safety, convenience, and cost in the management of insomnia. Chapter 2 addressed the types of insomnia in which diphenhydramine could be used. Chapter 3 examined the use of diphenhydramine as a hypnotic/sedative by age group. Chapter 4 considered the duration of diphenhydramine treatment for managing insomnia. Chapter 5 explored the evidence and levels of evidence on the use of diphenhydramine for managing short-term insomnia.
All questions were quantified by a Likert Scale, with 1= total disagreement, 2 = partial disagreement, 3 = neutral, 4 = partial agreement, and 5 = total agreement. An agreement of less than 60% of the votes was considered as no agreement, a supermajority between 60% and 74% was considered a weak agreement, a supermajority equal or greater than 75% as a strong agreement, and 100% as unanimous agreement. A consensus was defined as an interquartile range equal to or less than 1. The interquartile range (IQR) was calculated by finding the difference between the third quartile (Q3) and the first quartile (Q1) (IQR = Q3 − Q1). All the answers were treated anonymously. Statistical analyses and figures were performed using MATL–AB Online. Version R2022b (MathWorks, Natick, MA, USA).
3. Results and Discussion
In this section, we present the results of the literature review on various aspects of diphenhydramine use for short-term insomnia. Additionally, we provide an analysis of the frequency and agreement among experts regarding each question within the same dimensions explored in the literature.
3.1. Diphenhydramine Pharmacodynamics and Efficacy in Insomnia
Diphenhydramine is a first-generation antihistamine that was discovered in the 1940s . Since its introduction to the market, it has been widely used for the treatment of various allergic conditions, including allergic rhinitis, urticaria, and dermatitis . Diphenhydramine antagonizes the H1 histamine receptor. Histamine receptors have distinct roles and locations: H1 and H2 receptors are postsynaptic and excitatory, with H1 linked to phospholipase C and found in the hypothalamus and limbic regions, while H2 is coupled to adenylate cyclase and concentrated in the hippocampus, amygdala, and basal ganglia. In contrast, H3 receptors are presynaptic, inhibitory, and primarily located in the basal ganglia, regulating histamine and neurotransmitter release by inhibiting calcium channels .
H1 receptors are G-protein-coupled receptors (GPCRs) linked to the Gq pathway, which activates phospholipase C, leading to the inositol triphosphate (IP3) and diacylglycerol (DAG) signaling cascade, ultimately enhancing the neural activity. These receptors exhibit high basal activity and induce cortical desynchronization, a state associated with heightened brain activity and wakefulness [23,24]. Diphenhydramine acts as a negative allosteric modulator and inverse agonist of H1 receptors, blocking histamine’s action at these sites. By inhibiting histamine binding, it reduces neuronal excitation mediated by H1 receptors, decreasing the cortical activity and inducing sleepiness .
Diphenhydramine can easily cross the blood–brain barrier due to its lipophilic nature. Once in the brain, its sedative effect is enhanced by its ability to interact with other neurotransmitter systems other than the histaminergic. Although its affinity is lower than H1, diphenhydramine can also affect muscarinic acetylcholine receptors, which also play a role in regulating the sleep–wake cycle. The blockade of these receptors contributes to the secondary sedative effect .
For instance, Carruthers et al. demonstrated in healthy volunteers that a 50 mg dose of diphenhydramine exhibited a positive correlation between the plasma concentration and sedative effects . Similarly, Roth et al. compared the effects of diphenhydramine (50 mg TID) and loratadine (10 mg and 40 mg) in 16 healthy adults. Diphenhydramine significantly reduced sleep latency, but was associated with an impaired daytime performance. However, it is worth noting that diphenhydramine was used at high doses in this study and the patients took the medication every 8 h (including two daytime doses) instead of taking the medication at night, before going to sleep . Moreover, Boberly et al. recruited healthy young adult volunteers who received 50 to 75 mg of diphenhydramine. Self-reported sleep latency showed mild hypnotic effects, with no significant differences in subjective sleep parameters, and no deterioration in the psychomotor performance or rebound insomnia . Similar findings have been reported in healthy individuals [29,30].
Going deeper into diphenhydramine evidence for insomnia, Rickels et al. conducted a double-blind, crossover study to evaluate the effect of diphenhydramine on insomnia in adults. They compared 50 mg of diphenhydramine with a placebo in 111 patients with mild to moderate insomnia. Significant improvements were observed in sleep latency and restfulness with diphenhydramine. Furthermore, the authors recommended diphenhydramine as an over-the-counter sleep aid in the treatment of temporary mild to moderate insomnia . Accordingly, Morin et al. reported improvements in subjective sleep parameters and increased sleep efficiency in 184 patients with mild insomnia who received 50 mg of diphenhydramine twice daily . Schweitzer et al. compared drowsiness and performance levels between two antihistamines, diphenhydramine, and cetirizine. This study administered 50 mg of diphenhydramine, 10 mg/day of cetirizine, or a placebo three times daily for three days. Twelve atopic subjects received each treatment in a double-blind Latin square design. The main findings indicated that diphenhydramine, unlike cetirizine, caused acute decreases in alertness and performance. However, tolerance to its sedative effects developed by day three, suggesting that diphenhydramine may be useful for short-term insomnia .
Regarding studies evaluating the efficacy of diphenhydramine in different populations, in the elderly, Teutsch et al. compared the hypnotic effects of diphenhydramine and methapyrilene with those of pentobarbital in hospitalized veterans. The main findings indicated that 50 mg or 150 mg of diphenhydramine were no more effective than 60 mg of pentobarbital in treating insomnia, meaning diphenhydramine was no different to pentobarbital to induce sleep . Similar findings were obtained by Glass et al. . Furthermore, Stewart et al. conducted a randomized, double-blind, crossover clinical trial to test 50 mg of diphenhydramine and 15 mg of temazepam. The main results showed that diphenhydramine reduced sleep latency more effectively than the placebo, provided a longer sleep duration than temazepam on the fifth night, and both temazepam and diphenhydramine were associated with residual daytime drowsiness .
In the case of diphenhydramine use in children with sleep disorders, Russo et al. conducted a double-blind, placebo-controlled trial of diphenhydramine at 10 mg/kg. The main findings showed that diphenhydramine produced a significant reduction in sleep latency and night awakenings, with a marginal increase in sleep duration. Additionally, global weekly evaluations of the daytime performance favored diphenhydramine over the placebo .
Furthermore, Sunshine et al. evaluated the sedative effect of diphenhydramine in a group of 1295 postpartum women with sleep problems through a controlled, double-blind study. The patients were assigned to receive an oral dose of diphenhydramine hydrochloride (12.5, 25, or 50 mg), mepirizole fumarate (36, 72, or 144 mg), or a placebo. The hypnotic activity was clinically evaluated using both subjective and objective techniques. It was found that both mepirizole and diphenhydramine, at all doses, were effective hypnotics compared to the placebo, based on sleep latency, sleep duration, night-time awakenings, a global assessment, and morning alertness. Although a dose–response relationship was documented, it was also concluded that increasing the dose of these medications within the studied range produced only a minimal increase in efficacy .
Moreover, Kudo and Kurihara conducted a double-blind study of 144 psychiatric patients with insomnia, where diphenhydramine hydrochloride at doses of 12.5, 25, and 50 mg demonstrated clinical improvement in over 60% of patients after two weeks of treatment. Side effects were reported in 7.6% of participants but were mild, and no signs of drug dependence were observed. Treatment effectiveness was greater in patients without prior insomnia therapy, with a dose-dependent increase in hypnotic effects in this subgroup. These results suggest that diphenhydramine is effective for managing insomnia, with the optimal dose influenced by the patient’s treatment history . Please refer to Table 2 for a summary of the most important clinical studies evaluating the use of diphenhydramine in the management of insomnia.
Table 2.
Summary of the most important clinical studies evaluating the use of diphenhydramine in the management of insomnia.
| Reference | Population | Design | Doses | Main Findings | Safety |
:---: :---: :---: |
| Teutsch et al. (1975) | More than 100 elderly patients in VA hospitals | Comparative with placebo | 50 mg and 150 mg | It was not significantly different from pentobarbital for control of insomnia | No significant differences in adverse effects |
| Russo et al. (1976) | 50 children with sleep disorders | Placebo controlled | 10 mg/kg | Significantly reduced sleep latency and night awakenings | Significantly reduced sleep latency and night awakenings |
| Carruthers et al. (1978) | 6 healthy volunteers | Double blind, crossover | 50 mg | Positive correlation between plasma concentration and sedative effects | No specific adverse effects are detailed |
| Sunshine et al. (1978) | 1295 postpartum women with insomnia | Double-blind controlled study | 12.5, 25, and 50 mg | Effective hypnotics in comparison to placebo | No significant adverse events reported |
| Rickels et al. (1983) | 111 patients with mild to moderate insomnia | Double blind, crossover | 50 mg | Improved several sleep parameters, patients reported feeling more rested | More side effects reported with diphenhydramine |
| Stewart et al. (1987) | 17 nursing home residents with insomnia | Double blind, crossover | 50 mg | Shorter sleep latency and longer sleep duration than temazepam | Worse performance on neurological tests compared to placebo |
| Roth et al. (1987) | 16 healthy adults | Crossover | 50 mg (3 times a day) | No significant difference compared to loratadine | Daytime sedation and decreased performance |
| Borbély et al. (1988) | 10 young and healthy adults | Double blind, crossover | 50 mg and 75 mg | No significant differences in subjective sleep parameters compared to placebo | Diphenhydramine did not cause deterioration in psychomotor performance or rebound insomnia |
| Kudo and Kurihara (1990) | 144 psychiatric patients aged 15 to 82 years old with insomnia | Randomized, Double blind | 12.5, 25, and 50 mg | Diphenhydramine was effective in improving sleep quality in psychiatric patients | Well tolerated, no serious side effects during the trial |
| Roehrs et al. (1993) | 12 young and healthy men | Double blind, Latin square | 50 mg | Significant sedative effects for 6.5 h, similar to triazolam | Residual sedation for ethanol but not for diphenhydramine and triazolam |
| Schweitzer et al. (1994) | 12 atopic subjects | Double blind, crossover | 50 mg (3 times a day) | Decreased alertness and performance on day 1, tolerance developed by day 3 | Central nervous system depression only on the first day |
| Richardson et al. (2002) | 15 healthy volunteers aged 18–50 years | Randomized, double-blind, crossover | 50 mg (2 times a day) | Increased drowsiness on day 1, tolerance developed by day 4 | Performance decline reversed on day 4 |
| Morin et al. (2005) | 184 patients with mild insomnia | Multicenter, randomized, placebo-controlled | 50 mg (2 times a day) | Improvements in subjective sleep parameters, increased sleep efficiency in the first 14 days | There were no significant residual effects or serious adverse events. |
| Glass et al. (2008) | 25 elderly with insomnia | Latin Square Desing | 50 mg | Improvement only in the number of awakenings compared to placebo; no better than temazepam | Similar number of adverse events, one fall reported with temazepam |
| Moulin et al. (2022) | 27 healthy adult participants | Randomized, double-blind, placebo-controlled, crossover | 50 mg for 7 days | Improvement in sleep debt, natural supplement showed better efficacy in sleep parameters | No serious adverse effects |
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3.2. Pharmacokinetics of Diphenhydramine
Simons et al. investigated the pharmacokinetics of diphenhydramine in 21 subjects categorized into three groups: children, young adults, and elderly individuals. The participants were administered a diphenhydramine syrup at a dose of 1.25 mg/kg, and blood samples were collected over a 72 h period. The study revealed that the half-life (t½), area under the curve (AUC), and mean residence time (MRT) increased with age, whereas the clearance and volume of distribution decreased. Significant differences were observed in maximal plasma concentration (Cmax), but not in the time to the maximal concentration (Tmax) across age groups. The authors noted significant variations in the t½ and clearance rates between age groups . These findings suggest that elderly individuals experience prolonged drug exposure, highlighting the potential need for dose adjustments to achieve an effective and safe therapeutic steady state.
Additional studies have reported that the pharmacokinetics of diphenhydramine in children (2–17 years) were studied using a weight- and age-based dosing schedule (6.25–50 mg). Cmax and AUC increased by 90% to 140% across age groups, with a tmax of 1.5 h. Oral clearance increased with age, but no maturation effect was seen after allometric scaling. Mild somnolence was the most common side effect (95%) . The data are summarized in Table 3.
Table 3.
Diphenhydramine pharmacokinetic parameters by age group.
| Parameter | Children (8.9 ± 1.7 y.o.) | Young Adults (31.5 ± 10.4 y.o.) | Elderly (69.4 ± 4.3 y.o.) |
:---: :---: |
| Weight (kg) | 31.6 ± 6.8 | 70.3 ± 9.9 | 71.0 ± 11.4 |
| Dose (mg) | 39.5 ± 8.4 | 87.9 ± 12.4 | 86.0 ± 7.3 |
| Cmax (ng/mL) | 81.8 ± 30.2 | 133.2 ± 37.6 | 188.4 ± 54.5 |
| Tmax (h) | 1.3 ± 0.5 | 1.7 ± 1.0 | 1.7 ± 0.8 |
| t½ (h) | 5.4 ± 1.8 | 9.2 ± 2.5 | 13.5 ± 4.2 |
| Cl (mL/min/kg) | 49.2 ± 22.8 | 23.3 ± 9.4 | 11.7 ± 3.1 |
| Vdss (L/kg) | 17.9 ± 5.9 | 14.6 ± 4.0 | 10.2 ± 3.0 |
| Vd (L/kg) | 21.7 ± 6.6 | 17.4 ± 4.8 | 13.6 ± 6.3 |
| AUC (ng/mL/h) | 475 ± 137 | 1031 ± 437 | 1902 ± 572 |
| MRT (h) | 6.4 ± 1.6 | 11.3 ± 3.1 | 14.8 ± 2.8 |
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Cmax: Maximum plasma concentration; Tmax: Time to reach maximum plasma concentration; t½: Elimination half-life; Cl: Clearance (mL/min/kg); Vdss: Volume of distribution at steady state (L/kg); Vd: Volume of distribution (L/kg); AUC: Area under the concentration–time curve (ng/mL/h). Table adapted from .
3.3. Toxic Effects of Diphenhydramine
Diphenhydramine overdose can have toxic effects, such as increased sedation and antimuscarinic effects. Clinical signs and symptoms include drowsiness, hyperpyrexia, mydriasis, fever, flushing, agitation, tremor, dystonic reactions, hallucinations, and electrocardiographic changes on the EKG, such as prolonged QRS complexes and QT intervals, as well as the appearance of a Brugada-like syndrome [40,42]. High doses, particularly in children, may result in delirium, psychosis, arrhythmias, coma, or cardiovascular collapse. Differential diagnoses for diphenhydramine intoxication include tricyclic antidepressant overdose, acetaminophen overdose, hypoglycemia, and serotonin syndrome [26,43].
Regarding drug–drug interactions, contraindicated interactions have been reported with oxybates and potassium salts . The co-administration of diphenhydramine with epinephrine or norepinephrine should be avoided due to the risk of prolonged hypertension. Diphenhydramine has been reported to enhance the sedative and hypnotic effects of benzodiazepines and Z-drugs. Alcohol consumption is also discouraged while taking diphenhydramine. Concurrent use with monoamine oxidase inhibitors may intensify central nervous system depression and anticholinergic effects. Moreover, combining diphenhydramine with first- or second-generation antipsychotics (e.g., chlorpromazine, olanzapine) or antiparkinsonian agents (e.g., benztropine, trihexyphenidyl) increases the risk of additive anticholinergic effects [21,43].
Recent reports indicate a rising incidence of diphenhydramine poisoning, highlighting the need for the timely administration of physostigmine in selected cases [45,46]. Alerts have been issued regarding the “Benadryl challenge”, a social media trend linked to severe intoxications . Additionally, diphenhydramine has been implicated in suicide attempts . Altogether, these findings underscore the importance of not underestimating the drug’s potential for harm.
Together, there are many studies showing that diphenhydramine improves key parameters related to sleep including sleep onset latency and overall sleep quality. In addition, the rapid pharmacokinetic action of diphenhydramine allows for rapid sleep onset, which may be beneficial for patients with insomnia who need immediate intervention to promote rest. Unlike other sedative hypnotics, diphenhydramine has minimal abuse potential and a low risk of residual sedation as a side effect. These characteristics make diphenhydramine an appealing option for the short-term treatment of insomnia, particularly in adults without comorbidities, provided that the dose and duration are carefully managed to minimize potential risks. Its use should be avoided in individuals with cardiac or pulmonary conditions, or in those taking other sedatives, hypnotics, or anticholinergic agents. Use in the elderly is not recommended due to the heightened risk of severe anticholinergic effects with cumulative exposure .
3.4. Consensus Results
The following are the results of the consensus for each dimension analyzed; results on interquartile range across the rounds are showed in Figure 1. In Chapter 1, “Evaluation of the Use of Diphenhydramine for Insomnia: Efficacy, Safety, Convenience, and Cost”, experts evaluated the following statements:
For question 1, “Diphenhydramine is an effective medication for the management of acute insomnia”, the panel of experts unanimously agreed, giving a rating of 5/5 with an interquartile range of 0 (100% agreement). This indicates complete agreement and consensus on the premise. This unanimous consensus highlights a shared confidence in diphenhydramine’s efficacy in managing acute insomnia.
For question 2, “Diphenhydramine is a safe medication for the management of short-term insomnia”, 80% of the experts agreed, demonstrating strong agreement with this premise. The interquartile range was 0.5, reflecting a consensus. Frequency analysis revealed that one out of five experts were neutral, four out of six partially agreed, and one out of six fully agreed. These findings suggest a consensus regarding the safety of diphenhydramine for short-term use, although the neutral stance of one expert and partial agreements indicate a need for the further exploration of specific safety concerns.
For question 3, “If diphenhydramine were available in the Colombian market, do you consider this medication could be an accessible option for managing short-term insomnia?”, the panel showed full agreement (100%) on this statement, with a median value of five and an interquartile range of zero. These results indicate a unanimous consensus among the experts, affirming that diphenhydramine is perceived as an accessible option for managing short-term insomnia if made available in the Colombian market. This agreement reflects the experts’ confidence in its potential affordability and practicality for patients.
Figure 1.
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Interquartile range across rounds from expert’s panel.
Grouped bars display the interquartile range across three rounds of questions. Questions 1, 3, 5, and 12 have an interquartile range of zero, indicating strong consensus with no variability among expert responses. Questions 2, 6, 7, 8, 9, and 11 show consensuses with slight variability. In contrast, questions 4 and 9 did not reach consensus in the first round due to high variability in responses. Question 4 exhibited a decrease in the interquartile range, achieving a consensus by round two. However, question 9 did not reach a consensus across all three rounds.
For question 4, “Diphenhydramine is a convenient medication for most patients with short-term insomnia, regardless of their comorbidities or clinical situations, and therefore has the potential to be marketed as an over-the-counter medication for managing short-term insomnia”, 80% of experts agreed, showing strong agreement, but not a unanimous consensus. The median value was 4, with an interquartile range of 1.5, indicating slight variability in responses. Frequency analysis revealed that one out of six experts partially disagreed, two out of six partially agreed, and two out of six fully agreed. This variability suggests differing perspectives on the convenience of diphenhydramine, particularly regarding its suitability for patients with comorbidities or diverse clinical situations. The partial disagreement and variability highlight that while there is a general agreement, additional research or clarification may be necessary to address specific concerns.
In the second round, a consensus was achieved with an interquartile range of 0.25. Frequency analysis revealed that one of five experts remained neutral and four out five strongly agreed with the statement. The median value was 3.8, indicating that after the first round and subsequent revisions, experts agreed that diphenhydramine is a convenient medication for most patients with short-term insomnia and has the potential to be marketed as an over-the-counter medication.
In Chapter 2, “Type(s) of insomnia where diphenhydramine could be used”, the results were the following:
For question 5, “Diphenhydramine is a useful medication for short-term insomnia (less than 3 months in duration)”, the median value was five, and the interquartile range was zero, reflecting a unanimous agreement and consensus (100% agreement). Frequency analysis revealed that all five experts rated this statement with a five, further affirming the unanimity of the consensus.
For question 6, “Diphenhydramine is a useful medication for chronic insomnia (more than 3 months in duration)”, the median value was 1.6, and the interquartile range was 1, reflecting total disagreement and consensus (0% agreement) within the panel. Frequency analysis showed that two out of five experts rated it as one, and three out of five rated it as two. This result indicates that the panel does not recommend diphenhydramine for chronic insomnia.
In Chapter 3, “Use of diphenhydramine as a hypnotic-sedative by age group”, the results were the following:
For question 7, “Diphenhydramine is an effective and safe medication for children aged 7 and older”, the panel showed a median value of 4.2 and an interquartile range of 0.25, reflecting unanimous agreement and consensus (100% agreement). Frequency analysis revealed that four out five experts rated it as four, and one out of five rated it as five. This indicates a consistent and strong level of agreement with the statement.
For question 8, “Diphenhydramine is an effective and safe medication for young adults (18 to 65 years) for managing short-term insomnia”, the median value was 4.8, and the interquartile range was 0.25, reflecting a unanimous agreement and consensus (100% agreement). Frequency analysis showed that one out of five experts rated it as four, while four out of five rated it as five, demonstrating a high level of agreement with slight variability.
For question 9, “Diphenhydramine is an effective and safe medication for elderly individuals (65 years and older) for managing short-term insomnia”, the median value was 3, and the interquartile range was 1.5, reflecting no agreement and no consensus (20% agreement). Frequency analysis indicated that two out of five experts rated it as two, two out of five as three, and one out of five as five. This wide distribution of ratings underscores the lack of consensus and varying perspectives on this statement in the first round.
Since a consensus was not achieved in the first round, a debrief was conducted on the use of diphenhydramine for elderly individuals. However, rounds two and three showed an interquartile range of 1.25, indicating significant variability in responses, and thus, no consensus was reached. In round two, frequency analysis revealed that one out of five experts partially disagreed, one out of five remained neutral, and three out of five partially agreed, with a median value of 3.4. In round three, one out of five experts partially disagreed, one out of five experts remained neutral, two out of five partially agreed, and one out of five fully agreed, with a median value of 4.2. Although there was a slight shift toward agreement, a consensus was still not achieved.
In Chapter 4, “Duration of diphenhydramine treatment for managing insomnia”, the results were the following:
For question 10, “The maximum recommended duration for using diphenhydramine as a hypnotic/sedative for short-term insomnia should be around four weeks”, the median value was 4.6, and the interquartile range was 1, showing a strong agreement and tight consensus (100% agreement). Frequency analysis revealed that two out five experts rated it as four, while three out of five rated it as five. This indicates a shared belief in limiting the duration of diphenhydramine use, with a small degree of variability.
In Chapter 5, “Evidence and levels of evidence on the use of diphenhydramine for managing short-term insomnia”, the results were the following:
For question 11, “There is a sufficient body of clinical evidence to recommend the use of diphenhydramine in patients with short-term insomnia”, the panel’s answers showed a median value of 4.8 and an interquartile range of 0.25, reflecting a strong agreement and consensus (100% agreement). Frequency analysis showed that one out of five experts partially agreed and four out of five strongly agreed with the statement.
For question 12, “There is a sufficient level of clinical evidence to recommend the use of diphenhydramine in patients with short-term insomnia”, the median value was four, and the interquartile range was zero, reflecting a partial agreement and strong consensus (100% agreement). Frequency analysis revealed that all experts rated this statement as a four, emphasizing a unified agreement.
4. Discussion
As shown above, several studies have consistently demonstrated that diphenhydramine is effective for the treatment of short-term insomnia [25,31,32,34,38,50]. Therefore, it is important to compare its efficacy with that of other medications currently used for insomnia.
One such study, conducted by Stewart in 1987, evaluated the efficacy of diphenhydramine in comparison with temazepam. In the study, diphenhydramine was administered at a dose of 50 milligrams for five consecutive nights, with two nights of placebo between each five-day treatment period. Sleep-related metrics, including sleep quality, sleep onset latency, number of awakenings, and total sleep duration, were assessed to determine the effects of both treatments. The results indicated that diphenhydramine was as effective as temazepam as a hypnotic agent in older adults. Moreover, diphenhydramine significantly improved self-perceived sleep latency, and by the fifth night of treatment, the self-reported sleep duration was significantly longer with diphenhydramine than with temazepam. Regarding neurological adverse effects, neither diphenhydramine nor temazepam produced significant impairments .
In a similar study, Glass et al. compared the efficacy of temazepam and diphenhydramine. This trial was conducted over a 14-night treatment period. Both medications demonstrated hypnotic efficacy, but temazepam was more effective than diphenhydramine when compared with a placebo at the doses tested. The authors noted that this difference was offset by the increased risk of falls associated with temazepam use .
When comparing diphenhydramine with a non-benzodiazepine hypnotic, Katayose et al. evaluated the effects of diphenhydramine (50 milligrams), ketotifen (1 milligram), and the Z-drug zolpidem (10 milligrams). This study was a randomized, double-blind, placebo-controlled trial in which overall sleep quality, daytime sleepiness, and psychomotor performance were assessed. Among the most significant findings, diphenhydramine and zolpidem produced comparable effects on overall sleep quality. However, diphenhydramine significantly prolonged rapid eye movement (REM) sleep latency and reduced the percentage of REM sleep. Regarding daytime effects, diphenhydramine showed a tendency to increase next-day sedation and led to a significant reduction in the psychomotor performance. The authors concluded that both diphenhydramine and ketotifen significantly increased subjective and objective sleepiness while significantly impairing the next-day psychomotor performance, resulting in clinically relevant sedative/hypnotic carryover effects . Some other studies have shown that diphenhydramine can impact the next-day post-administration performance .
Regarding safety, Erb and Bschor conducted a systematic review of the literature from 1972 to 2012 and reported a clinical case providing evidence of the addictive potential of diphenhydramine. Their findings highlight the need for caution, particularly in patients with a history of substance use disorders . Other studies have also emphasized the importance of the careful use of diphenhydramine . There is a similar preoccupying scenario as there is substantial evidence demonstrating the addictive potential of benzodiazepines and Z-drugs [55,56]. Overall, the long-term use of hypnotic agents carries a significant risk of dependence. While benzodiazepines and Z-drugs have been extensively studied in this regard, fewer studies have addressed the potential for diphenhydramine dependence.
Diphenhydramine is associated with several disease-related interactions. In patients diagnosed with major depressive disorder who are also taking anxiolytics, sedatives, or hypnotics, the use of diphenhydramine may lead to episodes of disinhibition, aggressiveness, agitation, or hallucinations. Individuals with comorbid conditions such as prostatic hypertrophy, urinary retention or obstruction, glaucoma, or gastrointestinal obstruction are particularly susceptible to enhanced anticholinergic effects. Moreover, diphenhydramine can increase the viscosity of bronchial secretions, potentially leading to respiratory tract obstruction; thus, caution is advised in patients with asthma or chronic obstructive pulmonary disease. Although infrequent, cardiovascular side effects have been reported, including tachycardia, palpitations, electrocardiographic abnormalities, arrhythmias, hypotension, and hypertension, particularly in individuals with pre-existing cardiac conditions . Furthermore, studies have indicated that the higher cumulative use of anticholinergic medications, as diphenhydramine, is associated with an increased risk of dementia. However, to date, there is no conclusive evidence establishing diphenhydramine as a direct cause of dementia [58,59].
During pregnancy, diphenhydramine is not contraindicated, as current evidence does not suggest an increased risk of miscarriage or congenital anomalies. However, during breastfeeding, diphenhydramine may pass into breast milk in small amounts. While generally well tolerated, high doses may lead to irritability or alterations in sleep patterns in the nursing infant. Therefore, the lowest effective dose should be used and prolonged use should be avoided .
The expert consensus highlighted several points in favor of diphenhydramine. There was strong agreement regarding its efficacy, safety, and short duration of action, supporting its use exclusively for short-term insomnia. However, there was substantial disagreement regarding its use in chronic insomnia and in the elderly. These concerns align with available evidence, as the effectiveness of diphenhydramine has only been demonstrated for short-term insomnia. Furthermore, in older adults, all medications, including diphenhydramine, should be prescribed with caution or avoided due to potential risks. Regarding its availability as an over-the-counter medication, its approval by the Food and Drug Administration represents a significant milestone . Diphenhydramine is widely accessible and affordable, making it a practical option for many individuals.
5. Conclusions
Diphenhydramine at a dose of 50 mg before sleep has been shown to be effective for short-term insomnia and can be used safely in healthy young and adult individuals. However, diphenhydramine should be avoided in the elderly, as well as in individuals with concurrent cardiac or pulmonary conditions, or those taking sedative/hypnotic medications. When compared to other medications, diphenhydramine demonstrates a similar efficacy profile; however, next-day side effects, such as residual sedation and cognitive impairment, are frequently reported. The available evidence and expert consensus support its use as an over-the-counter medication option for short-term insomnia. Nevertheless, patients should always be informed about its potential adverse effects, including next-day drowsiness and impaired psychomotor performance. Additionally, while the risk of dependence is lower than that of other hypnotics, its addictive potential should not be overlooked. Given these factors, diphenhydramine remains a viable short-term treatment, provided that its risks and benefits are carefully considered.
Author Contributions
All authors met the authorship criteria established by the ICMJE. D.F.A.-S. was responsible for manuscript preparation, writing, questionnaire formulation, and final draft revisions. M.V. participated in the literature review, final draft revisions, and contributed to the expert consensus. K.P. participated in the literature review and contributed to the expert consensus. S.A. participated in the literature review and contributed to the expert consensus. J.E. participated in the literature review and contributed to the expert consensus. C.A.C.-O. contributed to manuscript preparation, the selection of papers for review, writing, questionnaire formulation, final draft revisions, and participation in the expert consensus. All authors have read and agreed to the published version of the manuscript.
Conflicts of Interest
The authors maintained full independence in the search and generation of scientific evidence, as well as in the formulation of the consensus and the writing of the manuscript.
Funding Statement
The project was funded by P&G, but the structure and content of this article were at the sole discretion and decision of the authors.
Footnotes
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
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Head of ulna
Ulnar styloid process
Human anatomy 2
ha2
Human body
General Anatomy
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Ulnar styloid process
Processus styloideus ulnae
Definition
IMAIOS
The lower extremity of the ulna is small, and presents two eminences; the lateral and larger is a rounded, articular eminence, termed the head of the ulna; the medial, narrower and more projecting, is a non-articular eminence, the styloid process.
The styloid process projects from the medial and back part of the bone; it descends a little lower than the head, and its rounded end affords attachment to the ulnar collateral ligament of the wrist-joint. The head is separated from the styloid process by a depression for the attachment of the apex of the triangular articular disk, and behind, by a shallow groove for the tendon of the Extensor carpi ulnaris.
References
This definition incorporates text from a public domain edition of Gray's Anatomy (20th U.S. edition of Gray's Anatomy of the Human Body, published in 1918 – from
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Human anatomy 2
Human body>
Musculoskeletal systems>
Skeletal system>
Bones of upper limb>
Bones of free part of upper limb>
Ulna>
Head of ulna>
Ulnar styloid process
Underlying structures: There are no anatomical children for this anatomical part
Human anatomy 1
Systemic anatomy>
Bones; Skeletal system>
Bones of upper limb>
Free part of upper limb>
Ulna>
Head of ulna>
Ulnar styloid process
Underlying structures: There are no anatomical children for this anatomical part
Comparative anatomy in animals
Lateral styloid process
Styloid process
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