id
stringlengths 2
8
| title
stringlengths 1
130
| text
stringlengths 0
252k
| formulas
listlengths 1
823
| url
stringlengths 38
44
|
|---|---|---|---|---|
14907
|
Inverse function
|
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by formula_0
For a function formula_1, its inverse formula_2 admits an explicit description: it sends each element formula_3 to the unique element formula_4 such that "f"("x") = "y".
As an example, consider the real-valued function of a real variable given by "f"("x") = 5"x" − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function formula_5 defined by formula_6
Definitions.
Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is "invertible" if there exists a function g from Y to X such that formula_7 for all formula_4 and formula_8 for all formula_3.
If f is invertible, then there is exactly one function g satisfying this property. The function g is called the inverse of f, and is usually denoted as "f" −1, a notation introduced by John Frederick William Herschel in 1813.
The function f is invertible if and only if it is bijective. This is because the condition formula_7 for all formula_4 implies that f is injective, and the condition formula_8 for all formula_3 implies that f is surjective.
The inverse function "f" −1 to f can be explicitly described as the function
formula_9.
Inverses and composition.
Recall that if f is an invertible function with domain X and codomain Y, then
formula_10, for every formula_11 and formula_12 for every formula_13.
Using the composition of functions, this statement can be rewritten to the following equations between functions:
formula_14 and formula_15
where id"X" is the identity function on the set X; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism.
Considering function composition helps to understand the notation "f" −1. Repeatedly composing a function "f": "X"→"X" with itself is called iteration. If f is applied n times, starting with the value x, then this is written as "f" "n"("x"); so "f" 2("x")
"f" ("f" ("x")), etc. Since "f" −1("f" ("x"))
"x", composing "f" −1 and "f" "n" yields "f" "n"−1, "undoing" the effect of one application of f.
Notation.
While the notation "f" −1("x") might be misunderstood, ("f"("x"))−1 certainly denotes the multiplicative inverse of "f"("x") and has nothing to do with the inverse function of f. The notation formula_16 might be used for the inverse function to avoid ambiguity with the multiplicative inverse.
In keeping with the general notation, some English authors use expressions like sin−1("x") to denote the inverse of the sine function applied to x (actually a partial inverse; see below). Other authors feel that this may be confused with the notation for the multiplicative inverse of sin ("x"), which can be denoted as (sin ("x"))−1. To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin ). For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin("x"). Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin ). For instance, the inverse of the hyperbolic sine function is typically written as arsinh("x"). The expressions like sin−1("x") can still be useful to distinguish the multivalued inverse from the partial inverse: formula_17. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the "f" −1 notation should be avoided.
Examples.
Squaring and square root functions.
The function "f": R → [0,∞) given by "f"("x") = "x"2 is not injective because formula_18 for all formula_19. Therefore, f is not invertible.
If the domain of the function is restricted to the nonnegative reals, that is, we take the function formula_20 with the same "rule" as before, then the function is bijective and so, invertible. The inverse function here is called the "(positive) square root function" and is denoted by formula_21.
Standard inverse functions.
The following table shows several standard functions and their inverses:
Formula for the inverse.
Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse formula_22 of an invertible function formula_23 has an explicit description as
formula_24.
This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if f is the function
formula_25
then to determine formula_26 for a real number y, one must find the unique real number x such that (2"x" + 8)3 = "y". This equation can be solved:
formula_27
Thus the inverse function "f" −1 is given by the formula
formula_28
Sometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if f is the function
formula_29
then f is a bijection, and therefore possesses an inverse function "f" −1. The formula for this inverse has an expression as an infinite sum:
formula_30
Properties.
Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations.
Uniqueness.
If an inverse function exists for a given function f, then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by f.
Symmetry.
There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y, then its inverse "f" −1 has domain Y and image X, and the inverse of "f" −1 is the original function f. In symbols, for functions "f":"X" → "Y" and "f"−1:"Y" → "X",
formula_31 and formula_32
This statement is a consequence of the implication that for f to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by
formula_33
The inverse of a composition of functions is given by
formula_34
Notice that the order of g and f have been reversed; to undo f followed by g, we must first undo g, and then undo f.
For example, let "f"("x") = 3"x" and let "g"("x") = "x" + 5. Then the composition "g" ∘ "f" is the function that first multiplies by three and then adds five,
formula_35
To reverse this process, we must first subtract five, and then divide by three,
formula_36
This is the composition
("f" −1 ∘ "g" −1)("x").
Self-inverses.
If X is a set, then the identity function on X is its own inverse:
formula_37
More generally, a function "f" : "X" → "X" is equal to its own inverse, if and only if the composition "f" ∘ "f" is equal to id"X". Such a function is called an involution.
Graph of the inverse.
If f is invertible, then the graph of the function
formula_38
is the same as the graph of the equation
formula_39
This is identical to the equation "y" = "f"("x") that defines the graph of f, except that the roles of x and y have been reversed. Thus the graph of "f" −1 can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line
"y" = "x".
Inverses and derivatives.
By the inverse function theorem, a continuous function of a single variable formula_40 (where formula_41) is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function
formula_42
is invertible, since the derivative
"f′"("x") = 3"x"2 + 1 is always positive.
If the function f is differentiable on an interval I and "f′"("x") ≠ 0 for each "x" ∈ "I", then the inverse "f" −1 is differentiable on "f"("I"). If "y" = "f"("x"), the derivative of the inverse is given by the inverse function theorem,
formula_43
Using Leibniz's notation the formula above can be written as
formula_44
This result follows from the chain rule (see the article on inverse functions and differentiation).
The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable multivariable function "f ": R"n" → R"n" is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible. In this case, the Jacobian of "f" −1 at "f"("p") is the matrix inverse of the Jacobian of f at p.
Generalizations.
Partial inverses.
Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function
formula_49
is not one-to-one, since "x"2 = (−"x")2. However, the function becomes one-to-one if we restrict to the domain "x" ≥ 0, in which case
formula_50
(If we instead restrict to the domain "x" ≤ 0, then the inverse is the negative of the square root of y.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:
formula_51
Sometimes, this multivalued inverse is called the full inverse of f, and the portions (such as √ and −√) are called "branches". The most important branch of a multivalued function (e.g. the positive square root) is called the "principal branch", and its value at y is called the "principal value" of "f" −1("y").
For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture).
These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since
formula_52
for every real x (and more generally sin("x" + 2π"n") = sin("x") for every integer n). However, the sine is one-to-one on the interval
[−,], and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between − and . The following table describes the principal branch of each inverse trigonometric function:
Left and right inverses.
Function composition on the left and on the right need not coincide. In general, the conditions
"x"" and
"x""
imply different properties of f. For example, let "f": R → [0, ∞) denote the squaring map, such that "f"("x") = "x"2 for all x in R, and let denote the square root map, such that "g"("x")
√ for all "x" ≥ 0. Then "f"("g"("x")) = "x" for all x in [0, ∞); that is, g is a right inverse to f. However, g is not a left inverse to f, since, e.g., "g"("f"(−1)) = 1 ≠ −1.
Left inverses.
If "f": "X" → "Y", a left inverse for f (or "retraction" of f ) is a function "g": "Y" → "X" such that composing f with g from the left gives the identity function formula_53 That is, the function g satisfies the rule
If "f"("x")
"y", then "g"("y")
"x".
The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image.
A function f with nonempty domain is injective if and only if it has a left inverse. An elementary proof runs as follows:
In classical mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}.
Right inverses.
A right inverse for f (or "section" of f ) is a function "h": "Y" → "X" such that
formula_54
That is, the function h satisfies the rule
If formula_55, then formula_56
Thus, "h"("y") may be any of the elements of X that map to y under f.
A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice).
If h is the right inverse of f, then f is surjective. For all formula_57, there is formula_58 such that formula_59.
If f is surjective, f has a right inverse h, which can be constructed as follows: for all formula_57, there is at least one formula_11 such that formula_60 (because f is surjective), so we choose one to be the value of "h"("y").
Two-sided inverses.
An inverse that is both a left and right inverse (a two-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse.
If formula_61 is a left inverse and formula_62 a right inverse of formula_63, for all formula_57, formula_64.
A function has a two-sided inverse if and only if it is bijective.
A bijective function f is injective, so it has a left inverse (if f is the empty function, formula_65 is its own left inverse). f is surjective, so it has a right inverse. By the above, the left and right inverse are the same.
If f has a two-sided inverse g, then g is a left inverse and right inverse of f, so f is injective and surjective.
Preimages.
If "f": "X" → "Y" is any function (not necessarily invertible), the preimage (or inverse image) of an element "y" ∈ "Y" is defined to be the set of all elements of X that map to y:
formula_66
The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f.
Similarly, if S is any subset of Y, the preimage of S, denoted formula_67, is the set of all elements of X that map to S:
formula_68
For example, take the function "f": R → R; "x" ↦ "x"2. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.
formula_69.
The preimage of a single element "y" ∈ "Y" – a singleton set {"y"} – is sometimes called the "fiber" of y. When Y is the set of real numbers, it is common to refer to "f" −1({"y"}) as a "level set".
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f^{-1} ."
},
{
"math_id": 1,
"text": "f\\colon X\\to Y"
},
{
"math_id": 2,
"text": "f^{-1}\\colon Y\\to X"
},
{
"math_id": 3,
"text": "y\\in Y"
},
{
"math_id": 4,
"text": "x\\in X"
},
{
"math_id": 5,
"text": "f^{-1}\\colon \\R\\to\\R"
},
{
"math_id": 6,
"text": "f^{-1}(y) = \\frac{y+7}{5} ."
},
{
"math_id": 7,
"text": "g(f(x))=x"
},
{
"math_id": 8,
"text": "f(g(y))=y"
},
{
"math_id": 9,
"text": "f^{-1}(y)=(\\text{the unique element }x\\in X\\text{ such that }f(x)=y)"
},
{
"math_id": 10,
"text": " f^{-1}\\left(f(x)\\right) = x"
},
{
"math_id": 11,
"text": "x \\in X"
},
{
"math_id": 12,
"text": " f\\left(f^{-1}(y)\\right) = y"
},
{
"math_id": 13,
"text": "y \\in Y "
},
{
"math_id": 14,
"text": " f^{-1} \\circ f = \\operatorname{id}_X"
},
{
"math_id": 15,
"text": "f \\circ f^{-1} = \\operatorname{id}_Y, "
},
{
"math_id": 16,
"text": "f^{\\langle -1\\rangle}"
},
{
"math_id": 17,
"text": "\\sin^{-1}(x) = \\{(-1)^n \\arcsin(x) + \\pi n : n \\in \\mathbb Z\\}"
},
{
"math_id": 18,
"text": "(-x)^2=x^2"
},
{
"math_id": 19,
"text": "x\\in\\R"
},
{
"math_id": 20,
"text": "f\\colon [0,\\infty)\\to [0,\\infty);\\ x\\mapsto x^2"
},
{
"math_id": 21,
"text": "x\\mapsto\\sqrt x"
},
{
"math_id": 22,
"text": "f^{-1} "
},
{
"math_id": 23,
"text": "f\\colon\\R\\to\\R"
},
{
"math_id": 24,
"text": "f^{-1}(y)=(\\text{the unique element }x\\in \\R\\text{ such that }f(x)=y)"
},
{
"math_id": 25,
"text": "f(x) = (2x + 8)^3 "
},
{
"math_id": 26,
"text": "f^{-1}(y) "
},
{
"math_id": 27,
"text": "\\begin{align}\n y & = (2x+8)^3 \\\\\n \\sqrt[3]{y} & = 2x + 8 \\\\\n\\sqrt[3]{y} - 8 & = 2x \\\\\n\\dfrac{\\sqrt[3]{y} - 8}{2} & = x .\n\\end{align}"
},
{
"math_id": 28,
"text": "f^{-1}(y) = \\frac{\\sqrt[3]{y} - 8} 2."
},
{
"math_id": 29,
"text": "f(x) = x - \\sin x ,"
},
{
"math_id": 30,
"text": " f^{-1}(y) =\n\\sum_{n=1}^\\infty\n \\frac{y^{n/3}}{n!} \\lim_{ \\theta \\to 0} \\left(\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d} \\theta^{\\,n-1}} \\left(\n \\frac \\theta { \\sqrt[3]{ \\theta - \\sin( \\theta )} } \\right)^n\n\\right).\n"
},
{
"math_id": 31,
"text": "f^{-1}\\circ f = \\operatorname{id}_X "
},
{
"math_id": 32,
"text": " f \\circ f^{-1} = \\operatorname{id}_Y."
},
{
"math_id": 33,
"text": "\\left(f^{-1}\\right)^{-1} = f."
},
{
"math_id": 34,
"text": "(g \\circ f)^{-1} = f^{-1} \\circ g^{-1}."
},
{
"math_id": 35,
"text": "(g \\circ f)(x) = 3x + 5."
},
{
"math_id": 36,
"text": "(g \\circ f)^{-1}(x) = \\tfrac13(x - 5)."
},
{
"math_id": 37,
"text": "{\\operatorname{id}_X}^{-1} = \\operatorname{id}_X."
},
{
"math_id": 38,
"text": "y = f^{-1}(x)"
},
{
"math_id": 39,
"text": "x = f(y) ."
},
{
"math_id": 40,
"text": "f\\colon A\\to\\mathbb{R}"
},
{
"math_id": 41,
"text": "A\\subseteq\\mathbb{R}"
},
{
"math_id": 42,
"text": "f(x) = x^3 + x"
},
{
"math_id": 43,
"text": "\\left(f^{-1}\\right)^\\prime (y) = \\frac{1}{f'\\left(x \\right)}. "
},
{
"math_id": 44,
"text": "\\frac{dx}{dy} = \\frac{1}{dy / dx}. "
},
{
"math_id": 45,
"text": " F = f(C) = \\tfrac95 C + 32 ;"
},
{
"math_id": 46,
"text": " C = f^{-1}(F) = \\tfrac59 (F - 32) ,"
},
{
"math_id": 47,
"text": "\n\\begin{align}\nf^{-1} (f(C)) = {} & f^{-1}\\left( \\tfrac95 C + 32 \\right) = \\tfrac59 \\left( (\\tfrac95 C + 32 ) - 32 \\right) = C, \\\\\n& \\text{for every value of } C, \\text{ and } \\\\[6pt]\nf\\left(f^{-1}(F)\\right) = {} & f\\left(\\tfrac59 (F - 32)\\right) = \\tfrac95 \\left(\\tfrac59 (F - 32)\\right) + 32 = F, \\\\\n& \\text{for every value of } F.\n\\end{align}\n"
},
{
"math_id": 48,
"text": "\\begin{align}\n f(\\text{Allan})&=2005 , \\quad & f(\\text{Brad})&=2007 , \\quad & f(\\text{Cary})&=2001 \\\\\n f^{-1}(2005)&=\\text{Allan} , \\quad & f^{-1}(2007)&=\\text{Brad} , \\quad & f^{-1}(2001)&=\\text{Cary}\n\\end{align}\n"
},
{
"math_id": 49,
"text": "f(x) = x^2"
},
{
"math_id": 50,
"text": "f^{-1}(y) = \\sqrt{y} . "
},
{
"math_id": 51,
"text": "f^{-1}(y) = \\pm\\sqrt{y} . "
},
{
"math_id": 52,
"text": "\\sin(x + 2\\pi) = \\sin(x)"
},
{
"math_id": 53,
"text": "g \\circ f = \\operatorname{id}_X\\text{.}"
},
{
"math_id": 54,
"text": "f \\circ h = \\operatorname{id}_Y . "
},
{
"math_id": 55,
"text": "\\displaystyle h(y) = x"
},
{
"math_id": 56,
"text": "\\displaystyle f(x) = y ."
},
{
"math_id": 57,
"text": "y \\in Y"
},
{
"math_id": 58,
"text": "x = h(y)"
},
{
"math_id": 59,
"text": "f(x) = f(h(y)) = y"
},
{
"math_id": 60,
"text": "f(x) = y"
},
{
"math_id": 61,
"text": "g"
},
{
"math_id": 62,
"text": "h"
},
{
"math_id": 63,
"text": "f"
},
{
"math_id": 64,
"text": "g(y) = g(f(h(y)) = h(y)"
},
{
"math_id": 65,
"text": "f \\colon \\varnothing \\to \\varnothing"
},
{
"math_id": 66,
"text": "f^{-1}(\\{y\\}) = \\left\\{ x\\in X : f(x) = y \\right\\} . "
},
{
"math_id": 67,
"text": "f^{-1}(S) "
},
{
"math_id": 68,
"text": "f^{-1}(S) = \\left\\{ x\\in X : f(x) \\in S \\right\\} . "
},
{
"math_id": 69,
"text": "f^{-1}(\\left\\{1,4,9,16\\right\\}) = \\left\\{-4,-3,-2,-1,1,2,3,4\\right\\}"
}
] |
https://en.wikipedia.org/wiki?curid=14907
|
1490758
|
Grim trigger
|
Trigger strategy
In game theory, grim trigger (also called the grim strategy or just grim) is a trigger strategy for a repeated game.
Initially, a player using grim trigger will cooperate, but as soon as the opponent defects (thus satisfying the trigger condition), the player using grim trigger will defect for the remainder of the iterated game. Since a single defect by the opponent triggers defection forever, grim trigger is the most strictly unforgiving of strategies in an iterated game.
In Robert Axelrod's book "The Evolution of Cooperation", grim trigger is called "Friedman", for a 1971 paper by James W. Friedman, which uses the concept.
The infinitely repeated prisoners' dilemma.
The infinitely repeated prisoners’ dilemma is a well-known example for the grim trigger strategy. The normal game for two prisoners is as follows:
In the prisoners' dilemma, each player has two choices in each stage:
If a player defects, he will be punished for the remainder of the game. In fact, both players are better off to stay silent (cooperate) than to betray the other, so playing (C, C) is the cooperative profile while playing (D, D), also the unique Nash equilibrium in this game, is the punishment profile.
In the grim trigger strategy, a player cooperates in the first round and in the subsequent rounds as long as his opponent does not defect from the agreement. Once the player finds that the opponent has betrayed in the previous game, he will then defect forever.
In order to evaluate the subgame perfect equilibrium (SPE) for the following grim trigger strategy of the game, strategy S* for players "i" and "j" is as follows:
Then, the strategy is an SPE only if the discount factor is formula_0. In other words, neither Player 1 or Player 2 is incentivized to defect from the cooperation profile if the discount factor is greater than one half.
To prove that the strategy is a SPE, cooperation should be the best response to the other player's cooperation, and the defection should be the best response to the other player's defection.
Step 1: Suppose that D is never played so far.
Then, C is better than D if formula_3.
Step 2: Suppose that someone has played D previously, then Player j will play D no matter what.
Since formula_6, playing D is optimal.
The preceding argument emphasizes that there is no incentive to deviate (no profitable deviation) from the cooperation profile if formula_7, and this is true for every subgame. Therefore, the strategy for the infinitely repeated prisoners’ dilemma game is a Subgame Perfect Nash equilibrium.
In iterated prisoner's dilemma strategy competitions, grim trigger performs poorly even without noise, and adding signal errors makes it even worse. Its ability to threaten permanent defection gives it a theoretically effective way to sustain trust, but because of its unforgiving nature and the inability to communicate this threat in advance, it performs poorly.
Grim trigger in international relations.
Under the grim trigger in international relations perspective, a nation cooperates only if its partner has never exploited it in the past. Because a nation will refuse to cooperate in all future periods once its partner defects once, the indefinite removal of cooperation becomes the threat that makes such strategy a limiting case.
Grim trigger in user-network interactions.
Game theory has recently been used in developing future communications systems, and the user in the user-network interaction game employing the grim trigger strategy is one of such examples. If the grim trigger is decided to be used in the user-network interaction game, the user stays in the network (cooperates) if the network maintains a certain quality, but punishes the network by stopping the interaction and leaving the network as soon as the user finds out the opponent defects. Antoniou et al. explains that “given such a strategy, the network has a stronger incentive to keep the promise given for a certain quality, since it faces the threat of losing its customer forever.”
Comparison with other strategies.
Tit for tat and grim trigger strategies are similar in nature in that both are trigger strategy where a player refuses to defect first if he has the ability to punish the opponent for defecting. The difference, however, is that grim trigger seeks maximal punishment for a single defection while tit for tat is more forgiving, offering one punishment for each defection.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\delta \\geq {\\frac{1}{2}}"
},
{
"math_id": 1,
"text": "(1-\\delta)[1+\\delta+\\delta^2+ ... ] = (1-\\delta) \\times \\frac{1}{1-\\delta} = 1"
},
{
"math_id": 2,
"text": "(1-\\delta)[2+0+0+ ... ] = 2(1-\\delta)"
},
{
"math_id": 3,
"text": "1 \\geq 2(1-\\delta)"
},
{
"math_id": 4,
"text": "(1-\\delta)[-1+\\delta \\times 0+\\delta^2 \\times 0 + ... ] = (1-\\delta) \\times -1 = \\delta - 1"
},
{
"math_id": 5,
"text": "(1-\\delta)[0+\\delta \\times 0+\\delta^2 \\times 0 + ... ] = 0"
},
{
"math_id": 6,
"text": " 0 \\leq \\delta \\leq 1"
},
{
"math_id": 7,
"text": "\\delta \\geq \\frac{1}{2}"
}
] |
https://en.wikipedia.org/wiki?curid=1490758
|
14909067
|
Perspective (geometry)
|
Term in geometry
Two figures in a plane are perspective from a point "O", called the center of perspectivity, if the lines joining corresponding points of the figures all meet at "O". Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions.
Terminology.
The line which goes through the points where the figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or archaically perspector. The figures are said to be perspective from this center.
Perspectivity.
If each of the perspective figures consists of all the points on a line (a range) then transformation of the points of one range to the other is called a "central perspectivity". A dual transformation, taking all the lines through a point (a pencil) to another pencil by means of an axis of perspectivity is called an "axial perspectivity".
Triangles.
An important special case occurs when the figures are triangles. Two triangles that are perspective from a point are said to be "centrally perspective" and are called a "central couple". Two triangles that are perspective from a line are called "axially perspective" and an "axial couple".
Notation.
Karl von Staudt introduced the notation formula_0 to indicate that triangles ABC and abc are perspective.
Related theorems and configurations.
Desargues' theorem states that a central couple of triangles is axial. The converse statement, that an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the real projective plane, and with suitable modifications for special cases, in the Euclidean plane. Projective planes in which central and axial perspectivity of triangles are equivalent are called "Desarguesian planes".
There are ten points associated with these two kinds of perspective: six on the two triangles, three on the axis of perspectivity, and one at the center of perspectivity. Dually, there are also ten lines associated with two perspective triangles: three sides of the triangles, three lines through the center of perspectivity, and the axis of perspectivity. These ten points and ten lines form an instance of the Desargues configuration.
If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of Pappus's (hexagon) theorem. When this happens, the nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of the Pappus configuration.
The Reye configuration is formed by four quadruply perspective tetrahedra in an analogous way to the Pappus configuration.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "ABC \\doublebarwedge abc"
}
] |
https://en.wikipedia.org/wiki?curid=14909067
|
1491119
|
Forward price
|
Future cost agreed upon by a buyer and seller of something to be traded at a later time
The forward price (or sometimes forward rate) is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, the forward price can be expressed in terms of the spot price and any dividends. For forwards on non-tradeables, pricing the forward may be a complex task.
Forward price formula.
If the underlying asset is tradable and a dividend exists, the forward price is given by:
formula_0
where
formula_1 is the forward price to be paid at time formula_2
formula_3 is the exponential function (used for calculating continuous compounding interests)
formula_4 is the risk-free interest rate
formula_5 is the convenience yield
formula_6 is the spot price of the asset (i.e. what it would sell for at time 0)
formula_7 is a dividend that is guaranteed to be paid at time formula_8 where formula_9
Proof of the forward price formula.
The two questions here are what price the short position (the seller of the asset) should offer to maximize his gain, and what price the long position (the buyer of the asset) should accept to maximize his gain?
At the very least we know that both do not want to lose any money in the deal.
The short position knows as much as the long position knows: the short/long positions are both aware of any schemes that they could partake on to gain a profit given some forward price.
So of course they will have to settle on a fair price or else the transaction cannot occur.
An economic articulation would be:
(fair price + future value of asset's dividends) − spot price of asset = cost of capital
forward price = spot price − cost of carry
The future value of that asset's dividends (this could also be coupons from bonds, monthly rent from a house, fruit from a crop, etc.) is calculated using the risk-free force of interest. This is because we are in a risk-free situation (the whole point of the forward contract is to get rid of risk or to at least reduce it) so why would the owner of the asset take any chances? He would reinvest at the risk-free rate (i.e. U.S. T-bills which are considered risk-free). The spot price of the asset is simply the market value at the instant in time when the forward contract is entered into.
So OUT − IN = NET GAIN and his net gain can only come from the opportunity cost of keeping the asset for that time period (he could have sold it and invested the money at the risk-free rate).
let
"K" = fair price
"C" = cost of capital
"S" = spot price of asset
"F" = future value of asset's dividend
"I" = present value of "F" (discounted using "r" )
"r" = risk-free interest rate compounded continuously
"T" = length of time from when the contract was entered into
Solving for fair price and substituting mathematics we get:
formula_10
where:
formula_11
formula_13
where "ci" is the "i"th dividend paid at time "t i".
Doing some reduction we end up with:
formula_14
Notice that implicit in the above derivation is the assumption that the underlying can be traded. This assumption does not hold for certain kinds of forwards.
Forward versus futures prices.
There is a difference between forward and futures prices when interest rates are stochastic. This difference disappears when interest rates are deterministic.
In the language of stochastic processes, the forward price is a martingale under the forward measure, whereas the futures price is a martingale under the risk-neutral measure. The forward measure and the risk neutral measure are the same when interest rates are deterministic.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " F = S_0 e^{(r-q)T} - \\sum_{i=1}^N D_i e^{(r-q)(T-t_i)} \\,"
},
{
"math_id": 1,
"text": "F"
},
{
"math_id": 2,
"text": "T"
},
{
"math_id": 3,
"text": "e^x"
},
{
"math_id": 4,
"text": "r"
},
{
"math_id": 5,
"text": "q"
},
{
"math_id": 6,
"text": "S_0"
},
{
"math_id": 7,
"text": "D_i"
},
{
"math_id": 8,
"text": "t_i"
},
{
"math_id": 9,
"text": "0< t_i < T."
},
{
"math_id": 10,
"text": " K = C + S - F \\,"
},
{
"math_id": 11,
"text": "C = S(e^{rT} - 1) \\,"
},
{
"math_id": 12,
"text": " e^{rT} = 1 + j \\,"
},
{
"math_id": 13,
"text": " F = c_1 e^{r(T - t_1)} + \\cdots + c_n e^{r(T - t_n)} "
},
{
"math_id": 14,
"text": " K = (S - I)e^{rT}. \\,"
}
] |
https://en.wikipedia.org/wiki?curid=1491119
|
149117
|
Thomas Bayes
|
British statistician (c. 1701 – 1761)
Thomas Bayes ( ; c. 1701 – 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem. Bayes never published what would become his most famous accomplishment; his notes were edited and published posthumously by Richard Price.
Biography.
Thomas Bayes was the son of London Presbyterian minister Joshua Bayes, and was possibly born in Hertfordshire. He came from a prominent nonconformist family from Sheffield. In 1719, he enrolled at the University of Edinburgh to study logic and theology. On his return around 1722, he assisted his father at the latter's chapel in London before moving to Tunbridge Wells, Kent, around 1734. There he was minister of the Mount Sion Chapel, until 1752.
He is known to have published two works in his lifetime, one theological and one mathematical:
Bayes was elected as a Fellow of the Royal Society in 1742. His nomination letter was signed by Philip Stanhope, Martin Folkes, James Burrow, Cromwell Mortimer, and John Eames. It is speculated that he was accepted by the society on the strength of the "Introduction to the Doctrine of Fluxions", as he is not known to have published any other mathematical work during his lifetime.
In his later years he took a deep interest in probability. Historian Stephen Stigler thinks that Bayes became interested in the subject while reviewing a work written in 1755 by Thomas Simpson, but George Alfred Barnard thinks he learned mathematics and probability from a book by Abraham de Moivre. Others speculate he was motivated to rebut David Hume's argument against believing in miracles on the evidence of testimony in "An Enquiry Concerning Human Understanding". His work and findings on probability theory were passed in manuscript form to his friend Richard Price after his death.
By 1755, he was ill, and by 1761, he had died in Tunbridge Wells. He was buried in Bunhill Fields burial ground in Moorgate, London, where many nonconformists lie.
In 2018, the University of Edinburgh opened a £45 million research centre connected to its informatics department named after its alumnus, Bayes.
In April 2021, it was announced that Cass Business School, whose City of London campus is on Bunhill Row, was to be renamed after Bayes.
Bayes' theorem.
Bayes's solution to a problem of inverse probability was presented in "An Essay Towards Solving a Problem in the Doctrine of Chances", which was read to the Royal Society in 1763 after Bayes's death. Richard Price shepherded the work through this presentation and its publication in the "Philosophical Transactions of the Royal Society of London" the following year. This was an argument for using a uniform prior distribution for a binomial parameter and not merely a general postulate. This essay gives the following theorem (stated here in present-day terminology).
Suppose a quantity "R" is uniformly distributed between 0 and 1. Suppose each of "X"1, ..., "X""n" is equal to either 1 or 0 and the conditional probability that any of them is equal to 1, given the value of "R", is "R". Suppose they are conditionally independent given the value of "R". Then the conditional probability distribution of "R", given the values of "X"1, ..., "X""n", is
formula_0
Thus, for example,
formula_1
This is a special case of the Bayes' theorem.
In the first decades of the eighteenth century, many problems concerning the probability of certain events, given specified conditions, were solved. For example: given a specified number of white and black balls in an urn, what is the probability of drawing a black ball? Or the converse: given that one or more balls has been drawn, what can be said about the number of white and black balls in the urn? These are sometimes called "inverse probability" problems.
Bayes's "Essay" contains his solution to a similar problem posed by Abraham de Moivre, author of "The Doctrine of Chances" (1718).
In addition, a paper by Bayes on asymptotic series was published posthumously.
Bayesianism.
Bayesian probability is the name given to several related interpretations of probability as an amount of epistemic confidence – the strength of beliefs, hypotheses etc. – rather than a frequency. This allows the application of probability to all sorts of propositions rather than just ones that come with a reference class. "Bayesian" has been used in this sense since about 1950. Since its rebirth in the 1950s, advancements in computing technology have allowed scientists from many disciplines to pair traditional Bayesian statistics with random walk techniques. The use of the Bayes theorem has been extended in science and in other fields.
Bayes himself might not have embraced the broad interpretation now called Bayesian, which was in fact pioneered and popularised by Pierre-Simon Laplace; it is difficult to assess Bayes's philosophical views on probability, since his essay does not go into questions of interpretation. There, Bayes defines "probability" of an event as "the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening" (Definition 5). In modern utility theory, the same definition would result by rearranging the definition of expected utility (the probability of an event times the payoff received in case of that event – including the special cases of buying risk for small amounts or buying security for big amounts) to solve for the probability. As Stigler points out, this is a subjective definition, and does not require repeated events; however, it does require that the event in question be observable, for otherwise it could never be said to have "happened". Stigler argues that Bayes intended his results in a more limited way than modern Bayesians. Given Bayes's definition of probability, his result concerning the parameter of a binomial distribution makes sense only to the extent that one can bet on its observable consequences.
The philosophy of Bayesian statistics is at the core of almost every modern estimation approach that includes conditioned probabilities, such as sequential estimation, probabilistic machine learning techniques, risk assessment, simultaneous localization and mapping, regularization or information theory. The rigorous axiomatic framework for probability theory as a whole, however, was developed 200 years later during the early and middle 20th century, starting with insightful results in ergodic theory by Plancherel in 1913.
See also.
<templatestyles src="Div col/styles.css"/>
Notes.
<templatestyles src="Reflist/styles.css" />
References.
Citations.
<templatestyles src="Reflist/styles.css" />
Sources.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\n\\frac {(n+1)!}{S!(n-S)!} r^S (1-r)^{n-S} \\, dr \\quad \\text{for }0\\le r\\le 1, \\text{ where } S=X_1+\\cdots+X_n.\n"
},
{
"math_id": 1,
"text": " \\Pr(R \\le r_0 \\mid X_1,\\ldots,X_n) = \\frac{(n+1)!}{S!(n-S)!} \\int_0^{r_0} r^S (1-r)^{n-S} \\, dr.\n"
}
] |
https://en.wikipedia.org/wiki?curid=149117
|
14913760
|
Scheffé's method
|
In statistics, Scheffé's method, named after American statistician Henry Scheffé, is a method for adjusting significance levels in a linear regression analysis to account for multiple comparisons. It is particularly useful in analysis of variance (a special case of regression analysis), and in constructing simultaneous confidence bands for regressions involving basis functions.
Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method. It works on similar principles as the Working–Hotelling procedure for estimating mean responses in regression, which applies to the set of all possible factor levels.
The method.
Let formula_0 be the means of some variable in formula_1 disjoint populations.
An arbitrary contrast is defined by
formula_2
where
formula_3
If formula_0 are all equal to each other, then all contrasts among them are 0. Otherwise, some contrasts differ from 0.
Technically there are infinitely many contrasts. The simultaneous confidence coefficient is exactly formula_4, whether the factor level sample sizes are equal or unequal. (Usually only a finite number of comparisons are of interest. In this case, Scheffé's method is typically quite conservative, and the family-wise error rate (experimental error rate) will generally be much smaller than formula_5.)
We estimate formula_6 by
formula_7
for which the estimated variance is
formula_8
where
It can be shown that the probability is formula_4 that all confidence limits of the type
formula_13
are simultaneously correct, where as usual formula_14 is the size of the whole population. Norman R. Draper and Harry Smith, in their 'Applied Regression Analysis' (see references), indicate that formula_1 should be in the equation in place of formula_15. The slip with formula_15 is a result of failing to allow for the additional effect of the constant term in many regressions. That the result based on formula_15 is wrong is readily seen by considering formula_16, as in a standard simple linear regression. That formula would then reduce to one with the usual formula_17-distribution, which is appropriate for predicting/estimating for a single value of the independent variable, not for constructing a confidence band for a range of values of the independent value. Also note that the formula is for dealing with the mean values for a range of independent values, not for comparing with individual values such as individual observed data values.
Denoting Scheffé significance in a table.
Frequently, subscript letters are used to indicate which values are significantly different using the Scheffé method. For example, when mean values of variables that have been analyzed using an ANOVA are presented in a table, they are assigned a different letter subscript based on a Scheffé contrast. Values that are not significantly different based on the post-hoc Scheffé contrast will have the same subscript and values that are significantly different will have different subscripts (i.e. 15a, 17a, 34b would mean that the first and second variables both differ from the third variable but not each other because they are both assigned the subscript "a").
Comparison with the Tukey–Kramer method.
If only a fixed number of pairwise comparisons are to be made, the Tukey–Kramer method will result in a more precise confidence interval. In the general case when many or all contrasts might be of interest, the Scheffé method is more appropriate and will give narrower confidence intervals in the case of a large number of comparisons.
References.
<templatestyles src="Reflist/styles.css" />
External links.
This article incorporates public domain material from
|
[
{
"math_id": 0,
"text": "\\mu_1, \\ldots , \\mu_r "
},
{
"math_id": 1,
"text": "r "
},
{
"math_id": 2,
"text": "C = \\sum_{i=1}^r c_i\\mu_i"
},
{
"math_id": 3,
"text": "\\sum_{i=1}^r c_i = 0."
},
{
"math_id": 4,
"text": "1- \\alpha "
},
{
"math_id": 5,
"text": "\\alpha "
},
{
"math_id": 6,
"text": "C "
},
{
"math_id": 7,
"text": "\\hat{C} = \\sum_{i=1}^r c_i\\bar{Y}_i"
},
{
"math_id": 8,
"text": "s_{\\hat{C}}^2 = \\hat{\\sigma}_e^2\\sum_{i=1}^r \\frac{c_i^2}{n_i},"
},
{
"math_id": 9,
"text": "n_i "
},
{
"math_id": 10,
"text": "i "
},
{
"math_id": 11,
"text": "\\mu_ i "
},
{
"math_id": 12,
"text": "\\hat{\\sigma}_e^2"
},
{
"math_id": 13,
"text": "\\hat{C}\\pm\\,s_\\hat{C}\\sqrt{\\left(r-1\\right)F_{\\alpha;r-1;N-r}} "
},
{
"math_id": 14,
"text": "N "
},
{
"math_id": 15,
"text": "r-1 "
},
{
"math_id": 16,
"text": "r=2 "
},
{
"math_id": 17,
"text": "t"
}
] |
https://en.wikipedia.org/wiki?curid=14913760
|
149183
|
Millennials
|
Generational cohort born 1981 to 1996
Millennials, also known as Generation Y or Gen Y, are the demographic cohort following Generation X and preceding Generation Z. Researchers and popular media use the early 1980s as starting birth years and the mid-1990s to early 2000s as ending birth years, with the generation typically being defined as people born from 1981 to 1996. Most Millennials are the children of Baby Boomers and older Generation X. In turn Millennials are often the parents of Generation Alpha.
As the first generation to grow up with the Internet, Millennials have been described as the first global generation. The generation is generally marked by elevated usage of and familiarity with the Internet, mobile devices, social media, and technology in general. The term "digital natives", which is now also applied to successive generations, was originally coined to describe this generation. Between the 1990s and 2010s, people from developing countries became increasingly well-educated, a factor that boosted economic growth in these countries. In contrast, Millennials across the world have suffered significant economic disruption since starting their working lives, with many facing high levels of youth unemployment in the wake of the Great Recession and the COVID-19 recession.
Millennials have been called the "Unluckiest Generation" as the average Millennial has experienced slower economic growth and more recessions since entering the workforce than any other generation in history. They have also been weighed down by student debt and childcare costs. Across the globe, Millenials and subsequent generations have postponed marriage or living together as a couple. Millennials were born at a time of declining fertility rates around the world, and continue to have fewer children than their predecessors. Those in developing countries will continue to constitute the bulk of global population growth. In developed countries, young people of the 2010s were less inclined to have sex compared to their predecessors when they were the same age. Millennials in the West are less likely to be religious than their predecessors, but may identify as spiritual.
Terminology and etymology.
Members of this demographic cohort are known as Millennials because the oldest became adults around the turn of the millennium. Authors William Strauss and Neil Howe, known for creating the Strauss–Howe generational theory, are widely credited with naming the Millennials. They coined the term in 1987, around the time children born in 1982 were entering kindergarten, and the media were first identifying their prospective link to the impending new millennium as the high school graduating class of 2000. They wrote about the cohort in their books "Generations: The History of America's Future, 1584 to 2069" (1991) and "Millennials Rising: The Next Great Generation" (2000).
In August 1993, an "Advertising Age" editorial coined the phrase "Generation Y" to describe teenagers of the day, then aged 13–19 (born 1974–1980), who were at the time defined as different from Generation X. However, the 1974–1980 cohort was later re-identified by most media sources as the last wave of Generation X, and by 2003 "Ad Age" had moved their Generation Y starting year up to 1982. According to journalist Bruce Horovitz, in 2012, "Ad Age" "threw in the towel by conceding that Millennials is a better name than Gen Y," and by 2014, a past director of data strategy at "Ad Age" said to NPR "the Generation Y label was a placeholder until we found out more about them."
Millennials are sometimes called "Echo Boomers", due to them often being the offspring of the Baby Boomers, the significant increase in birth rates from the early 1980s to mid-1990s, and their generation's large size relative to that of Boomers. In the United States, the echo boom's birth rates peaked in August 1990 and a twentieth-century trend toward smaller families in developed countries continued. Psychologist Jean Twenge described Millennials as "Generation Me" in her 2006 book "Generation Me: Why Today's Young Americans Are More Confident, Assertive, Entitled – and More Miserable Than Ever Before", while in 2013, "Time" magazine ran a cover story titled "Millennials: The Me Me Me Generation". Alternative names for this group proposed include the "Net Generation", "Generation 9/11", "Generation Next", and "The Burnout Generation".
Date and age range definitions.
Oxford Living Dictionaries describes a Millennial as a person "born between the early 1980s and the late 1990s." Merriam-Webster Dictionary defines Millennial as "a person born in the 1980s or 1990s". More detailed definitions in use are as follows:
Jonathan Rauch, senior fellow at the Brookings Institution, wrote for "The Economist" in 2018 that "generations are squishy concepts", but the 1981 to 1996 birth cohort is a "widely accepted" definition for Millennials. Reuters also state that the "widely accepted definition" is 1981–1996.
The Pew Research Center defines Millennials as the people born from 1981 to 1996, choosing these dates for "key political, economic and social factors", including the 11 September terrorist attacks, the 2003 invasion of Iraq, Great Recession, and Internet explosion. The United States Library of Congress explains that date ranges are 'subjective' and the traits of each cohort are generalized based around common economic, social, or political factors that happened during formative years. They acknowledge disagreements, complaints over date ranges, generation names, and the overgeneralized "personality" of each generation. They suggest that marketers and journalists use the different groupings to target their marketing to particular age groups. However, they cite Pew's 1981–1996 definition to define Millennials. Various media outlets and statistical organizations have cited Pew's definition including "Time" magazine, BBC News, "The New York Times", "The Guardian", the United States Bureau of Labor Statistics, and Statistics Canada.
The Brookings Institution defines the Millennial generation as people born from 1981 to 1996, as does Gallup, Federal Reserve Board, and the American Psychological Association. Encyclopædia Britannica defines Millennials as "the term used to describe a person born between 1981 and 1996, though different sources can vary by a year or two." Although the United States Census Bureau have said that "there is no official start and end date for when Millennials were born" and they do not officially define Millennials, a U.S. Census publication in 2022 noted that Millennials are "colloquially defined as the cohort born from 1981 to 1996", using this definition in a breakdown of Survey of Income and Program Participation (SIPP) data.
The Australian Bureau of Statistics uses the years 1981 to 1995 to define Millennials in a 2021 Census report. A report by Ipsos MORI describes the term 'Millennials' as a working title for the circa 15-year birth cohort born around 1980 to 1995, which has 'unique, defining traits'. Governmental institutions such as the UK Department of Health and Social Care have also used 1980 to 1995. Psychologist Jean Twenge defines millennials as those born from 1980 to 1994. Likewise, Australia's McCrindle Research uses the years 1980 to 1994 as Generation Y (millennial) birth years.
A 2023 report by the Population Reference Bureau defines Millennials as those born from 1981 to 1999. CNN reports that studies sometimes define Millennials as born between 1980 and 2000. A 2017 BBC report has also referred to this age range in reference to that used by National Records of Scotland. In the UK, the Resolution Foundation uses 1981–2000. The U.S. Government Accountability Office defines Millennials as those born between 1982 and 2000. Sociologist Elwood Carlson, who calls the generation "New Boomers", identified the birth years of 1983–2001, based on the upswing in births after 1983 and finishing with the "political and social challenges" that occurred after the 11 September terrorist acts. Author Neil Howe, co-creator of the Strauss–Howe generational theory, defines Millennials as being born from 1982 to 2004.
The cohorts born during the cusp years before and after Millennials have been identified as "microgenerations" with characteristics of both generations. Names given to these cuspers include "Xennials", "Generation Catalano", the "Oregon Trail Generation"; "Zennials" and "Zillennials", respectively. The term "Geriatric Millennial" gained popularity in 2021 to describe those born in the beginning half of the 1980s between 1980 and 1985. The term has since been used and discussed by various media outlets including Today, CTV News, HuffPost, news.com.au, "The Irish Times", and Business Insider.
Psychology.
Psychologist Jean Twenge, the author of the 2006 book "Generation Me", considers millennials, along with younger members of Generation X, to be part of what she calls "Generation Me". Twenge attributes millennials with the traits of confidence and tolerance, but also describes a sense of entitlement and narcissism, based on NPI surveys showing increased narcissism among millennials compared to preceding generations when they were teens and in their twenties. Psychologist Jeffrey Arnett of Clark University, Worcester has criticized Twenge's research on narcissism among millennials, stating "I think she is vastly misinterpreting or over-interpreting the data, and I think it's destructive". He doubts that the Narcissistic Personality Inventory really measures narcissism at all. Arnett says that not only are millennials less narcissistic, they're "an exceptionally generous generation that holds great promise for improving the world". A study published in 2017 in the journal "Psychological Science" found a small "decline" in narcissism among young people since the 1990s.
Authors William Strauss and Neil Howe argue that each generation has common characteristics that give it a specific character with four basic generational archetypes, repeating in a cycle. According to their hypothesis, they predicted millennials would become more like the "civic-minded" G.I. Generation with a strong sense of community both local and global. Strauss and Howe ascribe seven basic traits to the millennial cohort: special, sheltered, confident, team-oriented, conventional, pressured, and achieving. However, Arthur E. Levine, author of "When Hope and Fear Collide: A Portrait of Today's College Student", dismissed these generational images as "stereotypes". In addition, psychologist Jean Twenge says Strauss and Howe's assertions are overly deterministic, non-falsifiable, and unsupported by rigorous evidence.
Polling agency Ipsos-MORI warned that the word "millennials" is "misused to the point where it's often mistaken for just another meaningless buzzword" because "many of the claims made about millennial characteristics are simplified, misinterpreted or just plain wrong, which can mean real differences get lost" and that "[e]qually important are the similarities between other generations—the attitudes and behaviors that are staying the same are sometimes just as important and surprising."
Though it is often said that millennials ignore conventional advertising, they are in fact heavily influenced by it. They are particularly sensitive to appeals to transparency, to experiences rather than things, and flexibility.
A 2015 study by Microsoft found that 77% of respondents aged 18 to 24 said yes to the statement, "When nothing is occupying my attention, the first thing I do is reach for my phone," compared to just 10% for those aged 65 and over.
The term has been used to denote anxiety experienced by many Japanese Millennials struggling with a sense of and self-blaming, caused by a vast array of issues from unemployment, poverty, family problems, bullying, social withdrawal and mental ill-health.
Cognitive abilities.
Intelligence researcher James R. Flynn discovered that back in the 1950s, the gap between the vocabulary levels of adults and children was much smaller than it is in the early twenty-first century. Between 1953 and 2006, adult gains on the vocabulary subtest of the Wechsler IQ test were 17.4 points whereas the corresponding gains for children were only 4. He asserted that some of the reasons for this are the surge in interest in higher education and cultural changes. The number of Americans pursuing tertiary qualifications and cognitively demanding jobs has risen significantly since the 1950s. This boosted the level of vocabulary among adults. Back in the 1950s, children generally imitated their parents and adopted their vocabulary. This was no longer the case in the 2000s, when teenagers often developed their own subculture and as such were less likely to use adult-level vocabulary on their essays.
In a 2009 report, Flynn analyzed the results of the Raven's Progressive Matrices test for British fourteen-year-olds from 1980 to 2008. He discovered that their average IQ had dropped by more than two points during that time period. Among those in the higher half of the intelligence distribution, the decline was even more significant, six points. This is a clear case of the reversal of the Flynn effect, the apparent rise in IQ scores observed during the twentieth century. Flynn suspected that this was due to changes in British youth culture. He further noted that in the past, IQ gains had been correlated with socioeconomic class, but this was no longer true.
Psychologists Jean Twenge, W. Keith Campbell, and Ryne A. Sherman analyzed vocabulary test scores on the U.S. General Social Survey (formula_0) and found that after correcting for education, the use of sophisticated vocabulary has declined between the mid-1970s and the mid-2010s across all levels of education, from below high school to graduate school. Those with at least a bachelor's degree saw the steepest decline. Hence, the gap between people who never received a high-school diploma and a university graduate has shrunk from an average of 3.4 correct answers in the mid- to late-1970s to 2.9 in the early- to mid-2010s. Higher education offers little to no benefits to verbal ability. Because those with only a moderate level of vocabulary were more likely to be admitted to university than in the past, the average for degree holders declined. There are various explanations for this. Accepting high levels of immigrants, many of whom not particularly proficient in the English language, could lower the national adult average. Young people nowadays are much less likely to read for pleasure, thus reducing their levels of vocabulary. On the other hand, while the College Board has reported that SAT verbal scores were on the decline, these scores are an imperfect measure of the vocabulary level of the nation as a whole because the test-taking demographic has changed and because more students take the SAT in the 2010s than in the 1970s, which means there are more with limited ability who took it. Population aging is unconvincing because the effect is too weak.
Cultural identity.
In the United States.
A 2007 report by the National Endowment of the Arts stated that as a group, American adults were reading for pleasure less often than before. In particular, Americans aged 15 to 24 spent an average of two hours watching television and only seven minutes on reading. In 2002, only 52% of Americans between the ages of 18 and 24 voluntarily read books, down from 59% in 1992. Reading comprehension skills of American adults of all levels of education deteriorated between the early 1990s and the early 2000s, especially among those with advanced degrees. According to employers, almost three quarters of university graduates were "deficient" in English writing skills. Meanwhile, the reading scores of American tenth-graders proved mediocre, in fifteenth place out of 31 industrialized nations, and the number of twelfth-graders who had never read for pleasure doubled to 19%.
Publishers and booksellers observed that the sales of adolescent and young-adult fiction remained strong. This could be because older adults were buying titles intended for younger people, which inflated the market, and because there were fewer readers buying more books.
By the late 2010s, viewership of late-night American television among adults aged 18 to 49, the most important demographic group for advertisers, has fallen substantially despite an abundance of materials. This is due in part to the availability and popularity of streaming services. However, when delayed viewing within three days is taken into account, the top shows all saw their viewership numbers boosted. This development undermines the current business model of the television entertainment industry. "If the sky isn't exactly falling on the broadcast TV advertising model, it certainly seems to be a lot closer to the ground than it once was," wrote reporter Anthony Crupi for "Ad Age". Despite having the reputation for "killing" many things of value to the older generations, millennials and Generation Z are nostalgically preserving Polaroid cameras, vinyl records, needlepoint, and home gardening, to name a few. In fact, Millennials are a key cohort behind the vinyl revival. However, due to the COVID-19 pandemic in the early 2020s, certain items whose futures were in doubt due to a general lack of interest by millennials appear to be reviving with stronger sales than in previous years, such as canned food.
A 2019 poll by Ypulse found that among people aged 27 to 37, the musicians most representative of their generation were Taylor Swift, Beyoncé, the Backstreet Boys, Michael Jackson, Drake, and Eminem. (The last two were tied in fifth place.)
Since the 2000 U.S. Census, millennials have taken advantage of the possibility of selecting more than one racial group in abundance. In 2015, the Pew Research Center conducted research regarding generational identity that said a majority of millennials surveyed did not like the "millennial" label. It was discovered that millennials are less likely to strongly identify with the generational term when compared to Generation X or the baby boomers, with only 40% of those born between 1981 and 1997 identifying as millennials. Among older millennials, those born 1981–1988, Pew Research found that 43% personally identified as members of the older demographic cohort, Generation X, while only 35% identified as millennials. Among younger millennials (born 1989–1997), generational identity was not much stronger, with only 45% personally identifying as millennials. It was also found that millennials chose most often to define themselves with more negative terms such as self-absorbed, wasteful, or greedy.
Fred Bonner, a Samuel DeWitt Proctor Chair in Education at Rutgers University and author of "Diverse Millennial Students in College: Implications for Faculty and Student Affairs", believes that much of the commentary on the Millennial Generation may be partially correct, but overly general and that many of the traits they describe apply primarily to "white, affluent teenagers who accomplish great things as they grow up in the suburbs, who confront anxiety when applying to super-selective colleges, and who multitask with ease as their helicopter parents hover reassuringly above them." During class discussions, Bonner listened to black and Hispanic students describe how some or all of the so-called core traits did not apply to them. They often said that the "special" trait, in particular, is unrecognizable. Other socioeconomic groups often do not display the same attributes commonly attributed to millennials. "It's not that many diverse parents don't want to treat their kids as special," he says, "but they often don't have the social and cultural capital, the time and resources, to do that."
The University of Michigan's "Monitoring the Future" study of high school seniors (conducted continually since 1975) and the American Freshman Survey, conducted by UCLA's Higher Education Research Institute of new college students since 1966, showed an increase in the proportion of students who consider wealth a very important attribute, from 45% for Baby Boomers (surveyed between 1967 and 1985) to 70% for Gen Xers, and 75% for millennials. The percentage who said it was important to keep abreast of political affairs fell, from 50% for Baby Boomers to 39% for Gen Xers, and 35% for millennials. The notion of "developing a meaningful philosophy of life" decreased the most across generations, from 73% for Boomers to 45% for millennials. The willingness to be involved in an environmental cleanup program dropped from 33% for Baby Boomers to 21% for millennials.
In general and in other countries.
Political scientist Shirley Le Penne argues that for Millennials "pursuing a sense of belonging becomes a means of achieving a sense of being needed... Millennials experience belonging by seeking to impact the world." Educational psychologist Elza Venter believes Millennials are digital natives because they have grown up experiencing digital technology and have known it all their lives. Prensky coined the concept "digital natives" because the members of the generation are "native speakers of the digital language of computers, video games and the internet". This generation's older members use a combination of face-to-face communication and computer mediated communication, while its younger members use mainly electronic and digital technologies for interpersonal communication.
A 2013 survey of almost a thousand Britons aged 18 to 24 found that 62% had a favorable opinion of the British Broadcasting Corporation (BBC) and 70% felt proud of their national history. In 2017, research suggested nearly half of 18 to 34 year olds living in the UK had attended a live music event in the previous year.
<templatestyles src="Template:Quote_box/styles.css" />
Computer games and computer culture has led to a decrease in reading books. The tendency for teachers to now "teach to the test" has also led to a decrease in the capacity to think in lateral ways.
Richard House, Roehampton University
Having faced the full brunt of the Great Recession, Millennials in Europe tended to be pessimistic about the future direction of their countries, though there were significant differences, the Pew Research Center found in 2014. Millennials from countries with relatively healthy economies such as Germany and the United Kingdom were generally happier than their counterparts from struggling economies, such as Spain, Italy, and Greece. On the other hand, the young were more likely than the old to feel optimistic.
Millennials came of age in a time where the entertainment industry began to be affected by the Internet. Using artificial intelligence, Joan Serrà and his team at the Spanish National Research Council studied the massive Million Song Dataset and found that between 1955 and 2010, popular music has gotten louder, while the chords, melodies, and types of sounds used have become increasingly homogenized. Indeed, producers seem to be engaging in a "Loudness war", with the intention of attracting more and more audience members. Serrà and his colleagues wrote, "...old tune with slightly simpler chord progressions, new instrument sonorities that were in agreement with current tendencies, and recorded with modern techniques that allowed for increased loudness levels could be easily perceived as novel, fashionable, and groundbreaking." While the music industry has long been accused of producing songs that are louder and blander, this is the first time the quality of songs is comprehensively studied and measured. Additional research showed that within the past few decades, popular music has gotten slower; that majorities of listeners young and old preferred older songs rather than keeping up with new ones; that the language of popular songs were becoming more negative psychologically; and that lyrics were becoming simpler and more repetitive, approaching one-word sheets, something measurable by observing how efficiently lossless compression algorithms (such as the LZ algorithm) handled them.
In modern society, there are inevitably people who refuse to conform to the dominant culture and seek to do the exact opposite; given enough time, the anti-conformists will become more homogeneous with respect to their own subculture, making their behavior the opposite to any claims of counterculture. This synchronization occurs even if more than two choices are available, such as multiple styles of beard rather than whether or not to have a beard. Mathematician Jonathan Touboul of Brandeis University who studies how information propagation through society affects human behavior calls this the hipster effect.
Once a highly successful genre on radio and then television, soap operas—characterized by melodramatic plots focused on interpersonal affairs and cheap production value—has been declining in viewership since the 1990s. Experts believe that this is due to their failure to attract younger demographics, the tendency of modern audiences to have shorter attention spans, and the rise of reality television in the 1990s. Nevertheless, Internet streaming services do offer materials in the serial format, a legacy of soap operas. However, the availability of such on-demand platforms saw to it that soap operas would never again be the cultural phenomenon they were in the twentieth century, especially among the younger generations, not least because cliffhangers could no longer capture the imagination of the viewers the way they did in the past, when television shows were available as scheduled, not on demand.
Demographics.
Asia.
Chinese millennials are commonly called the post-80s and post-90s generations. At a 2015 conference in Shanghai organized by University of Southern California's US–China Institute, millennials in China were examined and contrasted with American millennials. Findings included millennials' marriage, childbearing, and child raising preferences, life and career ambitions, and attitudes towards volunteerism and activism. Due to the one-child policy introduced in the late 1970s, one-child households have become the norm in China, leading to rapid population aging, especially in the cities where the costs of living are much higher than in the countryside.
As a result of cultural ideals, government policy, and modern medicine, there has been severe gender imbalances in China and India. According to the United Nations, in 2018, there were 112 Chinese males aged 15 to 29 for every hundred females in that age group. That number in India was 111. China had a total of 34 million excess males and India 37 million, more than the entire population of Malaysia. Such a discrepancy fuels loneliness epidemics, human trafficking (from elsewhere in Asia, such as Cambodia and Vietnam), and prostitution, among other societal problems.
Singapore's birth rate has fallen below the replacement level of 2.1 since the 1980s before stabilizing by during the 2000s and 2010s. (It reached 1.14 in 2018, making it the lowest since 2010 and one of the lowest in the world.) Government incentives such as the baby bonus have proven insufficient to raise the birth rate. Singapore's experience mirrors those of Japan and South Korea.
Vietnam's median age in 2018 was 26 and rising. Between the 1970s and the late 2010s, life expectancy climbed from 60 to 76. It is now the second highest in Southeast Asia. Vietnam's fertility rate dropped from 5 in 1980 to 3.55 in 1990 and then to 1.95 in 2017. In that same year, 23% of the Vietnamese population was 15 years of age or younger, down from almost 40% in 1989. Other rapidly growing Southeast Asian countries, such as the Philippines, saw similar demographic trends.
Europe.
From about 1750 to 1950, most of Western Europe transitioned from having both high birth and death rates to low birth and death rates. By the late 1960s and 1970s, the average woman had fewer than two children, and, although demographers at first expected a "correction", such a rebound came only for a few countries. Despite a bump in the total fertility rates (TFR) of some European countries in the very late twentieth century (the 1980s and 1990s), especially France and Scandinavia, it returned to replacement level only in Sweden (reaching a TFR of 2.14 in 1990, up from 1.68 in 1980), along with Ireland and Iceland; the bump in Sweden was largely due to improving economic output and the generous, far-reaching family benefits granted by the Nordic welfare system, while in France it was mostly driven by older women realizing their dreams of motherhood. For Sweden, the increase in the fertility rate came with a rise in the birth rate (going from 11.7 in 1980 to 14.5 in 1990), which slowed down and then stopped for a brief period to the aging of the Swedish population caused by the decline in birth rates in the late 1970s and early 1980s. To this day, France and Sweden still have higher fertility rates than most of Europe, and both almost reached replacement level in 2010 (2.03 and 1.98 respectively).
At first, falling fertility is due to urbanization and decreased infant mortality rates, which diminished the benefits and increased the costs of raising children. In other words, it became more economically sensible to invest more in fewer children, as economist Gary Becker argued. (This is the first demographic transition.) Falling fertility then came from attitudinal shifts. By the 1960s, people began moving from traditional and communal values towards more expressive and individualistic outlooks due to access to and aspiration of higher education, and to the spread of lifestyle values once practiced only by a tiny minority of cultural elites. (This is the second demographic transition.) Although the momentous cultural changes of the 1960s leveled off by the 1990s, the social and cultural environment of the very late twentieth-century was quite different from that of the 1950s. Such changes in values have had a major effect on fertility. Member states of the European Economic Community saw a steady increase in not just divorce and out-of-wedlock births between 1960 and 1985 but also falling fertility rates. In 1981, a survey of countries across the industrialized world found that while more than half of people aged 65 and over thought that women needed children to be fulfilled, only 35% of those between the ages of 15 and 24 (younger Baby Boomers and older Generation X) agreed. In the early 1980s, East Germany, West Germany, Denmark, and the Channel Islands had some of the world's lowest fertility rates.
At the start of the twenty-first century, Europe suffers from an aging population. This problem is especially acute in Eastern Europe, whereas in Western Europe, it is alleviated by international immigration. In addition, an increasing number of children born in Europe has been born to non-European parents. Because children of immigrants in Europe tend to be about as religious as they are, this could slow the decline of religion (or the growth of secularism) in the continent as the twenty-first century progresses. In the United Kingdom, the number of foreign-born residents stood at 6% of the population in 1991. Immigration subsequently surged and has not fallen since (as of 2018). Research by the demographers and political scientists Eric Kaufmann, Roger Eatwell, and Matthew Goodwin suggest that such a fast ethno-demographic change is one of the key reasons behind public backlash in the form of national populism across the rich liberal democracies, an example of which is the 2016 United Kingdom European Union membership referendum (Brexit).
Italy is a country where the problem of an aging population is especially acute. The fertility rate dropped from about four in the 1960s down to 1.2 in the 2010s. This is not because young Italians do not want to procreate. Quite the contrary, having many children is an Italian ideal. But its economy has been floundering since the Great Recession of 2007–08, with the youth unemployment rate at a staggering 35% in 2019. Many Italians have moved abroad—150,000 did in 2018—and many are young people pursuing educational and economic opportunities. With the plunge in the number of births each year, the Italian population is expected to decline in the next five years. Moreover, the Baby Boomers are retiring in large numbers, and their numbers eclipse those of the young people taking care of them. Only Japan has an age structure more tilted towards the elderly.
Greece also suffers from a serious demographic problem as many young people are leaving the country in search of better opportunities elsewhere in the wake of the Great Recession. This brain drain and a rapidly aging population could spell disaster for the country.
Overall, E.U. demographic data shows that the number of people aged 18 to 33 in 2014 was 24% of the population, with a high of 28% for Poland and a low of 19% for Italy.
As a result of the shocks due to the decline and dissolution of the Soviet Union, Russia's birth rates began falling in the late 1980s while death rates have risen, especially among men. In the early 2000s, Russia had not only a falling birth rate but also a declining population despite having an improving economy. Between 1992 and 2002, Russia's population dropped from 149 million to 144 million. According to the "medium case scenario" of the U.N.'s Population Division, Russia could lose another 20 million people by the 2020s.
Europe's demographic reality contributes to its economic troubles. Because the European baby boomers failed to replace themselves, by the 2020s and 2030s, dozens of European nations will find their situation even tougher than before.
Oceania.
Australia's total fertility rate has fallen from above three in the post-war era, to about replacement level (2.1) in the 1970s to below that in the late 2010s. However, immigration has been offsetting the effects of a declining birthrate. In the 2010s, among the residents of Australia, 5% were born in the United Kingdom, 3% from China, 2% from India, and 1% from the Philippines. 84% of new arrivals in the fiscal year of 2016 were below 40 years of age, compared to 54% of those already in the country. Like other immigrant-friendly countries, such as Canada, the United Kingdom, and the United States, Australia's working-age population is expected to grow till about 2025. However, the ratio of people of working age to retirees (the dependency ratio) has gone from eight in the 1970s to about four in the 2010s. It could drop to two by the 2060s, depending in immigration levels. "The older the population is, the more people are on welfare benefits, we need more health care, and there's a smaller base to pay the taxes," Ian Harper of the Melbourne Business School told ABC News (Australia). While the government has scaled back plans to increase the retirement age, to cut pensions, and to raise taxes due to public opposition, demographic pressures continue to mount as the buffering effects of immigration are fading away.
North America.
Historically, the early Anglo-Protestant settlers in the seventeenth century were the most successful group, culturally, economically, and politically, and they maintained their dominance till the early twentieth century. Commitment to the ideals of the Enlightenment meant that they sought to assimilate newcomers from outside of the British Isles, but few were interested in adopting a pan-European identity for the nation, much less turning it into a global melting pot. But in the early 1900s, liberal progressives and modernists began promoting more inclusive ideals for what the national identity of the United States should be. While the more traditionalist segments of society continued to maintain their Anglo-Protestant ethnocultural traditions, universalism and cosmopolitanism started gaining favor among the elites. These ideals became institutionalized after the Second World War, and ethnic minorities started moving towards institutional parity with the once dominant Anglo-Protestants. The Immigration and Nationality Act of 1965 (also known as the Hart-Cellar Act), passed at the urging of President Lyndon B. Johnson, abolished national quotas for immigrants and replaced it with a system that admits a fixed number of persons per year based in qualities such as skills and the need for refuge. Immigration subsequently surged from elsewhere in North America (especially Canada and Mexico), Asia, Central America, and the West Indies. By the mid-1980s, most immigrants originated from Asia and Latin America. Some were refugees from Vietnam, Cuba, Haiti, and other parts of the Americas while others came illegally by crossing the long and largely undefended U.S.-Mexican border. At the same time, the postwar baby boom and subsequently falling fertility rate seemed to jeopardize America's social security system as the Baby Boomers retire in the twenty-first century. Provisional data from the Center for Disease Control and Prevention reveal that U.S. fertility rates have fallen below the replacement level of 2.1 since 1971. (In 2017, it fell to 1.765.)
Millennial population size varies, depending on the definition used. Using its own definition, the Pew Research Center estimated that millennials comprised 27% of the U.S. population in 2014. In the same year, using dates ranging from 1982 to 2004, Neil Howe revised the number to over 95 million people in the U.S. In a 2012 "Time" magazine article, it was estimated that there were approximately 80 million U.S. millennials. The United States Census Bureau, using birth dates ranging from 1982 to 2000, stated the estimated number of U.S. millennials in 2015 was 83.1 million people.
In 2017, fewer than 56% millennial were non-Hispanic whites, compared with more than 84% of Americans in their 70s and 80s, 57% had never been married, and 67% lived in a metropolitan area. According to the Brookings Institution, millennials are the "demographic bridge between the largely white older generations (pre-millennials) and much more racially diverse younger generations (post-millennials)."
By analyzing data from the U.S. Census Bureau, the Pew Research Center estimated that millennials, whom they define as people born between 1981 and 1996, outnumbered baby boomers, born from 1946 to 1964, for the first time in 2019. That year, there were 72.1 million millennials compared to 71.6 million baby boomers, who had previously been the largest living adult generation in the country. Data from the National Center for Health Statistics shows that about 62 million millennials were born in the United States, compared to 55 million members of Generation X, 76 million baby boomers, and 47 million from the Silent Generation. Between 1981 and 1996, an average of 3.9 million millennial babies were born each year, compared to 3.4 million average Generation X births per year between 1965 and 1980. But millennials continue to grow in numbers as a result of immigration and naturalization. In fact, millennials form the largest group of immigrants to the United States in the 2010s. Pew projected that the millennial generation would reach around 74.9 million in 2033, after which mortality would outweigh immigration. Yet 2020 would be the first time millennials (who are between the ages of 24 and 39) find their share of the electorate shrink as the leading wave of Generation Z (aged 18 to 23) became eligible to vote. In other words, their electoral power peaked in 2016. In absolute terms, however, the number of foreign-born millennials continues to increase as they become naturalized citizens. In fact, 10% of American voters were born outside the country by the 2020 election, up from 6% in 2000. The fact that people from different racial or age groups vote differently means that this demographic change will influence the future of the American political landscape. While younger voters hold significantly different views from their elders, they are considerably less likely to vote. Non-whites tend to favor candidates from the Democratic Party while whites by and large prefer the Republican Party.
As of the mid-2010s, the United States is one of the few developed countries that does "not" have a top-heavy population pyramid. In fact, as of 2016, the median age of the U.S. population was younger than that of all other rich nations except Australia, New Zealand, Cyprus, Ireland, and Iceland, whose combined population is only a fraction of the United States. This is because American baby boomers had a higher fertility rate compared to their counterparts from much of the developed world. Canada, Germany, Italy, Japan, and South Korea are all aging rapidly by comparison because their millennials are smaller in number than their parents. This demographic reality puts the United States at an advantage compared to many other major economies as the millennials reach middle age: the nation will still have a significant number of consumers, investors, and taxpayers.
According to the Pew Research Center, "Among men, only 4% of millennials [ages 21 to 36 in 2017] are veterans, compared with 47%" of men in their 70s and 80s, "many of whom came of age during the Korean War and its aftermath." Some of these former military service members are combat veterans, having fought in Afghanistan and/or Iraq. As of 2016, millennials are the majority of the total veteran population. According to the Pentagon in 2016, 19% of Millennials are interested in serving in the military, and 15% have a parent with a history of military service.
Economic prospects and trends.
Trends suggest developments in artificial intelligence and robotics will not result in mass unemployment, but can actually create high-skilled jobs. However, in order to take advantage of this situation, people need to hone skills that machines have not yet mastered, such as teamwork.
By analyzing data from the United Nations and the Global Talent Competitive Index, KDM Engineering found that As of 2019[ [update]], the top five countries for international high-skilled workers are Switzerland, Singapore, the United Kingdom, the United States, and Sweden. Factors taken into account included the ability to attract high-skilled foreign workers, business-friendliness, regulatory environment, the quality of education, and the standard of living. Switzerland is best at retaining talents due to its excellent quality of life. Singapore is home to a world-class environment for entrepreneurs. And the United States offers the most opportunity for growth due to the sheer size of its economy and the quality of higher education and training. As of 2019, these are also some of the world's most competitive economies, according to the World Economic Forum (WEF). In order to determine a country or territory's economic competitiveness, the WEF considers factors such as the trustworthiness of public institutions, the quality of infrastructure, macro-economic stability, the quality of healthcare, business dynamism, labor market efficiency, and innovation capacity.
From 2000–2020, before the COVID pandemic, economic activities tended to concentrate in the large metropolitan areas, such as San Francisco, New York, London, Tokyo and Sydney. Productivity increased enormously as knowledge workers agglomerated. The pandemic led to an increase in remote work, more so in developed countries, aided by technology.
Using a variety of measures, economists have reached the conclusion that the rate of innovation and entrepreneurship has been declining across the Western world between the early 1990s and early 2010s, when it leveled off. In the case of the U.S., one of the most complex economies in existence, economist Nicholas Kozeniauskas explained that "the decline in entrepreneurship is concentrated among the smart" as the share of entrepreneurs with university degrees in that country more than halved between the mid-1980s and the mid-2010s. There are many possible reasons for this: population aging, market concentration, and zombie firms (those with low productivity but are kept alive by subsidies). While employment has become more stable and more suitable, modern economies are so complex they are essentially ossified, making them vulnerable to disruptions.
Education.
Global trends.
From the late 1990s to the late 2010s, education transformed the economic realities of countries worldwide. As the people from developing nations became better educated, they close the gap between them and the developed world. Hence Westerners lost their relative advantage in education, as the world saw more people with high-school diplomas than ever before. The number of people with Bachelor's degree and advanced degrees grew significantly as well. Westerners who only passed secondary school had their income cut in real terms during that same period while those with university degrees had incomes that barely increased on average. The fact that many jobs are suitable for remote work due to modern technology further eroded the relative advantage of education in the Western world, resulting in a backlash against immigration and globalization.
As more and more women became educated in the developing world, more leave the rural areas for the cities, enter the work force and compete with men, sparking resentment among men in those countries.
For information on public support for higher education (for domestic students) in the OECD in 2011, see chart below.
In Europe.
In Sweden, universities are tuition-free, as is the case in Norway, Denmark, Iceland, and Finland. However, Swedish students typically graduate very indebted due to the high cost of living in their country, especially in the large cities such as Stockholm. The ratio of debt to expected income after graduation for Swedes was about 80% in 2013. In the U.S., despite incessant talk of student debt reaching epic proportions, that number stood at 60%. Moreover, about seven out of eight Swedes graduate with debt, compared to one half in the U.S. In the 2008–09 academic year, virtually all Swedish students take advantage of state-sponsored financial aid packages from a govern agency known as the Centrala Studiestödsnämnden (CSN), which include low-interest loans with long repayment schedules (25 years or until the student turns 60). In Sweden, student aid is based on their own earnings whereas in some other countries, such as Germany or the United States, such aid is premised on parental income as parents are expected to help foot the bill for their children's education. In the 2008–09 academic year, Australia, Austria, Japan, the Netherlands, and New Zealand saw an increase in both the average tuition fees of their public universities for full-time domestic students and the percentage of students taking advantage of state-sponsored student aid compared to 1995. In the United States, there was an increase in the former but not the latter.
In 2005, judges in Karlsruhe, Germany, struck down a ban on university fees as unconstitutional on the grounds that it violated the constitutional right of German states to regulate their own higher education systems. This ban was introduced in order to ensure equality of access to higher education regardless of socioeconomic class. Bavarian Science Minister Thomas Goppel told the Associated Press, "Fees will help to preserve the quality of universities." Supporters of fees argued that they would help ease the financial burden on universities and would incentivize students to study more efficiently, despite not covering the full cost of higher education, an average of €8,500 as of 2005. Opponents believed fees would make it more difficult for people to study and graduate on time. Germany also suffered from a brain drain, as many bright researchers moved abroad while relatively few international students were interested in coming to Germany. This has led to the decline of German research institutions.
In the 1990s, due to a combination of financial hardship and the fact that universities elsewhere charged tuition, British universities pressed the government to allow them to take in fees. A nominal tuition fee of £1,000 was introduced in autumn 1998. Because not all parents would be able to pay all the fees in one go, monthly payment options, loans, and grants were made available. Some were concerned that making people pay for higher education may deter applicants. This turned out not to be the case. The number of applications fell by only 3% in 1998, and mainly due to mature students rather than 18-year-olds.
In 2012, £9,000 worth of student fees were introduced. Despite this, the number of people interested in pursuing higher education grew at a faster rate than the UK population. In 2017, almost half of young people in England had received higher education by the age of 30. Prime Minister Tony Blair introduced the goal of having half of young Britons having a university degree in 1999, though the 2010 deadline was missed. What the Prime Minister did not realize, however, is that an oversupply of young people with high levels of education historically precipitated periods of political instability and unrest in various societies, from early modern Western Europe and late Tokugawa Japan to the Soviet Union, modern Iran, and the United States. In any case, demand for higher education in the United Kingdom has remained strong throughout the early 21st century, driven by the need for high-skilled workers from both the public and private sectors. There has been, however, a widening gender gap. As of 2017, women were more likely to attend or to have attended university than men, by 55% against 43%, a difference of 12 percentage points.
Oceania.
In Australia, university tuition fees were introduced in 1989. Regardless, the number of applicants has risen considerably. By the 1990s, students and their families were expected to pay 37% of the cost, up from a quarter in the late 1980s. The most expensive subjects were law, medicine, and dentistry, followed by the natural sciences, and then by the arts and social studies. Under the new funding scheme, the Government of Australia also capped the number of people eligible for higher education, enabling schools to recruits more well-financed (though not necessarily bright) students.
North America.
According to the Pew Research Center, 53% of American millennials attended or were enrolled in university in 2002. For comparison, the number of young people attending university was 44% in 1986. By the 2020s, 39% of millennials had at least a bachelor's degree, more than the Baby Boomers at 25%, the Economist reports.
In the United States today, high school students are generally encouraged to attend college or university after graduation while the options of technical school and vocational training are often neglected. Historically, high schools separated students on career tracks, with programs aimed at students bound for higher education and those bound for the workforce. Students with learning disabilities or behavioral issues were often directed towards vocational or technical schools. All this changed in the late 1980s and early 1990s thanks to a major effort in the large cities to provide more abstract academic education to everybody. The mission of high schools became preparing students for college, referred to as "high school to Harvard." However, this program faltered in the 2010s, as institutions of higher education came under heightened skepticism due to high costs and disappointing results. People became increasingly concerned about debts and deficits. No longer were promises of educating "citizens of the world" or estimates of economic impact coming from abstruse calculations sufficient. Colleges and universities found it necessary to prove their worth by clarifying how much money from which industry and company funded research, and how much it would cost to attend.
Because jobs (that suited what one studied) were so difficult to find in the few years following the Great Recession, the value of getting a liberal arts degree and studying the humanities at an American university came into question, their ability to develop a well-rounded and broad-minded individual notwithstanding. As of 2019, the total college debt has exceeded US$1.5 trillion, and two out of three college graduates are saddled with debt. The average borrower owes US$37,000, up US$10,000 from ten years before. A 2019 survey by TD Ameritrade found that over 18% of millennials (and 30% of Generation Z) said they have considered taking a gap year between high school and college.
In 2019, the Federal Reserve Bank of St. Louis published research (using data from the 2016 "Survey of Consumer Finances") demonstrating that after controlling for race and age cohort families with heads of household with post-secondary education who were born before 1980 there have been wealth and income premiums, while for families with heads of household with post-secondary education but born after 1980 the wealth premium has weakened to point of statistical insignificance (in part because of the rising cost of college) and the income premium while remaining positive has declined to historic lows (with more pronounced downward trajectories with heads of household with postgraduate degrees). Quantitative historian Peter Turchin noted that the United States was overproducing university graduates—he termed this elite overproduction—in the 2000s and predicted, using historical trends, that this would be one of the causes of political instability in the 2020s, alongside income inequality, stagnating or declining real wages, growing public debt. According to Turchin, intensifying competition among graduates, whose numbers were larger than what the economy could absorb, leads to political polarization, social fragmentation, and even violence as many become disgruntled with their dim prospects despite having attained a high level of education. He warned that the turbulent 1960s and 1970s could return, as having a massive young population with university degrees was one of the key reasons for the instability of the past.
According to the American Academy of Arts and Sciences, students were turning away from liberal arts programs. Between 2012 and 2015, the number of graduates in the humanities dropped from 234,737 to 212,512. Consequently, many schools have relinquished these subjects, dismissed faculty members, or closed completely. Data from the National Center for Education Statistics revealed that between 2008 and 2017, the number of people majoring in English plummeted by just over a quarter. At the same time, those in philosophy and religion fell 22% and those who studied foreign languages dropped 16%. Meanwhile, the number of university students majoring in homeland security, science, technology, engineering, and mathematics (STEM), and healthcare skyrocketed. (See figure below.)
According to the U.S. Department of Education, people with technical or vocational trainings are slightly more likely to be employed than those with a bachelor's degree and significantly more likely to be employed in their fields of specialty. The United States currently suffers from a shortage of skilled tradespeople.
Despite the fact that educators and political leaders, such as President Barack Obama, have been trying to years to improve the quality of STEM education in the United States, and that various polls have demonstrated that more students are interested in these subjects, graduating with a STEM degree is a different kettle of fish altogether. According to "The Atlantic", 48% of students majoring in STEM dropped out of their programs between 2003 and 2009. Data collected by the University of California, Los Angeles, (UCLA) in 2011 showed that although these students typically came in with excellent high school GPAs and SAT scores, among science and engineering students, including pre-medical students, 60% changed their majors or failed to graduate, twice the attrition rate of all other majors combined. Despite their initial interest in secondary school, many university students find themselves overwhelmed by the reality of a rigorous STEM education. Some are mathematically unskilled, while others are simply lazy. The National Science Board raised the alarm all the way back in the mid-1980s that students often forget why they wanted to be scientists and engineers in the first place. Many bright students had an easy time in high school and failed to develop good study habits. In contrast, Chinese, Indian, and Singaporean students are exposed to mathematics and science at a high level from a young age. Moreover, according education experts, many mathematics schoolteachers were not as well-versed in their subjects as they should be, and might well be uncomfortable with mathematics. Given two students who are equally prepared, the one who goes to a more prestigious university is less likely to graduate with a STEM degree than the one who attends a less difficult school. Competition can defeat even the top students. Meanwhile, grade inflation is a real phenomenon in the humanities, giving students an attractive alternative if their STEM ambitions prove too difficult to achieve. Whereas STEM classes build on top of each other—one has to master the subject matter before moving to the next course—and have black and white answers, this is not the case in the humanities, where things are a lot less clear-cut.
In 2015, educational psychologist Jonathan Wai analyzed average test scores from the Army General Classification Test in 1946 (10,000 students), the Selective Service College Qualification Test in 1952 (38,420), Project Talent in the early 1970s (400,000), the Graduate Record Examination between 2002 and 2005 (over 1.2 million), and the SAT Math and Verbal in 2014 (1.6 million). Wai identified one consistent pattern: those with the highest test scores tended to pick the physical sciences and engineering as their majors while those with the lowest were more likely to choose education. (See figure below.)
During the 2010s, the mental health of American graduate students in general was in a state of crisis.
Historical knowledge.
A February 2018 survey of 1,350 individuals found that 66% of the American millennials (and 41% of all U.S. adults) surveyed did not know what Auschwitz was, while 41% incorrectly claimed that 2 million Jews or fewer were killed during the Holocaust, and 22% said that they had never heard of the Holocaust. Over 95% of American millennials were unaware that a portion of the Holocaust occurred in the Baltic states, which lost over 90% of their pre-war Jewish population, and 49% were not able to name a single Nazi concentration camp or ghetto in German-occupied Europe. However, at least 93% surveyed believed that teaching about the Holocaust in school is important and 96% believed the Holocaust happened.
The YouGov survey found that 42% of American millennials have never heard of Mao Zedong, who ruled China from 1949 to 1976 and was responsible for the deaths of 20–45 million people; another 40% are unfamiliar with Che Guevara.
Health and welfare.
According to a 2018 report from Cancer Research UK, millennials in the United Kingdom are on track to have the highest rates of overweight and obesity, with current data trends indicating millennials will overtake the Baby boomer generation in this regard, making millennials the heaviest generation since current records began. Cancer Research UK reports that more than 70% of millennials will be overweight or obese by ages 35–45, in comparison to 50% of Baby boomers who were overweight or obese at the same ages.
Even though the majority of strokes affect people aged 65 or older and the probability of having a stroke doubles only every decade after the age of 55, anyone can suffer from a stroke at any age. A stroke occurs when the blood supply to the brain is disrupted, causing neurons to die within minutes, leading to irreparable brain damage, disability, or even death. Statistics from the Centers for Disease Control and Prevention (CDC), strokes are the fifth leading cause of death and a major factor behind disability in the United States. According to the National Strokes Association, the risk of having a stroke is increasing among young adults (those in their 20s and 30s) and even adolescents. During the 2010s, there was a 44% increase in the number of young people hospitalized for strokes. Health experts believe this development is due to a variety of reasons related to lifestyle choices, including obesity, smoking, alcoholism, and physical inactivity. Obesity is also linked to hypertension, diabetes, and high cholesterol levels. CDC data reveals that during the mid-2000s, about 28% of young Americans were obese; this number rose to 36% a decade later. Up to 80% of strokes can be prevented by making healthy lifestyle choices while the rest are due to factors beyond a person's control, namely age and genetic defects (such as congenital heart disease). In addition, between 30% and 40% of young patients suffered from cryptogenic strokes, or those with unknown causes.
According to a 2019 report from the American College of Cardiology, the prevalence of heart attacks among Americans under the age of 40 increased by an average rate of two percent per year in the previous decade. About one in five patients suffered from a heart attack came from this age group. This is despite the fact that Americans in general were less likely to suffer from heart attacks than before, due in part to a decline in smoking. The consequences of having a heart attack were much worse for young patients who also had diabetes. Besides the common risk factors of heart attacks, namely diabetes, high blood pressure, and family history, young patients also reported marijuana and cocaine intake, but less alcohol consumption.
Drug addiction and overdoses adversely affect millennials more than prior generations with overdose deaths among millennials increasing by 108% from 2006 to 2015. In the United States, millennials and older zoomers represented a majority of all opioid overdose deaths in 2021. The leading cause of death for people aged 25–44 in 2021 were drug overdoses (classified as poisonings by the Centers for Disease Control and Prevention) with overdose deaths being triple that of the second and third leading causes of death; suicide and traffic accidents, respectively. This represents a major shift as traffic accidents typically constituted a majority of accidental deaths for prior generations.
Millennials struggle with dental and oral health. More than 30% of young adults have untreated tooth decay (the highest of any age group), 35% have trouble biting and chewing, and some 38% of this age group find life in general "less satisfying" due to teeth and mouth problems.
Sports and fitness.
Fewer American millennials follow sports than their Generation X predecessors, with a McKinsey survey finding that 38 percent of millennials in contrast to 45 percent of Generation X are committed sports fans. However, the trend is not uniform across all sports; the gap disappears for National Basketball Association, Ultimate Fighting Championship, English Premier League and college sports. For example, a survey in 2013 found that engagement with mixed martial arts had increased in the 21st century and was more popular than boxing and wrestling for Americans aged 18 to 34 years old, in contrast to those aged 35 and over who preferred boxing. In the United States, while the popularity of American football and the National Football League has declined among millennials, the popularity of Association football and Major League Soccer has increased more among millennials than for any other generation, and as of 2018 was the second most popular sport among those aged 18 to 34.
Regarding the sports participation by millennials, activities that are popular or emerging among millennials including boxing, cycling, running, and swimming, while other sports including golf are facing decline among millennials. The Physical Activity Council's 2018 Participation Report found that in the U.S., millennials were more likely than other generations to participate in water sports such as stand up paddling, board-sailing and surfing. According to the survey of 30,999 Americans, which was conducted in 2017, approximately half of U.S. millennials participated in high caloric activities while approximately one quarter were sedentary. The 2018 report from the Physical Activity Council found millennials were more active than Baby Boomers in 2017. Thirty-five percent of both millennials and Generation X were reported to be "active to a healthy level", with millennial's activity level reported as higher overall than that of Generation X in 2017.
Political views and participation.
Millennials are reshaping political discourse, showing evolving attitudes towards governance, social issues, and economic policies. Their increasing political participation and distinct generational identity signify a transformative phase in contemporary politics, with potential long-term implications for national and global political trends.
American millennials exhibit a complex spectrum of political views, paralleling broader generational shifts in attitudes towards social, economic, and political issues. Surveys indicate a significant portion of millennials' political views align with their parents, though a notable fraction express more liberal tendencies. Key issues for US millennials include support for same-sex marriage, varying attitudes towards the LGBT community, and a more moderate stance on political ideologies compared to older generations. Millennials in the United States demonstrate increasing skepticism towards capitalism, with a preference for socialism seen in younger segments of the demographic. Canadian millennials played a crucial role in the election of Justin Trudeau, driven by social and economic liberal values. Despite historically low political participation, the 2015 federal election saw a surge in youth voter turnout, influenced by Trudeau's progressive campaign promises.
British millennials, characterized by a relative political disengagement in their early years, have shown liberal tendencies on social and economic matters, favoring individual liberty and limited government intervention. Significant political moments like the Brexit referendum mobilized young voters, displaying a strong preference for remaining in the European Union, highlighting generational divides in political priorities and attitudes.
Across Europe, millennials are part of a larger shift towards post-materialist values, emphasizing environmentalism, social liberalism, and global citizenship. This generational shift is contributing to changing political landscapes, challenging traditional party alignments and contributing to the rise of new political movements.French millennials, while exempt from mandatory military service, still engage in a Defense and Citizenship Day, reflecting continued engagement with national civic duties. A significant majority support the reintroduction of some form of national service, reflecting broader desires for national cohesion and integration.
Preferred modes of transport.
Millennials in the U.S. were initially not keen on getting a driver's license or owning a vehicle thanks to new licensing laws and the state of the economy when they came of age, but the oldest among them have already begun buying cars in great numbers. In 2016, millennials purchased more cars and trucks than any living generation except the Baby Boomers; in fact, millennials overtook Baby Boomers in car ownership in California that year. A working paper by economists Christopher Knittel and Elizabeth Murphy then at the Massachusetts Institute of Technology and the National Bureau of Economic Research analyzed data from the U.S. Department of Transportation's National Household Transportation Survey, the U.S. Census Bureau, and American Community Survey in order to compare the driving habits of the Baby Boomers, Generation X, and the oldest millennials (born between 1980 and 1984). That found that on the surface, the popular story is true: American millennials on average own 0.4 fewer cars than their elders. But when various factors—including income, marital status, number of children, and geographical location—were taken into account, such a distinction ceased to be. In addition, once those factors are accounted for, millennials actually drive longer distances than the Baby Boomers. Economic forces, namely low gasoline prices, higher income, and suburban growth, result in millennials having an attitude towards cars that is no different from that of their predecessors. An analysis of the National Household Travel Survey by the State Smart Transportation Initiative revealed that higher-income millennials drive less than their peers probably because they are able to afford the higher costs of living in large cities, where they can take advantage of alternative modes of transportation, including public transit and ride-hailing services.
According to the Pew Research Center, young people are more likely to ride public transit. In 2016, 21% of adults aged 18 to 21 took public transit on a daily, almost daily, or weekly basis. By contrast, this number of all U.S. adults was 11%. Nationwide, about three quarters of American commuters drive their own cars. Also according to Pew, 51% of U.S. adults aged 18 to 29 used Lyft or Uber in 2018 compared to 28% in 2015. That number for all U.S. adults were 15% in 2015 and 36% in 2018. In general, users tend to be urban residents, young (18–29), university graduates, and high income earners ($75,000 a year or more).
Religious beliefs.
Millennials often describe themselves as "spiritual but not religious" and will sometimes turn to astrology, meditation or mindfulness techniques possibly to seek meaning or a sense of control. According to 2015 analysis of the European Values Study in the "Handbook of Children and Youth Studies" "the majority of young respondents in Europe claimed that they belonged to a Christian denomination", and "in most countries, the majority of young people believe in God". However, according to the same analysis a "dramatic decline" in religious affiliation among young respondents happened in Great Britain, Sweden, France, Italy and Denmark. By contrast an increase in religious affiliation happened among young respondents in Russia, Ukraine, and Romania.
According to a 2013 YouGov poll of almost a thousand Britons between the ages of 18 and 24, 56% said they had never attended a place of worship, other than for a wedding or a funeral. 25% said they believed in God, 19% in a "spiritual greater power" while 38% said they did not believe in God nor any other "greater spiritual power". The poll also found that 14% thought religion was a "cause of good" in the world while 41% thought religion was "the cause of evil". 34% answered "neither". The British Social Attitudes Survey found that 71% of British 18–24 year-olds were not religious, with just 3% affiliated to the once-dominant Church of England, and 5% say they are Catholics, and 14% say they belong to other Christian denomination.
In the U.S., millennials are the least likely to be religious when compared to older generations. There is a trend towards irreligion that has been increasing since the 1940s. According to a 2012 study by Pew Research, 32 percent of Americans aged 18–29 are irreligious, as opposed to 21 percent aged 30–49, 15 percent aged 50–64, and only 9 percent born aged 65 and above. A 2005 study looked at 1,385 people aged 18 to 25 and found that more than half of those in the study said that they pray regularly before a meal. One-third said that they discussed religion with friends, attended religious services, and read religious material weekly. Twenty-three percent of those studied did not identify themselves as religious practitioners. A 2010 Pew Research Center study on millennials shows that of those between 18 and 29 years old, only 3% of these emerging adults self-identified as "atheists" and only 4% self-identified as "agnostics". While 68% of those between 18 and 29 years old self-identified as "Christians" (43% self-identified as Protestants and 22% self-identified as Catholics). Overall, 25% of millennials are "Nones" and 75% are religiously affiliated. In 2011, social psychologists Jason Weeden, Adam Cohen, and Douglas Kenrick analyzed survey data sets from the American general public and university undergraduates and discovered that sociosexual tendencies—that is, mating strategies—play a more important role in determining the level of religiousness than any other social variables. In fact, when controlled for family structure and sexual attitudes, variables such as age, sex, and moral beliefs on sexuality substantially drop in significance in determining religiosity. In the context of the United States, religiousness facilitates seeking and maintaining high-fertility, marriage-oriented, heterosexual monogamous relationships. As such, the central goals of religious attendance are reproduction and child-rearing. However, this Reproductive Religiosity Model does not necessarily apply to other countries. In Singapore, for example, they found no relationships between the religiousness of Buddhists and their attitudes towards sexuality.
A 2016 U.S. study found that church attendance during young adulthood was 41% among Generation Z, 18% for the millennials, 21% for Generation X, and 26% for the Baby Boomers when they were at the same age. A 2016 survey by Barna and Impact 360 Institute on about 1,500 Americans aged 13 and up suggests that the proportion of atheists and agnostics was 21% among Generation Z, 15% for millennials, 13% for Generation X, and 9% for Baby Boomers. 59% of Generation Z were Christians (including Catholics), as were 65% for the millennials, 65% for Generation X, and 75% for the Baby Boomers. 41% of teens believed that science and the Bible are fundamentally at odds with one another, with 27% taking the side of science and 17% picking religion. For comparison, 45% of millennials, 34% of Generation X, and 29% of the Baby Boomers believed such a conflict exists. 31% of Generation Z believed that science and religion refer to different aspects of reality, on par with millennials and Generation X (both 30%), and above the Baby Boomers (25%). 28% of Generation Z thought that science and religion are complementary, compared to 25% of millennials, 36% of Generation X, and 45% for Baby Boomers.
Social tendencies.
Social circles.
In March 2014, the Pew Research Center issued a report about how "millennials in adulthood" are "detached from institutions and networked with friends". The report said millennials are somewhat more upbeat than older adults about America's future, with 49% of millennials saying the country's best years are ahead, though they're the first in the modern era to have higher levels of student loan debt and unemployment.
Courtship behavior.
In many countries, people have since the mid-twentieth century been increasingly looking for mates of the same socioeconomic status and educational attainment. The phenomenon of preferring mates with characteristics similar to one's own is known as assortative mating. Part of the reason growing economic and educational assortative mating was economic in nature. Innovations which became commercially available in the late twentieth century such as the washing machine and frozen food reduced the amount of time people needed to spend on housework, which diminished the importance of domestic skills. Moreover, by the early 2000s, it was less feasible for a couple with one spouse having no more than a high-school diploma to earn about the national average; on the other hand, couples both of whom had at least a bachelor's degree could expect to make a significant amount above the national average. People thus had a clear economic incentive to seek out a mate with at least as high a level of education in order to maximize their potential income. Another incentive for this kind of assortative mating lies in the future of the offspring. People have since the mid-twentieth century increasingly wanted intelligent and well-educated children, and marrying bright people who make a lot of money goes a long way in achieving that goal. Couples in the early twenty-first century tend to hold egalitarian rather than traditional views on gender roles. Modern marriage is more about companionship rather than bread-winning for the man and homemaking for the woman. American and Chinese youths are increasingly choosing whether or not to marry according to their personal preferences rather than family, societal, or religious expectations.
As of 2016, 54% of Russian millennials were married.
According to the Chinese National Bureau of Statistics, the number of people getting married for the first time went from 23.8 million in 2013 to 13.9 million in 2019, a 41% drop. Meanwhile, the marriage rate continued its decline, 6.6 per 1,000 people, a 33% drop compared to 2013. These trends are due to multiple reasons. The one-child policy, introduced in 1979, has curbed the number of young people in China. On top of that, the traditional preference for sons has resulted in a marked gender imbalance; as of 2021, China has over 30 million "surplus" men.
In the 1990s, the Chinese government reformed higher education in order to increase access, whereupon significantly more young people, a slight majority of whom being women, have received a university degree. Consequently, many young women are now gainfully employed and financially secure. Traditional views on gender roles dictate that women be responsible for housework and childcare, regardless of their employment status. Workplace discrimination against women (with families) is commonplace; for example, an employer might be more skeptical towards a married woman with one child, fearing she might have another (as the one-child policy was rescinded in 2016) and take more maternity leave. Altogether, there is less incentive for young women to marry.
For young Chinese couples in general, the cost of living, especially the cost of housing in the big cities, is a serious obstacle to marriage. In addition, Chinese millennials are less keen on marrying than their predecessors as a result of cultural change.
Writing for "The Atlantic" in 2018, Kate Julian reported that among the countries that kept track of the sexual behavior of their citizens—Australia, Finland, Japan, the Netherlands, Sweden, the United Kingdom, and the United States—all saw a decline in the frequency of sexual intercourse among teenagers and young adults. Although experts disagree on the methodology of data analysis, they do believe that young people today are less sexually engaged than their elders, such as the baby boomers, when they were their age. This is despite the fact that online dating platforms allow for the possibility of casual sex, the wide availability of contraception, and the relaxation of attitudes towards sex outside of marriage.
A 2020 study published in the Journal of the American Medical Association (JAMA) by researchers from Indiana University in the United States and the Karolinska Institutet from Sweden found that during the first two decades of the twenty-first century, young Americans had sexual intercourse less frequently than in the past. Among men aged 18 to 24, the share of the sexually inactive increased from 18.9% between 2000 and 2002 to 30.9% between 2016 and 2018. Women aged 18 to 34 had sex less often as well. Reasons for this trend are manifold. People who were unemployed, only had part-time jobs, and students were the most likely to forego sexual experience while those who had higher income were stricter in mate selection. Psychologist Jean Twenge, who did not participate in the study, suggested that this might be due to "a broader cultural trend toward delayed development", meaning various adult activities are postponed. She noted that being economically dependent on one's parents discourages sexual intercourse. Other researchers noted that the rise of the Internet, computer games, and social media could play a role, too, since older and married couples also had sex less often. In short, people had many options. A 2019 study by the London School of Hygiene and Tropical Medicine found a similar trend in the United Kingdom. Although this trend precedes the COVID-19 pandemic, fear of infection is likely to fuel the trend the future, study co-author Peter Ueda told Reuters.
In a 2019 poll, the Pew Research Center found that about 47% American adults believed dating had become more difficult within the last decade or so, while only 19% said it became easier and 33% thought it was the same. Majorities of both men (65%) and women (43%) agreed that the #MeToo movement posed challenges for the dating market while 24% and 38%, respectively, thought it made no difference. In all, one in two of single adults were not looking for a romantic relationship. Among the rest, 10% were only interested in casual relationships, 14% wanted committed relationships only, and 26% were open to either kind. Among younger people (18 to 39), 27% wanted a committed relationship only, 15% casual dates only, and 58% either type of relationship. For those between the ages of 18 and 49, the top reasons for their decision to avoid dating were having more important priorities in life (61%), preferring being single (41%), being too busy (29%), and pessimism about their chances of success (24%).
While most Americans found their romantic partners with the help of friends and family, younger adults were more likely to encounter them online than their elders, with 21% of those aged 18 to 29 and 15% of those aged 30 to 49 saying they met their current partners this way. For comparison, only 8% of those aged 50 to 64 and 5% of those aged 65 and over did the same. People aged 18 to 29 were most likely to have met their current partners in school while adults aged 50 and up were more likely to have met their partners at work. Among those in the 18 to 29 age group, 41% were single, including 51% of men and 32% of women. Among those in the 30 to 49 age group, 23% were single, including 27% of men and 19% of women. This reflects the general trend across the generations that men tend to marry later (and die earlier) than women.
Most single people, regardless of whether or not they were interested in dating, felt little to no pressure from their friends and family to seek a romantic partner. Young people, however, were under significant pressure compared to the sample average or older age groups. 53% of single people aged 18 to 29 thought there was at least some pressure from society on them to find a partner, compared to 42% for people aged 30 to 49, 32% for people aged 50 to 64, and 21% for people aged 50 to 64.
Family life and offspring.
According to the Brookings Institution, the number of American mothers who never married ballooned between 1968, when they were extremely rare, and 2008, when they became much more common, especially among the less educated. In particular, in 2008, the number of mothers who never married with at least 16 years of education was 3.3%, compared to 20.1% of those who never graduated from high school. Unintended pregnancies were also higher among the less educated.
Research by the Urban Institute conducted in 2014, projected that if current trends continue, millennials will have a lower marriage rate compared to previous generations, predicting that by age 40, 31% of millennial women will remain single, approximately twice the share of their single Gen X counterparts. The data showed similar trends for males. A 2016 study from Pew Research showed millennials delay some activities considered rites of passage of adulthood with data showing young adults aged 18–34 were more likely to live with parents than with a relationship partner, an unprecedented occurrence since data collection began in 1880. Data also showed a significant increase in the percentage of young adults living with parents compared to the previous demographic cohort, Generation X, with 23% of young adults aged 18–34 living with parents in 2000, rising to 32% in 2014. Additionally, in 2000, 43% of those aged 18–34 were married or living with a partner, with this figure dropping to 32% in 2014. High student debt is described as one reason for continuing to live with parents, but may not be the dominant factor for this shift as the data shows the trend is stronger for those without a college education. Richard Fry, a senior economist for Pew Research said of millennials, "they're the group much more likely to live with their parents," further stating that "they're concentrating more on school, careers and work and less focused on forming new families, spouses or partners and children."
According to a cross-generational study comparing millennials to Generation X conducted at the Wharton School at the University of Pennsylvania, more than half of millennial undergraduates surveyed do not plan to have children. The researchers compared surveys of the Wharton graduating class of 1992 and 2012. In 1992, 78% of women planned to eventually have children dropping to 42% in 2012. The results were similar for male students. The research revealed among both genders the proportion of undergraduates who reported they eventually planned to have children had dropped in half over the course of a generation. "Quest" reported in March 2020 that, in Belgium, 11% of women and 16% of men between the ages of 25 and 35 did not want children and that in the Netherlands, 10% of 30-year-old women polled had decided against having children or having more children. A 2019 study revealed that among 191 Swedish men aged 20 to 50, 39 were not fathers and did not want to have children in the future (20.4%). Desire to have (more) children was not related to level of education, country of birth, sexual orientation or relationship status. Some Swedish men "passively" choose not to have children because they feel their life is already good as it is without bringing children to the world, and because they do not face the same amount of social pressure to have children the way voluntarily childless women do.
But as their economic prospects improve, most millennials in the United States say they desire marriage, children, and home ownership. Geopolitical analyst Peter Zeihan argued that because of the size of the millennial cohort relative to the size of the U.S. population and because they are having children, the United States will continue to maintain an economic advantage over most other developed nations, whose millennial cohorts are not only smaller than those of their elders but also do not have as high a fertility rate. The prospects of any given country is constrained by its demography. Psychologist Jean Twenge and a colleague's analysis of data from the General Social Survey of 40,000 Americans aged 30 and over from the 1970s to the 2010s suggests that socioeconomic status (as determined by factors such as income, educational attainment, and occupational prestige), marriage, and happiness are positive correlated and that these relationships are independent of cohort or age. However, the data cannot tell whether marriage causes happiness or the other way around; correlation does not mean causation.
In the United States, between the late 1970s and the late 2010s, the shares of people who were married declined among the lower class (from 60% down to 33%) and the middle class (84% down to 66%), but remained steady among the upper class (~80%). In fact, it was the lower and middle classes that were driving the U.S. marriage rate down. Among Americans aged 25 to 39, the divorce rate per 1,000 married persons dropped from 30 to 24 between 1990 and 2015. For comparison, among those aged 50 and up, the divorce rate went from 5 in 1990 to 10 in 2015; that among people aged 40 to 49 increased from 18 to 21 per 1,000 married persons. In general, the level of education is a predictor of marriage and income. University graduates are more likely to get married and less likely to divorce.
Demographer and futurist Mark McCrindle suggested the name "Generation Alpha" (or Generation formula_1) for the offspring of a majority of millennials, people born after Generation Z, noting that scientific disciplines often move to the Greek alphabet after exhausting the Roman alphabet. By 2016, the cumulative number of American women of the millennial generation who had given birth at least once reached 17.3 million. Globally, there are some two and a half million people belonging to Generation Alpha born every week and their number is expected to reach two billion by 2025. However, most of the human population growth in the 2010s comes from Africa and Asia, as nations in Europe and the Americas tend to have too few children to replace themselves. According to the United Nations, the global annual rate of growth has been declining steadily since the late twentieth century, dropping to about one percent in 2019. They also discovered that fertility rates were falling faster in the developing world than previously thought, and subsequently revised their projection of human population in 2050 down to 9.7 billion. Fertility rates have been falling around the world thanks to rising standards of living, better access to contraceptives, and improved educational and economic opportunities. The global average fertility rate was 2.4 in 2017, down from 4.7 in 1950.
Effects of intensifying assortative mating (discussed in the previous section) will likely be seen in the next generation, as parental income and educational level are positively correlated with children's success. In the United States, children from families in the highest income quintile are the most likely to live with married parents (94% in 2018), followed by children of the middle class (74%) and the bottom quintile (35%).
Living in the digital age, Millennial parents have taken plenty of photographs of their children, and have chosen both digital storage (e.g. Dropbox) or physical photo albums to preserve their memories. Many Millennial parents document the childhood and growth of their children on social media platforms such as Instagram and Facebook.
Workplace attitudes.
In 2008, author Ron Alsop called the millennials "Trophy Kids", a term that reflects a trend in competitive sports, as well as many other aspects of life, where mere participation is frequently enough for a reward. It has been reported that this is an issue in corporate environments. Some employers are concerned that millennials have too great expectations from the workplace. Some studies predict they will switch jobs frequently, holding many more jobs than Gen Xers due to their great expectations. Psychologist Jean Twenge reports data suggesting there are differences between older and younger millennials regarding workplace expectations, with younger millennials being "more practical" and "more attracted to industries with steady work and are more likely to say they are willing to work overtime" which Twenge attributes to younger millennials coming of age following the financial crisis of 2007–2008.
In 2010 the "Journal of Business and Psychology", contributors Myers and Sadaghiani find millennials "expect close relationships and frequent feedback from supervisors" to be a main point of differentiation. Multiple studies observe millennials' associating job satisfaction with free flow of information, strong connectivity to supervisors, and more immediate feedback. Hershatter and Epstein, researchers from Emory University, argue many of these traits can be linked to millennials entering the educational system on the cusp of academic reform, which created a much more structured educational system. Some argue in the wake of these reforms, such as the No Child Left Behind Act, millennials have increasingly sought the aid of mentors and advisers, leading to 66% of millennials seeking a flat work environment.
Hershatter and Epstein also stress a growing importance on work-life balance. Studies show nearly one-third of students' top priority is to "balance personal and professional life". The Brain Drain Study shows nearly 9 out of 10 millennials place an importance on work-life balance, with additional surveys demonstrating the generation to favor familial over corporate values. Studies also show a preference for work-life balance, which contrasts to the Baby Boomers' work-centric attitude.
There is also a contention that the major differences are found solely between millennials and Generation X. Researchers from the University of Missouri and The University of Tennessee conducted a study based on measurement equivalence to determine if such a difference does in fact exist. The study looked at 1,860 participants who had completed the Multidimensional Work Ethic Profile (MWEP), a survey aimed at measuring identification with work-ethic characteristics, across a 12-year period spanning from 1996 to 2008. The results of the findings suggest the main difference in work ethic sentiments arose between the two most recent generational cohorts, Generation X and millennials, with relatively small variances between the two generations and their predecessor, the Baby Boomers.
A meta study conducted by researchers from The George Washington University and The U.S. Army Research Institute for the Behavioral and Social Sciences questions the validity of workplace differences across any generational cohort. According to the researchers, disagreement in which events to include when assigning generational cohorts, as well as varied opinions on which age ranges to include in each generational category are the main drivers behind their skepticism. The analysis of 20 research reports focusing on the three work-related factors of job satisfaction, organizational commitment and intent to turn over proved any variation was too small to discount the impact of employee tenure and aging of individuals. Newer research shows that millennials change jobs for the same reasons as other generations—namely, more money and a more innovative work environment. They look for versatility and flexibility in the workplace, and strive for a strong work–life balance in their jobs and have similar career aspirations to other generations, valuing financial security and a diverse workplace just as much as their older colleagues.
Data also suggests millennials are driving a shift towards the public service sector. In 2010, Myers and Sadaghiani published research in the "Journal of Business and Psychology" stating heightened participation in the Peace Corps and AmeriCorps as a result of millennials, with volunteering being at all-time highs. Volunteer activity between 2007 and 2008 show the millennial age group experienced almost three-times the increase of the overall population, which is consistent with a survey of 130 college upperclassmen depicting an emphasis on altruism in their upbringing. This has led, according to a Harvard University Institute of Politics, six out of ten millennials to consider a career in public service.
The 2014 Brookings publication shows a generational adherence to corporate social responsibility, with the National Society of High School Scholars (NSHSS) 2013 survey and Universum's 2011 survey, depicting a preference to work for companies engaged in the betterment of society. Millennials' shift in attitudes has led to data depicting 64% of millennials would take a 60% pay cut to pursue a career path aligned with their passions, and financial institutions have fallen out of favor with banks comprising 40% of the generation's least liked brands.
Use of digital technology.
Marc Prensky coined the term "digital native" to describe "K through college" students in 2001, explaining they "represent the first generations to grow up with this new technology". In their 2007 book "Connecting to the Net.Generation: What Higher Education Professionals Need to Know About Today's Students", authors Reynol Junco and Jeanna Mastrodicasa expanded on the work of William Strauss and Neil Howe to include research-based information about the personality profiles of millennials, especially as it relates to higher education. They conducted a large-sample (7,705) research study of college students. They found that Net Generation college students, born 1982 onwards, were frequently in touch with their parents and they used technology at higher rates than people from other generations. In their survey, they found that 97% of these students owned a computer, 94% owned a mobile phone, and 56% owned an MP3 player. They also found that students spoke with their parents an average of 1.5 times a day about a wide range of topics. Other findings in the Junco and Mastrodicasa survey revealed 76% of students used instant messaging, 92% of those reported multitasking while instant messaging, 40% of them used television to get most of their news, and 34% of students surveyed used the Internet as their primary news source.
One of the most popular forms of media use by millennials is social networking. Millennials use social networking sites, such as Facebook and Twitter, to create a different sense of belonging, make acquaintances, and to remain connected with friends. In 2010, research was published in the Elon Journal of Undergraduate Research which claimed that students who used social media and decided to quit showed the same withdrawal symptoms of a drug addict who quit their stimulant. In the 2014 PBS "Frontline" episode "Generation Like" there is discussion about millennials, their dependence on technology, and the ways the social media sphere is commoditized. Some millennials enjoy having hundreds of channels from cable TV. However, some other millennials do not even have a TV, so they watch media over the Internet using smartphones and tablets. Jesse Singal of "New York" magazine argues that this technology has created a rift within the generation; older millennials, defined here as those born 1988 and earlier, came of age prior to widespread usage and availability of smartphones, in contrast to younger millennials, those born in 1989 and later, who were exposed to this technology in their teen years.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "n = 29,912"
},
{
"math_id": 1,
"text": "\\alpha"
}
] |
https://en.wikipedia.org/wiki?curid=149183
|
14919
|
ISBN
|
Unique numeric book identifier since 1970
The International Standard Book Number (ISBN) is a numeric commercial book identifier that is intended to be unique. Publishers purchase or receive ISBNs from an affiliate of the International ISBN Agency.
A different ISBN is assigned to each separate edition and variation of a publication, but not to a simple reprinting of an existing item. For example, an e-book, a paperback and a hardcover edition of the same book must each have a different ISBN, but an unchanged reprint of the hardcover edition keeps the same ISBN. The ISBN is ten digits long if assigned before 2007, and thirteen digits long if assigned on or after 1 January 2007. The method of assigning an ISBN is nation-specific and varies between countries, often depending on how large the publishing industry is within a country.
The first version of the ISBN identification format was devised in 1967, based upon the 9-digit Standard Book Numbering (SBN) created in 1966. The 10-digit ISBN format was developed by the International Organization for Standardization (ISO) and was published in 1970 as international standard ISO 2108 (any 9-digit SBN can be converted to a 10-digit ISBN by prefixing it with a zero).
Privately published books sometimes appear without an ISBN. The International ISBN Agency sometimes assigns ISBNs to such books on its own initiative.
A separate identifier code of a similar kind, the International Standard Serial Number (ISSN), identifies periodical publications such as magazines and newspapers. The International Standard Music Number (ISMN) covers musical scores.
History.
The Standard Book Number (SBN) is a commercial system using nine-digit code numbers to identify books. In 1965, British bookseller and stationers WHSmith announced plans to implement a standard numbering system for its books. They hired consultants to work on their behalf, and the system was devised by Gordon Foster, emeritus professor of statistics at Trinity College Dublin. The International Organization for Standardization (ISO) Technical Committee on Documentation sought to adapt the British SBN for international use. The ISBN identification format was conceived in 1967 in the United Kingdom by David Whitaker (regarded as the "Father of the ISBN") and in 1968 in the United States by Emery Koltay (who later became director of the U.S. ISBN agency R. R. Bowker).
The 10-digit ISBN format was developed by the ISO and was published in 1970 as international standard ISO 2108. The United Kingdom continued to use the nine-digit SBN code until 1974. ISO has appointed the International ISBN Agency as the registration authority for ISBN worldwide and the ISBN Standard is developed under the control of ISO Technical Committee 46/Subcommittee 9 TC 46/SC 9. The ISO on-line facility only refers back to 1978.
An SBN may be converted to an ISBN by prefixing the digit "0". For example, the second edition of "Mr. J. G. Reeder Returns", published by Hodder in 1965, has "SBN 340 01381 8", where "340" indicates the publisher, "01381" is the serial number assigned by the publisher, and "8" is the check digit. By prefixing a zero, this can be converted to ; the check digit does not need to be re-calculated. Some publishers, such as Ballantine Books, would sometimes use 12-digit SBNs where the last three digits indicated the price of the book; for example, "Woodstock Handmade Houses" had a 12-digit Standard Book Number of 345-24223-8-595 (valid SBN: 345-24223-8, ISBN: 0-345-24223-8), and it cost US$.
Since 1 January 2007, ISBNs have contained thirteen digits, a format that is compatible with "Bookland" European Article Numbers, which have 13 digits.
The United States, with 3.9 million registered ISBNs in 2020, was by far the biggest user of the ISBN identifier in 2020, followed by the Republic of Korea (329,582), Germany (284,000), China (263,066), the UK (188,553) and Indonesia (144,793). Lifetime ISBNs registered in the United States are over 39 million as of 2020.
Overview.
A separate ISBN is assigned to each edition and variation (except reprintings) of a publication. For example, an ebook, audiobook, paperback, and hardcover edition of the same book must each have a different ISBN assigned to it. The ISBN is thirteen digits long if assigned on or after 1 January 2007, and ten digits long if assigned before 2007. An International Standard Book Number consists of four parts (if it is a 10-digit ISBN) or five parts (for a 13-digit ISBN).
Section 5 of the International ISBN Agency's official user manual describes the structure of the 13-digit ISBN, as follows:
A 13-digit ISBN can be separated into its parts ("prefix element", "registration group", "registrant", "publication" and "check digit"), and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts ("registration group", "registrant", "publication" and "check digit") of a 10-digit ISBN is also done with either hyphens or spaces. Figuring out how to correctly separate a given ISBN is complicated, because most of the parts do not use a fixed number of digits.
Issuing process.
ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for that country or territory regardless of the publication language. The ranges of ISBNs assigned to any particular country are based on the publishing profile of the country concerned, and so the ranges will vary depending on the number of books and the number, type, and size of publishers that are active. Some ISBN registration agencies are based in national libraries or within ministries of culture and thus may receive direct funding from the government to support their services. In other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded.
A full directory of ISBN agencies is available on the International ISBN Agency website. A list for a few countries is given below:
Registration group element.
The ISBN registration group element is a 1-to-5-digit number that is valid within a single prefix element (i.e. one of 978 or 979), and can be separated between hyphens, such as "978-1-...". Registration groups have primarily been allocated within the 978 prefix element. The single-digit registration groups within the 978-prefix element are: 0 or 1 for English-speaking countries; 2 for French-speaking countries; 3 for German-speaking countries; 4 for Japan; 5 for Russian-speaking countries; and 7 for People's Republic of China. Example 5-digit registration groups are 99936 and 99980, for Bhutan. The allocated registration groups are: 0–5, 600–631, 65, 7, 80–94, 950–989, 9910–9989, and 99901–99993. Books published in rare languages typically have longer group elements.
Within the 979 prefix element, the registration group 0 is reserved for compatibility with International Standard Music Numbers (ISMNs), but such material is not actually assigned an ISBN. The registration groups within prefix element 979 that have been assigned are 8 for the United States of America, 10 for France, 11 for the Republic of Korea, and 12 for Italy.
The original 9-digit standard book number (SBN) had no registration group identifier, but prefixing a zero to a 9-digit SBN creates a valid 10-digit ISBN.
Registrant element.
The national ISBN agency assigns the registrant element (cf. ) and an accompanying series of ISBNs within that registrant element to the publisher; the publisher then allocates one of the ISBNs to each of its books. In most countries, a book publisher is not legally required to assign an ISBN, although most large bookstores only handle publications that have ISBNs assigned to them.
The International ISBN Agency maintains the details of over one million ISBN prefixes and publishers in the Global Register of Publishers. This database is freely searchable over the internet.
Publishers receive blocks of ISBNs, with larger blocks allotted to publishers expecting to need them; a small publisher may receive ISBNs of one or more digits for the registration group identifier, several digits for the registrant, and a single digit for the publication element. Once that block of ISBNs is used, the publisher may receive another block of ISBNs, with a different registrant element. Consequently, a publisher may have different allotted registrant elements. There also may be more than one registration group identifier used in a country. This might occur once all the registrant elements from a particular registration group have been allocated to publishers.
By using variable block lengths, registration agencies are able to customise the allocations of ISBNs that they make to publishers. For example, a large publisher may be given a block of ISBNs where fewer digits are allocated for the registrant element and many digits are allocated for the publication element; likewise, countries publishing many titles have few allocated digits for the registration group identifier and many for the registrant and publication elements. Here are some sample ISBN-10 codes, illustrating block length variations.
English-language pattern.
English-language registration group elements are 0 and 1 (2 of more than 220 registration group elements). These two registration group elements are divided into registrant elements in a systematic pattern, which allows their length to be determined, as follows:
Check digits.
A check digit is a form of redundancy check used for error detection, the decimal equivalent of a binary check bit. It consists of a single digit computed from the other digits in the number. The method for the 10-digit ISBN is an extension of that for SBNs, so the two systems are compatible; an SBN prefixed with a zero (the 10-digit ISBN) will give the same check digit as the SBN without the zero. The check digit is base eleven, and can be an integer between 0 and 9, or an 'X'. The system for 13-digit ISBNs is not compatible with SBNs and will, in general, give a different check digit from the corresponding 10-digit ISBN, so does not provide the same protection against transposition. This is because the 13-digit code was required to be compatible with the EAN format, and hence could not contain the letter 'X'.
ISBN-10 check digits.
According to the 2001 edition of the International ISBN Agency's official user manual, the ISBN-10 check digit (which is the last digit of the 10-digit ISBN) must range from 0 to 10 (the symbol 'X' is used for 10), and must be such that the sum of the ten digits, each multiplied by its (integer) weight, descending from 10 to 1, is a multiple of 11. That is, if x is the ith digit, then x10 must be chosen such that:
<templatestyles src="Block indent/styles.css"/>formula_0
For example, for an ISBN-10 of 0-306-40615-2:
<templatestyles src="Block indent/styles.css"/>formula_1
Formally, using modular arithmetic, this is rendered
<templatestyles src="Block indent/styles.css"/>formula_2
It is also true for ISBN-10s that the sum of all ten digits, each multiplied by its weight in "ascending" order from 1 to 10, is a multiple of 11. For this example:
<templatestyles src="Block indent/styles.css"/>formula_3
Formally, this is rendered
<templatestyles src="Block indent/styles.css"/>formula_4
The two most common errors in handling an ISBN (e.g. when typing it or writing it down) are a single altered digit or the transposition of adjacent digits. It can be proven mathematically that all pairs of valid ISBN-10s differ in at least two digits. It can also be proven that there are no pairs of valid ISBN-10s with eight identical digits and two transposed digits (these proofs are true because the ISBN is less than eleven digits long and because 11 is a prime number). The ISBN check digit method therefore ensures that it will always be possible to detect these two most common types of error, i.e., if either of these types of error has occurred, the result will never be a valid ISBN—the sum of the digits multiplied by their weights will never be a multiple of 11. However, if the error were to occur in the publishing house and remain undetected, the book would be issued with an invalid ISBN.
In contrast, it is possible for other types of error, such as two altered non-transposed digits, or three altered digits, to result in a valid ISBN (although it is still unlikely).
ISBN-10 check digit calculation.
Each of the first nine digits of the 10-digit ISBN—excluding the check digit itself—is multiplied by its (integer) weight, descending from 10 to 2, and the sum of these nine products found. The value of the check digit is simply the one number between 0 and 10 which, when added to this sum, means the total is a multiple of 11.
For example, the check digit for an ISBN-10 of 0-306-40615-"?" is calculated as follows:
<templatestyles src="Block indent/styles.css"/>formula_5
Adding 2 to 130 gives a multiple of 11 (because 132 = 12×11)—this is the only number between 0 and 10 which does so. Therefore, the check digit has to be 2, and the complete sequence is ISBN 0-306-40615-2. If the value of formula_6 required to satisfy this condition is 10, then an 'X' should be used.
Alternatively, modular arithmetic is convenient for calculating the check digit using modulus 11. The remainder of this sum when it is divided by 11 (i.e. its value modulo 11), is computed. This remainder plus the check digit must equal either 0 or 11. Therefore, the check digit is (11 minus the remainder of the sum of the products modulo 11) modulo 11. Taking the remainder modulo 11 a second time accounts for the possibility that the first remainder is 0. Without the second modulo operation, the calculation could result in a check digit value of 11 − 0
11, which is invalid. (Strictly speaking, the "first" "modulo 11" is not needed, but it may be considered to simplify the calculation.)
For example, the check digit for the ISBN of 0-306-40615-"?" is calculated as follows:
<templatestyles src="Block indent/styles.css"/>formula_7
Thus the check digit is 2.
It is possible to avoid the multiplications in a software implementation by using two accumulators. Repeatedly adding codice_0 into codice_1 computes the necessary multiples:
// Returns ISBN error syndrome, zero for a valid ISBN, non-zero for an invalid one.
// digits[i] must be between 0 and 10.
int CheckISBN(int const digits[10]) {
int i, s = 0, t = 0;
for (i = 0; i < 10; ++i) {
t += digits[i];
s += t;
return s % 11;
The modular reduction can be done once at the end, as shown above (in which case codice_1 could hold a value as large as 496, for the invalid ISBN 99999-999-9-X), or codice_1 and codice_0 could be reduced by a conditional subtract after each addition.
ISBN-13 check digit calculation.
Appendix 1 of the International ISBN Agency's official user manual describes how the 13-digit ISBN check digit is calculated. The ISBN-13 check digit, which is the last digit of the ISBN, must range from 0 to 9 and must be such that the sum of all the thirteen digits, each multiplied by its (integer) weight, alternating between 1 and 3, is a multiple of 10. As ISBN-13 is a subset of EAN-13, the algorithm for calculating the check digit is exactly the same for both.
Formally, using modular arithmetic, this is rendered:
<templatestyles src="Block indent/styles.css"/>formula_8
The calculation of an ISBN-13 check digit begins with the first twelve digits of the 13-digit ISBN (thus excluding the check digit itself). Each digit, from left to right, is alternately multiplied by 1 or 3, then those products are summed modulo 10 to give a value ranging from 0 to 9. Subtracted from 10, that leaves a result from 1 to 10. A zero replaces a ten, so, in all cases, a single check digit results.
For example, the ISBN-13 check digit of 978-0-306-40615-"?" is calculated as follows:
s = 9×1 + 7×3 + 8×1 + 0×3 + 3×1 + 0×3 + 6×1 + 4×3 + 0×1 + 6×3 + 1×1 + 5×3
= 9 + 21 + 8 + 0 + 3 + 0 + 6 + 12 + 0 + 18 + 1 + 15
= 93
93 / 10 = 9 remainder 3
10 – 3 = 7
Thus, the check digit is 7, and the complete sequence is ISBN 978-0-306-40615-7.
In general, the ISBN check digit is calculated as follows.
Let
<templatestyles src="Block indent/styles.css"/>formula_9
Then
<templatestyles src="Block indent/styles.css"/>formula_10
This check system—similar to the UPC check digit formula—does not catch all errors of adjacent digit transposition. Specifically, if the difference between two adjacent digits is 5, the check digit will not catch their transposition. For instance, the above example allows this situation with the 6 followed by a 1. The correct order contributes 3 × 6 + 1 × 1
19 to the sum; while, if the digits are transposed (1 followed by a 6), the contribution of those two digits will be 3 × 1 + 1 × 6
9. However, 19 and 9 are congruent modulo 10, and so produce the same, final result: both ISBNs will have a check digit of 7. The ISBN-10 formula uses the prime modulus 11 which avoids this blind spot, but requires more than the digits 0–9 to express the check digit.
Additionally, if the sum of the 2nd, 4th, 6th, 8th, 10th, and 12th digits is tripled then added to the remaining digits (1st, 3rd, 5th, 7th, 9th, 11th, and 13th), the total will always be divisible by 10 (i.e., end in 0).
ISBN-10 to ISBN-13 conversion.
A 10-digit ISBN is converted to a 13-digit ISBN by prepending "978" to the ISBN-10 and recalculating the final checksum digit using the ISBN-13 algorithm. The reverse process can also be performed, but not for numbers commencing with a prefix other than 978, which have no 10-digit equivalent.
Errors in usage.
Publishers and libraries have varied policies about the use of the ISBN check digit. Publishers sometimes fail to check the correspondence of a book title and its ISBN before publishing it; that failure causes book identification problems for libraries, booksellers, and readers. For example, is shared by two books—"Ninja gaiden: a novel based on the best-selling game by Tecmo" (1990) and "Wacky laws" (1997), both published by Scholastic.
Most libraries and booksellers display the book record for an invalid ISBN issued by the publisher. The Library of Congress catalogue contains books published with invalid ISBNs, which it usually tags with the phrase "Cancelled ISBN". The International Union Library Catalog (a.k.a., WorldCat OCLC—Online Computer Library Center system) often indexes by invalid ISBNs, if the book is indexed in that way by a member library.
eISBN.
Only the term "ISBN" should be used; the terms "eISBN" and "e-ISBN" have historically been sources of confusion and should be avoided. If a book exists in one or more digital (e-book) formats, each of those formats must have its own ISBN. In other words, each of the three separate EPUB, Amazon Kindle, and PDF formats of a particular book will have its own specific ISBN. They should not share the ISBN of the paper version, and there is no generic "eISBN" which encompasses all the e-book formats for a title.
EAN format used in barcodes, and upgrading.
The barcodes on a book's back cover (or inside a mass-market paperback book's front cover) are EAN-13; they may have a separate barcode encoding five digits called an EAN-5 for the currency and the recommended retail price. For 10-digit ISBNs, the number "978", the Bookland "country code", is prefixed to the ISBN in the barcode data, and the check digit is recalculated according to the EAN-13 formula (modulo 10, 1× and 3× weighting on alternating digits).
Partly because of an expected shortage in certain ISBN categories, the International Organization for Standardization (ISO) decided to migrate to a 13-digit ISBN (ISBN-13). The process began on 1 January 2005 and was planned to conclude on 1 January 2007. As of 2011[ [update]], all the 13-digit ISBNs began with 978. As the 978 ISBN supply is exhausted, the 979 prefix was introduced. Part of the 979 prefix is reserved for use with the Musicland code for musical scores with an ISMN. The 10-digit ISMN codes differed visually as they began with an "M" letter; the bar code represents the "M" as a zero, and for checksum purposes it counted as a 3. All ISMNs are now thirteen digits commencing 979-0; 979-1 to 979-9 will be used by ISBN.
Publisher identification code numbers are unlikely to be the same in the 978 and 979 ISBNs, likewise, there is no guarantee that language area code numbers will be the same. Moreover, the 10-digit ISBN check digit generally is not the same as the 13-digit ISBN check digit. Because the GTIN-13 is part of the Global Trade Item Number (GTIN) system (that includes the GTIN-14, the GTIN-12, and the GTIN-8), the 13-digit ISBN falls within the 14-digit data field range.
Barcode format compatibility is maintained, because (aside from the group breaks) the ISBN-13 barcode format is identical to the EAN barcode format of existing 10-digit ISBNs. So, migration to an EAN-based system allows booksellers the use of a single numbering system for both books and non-book products that is compatible with existing ISBN based data, with only minimal changes to information technology systems. Hence, many booksellers (e.g., Barnes & Noble) migrated to EAN barcodes as early as March 2005. Although many American and Canadian booksellers were able to read EAN-13 barcodes before 2005, most general retailers could not read them. The upgrading of the UPC barcode system to full EAN-13, in 2005, eased migration to the ISBN in North America.
Explanatory notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sum_{i = 1}^{10} (11-i)x_i \\equiv 0 \\pmod{11}."
},
{
"math_id": 1,
"text": "\n\\begin{align}\n s &= (0\\times 10) + (3\\times 9) + (0\\times 8) + (6\\times 7) + (4\\times 6) + (0\\times 5) + (6\\times 4) + (1\\times 3) + (5\\times 2) + (2\\times 1) \\\\\n &= 0 + 27 + 0 + 42 + 24 + 0 + 24 + 3 + 10 + 2\\\\\n &= 132 = 12\\times 11.\n\\end{align}\n"
},
{
"math_id": 2,
"text": "(10x_1+9x_2+8x_3+7x_4+6x_5+5x_6+4x_7+3x_8+2x_9+x_{10})\\equiv 0 \\pmod{11}."
},
{
"math_id": 3,
"text": "\n\\begin{align}\n s &= (0\\times 1) + (3\\times 2) + (0\\times 3) + (6\\times 4) + (4\\times 5) + (0\\times 6) + (6\\times 7) + (1\\times 8) + (5\\times 9) + (2\\times 10) \\\\\n &= 0 + 6 + 0 + 24 + 20 + 0 + 42 + 8 + 45 + 20\\\\\n &= 165 = 15\\times 11.\n\\end{align}\n"
},
{
"math_id": 4,
"text": "(x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 + 6x_6 + 7x_7 + 8x_8 + 9x_9 + 10x_{10})\\equiv 0 \\pmod{11}."
},
{
"math_id": 5,
"text": "\n\\begin{align}\n s &= (0\\times 10)+(3\\times 9)+(0\\times 8)+(6\\times 7)+(4\\times 6)+(0\\times 5)+(6\\times 4)+(1\\times 3)+(5\\times 2)\\\\\n &= 130.\n\\end{align}\n"
},
{
"math_id": 6,
"text": "x_{10}"
},
{
"math_id": 7,
"text": "\n\\begin{align}\n s &= (11 - ( ( (0\\times 10)+(3\\times 9)+(0\\times 8)+(6\\times 7)+(4\\times 6)+(0\\times 5)+(6\\times 4)+(1\\times 3)+(5\\times 2) ) \\,\\bmod\\, 11 ) ) \\,\\bmod\\, 11\\\\\n &= (11 - ( (0 + 27 + 0 + 42 + 24 + 0 + 24 + 3 + 10 ) \\,\\bmod\\, 11) ) \\,\\bmod\\, 11\\\\\n &= (11-((130) \\,\\bmod\\, 11))\\,\\bmod\\, 11 \\\\\n &= (11-(9))\\,\\bmod\\, 11 \\\\\n &= (2)\\,\\bmod\\, 11 \\\\\n &= 2\n\\end{align}\n"
},
{
"math_id": 8,
"text": "(x_1 + 3x_2 + x_3 + 3x_4 + x_5 + 3x_6 + x_7 + 3x_8 + x_9 + 3x_{10} + x_{11} + 3x_{12} + x_{13} ) \\equiv 0 \\pmod{10}."
},
{
"math_id": 9,
"text": "r = \\big(10 - \\big(x_1 + 3x_2 + x_3 + 3x_4 + \\cdots + x_{11} + 3x_{12}\\big) \\bmod 10\\big)."
},
{
"math_id": 10,
"text": "\n x_{13} = \\begin{cases}\n r, & r < 10, \\\\\n 0, & r = 10.\n\\end{cases}\n"
}
] |
https://en.wikipedia.org/wiki?curid=14919
|
14920509
|
Markov's principle
|
Markov's principle (also known as the Leningrad principle), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well.
History.
The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function.
In computability theory.
In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. These statements are provable in classical logic.
In intuitionistic logic.
In predicate logic, a predicate "P" over some domain is called "decidable" if for every "x" in the domain, either "P"("x") holds, or the negation of "P"("x") holds. This is not trivially true constructively.
Markov's principle then states: For a decidable predicate "P" over the natural numbers, if "P" cannot be false for all natural numbers "n", then it is true for some "n". Written using quantifiers:
formula_0
Markov's rule.
Markov's rule is the formulation of Markov's principle as a rule. It states that formula_1 is derivable as soon as formula_2 is, for formula_3 decidable. Formally,
formula_4
Anne Troelstra proved that it is an admissible rule in Heyting arithmetic. Later, the logician Harvey Friedman showed that Markov's rule is an admissible rule in first-order intuitionistic logic, Heyting arithmetic, and various other intuitionistic theories, using the Friedman translation.
In Heyting arithmetic.
Markov's principle is equivalent in the language of arithmetic to:
formula_5
for formula_6 a total recursive function on the natural numbers.
In the presence of Church's thesis principle, the principle is equivalent to its form for primitive recursive functions. Using Kleene's T predicate, the latter may be expressed as
formula_7
Realizability.
If constructive arithmetic is translated using realizability into a classical meta-theory that proves the formula_8-consistency of the relevant classical theory (for example, Peano arithmetic if we are studying Heyting arithmetic), then Markov's principle is justified: a realizer is the constant function that takes a realization that formula_3 is not everywhere false to the unbounded search that successively checks if formula_9 is true. If formula_3 is not everywhere false, then by formula_8-consistency there must be a term for which formula_3 holds, and each term will be checked by the search eventually. If however formula_3 does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), formula_3 must either hold nowhere or not hold nowhere, therefore this constant function is a realizer.
If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified. Indeed, for first-order arithmetic, Markov's principle exactly captures the difference between a constructive and classical meta-theory. Specifically, a statement is provable in Heyting arithmetic with extended Church's thesis if and only if there is a number that provably realizes it in Heyting arithmetic; and it is provable in Heyting arithmetic with extended Church's thesis "and Markov's principle" if and only if there is a number that provably realizes it in Peano arithmetic.
In constructive analysis.
Markov's principle is equivalent, in the language of real analysis, to the following principles:
Modified realizability does not justify Markov's principle, even if classical logic is used in the meta-theory: there is no realizer in the language of simply typed lambda calculus as this language is not Turing-complete and arbitrary loops cannot be defined in it.
Weak Markov's principle.
The weak Markov's principle is a weaker form of the principle. It may be stated in the language of analysis, as a conditional statement for the positivity of a real number:
formula_10
This form can be justified by Brouwer's continuity principles, whereas the stronger form contradicts them. Thus the weak Markov principle can be derived from intuitionistic, realizability, and classical reasoning, in each case for different reasons, but it is not valid in the general constructive sense of Bishop, nor provable in the set theory formula_11.
To understand what the principle is about, it helps to inspect a stronger statement. The following expresses that any real number formula_12, such that no non-positive formula_13 is not below it, is positive:
formula_14
where formula_15 denotes the negation of formula_16. This is a stronger implication because the antecedent is looser. Note that here a logically positive statement is concluded from a logically negative one. It is implied by the weak Markov's principle when elevating the De Morgan's law for formula_17 to an equivalence.
Assuming classical double-negation elimination, the weak Markov's principle becomes trivial, expressing that a number larger than all non-positive numbers is positive.
Extensionality of functions.
A function formula_18 between metric spaces is called "strongly extensional" if formula_19 implies formula_20, which is classically just the contraposition of the function preserving equality. Markov's principle can be shown to be equivalent to the proposition that all functions between arbitrary metric spaces are strongly extensional, while the weak Markov's principle is equivalent to the proposition that all functions from complete metric spaces to metric spaces are strongly extensional.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\Big(\\forall n \\big(P(n) \\vee \\neg P(n)\\big) \\wedge \\neg \\forall n\\, \\neg P(n)\\Big) \\rightarrow \\exists n\\, P(n)"
},
{
"math_id": 1,
"text": "\\exists n\\;P(n)"
},
{
"math_id": 2,
"text": "\\neg \\neg \\exists n\\;P(n)"
},
{
"math_id": 3,
"text": "P"
},
{
"math_id": 4,
"text": "\\forall n \\big(P(n)\\lor \\neg P(n)\\big),\\ \\neg \\neg \\exists n\\;P(n)\\ \\ \\vdash\\ \\ \\exists n\\;P(n)"
},
{
"math_id": 5,
"text": "\\neg \\neg \\exists n\\;f(n)=0 \\rightarrow \\exists n\\;f(n)=0"
},
{
"math_id": 6,
"text": "f"
},
{
"math_id": 7,
"text": "\\forall e\\;\\forall x\\;\\big(\\neg \\neg \\exists w\\; T_1(e, x, w) \\rightarrow \\exists w\\; T_1(e, x, w)\\big)"
},
{
"math_id": 8,
"text": "\\omega"
},
{
"math_id": 9,
"text": "P(0), P(1), P(2),\\dots"
},
{
"math_id": 10,
"text": "\\forall (x\\in\\mathbb{R})\\, \\Big(\\forall(y\\in\\mathbb{R}) \\big(\\neg\\neg(0 < y) \\lor \\neg\\neg(y < x)\\big)\\Big) \\,\\to\\, (0 < x)."
},
{
"math_id": 11,
"text": "{\\mathsf {IZF}}"
},
{
"math_id": 12,
"text": "x"
},
{
"math_id": 13,
"text": "y"
},
{
"math_id": 14,
"text": "\\nexists(y \\le 0)\\, x \\le y \\,\\to\\, (0 < x),"
},
{
"math_id": 15,
"text": "x \\leq y"
},
{
"math_id": 16,
"text": "y < x"
},
{
"math_id": 17,
"text": "\\neg A\\lor \\neg B"
},
{
"math_id": 18,
"text": "f: X \\to Y"
},
{
"math_id": 19,
"text": "d(f(x),f(y)) > 0 "
},
{
"math_id": 20,
"text": "d(x,y) > 0"
}
] |
https://en.wikipedia.org/wiki?curid=14920509
|
149215
|
Hilbert's Nullstellensatz
|
Relation between algebraic varieties and polynomial ideals
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem).
Formulation.
Let formula_0 be a field (such as the rational numbers) and formula_1 be an algebraically closed field extension of formula_0 (such as the complex numbers). Consider the polynomial ring formula_2 and let formula_3 be an ideal in this ring. The algebraic set formula_4 defined by this ideal consists of all formula_5-tuples formula_6 in formula_7 such that formula_8 for all formula_9 in formula_3. Hilbert's Nullstellensatz states that if "p" is some polynomial in formula_2 that vanishes on the algebraic set formula_4, i.e. formula_10 for all formula_11 in formula_4, then there exists a natural number formula_12 such that formula_13 is in formula_3.
An immediate corollary is the weak Nullstellensatz: The ideal formula_14 contains 1 if and only if the polynomials in "I" do not have any common zeros in "Kn". The weak Nullstellensatz may also be formulated as follows: if "I" is a proper ideal in formula_15 then V("I") cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of "k". This is the reason for the name of the theorem, the full version of which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here; for example, the elements of the proper ideal ("X"2 + 1) in formula_16 do not have a common zero in formula_17
With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as
formula_18
for every ideal "J". Here, formula_19 denotes the radical of "J" and I("U") is the ideal of all polynomials that vanish on the set "U".
In this way, taking formula_20 we obtain an order-reversing bijective correspondence between the algebraic sets in "K""n" and the radical ideals of formula_21 In fact, more generally, one has a Galois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators.
As a particular example, consider a point formula_22. Then formula_23. More generally,
formula_24
Conversely, every maximal ideal of the polynomial ring formula_25 (note that formula_1 is algebraically closed) is of the form formula_26 for some formula_27.
As another example, an algebraic subset "W" in "K""n" is irreducible (in the Zariski topology) if and only if formula_28 is a prime ideal.
Proofs.
There are many known proofs of the theorem. Some are non-constructive, such as the first one. Others are constructive, as based on algorithms for expressing 1 or "pr" as a linear combination of the generators of the ideal.
Using Zariski's lemma.
Zariski's lemma asserts that if a field is finitely generated as an associative algebra over a field "k", then it is a finite field extension of "k" (that is, it is also finitely generated as a vector space).
Here is a sketch of a proof using this lemma.
Let formula_29 ("k" algebraically closed field), "I" an ideal of "A," and "V" the common zeros of "I" in formula_30. Clearly, formula_31. Let formula_32. Then formula_33 for some prime ideal formula_34 in "A". Let formula_35 and formula_36 a maximal ideal in formula_37. By Zariski's lemma, formula_38 is a finite extension of "k"; thus, is "k" since "k" is algebraically closed. Let formula_39 be the images of formula_40 under the natural map formula_41 passing through formula_37. It follows that formula_42 and formula_43.
Using resultants.
The following constructive proof of the weak form is one of the oldest proofs (the strong form results from the Rabinowitsch trick, which is also constructive).
The resultant of two polynomials depending on a variable x and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in x, every zero (in the other variables) of the resultant may be extended into a common zero of the two polynomials.
The proof is as follows.
If the ideal is principal, generated by a non-constant polynomial p that depends on x, one chooses arbitrary values for the other variables. The fundamental theorem of algebra asserts that this choice can be extended to a zero of p.
In the case of several polynomials formula_44 a linear change of variables allows to suppose that formula_45 is monic in the first variable x. Then, one introduces formula_46 new variables formula_47 and one considers the resultant
formula_48
As R is in the ideal generated by formula_44 the same is true for the coefficients in R of the monomials in formula_49 So, if 1 is in the ideal generated by these coefficients, it is also in the ideal generated by formula_50 On the other hand, if these coefficients have a common zero, this zero can be extended to a common zero of formula_44 by the above property of the resultant.
This proves the weak Nullstellensatz by induction on the number of variables.
Using Gröbner bases.
A Gröbner basis is an algorithmic concept that was introduced in 1973 by Bruno Buchberger. It is presently fundamental in computational geometry. A Gröbner basis is a special generating set of an ideal from which most properties of the ideal can easily be extracted. Those that are related to the Nullstellensatz are the following:
Generalizations.
The Nullstellensatz is subsumed by a systematic development of the theory of Jacobson rings, which are those rings in which every radical ideal is an intersection of maximal ideals. Given Zariski's lemma, proving the Nullstellensatz amounts to showing that if "k" is a field, then every finitely generated "k"-algebra "R" (necessarily of the form formula_51) is Jacobson. More generally, one has the following theorem:
Let formula_37 be a Jacobson ring. If formula_52 is a finitely generated "R"-algebra, then formula_52 is a Jacobson ring. Furthermore, if formula_53 is a maximal ideal, then formula_54 is a maximal ideal of formula_55, and formula_56 is a finite extension of formula_38.
Other generalizations proceed from viewing the Nullstellensatz in scheme-theoretic terms as saying that for any field "k" and nonzero finitely generated "k"-algebra "R", the morphism formula_57 admits a section étale-locally (equivalently, after base change along formula_58 for some finite field extension formula_59). In this vein, one has the following theorem:
Any faithfully flat morphism of schemes formula_60 locally of finite presentation admits a "quasi-section", in the sense that there exists a faithfully flat and locally quasi-finite morphism formula_61 locally of finite presentation such that the base change formula_62 of formula_63 along formula_64 admits a section. Moreover, if formula_65 is quasi-compact (resp. quasi-compact and quasi-separated), then one may take formula_66 to be affine (resp. formula_66 affine and formula_64 quasi-finite), and if formula_63 is smooth surjective, then one may take formula_64 to be étale.
Serge Lang gave an extension of the Nullstellensatz to the case of infinitely many generators:
Let formula_67 be an infinite cardinal and let formula_68 be an algebraically closed field whose transcendence degree over its prime subfield is strictly greater than formula_69. Then for any set formula_70 of cardinality formula_67, the polynomial ring formula_71 satisfies the Nullstellensatz, i.e., for any ideal formula_72 we have that formula_73.
Effective Nullstellensatz.
In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial g belongs or not to an ideal generated, say, by "f"1, ..., "fk"; we have "g"
"f r" in the strong version, "g"
1 in the weak form. This means the existence or the non-existence of polynomials "g"1, ..., "gk" such that "g"
"f"1"g"1 + ... + "fkgk". The usual proofs of the Nullstellensatz are not constructive, non-effective, in the sense that they do not give any way to compute the "gi".
It is thus a rather natural question to ask if there is an effective way to compute the "gi" (and the exponent r in the strong form) or to prove that they do not exist. To solve this problem, it suffices to provide an upper bound on the total degree of the "gi": such a bound reduces the problem to a finite system of linear equations that may be solved by usual linear algebra techniques. Any such upper bound is called an effective Nullstellensatz.
A related problem is the ideal membership problem, which consists in testing if a polynomial belongs to an ideal. For this problem also, a solution is provided by an upper bound on the degree of the "gi". A general solution of the ideal membership problem provides an effective Nullstellensatz, at least for the weak form.
In 1925, Grete Hermann gave an upper bound for ideal membership problem that is doubly exponential in the number of variables. In 1982 Mayr and Meyer gave an example where the "gi" have a degree that is at least double exponential, showing that every general upper bound for the ideal membership problem is doubly exponential in the number of variables.
Since most mathematicians at the time assumed the effective Nullstellensatz was at least as hard as ideal membership, few mathematicians sought a bound better than double-exponential. In 1987, however, W. Dale Brownawell gave an upper bound for the effective Nullstellensatz that is simply exponential in the number of variables. Brownawell's proof relied on analytic techniques valid only in characteristic 0, but, one year later, János Kollár gave a purely algebraic proof, valid in any characteristic, of a slightly better bound.
In the case of the weak Nullstellensatz, Kollár's bound is the following:
Let "f"1, ..., "fs" be polynomials in "n" ≥ 2 variables, of total degree "d"1 ≥ ... ≥ "ds". If there exist polynomials "gi" such that "f"1"g"1 + ... + "fsgs"
1, then they can be chosen such that
formula_74
This bound is optimal if all the degrees are greater than 2.
If d is the maximum of the degrees of the "fi", this bound may be simplified to
formula_75
An improvement due to M. Sombra is
formula_76
His bound improves Kollár's as soon as at least two of the degrees that are involved are lower than 3.
Projective Nullstellensatz.
We can formulate a certain correspondence between homogeneous ideals of polynomials and algebraic subsets of a projective space, called the projective Nullstellensatz, that is analogous to the affine one. To do that, we introduce some notations. Let formula_77 The homogeneous ideal,
formula_78
is called the "maximal homogeneous ideal" (see also irrelevant ideal). As in the affine case, we let: for a subset formula_79 and a homogeneous ideal "I" of "R",
formula_80
By formula_81 we mean: for every homogeneous coordinates formula_82 of a point of "S" we have formula_83. This implies that the homogeneous components of "f" are also zero on "S" and thus that formula_84 is a homogeneous ideal. Equivalently, formula_84 is the homogeneous ideal generated by homogeneous polynomials "f" that vanish on "S". Now, for any homogeneous ideal formula_85, by the usual Nullstellensatz, we have:
formula_86
and so, like in the affine case, we have:
There exists an order-reversing one-to-one correspondence between proper homogeneous radical ideals of "R" and subsets of formula_87 of the form formula_88 The correspondence is given by formula_89 and formula_90
Analytic Nullstellensatz (Rückert’s Nullstellensatz).
The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex "n"-space formula_91 Precisely, for each open subset formula_92 let formula_93 denote the ring of holomorphic functions on "U"; then formula_94 is a "sheaf" on formula_91 The stalk formula_95 at, say, the origin can be shown to be a Noetherian local ring that is a unique factorization domain.
If formula_96 is a germ represented by a holomorphic function formula_97, then let formula_98 be the equivalence class of the set
formula_99
where two subsets formula_100 are considered equivalent if formula_101 for some neighborhood "U" of 0. Note formula_98 is independent of a choice of the representative formula_102 For each ideal formula_103 let formula_104 denote formula_105 for some generators formula_106 of "I". It is well-defined; i.e., is independent of a choice of the generators.
For each subset formula_107, let
formula_108
It is easy to see that formula_109 is an ideal of formula_110 and that formula_111 if formula_112 in the sense discussed above.
The analytic Nullstellensatz then states: for each ideal formula_113,
formula_114
where the left-hand side is the radical of "I".
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "k"
},
{
"math_id": 1,
"text": "K"
},
{
"math_id": 2,
"text": "k[X_1, \\ldots, X_n]"
},
{
"math_id": 3,
"text": "I"
},
{
"math_id": 4,
"text": "\\mathrm V(I)"
},
{
"math_id": 5,
"text": "n"
},
{
"math_id": 6,
"text": "\\mathbf x = (x_1, \\dots, x_n)"
},
{
"math_id": 7,
"text": "K^n"
},
{
"math_id": 8,
"text": "f(\\mathbf x) = 0"
},
{
"math_id": 9,
"text": "f"
},
{
"math_id": 10,
"text": "p(\\mathbf x) = 0"
},
{
"math_id": 11,
"text": "\\mathbf x"
},
{
"math_id": 12,
"text": "r"
},
{
"math_id": 13,
"text": "p^r"
},
{
"math_id": 14,
"text": "I \\subseteq k[X_1, \\ldots, X_n]"
},
{
"math_id": 15,
"text": "k[X_1, \\ldots, X_n],"
},
{
"math_id": 16,
"text": "\\R[X]"
},
{
"math_id": 17,
"text": "\\R."
},
{
"math_id": 18,
"text": "\\hbox{I}(\\hbox{V}(J))=\\sqrt{J}"
},
{
"math_id": 19,
"text": "\\sqrt{J}"
},
{
"math_id": 20,
"text": "k = K"
},
{
"math_id": 21,
"text": "K[X_1, \\ldots, X_n]."
},
{
"math_id": 22,
"text": "P = (a_1, \\dots, a_n) \\in K^n"
},
{
"math_id": 23,
"text": "I(P) = (X_1 - a_1, \\ldots, X_n - a_n)"
},
{
"math_id": 24,
"text": "\\sqrt{I} = \\bigcap_{(a_1, \\dots, a_n) \\in V(I)} (X_1 - a_1, \\dots, X_n - a_n)."
},
{
"math_id": 25,
"text": "K[X_1,\\ldots,X_n]"
},
{
"math_id": 26,
"text": "(X_1 - a_1, \\ldots, X_n - a_n)"
},
{
"math_id": 27,
"text": "a_1,\\ldots,a_n \\in K"
},
{
"math_id": 28,
"text": "I(W)"
},
{
"math_id": 29,
"text": "A = k[t_1, \\ldots, t_n]"
},
{
"math_id": 30,
"text": "k^n"
},
{
"math_id": 31,
"text": "\\sqrt{I} \\subseteq I(V)"
},
{
"math_id": 32,
"text": "f \\not\\in \\sqrt{I}"
},
{
"math_id": 33,
"text": "f \\not\\in \\mathfrak{p}"
},
{
"math_id": 34,
"text": "\\mathfrak{p}\\supseteq I"
},
{
"math_id": 35,
"text": "R = (A/\\mathfrak{p}) [f^{-1}]"
},
{
"math_id": 36,
"text": "\\mathfrak{m}"
},
{
"math_id": 37,
"text": "R"
},
{
"math_id": 38,
"text": "R/\\mathfrak{m}"
},
{
"math_id": 39,
"text": "x_i"
},
{
"math_id": 40,
"text": "t_i"
},
{
"math_id": 41,
"text": "A \\to k"
},
{
"math_id": 42,
"text": "x = (x_1, \\ldots, x_n) \\in V"
},
{
"math_id": 43,
"text": "f(x) \\ne 0"
},
{
"math_id": 44,
"text": "p_1,\\ldots, p_n,"
},
{
"math_id": 45,
"text": "p_1"
},
{
"math_id": 46,
"text": "n-1"
},
{
"math_id": 47,
"text": "u_2, \\ldots, u_n,"
},
{
"math_id": 48,
"text": "R=\\operatorname{Res}_x(p_1,u_2p_2+\\cdots +u_np_n)."
},
{
"math_id": 49,
"text": "u_2, \\ldots, u_n."
},
{
"math_id": 50,
"text": "p_1,\\ldots, p_n."
},
{
"math_id": 51,
"text": "R = k[t_1,\\cdots,t_n]/I"
},
{
"math_id": 52,
"text": "S"
},
{
"math_id": 53,
"text": "\\mathfrak{n}\\subseteq S"
},
{
"math_id": 54,
"text": "\\mathfrak{m} := \\mathfrak{n} \\cap R"
},
{
"math_id": 55,
"text": "R"
},
{
"math_id": 56,
"text": "S/\\mathfrak{n}"
},
{
"math_id": 57,
"text": "\\mathrm{Spec} \\, R \\to \\mathrm{Spec} \\, k"
},
{
"math_id": 58,
"text": "\\mathrm{Spec} \\, L \\to \\mathrm{Spec} \\, k"
},
{
"math_id": 59,
"text": "L/k"
},
{
"math_id": 60,
"text": "f: Y \\to X"
},
{
"math_id": 61,
"text": "g: X' \\to X"
},
{
"math_id": 62,
"text": "f': Y \\times_X X' \\to X'"
},
{
"math_id": 63,
"text": "f"
},
{
"math_id": 64,
"text": "g"
},
{
"math_id": 65,
"text": "X"
},
{
"math_id": 66,
"text": "X'"
},
{
"math_id": 67,
"text": "\\kappa"
},
{
"math_id": 68,
"text": "K"
},
{
"math_id": 69,
"text": "\\kappa"
},
{
"math_id": 70,
"text": "S"
},
{
"math_id": 71,
"text": "A = K[x_i]_{i \\in S}"
},
{
"math_id": 72,
"text": "J \\sub A"
},
{
"math_id": 73,
"text": "\\sqrt{J} = \\hbox{I} (\\hbox{V} (J))"
},
{
"math_id": 74,
"text": "\\deg(f_ig_i) \\le \\max(d_s,3)\\prod_{j=1}^{\\min(n,s)-1}\\max(d_j,3)."
},
{
"math_id": 75,
"text": "\\max(3,d)^{\\min(n,s)}."
},
{
"math_id": 76,
"text": "\\deg(f_ig_i) \\le 2d_s\\prod_{j=1}^{\\min(n,s)-1}d_j."
},
{
"math_id": 77,
"text": "R = k[t_0, \\ldots, t_n]."
},
{
"math_id": 78,
"text": "R_+ = \\bigoplus_{d \\geqslant 1} R_d"
},
{
"math_id": 79,
"text": "S \\subseteq \\mathbb{P}^n"
},
{
"math_id": 80,
"text": "\\begin{align}\n\\operatorname{I}_{\\mathbb{P}^n}(S) &= \\{ f \\in R_+ \\mid f = 0 \\text{ on } S \\}, \\\\\n\\operatorname{V}_{\\mathbb{P}^n}(I) &= \\{ x \\in \\mathbb{P}^n \\mid f(x) = 0 \\text{ for all } f \\in I \\}.\n\\end{align}"
},
{
"math_id": 81,
"text": "f = 0 \\text{ on } S"
},
{
"math_id": 82,
"text": "(a_0 : \\cdots : a_n)"
},
{
"math_id": 83,
"text": "f(a_0,\\ldots, a_n)=0"
},
{
"math_id": 84,
"text": "\\operatorname{I}_{\\mathbb{P}^n}(S)"
},
{
"math_id": 85,
"text": "I \\subseteq R_+"
},
{
"math_id": 86,
"text": "\\sqrt{I} = \\operatorname{I}_{\\mathbb{P}^n}(\\operatorname{V}_{\\mathbb{P}^n}(I)),"
},
{
"math_id": 87,
"text": "\\mathbb{P}^n"
},
{
"math_id": 88,
"text": "\\operatorname{V}_{\\mathbb{P}^n}(I)."
},
{
"math_id": 89,
"text": "\\operatorname{I}_{\\mathbb{P}^n}"
},
{
"math_id": 90,
"text": "\\operatorname{V}_{\\mathbb{P}^n}."
},
{
"math_id": 91,
"text": "\\Complex^n."
},
{
"math_id": 92,
"text": "U \\subseteq \\Complex^n,"
},
{
"math_id": 93,
"text": "\\mathcal{O}_{\\Complex^n}(U)"
},
{
"math_id": 94,
"text": "\\mathcal{O}_{\\Complex^n}"
},
{
"math_id": 95,
"text": "\\mathcal{O}_{\\Complex^n, 0}"
},
{
"math_id": 96,
"text": "f \\in \\mathcal{O}_{\\Complex ^n, 0}"
},
{
"math_id": 97,
"text": "\\widetilde{f}: U \\to \\Complex "
},
{
"math_id": 98,
"text": "V_0(f)"
},
{
"math_id": 99,
"text": "\\left \\{ z \\in U \\mid \\widetilde{f}(z) = 0 \\right\\},"
},
{
"math_id": 100,
"text": "X, Y \\subseteq \\Complex^n"
},
{
"math_id": 101,
"text": "X \\cap U = Y \\cap U"
},
{
"math_id": 102,
"text": "\\widetilde{f}."
},
{
"math_id": 103,
"text": "I \\subseteq \\mathcal{O}_{\\Complex^n,0},"
},
{
"math_id": 104,
"text": "V_0(I)"
},
{
"math_id": 105,
"text": "V_0(f_1) \\cap \\dots \\cap V_0(f_r)"
},
{
"math_id": 106,
"text": "f_1, \\ldots, f_r"
},
{
"math_id": 107,
"text": "X \\subseteq \\Complex ^n"
},
{
"math_id": 108,
"text": "I_0(X) = \\left \\{ f \\in \\mathcal{O}_{\\Complex^n,0} \\mid V_0(f) \\supset X \\right \\}."
},
{
"math_id": 109,
"text": "I_0(X)"
},
{
"math_id": 110,
"text": "\\mathcal{O}_{\\Complex ^n, 0}"
},
{
"math_id": 111,
"text": "I_0(X) = I_0(Y)"
},
{
"math_id": 112,
"text": "X \\sim Y"
},
{
"math_id": 113,
"text": "I \\subseteq \\mathcal{O}_{\\Complex ^n, 0}"
},
{
"math_id": 114,
"text": "\\sqrt{I} = I_0(V_0(I))"
}
] |
https://en.wikipedia.org/wiki?curid=149215
|
149217
|
Bézout's theorem
|
Number of intersection points of algebraic curves and hypersurfaces
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that "in general" the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout.
In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two plane algebraic curves of degrees formula_0 and formula_1 have no component in common, they have formula_2 intersection points, counted with their multiplicity, and including points at infinity and points with complex coordinates.
In its modern formulation, the theorem states that, if N is the number of common points over an algebraically closed field of n projective hypersurfaces defined by homogeneous polynomials in "n" + 1 indeterminates, then N is either infinite, or equals the product of the degrees of the polynomials. Moreover, the finite case occurs almost always.
In the case of two variables and in the case of affine hypersurfaces, if multiplicities and points at infinity are not counted, this theorem provides only an upper bound of the number of points, which is almost always reached. This bound is often referred to as the Bézout bound.
Bézout's theorem is fundamental in computer algebra and effective algebraic geometry, by showing that most problems have a computational complexity that is at least exponential in the number of variables. It follows that in these areas, the best complexity that can be hoped for will occur with algorithms that have a complexity that is polynomial in the Bézout bound.
History.
In the case of plane curves, Bézout's theorem was essentially stated by Isaac Newton in his proof of Lemma 28 of volume 1 of his "Principia" in 1687, where he claims that two curves have a number of intersection points given by the product of their degrees.
The general theorem was later published in 1779 in Étienne Bézout's "Théorie générale des équations algébriques". He supposed the equations to be "complete", which in modern terminology would translate to generic. Since with generic polynomials, there are no points at infinity, and all multiplicities equal one, Bézout's formulation is correct, although his proof does not follow the modern requirements of rigor. This and the fact that the concept of intersection multiplicity was outside the knowledge of his time led to a sentiment expressed by some authors that his proof was neither correct nor the first proof to be given.
The proof of the statement that includes multiplicities requires an accurate definition of the intersection multiplicities, and was therefore not possible before the 20th century. The definitions of multiplicities that was given during the first half of the 20th century involved continuous and infinitesimal deformations. It follows that the proofs of this period apply only over the field of complex numbers. It is only in 1958 that Jean-Pierre Serre gave a purely algebraic definition of multiplicities, which led to a proof valid over any algebraically closed field.
Modern studies related to Bézout's theorem obtained different upper bounds to system of polynomials by using other properties of the polynomials, such as the Bernstein–Kushnirenko theorem, or generalized it to a large class of functions, such as Nash functions.
Statement.
Plane curves.
Suppose that "X" and "Y" are two plane projective curves defined over a field "F" that do not have a common component (this condition means that "X" and "Y" are defined by polynomials, without common divisor of positive degree). Then the total number of intersection points of "X" and "Y" with coordinates in an algebraically closed field "E" that contains "F", counted with their multiplicities, is equal to the product of the degrees of "X" and "Y".
General case.
The generalization in higher dimension may be stated as:
Let "n" projective hypersurfaces be given in a projective space of dimension "n" over an algebraically closed field, which are defined by "n" homogeneous polynomials in "n" + 1 variables, of degrees formula_3 Then either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product formula_4 If the hypersurfaces are in relative general position, then there are formula_5 intersection points, all with multiplicity 1.
There are various proofs of this theorem, which either are expressed in purely algebraic terms, or use the language of algebraic geometry. Three algebraic proofs are sketched below.
Bézout's theorem has been generalized as the so-called multi-homogeneous Bézout theorem.
Affine case.
The affine case of the theorem is the following statement, that was proven in 1983 by David Masser and Gisbert Wüstholz.
Consider n affine hypersurfaces that are defined over an algebraically closed field by n polynomials in n variables, of degrees formula_3 Then either the number of intersection points is infinite, or the number of intersection points, counted with their multiplicities, is at most the product formula_4 If the hypersurfaces are in relative general position, then there are exactly formula_5 intersection points, all with multiplicity 1.
This version is not a direct consequence of the general case, because it is possible to have a finite number of intersection points in the affine space, with infinitely many intersection points at infinity. The above statement is a special case of a more general statement, which is the result that Masser and Wüstholz proved.
For stating the general result, one has to recall that the intersection points form an algebraic set, and that there is a finite number of intersection points if and only if all component of the intersection have a zero dimension (an algebraic set of positive dimension has an infinity of points over an algebraically closed field). An intersection point is said "isolated" if it does not belong to a component of positive dimension of the intersection; the terminology make sense, since an isolated intersection point has neighborhoods (for Zariski topology or for the usual topology in the case of complex hypersurfaces) that does not contain any other intersection point.
Consider n projective hypersurfaces that are defined over an algebraically closed field by n homogeneous polynomials in formula_6 variables, of degrees formula_3 Then, the sum of the multiplicities of their isolated intersection points is at most the product formula_4 The result remains valid for any number m of hypersurfaces, if one sets formula_7 in the case formula_8 and, otherwise, if one orders the degrees for having formula_9 That is, there is no isolated intersection point if formula_8 and, otherwise, the bound is the product of the smallest degree and the formula_10 largest degrees.
Examples (plane curves).
Two lines.
The equation of a line in a Euclidean plane is linear, that is, it equates a polynomial of degree one to zero. So, the Bézout bound for two lines is 1, meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are parallel and meet at a point at infinity.
One can verify this with equations. The equation of a first line can be written in slope-intercept form formula_11 or, in projective coordinates formula_12 (if the line is vertical, one may exchange x and y). If the equation of a second line is (in projective coordinates) formula_13 by substituting formula_14 for y in it, one gets formula_15 If formula_16 one gets the x-coordinate of the intersection point by solving the latter equation in x and putting "t" = 1.
If formula_17 that is formula_18 the two line are parallel as having the same slope. If formula_19 they are distinct, and the substituted equation gives "t" = 0. This gives the point at infinity of projective coordinates (1, "s", 0).
A line and a curve.
As above, one may write the equation of the line in projective coordinates as formula_20 If curve is defined in projective coordinates by a homogeneous polynomial formula_21 of degree n, the substitution of y provides a homogeneous polynomial of degree n in x and t. The fundamental theorem of algebra implies that it can be factored in linear factors. Each factor gives the ratio of the x and t coordinates of an intersection point, and the multiplicity of the factor is the multiplicity of the intersection point.
If t is viewed as the "coordinate of infinity", a factor equal to t represents an intersection point at infinity.
If at least one partial derivative of the polynomial p is not zero at an intersection point, then the tangent of the curve at this point is defined (see ), and the intersection multiplicity is greater than one if and only if the line is tangent to the curve. If all partial derivatives are zero, the intersection point is a singular point, and the intersection multiplicity is at least two.
Two conic sections.
Two conic sections generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:
Multiplicity.
The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality.
Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed. For example, a tangent to a curve is a line that cuts the curve at a point that splits in several points if the line is slightly moved. This number is two in general (ordinary points), but may be higher (three for inflection points, four for undulation points, etc.). This number is the "multiplicity of contact" of the tangent.
This definition of a multiplicities by deformation was sufficient until the end of the 19th century, but has several problems that led to more convenient modern definitions: Deformations are difficult to manipulate; for example, in the case of a root of a univariate polynomial, for proving that the multiplicity obtained by deformation equals the multiplicity of the corresponding linear factor of the polynomial, one has to know that the roots are continuous functions of the coefficients. Deformations cannot be used over fields of positive characteristic. Moreover, there are cases where a convenient deformation is difficult to define (as in the case of more than two plane curves having a common intersection point), and even cases where no deformation is possible.
Currently, following Jean-Pierre Serre, a multiplicity is generally defined as the length of a local ring associated with the point where the multiplicity is considered. Most specific definitions can be shown to be special case of Serre's definition.
In the case of Bézout's theorem, the general intersection theory can be avoided, as there are proofs (see below) that associate to each input data for the theorem a polynomial in the coefficients of the equations, which factorizes into linear factors, each corresponding to a single intersection point. So, the multiplicity of an intersection point is the multiplicity of the corresponding factor. The proof that this multiplicity equals the one that is obtained by deformation, results then from the fact that the intersection points and the factored polynomial depend continuously on the roots.
Proofs.
Using the resultant (plane curves).
Let P and Q be two homogeneous polynomials in the indeterminates "x", "y", "t" of respective degrees p and q. Their zeros are the homogeneous coordinates of two projective curves. Thus the homogeneous coordinates of their intersection points are the common zeros of P and Q.
By collecting together the powers of one indeterminate, say y, one gets univariate polynomials whose coefficients are homogeneous polynomials in x and t.
For technical reasons, one must change of coordinates in order that the degrees in y of P and Q equal their total degrees (p and q), and each line passing through two intersection points does not pass through the point (0, 1, 0) (this means that no two point have the same Cartesian x-coordinate.
The resultant "R"("x" ,"t") of P and Q with respect to y is a homogeneous polynomial in x and t that has the following property: formula_24 with formula_25 if and only if it exist formula_26 such that formula_27 is a common zero of P and Q (see ). The above technical condition ensures that formula_26 is unique. The first above technical condition means that the degrees used in the definition of the resultant are p and q; this implies that the degree of R is pq (see ).
As R is a homogeneous polynomial in two indeterminates, the fundamental theorem of algebra implies that R is a product of pq linear polynomials. If one defines the multiplicity of a common zero of P and Q as the number of occurrences of the corresponding factor in the product, Bézout's theorem is thus proved.
For proving that the intersection multiplicity that has just been defined equals the definition in terms of a deformation, it suffices to remark that the resultant and thus its linear factors are continuous functions of the coefficients of P and Q.
Proving the equality with other definitions of intersection multiplicities relies on the technicalities of these definitions and is therefore outside the scope of this article.
Using U-resultant.
In the early 20th century, Francis Sowerby Macaulay introduced the multivariate resultant (also known as "Macaulay's resultant") of n homogeneous polynomials in n indeterminates, which is generalization of the usual resultant of two polynomials. Macaulay's resultant is a polynomial function of the coefficients of n homogeneous polynomials that is zero if and only the polynomials have a nontrivial (that is some component is nonzero) common zero in an algebraically closed field containing the coefficients.
The U-resultant is a particular instance of Macaulay's resultant, introduced also by Macaulay. Given n homogeneous polynomials formula_28 in "n" + 1 indeterminates formula_29 the U-resultant is the resultant of formula_30 and formula_31 where the coefficients formula_32 are auxiliary indeterminates. The U-resultant is a homogeneous polynomial in formula_33 whose degree is the product of the degrees of the formula_34
Although a multivariate polynomial is generally irreducible, the U-resultant can be factorized into linear (in the formula_35) polynomials over an algebraically closed field containing the coefficients of the formula_34 These linear factors correspond to the common zeros of the formula_36 in the following way: to each common zero formula_37 corresponds a linear factor formula_38 and conversely.
This proves Bézout's theorem, if the multiplicity of a common zero is defined as the multiplicity of the corresponding linear factor of the U-resultant. As for the preceding proof, the equality of this multiplicity with the definition by deformation results from the continuity of the U-resultant as a function of the coefficients of the formula_34
This proof of Bézout's theorem seems the oldest proof that satisfies the modern criteria of rigor.
Using the degree of an ideal.
Bézout's theorem can be proved by recurrence on the number of polynomials
by using the following theorem.
"Let V be a projective algebraic set of dimension formula_39 and degree formula_0, and H be a hypersurface (defined by a single polynomial) of degree formula_1, that does not contain any irreducible component of V; under these hypotheses, the intersection of V and H has dimension formula_40 and degree formula_41"
For a (sketched) proof using Hilbert series, see .
Beside allowing a conceptually simple proof of Bézout's theorem, this theorem is fundamental for intersection theory, since this theory is essentially devoted to the study of intersection multiplicities when the hypotheses of the above theorem do not apply.
|
[
{
"math_id": 0,
"text": "d_1"
},
{
"math_id": 1,
"text": "d_2"
},
{
"math_id": 2,
"text": "d_1d_2"
},
{
"math_id": 3,
"text": "d_1, \\ldots,d_n."
},
{
"math_id": 4,
"text": "d_1 \\cdots d_n."
},
{
"math_id": 5,
"text": "d_1 \\cdots d_n"
},
{
"math_id": 6,
"text": "n+1"
},
{
"math_id": 7,
"text": "d_{m+1}=0"
},
{
"math_id": 8,
"text": "m<n,"
},
{
"math_id": 9,
"text": "d_2\\ge d_3\\ge\\cdots \\ge d_m \\ge d_1."
},
{
"math_id": 10,
"text": "n-1"
},
{
"math_id": 11,
"text": "y=sx+m"
},
{
"math_id": 12,
"text": "y=sx+mt"
},
{
"math_id": 13,
"text": "ax+by+ct=0,"
},
{
"math_id": 14,
"text": "sx+mt"
},
{
"math_id": 15,
"text": "(a+bs)x + (c+bm)t=0."
},
{
"math_id": 16,
"text": "a+bs\\ne 0, "
},
{
"math_id": 17,
"text": "a+bs= 0, "
},
{
"math_id": 18,
"text": "s=-a/b,"
},
{
"math_id": 19,
"text": "m\\ne -c/b,"
},
{
"math_id": 20,
"text": "y=sx+mt."
},
{
"math_id": 21,
"text": "p(x,y,t)"
},
{
"math_id": 22,
"text": "(x-a)^2+(y-b)^2 = r^2"
},
{
"math_id": 23,
"text": "(x-az)^2+(y-bz)^2 - r^2z^2 = 0,"
},
{
"math_id": 24,
"text": "R(\\alpha,\\tau)=0"
},
{
"math_id": 25,
"text": "(\\alpha, \\tau)\\ne (0,0)"
},
{
"math_id": 26,
"text": "\\beta"
},
{
"math_id": 27,
"text": "(\\alpha, \\beta, \\tau)"
},
{
"math_id": 28,
"text": "f_1,\\ldots,f_n"
},
{
"math_id": 29,
"text": "x_0, \\ldots, x_n,"
},
{
"math_id": 30,
"text": "f_1,\\ldots,f_n,"
},
{
"math_id": 31,
"text": "U_0x_0+\\cdots +U_nx_n,"
},
{
"math_id": 32,
"text": "U_0, \\ldots, U_n"
},
{
"math_id": 33,
"text": "U_0, \\ldots, U_n,"
},
{
"math_id": 34,
"text": "f_i."
},
{
"math_id": 35,
"text": "U_i"
},
{
"math_id": 36,
"text": "f_i"
},
{
"math_id": 37,
"text": "(\\alpha_0, \\ldots, \\alpha_n)"
},
{
"math_id": 38,
"text": "(\\alpha_0 U_0 + \\cdots + \\alpha_n U_n),"
},
{
"math_id": 39,
"text": "\\delta"
},
{
"math_id": 40,
"text": "\\delta-1"
},
{
"math_id": 41,
"text": "d_1d_2."
}
] |
https://en.wikipedia.org/wiki?curid=149217
|
14921864
|
Summability criterion
|
In election science, a voting method satisfies the summability criterion if it is possible to tally election results locally by precinct, then calculate the results by adding up all the votes. More formally, the compilation or summation complexity of a voting system measures the difficulty of vote counting for individual precincts, and is equal to the smallest number of bits needed to summarize all the votes. A voting method is called summable if the number of bits grows as a polynomial function of the number of candidates.
Often, a group has to accept a decision, but not all the votes can be gathered together in a single location. In such a situation, we need to take the votes of the present voters and summarize them such that, when the other votes arrive, we can determine the winner. The compilation complexity of a voting-rule is the smallest number of bits required for the summary.
A key advantage of low compilation complexity is it makes it easier to verify voting outcomes. Low compilation complexity lets us summarize the outcome in each voting-station separately, which is easy to verify by having representatives from each party count the ballots in each polling station. Then, any voter can verify the final outcome by summing up the results from the 1000 voting stations. This verifiability is important so that the public trusts and accepts the results. The publicly-released information from each precinct can be used by independent election auditors to identify any evidence of electoral fraud with statistical techniques.
Compilation complexity is also algorithmically useful for computing the backward induction winner in Stackelberg voting games.
Definitions.
Let "r" be a voting rule: a function that takes as input a list of "n" ranked ballots, representing the preferences of "n" voters, and returns an outcome. There are some "k"<"n" voters whose votes are "known". A "compilation function" is a function "f" that takes as input a list of "k" ranked ballots and returns some output such that, given any number "u" := "n"-"k" of additional ranked ballots, the output of r on the entire set of ballots can be computed exactly.
The compilation complexity of a rule r is the worst-case number of bits in the output of the most efficient compilation function "f". This number is typically a function of "n" (the number of voters), "k" (the number of known votes), and "c" (the number of candidates). However, we focus on "c" alone for simplicity, as we are usually interested in the case with a very large number of unknown votes.
Compilation complexity of single-winner voting rules.
The number of possible ballots for any ranked voting rule is formula_0, providing an upper bound on the complexity. However, most rules have a much smaller compilation complexity.
Positional voting.
In positional voting systems like plurality or Borda, any set of votes can be summarized by recording the total score of each candidate (e.g. the number of times a candidate appears first in plurality). The winner can then be found by adding the scores in each precinct giving a bound of formula_1. A similar argument applies for score voting and approval voting.
Voting rules based on weighted majority graph.
The "weighted majority graph" of a voter profile is a directed graph in which the nodes are the candidates, and there is a directed edge from "x" to "y" iff a majority of voters prefer "x" to "y". The weight of this edge is the number of voters who prefer "x" to "y". Many rules are based only on the majority graph; the number of equivalence classes of such rules is at most the number of possible weighted majority graphs. This number is denoted by T("k","c") - the number of weighted tournaments on "c" vertices that can be obtained from "k" voters. Therefore, the compilation complexity is at most log(T("k","c")). An upper bound on log(T("k","c")) is formula_2, since it is sufficient to keep, for each pair of candidates x,y, the number of voters who prefer x to y, and this number is between 0 and "k".
Voting rules with runoff.
The compilation complexity of two-round voting (the contingent vote) is in formula_3. Note that this is higher than the compilation complexity of Borda voting, though the communication complexity of two-round voting is "lower" than that of Borda voting.
The compilation complexity of the single transferable vote is in formula_4, making it non-summable.
STAR voting is also in formula_3.
Bucklin's rule.
For Bucklin voting the compilation complexity is formula_5. For the closely-related highest median voting rules, the complexity for a ballot including formula_6 possible ratings is formula_7.
Compilation complexity of multi-winner voting rules.
Karia and Lang study the compilation complexity of several multiwinner voting rules, with either ranked ballots or approval ballots. For example:
|
[
{
"math_id": 0,
"text": "\\Theta(c!)"
},
{
"math_id": 1,
"text": "\\Theta(c)"
},
{
"math_id": 2,
"text": "\\frac{c (c-1)}{2}"
},
{
"math_id": 3,
"text": "\\Theta(c^2)"
},
{
"math_id": 4,
"text": "\\Theta\\left(2^c c\\right)"
},
{
"math_id": 5,
"text": "\\Theta(c^2) "
},
{
"math_id": 6,
"text": "k "
},
{
"math_id": 7,
"text": "\\Theta(c k) "
},
{
"math_id": 8,
"text": "\\Theta(c \\log{k c})"
}
] |
https://en.wikipedia.org/wiki?curid=14921864
|
14921993
|
Explained variation
|
Concept in mathematical modelling
In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be used.
The complementary part of the total variation is called unexplained or residual variation; likewise, when discussing variance as such, this is referred to as unexplained or residual variance.
Definition in terms of information gain.
Information gain by better modelling.
Following Kent (1983), we use the Fraser information (Fraser 1965)
formula_0
where formula_1 is the probability density of a random variable formula_2, and formula_3 with formula_4 (formula_5) are two families of parametric models. Model family 0 is the simpler one, with a restricted parameter space formula_6.
Parameters are determined by maximum likelihood estimation,
formula_7
The information gain of model 1 over model 0 is written as
formula_8
where a factor of 2 is included for convenience. Γ is always nonnegative; it measures the extent to which the best model of family 1 is better than the best model of family 0 in explaining "g"("r").
Information gain by a conditional model.
Assume a two-dimensional random variable formula_9 where "X" shall be considered as an explanatory variable, and "Y" as a dependent variable. Models of family 1 "explain" "Y" in terms of "X",
formula_10,
whereas in family 0, "X" and "Y" are assumed to be independent. We define the randomness of "Y" by formula_11, and the randomness of "Y", given "X", by formula_12. Then,
formula_13
can be interpreted as proportion of the data dispersion which is "explained" by "X".
Special cases and generalized usage.
Linear regression.
The fraction of variance unexplained is an established concept in the context of linear regression. The usual definition of the coefficient of determination is based on the fundamental concept of explained variance.
Correlation coefficient as measure of explained variance.
Let "X" be a random vector, and "Y" a random variable that is modeled by a normal distribution with centre formula_14. In this case, the above-derived proportion of explained variation formula_15 equals the squared correlation coefficient formula_16.
Note the strong model assumptions: the centre of the "Y" distribution must be a linear function of "X", and for any given "x", the "Y" distribution must be normal. In other situations, it is generally not justified to interpret formula_16 as proportion of explained variance.
In principal component analysis.
Explained variance is routinely used in principal component analysis. The relation to the Fraser–Kent information gain remains to be clarified.
Criticism.
As the fraction of "explained variance" equals the squared correlation coefficient formula_16, it shares all the disadvantages of the latter: it reflects not only the quality of the regression, but also the distribution of the independent (conditioning) variables.
In the words of one critic: "Thus formula_16 gives the 'percentage of variance explained' by the regression, an expression that, for most social scientists, is of doubtful meaning but great rhetorical value. If this number is large, the regression gives a good fit, and there is little point in searching for additional variables. Other regression equations on different data sets are said to be less satisfactory or less powerful if their formula_16 is lower. Nothing about formula_16 supports these claims". And, after constructing an example where formula_16 is enhanced just by jointly considering data from two different populations: "'Explained variance' explains nothing."
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "F(\\theta) = \\int \\textrm{d}r\\,g(r)\\,\\ln f(r;\\theta)"
},
{
"math_id": 1,
"text": "g(r)"
},
{
"math_id": 2,
"text": "R\\,"
},
{
"math_id": 3,
"text": "f(r;\\theta)\\,"
},
{
"math_id": 4,
"text": "\\theta\\in\\Theta_i"
},
{
"math_id": 5,
"text": "i=0,1\\,"
},
{
"math_id": 6,
"text": "\\Theta_0\\subset\\Theta_1"
},
{
"math_id": 7,
"text": "\\theta_i = \\operatorname{arg max}_{\\theta\\in\\Theta_i} F(\\theta)."
},
{
"math_id": 8,
"text": "\\Gamma(\\theta_1:\\theta_0) = 2 [ F(\\theta_1)-F(\\theta_0) ]\\,"
},
{
"math_id": 9,
"text": "R=(X,Y)"
},
{
"math_id": 10,
"text": "f(y\\mid x;\\theta)"
},
{
"math_id": 11,
"text": "D(Y)=\\exp[-2F(\\theta_0)]"
},
{
"math_id": 12,
"text": "D(Y\\mid X)=\\exp[-2F(\\theta_1)]"
},
{
"math_id": 13,
"text": "\\rho_C^2 = 1-D(Y\\mid X)/D(Y)"
},
{
"math_id": 14,
"text": "\\mu=\\Psi^\\textrm{T}X"
},
{
"math_id": 15,
"text": "\\rho_C^2"
},
{
"math_id": 16,
"text": "R^2"
}
] |
https://en.wikipedia.org/wiki?curid=14921993
|
14922
|
If and only if
|
Logical connective
<templatestyles src="Template:Quote_box/styles.css" />
Logical symbols representing "iff"
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, "P if and only if Q" means that "P" is true whenever "Q" is true, and the only case in which "P" is true is if "Q" is also true, whereas in the case of "P if Q", there could be other scenarios where "P" is true and "Q" is false.
In writing, phrases commonly used as alternatives to P "if and only if" Q include: "Q is necessary and sufficient for P", "for P it is necessary and sufficient that Q", "P is equivalent (or materially equivalent) to Q" (compare with material implication), "P precisely if Q", "P precisely (or exactly) when Q", "P exactly in case Q", and "P just in case Q". Some authors regard "iff" as unsuitable in formal writing; others consider it a "borderline case" and tolerate its use. In logical formulae, logical symbols, such as formula_0 and formula_1, are used instead of these phrases; see below.
Definition.
The truth table of "P" formula_0 "Q" is as follows:
It is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.
Usage.
Notation.
The corresponding logical symbols are "formula_0", "formula_1", and formula_2, and sometimes "iff". These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's Polish notation, it is the prefix symbol formula_3.
Another term for the logical connective, i.e., the symbol in logic formulas, is exclusive nor.
In TeX, "if and only if" is shown as a long double arrow: formula_4 via command \iff or \Longleftrightarrow.
Proofs.
In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts—that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have been shown to be both true, or both false.
Origin of iff and pronunciation.
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book "General Topology". Its invention is often credited to Paul Halmos, who wrote "I invented 'iff,' for 'if and only if'—but I could never believe I was really its first inventor."
It is somewhat unclear how "iff" was meant to be pronounced. In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". However, in the preface of "General Topology", Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and euphony demands something less I use Halmos' 'iff'". The authors of one discrete mathematics textbook suggest: "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if'", implying that "iff" could be pronounced as .
Usage in definitions.
Conventionally, definitions are "if and only if" statements; some texts — such as Kelley's "General Topology" — follow this convention, and use "if and only if" or "iff" in definitions of new terms. However, this usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention of interpreting "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover"). Moreover, in the case of a recursive definition, the "only if" half of the definition is interpreted as a sentence in the metalanguage stating that the sentences in the definition of a predicate are the "only sentences" determining the extension of the predicate.
In terms of Euler diagrams.
Euler diagrams show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and "P→Q" all mean that P is a subset, either proper or improper, of Q. "P if Q", "if Q then P", and Q→P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.
More general usage.
"Iff" is used outside the field of logic as well. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for "if and only if", indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, "if" is more often used than "iff" in statements of definition).
The elements of "X" are "all and only" the elements of "Y" means: "For any "z" in the domain of discourse, "z" is in "X" if and only if "z" is in "Y"."
When "if" means "if and only if".
In their , Russell and Norvig note (page 282), in effect, that it is often more natural to express "if and only if" as "if" together with a "database (or logic programming) semantics". They give the example of the English sentence "Richard has two brothers, Geoffrey and John".
In a database or logic program, this could be represented simply by two sentences:
Brother(Richard, Geoffrey).
Brother(Richard, John).
The database semantics interprets the database (or program) as containing "all" and "only" the knowledge relevant for problem solving in a given domain. It interprets "only if" as expressing in the metalanguage that the sentences in the database represent the "only" knowledge that should be considered when drawing conclusions from the database.
In first-order logic (FOL) with the standard semantics, the same English sentence would need to be represented, using "if and only if", with "only if" interpreted in the object language, in some such form as:
formula_5 X(Brother(Richard, X) iff X = Geoffrey or X = John).
Geoffrey ≠ John.
Compared with the standard semantics for FOL, the database semantics has a more efficient implementation. Instead of reasoning with sentences of the form:
"conclusion iff conditions"
it uses sentences of the form:
"conclusion if conditions"
to reason forwards from "conditions" to "conclusions" or backwards from "conclusions" to "conditions".
The database semantics is analogous to the legal principle expressio unius est exclusio alterius (the express mention of one thing excludes all others). Moreover, it underpins the application of logic programming to the representation of legal texts and legal reasoning.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\leftrightarrow"
},
{
"math_id": 1,
"text": "\\Leftrightarrow"
},
{
"math_id": 2,
"text": "\\equiv"
},
{
"math_id": 3,
"text": "E"
},
{
"math_id": 4,
"text": "\\iff"
},
{
"math_id": 5,
"text": "\\forall "
}
] |
https://en.wikipedia.org/wiki?curid=14922
|
14922523
|
Volume contraction
|
Decrease in the volume of body fluid
Volume contraction is a decrease in the volume of body fluid, including the dissolved substances that maintain osmotic balance (osmolytes). The loss of the water component of body fluid is specifically termed dehydration.
By body fluid compartment.
Volume contraction is more or less a loss of extracellular fluid (ECF) and/or intracellular fluid (ICF).
ECF volume contraction.
Volume contraction of extracellular fluid is directly coupled to and almost proportional to volume contraction of blood plasma, which is termed hypovolemia. Thus, it primarily affects the circulatory system, potentially causing hypovolemic shock.
ECF volume contraction or hypovolemia is usually the type of volume contraction of primary concern in emergency, since ECF is approximately half the volume of ICF and is the first to be affected in e.g. bleeding. Volume contraction is sometimes even used synonymously with hypovolemia.
ICF volume contraction.
Volume contraction of intracellular fluid may occur after substantial fluid loss, since it is much larger than ECF volume, or loss of potassium (K+) "see section below".
ICF volume contraction may cause disturbances in various organs throughout the body.
Dependence on lost solutes.
Na+ loss approximately correlates with fluid loss from ECF, since Na+ has a much higher concentration in ECF than ICF. In contrast, K+ has a much higher concentration in ICF than ECF, and therefore its loss rather correlates with fluid loss from ICF, since K+ loss from ECF causes the K+ in ICF to diffuse out of the cells, dragging water with it by osmosis.
Estimation.
When the body loses fluids, the amount lost from ICF and ECF, respectively, can be estimated by measuring volume and amount of substance of sodium (Na+) and potassium (K+) in the lost fluid, as well as estimating the body composition of the person.
1. To calculate an estimation, the total amount of substance in the body before the loss is first estimated:
formula_0
where:
2. The total amount of substance in the body after the loss is then estimated:
formula_1
where:
3. The new osmolarity becomes:
formula_2
where:
4. This osmolarity is evenly distributed in the body, and is used to estimate the new volumes of ICF and ECF, respectively:
formula_3
where:
In homologous manner:
formula_4
where:
5. The volume of lost fluid from each compartment:
formula_5
formula_6
where:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " n_b = Osm_b \\times TBW_b "
},
{
"math_id": 1,
"text": " n_a = n_b - n_{lost Na^+} - n_{lost K^+} "
},
{
"math_id": 2,
"text": " Osm_a = \\frac{n_a}{TBW_b - V_{lost}} "
},
{
"math_id": 3,
"text": " V_{ICF a} = \\frac{n_{ICF a}}{Osm_a} = \\frac{V_{ICF b} \\times Osm_b - n_{lost K^+}}{Osm_a} "
},
{
"math_id": 4,
"text": " V_{ECF a} = \\frac{n_{ECF a}}{Osm_a} = \\frac{V_{ECF b} \\times Osm_b - n_{lost Na^+}}{Osm_a} "
},
{
"math_id": 5,
"text": " V_{lost ICF} = V_{ICF b} - V_{ICF a} "
},
{
"math_id": 6,
"text": " V_{lost ECF} = V_{ECF b} - V_{ECF a} "
}
] |
https://en.wikipedia.org/wiki?curid=14922523
|
14922915
|
Rec. 709
|
Standard for HDTV image encoding and signal characteristics
Rec. 709, also known as Rec.709, BT.709, and ITU 709, is a standard developed by ITU-R for image encoding and signal characteristics of high-definition television.
The most recent version is BT.709-6 released in 2015. BT.709-6 defines the picture characteristics as having a (widescreen) aspect ratio of 16:9, 1080 active lines per picture, 1920 samples per line, and a square pixel aspect ratio.
The first version of the standard was approved by the CCIR as Rec.709 in 1990 (there was also CCIR Rec. XA/11 MOD F in 1989), with the stated goal of a worldwide HDTV standard. The ITU superseded the CCIR in 1992, and subsequently released BT.709-1 in November 1993. These early versions still left many unanswered questions, and the lack of consensus toward a worldwide HDTV standard was evident. So much so, some early HDTV systems such as 1035i30 and 1152i25 were still a part of the standard as late as 2002 in BT.709-5.
Technical details.
The standard is freely available at the ITU website, and that document should be used as the authoritative reference. The essentials are summarized below.
Image definition.
Recommendation ITU-R BT.709-6 defines a common image format (CIF) where picture characteristics are independent of the frame rate. The image is 1920x1080 pixels, for a total pixel count of 2,073,600.
Previous versions of BT.709 included legacy systems such as 1035i30 and 1152i25 HDTV systems. These are now obsolete, and replaced by the system defined in the 2015 ITU BT.709-6.
Frame rates.
BT.709 offers over a variety of frame rates and scanning schemes, which along with separating the picture size from frame rate has provided the flexibility for BT.709 to become the worldwide standard for HDTV. This allows manufacturers to create a single television set or display for all markets world-wide.
BT.709-6 specifies the following frame rates, where P indicates a progressively scanned frame, PsF indicates progressive segmented frames, and I indicates interlaced:
match the frame rate used for theatrical motion pictures. The fractional rates are included for compatibility with the "pull-down" rates used with NTSC.
regions that formerly used 50 Hz systems such as PAL or SECAM. There are no fractional rates as PAL and SECAM did not have the pull-down issue of NTSC.
regions that formerly used 60 Hz systems such as NTSC. Here again, the fractional rates are for compatibility with legacy NTSC pull-down rates.
Image capture, encoding, and distribution.
Per BT.709, cameras may capture in either progressive or interlaced form. Video captured as progressive can be recorded, broadcast, or streamed as progressive or as progressive segmented frame (PsF). Video captured using an interlaced mode must be distributed as interlace unless a de-interlace process is applied in post production.
In cases where a progressive captured image is distributed in segmented frame mode, segment/field frequency must be twice the frame rate. Thus 30/PsF has the same field rate as 60/I.
Primary chromaticities.
Note that red and blue and yG are the same as the EBU Tech 3213 (PAL) primaries while the xG is halfway between EBU Tech 3213's xG and SMPTE C's xG (PAL and NTSC are two types of BT.601-6). In coverage of the CIE 1931 color space the Rec. 709 color space (and the derivative sRGB color space) is almost identical to Rec. 601 and covers 35.9%. It also covers 33.24% of CIE 1976 u’v’ and 33.5% of CIE 1931 xy. White point is D65 as specified in 2° standard observer.
Transfer characteristics.
Rec. 709 specifies a non-linear OETF (opto-electrical transfer function) which is known as the "camera gamma" and which describes how HDTV camera encodes the linear scene light into a non-linear electrical signal value. Rec. 709 doesn't specify the display EOTF (electro-optical transfer function) which describes how HDTV displays should convert the non-linear electrical signal into linear displayed light, that was done in ITU-R BT.1886. See ITU-R BT.2087 for a detailed description of the options for color conversion from Rec. 709 to Rec. 2020.
Rec. 709 OETF is as follows, close to 1/1.9 – 1/2.0 pure gamma:
formula_0
where
Rec. 709 OETF is linear in the bottom part and then a power function with a gamma 0.45 (about 1/2.222..., which is different from sRGB approximation of 2.2) for the rest of the range. The overall OETF approximate to a pure power function with a gamma 0.50 – 0.53 (about 1/1.9 – 1/2.0). Using any pure gamma as OETF is impossible, because compression into nonlinear values will remove a lot of immediately near black shadows. Thus linear segment was invented and a gamma of 0.45 has been used for the power segment. Old CRTs had a EOTF of 2.35 pure gamma and thus the corresponding correction of 709 OETF to get EOTF linear image (if 1.2 end-to-end gamma is assumed) was a pure gamma of 1.2 / 2.35 = 0.51 = 1/1.9608. It was used in such way by Apple until Display P3 devices came into existence.
In typical production practice the encoding function of image sources (OETF) is adjusted so that the final picture has the desired aesthetic look, as viewed on a reference monitor with a gamma of 2.4 (per ITU-R BT.1886) in a dim reference viewing environment (per ITU-R Rec. BT.2035 it is 10 lux of D65 or D93 in Japan).
Rec. 709 inverse OETF describes the conversion of the non-linear electrical signal value into the linear scene luminance. It is as follows:
formula_5
The display EOTF of HDTV (sometimes referred as the "display gamma"), is not the inverse of the camera OETF. The EOTF is not specified in Rec. 709. It is discussed in EBU Tech 3320 and specified in ITU-R BT.1886 as an equivalent gamma of 2.4, that is deviating from it in black region depending on how deep the black is. This is a higher gamma than the approximately gamma 2.0 of Rec. 709 OETF. The resulting end-to-end system gamma (OOTF) of HD television system is about 1.2 and it has been deliberately designed to provide compensation for the dim surround effect.
Rec. 709 and sRGB share the same primary chromaticities and white point chromaticity; however, sRGB is explicitly output (display) referred with an equivalent gamma of 2.2 (the actual function is also piecewise to avoid near black issues). Display P3 uses sRGB EOTF with its linear segment, a change of that segment from 709 is needed by either using parametric curve encoding of ICC v4 or by using slope limit.
Digital representation.
Rec. 709 defines an "R’G’B’" encoding and a "Y’C"B"C"R encoding, each with either 8 bits or 10 bits per sample in each color channel. In the 8-bit encoding the "R’", "B’", "G’", and "Y’" channels have a nominal range of [16..235], and the "C"B and "C"R channels have a nominal range of [16..240] with 128 as the neutral value. So in limited range "R’G’B’" reference black is (16, 16, 16) and reference white is (235, 235, 235), and in "Y’C"B"C"R reference black is (16, 128, 128) and reference white is (235, 128, 128). Values outside the nominal ranges are allowed, but typically they would be clamped for broadcast or for display (except for Superwhite and xvYCC). Values 0 and 255 are reserved as timing references (SAV and EAV), and may not contain color data (for 8 bits, for 10 bits more values are reserved and for 12 bits even more, no values are reserved in files or RGB mode or full range YCbCr digital modes like sYCC or opYCC). Rec. 709's 10-bit encoding uses nominal values four times those of the 8-bit encoding, to ease the conversion it uses simple padding for reference values, for example 240 is just padded by two trailing zeroes and gives 960 for 10 bit maximum chroma. Rec. 709's nominal ranges are the same as those defined in ITU Rec. 601.
Standards conversion.
Conversion between different standards of video frame rates and color encoding has always been a challenge for content producers distributing through regions with different standards and requirements. While BT.709 has eased the compatibility issue in terms of the consumer and television set manufacturer, broadcast facilities still use a particular frame rate based on region, such as 29.97 in North America, or 25 in Europe meaning that broadcast content still requires at least frame rate conversion.
Converting standard definition.
The vast legacy library of standard-definition programs and content presents further challenges. NTSC, PAL, and SECAM are all interlaced formats in a 4:3 aspect ratio, and at a relatively low resolution. Scaling them up to HD resolution with a 16:9 aspect ratio presents a number of challenges.
First is the potential for distracting motion artifacts due to interlaced video content. The solution is to either up-convert only to an interlaced BT.709 format at the same field rate, and scale the fields independently, or use motion processing to remove the inter-field motion and deinterlace, creating progressive frames. In the latter case, motion processing can introduce artifacts and can be slow to process.
Second is the issue of accommodating the SD 4:3 aspect ratio into the HD 16:9 frame. Cropping the top and/or bottom of the standard-definition frame may or may not work, depending on if the composition allows it and if there are graphics or titles that would be cut off. Alternately, pillar-boxing can show the entire 4:3 image by leaving black borders on the left and right. Sometimes this black is filled with a stretched and blurred form of the image.
In addition, the SMPTE C RGB primaries used in North American standard definition are different than those of BT.709 (SMPTE C is commonly referred to as NTSC, however it is a different set of primaries and a different white point than the 1953 NTSC). The red and blue primaries for PAL and SECAM are the same as BT.709, with a change in the green primary. Converting the image precisely requires a LUT (lookup table) or a color managed workflow to convert the colors to the new colorspace. However in practice this is often ignored, except in mpv, because even if the player is color managed (most of them are not, including VLC), it can see BT.709 or BT.2020 primaries only.
Luma coefficients.
When encoding "Y’C"B"C"R video, BT.709 creates gamma-encoded luma ("Y’") using matrix coefficients 0.2126, 0.7152, and 0.0722 (together they add to 1). BT.709-1 used slightly different 0.2125, 0.7154, 0.0721 (changed to standard ones in BT.709-2). Although worldwide agreement on a single R’G’B’ system was achieved with Rec. 709, adoption of different luma coefficients (as those are derived from primaries and white point) for "Y’C"B"C"R requires the use of different luma-chroma decoding for standard definition and high definition.
Conversion software and hardware.
These problems can be handled with video processing software which can be slow, or hardware solutions which allow for realtime conversion, and often with quality improvements.
Film retransfer.
A more ideal solution is to go back to original film elements for projects that originated on film. Due to the legacy issues of international distribution, many television programs that shot on film used a traditional negative cutting process, and then had a single film master that could be telecined for different formats. These projects can re-telecine their cut negative masters to a BT.709 master at a reasonable cost, and gain the benefit of the full resolution of film.
On the other hand for projects that originated on film, but completed their online master using video online methods would need to re-telecine the individual needed film takes and then re-assemble, a significantly greater amount of labor and machine time is required in this case, versus a telecine for a conformed negative. In this case, to enjoy the benefits of the film original would entail much higher costs to conform the film originals to a new HD master.
Relationship to sRGB.
sRGB was created after the early development of Rec.709. The creators of sRGB chose to use the same primaries and white point as Rec.709, but changed the tone response curve (sometimes referred to as gamma) to better suit the intended use in offices and brighter conditions than television viewing in a dark living room.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "V=\\begin{cases}\n4.500L & L < 0.018\\\\\n1.099 L^{0.45} - 0.099 & L \\ge 0.018\n\\end{cases}\n"
},
{
"math_id": 1,
"text": "V"
},
{
"math_id": 2,
"text": "\\left[0, 1 \\right]"
},
{
"math_id": 3,
"text": "L"
},
{
"math_id": 4,
"text": "\\begin{cases}\n4.5\\beta = \\alpha\\beta^{0.45} -\\alpha +1 \\\\\n4.5 = 0.45 \\alpha\\beta^{-0.55}\n\\end{cases}\n"
},
{
"math_id": 5,
"text": "L=\\begin{cases}\n\\dfrac{V}{4.5} & V < 0.081\\\\\n\\left ( \\dfrac{ V+0.099 }{ 1.099} \\right ) ^{\\frac{1}{0.45} } & V \\ge 0.081\n\\end{cases}\n"
}
] |
https://en.wikipedia.org/wiki?curid=14922915
|
14923880
|
Expectation propagation
|
Method to approximate a probability distribution
Expectation propagation (EP) is a technique in Bayesian machine learning.
EP finds approximations to a probability distribution. It uses an iterative approach that uses the factorization structure of the target distribution. It differs from other Bayesian approximation approaches such as variational Bayesian methods.
More specifically, suppose we wish to approximate an intractable probability distribution formula_0 with a tractable distribution formula_1. Expectation propagation achieves this approximation by minimizing the Kullback-Leibler divergence formula_2. Variational Bayesian methods minimize formula_3 instead.
If formula_1 is a Gaussian formula_4, then formula_2 is minimized with formula_5 and formula_6 being equal to the mean of formula_0 and the covariance of formula_0, respectively; this is called moment matching.
Applications.
Expectation propagation via moment matching plays a vital role in approximation for indicator functions that appear when deriving the message passing equations for TrueSkill.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "p(\\mathbf{x})"
},
{
"math_id": 1,
"text": "q(\\mathbf{x})"
},
{
"math_id": 2,
"text": "\\mathrm{KL}(p||q)"
},
{
"math_id": 3,
"text": "\\mathrm{KL}(q||p)"
},
{
"math_id": 4,
"text": "\\mathcal{N}(\\mathbf{x}|\\mu, \\Sigma)"
},
{
"math_id": 5,
"text": "\\mu"
},
{
"math_id": 6,
"text": "\\Sigma"
}
] |
https://en.wikipedia.org/wiki?curid=14923880
|
14927756
|
X-Machine Testing
|
The (Stream) X-Machine Testing Methodology is a "complete" functional testing approach to software- and hardware testing that exploits the scalability of the Stream X-Machine model of computation.
Using this methodology, it is likely to identify a finite test-set that exhaustively determines whether the tested system's implementation matches its specification. This goal is achieved by a divide-and-conquer approach, in which the design is decomposed by refinement into a collection of Stream X-Machines, which are implemented as separate modules, then tested bottom-up. At each integration stage, the testing method guarantees that the tested components are correctly integrated.
The methodology overcomes formal undecidability limitations by requiring that certain design for test principles are followed during specification and implementation. The resulting scalability means that practical software and hardware systems consisting of hundreds of thousands of states and millions of transitions have been tested successfully.
Motivation.
Much software testing is merely hopeful, seeking to exercise the software system in various ways to see whether any faults can be detected. Testing may indeed reveal some faults, but can never guarantee that the system is correct, once testing is over.
Functional testing methods seek to improve on this situation, by developing a formal specification describing the intended behaviour of the system, against which the implementation is later tested (a kind of conformance testing). The specification can be validated against the user-requirements and later proven to be consistent and complete by mathematical reasoning (eliminating any logical design flaws). Complete functional testing methods exploit the specification systematically, generating test-sets which exercise the implemented software system "exhaustively", to determine whether it conforms to the specification. In particular:
This level of testing can be difficult to achieve, since software systems are extremely complex, with hundreds of thousands of states and millions of transitions. What is needed is a way of breaking down the specification and testing problem into parts which can be addressed separately.
Scalable, Abstract Specifications.
Mike Holcombe first proposed using Samuel Eilenberg's theoretical X-machine model as the basis for software specification in the late 1980s.
This is because the model cleanly separates the "control flow" of a system from the "processing" carried out by the system. At a given level of abstraction, the system can be viewed as a simple finite-state machine consisting of a few states and transitions. The more complex processing is delegated to the "processing functions" on the transitions, which modify the underlying fundamental data type "X". Later, each processing function may be separately exposed and characterized by another X-machine, modelling the behaviour of that system operation.
This supports a divide-and-conquer approach, in which the overall system architecture is specified first, then each major system operation is specified next, followed by subroutines, and so forth. At each step, the level of complexity is manageable, because of the independence of each layer. In particular, it is easy for software engineers to validate the simple finite-state machines against user requirements.
Incrementally Testable Specifications.
Gilbert Laycock first proposed a particular kind of X-machine, the Stream X-Machine, as the basis for the testing method. The advantage of this variant was the way in which testing could be controlled. In a Stream X-Machine, the fundamental data type has a particular form: "X" = "Out"* × "Mem" × "In"*, where "In"* is an input stream, "Out"* is an output stream, and "Mem" is the internal memory. The transitions of a Stream X-Machine are labelled with processing functions of the form φ: "Mem" × "In" → "Out" × "Mem", that is, they consume one input from the input stream, possibly modify memory, and produce one output on the output stream (see the associated article for more details).
The benefits for testing are that software systems designed in this way are "observable" at each step. For each input, the machine takes one step, producing an output, such that input/output pairs may be matched exactly. This contrasts with other approaches in which the system "runs to completion" (taking multiple steps) before any observation is made. Furthermore, layered Stream X-Machines offer a convenient abstraction. At each level, the tester may "forget" about the details of the processing functions and consider the (sub-)system just as a simple finite-state machine. Powerful methods exist for testing systems that conform to finite-state specifications, such as Chow's W-method.
Specification Method.
When following the (Stream) X-Machine methodology, the first stage is to identify the various types of data to be processed. For example, a word processor will use basic types "Character" (keyboard input), "Position" (mouse cursor position) and "Command" (mouse or menu command). There may be other constructed types, such as "Text" ::= "Character"* (a sequence of characters), "Selection" ::= "Position" × "Position" (the start and end of the selection) and "Document" ::= "Text" × "Selection" × "Boolean" (the text, a possible selection, and a flag to signal if the document has been modified).
High-Level Specification.
The top-level specification is a Stream X-Machine describing the main user interaction with the system. For example, the word processor will exist in a number of states, in which keystrokes and commands will have different effects. Suppose that this word processor exists in the states {Writing, Selecting, Filing, Editing}. We expect the word processor to start in the initial Writing state, but to move to the Selecting state if either the mouse is "dragged", or the "shift-key" is held down. Once the selection is established, it should return to the Writing state. Likewise, if a menu option is chosen, this should enter the Editing or Filing state. In these states, certain keystrokes may have different meanings. The word processor eventually returns to the Writing state, when any menu command has finished. This state machine is designed and labelled informally with the various actions that cause it to change state.
The input, memory and output types for the top-level machine are now formalised. Suppose that the memory type of the simple word processor is the type "Document" defined above. This treats a document as a text string, with two positions marking a possible selection and a flag to indicate modification since the last "save"-command. A more complex word processor might support undoable editing, with a sequence of document states: "Document" ::= ("Text" × "Selection")*, which are collapsed to one document every time a "save"-command is performed.
Suppose that the input type for the machine is: "In" ::= "Command" × "Character" × "Position". This recognises that every interaction could be a simple character insertion, a menu command or a cursor placement. Any given interaction is a 3-tuple, but some places may be empty. For example, ("Insert", 'a', ε) would represent typing the character 'a'; while ("Position", ε, 32) would mean placing the cursor between characters 32 and 33; and ("Select", ε, 32) would mean selecting the text between the current cursor position and the place between characters 32 and 33.
The output type for the machine is designed so that it is possible to determine from the output "which" processing function was executed, in response to a given input. This relates to the property of "output distinguishability", described below.
Low-Level Specification.
If a system is complex, then it will most likely be decomposed into several Stream X-Machines. The most common kind of refinement is to take each of the major processing functions (which were the labels on the high-level machine) and treat these as separate Stream X-Machines. In this case, the input, memory and output types for the low-level machines will be different from those defined for the high-level machine. Either, this is treated as an expansion of the data sets used at the high level, or there is a translation from more abstract data sets at the high level into more detailed data sets at the lower level. For example, a command "Select" at the high level could be decomposed into three events: "MouseDown", "MouseMove", "MouseUp" at the lower level.
Ipate and Holcombe mention several kinds of refinement, including "functional refinement", in which the behaviour of the processing functions is elaborated in more detail, and "state refinement", in which a simple state-space is partitioned into a more complex state-space. Ipate proves these two kinds of refinement to be eventually equivalent
Systems are otherwise specified down to the level at which the designer is prepared to trust the primitive operations supported by the implementation environment. It is also possible to test small units exhaustively by other testing methods.
Design-For-Test Conditions.
The (Stream) X-Machine methodology requires the designer to observe certain design for test conditions. These are typically not too difficult to satisfy. For each Stream X-Machine in the specification, we must obtain:
A minimal machine is the machine with the fewest states and transitions for some given behaviour. Keeping the specification minimal simply ensures that the test sets are as small as possible. A deterministic machine is required for systems that are predictable. Otherwise, an implementation could make an arbitrary choice regarding which transition was taken. Some recent work has relaxed this assumption to allow testing of non-deterministic machines.
Test completeness is needed to ensure that the implementation is testable within tractable time. For example, if a system has distant, or hard-to-reach states that are only entered after memory has reached a certain limiting value, then special test inputs should be added to allow memory to be bypassed, forcing the state machine into the distant state. This allows all (abstract) states to be covered quickly during testing. Output distinguishability is the key property supporting the scalable testing method. It allows the tester to treat the processing functions φ as simple labels, whose detailed behaviour may be safely ignored, while testing the state machine of the next integration layer. The unique outputs are witness values, which guarantee that a particular function was invoked.
Testing Method.
The (Stream) X-Machine Testing Method assumes that both the design and the implementation can be considered as (a collection of) Stream X-Machines. For each pair of corresponding machines ("Spec", "Imp"), the purpose of testing is to determine whether the behaviour of "Imp", the machine of the implementation, exactly matches the behaviour of "Spec", the machine of the specification. Note that "Imp" need not be a minimal machine - it may have more states and transitions than "Spec" and still behave in an identical way.
To test all behaviours, it must be possible to drive a machine into all of its states, then attempt all possible transitions (those which should succeed, and those which should be blocked) to achieve full "positive" and "negative" testing (see above). For transitions which succeed, the destination state must also be verified. Note that if "Spec" and "Imp" have the same number of states, the above describes the smallest test-set that achieves the objective. If "Imp" has more states and transitions than "Spec", longer test sequences are needed to guarantee that "redundant" states in "Imp" also behave as expected.
Testing all States.
The basis for the test generation strategy is Tsun S. Chow's W-Method for testing finite-state automata, chosen because it supports the testing of redundant implementations. Chow's method assumes simple finite-state machines with observable inputs and outputs, but no directly observable states. To map onto Chow's formalism, the functions φi on the transitions of the Stream X-Machines are treated simply as labels (inputs, in Chow's terms) and the distinguishing outputs are used directly. (Later, a mapping from real inputs and memory ("mem", "in") is chosen to trigger each function φ, according to its domain).
To identify specific states in "Imp", Chow chooses a "characterization set, W", the smallest set of test sequences that uniquely characterizes each state in "Spec". That is, when starting in a given state, exercising the sequences in "W" should yield at least one observable difference, compared to starting in any other state.
To reach each state expected in "Spec", the tester constructs the "state cover, C", the smallest set of test sequences that reaches every state. This can be constructed by automatic breadth-first exploration of "Spec". The test-set which validates all the states of a minimal "Imp" is then: formula_0, where formula_1 denotes the "concatenated product" of the two sets. For example, if "C" = {⟨"a"⟩, ⟨"b"⟩} and "W" = {⟨"c"⟩, ⟨"d"⟩}, then formula_2.
Testing all Transitions.
The above test-set determines whether a minimal "Imp" has the same states as "Spec". To determine whether a minimal "Imp" also has the same transition behaviour as "Spec", the tester constructs the "transition cover, K". This is the smallest set of test sequences that reaches every state and then attempts every possible transition once, from that state. Now, the input alphabet consists of (the labels of) every function φ in Φ. Let us construct a set of length-1 test sequences, consisting of single functions chosen from Φ, and call this Φ1. The transition cover is defined as formula_3.
This will attempt every possible transition from every state. For those which succeed, we must validate the destination states. So, the smallest test-set "T"1 which completely validates the behaviour of a minimal "Imp" is given by: formula_4. This formula can be rearranged as:
formula_5,
where Φ0 is the set containing the empty sequence {}.
If "Imp" has more states than "Spec", the above test-set may not be sufficient to guarantee the conformant behaviour of replicated states in "Imp". So, sets of longer test sequences are chosen, consisting of all pairs of functions Φ2, all triples of functions Φ3 up to some limit Φ"k", when the tester is satisfied that "Imp" cannot contain chains of duplicated states longer than "k"-1. The final test formula is given by:
formula_6.
This test-set completely validates the behaviour of a non-minimal "Imp" in which chains of duplicated states are expected to be no longer than "k"-1. For most practical purposes, testing up to "k"=2, or "k"=3 is quite exhaustive, revealing all state-related faults in really poor implementations.
|
[
{
"math_id": 0,
"text": "C \\otimes W"
},
{
"math_id": 1,
"text": "\\otimes"
},
{
"math_id": 2,
"text": "C \\otimes W = \\{\\langle ac \\rangle, \\langle ad \\rangle, \\langle bc \\rangle, \\langle bd \\rangle \\}"
},
{
"math_id": 3,
"text": "K :=C \\cup C \\otimes \\Phi_{1}"
},
{
"math_id": 4,
"text": "T_{1} ::=C \\otimes W \\cup C \\otimes \\Phi_{1} \\otimes W"
},
{
"math_id": 5,
"text": "T_{1} ::=C \\otimes\\left(\\Phi_{0} \\cup \\Phi_{1}\\right) \\otimes W"
},
{
"math_id": 6,
"text": "T_{\\mathrm{k}} ::=C \\otimes\\left(\\Phi_{0} \\cup \\Phi_{1} \\ldots \\cup \\Phi_{\\mathrm{k}}\\right) \\otimes W"
}
] |
https://en.wikipedia.org/wiki?curid=14927756
|
1493236
|
Random permutation
|
Sequence where any order is equally likely
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards.
Generating random permutations.
Entry-by-entry brute force method.
One method of generating a random permutation of a set of size "n" uniformly at random (i.e., each of the "n"! permutations is equally likely to appear) is to generate a sequence by taking a random number between 1 and "n" sequentially, ensuring that there is no repetition, and interpreting this sequence ("x"1, ..., "x""n") as the permutation
formula_0
shown here in two-line notation.
This brute-force method will require occasional retries whenever the random number picked is a repeat of a number already selected. This can be avoided if, on the "i"th step (when "x"1, ..., "x""i" − 1 have already been chosen), one chooses a number "j" at random between 1 and "n" − "i" + 1 and sets "x""i" equal to the "j"th largest of the unchosen numbers.
Fisher-Yates shuffles.
A simple algorithm to generate a permutation of "n" items uniformly at random without retries, known as the Fisher–Yates shuffle, is to start with any permutation (for example, the identity permutation), and then go through the positions 0 through "n" − 2 (we use a convention where the first element has index 0, and the last element has index "n" − 1), and for each position "i" swap the element currently there with a randomly chosen element from positions "i" through "n" − 1 (the end), inclusive. It's easy to verify that any permutation of "n" elements will be produced by this algorithm with probability exactly 1/"n"!, thus yielding a uniform distribution over all such permutations.
unsigned uniform(unsigned m); /* Returns a random integer 0 <= uniform(m) <= m-1 with uniform distribution */
void initialize_and_permute(unsigned permutation[], unsigned n)
unsigned i;
for (i = 0; i <= n-2; i++) {
unsigned j = i+uniform(n-i); /* A random integer such that i ≤ j < n */
swap(permutation[i], permutation[j]); /* Swap the randomly picked element with permutation[i] */
Note that if the codice_0 function is implemented simply as codice_1 then a bias in the results is introduced if the number of return values of codice_2 is not a multiple of m, but this becomes insignificant if the number of return values of codice_2 is orders of magnitude greater than m.
Statistics on random permutations.
Fixed points.
The probability distribution of the number of fixed points in a uniformly distributed random permutation approaches a Poisson distribution with expected value 1 as "n" grows. In particular, it is an elegant application of the inclusion–exclusion principle to show that the probability that there are no fixed points approaches 1/"e". When "n" is big enough, the probability distribution of fixed points is almost the Poisson distribution with expected value 1. The first "n" moments of this distribution are exactly those of the Poisson distribution.
Randomness testing.
As with all random processes, the quality of the resulting distribution of an implementation of a randomized algorithm such as the Knuth shuffle (i.e., how close it is to the desired uniform distribution) depends on the quality of the underlying source of randomness, such as a pseudorandom number generator. There are many possible randomness tests for random permutations, such as some of the Diehard tests. A typical example of such a test is to take some permutation statistic for which the distribution is known and test whether the distribution of this statistic on a set of randomly generated permutations closely approximates the true distribution.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\begin{pmatrix}\n1 & 2 & 3 & \\cdots & n \\\\\nx_1 & x_2 & x_3 & \\cdots & x_n \\\\\n\\end{pmatrix},"
}
] |
https://en.wikipedia.org/wiki?curid=1493236
|
149326
|
Phylogenetic tree
|
Branching diagram of evolutionary relationships between organisms
A phylogenetic tree, phylogeny or evolutionary tree is a graphical representation which shows the evolutionary history between a set of species or taxa during a specific time. In other words, it is a branching diagram or a tree showing the evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical or genetic characteristics. In evolutionary biology, all life on Earth is theoretically part of a single phylogenetic tree, indicating common ancestry. Phylogenetics is the study of phylogenetic trees. The main challenge is to find a phylogenetic tree representing optimal evolutionary ancestry between a set of species or taxa. Computational phylogenetics (also phylogeny inference) focuses on the algorithms involved in finding optimal phylogenetic tree in the phylogenetic landscape.
Phylogenetic trees may be rooted or unrooted. In a "rooted" phylogenetic tree, each node with descendants represents the inferred most recent common ancestor of those descendants, and the edge lengths in some trees may be interpreted as time estimates. Each node is called a taxonomic unit. Internal nodes are generally called hypothetical taxonomic units, as they cannot be directly observed. Trees are useful in fields of biology such as bioinformatics, systematics, and phylogenetics. "Unrooted" trees illustrate only the relatedness of the leaf nodes and do not require the ancestral root to be known or inferred.
History.
The idea of a tree of life arose from ancient notions of a ladder-like progression from lower into higher forms of life (such as in the Great Chain of Being). Early representations of "branching" phylogenetic trees include a "paleontological chart" showing the geological relationships among plants and animals in the book "Elementary Geology", by Edward Hitchcock (first edition: 1840).
Charles Darwin featured a diagrammatic evolutionary "tree" in his 1859 book "On the Origin of Species". Over a century later, evolutionary biologists still use tree diagrams to depict evolution because such diagrams effectively convey the concept that speciation occurs through the adaptive and semirandom splitting of lineages.
The term "phylogenetic", or "phylogeny", derives from the two ancient greek words (), meaning "race, lineage", and (), meaning "origin, source".
Properties.
Rooted tree.
A rooted phylogenetic tree (see two graphics at top) is a directed tree with a unique node — the root — corresponding to the (usually imputed) most recent common ancestor of all the entities at the leaves of the tree. The root node does not have a parent node, but serves as the parent of all other nodes in the tree. The root is therefore a node of degree 2, while other internal nodes have a minimum degree of 3 (where "degree" here refers to the total number of incoming and outgoing edges).
The most common method for rooting trees is the use of an uncontroversial outgroup—close enough to allow inference from trait data or molecular sequencing, but far enough to be a clear outgroup. Another method is midpoint rooting, or a tree can also be rooted by using a non-stationary substitution model.
Unrooted tree.
Unrooted trees illustrate the relatedness of the leaf nodes without making assumptions about ancestry. They do not require the ancestral root to be known or inferred. Unrooted trees can always be generated from rooted ones by simply omitting the root. By contrast, inferring the root of an unrooted tree requires some means of identifying ancestry. This is normally done by including an outgroup in the input data so that the root is necessarily between the outgroup and the rest of the taxa in the tree, or by introducing additional assumptions about the relative rates of evolution on each branch, such as an application of the molecular clock hypothesis.
Bifurcating versus multifurcating.
Both rooted and unrooted trees can be either bifurcating or multifurcating. A rooted bifurcating tree has exactly two descendants arising from each interior node (that is, it forms a binary tree), and an unrooted bifurcating tree takes the form of an unrooted binary tree, a free tree with exactly three neighbors at each internal node. In contrast, a rooted multifurcating tree may have more than two children at some nodes and an unrooted multifurcating tree may have more than three neighbors at some nodes.
Labeled versus unlabeled.
Both rooted and unrooted trees can be either labeled or unlabeled. A labeled tree has specific values assigned to its leaves, while an unlabeled tree, sometimes called a tree shape, defines a topology only. Some sequence-based trees built from a small genomic locus, such as Phylotree, feature internal nodes labeled with inferred ancestral haplotypes.
Enumerating trees.
The number of possible trees for a given number of leaf nodes depends on the specific type of tree, but there are always more labeled than unlabeled trees, more multifurcating than bifurcating trees, and more rooted than unrooted trees. The last distinction is the most biologically relevant; it arises because there are many places on an unrooted tree to put the root. For bifurcating labeled trees, the total number of rooted trees is:
formula_0 for formula_1, formula_2 represents the number of leaf nodes.
For bifurcating labeled trees, the total number of unrooted trees is:
formula_3 for formula_4.
Among labeled bifurcating trees, the number of unrooted trees with formula_2 leaves is equal to the number of rooted trees with formula_5 leaves.
The number of rooted trees grows quickly as a function of the number of tips. For 10 tips, there are more than formula_6 possible bifurcating trees, and the number of multifurcating trees rises faster, with ca. 7 times as many of the latter as of the former.
Special tree types.
Dendrogram.
A dendrogram is a general name for a tree, whether phylogenetic or not, and hence also for the diagrammatic representation of a phylogenetic tree.
Cladogram.
A cladogram only represents a branching pattern; i.e., its branch lengths do not represent time or relative amount of character change, and its internal nodes do not represent ancestors.
Phylogram.
A phylogram is a phylogenetic tree that has branch lengths proportional to the amount of character change.
Chronogram.
A chronogram is a phylogenetic tree that explicitly represents time through its branch lengths.
Dahlgrenogram.
A Dahlgrenogram is a diagram representing a cross section of a phylogenetic tree.
Phylogenetic network.
A phylogenetic network is not strictly speaking a tree, but rather a more general graph, or a directed acyclic graph in the case of rooted networks. They are used to overcome some of the limitations inherent to trees.
Spindle diagram.
A spindle diagram, or bubble diagram, is often called a romerogram, after its popularisation by the American palaeontologist Alfred Romer.
It represents taxonomic diversity (horizontal width) against geological time (vertical axis) in order to reflect the variation of abundance of various taxa through time.
A spindle diagram is not an evolutionary tree: the taxonomic spindles obscure the actual relationships of the parent taxon to the daughter taxon and have the disadvantage of involving the paraphyly of the parental group.
This type of diagram is no longer used in the form originally proposed.
Coral of life.
Darwin also mentioned that the "coral" may be a more suitable metaphor than the "tree". Indeed, phylogenetic corals are useful for portraying past and present life, and they have some advantages over trees (anastomoses allowed, etc.).
Construction.
Phylogenetic trees composed with a nontrivial number of input sequences are constructed using computational phylogenetics methods. Distance-matrix methods such as neighbor-joining or UPGMA, which calculate genetic distance from multiple sequence alignments, are simplest to implement, but do not invoke an evolutionary model. Many sequence alignment methods such as ClustalW also create trees by using the simpler algorithms (i.e. those based on distance) of tree construction. Maximum parsimony is another simple method of estimating phylogenetic trees, but implies an implicit model of evolution (i.e. parsimony). More advanced methods use the optimality criterion of maximum likelihood, often within a Bayesian framework, and apply an explicit model of evolution to phylogenetic tree estimation. Identifying the optimal tree using many of these techniques is NP-hard, so heuristic search and optimization methods are used in combination with tree-scoring functions to identify a reasonably good tree that fits the data.
Tree-building methods can be assessed on the basis of several criteria:
Tree-building techniques have also gained the attention of mathematicians. Trees can also be built using T-theory.
File formats.
Trees can be encoded in a number of different formats, all of which must represent the nested structure of a tree. They may or may not encode branch lengths and other features. Standardized formats are critical for distributing and sharing trees without relying on graphics output that is hard to import into existing software. Commonly used formats are
Limitations of phylogenetic analysis.
Although phylogenetic trees produced on the basis of sequenced genes or genomic data in different species can provide evolutionary insight, these analyses have important limitations. Most importantly, the trees that they generate are not necessarily correct – they do not necessarily accurately represent the evolutionary history of the included taxa. As with any scientific result, they are subject to falsification by further study (e.g., gathering of additional data, analyzing the existing data with improved methods). The data on which they are based may be noisy; the analysis can be confounded by genetic recombination, horizontal gene transfer, hybridisation between species that were not nearest neighbors on the tree before hybridisation takes place, and conserved sequences.
Also, there are problems in basing an analysis on a single type of character, such as a single gene or protein or only on morphological analysis, because such trees constructed from another unrelated data source often differ from the first, and therefore great care is needed in inferring phylogenetic relationships among species. This is most true of genetic material that is subject to lateral gene transfer and recombination, where different haplotype blocks can have different histories. In these types of analysis, the output tree of a phylogenetic analysis of a single gene is an estimate of the gene's phylogeny (i.e. a gene tree) and not the phylogeny of the taxa (i.e. species tree) from which these characters were sampled, though ideally, both should be very close. For this reason, serious phylogenetic studies generally use a combination of genes that come from different genomic sources (e.g., from mitochondrial or plastid vs. nuclear genomes), or genes that would be expected to evolve under different selective regimes, so that homoplasy (false homology) would be unlikely to result from natural selection.
When extinct species are included as terminal nodes in an analysis (rather than, for example, to constrain internal nodes), they are considered not to represent direct ancestors of any extant species. Extinct species do not typically contain high-quality DNA.
The range of useful DNA materials has expanded with advances in extraction and sequencing technologies. Development of technologies able to infer sequences from smaller fragments, or from spatial patterns of DNA degradation products, would further expand the range of DNA considered useful.
Phylogenetic trees can also be inferred from a range of other data types, including morphology, the presence or absence of particular types of genes, insertion and deletion events – and any other observation thought to contain an evolutionary signal.
Phylogenetic networks are used when bifurcating trees are not suitable, due to these complications which suggest a more reticulate evolutionary history of the organisms sampled.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " (2n-3)!! = \\frac{(2n-3)!}{2^{n-2}(n-2)!} "
},
{
"math_id": 1,
"text": "n \\ge 2"
},
{
"math_id": 2,
"text": "n"
},
{
"math_id": 3,
"text": " (2n-5)!! = \\frac{(2n-5)!}{2^{n-3}(n-3)!} "
},
{
"math_id": 4,
"text": "n \\ge 3"
},
{
"math_id": 5,
"text": "n-1"
},
{
"math_id": 6,
"text": "34 \\times 10^6"
}
] |
https://en.wikipedia.org/wiki?curid=149326
|
14933760
|
Certificate (complexity)
|
In computational complexity theory, a certificate (also called a witness) is a string that certifies the answer to a computation, or certifies the membership of some string in a language. A certificate is often thought of as a solution path within a verification process, which is used to check whether a problem gives the answer "Yes" or "No".
In the decision tree model of computation, certificate complexity is the minimum number of the formula_0 input variables of a decision tree that need to be assigned a value in order to definitely establish the value of the Boolean function formula_1.
Use in definitions.
The notion of certificate is used to define semi-decidability: a formal language formula_2 is semi-decidable if there is a two-place predicate relationformula_3 such that formula_4 is computable, and such that for all formula_5:
x ∈ L ⇔ there exists y such that R(x, y)
Certificates also give definitions for some complexity classes which can alternatively be characterised in terms of nondeterministic Turing machines. A language formula_2 is in NP if and only if there exists a polynomial formula_6 and a polynomial-time bounded Turing machine formula_7 such that every word formula_5 is in the language formula_2 precisely if there exists a certificate formula_8 of length at most formula_9 such that formula_7 accepts the pair formula_10. The class co-NP has a similar definition, except that there are certificates for the words "not" in the language.
The class NL has a certificate definition: a problem in the language has a certificate of polynomial length, which can be verified by a deterministic logarithmic-space bounded Turing machine that can read each bit of the certificate once only. Alternatively, the deterministic logarithmic-space Turing machine in the statement above can be replaced by a bounded-error probabilistic constant-space Turing machine that is allowed to use only a constant number of random bits.
Examples.
The problem of determining, for a given graph formula_11 and number formula_12, if the graph contains an independent set of size formula_12 is in NP. Given a pair formula_13 in the language, a certificate is a set of formula_12 vertices which are pairwise not adjacent (and hence are an independent set of size formula_12).
A more general example, for the problem of determining if a given Turing machine accepts an input in a certain number of steps, is as follows:
Show L ∈ NP.
verifier:
gets string c = <M>, x, w such that |c| <= P(|w|)
check if c is an accepting computation of M on x with at most |w| steps
|c| <= O(|w|3)
if we have a computation of a TM with k steps the total size of the computation string is k2
Thus, «M>, x, w> ∈ L ⇔ there exists c <= a|w|3 such that «M>, x, w, c> ∈ V ∈ P
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "f"
},
{
"math_id": 2,
"text": "L"
},
{
"math_id": 3,
"text": "R \\subseteq \\Sigma^* \\times \\Sigma^*"
},
{
"math_id": 4,
"text": "R"
},
{
"math_id": 5,
"text": "x \\in \\Sigma^*"
},
{
"math_id": 6,
"text": "p"
},
{
"math_id": 7,
"text": "M"
},
{
"math_id": 8,
"text": "c"
},
{
"math_id": 9,
"text": "p(|x|)"
},
{
"math_id": 10,
"text": "(x, c)"
},
{
"math_id": 11,
"text": "G"
},
{
"math_id": 12,
"text": "k"
},
{
"math_id": 13,
"text": "(G, k)"
}
] |
https://en.wikipedia.org/wiki?curid=14933760
|
1493395
|
Pappus's hexagon theorem
|
Geometry theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that
It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes.
If one considers a pappian plane containing a hexagon as just described but with sides formula_3 and formula_8 parallel and also sides formula_9 and formula_6 parallel (so that the Pappus line formula_10 is the line at infinity), one gets the "affine version" of Pappus's theorem shown in the second diagram.
If the Pappus line formula_10 and the lines formula_11 have a point in common, one gets the so-called little version of Pappus's theorem.
The dual of this incidence theorem states that given one set of concurrent lines formula_12, and another set of concurrent lines formula_13, then the lines formula_14 defined by pairs of points resulting from pairs of intersections formula_15 and formula_16 and formula_17 and formula_18 are concurrent. ("Concurrent" means that the lines pass through one point.)
Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem.
The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of formula_19 and formula_20. This configuration is self dual. Since, in particular, the lines formula_21 have the properties of the lines formula_22 of the dual theorem, and collinearity of formula_2 is equivalent to concurrence of formula_21, the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.
Proof: affine form.
If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique.
Because of the parallelity in an affine plane one has to distinct two cases: formula_23 and formula_24. The key for a simple proof is the possibility for introducing a "suitable" coordinate system:
Case 1: The lines formula_11 intersect at point formula_25.
In this case coordinates are introduced, such that formula_26 (see diagram).
formula_27 have the coordinates formula_28.
From the parallelity of the lines formula_29 one gets formula_30 and the parallelity of the lines formula_31 yields formula_32. Hence line formula_33 has slope formula_34 and is parallel line formula_35.
Case 2: formula_36 (little theorem).
In this case the coordinates are chosen such that formula_37. From the parallelity of formula_38 and formula_39 one gets formula_40 and formula_41, respectively, and at least the parallelity formula_42.
Proof with homogeneous coordinates.
Choose homogeneous coordinates with
formula_43.
On the lines formula_44, given by formula_45, take the points formula_46 to be
formula_47
for some formula_48. The three lines formula_49 are formula_50, so they pass through the same point formula_51 if and only if formula_52. The condition for the three lines formula_53 and formula_54 with equations formula_55 to pass through the same point formula_56 is formula_57. So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so formula_58. Equivalently, formula_59 are collinear.
The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem. In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes.
The proof is invalid if formula_60 happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference.
Dual theorem.
Because of the principle of duality for projective planes the dual theorem of Pappus is true:
If 6 lines formula_61 are chosen alternately from two pencils with centers formula_62, the lines
formula_63
formula_64
formula_65
are concurrent, that means: they have a point formula_66 in common.<br>
The left diagram shows the projective version, the right one an affine version, where the points
formula_62 are points at infinity. If point formula_66 is on the line formula_67 than one gets the "dual little theorem" of Pappus' theorem.
If in the affine version of the dual "little theorem" point formula_66 is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane. The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too:
Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms "connect, intersect" and "parallel", the statement is affinely invariant, and one can introduce coordinates such that
formula_68 (see right diagram). The starting point of the sequence of chords is formula_69 One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point.
Other statements of the theorem.
In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:
formula_70
That is, if formula_71 are lines, then Pappus's theorem states that formula_72 must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when formula_73 "etc." are triples of concurrent lines.
Origins.
In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of Pappus's "Collection". These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's "Porisms."
The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ).
Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J.
Also KL is drawn parallel to AZ.
Then
KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).
These proportions might be written today as equations:
KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB).
The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular
(J, G; D, B) = (J, Z; H, E).
It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X.
Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.
What we showed originally can be written as (J, ∞; K, L) = (J, G; D, B), with ∞ taking the place of the (nonexistent) intersection of JK and AG. Pappus shows this, in effect, in Lemma XI, whose diagram, however, has different lettering:
What Pappus shows is DE.ZH : EZ.HD :: GB : BE, which we may write as
(D, Z; E, H) = (∞, B; E, G).
The diagram for Lemma XII is:
The diagram for Lemma XIII is the same, but BA and DG, extended, meet at N. In any case, considering straight lines through G as cut by the three straight lines through A, (and accepting that equations of cross ratios remain valid after permutation of the entries,) we have by Lemma III or XI
(G, J; E, H) = (G, D; ∞ Z).
Considering straight lines through D as cut by the three straight lines through B, we have
(L, D; E, K) = (G, D; ∞ Z).
Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear.
Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "A, B, C,"
},
{
"math_id": 1,
"text": "a,b,c,"
},
{
"math_id": 2,
"text": "X,Y,Z"
},
{
"math_id": 3,
"text": "Ab"
},
{
"math_id": 4,
"text": "aB, Ac"
},
{
"math_id": 5,
"text": "aC, Bc"
},
{
"math_id": 6,
"text": "bC"
},
{
"math_id": 7,
"text": "AbCaBc"
},
{
"math_id": 8,
"text": "aB"
},
{
"math_id": 9,
"text": "Bc"
},
{
"math_id": 10,
"text": "u"
},
{
"math_id": 11,
"text": "g,h"
},
{
"math_id": 12,
"text": "A, B, C"
},
{
"math_id": 13,
"text": "a, b, c"
},
{
"math_id": 14,
"text": "x, y, z"
},
{
"math_id": 15,
"text": "A\\cap b"
},
{
"math_id": 16,
"text": "a\\cap B, \\; A\\cap c"
},
{
"math_id": 17,
"text": "a\\cap C, \\;B\\cap c"
},
{
"math_id": 18,
"text": "b\\cap C"
},
{
"math_id": 19,
"text": "ABC"
},
{
"math_id": 20,
"text": "abc"
},
{
"math_id": 21,
"text": "Bc, bC, XY"
},
{
"math_id": 22,
"text": "x,y,z"
},
{
"math_id": 23,
"text": "g \\not\\parallel h"
},
{
"math_id": 24,
"text": "g \\parallel h"
},
{
"math_id": 25,
"text": "S=g\\cap h"
},
{
"math_id": 26,
"text": "\\;S=(0,0), \\; A=(0,1), \\;c=(1,0)\\;"
},
{
"math_id": 27,
"text": "B,C"
},
{
"math_id": 28,
"text": "\\;B=(0,\\gamma),\\; C=(0,\\delta), \\; \\gamma,\\delta \\notin \\{0,1\\}"
},
{
"math_id": 29,
"text": "Bc,\\; Cb"
},
{
"math_id": 30,
"text": "b=(\\tfrac{\\delta}{\\gamma},0)"
},
{
"math_id": 31,
"text": "Ab, Ba"
},
{
"math_id": 32,
"text": "a=(\\delta,0)"
},
{
"math_id": 33,
"text": "Ca"
},
{
"math_id": 34,
"text": "-1"
},
{
"math_id": 35,
"text": "Ac"
},
{
"math_id": 36,
"text": "g\\parallel h \\ "
},
{
"math_id": 37,
"text": "\\;c=(0,0), \\;b=(1,0),\\; A=(0,1), \\;B=(\\gamma,1),\\;\\gamma\\ne 0"
},
{
"math_id": 38,
"text": "Ab\\parallel Ba"
},
{
"math_id": 39,
"text": " cB\\parallel bC"
},
{
"math_id": 40,
"text": "\\;C=(\\gamma+1,1)\\;"
},
{
"math_id": 41,
"text": " \\;a=(\\gamma+1,0)\\;"
},
{
"math_id": 42,
"text": "\\;Ac\\parallel Ca\\;"
},
{
"math_id": 43,
"text": "C = (1, 0, 0), \\; c= (0, 1, 0),\\; X = (0, 0, 1), \\; A = (1, 1, 1)"
},
{
"math_id": 44,
"text": "AC, Ac, AX"
},
{
"math_id": 45,
"text": "x_2 = x_3,\\; x_1 =x_3, \\; x_2 = x_1"
},
{
"math_id": 46,
"text": "B, Y, b"
},
{
"math_id": 47,
"text": "B = (p, 1, 1),\\; Y = (1, q, 1),\\; b = (1, 1, r)"
},
{
"math_id": 48,
"text": "p, q, r"
},
{
"math_id": 49,
"text": "XB, CY, cb"
},
{
"math_id": 50,
"text": "x_1 = x_2 p,\\; x_2= x_3 q,\\; x_3 = x_1 r"
},
{
"math_id": 51,
"text": "a"
},
{
"math_id": 52,
"text": "rqp = 1"
},
{
"math_id": 53,
"text": "Cb, cB"
},
{
"math_id": 54,
"text": "XY"
},
{
"math_id": 55,
"text": " x_2 = x_1 q,\\; x_1 = x_3 p ,\\; x_3 = x_2 r"
},
{
"math_id": 56,
"text": "Z"
},
{
"math_id": 57,
"text": "rpq =1"
},
{
"math_id": 58,
"text": "pq = qp"
},
{
"math_id": 59,
"text": "X, Y, Z"
},
{
"math_id": 60,
"text": "C, c, X"
},
{
"math_id": 61,
"text": "A,b,C,a,B,c "
},
{
"math_id": 62,
"text": "G,H"
},
{
"math_id": 63,
"text": " X:= (A\\cap b) (a\\cap B), "
},
{
"math_id": 64,
"text": " Y:= (c\\cap A) (C\\cap a), "
},
{
"math_id": 65,
"text": " Z:= (b\\cap C) (B\\cap c)"
},
{
"math_id": 66,
"text": "U"
},
{
"math_id": 67,
"text": "GH"
},
{
"math_id": 68,
"text": "P=(0,0), \\; Q=(1,0), \\; R=(0,1)"
},
{
"math_id": 69,
"text": "(0,\\lambda)."
},
{
"math_id": 70,
"text": "\\left|\\begin{matrix}\nA & B & C \\\\\na & b & c \\\\\nX & Y & Z \\end{matrix}\n\\right|"
},
{
"math_id": 71,
"text": "\\ ABC, abc, AbZ, BcX, CaY, XbC, YcA, ZaB\\ "
},
{
"math_id": 72,
"text": "XYZ"
},
{
"math_id": 73,
"text": "(A,B,C)"
},
{
"math_id": 74,
"text": "\\; AB, CD,\\;"
},
{
"math_id": 75,
"text": "EF"
},
{
"math_id": 76,
"text": "DE, FA, "
},
{
"math_id": 77,
"text": "BC"
},
{
"math_id": 78,
"text": " AD, BE,"
},
{
"math_id": 79,
"text": "CF"
}
] |
https://en.wikipedia.org/wiki?curid=1493395
|
149353
|
Computational biology
|
Branch of biology
Computational biology refers to the use of data analysis, mathematical modeling and computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and big data, the field also has foundations in applied mathematics, chemistry, and genetics. It differs from biological computing, a subfield of computer science and engineering which uses bioengineering to build computers.
History.
Bioinformatics, the analysis of informatics processes in biological systems, began in the early 1970s. At this time, research in artificial intelligence was using network models of the human brain in order to generate new algorithms. This use of biological data pushed biological researchers to use computers to evaluate and compare large data sets in their own field.
By 1982, researchers shared information via punch cards. The amount of data grew exponentially by the end of the 1980s, requiring new computational methods for quickly interpreting relevant information.
Perhaps the best-known example of computational biology, the Human Genome Project, officially began in 1990. By 2003, the project had mapped around 85% of the human genome, satisfying its initial goals. Work continued, however, and by 2021 level " a complete genome" was reached with only 0.3% remaining bases covered by potential issues. The missing Y chromosome was added in January 2022.
Since the late 1990s, computational biology has become an important part of biology, leading to numerous subfields. Today, the International Society for Computational Biology recognizes 21 different 'Communities of Special Interest', each representing a slice of the larger field. In addition to helping sequence the human genome, computational biology has helped create accurate models of the human brain, map the 3D structure of genomes, and model biological systems.
Applications.
Anatomy.
Computational anatomy is the study of anatomical shape and form at the visible or gross anatomical formula_0 scale of morphology. It involves the development of computational mathematical and data-analytical methods for modeling and simulating biological structures. It focuses on the anatomical structures being imaged, rather than the medical imaging devices. Due to the availability of dense 3D measurements via technologies such as magnetic resonance imaging, computational anatomy has emerged as a subfield of medical imaging and bioengineering for extracting anatomical coordinate systems at the morpheme scale in 3D.
The original formulation of computational anatomy is as a generative model of shape and form from exemplars acted upon via transformations. The diffeomorphism group is used to study different coordinate systems via coordinate transformations as generated via the Lagrangian and Eulerian velocities of flow from one anatomical configuration in formula_1 to another. It relates with shape statistics and morphometrics, with the distinction that diffeomorphisms are used to map coordinate systems, whose study is known as diffeomorphometry.
Data and modeling.
Mathematical biology is the use of mathematical models of living organisms to examine the systems that govern structure, development, and behavior in biological systems. This entails a more theoretical approach to problems, rather than its more empirically-minded counterpart of experimental biology. Mathematical biology draws on discrete mathematics, topology (also useful for computational modeling), Bayesian statistics, linear algebra and Boolean algebra.
These mathematical approaches have enabled the creation of databases and other methods for storing, retrieving, and analyzing biological data, a field known as bioinformatics. Usually, this process involves genetics and analyzing genes.
Gathering and analyzing large datasets have made room for growing research fields such as data mining, and computational biomodeling, which refers to building computer models and visual simulations of biological systems. This allows researchers to predict how such systems will react to different environments, which is useful for determining if a system can "maintain their state and functions against external and internal perturbations". While current techniques focus on small biological systems, researchers are working on approaches that will allow for larger networks to be analyzed and modeled. A majority of researchers believe this will be essential in developing modern medical approaches to creating new drugs and gene therapy. A useful modeling approach is to use Petri nets via tools such as esyN.
Along similar lines, until recent decades theoretical ecology has largely dealt with analytic models that were detached from the statistical models used by empirical ecologists. However, computational methods have aided in developing ecological theory via simulation of ecological systems, in addition to increasing application of methods from computational statistics in ecological analyses.
Systems Biology.
Systems biology consists of computing the interactions between various biological systems ranging from the cellular level to entire populations with the goal of discovering emergent properties. This process usually involves networking cell signaling and metabolic pathways. Systems biology often uses computational techniques from biological modeling and graph theory to study these complex interactions at cellular levels.
Evolutionary biology.
Computational biology has assisted evolutionary biology by:
Genomics.
Computational genomics is the study of the genomes of cells and organisms. The Human Genome Project is one example of computational genomics. This project looks to sequence the entire human genome into a set of data. Once fully implemented, this could allow for doctors to analyze the genome of an individual patient. This opens the possibility of personalized medicine, prescribing treatments based on an individual's pre-existing genetic patterns. Researchers are looking to sequence the genomes of animals, plants, bacteria, and all other types of life.
One of the main ways that genomes are compared is by sequence homology. Homology is the study of biological structures and nucleotide sequences in different organisms that come from a common ancestor. Research suggests that between 80 and 90% of genes in newly sequenced prokaryotic genomes can be identified this way.
Sequence alignment is another process for comparing and detecting similarities between biological sequences or genes. Sequence alignment is useful in a number of bioinformatics applications, such as computing the longest common subsequence of two genes or comparing variants of certain diseases.
An untouched project in computational genomics is the analysis of intergenic regions, which comprise roughly 97% of the human genome. Researchers are working to understand the functions of non-coding regions of the human genome through the development of computational and statistical methods and via large consortia projects such as ENCODE and the Roadmap Epigenomics Project.
Understanding how individual genes contribute to the biology of an organism at the molecular, cellular, and organism levels is known as gene ontology. The Gene Ontology Consortium's mission is to develop an up-to-date, comprehensive, computational model of biological systems, from the molecular level to larger pathways, cellular, and organism-level systems. The Gene Ontology resource provides a computational representation of current scientific knowledge about the functions of genes (or, more properly, the protein and non-coding RNA molecules produced by genes) from many different organisms, from humans to bacteria.
3D genomics is a subsection in computational biology that focuses on the organization and interaction of genes within a eukaryotic cell. One method used to gather 3D genomic data is through Genome Architecture Mapping (GAM). GAM measures 3D distances of chromatin and DNA in the genome by combining cryosectioning, the process of cutting a strip from the nucleus to examine the DNA, with laser microdissection. A nuclear profile is simply this strip or slice that is taken from the nucleus. Each nuclear profile contains genomic windows, which are certain sequences of nucleotides - the base unit of DNA. GAM captures a genome network of complex, multi enhancer chromatin contacts throughout a cell.
Neuroscience.
Computational neuroscience is the study of brain function in terms of the information processing properties of the nervous system. A subset of neuroscience, it looks to model the brain to examine specific aspects of the neurological system. Models of the brain include:
It is the work of computational neuroscientists to improve the algorithms and data structures currently used to increase the speed of such calculations.
Computational neuropsychiatry is an emerging field that uses mathematical and computer-assisted modeling of brain mechanisms involved in mental disorders. Several initiatives have demonstrated that computational modeling is an important contribution to understand neuronal circuits that could generate mental functions and dysfunctions.
Pharmacology.
Computational pharmacology is "the study of the effects of genomic data to find links between specific genotypes and diseases and then screening drug data". The pharmaceutical industry requires a shift in methods to analyze drug data. Pharmacologists were able to use Microsoft Excel to compare chemical and genomic data related to the effectiveness of drugs. However, the industry has reached what is referred to as the Excel barricade. This arises from the limited number of cells accessible on a spreadsheet. This development led to the need for computational pharmacology. Scientists and researchers develop computational methods to analyze these massive data sets. This allows for an efficient comparison between the notable data points and allows for more accurate drugs to be developed.
Analysts project that if major medications fail due to patents, that computational biology will be necessary to replace current drugs on the market. Doctoral students in computational biology are being encouraged to pursue careers in industry rather than take Post-Doctoral positions. This is a direct result of major pharmaceutical companies needing more qualified analysts of the large data sets required for producing new drugs.
Similarly, computational oncology aims to determine the future mutations in cancer through algorithmic approaches. Research in this field has led to the use of high-throughput measurement that millions of data points using robotics and other sensing devices. This data is collected from DNA, RNA, and other biological structures. Areas of focus include determining the characteristics of tumors, analyzing molecules that are deterministic in causing cancer, and understanding how the human genome relates to the causation of tumors and cancer.
Techniques.
Computational biologists use a wide range of software and algorithms to carry out their research.
Unsupervised Learning.
Unsupervised learning is a type of algorithm that finds patterns in unlabeled data. One example is k-means clustering, which aims to partition "n" data points into "k" clusters, in which each data point belongs to the cluster with the nearest mean. Another version is the k-medoids algorithm, which, when selecting a cluster center or cluster centroid, will pick one of its data points in the set, and not just an average of the cluster.
The algorithm follows these steps:
One example of this in biology is used in the 3D mapping of a genome. Information of a mouse's HIST1 region of chromosome 13 is gathered from Gene Expression Omnibus. This information contains data on which nuclear profiles show up in certain genomic regions. With this information, the Jaccard distance can be used to find a normalized distance between all the loci.
Graph Analytics.
Graph analytics, or network analysis, is the study of graphs that represent connections between different objects. Graphs can represent all kinds of networks in biology such as protein-protein interaction networks, regulatory networks, Metabolic and biochemical networks and much more. There are many ways to analyze these networks. One of which is looking at centrality in graphs. Finding centrality in graphs assigns nodes rankings to their popularity or centrality in the graph. This can be useful in finding which nodes are most important. For example, given data on the activity of genes over a time period, degree centrality can be used to see what genes are most active throughout the network, or what genes interact with others the most throughout the network. This contributes to the understanding of the roles certain genes play in the network.
There are many ways to calculate centrality in graphs all of which can give different kinds of information on centrality. Finding centralities in biology can be applied in many different circumstances, some of which are gene regulatory, protein interaction and metabolic networks.
Supervised Learning.
Supervised learning is a type of algorithm that learns from labeled data and learns how to assign labels to future data that is unlabeled. In biology supervised learning can be helpful when we have data that we know how to categorize and we would like to categorize more data into those categories.
A common supervised learning algorithm is the random forest, which uses numerous decision trees to train a model to classify a dataset. Forming the basis of the random forest, a decision tree is a structure which aims to classify, or label, some set of data using certain known features of that data. A practical biological example of this would be taking an individual's genetic data and predicting whether or not that individual is predisposed to develop a certain disease or cancer. At each internal node the algorithm checks the dataset for exactly one feature, a specific gene in the previous example, and then branches left or right based on the result. Then at each leaf node, the decision tree assigns a class label to the dataset. So in practice, the algorithm walks a specific root-to-leaf path based on the input dataset through the decision tree, which results in the classification of that dataset. Commonly, decision trees have target variables that take on discrete values, like yes/no, in which case it is referred to as a classification tree, but if the target variable is continuous then it is called a regression tree. To construct a decision tree, it must first be trained using a training set to identify which features are the best predictors of the target variable.
Open source software.
Open source software provides a platform for computational biology where everyone can access and benefit from software developed in research. PLOS cites four main reasons for the use of open source software:
Research.
There are several large conferences that are concerned with computational biology. Some notable examples are Intelligent Systems for Molecular Biology, European Conference on Computational Biology and Research in Computational Molecular Biology.
There are also numerous journals dedicated to computational biology. Some notable examples include Journal of Computational Biology and PLOS Computational Biology, a peer-reviewed open access journal that has many notable research projects in the field of computational biology. They provide reviews on software, tutorials for open source software, and display information on upcoming computational biology conferences. Other journals relevant to this field include Bioinformatics, Computers in Biology and Medicine, BMC Bioinformatics, Nature Methods, Nature Communications, Scientific Reports, PLOS One, etc.
Related fields.
Computational biology, bioinformatics and mathematical biology are all interdisciplinary approaches to the life sciences that draw from quantitative disciplines such as mathematics and information science. The NIH describes computational/mathematical biology as the use of computational/mathematical approaches to address theoretical and experimental questions in biology and, by contrast, bioinformatics as the application of information science to understand complex life-sciences data.
Specifically, the NIH defines
<templatestyles src="Template:Blockquote/styles.css" />
<templatestyles src="Template:Blockquote/styles.css" />
While each field is distinct, there may be significant overlap at their interface, so much so that to many, bioinformatics and computational biology are terms that are used interchangeably.
The terms computational biology and evolutionary computation have a similar name, but are not to be confused. Unlike computational biology, evolutionary computation is not concerned with modeling and analyzing biological data. It instead creates algorithms based on the ideas of evolution across species. Sometimes referred to as genetic algorithms, the research of this field can be applied to computational biology. While evolutionary computation is not inherently a part of computational biology, computational evolutionary biology is a subfield of it.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " 50-100 \\mu "
},
{
"math_id": 1,
"text": "{\\mathbb R}^3"
}
] |
https://en.wikipedia.org/wiki?curid=149353
|
149401
|
Jean-André Deluc
|
Swiss geologist and meteorologist (1727–1817)
Jean-André Deluc or de Luc (8 February 1727 – 7 November 1817) was a geologist, natural philosopher and meteorologist from the Republic of Geneva. He also devised measuring instruments.
Biography.
Jean-André Deluc was born in Geneva. His family had come to the Republic of Geneva from Lucca, Italy, in the 15th century. His mother was Françoise Huaut. His father, Jacques-François Deluc, had written in refutation of Bernard Mandeville and other rationalistic writers, but he was also a decided supporter of Jean-Jacques Rousseau.
As a student of Georges-Louis Le Sage, Jean-André Deluc received a basic education in mathematics and in natural philosophy. He engaged early in business, which occupied a large part of his first adult years, with the exception of scientific investigation in the Alps. With the help of his brother Guillaume-Antoine, he built a splendid collection of mineralogy and fossils.
Deluc also took part in politics. In 1768, sent on an embassy to the duc de Choiseul in Paris, he succeeded in gaining the duke's friendship. In 1770 he became a member of the Council of Two Hundred in Geneva.
Three years later, business reverses forced him to leave his native town; he returned, briefly, only once. The change freed him for non-scientific pursuits; with little regret he moved to Great Britain in 1773, where he was appointed reader to Queen Charlotte, a position he held for forty-four years and that afforded him both leisure and income.
Despite his duties at court, he was given leave to make several tours of Switzerland, France, Holland and Germany. At the beginning of his German tour (1798–1804), he was distinguished with an honorary professorship of philosophy and geology at the University of Göttingen, which helped to cover diplomatic missions for the king George III. Back to Britain, he undertook a geological tour of the country (1804–1807).
In 1773 Deluc was made a fellow of the Royal Society; he was a correspondent of the French Academy of Sciences and a member of several other learned societies. He died at Windsor, Berkshire, England, in 1817, after nearly 70 years of research. Deluc, an impact crater on the Moon, was given his name.
Scientific contributions.
Observations and theory.
Deluc's main interests were geology and meteorology; Georges Cuvier mentions him as an authority on the former subject. His major geological work, "Lettres physiques et morales sur les montagnes et sur l'histoire de la terre et de l'homme" (6 vol., 1778–1780), was dedicated to Queen Charlotte. He published volumes on geological travels: in northern Europe (1810), in England (1811), and in France, Switzerland and Germany (1813).
Deluc noticed the disappearance of heat in the thawing of ice about the same time that Joseph Black made it the foundation of his hypothesis of latent heat. He ascertained that water was densest at about 5 °C (and not at the freezing temperature). He was the originator of the theory, later reactivated by John Dalton, that the quantity of water vapour contained in any space is independent of the presence or density of the air, or of any other elastic fluid.
His book "Lettres sur l'histoire physique de la terre" (Paris, 1798), addressed to Johann Friedrich Blumenbach, develops a theory of the Earth divided into six periods modelled on the six days of Creation. It contains an essay on the existence of a General Principle of Morality and gives an interesting account of conversations with Voltaire and Rousseau. Deluc was an ardent admirer of Francis Bacon, on whose writings he published two works: "Bacon tel qu'il est" (Berlin, 1800), showing the bad faith of the French translator, who had omitted many passages favorable to revealed religion, and "Précis de la philosophie de Bacon" (2 vols 8vo, Paris, 1802), giving an interesting view of the progress of natural science. "Lettres sur le christianisme" (Berlin and Hanover, 1803) was a controversial correspondence with Wilhelm Abraham Teller of Berlin in regard to the Mosaic cosmogony. His "Traité élémentaire de géologie" (Paris, 1809, translated into English by Henry de la Fite the same year) was principally intended as a refutation of James Hutton and John Playfair. They had shown that geology was driven by the operation of internal heat and erosion, but their system required much more time than Deluc's Mosaic variety of neptunism allowed.
Many other papers were in the "Journal de Physique", in the "Philosophical Transactions" and in the "Philosophical Magazine".
Instruments.
Deluc dedicated a large part of his activity to perfecting or inventing measuring instruments.
He devised a portable barometer for use in geological expeditions. His "Recherches sur les modifications de l'atmosphère" (2 vols. 4to, Geneva, 1772; 2nd ed., 4 vols. Paris, 1784) contain experiments on moisture, evaporation and the indications of hygrometers and thermometers. He applied the barometer to the determination of heights. The "Philosophical Transactions" published his account of a new hygrometer, which resembled a mercurial thermometer, with an ivory bulb, which expanded by moisture, and caused the mercury to descend. He later devised a whalebone hygrometer which sparked a bitter controversy with Horace-Bénédict de Saussure, himself inventor of a hair hygrometer. He gave the first correct rules for measuring heights with the help of a barometer.
Based on his experiments in 1772, Deluc advocated the use of mercury, instead of alcohol or other fluids, in thermometers, as its volume varies the most linearly with the method of mixtures. In detail, if two portions of water of equal masses A, B were mixed, and let the resulting water be C, and if we immerse a thermometer in A, B, C, we obtain lengths formula_0. Deluc expected that formula_1, and similarly for other ratios of mixtures. He found that thermometers made using mercury allowed the closest fit to his expectation of linearity.
In 1809 he sent a long paper to the Royal Society on separating the chemical from the electrical effect of the dry pile, a form of Voltaic pile, with a description of the electric column and aerial electroscope, in which he advanced opinions contradicting the latest discoveries of the day; they were deemed inappropriate to admit into the "Transactions". The dry column described by Deluc was constructed by various scientists and his improvement of the dry pile has been regarded as his most important work, although he was not in fact its inventor. He was a valued mentor to the young Francis Ronalds, who published several papers on dry piles in 1814–15.
Scriptural and observational data.
The last decades of Deluc's life were occupied with theological considerations. In his controversy with Hutton, "while never arguing that Hutton was an atheist, Deluc did accuse him of failing to counter atheism sufficiently".
He took care in reconciling observational data and the Scriptures considered as a description of the history of the world. In his "Lettres physiques et morales" he explained the six days of the creation as epochs preceding the current state of the globe, and attributed the deluge to the filling up of cavities in the interior of the earth.
The subject is discussed at length by Martina Kölbl-Ebert in "Geology and Religion".
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "l_A, l_B, l_C"
},
{
"math_id": 1,
"text": "l_C = \\frac 12 (l_A + l_B)"
}
] |
https://en.wikipedia.org/wiki?curid=149401
|
14940135
|
Japan Crude Cocktail
|
Informal nickname given to the pricing index of Crude Oil
The Japan Crude Cocktail (JCC) is the informal nickname given to the pricing index of Crude Oil used in most East Asian countries. The JCC is the average price of customs-cleared crude oil imports into Japan and is published by the Petroleum Association of Japan. The official name of the JCC is the Japan Customs-cleared Crude Oil Price.
The valuation of the JCC closely reflects the market state of supply and demand. Clear fluctuations in JCC pricing can be linked to distinct events such as the 2007-08 Global Financial Crisis and the 2011 Fukushima Disaster.The JCC was historically the main index to price Liquified Natural Gas (LNG) contracts as no global benchmark existed. However, as JCC is based on oil prices as opposed to gas, there has been a rise in objection to its use. In Europe and most North American nations, LNG pricing has shifted from use of the JCC to gas-based indexes (e.g. Henry Hub).
Historical Development.
Following the end of World War II, Japan's rapid industrialisation made them one of the largest energy consumers. However, Japan's limited domestic resources of fuel means the country relies heavily on imports. In the 1950s, the Japan Crude Cocktail was formulated to create a pricing index of customs-cleared crude oil imports.
The Customs-cleared crude oils applicable have the following Harmonised System codes; 2709.00 100, 2709.00 900, 2710.19 162, 2710.19 164, 2710.19 166, 2710.19 169, 2710.19 172, 2710.19 174, 2710.19.179.
Petroleum Association of Japan.
This Petroleum Association of Japan (PAJ) was established in 1955, around the time of the JCC development. The PAJ consists of 11 refiners and primary distributors in Japan. They are responsible for providing central information on the petroleum industry for the public, as well as advocating, researching and enhancing communication among the public and oil companies.
The 11 refiners and primary distributors of the PAJ are Idemitsu Kosan, Cosmo Oil Marketing, TOA Oil, ENEOS, KASHIMA Oil, Kygnus Sekiyu, Taiyo Oil Company, Showa Yokkaichi Sekiyu, Fuji Oil Company, Seibu Oil and Cosmo Oil Company.
The PAJ publish a range of oil statistics monthly on their website, including the JCC.
Users.
LNG Contracts.
Liquified Natural Gas (LNG) contracts are legal agreements for the sale and purchase of LNG between sellers and buyers LNG is an energy commodity, just like crude oil, allowing for a price relationship between the two.
LNG contract pricing has been based on the JCC index since the 1970s. LNG was initially introduced in Japan as an energy substitute to reduce the country's high dependence on oil. In turn, this guaranteed that gas could remain competitive with oil by having similar pricing. The JCC is used as the pricing index in LNG contracts mainly in Asia Pacific (APAC) and Europe, the Middle East and Africa (EMEA). For example, the index is heavily used in China, South Korea, India, Taiwan.
JCC Formula.
JCC.
The JCC pricing index is based on the average price of customs-cleared crude oil imports into Japan. The Ministry of Finance sector within the Japanese government publishes the data used to calculate the JCC each month.formula_0$"X" = The total value of customs cleared crude oil imports
"Y" = The total volume of customs cleared crude oil imports
LNG Contracts.
The JCC figure is then used to generate the price of LNG contracts using the following formula.
formula_1
"PLNG" = The Cost, Insurance and Freight (CIF) price of LNG
"a" = The extent to which the PLNG changes in response to changes in the JCC, also known as the ‘slope’
"b" = The additional freight charges, discounts or premiums incurred. This constant term is determined through contract negotiations.
Pricing, 1980-2019.
The Japan Crude Cocktail prices are given in USD per barrel (USD/bbl). JCC prices are available from the Organisation for Economic Co-operation and Development (OECD).
The JCC has had continuous price fluctuations over its history, whilst still steadily increasing. From the raw data and graph, the two major spikes in prices can be seen in 2008 and 2011–12. A comparatively dramatic price low can be observed in 2016.
Price Variations.
First Oil Crisis 1973.
The Yom Kippur War between Israel and Arab nation resulted in the first oil crisis internationally. In October 1973, the Organisation of Petroleum Exporting Countries’ (OPEC) started an oil embargo, targeted at nations thought to be supporting Israel during the conflict. The countries primarily targeted were Japan, US, Canada, UK and the Netherlands. Originally, oil exports from Arab nations were to be reduced by 5% monthly until Israel met the demands of the Arab nations. However, in December, a full embargo was put in place, restricting all trade of oil. Following the events of this first oil crisis, Japan became more aware to their vulnerability to external oil prices and began focusing on increasing their self-dependence on their own natural resources and energy sources. Their whole oil industry was redesigned.
The first oil crisis resulted in recession, inflation and a trade deficit in Japan as all purchasing power was removed from Japan and given to the Arab oil producing nations. In regard to the JCC, the ban on supply heavily increased import prices. Crude oil import prices rose by 300% globally. In 1970, almost 85% of Japan's crude oil imports came from the Middle East.
Global Financial Crisis 2007-2008.
The Global Financial Crisis 2007-2008 (GFC) was one of the greatest periods of extreme economic stress since the 1930s. Thousands of businesses suffered heavy losses and/or bankruptcy, with millions of people losing their jobs. The intense period of recession's influential impact on the JCC is highly notable, observed in the first spike seen in 'Crude oil import Prices in Japan 1980-2019' graph.
Worldwide, oil demand heavily relies upon income rather than market demand. When Gross Domestic Product (GDP) is in elevated periods, disposable income is greater, allowing for increased purchase of goods. This creates dependent relationship oil consumption growth rates have with GDP growth rates. It can be observed that in periods of high GDP growth rate, oil consumption growth rates soon also increase. The same can be said for decreases in GDP grow rates.
The period of 2008-2009 saw a decline in both GDP growth rate and oil consumption growth rate in Japan. JCC spiked in 2008 at US$100.98/bbl before dramatically declining in 2009 to $61.29. This aligns with the economic concepts of recession. In times of recession, demand dramatically reduces, driving down prices for suppliers
In the recent years leading to the GFC, China's crude oil imports exponentially increased. With greater economic resources, China began to dominate the consumption of crude oil, reducing crude oil supply available and thus consumption for other nations. From 2005 to 2007, Japan's crude oil consumption per day reduced by 318,000 barrels. Limited supply, drives prices up, reflected in the 2005-2007 JCC increased from US$51.57/bbl to US$70.09/bbl.
Fukushima Disaster 2011.
In March 2011, the Tohoku earthquake and tsunami hit the east coast of Japan. The generators at the Fukushima Nuclear Power plant nearby shut down automatically after detecting the earthquake. The shortly followed tsunami flooded the nuclear power plants, destroying the reactors and releasing radiation. This destruction of the nuclear plant is known as the Fukushima Daiichi Nuclear Disaster.
All 50 reactors were recorded as offline by 2012. One fifth of Japan's energy generating capacity was lost as a result. The loss of domestic energy sources created immense pressure on the Japanese economy as demand for LNG imports rose substantially. In the initial years and immediately after the disaster, LNG imports accounted for 40% of Japan's power. The sudden steep incline in demand rapidly drove prices. From 2010 to 2012, JCC rose from US$79.43/bbl to US$114.75/bbl.
It wasn't until 2015 when some of Japan's nuclear plants were restarted, that JCC prices to return to a lower price.
COVID-19 Pandemic.
In 2020, the COVID-19 exponentially infected all parts of the globe, creating an international health outbreak and pandemic. Many nations were sent into lockdown as a result of the health risks. The World Bank stated that the "global economic shock of the COVID-19 pandemic has driven most commodity prices down and is expected to result in substantially lower prices over 2020". This is due to the mass mandatory quarantine that restricted any movement of people, domestically and internationally. Lockdown created a halt in economic activity as people were unable to leave home to attend jobs or to support the economy (e.g. tourism, travel, shopping), affecting both supply and demand. Inability for movement impacted the oil industry heavily as travel restrictions correlate to a cut in demand for jet fuel. The closure of factories and reduced commute amid the crisis further reduced need for fuels.
In Japan, crude oil imports dropped by 25% in 2020, with only 2.28 million barrels per day imported in May 2020. This figure is the lowest it has been in 53 years. This is a direct result of the abrupt freeze of production and supply amid the pandemic. Price of the JCC during the pandemic can be observed in the table generated below. Prices fell to US$24.55/bb in June 2020 as demand for crude oil in Japan dramatically declined.
Future Directions of JCC.
The JCC price index was established in the early 1950s to reflect the market fundamentals of the time. However, as these have shifted over time, the JCC has failed to keep up to date with these changes. After the impacts of the Fukushima Disaster, JCC reached an all-time high of US$114.75/bbl in 2014. Since this event, there has been growing unrest amongst users of the index.
The main concern for users is that JCC is predominately used for pricing LNG contracts. Oil-based LNG contracts are priced considerably higher than alternatives as they reflect the price of oil imports. In European and North American nations, LNG pricing has already shifted from an oil-based pricing index to hub-based pricing such as the Henry Hub and Dutch Title Transfer. However, most East Asian nations continue to use the JCC to price LNG. cheaper alternative. In 2014, senior gas analyst at the Institute of Energy Economics, Hiroshi Hashimoto, stated “the aim is to link 100% to Henry Hub prices, rather than JCC as has been the custom globally”.
In addition to the cheaper prices that a gas-based index would offer, the shift would also remove the current time lag between crude oil and LNG prices. The LNG pricing formula in Japan is . (See JCC formula subheading). JCC prices take on average 5 months to be reflected in LNG contract prices. This is due to the fact that the JCC is derived from the price of crude oil when imported into Japan. However, this figure can greatly change from the international market price on the day of transportation. By shifting to a gas-based pricing index, LNG contracts can be based on current market value, remaining competitive with other indexes.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "JCC = \\frac{\\$X}{Y} "
},
{
"math_id": 1,
"text": "P_{LNG} = a \\times JCC + b "
}
] |
https://en.wikipedia.org/wiki?curid=14940135
|
14941140
|
Mittenpunkt
|
Triangle center: symmedian point of the triangle's excentral triangle
In geometry, the (from German: "middle point") of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It was identified in 1836 by Christian Heinrich von Nagel as the symmedian point of the excentral triangle of the given triangle.
Coordinates.
The mittenpunkt has trilinear coordinates
formula_0
where a, b, and c are the side lengths of the given triangle.
Expressed instead in terms of the angles A, B, and C, the trilinears are
formula_1
The barycentric coordinates are
formula_2
Collinearities.
The mittenpunkt is at the intersection of the line connecting the centroid and the Gergonne point, the line connecting the incenter and the symmedian point and the line connecting the orthocenter with the Spieker center, thus establishing three collinearities involving the mittenpunkt.
Related figures.
The three lines connecting the excenters of the given triangle to the corresponding edge midpoints all meet at the mittenpunkt; thus, it is the center of perspective of the excentral triangle and the median triangle, with the corresponding axis of perspective being the trilinear polar of the Gergonne point. The mittenpunkt is also the centroid of the Mandart inellipse of the given triangle, the ellipse tangent to the triangle at its extouch points.
Notes.
The Mittenpunkt also serves as the Gergonne point of the Medial triangle.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "(b+c-a): (c+a-b ):(a+b-c)"
},
{
"math_id": 1,
"text": "\\cot \\frac{A}{2} : \\cot \\frac{B}{2} : \\cot \\frac{C}{2}=(\\csc A+\\cot A):(\\csc B+\\cot B):(\\csc C+\\cot C)."
},
{
"math_id": 2,
"text": "a(b+c-a):b(c+a-b):c(a+b-c) = (1+\\cos A):(1+\\cos B):(1+\\cos C)."
}
] |
https://en.wikipedia.org/wiki?curid=14941140
|
1494164
|
Dynamic inconsistency
|
When a decision-maker's future preferences can contradict earlier preferences
In economics, dynamic inconsistency or time inconsistency is a situation in which a decision-maker's preferences change over time in such a way that a preference can become inconsistent at another point in time. This can be thought of as there being many different "selves" within decision makers, with each "self" representing the decision-maker at a different point in time; the inconsistency occurs when not all preferences are aligned.
The term "dynamic inconsistency" is more closely affiliated with game theory, whereas "time inconsistency" is more closely affiliated with behavioral economics.
In game theory.
In the context of game theory, dynamic inconsistency is a situation in a dynamic game where a player's best plan for some future period will not be optimal when that future period arrives. A dynamically inconsistent game is subgame imperfect. In this context, the inconsistency is primarily about commitment and credible threats. This manifests itself through a violation of Bellman's Principle of Optimality by the leader or dominant player, as shown in Simaan and Cruz (1973a, 1973b).
For example, a firm might want to commit itself to dramatically dropping the price of a product it sells if a rival firm enters its market. If this threat were credible, it would discourage the rival from entering. However, the firm might not be able to commit its future self to taking such an action because if the rival does in fact end up entering, the firm's future self might determine that, given the fact that the rival is now actually in the market and there is no point in trying to discourage entry, it is now not in its interest to dramatically drop the price. As such, the threat would not be credible. The present self of the firm has preferences that would have the future self be committed to the threat, but the future self has preferences that have it not carry out the threat. Hence, the dynamic inconsistency.
In behavioral economics.
In the context of behavioral economics, time inconsistency is related to how each different self of a decision-maker may have different preferences over current and future choices.
Consider, for example, the following question:
When this question is asked, to be time-consistent, one must make the same choice for (b) as for (a). According to George Loewenstein and Drazen Prelec, however, people are not always consistent. People tend to choose "500 dollars today" and "505 dollars 366 days later", which is different from the time-consistent answer.
One common way in which selves may differ in their preferences is they may be modeled as all holding the view that "now" has especially high value compared to any future time. This is sometimes called the "immediacy effect" or "temporal discounting". As a result, the present self will care too much about itself and not enough about its future selves. The self control literature relies heavily on this type of time inconsistency, and it relates to a variety of topics including procrastination, addiction, efforts at weight loss, and saving for retirement.
Time inconsistency basically means that there is disagreement between a decision-maker's different selves about what actions should be taken. Formally, consider an economic model with different mathematical weightings placed on the utilities of each self. Consider the possibility that for any given self, the weightings that self places on all the utilities could differ from the weightings that another given self places on all the utilities. The important consideration now is the relative weighting between two particular utilities. Will this relative weighting be the same for one given self as it is for a different given self? If it is, then we have a case of time consistency. If the relative weightings of all pairs of utilities are all the same for all given selves, then the decision-maker has time-consistent preferences. If there exists a case of one relative weighting of utilities where one self has a different relative weighting of those utilities than another self has, then we have a case of time inconsistency and the decision-maker will be said to have time-inconsistent preferences.
It is common in economic models that involve decision-making over time to assume that decision-makers are exponential discounters. Exponential discounting posits that the decision maker assigns future utility of any good according to the formula
formula_0
where formula_1 is the present, formula_2 is the utility assigned to the good if it were consumed immediately, and formula_3 is the "discount factor", which is the same for all goods and constant over time. Mathematically, it is the unique continuous function that satisfies the equation
formula_4
that is, the ratio of utility values for a good at two different moments of time only depends on the interval between these times, but not on their choice. (If you're willing to pay 10% over list price to buy a new phone today instead of paying list price and having it delivered in a week, you'd also be willing to pay extra 10% to get it one week sooner if you were ordering it six months in advance.)
If formula_3 is the same for all goods, then it is also the case that
formula_5
that is, if good A is assigned higher utility than good B at time formula_6, that relationship also holds at all other times. (If you'd rather eat broccoli than cake tomorrow for lunch, you'll also pick broccoli over cake if you're hungry right now.)
Exponential discounting yields time-consistent preferences. Exponential discounting and, more generally, time-consistent preferences are often assumed in rational choice theory, since they imply that all of a decision-maker's selves will agree with the choices made by each self. Any decision that the individual makes for himself in advance will remain valid (i.e., an optimal choice) as time advances, unless utilities themselves change.
However, empirical research makes a strong case that time inconsistency is, in fact, standard in human preferences. This would imply disagreement by people's different selves on decisions made and a rejection of the time consistency aspect of rational choice theory.
For example, consider having the choice between getting the day off work tomorrow or getting a day and a half off work one month from now. Suppose you would choose one day off tomorrow. Now suppose that you were asked to make that same choice ten years ago. That is, you were asked then whether you would prefer getting one day off in ten years or getting one and a half days off in ten years and one month. Suppose that then you would have taken the day and a half off. This would be a case of time inconsistency because your relative preferences for tomorrow versus one month from now would be different at two different points in time—namely now versus ten years ago. The decision made ten years ago indicates a preference for delayed gratification, but the decision made just before the fact indicates a preference for immediate pleasure.
More generally, humans have a systematic tendency to switch towards "vices" (products or activities which are pleasant in the short term) from "virtues" (products or activities which are seen as valuable in the long term) as the moment of consumption approaches, even if this involves changing decisions made in advance.
One way that time-inconsistent preferences have been formally introduced into economic models is by first giving the decision-maker standard exponentially discounted preferences, and then adding another term that heavily discounts any time that is not now. Preferences of this sort have been called "present-biased preferences". The hyperbolic discounting model is another commonly used model that allows one to obtain more realistic results with regard to human decision-making.
A different form of dynamic inconsistency arises as a consequence of "projection bias" (not to be confused with a defense mechanism of the same name). Humans have a tendency to mispredict their future marginal utilities by assuming that they will remain at present levels. This leads to inconsistency as marginal utilities (for example, tastes) change over time in a way that the individual did not expect. For example, when individuals are asked to choose between a piece of fruit and an unhealthy snack (such as a candy bar) for a future meal, the choice is strongly affected by their "current" level of hunger. Individuals may become addicted to smoking or drugs because they underestimate future marginal utilities of these habits (such as craving for cigarettes) once they become addicted.
In media studies.
Theories of media choice have not explicitly dealt with choice inconsistency as it was defined by behavioral economics. However, an article by Gui et al. (2021) draws on behavioral economics literature to address blind spots in theorization of inconsistent media selection in media studies. It also highlights that inconsistent choice is even more frequent and relevant in the digital environment, as higher stimulation and multitasking makes it easier to opt for immediate gratification even in the presence of different long-term preferences.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "U(t) = U(0) e^{-\\rho t}"
},
{
"math_id": 1,
"text": "t=0"
},
{
"math_id": 2,
"text": "U(0)"
},
{
"math_id": 3,
"text": "\\rho"
},
{
"math_id": 4,
"text": " U(t_1)/U(t_2) = U(t_1+c)/U(t_2+c);"
},
{
"math_id": 5,
"text": " U_A(t_1)/U_B(t_1) = U_A(t_2)/U_B(t_2);"
},
{
"math_id": 6,
"text": "t_1"
}
] |
https://en.wikipedia.org/wiki?curid=1494164
|
14941864
|
Reverberation mapping
|
Astrophysical technique
Reverberation mapping (or Echo mapping) is an astrophysical technique for measuring the structure of the broad-line region (BLR) around a supermassive black hole at the center of an active galaxy, and thus estimating the hole's mass. It is considered a "primary" mass estimation technique, i.e., the mass is measured directly from the motion that its gravitational force induces in the nearby gas.
Newton's law of gravity defines a direct relation between the mass of a central object and the speed of a smaller object in orbit around the central mass. Thus, for matter orbiting a black hole, the black-hole mass formula_0 is related by the formula
formula_1
to the RMS velocity Δ"V" of gas moving near the black hole in the broad emission-line region, measured from the Doppler broadening of the gaseous emission lines. In this formula, "R"BLR is the radius of the broad-line region; "G" is the constant of gravitation; and "f" is a poorly known "form factor" that depends on the shape of the BLR.
While Δ"V" can be measured directly using spectroscopy, the necessary determination of "R"BLR is much less straightforward. This is where reverberation mapping comes into play. It utilizes the fact that the emission-line fluxes vary strongly in response to changes in the continuum, i.e., the light from the accretion disk near the black hole. Put simply, if the brightness of the accretion disk varies, the emission lines, which are excited in response to the accretion disk's light, will "reverberate", that is, vary in response. But it will take some time for light from the accretion disk to reach the broad-line region. Thus, the emission-line response is delayed with respect to changes in the continuum. Assuming that this delay is solely due to light travel times, the distance traveled by the light, corresponding to the radius of the broad emission-line region, can be measured.
Only a small handful (less than 40) of active galactic nuclei have been accurately "mapped" in this way. An alternative approach is to use an empirical correlation between "R"BLR and the continuum luminosity.
Another uncertainty is the value of "f". In principle, the response of the BLR to variations in the continuum could be used to map out the three-dimensional structure of the BLR. In practice, the amount and quality of data required to carry out such a deconvolution is prohibitive. Until about 2004, "f" was estimated ab initio based on simple models for the structure of the BLR. More recently, the value of "f" has been determined so as to bring the M–sigma relation for active galaxies into the best possible agreement with the M–sigma relation for quiescent galaxies. When "f" is determined in this way, reverberation mapping becomes a "secondary", rather than "primary", mass estimation technique.
References and notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "M_\\bullet"
},
{
"math_id": 1,
"text": " \nGM_\\bullet = f \\, R_\\text{BLR} \\, (\\Delta V)^2 \n"
}
] |
https://en.wikipedia.org/wiki?curid=14941864
|
14943165
|
Weighted space
|
In functional analysis, a weighted space is a space of functions under a "weighted norm", which is a finite norm (or semi-norm) that involves multiplication by a particular function referred to as the "weight".
Weights can be used to expand or reduce a space of considered functions. For example, in the space of functions from a set formula_0 to formula_1 under the norm formula_2 defined by: formula_3, functions that have infinity as a limit point are excluded. However, the weighted norm formula_4 is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm formula_5 is finite for many fewer functions.
When the weight is of the form formula_6, the weighted space is called "polynomial-weighted".
|
[
{
"math_id": 0,
"text": "U\\subset\\mathbb{R}"
},
{
"math_id": 1,
"text": "\\mathbb{R}"
},
{
"math_id": 2,
"text": "\\|\\cdot\\|_U"
},
{
"math_id": 3,
"text": "\\|f\\|_U=\\sup_{x\\in U}{|f(x)|}"
},
{
"math_id": 4,
"text": "\\|f\\|=\\sup_{x\\in U}{\\left|f(x)\\tfrac{1}{1+x^2}\\right|}"
},
{
"math_id": 5,
"text": "\\|f\\|=\\sup_{x\\in U}{\\left|f(x)(1 + x^4)\\right|}"
},
{
"math_id": 6,
"text": "\\tfrac{1}{1+x^m}"
}
] |
https://en.wikipedia.org/wiki?curid=14943165
|
14943561
|
Pivotal altitude
|
Pivotal altitude is the height for a given ground speed at which the line of sight from the cockpit
directly parallel to the lateral axis of the aircraft will remain stationary on an object on the ground. A good rule of thumb for estimating the pivotal altitude is to square the groundspeed, then divide by 15 (if the groundspeed is in miles per hour) or divide by 11.3 (if the groundspeed is in knots), and then add the mean sea level (MSL) altitude of the ground reference. The pivotal altitude is the altitude at which, for a given groundspeed, the projection of the visual
reference line to the pylon appears to pivot. The pivotal altitude does not vary with the angle of bank unless the bank is steep enough to affect the groundspeed.
A rule of thumb for calculating the pivotal altitude H in feet, given the speed in knots formula_0, is
formula_1
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "v"
},
{
"math_id": 1,
"text": " H = \\frac {v^2}{11.3}"
}
] |
https://en.wikipedia.org/wiki?curid=14943561
|
14943683
|
Szász–Mirakyan operator
|
In functional analysis, a discipline within mathematics, the Szász–Mirakyan operators (also spelled "Mirakjan" and "Mirakian") are generalizations of Bernstein polynomials to infinite intervals, introduced by Otto Szász in 1950 and G. M. Mirakjan in 1941. They are defined by
formula_0
where formula_1 and formula_2.
Basic results.
In 1964, Cheney and Sharma showed that if formula_3 is convex and non-linear, the sequence formula_4 decreases with formula_5 (formula_6). They also showed that if formula_3 is a polynomial of degree formula_7, then so is formula_8 for all formula_5.
A converse of the first property was shown by Horová in 1968 (Altomare & Campiti 1994:350).
Theorem on convergence.
In Szász's original paper, he proved the following as Theorem 3 of his paper:
If formula_3 is continuous on formula_9, having a finite limit at infinity, then formula_8 converges uniformly to formula_3 as formula_10.
This is analogous to a theorem stating that Bernstein polynomials approximate continuous functions on [0,1].
Generalizations.
A Kantorovich-type generalization is sometimes discussed in the literature. These generalizations are also called the Szász–Mirakjan–Kantorovich operators.
In 1976, C. P. May showed that the Baskakov operators can reduce to the Szász–Mirakyan operators.
|
[
{
"math_id": 0,
"text": "\\left[\\mathcal{S}_n(f)\\right](x) := e^{-nx}\\sum_{k=0}^\\infty{\\frac{(nx)^k}{k!}f\\left(\\tfrac{k}{n}\\right)}"
},
{
"math_id": 1,
"text": "x\\in[0,\\infty)\\subset\\mathbb{R}"
},
{
"math_id": 2,
"text": "n\\in\\mathbb{N}"
},
{
"math_id": 3,
"text": "f"
},
{
"math_id": 4,
"text": "(\\mathcal{S}_n(f))_{n\\in\\mathbb{N}}"
},
{
"math_id": 5,
"text": "n"
},
{
"math_id": 6,
"text": "\\mathcal{S}_n(f)\\geq f"
},
{
"math_id": 7,
"text": "\\leq m"
},
{
"math_id": 8,
"text": "\\mathcal{S}_n(f)"
},
{
"math_id": 9,
"text": "[0,\\infty)"
},
{
"math_id": 10,
"text": "n\\rightarrow\\infty"
}
] |
https://en.wikipedia.org/wiki?curid=14943683
|
14943850
|
Szász–Mirakjan–Kantorovich operator
|
In functional analysis, a discipline within mathematics, the Szász–Mirakjan–Kantorovich operators are defined by
formula_0
where formula_1 and formula_2.
|
[
{
"math_id": 0,
"text": "[\\mathcal{T}_n(f)](x)=ne^{-nx}\\sum_{k=0}^\\infty{\\frac{(nx)^k}{k!}\\int_{k/n}^{(k+1)/n}f(t)\\,dt}"
},
{
"math_id": 1,
"text": "x\\in[0,\\infty)\\subset\\mathbb{R}"
},
{
"math_id": 2,
"text": "n\\in\\mathbb{N}"
}
] |
https://en.wikipedia.org/wiki?curid=14943850
|
1494479
|
Generalized extreme value distribution
|
Family of probability distributions
In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.
In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after Ronald Fisher and L. H. C. Tippett who recognised three different forms outlined below. However usage of this name is sometimes restricted to mean the special case of the Gumbel distribution. The origin of the common functional form for all 3 distributions dates back to at least Jenkinson, A. F. (1955), though allegedly it could also have been given by von Mises, R. (1936).
Specification.
Using the standardized variable formula_0 where formula_1 the location parameter, can be any real number, and formula_2 is the scale parameter; the cumulative distribution function of the GEV distribution is then
formula_3
where formula_4 the shape parameter, can be any real number. Thus, for formula_5 the expression is valid for formula_6 while for formula_7 it is valid for formula_8 In the first case, formula_9 is the negative, lower end-point, where formula_10 is in the second case, formula_9 is the positive, upper end-point, where formula_11 is 1. For formula_12 the second expression is formally undefined and is replaced with the first expression, which is the result of taking the limit of the second, as formula_13 in which case formula_14 can be any real number.
In the special case of formula_15 so formula_16 and formula_17 for whatever values formula_18 and formula_19 might have.
The probability density function of the standardized distribution is
formula_20
again valid for formula_21 in the case formula_5 and for formula_22 in the case formula_23 The density is zero outside of the relevant range. In the case formula_12 the density is positive on the whole real line.
Since the cumulative distribution function is invertible, the quantile function for the GEV distribution has an explicit expression, namely
formula_24
and therefore the quantile density function, formula_25 is
formula_26
valid for formula_2 and for any real formula_27
Summary statistics.
Some simple statistics of the distribution are:
formula_28 for formula_29
formula_30
formula_31
The skewness is for ξ>0
formula_32
For ξ < 0, the sign of the numerator is reversed.
The excess kurtosis is:
formula_33
where formula_34 formula_35 and formula_36 is the gamma function.
Link to Fréchet, Weibull, and Gumbel families.
The shape parameter formula_18 governs the tail behavior of the distribution. The sub-families defined by three cases: formula_37 formula_5 and formula_38 these correspond, respectively, to the "Gumbel", "Fréchet", and "Weibull" families, whose cumulative distribution functions are displayed below.
formula_41
Let formula_44 and formula_45
formula_46
Let formula_49 and formula_50
formula_51
The subsections below remark on properties of these distributions.
Modification for minima rather than maxima.
The theory here relates to data maxima and the distribution being discussed is an extreme value distribution for maxima. A generalised extreme value distribution for data minima can be obtained, for example by substituting formula_52 for formula_53 in the distribution function, and subtracting the cumulative distribution from one: That is, replace formula_54 with formula_55 . Doing so yields yet another family of distributions.
Alternative convention for the Weibull distribution.
The ordinary Weibull distribution arises in reliability applications and is obtained from the distribution here by using the variable formula_56 which gives a strictly positive support, in contrast to the use in the formulation of extreme value theory here. This arises because the ordinary Weibull distribution is used for cases that deal with data "minima" rather than data maxima. The distribution here has an addition parameter compared to the usual form of the Weibull distribution and, in addition, is reversed so that the distribution has an upper bound rather than a lower bound. Importantly, in applications of the GEV, the upper bound is unknown and so must be estimated, whereas when applying the ordinary Weibull distribution in reliability applications the lower bound is usually known to be zero.
Ranges of the distributions.
Note the differences in the ranges of interest for the three extreme value distributions: Gumbel is unlimited, Fréchet has a lower limit, while the reversed Weibull has an upper limit.
More precisely, Extreme Value Theory (Univariate Theory) describes which of the three is the limiting law according to the initial law X and in particular depending on its tail.
Distribution of log variables.
One can link the type I to types II and III in the following way: If the cumulative distribution function of some random variable formula_57 is of type II, and with the positive numbers as support, i.e. formula_58 then the cumulative distribution function of formula_59 is of type I, namely formula_60 Similarly, if the cumulative distribution function of formula_57 is of type III, and with the negative numbers as support, i.e. formula_61 then the cumulative distribution function of formula_62 is of type I, namely formula_63
Link to logit models (logistic regression).
Multinomial logit models, and certain other types of logistic regression, can be phrased as latent variable models with error variables distributed as Gumbel distributions (type I generalized extreme value distributions). This phrasing is common in the theory of discrete choice models, which include logit models, probit models, and various extensions of them, and derives from the fact that the difference of two type-I GEV-distributed variables follows a logistic distribution, of which the logit function is the quantile function. The type-I GEV distribution thus plays the same role in these logit models as the normal distribution does in the corresponding probit models.
Properties.
The cumulative distribution function of the generalized extreme value distribution solves the stability postulate equation. The generalized extreme value distribution is a special case of a max-stable distribution, and is a transformation of a min-stable distribution.
Applications.
Example for Normally distributed variables.
Let formula_64 be i.i.d. normally distributed random variables with mean 0 and variance 1.
The Fisher–Tippett–Gnedenko theorem tells us that
formula_65
where
formula_66
This allow us to estimate e.g. the mean of formula_67
from the mean of the GEV distribution:
formula_68
where formula_69 is the Euler–Mascheroni constant.
Note that formula_84
Related distributions.
Proofs.
4. Let formula_85 then the cumulative distribution of formula_86 is:
formula_87
which is the cdf for formula_88
5. Let formula_89 then the cumulative distribution of formula_90 is:
formula_91
which is the cumulative distribution of formula_92
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\ s \\equiv \\frac{\\ x - \\mu\\ }{\\sigma}\\ ,"
},
{
"math_id": 1,
"text": "\\ \\mu\\ ,"
},
{
"math_id": 2,
"text": "\\ \\sigma > 0\\ "
},
{
"math_id": 3,
"text": " F(\\ s;\\ \\xi\\ ) = \\begin{cases} \\exp\\! \\Bigl( -e^{-s} \\Bigr) & ~~ \\text{ for } ~~ \\xi = 0\\ , \\\\ {} \\\\\n\\exp\\! \\Bigl( - \\bigl( 1 + \\xi s \\bigr)^{-\\tfrac{\\ 1\\ }{ \\xi } }\\Bigr) & ~~ \\text{ for } ~~ \\xi \\neq 0 ~~ \\text{ and } ~~ \\xi\\ s > -1\\ , \\\\ {} \\\\\n0 & ~~ \\text{ for } ~~ \\xi > 0 ~~ \\text{ and } ~~ s \\le -\\tfrac{\\ 1\\ }{\\xi}\\ , \\\\ {} \\\\\n1 & ~~ \\text{ for } ~~ \\xi < 0 ~~ \\text{ and } ~~ s \\ge \\tfrac{ 1 }{\\ |\\ \\xi\\ |\\ }\\ ; \\end{cases}"
},
{
"math_id": 4,
"text": "\\ \\xi\\ ,"
},
{
"math_id": 5,
"text": "\\ \\xi > 0\\ ,"
},
{
"math_id": 6,
"text": "\\ s > -\\tfrac{\\ 1\\ }{\\xi}\\ ,"
},
{
"math_id": 7,
"text": "\\ \\xi < 0\\ "
},
{
"math_id": 8,
"text": "\\ s < - \\tfrac{\\ 1\\ }{\\xi} ~."
},
{
"math_id": 9,
"text": "\\ -\\tfrac{\\ 1\\ }{\\xi}\\ "
},
{
"math_id": 10,
"text": "\\ F\\ "
},
{
"math_id": 11,
"text": "F"
},
{
"math_id": 12,
"text": "\\ \\xi = 0\\ "
},
{
"math_id": 13,
"text": "\\ \\xi \\to 0\\ "
},
{
"math_id": 14,
"text": "\\ s\\ "
},
{
"math_id": 15,
"text": "\\ x = \\mu\\ ,"
},
{
"math_id": 16,
"text": "\\ s = 0\\ "
},
{
"math_id": 17,
"text": "\\ F(\\ 0;\\ \\xi\\ ) = e^{-1}\\ \\approx 0.368\\ "
},
{
"math_id": 18,
"text": "\\ \\xi\\ "
},
{
"math_id": 19,
"text": "\\ \\sigma\\ "
},
{
"math_id": 20,
"text": "f(\\ s;\\ \\xi\\ ) = \\begin{cases} e^{-s} \\exp\\! \\Bigl( -e^{-s} \\Bigr) & ~~ \\text{ for } ~~ \\xi = 0 , \\\\ {} \\\\\n\\Bigl(\\ 1 + \\xi s\\ \\Bigr)^{-\\left( 1 + \\tfrac{\\ 1\\ }{\\xi} \\right)}\\ \\exp\\! \\Bigl( -\\left( 1 + \\xi s \\right)^{\\tfrac{\\ -1\\ }{\\xi}} \\Bigr) & ~~ \\text{ for } ~~ \\xi \\neq 0 ~~ \\text{ and } ~~ \\xi\\ s > -1\\ , \\\\ {} \\\\\n0 & ~~ \\text{ otherwise; } \\end{cases}"
},
{
"math_id": 21,
"text": "\\ s > -\\tfrac{\\ 1\\ }{\\xi}\\ "
},
{
"math_id": 22,
"text": "\\ s < -\\tfrac{\\ 1\\ }{\\xi}\\ "
},
{
"math_id": 23,
"text": "\\ \\xi < 0 ~."
},
{
"math_id": 24,
"text": "\\ Q(\\ p;\\ \\mu,\\ \\sigma,\\ \\xi\\ ) = \\begin{cases}\n\\mu - \\sigma\\ \\ln\\! \\Bigl( -\\ln(p)\\ \\Bigr) & ~ \\text{ for } ~ \\xi = 0 ~ \\text{ and } ~ p \\in (\\ 0\\ ,\\ 1\\ )\\ , \\\\ {} \\\\\n\\mu + \\displaystyle{ \\frac{\\ \\sigma\\ }{\\ \\xi\\ }} \\left( \\Bigl( -\\ln(p)\\ \\Bigr)^{-\\xi} - 1 \\right) & ~ \\text{ for } ~ \\xi > 0 ~ \\text{ and } ~ p \\in [\\ 0\\ ,\\ 1\\ )\\ , \\\\\n {} & ~~ \\text{ or } ~ \\, \\xi < 0 ~ \\text{ and } ~ p \\in \\ (\\ 0\\ ,\\ 1\\ ]\\ ; \\end{cases}"
},
{
"math_id": 25,
"text": "\\ q \\equiv \\frac{\\;\\mathrm{d}\\ Q\\;}{\\mathrm{d}\\ p}\\ ,"
},
{
"math_id": 26,
"text": "\\ q(\\ p;\\ \\sigma,\\ \\xi\\ ) = \\frac{\\sigma}{\\ \\Bigl( - \\ln(p)\\ \\Bigr)^{\\xi + 1}\\ p \\;} \\quad \\text{ for } ~~ p \\in (\\ 0\\ ,\\ 1\\ )\\ ,"
},
{
"math_id": 27,
"text": "\\ \\xi ~."
},
{
"math_id": 28,
"text": "\\operatorname{\\mathbb E}(X) = \\mu + (g_1-1)\\frac{\\sigma}{\\xi}"
},
{
"math_id": 29,
"text": "\\xi < 1"
},
{
"math_id": 30,
"text": "\\operatorname{Var}(X) = (g_2-g_1^2)\\frac{\\sigma^2}{\\xi^2} ,"
},
{
"math_id": 31,
"text": "\\operatorname{Mode}(X) = \\mu+\\frac{\\sigma}{\\xi}[(1+\\xi)^{-\\xi}-1] ."
},
{
"math_id": 32,
"text": "\\operatorname{skewness}(X) = \\frac{g_3-3g_2g_1+2g_1^3}{(g_2-g_1^2)^{3/2}} "
},
{
"math_id": 33,
"text": "\\operatorname{kurtosis\\ excess}(X) = \\frac{g_4-4g_3g_1+6g_2g_1^2-3g_1^4}{(g_2-g_1^2)^2}-3 ~."
},
{
"math_id": 34,
"text": "\\ g_k = \\Gamma(1 - k\\ \\xi)\\ ,"
},
{
"math_id": 35,
"text": "\\ k=1,2,3,4\\ ,"
},
{
"math_id": 36,
"text": "\\ \\Gamma(t)\\ "
},
{
"math_id": 37,
"text": "\\ \\xi = 0\\ ,"
},
{
"math_id": 38,
"text": "\\ \\xi < 0\\ ;"
},
{
"math_id": 39,
"text": "~ \\xi = 0\\ , \\quad"
},
{
"math_id": 40,
"text": " \\quad x \\in \\Bigl(\\ -\\infty\\ ,\\ +\\infty\\ \\Bigr)\\ :"
},
{
"math_id": 41,
"text": " F(\\ x;\\ \\mu,\\ \\sigma,\\ 0\\ ) = \\exp \\left( - \\exp \\left( -\\frac{\\ x - \\mu\\ }{\\sigma} \\right) \\right) ~. "
},
{
"math_id": 42,
"text": "~ \\xi > 0\\ , \\quad "
},
{
"math_id": 43,
"text": " \\quad x \\in \\left(\\ \\mu - \\tfrac{\\sigma}{\\ \\xi\\ }\\ ,\\ +\\infty\\ \\right)\\ :"
},
{
"math_id": 44,
"text": "\\quad \\alpha \\equiv \\tfrac{\\ 1\\ }{ \\xi } > 0 \\quad "
},
{
"math_id": 45,
"text": " \\quad y \\equiv 1 + \\tfrac{\\xi}{\\sigma} (x-\\mu)\\ ;"
},
{
"math_id": 46,
"text": " F(\\ x;\\ \\mu,\\ \\sigma,\\ \\xi\\ ) = \\begin{cases} 0 & y \\leq 0 \\quad \\mathsf{~ or\\ equiv. ~} \\quad x \\leq \\mu - \\tfrac{\\sigma}{\\ \\xi\\ } \\\\ \\exp\\left( -\\frac{1}{~ y^\\alpha\\ } \\right) & y > 0 \\quad \\mathsf{~ or\\ equiv. ~} \\quad x > \\mu - \\tfrac{\\sigma}{\\ \\xi\\ } ~. \\end{cases}"
},
{
"math_id": 47,
"text": "~ \\xi < 0\\ , \\quad "
},
{
"math_id": 48,
"text": " \\quad x \\in \\left( -\\infty\\ ,\\ \\mu + \\tfrac{ \\sigma }{\\ |\\ \\xi\\ |\\ }\\ \\right)\\ :"
},
{
"math_id": 49,
"text": " \\quad \\alpha \\equiv - \\tfrac{1}{\\ \\xi\\ } > 0 \\quad "
},
{
"math_id": 50,
"text": " \\quad y \\equiv 1 - \\tfrac{\\ |\\ \\xi\\ |\\ }{\\sigma} (x - \\mu)\\ ;"
},
{
"math_id": 51,
"text": " F(\\ x;\\ \\mu,\\ \\sigma,\\ \\xi\\ ) = \\begin{cases} \\exp\\left( -y^{\\alpha} \\right) & y > 0 \\quad \\mathsf{~ or\\ equiv. ~} \\quad x < \\mu + \\tfrac{ \\sigma }{\\ |\\ \\xi\\ |\\ } \\\\ 1 & y \\leq 0 \\quad \\mathsf{~ or\\ equiv. ~} \\quad x \\geq \\mu + \\tfrac{ \\sigma }{\\ |\\ \\xi\\ |\\ } ~. \\end{cases}"
},
{
"math_id": 52,
"text": "\\ -x\\;"
},
{
"math_id": 53,
"text": "\\;x\\;"
},
{
"math_id": 54,
"text": "\\ F(x)\\ "
},
{
"math_id": 55,
"text": "\\ 1 - F(-x)\\ "
},
{
"math_id": 56,
"text": "\\ t = \\mu - x\\ ,"
},
{
"math_id": 57,
"text": "\\ X\\ "
},
{
"math_id": 58,
"text": "\\ F(\\ x;\\ 0,\\ \\sigma,\\ \\alpha\\ )\\ ,"
},
{
"math_id": 59,
"text": "\\ln X"
},
{
"math_id": 60,
"text": "\\ F(\\ x;\\ \\ln \\sigma,\\ \\tfrac{1}{\\ \\alpha\\ },\\ 0\\ ) ~."
},
{
"math_id": 61,
"text": "\\ F(\\ x;\\ 0,\\ \\sigma,\\ -\\alpha\\ )\\ ,"
},
{
"math_id": 62,
"text": "\\ \\ln (-X)\\ "
},
{
"math_id": 63,
"text": "\\ F(\\ x;\\ -\\ln \\sigma,\\ \\tfrac{\\ 1\\ }{\\alpha},\\ 0\\ ) ~."
},
{
"math_id": 64,
"text": "\\ \\left\\{\\ X_i\\ \\big|\\ 1 \\le i \\le n\\ \\right\\}\\ "
},
{
"math_id": 65,
"text": "\\ \\max \\{\\ X_i\\ \\big|\\ 1 \\le i \\le n\\ \\} \\sim GEV(\\mu_n, \\sigma_n, 0)\\ ,"
},
{
"math_id": 66,
"text": "\n\\begin{align}\n \\mu_n &= \\Phi^{-1}\\left( 1 - \\frac{\\ 1\\ }{ n } \\right) \\\\\n \\sigma_n &= \\Phi^{-1}\\left( 1 - \\frac{ 1 }{\\ n\\ \\mathrm{e}\\ } \\right)- \\Phi^{-1}\\left(1-\\frac{\\ 1\\ }{ n } \\right) ~.\n\\end{align}\n"
},
{
"math_id": 67,
"text": "\\ \\max \\{\\ X_i\\ \\big|\\ 1 \\le i \\le n\\ \\}\\ "
},
{
"math_id": 68,
"text": "\n\\begin{align}\n\\operatorname{\\mathbb E}\\left\\{\\ \\max\\left\\{\\ X_i\\ \\big|\\ 1 \\le i \\le n\\ \\right\\}\\ \\right\\}\n& \\approx \\mu_n + \\gamma_{\\mathsf E}\\ \\sigma_n \\\\\n&= (1 - \\gamma_{\\mathsf E})\\ \\Phi^{-1}\\left( 1 - \\frac{\\ 1\\ }{ n } \\right) + \\gamma_{\\mathsf E}\\ \\Phi^{-1}\\left( 1 - \\frac{1}{\\ e\\ n\\ } \\right) \\\\\n&= \\sqrt{\\log \\left(\\frac{ n^2 }{\\ 2 \\pi\\ \\log \\left(\\frac{n^2}{2\\pi} \\right)\\ }\\right) ~}\\ \\cdot\\ \\left(1 + \\frac{ \\gamma }{\\ \\log n\\ } + \\mathcal{o} \\left(\\frac{ 1 }{\\ \\log n\\ } \\right) \\right)\\ ,\n\\end{align}\n"
},
{
"math_id": 69,
"text": "\\ \\gamma_{\\mathsf E}\\ "
},
{
"math_id": 70,
"text": "\\ X \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,\\xi)\\ "
},
{
"math_id": 71,
"text": "\\ m X + b \\sim \\textrm{GEV}(m \\mu+b,\\ m\\sigma,\\ \\xi)\\ "
},
{
"math_id": 72,
"text": "\\ X \\sim \\textrm{Gumbel}(\\mu,\\ \\sigma)\\ "
},
{
"math_id": 73,
"text": "\\ X \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)\\ "
},
{
"math_id": 74,
"text": "\\ X \\sim \\textrm{Weibull}(\\sigma,\\,\\mu)\\ "
},
{
"math_id": 75,
"text": "\\ \\mu\\left(1-\\sigma\\log \\tfrac{X}{\\sigma} \\right) \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)\\ "
},
{
"math_id": 76,
"text": "\\ \\sigma \\exp (-\\tfrac{X-\\mu}{\\mu \\sigma} ) \\sim \\textrm{Weibull}(\\sigma,\\,\\mu)\\ "
},
{
"math_id": 77,
"text": "\\ X \\sim \\textrm{Exponential}(1)\\ "
},
{
"math_id": 78,
"text": "\\ \\mu - \\sigma \\log X \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)\\ "
},
{
"math_id": 79,
"text": "\\ X \\sim \\mathrm{Gumbel}(\\alpha_X, \\beta)\\ "
},
{
"math_id": 80,
"text": "\\ Y \\sim \\mathrm{Gumbel}(\\alpha_Y, \\beta)\\ "
},
{
"math_id": 81,
"text": "\\ X-Y \\sim \\mathrm{Logistic}(\\alpha_X-\\alpha_Y,\\beta)\\ "
},
{
"math_id": 82,
"text": "\\ Y \\sim \\mathrm{Gumbel}(\\alpha, \\beta)\\ "
},
{
"math_id": 83,
"text": "\\ X+Y \\nsim \\mathrm{Logistic}(2 \\alpha,\\beta)\\ "
},
{
"math_id": 84,
"text": "\\ \\operatorname{\\mathbb E}\\{\\ X + Y\\ \\} = 2\\alpha+2\\beta\\gamma \\neq 2\\alpha = \\operatorname{\\mathbb E}\\left\\{\\ \\operatorname{Logistic}(2 \\alpha,\\beta)\\ \\right\\} ~."
},
{
"math_id": 85,
"text": "\\ X \\sim \\textrm{ Weibull }(\\sigma,\\,\\mu)\\ ,"
},
{
"math_id": 86,
"text": "\\ g(x) = \\mu\\left(1-\\sigma\\log\\frac{X}{\\sigma} \\right)\\ "
},
{
"math_id": 87,
"text": "\n\\begin{align}\n\\operatorname{\\mathbb P}\\left\\{\\ \\mu \\left(1-\\sigma\\log\\frac{\\ X\\ }{ \\sigma } \\right) < x\\ \\right\\} &= \\operatorname{\\mathbb P}\\left\\{\\ \\log\\frac{X}{\\sigma} > \\frac{1 - x/\\mu}{\\sigma}\\ \\right\\} \\\\ {} \\\\\n& \\mathsf{\\ Since\\ the\\ logarithm\\ is\\ always\\ increasing:\\ } \\\\ {} \\\\\n&= \\operatorname{\\mathbb P}\\left\\{\\ X > \\sigma \\exp\\left[ \\frac{1 - x/\\mu}{\\sigma} \\right]\\ \\right\\} \\\\\n&= \\exp\\left( - \\left(\\cancel{\\sigma} \\exp\\left[ \\frac{1 - x/\\mu}{\\sigma} \\right] \\cdot \\cancel{\\frac{1}{\\sigma}} \\right)^\\mu \\right) \\\\\n&= \\exp\\left( - \\left( \\exp\\left[ \\frac{\\cancelto{\\mu}{1} - x/\\cancel{\\mu}}{\\sigma} \\right] \\right)^\\cancel{\\mu} \\right) \\\\\n&= \\exp\\left( - \\exp\\left[ \\frac{\\mu - x}{\\sigma} \\right] \\right) \\\\\n&= \\exp\\left( - \\exp\\left[ - s \\right] \\right), \\quad s = \\frac{x - \\mu}{\\sigma}\\ ,\n\\end{align}\n"
},
{
"math_id": 88,
"text": "\\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0) ~."
},
{
"math_id": 89,
"text": "\\ X \\sim \\textrm{Exponential}(1)\\ ,"
},
{
"math_id": 90,
"text": "\\ g(X) = \\mu - \\sigma \\log X\\ "
},
{
"math_id": 91,
"text": "\n\\begin{align}\n\\operatorname{\\mathbb P}\\left\\{\\ \\mu - \\sigma \\log X < x\\ \\right\\} &= \\operatorname{\\mathbb P}\\left\\{\\ \\log X > \\frac{\\mu - x}{\\sigma}\\ \\right\\} \\\\ {} \\\\\n& \\mathsf{\\ Since\\ the\\ logarithm\\ is\\ always\\ increasing:\\ } \\\\ {} \\\\\n&= \\operatorname{\\mathbb P}\\left\\{\\ X > \\exp\\left( \\frac{\\ \\mu - x\\ }{ \\sigma } \\right)\\ \\right\\} \\\\\n&= \\exp\\left[- \\exp\\left( \\frac{\\ \\mu - x\\ }{ \\sigma } \\right) \\right] \\\\\n&= \\exp\\left[- \\exp(-s) \\right]\\ , \\quad ~\\mathsf{ where }~ \\quad s \\equiv \\frac{x - \\mu}{\\sigma}\\ ;\n\\end{align}\n"
},
{
"math_id": 92,
"text": "\\ \\operatorname{GEV}(\\mu, \\sigma, 0) ~."
}
] |
https://en.wikipedia.org/wiki?curid=1494479
|
14947251
|
Extraction ratio
|
Extraction ratio is a measure in renal physiology, primarily used to calculate renal plasma flow in order to evaluate renal function. It measures the percentage of the compound entering the kidney that was excreted into the final urine.
Measured in concentration in blood plasma, it may thus be expressed as:
formula_0
, where Pa is the concentration in renal artery, and Pv is the concentration in the renal vein.
For instance, para aminohippuric acid (PAH) is almost completely excreted in the final urine, and thus almost none is found in the venous return (Pv ~0). Therefore, the extraction ratio of PAH ~1. This is why PAH is used in PAH clearance to estimate renal plasma flow.
Hepatic extraction ratio.
The "Hepatic Extraction Ratio" is a similar measurement for clearance of a substance (usually a pharmacological drug) by the liver. It is defined as the fraction of drug removed from blood by the liver, and depends on 3 factors— the hepatic blood flow, the uptake into the hepatocytes, and the enzyme metabolic capacity. Examples of drugs with a high hepatic extraction ratio include propranolol, opiates, and lignocaine.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Extraction \\ ratio = \\frac{P_a - P_v}{P_a}"
}
] |
https://en.wikipedia.org/wiki?curid=14947251
|
14947424
|
PAH clearance
|
Para-aminohippurate (PAH) clearance is a method used in renal physiology to measure renal plasma flow, which is a measure of renal function.
PAH is completely removed from blood that passes through the kidneys (PAH undergoes both glomerular filtration and tubular secretion), and therefore the rate at which the kidneys can clear PAH from the blood reflects total renal plasma flow.
The concentration of PAH is measured in one arterial blood sample (PPAH) and one urine sample(UPAH). The urine flow (V) is also measured. Renal perfusion flow is then calculated by:
formula_0
What is calculated is the effective renal plasma flow (eRPF). However, since the renal extraction ratio of PAH almost equals 1, then eRPF almost equals RPF.
Precision.
The renal extraction ratio of PAH in a normal individual is approximately 0.92, and thus not exactly 1.0. Thus, this method usually underestimates RPF by approximately 10%. This margin of error is generally acceptable considering the ease with which eRPF is measured.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "RPF = \\frac{U_{PAH}}{P_{PAH}} V"
}
] |
https://en.wikipedia.org/wiki?curid=14947424
|
14948065
|
Color–flavor locking
|
Phenomenon in high-density strange matter
Color–flavor locking (CFL) is a phenomenon that is expected to occur in ultra-high-density strange matter, a form of quark matter. The quarks form Cooper pairs, whose color properties are correlated with their flavor properties in a one-to-one correspondence between three color pairs and three flavor pairs. According to the Standard Model of particle physics, the color-flavor-locked phase is the highest-density phase of three-flavor colored matter.
Color-flavor-locked Cooper pairing.
If each quark is represented as formula_0, with color index formula_1 taking values 1, 2, 3 corresponding to red, green, and blue, and flavor index formula_2 taking values 1, 2, 3 corresponding to up, down, and strange, then the color-flavor-locked pattern of Cooper pairing is
formula_3
This means that a Cooper pair of an up quark and a down quark must have colors red and green, and so on. This pairing pattern is special because it leaves a large unbroken symmetry group.
Physical properties.
The CFL phase has several remarkable properties.
There are several variants of the CFL phase, representing distortions of the pairing structure in response to external stresses such as a difference between the mass of the strange quark and the mass of the up and down quarks.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\psi^\\alpha_i"
},
{
"math_id": 1,
"text": "\\alpha"
},
{
"math_id": 2,
"text": "i"
},
{
"math_id": 3,
"text": "\\langle \\psi^\\alpha_i C \\gamma_5 \\psi^\\beta_j \\rangle\n\\propto \\delta^\\alpha_i\\delta^\\beta_j - \\delta^\\alpha_j\\delta^\\beta_i \n= \\epsilon^{\\alpha\\beta A}\\epsilon_{ij A} \n"
}
] |
https://en.wikipedia.org/wiki?curid=14948065
|
1495467
|
Maximum length sequence
|
Type of pseudorandom binary sequence
A maximum length sequence (MLS) is a type of pseudorandom binary sequence.
They are bit sequences generated using maximal linear-feedback shift registers and are so called because they are periodic and reproduce every binary sequence (except the zero vector) that can be represented by the shift registers (i.e., for length-"m" registers they produce a sequence of length 2"m" − 1). An MLS is also sometimes called an n-sequence or an m-sequence. MLSs are spectrally flat, with the exception of a near-zero DC term.
These sequences may be represented as coefficients of irreducible polynomials in a polynomial ring over Z/2Z.
Practical applications for MLS include measuring impulse responses (e.g., of room reverberation or arrival times from towed sources in the ocean). They are also used as a basis for deriving pseudo-random sequences in digital communication systems that employ direct-sequence spread spectrum and frequency-hopping spread spectrum transmission systems, and in the efficient design of some fMRI experiments.
Generation.
MLS are generated using maximal linear-feedback shift registers. An MLS-generating system with a shift register of length 4 is shown in Fig. 1. It can be expressed using the following recursive relation:
formula_0
where "n" is the time index and formula_1 represents modulo-2 addition. For bit values 0 = FALSE or 1 = TRUE, this is equivalent to the XOR operation.
As MLS are periodic and shift registers cycle through every possible binary value (with the exception of the zero vector), registers can be initialized to any state, with the exception of the zero vector.
Polynomial interpretation.
A polynomial over GF(2) can be associated with the linear-feedback shift register. It has degree of the length of the shift register, and has coefficients that are either 0 or 1, corresponding to the taps of the register that feed the xor gate. For example, the polynomial corresponding to Figure 1 is formula_2.
A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial be primitive.
Implementation.
MLS are inexpensive to implement in hardware or software, and relatively low-order feedback shift registers can generate long sequences; a sequence generated using a shift register of length 20 is 220 − 1 samples long (1,048,575 samples).
Properties of maximum length sequences.
MLS have the following properties, as formulated by Solomon Golomb.
Balance property.
The occurrence of 0 and 1 in the sequence should be approximately the same. More precisely, in a maximum length sequence of length formula_3 there are formula_4 ones and formula_5 zeros. The number of ones equals the number of zeros plus one, since the state containing only zeros cannot occur.
Run property.
A "run" is a sub-sequence of consecutive "1"s or consecutive "0"s within the MLS concerned. The number of runs is the number of such sub-sequences.
Of all the "runs" (consisting of "1"s or "0"s) in the sequence :
Correlation property.
The circular autocorrelation of an MLS is a Kronecker delta function (with DC offset and time delay, depending on implementation). For the ±1 convention, i.e., bit value 1 is assigned formula_6 and bit value 0 formula_7, mapping XOR to the negative of the product:
formula_8
where formula_9 represents the complex conjugate and formula_10 represents a circular shift.
The linear autocorrelation of an MLS approximates a Kronecker delta.
Extraction of impulse responses.
If a linear time invariant (LTI) system's impulse response is to be measured using a MLS, the response can be extracted from the measured system output "y"["n"] by taking its circular cross-correlation with the MLS. This is because the autocorrelation of a MLS is 1 for zero-lag, and nearly zero (−1/"N" where "N" is the sequence length) for all other lags; in other words, the autocorrelation of the MLS can be said to approach unit impulse function as MLS length increases.
If the impulse response of a system is "h"["n"] and the MLS is "s"["n"], then
formula_11
Taking the cross-correlation with respect to "s"["n"] of both sides,
formula_12
and assuming that φ"ss" is an impulse (valid for long sequences)
formula_13
Any signal with an impulsive autocorrelation can be used for this purpose, but signals with high crest factor, such as the impulse itself, produce impulse responses with poor signal-to-noise ratio. It is commonly assumed that the MLS would then be the ideal signal, as it consists of only full-scale values and its digital crest factor is the minimum, 0 dB. However, after analog reconstruction, the sharp discontinuities in the signal produce strong intersample peaks, degrading the crest factor by 4-8 dB or more, increasing with signal length, making it worse than a sine sweep. Other signals have been designed with minimal crest factor, though it is unknown if it can be improved beyond 3 dB.
Relationship to Hadamard transform.
Cohn and Lempel showed the relationship of the MLS to the Hadamard transform. This relationship allows the correlation of an MLS to be computed in a fast algorithm similar to the FFT.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\begin{cases}\na_3[n+1] = a_0[n] + a_1[n]\\\\\na_2[n+1] = a_3[n] \\\\\na_1[n+1] = a_2[n] \\\\\na_0[n+1] = a_1[n] \\\\\n\\end{cases}\n"
},
{
"math_id": 1,
"text": "+"
},
{
"math_id": 2,
"text": "x^4+x+1"
},
{
"math_id": 3,
"text": "2^n-1"
},
{
"math_id": 4,
"text": "2^{n-1}"
},
{
"math_id": 5,
"text": "2^{n-1}-1"
},
{
"math_id": 6,
"text": "s = +1"
},
{
"math_id": 7,
"text": "s = -1"
},
{
"math_id": 8,
"text": "R(n)=\\frac 1 N \\sum_{m=1}^N s[m]\\, s^*[m+n]_N = \\begin{cases}\n1 &\\text{if } n = 0, \\\\\n-\\frac 1 N &\\text{if } 0 < n < N. \\end{cases}"
},
{
"math_id": 9,
"text": "s^*"
},
{
"math_id": 10,
"text": "[m+n]_N"
},
{
"math_id": 11,
"text": "y[n] = (h*s)[n].\\,"
},
{
"math_id": 12,
"text": "{\\phi}_{sy} = h[n]*{\\phi}_{ss}\\,"
},
{
"math_id": 13,
"text": "h[n] = {\\phi}_{sy}.\\,"
}
] |
https://en.wikipedia.org/wiki?curid=1495467
|
14954860
|
Hanner's inequalities
|
Mathematical results
In mathematics, Hanner's inequalities are results in the theory of "L""p" spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of "L""p" spaces for "p" ∈ (1, +∞) than the approach proposed by James A. Clarkson in 1936.
Statement of the inequalities.
Let "f", "g" ∈ "L""p"("E"), where "E" is any measure space. If "p" ∈ [1, 2], then
formula_0
The substitutions "F" = "f" + "g" and "G" = "f" − "g" yield the second of Hanner's inequalities:
formula_1
For "p" ∈ [2, +∞) the inequalities are reversed (they remain non-strict).
Note that for formula_2 the inequalities become equalities which are both the parallelogram rule.
|
[
{
"math_id": 0,
"text": "\\|f+g\\|_p^p + \\|f-g\\|_p^p \\geq \\big( \\|f\\|_p + \\|g\\|_p \\big)^p + \\big| \\|f\\|_p-\\|g\\|_p \\big|^p."
},
{
"math_id": 1,
"text": "2^p \\big( \\|F\\|_p^p + \\|G\\|_p^p \\big) \\geq \\big( \\|F+G\\|_p + \\|F-G\\|_p \\big)^p + \\big| \\|F+G\\|_p-\\|F-G\\|_p \\big|^p."
},
{
"math_id": 2,
"text": "p = 2"
}
] |
https://en.wikipedia.org/wiki?curid=14954860
|
149568
|
Pafnuty Chebyshev
|
Russian mathematician (1821–1894)
Pafnuty Lvovich Chebyshev (Russian: , ) (16 May [O.S. 4 May] 1821 – 8 December [O.S. 26 November] 1894) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias.
Transcription.
The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian), and Chebychev, a mixture between English and French transliterations considered erroneous. It is one of the most well known data-retrieval nightmares in mathematical literature. Currently, the English transliteration "Chebyshev" has gained widespread acceptance, except by the French, who prefer "Tchebychev." The correct transliteration according to ISO 9 is "Čebyšëv". The American Mathematical Society adopted the transcription "Chebyshev" in its Mathematical Reviews.
His first name comes from the Greek Paphnutius (Παφνούτιος), which in turn takes its origin in the Coptic Paphnuty (Ⲡⲁⲫⲛⲟⲩϯ), meaning "that who belongs to God" or simply "the man of God".
Biography.
Early years.
One of nine children, Chebyshev was born in the village of Okatovo in the district of Borovsk, province of Kaluga. His father, Lev Pavlovich, was a Russian nobleman and wealthy landowner. Pafnuty Lvovich was first educated at home by his mother Agrafena Ivanovna Pozniakova (in reading and writing) and by his cousin Avdotya Kvintillianovna Sukhareva (in French and arithmetic). Chebyshev mentioned that his music teacher also played an important role in his education, for she "raised his mind to exactness and analysis".
Trendelenburg's gait affected Chebyshev's adolescence and development. From childhood, he limped and walked with a stick and so his parents abandoned the idea of his becoming an officer in the family tradition. His disability prevented his playing many children's games and he devoted himself instead to mathematics.
In 1832, the family moved to Moscow, mainly to attend to the education of their eldest sons (Pafnuty and Pavel, who would become lawyers). Education continued at home and his parents engaged teachers of excellent reputation, including (for mathematics and physics) the senior Moscow University teacher Platon Pogorelsky, who had taught, among others, the future writer Ivan Turgenev.
University studies.
In summer 1837, Chebyshev passed the registration examinations and, in September of that year, began his mathematical studies at the second philosophical department of Moscow University. His teachers included N.D. Brashman, N.E. Zernov and D.M. Perevoshchikov of whom it seems clear that Brashman had the greatest influence on Chebyshev. Brashman instructed him in practical mechanics and probably showed him the work of French engineer J.V. Poncelet.
In 1841 Chebyshev was awarded the silver medal for his work "calculation of the roots of equations" which he had finished in 1838. In this, Chebyshev derived an approximating algorithm for the solution of algebraic equations of "nth" degree based on Newton's method. In the same year, he finished his studies as "most outstanding candidate".
In 1841, Chebyshev's financial situation changed drastically. There was famine in Russia, and his parents were forced to leave Moscow. Although they could no longer support their son, he decided to continue his mathematical studies and prepared for the master examinations, which lasted six months. Chebyshev passed the final examination in October 1843 and, in 1846, defended his master thesis "An Essay on the Elementary Analysis of the Theory of Probability." His biographer Prudnikov suggests that Chebyshev was directed to this subject after learning of recently published books on probability theory or on the revenue of the Russian insurance industry.
Adult years.
In 1847, Chebyshev promoted his thesis "pro venia legendi" "On integration with the help of logarithms" at St Petersburg University and thus obtained the right to teach there as a lecturer. At that time some of Leonhard Euler's works were rediscovered by P. N. Fuss and were being edited by Viktor Bunyakovsky, who encouraged Chebyshev to study them. This would come to influence Chebyshev's work. In 1848, he submitted his work "The Theory of Congruences" for a doctorate, which he defended in May 1849. He was elected an extraordinary professor at St Petersburg University in 1850, ordinary professor in 1860 and, after 25 years of lectureship, he became merited professor in 1872. In 1882 he left the university and devoted his life to research.
During his lectureship at the university (1852–1858), Chebyshev also taught practical mechanics at the Alexander Lyceum in Tsarskoe Selo (now Pushkin), a southern suburb of St Petersburg.
His scientific achievements were the reason for his election as junior academician (adjunkt) in 1856. Later, he became an extraordinary (1856) and in 1858 an ordinary member of the Imperial Academy of Sciences. In the same year he became an honorary member of Moscow University. He accepted other honorary appointments and was decorated several times. In 1856, Chebyshev became a member of the scientific committee of the ministry of national education. In 1859, he became an ordinary member of the ordnance department of the academy with the adoption of the headship of the commission for mathematical questions according to ordnance and experiments related to ballistics. The Paris academy elected him corresponding member in 1860 and full foreign member in 1874. In 1893, he was elected honorable member of the St. Petersburg Mathematical Society, which had been founded three years earlier.
Chebyshev died in St Petersburg on 26 November 1894.
Mathematical contributions.
Chebyshev is known for his work in the fields of probability, statistics, mechanics, and number theory. The Chebyshev inequality states that if formula_0 is a random variable with standard deviation "σ" > 0, then the probability that the outcome of formula_0 is no less than formula_1 away from its mean is no more than formula_2:
formula_3
The Chebyshev inequality is used to prove the weak law of large numbers.
The Bertrand–Chebyshev theorem (1845, 1852) states that for any formula_4, there exists a prime number formula_5 such that formula_6. This is a consequence of the Chebyshev inequalities for the number formula_7 of prime numbers less than formula_8, which state that formula_7 is of the order of formula_9. A more precise form is given by the celebrated prime number theorem: the "quotient" of the two expressions approaches 1.0 as formula_8 tends to infinity.
Chebyshev is also known for the Chebyshev polynomials and the Chebyshev bias – the difference between the number of primes that are congruent to 3 (modulo 4) and 1 (modulo 4).
Chebyshev was the first person to think systematically in terms of random variables and their moments and expectations.
Legacy.
Chebyshev is considered to be a founding father of Russian mathematics. Among his well-known students were the mathematicians Dmitry Grave, Aleksandr Korkin, Aleksandr Lyapunov, and Andrei Markov. According to the Mathematics Genealogy Project, Chebyshev has 16,874 mathematical "descendants" as of February 2024.
The lunar crater "Chebyshev" and the asteroid 2010 Chebyshev were named to honor his major achievements in the mathematical realm.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "a\\sigma"
},
{
"math_id": 2,
"text": "1/a^2"
},
{
"math_id": 3,
"text": "\\Pr(|X - {\\mathbf E}(X)| \\ge a\\ )\\le \\frac {\\sigma^2}{a^2}."
},
{
"math_id": 4,
"text": "n > 3"
},
{
"math_id": 5,
"text": "p"
},
{
"math_id": 6,
"text": "n < p < 2n"
},
{
"math_id": 7,
"text": "\\pi(n)"
},
{
"math_id": 8,
"text": "n"
},
{
"math_id": 9,
"text": "n/\\log(n)"
}
] |
https://en.wikipedia.org/wiki?curid=149568
|
1495744
|
Intrinsic equation
|
Equation which defines a curve independently of a coordinate system
In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system.
The intrinsic quantities used most often are arc length formula_0, tangential angle formula_1, curvature formula_2 or radius of curvature, and, for 3-dimensional curves, torsion formula_3. Specifically:
The equation of a circle (including a line) for example is given by the equation formula_4 where formula_5 is the arc length, formula_2 the curvature and formula_6 the radius of the circle.
These coordinates greatly simplify some physical problem. For elastic rods for example, the potential energy is given by
formula_7
where formula_8 is the bending modulus formula_9. Moreover, as formula_10, elasticity of rods can be given a simple variational form.
|
[
{
"math_id": 0,
"text": " s "
},
{
"math_id": 1,
"text": " \\theta "
},
{
"math_id": 2,
"text": "\\kappa"
},
{
"math_id": 3,
"text": "\\tau "
},
{
"math_id": 4,
"text": "\\kappa(s) = \\tfrac{1}{r}"
},
{
"math_id": 5,
"text": "s"
},
{
"math_id": 6,
"text": "r"
},
{
"math_id": 7,
"text": "E= \\int_0^L B \\kappa^2(s)ds "
},
{
"math_id": 8,
"text": "B"
},
{
"math_id": 9,
"text": "EI"
},
{
"math_id": 10,
"text": "\\kappa(s) = d\\theta/ds"
}
] |
https://en.wikipedia.org/wiki?curid=1495744
|
14962
|
Identity element
|
Specific element of an algebraic structure
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term "identity element" is often shortened to "identity" (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Definitions.
Let ("S", ∗) be a set S equipped with a binary operation ∗. Then an element e of S is called a <templatestyles src="Template:Visible anchor/styles.css" />left identity if "e" ∗ "s" = "s" for all s in S, and a <templatestyles src="Template:Visible anchor/styles.css" />right identity if "s" ∗ "e" = "s" for all s in S. If e is both a left identity and a right identity, then it is called a <templatestyles src="Template:Visible anchor/styles.css" />two-sided identity, or simply an <templatestyles src="Template:Visible anchor/styles.css" />identity.
An identity with respect to addition is called an <templatestyles src="Template:Visible anchor/styles.css" />additive identity (often denoted as 0) and an identity with respect to multiplication is called a <templatestyles src="Template:Visible anchor/styles.css" />multiplicative identity (often denoted as 1). These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a group for example, the identity element is sometimes simply denoted by the symbol formula_0. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called <templatestyles src="Template:Visible anchor/styles.css" />unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit.
Properties.
In the example "S" = {"e,f"} with the equalities given, "S" is a semigroup. It demonstrates the possibility for ("S", ∗) to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity.
To see this, note that if l is a left identity and r is a right identity, then "l" = "l" ∗ "r" = "r". In particular, there can never be more than one two-sided identity: if there were two, say e and f, then "e" ∗ "f" would have to be equal to both e and f.
It is also quite possible for ("S", ∗) to have "no" identity element, such as the case of even integers under the multiplication operation. Another common example is the cross product of vectors, where the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive semigroup of positive natural numbers.
Notes and references.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "e"
}
] |
https://en.wikipedia.org/wiki?curid=14962
|
1496229
|
Pentadecagon
|
Polygon with 15 edges
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
Regular pentadecagon.
A "regular pentadecagon" is represented by Schläfli symbol {15}.
A regular pentadecagon has interior angles of 156°, and with a side length "a", has an area given by
formula_0
Construction.
As 15 = 3 × 5, a product of distinct Fermat primes, a regular pentadecagon is constructible using compass and straightedge:
The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's "Elements".
Compare the construction according to Euclid in this image: Pentadecagon
In the construction for given circumcircle: formula_1 is a side of equilateral triangle and formula_2 is a side of a regular pentagon.
The point formula_3 divides the radius formula_4 in golden ratio: formula_5
Compared with the first animation (with green lines) are in the following two images the two circular arcs (for angles 36° and 24°) rotated 90° counterclockwise shown. They do not use the segment formula_6, but rather they use segment formula_7 as radius formula_8 for the second circular arc (angle 36°).
A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here formula_9 which is divided according to the golden ratio:
formula_10
Circumradius formula_11 Side length formula_12 Angle formula_13
formula_14
Symmetry.
The "regular pentadecagon" has Dih15 dihedral symmetry, order 30, represented by 15 lines of reflection. Dih15 has 3 dihedral subgroups: Dih5, Dih3, and Dih1. And four more cyclic symmetries: Z15, Z5, Z3, and Z1, with Zn representing π/"n" radian rotational symmetry.
On the pentadecagon, there are 8 distinct symmetries. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. He gives r30 for the full reflective symmetry, Dih15. He gives d (diagonal) with reflection lines through vertices, p with reflection lines through edges (perpendicular), and for the odd-sided pentadecagon i with mirror lines through both vertices and edges, and g for cyclic symmetry. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular pentadecagons. Only the g15 subgroup has no degrees of freedom but can be seen as directed edges.
Pentadecagrams.
There are three regular star polygons: {15/2}, {15/4}, {15/7}, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, fourth, or seventh vertex respectively.
There are also three regular star figures: {15/3}, {15/5}, {15/6}, the first being a compound of three pentagons, the second a compound of five equilateral triangles, and the third a compound of three pentagrams.
The compound figure {15/3} can be loosely seen as the two-dimensional equivalent of the 3D compound of five tetrahedra.
Isogonal pentadecagons.
Deeper truncations of the regular pentadecagon and pentadecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths.
Petrie polygons.
The regular pentadecagon is the Petrie polygon for some higher-dimensional polytopes, projected in a skew orthogonal projection:
Uses.
<br>A regular triangle, decagon, and pentadecagon can completely fill a plane vertex. However, due to the triangle's odd number of sides, the figures cannot alternate around the triangle, so the vertex cannot produce a semiregular tiling.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\n \\begin{align} A = \\frac{15}{4}a^2 \\cot \\frac{\\pi}{15} & = \\frac{15}{4}\\sqrt{7+2\\sqrt{5}+2\\sqrt{15+6\\sqrt{5}}}a^2 \\\\\n & = \\frac{15a^2}{8} \\left( \\sqrt{3}+\\sqrt{15}+\n \\sqrt{2}\\sqrt{5+\\sqrt{5}} \n \\right) \\\\\n & \\simeq 17.6424\\,a^2.\n \\end{align}"
},
{
"math_id": 1,
"text": " \\overline{FG} = \\overline{CF}\\text{,} \\; \\overline{AH} = \\overline{GM}\\text{,} \\; |E_1E_6|"
},
{
"math_id": 2,
"text": "|E_2E_5|"
},
{
"math_id": 3,
"text": "H"
},
{
"math_id": 4,
"text": "\\overline{AM}"
},
{
"math_id": 5,
"text": "\\frac{\\overline{AH}}{\\overline{HM}} = \\frac{\\overline{AM}}{\\overline{AH}} = \\frac{1+ \\sqrt{5}}{2} = \\Phi \\approx 1.618 \\text{.}"
},
{
"math_id": 6,
"text": " \\overline{CG}"
},
{
"math_id": 7,
"text": "\\overline{MG}"
},
{
"math_id": 8,
"text": "\\overline{AH}"
},
{
"math_id": 9,
"text": "\\overline{FE_2}\\text{,}"
},
{
"math_id": 10,
"text": "\\frac{\\overline{E_1 E_2}}{\\overline{E_1 F}} = \\frac{\\overline{E_2 F}}{\\overline{E_1 E_2}} = \\frac{1+ \\sqrt{5}}{2} = \\Phi \\approx 1.618 \\text{.}"
},
{
"math_id": 11,
"text": "\\overline{E_2 M} = R\\;;\\;\\;"
},
{
"math_id": 12,
"text": "\\overline{E_1 E_2} = a\\;;\\;\\;"
},
{
"math_id": 13,
"text": " D E_1M = ME_2D = 78^\\circ"
},
{
"math_id": 14,
"text": "\\begin{align}\nR &= a \\cdot \\frac{1}{2} \\cdot \\left(\\sqrt{5 + 2 \\cdot \\sqrt{5}} + \\sqrt{3} \\right)= \\frac{1}{2} \\cdot \\sqrt{8+ 2 \\cdot \\sqrt{5}+2\\sqrt{15 + 6 \\cdot \\sqrt{5}}}\\cdot a\\\\\n &= \\frac {\\sin (78^\\circ)}{ \\sin (24^\\circ)} \\cdot a \\approx 2.40486\\cdot a\n\\end{align}"
},
{
"math_id": 15,
"text": "\\overline{CG} = R "
}
] |
https://en.wikipedia.org/wiki?curid=1496229
|
149626
|
Wilhelm Wien
|
German physicist (1864–1928)
Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody at any temperature from the emission at any one reference temperature.
He also formulated an expression for the black-body radiation, which is correct in the photon-gas limit. His arguments were based on the notion of adiabatic invariance, and were instrumental for the formulation of quantum mechanics. Wien received the 1911 Nobel Prize for his work on heat radiation.
He was a cousin of Max Wien, inventor of the Wien bridge.
Biography.
Early years.
Wien was born at Gaffken (now in Baltiysky District) near Fischhausen in the Province of Prussia as the son of landowner Carl Wien. In 1866, his family moved to Drachenstein near Rastenburg (now Kętrzyn, Poland).
In 1879, Wien went to school in Rastenburg and from 1880 to 1882 he attended the city school of Heidelberg. In 1882 he attended the University of Göttingen and the University of Berlin. From 1883 to 1885, he worked in the laboratory of Hermann von Helmholtz and, in 1886, he received his Ph.D. with a thesis on the diffraction of light upon metals and on the influence of various materials upon the color of refracted light. From 1896 to 1899, Wien lectured at RWTH Aachen University. He became twice successor of Wilhelm Conrad Röntgen, in 1900 at the University of Würzburg and in 1919 at the University of Munich. Wien was very active in science politics representing conservative and nationalistic positions though being not as extreme as sharing the attitude of those going to develop the "Deutsche Physik". He appreciated both Albert Einstein and relativity.
Career.
In 1896 Wien empirically determined a distribution law of blackbody radiation, later named after him: Wien's law. Max Planck, who was a colleague of Wien's, did not believe in empirical laws, so using electromagnetism and thermodynamics, he proposed a theoretical basis for Wien's law, which became the Wien–Planck law. However, Wien's law was only valid at high frequencies, and underestimated the radiancy at low frequencies. Planck corrected the theory and proposed what is now called Planck's law, which led to the development of quantum theory. However, Wien's other empirical formulation formula_0, called Wien's displacement law, is still very useful, as it relates the peak wavelength emitted by a body ("λ"max), to the temperature of the body (T). In 1900 (following the work of George Frederick Charles Searle), he assumed that the entire mass of matter is of electromagnetic origin and proposed the formula formula_1 for the relation between electromagnetic mass and electromagnetic energy.
Wien developed the Wien filter (also known as velocity selector) in 1898 for the study of anode rays. It is a device consisting of perpendicular electric and magnetic fields that can be used as a velocity filter for charged particles, for example in electron microscopes and spectrometers. It is used in accelerator mass spectrometry to select particles based on their speed. The device is composed of orthogonal electric and magnetic fields, such that particles with the correct speed will be unaffected while other particles will be deflected. It can be configured as a charged particle energy analyzer, monochromator, or mass spectrometer.
While studying streams of ionized gas, Wien, in 1898, identified a positive particle equal in mass to the hydrogen atom. Wien, with this work, laid the foundation of mass spectrometry. J. J. Thomson refined Wien's apparatus and conducted further experiments in 1913 then, after work by Ernest Rutherford in 1919, Wien's particle was accepted and named the proton.
In 1911, Wien was awarded the Nobel Prize in Physics "for his discoveries regarding the laws governing the radiation of heat". He delivered the Ernest Kempton Adams Lecture at Columbia University in 1913.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\lambda_{\\mathrm{max}} T = \\mathrm{constant}"
},
{
"math_id": 1,
"text": "m=(4/3)E/c^2"
}
] |
https://en.wikipedia.org/wiki?curid=149626
|
1496379
|
Abelian von Neumann algebra
|
In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
The prototypical example of an abelian von Neumann algebra is the algebra "L"∞("X", μ) for μ a σ-finite measure on "X" realized as an algebra of operators on the Hilbert space "L"2("X", μ) as follows: Each "f" ∈ "L"∞("X", μ) is identified with the multiplication operator
formula_0
Of particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants.
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces ("X", μ) is a standard measure space (that is "X" − "N" is a standard Borel space for some null set "N" and μ is a σ-finite measure) then "L"2("X", μ) is separable.
Classification.
The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to "L"∞("X") for some standard measure space ("X", μ) and conversely, for every standard measure space "X", "L"∞("X") is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism.
In fact we can state this more precisely as follows:
Theorem. Any abelian von Neumann algebra of operators on a separable Hilbert space is *-isomorphic to exactly one of the following
The isomorphism can be chosen to preserve the weak operator topology.
In the above list, the interval [0,1] has Lebesgue measure and the sets {1, 2, ..., "n"} and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of Maharam's classification theorem for separable measure algebras. The version of Maharam's classification theorem that is most useful involves a point realization of the equivalence, and is somewhat of a folk theorem.
Although every standard measure space is isomorphic to one of the above and the list is exhaustive in this sense, there is a more canonical choice for the measure space in the case of abelian von Neumann algebras "A": The set of all projectors is a formula_6-complete Boolean algebra, that is a pointfree formula_6-algebra. In the special case formula_7 one recovers the abstract formula_6-algebra formula_8. This pointfree approach can be turned into a duality theorem analogue to Gelfand-duality between the category of abelian von Neumann algebras and the category of abstract formula_6-algebras.
Let μ and ν be non-atomic probability measures on standard Borel spaces "X" and "Y" respectively. Then there is a μ null subset "N" of "X", a ν null subset "M" of "Y" and a Borel isomorphism
formula_9
which carries μ into ν.
Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work.
In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras "L"∞("X", μ), the following topologies agree on norm bounded sets:
However, for an abelian von Neumann algebra "A" the realization of "A" as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of "A" is given by spectral multiplicity theory and requires the use of direct integrals.
Spatial isomorphism.
Using direct integral theory, it can be shown that the abelian von Neumann algebras of the form "L"∞("X", μ) acting as operators on "L"2("X", μ) are all maximal abelian. This means that they cannot be extended to properly larger abelian algebras. They are also referred to as "Maximal abelian self-adjoint algebras" (or M.A.S.A.s). Another phrase used to describe them is abelian von Neumann algebras of "uniform multiplicity 1"; this description makes sense only in relation to multiplicity theory described below.
Von Neumann algebras "A" on "H", "B" on "K" are "spatially isomorphic" (or "unitarily isomorphic") if and only if there is a unitary operator "U": "H" → "K" such that
formula_10
In particular spatially isomorphic von Neumann algebras are algebraically isomorphic.
To describe the most general abelian von Neumann algebra on a separable Hilbert space "H" up to spatial isomorphism, we need to refer the direct integral decomposition of "H". The details of this decomposition are discussed in decomposition of abelian von Neumann algebras. In particular:
Theorem Any abelian von Neumann algebra on a separable Hilbert space "H" is spatially isomorphic to "L"∞("X", μ) acting on
formula_11
for some measurable family of Hilbert spaces {"H""x"}"x" ∈ "X".
Note that for abelian von Neumann algebras acting on such direct integral spaces, the equivalence of the weak operator topology, the ultraweak topology and the weak* topology on norm bounded sets still hold.
Point and spatial realization of automorphisms.
Many problems in ergodic theory reduce to problems about automorphisms of abelian von Neumann algebras. In that regard, the following results are useful:
Theorem. Suppose μ, ν are standard measures on "X", "Y" respectively. Then any involutive isomorphism
formula_12
which is weak*-bicontinuous corresponds to a point transformation in the following sense: There are Borel null subsets "M" of "X" and "N" of "Y" and a Borel isomorphism
formula_13
such that
formula_14
Note that in general we cannot expect η to carry μ into ν.
The next result concerns unitary transformations which induce a weak*-bicontinuous isomorphism between abelian von Neumann algebras.
Theorem. Suppose μ, ν are standard measures on "X", "Y" and
formula_15
for measurable families of Hilbert spaces {"H""x"}"x" ∈ "X", {"K""y"}"y" ∈ "Y". If "U" : "H" → "K" is a unitary such that
formula_16
then there is an almost everywhere defined Borel point transformation η : "X" → "Y" as in the previous theorem and a measurable family {"Ux"}"x" ∈ "X" of unitary operators
formula_17
such that
formula_18
where the expression in square root sign is the Radon–Nikodym derivative of μ η−1 with respect to ν. The statement follows combining the theorem on point realization of automorphisms stated above with the theorem characterizing the algebra of diagonalizable operators stated in the article on direct integrals.
|
[
{
"math_id": 0,
"text": " \\psi \\mapsto f \\psi. "
},
{
"math_id": 1,
"text": "\\ell^\\infty(\\{1,2, \\ldots, n\\}), \\quad n \\geq 1 "
},
{
"math_id": 2,
"text": "\\ell^\\infty(\\mathbf{N}) "
},
{
"math_id": 3,
"text": "L^\\infty([0,1]) "
},
{
"math_id": 4,
"text": "L^\\infty([0,1] \\cup \\{1,2, \\ldots, n\\}), \\quad n \\geq 1 "
},
{
"math_id": 5,
"text": "L^\\infty([0,1] \\cup \\mathbf{N}). "
},
{
"math_id": 6,
"text": "\\sigma"
},
{
"math_id": 7,
"text": "A=L^\\infty(X,\\mathfrak{A},\\mu)"
},
{
"math_id": 8,
"text": "\\mathfrak{A}/\\{A \\mid \\mu(A)=0\\}"
},
{
"math_id": 9,
"text": " \\phi: X \\setminus N \\rightarrow Y \\setminus M, \\quad "
},
{
"math_id": 10,
"text": " U A U^* = B."
},
{
"math_id": 11,
"text": " \\int_X^\\oplus H(x) \\, d \\mu(x) "
},
{
"math_id": 12,
"text": " \\Phi: L^\\infty(X, \\mu) \\rightarrow L^\\infty(Y, \\nu) "
},
{
"math_id": 13,
"text": " \\eta: X \\setminus M \\rightarrow Y \\setminus N "
},
{
"math_id": 14,
"text": " \\Phi (f) = f \\circ \\eta^{-1}. "
},
{
"math_id": 15,
"text": " H = \\int_X^\\oplus H_x d \\mu(x), \\quad K = \\int_Y^\\oplus K_y d \\nu(y) "
},
{
"math_id": 16,
"text": " U \\, L^\\infty(X, \\mu) \\, U^* = L^\\infty(Y, \\nu) "
},
{
"math_id": 17,
"text": " U_x: H_x \\rightarrow K_{\\eta(x)} "
},
{
"math_id": 18,
"text": " U \\bigg(\\int_X^\\oplus \\psi_x d \\mu(x) \\bigg)= \\int_Y^\\oplus \\sqrt{ \\frac{d (\\mu \\circ \\eta^{-1})}{d \\nu}(y)} \\ U_{\\eta^{-1}(y)} \\bigg(\\psi_{\\eta^{-1}(y)}\\bigg) d \\nu(y),"
}
] |
https://en.wikipedia.org/wiki?curid=1496379
|
1496597
|
Friction stir welding
|
Using a spinning tool to mix metal workpieces together at the joint, without melting them
Friction stir welding (FSW) is a solid-state joining process that uses a non-consumable tool to join two facing workpieces without melting the workpiece material. Heat is generated by friction between the rotating tool and the workpiece material, which leads to a softened region near the FSW tool. While the tool is traversed along the joint line, it mechanically intermixes the two pieces of metal, and forges the hot and softened metal by the mechanical pressure, which is applied by the tool, much like joining clay, or dough. It is primarily used on wrought or extruded aluminium and particularly for structures which need very high weld strength. FSW is capable of joining aluminium alloys, copper alloys, titanium alloys, mild steel, stainless steel and magnesium alloys. More recently, it was successfully used in welding of polymers. In addition, joining of dissimilar metals, such as aluminium to magnesium alloys, has been recently achieved by FSW. Application of FSW can be found in modern shipbuilding, trains, and aerospace applications.
The concept was patented in the Soviet Union by Yu. Klimenko in 1967, but it wasn't developed into a commercial technology at that time. It was experimentally proven and commercialized at The Welding Institute (TWI) in the UK in 1991. TWI held patents on the process, the first being the most descriptive.
Principle of operation.
Friction stir welding is performed with a rotating cylindrical tool which has a profiled pin (also known as a probe) having a diameter smaller than the diameter of its shoulder. During welding the tool is fed into a butt joint between two clamped workpieces, until the probe pierces into the workpiece and its shoulder touches the surface of the workpieces. The probe is slightly shorter than the weld depth required, with the tool shoulder riding atop the work surface. After a short dwell time, the tool is moved forward along the joint line at the pre-set welding speed.
Frictional heat is generated between the wear-resistant tool and the work pieces. This heat, along with that generated by the mechanical mixing process and the adiabatic heat within the material, cause the stirred materials to soften without melting. As the tool is moved forward, a special profile on the probe forces plasticised material from the leading face to the rear, where the high forces assist in a forged consolidation of the weld.
This process of the tool traversing along the weld line in a plasticised tubular shaft of metal results in severe solid-state deformation involving dynamic recrystallization of the base material.
Micro-structural features.
The solid-state nature of the FSW process, combined with its unusual tool shape and asymmetric speed profile, results in a highly characteristic micro-structure. The micro-structure can be broken up into the following zones:
Advantages and limitations.
The solid-state nature of FSW leads to several advantages over fusion welding methods, as problems associated with cooling from the liquid phase are avoided. Issues such as porosity, solute redistribution, solidification cracking and liquation cracking do not arise during FSW. In general, FSW has been found to produce a low concentration of defects and is very tolerant to variations in parameters and materials.
Nevertheless, FSW is associated with a number of unique defects if it isn't done properly. Insufficient weld temperatures, due to low rotational speeds or high traverse speeds, for example, mean that the weld material is unable to accommodate the extensive deformation during welding. This may result in long, tunnel-like defects running along the weld, which may occur on the surface or subsurface. Low temperatures may also limit the forging action of the tool and so reduce the continuity of the bond between the material from each side of the weld. The light contact between the material has given rise to the name "kissing bond". This defect is particularly worrying, since it is very difficult to detect using nondestructive methods such as X-ray or ultrasonic testing. If the pin is not long enough or the tool rises out of the plate, then the interface at the bottom of the weld may not be disrupted and forged by the tool, resulting in a lack-of-penetration defect. This is essentially a notch in the material, which can be a potential source of fatigue cracks.
A number of potential advantages of FSW over conventional fusion-welding processes have been identified:
However, some disadvantages of the process have been identified:
Important welding parameters.
Tool design.
The design of the tool is a critical factor, as a good tool can improve both the quality of the weld and the maximal possible welding speed.
It is desirable that the tool material be sufficiently strong, tough, and hard wearing at the welding temperature. Further, it should have a good oxidation resistance and a low thermal conductivity to minimise heat loss and thermal damage to the machinery further up the drive train. Hot-worked tool steel such as AISI H13 has proven perfectly acceptable for welding aluminium alloys within thickness ranges of 0.5–50 mm but more advanced tool materials are necessary for more demanding applications such as highly abrasive metal matrix composites or higher-melting-point materials such as steel or titanium.
Improvements in tool design have been shown to cause substantial improvements in productivity and quality. TWI has developed tools specifically designed to increase the penetration depth and thus increasing the plate thicknesses that can be successfully welded. An example is the "whorl" design that uses a tapered pin with re-entrant features or a variable-pitch thread to improve the downwards flow of material. Additional designs include the Triflute and Trivex series. The Triflute design has a complex system of three tapering, threaded re-entrant flutes that appear to increase material movement around the tool. The Trivex tools use a simpler, non-cylindrical, pin and have been found to reduce the forces acting on the tool during welding.
The majority of tools have a concave shoulder profile, which acts as an escape volume for the material displaced by the pin, prevents material from extruding out of the sides of the shoulder and maintains downwards pressure and hence good forging of the material behind the tool. The Triflute tool uses an alternative system with a series of concentric grooves machined into the surface, which are intended to produce additional movement of material in the upper layers of the weld.
Widespread commercial applications of friction stir welding process for steels and other hard alloys such as titanium alloys will require the development of cost-effective and durable tools. Material selection, design and cost are important considerations in the search for commercially useful tools for the welding of hard materials. Work is continuing to better understand the effects of tool material's composition, structure, properties and geometry on their performance, durability and cost.
Tool rotation and traverse speeds.
There are two tool speeds to be considered in friction-stir welding; how fast the tool rotates and how quickly it traverses along the interface. These two parameters have considerable importance and must be chosen with care to ensure a successful and efficient welding cycle. The relationship between the rotation speed, the welding speed and the heat input during welding is complex, but in general, it can be said that increasing the rotation speed or decreasing the traverse speed will result in a hotter weld. In order to produce a successful weld, it is necessary that the material surrounding the tool is hot enough to enable the extensive plastic flow required and minimize the forces acting on the tool. If the material is too cold, then voids or other flaws may be present in the stir zone and in extreme cases the tool may break.
Excessively high heat input, on the other hand, may be detrimental to the final properties of the weld. Theoretically, this could even result in defects due to the liquation of low-melting-point phases (similar to liquation cracking in fusion welds). These competing demands lead onto the concept of a "processing window": the range of processing parameters viz. tool rotation and traverse speed, that will produce a good quality weld. Within this window the resulting weld will have a sufficiently high heat input to ensure adequate material plasticity but not so high that the weld properties are excessively deteriorated.
Tool tilt and plunge depth.
The plunge depth is defined as the depth of the lowest point of the shoulder below the surface of the welded plate and has been found to be a critical parameter for ensuring weld quality. Plunging the shoulder below the plate surface increases the pressure below the tool and helps ensure adequate forging of the material at the rear of the tool. Tilting the tool by 2–4 degrees, such that the rear of the tool is lower than the front, has been found to assist this forging process. The plunge depth needs to be correctly set, both to ensure the necessary downward pressure is achieved and to ensure that the tool fully penetrates the weld. Given the high loads required, the welding machine may deflect and so reduce the plunge depth compared to the nominal setting, which may result in flaws in the weld. On the other hand, an excessive plunge depth may result in the pin rubbing on the backing plate surface or a significant undermatch of the weld thickness compared to the base material. Variable-load welders have been developed to automatically compensate for changes in the tool displacement, while TWI have demonstrated a roller system that maintains the tool position above the weld plate.
Welding forces.
During welding, a number of forces will act on the tool:
In order to prevent tool fracture and to minimize excessive wear and tear on the tool and associated machinery, the welding cycle is modified so that the forces acting on the tool are as low as possible, and abrupt changes are avoided. In order to find the best combination of welding parameters, it is likely that a compromise must be reached, since the conditions that favour low forces (e.g. high heat input, low travel speeds) may be undesirable from the point of view of productivity and weld properties.
Flow of material.
Early work on the mode of material flow around the tool used inserts of a different alloy, which had a different contrast to the normal material when viewed through a microscope, in an effort to determine where material was moved as the tool passed.
The data was interpreted as representing a form of in-situ extrusion, where the tool, backing plate and cold base material form the "extrusion chamber", through which the hot, plasticised material is forced. In this model the rotation of the tool draws little or no material around the front of the probe; instead, the material parts in front of the pin and passes down either side. After the material has passed the probe, the side pressure exerted by the "die" forces the material back together, and consolidation of the joint occurs, as the rear of the tool shoulder passes overhead and the large down force forges the material.
More recently, an alternative theory has been advanced that advocates considerable material movement in certain locations. This theory holds that some material does rotate around the probe, for at least one rotation, and it is this material movement that produces the "onion-ring" structure in the stir zone. The researchers used a combination of thin copper strip inserts and a "frozen pin" technique, where the tool is rapidly stopped in place. They suggested that material motion occurs by two processes:
The primary advantage of this explanation is that it provides a plausible explanation for the production of the onion-ring structure.
The marker technique for friction stir welding provides data on the initial and final positions of the marker in the welded material. The flow of material is then reconstructed from these positions. Detailed material flow field during friction stir welding can also be calculated from theoretical considerations based on fundamental scientific principles. Material flow calculations are routinely used in numerous engineering applications. Calculation of material flow fields in friction stir welding can be undertaken both using comprehensive numerical simulations or simple but insightful analytical equations. The comprehensive models for the calculation of material flow fields also provide important information such as geometry of the stir zone and the torque on the tool. The numerical simulations have shown the ability to correctly predict the results from marker experiments and the stir zone geometry observed in friction stir welding experiments.
Generation and flow of heat.
For any welding process, it is, in general, desirable to increase the travel speed and minimise the heat input, as this will increase productivity and possibly reduce the impact of welding on the mechanical properties of the weld. At the same time, it is necessary to ensure that the temperature around the tool is sufficiently high to permit adequate material flow and prevent flaws or tool damage.
When the traverse speed is increased, for a given heat input, there is less time for heat to conduct ahead of the tool, and the thermal gradients are larger. At some point the speed will be so high that the material ahead of the tool will be too cold, and the flow stress too high, to permit adequate material movement, resulting in flaws or tool fracture. If the "hot zone" is too large, then there is scope to increase the traverse speed and hence productivity.
The welding cycle can be split into several stages, during which the heat flow and thermal profile will be different:
Heat generation during friction-stir welding arises from two main sources: friction at the surface of the tool and the deformation of the material around the tool. The heat generation is often assumed to occur predominantly under the shoulder, due to its greater surface area, and to be equal to the power required to overcome the contact forces between the tool and the workpiece. The contact condition under the shoulder can be described by sliding friction, using a friction coefficient μ and interfacial pressure "P", or sticking friction, based on the interfacial shear strength at an appropriate temperature and strain rate. Mathematical approximations for the total heat generated by the tool shoulder "Q"total have been developed using both sliding and sticking friction models:
formula_0 (sliding)
formula_1 (sticking)
where ω is the angular velocity of the tool, "R"shoulder is the radius of the tool shoulder, and "R"pin is that of the pin. Several other equations have been proposed to account for factors such as the pin, but the general approach remains the same.
A major difficulty in applying these equations is determining suitable values for the friction coefficient or the interfacial shear stress. The conditions under the tool are both extreme and very difficult to measure. To date, these parameters have been used as "fitting parameters", where the model works back from measured thermal data to obtain a reasonable simulated thermal field. While this approach is useful for creating process models to predict, for example, residual stresses, it is less useful for providing insights into the process itself.
Applications.
The FSW process has initially been patented by TWI in most industrialised countries and licensed for over 183 users. Friction stir welding and its variants – friction stir spot welding and friction stir processing – are used for the following industrial applications: shipbuilding and offshore,
aerospace, automotive, rolling stock for railways, general fabrication, robotics, and computers.
Shipbuilding and offshore.
Two Scandinavian aluminium extrusion companies were the first to apply FSW commercially to the manufacture of fish freezer panels at Sapa in 1996, as well as deck panels and helicopter landing platforms at Marine Aluminium Aanensen. Marine Aluminium Aanensen subsequently merged with Hydro Aluminium Maritime to become Hydro Marine Aluminium. Some of these freezer panels are now produced by Riftec and Bayards. In 1997 two-dimensional friction stir welds in the hydrodynamically flared bow section of the hull of the ocean viewer vessel "The Boss" were produced at Research Foundation Institute with the first portable FSW machine. The "Super Liner Ogasawara" at Mitsui Engineering and Shipbuilding is the largest friction stir welded ship so far. The "Sea Fighter" of Nichols Bros and the "Freedom"-class Littoral Combat Ships contain prefabricated panels by the FSW fabricators Advanced Technology and Friction Stir Link, Inc. respectively. The "Houbei"-class missile boat has friction stir welded rocket launch containers of China Friction Stir Centre. HMNZS "Rotoiti" in New Zealand has FSW panels made by Donovans in a converted milling machine. Various companies apply FSW to armor plating for amphibious assault ships.
Aerospace.
United Launch Alliance applies FSW to the Delta II, Delta IV, Atlas V, and the new Vulcan expendable launch vehicles along with their Cryogenic Upper Stages, and the first of these with a friction stir welded interstage module was launched in 1999. The process was also used for the Space Shuttle external tank, for Ares I until the project was canceled in 2012, the SLS Core which replaced the Ares, and for the Orion Crew Vehicle test article and the current model of the Orion at NASA, as well as Falcon 1 and Falcon 9 rockets at SpaceX. The toe nails for ramp of Boeing C-17 Globemaster III cargo aircraft by Advanced Joining Technologies and the cargo barrier beams for the Boeing 747 Large Cargo Freighter were the first commercially produced aircraft parts. FAA-approved wings and fuselage panels of the Eclipse 500 aircraft were made at Eclipse Aviation, and this company delivered 259 friction stir welded business jets, before they were forced into Chapter 7 liquidation. Floor panels for Airbus A400M military aircraft are now made by Pfalz Flugzeugwerke and Embraer used FSW for the Legacy 450 and 500 Jets Friction stir welding also is employed for fuselage panels on the Airbus A380. BRÖTJE-Automation uses friction stir welding for gantry production machines developed for the aerospace sector, as well as other industrial applications.
Automotive.
Aluminium engine cradles and suspension struts for stretched Lincoln Town Cars were the first automotive parts that were friction stir welded at Tower Automotive, who use the process also for the engine tunnel of the Ford GT. A spin-off of this company is called Friction Stir Link, Inc. and successfully exploits the FSW process, e.g. for the flatbed trailer "Revolution" of Fontaine Trailers. In Japan FSW is applied to suspension struts at Showa Denko and for joining of aluminium sheets to galvanized steel brackets for the boot (trunk) lid of the Mazda MX-5. Friction stir spot welding is successfully used for the bonnet (hood) and rear doors of the Mazda RX-8 and the boot lid of the Toyota Prius. Wheels are friction stir welded at Simmons Wheels, UT Alloy Works and Fundo. Rear seats for the Volvo V70 are friction stir welded at Sapa, HVAC pistons at Halla Climate Control and exhaust gas recirculation coolers at Pierburg. Tailor welded blanks are friction stir welded for the Audi R8 at Riftec. The B-column of the Audi R8 Spider is friction stir welded from two extrusions at Hammerer Aluminium Industries in Austria. The front subframe of the 2013 Honda Accord was friction stir welded to join aluminum and steel halves.
Railways.
Since 1997 roof panels were made from aluminium extrusions at Hydro Marine Aluminium with a bespoke 25 m long FSW machine, e.g. for DSB class SA-SD trains of Alstom LHB. Curved side and roof panels for the Victoria line trains of London Underground, side panels for Bombardier Electrostar trains at Sapa Group and side panels for Alstom's British Rail Class 390 Pendolino trains are made at Sapa Group. Japanese commuter and express A-trains, and British Rail Class 395 trains are friction stir welded by Hitachi, while Kawasaki applies friction stir spot welding to roof panels and Sumitomo Light Metal produces Shinkansen floor panels. Innovative FSW floor panels are made by Hammerer Aluminium Industries in Austria for the Stadler Kiss double decker rail cars, to obtain an internal height of 2 m on both floors and for the new car bodies of the Wuppertal Suspension Railway.
Heat sinks for cooling high-power electronics of locomotives are made at Sykatek, EBG, Austerlitz Electronics, EuroComposite, Sapa and , and are the most common application of FSW due to the excellent heat transfer.
Fabrication.
Façade panels and cathode sheets are friction stir welded at AMAG and Hammerer Aluminium Industries, including friction stir lap welds of copper to aluminium. Bizerba meat slicers, Ökolüfter HVAC units and Siemens X-ray vacuum vessels are friction stir welded at Riftec. Vacuum valves and vessels are made by FSW at Japanese and Swiss companies. FSW is also used for the encapsulation of nuclear waste at SKB in 50-mm-thick copper canisters. Pressure vessels from ø1 m semispherical forgings of 38.1 mm thick aluminium alloy 2219 at Advanced Joining Technologies and Lawrence Livermore Nat Lab. Friction stir processing is applied to ship propellers at Friction Stir Link, Inc. and to hunting knives by DiamondBlade. Bosch uses it in Worcester for the production of heat exchangers.
Robotics.
KUKA Robot Group has adapted its KR500-3MT heavy-duty robot for friction stir welding via the DeltaN FS tool. The system made its first public appearance at the EuroBLECH show in November 2012.
Personal computers.
Apple applied friction stir welding on the 2012 iMac to effectively join the bottom to the back of the device.
Joining of aluminum 3D printing material.
FSW is proven able to be used as one of the methods to join the metal 3D printing materials. By using proper FSW tools and correct parameter setting a sound and defect-free weld can be produced in order to joint the metal 3D printing materials. Besides, the FSW tools must be harder than the materials that need to weld. The most important parameters in FSW are the rotation of probe, traverse speed, spindle tilt angle and target depth. The weld joint efficiency of FSW on the 3D printing metal can reach up to 83.3% compared to its base materials strength.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Q_\\text{total} = \\frac{2}{3} \\pi P \\mu \\omega \\left(R_\\text{shoulder}^3 - R_\\text{pin}^3\\right)"
},
{
"math_id": 1,
"text": "Q_\\text{total} = \\frac{2}{3} \\pi \\tau \\omega \\left(R_\\text{shoulder}^3 - R_\\text{pin}^3\\right)"
}
] |
https://en.wikipedia.org/wiki?curid=1496597
|
1496726
|
Maximal compact subgroup
|
Concept in topology
In mathematics, a maximal compact subgroup "K" of a topological group "G" is a subgroup "K" that is a compact space, in the subspace topology, and maximal amongst such subgroups.
Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are "not" in general unique, but are unique up to conjugation – they are essentially unique.
Example.
An example would be the subgroup O(2), the orthogonal group, inside the general linear group GL(2, R). A related example is the circle group SO(2) inside SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as any inner product has an associated orthogonal group, and the essential uniqueness corresponds to the essential uniqueness of the inner product.
Definition.
A maximal compact subgroup is a maximal subgroup amongst compact subgroups – a "maximal (compact subgroup)" – rather than being (alternate possible reading) a maximal subgroup that happens to be compact; which would probably be called a "compact (maximal subgroup)", but in any case is not the intended meaning (and in fact maximal proper subgroups are not in general compact).
Existence and uniqueness.
The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected locally compact group) admits maximal compact subgroups and that they are all conjugate to one another. For a semisimple Lie group uniqueness is a consequence of the Cartan fixed point theorem, which asserts that if a compact group acts by isometries on a complete simply connected negatively curved Riemannian manifold then it has a fixed point.
Maximal compact subgroups of connected Lie groups are usually "not" unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups "K" and "L", there is an element "g" ∈ "G" such that "gKg"−1 = "L". Hence a maximal compact subgroup is essentially unique, and people often speak of "the" maximal compact subgroup.
For the example of the general linear group GL("n", R), this corresponds to the fact that "any" inner product on R"n" defines a (compact) orthogonal group (its isometry group) – and that it admits an orthonormal basis: the change of basis defines the conjugating element conjugating the isometry group to the classical orthogonal group O("n", R).
Proofs.
For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in and . and discuss the extension to connected Lie groups and connected locally compact groups.
For semisimple groups, existence is a consequence of the existence of a compact real form of the noncompact semisimple Lie group and the corresponding Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding Riemannian symmetric space "G"/"K" has negative curvature and
Cartan's fixed point theorem. showed that the derivative of the exponential map at any point of "G"/"K" satisfies |d exp "X"| ≥ |X|. This implies that "G"/"K" is a Hadamard space, i.e. a complete metric space satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the Bruhat-Tits fixed point theorem. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its circumcenter. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.
Proof of uniqueness for semisimple groups.
also related the general problem for semisimple groups to the case of GL("n", R). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in .
Let formula_0 be a real semisimple Lie algebra with Cartan involution σ. Thus the fixed point subgroup of σ is the maximal compact subgroup "K" and there is an eigenspace decomposition
formula_1
where formula_2, the Lie algebra of "K", is the +1 eigenspace. The Cartan decomposition gives
formula_3
If "B" is the Killing form on formula_0 given by "B"("X","Y") = Tr (ad X)(ad Y), then
formula_4
is a real inner product on formula_0. Under the adjoint representation, "K" is the subgroup of "G" that preserves this inner product.
If "H" is another compact subgroup of "G", then averaging the inner product over "H" with respect to the Haar measure gives an inner product invariant under "H". The operators Ad "p" with "p" in "P" are positive symmetric operators. This new inner produst can be written as
formula_5
where "S" is a positive symmetric operator on formula_0 such that
Ad("h")"t""S" Ad "h" = "S" for "h" in "H" (with the transposes computed with respect to the inner product). Moreover, for "x" in "G",
formula_6
So for "h" in "H",
formula_7
For "X" in formula_8 define
formula_9
If "e""i" is an orthonormal basis of eigenvectors for "S" with "Se""i" = λ"i" "e""i", then
formula_10
so that "f" is strictly positive and tends to ∞ as |"X"| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad "X" and each non-zero eigenvalue occurs with its negative, since i ad "X" is a "skew-adjoint operator" on the compact real form formula_11.
So "f" has a global minimum at "Y" say. This minimum is unique, because if "Z" were another then
formula_12
where "X" in formula_8 is defined by the Cartan decomposition
formula_13
If "f""i" is an orthonormal basis of eigenvectors of ad "X" with corresponding real eigenvalues μ"i", then
formula_14
Since the right hand side is a positive combination of exponentials, the real-valued function "g" is strictly convex if "X" ≠ 0, so has a unique minimum. On the other hand, it has local minima at "t" = 0 and "t" = 1, hence "X" = 0 and "p" = exp "Y" is the unique global minimum. By construction
"f"("x") = "f"(σ("h")"xh"−1) for "h" in "H", so that "p" = σ("h")"ph"−1 for "h" in "H". Hence σ("h")= "php"−1. Consequently, if "g" = exp "Y"/2, "gHg"−1 is fixed by σ and therefore lies in "K".
Applications.
Representation theory.
Maximal compact subgroups play a basic role in the representation theory when "G" is not compact. In that case a maximal compact subgroup "K" is a compact Lie group (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier.
The operations relating the representation theories of "G" and "K" are restricting representations from "G" to "K", and inducing representations from "K" to "G", and these are quite well understood; their theory includes that of spherical functions.
Topology.
The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup "K". To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact "K" and a Euclidean space – "G" = "K" × R"d" – thus in particular "K" is a deformation retract of "G," and is homotopy equivalent, and thus they have the same homotopy groups. Indeed, the inclusion formula_15 and the deformation retraction formula_16 are homotopy equivalences.
For the general linear group, this decomposition is the QR decomposition, and the deformation retraction is the Gram-Schmidt process. For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of "G" as "G" = "KAN" in which "K" occurs in a product with a contractible subgroup "AN".
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathfrak{g}"
},
{
"math_id": 1,
"text": "\\displaystyle{\\mathfrak{g}=\\mathfrak{k}\\oplus \\mathfrak{p},}"
},
{
"math_id": 2,
"text": "\\mathfrak{k}"
},
{
"math_id": 3,
"text": "\\displaystyle{G=K\\cdot \\exp \\mathfrak{p} = K\\cdot P = P\\cdot K.}"
},
{
"math_id": 4,
"text": "\\displaystyle{(X,Y)_\\sigma=-B(X,\\sigma(Y))}"
},
{
"math_id": 5,
"text": "(S\\cdot X,Y)_\\sigma,"
},
{
"math_id": 6,
"text": "\\displaystyle{\\mathrm{Ad}\\, \\sigma(x)=(\\mathrm{Ad}\\,(x)^{-1})^t.}"
},
{
"math_id": 7,
"text": "\\displaystyle{S\\circ \\mathrm{Ad}(\\sigma(h))= \\mathrm{Ad}(h)\\circ S.}"
},
{
"math_id": 8,
"text": "\\mathfrak{p}"
},
{
"math_id": 9,
"text": "\\displaystyle{f(e^X)=\\mathrm{Tr}\\, \\mathrm{Ad}(e^X) S.}"
},
{
"math_id": 10,
"text": "\\displaystyle{f(e^X)=\\sum \\lambda_i (\\mathrm{Ad}(e^X)e_i,e_i)_\\sigma \\ge (\\min \\lambda_i)\\cdot \\mathrm{Tr}\\,e^{\\mathrm{ad}\\,X},}"
},
{
"math_id": 11,
"text": "\\mathfrak{k}\\oplus i\\mathfrak{p}"
},
{
"math_id": 12,
"text": "\\displaystyle{e^Z=e^{Y/2} e^X e^{Y/2},}"
},
{
"math_id": 13,
"text": "\\displaystyle{e^{Z/2}e^{-Y/2}=k\\cdot e^{X/2}.}"
},
{
"math_id": 14,
"text": "\\displaystyle{g(t)= f(e^{Y/2} e^{tX} e^{Y/2})= \\sum e^{\\mu_i t} \\|Ad(e^{Y/2})f_i\\|^2_\\sigma.}"
},
{
"math_id": 15,
"text": "K \\hookrightarrow G"
},
{
"math_id": 16,
"text": "G \\twoheadrightarrow K"
}
] |
https://en.wikipedia.org/wiki?curid=1496726
|
14967282
|
Quantum imaging
|
Quantum imaging is a new sub-field of quantum optics that exploits quantum correlations such as quantum entanglement of the electromagnetic field in order to image objects with a resolution or other imaging criteria that is beyond what is possible in classical optics. Examples of quantum imaging are quantum ghost imaging, quantum lithography, imaging with undetected photons, sub-shot-noise imaging, and quantum sensing. Quantum imaging may someday be useful for storing patterns of data in quantum computers and transmitting large amounts of highly secure encrypted information. Quantum mechanics has shown that light has inherent “uncertainties” in its features, manifested as moment-to-moment fluctuations in its properties. Controlling these fluctuations—which represent a sort of “noise”—can improve detection of faint objects, produce better amplified images, and allow workers to more accurately position laser beams.
Quantum imaging methods.
Quantum imaging can be done in different methods. One method uses scattered light from a free-electron laser. This method converts the light to quasi-monochromatic pseudo-thermal light. Another method known as interaction-free imaging is used to locate an object without absorbing photons. One more method of quantum imaging is known as ghost imaging. This process uses a photon pair to define an image. The image is created by correlations between the two photons, the stronger the correlations the greater the resolution.
Quantum lithography is a type of quantum imaging that focuses on aspects of photons to surpass the limits of classical lithography. Using entangled light, the effective resolution becomes a factor of N lesser than the Rayleigh limit of formula_0. Another study determines that waves created by Raman pulses have narrower peaks and have a width that is four times smaller than the diffraction limit in classical lithography. Quantum lithography has potential applications in communications and computing.
Another type of quantum imaging is called quantum metrology, or quantum sensing. The goal of these processes is to achieve higher levels of accuracy than equivalent measurements from classical optics. They take advantage of quantum properties of individual particles or quantum systems to create units of measurement. By doing this, quantum metrology enhances the limits of accuracy beyond classical attempts.
Photonics.
In photonics and quantum optics, quantum sensors are often built on continuous variable systems, i.e., quantum systems characterized by continuous degrees of freedom such as position and momentum quadratures. The basic working mechanism typically relies on using optical states of light which have squeezing or two-mode entanglement. These states are particularly sensitive to record physical transformations that are finally detected by interferometric measurements.
In Practice.
Absolute Photon Sources.
Many of the procedures for executing quantum metrology require certainty in the measurement of light. An absolute photon source is knowing the origin of the photon which helps determine which measurements relate for the sample being imaged. The best methods for approaching an absolute photon source is through spontaneous parametric down-conversion (SPDC). Coincidence measurements are a key component for reducing noise from the environment by factoring in the amount of the incident photons registered with respect to the photon number. However, this is not a perfected system as error can still exist through inaccurate detection of the photons.
Types of Quantum Metrology.
Quantum Ellipsometry.
Classical ellipsometry is a thin film material characterization methodology used to determine reflectivity, phase shift, and thickness resulting from light shining on a material. Though, it can only be effectively used if the properties are well known for the user to reference and calibrate. Quantum ellipsometry has the distinct advantage of not requiring the properties of the material to be well-defined for calibration. This is because any detected photons will already have a relative phase relation with another detected photon assuring the measured light is from the material being studied.
Quantum Optical Coherence Tomography (QOCT).
Optical coherence tomography uses Michelson interferometry with a distance adjustable mirror. Coherent light passes through a beam splitter where one path hits the mirror then the detector and the other hits a sample then reflects into the detector. The quantum analogue uses the same premise with entangle photons and a Hong–Ou–Mandel interferometer. Coincidence counting of the detected photons permits more recognizable interference leading to less noise and higher resolution.
Real-world applications.
As research in quantum imaging continues, more and more real-world methods arise. Two important ones are ghost imaging and quantum illumination. Ghost imaging takes advantage of two light detectors to create an image of an object that is not directly visible to the naked eye. The first detector is a multi-pixel detector that does not view the subject object while the second, a single-pixel (bucket) detector, views the object. The performance is measured through the resolution and signal-to-noise ratio (SNR). SNRs are important to determine how well an image looks as a result of ghost imaging. On the other hand, resolution and the attention to detail is determined by the number of “specks” in the image. Ghost imaging is important as it allows an image to be produced when a traditional camera is not sufficient.
Quantum Illumination was first introduced by Seth Lloyd and collaborators at MIT in 2008 and takes advantage of quantum states of light. The basic setup is through target detection in which a sender prepares two entangled system, signal and idler. The idler is kept in place while the signal is sent to check out an object with a low-reflective rate and high noise background. A reflection of the object is sent back and then the idler and reflected signal combined to create a joint measurement to tell the sender one of two possibilities: an object is present or an object is absent. A key feature of quantum illumination is entanglement between the idler and reflected signal is lost completely. Therefore, it is heavily reliant on the presence of entanglement in the initial idler-signal system.
Current uses.
Quantum imaging is expected to have a lot of potential to expand. In the future, it could be used to store patterns of data in quantum computers and allow communication through highly encrypted information . Quantum imaging techniques can allow improvement in detection of faint objects, amplified images, and accurate position of lasers. Today, quantum imaging (mostly ghost imaging) is studied and tested in areas of military and medical use. The military aims to use ghost imaging to detect enemies and objects in situations where the naked eye and traditional cameras fail. For example, if an enemy or object is hidden in a cloud of smoke or dust, ghost imaging can help an individual to know where a person is located and if they are an ally or foe. In the medical field, imaging is used to increase the accuracy and lessen the amount of radiation exposed to a patient during x-rays. Ghost imaging could allow doctors to look at a part of the human body without having direct contact with it, therefore, lowering the amount of direct radiation to the patient . Similar to the military, it is used to look at objects that cannot be seen with the human eye such as bones and organs with a light with beneficial properties.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\Delta\nx\n=\n\\frac{\\lambda}{2}"
}
] |
https://en.wikipedia.org/wiki?curid=14967282
|
1496736
|
Heptatonic scale
|
Musical scale with seven pitches
A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include:
Indian classical theory postulates seventy-two seven-tone scale types, collectively called "melakarta" or "thaat", whereas others postulate twelve or ten (depending on the theorist) seven-tone scale types.
Several heptatonic scales in Western, Roman, Spanish, Hungarian, and Greek music can be analyzed as juxtapositions of tetrachords. All heptatonic scales have all intervals present in their interval vector analysis, and thus all heptatonic scales are both hemitonic and tritonic. There is a special affinity for heptatonic scales in the Western key signature system.
Diatonic scale.
A diatonic scale is any seven-note scale constructed sequentially using only whole tones and half tones, repeating at the octave, having a tonal center, and comprising only one tritone interval between any two scale members, which ensures that the half tone intervals are as far apart as possible. In Western music, there are seven such scales, and they are commonly known as the modes of the major scale (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian).
Melodic minor scale.
In traditional classical theory, the melodic minor scale has two forms, as noted above, an ascending form and a descending form. Although each of these forms of itself comprises seven pitches, together they comprise nine, which might seem to call into question the scale's status as a heptatonic scale. In certain twentieth-century music, however, it became common systematically to use the ascending form for both ascending and descending passages. Such a use has been notably ascribed to the works of Béla Bartók and to bop and post-bop jazz practice. The traditional descending form of the melodic minor scale is equivalent to the natural minor scale in both pitch collection (which is diatonic) and tonal center.
Harmonic minor scale.
The harmonic minor scale is so called because in tonal music of the common practice period (from approximately 1600 to approximately 1900) chords or harmonies are derived from it more than from the natural minor scale or the melodic minor scale. The augmented second between its sixth degree and its raised seventh degree (the "leading tone"), traditionally considered undesirable in melodic progression, is avoided by placing these pitches in different voices in adjacent chords, as in this progression: F A♭ D, F G B, F A♭ C (ii°b–V7d–iv in C minor). The A♭ in the middle voice does not ascend to B, and the B in the upper voice does not descend to A♭.
Heptatonia prima and secunda.
The names "heptatonia prima" and "heptatonia secunda" apply to seven-note scales that can be formed using five tones (t) and two semi-tones (s), (also called whole-steps and half-steps), but without two semi-tones in succession. Throughout history and to the present day, some have occurred much more commonly than others, namely Ionian (also called the major scale), Aeolian (also called the natural minor scale), melodic ascending minor, Dorian, Mixolydian, Lydian, Lydian dominant, Aeolian dominant, and altered scales.
Heptatonia prima.
In these scales the semi-tones are maximally separated. They are known most commonly as the diatonic modes. Beginning on keynote C and working up the notes of the 'natural minor' scale (A, B, C, D, E, F, G, A), the seven modes are:
Heptatonia secunda.
While the diatonic modes have two and three tones on either side of each semitone, the heptatonia secunda modes have one and four.
These are sometimes called modes of the melodic ascending minor since that is the most commonly used scale of this type, but other modes can be produced by starting on the different scale notes in turn.
Thus starting on keynote A as above and following the notes of the ascending melodic minor (A, B, C, D, E, F♯, G♯) yields these seven modes:
These modes are more awkward to use than those of the diatonic scales due to the four tones in a row yielding augmented intervals on one hand while the one tone between two semitones gives rise to diminished intervals on the other. For example, the last two modes listed above both have 'Locrian' diminished triads built on their tonics, giving them unstable tonality, while the third mode not only has an augmented fourth a la the Lydian mode but also an augmented fifth making the dominant and subdominant essentially unusable.
Heptatonia tertia.
The last group of seven-note tone/semitone scales is "heptatonia tertia", and consists of scales with two adjacent semitones—which amounts to a whole-tone scale, but with an additional note somewhere in its sequence, e.g., B C D E F♯ G♯ A♯. One such example is the Neapolitan major scale.
Other heptatonic scales.
If the interval of the augmented second is used, many other scales become possible. These include
Gypsy I-♭II-III-IV-V-♭VI-VII
Hungarian I-II-♭III-♯IV-V-♭VI-VII
The scales are symmetrical about the tonic and dominant respectively and the names are sometimes used interchangeably.
The double harmonic scale, also known as the Byzantine or Hungarian, scale, contains the notes C D E♭ F♯ G A♭ B C.
Phrygian dominant or dominant harmonic minor I-♭II-III-IV-V-♭VI-♭VII
This differs from the Phrygian in having a major third. It may also be considered built on the dominant of the harmonic minor scale.
Neapolitan minor differs from the Phrygian in having a major seventh.
Verdi's Scala Enigmatica I-♭II-III-♯IV-♯V-♯VI-VII i.e. G A♭ B C♯ D♯ E♯ F♯, which is similar to the heptatonia tertia mentioned above, differing only in that the second degree here is flattened.
Melakarta.
Melakarta is a South Indian classical method of organizing Raagas based on their unique heptatonic scales. The postulated number of melakarta derives from arithmetical calculation and not from Carnatic practice, which uses far fewer scale forms. Seven-pitch melakarta are considered subsets of a twelve-pitch scale roughly analogous to the Western chromatic scale. The first and fifth melakarta tones, corresponding to the first and eighth chromatic tones, are invariable in inflection, and the fourth melakarta tone, corresponding to the sixth or seventh chromatic tone, is allowed one of two inflections only, a natural () position and a raised () position. The second and third melakarta tones can be picked from the 4 chromatic tones (second through fifth), and similarly for the sixth and seventh. Thus the number of possible forms is equal to twice the square of the number of ways a two-membered subset can be extracted from a four-membered set:
formula_0
Thaat.
Hindustani heptatonic theory additionally stipulates that the second, third, sixth and seventh degrees of heptatonic scale forms ("saptak") are also allowed only two inflections each, in this case, one natural position, and one lowered ("komal") position. Arithmetically this produces 25, or thirty-two, possibilities, but Hindustani theory, in contradistinction to Carnatic theory, excludes scale forms not commonly used.
Chinese Gongche notation.
Gongche notation heptatonic scale gives a do, re, mi, (between fa and fa♯), sol, la, (between ti♭ and ti) heptatonic scale.
|
[
{
"math_id": 0,
"text": "2\\cdot\\left(\\frac{4!}{2!\\cdot2!}\\right)^2 = 2\\cdot6^2 =72"
}
] |
https://en.wikipedia.org/wiki?curid=1496736
|
14968
|
Regular icosahedron
|
Convex polyhedron with 20 triangular faces
In geometry, the regular icosahedron (or simply "icosahedron") is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.
Many polyhedrons are constructed from the regular icosahedron. For example, most of the Kepler–Poinsot polyhedron is constructed by faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron has many relations with other Platonic solids, one of them is the regular dodecahedron as its dual polyhedron and has the historical background on the comparison mensuration. It also has many relations with other polytopes.
The appearance of regular icosahedron can be found in nature, such as the virus with icosahedral-shaped shells and radiolarians. Other applications of the regular icosahedron are the usage of its net in cartography, twenty-sided dice that may have been found in ancient times and role-playing games.
Construction.
The regular icosahedron can be constructed like other gyroelongated bipyramids, started from a pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its faces. These pyramids cover the pentagonal faces, replacing them with five equilateral triangles, such that the resulting polyhedron has 20 equilateral triangles as its faces. This process construction is known as the gyroelongation.
Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them. Because of the constructions above, the regular icosahedron is Platonic solid, a family of polyhedra with regular faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the regular icosahedron.
The regular icosahedron can also be constructed starting from a regular octahedron. All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles. This process is known as snub, and the regular icosahedron is also known as snub octahedron.
One possible system of Cartesian coordinate for the vertices of a regular icosahedron, giving the edge length 2, is:
formula_0
where formula_1 denotes the golden ratio.
Properties.
Mensuration.
The insphere of a convex polyhedron is a sphere inside the polyhedron, touching every face. The circumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The midsphere of a convex polyhedron is a sphere tangent to every edge. Therefore, given that the edge length formula_2 of a regular icosahedron, the radius of insphere (inradius) formula_3, the radius of circumsphere (circumradius) formula_4, and the radius of midsphere (midradius) formula_5 are, respectively:
formula_6
The surface area of polyhedra is the sum of its every face. Therefore, the surface area of regular icosahedra formula_7 equals the area of 20 equilateral triangles. The volume of a regular icosahedron formula_8 is obtained by calculating the volume of all pyramids with the base of triangular faces and the height with the distance from a triangular face's centroid to the center inside the regular icosahedron, the circumradius of a regular icosahedron; alternatively, it can be ascertained by slicing it off into two regular pentagonal pyramids and a pentagonal antiprism, and adding up their volume. The expressions of both are:
formula_9
A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers. As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).
The dihedral angle of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces is approximately 38.2°. The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is 100.8°, and the dihedral angle of a pentagonal pyramid between the same faces is 37.4°. Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is 37.4° + 100.8° = 138.2°.
Symmetry.
The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters. This non-abelian simple group is the only non-trivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general quintic equation is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the Abel–Ruffini theorem uses this simple fact, and Felix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.
The full symmetry group of the icosahedron (including reflections) is known as the full icosahedral group. It is isomorphic to the product of the rotational symmetry group and the group formula_10 of size two, which is generated by the reflection through the center of the icosahedron.
Icosahedral graph.
Every Platonic graph, including the icosahedral graph, is a polyhedral graph. This means that they are planar graphs, graphs that can be drawn in the plane without crossing its edges; and they are 3-vertex-connected, meaning that the removal of any two of its vertices leaves a connected subgraph. According to Steinitz theorem, the icosahedral graph endowed with these heretofore properties represents the skeleton of a regular icosahedron.
The icosahedral graph is Hamiltonian, meaning that it contains a Hamiltonian cycle, or a cycle that visits each vertex exactly once.
Related polyhedra.
In other Platonic solids.
Aside from comparing the mensuration between the regular icosahedron and regular dodecahedron, they are dual to each other. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.
An icosahedron can be inscribed in an octahedron by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections. Because the golden sections are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.
An icosahedron of edge length formula_11 can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges. Because there are five times as many icosahedron edges as cube faces, there are five ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the golden ratio.
Stellation.
The icosahedron has a large number of stellations. stated 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular Kepler–Poinsot polyhedron. Three are regular compound polyhedra.
Facetings.
The small stellated dodecahedron, great dodecahedron, and great icosahedron are three facetings of the regular icosahedron. They share the same vertex arrangement. They all have 30 edges. The regular icosahedron and great dodecahedron share the same edge arrangement but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles).
Diminishment.
The Johnson solids are the polyhedra whose faces are all regular, but not uniform. This means they do not include the Archimedean solids, Catalan solids, prisms, and antiprisms. Some of them are constructed involving the removal of the part of a regular icosahedron, a process known as "diminishment". They are gyroelongated pentagonal pyramid, metabidiminished icosahedron, and tridiminished icosahedron, which remove one, two, and three pentagonal pyramids from the icosahedron, respectively. The similar dissected regular icosahedron has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces.
Relations to the 600-cell and other 4-polytopes.
The icosahedron is the dimensional analogue of the 600-cell, a regular 4-dimensional polytope. The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell.
The unit-radius 600-cell has tetrahedral cells of edge length formula_12, 20 of which meet at each vertex to form an icosahedral pyramid (a 4-pyramid with an icosahedron as its base). Thus the 600-cell contains 120 icosahedra of edge length formula_12. The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as interior features formed by its unit-length chords. In the unit-radius 120-cell (another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron, and in a cube).
A semiregular 4-polytope, the snub 24-cell, has icosahedral cells.
Relations to other uniform polytopes.
As mentioned above, the regular icosahedron is unique among the Platonic solids in possessing a dihedral angle is approximately formula_13. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in "n" dimensions, at least three facets must meet at a peak and leave a positive defect for folding in "n"-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex regular polychora.
There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with formula_14-symmetry, i.e. have different planes of symmetry from the tetrahedron.
Appearances.
Dice are the common objects with the different polyhedron, one of them is the regular icosahedron. The twenty-sided dice was found in many ancient times. One example is the dice from the Ptolemaic of Egypt, which was later the Greek letters inscribed on the faces in the period of Greece and Roman.
Another example was found in the treasure of Tipu Sultan, which was made out of golden and with numbers written on each face. In several roleplaying games, such as "Dungeons & Dragons", the twenty-sided die (labeled as d20) is commonly used in determining success or failure of an action. It may be numbered from "0" to "9" twice, in which form it usually serves as a ten-sided die (d10); most modern versions are labeled from "1" to "20". "Scattergories" is another board game, where the player names the categories in the card with given the set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.
The regular icosahedron may also appear in many fields of science as follows:
As mentioned above, the regular icosahedron is one of the five Platonic solids. The regular polyhedra have been known since antiquity, but are named after Plato who, in his "Timaeus" dialogue, identified these with the five "elements", whose elementary units were attributed these shapes: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube) and the shape of the universe as a whole (dodecahedron). Euclid's "Elements" defined the Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length. Following their identification with the elements by Plato, Johannes Kepler in his "Harmonices Mundi" sketched each of them, in particular, the regular icosahedron. In his "Mysterium Cosmographicum", he also proposed a model of the Solar System based on the placement of Platonic solids in a concentric sequence of increasing radius of the inscribed and circumscribed spheres whose radii gave the distance of the six known planets from the common center. The ordering of the solids, from innermost to outermost, consisted of: regular octahedron, regular icosahedron, regular dodecahedron, regular tetrahedron, and cube.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": " \\left(0, \\pm 1, \\pm \\varphi \\right), \\left(\\pm 1, \\pm \\varphi, 0 \\right), \\left(\\pm \\varphi, 0, \\pm 1 \\right), "
},
{
"math_id": 1,
"text": "\\varphi = (1 + \\sqrt{5})/2 "
},
{
"math_id": 2,
"text": " a "
},
{
"math_id": 3,
"text": " r_I "
},
{
"math_id": 4,
"text": " r_C "
},
{
"math_id": 5,
"text": " r_M "
},
{
"math_id": 6,
"text": "\n r_I = \\frac{\\varphi^2 a}{2 \\sqrt{3}} \\approx 0.756a, \\qquad\n r_C = \\frac{\\sqrt{\\varphi^2 + 1}}{2}a \\approx 0.951a, \\qquad\n r_M = \\frac{\\varphi}{2}a \\approx 0.809a. \n"
},
{
"math_id": 7,
"text": " A "
},
{
"math_id": 8,
"text": " V "
},
{
"math_id": 9,
"text": "\n A = 5\\sqrt{3}a^2 \\approx 8.660a^2, \\qquad\n V = \\frac{5 \\varphi^2}{6}a^3 \\approx 2.182a^3. \n"
},
{
"math_id": 10,
"text": "C_2"
},
{
"math_id": 11,
"text": "\\frac{1}{\\varphi} \\approx 0.618"
},
{
"math_id": 12,
"text": "\\frac{1}{\\varphi}"
},
{
"math_id": 13,
"text": " 138.19^\\circ "
},
{
"math_id": 14,
"text": " T_h "
},
{
"math_id": 15,
"text": "n"
},
{
"math_id": 16,
"text": "n = 12"
}
] |
https://en.wikipedia.org/wiki?curid=14968
|
1497050
|
X-13ARIMA-SEATS
|
Statistical algorithms for time series data analysis
X-13ARIMA-SEATS, successor to X-12-ARIMA and X-11, is a set of statistical methods for seasonal adjustment and other descriptive analysis of time series data that are implemented in the U.S. Census Bureau's software package. These methods are or have been used by Statistics Canada, Australian Bureau of Statistics, and the statistical offices of many other countries.
X-12-ARIMA can be used together with many statistical packages, such as SAS in its econometric and time series (ETS) package, R in its (seasonal) package, Gretl or EViews which provides a graphical user interface for X-12-ARIMA, and NumXL which avails X-12-ARIMA functionality in Microsoft Excel. There is also a version for MATLAB.
Notable statistical agencies presently using X-12-ARIMA for seasonal adjustment include Statistics Canada, the U.S. Bureau of Labor Statistics and Census and Statistics Department (Hong Kong). The Brazilian Institute of Geography and Statistics uses X-13-ARIMA.
X-12-ARIMA was the successor to X-11-ARIMA; the current version is X-13ARIMA-SEATS.
X-13-ARIMA-SEATS's source code can be found on the Census Bureau's website.
Methods.
The default method for seasonal adjustment is based on the X-11 algorithm. It is assumed that the observations in a time series, formula_0, can be decomposed additively,
formula_1
or multiplicatively,
formula_2
In this decomposition, formula_3 is the trend (or the "trend cycle" because it also includes cyclical movements such as business cycles) component, formula_4 is the seasonal component, and formula_5 is the irregular (or random) component. The goal is to estimate each of the three components and then remove the seasonal component from the time series, producing a seasonally adjusted time series.
The decomposition is accomplished through the iterative application of centered moving averages. For an additive decomposition of a monthly time series, for example, the algorithm follows the following pattern:
The method also includes a number of tests, diagnostics and other statistics for evaluating the quality of the seasonal adjustments.
Copyright and conditions.
The software is US government work, and those are in the public domain (in the US); for this software copyright has also been granted for other countries; the "User agrees to make a good faith effort to use the Software in a way that does not cause damage, harm, or embarrassment to the United States/Commerce."
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Y_{t}"
},
{
"math_id": 1,
"text": "\n \\begin{align}\n \\textit{Y}_{t} &= {T}_{t} + {S}_{t} + {I}_{t}\n \\end{align}\n"
},
{
"math_id": 2,
"text": "\n \\begin{align}\n \\textit{Y}_{t} &= {T}_{t} \\times {S}_{t} \\times {I}_{t}.\n \\end{align}\n"
},
{
"math_id": 3,
"text": "T_{t}"
},
{
"math_id": 4,
"text": "S_{t}"
},
{
"math_id": 5,
"text": "I_{t}"
},
{
"math_id": 6,
"text": "t-6"
},
{
"math_id": 7,
"text": "t+6"
},
{
"math_id": 8,
"text": "t-24, t-12, t, t+12, t+24"
}
] |
https://en.wikipedia.org/wiki?curid=1497050
|
1497098
|
Kaprekar's routine
|
Iterative algorithm
In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers.
As an example, starting with the number 8991 in base 10:
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174
6174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations. The algorithm runs on any natural number in any given number base.
Definition and properties.
The algorithm is as follows:
The sequence is called a Kaprekar sequence and the function formula_5 is the Kaprekar mapping. Some numbers map to themselves; these are the fixed points of the Kaprekar mapping, and are called Kaprekar's constants. Zero is a Kaprekar's constant for all bases formula_1, and so is called a trivial Kaprekar's constant. All other Kaprekar's constants are nontrivial Kaprekar's constants.
For example, in base 10, starting with 3524,
formula_6
formula_7
formula_8
formula_9
with 6174 as a Kaprekar's constant.
All Kaprekar sequences will either reach one of these fixed points or will result in a repeating cycle. Either way, the end result is reached in a fairly small number of steps.
Note that the numbers formula_2 and formula_3 have the same digit sum and hence the same remainder modulo formula_10. Therefore, each number in a Kaprekar sequence of base formula_1 numbers (other than possibly the first) is a multiple of formula_10.
When leading zeroes are retained, only repdigits lead to the trivial Kaprekar's constant.
Families of Kaprekar's constants.
In base 4, it can easily be shown that all numbers of the form 3021, 310221, 31102221, 3...111...02...222...1 (where the length of the "1" sequence and the length of the "2" sequence are the same) are fixed points of the Kaprekar mapping.
In base 10, it can easily be shown that all numbers of the form 6174, 631764, 63317664, 6...333...17...666...4 (where the length of the "3" sequence and the length of the "6" sequence are the same) are fixed points of the Kaprekar mapping.
"b" = 2"k".
It can be shown that all natural numbers
formula_11
are fixed points of the Kaprekar mapping in even base "b" = 2"k" for all natural numbers n.
<templatestyles src="Math_proof/styles.css" />Proof
formula_12
formula_13
formula_14
Citations.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "b"
},
{
"math_id": 2,
"text": "\\alpha"
},
{
"math_id": 3,
"text": "\\beta"
},
{
"math_id": 4,
"text": "\\alpha -\\beta"
},
{
"math_id": 5,
"text": "K_b(n) = \\alpha - \\beta"
},
{
"math_id": 6,
"text": "K_{10}(3524) = 5432 - 2345 = 3087"
},
{
"math_id": 7,
"text": "K_{10}(3087) = 8730 - 378 = 8352"
},
{
"math_id": 8,
"text": "K_{10}(8352) = 8532 - 2358 = 6174"
},
{
"math_id": 9,
"text": "K_{10}(6174) = 7641 - 1467 = 6174"
},
{
"math_id": 10,
"text": "b - 1"
},
{
"math_id": 11,
"text": "m = (k) b^{2n + 3} \\left(\\sum_{i = 0}^{n - 1} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b \\left(\\sum_{i = 0}^{n - 1} b^i\\right) + (k)"
},
{
"math_id": 12,
"text": "\\alpha = (2k - 1) b^{2n + 2} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) \\left(\\sum_{i = 0}^{n} b^i\\right)"
},
{
"math_id": 13,
"text": "\\beta = (k - 1) b^{2n + 2} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (2k - 1) \\left(\\sum_{i = 0}^{n} b^i\\right)"
},
{
"math_id": 14,
"text": "\n\\begin{align}\nK_b(m) & = \\alpha - \\beta \\\\\n& = ((2k - 1) - (k - 1)) b^{2n + 2} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - k) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + ((k - 1) - (2k - 1)) \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 2} \\left(\\sum_{i = 0}^{n} b^i\\right) - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + b^{2n + 2} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k) b^{2n + 1} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{2n + 1} + b^{2n + 1} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{2n + 1 - 1} \\left(\\sum_{i = 0}^{1} b^i\\right) + b^{2n + 1 - 1} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{2n + 1 - n} \\left(\\sum_{i = 0}^{n} b^i\\right) + b^{2n + 1 - n} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + b^{n + 1} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (2k) b^{n} - k \\left(\\sum_{i = 0}^{n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + k b^{n} - k \\left(\\sum_{i = 0}^{n - 1} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{n + 1 - 1} + b^{n + 1 - 1} - k \\left(\\sum_{i = 0}^{n - n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{n + 1 - n} \\left(\\sum_{i = 0}^{n} b^i\\right) + b^{n + 1 - n} - k \\left(\\sum_{i = 0}^{n - n} b^i\\right) \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b \\left(\\sum_{i = 0}^{n} b^i\\right) + b - k \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b \\left(\\sum_{i = 0}^{n} b^i\\right) + 2k - k \\\\\n& = k b^{2n + 3} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b^{2n + 2} + (2k - 1) b^{n + 1} \\left(\\sum_{i = 0}^{n} b^i\\right) + (k - 1) b \\left(\\sum_{i = 0}^{n} b^i\\right) + k \\\\\n& = m \\\\\n\\end{align}\n"
}
] |
https://en.wikipedia.org/wiki?curid=1497098
|
14972
|
Idempotence
|
Property of operations
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).
The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + "potence" (same + power).
Definition.
An element formula_0 of a set formula_1 equipped with a binary operator formula_2 is said to be "idempotent" under formula_2 if
formula_3.
The "binary operation" formula_2 is said to be "idempotent" if
formula_3 for all formula_4.
Examples.
Idempotent functions.
In the monoid formula_37 of the functions from a set formula_23 to itself (see set exponentiation) with function composition formula_38, idempotent elements are the functions formula_39 such that formula_40, that is such that formula_41 for all formula_42 (in other words, the image formula_43 of each element formula_42 is a fixed point of formula_44). For example:
If the set formula_23 has formula_48 elements, we can partition it into formula_49 chosen fixed points and formula_50 non-fixed points under formula_44, and then formula_51 is the number of different idempotent functions. Hence, taking into account all possible partitions,
formula_52
is the total number of possible idempotent functions on the set. The integer sequence of the number of idempotent functions as given by the sum above for "n" = 0, 1, 2, 3, 4, 5, 6, 7, 8, ... starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, ... (sequence in the OEIS).
Neither the property of being idempotent nor that of being not is preserved under function composition. As an example for the former, formula_54 mod 3 and formula_55 are both idempotent, but formula_53 is not, although formula_56 happens to be. As an example for the latter, the negation function formula_57 on the Boolean domain is not idempotent, but formula_58 is. Similarly, unary negation formula_59 of real numbers is not idempotent, but formula_60 is. In both cases, the composition is simply the identity function, which is idempotent.
Computer science meaning.
In computer science, the term "idempotence" may have a different meaning depending on the context in which it is applied:
This is a very useful property in many situations, as it means that an operation can be repeated or retried as often as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was already performed or not.
Computer science examples.
A function looking up a customer's name and address in a database is typically idempotent, since this will not cause the database to change. Similarly, a request for changing a customer's address to XYZ is typically idempotent, because the final address will be the same no matter how many times the request is submitted. However, a customer request for placing an order is typically not idempotent since multiple requests will lead to multiple orders being placed. A request for canceling a particular order is idempotent because no matter how many requests are made the order remains canceled.
A sequence of idempotent subroutines where at least one subroutine is different from the others, however, is not necessarily idempotent if a later subroutine in the sequence changes a value that an earlier subroutine depends on—"idempotence is not closed under sequential composition". For example, suppose the initial value of a variable is 3 and there is a subroutine sequence that reads the variable, then changes it to 5, and then reads it again. Each step in the sequence is idempotent: both steps reading the variable have no side effects and the step changing the variable to 5 will always have the same effect no matter how many times it is executed. Nonetheless, executing the entire sequence once produces the output (3, 5), but executing it a second time produces the output (5, 5), so the sequence is not idempotent.
int x = 3;
int main() {
sequence(); // prints "3\n5\n"
sequence(); // prints "5\n5\n"
return 0;
In the Hypertext Transfer Protocol (HTTP), idempotence and safety are the major attributes that separate HTTP methods. Of the major HTTP methods, GET, PUT, and DELETE should be implemented in an idempotent manner according to the standard, but POST doesn't need to be. GET retrieves the state of a resource; PUT updates the state of a resource; and DELETE deletes a resource. As in the example above, reading data usually has no side effects, so it is idempotent (in fact "nullipotent"). Updating and deleting a given data are each usually idempotent as long as the request uniquely identifies the resource and only that resource again in the future. PUT and DELETE with unique identifiers reduce to the simple case of assignment to a variable of either a value or the null-value, respectively, and are idempotent for the same reason; the end result is always the same as the result of the initial execution, even if the response differs.
Violation of the unique identification requirement in storage or deletion typically causes violation of idempotence. For example, storing or deleting a given set of content without specifying a unique identifier: POST requests, which do not need to be idempotent, often do not contain unique identifiers, so the creation of the identifier is delegated to the receiving system which then creates a corresponding new record. Similarly, PUT and DELETE requests with nonspecific criteria may result in different outcomes depending on the state of the system - for example, a request to delete the most recent record. In each case, subsequent executions will further modify the state of the system, so they are not idempotent.
In event stream processing, idempotence refers to the ability of a system to produce the same outcome, even if the same file, event or message is received more than once.
In a load–store architecture, instructions that might possibly cause a page fault are idempotent. So if a page fault occurs, the operating system can load the page from disk and then simply re-execute the faulted instruction. In a processor where such instructions are not idempotent, dealing with page faults is much more complex.
When reformatting output, pretty-printing is expected to be idempotent. In other words, if the output is already "pretty", there should be nothing to do for the pretty-printer.
In service-oriented architecture (SOA), a multiple-step orchestration process composed entirely of idempotent steps can be replayed without side-effects if any part of that process fails.
Many operations that are idempotent often have ways to "resume" a process if it is interrupted – ways that finish much faster than starting all over from the beginning. For example, resuming a file transfer,
synchronizing files, creating a software build, installing an application and all of its dependencies with a package manager, etc.
Applied examples.
Applied examples that many people could encounter in their day-to-day lives include elevator call buttons and crosswalk buttons. The initial activation of the button moves the system into a requesting state, until the request is satisfied. Subsequent activations of the button between the initial activation and the request being satisfied have no effect, unless the system is designed to adjust the time for satisfying the request based on the number of activations.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x"
},
{
"math_id": 1,
"text": "S"
},
{
"math_id": 2,
"text": "\\cdot"
},
{
"math_id": 3,
"text": "x\\cdot x=x"
},
{
"math_id": 4,
"text": "x\\in S"
},
{
"math_id": 5,
"text": "(\\mathbb{N}, \\times)"
},
{
"math_id": 6,
"text": "0"
},
{
"math_id": 7,
"text": "1"
},
{
"math_id": 8,
"text": "0\\times 0 = 0"
},
{
"math_id": 9,
"text": "1\\times 1=1"
},
{
"math_id": 10,
"text": "(\\mathbb{N}, +)"
},
{
"math_id": 11,
"text": "(M, \\cdot)"
},
{
"math_id": 12,
"text": "e"
},
{
"math_id": 13,
"text": "a"
},
{
"math_id": 14,
"text": "e\\cdot e=e"
},
{
"math_id": 15,
"text": "a\\cdot a=a"
},
{
"math_id": 16,
"text": "(G, \\cdot)"
},
{
"math_id": 17,
"text": "G"
},
{
"math_id": 18,
"text": "x\\cdot x=x\\cdot e"
},
{
"math_id": 19,
"text": "x=e"
},
{
"math_id": 20,
"text": "(\\mathcal{P}(E), \\cup)"
},
{
"math_id": 21,
"text": "(\\mathcal{P}(E), \\cap)"
},
{
"math_id": 22,
"text": "\\mathcal{P}(E)"
},
{
"math_id": 23,
"text": "E"
},
{
"math_id": 24,
"text": "\\cup"
},
{
"math_id": 25,
"text": "\\cap"
},
{
"math_id": 26,
"text": "x\\cup x=x"
},
{
"math_id": 27,
"text": "x\\in \\mathcal{P}(E)"
},
{
"math_id": 28,
"text": "x\\cap x=x"
},
{
"math_id": 29,
"text": "(\\{0, 1\\}, \\vee)"
},
{
"math_id": 30,
"text": "(\\{0, 1\\}, \\wedge)"
},
{
"math_id": 31,
"text": "\\vee"
},
{
"math_id": 32,
"text": "\\wedge"
},
{
"math_id": 33,
"text": "x\\vee x=x"
},
{
"math_id": 34,
"text": "x\\in \\{0, 1\\}"
},
{
"math_id": 35,
"text": "x\\wedge x=x"
},
{
"math_id": 36,
"text": "\\mathbb{Z}"
},
{
"math_id": 37,
"text": "(E^E, \\circ)"
},
{
"math_id": 38,
"text": "\\circ"
},
{
"math_id": 39,
"text": "f\\colon E\\to E"
},
{
"math_id": 40,
"text": "f\\circ f = f"
},
{
"math_id": 41,
"text": "f(f(x))=f(x)"
},
{
"math_id": 42,
"text": "x\\in E"
},
{
"math_id": 43,
"text": "f(x)"
},
{
"math_id": 44,
"text": "f"
},
{
"math_id": 45,
"text": "\\operatorname{abs}\\circ \\operatorname{abs}=\\operatorname{abs}"
},
{
"math_id": 46,
"text": "\\operatorname{abs}(\\operatorname{abs}(x))=\\operatorname{abs}(x)"
},
{
"math_id": 47,
"text": "\\mathrm{Re}(z)"
},
{
"math_id": 48,
"text": "n"
},
{
"math_id": 49,
"text": "k"
},
{
"math_id": 50,
"text": "n-k"
},
{
"math_id": 51,
"text": "k^{n-k}"
},
{
"math_id": 52,
"text": "\\sum_{k=0}^n {n \\choose k} k^{n-k}"
},
{
"math_id": 53,
"text": "f\\circ g"
},
{
"math_id": 54,
"text": "f(x)=x"
},
{
"math_id": 55,
"text": "g(x)=\\max(x, 5)"
},
{
"math_id": 56,
"text": "g\\circ f"
},
{
"math_id": 57,
"text": "\\neg"
},
{
"math_id": 58,
"text": "\\neg\\circ\\neg"
},
{
"math_id": 59,
"text": "-(\\cdot)"
},
{
"math_id": 60,
"text": "-(\\cdot)\\circ -(\\cdot)"
}
] |
https://en.wikipedia.org/wiki?curid=14972
|
14973218
|
Toroid
|
Surface of revolution with a hole in the middle
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
The term "toroid" is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A "g"-holed "toroid" can be seen as approximating the surface of a torus having a topological genus, "g", of 1 or greater. The Euler characteristic χ of a "g" holed toroid is 2(1-"g").
The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and should not be confused with toroids.
Toroidal structures occur in both natural and synthetic materials.
Equations.
A toroid is specified by the radius of revolution "R" measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference "C" and area "A" of the section):
Square toroid.
The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.
formula_0
formula_1
Circular toroid.
The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.
formula_2
formula_3
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "V = 2 \\pi R A"
},
{
"math_id": 1,
"text": "S = 2 \\pi R C"
},
{
"math_id": 2,
"text": "V = 2 \\pi^2 r^2 R"
},
{
"math_id": 3,
"text": "S = 4 \\pi^2 r R"
}
] |
https://en.wikipedia.org/wiki?curid=14973218
|
14973487
|
Toroidal inductors and transformers
|
Type of electrical device
Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal (ring or donut) shape. They are passive electronic components, consisting of a circular ring or donut shaped magnetic core of ferromagnetic material such as laminated iron, iron powder, or ferrite, around which wire is wound.
Although closed-core inductors and transformers often use cores with a rectangular shape, the use of toroidal-shaped cores sometimes provides superior electrical performance. The advantage of the toroidal shape is that, due to its symmetry, the amount of magnetic flux that escapes outside the core (leakage flux) can be made low, potentially making it more efficient and making it emit less electromagnetic interference (EMI).
Toroidal inductors and transformers are used in a wide range of electronic circuits: power supplies, inverters, and amplifiers, which in turn are used in the vast majority of electrical equipment: TVs, radios, computers, and audio systems.
Advantages of toroidal windings.
In general, a toroidal inductor/transformer is more compact than other shaped cores because they are made of fewer materials and include a centering washer, nuts, and bolts resulting in up to a 50% lighter weight design. This is especially the case for power devices.
Because the toroid is a closed-loop core, it will have a higher magnetic field and thus higher inductance and Q factor than an inductor of the same mass with a straight core (solenoid coils). This is because most of the magnetic field is contained within the core. By comparison, with an inductor with a straight core, the magnetic field emerging from one end of the core has a long path through air to enter the other end.
In addition, because the windings are relatively short and wound in a closed magnetic field, a toroidal transformer will have a lower secondary impedance which will increase efficiency, electrical performance and reduce effects such as distortion and fringing.
Due to the symmetry of a toroid, little magnetic flux escapes from the core (leakage flux). Thus, a toroidal inductor/transformer, radiates less electromagnetic interference (EMI) to adjacent circuits and is an ideal choice for highly concentrated environments. Manufacturers have adopted toroidal coils in recent years to comply with increasingly strict international standards limiting the amount of electromagnetic field consumer electronics can produce.
Total B field confinement by toroidal inductors.
In some circumstances, the current in the winding of a toroidal inductor contributes only to the B field inside the windings. It does not contribute to the magnetic B field outside the windings. This is a consequence of symmetry and Ampère's circuital law.
Sufficient conditions for total internal confinement of the B field.
The absence of circumferential current (the path of circumferential current is indicated by the red arrow in figure 3 of this section) and the axially symmetric layout of the conductors and magnetic materials are sufficient conditions for total internal confinement of the B field. (Some authors prefer to use the H field). Because of the symmetry, the lines of B flux must form circles of constant intensity centered on the axis of symmetry. The only lines of B flux that encircle any current are those that are inside the toroidal winding. Therefore, from Ampere's circuital law, the intensity of the B field must be zero outside the windings.
Figure 3 of this section shows the most common toroidal winding. It fails both requirements for total B field confinement. Looking out from the axis, sometimes the winding is on the inside of the core and sometimes on the outside of the core. It is not axially symmetric in the near region. However, at points a distance of several times the winding spacing, the toroid does look symmetric. There is still the problem of the circumferential current. No matter how many times the winding encircles the core and no matter how thin the wire, this toroidal inductor will still include a one coil loop in the plane of the toroid. This winding will also produce and be susceptible to an E field in the plane of the inductor.
Figures 4-6 show different ways to neutralize the circumferential current. Figure 4 is the simplest and has the advantage that the return wire can be added after the inductor is bought or built.
E field in the plane of the toroid.
There will be a distribution of potential along the winding. This can lead to an E-Field in the plane of the toroid and also a susceptibility to an E field in the plane of the toroid, as shown in figure 7. This can be mitigated by using a return winding, as shown in Figure 8. With this winding, each place the winding crosses itself; the two parts will be at equal and opposite polarity, which substantially reduces the E field generated in the plane.
Toroidal inductor/transformer and magnetic vector potential.
See Feynman chapter 14 and 15 for a general discussion of magnetic vector potential. See Feynman page 15-11 for a diagram of the magnetic vector potential around a long thin solenoid which also exhibits total internal confinement of the B field, at least in the infinite limit.
The A field is accurate when using the assumption formula_0. This would be true under the following assumptions:
Number 4 will be presumed for the rest of this section and may be referred to the "quasi-static condition".
Although the axially symmetric toroidal inductor with no circumferential current totally confines the B field within the windings, the A field (magnetic vector potential) is not confined. Arrow #1 in the picture depicts the vector potential on the axis of symmetry. Radial current sections a and b are equal distances from the axis but pointed in opposite directions, so they will cancel. Likewise, segments c and d cancel. All the radial current segments cancel. The situation for axial currents is different. The axial current on the outside of the toroid is pointed down and the axial current on the inside of the toroid is pointed up. Each axial current segment on the outside of the toroid can be matched with an equal but oppositely directed segment on the inside of the toroid. The segments on the inside are closer than the segments on the outside to the axis, therefore there is a net upward component of the A field along the axis of symmetry.
Since the equations formula_3, and formula_4 (assuming quasi-static conditions, i.e. formula_5 ) have the same form, then the lines and contours of A relate to B like the lines and contours of B relate to j. Thus, a depiction of the A field around a loop of B flux (as would be produced in a toroidal inductor) is qualitatively the same as the B field around a loop of current. The figure to the left is an artist's depiction of the A field around a toroidal inductor. The thicker lines indicate paths of higher average intensity (shorter paths have higher intensity so that the path integral is the same). The lines are just drawn to look good and impart general look of the A field.
Toroidal transformer action in the presence of total B field confinement.
The E and B fields can be computed from the A and formula_6 (scalar electric potential) fields
formula_7 and :formula_8 and so even if the region outside the windings is devoid of B field, it is filled with non-zero E field.
The quantity formula_9 is responsible for the desirable magnetic field coupling between primary and secondary while the quantity formula_10 is responsible for the undesirable electric field coupling between primary and secondary. Transformer designers attempt to minimize the electric field coupling. For the rest of this section, formula_11 will assumed to be zero unless otherwise specified.
Stokes theorem applies, so that the path integral of A is equal to the enclosed B flux, just as the path integral B is equal to a constant times the enclosed current
The path integral of E along the secondary winding gives the secondary's induced EMF (Electro-Motive Force).
formula_12
which says the EMF is equal to the time rate of change of the B flux enclosed by the winding, which is the usual result.
Toroidal transformer Poynting vector coupling from primary to secondary in the presence of total B field confinement.
Explanation of the figure.
This figure shows the half section of a toroidal transformer. Quasi-static conditions are assumed, so the phase of each field is the same everywhere. The transformer, its windings and all things are distributed symmetrically about the axis of symmetry. The windings are such that there is no circumferential current. The requirements are met for full internal confinement of the B field due to the primary current. The core and primary winding are represented by the gray-brown torus. The primary winding is not shown, but the current in the winding at the cross-section surface is shown as gold (or orange) ellipses. The B field caused by the primary current is confined to the region enclosed by the primary winding (i.e. the core). Blue dots on the left-hand cross-section indicate that lines of B flux in the core come out of the left-hand cross-section. On the other cross-section, blue plus signs indicate that the B flux enters there. The E field sourced from the primary currents is shown as green ellipses. The secondary winding is shown as a brown line coming directly down the axis of symmetry. In standard practice, the two ends of the secondary are connected with a long wire that stays well away from the torus, but to maintain the absolute axial symmetry, the entire apparatus is envisioned as being inside a perfectly conductive sphere with the secondary wire "grounded" to the inside of the sphere at each end. The secondary is made of resistance wire, so there is no separate load. The E field along the secondary causes current in the secondary (yellow arrows), which causes a B field around the secondary (shown as blue ellipses). This B field fills space, including inside the transformer core, so in the end, there is a continuous non-zero B field from the primary to the secondary, if the secondary is not open-circuited. The cross product of the E field (sourced from primary currents) and the B field (sourced from the secondary currents) forms the Poynting vector, which points from the primary toward the secondary.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "bf{A} = 0"
},
{
"math_id": 1,
"text": "\\rho = 0 \\, "
},
{
"math_id": 2,
"text": " \\frac{1}{c^2}\\frac{\\partial\\phi}{\\partial t} "
},
{
"math_id": 3,
"text": "\\nabla \\times \\mathbf{A} = \\mathbf{B} \\ "
},
{
"math_id": 4,
"text": "\\nabla \\times \\mathbf{B} = \\mu_0\\mathbf{j} \\ "
},
{
"math_id": 5,
"text": "\\frac{\\partial E}{\\partial t}\\rightarrow 0"
},
{
"math_id": 6,
"text": " \\phi \\, "
},
{
"math_id": 7,
"text": "\\mathbf{B} = \\nabla \\times \\mathbf{A}."
},
{
"math_id": 8,
"text": "\\mathbf{E} = - \\nabla \\phi - \\frac { \\partial \\mathbf{A} } { \\partial t } "
},
{
"math_id": 9,
"text": " \\frac { \\partial \\mathbf{A} } { \\partial t }"
},
{
"math_id": 10,
"text": " \\nabla \\phi \\, "
},
{
"math_id": 11,
"text": " \\nabla \\phi \\, "
},
{
"math_id": 12,
"text": "\\mathbf{EMF} = \\oint_{path} \\mathbf{E} \\cdot {\\rm d}l = -\\oint_{path} \\frac { \\partial \\mathbf{A} } { \\partial t } \\cdot {\\rm d}l = -\\frac { \\partial } { \\partial t } \\oint_{path} \\mathbf{A}\\cdot {\\rm d}l = -\\frac { \\partial } { \\partial t } \\int_{surface} \\mathbf{B}\\cdot {\\rm d}s "
}
] |
https://en.wikipedia.org/wiki?curid=14973487
|
14974602
|
Mandart inellipse
|
Inellipse tangent where the triangle's excircles touch its sides
In geometry, the Mandart inellipse of a triangle is an ellipse that is inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitters). The Mandart inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century.
Parameters.
As an inconic, the Mandart inellipse is described by the parameters
formula_0
where "a", "b", and "c" are sides of the given triangle.
Related points.
The center of the Mandart inellipse is the mittenpunkt of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the Nagel point of the triangle.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x:y:z=\\frac{a}{b+c-a}:\\frac{b}{a+c-b}:\\frac{c}{a+b-c}"
}
] |
https://en.wikipedia.org/wiki?curid=14974602
|
1497569
|
Consistent estimator
|
Statistical estimator converging in probability to a true parameter as sample size increases
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter "θ"0—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to "θ"0. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to "θ"0 converges to one.
In practice one constructs an estimator as a function of an available sample of size "n", and then imagines being able to keep collecting data and expanding the sample "ad infinitum". In this way one would obtain a sequence of estimates indexed by "n", and consistency is a property of what occurs as the sample size “grows to infinity”. If the sequence of estimates can be mathematically shown to converge in probability to the true value "θ"0, it is called a consistent estimator; otherwise the estimator is said to be inconsistent.
Consistency as defined here is sometimes referred to as weak consistency. When we replace convergence in probability with almost sure convergence, then the estimator is said to be strongly consistent. Consistency is related to bias; see bias versus consistency.
Definition.
Formally speaking, an estimator "Tn" of parameter "θ" is said to be weakly consistent, if it converges in probability to the true value of the parameter:
formula_0
i.e. if, for all "ε" > 0
formula_1
An estimator "Tn" of parameter "θ" is said to be strongly consistent, if it converges almost surely to the true value of the parameter:
formula_2
A more rigorous definition takes into account the fact that "θ" is actually unknown, and thus, the convergence in probability must take place for every possible value of this parameter. Suppose {"pθ": "θ" ∈ Θ} is a family of distributions (the parametric model), and "Xθ" = {"X"1, "X"2, … : "Xi" ~ "pθ"} is an infinite sample from the distribution "pθ". Let { "Tn"("Xθ") } be a sequence of estimators for some parameter "g"("θ"). Usually, "Tn" will be based on the first "n" observations of a sample. Then this sequence {"Tn"} is said to be (weakly) consistent if
formula_3
This definition uses "g"("θ") instead of simply "θ", because often one is interested in estimating a certain function or a sub-vector of the underlying parameter. In the next example, we estimate the location parameter of the model, but not the scale:
Examples.
Sample mean of a normal random variable.
Suppose one has a sequence of statistically independent observations {"X"1, "X"2, ...} from a normal "N"("μ", "σ"2) distribution. To estimate "μ" based on the first "n" observations, one can use the sample mean: "Tn" = ("X"1 + ... + "Xn")/"n". This defines a sequence of estimators, indexed by the sample size "n".
From the properties of the normal distribution, we know the sampling distribution of this statistic: "T""n" is itself normally distributed, with mean "μ" and variance "σ"2/"n". Equivalently, formula_4 has a standard normal distribution:
formula_5
as "n" tends to infinity, for any fixed "ε" > 0. Therefore, the sequence "Tn" of sample means is consistent for the population mean "μ" (recalling that formula_6 is the cumulative distribution of the normal distribution).
Establishing consistency.
The notion of asymptotic consistency is very close, almost synonymous to the notion of convergence in probability. As such, any theorem, lemma, or property which establishes convergence in probability may be used to prove the consistency. Many such tools exist:
formula_7
the most common choice for function "h" being either the absolute value (in which case it is known as Markov inequality), or the quadratic function (respectively Chebyshev's inequality).
formula_8
formula_9
formula_10
Bias versus consistency.
Unbiased but not consistent.
An estimator can be unbiased but not consistent. For example, for an iid sample {"x"..., "x"} one can use "T"("X") = "x" as the estimator of the mean E["X"]. Note that here the sampling distribution of "T" is the same as the underlying distribution (for any "n," as it ignores all points but the last), so E["T"("X")] = E["X"] and it is unbiased, but it does not converge to any value.
However, if a sequence of estimators is unbiased "and" converges to a value, then it is consistent, as it must converge to the correct value.
Biased but consistent.
Alternatively, an estimator can be biased but consistent. For example, if the mean is estimated by formula_11 it is biased, but as formula_12, it approaches the correct value, and so it is consistent.
Important examples include the sample variance and sample standard deviation. Without Bessel's correction (that is, when using the sample size formula_13 instead of the degrees of freedom formula_14), these are both negatively biased but consistent estimators. With the correction, the corrected sample variance is unbiased, while the corrected sample standard deviation is still biased, but less so, and both are still consistent: the correction factor converges to 1 as sample size grows.
Here is another example. Let formula_15 be a sequence of estimators for formula_16.
formula_17
We can see that formula_18, formula_19, and the bias does not converge to zero.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\n \\underset{n\\to\\infty}{\\operatorname{plim}}\\;T_n = \\theta.\n "
},
{
"math_id": 1,
"text": "\n \\lim_{n\\to\\infty}\\Pr\\big(|T_n-\\theta| > \\varepsilon\\big) = 0.\n "
},
{
"math_id": 2,
"text": "\n \\Pr\\big(\\lim_{n\\to\\infty}T_n = \\theta\\big) = 1.\n "
},
{
"math_id": 3,
"text": "\n \\underset{n\\to\\infty}{\\operatorname{plim}}\\;T_n(X^{\\theta}) = g(\\theta),\\ \\ \\text{for all}\\ \\theta\\in\\Theta.\n "
},
{
"math_id": 4,
"text": "\\scriptstyle (T_n-\\mu)/(\\sigma/\\sqrt{n})"
},
{
"math_id": 5,
"text": "\n \\Pr\\!\\left[\\,|T_n-\\mu|\\geq\\varepsilon\\,\\right] = \n \\Pr\\!\\left[ \\frac{\\sqrt{n}\\,\\big|T_n-\\mu\\big|}{\\sigma} \\geq \\sqrt{n}\\varepsilon/\\sigma \\right] = \n 2\\left(1-\\Phi\\left(\\frac{\\sqrt{n}\\,\\varepsilon}{\\sigma}\\right)\\right) \\to 0\n "
},
{
"math_id": 6,
"text": "\\Phi"
},
{
"math_id": 7,
"text": "\n \\Pr\\!\\big[h(T_n-\\theta)\\geq\\varepsilon\\big] \\leq \\frac{\\operatorname{E}\\big[h(T_n-\\theta)\\big]}{h(\\varepsilon)},\n "
},
{
"math_id": 8,
"text": "\n T_n\\ \\xrightarrow{p}\\ \\theta\\ \\quad\\Rightarrow\\quad g(T_n)\\ \\xrightarrow{p}\\ g(\\theta)\n "
},
{
"math_id": 9,
"text": "\\begin{align}\n & T_n + S_n \\ \\xrightarrow{d}\\ \\alpha+\\beta, \\\\\n & T_n S_n \\ \\xrightarrow{d}\\ \\alpha \\beta, \\\\\n & T_n / S_n \\ \\xrightarrow{d}\\ \\alpha/\\beta, \\text{ provided that }\\beta\\neq0\n \\end{align}"
},
{
"math_id": 10,
"text": "\\frac{1}{n}\\sum_{i=1}^n g(X_i) \\ \\xrightarrow{p}\\ \\operatorname{E}[\\,g(X)\\,]"
},
{
"math_id": 11,
"text": "{1 \\over n} \\sum x_i + {1 \\over n}"
},
{
"math_id": 12,
"text": "n \\rightarrow \\infty"
},
{
"math_id": 13,
"text": "n"
},
{
"math_id": 14,
"text": "n-1"
},
{
"math_id": 15,
"text": "T_n"
},
{
"math_id": 16,
"text": "\\theta"
},
{
"math_id": 17,
"text": "\\Pr(T_n) = \\begin{cases}\n 1 - 1/n, & \\mbox{if }\\, T_n = \\theta \\\\\n 1/n, & \\mbox{if }\\, T_n = n\\delta + \\theta\n\\end{cases}"
},
{
"math_id": 18,
"text": "T_n \\xrightarrow{p} \\theta"
},
{
"math_id": 19,
"text": "\\operatorname{E}[T_n] = \\theta + \\delta "
}
] |
https://en.wikipedia.org/wiki?curid=1497569
|
14975781
|
Rabinowitsch trick
|
In mathematics, the Rabinowitsch trick, introduced by ,
is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called "weak" Nullstellensatz), by introducing an extra variable.
The Rabinowitsch trick goes as follows. Let "K" be an algebraically closed field. Suppose the polynomial "f" in "K"["x"1..."x""n"] vanishes whenever all polynomials "f"1...,"f""m" vanish. Then the polynomials "f"1...,"f""m", 1 − "x"0"f" have no common zeros (where we have introduced a new variable "x"0), so by the weak Nullstellensatz for "K"["x"0, ..., "x""n"] they generate the unit ideal of "K"["x"0 ..., "x""n"]. Spelt out, this means there are polynomials formula_0 such that
formula_1
as an equality of elements of the polynomial ring formula_2. Since formula_3 are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting formula_4 that
formula_5
as elements of the field of rational functions formula_6, the field of fractions of the polynomial ring formula_7. Moreover, the only expressions that occur in the denominators of the right hand side are "f" and powers of "f", so rewriting that right hand side to have a common denominator results in an equality on the form
formula_8
for some natural number "r" and polynomials formula_9. Hence
formula_10
which literally states that formula_11 lies in the ideal generated by "f"1...,"f""m". This is the full version of the Nullstellensatz for "K"["x"1...,"x""n"].
|
[
{
"math_id": 0,
"text": "g_0,g_1,\\dots,g_m \\in K[x_0,x_1,\\dots,x_n]"
},
{
"math_id": 1,
"text": "1 = g_0(x_0,x_1,\\dots,x_n) (1 - x_0 f(x_1,\\dots,x_n)) + \\sum_{i=1}^m g_i(x_0,x_1,\\dots,x_n) f_i(x_1,\\dots,x_n)"
},
{
"math_id": 2,
"text": "K[x_0,x_1,\\dots,x_n]"
},
{
"math_id": 3,
"text": "x_0,x_1,\\dots,x_n"
},
{
"math_id": 4,
"text": " x_0 = 1/f(x_1,\\dots,x_n) "
},
{
"math_id": 5,
"text": "1 = \\sum_{i=1}^m g_i(1/f(x_1,\\dots,x_n),x_1,\\dots,x_n) f_i(x_1,\\dots,x_n)"
},
{
"math_id": 6,
"text": "K(x_1,\\dots,x_n)"
},
{
"math_id": 7,
"text": "K[x_1,\\dots,x_n]"
},
{
"math_id": 8,
"text": " 1 = \\frac{ \\sum_{i=1}^m h_i(x_1,\\dots,x_n) f_i(x_1,\\dots,x_n) }{f(x_1,\\dots,x_n)^r}"
},
{
"math_id": 9,
"text": "h_1,\\dots,h_m \\in K[x_1,\\dots,x_n]"
},
{
"math_id": 10,
"text": " f(x_1,\\dots,x_n)^r = \\sum_{i=1}^m h_i(x_1,\\dots,x_n) f_i(x_1,\\dots,x_n), "
},
{
"math_id": 11,
"text": "f^r"
}
] |
https://en.wikipedia.org/wiki?curid=14975781
|
1498040
|
Rydberg atom
|
Excited atomic quantum state with high principal quantum number (n)
A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, "n". The higher the value of "n", the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculiar properties including an exaggerated response to electric and magnetic fields, long decay periods and electron wavefunctions that approximate, under some conditions, classical orbits of electrons about the nuclei. The core electrons shield the outer electron from the electric field of the nucleus such that, from a distance, the electric potential looks identical to that experienced by the electron in a hydrogen atom.
In spite of its shortcomings, the Bohr model of the atom is useful in explaining these properties. Classically, an electron in a circular orbit of radius "r", about a hydrogen nucleus of charge +"e", obeys Newton's second law:
formula_0
where "k" = 1/(4πε0).
Orbital momentum is quantized in units of "ħ":
formula_1.
Combining these two equations leads to Bohr's expression for the orbital radius in terms of the principal quantum number, "n":
formula_2
It is now apparent why Rydberg atoms have such peculiar properties: the radius of the orbit scales as "n"2 (the "n" = 137 state of hydrogen has an atomic radius ~1 μm) and the geometric cross-section as "n"4. Thus, Rydberg atoms are extremely large, with loosely bound valence electrons, easily perturbed or ionized by collisions or external fields.
Because the binding energy of a Rydberg electron is proportional to 1/"r" and hence falls off like 1/"n"2, the energy level spacing falls off like 1/"n"3 leading to ever more closely spaced levels converging on the first ionization energy. These closely spaced Rydberg states form what is commonly referred to as the "Rydberg series". Figure 2 shows some of the energy levels of the lowest three values of orbital angular momentum in lithium.
History.
The existence of the Rydberg series was first demonstrated in 1885 when Johann Balmer discovered a simple empirical formula for the wavelengths of light associated with transitions in atomic hydrogen. Three years later, the Swedish physicist Johannes Rydberg presented a generalized and more intuitive version of Balmer's formula that came to be known as the Rydberg formula. This formula indicated the existence of an infinite series of ever more closely spaced discrete energy levels converging on a finite limit.
This series was qualitatively explained in 1913 by Niels Bohr with his semiclassical model of the hydrogen atom in which quantized values of angular momentum lead to the observed discrete energy levels. A full quantitative derivation of the observed spectrum was derived by Wolfgang Pauli in 1926 following development of quantum mechanics by Werner Heisenberg and others.
Methods of production.
The only truly stable state of a hydrogen-like atom is the ground state with "n" = 1. The study of Rydberg states requires a reliable technique for exciting ground state atoms to states with a large value of "n".
Electron impact excitation.
Much early experimental work on Rydberg atoms relied on the use of collimated beams of fast electrons incident on ground-state atoms. Inelastic scattering processes can use the electron kinetic energy to increase the atoms' internal energy exciting to a broad range of different states including many high-lying Rydberg states,
formula_3
Because the electron can retain any arbitrary amount of its initial kinetic energy, this process results in a population with a broad spread of different energies.
Charge exchange excitation.
Another mainstay of early Rydberg atom experiments relied on charge exchange between a beam of ions and a population of neutral atoms of another species, resulting in the formation of a beam of highly excited atoms,
formula_4
Again, because the kinetic energy of the interaction can contribute to the final internal energies of the constituents, this technique populates a broad range of energy levels.
Optical excitation.
The arrival of tunable dye lasers in the 1970s allowed a much greater level of control over populations of excited atoms. In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. The problem of producing single state, mono-energetic populations of Rydberg atoms thus becomes the somewhat simpler problem of precisely controlling the frequency of the laser output,
formula_5
This form of direct optical excitation is generally limited to experiments with the alkali metals, because the ground state binding energy in other species is generally too high to be accessible with most laser systems.
For atoms with a large valence electron binding energy (equivalent to a large first ionization energy), the excited states of the Rydberg series are inaccessible with conventional laser systems. Initial collisional excitation can make up the energy shortfall allowing optical excitation to be used to select the final state. Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state.
Hydrogenic potential.
An atom in a Rydberg state has a valence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almost hydrogenic Coulomb potential, "U"C, from a compact ion core consisting of a nucleus with "Z" protons and the lower electron shells filled with "Z"-1 electrons. An electron in the spherically symmetric Coulomb potential has potential energy:
formula_6
The similarity of the effective potential "seen" by the outer electron to the hydrogen potential is a defining characteristic of Rydberg states and explains why the electron wavefunctions approximate to classical orbits in the limit of the correspondence principle. In other words, the electron's orbit resembles the orbit of planets inside a solar system, similar to what was seen in the obsolete but visually useful Bohr and Rutherford models of the atom.
There are three notable exceptions that can be characterized by the additional term added to the potential energy:
formula_7
formula_8
where "α"d is the dipole polarizability. Figure 3 shows how the polarization term modifies the potential close to the nucleus.
Quantum-mechanical details.
Quantum-mechanically, a state with abnormally high "n" refers to an atom in which the valence electron(s) have been excited into a formerly unpopulated electron orbital with higher energy and lower binding energy. In hydrogen the binding energy is given by:
formula_9
where Ry = 13.6 eV is the Rydberg constant. The low binding energy at high values of "n" explains why Rydberg states are susceptible to ionization.
Additional terms in the potential energy expression for a Rydberg state, on top of the hydrogenic Coulomb potential energy require the introduction of a quantum defect, "δ""ℓ", into the expression for the binding energy:
formula_10
Electron wavefunctions.
The long lifetimes of Rydberg states with high orbital angular momentum can be explained in terms of the overlapping of wavefunctions. The wavefunction of an electron in a high "ℓ" state (high angular momentum, “circular orbit”) has very little overlap with the wavefunctions of the inner electrons and hence remains relatively unperturbed.
The three exceptions to the definition of a Rydberg atom as an atom with a hydrogenic potential, have an alternative, quantum mechanical description that can be characterized by the additional term(s) in the atomic Hamiltonian:
In external fields.
The large separation between the electron and ion-core in a Rydberg atom makes possible an extremely large electric dipole moment, d. There is an energy associated with the presence of an electric dipole in an electric field, F, known in atomic physics as a Stark shift,
formula_11
Depending on the sign of the projection of the dipole moment onto the local electric field vector, a state may have energy that increases or decreases with field strength (low-field and high-field seeking states respectively). The narrow spacing between adjacent "n"-levels in the Rydberg series means that states can approach degeneracy even for relatively modest field strengths. The theoretical field strength at which a crossing would occur assuming no coupling between the states is given by the Inglis–Teller limit,
formula_12
In the hydrogen atom, the pure 1/"r" Coulomb potential does not couple Stark states from adjacent "n"-manifolds resulting in real crossings as shown in figure 5. The presence of additional terms in the potential energy can lead to coupling resulting in avoided crossings as shown for lithium in figure 6.
Applications and further research.
Precision measurements of trapped Rydberg atoms.
The radiative decay lifetimes of atoms in metastable states to the ground state are important to understanding astrophysics observations and tests of the standard model.
Investigating diamagnetic effects.
The large sizes and low binding energies of Rydberg atoms lead to a high magnetic susceptibility, formula_13. As diamagnetic effects scale with the area of the orbit and the area is proportional to the radius squared ("A" ∝ "n"4), effects impossible to detect in ground state atoms become obvious in Rydberg atoms, which demonstrate very large diamagnetic shifts.
Rydberg atoms exhibit strong electric-dipole coupling of the atoms to electromagnetic fields and has been used to detect radio communications.
In plasmas.
Rydberg atoms form commonly in plasmas due to the recombination of electrons and positive ions; low energy recombination results in fairly stable Rydberg atoms, while recombination of electrons and positive ions with high kinetic energy often form autoionising Rydberg states. Rydberg atoms’ large sizes and susceptibility to perturbation and ionisation by electric and magnetic fields, are an important factor determining the properties of plasmas.
Condensation of Rydberg atoms forms Rydberg matter, most often observed in form of long-lived clusters. The de-excitation is significantly impeded in Rydberg matter by exchange-correlation effects in the non-uniform electron liquid formed on condensation by the collective valence electrons, which causes extended lifetime of clusters.
In astrophysics (Radio recombination lines).
Rydberg atoms occur in space due to the dynamic equilibrium between photoionization by hot stars and recombination with electrons, which at these very low densities usually proceeds via the electron re-joining the atom in a very high "n" state, and then gradually dropping through the energy levels to the ground state, giving rise to a sequence of recombination spectral lines spread across the electromagnetic spectrum. The very small differences in energy between Rydberg states differing in "n" by one or a few means that photons emitted in transitions between such states have low frequencies and long wavelengths, even up to radio waves. The first detection of such a radio recombination line (RRL) was by Soviet radio astronomers in 1964; the line, designated H90α, was emitted by hydrogen atoms in the "n" = 90 state. Today, Rydberg atoms of hydrogen, helium and carbon in space are routinely observed via RRLs, the brightest of which are the H"n"α lines corresponding to transitions from "n"+1 to "n." Weaker lines, H"nβ and" H"n"γ, with "Δn" = 2 and 3 are also observed. Corresponding lines for helium and carbon are He"n"α, C"n"α, and so on. The discovery of lines with "n" > 100 was surprising, as even in the very low densities of interstellar space, many orders of magnitude lower than the best laboratory vacuums attainable on Earth, it had been expected that such highly-excited atoms would be frequently destroyed by collisions, rendering the lines unobservable. Improved theoretical analysis showed that this effect had been overestimated, although collisional broadening does eventually limit detectability of the lines at very high "n".. The record wavelength for hydrogen is λ = 73 cm for H253α, implying atomic diameters of a few microns, and for carbon, λ = 18 metres, from C732α, from atoms with a diameter of 57 micron.
RRLs from hydrogen and helium are produced in highly ionized regions (H II regions and the Warm Ionised Medium). Carbon has a lower ionization energy than hydrogen, and so singly-ionized carbon atoms, and the corresponding recombining Rydberg states, exist further from the ionizing stars, in so-called C II regions which form thick shells around H II regions. The larger volume partially compensates for the low abundance of C compared to H, making the carbon RRLs detectable.
In the absence of collisional broadening, the wavelengths of RRLs are modified only by the Doppler effect, so the measured wavelength, formula_14, is usually converted to radial velocity, formula_15, where formula_16 is the rest-frame wavelength. H II regions in our Galaxy can have radial velocities up to ±150 km/s, due to their motion relative to Earth as both orbit the centre of the Galaxy. These motions are regular enough that formula_17 can be used to estimate the position of the H II region on the line of sight and so its 3D position in the Galaxy. Because all astrophysical Rydberg atoms are hydrogenic, the frequencies of transitions for H, He, and C are given by the same formula, except for the slightly different reduced mass of the valence electron for each element. This gives helium and carbon lines apparent Doppler shifts of −100 and −140 km/s, respectively, relative to the corresponding hydrogen line.
RRLs are used to detect ionized gas in distant regions of our Galaxy, and also in external galaxies, because the radio photons are not absorbed by interstellar dust, which blocks photons from the more familiar optical transitions. They are also used to measure the temperature of the ionized gas, via the ratio of line intensity to the continuum bremsstrahlung emission from the plasma. Since the temperature of H II regions is regulated by line emission from heavier elements such as C, N, and O, recombination lines also indirectly measure their abundance (metallicity).
RRLs are spread across the radio spectrum with relatively small intervals in wavelength between them, so they frequently occur in radio spectral observations primarily targeted at other spectral lines. For instance, H166α, H167α, and H168α are very close in wavelength to the 21-cm line from neutral hydrogen. This allows radio astronomers to study both the neutral and the ionized interstellar medium from the same set of observations. Since RRLs are numerous and weak, common practice is to average the velocity spectra of several neighbouring lines, to improve sensitivity.
There are a variety of other potential applications of Rydberg atoms in cosmology and astrophysics.
Strongly interacting systems.
Due to their large size, Rydberg atoms can exhibit very large electric dipole moments. Calculations using perturbation theory show that this results in strong interactions between two close Rydberg atoms. Coherent control of these interactions combined with their relatively long lifetime makes them a suitable candidate to realize a quantum computer. In 2010 two-qubit gates were achieved experimentally. Strongly interacting Rydberg atoms also feature quantum critical behavior, which makes them interesting to study on their own.
Current research directions.
Since 2000's Rydberg atoms research encompasses broadly five directions: sensing, quantum optics, quantum computation, quantum simulation and quantum matters. High electric dipole moments between Rydberg atomic states are used for radio frequency and terahertz sensing and imaging, including non-demolition measurements of individual microwave photons. Electromagnetically induced transparency was used in combination with strong interactions between two atoms excited in Rydberg state to provide medium that exhibits strongly nonlinear behaviour at the level of individual optical photons. The tuneable interaction between Rydberg states, enabled also first quantum simulation experiments.
In October 2018, the United States Army Research Laboratory publicly discussed efforts to develop a super wideband radio receiver using Rydberg atoms. In March 2020, the laboratory announced that its scientists analysed the Rydberg sensor's sensitivity to oscillating electric fields over an enormous range of frequencies—from 0 to 1012 Hertz (the spectrum to 0.3mm wavelength). The Rydberg sensor can reliably detect signals over the entire spectrum and compare favourably with other established electric field sensor technologies, such as electro-optic crystals and dipole antenna-coupled passive electronics.
Classical simulation.
A simple 1/"r" potential results in a closed Keplerian elliptical orbit. In the presence of an external electric field Rydberg atoms can obtain very large electric dipole moments making them extremely susceptible to perturbation by the field. Figure 7 shows how application of an external electric field (known in atomic physics as a Stark field) changes the geometry of the potential, dramatically changing the behaviour of the electron. A Coulombic potential does not apply any torque as the force is always antiparallel to the position vector (always pointing along a line running between the electron and the nucleus):
formula_18,
formula_19.
With the application of a static electric field, the electron feels a continuously changing torque. The resulting trajectory becomes progressively more distorted over time, eventually going through the full range of angular momentum from "L" = "L"MAX, to a straight line "L" = 0, to the initial orbit in the opposite sense
"L" = −"L"MAX.
The time period of the oscillation in angular momentum (the time to complete the trajectory in figure 8), almost exactly matches the quantum mechanically predicted period for the wavefunction to return to its initial state, demonstrating the classical nature of the Rydberg atom.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathbf{F}=m\\mathbf{a} \\Rightarrow { ke^2 \\over r^2}={mv^2 \\over r}"
},
{
"math_id": 1,
"text": " mvr=n\\hbar "
},
{
"math_id": 2,
"text": " r={n^2\\hbar^2 \\over ke^2m}. "
},
{
"math_id": 3,
"text": " e^- + A \\rarr A^* + e^- ."
},
{
"math_id": 4,
"text": " A^+ + B \\rarr A^* + B^+. "
},
{
"math_id": 5,
"text": " A + \\gamma \\rarr A^*."
},
{
"math_id": 6,
"text": "U_\\text{C} = -\\dfrac{e^2}{4\\pi\\varepsilon_0r}. "
},
{
"math_id": 7,
"text": "U_{ee} = \\dfrac{e^2}{4\\pi\\varepsilon_0}\\sum_{i < j}\\dfrac{1}{|\\mathbf{r}_i - \\mathbf{r}_j|}."
},
{
"math_id": 8,
"text": "U_\\text{pol} = -\\frac{e^2\\alpha_\\text{d}}{(4\\pi\\varepsilon_0)^2r^4}, "
},
{
"math_id": 9,
"text": " E_\\text{B} = -\\frac{\\rm Ry}{n^2}, "
},
{
"math_id": 10,
"text": "E_\\text{B} = -\\frac{\\rm Ry}{(n-\\delta_l)^2}. "
},
{
"math_id": 11,
"text": "E_\\text{S} = -\\mathbf{d}\\cdot\\mathbf{F}."
},
{
"math_id": 12,
"text": "F_\\text{IT} = \\dfrac{e}{12\\pi\\varepsilon_0a_0^2n^5}."
},
{
"math_id": 13,
"text": "\\chi"
},
{
"math_id": 14,
"text": "\\lambda"
},
{
"math_id": 15,
"text": "v \\approx c(\\lambda -\\lambda_0)/\\lambda_0"
},
{
"math_id": 16,
"text": "\\lambda_0"
},
{
"math_id": 17,
"text": "v"
},
{
"math_id": 18,
"text": "|\\mathbf{\\tau}|=|\\mathbf{r} \\times \\mathbf{F}|=|\\mathbf{r}||\\mathbf{F}|\\sin\\theta "
},
{
"math_id": 19,
"text": "\\theta=\\pi \\Rightarrow \\mathbf{\\tau}=0 "
}
] |
https://en.wikipedia.org/wiki?curid=1498040
|
1498076
|
Unit in the last place
|
Floating-point accuracy metric
In computer science and numerical analysis, unit in the last place or unit of least precision (ulp) is the spacing between two consecutive floating-point numbers, i.e., the value the "least significant digit" (rightmost digit) represents if it is 1. It is used as a measure of accuracy in numeric calculations.
Definition.
The most common definition is: In radix formula_0 with precision formula_1, if formula_2, then where formula_3 is the minimal exponent of the normal numbers. In particular, formula_4 for normal numbers, and formula_5 for subnormals.
Another definition, suggested by John Harrison, is slightly different: formula_6 is the distance between the two closest "straddling" floating-point numbers formula_7 and formula_0 (i.e., satisfying formula_8 and formula_9), assuming that the exponent range is not upper-bounded. These definitions differ only at signed powers of the radix.
The IEEE 754 specification—followed by all modern floating-point hardware—requires that the result of an elementary arithmetic operation (addition, subtraction, multiplication, division, and square root since 1985, and FMA since 2008) be correctly rounded, which implies that in rounding to nearest, the rounded result is within 0.5 ulp of the mathematically exact result, using John Harrison's definition; conversely, this property implies that the distance between the rounded result and the mathematically exact result is minimized (but for the halfway cases, it is satisfied by two consecutive floating-point numbers). Reputable numeric libraries compute the basic transcendental functions to between 0.5 and about 1 ulp. Only a few libraries compute them within 0.5 ulp, this problem being complex due to the Table-maker's dilemma.
Since the 2010s, advances in floating-point mathematics have allowed correctly rounded functions to be almost as fast in average as these earlier, less accurate functions. A correctly rounded function would also be fully reproducible. which theoretically would only produce one incorrect rounding out of 1000 random floating-point inputs.
Examples.
Example 1.
Let formula_10 be a positive floating-point number and assume that the active rounding mode is round to nearest, ties to even, denoted formula_11. If formula_12, then formula_13. Otherwise, formula_14 or formula_15, depending on the value of the least significant digit and the exponent of formula_10. This is demonstrated in the following Haskell code typed at an interactive prompt:
> until (\x -> x == x+1) (+1) 0 :: Float
1.6777216e7
> it-1
1.6777215e7
> it+1
1.6777216e7
Here we start with 0 in single precision (binary32) and repeatedly add 1 until the operation does not change the value. Since the significand for a single-precision number contains 24 bits, the first integer that is not exactly representable is 224+1, and this value rounds to 224 in round to nearest, ties to even. Thus the result is equal to 224.
Example 2.
The following example in Java approximates π as a floating point value by finding the two double values bracketing formula_16: formula_17.
// π with 20 decimal digits
BigDecimal π = new BigDecimal("3.14159265358979323846");
// truncate to a double floating point
double p0 = π.doubleValue();
// -> 3.141592653589793 (hex: 0x1.921fb54442d18p1)
// p0 is smaller than π, so find next number representable as double
double p1 = Math.nextUp(p0);
// -> 3.1415926535897936 (hex: 0x1.921fb54442d19p1)
Then formula_18 is determined as formula_19.
// ulp(π) is the difference between p1 and p0
BigDecimal ulp = new BigDecimal(p1).subtract(new BigDecimal(p0));
// -> 4.44089209850062616169452667236328125E-16
// (this is precisely 2**(-51))
// same result when using the standard library function
double ulpMath = Math.ulp(p0);
// -> 4.440892098500626E-16 (hex: 0x1.0p-51)
Example 3.
Another example, in Python, also typed at an interactive prompt, is:
»> x = 1.0
»> p = 0
»> while x != x + 1:
... x = x * 2
... p = p + 1
»> x
9007199254740992.0
»> p
53
»> x + 2 + 1
9007199254740996.0
In this case, we start with codice_0 and repeatedly double it until codice_1. Similarly to Example 1, the result is 253 because the double-precision floating-point format uses a 53-bit significand.
Language support.
The Boost C++ libraries provides the functions codice_2, codice_3, codice_4
and codice_5 to obtain nearby (and distant) floating-point values, and codice_6 to calculate the floating-point distance between two doubles.
The C language library provides functions to calculate the next floating-point number in some given direction: codice_7 and codice_8 for codice_9, codice_10 and codice_11 for codice_12, codice_13 and codice_14 for codice_15, declared in codice_16. It also provides the macros codice_17, codice_18, codice_19, which represent the positive difference between 1.0 and the next greater representable number in the corresponding type (i.e. the ulp of one).
The Java standard library provides the functions and . They were introduced with Java 1.5.
The Swift standard library provides access to the next floating-point number in some given direction via the instance properties codice_20 and codice_21. It also provides the instance property codice_22 and the type property codice_23 (which corresponds to C macros like codice_17) for Swift's floating-point types.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "b"
},
{
"math_id": 1,
"text": "p"
},
{
"math_id": 2,
"text": "b^e \\le |x| < b^{e+1}"
},
{
"math_id": 3,
"text": "e_\\min"
},
{
"math_id": 4,
"text": "\\operatorname{ulp}(x) = b^{e - p + 1}"
},
{
"math_id": 5,
"text": "\\operatorname{ulp}(x) = b^{e_\\min - p + 1}"
},
{
"math_id": 6,
"text": "\\operatorname{ulp}(x)"
},
{
"math_id": 7,
"text": "a"
},
{
"math_id": 8,
"text": "a \\le x \\le b"
},
{
"math_id": 9,
"text": "a \\neq b"
},
{
"math_id": 10,
"text": "x"
},
{
"math_id": 11,
"text": "\\operatorname{RN}"
},
{
"math_id": 12,
"text": "\\operatorname{ulp}(x) \\le 1"
},
{
"math_id": 13,
"text": "\\operatorname{RN} (x + 1) > x"
},
{
"math_id": 14,
"text": "\\operatorname{RN} (x + 1) = x"
},
{
"math_id": 15,
"text": "\\operatorname{RN} (x + 1) = x + \\operatorname{ulp}(x)"
},
{
"math_id": 16,
"text": "\\pi"
},
{
"math_id": 17,
"text": "p_0 < \\pi < p_1"
},
{
"math_id": 18,
"text": "\\operatorname{ulp}(\\pi)"
},
{
"math_id": 19,
"text": "\\operatorname{ulp}(\\pi) = p_1 - p_0"
}
] |
https://en.wikipedia.org/wiki?curid=1498076
|
14982946
|
Turán number
|
In mathematics, the Turán number T("n","k","r") for "r"-uniform hypergraphs of order "n" is the smallest number of "r"-edges such that every induced subgraph on "k" vertices contains an edge. This number was determined for "r" = 2 by , and the problem for general "r" was introduced in . The paper gives a survey of Turán numbers.
Definitions.
Fix a set "X" of "n" vertices. For given "r", an "r"-edge or block is a set of "r" vertices. A set of blocks is called a Turán ("n","k","r") system ("n" ≥ "k" ≥ "r") if every "k"-element subset of "X" contains a block.
The Turán number T("n","k","r") is the minimum size of such a system.
Example.
The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7.
Relations to other combinatorial designs.
It can be shown that
formula_0
Equality holds if and only if there exists a Steiner system S("n" - "k", "n" - "r", "n").
An ("n","r","k","r")-lotto design is an ("n", "k", "r")-Turán system. Thus, T("n","k", "r") = L("n","r","k","r").
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "T(n,k,r) \\geq \\binom{n}{r} {\\binom{k}{r}}^{-1}."
}
] |
https://en.wikipedia.org/wiki?curid=14982946
|
14983028
|
Secondary flow
|
Relatively minor flow superimposed on the primary flow by inviscid assumptions
In fluid dynamics, flow can be decomposed into primary flow plus secondary flow, a relatively weaker flow pattern superimposed on the stronger primary flow pattern. The primary flow is often chosen to be an exact solution to simplified or approximated governing equations, such as potential flow around a wing or geostrophic current or wind on the rotating Earth. In that case, the secondary flow usefully spotlights the effects of complicated real-world terms neglected in those approximated equations. For instance, the consequences of viscosity are spotlighted by secondary flow in the viscous boundary layer, resolving the tea leaf paradox. As another example, if the primary flow is taken to be a balanced flow approximation with net force equated to zero, then the secondary circulation helps spotlight acceleration due to the mild imbalance of forces. A smallness assumption about secondary flow also facilitates linearization.
In engineering, secondary flow also identifies an additional flow path.
Examples of secondary flows.
Wind near ground level.
The basic principles of physics and the Coriolis effect define an approximate geostrophic wind or gradient wind, balanced flows that are parallel to the isobars. Measurements of wind speed and direction at heights well above ground level confirm that wind matches these approximations quite well. However, nearer the Earth's surface, the wind speed is less than predicted by the barometric pressure gradient, and the wind direction is partly across the isobars rather than parallel to them. This flow of air across the isobars is a "secondary flow"., a difference from the primary flow which is parallel to the isobars. Interference by surface roughness elements such as terrain, waves, trees and buildings cause drag on the wind and prevent the air from accelerating to the speed necessary to achieve balanced flow. As a result, the wind direction near ground level is partly parallel to the isobars in the region, and partly across the isobars in the direction from higher pressure to lower pressure.
As a result of the slower wind speed at the earth's surface, in a region of low pressure the barometric pressure is usually significantly higher at the surface than would be expected, given the barometric pressure at mid altitudes, due to Bernoulli's principle. Hence, the secondary flow toward the center of a region of low pressure is also drawn upward by the significantly lower pressure at mid altitudes. This slow, widespread ascent of the air in a region of low pressure can cause widespread cloud and rain if the air is of sufficiently high relative humidity.
In a region of high pressure (an anticyclone) the secondary flow includes a slow, widespread descent of air from mid altitudes toward ground level, and then outward across the isobars. This descent causes a reduction in relative humidity and explains why regions of high pressure usually experience cloud-free skies for many days.
Tropical cyclones.
The flow around a tropical cyclone is often well approximated as parallel to circular isobars, such as in a vortex. A strong pressure gradient draws air toward the center of the cyclone, a centripetal force nearly balanced by Coriolis and centrifugal forces in gradient wind balance. The viscous secondary flow near the Earth's surface converges toward the center of the cyclone, ascending in the eyewall to satisfy mass continuity. As the secondary flow is drawn upward the air cools as its pressure falls, causing extremely heavy rainfall and releasing latent heat which is an important driver of the storm's energy budget.
Tornadoes and dust devils.
Tornadoes and dust devils display localised vortex flow. Their fluid motion is similar to tropical cyclones but on a much smaller scale so that the Coriolis effect is not significant. The primary flow is circular around the vertical axis of the tornado or dust devil. As with all vortex flow, the speed of the flow is fastest at the core of the vortex. In accordance with Bernoulli's principle where the wind speed is fastest the air pressure is lowest; and where the wind speed is slowest the air pressure is highest. Consequently, near the center of the tornado or dust devil the air pressure is low. There is a pressure gradient toward the center of the vortex. This gradient, coupled with the slower speed of the air near the earth's surface, causes a "secondary flow" toward the center of the tornado or dust devil, rather than in a purely circular pattern.
The slower speed of the air at the surface prevents the air pressure from falling as low as would normally be expected from the air pressure at greater heights. This is compatible with Bernoulli's principle. The secondary flow is toward the center of the tornado or dust devil, and is then drawn upward by the significantly lower pressure several thousands of feet above the surface in the case of a tornado, or several hundred feet in the case of a dust devil. Tornadoes can be very destructive and the secondary flow can cause debris to be swept into a central location and carried to low altitudes.
Dust devils can be seen by the dust stirred up at ground level, swept up by the secondary flow and concentrated in a central location. The accumulation of dust then accompanies the secondary flow upward into the region of intense low pressure that exists outside the influence of the ground.
Circular flow in a bowl or cup.
When water in a circular bowl or cup is moving in circular motion the water displays free-vortex flow – the water at the center of the bowl or cup spins at relatively high speed, and the water at the perimeter spins more slowly. The water is a little deeper at the perimeter and a little more shallow at the center, and the surface of the water is not flat but displays the characteristic depression toward the axis of the spinning fluid. At any elevation within the water the pressure is a little greater near the perimeter of the bowl or cup where the water is a little deeper, than near the center. The water pressure is a little greater where the water speed is a little slower, and the pressure is a little less where the speed is faster, and this is consistent with Bernoulli's principle.
There is a pressure gradient from the perimeter of the bowl or cup toward the center. This pressure gradient provides the centripetal force necessary for the circular motion of each parcel of water. The pressure gradient also accounts for a "secondary flow" of the boundary layer in the water flowing across the floor of the bowl or cup. The slower speed of the water in the boundary layer is unable to balance the pressure gradient. The boundary layer spirals inward toward the axis of circulation of the water. On reaching the center the secondary flow is then upward toward the surface, progressively mixing with the primary flow. Near the surface there may also be a slow secondary flow outward toward the perimeter.
The secondary flow along the floor of the bowl or cup can be seen by sprinkling heavy particles such as sugar, sand, rice or tea leaves into the water and then setting the water in circular motion by stirring with a hand or spoon. The boundary layer spirals inward and sweeps the heavier solids into a neat pile in the center of the bowl or cup. With water circulating in a bowl or cup, the primary flow is purely circular and might be expected to fling heavy particles outward to the perimeter. Instead, heavy particles can be seen to congregate in the center as a result of the secondary flow along the floor.
River bends.
Water flowing through a bend in a river must follow curved streamlines to remain within the banks of the river. The water surface is slightly higher near the concave bank than near the convex bank. (The "concave bank" has the greater radius. The "convex bank" has the smaller radius.) As a result, at any elevation within the river, water pressure is slightly higher near the concave bank than near the convex bank. A pressure gradient results from the concave bank toward the other bank. Centripetal forces are necessary for the curved path of each parcel of water, which is provided by the pressure gradient.
The primary flow around the bend approximates a free vortex – fastest speed where the radius of curvature of the stream itself is smallest and slowest speed where the radius is largest. The higher pressure near the concave (outer) bank is accompanied by slower water speed, and the lower pressure near the convex bank is accompanied by faster water speed, and all this is consistent with Bernoulli's principle.
A "secondary flow" is produced in the boundary layer along the floor of the river bed. The boundary layer is not moving fast enough to balance the pressure gradient and so its path is partly downstream and partly across the stream from the concave bank toward the convex bank, driven by the pressure gradient. The secondary flow is then upward toward the surface where it mixes with the primary flow or moves slowly across the surface, back toward the concave bank. This motion is called helicoidal flow.
On the floor of the river bed the secondary flow sweeps sand, silt and gravel across the river and deposits the solids near the convex bank, in similar fashion to sugar or tea leaves being swept toward the center of a bowl or cup as described above. This process can lead to accentuation or creation of D-shaped islands, meanders through creation of cut banks and opposing point bars which in turn may result in an oxbow lake. The convex (inner) bank of river bends tends to be shallow and made up of sand, silt and fine gravel; the concave (outer) bank tends to be steep and elevated due to heavy erosion.
Turbomachinery.
Different definitions have been put forward for secondary flow in turbomachinery, such as "Secondary flow in broad terms means flow at right angles to intended primary flow".
Secondary flows occur in the main, or primary, flowpath in turbomachinery compressors and turbines (see also unrelated use of term for flow in the secondary air system of a gas turbine engine). They are always present when a wall boundary layer is turned through an angle by a curved surface. They are a source of total pressure loss and limit the efficiency that can be achieved for the compressor or turbine. Modelling the flow enables blade, vane and end-wall surfaces to be shaped to reduce the losses.
Secondary flows occur throughout the impeller in a centrifugal compressor but are less marked in axial compressors due to shorter passage lengths. Flow turning is low in axial compressors but boundary layers are thick on the annulus walls which gives significant secondary flows. Flow turning in turbine blading and vanes is high and generates strong secondary flow.
Secondary flows also occur in pumps for liquids and include inlet prerotation, or intake vorticity, tip clearance flow (tip leakage), flow separation when operating away from the design condition, and secondary vorticity.
The following, from Dixon, shows the secondary flow generated by flow turning in an axial compressor blade or stator passage. Consider flow with an approach velocity c1. The velocity profile will be non-uniform due to friction between the annulus wall and the fluid. The vorticity of this boundary layer is normal to the approach velocity formula_0 and of magnitude
formula_1 where "z" is the distance to the wall.
As the vorticity of each blade onto each other will be of opposite directions, a secondary vorticity will be generated. If the deflection angle, e, between the guide vanes is small, the magnitude of the secondary vorticity is represented as
formula_2
This secondary flow will be the integrated effect of the distribution of secondary vorticity along the blade length.
Gas turbine engines.
Gas turbine engines have a power-producing primary airflow passing through the compressor. They also have a substantial (25% of core flow in a Pratt & Whitney PW2000) secondary flow obtained from the primary flow and which is pumped from the compressor and used by the secondary air system. Like the secondary flow in turbomachinery this secondary flow is also a loss to the power-producing capability of the engine.
Air-breathing propulsion systems.
Thrust-producing flow which passes through an engines thermal cycle is called primary airflow. Using only cycle flow was relatively short-lived as the turbojet engine. Airflow through a propeller or a turbomachine fan is called secondary flow and is not part of the thermal cycle. This use of secondary flow reduces losses and increases the overall efficiency of the propulsion system. The secondary flow may be many times that through the engine.
Supersonic air-breathing propulsion systems.
During the 1960s cruising at speeds between Mach 2 to 3 was pursued for commercial and military aircraft. Concorde, North American XB-70 and Lockheed SR-71 used ejector-type supersonic nozzles which had a secondary flow obtained from the inlet upstream of the engine compressor. The secondary flow was used to purge the engine compartment, cool the engine case, cool the ejector nozzle and cushion the primary expansion. The secondary flow was ejected by the pumping action of the primary gas flow through the engine nozzle and the ram pressure in the inlet.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "c_{1}"
},
{
"math_id": 1,
"text": "w_1=\\frac{dc_1}{dz},"
},
{
"math_id": 2,
"text": "w_s = -2e \\left(\\frac{dc_1}{dz}\\right)"
}
] |
https://en.wikipedia.org/wiki?curid=14983028
|
149848
|
Combinatory logic
|
Logical formalism using combinators instead of variables
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators, which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.
In mathematics.
Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic. While the expressive power of combinatory logic typically exceeds that of first-order logic, the expressive power of predicate functor logic is identical to that of first order logic (Quine 1960, 1966, 1976).
The original inventor of combinatory logic, Moses Schönfinkel, published nothing on combinatory logic after his original 1924 paper. Haskell Curry rediscovered the combinators while working as an instructor at Princeton University in late 1927. In the late 1930s, Alonzo Church and his students at Princeton invented a rival formalism for functional abstraction, the lambda calculus, which proved more popular than combinatory logic. The upshot of these historical contingencies was that until theoretical computer science began taking an interest in combinatory logic in the 1960s and 1970s, nearly all work on the subject was by Haskell Curry and his students, or by Robert Feys in Belgium. Curry and Feys (1958), and Curry "et al." (1972) survey the early history of combinatory logic. For a more modern treatment of combinatory logic and the lambda calculus together, see the book by Barendregt, which reviews the models Dana Scott devised for combinatory logic in the 1960s and 1970s.
In computing.
In computer science, combinatory logic is used as a simplified model of computation, used in computability theory and proof theory. Despite its simplicity, combinatory logic captures many essential features of computation.
Combinatory logic can be viewed as a variant of the lambda calculus, in which lambda expressions (representing functional abstraction) are replaced by a limited set of "combinators", primitive functions without free variables. It is easy to transform lambda expressions into combinator expressions, and combinator reduction is much simpler than lambda reduction. Hence combinatory logic has been used to model some non-strict functional programming languages and hardware. The purest form of this view is the programming language Unlambda, whose sole primitives are the S and K combinators augmented with character input/output. Although not a practical programming language, Unlambda is of some theoretical interest.
Combinatory logic can be given a variety of interpretations. Many early papers by Curry showed how to translate axiom sets for conventional logic into combinatory logic equations. Dana Scott in the 1960s and 1970s showed how to marry model theory and combinatory logic.
Summary of lambda calculus.
Lambda calculus is concerned with objects called "lambda-terms", which can be represented by the following three forms of strings:
where &NoBreak;&NoBreak; is a variable name drawn from a predefined infinite set of variable names, and &NoBreak;&NoBreak; and &NoBreak;&NoBreak; are lambda-terms.
Terms of the form &NoBreak;&NoBreak; are called "abstractions". The variable "v" is called the formal parameter of the abstraction, and &NoBreak;&NoBreak; is the "body" of the abstraction. The term &NoBreak;&NoBreak; represents the function which, applied to an argument, binds the formal parameter "v" to the argument and then computes the resulting value of &NoBreak;&NoBreak;— that is, it returns &NoBreak;&NoBreak;, with every occurrence of "v" replaced by the argument.
Terms of the form &NoBreak;&NoBreak; are called "applications". Applications model function invocation or execution: the function represented by &NoBreak;&NoBreak; is to be invoked, with &NoBreak;&NoBreak; as its argument, and the result is computed. If &NoBreak;&NoBreak; (sometimes called the "applicand") is an abstraction, the term may be "reduced": &NoBreak;&NoBreak;, the argument, may be substituted into the body of &NoBreak;&NoBreak; in place of the formal parameter of &NoBreak;&NoBreak;, and the result is a new lambda term which is "equivalent" to the old one. If a lambda term contains no subterms of the form &NoBreak;&NoBreak; then it cannot be reduced, and is said to be in normal form.
The expression &NoBreak;&NoBreak; represents the result of taking the term E and replacing all free occurrences of v in it with a. Thus we write
&NoBreak;&NoBreak;
By convention, we take &NoBreak;&NoBreak; as shorthand for &NoBreak;&NoBreak; (i.e., application is left associative).
The motivation for this definition of reduction is that it captures the essential behavior of all mathematical functions. For example, consider the function that computes the square of a number. We might write
The square of "x" is &NoBreak;&NoBreak;
(Using "&NoBreak;&NoBreak;" to indicate multiplication.) "x" here is the formal parameter of the function. To evaluate the square for a particular argument, say 3, we insert it into the definition in place of the formal parameter:
The square of 3 is &NoBreak;&NoBreak;
To evaluate the resulting expression &NoBreak;&NoBreak;, we would have to resort to our knowledge of multiplication and the number 3. Since any computation is simply a composition of the evaluation of suitable functions on suitable primitive arguments, this simple substitution principle suffices to capture the essential mechanism of computation.
Moreover, in lambda calculus, notions such as '3' and '&NoBreak;&NoBreak;' can be represented without any need for externally defined primitive operators or constants. It is possible to identify terms in lambda calculus, which, when suitably interpreted, behave like the number 3 and like the multiplication operator, q.v. Church encoding.
Lambda calculus is known to be computationally equivalent in power to many other plausible models for computation (including Turing machines); that is, any calculation that can be accomplished in any of these other models can be expressed in lambda calculus, and vice versa. According to the Church–Turing thesis, both models can express any possible computation.
It is perhaps surprising that lambda-calculus can represent any conceivable computation using only the simple notions of function abstraction and application based on simple textual substitution of terms for variables. But even more remarkable is that abstraction is not even required. "Combinatory logic" is a model of computation equivalent to lambda calculus, but without abstraction. The advantage of this is that evaluating expressions in lambda calculus is quite complicated because the semantics of substitution must be specified with great care to avoid variable capture problems. In contrast, evaluating expressions in combinatory logic is much simpler, because there is no notion of substitution.
Combinatory calculi.
Since abstraction is the only way to manufacture functions in the lambda calculus, something must replace it in the combinatory calculus. Instead of abstraction, combinatory calculus provides a limited set of primitive functions out of which other functions may be built.
Combinatory terms.
A combinatory term has one of the following forms:
The primitive functions are "combinators", or functions that, when seen as lambda terms, contain no free variables.
To shorten the notations, a general convention is that &NoBreak;&NoBreak;, or even &NoBreak;&NoBreak;, denotes the term &NoBreak;&NoBreak;. This is the same general convention (left-associativity) as for multiple application in lambda calculus.
Reduction in combinatory logic.
In combinatory logic, each primitive combinator comes with a reduction rule of the form
("P" "x"1 ... "xn") = "E"
where "E" is a term mentioning only variables from the set {"x"1 ... "xn"}. It is in this way that primitive combinators behave as functions.
Examples of combinators.
The simplest example of a combinator is I, the identity combinator, defined by
(I "x") = "x"
for all terms "x". Another simple combinator is K, which manufactures constant functions: (K "x") is the function which, for any argument, returns "x", so we say
((K "x") "y") = "x"
for all terms "x" and "y". Or, following the convention for multiple application,
(K "x" "y") = "x"
A third combinator is S, which is a generalized version of application:
(S "x y z") = ("x z" ("y z"))
S applies "x" to "y" after first substituting "z" into
each of them. Or put another way, "x" is applied to "y" inside the environment "z".
Given S and K, I itself is unnecessary, since it can be built from the other two:
((S K K) "x")
= (S K K "x")
= (K "x" (K "x"))
= "x"
for any term "x". Note that although ((S K K)
"x") = (I "x") for any "x", (S K K)
itself is not equal to I. We say the terms are extensionally equal. Extensional equality captures the mathematical notion of the equality of functions: that two functions are "equal" if they always produce the same results for the same arguments. In contrast, the terms themselves, together with the reduction of primitive combinators, capture the notion of "intensional equality" of functions: that two functions are "equal" only if they have identical implementations up to the expansion of primitive combinators. There are many ways to implement an identity function; (S K K) and I are among these ways. (S K S) is yet another. We will use the word "equivalent" to indicate extensional equality, reserving "equal" for identical combinatorial terms.
A more interesting combinator is the fixed point combinator or Y combinator, which can be used to implement recursion.
Completeness of the S-K basis.
S and K can be composed to produce combinators that are extensionally equal to "any" lambda term, and therefore, by Church's thesis, to any computable function whatsoever. The proof is to present a transformation, "T"[ ], which converts an arbitrary lambda term into an equivalent combinator.
"T"[ ] may be defined as follows:
Note that "T"[ ] as given is not a well-typed mathematical function, but rather a term rewriter: Although it eventually yields a combinator, the transformation may generate intermediary expressions that are neither lambda terms nor combinators, via rule (5).
This process is also known as "abstraction elimination". This definition is exhaustive: any lambda expression will be subject to exactly one of these rules (see Summary of lambda calculus above).
It is related to the process of "bracket abstraction", which takes an expression "E" built from variables and application and produces a combinator expression [x]E in which the variable x is not free, such that ["x"]"E x" = "E" holds. A very simple algorithm for bracket abstraction is defined by induction on the structure of expressions as follows:
Bracket abstraction induces a translation from lambda terms to combinator expressions, by interpreting lambda-abstractions using the bracket abstraction algorithm.
Conversion of a lambda term to an equivalent combinatorial term.
For example, we will convert the lambda term "λx"."λy".("y" "x") to a combinatorial term:
"T"["λx"."λy".("y" "x")]
= "T"["λx"."T"["λy".("y" "x")]] (by 5)
= "T"["λx".(S "T"["λy"."y"] "T"["λy"."x"])] (by 6)
= "T"["λx".(S I "T"["λy"."x"])] (by 4)
= "T"["λx".(S I (K "T"["x"]))] (by 3)
= "T"["λx".(S I (K "x"))] (by 1)
= (S "T"["λx".(S I)] "T"["λx".(K "x")]) (by 6)
= (S (K (S I)) "T"["λx".(K "x")]) (by 3)
= (S (K (S I)) (S "T"["λx".K] "T"["λx"."x"])) (by 6)
= (S (K (S I)) (S (K K) "T"["λx"."x"])) (by 3)
= (S (K (S I)) (S (K K) I)) (by 4)
If we apply this combinatorial term to any two terms "x" and "y" (by feeding them in a queue-like fashion into the combinator 'from the right'), it reduces as follows:
(S (K (S I)) (S (K K) I) x y)
= (K (S I) x (S (K K) I x) y)
= (S I (S (K K) I x) y)
= (I y (S (K K) I x y))
= (y (S (K K) I x y))
= (y (K K x (I x) y))
= (y (K (I x) y))
= (y (I x))
= (y x)
The combinatory representation, (S (K (S I)) (S (K K) I)) is much longer than the representation as a lambda term, "λx"."λy".(y x). This is typical. In general, the "T"[ ] construction may expand a lambda term of length "n" to a combinatorial term of length Θ("n"3).
Explanation of the "T"[ ] transformation.
The "T"[ ] transformation is motivated by a desire to eliminate abstraction. Two special cases, rules 3 and 4, are trivial: "λx"."x" is clearly equivalent to I, and "λx"."E" is clearly equivalent to (K "T"["E"]) if "x" does not appear free in "E".
The first two rules are also simple: Variables convert to themselves, and applications, which are allowed in combinatory terms, are converted to combinators simply by converting the applicand and the argument to combinators.
It is rules 5 and 6 that are of interest. Rule 5 simply says that to convert a complex abstraction to a combinator, we must first convert its body to a combinator, and then eliminate the abstraction. Rule 6 actually eliminates the abstraction.
"λx".("E"₁ "E"₂) is a function which takes an argument, say "a", and substitutes it into the lambda term ("E"₁ "E"₂) in place of "x", yielding ("E"₁ "E"₂)["x" : = "a"]. But substituting "a" into ("E"₁ "E"₂) in place of "x" is just the same as substituting it into both "E"₁ and "E"₂, so
("E"₁ "E"₂)["x" := "a"] = ("E"₁["x" := "a"] "E"₂["x" := "a"])
("λx".("E"₁ "E"₂) "a") = (("λx"."E"₁ "a") ("λx"."E"₂ "a"))
= (S "λx"."E"₁ "λx"."E"₂ "a")
= ((S "λx"."E₁" "λx"."E"₂) "a")
By extensional equality,
"λx".("E"₁ "E"₂) = (S "λx"."E"₁ "λx"."E"₂)
Therefore, to find a combinator equivalent to "λx".("E"₁ "E"₂), it is sufficient to find a combinator equivalent to (S "λx"."E"₁ "λx"."E"₂), and
(S "T"["λx"."E"₁] "T"["λx"."E"₂])
evidently fits the bill. "E"₁ and "E"₂ each contain strictly fewer applications than ("E"₁ "E"₂), so the recursion must terminate in a lambda term with no applications at all—either a variable, or a term of the
form "λx"."E".
Simplifications of the transformation.
η-reduction.
The combinators generated by the "T"[ ] transformation can be made smaller if we take into account the "η-reduction" rule:
"T"["λx".("E" "x")] = "T"["E"] (if "x" is not free in "E")
"λx".("E" x) is the function which takes an argument, "x", and applies the function "E" to it; this is extensionally equal to the function "E" itself. It is therefore sufficient to convert "E" to combinatorial form.
Taking this simplification into account, the example above becomes:
"T"["λx"."λy".("y" "x")]
= (S (K (S I)) "T"["λx".(K "x")])
= (S (K (S I)) K) (by η-reduction)
This combinator is equivalent to the earlier, longer one:
(S (K (S I)) K "x y")
= (K (S I) "x" (K "x") "y")
= (S I (K "x") "y")
= (I "y" (K "x y"))
= ("y" (K "x y"))
= ("y x")
Similarly, the original version of the "T"[ ] transformation transformed the identity function "λf"."λx".("f" "x") into (S (S (K S) (S (K K) I)) (K I)). With the η-reduction rule, "λf"."λx".("f" "x") is
transformed into I.
One-point basis.
There are one-point bases from which every combinator can be composed extensionally equal to "any" lambda term. The simplest example of such a basis is {X} where:
X ≡ "λx".((xS)K)
It is not difficult to verify that:
X (X (X X)) =β K and
X (X (X (X X))) =β S.
Since {K, S} is a basis, it follows that {X} is a basis too. The Iota programming language uses X as its sole combinator.
Another simple example of a one-point basis is:
X' ≡ "λx".(x K S K) with
(X' X') X' =β K and
X' (X' X') =β S
In fact, there exist infinitely many such bases.
Combinators B, C.
In addition to S and K, included two combinators which are now called B and C, with the following reductions:
(C "f" "g" "x") = (("f" "x") "g")
(B "f" "g" "x") = ("f" ("g" "x"))
He also explains how they in turn can be expressed using only S and K:
B = (S (K S) K)
C = (S (S (K (S (K S) K)) S) (K K))
These combinators are extremely useful when translating predicate logic or lambda calculus into combinator expressions. They were also used by Curry, and much later by David Turner, whose name has been associated with their computational use. Using them, we can extend the rules for the transformation as follows:
Using B and C combinators, the transformation of "λx"."λy".("y" "x") looks like this:
"T"["λx"."λy".("y" "x")]
= "T"["λx"."T"["λy".("y" "x")]]
= "T"["λx".(C "T"["λy"."y"] "x")] (by rule 7)
= "T"["λx".(C I "x")]
= (C I) (η-reduction)
formula_0 (traditional canonical notation: formula_1)
formula_2 (traditional canonical notation: formula_3)
And indeed, (C I "x" "y") does reduce to ("y" "x"):
(C I "x" "y")
= (I "y" "x")
= ("y" "x")
The motivation here is that B and C are limited versions of S. Whereas S takes a value and substitutes it into both the applicand and its argument before performing the application, C performs the
substitution only in the applicand, and B only in the argument.
The modern names for the combinators come from Haskell Curry's doctoral thesis of 1930 (see B, C, K, W System). In Schönfinkel's original paper, what we now call S, K, I, B and C were called S, C, I, Z, and T respectively.
The reduction in combinator size that results from the new transformation rules can also be achieved without introducing B and C, as demonstrated in Section 3.2 of .
CLK versus CLI calculus.
A distinction must be made between the CLK as described in this article and the CLI calculus. The distinction corresponds to that between the λK and the λI calculus. Unlike the λK calculus, the λI calculus restricts abstractions to:
"λx"."E" where "x" has at least one free occurrence in "E".
As a consequence, combinator K is not present in the λI calculus nor in the CLI calculus. The constants of CLI are: I, B, C and S, which form a basis from which all CLI terms can be composed (modulo equality). Every λI term can be converted into an equal CLI combinator according to rules similar to those presented above for the conversion of λK terms into CLK combinators. See chapter 9 in Barendregt (1984).
Reverse conversion.
The conversion "L"[ ] from combinatorial terms to lambda terms is trivial:
"L"[I] = "λx"."x"
"L"[K] = "λx"."λy"."x"
"L"[C] = "λx"."λy"."λz".("x" "z" "y")
"L"[B] = "λx"."λy"."λz".("x" ("y" "z"))
"L"[S] = "λx"."λy"."λz".("x" "z" ("y" "z"))
"L"[("E₁" "E₂")] = ("L"["E₁"] "L"["E₂"])
Note, however, that this transformation is not the inverse
transformation of any of the versions of "T"[ ] that we have seen.
Undecidability of combinatorial calculus.
A normal form is any combinatory term in which the primitive combinators that occur, if any, are not applied to enough arguments to be simplified. It is undecidable whether a general combinatory term has a normal form; whether two combinatory terms are equivalent, etc. This can be shown in a similar way as for the corresponding problems for lambda terms.
Undefinability by predicates.
The undecidable problems above (equivalence, existence of normal form, etc.) take as input syntactic representations of terms under a suitable encoding (e.g., Church encoding). One may also consider a toy trivial computation model where we "compute" properties of terms by means of combinators applied directly to the terms themselves as arguments, rather than to their syntactic representations. More precisely, let a "predicate" be a combinator that, when applied, returns either T or F (where T and F represent the conventional Church encodings of true and false, "λx"."λy"."x" and "λx"."λy"."y", transformed into combinatory logic; the combinatory versions have T = K and F = (K I)). A predicate N is "nontrivial" if there are two arguments "A" and "B" such that N "A" = T and N "B" = F. A combinator N is "complete" if N"M" has a normal form for every argument "M". An analogue of Rice's theorem for this toy model then says that every complete predicate is trivial. The proof of this theorem is rather simple.
<templatestyles src="Math_proof/styles.css" />Proof
By reductio ad absurdum. Suppose there is a complete non trivial predicate, say N. Because N is supposed to be non trivial there are combinators "A" and "B" such that
(N "A") = T and
(N "B") = F.
Define NEGATION ≡ "λx".(if (N "x") then "B" else "A") ≡ "λx".((N "x") "B" "A")
Define ABSURDUM ≡ (Y NEGATION)
Fixed point theorem gives: ABSURDUM = (NEGATION ABSURDUM), for
ABSURDUM ≡ (Y NEGATION) = (NEGATION (Y NEGATION)) ≡ (NEGATION ABSURDUM).
Because N is supposed to be complete either:
Hence (N ABSURDUM) is neither T nor F, which contradicts the presupposition that N would be a complete non trivial predicate. Q.E.D.
From this undefinability theorem it immediately follows that there is no complete predicate that can discriminate between terms that have a normal form and terms that do not have a normal form. It also follows that there is no complete predicate, say EQUAL, such that:
(EQUAL "A B") = T if "A" = "B" and
(EQUAL "A B") = F if "A" ≠ "B".
If EQUAL would exist, then for all "A", "λx."(EQUAL "x A") would have to be a complete non trivial predicate.
However, note that it also immediately follows from this undefinability theorem that many properties of terms that are obviously decidable are not definable by complete predicates either: e.g., there is no predicate that could tell whether the first primitive function letter occurring in a term is a K. This shows that definability by predicates is a not a reasonable model of decidability.
Applications.
Compilation of functional languages.
David Turner used his combinators to implement the SASL programming language.
Kenneth E. Iverson used primitives based on Curry's combinators in his J programming language, a successor to APL. This enabled what Iverson called tacit programming, that is, programming in functional expressions containing no variables, along with powerful tools for working with such programs. It turns out that tacit programming is possible in any APL-like language with user-defined operators.
Logic.
The Curry–Howard isomorphism implies a connection between logic and programming: every proof of a theorem of intuitionistic logic corresponds to a reduction of a typed lambda term, and conversely. Moreover, theorems can be identified with function type signatures. Specifically, a typed combinatory logic corresponds to a Hilbert system in proof theory.
The K and S combinators correspond to the axioms
AK: "A" → ("B" → "A"),
AS: ("A" → ("B" → "C")) → (("A" → "B") → ("A" → "C")),
and function application corresponds to the detachment (modus ponens) rule
MP: from "A" and "A" → "B" infer "B".
The calculus consisting of AK, AS, and MP is complete for the implicational fragment of the intuitionistic logic, which can be seen as follows. Consider the set "W" of all deductively closed sets of formulas, ordered by inclusion. Then formula_4 is an intuitionistic Kripke frame, and we define a model formula_5 in this frame by
formula_6
This definition obeys the conditions on satisfaction of →: on one hand, if formula_7, and formula_8 is such that formula_9 and formula_10, then formula_11 by modus ponens. On the other hand, if formula_12, then formula_13 by the deduction theorem, thus the deductive closure of formula_14 is an element formula_8 such that formula_9, formula_10, and formula_15.
Let "A" be any formula which is not provable in the calculus. Then "A" does not belong to the deductive closure "X" of the empty set, thus formula_16, and "A" is not intuitionistically valid.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "= \\mathbf{C}_{*}"
},
{
"math_id": 1,
"text": "\\mathbf{X}_{*} = \\mathbf{X I}"
},
{
"math_id": 2,
"text": "= \\mathbf{I}'"
},
{
"math_id": 3,
"text": "\\mathbf{X}' = \\mathbf{C X}"
},
{
"math_id": 4,
"text": "\\langle W,\\subseteq\\rangle"
},
{
"math_id": 5,
"text": "\\Vdash"
},
{
"math_id": 6,
"text": "X\\Vdash A\\iff A\\in X."
},
{
"math_id": 7,
"text": "X\\Vdash A\\to B"
},
{
"math_id": 8,
"text": "Y\\in W"
},
{
"math_id": 9,
"text": "Y\\supseteq X"
},
{
"math_id": 10,
"text": "Y\\Vdash A"
},
{
"math_id": 11,
"text": "Y\\Vdash B"
},
{
"math_id": 12,
"text": "X\\not\\Vdash A\\to B"
},
{
"math_id": 13,
"text": "X,A\\not\\vdash B"
},
{
"math_id": 14,
"text": "X\\cup\\{A\\}"
},
{
"math_id": 15,
"text": "Y\\not\\Vdash B"
},
{
"math_id": 16,
"text": "X\\not\\Vdash A"
}
] |
https://en.wikipedia.org/wiki?curid=149848
|
149861
|
Work (physics)
|
Process of energy transfer to an object via force application through displacement
<templatestyles src="Hlist/styles.css"/>
In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do "positive work" if it has a component in the direction of the displacement of the point of application. A force does "negative work" if it has a component opposite to the direction of the displacement at the point of application of the force.
For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction.
Both force and displacement are vectors. The work done is given by the dot product of the two vectors, where the result is a scalar. When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by:
formula_0
If the force is variable, then work is given by the line integral:
formula_1
where formula_2 is the tiny change in displacement vector.
Work is a scalar quantity, so it has only magnitude and no direction. Work transfers energy from one place to another, or one form to another. The SI unit of work is the joule (J), the same unit as for energy.
History.
The ancient Greek understanding of physics was limited to the statics of simple machines (the balance of forces), and did not include dynamics or the concept of work. During the Renaissance the dynamics of the "Mechanical Powers", as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in "Le Meccaniche" ("On Mechanics"), in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain that simple machines do not create energy, only transform it.
Early concepts of work.
Although "work" was not formally used until 1826, similar concepts existed before then. Early names for the same concept included "moment of activity, quantity of action, latent live force, dynamic effect, efficiency", and even "force". In 1637, the French philosopher René Descartes wrote:
<templatestyles src="Template:Blockquote/styles.css" />Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet.
In 1686, the German philosopher Gottfried Leibniz wrote:
<templatestyles src="Template:Blockquote/styles.css" />The same force ["work" in modern terms] is necessary to raise body A of 1 pound (libra) to a height of 4 yards (ulnae), as is necessary to raise body B of 4 pounds to a height of 1 yard.
In 1759, John Smeaton described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion." Smeaton continues that this quantity can be calculated if "the weight raised is multiplied by the height to which it can be raised in a given time," making this definition remarkably similar to Coriolis's.
Etymology.
According to the 1957 physics textbook by Max Jammer, the term "work" was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis as "weight "lifted" through a height", which is based on the use of early steam engines to lift buckets of water out of flooded ore mines. According to Rene Dugas, French engineer and historian, it is to Solomon of Caux "that we owe the term "work" in the sense that it is used in mechanics now".
Units.
The SI unit of work is the joule (J), named after English physicist James Prescott Joule (1818-1889), which is defined as the work required to exert a force of one newton through a displacement of one metre.
The dimensionally equivalent newton-metre (N⋅m) is sometimes used as the measuring unit for work, but this can be confused with the measurement unit of torque. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton-metres is a torque measurement, or a measurement of work.
Another unit for work is the foot-pound, which comes from the English system of measurement. As the unit name suggests, it is the product of pounds for the unit of force and feet for the unit of displacement. One joule is equivalent to 0.07376 ft-lbs.
Non-SI units of work include the newton-metre, erg, the foot-pound, the foot-poundal, the kilowatt hour, the litre-atmosphere, and the horsepower-hour. Due to work having the same physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU and calorie, are used as a measuring unit.
Work and energy.
The work W done by a constant force of magnitude F on a point that moves a displacement s in a straight line in the direction of the force is the product
formula_3
For example, if a force of 10 newtons ("F" = 10 N) acts along a point that travels 2 metres ("s" = 2 m), then "W" = "Fs" = (10 N) (2 m) = 20 J. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity.
The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance.
Work is closely related to energy. Energy shares the same unit of measurement with work (Joules) because the energy from the object doing work is transferred to the other objects it interacts with when work is being done. The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force. Thus, if the net work is positive, then the particle's kinetic energy increases by the amount of the work. If the net work done is negative, then the particle's kinetic energy decreases by the amount of work.
From Newton's second law, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy "E"k corresponding to the linear velocity and angular velocity of that body,
formula_4
The work of forces generated by a potential function is known as potential energy and the forces are said to be conservative. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to "minus" the change of potential energy "E"p of the object,
formula_5
These formulas show that work is the energy associated with the action of a force, so work subsequently possesses the physical dimensions, and units, of energy.
The work/energy principles discussed here are identical to electric work/energy principles.
Constraint forces.
Constraint forces determine the object's displacement in the system, limiting it within a range. For example, in the case of a slope plus gravity, the object is "stuck to" the slope and, when attached to a taut string, it cannot move in an outwards direction to make the string any 'tauter'. It eliminates all displacements in that direction, that is, the velocity in the direction of the constraint is limited to 0, so that the constraint forces do not perform work on the system.
For a mechanical system, constraint forces eliminate movement in directions that characterize the constraint. Thus the virtual work done by the forces of constraint is zero, a result which is only true if friction forces are excluded.
Fixed, frictionless constraint forces do not perform work on the system, as the angle between the motion and the constraint forces is always 90°. Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping.
For example, in a pulley system like the Atwood machine, the internal forces on the rope and at the supporting pulley do no work on the system. Therefore, work need only be computed for the gravitational forces acting on the bodies. Another example is the centripetal force exerted "inwards" by a string on a ball in uniform circular motion "sideways" constrains the ball to circular motion restricting its movement away from the centre of the circle. This force does zero work because it is perpendicular to the velocity of the ball.
The magnetic force on a charged particle is F = "q"v × B, where q is the charge, v is the velocity of the particle, and B is the magnetic field. The result of a cross product is always perpendicular to both of the original vectors, so F ⊥ v. The dot product of two perpendicular vectors is always zero, so the work "W" = F ⋅ v = 0, and the magnetic force does not do work. It can change the direction of motion but never change the speed.
Mathematical calculation.
For moving objects, the quantity of work/time (power) is integrated along the trajectory of the point of application of the force. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector), and the velocity vector of the point of application. This scalar product of force and velocity is known as instantaneous power. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.
Work is the result of a force on a point that follows a curve X, with a velocity v, at each instant. The small amount of work "δW" that occurs over an instant of time "dt" is calculated as
formula_6
where the F ⋅ v is the power over the instant "dt". The sum of these small amounts of work over the trajectory of the point yields the work,
formula_7
where "C" is the trajectory from x("t"1) to x("t"2). This integral is computed along the trajectory of the particle, and is therefore said to be "path dependent".
If the force is always directed along this line, and the magnitude of the force is "F", then this integral simplifies to
formula_8
where s is displacement along the line. If F is constant, in addition to being directed along the line, then the integral simplifies further to
formula_9
where "s" is the displacement of the point along the line.
This calculation can be generalized for a constant force that is not directed along the line, followed by the particle. In this case the dot product F ⋅ "d"s = "F" cos "θ" "ds", where θ is the angle between the force vector and the direction of movement, that is
formula_10
When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. Thus, no work can be performed by gravity on a planet with a circular orbit (this is ideal, as all orbits are slightly elliptical). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge.
Work done by a variable force.
Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of force can be described by the scalar quantity called "scalar tangential component" ("F" cos("θ"), where θ is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows:
<templatestyles src="Block indent/styles.css"/>"Work done by a variable force is the line integral of its scalar tangential component along the path of its application point."
If the force varies (e.g. compressing a spring) we need to use calculus to find the work done. If the force as a variable of x is given by F(x), then the work done by the force along the x-axis from x1 to x2 is:
formula_11
Thus, the work done for a variable force can be expressed as a definite integral of force over displacement.
If the displacement as a variable of time is given by ∆x(t), then work done by the variable force from t1 to t2 is:
formula_12
Thus, the work done for a variable force can be expressed as a definite integral of power over time.
Torque and rotation.
A force couple results from equal and opposite forces, acting on two different points of a rigid body. The sum (resultant) of these forces may cancel, but their effect on the body is the couple or torque T. The work of the torque is calculated as
formula_13
where the T ⋅ ω is the power over the instant "dt". The sum of these small amounts of work over the trajectory of the rigid body yields the work,
formula_14
This integral is computed along the trajectory of the rigid body with an angular velocity ω that varies with time, and is therefore said to be "path dependent".
If the angular velocity vector maintains a constant direction, then it takes the form,
formula_15
where formula_16 is the angle of rotation about the constant unit vector S. In this case, the work of the torque becomes,
formula_17
where "C" is the trajectory from formula_18 to formula_19. This integral depends on the rotational trajectory formula_20, and is therefore path-dependent.
If the torque formula_21 is aligned with the angular velocity vector so that,
formula_22
and both the torque and angular velocity are constant, then the work takes the form,
formula_23
This result can be understood more simply by considering the torque as arising from a force of constant magnitude "F", being applied perpendicularly to a lever arm at a distance formula_24, as shown in the figure. This force will act through the distance along the circular arc formula_25, so the work done is
formula_26
Introduce the torque "τ" = "Fr", to obtain
formula_27
as presented above.
Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.
Work and potential energy.
The scalar product of a force F and the velocity v of its point of application defines the power input to a system at an instant of time. Integration of this power over the trajectory of the point of application, "C" = x("t"), defines the work input to the system by the force.
Path dependence.
Therefore, the work done by a force F on an object that travels along a curve "C" is given by the line integral:
formula_28
where "dx"("t") defines the trajectory "C" and v is the velocity along this trajectory.
In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent.
The time derivative of the integral for work yields the instantaneous power,
formula_29
Path independence.
If the work for an applied force is independent of the path, then the work done by the force, by the gradient theorem, defines a potential function which is evaluated at the start and end of the trajectory of the point of application. This means that there is a potential function "U"(x), that can be evaluated at the two points x("t"1) and x("t"2) to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is
formula_30
The function "U"(x) is called the potential energy associated with the applied force. The force derived from such a potential function is said to be conservative. Examples of forces that have potential energies are gravity and spring forces.
In this case, the gradient of work yields
formula_31
and the force F is said to be "derivable from a potential."
Because the potential U defines a force F at every point x in space, the set of forces is called a force field. The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity V of the body, that is
formula_32
Work by gravity.
In the absence of other forces, gravity results in a constant downward acceleration of every freely moving object. Near Earth's surface the acceleration due to gravity is "g" = 9.8 m⋅s−2 and the gravitational force on an object of mass "m" is Fg = "mg". It is convenient to imagine this gravitational force concentrated at the center of mass of the object.
If an object with weight "mg" is displaced upwards or downwards a vertical distance "y"2 − "y"1, the work "W" done on the object is:
formula_33
where "Fg" is weight (pounds in imperial units, and newtons in SI units), and Δ"y" is the change in height "y". Notice that the work done by gravity depends only on the vertical movement of the object. The presence of friction does not affect the work done on the object by its weight.
In space.
The force of gravity exerted by a mass M on another mass m is given by
formula_34
where r is the position vector from M to m and r̂ is the unit vector in the direction of r.
Let the mass m move at the velocity v; then the work of gravity on this mass as it moves from position r("t"1) to r("t"2) is given by
formula_35
Notice that the position and velocity of the mass m are given by
formula_36
where e"r" and e"t" are the radial and tangential unit vectors directed relative to the vector from M to m, and we use the fact that formula_37 Use this to simplify the formula for work of gravity to,
formula_38
This calculation uses the fact that
formula_39
The function
formula_40
is the gravitational potential function, also known as gravitational potential energy. The negative sign follows the convention that work is gained from a loss of potential energy.
Work by a spring.
Consider a spring that exerts a horizontal force F = (−"kx", 0, 0) that is proportional to its deflection in the "x" direction independent of how a body moves. The work of this spring on a body moving along the space with the curve X("t") = ("x"("t"), "y"("t"), "z"("t")), is calculated using its velocity, v = ("v"x, "v"y, "v"z), to obtain
formula_41
For convenience, consider contact with the spring occurs at "t" = 0, then the integral of the product of the distance x and the x-velocity, "xv"x"dt", over time t is "x"2. The work is the product of the distance times the spring force, which is also dependent on distance; hence the "x"2 result.
Work by a gas.
The work formula_42 done by a body of gas on its surroundings is:
formula_43
where P is pressure, V is volume, and a and b are initial and final volumes.
Work–energy principle.
The principle of work and kinetic energy (also known as the work–energy principle) states that "the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle." That is, the work "W" done by the resultant force on a particle equals the change in the particle's kinetic energy formula_44,
formula_45
where formula_46 and formula_47 are the speeds of the particle before and after the work is done, and m is its mass.
The derivation of the "work–energy principle" begins with Newton's second law of motion and the resultant force on a particle. Computation of the scalar product of the force with the velocity of the particle evaluates the instantaneous power added to the system.
(Constraints define the direction of movement of the particle by ensuring there is no component of velocity in the direction of the constraint force. This also means the constraint forces do not add to the instantaneous power.) The time integral of this scalar equation yields work from the instantaneous power, and kinetic energy from the scalar product of acceleration with velocity. The fact that the work–energy principle eliminates the constraint forces underlies Lagrangian mechanics.
This section focuses on the work–energy principle as it applies to particle dynamics. In more general systems work can change the potential energy of a mechanical device, the thermal energy in a thermal system, or the electrical energy in an electrical device. Work transfers energy from one place to another or one form to another.
Derivation for a particle moving along a straight line.
In the case the resultant force F is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration "a" along a straight line. The relation between the net force and the acceleration is given by the equation "F" = "ma" (Newton's second law), and the particle displacement s can be expressed by the equation
formula_48
which follows from formula_49 (see Equations of motion).
The work of the net force is calculated as the product of its magnitude and the particle displacement. Substituting the above equations, one obtains:
formula_50
Other derivation:
formula_51
In the general case of rectilinear motion, when the net force F is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle:
formula_52
General derivation of the work–energy principle for a particle.
For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. It is known as the work–energy principle:
formula_53
The identity formula_54 requires some algebra.
From the identity formula_55 and definition formula_56
it follows
formula_57
The remaining part of the above derivation is just simple calculus, same as in the preceding rectilinear case.
Derivation for a particle in constrained movement.
In particle dynamics, a formula equating work applied to a system to its change in kinetic energy is obtained as a first integral of Newton's second law of motion. It is useful to notice that the resultant force used in Newton's laws can be separated into forces that are applied to the particle and forces imposed by constraints on the movement of the particle. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle.
To see this, consider a particle P that follows the trajectory X("t") with a force F acting on it. Isolate the particle from its environment to expose constraint forces R, then Newton's Law takes the form
formula_58
where m is the mass of the particle.
Vector formulation.
Note that n dots above a vector indicates its nth time derivative.
The scalar product of each side of Newton's law with the velocity vector yields
formula_59
because the constraint forces are perpendicular to the particle velocity. Integrate this equation along its trajectory from the point X("t"1) to the point X("t"2) to obtain
formula_60
The left side of this equation is the work of the applied force as it acts on the particle along the trajectory from time "t"1 to time "t"2. This can also be written as
formula_61
This integral is computed along the trajectory X("t") of the particle and is therefore path dependent.
The right side of the first integral of Newton's equations can be simplified using the following identity
formula_62
(see product rule for derivation). Now it is integrated explicitly to obtain the change in kinetic energy,
formula_63
where the kinetic energy of the particle is defined by the scalar quantity,
formula_64
Tangential and normal components.
It is useful to resolve the velocity and acceleration vectors into tangential and normal components along the trajectory X("t"), such that
formula_65
where
formula_66
Then, the scalar product of velocity with acceleration in Newton's second law takes the form
formula_67
where the kinetic energy of the particle is defined by the scalar quantity,
formula_68
The result is the work–energy principle for particle dynamics,
formula_69
This derivation can be generalized to arbitrary rigid body systems.
Moving in a straight line (skid to a stop).
Consider the case of a vehicle moving along a straight horizontal trajectory under the action of a driving force and gravity that sum to F. The constraint forces between the vehicle and the road define R, and we have
formula_70
For convenience let the trajectory be along the X-axis, so X = ("d", 0) and the velocity is V = ("v", 0), then R ⋅ V = 0, and F ⋅ V = "F"x"v", where "F"x is the component of F along the X-axis, so
formula_71
Integration of both sides yields
formula_72
If "F"x is constant along the trajectory, then the integral of velocity is distance, so
formula_73
As an example consider a car skidding to a stop, where "k" is the coefficient of friction and "W" is the weight of the car. Then the force along the trajectory is "F"x = −"kW". The velocity "v" of the car can be determined from the length s of the skid using the work–energy principle,
formula_74
This formula uses the fact that the mass of the vehicle is "m" = "W"/"g".
Coasting down an inclined surface (gravity racing).
Consider the case of a vehicle that starts at rest and coasts down an inclined surface (such as mountain road), the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity "V", of say 60 mph (88 fps). Rolling resistance and air drag will slow the vehicle down so the actual distance will be greater than if these forces are neglected.
Let the trajectory of the vehicle following the road be X("t") which is a curve in three-dimensional space. The force acting on the vehicle that pushes it down the road is the constant force of gravity F = (0, 0, "W"), while the force of the road on the vehicle is the constraint force R. Newton's second law yields,
formula_75
The scalar product of this equation with the velocity, V = ("v"x, "v"y, "v"z), yields
formula_76
where "V" is the magnitude of V. The constraint forces between the vehicle and the road cancel from this equation because R ⋅ V = 0, which means they do no work.
Integrate both sides to obtain
formula_77
The weight force "W" is constant along the trajectory and the integral of the vertical velocity is the vertical distance, therefore,
formula_78
Recall that V("t"1)=0. Notice that this result does not depend on the shape of the road followed by the vehicle.
In order to determine the distance along the road assume the downgrade is 6%, which is a steep road. This means the altitude decreases 6 feet for every 100 feet traveled—for angles this small the sin and tan functions are approximately equal. Therefore, the distance s in feet down a 6% grade to reach the velocity V is at least
formula_79
This formula uses the fact that the weight of the vehicle is "W" = "mg".
Work of forces acting on a rigid body.
The work of forces acting at various points on a single rigid body can be calculated from the work of a resultant force and torque. To see this, let the forces F1, F2, ..., Fn act on the points X1, X2, ..., X"n" in a rigid body.
The trajectories of X"i", "i" = 1, ..., "n" are defined by the movement of the rigid body. This movement is given by the set of rotations ["A"("t")] and the trajectory d("t") of a reference point in the body. Let the coordinates x"i" "i" = 1, ..., "n" define these points in the moving rigid body's reference frame "M", so that the trajectories traced in the fixed frame "F" are given by
formula_80
The velocity of the points X"i" along their trajectories are
formula_81
where ω is the angular velocity vector obtained from the skew symmetric matrix
formula_82
known as the angular velocity matrix.
The small amount of work by the forces over the small displacements "δr"i" can be determined by approximating the displacement by "δr = v"δt" so
formula_83
or
formula_84
This formula can be rewritten to obtain
formula_85
where F and T are the resultant force and torque applied at the reference point d of the moving frame "M" in the rigid body.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " W = F s \\cos{\\theta}"
},
{
"math_id": 1,
"text": "W = \\int \\vec{F} \\cdot d\\vec{s}\n"
},
{
"math_id": 2,
"text": "d\\vec{s}\n"
},
{
"math_id": 3,
"text": " W = \\vec{F} \\cdot \\vec{s}\n"
},
{
"math_id": 4,
"text": " W = \\Delta E_\\text{k}."
},
{
"math_id": 5,
"text": " W = -\\Delta E_\\text{p}."
},
{
"math_id": 6,
"text": " \\delta W = \\mathbf{F} \\cdot d\\mathbf{s} = \\mathbf{F} \\cdot \\mathbf{v}dt "
},
{
"math_id": 7,
"text": " W = \\int_{t_1}^{t_2}\\mathbf{F} \\cdot \\mathbf{v} \\, dt = \\int_{t_1}^{t_2}\\mathbf{F} \\cdot \\tfrac{d\\mathbf{s}}{dt} \\, dt =\\int_C \\mathbf{F} \\cdot d\\mathbf{s},"
},
{
"math_id": 8,
"text": " W = \\int_C F\\,ds"
},
{
"math_id": 9,
"text": " W = \\int_C F\\,ds = F\\int_C ds = Fs"
},
{
"math_id": 10,
"text": "W = \\int_C \\mathbf{F} \\cdot d\\mathbf{s} = Fs\\cos\\theta."
},
{
"math_id": 11,
"text": "W = \\lim_{\\Delta\\mathbf{x} \\to 0}\\sum_{x_1}^{x_2}\\mathbf{F(x)}\\Delta\\mathbf{x} = \\int_{x_1}^{x_2}\\mathbf{F(x)}d\\mathbf{x}."
},
{
"math_id": 12,
"text": "W = \\int_{t_1}^{t_2}\\mathbf{F}(t)\\cdot \\mathbf{v}(t)dt = \\int_{t_1}^{t_2}P(t)dt."
},
{
"math_id": 13,
"text": " \\delta W = \\mathbf{T} \\cdot \\boldsymbol{\\omega} \\, dt,"
},
{
"math_id": 14,
"text": " W = \\int_{t_1}^{t_2} \\mathbf{T} \\cdot \\boldsymbol{\\omega} \\, dt."
},
{
"math_id": 15,
"text": " \\boldsymbol{\\omega} = \\dot{\\phi}\\mathbf{S},"
},
{
"math_id": 16,
"text": "\\phi"
},
{
"math_id": 17,
"text": "W = \\int_{t_1}^{t_2} \\mathbf{T} \\cdot \\boldsymbol{\\omega} \\, dt = \\int_{t_1}^{t_2} \\mathbf{T} \\cdot \\mathbf{S} \\frac{d\\phi}{dt} dt = \\int_C\\mathbf{T}\\cdot \\mathbf{S} \\, d\\phi,"
},
{
"math_id": 18,
"text": "\\phi (t_{1})"
},
{
"math_id": 19,
"text": "\\phi (t_{2})"
},
{
"math_id": 20,
"text": "\\phi (t)"
},
{
"math_id": 21,
"text": "\\tau"
},
{
"math_id": 22,
"text": " \\mathbf{T} = \\tau \\mathbf{S},"
},
{
"math_id": 23,
"text": "W = \\int_{t_1}^{t_2} \\tau \\dot{\\phi} \\, dt = \\tau(\\phi_2 - \\phi_1)."
},
{
"math_id": 24,
"text": "r"
},
{
"math_id": 25,
"text": "l=s=r\\phi"
},
{
"math_id": 26,
"text": " W = F s = F r \\phi ."
},
{
"math_id": 27,
"text": " W = F r \\phi = \\tau \\phi ,"
},
{
"math_id": 28,
"text": " W = \\int_C \\mathbf{F} \\cdot d\\mathbf{x} = \\int_{t_1}^{t_2}\\mathbf{F}\\cdot \\mathbf{v}dt,"
},
{
"math_id": 29,
"text": "\\frac{dW}{dt} = P(t) = \\mathbf{F}\\cdot \\mathbf{v} ."
},
{
"math_id": 30,
"text": " W = \\int_C \\mathbf{F} \\cdot d\\mathbf{x} = \\int_{\\mathbf{x}(t_1)}^{\\mathbf{x}(t_2)} \\mathbf{F} \\cdot d\\mathbf{x} = U(\\mathbf{x}(t_1))-U(\\mathbf{x}(t_2))."
},
{
"math_id": 31,
"text": " \\nabla W = -\\nabla U = -\\left(\\frac{\\partial U}{\\partial x}, \\frac{\\partial U}{\\partial y}, \\frac{\\partial U}{\\partial z}\\right) = \\mathbf{F},"
},
{
"math_id": 32,
"text": "P(t) = -\\nabla U \\cdot \\mathbf{v} = \\mathbf{F}\\cdot\\mathbf{v}."
},
{
"math_id": 33,
"text": "W = F_g (y_2 - y_1) = F_g\\Delta y = mg\\Delta y"
},
{
"math_id": 34,
"text": " \\mathbf{F} = -\\frac{GMm}{r^2} \\hat\\mathbf{r} = -\\frac{GMm}{r^3}\\mathbf{r},"
},
{
"math_id": 35,
"text": " W = -\\int^{\\mathbf{r}(t_2)}_{\\mathbf{r}(t_1)} \\frac{GMm}{r^3} \\mathbf{r} \\cdot d\\mathbf{r} = -\\int^{t_2}_{t_1} \\frac{GMm}{r^3}\\mathbf{r} \\cdot \\mathbf{v} \\, dt."
},
{
"math_id": 36,
"text": " \\mathbf{r} = r\\mathbf{e}_r, \\qquad\\mathbf{v} = \\frac{d\\mathbf{r}}{dt} = \\dot{r}\\mathbf{e}_r + r\\dot{\\theta}\\mathbf{e}_t,"
},
{
"math_id": 37,
"text": " d \\mathbf{e}_r / dt = \\dot{\\theta}\\mathbf{e}_t. "
},
{
"math_id": 38,
"text": " W = -\\int^{t_2}_{t_1}\\frac{GmM}{r^3}(r\\mathbf{e}_r) \\cdot \\left(\\dot{r}\\mathbf{e}_r + r\\dot{\\theta}\\mathbf{e}_t\\right) dt = -\\int^{t_2}_{t_1}\\frac{GmM}{r^3}r\\dot{r}dt = \\frac{GMm}{r(t_2)}-\\frac{GMm}{r(t_1)}."
},
{
"math_id": 39,
"text": " \\frac{d}{dt}r^{-1} = -r^{-2}\\dot{r} = -\\frac{\\dot{r}}{r^2}."
},
{
"math_id": 40,
"text": " U = -\\frac{GMm}{r}, "
},
{
"math_id": 41,
"text": " W=\\int_0^t\\mathbf{F}\\cdot\\mathbf{v}dt =-\\int_0^tkx v_x dt = -\\frac{1}{2}kx^2. "
},
{
"math_id": 42,
"text": "W"
},
{
"math_id": 43,
"text": " W = \\int_a^b P \\, dV "
},
{
"math_id": 44,
"text": "E_\\text{k}"
},
{
"math_id": 45,
"text": " W = \\Delta E_\\text{k} = \\frac{1}{2}mv_2^2 - \\frac{1}{2}mv_1^2"
},
{
"math_id": 46,
"text": "v_1"
},
{
"math_id": 47,
"text": "v_2"
},
{
"math_id": 48,
"text": "s = \\frac{v_2^2 - v_1^2}{2a}"
},
{
"math_id": 49,
"text": "v_2^2 = v_1^2 + 2as"
},
{
"math_id": 50,
"text": " W = Fs = mas = ma\\frac{v_2^2-v_1^2}{2a} = \\frac{mv_2^2}{2}- \\frac{mv_1^2}{2} = \\Delta E_\\text{k}"
},
{
"math_id": 51,
"text": " W = Fs = mas = m\\frac{v_2^2 - v_1^2}{2s}s = \\frac{1}{2}mv_2^2-\\frac{1}{2}mv_1^2 = \\Delta E_\\text{k}"
},
{
"math_id": 52,
"text": " W = \\int_{t_1}^{t_2} \\mathbf{F}\\cdot \\mathbf{v}dt = \\int_{t_1}^{t_2} F \\,v \\, dt = \\int_{t_1}^{t_2} ma \\,v \\, dt = m \\int_{t_1}^{t_2} v \\,\\frac{dv}{dt}\\,dt = m \\int_{v_1}^{v_2} v\\,dv = \\tfrac12 m \\left(v_2^2 - v_1^2\\right) ."
},
{
"math_id": 53,
"text": " W = \\int_{t_1}^{t_2} \\mathbf{F}\\cdot \\mathbf{v}dt = m \\int_{t_1}^{t_2} \\mathbf{a} \\cdot \\mathbf{v}dt = \\frac{m}{2} \\int_{t_1}^{t_2} \\frac{d v^2}{dt}\\,dt = \\frac{m}{2} \\int_{v^2_1}^{v^2_2} d v^2 = \\frac{mv_2^2}{2} - \\frac{mv_1^2}{2} = \\Delta E_\\text{k} "
},
{
"math_id": 54,
"text": "\\mathbf{a} \\cdot \\mathbf{v} = \\frac{1}{2} \\frac{d v^2}{dt}"
},
{
"math_id": 55,
"text": "v^2 = \\mathbf{v} \\cdot \\mathbf{v}"
},
{
"math_id": 56,
"text": "\\mathbf{a} = \\frac{d \\mathbf{v}}{dt} "
},
{
"math_id": 57,
"text": " \\frac{d v^2}{dt} = \\frac{d (\\mathbf{v} \\cdot \\mathbf{v})}{dt} = \\frac{d \\mathbf{v}}{dt} \\cdot \\mathbf{v} + \\mathbf{v} \\cdot \\frac{d \\mathbf{v}}{dt} = 2 \\frac{d \\mathbf{v}}{dt} \\cdot \\mathbf{v} = 2 \\mathbf{a} \\cdot \\mathbf{v} ."
},
{
"math_id": 58,
"text": " \\mathbf{F} + \\mathbf{R} = m \\ddot{\\mathbf{X}}, "
},
{
"math_id": 59,
"text": " \\mathbf{F}\\cdot\\dot{\\mathbf{X}} = m\\ddot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}},"
},
{
"math_id": 60,
"text": " \\int_{t_1}^{t_2} \\mathbf{F} \\cdot \\dot{\\mathbf{X}} dt = m \\int_{t_1}^{t_2} \\ddot{\\mathbf{X}} \\cdot \\dot{\\mathbf{X}} dt. "
},
{
"math_id": 61,
"text": " W = \\int_{t_1}^{t_2} \\mathbf{F}\\cdot\\dot{\\mathbf{X}} dt = \\int_{\\mathbf{X}(t_1)}^{\\mathbf{X}(t_2)} \\mathbf{F}\\cdot d\\mathbf{X}. "
},
{
"math_id": 62,
"text": " \\frac{1}{2}\\frac{d}{dt}(\\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}}) = \\ddot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}, "
},
{
"math_id": 63,
"text": "\\Delta K = m\\int_{t_1}^{t_2}\\ddot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}dt = \\frac{m}{2}\\int_{t_1}^{t_2}\\frac{d}{dt} (\\dot{\\mathbf{X}} \\cdot \\dot{\\mathbf{X}}) dt = \\frac{m}{2} \\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}}(t_2) - \\frac{m}{2} \\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}} (t_1) = \\frac{1}{2}m \\Delta \\mathbf{v}^2 , "
},
{
"math_id": 64,
"text": " K = \\frac{m}{2} \\dot{\\mathbf{X}} \\cdot \\dot{\\mathbf{X}} =\\frac{1}{2} m {\\mathbf{v}^2}"
},
{
"math_id": 65,
"text": " \\dot{\\mathbf{X}}=v \\mathbf{T}\\quad\\text{and}\\quad \\ddot{\\mathbf{X}}=\\dot{v}\\mathbf{T} + v^2\\kappa \\mathbf{N},"
},
{
"math_id": 66,
"text": " v=|\\dot{\\mathbf{X}}|=\\sqrt{\\dot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}}."
},
{
"math_id": 67,
"text": " \\Delta K = m\\int_{t_1}^{t_2}\\dot{v}v \\, dt = \\frac{m}{2} \\int_{t_1}^{t_2} \\frac{d}{dt}v^2 \\, dt = \\frac{m}{2} v^2(t_2) - \\frac{m}{2} v^2(t_1),"
},
{
"math_id": 68,
"text": " K = \\frac{m}{2} v^2 = \\frac{m}{2} \\dot{\\mathbf{X}} \\cdot \\dot{\\mathbf{X}}. "
},
{
"math_id": 69,
"text": " W = \\Delta K. "
},
{
"math_id": 70,
"text": " \\mathbf{F} + \\mathbf{R} = m\\ddot{\\mathbf{X}}. "
},
{
"math_id": 71,
"text": " F_x v = m\\dot{v}v."
},
{
"math_id": 72,
"text": " \\int_{t_1}^{t_2}F_x v dt = \\frac{m}{2} v^2(t_2) - \\frac{m}{2} v^2(t_1). "
},
{
"math_id": 73,
"text": " F_x (d(t_2)-d(t_1)) = \\frac{m}{2} v^2(t_2) - \\frac{m}{2} v^2(t_1). "
},
{
"math_id": 74,
"text": "kWs = \\frac{W}{2g} v^2,\\quad\\text{or}\\quad v = \\sqrt{2ksg}."
},
{
"math_id": 75,
"text": " \\mathbf{F} + \\mathbf{R} = m \\ddot{\\mathbf{X}}. "
},
{
"math_id": 76,
"text": " W v_z = m\\dot{V}V,"
},
{
"math_id": 77,
"text": " \\int_{t_1}^{t_2} W v_z dt = \\frac{m}{2} V^2(t_2) - \\frac{m}{2} V^2 (t_1). "
},
{
"math_id": 78,
"text": " W \\Delta z = \\frac{m}{2}V^2. "
},
{
"math_id": 79,
"text": " s = \\frac{\\Delta z}{0.06}= 8.3\\frac{V^2}{g},\\quad\\text{or}\\quad s=8.3\\frac{88^2}{32.2}\\approx 2000\\mathrm{ft}."
},
{
"math_id": 80,
"text": " \\mathbf{X}_i(t)= [A(t)]\\mathbf{x}_i + \\mathbf{d}(t)\\quad i=1,\\ldots, n. "
},
{
"math_id": 81,
"text": "\\mathbf{V}_i = \\boldsymbol{\\omega}\\times(\\mathbf{X}_i-\\mathbf{d}) + \\dot{\\mathbf{d}},"
},
{
"math_id": 82,
"text": " [\\Omega] = \\dot{A}A^\\mathsf{T},"
},
{
"math_id": 83,
"text": " \\delta W = \\mathbf{F}_1\\cdot\\mathbf{V}_1\\delta t+\\mathbf{F}_2\\cdot\\mathbf{V}_2\\delta t + \\ldots + \\mathbf{F}_n\\cdot\\mathbf{V}_n\\delta t"
},
{
"math_id": 84,
"text": " \\delta W = \\sum_{i=1}^n \\mathbf{F}_i\\cdot (\\boldsymbol{\\omega}\\times(\\mathbf{X}_i-\\mathbf{d}) + \\dot{\\mathbf{d}})\\delta t. "
},
{
"math_id": 85,
"text": " \\delta W = \\left(\\sum_{i=1}^n \\mathbf{F}_i\\right)\\cdot\\dot{\\mathbf{d}}\\delta t + \\left(\\sum_{i=1}^n \\left(\\mathbf{X}_i-\\mathbf{d}\\right)\\times\\mathbf{F}_i\\right) \\cdot \\boldsymbol{\\omega}\\delta t = \\left(\\mathbf{F}\\cdot\\dot{\\mathbf{d}}+ \\mathbf{T} \\cdot \\boldsymbol{\\omega}\\right)\\delta t, "
}
] |
https://en.wikipedia.org/wiki?curid=149861
|
1498625
|
Mellin inversion theorem
|
In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under
which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.
Method.
If formula_0 is analytic in the strip formula_1,
and if it tends to zero uniformly as formula_2 for any real value "c" between "a" and "b", with its integral along such a line converging absolutely, then if
formula_3
we have that
formula_4
Conversely, suppose formula_5 is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral
formula_6
is absolutely convergent when formula_1. Then formula_7 is recoverable via the inverse Mellin transform from its Mellin transform formula_8. These results can be obtained by relating the Mellin transform to the Fourier transform by a change of variables and then applying an appropriate version of the Fourier inversion theorem.
Boundedness condition.
The boundedness condition on formula_0 can be strengthened if
formula_5 is continuous. If formula_0 is analytic in the strip formula_1, and if formula_9, where "K" is a positive constant, then formula_5 as defined by the inversion integral exists and is continuous; moreover the Mellin transform of formula_7 is formula_8 for at least formula_1.
On the other hand, if we are willing to accept an original formula_7 which is a
generalized function, we may relax the boundedness condition on
formula_8 to
simply make it of polynomial growth in any closed strip contained in the open strip formula_1.
We may also define a Banach space version of this theorem. If we call by
formula_10 the weighted Lp space of complex valued functions formula_7 on the positive reals such that
formula_11
where ν and "p" are fixed real numbers with formula_12, then if formula_5
is in formula_10 with formula_13, then
formula_0 belongs to formula_14 with formula_15 and
formula_16
Here functions, identical everywhere except on a set of measure zero, are identified.
Since the two-sided Laplace transform can be defined as
formula_17
these theorems can be immediately applied to it also.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\varphi(s)"
},
{
"math_id": 1,
"text": "a < \\Re(s) < b"
},
{
"math_id": 2,
"text": " \\Im(s) \\to \\pm \\infty "
},
{
"math_id": 3,
"text": "f(x)= \\{ \\mathcal{M}^{-1} \\varphi \\} = \\frac{1}{2 \\pi i} \\int_{c-i \\infty}^{c+i \\infty} x^{-s} \\varphi(s)\\, ds"
},
{
"math_id": 4,
"text": "\\varphi(s)= \\{ \\mathcal{M} f \\} = \\int_0^{\\infty} x^{s-1} f(x)\\,dx."
},
{
"math_id": 5,
"text": "f(x)"
},
{
"math_id": 6,
"text": "\\varphi(s)=\\int_0^{\\infty} x^{s-1} f(x)\\,dx"
},
{
"math_id": 7,
"text": "f"
},
{
"math_id": 8,
"text": "\\varphi"
},
{
"math_id": 9,
"text": "|\\varphi(s)| < K |s|^{-2}"
},
{
"math_id": 10,
"text": "L_{\\nu, p}(R^{+})"
},
{
"math_id": 11,
"text": "\\|f\\| = \\left(\\int_0^\\infty |x^\\nu f(x)|^p\\, \\frac{dx}{x}\\right)^{1/p} < \\infty"
},
{
"math_id": 12,
"text": "p>1"
},
{
"math_id": 13,
"text": "1 < p \\le 2"
},
{
"math_id": 14,
"text": "L_{\\nu, q}(R^{+})"
},
{
"math_id": 15,
"text": "q = p/(p-1)"
},
{
"math_id": 16,
"text": "f(x)=\\frac{1}{2 \\pi i} \\int_{\\nu-i \\infty}^{\\nu+i \\infty} x^{-s} \\varphi(s)\\,ds."
},
{
"math_id": 17,
"text": " \\left\\{\\mathcal{B} f\\right\\}(s) = \\left\\{\\mathcal{M} f(- \\ln x) \\right\\}(s)"
}
] |
https://en.wikipedia.org/wiki?curid=1498625
|
14986442
|
History of statistics
|
Statistics, in the modern sense of the word, began evolving in the 18th century in response to the novel needs of industrializing sovereign states.
In early times, the meaning was restricted to information about states, particularly demographics such as population. This was later extended to include all collections of information of all types, and later still it was extended to include the analysis and interpretation of such data. In modern terms, "statistics" means both sets of collected information, as in national accounts and temperature record, and analytical work which requires statistical inference. Statistical activities are often associated with models expressed using probabilities, hence the connection with probability theory. The large requirements of data processing have made statistics a key application of computing. A number of statistical concepts have an important impact on a wide range of sciences. These include the design of experiments and approaches to statistical inference such as Bayesian inference, each of which can be considered to have their own sequence in the development of the ideas underlying modern statistics.
Introduction.
By the 18th century, the term "statistics" designated the systematic collection of demographic and economic data by states. For at least two millennia, these data were mainly tabulations of human and material resources that might be taxed or put to military use. In the early 19th century, collection intensified, and the meaning of "statistics" broadened to include the discipline concerned with the collection, summary, and analysis of data. Today, data is collected and statistics are computed and widely distributed in government, business, most of the sciences and sports, and even for many pastimes. Electronic computers have expedited more elaborate statistical computation even as they have facilitated the collection and aggregation of data. A single data analyst may have available a set of data-files with millions of records, each with dozens or hundreds of separate measurements. These were collected over time from computer activity (for example, a stock exchange) or from computerized sensors, point-of-sale registers, and so on. Computers then produce simple, accurate summaries, and allow more tedious analyses, such as those that require inverting a large matrix or perform hundreds of steps of iteration, that would never be attempted by hand. Faster computing has allowed statisticians to develop "computer-intensive" methods which may look at all permutations, or use randomization to look at 10,000 permutations of a problem, to estimate answers that are not easy to quantify by theory alone.
The term "mathematical statistics" designates the mathematical theories of probability and statistical inference, which are used in statistical practice. The relation between statistics and probability theory developed rather late, however. In the 19th century, statistics increasingly used probability theory, whose initial results were found in the 17th and 18th centuries, particularly in the analysis of games of chance (gambling). By 1800, astronomy used probability models and statistical theories, particularly the method of least squares. Early probability theory and statistics was systematized in the 19th century and statistical reasoning and probability models were used by social scientists to advance the new sciences of experimental psychology and sociology, and by physical scientists in thermodynamics and statistical mechanics. The development of statistical reasoning was closely associated with the development of inductive logic and the scientific method, which are concerns that move statisticians away from the narrower area of mathematical statistics. Much of the theoretical work was readily available by the time computers were available to exploit them. By the 1970s, Johnson and Kotz produced a four-volume "Compendium on Statistical Distributions" (1st ed., 1969–1972), which is still an invaluable resource.
Applied statistics can be regarded as not a field of mathematics but an autonomous mathematical science, like computer science and operations research. Unlike mathematics, statistics had its origins in public administration. Applications arose early in demography and economics; large areas of micro- and macro-economics today are "statistics" with an emphasis on time-series analyses. With its emphasis on learning from data and making best predictions, statistics also has been shaped by areas of academic research including psychological testing, medicine and epidemiology. The ideas of statistical testing have considerable overlap with decision science. With its concerns with searching and effectively presenting data, statistics has overlap with information science and computer science.
"Look up "statistics" in Wiktionary, the free dictionary."
Etymology.
The term "statistics" is ultimately derived from the Neo-Latin ("council of state") and the Italian word ("statesman" or "politician"). The German , first introduced by Gottfried Achenwall (1749), originally designated the analysis of data about the state, signifying the "science of state" (then called "political arithmetic" in English). It acquired the meaning of the collection and classification of data generally in the early 19th century. It was introduced into English in 1791 by Sir John Sinclair when he published the first of 21 volumes titled "Statistical Account of Scotland".
Origins in probability theory.
Basic forms of statistics have been used since the beginning of civilization. Early empires often collated censuses of the population or recorded the trade in various commodities. The Han dynasty and the Roman Empire were some of the first states to extensively gather data on the size of the empire's population, geographical area and wealth.
The use of statistical methods dates back to at least the 5th century BCE. The historian Thucydides in his "History of the Peloponnesian War" describes how the Athenians calculated the height of the wall of Platea by counting the number of bricks in an unplastered section of the wall sufficiently near them to be able to count them. The count was repeated several times by a number of soldiers. The most frequent value (in modern terminology – the mode) so determined was taken to be the most likely value of the number of bricks. Multiplying this value by the height of the bricks used in the wall allowed the Athenians to determine the height of the ladders necessary to scale the walls.
The Trial of the Pyx is a test of the purity of the coinage of the Royal Mint which has been held on a regular basis since the 12th century. The Trial itself is based on statistical sampling methods. After minting a series of coins – originally from ten pounds of silver – a single coin was placed in the Pyx – a box in Westminster Abbey. After a given period – now once a year – the coins are removed and weighed. A sample of coins removed from the box are then tested for purity.
The "Nuova Cronica", a 14th-century history of Florence by the Florentine banker and official Giovanni Villani, includes much statistical information on population, ordinances, commerce and trade, education, and religious facilities and has been described as the first introduction of statistics as a positive element in history, though neither the term nor the concept of statistics as a specific field yet existed.
The arithmetic mean, although a concept known to the Greeks, was not generalised to more than two values until the 16th century. The invention of the decimal system by Simon Stevin in 1585 seems likely to have facilitated these calculations. This method was first adopted in astronomy by Tycho Brahe who was attempting to reduce the errors in his estimates of the locations of various celestial bodies.
The idea of the median originated in Edward Wright's book on navigation ("Certaine Errors in Navigation") in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations. The difference between the mean and the median was noticed in 1669 by Chistiaan Huygens in the context of using Graunt's tables.
The term 'statistic' was introduced by the Italian scholar Girolamo Ghilini in 1589 with reference to this science. The birth of statistics is often dated to 1662, when John Graunt, along with William Petty, developed early human statistical and census methods that provided a framework for modern demography. He produced the first life table, giving probabilities of survival to each age. His book "Natural and Political Observations Made upon the Bills of Mortality" used analysis of the mortality rolls to make the first statistically based estimation of the population of London. He knew that there were around 13,000 funerals per year in London and that three people died per eleven families per year. He estimated from the parish records that the average family size was 8 and calculated that the population of London was about 384,000; this is the first known use of a ratio estimator. Laplace in 1802 estimated the population of France with a similar method; see for details.
Although the original scope of statistics was limited to data useful for governance, the approach was extended to many fields of a scientific or commercial nature during the 19th century. The mathematical foundations for the subject heavily drew on the new probability theory, pioneered in the 16th century by Gerolamo Cardano, Pierre de Fermat and Blaise Pascal. Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's "Ars Conjectandi" (posthumous, 1713) and Abraham de Moivre's "The Doctrine of Chances" (1718) treated the subject as a branch of mathematics. In his book Bernoulli introduced the idea of representing complete certainty as one and probability as a number between zero and one.
A key early application of statistics in the 18th century was to the human sex ratio at birth. John Arbuthnot studied this question in 1710. Arbuthnot examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. Considering more male or more female births as equally likely, the probability of the observed outcome is 0.5^82, or about 1 in 4,8360,0000,0000,0000,0000,0000; in modern terms, the "p"-value. This is vanishingly small, leading Arbuthnot that this was not due to chance, but to divine providence: "From whence it follows, that it is Art, not Chance, that governs." This is and other work by Arbuthnot is credited as "the first use of significance tests" the first example of reasoning about statistical significance and moral certainty, and "... perhaps the first published report of a nonparametric test ...", specifically the sign test; see details at .
The formal study of theory of errors may be traced back to Roger Cotes' "Opera Miscellanea" (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given. Simpson discussed several possible distributions of error. He first considered the uniform distribution and then the discrete symmetric triangular distribution followed by the continuous symmetric triangle distribution. Tobias Mayer, in his study of the libration of the moon ("", Nuremberg, 1750), invented the first formal method for estimating the unknown quantities by generalized the averaging of observations under identical circumstances to the averaging of groups of similar equations.
Roger Joseph Boscovich in 1755 based in his work on the shape of the earth proposed in his book "De Litteraria expeditione per pontificiam ditionem ad dimetiendos duos meridiani gradus a PP. Maire et Boscovicli" that the true value of a series of observations would be that which minimises the sum of absolute errors. In modern terminology this value is the median. The first example of what later became known as the normal curve was studied by Abraham de Moivre who plotted this curve on November 12, 1733. de Moivre was studying the number of heads that occurred when a 'fair' coin was tossed.
In 1763 Richard Price transmitted to the Royal Society Thomas Bayes proof of a rule for using a binomial distribution to calculate a posterior probability on a prior event.
In 1765 Joseph Priestley invented the first timeline charts.
Johann Heinrich Lambert in his 1765 book "Anlage zur Architectonic" proposed the semicircle as a distribution of errors:
formula_0
with -1 < "x" < 1.
Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve and deduced a formula for the mean of three observations.
Laplace in 1774 noted that the frequency of an error could be expressed as an exponential function of its magnitude once its sign was disregarded. This distribution is now known as the Laplace distribution. Lagrange proposed a parabolic fractal distribution of errors in 1776.
Laplace in 1778 published his second law of errors wherein he noted that the frequency of an error was proportional to the exponential of the square of its magnitude. This was subsequently rediscovered by Gauss (possibly in 1795) and is now best known as the normal distribution which is of central importance in statistics. This distribution was first referred to as the "normal" distribution by C. S. Peirce in 1873 who was studying measurement errors when an object was dropped onto a wooden base. He chose the term "normal" because of its frequent occurrence in naturally occurring variables.
Lagrange also suggested in 1781 two other distributions for errors – a raised cosine distribution and a logarithmic distribution.
Laplace gave (1781) a formula for the law of facility of error (a term due to Joseph Louis Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
In 1786 William Playfair (1759–1823) introduced the idea of graphical representation into statistics. He invented the line chart, bar chart and histogram and incorporated them into his works on economics, the "Commercial and Political Atlas". This was followed in 1795 by his invention of the pie chart and circle chart which he used to display the evolution of England's imports and exports. These latter charts came to general attention when he published examples in his "Statistical Breviary" in 1801.
Laplace, in an investigation of the motions of Saturn and Jupiter in 1787, generalized Mayer's method by using different linear combinations of a single group of equations.
In 1791 Sir John Sinclair introduced the term 'statistics' into English in his Statistical Accounts of Scotland.
In 1802 Laplace estimated the population of France to be 28,328,612. He calculated this figure using the number of births in the previous year and census data for three communities. The census data of these communities showed that they had 2,037,615 persons and that the number of births were 71,866. Assuming that these samples were representative of France, Laplace produced his estimate for the entire population.
The method of least squares, which was used to minimize errors in data measurement, was published independently by Adrien-Marie Legendre (1805), Robert Adrain (1808), and Carl Friedrich Gauss (1809). Gauss had used the method in his famous 1801 prediction of the location of the dwarf planet Ceres. The observations that Gauss based his calculations on were made by the Italian monk Piazzi.
The method of least squares was preceded by the use a median regression slope. This method minimizing the sum of the absolute deviances. A method of estimating this slope was invented by Roger Joseph Boscovich in 1760 which he applied to astronomy.
The term "probable error" (') – the median deviation from the mean – was introduced in 1815 by the German astronomer Frederik Wilhelm Bessel. Antoine Augustin Cournot in 1843 was the first to use the term "median" (') for the value that divides a probability distribution into two equal halves.
Other contributors to the theory of errors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for formula_1, the "probable error" of a single observation was widely used and inspired early robust statistics (resistant to outliers: see Peirce's criterion).
In the 19th century authors on statistical theory included Laplace, S. Lacroix (1816), Littrow (1833), Dedekind (1860), Helmert (1872), Laurent (1873), Liagre, Didion, De Morgan and Boole.
Gustav Theodor Fechner used the median ("Centralwerth") in sociological and psychological phenomena. It had earlier been used only in astronomy and related fields. Francis Galton used the English term "median" for the first time in 1881 having earlier used the terms "middle-most value" in 1869 and the "medium" in 1880.
Adolphe Quetelet (1796–1874), another important founder of statistics, introduced the notion of the "average man" ("l'homme moyen") as a means of understanding complex social phenomena such as crime rates, marriage rates, and suicide rates.
The first tests of the normal distribution were invented by the German statistician Wilhelm Lexis in the 1870s. The only data sets available to him that he was able to show were normally distributed were birth rates.
Development of modern statistics.
Although the origins of statistical theory lie in the 18th-century advances in probability, the modern field of statistics only emerged in the late-19th and early-20th century in three stages. The first wave, at the turn of the century, was led by the work of Francis Galton and Karl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. The second wave of the 1910s and 20s was initiated by William Sealy Gosset, and reached its culmination in the insights of Ronald Fisher. This involved the development of better design of experiments models, hypothesis testing and techniques for use with small data samples. The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between Egon Pearson and Jerzy Neyman in the 1930s. Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology.
The first statistical bodies were established in the early 19th century. The Royal Statistical Society was founded in 1834 and Florence Nightingale, its first female member, pioneered the application of statistical analysis to health problems for the furtherance of epidemiological understanding and public health practice. However, the methods then used would not be considered as modern statistics today.
The Oxford scholar Francis Ysidro Edgeworth's book, "Metretike: or The Method of Measuring Probability and Utility" (1887) dealt with probability as the basis of inductive reasoning, and his later works focused on the 'philosophy of chance'. His first paper on statistics (1883) explored the law of error (normal distribution), and his "Methods of Statistics" (1885) introduced an early version of the t distribution, the Edgeworth expansion, the Edgeworth series, the method of variate transformation and the asymptotic theory of maximum likelihood estimates.
The Norwegian Anders Nicolai Kiær introduced the concept of stratified sampling in 1895. Arthur Lyon Bowley introduced new methods of data sampling in 1906 when working on social statistics. Although statistical surveys of social conditions had started with Charles Booth's "Life and Labour of the People in London" (1889–1903) and Seebohm Rowntree's "Poverty, A Study of Town Life" (1901), Bowley's, key innovation consisted of the use of random sampling techniques. His efforts culminated in his "New Survey of London Life and Labour".
Francis Galton is credited as one of the principal founders of statistical theory. His contributions to the field included introducing the concepts of standard deviation, correlation, regression and the application of these methods to the study of the variety of human characteristics – height, weight, eyelash length among others. He found that many of these could be fitted to a normal curve distribution.
Galton submitted a paper to "Nature" in 1907 on the usefulness of the median. He examined the accuracy of 787 guesses of the weight of an ox at a country fair. The actual weight was 1208 pounds: the median guess was 1198. The guesses were markedly non-normally distributed (cf. Wisdom of the Crowd).
Galton's publication of "Natural Inheritance" in 1889 sparked the interest of a brilliant mathematician, Karl Pearson, then working at University College London, and he went on to found the discipline of mathematical statistics. He emphasised the statistical foundation of scientific laws and promoted its study and his laboratory attracted students from around the world attracted by his new methods of analysis, including Udny Yule. His work grew to encompass the fields of biology, epidemiology, anthropometry, medicine and social history. In 1901, with Walter Weldon, founder of biometry, and Galton, he founded the journal "Biometrika" as the first journal of mathematical statistics and biometry.
His work, and that of Galton, underpins many of the 'classical' statistical methods which are in common use today, including the Correlation coefficient, defined as a product-moment; the method of moments for the fitting of distributions to samples; Pearson's system of continuous curves that forms the basis of the now conventional continuous probability distributions; Chi distance a precursor and special case of the Mahalanobis distance and P-value, defined as the probability measure of the complement of the ball with the hypothesized value as center point and chi distance as radius. He also introduced the term 'standard deviation'.
He also founded the statistical hypothesis testing theory, Pearson's chi-squared test and principal component analysis. In 1911 he founded the world's first university statistics department at University College London.
The second wave of mathematical statistics was pioneered by Ronald Fisher who wrote two textbooks, "Statistical Methods for Research Workers", published in 1925 and "The Design of Experiments" in 1935, that were to define the academic discipline in universities around the world. He also systematized previous results, putting them on a firm mathematical footing. In his 1918 seminal paper "The Correlation between Relatives on the Supposition of Mendelian Inheritance", the first use to use the statistical term, variance. In 1919, at Rothamsted Experimental Station he started a major study of the extensive collections of data recorded over many years. This resulted in a series of reports under the general title "Studies in Crop Variation." In 1930 he published "The Genetical Theory of Natural Selection" where he applied statistics to evolution.
Over the next seven years, he pioneered the principles of the design of experiments (see below) and elaborated his studies of analysis of variance. He furthered his studies of the statistics of small samples. Perhaps even more important, he began his systematic approach of the analysis of real data as the springboard for the development of new statistical methods. He developed computational algorithms for analyzing data from his balanced experimental designs. In 1925, this work resulted in the publication of his first book, "Statistical Methods for Research Workers". This book went through many editions and translations in later years, and it became the standard reference work for scientists in many disciplines. In 1935, this book was followed by "The Design of Experiments", which was also widely used.
In addition to analysis of variance, Fisher named and promoted the method of maximum likelihood estimation. Fisher also originated the concepts of sufficiency, ancillary statistics, Fisher's linear discriminator and Fisher information. His article "On a distribution yielding the error functions of several well known statistics" (1924) presented Pearson's chi-squared test and William Sealy Gosset's t in the same framework as the Gaussian distribution, and his own parameter in the analysis of variance Fisher's z-distribution (more commonly used decades later in the form of the F distribution).
The 5% level of significance appears to have been introduced by Fisher in 1925. Fisher stated that deviations exceeding twice the standard deviation are regarded as significant. Before this deviations exceeding three times the probable error were considered significant. For a symmetrical distribution the probable error is half the interquartile range. For a normal distribution the probable error is approximately 2/3 the standard deviation. It appears that Fisher's 5% criterion was rooted in previous practice.
Other important contributions at this time included Charles Spearman's rank correlation coefficient that was a useful extension of the Pearson correlation coefficient. William Sealy Gosset, the English statistician better known under his pseudonym of "Student", introduced Student's t-distribution, a continuous probability distribution useful in situations where the sample size is small and population standard deviation is unknown.
Egon Pearson (Karl's son) and Jerzy Neyman introduced the concepts of "Type II" error, power of a test and confidence intervals. Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling.
Design of experiments.
In 1747, while serving as surgeon on HM Bark "Salisbury", James Lind carried out a controlled experiment to develop a cure for scurvy. In this study his subjects' cases "were as similar as I could have them", that is he provided strict entry requirements to reduce extraneous variation. The men were paired, which provided blocking. From a modern perspective, the main thing that is missing is randomized allocation of subjects to treatments.
Lind is today often described as a one-factor-at-a-time experimenter. Similar one-factor-at-a-time (OFAT) experimentation was performed at the Rothamsted Research Station in the 1840s by Sir John Lawes to determine the optimal inorganic fertilizer for use on wheat.
A theory of statistical inference was developed by Charles S. Peirce in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883), two publications that emphasized the importance of randomization-based inference in statistics. In another study, Peirce randomly assigned volunteers to a blinded, repeated-measures design to evaluate their ability to discriminate weights.
Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s. Peirce also contributed the first English-language publication on an optimal design for regression-models in 1876. A pioneering optimal design for polynomial regression was suggested by Gergonne in 1815. In 1918 Kirstine Smith published optimal designs for polynomials of degree six (and less).
The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, was pioneered by Abraham Wald in the context of sequential tests of statistical hypotheses. Surveys are available of optimal sequential designs, and of adaptive designs. One specific type of sequential design is the "two-armed bandit", generalized to the multi-armed bandit, on which early work was done by Herbert Robbins in 1952.
The term "design of experiments" (DOE) derives from early statistical work performed by Sir Ronald Fisher. He was described by Anders Hald as "a genius who almost single-handedly created the foundations for modern statistical science." Fisher initiated the principles of design of experiments and elaborated on his studies of "analysis of variance". Perhaps even more important, Fisher began his systematic approach to the analysis of real data as the springboard for the development of new statistical methods. He began to pay particular attention to the labour involved in the necessary computations performed by hand, and developed methods that were as practical as they were founded in rigour. In 1925, this work culminated in the publication of his first book, "Statistical Methods for Research Workers". This went into many editions and translations in later years, and became a standard reference work for scientists in many disciplines.
A methodology for designing experiments was proposed by Ronald A. Fisher, in his innovative book "The Design of Experiments" (1935) which also became a standard. As an example, he described how to test the hypothesis that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. While this sounds like a frivolous application, it allowed him to illustrate the most important ideas of experimental design: see Lady tasting tea.
Agricultural science advances served to meet the combination of larger city populations and fewer farms. But for crop scientists to take due account of widely differing geographical growing climates and needs, it was important to differentiate local growing conditions. To extrapolate experiments on local crops to a national scale, they had to extend crop sample testing economically to overall populations. As statistical methods advanced (primarily the efficacy of designed experiments instead of one-factor-at-a-time experimentation), representative factorial design of experiments began to enable the meaningful extension, by inference, of experimental sampling results to the population as a whole. But it was hard to decide how representative was the crop sample chosen. Factorial design methodology showed how to estimate and correct for any random variation within the sample and also in the data collection procedures.
Bayesian statistics.
The term "Bayesian" refers to Thomas Bayes (1702–1761), who proved that probabilistic limits could be placed on an unknown event. However it was Pierre-Simon Laplace (1749–1827) who introduced (as principle VI) what is now called Bayes' theorem and applied it to celestial mechanics, medical statistics, reliability, and jurisprudence. When insufficient knowledge was available to specify an informed prior, Laplace used uniform priors, according to his "principle of insufficient reason". Laplace assumed uniform priors for mathematical simplicity rather than for philosophical reasons. Laplace also introduced primitive versions of conjugate priors and the theorem of von Mises and Bernstein, according to which the posteriors corresponding to initially differing priors ultimately agree, as the number of observations increases. This early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called "inverse probability" (because it infers backwards from observations to parameters, or from effects to causes).
After the 1920s, inverse probability was largely supplanted by a collection of methods that were developed by Ronald A. Fisher, Jerzy Neyman and Egon Pearson. Their methods came to be called frequentist statistics. Fisher rejected the Bayesian view, writing that "the theory of inverse probability is founded upon an error, and must be wholly rejected". At the end of his life, however, Fisher expressed greater respect for the essay of Bayes, which Fisher believed to have anticipated his own, fiducial approach to probability; Fisher still maintained that Laplace's views on probability were "fallacious rubbish". Neyman started out as a "quasi-Bayesian", but subsequently developed confidence intervals (a key method in frequentist statistics) because "the whole theory would look nicer if it were built from the start without reference to Bayesianism and priors".
The word "Bayesian" appeared around 1950, and by the 1960s it became the term preferred by those dissatisfied with the limitations of frequentist statistics.
In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise to "objective" and "subjective" currents in Bayesian practice. In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed. No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.
In the further development of Laplace's ideas, subjective ideas predate objectivist positions. The idea that 'probability' should be interpreted as 'subjective degree of belief in a proposition' was proposed, for example, by John Maynard Keynes in the early 1920s. This idea was taken further by Bruno de Finetti in Italy ("", 1930) and Frank Ramsey in Cambridge ("The Foundations of Mathematics", 1931). The approach was devised to solve problems with the frequentist definition of probability but also with the earlier, objectivist approach of Laplace. The subjective Bayesian methods were further developed and popularized in the 1950s by L.J. Savage.
Objective Bayesian inference was further developed by Harold Jeffreys at the University of Cambridge. His book "Theory of Probability" first appeared in 1939 and played an important role in the revival of the Bayesian view of probability. In 1957, Edwin Jaynes promoted the concept of maximum entropy for constructing priors, which is an important principle in the formulation of objective methods, mainly for discrete problems. In 1965, Dennis Lindley's two-volume work "Introduction to Probability and Statistics from a Bayesian Viewpoint" brought Bayesian methods to a wide audience. In 1979, José-Miguel Bernardo introduced reference analysis, which offers a general applicable framework for objective analysis. Other well-known proponents of Bayesian probability theory include I.J. Good, B.O. Koopman, Howard Raiffa, Robert Schlaifer and Alan Turing.
In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications. Despite growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics. Nonetheless, Bayesian methods are widely accepted and used, such as for example in the field of machine learning.
Important contributors to statistics.
<templatestyles src="Col-begin/styles.css"/>
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " f(x) = \\frac{ 1 }{ 2 } \\sqrt{ ( 1 - x^2 ) } "
},
{
"math_id": 1,
"text": "r"
}
] |
https://en.wikipedia.org/wiki?curid=14986442
|
1498680
|
Itô calculus
|
Calculus of stochastic differential equations
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations.
The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes:
formula_0
where "H" is a locally square-integrable process adapted to the filtration generated by "X" , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular t is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any point and has infinite variation over every time interval.
The main insight is that the integral can be defined as long as the integrand "H" is adapted, which loosely speaking means that its value at time t can only depend on information available up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to t and constructs Riemann sums. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. It is crucial which point in each of the small intervals is used to compute the value of the function. The limit then is taken in probability as the mesh of the partition is going to zero. Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions. Typically, the left end of the interval is used.
Important results of Itô calculus include the integration by parts formula and Itô's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms.
In mathematical finance, the described evaluation strategy of the integral is conceptualized as that we are first deciding what to do, then observing the change in the prices. The integrand is how much stock we hold, the integrator represents the movement of the prices, and the integral is how much money we have in total including what our stock is worth, at any given moment. The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often, geometric Brownian motion (see Black–Scholes). Then, the Itô stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount "Ht" of the stock at time "t". In this situation, the condition that "H" is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time. This prevents the possibility of unlimited gains through clairvoyance: buying the stock just before each uptick in the market and selling before each downtick. Similarly, the condition that "H" is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums .
Notation.
The process "Y" defined before as
formula_1
is itself a stochastic process with time parameter "t", which is also sometimes written as "Y" = "H" · "X" . Alternatively, the integral is often written in differential form "dY" = "H" "dX", which is equivalent to "Y" − "Y"0 = "H" · "X". As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given
formula_2
The σ-algebra "formula_3" represents the information available up until time t, and a process "X" is adapted if "Xt" is formula_3-measurable. A Brownian motion "B" is understood to be an formula_3-Brownian motion, which is just a standard Brownian motion with the properties that "B""t" is formula_3-measurable and that "B""t"+"s" − "B""t" is independent of formula_3 for all "s","t" ≥ 0 .
Integration with respect to Brownian motion.
The Itô integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that "B" is a Wiener process (Brownian motion) and that "H" is a right-continuous (càdlàg), adapted and locally bounded process. If formula_4 is a sequence of partitions of [0, "t"] with mesh width going to zero, then the Itô integral of "H" with respect to "B" up to time t is a random variable
formula_5
It can be shown that this limit converges in probability.
For some applications, such as martingale representation theorems and local times, the integral is needed for processes that are not continuous. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes. If "H" is any predictable process such that ∫0"t" "H"2 "ds" < ∞ for every "t" ≥ 0 then the integral of "H" with respect to "B" can be defined, and "H" is said to be "B"-integrable. Any such process can be approximated by a sequence "Hn" of left-continuous, adapted and locally bounded processes, in the sense that
formula_6
in probability. Then, the Itô integral is
formula_7
where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itô isometry
formula_8
which holds when "H" is bounded or, more generally, when the integral on the right hand side is finite.
Itô processes.
An Itô process is defined to be an adapted stochastic process that can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,
formula_9
Here, "B" is a Brownian motion and it is required that σ is a predictable "B"-integrable process, and μ is predictable and (Lebesgue) integrable. That is,
formula_10
for each t. The stochastic integral can be extended to such Itô processes,
formula_11
This is defined for all locally bounded and predictable integrands. More generally, it is required that "Hσ" be "B"-integrable and "Hμ" be Lebesgue integrable, so that
formula_12
Such predictable processes "H" are called "X"-integrable.
An important result for the study of Itô processes is Itô's lemma. In its simplest form, for any twice continuously differentiable function "f" on the reals and Itô process "X" as described above, it states that formula_13 is itself an Itô process satisfying
formula_14
This is the stochastic calculus version of the change of variables formula and chain rule. It differs from the standard result due to the additional term involving the second derivative of "f", which comes from the property that Brownian motion has non-zero quadratic variation.
Semimartingales as integrators.
The Itô integral is defined with respect to a semimartingale "X". These are processes which can be decomposed as "X" = "M" + "A" for a local martingale "M" and finite variation process "A". Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process "H" the integral "H" · "X" exists, and can be calculated as a limit of Riemann sums. Let π"n" be a sequence of partitions of [0, "t"] with mesh going to zero,
formula_15
This limit converges in probability. The stochastic integral of left-continuous processes is general enough for studying much of stochastic calculus. For example, it is sufficient for applications of Itô's Lemma, changes of measure via Girsanov's theorem, and for the study of stochastic differential equations. However, it is inadequate for other important topics such as martingale representation theorems and local times.
The integral extends to all predictable and locally bounded integrands, in a unique way, such that the dominated convergence theorem holds. That is, if "Hn" → "H" and for a locally bounded process "J", then
formula_16
in probability. The uniqueness of the extension from left-continuous to predictable integrands is a result of the monotone class lemma.
In general, the stochastic integral "H" · "X" can be defined even in cases where the predictable process "H" is not locally bounded. If "K" = 1 / (1 + |"H"|) then "K" and "KH" are bounded. Associativity of stochastic integration implies that "H" is "X"-integrable, with integral "H" · "X" = "Y", if and only if "Y"0 = 0 and "K" · "Y" = ("KH") · "X". The set of "X"-integrable processes is denoted by "L"("X").
Properties.
The following properties can be found in works such as and :
Integration by parts.
As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itô integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itô calculus deals with processes with non-zero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If "X" and "Y" are semimartingales then
formula_19
where ["X", "Y"] is the quadratic covariation process.
The result is similar to the integration by parts theorem for the Riemann–Stieltjes integral but has an additional quadratic variation term.
Itô's lemma.
Itô's lemma is the version of the chain rule or change of variables formula which applies to the Itô integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous n-dimensional semimartingale "X" = ("X"1...,"X""n") and twice continuously differentiable function "f" from R"n" to R, it states that "f"("X") is a semimartingale and,
formula_20
This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation ["X""i","X""j" ]. The formula can be generalized to include an explicit time-dependence in formula_21 and in other ways (see Itô's lemma).
Martingale integrators.
Local martingales.
An important property of the Itô integral is that it preserves the local martingale property. If "M" is a local martingale and "H" is a locally bounded predictable process then "H" · "M" is also a local martingale. For integrands which are not locally bounded, there are examples where "H" · "M" is not a local martingale. However, this can only occur when "M" is not continuous. If "M" is a continuous local martingale then a predictable process "H" is "M"-integrable if and only if
formula_22
for each t, and "H" · "M" is always a local martingale.
The most general statement for a discontinuous local martingale "M" is that if ("H"2 · ["M"])1/2 is locally integrable then "H" · "M" exists and is a local martingale.
Square integrable martingales.
For bounded integrands, the Itô stochastic integral preserves the space of "square integrable" martingales, which is the set of càdlàg martingales "M" such that E["M""t"2] is finite for all t. For any such square integrable martingale "M", the quadratic variation process ["M"] is integrable, and the Itô isometry states that
formula_23
This equality holds more generally for any martingale "M" such that "H"2 · ["M"]"t" is integrable. The Itô isometry is often used as an important step in the construction of the stochastic integral, by defining "H" · "M" to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes.
"p"-Integrable martingales.
For any "p" > 1, and bounded predictable integrand, the stochastic integral preserves the space of "p"-integrable martingales. These are càdlàg martingales such that is finite for all t. However, this is not always true in the case where "p" = 1. There are examples of integrals of bounded predictable processes with respect to martingales which are not themselves martingales.
The maximum process of a càdlàg process "M" is written as "M*t" = sup"s" ≤"t" |"Ms"|. For any "p" ≥ 1 and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales "M" such that E[("M*t")"p"] is finite for all t. If "p" > 1 then this is the same as the space of "p"-integrable martingales, by Doob's inequalities.
The Burkholder–Davis–Gundy inequalities state that, for any given "p" ≥ 1, there exist positive constants "c", "C" that depend on "p", but not "M" or on t such that
formula_24
for all càdlàg local martingales "M". These are used to show that if ("M*t")"p" is integrable and "H" is a bounded predictable process then
formula_25
and, consequently, "H" · "M" is a "p"-integrable martingale. More generally, this statement is true whenever ("H"2 · ["M"])"p"/2 is integrable.
Existence of the integral.
Proofs that the Itô integral is well defined typically proceed by first looking at very simple integrands, such as piecewise constant, left continuous and adapted processes where the integral can be written explicitly. Such "simple predictable" processes are linear combinations of terms of the form "Ht" = "A"1{"t" > "T"} for stopping times "T" and "FT"-measurable random variables "A", for which the integral is
formula_26
This is extended to all simple predictable processes by the linearity of "H" · "X" in "H".
For a Brownian motion "B", the property that it has independent increments with zero mean and variance Var("Bt") = "t" can be used to prove the Itô isometry for simple predictable integrands,
formula_27
By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying
formula_28
in such way that the Itô isometry still holds. It can then be extended to all "B"-integrable processes by localization. This method allows the integral to be defined with respect to any Itô process.
For a general semimartingale "X", the decomposition "X" = "M" + "A" into a local martingale "M" plus a finite variation process "A" can be used. Then, the integral can be shown to exist separately with respect to "M" and "A" and combined using linearity, "H" · "X" = "H" · "M" + "H" · "A", to get the integral with respect to "X". The standard Lebesgue–Stieltjes integral allows integration to be defined with respect to finite variation processes, so the existence of the Itô integral for semimartingales will follow from any construction for local martingales.
For a càdlàg square integrable martingale "M", a generalized form of the Itô isometry can be used. First, the Doob–Meyer decomposition theorem is used to show that a decomposition "M"2 = "N" + ⟨"M"⟩ exists, where "N" is a martingale and is a right-continuous, increasing and predictable process starting at zero. This uniquely defines , which is referred to as the "predictable quadratic variation" of "M". The Itô isometry for square integrable martingales is then
formula_29
which can be proved directly for simple predictable integrands. As with the case above for Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying "E"["H"2 · ⟨"M"⟩"t"] < ∞. This method can be extended to all local square integrable martingales by localization. Finally, the Doob–Meyer decomposition can be used to decompose any local martingale into the sum of a local square integrable martingale and a finite variation process, allowing the Itô integral to be constructed with respect to any semimartingale.
Many other proofs exist which apply similar methods but which avoid the need to use the Doob–Meyer decomposition theorem, such as the use of the quadratic variation ["M"] in the Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô isometry. The latter applies directly to local martingales without having to first deal with the square integrable martingale case.
Alternative proofs exist only making use of the fact that "X" is càdlàg, adapted, and the set {"H" · "Xt": |"H"| ≤ 1 is simple previsible} is bounded in probability for each time "t", which is an alternative definition for "X" to be a semimartingale. A continuous linear extension can be used to construct the integral for all left-continuous and adapted integrands with right limits everywhere (caglad or L-processes). This is general enough to be able to apply techniques such as Itô's lemma . Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands .
Differentiation in Itô calculus.
The Itô calculus is first and foremost defined as an integral calculus as outlined above. However, there are also different notions of "derivative" with respect to Brownian motion:
Malliavin derivative.
Malliavin calculus provides a theory of differentiation for random variables defined over Wiener space, including an integration by parts formula .
Martingale representation.
The following result allows to express martingales as Itô integrals: if "M" is a square-integrable martingale on a time interval [0, "T"] with respect to the filtration generated by a Brownian motion "B", then there is a unique adapted square integrable process formula_30 on [0, "T"] such that
formula_31
almost surely, and for all "t" ∈ [0, "T"] . This representation theorem can be interpreted formally as saying that α is the "time derivative" of "M" with respect to Brownian motion "B", since α is precisely the process that must be integrated up to time t to obtain "M""t" − "M"0, as in deterministic calculus.
Itô calculus for physicists.
In physics, usually stochastic differential equations (SDEs), such as Langevin equations, are used, rather than stochastic integrals. Here an Itô stochastic differential equation (SDE) is often formulated via
formula_32
where formula_33 is Gaussian white noise with
formula_34
and Einstein's summation convention is used.
If formula_35 is a function of the "xk", then Itô's lemma has to be used:
formula_36
An Itô SDE as above also corresponds to a Stratonovich SDE which reads
formula_37
SDEs frequently occur in physics in Stratonovich form, as limits of stochastic differential equations driven by colored noise if the correlation time of the noise term approaches zero.
For a recent treatment of different interpretations of stochastic differential equations see for example .
See also.
<templatestyles src="Div col/styles.css"/>
References.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "Y_t = \\int_0^t H_s\\,dX_s,"
},
{
"math_id": 1,
"text": "Y_t = \\int_0^t H\\,dX\\equiv\\int_0^t H_s\\,dX_s ,"
},
{
"math_id": 2,
"text": "(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t\\ge 0},\\mathbb{P}) ."
},
{
"math_id": 3,
"text": "\\mathcal{F}_t"
},
{
"math_id": 4,
"text": "\\{\\pi_n\\}"
},
{
"math_id": 5,
"text": "\\int_0^t H \\,d B =\\lim_{n\\rightarrow\\infty} \\sum_{[t_{i-1},t_i]\\in\\pi_n}H_{t_{i-1}}(B_{t_i}-B_{t_{i-1}})."
},
{
"math_id": 6,
"text": " \\int_0^t (H-H_n)^2\\,ds\\to 0"
},
{
"math_id": 7,
"text": "\\int_0^t H\\,dB = \\lim_{n\\to\\infty}\\int_0^t H_n\\,dB"
},
{
"math_id": 8,
"text": "\\mathbb{E}\\left[ \\left(\\int_0^t H_s \\, dB_s\\right)^2\\right] = \\mathbb{E} \\left[ \\int_0^t H_s^2\\,ds\\right ]"
},
{
"math_id": 9,
"text": "X_t=X_0+\\int_0^t\\sigma_s\\,dB_s + \\int_0^t\\mu_s\\,ds."
},
{
"math_id": 10,
"text": "\\int_0^t(\\sigma_s^2+|\\mu_s|)\\,ds<\\infty"
},
{
"math_id": 11,
"text": "\\int_0^t H\\,dX =\\int_0^t H_s\\sigma_s\\,dB_s + \\int_0^t H_s\\mu_s\\,ds."
},
{
"math_id": 12,
"text": "\\int_0^t \\left(H^2 \\sigma^2 + |H\\mu| \\right) ds < \\infty."
},
{
"math_id": 13,
"text": " Y_t=f(X_t) "
},
{
"math_id": 14,
"text": "d Y_t = f^\\prime(X_t) \\mu_t\\,d t + \\tfrac{1}{2} f^{\\prime\\prime} (X_t) \\sigma_t^2 \\, d t \n+ f^\\prime(X_t) \\sigma_t \\,dB_t ."
},
{
"math_id": 15,
"text": "\\int_0^t H\\,dX = \\lim_{n\\to\\infty} \\sum_{t_{i-1},t_i\\in\\pi_n}H_{t_{i-1}}(X_{t_i}-X_{t_{i-1}})."
},
{
"math_id": 16,
"text": "\\int_0^t H_n \\,dX \\to \\int_0^t H \\,dX, "
},
{
"math_id": 17,
"text": " J\\cdot (K\\cdot X) = (JK)\\cdot X"
},
{
"math_id": 18,
"text": "[H\\cdot X] = H^2\\cdot[X]"
},
{
"math_id": 19,
"text": "X_t Y_t = X_0 Y_0 + \\int_0^t X_{s-} \\, dY_s + \\int_0^t Y_{s-} \\, dX_s + [X,Y]_t "
},
{
"math_id": 20,
"text": "df(X_t)= \\sum_{i=1}^n f_{i}(X_t)\\,dX^i_t + \\frac{1}{2} \\sum_{i,j=1}^n f_{i,j}(X_{t}) \\, d[X^i,X^j]_t."
},
{
"math_id": 21,
"text": "f,"
},
{
"math_id": 22,
"text": "\\int_0^t H^2 \\, d[M] <\\infty,"
},
{
"math_id": 23,
"text": "\\mathbb{E}\\left [(H\\cdot M_t)^2\\right ]=\\mathbb{E}\\left [\\int_0^t H^2\\,d[M]\\right ]."
},
{
"math_id": 24,
"text": "c\\mathbb{E} \\left [ [M]_t^{\\frac{p}{2}} \\right ] \\le \\mathbb{E}\\left [(M^*_t)^p \\right ]\\le C\\mathbb{E}\\left [ [M]_t^{\\frac{p}{2}} \\right ]"
},
{
"math_id": 25,
"text": "\\mathbb{E}\\left [ ((H\\cdot M)_t^*)^p \\right ] \\le C\\mathbb{E}\\left [(H^2\\cdot[M]_t)^{\\frac{p}{2}} \\right ] < \\infty"
},
{
"math_id": 26,
"text": "H\\cdot X_t\\equiv \\mathbf{1}_{\\{t>T\\}}A(X_t-X_T)."
},
{
"math_id": 27,
"text": " \\mathbb{E} \\left [ (H\\cdot B_t)^2\\right ] = \\mathbb{E} \\left [\\int_0^tH_s^2\\,ds\\right ]."
},
{
"math_id": 28,
"text": " \\mathbb{E} \\left[ \\int_0^t H^2 \\, ds \\right ] < \\infty,"
},
{
"math_id": 29,
"text": "\\mathbb{E} \\left [(H\\cdot M_t)^2\\right ]= \\mathbb{E} \\left [\\int_0^tH^2_s\\,d\\langle M\\rangle_s\\right],"
},
{
"math_id": 30,
"text": "\\alpha"
},
{
"math_id": 31,
"text": "M_{t} = M_{0} + \\int_{0}^{t} \\alpha_{s} \\, \\mathrm{d} B_{s}"
},
{
"math_id": 32,
"text": " \\dot{x}_k = h_k + g_{kl} \\xi_l,"
},
{
"math_id": 33,
"text": "\\xi_j"
},
{
"math_id": 34,
"text": "\\langle\\xi_k(t_1)\\,\\xi_l(t_2)\\rangle = \\delta_{kl}\\delta(t_1-t_2)"
},
{
"math_id": 35,
"text": "y = y(x_k)"
},
{
"math_id": 36,
"text": " \\dot{y}=\\frac{\\partial y}{\\partial x_j}\\dot{x}_j+\\frac{1}{2}\\frac{\\partial^2 y}{\\partial x_k \\, \\partial x_l} g_{km}g_{ml}. "
},
{
"math_id": 37,
"text": " \\dot{x}_k = h_k + g_{kl} \\xi_l - \\frac{1}{2} \\frac{\\partial g_{kl}}{\\partial {x_m}} g_{ml}."
}
] |
https://en.wikipedia.org/wiki?curid=1498680
|
1498759
|
Ramsey problem
|
The Ramsey problem, or Ramsey pricing, or Ramsey–Boiteux pricing, is a second-best policy problem concerning what prices a public monopoly should charge for the various products it sells in order to maximize social welfare (the sum of producer and consumer surplus) while earning enough revenue to cover its fixed costs.
Under Ramsey pricing, the price markup over marginal cost is inverse to the price elasticity of demand and the Price elasticity of supply: the more elastic the product's demand or supply, the smaller the markup. Frank P. Ramsey found this 1927 in the context of Optimal taxation: the more elastic the demand or supply, the smaller the optimal tax. The rule was later applied by Marcel Boiteux (1956) to natural monopolies (industries with decreasing average cost). A natural monopoly earns negative profits if it sets price equals to marginal cost, so it must set prices for some or all of the products it sells to above marginal cost if it is to be viable without government subsidies. Ramsey pricing says to mark up most the goods with the least elastic (that is, least price-sensitive) demand or supply.
Description.
In a first-best world, without the need to earn enough revenue to cover fixed costs, the optimal solution would be to set the price for each product equal to its marginal cost. If the average cost curve is declining where the demand curve crosses it however, as happens when the fixed cost is large, this would result in a price less than average cost, and the firm could not survive without subsidy. The Ramsey problem is to decide exactly how much to raise each product's price above its marginal cost so the firm's revenue equals its total cost. If there is just one product, the problem is simple: raise the price to where it equals average cost. If there are two products, there is leeway to raise one product's price more and the other's less, so long as the firm can break even overall.
The principle is applicable to pricing of goods that the government is the sole supplier of (public utilities) or regulation of natural monopolies, such as telecommunications firms, where it is efficient for only one firm to operate but the government regulates its prices so it does not earn above-market profits.
In practice, government regulators are concerned with more than maximizing the sum of producer and consumer surplus. They may wish to put more weight on the surplus of politically powerful consumers, or they may wish to help the poor by putting more weight on their surplus. Moreover, many people will see Ramsey pricing as unfair, especially if they do not understand why it maximizes total surplus. In some contexts, Ramsey pricing is a form of price discrimination because the two products with different elasticities of demand are one physically identical product sold to two different groups of customers, e.g., electricity to residential customers and to commercial customers. Ramsey pricing says to charge whichever group has less elastic demand a higher price in order to maximize overall social welfare. Customers sometimes object to it on that basis, since they care about their own individual welfare, not social welfare. Customers who are charged more may consider unfair, especially they, with less elastic demand, would say they "need" the good more. In such situations regulators may further limit an operator’s ability to adopt Ramsey prices.
Formal presentation and solution.
Consider the problem of a regulator seeking to set prices formula_0 for a multiproduct monopolist with costs formula_1 where formula_2 is the output of good "i" and formula_3 is the price. Suppose that the products are sold in separate markets so demands are independent, and demand for good "i" is formula_4 with inverse demand function formula_5 Total revenue is formula_6
Total welfare is given by
formula_7
The problem is to maximize formula_8 by choice of the subject to the requirement that profit formula_9 equal some fixed value formula_10. Typically, the fixed value is zero, which is to say that the regulator wants to maximize welfare subject to the constraint that the firm not lose money. The constraint can be stated generally as:
formula_11
This problem may be solved using the Lagrange multiplier technique to yield the optimal output values, and backing out the optimal prices. The first order conditions on formula_12 are
formula_13
where formula_14 is a Lagrange multiplier, "C""i"(q) is the partial derivative of "C"(q) with respect to "q""i", evaluated at q, and formula_15 is the elasticity of demand for good formula_16
Dividing by formula_17 and rearranging yields
formula_18
where formula_19. That is, the price margin compared to marginal cost for good formula_20 is again inversely proportional to the elasticity of demand. Note that the Ramsey mark-up is smaller than the ordinary monopoly markup of the Lerner Rule which has formula_21, since formula_22 (the fixed-profit requirement, formula_23 is non-binding). The Ramsey-price setting monopoly is in a second-best equilibrium, between ordinary monopoly and perfect competition.
Ramsey condition.
An easier way to solve this problem in a two-output context is the Ramsey condition. According to Ramsey, as to minimize deadweight losses, one must increase prices to rigid and elastic demands/supplies in the same proportion, in relation to the prices that would be charged at the first-best solution (price equal to marginal cost).
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\left(p_1,\\ldots,p_N\\right) "
},
{
"math_id": 1,
"text": "C(q_1,q_2,\\ldots,q_N) =C( \\mathbf{q}), "
},
{
"math_id": 2,
"text": "q_{i}"
},
{
"math_id": 3,
"text": "p_{i}"
},
{
"math_id": 4,
"text": "q_{i}\\left( p_{i}\\right) , "
},
{
"math_id": 5,
"text": "p_i(q)."
},
{
"math_id": 6,
"text": "R\\left( \\mathbf{p,q}\\right) =\\sum_i p_i q_i (p_i)."
},
{
"math_id": 7,
"text": "W\\left( \\mathbf{p,q}\\right) =\\sum_i \\left( \\int\\limits_0^{q_i(p_i) }p_i( q) dq\\right) -C\\left( \\mathbf{q}\\right). "
},
{
"math_id": 8,
"text": "W\\left( \\mathbf{p,q}\\right) "
},
{
"math_id": 9,
"text": "\\Pi = R-C "
},
{
"math_id": 10,
"text": "\\Pi^* "
},
{
"math_id": 11,
"text": "R( \\mathbf{p,q}) -C( \\mathbf{q}) \\geq \\Pi^*"
},
{
"math_id": 12,
"text": "\\mathbf{q} "
},
{
"math_id": 13,
"text": "\\begin{align}\n p_i - C_i \\left(\\mathbf{q}\\right) &= -\\lambda \\left( \\frac{\\partial R}{\\partial q_{i}} - C_{i}\\left( \\mathbf{q}\\right) \\right) \\\\\n &= -\\lambda \\left( p_i \\left( 1 - \\frac{1}{Elasticity_i}\\right) - C_i \\left(\\mathbf{q}\\right) \\right)\n\\end{align}"
},
{
"math_id": 14,
"text": "\\lambda "
},
{
"math_id": 15,
"text": "Elasticity_i= -\\frac{\\partial q_i}{\\partial p_i}\\frac{p_i}{q_i} "
},
{
"math_id": 16,
"text": "i. "
},
{
"math_id": 17,
"text": "p_i "
},
{
"math_id": 18,
"text": "\\frac{p_i - C_i\\left( \\mathbf{q}\\right) }{p_i}=\\frac{k}{Elasticity_i}"
},
{
"math_id": 19,
"text": "k=\\frac{\\lambda }{1+\\lambda}< 1. "
},
{
"math_id": 20,
"text": "i"
},
{
"math_id": 21,
"text": "k=1 "
},
{
"math_id": 22,
"text": "\\lambda=1 "
},
{
"math_id": 23,
"text": "\\Pi^* = R-C "
}
] |
https://en.wikipedia.org/wiki?curid=1498759
|
14988066
|
Knaster–Kuratowski fan
|
Topological space that becomes totally disconnected with the removal of a single point
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point makes it totally disconnected. It is also known as Cantor's leaky tent or Cantor's teepee (after Georg Cantor), depending on the presence or absence of the apex.
Let formula_0 be the Cantor set, let formula_1 be the point formula_2, and let formula_3, for formula_4, denote the line segment connecting formula_5 to formula_1. If formula_4 is an endpoint of an interval deleted in the Cantor set, let formula_6; for all other points in formula_0 let formula_7; the Knaster–Kuratowski fan is defined as formula_8 equipped with the subspace topology inherited from the standard topology on formula_9.
The fan itself is connected, but becomes totally disconnected upon the removal of formula_1.
|
[
{
"math_id": 0,
"text": "C"
},
{
"math_id": 1,
"text": "p"
},
{
"math_id": 2,
"text": "\\left(\\tfrac1{2},\\tfrac1{2}\\right)\\in\\mathbb R^2"
},
{
"math_id": 3,
"text": "L(c)"
},
{
"math_id": 4,
"text": "c \\in C"
},
{
"math_id": 5,
"text": "(c,0)"
},
{
"math_id": 6,
"text": "X_{c} = \\{ (x,y) \\in L(c) : y \\in \\mathbb{Q} \\}"
},
{
"math_id": 7,
"text": "X_{c} = \\{ (x,y) \\in L(c) : y \\notin \\mathbb{Q} \\}"
},
{
"math_id": 8,
"text": "\\bigcup_{c \\in C} X_{c}"
},
{
"math_id": 9,
"text": "\\mathbb{R}^2"
}
] |
https://en.wikipedia.org/wiki?curid=14988066
|
14988352
|
Blum–Shub–Smale machine
|
Model of computation over real numbers
In computation theory, the Blum–Shub–Smale machine, or BSS machine, is a model of computation introduced by Lenore Blum, Michael Shub and Stephen Smale, intended to describe computations over the real numbers. Essentially, a BSS machine is a Random Access Machine with registers that can store arbitrary real numbers and that can compute rational functions over reals in a single time step. It is closely related to the Real RAM model.
BSS machines are more powerful than Turing machines, because the latter are by definition restricted to a finite set of symbols. A Turing machine can represent a countable set (such as the rational numbers) by strings of symbols, but this does not extend to the uncountable real numbers.
Definition.
A BSS machine M is given by a list formula_0 of formula_1 instructions (to be described below), indexed formula_2. A configuration of M is a tuple formula_3, where formula_4 is the index of the instruction to be executed next, formula_5 and formula_6 are registers holding non-negative integers, and formula_7 is a list of real numbers, with all but finitely many being zero. The list formula_8 is thought of as holding the contents of all registers of M. The computation begins with configuration formula_9 and ends whenever formula_10; the final content of formula_8 is said to be the output of the machine.
The instructions of M can be of the following types:
Theory.
Blum, Shub and Smale defined the complexity classes P (polynomial time) and NP (nondeterministic polynomial time) in the BSS model. Here NP is defined by adding an existentially-quantified input to a problem. They give a problem which is NP-complete for the class NP so defined: existence of roots of quartic polynomials. This is an analogue of the Cook-Levin Theorem for real numbers.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "I"
},
{
"math_id": 1,
"text": "N+1"
},
{
"math_id": 2,
"text": "0, 1, \\dots, N"
},
{
"math_id": 3,
"text": "(k,r,w,x)"
},
{
"math_id": 4,
"text": "k"
},
{
"math_id": 5,
"text": "r"
},
{
"math_id": 6,
"text": "w"
},
{
"math_id": 7,
"text": "x=(x_0,x_1,\\ldots)"
},
{
"math_id": 8,
"text": "x"
},
{
"math_id": 9,
"text": "(0,0,0,x)"
},
{
"math_id": 10,
"text": "k=N"
},
{
"math_id": 11,
"text": "x_{0} := g_{k}(x)"
},
{
"math_id": 12,
"text": "g_{k}"
},
{
"math_id": 13,
"text": "r := 0"
},
{
"math_id": 14,
"text": "r := r + 1"
},
{
"math_id": 15,
"text": "k+1"
},
{
"math_id": 16,
"text": "l"
},
{
"math_id": 17,
"text": "x_0 \\geq 0"
},
{
"math_id": 18,
"text": "x_r, x_w"
},
{
"math_id": 19,
"text": "x_r"
},
{
"math_id": 20,
"text": "x_w"
}
] |
https://en.wikipedia.org/wiki?curid=14988352
|
14989249
|
Turán's inequalities
|
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán (1950) (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.
If formula_0 is the formula_1th Legendre polynomial, Turán's inequalities state that
formula_2
For formula_3, the formula_1th Hermite polynomial, Turán's inequalities are
formula_4
whilst for Chebyshev polynomials they are
formula_5
|
[
{
"math_id": 0,
"text": "P_n"
},
{
"math_id": 1,
"text": "n"
},
{
"math_id": 2,
"text": "\\,\\! P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)\\ \\text{for}\\ -1<x<1."
},
{
"math_id": 3,
"text": "H_n"
},
{
"math_id": 4,
"text": "H_n(x)^2 - H_{n-1}(x)H_{n+1}(x)= (n-1)!\\cdot \\sum_{i=0}^{n-1}\\frac{2^{n-i}}{i!}H_i(x)^2>0 ,"
},
{
"math_id": 5,
"text": "T_n(x)^2 - T_{n-1}(x)T_{n+1}(x)= 1-x^2>0 \\ \\text{for}\\ -1<x<1 ."
}
] |
https://en.wikipedia.org/wiki?curid=14989249
|
14989517
|
Total offense
|
American football statistic
Total offense (or total offence) is a gridiron football statistic representing the total number of yards rushing and yards passing by a player or team. Total offense differs from yards from scrimmage, which gives credit for passing yardage to the person receiving the football rather than the person throwing the football. In football, progress is measured by advancing the football towards the opposing team's goal line. The team on offense can make progress during the play by advancing the ball from the line of scrimmage.
When the offensive team advances the ball by rushing the football, the player who carries the ball is given credit for the net gain, measured in rushing yards. When the offensive team advances the ball by pass reception, the player who throws the ball earns passing yards and the player who receives the ball earns receiving yards. The total of rushing yards and passing yards (but not receiving yards) is known as total offense. Although the ball may also be advanced by penalty, these yards do not contribute to total offense. Progress lost via quarterback sacks are classified differently, depending upon the league and/or level of football.
In the National Football League (NFL), the formula for a quarterback's total offense is:
formula_0
When defenses are measured on total offense allowed, it is called total defense.
Some definitions of individual total offense give credit to both the passer and receiver for passing yards. Thus, if a quarterback catches a pass in a trick play, or a non-quarterback throws a pass, some statistical issues arise.
Steve McNair holds the NCAA career and single-season total offense/game records. Case Keenum, B. J. Symons, and David Klingler hold the total offense career, single-season and single game records.
In the NFL, Patrick Mahomes holds the single season record for total offense with 5,420 yards in 2022. Tom Brady holds the NFL's career record with 86,761 yards.
NCAA definition.
The National Collegiate Athletic Association (NCAA) defines total offense as the total of net gain rushing and net gain forward passing: receiving and runback yards are not included in total offense. (at pg. 206).
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Total Offense = Passing Yards + Rushing Yards - Sack Yards"
}
] |
https://en.wikipedia.org/wiki?curid=14989517
|
149896
|
Emmy Noether
|
German mathematician (1882–1935)
Amalie Emmy Noether (, ; ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of "Privatdozent".
Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the "Noether Boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, "Moderne Algebra". By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world. The following year, Germany's Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania. There, she taught graduate and post-doctoral women including Marie Johanna Weiss, Ruth Stauffer, Grace Shover Quinn, and Olga Taussky-Todd. At the same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey. Noether died on 14 April 1935 at the age of 53.
Noether's mathematical work has been divided into three "epochs". In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic 1921 paper "Idealtheorie in Ringbereichen" ("Theory of Ideals in Ring Domains"), Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are named "Noetherian" in her honor. In the third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology.
Early life.
Emmy Noether was born on 23 March 1882. She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from Jewish merchant families. Her first name was "Amalie", but she began using her middle name at a young age and she invariably used the name "Emmy Noether" in her adult life and her publications.
In her youth, Noether did not stand out academically although she was known for being clever and friendly. She was near-sighted and talked with a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age. She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, although she loved to dance.
She had three younger brothers. The eldest, Alfred Noether, was born in 1883 and was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether was born in 1884, studied in Munich and made contributions to applied mathematics. He was executed in the Soviet Union in 1941. The youngest, Gustav Robert Noether, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.
Education.
Noether showed early proficiency in French and English. In the spring of 1900, she took the examination for teachers of these languages and received an overall score of "sehr gut" (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen, at which her father was a professor.
This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing mixed-sex education would "overthrow all academic order". One of just two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903, she passed the graduation exam at a "Realgymnasium" in Nuremberg.
During the 1903–1904 winter semester, she studied at the University of Göttingen, attending lectures given by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert.
In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded. Noether returned to Erlangen and officially reentered the university in October 1904, declaring her intention to focus solely on mathematics. She was one of six women in her year (two auditors) and the only woman in her chosen school. Under the supervision of Paul Gordan, she wrote her dissertation, "Über die Bildung des Formensystems der ternären biquadratischen Form" ("On Complete Systems of Invariants for Ternary Biquadratic Forms"), in 1907, graduating "summa cum laude" later that year. Gordan was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked-out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert. Although it had been well received, Noether later described her thesis and some subsequent similar papers she produced as "crap".
University of Erlangen.
From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without pay, occasionally substituting for her father, Max Noether, when he was too ill to lecture. In 1910 and 1911 she published an extension of her thesis work from three variables to "n" variables.
Gordan retired in 1910, and Noether taught under his successors, Erhard Schmidt and Ernst Fischer, who took over from the former in 1911. According to her colleague Hermann Weyl and her biographer Auguste Dick, Fischer was an important influence on Noether, in particular by introducing her to the work of David Hilbert. Noether and Fischer shared lively enjoyment of mathematics and would often discuss lectures long after they were over; Noether is known to have sent postcards to Fischer continuing her train of mathematical thoughts.
From 1913 to 1916 Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields of rational functions and the invariants of finite groups. This phase marked Noether's first exposure to abstract algebra, the field to which she would make groundbreaking contributions.
In Erlangen, Noether advised two doctoral students: Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916. Despite Noether's significant role, they were both officially under the supervision of her father. Following the completion of his doctorate, Falckenberg spent time in Braunschweig and Königsberg before becoming a professor at the University of Giessen while Seidelmann became a professor in Munich.
University of Göttingen.
Habilitation and Noether's theorem.
In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein. Their effort to recruit her was initially blocked by the philologists and historians among the philosophical faculty, who insisted that women should not become "privatdozenten". In a joint department meeting on the matter, one faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" Hilbert, who believed Noether's qualifications were the only important issue and that the sex of the candidate was irrelevant, objected with indignation and scolded those protesting her habilitation. Although his exact words have not been preserved, his objection is often said to have included the remark that the university was "not a bathhouse." According to Pavel Alexandrov's recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to her social-democratic political beliefs and Jewish ancestry.
Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death is unknown. At about the same time, Noether's father retired and her brother joined the German Army to serve in World War I. She returned to Erlangen for several weeks, mostly to care for her aging father.
During her first years teaching at Göttingen she did not have an official position and was not paid. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance".
Soon after arriving at Göttingen, she demonstrated her capabilities by proving the theorem now known as Noether's theorem which shows that a conservation law is associated with any differentiable symmetry of a physical system. The paper, "Invariante Variationsprobleme", was presented by a colleague, Felix Klein, on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society. American physicists Leon M. Lederman and Christopher T. Hill argue in their book "Symmetry and the Beautiful Universe" that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem".
When World War I ended, the German Revolution of 1918–1919 brought a significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed Noether to proceed with her "habilitation" (eligibility for tenure). Her oral examination was held in late May, and she successfully delivered her "habilitation" lecture in June 1919. Noether became a "privatdozent", and she delivered that fall semester the first lectures listed under her own name. She was still not paid for her work.
Three years later, she received a letter from Otto Boelitz, the Prussian Minister for Science, Art, and Public Education, in which he conferred on her the title of "nicht beamteter ausserordentlicher Professor" (an untenured professor with limited internal administrative rights and functions). This was an unpaid "extraordinary" professorship, not the higher "ordinary" professorship, which was a civil-service position. Although it recognized the importance of her work, the position still provided no salary. Noether was not paid for her lectures until she was appointed to the special position of "Lehrbeauftragte für Algebra" a year later.
Work in abstract algebra.
Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she is best remembered for her contributions to abstract algebra. In his introduction to Noether's "Collected Papers", Nathan Jacobson wrote thatThe development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.
Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about the theory of ideals in which they defined left and right ideals in a ring.
The following year she published the paper "Idealtheorie in Ringbereichen", analyzing ascending chain conditions with regards to (mathematical) ideals, in which she proved the Lasker-Noether theorem in its full generality. Noted algebraist Irving Kaplansky called this work "revolutionary". The publication gave rise to the term "Noetherian" for objects which satisfy the ascending chain condition.
In 1924, a young Dutch mathematician, Bartel Leendert van der Waerden, arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison". After returning to Amsterdam, he wrote "Moderne Algebra", a central two-volume text in the field; its second volume, published in 1931, borrowed heavily from Noether's work. Although Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin and E. Noether". Beginning in 1927, Noether worked closely with Emil Artin, Richard Brauer and Helmut Hasse on noncommutative algebras.
Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were the first of several in 1923. Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends. He began referring to her as "der Noether", using the masculine German article as a term of endearment to show his respect. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship to Princeton University for the 1927–1928 academic year from the Rockefeller Foundation.
Graduate students.
In Göttingen, Noether supervised more than a dozen doctoral students, though most were together with Edmund Landau and others as she was not allowed to supervise dissertations on her own. Her first was Grete Hermann, who defended her dissertation in February 1925. Although best remembered for her work on the foundations of quantum mechanics, her dissertation was considered an important contribution to ideal theory. Hermann later spoke reverently of her "dissertation-mother".
Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Nöether, though the latter died of tuberculosis shortly before his defense. Grell defended his thesis in 1926 and went on to work at the University of Jena and the University of Halle, before losing his teaching license in 1935 due to accusations of homosexual acts. He was later reinstated and became a professor at Humboldt University in 1948.
Noether then supervised Werner Weber and Jakob Levitzki, who both defended their theses in 1929. Weber, who was considered only a modest mathematician, would later take part in driving Jewish mathematicians out of Göttingen. Levitzki worked first at Yale University and then at the Hebrew University of Jerusalem in Palestine, making significant contributions (in particular Levitzky's theorem and the Hopkins–Levitzki theorem) to ring theory.
Other Noether Boys included Max Deuring, Hans Fitting, Ernst Witt, Chiungtze C. Tsen and Otto Schilling. Deuring, who had been considered the most promising of Noether's students, was awarded his doctorate in 1930. He worked in Hamburg, Marden and Göttingen and is known for his contributions to arithmetic geometry. Fitting graduated in 1931 with a thesis on abelian groups and is remembered for his work in group theory, particularly Fitting's theorem and the Fitting lemma. He died at the age of 31 from a bone disease.
Witt was initially supervised by Noether, but her position was revoked in April 1933 and he was assigned to Gustav Herglotz instead. He received his PhD in July 1933 with a thesis on the Riemann-Roch Theorem and zeta-functions, and went on to make several contributions that now bear his name. Tsen, best remembered for proving Tsen's theorem, received his doctorate in December of the same year. He returned to China in 1935 and started teaching at National Chekiang University, but died only five years later. Schilling also began studying under Noether but was forced to find a new advisor due to Noether's emigration. Under Helmut Hasse, he completed his PhD in 1934 at the University of Marburg. He later worked as a post doc at Trinity College, Cambridge before moving to the United States.
Noether's other students were Wilhelm Dörnte, who received his doctorate in 1927 with a thesis on groups, Werner Vorbeck, who did so in 1935 with a thesis on splitting fields, and Wolfgang Wichmann, who did so 1936 with a thesis on p-adic theory. There is no information about the first two, but it is known that Wichmann supported a student initiative that unsuccessfully attempted to revoke Noether's dismissal and died as a soldier on the Eastern Front during World War II.
Noether school.
Noether developed a close circle of mathematicians beyond just her doctoral students who shared Noether's approach to abstract algebra and contributed to the field's development, a group often referred to as the Noether school. An example of this is her close work with Wolfgang Krull, who greatly advanced commutative algebra with his "Hauptidealsatz" and his dimension theory for commutative rings. Another is Gottfried Köthe, who contributed to the development of the theory of hypercomplex quantities using Noether and Krull's methods.
In addition to her mathematical insight, Noether was respected for her consideration of others. Although she sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to name her "a severe critic", but she combined this demand for accuracy with a nurturing attitude. In Noether's obituary, Van der Waerden described her asCompletely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all.
Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and lectured at a local coffee house. Later, after Nazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.
Influential lectures.
Noether's frugal lifestyle at first was due to her being denied pay for her work. However, even after the university began paying her a small salary in 1923, she continued to live a simple and modest life. She was paid more generously later in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether.
Biographers suggest that she was mostly unconcerned about appearance and manners, focusing on her studies. Olga Taussky-Todd, a distinguished algebraist taught by Noether, described a luncheon during which Noether, wholly engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed". Appearance-conscious students cringed as she retrieved the handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematical discussion she was having with other students.
Noether did not follow a lesson plan for her lectures. She spoke quickly and her lectures were considered difficult to follow by many, including Carl Ludwig Siegel and Paul Dubreil. Students who disliked her style often felt alienated. "Outsiders" who occasionally visited Noether's lectures usually spent only half an hour in the room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out."
She used her lectures as a spontaneous discussion time with her students, to think through and clarify important problems in mathematics. Some of her most important results were developed in these lectures, and the lecture notes of her students formed the basis for several important textbooks, such as those of van der Waerden and Deuring. Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her.
Several of her colleagues attended her lectures and she sometimes allowed others (including her students) to receive credit for her ideas, resulting in much of her work appearing in papers not under her name. Noether was recorded as having given at least five semester-long courses at Göttingen:
These courses often preceded major publications on the same subjects.
Moscow State University.
In the winter of 1928–1929, Noether accepted an invitation to Moscow State University, where she continued working with P.S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.
Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution. She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess". Hermann Weyl recalled that "During the wild times after the Revolution of 1918," Noether "sided more or less with the Social Democrats". She was from 1919 through 1922 a member of the Independent Social Democrats, a short-lived splinter party. In the words of logician and historian Colin McLarty, "she was not a Bolshevist but was not afraid to be called one."
Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933, he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry. Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.
Recognition.
In 1932 Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics. The prize included a monetary reward of 500 ℛ︁ℳ︁ and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to the Göttingen "Gesellschaft der Wissenschaften" (academy of sciences) and was never promoted to the position of "Ordentlicher Professor" (full professor).
Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style. Helmut Hasse dedicated an article to her in the "Mathematische Annalen", wherein he confirmed her suspicion that some aspects of noncommutative algebra are simpler than those of commutative algebra, by proving a noncommutative reciprocity law. This pleased her immensely. He also sent her a mathematical riddle, which he called the "mμν-riddle of syllables". She solved it immediately, but the riddle has been lost.
In September of the same year, Noether delivered a plenary address ("großer Vortrag") on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich. The congress was attended by 800 people, including Noether's colleagues Hermann Weyl, Edmund Landau, and Wolfgang Krull. There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.
Expulsion from Göttingen by Nazi Germany.
When Adolf Hitler became the German "Reichskanzler" in January 1933, Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by "privatdozent" and Noether's former student Werner Weber. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."
One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen." Several of Noether's colleagues, including Max Born and Richard Courant, also had their positions revoked.
Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later wrote that "Emmy Noether – her courage, her frankness, her unconcern about her own fate, her conciliatory spirit – was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace." Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization "Sturmabteilung" (SA), she showed no sign of agitation and, reportedly, even laughed about it later.
Refuge at Bryn Mawr and Princeton.
As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them. Albert Einstein and Hermann Weyl were appointed by the Institute for Advanced Study in Princeton, while others worked to find a sponsor required for legal immigration. Noether was contacted by representatives of two educational institutions: Bryn Mawr College, in the United States, and Somerville College at the University of Oxford, in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.
At Bryn Mawr, Noether met and befriended Anna Wheeler, who had studied at Göttingen just before Noether arrived there. Another source of support at the college was the Bryn Mawr president, Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr. Noether in action!"
During her time at Bryn Mawr, Noether formed a group, sometimes called the Noether girls, of four post-doctoral (Grace Shover Quinn, Marie Johanna Weiss, Olga Taussky-Todd, who all went on to have successful careers in mathematics) and doctoral students (Ruth Stauffer). They enthusiastically worked through van der Waerden's "Moderne Algebra I" and parts of Erich Hecke's "Theorie der algebraischen Zahlen" ("Theory of algebraic numbers"). Stauffer was Noether's only doctoral student in the United States, but Noether died shortly before she graduated. She took her examination with Richard Brauer and received her degree in June 1935, with a thesis concerning separable normal extensions. After her doctorate, Stauffer worked as a teacher for a short period and as a statistician for over 30 years.
In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen. She also worked with Abraham Albert and Harry Vandiver. However, she remarked about Princeton University that she was not welcome at "the men's university, where nothing female is admitted".
Her time in the United States was pleasant, as she was surrounded by supportive colleagues and absorbed in her favorite subjects. In the summer of 1934, she briefly returned to Germany to see Emil Artin and her brother Fritz. The latter, after having been forced out of his job at the Technische Hochschule Breslau, had accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk, in the Siberian Federal District of Russia. He was subsequently executed during the Medvedev Forest massacre.
Although many of her former colleagues had been forced out of the universities, she was able to use the library in Göttingen as a "foreign scholar". Without incident, Noether returned to the United States and her studies at Bryn Mawr.
Death.
In April 1935 doctors discovered a tumor in Noether's pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst "the size of a large cantaloupe". Two smaller tumors in her uterus appeared to be benign and were not removed to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April, Noether fell unconscious, her temperature soared to , and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located." She was 53.
A few days after Noether's death, her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies. In the months that followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.
Contributions to mathematics and physics.
Noether's work in abstract algebra and topology was influential in mathematics, while Noether's theorem has widespread consequences for theoretical physics and dynamical systems. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways. Her friend and colleague Hermann Weyl described her scholarly output in three epochs:
<templatestyles src="Template:Blockquote/styles.css" />(1) the period of relative dependence, 1907–1919
(2) the investigations grouped around the general theory of ideals 1920–1926
(3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics
In the first epoch (1907–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. Shortly after moving to Göttingen in 1915, she proved the two Noether's theorems, "one of the most important mathematical theorems ever proved in guiding the development of modern physics".
In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings. In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields. Although the results of Noether's first epoch were impressive and useful, her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der Waerden in their obituaries of her.
In these epochs, she was not merely applying ideas and methods of the earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory of ideals in rings, generalizing the earlier work of Richard Dedekind. She is also renowned for developing ascending chain conditions – a simple finiteness condition that yielded powerful results in her hands. Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as elimination theory and the algebraic varieties that had been studied by her father.
Historical context.
In the century from 1832 to Noether's death in 1935, the field of mathematics – specifically algebra – underwent a profound revolution whose reverberations are still being felt. Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equations, as well as on the related problem of constructing regular polygons using compass and straightedge. Beginning with Carl Friedrich Gauss's 1832 proof that prime numbers such as five can be factored in Gaussian integers, Évariste Galois's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), William Rowan Hamilton's description of quaternions in 1843, and Arthur Cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this new field, abstract algebra.
Background on abstract algebra and "begriffliche Mathematik" (conceptual mathematics).
Two of the most basic objects in abstract algebra are groups and rings.
A "group" consists of a set of elements and a single operation which combines a first and a second element and returns a third. The operation must satisfy certain constraints for it to determine a group: it must be closed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as by multiplying a number by one), and for every element there must be an inverse element.
A "ring" likewise, has a set of elements, but now has "two" operations. The first operation must make the set a commutative group, and the second operation is associative and distributive with respect to the first operation. It may or may not be commutative; this means that the result of applying the operation to a first and a second element is the same as to the second and first – the order of the elements does not matter. If every non-zero element has a multiplicative inverse (an element "x" such that "ax" = "xa" = 1), the ring is called a "division ring". A "field" is defined as a commutative division ring. For instance, the integers form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. Any pair of integers can be added or multiplied, always resulting in another integer, and the first operation, addition, is commutative, i.e., for any elements "a" and "b" in the ring, "a" + "b" = "b" + "a". The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that "a" combined with "b" might be different from "b" combined with "a". Examples of noncommutative rings include matrices and quaternions. The integers do not form a division ring, because the second operation cannot always be inverted; for example, there is no integer "a" such that 3"a" = 1.
The integers have additional properties which do not generalize to all commutative rings. An important example is the fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime numbers. Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called the Lasker–Noether theorem, for the ideals of many rings. Much of Noether's work lay in determining what properties "do" hold for all rings, in devising novel analogs of the old integer theorems, and in determining the minimal set of assumptions required to yield certain properties of rings.
Groups are frequently studied through "group representations". In their most general form, these consist of a choice of group, a set, and an "action" of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. Most often, the set is a vector space, and the group describes the symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. Rotations are a type of symmetry of space, because the laws of physics themselves do not pick out a preferred direction. Noether used these sorts of symmetries in her work on invariants in physics.
A powerful way of studying rings is through their "modules". A module consists of a choice of ring, another set, usually distinct from the underlying set of the ring and called the underlying set of the module, an operation on pairs of elements of the underlying set of the module, and an operation which takes an element of the ring and an element of the module and returns an element of the module.
The underlying set of the module and its operation must form a group. A module is a ring-theoretic version of a group representation: ignoring the second ring operation and the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. An important special case of this is an "algebra". An algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. This operation makes the second ring into a module over the first. Often the first ring is a field.
Words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For instance, the elements might be logical propositions, where the first combining operation is exclusive or and the second is logical conjunction. Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, van der Waerden recalled that
<templatestyles src="Template:Blockquote/styles.css" />The maxim by which Emmy Noether was guided throughout her work might be formulated as follows: "Any relationships between numbers, functions, and operations become transparent, generally applicable, and fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts."
This is the "begriffliche Mathematik" (purely conceptual mathematics) that was characteristic of Noether. This style of mathematics was consequently adopted by other mathematicians, especially in the (then new) field of abstract algebra.
First epoch (1908–1919).
Algebraic invariant theory.
Much of Noether's work in the first epoch of her career was associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. As an everyday example, if a rigid yardstick is rotated, the coordinates ("x"1, "y"1, "z"1) and ("x"2, "y"2, "z"2) of its endpoints change, but its length "L" given by the formula "L"2 = Δ"x"2 + Δ"y"2 + Δ"z"2 remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by Felix Klein's Erlangen program, according to which different types of geometry should be characterized by their invariants under transformations, e.g., the cross-ratio of projective geometry.
An example of an "invariant" is the discriminant "B"2 − 4 "AC" of a binary quadratic form x·A x + y·B x + y·C y, where x and y are vectors and "·" is the dot product or "inner product" for the vectors. A, B, and C are linear operators on the vectors – typically matrices.
The discriminant is called "invariant" because it is not changed by linear substitutions x → "a" x + "b" y and y → "c" x + "d" y with determinant "ad" − "bc" = 1. These substitutions form the special linear group "SL"2.
One can ask for all polynomials in A, B, and C that are unchanged by the action of "SL"2; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant.
More generally, one can ask for the invariants of homogeneous polynomials A0 "x"r "y"0 + ... + Ar x0 "y"r of higher degree, which will be certain polynomials in the coefficients A0, ..., Ar, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.
One of the main goals of invariant theory was to solve the "finite basis problem". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called "generators", and then, adding or multiplying the generators together. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.
Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables. He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables. Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.
Galois theory.
Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable "x" of degree "n", in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of "x", which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. For example, if the polynomial is "x"2 + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of "x" makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.
Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +"i" and −"i", where "i" is the imaginary unit, that is, "i" 2 = −1. More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.
The Galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial. (These transformations are called automorphisms.) The Galois group of consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends +"i" to −"i". Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a permutation of the "n" roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the Galois group.
In 1918, Noether published a paper on the inverse Galois problem. Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "Noether's problem", which asks whether the fixed field of a subgroup "G" of the permutation group "S""n" acting on the field "k"("x"1, ..., "x""n") always is a pure transcendental extension of the field "k". (She first mentioned this problem in a 1913 paper, where she attributed the problem to her colleague Fischer.) She showed this was true for "n" = 2, 3, or 4. In 1969, Richard Swan found a counter-example to Noether's problem, with "n" = 47 and "G" a cyclic group of order 47 (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.
Physics.
Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with Noether's first theorem, which she proved in 1915, but did not publish until 1918. She not only solved the problem for general relativity, but also determined the conserved quantities for "every" system of physical laws that possesses some continuous symmetry. Upon receiving her work, Einstein wrote to Hilbert:<templatestyles src="Template:Blockquote/styles.css" />Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.
For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the "physical laws" governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.
Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon:If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.
Second epoch (1920–1926).
Ascending and descending chain conditions.
In this epoch, Noether became famous for her deft use of ascending ("Teilerkettensatz") or descending ("Vielfachenkettensatz") chain conditions. A sequence of non-empty subsets "A"1, "A"2, "A"3, etc. of a set "S" is usually said to be "ascending", if each is a subset of the next
formula_0
Conversely, a sequence of subsets of "S" is called "descending" if each contains the next subset:
formula_1
A chain "becomes constant after a finite number of steps" if there is an "n" such that formula_2 for all "m" ≥ "n". A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects – and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage. For example, chain conditions can be used to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These conclusions often are crucial steps in a proof.
Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called "Noetherian" in her honor. By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps. A Noetherian space is a topological space in which every strictly ascending chain of open subspaces becomes constant after a finite number of steps; this definition makes the spectrum of a Noetherian ring a Noetherian topological space.
The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and, "mutatis mutandis", the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
Another application of such chain conditions is in Noetherian induction – also known as well-founded induction – which is a generalization of mathematical induction. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that "S" is a partially ordered set. One way of proving a statement about the objects of "S" is to assume the existence of a counterexample and deduce a contradiction, thereby proving the contrapositive of the original statement. The basic premise of Noetherian induction is that every non-empty subset of "S" contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the "minimal counterexample". In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.
Commutative rings, ideals, and modules.
Noether's paper, "Idealtheorie in Ringbereichen" ("Theory of Ideals in Ring Domains", 1921), is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring. Before her paper, most results in commutative algebra were restricted to special examples of commutative rings, such as polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the ascending chain condition on ideals, every ideal is finitely generated. In 1943, French mathematician Claude Chevalley coined the term, "Noetherian ring", to describe this property. A major result in Noether's 1921 paper is the Lasker–Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings. The Lasker–Noether theorem can be viewed as a generalization of the fundamental theorem of arithmetic which states that any positive integer can be expressed as a product of prime numbers, and that this decomposition is unique.
Noether's work "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern" ("Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields", 1927) characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domains: integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields. This paper also contains what now are called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic results on Noetherian and Artinian modules.
Elimination theory.
In 1923–1924, Noether applied her ideal theory to elimination theory in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about the factorization of polynomials could be carried over directly. Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method of resultants.
For illustration, a system of equations often can be written in the form
Mv = 0
where a matrix (or linear transform) M (without the variable x) times a vector v (that only has non-zero powers of x) is equal to the zero vector, 0. Hence, the determinant of the matrix M must be zero, providing a new equation in which the variable x has been eliminated.
Invariant theory of finite groups.
Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper, Noether found a solution to the finite basis problem for a finite group of transformations "G" acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called Noether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is coprime to formula_3 (the factorial of the order formula_4 of the group "G"). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number formula_4, but Noether was not able to determine whether this bound was correct when the characteristic of the field divides formula_3 but not formula_4. For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.
In her 1926 paper, Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford conjecture. In this paper Noether also introduced the "Noether normalization lemma", showing that a finitely generated domain "A" over a field "k" has a set {"x"1, ..., "x""n"} of algebraically independent elements such that "A" is integral over "k"["x"1, ..., "x""n"].
Topology.
As noted by Hermann Weyl in his obituary, Noether's contributions to topology illustrate her generosity with ideas and how her insights could transform entire fields of mathematics. In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their connectedness. An old joke is that "a topologist cannot distinguish a donut from a coffee mug", since they can be continuously deformed into one another.
Noether is credited with fundamental ideas that led to the development of algebraic topology from the earlier combinatorial topology, specifically, the idea of homology groups. According to Alexandrov, Noether attended lectures given by him and Heinz Hopf in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle" and he continues that,
<templatestyles src="Template:Blockquote/styles.css" />When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the groups of algebraic complexes and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous to zero; instead of the usual definition of Betti numbers, she suggested immediately defining the Betti group as the complementary (quotient) group of the group of all cycles by the subgroup of cycles homologous to zero. This observation now seems self-evident. But in those years (1925–1928) this was a completely new point of view.
Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others, and it became a frequent topic of discussion among the mathematicians of Göttingen. Noether observed that her idea of a Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject "bears the imprint of these remarks of Emmy Noether". Noether mentions her own topology ideas only as an aside in a 1926 publication, where she cites it as an application of group theory.
This algebraic approach to topology was also developed independently in Austria. In a 1926–1927 course given in Vienna, Leopold Vietoris defined a homology group, which was developed by Walther Mayer, into an axiomatic definition in 1928.
Third epoch (1927–1935).
Hypercomplex numbers and representation theory.
Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these earlier results and gave the first general representation theory of groups and algebras. This single work by Noether was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.
Briefly, Noether subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions.
Noncommutative algebra.
Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.
A paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras, which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their "Hauptsatz" ("main theorem"):every finite dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension.These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra "D" are splitting fields. This paper also contains the Skolem–Noether theorem, which states that any two embeddings of an extension of a field "k" into a finite-dimensional central simple algebra over "k" are conjugate. The Brauer–Noether theorem gives a characterization of the splitting fields of a central division algebra over a field.
Legacy.
Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is consistently ranked as one of the greatest mathematicians of the twentieth century. In his obituary, fellow algebraist B.L. van der Waerden says that her mathematical originality was "absolute beyond comparison", and Hermann Weyl said that Noether "changed the face of [abstract] algebra by her work". Mathematician and historian Jeremy Gray wrote that any textbook on abstract algebra bears the evidence of Noether's contributions: "Mathematicians simply do ring theory her way." During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians such as Pavel Alexandrov, Hermann Weyl, and Jean Dieudonné.
In a letter to "The New York Times", Albert Einstein wrote:
<templatestyles src="Template:Blockquote/styles.css" />In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.
Notes.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
Sources.
<templatestyles src="Refbegin/styles.css" />
Selected works by Emmy Noether.
<templatestyles src="Refbegin/styles.css" />
*
|
[
{
"math_id": 0,
"text": "A_{1} \\subset A_{2} \\subset A_{3} \\subset \\cdots."
},
{
"math_id": 1,
"text": "A_{1} \\supset A_{2} \\supset A_{3} \\supset \\cdots."
},
{
"math_id": 2,
"text": "A_n = A_m"
},
{
"math_id": 3,
"text": "\\left|G\\right|!"
},
{
"math_id": 4,
"text": "\\left|G\\right|"
}
] |
https://en.wikipedia.org/wiki?curid=149896
|
1499089
|
Hardy's theorem
|
In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.
Let formula_0 be a holomorphic function on the open ball centered at zero and radius formula_1 in the complex plane, and assume that formula_0 is not a constant function. If one defines
formula_2
for formula_3 then this function is strictly increasing and is a convex function of formula_4.
References.
"This article incorporates material from Hardy's theorem on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "f"
},
{
"math_id": 1,
"text": "R"
},
{
"math_id": 2,
"text": "I(r) = \\frac{1}{2\\pi} \\int_0^{2\\pi}\\! \\left| f(r e^{i\\theta}) \\right| \\,d\\theta"
},
{
"math_id": 3,
"text": "0< r < R,"
},
{
"math_id": 4,
"text": "\\log r"
}
] |
https://en.wikipedia.org/wiki?curid=1499089
|
1499165
|
Vertex separator
|
Set of graph nodes which separate a given pair of nodes if removed
In graph theory, a vertex subset &NoBreak;&NoBreak; is a vertex separator (or vertex cut, separating set) for nonadjacent vertices a and b if the removal of S from the graph separates a and b into distinct connected components.
Examples.
Consider a grid graph with r rows and c columns; the total number n of vertices is "r" × "c". For instance, in the illustration, "r" = 5, "c" = 8, and "n" = 40. If r is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if c is odd, there is a single central column, and otherwise there are two columns equally close to the center. Choosing S to be any of these central rows or columns, and removing S from the graph, partitions the graph into two smaller connected subgraphs A and B, each of which has at most vertices. If "r" ≤ "c" (as in the illustration), then choosing a central column will give a separator S with formula_0 vertices, and similarly if "c" ≤ "r" then choosing a central row will give a separator with at most formula_1 vertices. Thus, every grid graph has a separator S of size at most formula_2 the removal of which partitions it into two connected components, each of size at most .
To give another class of examples, every free tree T has a separator S consisting of a single vertex, the removal of which partitions T into two or more connected components, each of size at most . More precisely, there is always exactly one or exactly two vertices, which amount to such a separator, depending on whether the tree is centered or bicentered.
As opposed to these examples, not all vertex separators are "balanced", but that property is most useful for applications in computer science, such as the planar separator theorem.
Minimal separators.
Let S be an ("a","b")-separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a "minimal" ("a","b")-"separator" if no proper subset of S separates a and b. More generally, S is called a "minimal separator" if it is a minimal separator for some pair ("a","b") of nonadjacent vertices. Notice that this is different from "minimal separating set" which says that no proper subset of S is a minimal ("u","v")-separator for any pair of vertices ("u","v"). The following is a well-known result characterizing the minimal separators:
Lemma. A vertex separator S in G is minimal if and only if the graph "G" – "S", obtained by removing S from G, has two connected components "C"1 and "C"2 such that each vertex in S is both adjacent to some vertex in "C"1 and to some vertex in "C"2.
The minimal ("a","b")-separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ("a","b")-separator S can be regarded as a "predecessor" of another ("a","b")-separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ("a","b")-separators in G. Then S is a predecessor of T, in symbols formula_3, if for each "x" ∈ "S" \ "T", every path connecting x to b meets T. It follows from the definition that the predecessor relation yields a preorder on the set of all ("a","b")-separators. Furthermore, proved that the predecessor relation gives rise to a complete lattice when restricted to the set of "minimal" ("a","b")-separators in G.
|
[
{
"math_id": 0,
"text": "r \\leq \\sqrt{n}"
},
{
"math_id": 1,
"text": "\\sqrt{n}"
},
{
"math_id": 2,
"text": "\\sqrt{n},"
},
{
"math_id": 3,
"text": "S \\sqsubseteq_{a,b}^G T"
}
] |
https://en.wikipedia.org/wiki?curid=1499165
|
149926
|
Face (geometry)
|
Planar surface that forms part of the boundary of a solid object
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a "polyhedron".
In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).
Polygonal face.
In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane "tile".
For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells.
Number of polygonal faces of a polyhedron.
Any convex polyhedron's surface has Euler characteristic
formula_0
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.
"k"-face.
In higher-dimensional geometry, the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face. For example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself and the empty set, where the empty set is for consistency given a "dimension" of −1. For any n-polytope (n-dimensional polytope), −1 ≤ "k" ≤ "n".
For example, with this meaning, the faces of a cube comprise the cube itself (3-face), its (square) facets (2-faces), its (line segment) edges (1-faces), its (point) vertices (0-faces), and the empty set.
In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P. From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set.
In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement for convexity is relaxed. Abstract theory still requires that the set of faces include the polytope itself and the empty set.
An n-dimensional simplex (line segment ("n" = 1), triangle ("n" = 2), tetrahedron ("n" = 3), etc.), defined by "n" + 1 vertices, has a face for each subset of the vertices, from the empty set up through the set of all vertices. In particular, there are 2"n" + 1 faces in total. The number of them that are k-faces, for "k" ∈ {−1, 0, ..., "n"}, is the binomial coefficient formula_1.
There are specific names for k-faces depending on the value of k and, in some cases, how close k is to the dimensionality n of the polytope.
Vertex or 0-face.
Vertex is the common name for a 0-face.
Edge or 1-face.
Edge is the common name for a 1-face.
Face or 2-face.
The use of face in a context where a specific k is meant for a k-face but is not explicitly specified is commonly a 2-face.
Cell or 3-face.
A cell is a polyhedral element (3-face) of a 4-dimensional polytope or 3-dimensional tessellation, or higher. Cells are facets for 4-polytopes and 3-honeycombs.
Examples:
Facet or ("n" − 1)-face.
In higher-dimensional geometry, the facets (also called hyperfaces) of a n-polytope are the ("n" − 1)-faces (faces of dimension one less than the polytope itself). A polytope is bounded by its facets.
For example:
Ridge or ("n" − 2)-face.
In related terminology, the ("n" − 2)-"face"s of an n-polytope are called ridges (also subfacets). A ridge is seen as the boundary between exactly two facets of a polytope or honeycomb.
For example:
Peak or ("n" − 3)-face.
The ("n" − 3)-"face"s of an n-polytope are called peaks. A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb.
For example:
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "V - E + F = 2,"
},
{
"math_id": 1,
"text": "\\binom{n+1}{k+1}"
}
] |
https://en.wikipedia.org/wiki?curid=149926
|
14993253
|
Ruin theory
|
Theory in actuarial science and applied probability
In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
Classical model.
The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process or Poisson risk process) was introduced in 1903 by the Swedish actuary Filip Lundberg. Lundberg's work was republished in the 1930s by Harald Cramér.
The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate "formula_0" from customers and claims arrive according to a Poisson process formula_1 with intensity formula_2 and are independent and identically distributed non-negative random variables formula_3 with distribution formula_4 and mean "formula_5" (they form a compound Poisson process). So for an insurer who starts with initial surplus "formula_6", the aggregate assets formula_7 are given by:
formula_8
The central object of the model is to investigate the probability that the insurer's surplus level eventually falls below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as
formula_9,
where the time of ruin is formula_10 with the convention that formula_11. This can be computed exactly using the Pollaczek–Khinchine formula as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an M/G/1 queue)
formula_12
where formula_13 is the transform of the tail distribution of formula_14,
formula_15
and formula_16 denotes the formula_17-fold convolution.
In the case where the claim sizes are exponentially distributed, this simplifies to
formula_18
Sparre Andersen model.
E. Sparre Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.
formula_19
where the claim number process formula_20 is a renewal process and formula_21 are independent and identically distributed random variables.
The model furthermore assumes that formula_22 almost surely and that formula_20 and formula_21 are independent. The model is also known as the renewal risk model.
Expected discounted penalty function.
Michael R. Powers and Gerber and Shiu analyzed the behavior of the insurer's surplus through the expected discounted penalty function, which is commonly referred to as Gerber-Shiu function in the ruin literature and named after actuarial scientists Elias S.W. Shiu and Hans-Ulrich Gerber. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers.
In Powers' notation, this is defined as
formula_23,
where formula_24 is the discounting force of interest, formula_25 is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation formula_26 corresponds to the probability measure formula_27. The function is called expected discounted cost of insolvency by Powers.
In Gerber and Shiu's notation, it is given as
formula_28,
where formula_24 is the discounting force of interest and formula_29 is a penalty function capturing the economic costs to the insurer at the time of ruin (assumed to depend on the surplus prior to ruin formula_30 and the deficit at ruin formula_31), and the expectation formula_26 corresponds to the probability measure formula_27. Here the indicator function formula_32 emphasizes that the penalty is exercised only when ruin occurs.
It is quite intuitive to interpret the expected discounted penalty function. Since the function measures the actuarial present value of the penalty that occurs at formula_33, the penalty function is multiplied by the discounting factor formula_34, and then averaged over the probability distribution of the waiting time to formula_33. While Gerber and Shiu applied this function to the classical compound-Poisson model, Powers argued that an insurer's surplus is better modeled by a family of diffusion processes.
There are a great variety of ruin-related quantities that fall into the category of the expected discounted penalty function.
Other finance-related quantities belonging to the class of the expected discounted penalty function include the perpetual American put option, the contingent claim at optimal exercise time, and more.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "c > 0"
},
{
"math_id": 1,
"text": "N_t"
},
{
"math_id": 2,
"text": "\\lambda"
},
{
"math_id": 3,
"text": "\\xi_i"
},
{
"math_id": 4,
"text": "F"
},
{
"math_id": 5,
"text": "\\mu"
},
{
"math_id": 6,
"text": "x"
},
{
"math_id": 7,
"text": "X_t"
},
{
"math_id": 8,
"text": "X_t = x + ct - \\sum_{i=1}^{N_t} \\xi_i \\quad \\text{ for t} \\geq 0."
},
{
"math_id": 9,
"text": "\\psi(x)=\\mathbb{P}^x\\{\\tau<\\infty\\}"
},
{
"math_id": 10,
"text": "\\tau=\\inf\\{t>0 \\,:\\, X(t)<0\\}"
},
{
"math_id": 11,
"text": "\\inf\\varnothing=\\infty"
},
{
"math_id": 12,
"text": "\\psi(x)=\\left(1-\\frac{\\lambda \\mu}{c}\\right) \\sum_{n=0}^\\infty \\left(\\frac{\\lambda \\mu}{c}\\right)^n (1-F^{\\ast n}_l(x))"
},
{
"math_id": 13,
"text": "F_l"
},
{
"math_id": 14,
"text": "F"
},
{
"math_id": 15,
"text": "F_l(x) = \\frac{1}{\\mu} \\int_0^x \\left(1-F(u)\\right) \\text{d}u"
},
{
"math_id": 16,
"text": "\\cdot^{\\ast n}"
},
{
"math_id": 17,
"text": "n"
},
{
"math_id": 18,
"text": "\\psi(x) = \\frac{\\lambda \\mu}{c}e^{-\\left( \\frac{1}{\\mu}-\\frac{\\lambda}{c}\\right)x}."
},
{
"math_id": 19,
"text": "X_t = x + ct - \\sum_{i=1}^{N_t} \\xi_i \\quad \\text{ for }t \\geq 0,"
},
{
"math_id": 20,
"text": " (N_t)_{t\\geq 0} "
},
{
"math_id": 21,
"text": "(\\xi_i)_{i\\in\\mathbb{N}}"
},
{
"math_id": 22,
"text": " \\xi_i > 0 "
},
{
"math_id": 23,
"text": "m(x)=\\mathbb{E}^x[e^{-\\delta\\tau}K_{\\tau}]"
},
{
"math_id": 24,
"text": "\\delta"
},
{
"math_id": 25,
"text": "K_{\\tau}"
},
{
"math_id": 26,
"text": "\\mathbb{E}^x"
},
{
"math_id": 27,
"text": "\\mathbb{P}^x"
},
{
"math_id": 28,
"text": "m(x)=\\mathbb{E}^x[e^{-\\delta\\tau}w(X_{\\tau-},X_{\\tau})\\mathbb{I}(\\tau<\\infty)]"
},
{
"math_id": 29,
"text": "w(X_{\\tau-},X_{\\tau})"
},
{
"math_id": 30,
"text": "X_{\\tau-}"
},
{
"math_id": 31,
"text": "X_{\\tau}"
},
{
"math_id": 32,
"text": "\\mathbb{I}(\\tau<\\infty)"
},
{
"math_id": 33,
"text": "\\tau"
},
{
"math_id": 34,
"text": "e^{-\\delta\\tau}"
}
] |
https://en.wikipedia.org/wiki?curid=14993253
|
1499388
|
Hadamard three-circle theorem
|
In complex analysis, a branch of mathematics, the
Hadamard three-circle theorem is a result about the behavior of holomorphic functions.
Let formula_0 be a holomorphic function on the annulus
formula_1
Let formula_2 be the maximum of formula_3 on the circle formula_4 Then, formula_5 is a convex function of the logarithm formula_6 Moreover, if formula_0 is not of the form formula_7 for some constants formula_8 and formula_9, then formula_5 is strictly convex as a function of formula_6
The conclusion of the theorem can be restated as
formula_10
for any three concentric circles of radii formula_11
History.
A statement and proof for the theorem was given by J.E. Littlewood in 1912, but he attributes it to no one in particular, stating it as a known theorem. Harald Bohr and Edmund Landau attribute the theorem to Jacques Hadamard, writing in 1896; Hadamard published no proof.
Proof.
The three circles theorem follows from the fact that for any real "a", the function Re log("z""a""f"("z")) is harmonic between two circles, and therefore takes its maximum value on one of the circles. The theorem follows by choosing the constant "a" so that this harmonic function has the same maximum value on both circles.
The theorem can also be deduced directly from Hadamard's three-line theorem.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
"This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "f(z)"
},
{
"math_id": 1,
"text": "r_1\\leq\\left| z\\right| \\leq r_3."
},
{
"math_id": 2,
"text": "M(r)"
},
{
"math_id": 3,
"text": "|f(z)|"
},
{
"math_id": 4,
"text": "|z|=r."
},
{
"math_id": 5,
"text": "\\log M(r)"
},
{
"math_id": 6,
"text": "\\log (r)."
},
{
"math_id": 7,
"text": "cz^\\lambda"
},
{
"math_id": 8,
"text": "\\lambda"
},
{
"math_id": 9,
"text": "c"
},
{
"math_id": 10,
"text": "\\log\\left(\\frac{r_3}{r_1}\\right)\\log M(r_2)\\leq\n\\log\\left(\\frac{r_3}{r_2}\\right)\\log M(r_1)\n+\\log\\left(\\frac{r_2}{r_1}\\right)\\log M(r_3)"
},
{
"math_id": 11,
"text": "r_1<r_2<r_3."
}
] |
https://en.wikipedia.org/wiki?curid=1499388
|
14993993
|
Boussinesq approximation (water waves)
|
Approximation valid for weakly non-linear and fairly long waves
In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.
The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours.
While the Boussinesq approximation is applicable to fairly long waves – that is, when the wavelength is large compared to the water depth – the Stokes expansion is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).
Boussinesq approximation.
The essential idea in the Boussinesq approximation is the elimination of the vertical coordinate from the flow equations, while retaining some of the influences of the vertical structure of the flow under water waves. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation.
This elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave (or wave of translation). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.
The steps in the Boussinesq approximation are:
Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate.
As a result, the resulting partial differential equations are in terms of functions of the horizontal coordinates (and time).
As an example, consider potential flow over a horizontal bed in the formula_0 plane, with formula_1 the horizontal and formula_2 the vertical coordinate. The bed is located at formula_3, where formula_4 is the mean water depth. A Taylor expansion is made of the velocity potential formula_5 around the bed level formula_3:
formula_6
where formula_7 is the velocity potential at the bed. Invoking Laplace's equation for formula_8, as valid for incompressible flow, gives:
formula_9
since the vertical velocity formula_10 is zero at the – impermeable – horizontal bed formula_3. This series may subsequently be truncated to a finite number of terms.
Original Boussinesq equations.
Derivation.
For water waves on an incompressible fluid and irrotational flow in the formula_0 plane, the boundary conditions at the free surface elevation formula_11 are:
formula_12
where:
Now the Boussinesq approximation for the velocity potential formula_8, as given above, is applied in these boundary conditions. Further, in the resulting equations only the linear and quadratic terms with respect to formula_18 and formula_19 are retained (with formula_20 the horizontal velocity at the bed formula_3). The cubic and higher order terms are assumed to be negligible. Then, the following partial differential equations are obtained:
formula_21
This set of equations has been derived for a flat horizontal bed, "i.e." the mean depth formula_4 is a constant independent of position formula_1. When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations.
Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation formula_18:
formula_22
From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number.
In dimensionless quantities, using the water depth formula_4 and gravitational acceleration formula_17 for non-dimensionalization, this equation reads, after normalization:
formula_23
with:
Linear frequency dispersion.
Water waves of different wave lengths travel with different phase speeds, a phenomenon known as frequency dispersion. For the case of infinitesimal wave amplitude, the terminology is "linear frequency dispersion". The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation.
The linear frequency dispersion characteristics for the above set A of equations are:
formula_24
with:
The relative error in the phase speed formula_25 for set A, as compared with linear theory for water waves, is less than 4% for a relative wave number formula_29. So, in engineering applications, set A is valid for wavelengths formula_28 larger than 4 times the water depth formula_4.
The linear frequency dispersion characteristics of equation B are:
formula_30
The relative error in the phase speed for equation B is less than 4% for formula_31, equivalent to wave lengths formula_28 longer than 7 times the water depth formula_4, called fairly long waves.
For short waves with formula_32 equation B become physically meaningless, because there are no longer real-valued solutions of the phase speed.
The original set of two partial differential equations (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
The shallow water equations have a relative error in the phase speed less than 4% for wave lengths formula_28 in excess of 13 times the water depth formula_4.
Boussinesq-type equations and extensions.
There are an overwhelming number of mathematical models which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as "the" Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them Boussinesq-type equations. Strictly speaking, "the" Boussinesq equations is the above-mentioned set B, since it is used in the analysis in the remainder of his 1872 paper.
Some directions, into which the Boussinesq equations have been extended, are:
Further approximations for one-way wave propagation.
While the Boussinesq equations allow for waves traveling simultaneously in opposing directions, it is often advantageous to only consider waves traveling in one direction. Under small additional assumptions, the Boussinesq equations reduce to:
Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called cnoidal waves. These are approximate solutions of the Boussinesq equation.
Numerical models.
For the simulation of wave motion near coasts and harbours, numerical models – both commercial and academic – employing Boussinesq-type equations exist. Some commercial examples are the Boussinesq-type wave modules in MIKE 21 and SMS. Some of the free Boussinesq models are Celeris, COULWAVE, and FUNWAVE. Most numerical models employ finite-difference, finite-volume or finite element techniques for the discretization of the model equations. Scientific reviews and intercomparisons of several Boussinesq-type equations, their numerical approximation and performance are e.g. , and .
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "(x,z)"
},
{
"math_id": 1,
"text": "x"
},
{
"math_id": 2,
"text": "z"
},
{
"math_id": 3,
"text": "z=-h"
},
{
"math_id": 4,
"text": "h"
},
{
"math_id": 5,
"text": "\\varphi(x,z,t)"
},
{
"math_id": 6,
"text": "\n\\begin{align}\n \\varphi\\, =\\, & \n \\varphi_b\\, \n +\\, (z+h)\\, \\left[ \\frac{\\partial \\varphi}{\\partial z } \\right]_{z=-h}\\, \n +\\, \\frac{1}{2}\\, (z+h)^2\\, \\left[ \\frac{\\partial^2 \\varphi}{\\partial z^2} \\right]_{z=-h}\\, \n \\\\ &\n +\\, \\frac{1}{6}\\, (z+h)^3\\, \\left[ \\frac{\\partial^3 \\varphi}{\\partial z^3} \\right]_{z=-h}\\,\n +\\, \\frac{1}{24}\\, (z+h)^4\\, \\left[ \\frac{\\partial^4 \\varphi}{\\partial z^4} \\right]_{z=-h}\\,\n +\\, \\cdots,\n\\end{align}\n"
},
{
"math_id": 7,
"text": "\\varphi_b(x,t)"
},
{
"math_id": 8,
"text": "\\varphi"
},
{
"math_id": 9,
"text": "\n\\begin{align}\n \\varphi\\, =\\, &\n \\left\\{\\, \n \\varphi_b\\,\n -\\, \\frac{1}{2}\\, (z+h)^2\\, \\frac{\\partial^2 \\varphi_b}{\\partial x^2}\\,\n +\\, \\frac{1}{24}\\, (z+h)^4\\, \\frac{\\partial^4 \\varphi_b}{\\partial x^4}\\,\n +\\, \\cdots\\,\n \\right\\}\\, \n \\\\\n & +\\,\n \\left\\{\\,\n (z+h)\\, \\left[ \\frac{\\partial \\varphi}{\\partial z} \\right]_{z=-h}\\, \n -\\, \\frac16\\, (z+h)^3\\, \\frac{\\partial^2}{\\partial x^2} \\left[ \\frac{\\partial \\varphi}{\\partial z} \\right]_{z=-h}\\,\n +\\, \\cdots\\,\n \\right\\}\n \\\\\n =\\, &\n \\left\\{\\, \n \\varphi_b\\,\n -\\, \\frac{1}{2}\\, (z+h)^2\\, \\frac{\\partial^2 \\varphi_b}{\\partial x^2}\\,\n +\\, \\frac{1}{24}\\, (z+h)^4\\, \\frac{\\partial^4 \\varphi_b}{\\partial x^4}\\,\n +\\, \\cdots\\,\n \\right\\}, \n\\end{align}\n"
},
{
"math_id": 10,
"text": "\\partial\\varphi/\\partial z"
},
{
"math_id": 11,
"text": "z=\\eta(x,t)"
},
{
"math_id": 12,
"text": " \n \\begin{align}\n \\frac{\\partial \\eta}{\\partial t}\\, &+\\, u\\, \\frac{\\partial \\eta}{\\partial x}\\, -\\, w\\, =\\, 0 \n \\\\\n \\frac{\\partial \\varphi}{\\partial t}\\, &+\\, \\frac{1}{2}\\, \\left( u^2 + w^2 \\right)\\, +\\, g\\, \\eta\\, =\\, 0,\n \\end{align}\n"
},
{
"math_id": 13,
"text": "u"
},
{
"math_id": 14,
"text": "u=\\partial\\varphi/\\partial x"
},
{
"math_id": 15,
"text": "w"
},
{
"math_id": 16,
"text": "w=\\partial\\varphi/\\partial z"
},
{
"math_id": 17,
"text": "g"
},
{
"math_id": 18,
"text": "\\eta"
},
{
"math_id": 19,
"text": "u_b"
},
{
"math_id": 20,
"text": "u_b=\\partial\\varphi_b/\\partial x"
},
{
"math_id": 21,
"text": "\n\\begin{align}\n \\frac{\\partial \\eta}{\\partial t}\\, \n & +\\, \\frac{\\partial}{\\partial x}\\, \\left[ \\left( h + \\eta \\right)\\, u_b \\right]\\, \n =\\, \\frac{1}{6}\\, h^3\\, \\frac{\\partial^3 u_b}{\\partial x^3}, \n \\\\\n \\frac{\\partial u_b}{\\partial t}\\, \n & +\\, u_b\\, \\frac{\\partial u_b}{\\partial x}\\, \n +\\, g\\, \\frac{\\partial \\eta}{\\partial x}\\, \n =\\, \\frac{1}{2}\\, h^2\\, \\frac{\\partial^3 u_b}{\\partial t\\, \\partial x^2}.\n\\end{align}\n"
},
{
"math_id": 22,
"text": " \n \\frac{\\partial^2 \\eta}{\\partial t^2}\\, \n -\\, g h\\, \\frac{\\partial^2 \\eta}{\\partial x^2}\\, \n -\\, g h\\, \\frac{\\partial^2}{\\partial x^2} \n \\left( \n \\frac{3}{2}\\, \\frac{\\eta^2}{h}\\, \n +\\, \\frac{1}{3}\\, h^2\\, \\frac{\\partial^2 \\eta}{\\partial x^2} \n \\right)\\, =\\, 0.\n"
},
{
"math_id": 23,
"text": " \n \\frac{\\partial^2 \\psi}{\\partial \\tau^2}\\, \n -\\, \\frac{\\partial^2 \\psi}{\\partial \\xi^2}\\, \n -\\, \\frac{\\partial^2}{\\partial \\xi^2} \n \\left(\\, \n 3\\, \\psi^2\\, \n +\\, \\frac{\\partial^2 \\psi}{\\partial \\xi^2}\\,\n \\right)\\, =\\, 0,\n"
},
{
"math_id": 24,
"text": " c^2\\, =\\; g h\\, \\frac{ 1\\, +\\, \\frac{1}{6}\\, k^2 h^2 }{ 1\\, +\\, \\frac{1}{2}\\, k^2 h^2 }, "
},
{
"math_id": 25,
"text": "c"
},
{
"math_id": 26,
"text": "k"
},
{
"math_id": 27,
"text": "k=2\\pi/\\lambda"
},
{
"math_id": 28,
"text": "\\lambda"
},
{
"math_id": 29,
"text": "kh<\\pi/2"
},
{
"math_id": 30,
"text": " c^2\\, =\\, g h\\, \\left( 1\\, -\\, \\frac{1}{3}\\, k^2 h^2 \\right). "
},
{
"math_id": 31,
"text": "kh<2\\pi/7"
},
{
"math_id": 32,
"text": "k^2 h^2>3"
}
] |
https://en.wikipedia.org/wiki?curid=14993993
|
14994282
|
Sl2-triple
|
In the theory of Lie algebras, an "sl"2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra "sl"2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits.
Definition.
Elements {"e","h","f"} of a Lie algebra "g" form an "sl"2-triple if
formula_0
These commutation relations are satisfied by the generators
formula_1
of the Lie algebra "sl"2 of 2 by 2 matrices with zero trace. It follows that "sl"2-triples in "g" are in a bijective correspondence with the Lie algebra homomorphisms from "sl"2 into "g".
The alternative notation for the elements of an "sl"2-triple is {"H", "X", "Y"}, with "H" corresponding to "h", "X" corresponding to "e", and "Y" corresponding to "f". H is called a neutral, X is called a nilpositive, and Y is called a nilnegative.
Properties.
Assume that "g" is a finite dimensional Lie algebra over a field of characteristic zero.
From the representation theory of the Lie algebra "sl"2, one concludes that the Lie algebra "g" decomposes into a direct sum of finite-dimensional subspaces, each of which is isomorphic to "V"j, the ("j" + 1)-dimensional simple "sl"2-module with highest weight "j". The element "h" of the "sl"2-triple is semisimple, with the simple eigenvalues "j", "j" − 2, ..., −"j" on a submodule of "g" isomorphic to "V"j . The elements "e" and "f" move between different eigenspaces of "h", increasing the eigenvalue by 2 in case of "e" and decreasing it by 2 in case of "f". In particular, "e" and "f" are nilpotent elements of the Lie algebra "g".
Conversely, the "Jacobson–Morozov theorem" states that any nilpotent element "e" of a semisimple Lie algebra "g" can be included into an "sl"2-triple {"e","h","f"}, and all such triples are conjugate under the action of the group "Z""G"("e"), the centralizer of "e" in the adjoint Lie group "G" corresponding to the Lie algebra "g".
The semisimple element "h" of any "sl"2-triple containing a given nilpotent element "e" of "g" is called a characteristic of "e".
An "sl"2-triple defines a grading on "g" according to the eigenvalues of "h":
formula_2
The "sl"2-triple is called even if only even "j" occur in this decomposition, and odd otherwise.
If "g" is a semisimple Lie algebra, then "g"0 is a reductive Lie subalgebra of "g" (it is not semisimple in general). Moreover, the direct sum of the eigenspaces of "h" with non-negative eigenvalues is a parabolic subalgebra of "g" with the Levi component "g"0.
If the elements of an "sl"2-triple are regular, then their span is called a principal subalgebra.
|
[
{
"math_id": 0,
"text": " [h,e] = 2e, \\quad [h,f] = -2f, \\quad [e,f] = h. "
},
{
"math_id": 1,
"text": " h = \\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix}, \\quad \ne = \\begin{bmatrix}\n0 & 1\\\\\n0 & 0\n\\end{bmatrix}, \\quad \nf = \\begin{bmatrix}\n0 & 0\\\\\n1 & 0\n\\end{bmatrix} "
},
{
"math_id": 2,
"text": " g = \\bigoplus_{j\\in\\mathbb{Z}} g_j,\\quad [h, a]= ja {\\ \\ }\\textrm{ for } {\\ \\ } a\\in g_j."
}
] |
https://en.wikipedia.org/wiki?curid=14994282
|
14994781
|
Steinberg group (K-theory)
|
In algebraic K-theory, a field of mathematics, the Steinberg group formula_0 of a ring formula_1 is the universal central extension of the commutator subgroup of the stable general linear group of formula_1.
It is named after Robert Steinberg, and it is connected with lower formula_2-groups, notably formula_3 and formula_4.
Definition.
Abstractly, given a ring formula_1, the Steinberg group formula_0 is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).
Presentation using generators and relations.
A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form formula_5, where formula_6 is the identity matrix, formula_7 is the matrix with formula_8 in the formula_9-entry and zeros elsewhere, and formula_10 — satisfy the following relations, called the Steinberg relations:
formula_11
The unstable Steinberg group of order formula_12 over formula_1, denoted by formula_13, is defined by the generators formula_14, where formula_15 and formula_16, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by formula_0, is the direct limit of the system formula_17. It can also be thought of as the Steinberg group of infinite order.
Mapping formula_18 yields a group homomorphism formula_19. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.
Interpretation as a fundamental group.
The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of formula_20.
Relation to "K"-theory.
"K"1.
formula_21 is the cokernel of the map formula_19, as formula_22 is the abelianization of formula_23 and the mapping formula_24 is surjective onto the commutator subgroup.
"K"2.
formula_25 is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher formula_2-groups.
It is also the kernel of the mapping formula_19. Indeed, there is an exact sequence
formula_26
Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: formula_27.
"K"3.
showed that formula_28.
|
[
{
"math_id": 0,
"text": " \\operatorname{St}(A) "
},
{
"math_id": 1,
"text": " A "
},
{
"math_id": 2,
"text": " K "
},
{
"math_id": 3,
"text": " K_{2} "
},
{
"math_id": 4,
"text": " K_{3} "
},
{
"math_id": 5,
"text": " {e_{pq}}(\\lambda) := \\mathbf{1} + {a_{pq}}(\\lambda) "
},
{
"math_id": 6,
"text": " \\mathbf{1} "
},
{
"math_id": 7,
"text": " {a_{pq}}(\\lambda) "
},
{
"math_id": 8,
"text": " \\lambda "
},
{
"math_id": 9,
"text": " (p,q) "
},
{
"math_id": 10,
"text": " p \\neq q "
},
{
"math_id": 11,
"text": "\n\\begin{align}\ne_{ij}(\\lambda) e_{ij}(\\mu) & = e_{ij}(\\lambda+\\mu); && \\\\\n\\left[ e_{ij}(\\lambda),e_{jk}(\\mu) \\right] & = e_{ik}(\\lambda \\mu), && \\text{for } i \\neq k; \\\\\n\\left[ e_{ij}(\\lambda),e_{kl}(\\mu) \\right] & = \\mathbf{1}, && \\text{for } i \\neq l \\text{ and } j \\neq k.\n\\end{align}\n"
},
{
"math_id": 12,
"text": " r "
},
{
"math_id": 13,
"text": " {\\operatorname{St}_{r}}(A) "
},
{
"math_id": 14,
"text": " {x_{ij}}(\\lambda) "
},
{
"math_id": 15,
"text": " 1 \\leq i \\neq j \\leq r "
},
{
"math_id": 16,
"text": " \\lambda \\in A "
},
{
"math_id": 17,
"text": " {\\operatorname{St}_{r}}(A) \\to {\\operatorname{St}_{r + 1}}(A) "
},
{
"math_id": 18,
"text": " {x_{ij}}(\\lambda) \\mapsto {e_{ij}}(\\lambda) "
},
{
"math_id": 19,
"text": " \\varphi: \\operatorname{St}(A) \\to {\\operatorname{GL}_{\\infty}}(A) "
},
{
"math_id": 20,
"text": "\\operatorname{GL}(A)"
},
{
"math_id": 21,
"text": " {K_{1}}(A) "
},
{
"math_id": 22,
"text": " K_{1} "
},
{
"math_id": 23,
"text": " {\\operatorname{GL}_{\\infty}}(A) "
},
{
"math_id": 24,
"text": " \\varphi "
},
{
"math_id": 25,
"text": " {K_{2}}(A) "
},
{
"math_id": 26,
"text": " 1 \\to {K_{2}}(A) \\to \\operatorname{St}(A) \\to {\\operatorname{GL}_{\\infty}}(A) \\to {K_{1}}(A) \\to 1. "
},
{
"math_id": 27,
"text": " {K_{2}}(A) = {H_{2}}(E(A);\\mathbb{Z}) "
},
{
"math_id": 28,
"text": " {K_{3}}(A) = {H_{3}}(\\operatorname{St}(A);\\mathbb{Z}) "
}
] |
https://en.wikipedia.org/wiki?curid=14994781
|
1499590
|
Projection-slice theorem
|
Theorem in mathematics
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:
In operator terms, if
then
formula_0
This idea can be extended to higher dimensions.
This theorem is used, for example, in the analysis of medical
CT scans where a "projection" is an x-ray
image of an internal organ. The Fourier transforms of these images are
seen to be slices through the Fourier transform of the 3-dimensional
density of the internal organ, and these slices can be interpolated to build
up a complete Fourier transform of that density. The inverse Fourier transform
is then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem.
The projection-slice theorem in "N" dimensions.
In "N" dimensions, the projection-slice theorem states that the
Fourier transform of the projection of an "N"-dimensional function
"f"(r) onto an "m"-dimensional linear submanifold
is equal to an "m"-dimensional slice of the "N"-dimensional Fourier transform of that
function consisting of an "m"-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms:
formula_1
The generalized Fourier-slice theorem.
In addition to generalizing to "N" dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis. For convenience of notation, we consider the change of basis to be represented as "B", an "N"-by-"N" invertible matrix operating on "N"-dimensional column vectors. Then the generalized Fourier-slice theorem can be stated as
formula_2
where formula_3 is the transpose of the inverse of the change of basis transform.
Proof in two dimensions.
The projection-slice theorem is easily proven for the case of two dimensions.
Without loss of generality, we can take the projection line to be the "x"-axis.
There is no loss of generality because if we use a shifted and rotated line, the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform results. The rotated function is the Fourier pair of the rotated Fourier transform, for which the theorem again holds.
If "f"("x", "y") is a two-dimensional function, then the projection of "f"("x", "y") onto the "x" axis is "p"("x") where
formula_4
The Fourier transform of formula_5 is
formula_6
The slice is then formula_7
formula_8
formula_9
formula_10
which is just the Fourier transform of "p"("x"). The proof for higher dimensions is easily generalized from the above example.
The FHA cycle.
If the two-dimensional function "f"(r) is circularly symmetric, it may be represented as "f"("r"), where "r" = |r|. In this case the projection onto any projection line
will be the Abel transform of "f"("r"). The two-dimensional Fourier transform
of "f"(r) will be a circularly symmetric function given by the zeroth-order Hankel transform of "f"("r"), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or
formula_11
where "A"1 represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, "F"1 represents the 1-D Fourier-transform
operator, and "H" represents the zeroth-order Hankel-transform operator.
Extension to fan beam or cone-beam CT.
The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.
References.
<templatestyles src="Reflist/styles.css" />
External links.
|
[
{
"math_id": 0,
"text": "F_1 P_1 = S_1 F_2."
},
{
"math_id": 1,
"text": "F_mP_m=S_mF_N.\\,"
},
{
"math_id": 2,
"text": "F_m P_m B = S_m \\frac{B^{-T}}{|B^{-T}|} F_N"
},
{
"math_id": 3,
"text": "B^{-T}=(B^{-1})^T"
},
{
"math_id": 4,
"text": "p(x)=\\int_{-\\infty}^\\infty f(x,y)\\,dy."
},
{
"math_id": 5,
"text": "f(x,y)"
},
{
"math_id": 6,
"text": "\nF(k_x,k_y)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty\nf(x,y)\\,e^{-2\\pi i(xk_x+yk_y)}\\,dxdy.\n"
},
{
"math_id": 7,
"text": "s(k_x)"
},
{
"math_id": 8,
"text": "s(k_x)=F(k_x,0)\n=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty f(x,y)\\,e^{-2\\pi ixk_x}\\,dxdy\n"
},
{
"math_id": 9,
"text": "=\\int_{-\\infty}^\\infty\n\\left[\\int_{-\\infty}^\\infty f(x,y)\\,dy\\right]\\,e^{-2\\pi ixk_x} dx\n"
},
{
"math_id": 10,
"text": "=\\int_{-\\infty}^\\infty p(x)\\,e^{-2\\pi ixk_x} dx\n"
},
{
"math_id": 11,
"text": "F_1 A_1 = H,"
}
] |
https://en.wikipedia.org/wiki?curid=1499590
|
1499595
|
Abel transform
|
Integral transform used in various branches of mathematics
In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function "f"("r") is given by
formula_0
Assuming that "f"("r") drops to zero more quickly than 1/"r", the inverse Abel transform is given by
formula_1
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance "y" from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance "r" from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as algebraic reconstruction technique (ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed.
In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
Geometrical interpretation.
In two dimensions, the Abel transform "F"("y") can be interpreted as the projection of a circularly symmetric function "f"("r") along a set of parallel lines of sight at a distance "y" from the origin. Referring to the figure on the right, the observer (I) will see
formula_2
where "f"("r") is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at "x" = ∞, so that the limits of integration are ±∞, and all lines of sight are parallel to the "x" axis.
Realizing that the radius "r" is related to "x" and "y" as "r"2 = "x"2 + "y"2, it follows that
formula_3
for "x" > 0. Since "f"("r") is an even function in "x", we may write
formula_4
which yields the Abel transform of "f"("r").
The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function "f"("ρ", "z"), where "ρ"2 = "x"2 + "y"2 is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the "z" axis. Without loss of generality, we can take that plane to be the "yz" plane, so that
formula_5
which is just the Abel transform of "f"("ρ", "z") in "ρ" and "y".
A particular type of axial symmetry is spherical symmetry. In this case, we have a function "f"("r"), where "r"2 = "x"2 + "y"2 + "z"2.
The projection onto, say, the "yz" plane will then be circularly symmetric and expressible as "F"("s"), where "s"2 = "y"2 + "z"2. Carrying out the integration, we have
formula_6
which is again, the Abel transform of "f"("r") in "r" and "s".
Verification of the inverse Abel transform.
Assuming formula_7 is continuously differentiable and formula_7, formula_8 drop to zero faster than formula_9, we can integrate by parts by setting formula_10 and formula_11 to find
formula_12
Differentiating formally,
formula_13
Now substitute this into the inverse Abel transform formula:
formula_14
By Fubini's theorem, the last integral equals
formula_15
Generalization of the Abel transform to discontinuous "F"("y").
Consider the case where formula_16 is discontinuous at formula_17, where it abruptly changes its value by a finite amount formula_18. That is, formula_19 and formula_18 are defined by formula_20. Such a situation is encountered in tethered polymers (Polymer brush) exhibiting a vertical phase separation, where formula_16 stands for the polymer density profile and formula_21 is related to the spatial distribution of terminal, non-tethered monomers of the polymers.
The Abel transform of a function "f"("r") is under these circumstances again given by:
formula_22
Assuming "f"("r") drops to zero more quickly than 1/"r", the inverse Abel transform is however given by
formula_23
where formula_24 is the Dirac delta function and formula_25 the Heaviside step function. The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous formula_16, and it reduces to the classical Abel transform when formula_26. If formula_16 has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains "n" additional terms, each of them corresponding to one of the "n" discontinuities.
Relationship to other integral transforms.
Relationship to the Fourier and Hankel transforms.
The Abel transform is one member of the FHA cycle of integral operators. For example, in two dimensions, if we define "A" as the Abel transform operator, "F" as the Fourier transform operator and "H" as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that
formula_27
In other words, applying the Abel transform to a 1-dimensional function and
then applying the Fourier transform to that result is the same as applying
the Hankel transform to that function. This concept can be extended to higher
dimensions.
Relationship to the Radon transform.
Abel transform can be viewed as the Radon transform of an isotropic 2D function "f"("r"). As "f"("r") is isotropic, its Radon transform is the same at different angles of the viewing axis. Thus, the Abel transform is a function of the distance along the viewing axis only.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "F(y) = 2 \\int_y^\\infty \\frac{f(r)r}{\\sqrt{r^2 - y^2}} \\,dr."
},
{
"math_id": 1,
"text": "f(r) = -\\frac{1}{\\pi} \\int_r^\\infty \\frac{dF}{dy} \\,\\frac{dy}{\\sqrt{y^2 - r^2}}."
},
{
"math_id": 2,
"text": "F(y) = \\int_{-\\infty}^\\infty f\\left(\\sqrt{x^2 + y^2}\\right) \\,dx,"
},
{
"math_id": 3,
"text": "dx = \\frac{r\\,dr}{\\sqrt{r^2 - y^2}}"
},
{
"math_id": 4,
"text": "\nF(y) = 2 \\int_0^\\infty f\\left(\\sqrt{x^2 + y^2}\\right) \\,dx =\n 2 \\int_{|y|}^\\infty f(r)\\,\\frac{r\\,dr}{\\sqrt{r^2 - y^2}},\n"
},
{
"math_id": 5,
"text": "\nF(y, z) = \\int_{-\\infty}^\\infty f(\\rho, z) \\,dx =\n 2 \\int_y^\\infty \\frac{f(\\rho, z) \\rho \\,d\\rho}{\\sqrt{\\rho^2 - y^2}},\n"
},
{
"math_id": 6,
"text": "\nF(s) = \\int_{-\\infty}^\\infty f(r) \\,dx =\n 2 \\int_s^\\infty \\frac{f(r) r \\,dr}{\\sqrt{r^2 - s^2}},"
},
{
"math_id": 7,
"text": "f"
},
{
"math_id": 8,
"text": "f'"
},
{
"math_id": 9,
"text": "1/r"
},
{
"math_id": 10,
"text": "u=f(r)"
},
{
"math_id": 11,
"text": " v'=r/\\sqrt{r^2-y^2} "
},
{
"math_id": 12,
"text": "F(y) = -2 \\int_y^\\infty f'(r) \\sqrt{r^2-y^2} \\, dr."
},
{
"math_id": 13,
"text": "F'(y) = 2 y \\int_y^\\infty \\frac{f'(r)}{\\sqrt{r^2-y^2}} \\, dr."
},
{
"math_id": 14,
"text": "-\\frac{1}{\\pi} \\int_r^\\infty \\frac{F'(y)}{\\sqrt{y^2-r^2}} \\, dy = \\int_r^\\infty \\int_y^\\infty \\frac{-2 y}{\\pi \\sqrt{(y^2-r^2) (s^2-y^2)}} f'(s) \\, ds dy."
},
{
"math_id": 15,
"text": "\\int_r^\\infty \\int_r^s \\frac{-2 y}{\\pi \\sqrt{(y^2-r^2) (s^2-y^2)}} \\, dy f'(s) \\,ds = \\int_r^\\infty (-1) f'(s) \\, ds = f(r)."
},
{
"math_id": 16,
"text": "F(y)"
},
{
"math_id": 17,
"text": "y=y_\\Delta"
},
{
"math_id": 18,
"text": "\\Delta F"
},
{
"math_id": 19,
"text": "y_\\Delta"
},
{
"math_id": 20,
"text": "\\Delta F \\equiv \\lim_{\\epsilon\\rightarrow 0} [ F(y_\\Delta-\\epsilon) - F(y_\\Delta+\\epsilon) ]"
},
{
"math_id": 21,
"text": "f(r)"
},
{
"math_id": 22,
"text": "F(y)=2\\int_y^\\infty \\frac{f(r)r\\,dr}{\\sqrt{r^2-y^2}}."
},
{
"math_id": 23,
"text": "\nf(r)=\\left[ \\frac{1}{2}\\delta(r-y_\\Delta)\\sqrt{1-(y_\\Delta/r)^2} - \\frac{1}{\\pi} \\frac{H(y_\\Delta-r)}{\\sqrt{y_\\Delta^2-r^2}} \\right] \\Delta F-\\frac{1}{\\pi}\\int_r^\\infty\\frac{d F}{dy}\\frac{dy}{\\sqrt{y^2-r^2}}.\n"
},
{
"math_id": 24,
"text": "\\delta"
},
{
"math_id": 25,
"text": "H(x)"
},
{
"math_id": 26,
"text": "\\Delta F=0"
},
{
"math_id": 27,
"text": "FA = H."
}
] |
https://en.wikipedia.org/wiki?curid=1499595
|
1499625
|
Hyperbolic motion (relativity)
|
Motion of an object with constant proper acceleration in special relativity.
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram whose coordinates represent a suitable inertial (non-accelerated) frame. This motion has several interesting features, among them that it is possible to outrun a photon if given a sufficient head start, as may be concluded from the diagram.
History.
Hermann Minkowski (1908) showed the relation between a point on a worldline and the magnitude of four-acceleration and a "curvature hyperbola" (). In the context of Born rigidity, Max Born (1909) subsequently coined the term "hyperbolic motion" () for the case of constant magnitude of four-acceleration, then provided a detailed description for charged particles in hyperbolic motion, and introduced the corresponding "hyperbolically accelerated reference system" (). Born's formulas were simplified and extended by Arnold Sommerfeld (1910). For early reviews see the textbooks by Max von Laue (1911, 1921) or Wolfgang Pauli (1921). See also Galeriu (2015) or Gourgoulhon (2013), and Acceleration (special relativity)#History.
Worldline.
The proper acceleration formula_1 of a particle is defined as the acceleration that a particle "feels" as it accelerates from one inertial reference frame to another. If the proper acceleration is directed parallel to the line of motion, it is related to the ordinary three-acceleration in special relativity formula_2 by
formula_3
where formula_4 is the instantaneous speed of the particle, formula_5 the Lorentz factor, formula_6 is the speed of light, and formula_7 is the coordinate time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time formula_7 as well as proper time formula_8. For simplification, all initial values for time, location, and velocity can be set to 0, thus:
This gives formula_9, which is a hyperbola in time T and the spatial location variable formula_0. In this case, the accelerated object is located at formula_10 at time formula_11. If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:
formula_12
Rapidity.
The worldline for hyperbolic motion (which from now on will be written as a function of proper time) can be simplified in several ways. For instance, the expression
formula_13
can be subjected to a spatial shift of amount formula_14, thus
formula_15,
by which the observer is at position formula_16 at time formula_11. Furthermore, by setting formula_17 and introducing the rapidity formula_18, the equations for hyperbolic motion reduce to
with the hyperbola formula_19.
Charged particles in hyperbolic motion.
Born (1909), Sommerfeld (1910), von Laue (1911), Pauli (1921) also formulated the equations for the electromagnetic field of charged particles in hyperbolic motion. This was extended by Hermann Bondi & Thomas Gold (1955) and Fulton & Rohrlich (1960)
formula_20
This is related to the controversially discussed question, whether charges in perpetual hyperbolic motion do radiate or not, and whether this is consistent with the equivalence principle – even though it is about an ideal situation, because perpetual hyperbolic motion is not possible. While early authors such as Born (1909) or Pauli (1921) argued that no radiation arises, later authors such as Bondi & Gold and Fulton & Rohrlich showed that radiation does indeed arise.
Proper reference frame.
In equation (2) for hyperbolic motion, the expression formula_21 was constant, whereas the rapidity formula_22 was variable. However, as pointed out by Sommerfeld, one can define formula_21 as a variable, while making formula_22 constant. This means, that the equations become transformations indicating the simultaneous rest shape of an accelerated body with hyperbolic coordinates formula_23 as seen by a comoving observer
formula_24
By means of this transformation, the proper time becomes the time of the hyperbolically accelerated frame. These coordinates, which are commonly called Rindler coordinates (similar variants are called Kottler-Møller coordinates or Lass coordinates), can be seen as a special case of Fermi coordinates or Proper coordinates, and are often used in connection with the Unruh effect. Using these coordinates, it turns out that observers in hyperbolic motion possess an apparent event horizon, beyond which no signal can reach them.
Special conformal transformation.
A lesser known method for defining a reference frame in hyperbolic motion is the employment of the special conformal transformation, consisting of an inversion, a translation, and another inversion. It is commonly interpreted as a gauge transformation in Minkowski space, though some authors alternatively use it as an acceleration transformation (see Kastrup for a critical historical survey). It has the form
formula_25
Using only one spatial dimension by formula_26, and further simplifying by setting formula_27, and using the acceleration formula_28, it follows
formula_29
with the hyperbola formula_30. It turns out that at formula_31 the time becomes singular, to which Fulton & Rohrlich & Witten remark that one has to stay away from this limit, while Kastrup (who is very critical of the acceleration interpretation) remarks that this is one of the strange results of this interpretation.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "\\alpha"
},
{
"math_id": 2,
"text": "a=du/dT"
},
{
"math_id": 3,
"text": "\\alpha=\\gamma^3 a=\\frac{1}{\\left(1-u^2/c^2\\right)^{3/2}}\\frac{du}{dT},"
},
{
"math_id": 4,
"text": "u"
},
{
"math_id": 5,
"text": "\\gamma"
},
{
"math_id": 6,
"text": "c"
},
{
"math_id": 7,
"text": "T"
},
{
"math_id": 8,
"text": "\\tau"
},
{
"math_id": 9,
"text": "\\left(X+c^{2}/\\alpha\\right)^{2}-c^{2}T^{2}=c^{4}/\\alpha^{2}"
},
{
"math_id": 10,
"text": "X=0"
},
{
"math_id": 11,
"text": "T=0"
},
{
"math_id": 12,
"text": "{\\scriptstyle \\begin{array}{c|c}\n\\begin{align}u(T) & =\\frac{u_{0}\\gamma_{0}+\\alpha T}{\\sqrt{1+\\left(\\frac{u_{0}\\gamma_{0}+\\alpha T}{c}\\right)^{2}}}\\quad\\\\\n & =c\\tanh\\left\\{ \\operatorname{arsinh}\\left(\\frac{u_{0}\\gamma_{0}+\\alpha T}{c}\\right)\\right\\} \\\\\nX(T) & =X_{0}+\\frac{c^{2}}{\\alpha}\\left(\\sqrt{1+\\left(\\frac{u_{0}\\gamma_{0}+\\alpha T}{c}\\right)^{2}}-\\gamma_{0}\\right)\\\\\n & =X_{0}+\\frac{c^{2}}{\\alpha}\\left\\{ \\cosh\\left[\\operatorname{arsinh}\\left(\\frac{u_{0}\\gamma_{0}+\\alpha T}{c}\\right)\\right]-\\gamma_{0}\\right\\} \\\\\nc\\tau(T) & =c\\tau_{0}+\\frac{c^{2}}{\\alpha}\\ln\\left(\\frac{\\sqrt{c^{2}+\\left(u_{0}\\gamma_{0}+\\alpha T\\right){}^{2}}+u_{0}\\gamma_{0}+\\alpha T}{\\left(c+u_{0}\\right)\\gamma_{0}}\\right)\\\\\n & =c\\tau_{0}+\\frac{c^{2}}{\\alpha}\\left\\{ \\operatorname{arsinh}\\left(\\frac{u_{0}\\gamma_{0}+\\alpha T}{c}\\right)-\\operatorname{artanh}\\left(\\frac{u_{0}}{c}\\right)\\right\\} \n\\end{align}\n & \\begin{align}u(\\tau) & =c\\tanh\\left\\{ \\operatorname{artanh}\\left(\\frac{u_{0}}{c}\\right)+\\frac{\\alpha\\tau}{c}\\right\\} \\\\\n\\\\\nX(\\tau) & =X_{0}+\\frac{c^{2}}{\\alpha}\\left\\{ \\cosh\\left[\\operatorname{artanh}\\left(\\frac{u_{0}}{c}\\right)+\\frac{\\alpha\\tau}{c}\\right]-\\gamma_{0}\\right\\} \\\\\n\\\\\ncT(\\tau) & =cT_{0}+\\frac{c^{2}}{\\alpha}\\left\\{ \\sinh\\left[\\operatorname{artanh}\\left(\\frac{u_{0}}{c}\\right)+\\frac{\\alpha\\tau}{c}\\right]-\\frac{u_{0}\\gamma_{0}}{c}\\right\\} \n\\end{align}\n\\end{array}}"
},
{
"math_id": 13,
"text": "X=\\frac{c^{2}}{\\alpha}\\left(\\cosh\\frac{\\alpha\\tau}{c}-1\\right)"
},
{
"math_id": 14,
"text": "c^2/\\alpha"
},
{
"math_id": 15,
"text": "X=\\frac{c^{2}}{\\alpha}\\cosh\\frac{\\alpha\\tau}{c}"
},
{
"math_id": 16,
"text": "X=c^2/\\alpha"
},
{
"math_id": 17,
"text": "x=c^2/\\alpha"
},
{
"math_id": 18,
"text": "\\eta=\\operatorname{artanh}\\frac{u}{c}=\\frac{\\alpha\\tau}{c}"
},
{
"math_id": 19,
"text": "X^{2}-c^{2}T^{2}=x^{2}"
},
{
"math_id": 20,
"text": "\\begin{align}E_{\\rho'}'= & \\frac{\\left(8e/\\alpha^{2}\\right)\\rho'z'}{\\xi^{\\prime3}}\\\\\nE_{z'}'= & \\frac{-\\left(4e/\\alpha^{2}\\right)1/\\alpha^{2}+t^{\\prime2}+\\rho^{\\prime2}-z^{\\prime2}}{\\xi^{\\prime3}}\\\\\nE_{\\varphi'}'= & H_{\\varphi'}'=H_{z'}'=0\\\\\nH_{\\varphi'}'= & \\frac{\\left(8e/\\alpha^{2}\\right)\\rho't'}{\\xi^{\\prime3}}\\\\\n\\xi'= & \\sqrt{\\left(1/\\alpha^{2}+t^{\\prime2}-\\rho^{\\prime2}-z^{\\prime2}\\right)^{2}+\\left(2\\rho'/\\alpha\\right)^{2}}\n\\end{align}"
},
{
"math_id": 21,
"text": "x"
},
{
"math_id": 22,
"text": "\\eta"
},
{
"math_id": 23,
"text": "(x,y,z,\\eta)"
},
{
"math_id": 24,
"text": "cT=x\\sinh\\eta,\\quad X=x\\cosh\\eta,\\quad Y=y,\\quad Z=z"
},
{
"math_id": 25,
"text": "X^{\\mu}=\\frac{x^{\\mu}-a^{\\mu}x^{2}}{1-2ax+a^{2}x^{2}}"
},
{
"math_id": 26,
"text": "x^{\\mu}=(t,x)"
},
{
"math_id": 27,
"text": "x=0"
},
{
"math_id": 28,
"text": "a^{\\mu}=(0,-\\alpha/2)"
},
{
"math_id": 29,
"text": "T=\\frac{t}{1-\\frac{1}{4}\\alpha{}^{2}t^{2}},\\quad X=\\frac{-\\alpha t^{2}}{2\\left(1-\\frac{1}{4}\\alpha{}^{2}t^{2}\\right)}"
},
{
"math_id": 30,
"text": "\\left(X-1/\\alpha\\right)^{2}-T^{2}=1/\\alpha^{2}"
},
{
"math_id": 31,
"text": "t=\\pm(x+2/\\alpha)"
}
] |
https://en.wikipedia.org/wiki?curid=1499625
|
149968
|
Row and column vectors
|
Matrix consisting of a single row or column
In linear algebra, a column vector with &NoBreak;&NoBreak; elements is an formula_0 matrix consisting of a single column of &NoBreak;&NoBreak; entries, for example,
formula_1
Similarly, a row vector is a formula_2 matrix for some &NoBreak;&NoBreak;, consisting of a single row of &NoBreak;&NoBreak; entries,
formula_3
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector:
formula_4
and
formula_5
The set of all row vectors with n entries in a given field (such as the real numbers) forms an n-dimensional vector space; similarly, the set of all column vectors with m entries forms an m-dimensional vector space.
The space of row vectors with n entries can be regarded as the dual space of the space of column vectors with n entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.
Notation.
To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.
formula_6
or
formula_7
Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons (see alternative notation 2 in the table below).
Operations.
Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix.
The dot product of two column vectors a, b, considered as elements of a coordinate space, is equal to the matrix product of the transpose of a with b,
formula_8
By the symmetry of the dot product, the dot product of two column vectors a, b is also equal to the matrix product of the transpose of b with a,
formula_9
The matrix product of a column and a row vector gives the outer product of two vectors a, b, an example of the more general tensor product. The matrix product of the column vector representation of a and the row vector representation of b gives the components of their dyadic product,
formula_10
which is the transpose of the matrix product of the column vector representation of b and the row vector representation of a,
formula_11
Matrix transformations.
An "n" × "n" matrix M can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector v, the product v"M" is another row vector p:
formula_12
Another "n" × "n" matrix Q can act on p,
formula_13
Then one can write t = pQ" = vMQ", so the matrix product transformation MQ maps v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs.
When a column vector is transformed to another column vector under an "n" × "n" matrix action, the operation occurs to the left,
formula_14
leading to the algebraic expression "QM" vT for the composed output from vT input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "m \\times 1"
},
{
"math_id": 1,
"text": "\\boldsymbol{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_m \\end{bmatrix}."
},
{
"math_id": 2,
"text": "1 \\times n"
},
{
"math_id": 3,
"text": "\\boldsymbol a = \\begin{bmatrix} a_1 & a_2 & \\dots & a_n \\end{bmatrix}. "
},
{
"math_id": 4,
"text": "\\begin{bmatrix} x_1 \\; x_2 \\; \\dots \\; x_m \\end{bmatrix}^{\\rm T} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_m \\end{bmatrix}"
},
{
"math_id": 5,
"text": "\\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_m \\end{bmatrix}^{\\rm T} = \\begin{bmatrix} x_1 \\; x_2 \\; \\dots \\; x_m \\end{bmatrix}."
},
{
"math_id": 6,
"text": "\\boldsymbol{x} = \\begin{bmatrix} x_1 \\; x_2 \\; \\dots \\; x_m \\end{bmatrix}^{\\rm T}"
},
{
"math_id": 7,
"text": "\\boldsymbol{x} = \\begin{bmatrix} x_1, x_2, \\dots, x_m \\end{bmatrix}^{\\rm T}"
},
{
"math_id": 8,
"text": "\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a}^\\intercal \\mathbf{b} = \\begin{bmatrix}\n a_1 & \\cdots & a_n\n\\end{bmatrix} \\begin{bmatrix} \n b_1 \\\\ \\vdots \\\\ b_n\n\\end{bmatrix} = a_1 b_1 + \\cdots + a_n b_n \\,, "
},
{
"math_id": 9,
"text": "\\mathbf{b} \\cdot \\mathbf{a} = \\mathbf{b}^\\intercal \\mathbf{a} = \\begin{bmatrix}\n b_1 & \\cdots & b_n\n\\end{bmatrix}\\begin{bmatrix} \n a_1 \\\\ \\vdots \\\\ a_n\n\\end{bmatrix} = a_1 b_1 + \\cdots + a_n b_n\\,. "
},
{
"math_id": 10,
"text": "\\mathbf{a} \\otimes \\mathbf{b} = \\mathbf{a} \\mathbf{b}^\\intercal = \\begin{bmatrix}\n a_1 \\\\ a_2 \\\\ a_3\n\\end{bmatrix}\\begin{bmatrix} \n b_1 & b_2 & b_3\n\\end{bmatrix} = \\begin{bmatrix} \na_1 b_1 & a_1 b_2 & a_1 b_3 \\\\\na_2 b_1 & a_2 b_2 & a_2 b_3 \\\\\na_3 b_1 & a_3 b_2 & a_3 b_3 \\\\\n\\end{bmatrix} \\,, "
},
{
"math_id": 11,
"text": "\\mathbf{b} \\otimes \\mathbf{a} = \\mathbf{b} \\mathbf{a}^\\intercal = \\begin{bmatrix}\n b_1 \\\\ b_2 \\\\ b_3\n\\end{bmatrix}\\begin{bmatrix} \n a_1 & a_2 & a_3\n\\end{bmatrix} = \\begin{bmatrix} \nb_1 a_1 & b_1 a_2 & b_1 a_3 \\\\\nb_2 a_1 & b_2 a_2 & b_2 a_3 \\\\\nb_3 a_1 & b_3 a_2 & b_3 a_3 \\\\\n\\end{bmatrix} \\,. "
},
{
"math_id": 12,
"text": "\\mathbf{v} M = \\mathbf{p} \\,."
},
{
"math_id": 13,
"text": " \\mathbf{p} Q = \\mathbf{t} \\,. "
},
{
"math_id": 14,
"text": " \\mathbf{p}^\\mathrm{T} = M \\mathbf{v}^\\mathrm{T} \\,,\\quad \\mathbf{t}^\\mathrm{T} = Q \\mathbf{p}^\\mathrm{T},"
}
] |
https://en.wikipedia.org/wiki?curid=149968
|
14996853
|
Vaughan's identity
|
Identity that estimates sums in analytic number theory involving the von Mangoldt function
In mathematics and analytic number theory, Vaughan's identity is an identity found by R. C. Vaughan (1977) that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate summatory functions of the form
formula_0
where "f" is some arithmetic function of the natural integers "n", whose values in applications are often roots of unity, and Λ is the von Mangoldt function.
Procedure for applying the method.
The motivation for Vaughan's construction of his identity is briefly discussed at the beginning of Chapter 24 in Davenport. For now, we will skip over most of the technical details motivating the identity and its usage in applications, and instead focus on the setup of its construction by parts. Following from the reference, we construct four distinct sums based on the expansion of the logarithmic derivative of the Riemann zeta function in terms of functions which are partial Dirichlet series respectively truncated at the upper bounds of formula_1 and formula_2, respectively. More precisely, we define formula_3 and formula_4, which leads us to the exact identity that
formula_5
This last expansion implies that we can write
formula_6
where the component functions are defined to be
formula_7
We then define the corresponding summatory functions for formula_8 to be
formula_9
so that we can write
formula_10
Finally, at the conclusion of a multi-page argument of technical and at times delicate estimations of these sums, we obtain the following form of Vaughan's identity when we assume that formula_11, formula_12, and formula_13:
formula_14
It is remarked that in some instances sharper estimates can be obtained from Vaughan's identity by treating the component sum formula_15 more carefully by expanding it in the form of
formula_16
The optimality of the upper bound obtained by applying Vaughan's identity appears to be application-dependent with respect to the best functions formula_17 and formula_18 we can choose to input into equation (V1). See the applications cited in the next section for specific examples that arise in the different contexts respectively considered by multiple authors.
formula_19
Applications.
In particular, we obtain an asymptotic upper bound for these sums (typically evaluated at irrational formula_20) whose rational approximations satisfy
formula_21
of the form
formula_22
The argument for this estimate follows from Vaughan's identity by proving by a somewhat intricate argument that
formula_23
and then deducing the first formula above in the non-trivial cases when formula_24 and with formula_25.
Generalizations.
Vaughan's identity was generalized by .
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sum_{n \\leq N} f(n)\\Lambda(n)"
},
{
"math_id": 1,
"text": "U"
},
{
"math_id": 2,
"text": "V"
},
{
"math_id": 3,
"text": "F(s) = \\sum_{m \\leq U} \\Lambda(m) m^{-s}"
},
{
"math_id": 4,
"text": "G(s) = \\sum_{d \\leq V} \\mu(d) d^{-s}"
},
{
"math_id": 5,
"text": "-\\frac{\\zeta^{\\prime}(s)}{\\zeta(s)} = F(s) - \\zeta(s) F(s) G(s) - \\zeta^{\\prime}(s) G(s) + \\left(-\\frac{\\zeta^{\\prime}(s)}{\\zeta(s)} - F(s)\\right) (1-\\zeta(s) G(s)). "
},
{
"math_id": 6,
"text": "\\Lambda(n) = a_1(n) + a_2(n) + a_3(n) + a_4(n), "
},
{
"math_id": 7,
"text": "\\begin{align} \na_1(n) & := \\Biggl\\{\\begin{matrix} \\Lambda(n), & \\text{ if } n \\leq U; \\\\ 0, & \\text{ if } n > U\\end{matrix} \\\\ \na_2(n) & := - \\sum_{\\stackrel{mdr = n}{\\stackrel{m \\leq U}{d \\leq V}}} \\Lambda(m) \\mu(d) \\\\ \na_3(n) & := \\sum_{\\stackrel{hd=n}{d \\leq V}} \\mu(d) \\log(h) \\\\ \na_4(n) & := -\\sum_{\\stackrel{mk=n}{\\stackrel{m > U}{k > 1}}} \\Lambda(m) \\left(\\sum_{\\stackrel{d|k}{d \\leq V}} \\mu(d)\\right). \n\\end{align} \n"
},
{
"math_id": 8,
"text": "1 \\leq i \\leq 4"
},
{
"math_id": 9,
"text": "S_i(N) := \\sum_{n \\leq N} f(n) a_i(n), "
},
{
"math_id": 10,
"text": "\\sum_{n \\leq N} f(n) \\Lambda(n) = S_1(N) + S_2(N) + S_3(N) + S_4(N). "
},
{
"math_id": 11,
"text": "|f(n)| \\leq 1,\\ \\forall n"
},
{
"math_id": 12,
"text": "U,V \\geq 2"
},
{
"math_id": 13,
"text": "UV \\leq N"
},
{
"math_id": 14,
"text": "\\sum_{n \\leq N} f(n) \\Lambda(n) \\ll U + (\\log N) \\times \\sum_{t\\leq UV}\\left(\\max_{w} \\left|\\sum_{w \\leq r \\leq \\frac{N}{t}} f(rt)\\right|\\right) + \n \\sqrt{N} (\\log N)^3 \\times \\max_{U \\leq M \\leq N/V} \\max_{V \\leq j \\leq N/M}\\left(\\sum_{V < k \\leq N/M} \\left| \n \\sum_{\\stackrel{M < m \\leq 2M}{\\stackrel{m \\leq N/k}{m \\leq N/j}}} f(mj) \\bar{f(mk)}\\right|\\right)^{1/2} \\mathbf{(V1)}."
},
{
"math_id": 15,
"text": "S_2"
},
{
"math_id": 16,
"text": "S_2 = \\sum_{t \\leq UV} \\longmapsto \\sum_{t \\leq U} + \\sum_{U < t \\leq UV} =: S_2^{\\prime} + S_2^{\\prime\\prime}. "
},
{
"math_id": 17,
"text": "U = f_U(N)"
},
{
"math_id": 18,
"text": "V = f_V(N)"
},
{
"math_id": 19,
"text": "S(\\alpha) := \\sum_{n \\leq N} \\Lambda(n) e\\left(n\\alpha\\right)."
},
{
"math_id": 20,
"text": "\\alpha \\in \\mathbb{R} \\setminus \\mathbb{Q}"
},
{
"math_id": 21,
"text": "\\left| \\alpha - \\frac{a}{q} \\right| \\leq \\frac{1}{q^2}, (a, q) = 1, "
},
{
"math_id": 22,
"text": "S(\\alpha) \\ll \\left(\\frac{N}{\\sqrt{q}} + N^{4/5} + \\sqrt{Nq}\\right) (\\log N)^4. "
},
{
"math_id": 23,
"text": "S(\\alpha) \\ll \\left(UV + q + \\frac{N}{\\sqrt{U}} + \\frac{N}{\\sqrt{V}} + \\frac{N}{\\sqrt{q}} + \\sqrt{Nq}\\right) (\\log(qN))^4, "
},
{
"math_id": 24,
"text": "q \\leq N"
},
{
"math_id": 25,
"text": "U = V = N^{2/5}"
}
] |
https://en.wikipedia.org/wiki?curid=14996853
|
149984
|
Wiener process
|
Stochastic process generalizing Brownian motion
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and disturbances in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.
Characterisations of the Wiener process.
The Wiener process "formula_0" is characterised by the following properties:
That the process has independent increments means that if 0 ≤ "s"1 < "t"1 ≤ "s"2 < "t"2 then "W""t"1 − "W""s"1 and "W""t"2 − "W""s"2 are independent random variables, and the similar condition holds for "n" increments.
An alternative characterisation of the Wiener process is the so-called "Lévy characterisation" that says that the Wiener process is an almost surely continuous martingale with "W"0 = 0 and quadratic variation ["W""t", "W""t"] = "t" (which means that "W""t"2 − "t" is also a martingale).
A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent "N"(0, 1) random variables. This representation can be obtained using the Karhunen–Loève theorem.
Another characterisation of a Wiener process is the definite integral (from time zero to time "t") of a zero mean, unit variance, delta correlated ("white") Gaussian process.
The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes). Unlike the random walk, it is scale invariant, meaning that
formula_13
is a Wiener process for any nonzero constant α. The Wiener measure is the probability law on the space of continuous functions "g", with "g"(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.
Wiener process as a limit of random walk.
Let formula_14 be i.i.d. random variables with mean 0 and variance 1. For each "n", define a continuous time stochastic process
formula_15
This is a random step function. Increments of formula_16 are independent because the formula_17 are independent. For large "n", formula_18 is close to formula_19 by the central limit theorem. Donsker's theorem asserts that as formula_20, formula_16 approaches a Wiener process, which explains the ubiquity of Brownian motion.
Properties of a one-dimensional Wiener process.
Basic properties.
The unconditional probability density function follows a normal distribution with mean = 0 and variance = "t", at a fixed time t:
formula_21
The expectation is zero:
formula_22
The variance, using the computational formula, is t:
formula_23
These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus
formula_24
Covariance and correlation.
The covariance and correlation (where formula_25):
formula_26
These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that formula_27.
formula_28
Substituting
formula_29
we arrive at:
formula_30
Since formula_31 and formula_32 are independent,
formula_33
Thus
formula_34
A corollary useful for simulation is that we can write, for "t"1 < "t"2:
formula_35
where Z is an independent standard normal variable.
Wiener representation.
Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If formula_36 are independent Gaussian variables with mean zero and variance one, then
formula_37
and
formula_38
represent a Brownian motion on formula_39. The scaled process
formula_40
is a Brownian motion on formula_41 (cf. Karhunen–Loève theorem).
Running maximum.
The joint distribution of the running maximum
formula_42
and "Wt" is
formula_43
To get the unconditional distribution of formula_44, integrate over −∞ < "w" ≤ "m":
formula_45
the probability density function of a Half-normal distribution. The expectation is
formula_46
If at time formula_12 the Wiener process has a known value formula_47, it is possible to calculate the conditional probability distribution of the maximum in interval formula_48 (cf. Probability distribution of extreme points of a Wiener stochastic process). The cumulative probability distribution function of the maximum value, conditioned by the known value formula_0, is:
formula_49
Self-similarity.
Brownian scaling.
For every "c" > 0 the process formula_50 is another Wiener process.
Time reversal.
The process formula_51 for 0 ≤ "t" ≤ 1 is distributed like "Wt" for 0 ≤ "t" ≤ 1.
Time inversion.
The process formula_52 is another Wiener process.
Projective invariance.
Consider a Wiener process formula_53, formula_54, conditioned so that formula_55 (which holds almost surely) and as usual formula_56. Then the following are all Wiener processes :
formula_57
Thus the Wiener process is invariant under the projective group PSL(2,R), being invariant under the generators of the group. The action of an element formula_58 is
formula_59
which defines a group action, in the sense that formula_60
Conformal invariance in two dimensions.
Let formula_53 be a two-dimensional Wiener process, regarded as a complex-valued process with formula_61. Let formula_62 be an open set containing 0, and formula_63 be associated Markov time:
formula_64
If formula_65 is a holomorphic function which is not constant, such that formula_66, then formula_67 is a time-changed Wiener process in formula_68 . More precisely, the process formula_69 is Wiener in formula_70 with the Markov time formula_71 where
formula_72
formula_73
formula_74
A class of Brownian martingales.
If a polynomial "p"("x", "t") satisfies the partial differential equation
formula_75
then the stochastic process
formula_76
is a martingale.
Example: formula_77 is a martingale, which shows that the quadratic variation of "W" on [0, "t"] is equal to t. It follows that the expected time of first exit of "W" from (−"c", "c") is equal to "c"2.
More generally, for every polynomial "p"("x", "t") the following stochastic process is a martingale:
formula_78
where "a" is the polynomial
formula_79
Example: formula_80 formula_81 the process
formula_82
is a martingale, which shows that the quadratic variation of the martingale formula_77 on [0, "t"] is equal to
formula_83
About functions "p"("xa", "t") more general than polynomials, see local martingales.
Some properties of sample paths.
The set of all functions "w" with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely.
Quantitative properties.
Law of the iterated logarithm.
formula_89
Modulus of continuity.
Local modulus of continuity:
formula_90
Global modulus of continuity (Lévy):
formula_91
Dimension doubling theorem.
The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost surely.
Local time.
The image of the Lebesgue measure on [0, "t"] under the map "w" (the pushforward measure) has a density "L""t". Thus,
formula_92
for a wide class of functions "f" (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density "Lt" is (more exactly, can and will be chosen to be) continuous. The number "Lt"("x") is called the local time at "x" of "w" on [0, "t"]. It is strictly positive for all "x" of the interval ("a", "b") where "a" and "b" are the least and the greatest value of "w" on [0, "t"], respectively. (For "x" outside this interval the local time evidently vanishes.) Treated as a function of two variables "x" and "t", the local time is still continuous. Treated as a function of "t" (while "x" is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of "w".
These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.
Information rate.
The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by
formula_93
Therefore, it is impossible to encode formula_94 using a binary code of less than formula_95 bits and recover it with expected mean squared error less than formula_70. On the other hand, for any formula_96, there exists formula_97 large enough and a binary code of no more than formula_98 distinct elements such that the expected mean squared error in recovering formula_94 from this code is at most formula_99.
In many cases, it is impossible to encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals formula_100 before applying a binary code to represent these samples, the optimal trade-off between code rate formula_101 and expected mean square error formula_70 (in estimating the continuous-time Wiener process) follows the parametric representation
formula_102
formula_103
where formula_104 and formula_105. In particular, formula_106 is the mean squared error associated only with the sampling operation (without encoding).
Related processes.
The stochastic process defined by
formula_107
is called a Wiener process with drift μ and infinitesimal variance σ2. These processes exhaust continuous Lévy processes, which means that they are the only continuous Lévy processes,
as a consequence of the Lévy–Khintchine representation.
Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion. In both cases a rigorous treatment involves a limiting procedure, since the formula "P"("A"|"B") = "P"("A" ∩ "B")/"P"("B") does not apply when "P"("B") = 0.
A geometric Brownian motion can be written
formula_108
It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.
The stochastic process
formula_109
is distributed like the Ornstein–Uhlenbeck process with parameters formula_110, formula_111, and formula_112.
The time of hitting a single point "x" > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers "x") is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals [0, "x"].
The local time "L" = ("Lxt")"x" ∈ R, "t" ≥ 0 of a Brownian motion describes the time that the process spends at the point "x". Formally
formula_113
where "δ" is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.
Brownian martingales.
Let "A" be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and "Xt" the conditional probability of "A" given the Wiener process on the time interval [0, "t"] (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, "t"] belongs to "A"). Then the process "Xt" is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all Brownian martingales are continuous. A Brownian martingale is, by definition, a martingale adapted to the Brownian filtration; and the Brownian filtration is, by definition, the filtration generated by the Wiener process.
Integrated Brownian motion.
The time-integral of the Wiener process
formula_114
is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to have the distribution "N"(0, "t"3/3), calculated using the fact that the covariance of the Wiener process is formula_115.
For the general case of the process defined by
formula_116
Then, for formula_117,
formula_118
formula_119
In fact, formula_120 is always a zero mean normal random variable. This allows for simulation of formula_121 given formula_120 by taking
formula_122
where "Z" is a standard normal variable and
formula_123
formula_124
The case of formula_125 corresponds to formula_126. All these results can be seen as direct consequences of Itô isometry.
The "n"-times-integrated Wiener process is a zero-mean normal variable with variance formula_127. This is given by the Cauchy formula for repeated integration.
Time change.
Every continuous martingale (starting at the origin) is a time changed Wiener process.
Example: 2"W""t" = "V"(4"t") where "V" is another Wiener process (different from "W" but distributed like "W").
Example. formula_128 where formula_129 and "V" is another Wiener process.
In general, if "M" is a continuous martingale then formula_130 where "A"("t") is the quadratic variation of "M" on [0, "t"], and "V" is a Wiener process.
Corollary. (See also Doob's martingale convergence theorems) Let "Mt" be a continuous martingale, and
formula_131
formula_132
Then only the following two cases are possible:
formula_133
formula_134
other cases (such as formula_135 formula_136 etc.) are of probability 0.
Especially, a nonnegative continuous martingale has a finite limit (as "t" → ∞) almost surely.
All stated (in this subsection) for martingales holds also for local martingales.
Change of measure.
A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.
Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.
Complex-valued Wiener process.
The complex-valued Wiener process may be defined as a complex-valued random process of the form formula_137 where formula_138 and formula_139 are independent Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify formula_140 with formula_141.
Self-similarity.
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.
Rotation invariance: for every complex number formula_142 such that formula_143 the process formula_144 is another complex-valued Wiener process.
Time change.
If formula_145 is an entire function then the process formula_146 is a time-changed complex-valued Wiener process.
Example: formula_147 where
formula_148
and formula_149 is another complex-valued Wiener process.
In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale formula_150 is not (here formula_138 and formula_139 are independent Wiener processes, as before).
Brownian sheet.
The Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter formula_12 while others define it for general dimensions.
See also.
<templatestyles src="Col-begin/styles.css"/>
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "W_t"
},
{
"math_id": 1,
"text": "W_0= 0"
},
{
"math_id": 2,
"text": "W"
},
{
"math_id": 3,
"text": "t>0,"
},
{
"math_id": 4,
"text": "W_{t+u} - W_t,"
},
{
"math_id": 5,
"text": "u \\ge 0,"
},
{
"math_id": 6,
"text": "W_s"
},
{
"math_id": 7,
"text": "s< t."
},
{
"math_id": 8,
"text": "W_{t+u} - W_t"
},
{
"math_id": 9,
"text": "0"
},
{
"math_id": 10,
"text": "u"
},
{
"math_id": 11,
"text": "W_{t+u} - W_t\\sim \\mathcal N(0,u)."
},
{
"math_id": 12,
"text": "t"
},
{
"math_id": 13,
"text": "\\alpha^{-1} W_{\\alpha^2 t}"
},
{
"math_id": 14,
"text": "\\xi_1, \\xi_2, \\ldots"
},
{
"math_id": 15,
"text": "W_n(t)=\\frac{1}{\\sqrt{n}}\\sum\\limits_{1\\leq k\\leq\\lfloor nt\\rfloor}\\xi_k, \\qquad t \\in [0,1]."
},
{
"math_id": 16,
"text": "W_n"
},
{
"math_id": 17,
"text": "\\xi_k"
},
{
"math_id": 18,
"text": "W_n(t)-W_n(s)"
},
{
"math_id": 19,
"text": "N(0,t-s)"
},
{
"math_id": 20,
"text": "n \\to \\infty"
},
{
"math_id": 21,
"text": "f_{W_t}(x) = \\frac{1}{\\sqrt{2 \\pi t}} e^{-x^2/(2t)}."
},
{
"math_id": 22,
"text": "\\operatorname E[W_t] = 0."
},
{
"math_id": 23,
"text": "\\operatorname{Var}(W_t) = t."
},
{
"math_id": 24,
"text": "W_t = W_t-W_0 \\sim N(0,t)."
},
{
"math_id": 25,
"text": "s \\leq t"
},
{
"math_id": 26,
"text": "\\begin{align}\n\\operatorname{cov}(W_s, W_t) &= s, \\\\\n\\operatorname{corr}(W_s,W_t) &= \\frac{\\operatorname{cov}(W_s,W_t)}{\\sigma_{W_s} \\sigma_{W_t}} = \\frac{s}{\\sqrt{st}} = \\sqrt{\\frac{s}{t}}.\n\\end{align}"
},
{
"math_id": 27,
"text": "t_1\\leq t_2"
},
{
"math_id": 28,
"text": "\\operatorname{cov}(W_{t_1}, W_{t_2}) = \\operatorname{E}\\left[(W_{t_1}-\\operatorname{E}[W_{t_1}]) \\cdot (W_{t_2}-\\operatorname{E}[W_{t_2}])\\right] = \\operatorname{E}\\left[W_{t_1} \\cdot W_{t_2} \\right]."
},
{
"math_id": 29,
"text": " W_{t_2} = ( W_{t_2} - W_{t_1} ) + W_{t_1} "
},
{
"math_id": 30,
"text": "\\begin{align}\n\\operatorname{E}[W_{t_1} \\cdot W_{t_2}] & = \\operatorname{E}\\left[W_{t_1} \\cdot ((W_{t_2} - W_{t_1})+ W_{t_1}) \\right] \\\\\n& = \\operatorname{E}\\left[W_{t_1} \\cdot (W_{t_2} - W_{t_1} )\\right] + \\operatorname{E}\\left[ W_{t_1}^2 \\right].\n\\end{align}"
},
{
"math_id": 31,
"text": " W_{t_1}=W_{t_1} - W_{t_0} "
},
{
"math_id": 32,
"text": " W_{t_2} - W_{t_1} "
},
{
"math_id": 33,
"text": " \\operatorname{E}\\left [W_{t_1} \\cdot (W_{t_2} - W_{t_1} ) \\right ] = \\operatorname{E}[W_{t_1}] \\cdot \\operatorname{E}[W_{t_2} - W_{t_1}] = 0."
},
{
"math_id": 34,
"text": "\\operatorname{cov}(W_{t_1}, W_{t_2}) = \\operatorname{E} \\left [W_{t_1}^2 \\right ] = t_1."
},
{
"math_id": 35,
"text": "W_{t_2} = W_{t_1}+\\sqrt{t_2-t_1}\\cdot Z"
},
{
"math_id": 36,
"text": "\\xi_n"
},
{
"math_id": 37,
"text": "W_t = \\xi_0 t+ \\sqrt{2}\\sum_{n=1}^\\infty \\xi_n\\frac{\\sin \\pi n t}{\\pi n}"
},
{
"math_id": 38,
"text": " W_t = \\sqrt{2} \\sum_{n=1}^\\infty \\xi_n \\frac{\\sin \\left(\\left(n - \\frac{1}{2}\\right) \\pi t\\right)}{ \\left(n - \\frac{1}{2}\\right) \\pi} "
},
{
"math_id": 39,
"text": "[0,1]"
},
{
"math_id": 40,
"text": "\\sqrt{c}\\, W\\left(\\frac{t}{c}\\right)"
},
{
"math_id": 41,
"text": "[0,c]"
},
{
"math_id": 42,
"text": " M_t = \\max_{0 \\leq s \\leq t} W_s "
},
{
"math_id": 43,
"text": " f_{M_t,W_t}(m,w) = \\frac{2(2m - w)}{t\\sqrt{2 \\pi t}} e^{-\\frac{(2m-w)^2}{2t}}, \\qquad m \\ge 0, w \\leq m."
},
{
"math_id": 44,
"text": "f_{M_t}"
},
{
"math_id": 45,
"text": "\\begin{align}\nf_{M_t}(m) & = \\int_{-\\infty}^m f_{M_t,W_t}(m,w)\\,dw = \\int_{-\\infty}^m \\frac{2(2m - w)}{t\\sqrt{2 \\pi t}} e^{-\\frac{(2m-w)^2}{2t}} \\,dw \\\\[5pt]\n& = \\sqrt{\\frac{2}{\\pi t}}e^{-\\frac{m^2}{2t}}, \\qquad m \\ge 0,\n\\end{align}"
},
{
"math_id": 46,
"text": " \\operatorname{E}[M_t] = \\int_0^\\infty m f_{M_t}(m)\\,dm = \\int_0^\\infty m \\sqrt{\\frac{2}{\\pi t}}e^{-\\frac{m^2}{2t}}\\,dm = \\sqrt{\\frac{2t}{\\pi}} "
},
{
"math_id": 47,
"text": "W_{t}"
},
{
"math_id": 48,
"text": "[0, t]"
},
{
"math_id": 49,
"text": "\\, F_{M_{W_t}} (m) = \\Pr \\left( M_{W_t} = \\max_{0 \\leq s \\leq t} W(s) \\leq m \\mid W(t) = W_t \\right) = \\ 1 -\\ e^{-2\\frac{m(m - W_t)}{t}}\\ \\, , \\,\\ \\ m > \\max(0,W_t)"
},
{
"math_id": 50,
"text": " V_t = (1 / \\sqrt c) W_{ct} "
},
{
"math_id": 51,
"text": " V_t = W_{1-t} - W_{1} "
},
{
"math_id": 52,
"text": " V_t = t W_{1/t} "
},
{
"math_id": 53,
"text": "W(t)"
},
{
"math_id": 54,
"text": "t\\in\\mathbb R"
},
{
"math_id": 55,
"text": "\\lim_{t\\to\\pm\\infty}tW(t)=0"
},
{
"math_id": 56,
"text": "W(0)=0"
},
{
"math_id": 57,
"text": "\n\\begin{array}{rcl}\nW_{1,s}(t) &=& W(t+s)-W(s), \\quad s\\in\\mathbb R\\\\\nW_{2,\\sigma}(t) &=& \\sigma^{-1/2}W(\\sigma t),\\quad \\sigma > 0\\\\\nW_3(t) &=& tW(-1/t).\n\\end{array}\n"
},
{
"math_id": 58,
"text": "g = \\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}"
},
{
"math_id": 59,
"text": "W_g(t) = (ct+d)W\\left(\\frac{at+b}{ct+d}\\right) - ctW\\left(\\frac{a}{c}\\right) - dW\\left(\\frac{b}{d}\\right),"
},
{
"math_id": 60,
"text": "(W_g)_h = W_{gh}."
},
{
"math_id": 61,
"text": "W(0)=0\\in\\mathbb C"
},
{
"math_id": 62,
"text": "D\\subset\\mathbb C"
},
{
"math_id": 63,
"text": "\\tau_D"
},
{
"math_id": 64,
"text": "\\tau_D = \\inf \\{ t\\ge 0 |W(t)\\not\\in D\\}."
},
{
"math_id": 65,
"text": "f:D\\to \\mathbb C"
},
{
"math_id": 66,
"text": "f(0)=0"
},
{
"math_id": 67,
"text": "f(W_t)"
},
{
"math_id": 68,
"text": "f(D)"
},
{
"math_id": 69,
"text": "Y(t)"
},
{
"math_id": 70,
"text": "D"
},
{
"math_id": 71,
"text": "S(t)"
},
{
"math_id": 72,
"text": "Y(t) = f(W(\\sigma(t)))"
},
{
"math_id": 73,
"text": "S(t) = \\int_0^t|f'(W(s))|^2\\,ds"
},
{
"math_id": 74,
"text": "\\sigma(t) = S^{-1}(t):\\quad t = \\int_0^{\\sigma(t)}|f'(W(s))|^2\\,ds."
},
{
"math_id": 75,
"text": "\\left( \\frac{\\partial}{\\partial t} - \\frac{1}{2} \\frac{\\partial^2}{\\partial x^2} \\right) p(x,t) = 0 "
},
{
"math_id": 76,
"text": " M_t = p ( W_t, t )"
},
{
"math_id": 77,
"text": " W_t^2 - t "
},
{
"math_id": 78,
"text": " M_t = p ( W_t, t ) - \\int_0^t a(W_s,s) \\, \\mathrm{d}s, "
},
{
"math_id": 79,
"text": " a(x,t) = \\left( \\frac{\\partial}{\\partial t} + \\frac 1 2 \\frac{\\partial^2}{\\partial x^2} \\right) p(x,t). "
},
{
"math_id": 80,
"text": " p(x,t) = \\left(x^2 - t\\right)^2, "
},
{
"math_id": 81,
"text": " a(x,t) = 4x^2; "
},
{
"math_id": 82,
"text": " \\left(W_t^2 - t\\right)^2 - 4 \\int_0^t W_s^2 \\, \\mathrm{d}s "
},
{
"math_id": 83,
"text": " 4 \\int_0^t W_s^2 \\, \\mathrm{d}s."
},
{
"math_id": 84,
"text": "\\epsilon > 0"
},
{
"math_id": 85,
"text": "w(t)"
},
{
"math_id": 86,
"text": "(\\tfrac 1 2 + \\epsilon)"
},
{
"math_id": 87,
"text": "(\\tfrac 1 2 - \\epsilon)"
},
{
"math_id": 88,
"text": "\\lim_{s \\to t} \\frac{|w(s)-w(t)|}{|s-t|} \\to \\infty."
},
{
"math_id": 89,
"text": " \\limsup_{t\\to+\\infty} \\frac{ |w(t)| }{ \\sqrt{ 2t \\log\\log t } } = 1, \\quad \\text{almost surely}. "
},
{
"math_id": 90,
"text": " \\limsup_{\\varepsilon \\to 0+} \\frac{ |w(\\varepsilon)| }{ \\sqrt{ 2\\varepsilon \\log\\log(1/\\varepsilon) } } = 1, \\qquad \\text{almost surely}. "
},
{
"math_id": 91,
"text": " \\limsup_{\\varepsilon\\to0+} \\sup_{0\\le s<t\\le 1, t-s\\le\\varepsilon}\\frac{|w(s)-w(t)|}{\\sqrt{ 2\\varepsilon \\log(1/\\varepsilon)}} = 1, \\qquad \\text{almost surely}. "
},
{
"math_id": 92,
"text": " \\int_0^t f(w(s)) \\, \\mathrm{d}s = \\int_{-\\infty}^{+\\infty} f(x) L_t(x) \\, \\mathrm{d}x "
},
{
"math_id": 93,
"text": "R(D) = \\frac{2}{\\pi^2 \\ln 2 D} \\approx 0.29D^{-1}."
},
{
"math_id": 94,
"text": "\\{w_t \\}_{t \\in [0,T]}"
},
{
"math_id": 95,
"text": "T R(D)"
},
{
"math_id": 96,
"text": " \\varepsilon>0"
},
{
"math_id": 97,
"text": "T"
},
{
"math_id": 98,
"text": "2^{TR(D)}"
},
{
"math_id": 99,
"text": "D - \\varepsilon"
},
{
"math_id": 100,
"text": "T_s"
},
{
"math_id": 101,
"text": "R(T_s,D)"
},
{
"math_id": 102,
"text": " R(T_s,D_\\theta) = \\frac{T_s}{2} \\int_0^1 \\log_2^+\\left[\\frac{S(\\varphi)- \\frac{1}{6}}{\\theta}\\right] d\\varphi, "
},
{
"math_id": 103,
"text": " D_\\theta = \\frac{T_s}{6} + T_s\\int_0^1 \\min\\left\\{S(\\varphi)-\\frac{1}{6},\\theta \\right\\} d\\varphi, "
},
{
"math_id": 104,
"text": "S(\\varphi) = (2 \\sin(\\pi \\varphi /2))^{-2}"
},
{
"math_id": 105,
"text": "\\log^+[x] = \\max\\{0,\\log(x)\\}"
},
{
"math_id": 106,
"text": "T_s/6"
},
{
"math_id": 107,
"text": " X_t = \\mu t + \\sigma W_t"
},
{
"math_id": 108,
"text": " e^{\\mu t-\\frac{\\sigma^2 t}{2}+\\sigma W_t}."
},
{
"math_id": 109,
"text": "X_t = e^{-t} W_{e^{2t}}"
},
{
"math_id": 110,
"text": "\\theta = 1"
},
{
"math_id": 111,
"text": "\\mu = 0"
},
{
"math_id": 112,
"text": "\\sigma^2 = 2"
},
{
"math_id": 113,
"text": "L^x(t) =\\int_0^t \\delta(x-B_t)\\,ds"
},
{
"math_id": 114,
"text": "W^{(-1)}(t) := \\int_0^t W(s) \\, ds"
},
{
"math_id": 115,
"text": " t \\wedge s = \\min(t, s)"
},
{
"math_id": 116,
"text": "V_f(t) = \\int_0^t f'(s)W(s) \\,ds=\\int_0^t (f(t)-f(s))\\,dW_s"
},
{
"math_id": 117,
"text": "a > 0"
},
{
"math_id": 118,
"text": "\\operatorname{Var}(V_f(t))=\\int_0^t (f(t)-f(s))^2 \\,ds"
},
{
"math_id": 119,
"text": "\\operatorname{cov}(V_f(t+a),V_f(t))=\\int_0^t (f(t+a)-f(s))(f(t)-f(s)) \\,ds"
},
{
"math_id": 120,
"text": "V_f(t)"
},
{
"math_id": 121,
"text": "V_f(t+a)"
},
{
"math_id": 122,
"text": "V_f(t+a)=A\\cdot V_f(t) +B\\cdot Z"
},
{
"math_id": 123,
"text": "A=\\frac{\\operatorname{cov}(V_f(t+a),V_f(t))}{\\operatorname{Var}(V_f(t))}"
},
{
"math_id": 124,
"text": "B^2=\\operatorname{Var}(V_f(t+a))-A^2\\operatorname{Var}(V_f(t))"
},
{
"math_id": 125,
"text": "V_f(t)=W^{(-1)}(t)"
},
{
"math_id": 126,
"text": "f(t)=t"
},
{
"math_id": 127,
"text": "\\frac{t}{2n+1}\\left ( \\frac{t^n}{n!} \\right )^2 "
},
{
"math_id": 128,
"text": " W_t^2 - t = V_{A(t)} "
},
{
"math_id": 129,
"text": " A(t) = 4 \\int_0^t W_s^2 \\, \\mathrm{d} s "
},
{
"math_id": 130,
"text": " M_t - M_0 = V_{A(t)} "
},
{
"math_id": 131,
"text": "M^-_\\infty = \\liminf_{t\\to\\infty} M_t,"
},
{
"math_id": 132,
"text": "M^+_\\infty = \\limsup_{t\\to\\infty} M_t. "
},
{
"math_id": 133,
"text": " -\\infty < M^-_\\infty = M^+_\\infty < +\\infty,"
},
{
"math_id": 134,
"text": "-\\infty = M^-_\\infty < M^+_\\infty = +\\infty; "
},
{
"math_id": 135,
"text": " M^-_\\infty = M^+_\\infty = +\\infty, "
},
{
"math_id": 136,
"text": " M^-_\\infty < M^+_\\infty < +\\infty "
},
{
"math_id": 137,
"text": "Z_t = X_t + i Y_t"
},
{
"math_id": 138,
"text": "X_t"
},
{
"math_id": 139,
"text": "Y_t"
},
{
"math_id": 140,
"text": "\\R^2"
},
{
"math_id": 141,
"text": "\\mathbb C"
},
{
"math_id": 142,
"text": "c"
},
{
"math_id": 143,
"text": "|c|=1"
},
{
"math_id": 144,
"text": "c \\cdot Z_t"
},
{
"math_id": 145,
"text": "f"
},
{
"math_id": 146,
"text": " f(Z_t) - f(0) "
},
{
"math_id": 147,
"text": " Z_t^2 = \\left(X_t^2 - Y_t^2\\right) + 2 X_t Y_t i = U_{A(t)} "
},
{
"math_id": 148,
"text": "A(t) = 4 \\int_0^t |Z_s|^2 \\, \\mathrm{d} s "
},
{
"math_id": 149,
"text": "U"
},
{
"math_id": 150,
"text": "2 X_t + i Y_t"
}
] |
https://en.wikipedia.org/wiki?curid=149984
|
1499906
|
Glass electrode
|
Electrode that is pH-sensitive
A glass electrode is a type of ion-selective electrode made of a doped glass membrane that is sensitive to a specific ion. The most common application of ion-selective glass electrodes is for the measurement of pH. The pH electrode is an example of a glass electrode that is sensitive to hydrogen ions. Glass electrodes play an important part in the instrumentation for chemical analysis, and physicochemical studies. The voltage of the glass electrode, relative to some reference value, is sensitive to changes in the activity of a certain type of ions.
History.
The first studies of glass electrodes (GE) found different sensitivities of different glasses to change the medium's acidity (pH), due to the effects of the alkali metal ions.
In 1906, M. Cremer, the father of Erika Cremer, determined that the electric potential that arises between parts of the fluid, located on opposite sides of the glass membrane is proportional to the concentration of acid (hydrogen ion concentration).
In 1909, S. P. L. Sørensen introduced the concept of pH, and in the same year F. Haber and Z. Klemensiewicz reported results of their research on the glass electrode in The Society of Chemistry in Karlsruhe.
In 1922, W. S. Hughes showed that the alkali-silicate glass electrodes are similar to hydrogen electrodes, reversible concerning H+.
In 1925, P. M. Tookey Kerridge developed the first glass electrode for analysis of blood samples and highlighted some of the practical problems with the equipment such as the high resistance of glass (50–150 MΩ). During her PhD, Kerridge developed a glass electrode aimed to measure small volume of solution. Her clever and careful design was a pioneering work in the making of glass electrodes.
Applications.
Glass electrodes are commonly used for pH measurements. There are also specialized ion-sensitive glass electrodes used for the determination of the concentration of lithium, sodium, ammonium, and other ions.
Glass electrodes find a wide diversity of uses in a large range of applications including research labs, control of industrial processes, analysis of foods and cosmetics, monitoring of environmental pollution, or soil acidity measurements... . Micro-electrodes are specifically designed for pH measurements on very small volumes of fluid, or for direct measurements in geochemical micro-environments, or in biochemical studies such as for determining the electrical potential of cell membrane.
Heavy duty electrodes withstanding several tens of bar of hydraulic pressure also allow measurements in water wells in deep aquifers, or to directly determine "in situ" the pH of pore water in deep clay formations. For long-term "in situ" measurements, it is critical to minimize the KCl leak from the reference electrode compartment (Ag / AgCl / KCl 3 M), and to use glycerol-free electrodes to avoid fuelling microbial growth, and to prevent unexpected but severe perturbations related to bacterial activity (pH decrease due to sulfate-reducing bacteria, or even methanogen bacteria).
Types.
All commercial electrodes respond to single-charged ions, such as H+, Na+, Ag+. The most common glass electrode is the pH-electrode. Only a few chalcogenide glass electrodes are presently known to be sensitive to double-charged ions, such as Pb2+, Cd2+, and some other divalent cations.
There are two main types of glass-forming systems:
Interfering ions.
Because of the ion-exchange nature of the glass membrane, it is possible for some other ions to concurrently interact with ion-exchange sites of the glass, and distort the linear dependence of the measured electrode potential on pH or other electrode functions. In some cases, it is possible to change the electrode function from one ion to another. For example, some silicate pPNA electrodes can be changed to pAg function by soaking in a silver salt solution.
Interference effects are commonly described by the semi-empirical Nicolsky-Shultz-Eisenman equation (also known as Nikolsky-Shultz-Eisenman equation), an extension to the Nernst equation. It is given by:
formula_0
where "E" is the electromotive force (emf), "E0" the standard electrode potential, "z" the ionic valency including the sign, "a" the activity, "i" the ion of interest, "j" the interfering ions and "k"ij is the selectivity coefficient quantifying the ion-exchange equilibrium between the ions "i" and "j". The smaller the selectivity coefficient, the less is the interference by "j".
To see the interfering effect of Na+ to a pH-electrode:
formula_1
Range of a pH glass electrode.
The pH range at constant concentration can be divided into 3 parts:
formula_2
where F is Faraday's constant (see Nernst equation).
The effect is usually noticeable at pH > 12, and at concentrations of lithium or sodium ions of 0.1 mol/L or more. Potassium ions usually cause less error than sodium ions.
Special electrodes exist for working in extreme pH ranges.
Construction.
A typical modern pH probe is a combination electrode, which combines both the glass and reference electrodes into one body. The combination electrode consists of the following parts (see the drawing):
The bottom of a pH electrode balloons out into a round thin glass bulb. The pH electrode is best thought of as a tube within a tube. The inner tube contains an unchanging 1×10−7 mol/L HCl solution. Also inside the inner tube is the cathode terminus of the reference probe. The anodic terminus wraps itself around the outside of the inner tube and ends with the same sort of reference probe as was on the inside of the inner tube. It is filled with a reference solution of KCl and has contact with the solution on the outside of the pH probe by way of a porous plug that serves as a salt bridge.
Galvanic cell schematic representation.
This section describes the functioning of two distinct types of electrodes as one unit which combines both the glass electrode and the reference electrode into one body. It deserves some explanation.
This device is essentially a galvanic cell that can be schematically represented as:
Internal electrode | Internal buffer solution || "Test Solution" || Reference solution | Reference electrode
Ag("s") | AgCl("s") | 0.1 M KCl("aq"), 1×10−7M H+ solution || "Test Solution" || KCl("aq") | AgCl("s") | Ag("s")
The double "pipe symbols" (||) indicate diffusive barriers – the glass membrane and the ceramic junction. The barriers prevent (glass membrane), or slow down (ceramic junction), the mixing of the different solutions.
In this schematic representation of the galvanic cell, one will note the symmetry between the left and the right members as seen from the center of the row occupied by the "Test Solution" (the solution whose pH must be measured). In other words, the glass membrane and the ceramic junction occupy both the same relative places in each electrode. By using the same electrodes on the left and right, any potentials generated at the interfaces cancel each other (in principle), resulting in the system voltage being dependent only on the interaction of the glass membrane and the test solution.
The measuring part of the electrode, the glass bulb on the bottom, is coated both inside and out with a ~10 nm layer of a hydrated gel. These two layers are separated by a layer of dry glass. The silica glass structure (that is, the conformation of its atomic structure) is shaped so that it allows Na+ ions some mobility. The metal cations (Na+) in the hydrated gel diffuse out of the glass and into solution while H+ from solution can diffuse into the hydrated gel. It is the hydrated gel which makes the pH electrode an ion-selective electrode.
H+ does not cross through the glass membrane of the pH electrode, it is the Na+ which crosses and leads to a change in free energy. When an ion diffuses from a region of activity to another region of activity, there is a free energy change and this is what the pH meter actually measures. The hydrated gel membrane is connected by Na+ transport and thus the concentration of H+ on the outside of the membrane is 'relayed' to the inside of the membrane by Na+.
All glass pH electrodes have extremely high electric resistance from 50 to 500 MΩ. Therefore, the glass electrode can be used only with a high input-impedance measuring device like a pH meter, or, more generically, a high input-impedance voltmeter which is called an electrometer.
Limitations.
The glass electrode has some inherent limitations due to the nature of its construction. Acid and alkaline errors are discussed above. An important limitation results from the existence of "asymmetry potentials" that are present at glass/liquid interfaces. The existence of these phenomena means that glass electrodes must always be calibrated before use; a common method of calibration involves the use of standard buffer solutions. Also, there is a slow deterioration due to diffusion into and out of the internal solution. These effects are masked when the electrode is calibrated against buffer solutions but deviations from ideal response are easily observed by means of a Gran plot. Typically, the slope of the electrode response decreases over a period of months.
Storage.
Between measurements any glass and membrane electrodes should be kept in a solution of its own ion. It is necessary to prevent the glass membrane from drying out because the performance is dependent on the existence of a hydrated layer, which forms slowly.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "E=E^0 + \\frac{RT}{z_iF} \\ln \\left [ a_i + \\sum_{j} \\left ( k_{ij}a_j^{z_i/z_j} \\right ) \\right ]"
},
{
"math_id": 1,
"text": "E=E^0 + \\frac{RT}{F} \\ln \\left ( a_{\\text{H}^+} + k_{\\text{H}^+,\\text{Na}^+}a_{\\text{Na}^+} \\right )"
},
{
"math_id": 2,
"text": "E=E^0 - \\frac{2.303RT}{F} \\text{pH}"
}
] |
https://en.wikipedia.org/wiki?curid=1499906
|
15000577
|
Mortgage yield
|
In finance, mortgage yield is a measure of yield of mortgage-backed bonds. It is also known as cash flow yield. The mortgage yield, or cash flow yield, of a mortgage-backed bond is the monthly compounded discount rate at which net present value of all future cash flows from the bond will be equal to the present price of the bond.
Formula.
When the coupon payments are made on a monthly basis, the mortgage yield can be calculated as:
formula_0
Where
"t" - the time of the cash flow
"n" - each instance of coupon payment
"r" - the discount rate
formula_1 - the net cash flow (the amount of cash) at time t.
Application.
Mortgage yields are primarily a tool for comparing mortgage bonds with conventional bonds. The difference between the mortgage-backed bond's yield (generally converted to semi-annually compounded yield to maturity) and a conventional bond is called the "yield spread" or "I-spread."
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mbox{Mortgage Yield: ri such that P} = \\sum_{n=1}^{N} \\frac{C(t)}{(1+ri/1200)^{t-1}}"
},
{
"math_id": 1,
"text": "C(t)"
}
] |
https://en.wikipedia.org/wiki?curid=15000577
|
150035
|
Church–Rosser theorem
|
Theorem in theoretical computer science
In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does not make a difference to the eventual result.
More precisely, if there are two distinct reductions or sequences of reductions that can be applied to the same term, then there exists a term that is reachable from both results, by applying (possibly empty) sequences of additional reductions. The theorem was proved in 1936 by Alonzo Church and J. Barkley Rosser, after whom it is named.
The theorem is symbolized by the adjacent diagram: If term "a" can be reduced to both "b" and "c", then there must be a further term "d" (possibly equal to either "b" or "c") to which both "b" and "c" can be reduced.
Viewing the lambda calculus as an abstract rewriting system, the Church–Rosser theorem states that the reduction rules of the lambda calculus are confluent. As a consequence of the theorem, a term in the lambda calculus has at most one normal form, justifying reference to ""the" normal form" of a given normalizable term.
History.
In 1936, Alonzo Church and J. Barkley Rosser proved that the theorem holds for β-reduction in the λI-calculus (in which every abstracted variable must appear in the term's body). The proof method is known as "finiteness of developments", and it has additional consequences such as the Standardization Theorem, which relates to a method in which reductions can be performed from left to right to reach a normal form (if one exists). The result for the pure untyped lambda calculus was proved by D. E. Schroer in 1965.
Pure untyped lambda calculus.
One type of reduction in the pure untyped lambda calculus for which the Church–Rosser theorem applies is β-reduction, in which a subterm of the form formula_0 is contracted by the substitution formula_1. If β-reduction is denoted by formula_2 and its reflexive, transitive closure by formula_3 then the Church–Rosser theorem is that:
formula_4
A consequence of this property is that two terms equal in formula_5 must reduce to a common term:
formula_6
The theorem also applies to η-reduction, in which a subterm formula_7 is replaced by formula_8. It also applies to βη-reduction, the union of the two reduction rules.
Proof.
For β-reduction, one proof method originates from William W. Tait and Per Martin-Löf. Say that a binary relation formula_9 satisfies the diamond property if:
formula_10
Then the Church–Rosser property is the statement that formula_3 satisfies the diamond property. We introduce a new reduction formula_11 whose reflexive transitive closure is formula_3 and which satisfies the diamond property. By induction on the number of steps in the reduction, it thus follows that formula_3 satisfies the diamond property.
The relation formula_11 has the formation rules:
The η-reduction rule can be proved to be Church–Rosser directly. Then, it can be proved that β-reduction and η-reduction commute in the sense that:
If formula_18 and formula_19 then there exists a term formula_20 such that formula_21 and formula_22.
Hence we can conclude that βη-reduction is Church–Rosser.
Normalisation.
A reduction rule that satisfies the Church–Rosser property has the property that every term "M" can have at most one distinct normal form, as follows: if "X" and "Y" are normal forms of "M" then by the Church–Rosser property, they both reduce to an equal term "Z". Both terms are already normal forms so formula_23.
If a reduction is strongly normalising (there are no infinite reduction paths) then a weak form of the Church–Rosser property implies the full property (see Newman's lemma). The weak property, for a relation formula_24, is:
formula_25 if formula_26 and formula_27 then there exists a term formula_20 such that formula_28 and formula_29.
Variants.
The Church–Rosser theorem also holds for many variants of the lambda calculus, such as the simply-typed lambda calculus, many calculi with advanced type systems, and Gordon Plotkin's beta-value calculus. Plotkin also used a Church–Rosser theorem to prove that the evaluation of functional programs (for both lazy evaluation and eager evaluation) is a function from programs to values (a subset of the lambda terms).
In older research papers, a rewriting system is said to be Church–Rosser, or to have the Church–Rosser property, when it is confluent.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "( \\lambda x . t) s"
},
{
"math_id": 1,
"text": " t [ x := s]"
},
{
"math_id": 2,
"text": " \\rightarrow_\\beta "
},
{
"math_id": 3,
"text": " \\twoheadrightarrow_\\beta "
},
{
"math_id": 4,
"text": "\\forall M, N_1, N_2 \\in \\Lambda: \\text{if}\\ M\\twoheadrightarrow_\\beta N_1 \\ \\text{and}\\ M\\twoheadrightarrow_\\beta N_2 \\ \\text{then}\\ \\exists X\\in \\Lambda: N_1\\twoheadrightarrow_\\beta X \\ \\text{and}\\ N_2\\twoheadrightarrow_\\beta X"
},
{
"math_id": 5,
"text": "\\lambda\\beta"
},
{
"math_id": 6,
"text": "\\forall M, N\\in \\Lambda: \\text{if}\\ \\lambda\\beta \\vdash M=N \\ \\text{then}\\ \\exists X: M \\twoheadrightarrow_\\beta X \\ \\text{and}\\ N\\twoheadrightarrow_\\beta X"
},
{
"math_id": 7,
"text": "\\lambda x.Sx"
},
{
"math_id": 8,
"text": "S"
},
{
"math_id": 9,
"text": " \\rightarrow "
},
{
"math_id": 10,
"text": "\\forall M, N_1, N_2 \\in \\Lambda: \\text{if}\\ M\\rightarrow N_1 \\ \\text{and}\\ M\\rightarrow N_2 \\ \\text{then}\\ \\exists X\\in \\Lambda: N_1\\rightarrow X \\ \\text{and}\\ N_2\\rightarrow X"
},
{
"math_id": 11,
"text": " \\rightarrow_{\\|} "
},
{
"math_id": 12,
"text": "M \\rightarrow_{\\|} M"
},
{
"math_id": 13,
"text": "M \\rightarrow_{\\|} M'"
},
{
"math_id": 14,
"text": "N \\rightarrow_{\\|} N'"
},
{
"math_id": 15,
"text": "\\lambda x.M \\rightarrow_{\\|} \\lambda x.M'"
},
{
"math_id": 16,
"text": "MN \\rightarrow_{\\|} M'N'"
},
{
"math_id": 17,
"text": "(\\lambda x. M)N \\rightarrow_{\\|} M'[x:=N']"
},
{
"math_id": 18,
"text": "M \\rightarrow_\\beta N_1"
},
{
"math_id": 19,
"text": "M \\rightarrow_\\eta N_2"
},
{
"math_id": 20,
"text": "X"
},
{
"math_id": 21,
"text": "N_1 \\rightarrow_\\eta X"
},
{
"math_id": 22,
"text": "N_2\\rightarrow_\\beta X"
},
{
"math_id": 23,
"text": "X=Z=Y"
},
{
"math_id": 24,
"text": "\\rightarrow"
},
{
"math_id": 25,
"text": "\\forall M, N_1, N_2\\in \\Lambda:"
},
{
"math_id": 26,
"text": "M\\rightarrow N_1"
},
{
"math_id": 27,
"text": "M\\rightarrow N_2"
},
{
"math_id": 28,
"text": "N_1\\twoheadrightarrow X"
},
{
"math_id": 29,
"text": "N_2 \\twoheadrightarrow X"
}
] |
https://en.wikipedia.org/wiki?curid=150035
|
150062
|
Goodstein's theorem
|
Theorem about natural numbers
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence (as defined below) eventually terminates at 0. Laurence Kirby and Jeff Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo-Fraenkel set theory). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example.
Kirby and Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time. Just like for Goodstein sequences, Kirby and Paris showed that it cannot be proven in Peano arithmetic alone.
Hereditary base-"n" notation.
Goodstein sequences are defined in terms of a concept called "hereditary base-"n" notation". This notation is very similar to usual base-"n" positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem.
To achieve the ordinary base-"n" notation, where "n" is a natural number greater than 1, an arbitrary natural number "m" is written as a sum of multiples of powers of "n":
formula_0
where each coefficient "ai" satisfies 0 ≤ "ai" < "n", and "ak" ≠ 0. For example, to achieve the base 2 notation, one writes
formula_1
Thus the base-2 representation of 35 is 100011, which means 25 + 2 + 1. Similarly, 100 represented in base-3 is 10201:
formula_2
Note that the exponents themselves are not written in base-"n" notation. For example, the expressions above include 25 and 34, and 5 > 2, 4 > 3.
To convert a base-n notation (which is a step in achieving base-"n" representation) to a hereditary base-"n" notation, first rewrite all of the exponents as a sum of powers of "n" (with the limitation on the coefficients 0 ≤ "ai" < "n"). Then rewrite any exponent inside the exponents in base-"n" notation (with the same limitation on the coefficients), and continue in this way until every number appearing in the expression (except the bases themselves) is written in base-"n" notation.
For example, while 35 in ordinary base-2 notation is 25 + 2 + 1, it is written in hereditary base-2 notation as
formula_3
using the fact that 5 = 221 + 1. Similarly, 100 in hereditary base-3 notation is
formula_4
Goodstein sequences.
The Goodstein sequence "G"("m") of a number "m" is a sequence of natural numbers. The first element in the sequence "G"("m") is "m" itself. To get the second, "G"("m")(2), write "m" in hereditary base-2 notation, change all the 2s to 3s, and then subtract 1 from the result. In general, the ("n" + 1)-st term, "G"("m")("n" + 1), of the Goodstein sequence of "m" is as follows:
Early Goodstein sequences terminate quickly. For example, "G"(3) terminates at the 6th step:
Later Goodstein sequences increase for a very large number of steps. For example, "G"(4) OEIS: starts as follows:
Elements of "G"(4) continue to increase for a while, but at base formula_5,
they reach the maximum of formula_6, stay there for the next formula_5 steps, and then begin their descent.
However, even "G"(4) doesn't give a good idea of just "how" quickly the elements of a Goodstein sequence can increase.
"G"(19) increases much more rapidly and starts as follows:
In spite of this rapid growth, Goodstein's theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.
Proof of Goodstein's theorem.
Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence "G"("m"), we construct a parallel sequence "P"("m") of ordinal numbers in Cantor normal form which is strictly decreasing and terminates. A common misunderstanding of this proof is to believe that "G"("m") goes to 0 "because" it is dominated by "P"("m"). Actually, the fact that "P"("m") dominates "G"("m") plays no role at all. The important point is: "G"("m")("k") exists if and only if "P"("m")("k") exists (parallelism), and comparison between two members of "G"("m") is preserved when comparing corresponding entries of "P"("m"). Then if "P"("m") terminates, so does "G"("m"). By infinite regress, "G"("m") must reach 0, which guarantees termination.
We define a function formula_7 which computes the hereditary base "k" representation of "u" and then replaces each occurrence of the base "k" with the first infinite ordinal number ω. For example, formula_8.
Each term "P"("m")("n") of the sequence "P"("m") is then defined as "f"("G"("m")("n"),"n+1"). For example, "G"(3)(1) = 3 = 21 + 20 and "P"(3)(1) = "f"(21 + 20,2) = ω1 + ω0 = ω + 1. Addition, multiplication and exponentiation of ordinal numbers are well defined.
We claim that formula_9:
Let formula_10 be "G"("m")("n") after applying the first,
"base-changing" operation in generating the next element of the Goodstein sequence,
but before the second "minus 1" operation in this generation.
Observe that formula_11.
Then formula_12. Now we apply the "minus 1" operation, and formula_13, as formula_14.
For example, formula_15 and formula_16, so formula_17 and formula_18, which is strictly smaller. Note that in order to calculate "f(G(m)(n),n+1)", we first need to write "G"("m")("n") in hereditary base "n"+1 notation, as for instance the expression formula_19 is not an ordinal.
Thus the sequence "P"("m") is strictly decreasing. As the standard order < on ordinals is well-founded, an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals terminates (and cannot be infinite). But "P"("m")("n") is calculated directly from "G"("m")("n"). Hence the sequence "G"("m") must terminate as well, meaning that it must reach 0.
While this proof of Goodstein's theorem is fairly easy, the "Kirby–Paris theorem", which shows that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countable nonstandard models of Peano arithmetic.
Extended Goodstein's theorem.
The above proof still works if the definition of the Goodstein sequence is changed so that the base-changing operation replaces each occurrence of the base "b" with "b" + 2 instead of "b" + 1.
More generally, let "b"1, "b"2, "b"3, ... be any non-decreasing sequence of integers with "b"1 ≥ 2.
Then let the ("n" + 1)-st
term "G"("m")("n" + 1) of the extended Goodstein sequence of "m" be as
follows:
An simple modification of the above proof shows that this sequence still terminates. For example, if "bn" = 4 and if "b""n"+1 = 9,
then
formula_20, hence the ordinal formula_21 is strictly greater than the ordinal formula_22
The extended version is in fact the one considered in Goodstein's original paper, where Goodstein proved that it is equivalent to the restricted ordinal theorem (i.e. the claim that transfinite induction below ε0 is valid), and gave a finitist proof for the case where formula_23 (equivalent to transfinite induction up to formula_24).
The extended Goodstein's theorem without any restriction on the sequence "bn" is not formalizable in Peano arithmetic (PA), since such an arbitrary infinite sequence cannot be represented in PA. This seems to be what kept Goodstein from claiming back in 1944 that the extended Goodstein's theorem is unprovable in PA due to Gödel's second incompleteness theorem and Gentzen's proof of the consistency of PA using ε0-induction. However, inspection of Gentzen's proof shows that it only needs the fact that there is no primitive recursive strictly decreasing infinite sequence of ordinals, so limiting "bn" to primitive recursive sequences would have allowed Goodstein to prove an unprovability result. Furthermore, with the relatively elementary technique of the Grzegorczyk hierarchy, it can be shown that every primitive recursive strictly decreasing infinite sequence of ordinals
can be "slowed down" so that it can be transformed to a Goodstein sequence where "b""n" = n + 1, thus giving an alternative proof to the same result Kirby and Paris proved.
Sequence length as a function of the starting value.
The Goodstein function, formula_25, is defined such that formula_26 is the length of the Goodstein sequence that starts with "n". (This is a total function since every Goodstein sequence terminates.) The extremely high growth rate of formula_27 can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions formula_28 in the Hardy hierarchy, and the functions formula_29 in the fast-growing hierarchy of Löb and Wainer:
formula_27 has approximately the same growth-rate as formula_30 (which is the same as that of formula_31); more precisely, formula_27 dominates formula_28 for every formula_32, and formula_30 dominates formula_33
(For any two functions formula_34, formula_35 is said to dominate formula_36 if formula_37 for all sufficiently large formula_38.)
formula_39
where formula_40 is the result of putting "n" in hereditary base-2 notation and then replacing all 2s with ω (as was done in the proof of Goodstein's theorem).
formula_43.
Some examples:
Application to computable functions.
Goodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine; thus the function which maps "n" to the number of steps required for the Goodstein sequence of "n" to terminate is computable by a particular Turing machine. This machine merely enumerates the Goodstein sequence of "n" and, when the sequence reaches "0", returns the length of the sequence. Because every Goodstein sequence eventually terminates, this function is total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmetic does not prove that this Turing machine computes a total function.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "m = a_k n^k + a_{k-1} n^{k-1} + \\cdots + a_0,"
},
{
"math_id": 1,
"text": "35 = 32 + 2 + 1 = 2^5 + 2^1 + 2^0."
},
{
"math_id": 2,
"text": "100 = 81 + 18 + 1 = 3^4 + 2 \\cdot 3^2 + 3^0."
},
{
"math_id": 3,
"text": "35 = 2^{2^{2^1}+1}+2^1+1, "
},
{
"math_id": 4,
"text": "100 = 3^{3^1+1} + 2 \\cdot 3^2 + 1."
},
{
"math_id": 5,
"text": "3 \\cdot 2^{402\\,653\\,209}"
},
{
"math_id": 6,
"text": "3 \\cdot 2^{402\\,653\\,210} - 1"
},
{
"math_id": 7,
"text": "f=f(u,k)"
},
{
"math_id": 8,
"text": "f(100,3)=f(3^{3^1+1}+2\\cdot3^2+1,3)=\\omega^{\\omega^1+1} + \\omega^2\\cdot2 + 1 = \\omega^{\\omega+1} + \\omega^2\\cdot2 + 1"
},
{
"math_id": 9,
"text": "f(G(m)(n),n+1) > f(G(m)(n+1),n+2)"
},
{
"math_id": 10,
"text": "G'(m)(n)"
},
{
"math_id": 11,
"text": "G(m)(n+1)= G'(m)(n)-1"
},
{
"math_id": 12,
"text": "f(G(m)(n),n+1) = f(G'(m)(n),n+2)"
},
{
"math_id": 13,
"text": "f(G'(m)(n),n+2) > f(G(m)(n+1),n+2)"
},
{
"math_id": 14,
"text": " G'(m)(n) = G(m)(n+1)+1"
},
{
"math_id": 15,
"text": "G(4)(1)=2^2"
},
{
"math_id": 16,
"text": "G(4)(2)=2\\cdot 3^2 + 2\\cdot 3+2"
},
{
"math_id": 17,
"text": "f(2^2,2)=\\omega^\\omega"
},
{
"math_id": 18,
"text": "f(2\\cdot 3^2 + 2\\cdot 3+2,3)=\n\\omega^2\\cdot2+\\omega\\cdot2+2"
},
{
"math_id": 19,
"text": "\\omega^\\omega-1"
},
{
"math_id": 20,
"text": "f(3 \\cdot 4^{4^4} + 4, 4) = 3 \\omega^{\\omega^\\omega} + \\omega= f(3\n\\cdot 9^{9^9} + 9, 9)"
},
{
"math_id": 21,
"text": "f(3 \\cdot 4^{4^4} +\n4, 4)"
},
{
"math_id": 22,
"text": "f\\big((3 \\cdot\n9^{9^9} + 9) - 1, 9\\big)."
},
{
"math_id": 23,
"text": "m \\le b_1^{b_1^{b_1}}"
},
{
"math_id": 24,
"text": "\\omega^{\\omega^\\omega}"
},
{
"math_id": 25,
"text": "\\mathcal{G}: \\mathbb{N} \\to \\mathbb{N} "
},
{
"math_id": 26,
"text": "\\mathcal{G}(n)"
},
{
"math_id": 27,
"text": "\\mathcal{G}"
},
{
"math_id": 28,
"text": "H_\\alpha"
},
{
"math_id": 29,
"text": "f_\\alpha"
},
{
"math_id": 30,
"text": "H_{\\epsilon_0}"
},
{
"math_id": 31,
"text": "f_{\\epsilon_0}"
},
{
"math_id": 32,
"text": "\\alpha < \\epsilon_0"
},
{
"math_id": 33,
"text": "\\mathcal{G}\\,\\!."
},
{
"math_id": 34,
"text": "f, g: \\mathbb{N} \\to \\mathbb{N} "
},
{
"math_id": 35,
"text": "f"
},
{
"math_id": 36,
"text": "g"
},
{
"math_id": 37,
"text": "f(n) > g(n)"
},
{
"math_id": 38,
"text": "n"
},
{
"math_id": 39,
"text": " \\mathcal{G}(n) = H_{R_2^\\omega(n+1)}(1) - 1, "
},
{
"math_id": 40,
"text": "R_2^\\omega(n)"
},
{
"math_id": 41,
"text": " n = 2^{m_1} + 2^{m_2} + \\cdots + 2^{m_k} "
},
{
"math_id": 42,
"text": " m_1 > m_2 > \\cdots > m_k, "
},
{
"math_id": 43,
"text": " \\mathcal{G}(n) = f_{R_2^\\omega(m_1)}(f_{R_2^\\omega(m_2)}(\\cdots(f_{R_2^\\omega(m_k)}(3))\\cdots)) - 2"
}
] |
https://en.wikipedia.org/wiki?curid=150062
|
15007551
|
List of gear nomenclature
|
This page lists the standard US nomenclature used in the description of mechanical gear construction and function, together with definitions of the terms. The terminology was established by the American Gear Manufacturers Association (AGMA), under accreditation from the American National Standards Institute (ANSI).
Addendum.
The addendum is the height by which a tooth of a gear projects beyond (outside for external, or inside for internal) the standard pitch circle or pitch line; also, the radial distance between the pitch diameter and the outside diameter.
Addendum angle.
Addendum angle in a bevel gear, is the angle between face cone and pitch cone.
Addendum circle.
The addendum circle coincides with the tops of the teeth of a gear and is concentric with the standard (reference) pitch circle and radially distant from it by the amount of the addendum. For external gears, the addendum circle lies on the outside cylinder while on internal gears the addendum circle lies on the internal cylinder.
Apex to back.
Apex to back, in a bevel gear or hypoid gear, is the distance in the direction of the axis from the apex of the pitch cone to a locating surface at the back of the blank.
Back angle.
The back angle of a bevel gear is the angle between an element of the back cone and a plane of rotation, and usually is equal to the pitch angle.
Back cone.
The back cone of a bevel or hypoid gear is an imaginary cone tangent to the outer ends of the teeth, with its elements perpendicular to those of the pitch cone. The surface of the gear blank at the outer ends of the teeth is customarily formed to such a back cone.
Back cone distance.
Back cone distance in a bevel gear is the distance along an element of the back cone from its apex to the pitch cone.
Backlash.
In mechanical engineering, backlash is the striking back of connected wheels in a piece of mechanism when pressure is applied. Another source defines it as the maximum distance through which one part of something can be moved without moving a connected part. It is also called lash or play. In the context of gears, backlash is clearance between mating components, or the amount of lost motion due to clearance or slackness when movement is reversed and contact is re-established. In a pair of gears, backlash is the amount of clearance between mated gear teeth.
Backlash is unavoidable for nearly all reversing mechanical couplings, although its effects can be negated. Depending on the application it may or may not be desirable. Reasons for requiring backlash include allowing for lubrication and thermal expansion, and to prevent jamming. Backlash may also result from manufacturing errors and deflection under load.
Base circle.
The base circle of an involute gear is the circle from which involute tooth profiles are derived.
Base cylinder.
The base cylinder corresponds to the base circle, and is the cylinder from which involute tooth surfaces are developed.
Base diameter.
The base diameter of an involute gear is the diameter of the base circle.
Bull gear.
The term bull gear is used to refer to the larger of two spur gears that are in engagement in any machine. The smaller gear is usually referred to as a pinion.
Center distance.
Center distance (operating) is the shortest distance between non-intersecting axes. It is measured along the mutual perpendicular to the axes, called the line of centers. It applies to spur gears, parallel axis or crossed axis helical gears, and worm gearing.
Central plane.
The central plane of a worm gear is perpendicular to the gear axis and contains the common perpendicular of the gear and worm axes. In the usual case with axes at right angles, it contains the worm axis.
Circular Pitch.
The Circular Pitch defines the width of one tooth and one gap measured on an arc on the pitch circle; in other words, this is the distance on the pitch circle from a point on one tooth to the corresponding point on the adjacent tooth. This is equal to π divided by the Diametral Pitch.
CP = Circular Pitch in inches
DP = Diametral Pitch
CP = 3.141 / DP
Composite action test.
The composite action test (double flank) is a method of inspection in which the work gear is rolled in tight double flank contact with a master gear or a specified gear, in order to determine (radial) composite variations (deviations). The composite action test must be made on a variable center distance composite action test device.
and this is composite action test for double flank
Cone distance.
Cone distance in a bevel gear is the general term for the distance along an element of the pitch cone from the apex to any given position in the teeth.
Outer cone distance in bevel gears is the distance from the apex of the pitch cone to the outer ends of the teeth. When not otherwise specified, the short term cone distance is understood to be outer cone distance.
Mean cone distance in bevel gears is the distance from the apex of the pitch cone to the middle of the face width.
Inner cone distance in bevel gears is the distance from the apex of the pitch cone to the inner ends of the teeth.
Conjugate gears.
Conjugate gears transmit uniform rotary motion from one shaft to another by means of gear teeth. The normals to the profiles of these teeth, at all points of contact, must pass through a fixed point in the common centerline of the two shafts. Usually conjugate gear tooth is made to suit the profile of other gear which is not made based on standard practice.
Crossed helical gear.
A crossed helical gear is a gear that operate on non-intersecting, non-parallel axes.
The term crossed helical gears has superseded the term "spiral gears". There is theoretically point contact between the teeth at any instant. They have teeth of the same or different helix angles, of the same or opposite hand. A combination of spur and helical or other types can operate on crossed axes.
Crossing point.
The crossing point is the point of intersection of bevel gear axes; also the apparent point of intersection of the axes in hypoid gears, crossed helical gears, worm gears, and offset face gears, when projected to a plane parallel to both axes.
Crown circle.
The crown circle in a bevel or hypoid gear is the circle of intersection of the back cone and face cone.
Crowned teeth.
Crowned teeth have surfaces modified in the lengthwise direction to produce localized contact or to prevent contact at their ends.
Diametral Pitch.
The Diametral Pitch (DP) is the number of teeth per inch of diameter of the pitch circle. The units of DP are inverse inches (1/in).
DP = Diametral Pitch
PD = Pitch Circle Diameter in inches
CP = Circular Pitch in inches
n = Number of Teeth
DP = n / PD
The Diametral Pitch (DP) is equal to π divided by the Circular Pitch (CP).
DP = 3.1416 / CP
Dedendum angle.
Dedendum angle in a bevel gear, is the angle between elements of the root cone and pitch cone.
Equivalent pitch radius.
Equivalent pitch radius is the radius of the pitch circle in a cross section of gear teeth in any plane other than a plane of rotation. It is properly the radius of curvature of the pitch surface in the given cross section. Examples of such sections are the transverse section of bevel gear teeth and the normal section of helical teeth.
Face (tip) angle.
Face (tip) angle in a bevel or hypoid gear, is the angle between an element of the face cone and its axis.
Face cone.
The face cone, also known as the tip cone is the imaginary surface that coincides with the tops of the teeth of a bevel or hypoid gear.
Face gear.
A face gear set typically consists of a disk-shaped gear, grooved on at least one face, in combination with a spur, helical, or conical pinion. A face gear has a planar pitch surface and a planar root surface, both of which are perpendicular to the axis of rotation. It can also be referred to as a face wheel, crown gear, crown wheel, contrate gear or contrate wheel.
Face width.
The face width of a gear is the length of teeth in an axial plane. For double helical, it does not include the gap.
Total face width is the actual dimension of a gear blank including the portion that exceeds the effective face width, or as in double helical gears where the total face width includes any distance or gap separating right hand and left hand helices.
For a cylindrical gear, effective face width is the portion that contacts the mating teeth. One member of a pair of gears may engage only a portion of its mate.
For a bevel gear, different definitions for effective face width are applicable.
Form diameter.
Form diameter is the diameter of a circle at which the trochoid (fillet curve) produced by the tooling intersects, or joins, the involute or specified profile. Although these terms are not preferred, it is also known as the true involute form diameter (TIF), start of involute diameter (SOI), or when undercut exists, as the undercut diameter. This diameter cannot be less than the base circle diameter.
Front angle.
The front angle, in a bevel gear, denotes the angle between an element of the front cone and a plane of rotation, and usually equals the pitch angle.
Front cone.
The front cone of a hypoid or bevel gear is an imaginary cone tangent to the inner ends of the teeth, with its elements perpendicular to those of the pitch cone. The surface of the gear blank at the inner ends of the teeth is customarily formed to such a front cone, but sometimes may be a plane on a pinion or a cylinder in a nearly flat gear.
Gear center.
A gear center is the center of the pitch circle.
Gear range.
The gear range is difference between the highest and lowest gear ratios and may be expressed as a percentage (e.g., 500%) or as a ratio (e.g., 5:1).
Heel.
The heel of a tooth on a bevel gear or pinion is the portion of the tooth surface near its outer end.
The toe of a tooth on a bevel gear or pinion is the portion of the tooth surface near its inner end.
Helical rack.
A helical rack has a planar pitch surface and teeth that are oblique to the direction of motion.
Helix angle.
Helix angle is the angle between the helical tooth face and an equivalent spur tooth face. For the same lead, the "helix angle" is greater for larger gear diameters. It is understood to be measured at the standard pitch diameter unless otherwise specified.
Hobbing.
Hobbing is a machining process for making gears, splines, and sprockets using a cylindrical tool with helical cutting teeth known as a hob.
Index deviation.
The displacement of any tooth flank from its theoretical position, relative to a datum tooth flank.
Distinction is made as to the direction and algebraic sign of this reading. A condition wherein the actual tooth flank position was nearer to the datum tooth flank, in the specified measuring path direction (clockwise or counterclockwise), than the theoretical position would be considered a minus (-) deviation. A condition wherein the actual tooth flank position was farther from the datum tooth flank, in the specified measuring path direction, than the theoretical position would be considered a plus (+) deviation.
The direction of tolerancing for index deviation along the arc of the tolerance diameter circle within the transverse plane.
Inside cylinder.
The inside cylinder is the surface that coincides with the tops of the teeth of an internal cylindrical gear.
Inside diameter.
Inside diameter is the diameter of the addendum circle of an internal gear, this is also known as minor diameter.
Involute polar angle.
Expressed as θ, the involute polar angle is the angle between a radius vector to a point, "P", on an involute curve and a radial line to the intersection, "A", of the curve with the base circle.
Involute roll angle.
Expressed as ε, the involute roll angle is the angle whose arc on the base circle of radius unity equals the tangent of the pressure angle at a selected point on the involute.
Involute teeth.
Involute teeth of spur gears, helical gears, and worms are those in which the profile in a transverse plane (exclusive of the fillet curve) is the involute of a circle.
Lands.
Bottom land.
The bottom land is the surface at the bottom of a gear tooth space adjoining the fillet.
Top land.
Top land is the (sometimes flat) surface of the top of a gear tooth.
Lead.
Lead is the axial advance of a helix gear tooth during one complete turn (360°), that is, the "Lead" is the axial travel (length along the axle) for one single complete helical revolution about the pitch diameter of the gear.
Lead angle is 90° to the helix angle between the helical tooth face and an equivalent spur tooth face. For the same lead, the "lead angle" is larger for smaller gear diameters. It is understood to be measured at the standard pitch diameter unless otherwise specified.
A spur gear tooth has a "lead angle" of 90°, and a "helix angle" of 0°.
See: Helix angle
Line of centers.
The line of centers connects the centers of the pitch circles of two engaging gears; it is also the common perpendicular of the axes in crossed helical gears and worm gears. When one of the gears is a rack, the line of centers is perpendicular to its pitch line.
Module.
The module is the measure of gear tooth size which is normally used for metric system gears. It is similar to the Diametral Pitch (DP), which is commonly used for UK system (inch measure) gears but they differ in the units used and in that they bear a reciprocal relationship. Module is the pitch circle diameter divided by the number of teeth. Module may also be applied to UK system gears, using inch units, but this usage is not in common use. Module is commonly expressed in units of millimeters (mm).
MM = Metric Module
PD = Pitch Circle Diameter in mm
n = Number of Teeth
MM = PD / n
UK system (inch measure) gears are more commonly specified with the Diametral Pitch (DP) which is the number of teeth per inch of diameter of the pitch circle. The units of DP are inverse inches (1/in).
DP = Diametral Pitch
PD = Pitch Circle Diameter in inches
n = Number of Teeth
DP = n / PD
When converting between module and DP there is an inverse relationship and normally a conversion between the two units of measure (inches and millimeter). Taking both of these into consideration the formulae for conversion are:
MM = 25.4 / DP
and
DP = 25.4 / MM
Mounting distance.
Mounting distance, for assembling bevel gears or hypoid gears, is the distance from the crossing point of the axes to a locating surface of a gear, which may be at either back or front.
Normal module.
Normal module is the value of the module in a normal plane of a helical gear or worm.
formula_0
Normal plane.
A normal plane is normal to a tooth surface at a pitch point, and perpendicular to the pitch plane. In a helical rack, a normal plane is normal to all the teeth it intersects. In a helical gear, however, a plane can be normal to only one tooth at a point lying in the plane surface. At such a point, the normal plane contains the line normal to the tooth surface.
Important positions of a normal plane in tooth measurement and tool design of helical teeth and worm threads are:
In a spiral bevel gear, one of the positions of a normal plane is at a mean point and the plane is normal to the tooth trace.
Offset.
Offset is the perpendicular distance between the axes of hypoid gears or offset face gears.
In the adjacent diagram, (a) and (b) are referred to as having an offset "below center", while those in (c) and (d) have an offset "above center". In determining the direction of offset, it is customary to look at the gear with the pinion at the right. For below center offset the pinion has a left hand spiral, and for above center offset the pinion has a right hand spiral.
Outside cylinder.
The outside (tip or addendum) cylinder is the surface that coincides with the tops of the teeth of an external cylindrical gear.
Outside diameter.
The outside diameter of a gear is the diameter of the addendum (tip) circle. In a bevel gear it is the diameter of the crown circle. In a throated worm gear it is the maximum diameter of the blank. The term applies to external gears, this is can also be known from major diameter.
Pinion.
A pinion is a round gear and usually refers to the smaller of two meshed gears.
Pitch angle.
Pitch angle in bevel gears is the angle between an element of a pitch cone and its axis. In external and internal bevel gears, the pitch angles are respectively less than and greater than 90 degrees.
Pitch circle.
A pitch circle (operating) is the curve of intersection of a pitch surface of revolution and a plane of rotation. It is the imaginary circle that rolls without slipping with a pitch circle of a mating gear.
These are the outlines of mating gears. Many important measurements are taken on and from this circle.
Pitch cone.
A pitch cone is the imaginary cone in a bevel gear that rolls without slipping on a pitch surface of another gear.
Pitch helix.
The pitch helix is the intersection of the tooth surface and the pitch cylinder of a helical gear or cylindrical worm.
Base helix.
The base helix of a helical, involute gear or involute worm lies on its base cylinder.
Base helix angle.
Base helix angle is the helix angle on the base cylinder of involute helical teeth or threads.
Base lead angle.
Base lead angle is the lead angle on the base cylinder. It is the complement of the base helix angle.
Outside helix.
The outside (tip or addendum) helix is the intersection of the tooth surface and the outside cylinder of a helical gear or cylindrical worm.
Outside helix angle.
Outside helix angle is the helix angle on the outside cylinder.
Outside lead angle.
Outside lead angle is the lead angle on the outside cylinder. It is the complement of the outside helix angle.
Normal helix.
A normal helix is a helix on the pitch cylinder, normal to the pitch helix.
Pitch line.
The pitch line corresponds, in the cross section of a rack, to the pitch circle (operating) in the cross section of a gear.
Pitch point.
The pitch point is the point of tangency of two pitch circles (or of a pitch circle and pitch line) and is on the line of centers.
Pitch surfaces.
Pitch surfaces are the imaginary planes, cylinders, or cones that roll together without slipping. For a constant velocity ratio, the pitch cylinders and pitch cones are circular.
Planes.
Pitch plane.
The pitch plane of a pair of gears is the plane perpendicular to the axial plane and tangent to the pitch surfaces. A pitch plane in an individual gear may be any plane tangent to its pitch surface.
The pitch plane of a rack or in a crown gear is the imaginary planar surface that rolls without slipping with a pitch cylinder or pitch cone of another gear. The pitch plane of a rack or crown gear is also the pitch surface.
Transverse plane.
The transverse plane is perpendicular to the axial plane and to the pitch plane. In gears with parallel axes, the transverse and the plane of rotation coincide.
Principal directions.
Principal directions are directions in the pitch plane, and correspond to the principal cross sections of a tooth.
The axial direction is a direction parallel to an axis.
The transverse direction is a direction within a transverse plane.
The normal direction is a direction within a normal plane.
Profile radius of curvature.
Profile radius of curvature is the radius of curvature of a tooth profile, usually at the pitch point or a point of contact. It varies continuously along the involute profile.
Radial composite deviation.
Tooth-to-tooth radial composite deviation (double flank) is the greatest change in center distance while the gear being tested is rotated through any angle of 360 degree/z during double flank composite action test.
Tooth-to-tooth radial composite tolerance (double flank) is the permissible amount of tooth-to-tooth radial composite deviation.
Total radial composite deviation (double flank) is the total change in center distance while the gear being tested is rotated one complete revolution during a double flank composite action test.
Total radial composite tolerance (double flank) is the permissible amount of total radial composite deviation.
Root angle.
Root angle in a bevel or hypoid gear, is the angle between an element of the root cone and its axis.
Root circle.
The root circle coincides with the bottoms of the tooth spaces.
Root cone.
The root cone is the imaginary surface that coincides with the bottoms of the tooth spaces in a bevel or hypoid gear.
Root cylinder.
The root cylinder is the imaginary surface that coincides with the bottoms of the tooth spaces in a cylindrical gear.
Shaft angle.
A shaft angle is the angle between the axes of two non-parallel gear shafts. In a pair of crossed helical gears, the shaft angle lies between the oppositely rotating portions of two shafts. This applies also in the case of worm gearing. In bevel gears, the shaft angle is the sum of the two pitch angles. In hypoid gears, the shaft angle is given when starting a design, and it does not have a fixed relation to the pitch angles and spiral angles.
Spiral gear.
See: Crossed helical gear.
Spur gear.
A spur gear has a cylindrical pitch surface and teeth that are parallel to the axis.
Spur rack.
A spur rack has a planar pitch surface and straight teeth that are at right angles to the direction of motion.
Standard pitch circle.
The standard pitch circle is the circle which intersects the involute at the point where the pressure angle is equal to the profile angle of the basic rack.
Standard pitch diameter.
The standard reference pitch diameter is the diameter of the standard pitch circle. In spur and helical gears, unless otherwise specified, the standard pitch diameter is related to the number of teeth and the standard transverse pitch. Standard reference pitch diameter can be estimated by taking average of gear teeth tips diameter and gear teeth base diameter.
The pitch diameter is useful in determining the spacing between gear centers because proper spacing of gears implies tangent pitch circles. The pitch diameters of two gears may be used to calculate the gear ratio in the same way the number of teeth is used.
formula_1
formula_2
Where formula_3 is the total number of teeth, formula_4 is the circular pitch, formula_5 is the diametrical pitch, and formula_6 is the helix angle for helical gears.
Standard reference pitch diameter.
The standard reference pitch diameter is the diameter of the standard pitch circle. In spur and helical gears, unless otherwise specified, the standard pitch diameter is related to the number of teeth and the standard transverse pitch. It is obtained as:
formula_7
formula_8
Test radius.
The test radius (R"r") is a number used as an arithmetic convention established to simplify the determination of the proper test distance between a master and a work gear for a composite action test. It is used as a measure of the effective size of a gear. The test radius of the master, plus the test radius of the work gear is the set up center distance on a composite action test device. Test radius is not the same as the operating pitch radii of two tightly meshing gears unless both are perfect and to basic or standard tooth thickness.
Throat diameter.
The throat diameter is the diameter of the addendum circle at the central plane of a worm gear or of a double-enveloping worm gear.
Throat form radius.
Throat form radius is the radius of the throat of an enveloping worm gear or of a double-enveloping worm, in an axial plane.
Tip radius.
Tip radius is the radius of the circular arc used to join a side-cutting edge and an end-cutting edge in gear cutting tools. Edge radius is an alternate term.
Tip relief.
Tip relief is a modification of a tooth profile whereby a small amount of material is removed near the tip of the gear tooth.
Tooth surface.
The tooth surface (flank) forms the side of a gear tooth.
It is convenient to choose one face of the gear as the reference face and to mark it with the letter “I”. The other non-reference face might be termed face “II”.
For an observer looking at the reference face, so that the tooth is seen with its tip uppermost, the right flank is on the right and the left flank is on the left. Right and left flanks are denoted by the letters “R” and “L” respectively.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "m_n=m_t \\cos \\beta \\, "
},
{
"math_id": 1,
"text": " d = \\frac{N}{P_d} = \\frac{pN}{\\pi} \\qquad \\text{spur gears}"
},
{
"math_id": 2,
"text": " d = \\frac{N}{P_{nd}\\cos \\psi } \\qquad \\text{helical gears}"
},
{
"math_id": 3,
"text": "N"
},
{
"math_id": 4,
"text": "p"
},
{
"math_id": 5,
"text": "P_d"
},
{
"math_id": 6,
"text": "\\psi"
},
{
"math_id": 7,
"text": " d = km = \\frac{zp}{\\pi} = z\\frac{m_n}{\\cos\\beta } "
},
{
"math_id": 8,
"text": " D = \\frac{N}{P_d} = \\frac{Np}{\\pi}= \\frac{N}{P_{nd} \\cos\\psi} "
}
] |
https://en.wikipedia.org/wiki?curid=15007551
|
1500869
|
Coefficient of determination
|
Indicator for how well data points fit a line or curve
In statistics, the coefficient of determination, denoted "R"2 or "r"2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.
There are several definitions of "R"2 that are only sometimes equivalent. One class of such cases includes that of simple linear regression where "r"2 is used instead of "R"2. When only an intercept is included, then "r"2 is simply the square of the sample correlation coefficient (i.e., "r") between the observed outcomes and the observed predictor values. If additional regressors are included, "R"2 is the square of the coefficient of multiple correlation. In both such cases, the coefficient of determination normally ranges from 0 to 1.
There are cases where "R"2 can yield negative values. This can arise when the predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data. Even if a model-fitting procedure has been used, "R"2 may still be negative, for example when linear regression is conducted without including an intercept, or when a non-linear function is used to fit the data. In cases where negative values arise, the mean of the data provides a better fit to the outcomes than do the fitted function values, according to this particular criterion.
The coefficient of determination can be more intuitively informative than MAE, MAPE, MSE, and RMSE in regression analysis evaluation, as the former can be expressed as a percentage, whereas the latter measures have arbitrary ranges. It also proved more robust for poor fits compared to SMAPE on the test datasets in the article.
When evaluating the goodness-of-fit of simulated ("Y"pred) vs. measured ("Y"obs) values, it is not appropriate to base this on the "R"2 of the linear regression (i.e., "Y"obs= "m"·"Y"pred + b). The "R"2 quantifies the degree of any linear correlation between "Y"obs and "Y"pred, while for the goodness-of-fit evaluation only one specific linear correlation should be taken into consideration: "Y"obs = 1·"Y"pred + 0 (i.e., the 1:1 line).
Definitions.
A data set has "n" values marked "y"1, ..., "y""n" (collectively known as "y""i" or as a vector y = ["y"1, ..., "y""n"]T), each associated with a fitted (or modeled, or predicted) value "f"1, ..., "f""n" (known as "f""i", or sometimes "ŷ""i", as a vector f).
Define the residuals as "e""i" = "y""i" − "f""i" (forming a vector e).
If formula_0 is the mean of the observed data:
formula_1
then the variability of the data set can be measured with two sums of squares formulas:
The most general definition of the coefficient of determination is
formula_4
In the best case, the modeled values exactly match the observed values, which results in formula_5 and "R"2 = 1. A baseline model, which always predicts "y", will have "R"2 = 0.
Relation to unexplained variance.
In a general form, "R"2 can be seen to be related to the fraction of variance unexplained (FVU), since the second term compares the unexplained variance (variance of the model's errors) with the total variance (of the data):
formula_6
As explained variance.
A larger value of "R"2 implies a more successful regression model.
Suppose "R"2 = 0.49. This implies that 49% of the variability of the dependent variable in the data set has been accounted for, and the remaining 51% of the variability is still unaccounted for.
For regression models, the regression sum of squares, also called the explained sum of squares, is defined as
formula_7
In some cases, as in simple linear regression, the total sum of squares equals the sum of the two other sums of squares defined above:
formula_8
See Partitioning in the general OLS model for a derivation of this result for one case where the relation holds. When this relation does hold, the above definition of "R"2 is equivalent to
formula_9
where "n" is the number of observations (cases) on the variables.
In this form "R"2 is expressed as the ratio of the explained variance (variance of the model's predictions, which is "SS"reg / "n") to the total variance (sample variance of the dependent variable, which is "SS"tot / "n").
This partition of the sum of squares holds for instance when the model values "ƒ""i" have been obtained by linear regression. A milder sufficient condition reads as follows: The model has the form
formula_10
where the "q""i" are arbitrary values that may or may not depend on "i" or on other free parameters (the common choice "q""i" = "x""i" is just one special case), and the coefficient estimates formula_11 and formula_12 are obtained by minimizing the residual sum of squares.
This set of conditions is an important one and it has a number of implications for the properties of the fitted residuals and the modelled values. In particular, under these conditions:
formula_13
As squared correlation coefficient.
In linear least squares multiple regression with an estimated intercept term, "R"2 equals the square of the Pearson correlation coefficient between the observed formula_14 and modeled (predicted) formula_15 data values of the dependent variable.
In a linear least squares regression with a single explanator but without an intercept term, this is also equal to the squared Pearson correlation coefficient of the dependent variable formula_14 and explanatory variable formula_16
It should not be confused with the correlation coefficient between two explanatory variables, defined as
formula_17
where the covariance between two coefficient estimates, as well as their standard deviations, are obtained from the covariance matrix of the coefficient estimates, formula_18.
Under more general modeling conditions, where the predicted values might be generated from a model different from linear least squares regression, an "R"2 value can be calculated as the square of the correlation coefficient between the original formula_14 and modeled formula_15 data values. In this case, the value is not directly a measure of how good the modeled values are, but rather a measure of how good a predictor might be constructed from the modeled values (by creating a revised predictor of the form "α" + "βƒ""i"). According to Everitt, this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables.
Interpretation.
"R"2 is a measure of the goodness of fit of a model. In regression, the "R"2 coefficient of determination is a statistical measure of how well the regression predictions approximate the real data points. An "R"2 of 1 indicates that the regression predictions perfectly fit the data.
Values of "R"2 outside the range 0 to 1 occur when the model fits the data worse than the worst possible least-squares predictor (equivalent to a horizontal hyperplane at a height equal to the mean of the observed data). This occurs when a wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvålseth is used (this is the equation used most often), "R"2 can be less than zero. If equation 2 of Kvålseth is used, "R"2 can be greater than one.
In all instances where "R"2 is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing "SS"res. In this case, "R"2 increases as the number of variables in the model is increased ("R"2 is monotone increasing with the number of variables included—it will never decrease). This illustrates a drawback to one possible use of "R"2, where one might keep adding variables (kitchen sink regression) to increase the "R"2 value. For example, if one is trying to predict the sales of a model of car from the car's gas mileage, price, and engine power, one can include probably irrelevant factors such as the first letter of the model's name or the height of the lead engineer designing the car because the "R"2 will never decrease as variables are added and will likely experience an increase due to chance alone.
This leads to the alternative approach of looking at the adjusted "R"2. The explanation of this statistic is almost the same as "R"2 but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, the "R"2 statistic can be calculated as above and may still be a useful measure. If fitting is by weighted least squares or generalized least squares, alternative versions of "R"2 can be calculated appropriate to those statistical frameworks, while the "raw" "R"2 may still be useful if it is more easily interpreted. Values for "R"2 can be calculated for any type of predictive model, which need not have a statistical basis.
In a multiple linear model.
Consider a linear model with more than a single explanatory variable, of the form
formula_19
where, for the "i"th case, formula_20 is the response variable, formula_21 are "p" regressors, and formula_22 is a mean zero error term. The quantities formula_23 are unknown coefficients, whose values are estimated by least squares. The coefficient of determination "R"2 is a measure of the global fit of the model. Specifically, "R"2 is an element of [0, 1] and represents the proportion of variability in "Y""i" that may be attributed to some linear combination of the regressors (explanatory variables) in "X".
"R"2 is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, "R"2 = 1 indicates that the fitted model explains all variability in formula_14, while "R"2 = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = formula_0) between the response variable and regressors). An interior value such as "R"2 = 0.7 may be interpreted as follows: "Seventy percent of the variance in the response variable can be explained by the explanatory variables. The remaining thirty percent can be attributed to unknown, lurking variables or inherent variability."
A caution that applies to "R"2, as to other statistical descriptions of correlation and association is that "correlation does not imply causation." In other words, while correlations may sometimes provide valuable clues in uncovering causal relationships among variables, a non-zero estimated correlation between two variables is not, on its own, evidence that changing the value of one variable would result in changes in the values of other variables. For example, the practice of carrying matches (or a lighter) is correlated with incidence of lung cancer, but carrying matches does not cause cancer (in the standard sense of "cause").
In case of a single regressor, fitted by least squares, "R"2 is the square of the Pearson product-moment correlation coefficient relating the regressor and the response variable. More generally, "R"2 is the square of the correlation between the constructed predictor and the response variable. With more than one regressor, the "R"2 can be referred to as the coefficient of multiple determination.
Inflation of "R"2.
In least squares regression using typical data, "R"2 is at least weakly increasing with an increase in number of regressors in the model. Because increases in the number of regressors increase the value of "R"2, "R"2 alone cannot be used as a meaningful comparison of models with very different numbers of independent variables. For a meaningful comparison between two models, an F-test can be performed on the residual sum of squares , similar to the F-tests in Granger causality, though this is not always appropriate. As a reminder of this, some authors denote "R"2 by "R""q"2, where "q" is the number of columns in "X" (the number of explanators including the constant).
To demonstrate this property, first recall that the objective of least squares linear regression is
formula_24
where "Xi" is a row vector of values of explanatory variables for case "i" and "b" is a column vector of coefficients of the respective elements of "Xi".
The optimal value of the objective is weakly smaller as more explanatory variables are added and hence additional columns of formula_25 (the explanatory data matrix whose "i"th row is "Xi") are added, by the fact that less constrained minimization leads to an optimal cost which is weakly smaller than more constrained minimization does. Given the previous conclusion and noting that formula_26 depends only on "y", the non-decreasing property of "R"2 follows directly from the definition above.
The intuitive reason that using an additional explanatory variable cannot lower the "R"2 is this: Minimizing formula_27 is equivalent to maximizing "R"2. When the extra variable is included, the data always have the option of giving it an estimated coefficient of zero, leaving the predicted values and the "R"2 unchanged. The only way that the optimization problem will give a non-zero coefficient is if doing so improves the "R"2.
The above gives an analytical explanation of the inflation of "R"2. Next, an example based on ordinary least square from a geometric perspective is shown below.
A simple case to be considered first:
formula_28
This equation describes the ordinary least squares regression model with one regressor. The prediction is shown as the red vector in the figure on the right. Geometrically, it is the projection of true value onto a model space in formula_29 (without intercept). The residual is shown as the red line.
formula_30
This equation corresponds to the ordinary least squares regression model with two regressors. The prediction is shown as the blue vector in the figure on the right. Geometrically, it is the projection of true value onto a larger model space in formula_31 (without intercept). Noticeably, the values of formula_32 and formula_32 are not the same as in the equation for smaller model space as long as formula_33 and formula_34 are not zero vectors. Therefore, the equations are expected to yield different predictions (i.e., the blue vector is expected to be different from the red vector). The least squares regression criterion ensures that the residual is minimized. In the figure, the blue line representing the residual is orthogonal to the model space in formula_31, giving the minimal distance from the space.
The smaller model space is a subspace of the larger one, and thereby the residual of the smaller model is guaranteed to be larger. Comparing the red and blue lines in the figure, the blue line is orthogonal to the space, and any other line would be larger than the blue one. Considering the calculation for "R"2, a smaller value of formula_26 will lead to a larger value of "R"2, meaning that adding regressors will result in inflation of "R"2.
Caveats.
"R"2 does not indicate whether:
Extensions.
Adjusted "R"2.
The use of an adjusted "R"2 (one common notation is formula_35, pronounced "R bar squared"; another is formula_36 or formula_37) is an attempt to account for the phenomenon of the "R"2 automatically increasing when extra explanatory variables are added to the model. There are many different ways of adjusting. By far the most used one, to the point that it is typically just referred to as adjusted "R", is the correction proposed by Mordecai Ezekiel.
The adjusted "R"2 is defined as
formula_38
where df"res" is the degrees of freedom of the estimate of the population variance around the model, and df"tot" is the degrees of freedom of the estimate of the population variance around the mean. df"res" is given in terms of the sample size "n" and the number of variables "p" in the model, df"res" = "n" − "p" − 1. df"tot" is given in the same way, but with "p" being unity for the mean, i.e. df"tot" = "n" − 1.
Inserting the degrees of freedom and using the definition of "R"2, it can be rewritten as:
formula_39
where "p" is the total number of explanatory variables in the model (excluding the intercept), and "n" is the sample size.
The adjusted "R"2 can be negative, and its value will always be less than or equal to that of "R"2. Unlike "R"2, the adjusted "R"2 increases only when the increase in "R"2 (due to the inclusion of a new explanatory variable) is more than one would expect to see by chance. If a set of explanatory variables with a predetermined hierarchy of importance are introduced into a regression one at a time, with the adjusted "R"2 computed each time, the level at which adjusted "R"2 reaches a maximum, and decreases afterward, would be the regression with the ideal combination of having the best fit without excess/unnecessary terms.
The adjusted "R"2 can be interpreted as an instance of the bias-variance tradeoff. When we consider the performance of a model, a lower error represents a better performance. When the model becomes more complex, the variance will increase whereas the square of bias will decrease, and these two metrices add up to be the total error. Combining these two trends, the bias-variance tradeoff describes a relationship between the performance of the model and its complexity, which is shown as a u-shape curve on the right. For the adjusted "R"2 specifically, the model complexity (i.e. number of parameters) affects the "R"2 and the term / frac and thereby captures their attributes in the overall performance of the model.
"R"2 can be interpreted as the variance of the model, which is influenced by the model complexity. A high "R"2 indicates a lower bias error because the model can better explain the change of Y with predictors. For this reason, we make fewer (erroneous) assumptions, and this results in a lower bias error. Meanwhile, to accommodate fewer assumptions, the model tends to be more complex. Based on bias-variance tradeoff, a higher complexity will lead to a decrease in bias and a better performance (below the optimal line). In "R"2, the term (1 − "R"2) will be lower with high complexity and resulting in a higher "R"2, consistently indicating a better performance.
On the other hand, the term/frac term is reversely affected by the model complexity. The term/frac will increase when adding regressors (i.e. increased model complexity) and lead to worse performance. Based on bias-variance tradeoff, a higher model complexity (beyond the optimal line) leads to increasing errors and a worse performance.
Considering the calculation of "R"2, more parameters will increase the "R"2 and lead to an increase in "R"2. Nevertheless, adding more parameters will increase the term/frac and thus decrease "R"2. These two trends construct a reverse u-shape relationship between model complexity and "R"2, which is in consistent with the u-shape trend of model complexity vs. overall performance. Unlike "R"2, which will always increase when model complexity increases, "R"2 will increase only when the bias eliminated by the added regressor is greater than the variance introduced simultaneously. Using "R"2 instead of "R"2 could thereby prevent overfitting.
Following the same logic, adjusted "R"2 can be interpreted as a less biased estimator of the population "R"2, whereas the observed sample "R"2 is a positively biased estimate of the population value. Adjusted "R"2 is more appropriate when evaluating model fit (the variance in the dependent variable accounted for by the independent variables) and in comparing alternative models in the feature selection stage of model building.
The principle behind the adjusted "R"2 statistic can be seen by rewriting the ordinary "R"2 as
formula_40
where formula_41 and formula_42 are the sample variances of the estimated residuals and the dependent variable respectively, which can be seen as biased estimates of the population variances of the errors and of the dependent variable. These estimates are replaced by statistically unbiased versions: formula_43 and formula_44.
Despite using unbiased estimators for the population variances of the error and the dependent variable, adjusted "R"2 is not an unbiased estimator of the population "R"2, which results by using the population variances of the errors and the dependent variable instead of estimating them. Ingram Olkin and John W. Pratt derived the minimum-variance unbiased estimator for the population "R"2, which is known as Olkin–Pratt estimator. Comparisons of different approaches for adjusting "R"2 concluded that in most situations either an approximate version of the Olkin–Pratt estimator or the exact Olkin–Pratt estimator should be preferred over (Ezekiel) adjusted "R"2.
Coefficient of partial determination.
The coefficient of partial determination can be defined as the proportion of variation that cannot be explained in a reduced model, but can be explained by the predictors specified in a full(er) model. This coefficient is used to provide insight into whether or not one or more additional predictors may be useful in a more fully specified regression model.
The calculation for the partial "R"2 is relatively straightforward after estimating two models and generating the ANOVA tables for them. The calculation for the partial "R"2 is
formula_45
which is analogous to the usual coefficient of determination:
formula_46
Generalizing and decomposing "R"2.
As explained above, model selection heuristics such as the adjusted "R"2 criterion and the F-test examine whether the total "R"2 sufficiently increases to determine if a new regressor should be added to the model. If a regressor is added to the model that is highly correlated with other regressors which have already been included, then the total "R"2 will hardly increase, even if the new regressor is of relevance. As a result, the above-mentioned heuristics will ignore relevant regressors when cross-correlations are high.
Alternatively, one can decompose a generalized version of "R"2 to quantify the relevance of deviating from a hypothesis. As Hoornweg (2018) shows, several shrinkage estimators – such as Bayesian linear regression, ridge regression, and the (adaptive) lasso – make use of this decomposition of "R"2 when they gradually shrink parameters from the unrestricted OLS solutions towards the hypothesized values. Let us first define the linear regression model as
formula_47
It is assumed that the matrix "X" is standardized with Z-scores and that the column vector formula_14 is centered to have a mean of zero. Let the column vector formula_32 refer to the hypothesized regression parameters and let the column vector formula_48 denote the estimated parameters. We can then define
formula_49
An "R"2 of 75% means that the in-sample accuracy improves by 75% if the data-optimized "b" solutions are used instead of the hypothesized formula_32 values. In the special case that formula_32 is a vector of zeros, we obtain the traditional "R"2 again.
The individual effect on "R"2 of deviating from a hypothesis can be computed with formula_50 ('R-outer'). This formula_51 times formula_51 matrix is given by
formula_52
where formula_53. The diagonal elements of formula_50 exactly add up to "R"2. If regressors are uncorrelated and formula_32 is a vector of zeros, then the formula_54 diagonal element of formula_55 simply corresponds to the "r"2 value between formula_56 and formula_14. When regressors formula_57 and formula_56 are correlated, formula_58 might increase at the cost of a decrease in formula_59. As a result, the diagonal elements of formula_55 may be smaller than 0 and, in more exceptional cases, larger than 1. To deal with such uncertainties, several shrinkage estimators implicitly take a weighted average of the diagonal elements of formula_55 to quantify the relevance of deviating from a hypothesized value. Click on the lasso for an example.
"R"2 in logistic regression.
In the case of logistic regression, usually fit by maximum likelihood, there are several choices of pseudo-"R"2.
One is the generalized "R"2 originally proposed by Cox & Snell, and independently by Magee:
formula_60
where formula_61 is the likelihood of the model with only the intercept, formula_62 is the likelihood of the estimated model (i.e., the model with a given set of parameter estimates) and "n" is the sample size. It is easily rewritten to:
formula_63
where "D" is the test statistic of the likelihood ratio test.
Nico Nagelkerke noted that it had the following properties:
However, in the case of a logistic model, where formula_64 cannot be greater than 1, "R"2 is between 0 and formula_65: thus, Nagelkerke suggested the possibility to define a scaled "R"2 as "R"2/"R"2max.
Comparison with norm of residuals.
Occasionally, the norm of residuals is used for indicating goodness of fit. This term is calculated as the square-root of the sum of squares of residuals:
formula_66
Both "R"2 and the norm of residuals have their relative merits. For least squares analysis "R"2 varies between 0 and 1, with larger numbers indicating better fits and 1 representing a perfect fit. The norm of residuals varies from 0 to infinity with smaller numbers indicating better fits and zero indicating a perfect fit. One advantage and disadvantage of "R"2 is the formula_67 term acts to normalize the value. If the "yi" values are all multiplied by a constant, the norm of residuals will also change by that constant but "R"2 will stay the same. As a basic example, for the linear least squares fit to the set of data:
"R"2 = 0.998, and norm of residuals = 0.302.
If all values of "y" are multiplied by 1000 (for example, in an SI prefix change), then "R"2 remains the same, but norm of residuals = 302.
Another single-parameter indicator of fit is the RMSE of the residuals, or standard deviation of the residuals. This would have a value of 0.135 for the above example given that the fit was linear with an unforced intercept.
History.
The creation of the coefficient of determination has been attributed to the geneticist Sewall Wright and was first published in 1921.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\bar{y}"
},
{
"math_id": 1,
"text": "\\bar{y}=\\frac{1}{n}\\sum_{i=1}^n y_i "
},
{
"math_id": 2,
"text": "SS_\\text{res}=\\sum_i (y_i - f_i)^2=\\sum_i e_i^2\\,"
},
{
"math_id": 3,
"text": "SS_\\text{tot}=\\sum_i (y_i - \\bar{y})^2"
},
{
"math_id": 4,
"text": "R^2 = 1 - {SS_{\\rm res}\\over SS_{\\rm tot}} "
},
{
"math_id": 5,
"text": "SS_\\text{res}=0"
},
{
"math_id": 6,
"text": "R^2 = 1 - \\text{FVU}"
},
{
"math_id": 7,
"text": "SS_\\text{reg}=\\sum_i (f_i -\\bar{y})^2"
},
{
"math_id": 8,
"text": "SS_\\text{res}+SS_\\text{reg}=SS_\\text{tot}"
},
{
"math_id": 9,
"text": "R^2 = \\frac{SS_\\text{reg}}{SS_\\text{tot}} = \\frac{SS_\\text{reg}/n}{SS_\\text{tot}/n}"
},
{
"math_id": 10,
"text": "f_i=\\widehat\\alpha+\\widehat\\beta q_i"
},
{
"math_id": 11,
"text": "\\widehat\\alpha"
},
{
"math_id": 12,
"text": "\\widehat\\beta"
},
{
"math_id": 13,
"text": "\\bar{f}=\\bar{y}.\\,"
},
{
"math_id": 14,
"text": "y"
},
{
"math_id": 15,
"text": "f"
},
{
"math_id": 16,
"text": "x."
},
{
"math_id": 17,
"text": "\\rho_{\\widehat\\alpha,\\widehat\\beta} = {\\operatorname{cov}\\left(\\widehat\\alpha,\\widehat\\beta\\right) \\over \\sigma_{\\widehat\\alpha} \\sigma_{\\widehat\\beta}},"
},
{
"math_id": 18,
"text": "(X^T X)^{-1}"
},
{
"math_id": 19,
"text": "Y_i = \\beta_0 + \\sum_{j=1}^p \\beta_j X_{i,j} + \\varepsilon_i,"
},
{
"math_id": 20,
"text": "{Y_i}"
},
{
"math_id": 21,
"text": "X_{i,1},\\dots,X_{i,p}"
},
{
"math_id": 22,
"text": "\\varepsilon_i"
},
{
"math_id": 23,
"text": "\\beta_0,\\dots,\\beta_p"
},
{
"math_id": 24,
"text": "\\min_b SS_\\text{res}(b) \\Rightarrow \\min_b \\sum_i (y_i - X_ib)^2\\,"
},
{
"math_id": 25,
"text": "X"
},
{
"math_id": 26,
"text": "SS_{tot}"
},
{
"math_id": 27,
"text": "SS_\\text{res}"
},
{
"math_id": 28,
"text": "Y=\\beta_0+\\beta_1\\cdot X_1+\\varepsilon\\,"
},
{
"math_id": 29,
"text": "\\mathbb{R}"
},
{
"math_id": 30,
"text": "Y=\\beta_0+\\beta_1\\cdot X_1+\\beta_2\\cdot X_2 + \\varepsilon\\,"
},
{
"math_id": 31,
"text": "\\mathbb{R}^2"
},
{
"math_id": 32,
"text": "\\beta_0"
},
{
"math_id": 33,
"text": "X_1"
},
{
"math_id": 34,
"text": "X_2"
},
{
"math_id": 35,
"text": "\\bar R^2"
},
{
"math_id": 36,
"text": "R^2_{\\text{a}}"
},
{
"math_id": 37,
"text": "R^2_{\\text{adj}}"
},
{
"math_id": 38,
"text": "\\bar R^2 = {1-{SS_\\text{res}/\\text{df}_\\text{res} \\over SS_\\text{tot}/\\text{df}_\\text{tot}}}"
},
{
"math_id": 39,
"text": "\\bar R^2 = 1-(1-R^2){n-1 \\over n-p-1}"
},
{
"math_id": 40,
"text": "R^2 = {1-{\\text{VAR}_\\text{res} \\over \\text{VAR}_\\text{tot}}}"
},
{
"math_id": 41,
"text": "\\text{VAR}_\\text{res} = SS_\\text{res}/n"
},
{
"math_id": 42,
"text": "\\text{VAR}_\\text{tot} = SS_\\text{tot}/n"
},
{
"math_id": 43,
"text": "\\text{VAR}_\\text{res} = SS_\\text{res}/(n-p)"
},
{
"math_id": 44,
"text": "\\text{VAR}_\\text{tot} = SS_\\text{tot}/(n-1)"
},
{
"math_id": 45,
"text": "\\frac{SS_\\text{ res, reduced} - SS_\\text{ res, full}}{SS_\\text{ res, reduced}},"
},
{
"math_id": 46,
"text": "\\frac{SS_\\text{tot} - SS_\\text{res}}{SS_\\text{tot}}."
},
{
"math_id": 47,
"text": "y=X\\beta+\\varepsilon."
},
{
"math_id": 48,
"text": "b"
},
{
"math_id": 49,
"text": "R^2=1-\\frac{(y-Xb)'(y-Xb)}{(y-X\\beta_0)'(y-X\\beta_0)}."
},
{
"math_id": 50,
"text": "R^\\otimes"
},
{
"math_id": 51,
"text": "p"
},
{
"math_id": 52,
"text": "R^{\\otimes}=(X'\\tilde y_0)(X'\\tilde y_0)' (X'X)^{-1}(\\tilde y_0'\\tilde y_0)^{-1},"
},
{
"math_id": 53,
"text": "\\tilde y_0=y-X\\beta_0"
},
{
"math_id": 54,
"text": "j^\\text{th}"
},
{
"math_id": 55,
"text": "R^{\\otimes}"
},
{
"math_id": 56,
"text": "x_j"
},
{
"math_id": 57,
"text": "x_i"
},
{
"math_id": 58,
"text": "R^\\otimes_{ii}"
},
{
"math_id": 59,
"text": "R^{\\otimes}_{jj}"
},
{
"math_id": 60,
"text": "R^2 = 1 - \\left({ \\mathcal{L}(0) \\over \\mathcal{L}(\\widehat{\\theta}) }\\right)^{2/n}"
},
{
"math_id": 61,
"text": "\\mathcal{L}(0)"
},
{
"math_id": 62,
"text": "{\\mathcal{L}(\\widehat{\\theta})}"
},
{
"math_id": 63,
"text": "R^2 = 1 - e^{\\frac{2}{n} (\\ln(\\mathcal{L}(0)) - \\ln(\\mathcal{L}(\\widehat{\\theta}))} = 1 - e^{-D/n}"
},
{
"math_id": 64,
"text": "\\mathcal{L}(\\widehat{\\theta})"
},
{
"math_id": 65,
"text": " R^2_\\max = 1- (\\mathcal{L}(0))^{2/n} "
},
{
"math_id": 66,
"text": "\\text{norm of residuals} = \\sqrt{SS_\\text{res}} = \\| e \\|. "
},
{
"math_id": 67,
"text": "SS_\\text{tot}"
}
] |
https://en.wikipedia.org/wiki?curid=1500869
|
15008875
|
Favard operator
|
Functional analysis operator
In functional analysis, a branch of mathematics, the Favard operators are defined by:
formula_0
where formula_1, formula_2. They are named after Jean Favard.
Generalizations.
A common generalization is:
formula_3
where formula_4 is a positive sequence that converges to 0. This reduces to the classical Favard operators when formula_5.
|
[
{
"math_id": 0,
"text": "[\\mathcal{F}_n(f)](x) = \\frac{1}{\\sqrt{n\\pi}} \\sum_{k=-\\infty}^\\infty {\\exp{\\left({-n {\\left({\\frac{k}{n}-x}\\right)}^2 }\\right)} f\\left(\\frac{k}{n}\\right)}"
},
{
"math_id": 1,
"text": "x\\in\\mathbb{R}"
},
{
"math_id": 2,
"text": "n\\in\\mathbb{N}"
},
{
"math_id": 3,
"text": "[\\mathcal{F}_n(f)](x) = \\frac{1}{n\\gamma_n\\sqrt{2\\pi}} \\sum_{k=-\\infty}^\\infty {\\exp{\\left({\\frac{-1}{2\\gamma_n^2} {\\left({\\frac{k}{n}-x}\\right)}^2 }\\right)} f\\left(\\frac{k}{n}\\right)}"
},
{
"math_id": 4,
"text": "(\\gamma_n)_{n=1}^\\infty"
},
{
"math_id": 5,
"text": "\\gamma_n^2=1/(2n)"
}
] |
https://en.wikipedia.org/wiki?curid=15008875
|
150102
|
Édouard Lucas
|
French mathematician (1842–1891)
François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.
Biography.
Lucas was born in Amiens and educated at the École Normale Supérieure. He worked in the Paris Observatory and later became a professor of mathematics at the Lycée Saint Louis and the Lycée Charlemagne in Paris.
Lucas served as an artillery officer in the French Army during the Franco-Prussian War of 1870–1871.
In 1875, Lucas posed a challenge to prove that the only solution of the Diophantine equation
formula_0
with "N" > 1 is when "N" = 24 and "M" = 70. This is known as the cannonball problem, since it can be visualized as the problem of taking a square arrangement of cannonballs on the ground and building a square pyramid out of them. It was not until 1918 that a proof (using elliptic functions) was found for this remarkable fact, which has relevance to the bosonic string theory in 26 dimensions. More recently, elementary proofs have been published.
He devised methods for testing the primality of numbers. In 1857, at age 15, Lucas began testing the primality of 2127 − 1, a number with 39 decimal digits, by hand, using Lucas sequences. In 1876, after 19 years of testing, he finally proved that 2127 − 1 was prime; this would remain the largest known Mersenne prime for three-quarters of a century. This may stand forever as the largest prime number proven by hand. Later Derrick Henry Lehmer refined Lucas's primality tests and obtained the Lucas–Lehmer primality test.
He worked on the development of the umbral calculus.
Lucas is credited as the first to publish the Kempner function.
Lucas was also interested in recreational mathematics. He found an elegant binary solution to the Baguenaudier puzzle. He also invented the Tower of Hanoi puzzle in 1883, which he marketed under the nickname "N. Claus de Siam", an anagram of "Lucas d'Amiens", and published for the first time a description of the dots and boxes game in 1889.
Lucas died in unusual circumstances. At the banquet of the annual congress of the "Association française pour l'avancement des sciences", a waiter dropped some crockery and a piece of broken plate cut Lucas on the cheek. He died a few days later of a severe skin inflammation, probably caused by sepsis, at 49 years old.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sum_{n=1}^{N} n^2 = M^2\\;"
}
] |
https://en.wikipedia.org/wiki?curid=150102
|
1501024
|
Dyadic transformation
|
Doubling map on the unit interval
The dyadic transformation (also known as the dyadic map, bit shift map, 2"x" mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation)
formula_0
formula_1
(where formula_2 is the set of sequences from formula_3) produced by the rule
formula_4
formula_5.
Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function
formula_6
The name "bit shift map" arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
The dyadic transformation provides an example of how a simple 1-dimensional map can give rise to chaos. This map readily generalizes to several others. An important one is the beta transformation, defined as formula_7. This map has been extensively studied by many authors. It was introduced by Alfréd Rényi in 1957, and an invariant measure for it was given by Alexander Gelfond in 1959 and again independently by Bill Parry in 1960.
Relation to the Bernoulli process.
The map can be obtained as a homomorphism on the Bernoulli process. Let formula_8 be the set of all semi-infinite strings of the letters formula_9 and formula_10. These can be understood to be the flips of a coin, coming up heads or tails. Equivalently, one can write formula_11 the space of all (semi-)infinite strings of binary bits. The word "infinite" is qualified with "semi-", as one can also define a different space formula_12 consisting of all doubly-infinite (double-ended) strings; this will lead to the Baker's map. The qualification "semi-" is dropped below.
This space has a natural shift operation, given by
formula_13
where formula_14 is an infinite string of binary digits. Given such a string, write
formula_15
The resulting formula_16 is a real number in the unit interval formula_17 The shift formula_10 induces a homomorphism, also called formula_10, on the unit interval. Since formula_18 one can easily see that formula_19 For the doubly-infinite sequence of bits formula_20 the induced homomorphism is the Baker's map.
The dyadic sequence is then just the sequence
formula_21
That is, formula_22
The Cantor set.
Note that the sum
formula_23
gives the Cantor function, as conventionally defined. This is one reason why the set formula_24 is sometimes called the Cantor set.
Rate of information loss and sensitive dependence on initial conditions.
One hallmark of chaotic dynamics is the loss of information as simulation occurs. If we start with information on the first "s" bits of the initial iterate, then after "m" simulated iterations ("m" < "s") we only have "s" − "m" bits of information remaining. Thus we lose information at the exponential rate of one bit per iteration. After "s" iterations, our simulation has reached the fixed point zero, regardless of the true iterate values; thus we have suffered a complete loss of information. This illustrates sensitive dependence on initial conditions—the mapping from the truncated initial condition has deviated exponentially from the mapping from the true initial condition. And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic.
Equivalent to the concept of information loss is the concept of information gain. In practice some real-world process may generate a sequence of values ("x""n") over time, but we may only be able to observe these values in truncated form. Suppose for example that "x"0 = 0.1001101, but we only observe the truncated value 0.1001. Our prediction for "x"1 is 0.001. If we wait until the real-world process has generated the true "x"1 value 0.001101, we will be able to observe the truncated value 0.0011, which is more accurate than our predicted value 0.001. So we have received an information gain of one bit.
Relation to tent map and logistic map.
The dyadic transformation is topologically semi-conjugate to the unit-height tent map. Recall that the unit-height tent map is given by
formula_25
The conjugacy is explicitly given by
formula_26
so that
formula_27
That is, formula_28 This is stable under iteration, as
formula_29
It is also conjugate to the chaotic "r" = 4 case of the logistic map.
The "r" = 4 case of the logistic map is formula_30; this is related to the bit shift map in variable "x" by
formula_31
There is also a semi-conjugacy between the dyadic transformation (here named angle doubling map) and the quadratic polynomial. Here, the map doubles angles measured in turns. That is, the map is given by
formula_32
Periodicity and non-periodicity.
Because of the simple nature of the dynamics when the iterates are viewed in binary notation, it is easy to categorize the dynamics based on the initial condition:
If the initial condition is irrational (as almost all points in the unit interval are), then the dynamics are non-periodic—this follows directly from the definition of an irrational number as one with a non-repeating binary expansion. This is the chaotic case.
If "x"0 is rational the image of "x"0 contains a finite number of distinct values within [0, 1) and the forward orbit of "x"0 is eventually periodic, with period equal to the period of the binary expansion of "x"0. Specifically, if the initial condition is a rational number with a finite binary expansion of "k" bits, then after "k" iterations the iterates reach the fixed point 0;
if the initial condition is a rational number with a "k"-bit transient ("k" ≥ 0) followed by a "q"-bit sequence ("q" > 1) that repeats itself infinitely, then after "k" iterations the iterates reach a cycle of length "q". Thus cycles of all lengths are possible.
For example, the forward orbit of 11/24 is:
formula_33
which has reached a cycle of period 2. Within any subinterval of [0, 1), no matter how small, there are therefore an infinite number of points whose orbits are eventually periodic, and an infinite number of points whose orbits are never periodic. This sensitive dependence on initial conditions is a characteristic of chaotic maps.
Periodicity via bit shifts.
The periodic and non-periodic orbits can be more easily understood not by working with the map formula_34 directly, but rather with the bit shift map formula_35 defined on the Cantor space formula_36.
That is, the homomorphism
formula_37
is basically a statement that the Cantor set can be mapped into the reals. It is a surjection: every dyadic rational has not one, but two distinct representations in the Cantor set. For example,
formula_38
This is just the binary-string version of the famous 0.999... = 1 problem. The doubled representations hold in general: for any given finite-length initial sequence formula_39 of length formula_40, one has
formula_41
The initial sequence formula_39 corresponds to the non-periodic part of the orbit, after which iteration settles down to all zeros (equivalently, all-ones).
Expressed as bit strings, the periodic orbits of the map can be seen to the rationals. That is, after an initial "chaotic" sequence of formula_39, a periodic orbit settles down into a repeating string formula_42 of length formula_43. It is not hard to see that such repeating sequences correspond to rational numbers. Writing
formula_44
one then clearly has
formula_45
Tacking on the initial non-repeating sequence, one clearly has a rational number. In fact, "every" rational number can be expressed in this way: an initial "random" sequence, followed by a cycling repeat. That is, the periodic orbits of the map are in one-to-one correspondence with the rationals.
This phenomenon is note-worthy, because something similar happens in many chaotic systems. For example, geodesics on compact manifolds can have periodic orbits that behave in this way.
Keep in mind, however, that the rationals are a set of measure zero in the reals. Almost all orbits are "not" periodic! The aperiodic orbits correspond to the irrational numbers. This property also holds true in a more general setting. An open question is to what degree the behavior of the periodic orbits constrain the behavior of the system as a whole. Phenomena such as Arnold diffusion suggest that the general answer is "not very much".
Density formulation.
Instead of looking at the orbits of individual points under the action of the map, it is equally worthwhile to explore how the map affects densities on the unit interval. That is, imagine sprinkling some dust on the unit interval; it is denser in some places than in others. What happens to this density as one iterates?
Write formula_46 as this density, so that formula_47. To obtain the action of formula_10 on this density, one needs to find all points formula_48 and write
formula_49
The denominator in the above is the Jacobian determinant of the transformation, here it is just the derivative of formula_10 and so formula_50. Also, there are obviously only two points in the preimage of formula_51, these are formula_52 and formula_53 Putting it all together, one gets
formula_54
By convention, such maps are denoted by formula_55 so that in this case, write
formula_56
The map formula_57 is a linear operator, as one easily sees that formula_58 and formula_59 for all functions formula_60 on the unit interval, and all constants formula_61.
Viewed as a linear operator, the most obvious and pressing question is: what is its spectrum? One eigenvalue is obvious: if formula_62 for all formula_16 then one obviously has formula_63 so the uniform density is invariant under the transformation. This is in fact the largest eigenvalue of the operator formula_57, it is the Frobenius–Perron eigenvalue. The uniform density is, in fact, nothing other than the invariant measure of the dyadic transformation.
To explore the spectrum of formula_57 in greater detail, one must first limit oneself to a suitable space of functions (on the unit interval) to work with. This might be the space of Lebesgue measurable functions, or perhaps the space of square integrable functions, or perhaps even just polynomials. Working with any of these spaces is surprisingly difficult, although a spectrum can be obtained.
Borel space.
A vast amount of simplification results if one instead works with the Cantor space formula_36, and functions formula_64 Some caution is advised, as the map formula_34 is defined on the unit interval of the real number line, assuming the natural topology on the reals. By contrast, the map formula_65 is defined on the Cantor space formula_11, which by convention is given a very different topology, the product topology. There is a potential clash of topologies; some care must be taken. However, as presented above, there is a homomorphism from the Cantor set into the reals; fortunately, it maps open sets into open sets, and thus preserves notions of continuity.
To work with the Cantor set formula_66, one must provide a topology for it; by convention, this is the product topology. By adjoining set-complements, it can be extended to a Borel space, that is, a sigma algebra. The topology is that of cylinder sets. A cylinder set has the generic form
formula_67
where the formula_68 are arbitrary bit values (not necessarily all the same), and the formula_69 are a finite number of specific bit-values scattered in the infinite bit-string. These are the open sets of the topology. The canonical measure on this space is the Bernoulli measure for the fair coin-toss. If there is just one bit specified in the string of arbitrary positions, the measure is 1/2. If there are two bits specified, the measure is 1/4, and so on. One can get fancier: given a real number formula_70 one can define a measure
formula_71
if there are formula_72 heads and formula_43 tails in the sequence. The measure with formula_73 is preferred, since it is preserved by the map
formula_74
So, for example, formula_75 maps to the interval formula_76 and formula_77 maps to the interval formula_78 and both of these intervals have a measure of 1/2. Similarly, formula_79 maps to the interval formula_80 which still has the measure 1/2. That is, the embedding above preserves the measure.
An alternative is to write
formula_81
which preserves the measure formula_82 That is, it maps such that the measure on the unit interval is again the Lebesgue measure.
Frobenius–Perron operator.
Denote the collection of all open sets on the Cantor set by formula_83 and consider the set formula_84 of all arbitrary functions formula_85 The shift formula_10 induces a pushforward
formula_86
defined by formula_87 This is again some function formula_88 In this way, the map formula_10 induces another map formula_57 on the space of all functions formula_88 That is, given some formula_89, one defines
formula_90
This linear operator is called the transfer operator or the "Ruelle–Frobenius–Perron operator". The largest eigenvalue is the Frobenius–Perron eigenvalue, and in this case, it is 1. The associated eigenvector is the invariant measure: in this case, it is the Bernoulli measure. Again, formula_91 when formula_92
Spectrum.
To obtain the spectrum of formula_57, one must provide a suitable set of basis functions for the space formula_93 One such choice is to restrict formula_84 to the set of all polynomials. In this case, the operator has a discrete spectrum, and the eigenfunctions are (curiously) the Bernoulli polynomials! (This coincidence of naming was presumably not known to Bernoulli.)
Indeed, one can easily verify that
formula_94
where the formula_95 are the Bernoulli polynomials. This follows because the Bernoulli polynomials obey the identity
formula_96
Note that formula_97
Another basis is provided by the Haar basis, and the functions spanning the space are the Haar wavelets. In this case, one finds a continuous spectrum, consisting of the unit disk on the complex plane. Given formula_98 in the unit disk, so that formula_99, the functions
formula_100
obey
formula_101
for formula_102 This is a complete basis, in that every integer can be written in the form formula_103 The Bernoulli polynomials are recovered by setting formula_104 and formula_105
A complete basis can be given in other ways, as well; they may be written in terms of the Hurwitz zeta function. Another complete basis is provided by the Takagi function. This is a fractal, differentiable-nowhere function. The eigenfunctions are explicitly of the form
formula_106
where formula_107 is the triangle wave. One has, again,
formula_108
All of these different bases can be expressed as linear combinations of one-another. In this sense, they are equivalent.
The fractal eigenfunctions show an explicit symmetry under the fractal groupoid of the modular group; this is developed in greater detail in the article on the Takagi function (the blancmange curve). Perhaps not a surprise; the Cantor set has exactly the same set of symmetries (as do the continued fractions.) This then leads elegantly into the theory of elliptic equations and modular forms.
Relation to the Ising model.
The Hamiltonian of the zero-field one-dimensional Ising model of formula_109 spins with periodic boundary conditions can be written as
formula_110
Letting formula_111 be a suitably chosen normalization constant and formula_112 be the inverse temperature for the system, the partition function for this model is given by
formula_113
We can implement the renormalization group by integrating out every other spin. In so doing, one finds that formula_114 can also be equated with the partition function for a smaller system with but formula_115 spins,
formula_116
provided we replace formula_111 and formula_117 with renormalized values formula_118 and formula_119 satisfying the equations
formula_120
formula_121
Suppose now that we allow formula_117 to be complex and that formula_122 for some formula_123. In that case we can introduce a parameter formula_124 related to formula_117 via the equation
formula_125
and the resulting renormalization group transformation for formula_126 will be precisely the dyadic map:
formula_127
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "T: [0, 1) \\to [0, 1)^\\infty"
},
{
"math_id": 1,
"text": "x \\mapsto (x_0, x_1, x_2, \\ldots)"
},
{
"math_id": 2,
"text": "[0, 1)^\\infty"
},
{
"math_id": 3,
"text": "[0, 1)"
},
{
"math_id": 4,
"text": "x_0 = x"
},
{
"math_id": 5,
"text": "\\text{for all } n \\ge 0,\\ x_{n+1} = (2 x_n) \\bmod 1"
},
{
"math_id": 6,
"text": "T(x)=\\begin{cases}2x & 0 \\le x < \\frac{1}{2} \\\\2x-1 & \\frac{1}{2} \\le x < 1. \\end{cases}"
},
{
"math_id": 7,
"text": "T_\\beta (x)=\\beta x\\bmod 1"
},
{
"math_id": 8,
"text": "\\Omega = \\{H,T\\}^{\\mathbb{N}}"
},
{
"math_id": 9,
"text": "H"
},
{
"math_id": 10,
"text": "T"
},
{
"math_id": 11,
"text": "\\Omega = \\{0,1\\}^{\\mathbb{N}}"
},
{
"math_id": 12,
"text": "\\{0,1\\}^{\\mathbb{Z}}"
},
{
"math_id": 13,
"text": "T(b_0, b_1, b_2, \\dots) = (b_1, b_2, \\dots)"
},
{
"math_id": 14,
"text": "(b_0, b_1, \\dots)"
},
{
"math_id": 15,
"text": "x = \\sum_{n=0}^\\infty \\frac{b_n}{2^{n+1}}."
},
{
"math_id": 16,
"text": "x"
},
{
"math_id": 17,
"text": "0 \\le x \\le 1."
},
{
"math_id": 18,
"text": "T(b_0, b_1, b_2, \\dots) = (b_1, b_2, \\dots),"
},
{
"math_id": 19,
"text": "T(x)=2x\\bmod 1."
},
{
"math_id": 20,
"text": "\\Omega = 2^{\\mathbb{Z}},"
},
{
"math_id": 21,
"text": "(x, T(x), T^2(x), T^3(x), \\dots)"
},
{
"math_id": 22,
"text": "x_n = T^n(x)."
},
{
"math_id": 23,
"text": "y=\\sum_{n=0}^\\infty \\frac{b_n}{3^{n+1}}"
},
{
"math_id": 24,
"text": "\\{H,T\\}^\\mathbb{N}"
},
{
"math_id": 25,
"text": "x_{n+1} = f_1(x_n) = \\begin{cases}\n x_n & \\mathrm{for}~~ x_n \\le 1/2 \\\\ \n 1-x_n & \\mathrm{for}~~ x_n \\ge 1/2 \n\\end{cases}"
},
{
"math_id": 26,
"text": "S(x)=\\sin \\pi x"
},
{
"math_id": 27,
"text": "f_1 = S^{-1} \\circ T \\circ S"
},
{
"math_id": 28,
"text": "f_1(x) = S^{-1}(T(S(x)))."
},
{
"math_id": 29,
"text": "f_1^n = f_1\\circ\\cdots\\circ f_1 = S^{-1} \\circ T \\circ S \\circ S^{-1} \\circ \\cdots \\circ T \\circ S = S^{-1} \\circ T^n \\circ S"
},
{
"math_id": 30,
"text": "z_{n+1}=4z_n(1-z_n)"
},
{
"math_id": 31,
"text": "z_n =\\sin^2 (2 \\pi x_n)."
},
{
"math_id": 32,
"text": "\\theta\\mapsto 2\\theta\\bmod 2\\pi."
},
{
"math_id": 33,
"text": "\\frac{11}{24} \\mapsto \\frac{11}{12} \\mapsto \\frac{5}{6} \\mapsto \\frac{2}{3} \\mapsto \\frac{1}{3} \\mapsto \\frac{2}{3} \\mapsto \\frac{1}{3} \\mapsto \\cdots, "
},
{
"math_id": 34,
"text": "T(x)=2x\\bmod 1"
},
{
"math_id": 35,
"text": "T(b_0,b_1,b_2,\\dots) = (b_1, b_2,\\dots)"
},
{
"math_id": 36,
"text": "\\Omega=\\{0,1\\}^\\mathbb{N}"
},
{
"math_id": 37,
"text": "x=\\sum_{n=0}^\\infty \\frac{b_n}{2^{n+1}}"
},
{
"math_id": 38,
"text": "0.1000000\\dots = 0.011111\\dots"
},
{
"math_id": 39,
"text": "b_0,b_1,b_2,\\dots,b_{k-1}"
},
{
"math_id": 40,
"text": "k"
},
{
"math_id": 41,
"text": "b_0,b_1,b_2,\\dots,b_{k-1},1,0,0,0,\\dots = b_0,b_1,b_2,\\dots,b_{k-1},0,1,1,1,\\dots"
},
{
"math_id": 42,
"text": "b_k,b_{k+1},b_{k+2},\\dots,b_{k+m-1}"
},
{
"math_id": 43,
"text": "m"
},
{
"math_id": 44,
"text": "y = \\sum_{j=0}^{m-1} b_{k+j}2^{-j-1}"
},
{
"math_id": 45,
"text": "\\sum_{j=0}^\\infty b_{k+j}2^{-j-1} = y\\sum_{j=0}^\\infty 2^{-jm} = \\frac{y}{1-2^{-m}}"
},
{
"math_id": 46,
"text": "\\rho:[0,1]\\to\\mathbb{R}"
},
{
"math_id": 47,
"text": "x\\mapsto\\rho(x)"
},
{
"math_id": 48,
"text": "y=T^{-1}(x)"
},
{
"math_id": 49,
"text": "\\rho(x) \\mapsto \\sum_{y=T^{-1}(x)} \\frac{\\rho(y)}{|T^\\prime(y)|}"
},
{
"math_id": 50,
"text": "T^\\prime(y)=2"
},
{
"math_id": 51,
"text": "T^{-1}(x)"
},
{
"math_id": 52,
"text": "y=x/2"
},
{
"math_id": 53,
"text": "y=(x+1)/2."
},
{
"math_id": 54,
"text": "\\rho(x) \\mapsto \\frac{1}{2}\\rho\\!\\left(\\frac{x}{2}\\right) + \\frac{1}{2}\\rho\\!\\left(\\frac{x+1}{2}\\right)"
},
{
"math_id": 55,
"text": "\\mathcal{L}"
},
{
"math_id": 56,
"text": "\\left[\\mathcal {L}_T\\rho\\right](x) = \\frac{1}{2}\\rho\\!\\left(\\frac{x}{2}\\right) + \\frac{1}{2}\\rho\\!\\left(\\frac{x+1}{2}\\right)"
},
{
"math_id": 57,
"text": "\\mathcal{L}_T"
},
{
"math_id": 58,
"text": "\\mathcal{L}_T(f+g)= \\mathcal{L}_T(f) + \\mathcal{L}_T(g)"
},
{
"math_id": 59,
"text": "\\mathcal{L}_T(af)= a\\mathcal{L}_T(f)"
},
{
"math_id": 60,
"text": "f,g"
},
{
"math_id": 61,
"text": "a"
},
{
"math_id": 62,
"text": "\\rho(x)=1"
},
{
"math_id": 63,
"text": "\\mathcal{L}_T\\rho=\\rho"
},
{
"math_id": 64,
"text": "\\rho:\\Omega\\to\\mathbb{R}."
},
{
"math_id": 65,
"text": "T(b_0, b_1, b_2, \\dots)=(b_1, b_2, \\dots)"
},
{
"math_id": 66,
"text": "\\Omega=\\{0,1\\}^{\\mathbb{N}}"
},
{
"math_id": 67,
"text": "(*,*,*,\\dots,*,b_k,b_{k+1},*,\\dots, *,b_m,*,\\dots)"
},
{
"math_id": 68,
"text": "*"
},
{
"math_id": 69,
"text": "b_k, b_m, \\dots"
},
{
"math_id": 70,
"text": "0 < p < 1"
},
{
"math_id": 71,
"text": "\\mu_p( *,\\dots,*,b_k,*,\\dots) = p^n(1-p)^m"
},
{
"math_id": 72,
"text": "n"
},
{
"math_id": 73,
"text": "p=1/2"
},
{
"math_id": 74,
"text": "(b_0, b_1, b_2, \\dots) \\mapsto x = \\sum_{n=0}^\\infty \\frac{b_n}{2^{n+1}}."
},
{
"math_id": 75,
"text": "(0,*,\\cdots)"
},
{
"math_id": 76,
"text": "[0,1/2]"
},
{
"math_id": 77,
"text": "(1,*,\\dots)"
},
{
"math_id": 78,
"text": "[1/2,1]"
},
{
"math_id": 79,
"text": "(*,0,*,\\dots)"
},
{
"math_id": 80,
"text": "[0,1/4]\\cup[1/2,3/4]"
},
{
"math_id": 81,
"text": "(b_0, b_1, b_2, \\dots) \\mapsto x = \\sum_{n=0}^\\infty \\left[b_n p^{n+1} + (1-b_n)(1-p)^{n+1}\\right]"
},
{
"math_id": 82,
"text": "\\mu_p."
},
{
"math_id": 83,
"text": "\\mathcal{B}"
},
{
"math_id": 84,
"text": "\\mathcal{F}"
},
{
"math_id": 85,
"text": "f:\\mathcal{B}\\to\\mathbb{R}."
},
{
"math_id": 86,
"text": "f\\circ T^{-1}"
},
{
"math_id": 87,
"text": "\\left(f \\circ T^{-1}\\right)\\!(x) = f(T^{-1}(x))."
},
{
"math_id": 88,
"text": "\\mathcal{B}\\to\\mathbb{R}."
},
{
"math_id": 89,
"text": "f:\\mathcal{B}\\to\\mathbb{R}"
},
{
"math_id": 90,
"text": "\\mathcal{L}_T f = f \\circ T^{-1}"
},
{
"math_id": 91,
"text": "\\mathcal{L}_T(\\rho)= \\rho"
},
{
"math_id": 92,
"text": "\\rho(x)=1."
},
{
"math_id": 93,
"text": "\\mathcal{F}."
},
{
"math_id": 94,
"text": "\\mathcal{L}_T B_n= 2^{-n}B_n"
},
{
"math_id": 95,
"text": "B_n"
},
{
"math_id": 96,
"text": "\\frac{1}{2}B_n\\!\\left(\\frac{y}{2}\\right) + \\frac{1}{2}B_n\\!\\left(\\frac{y+1}{2}\\right) = 2^{-n}B_n(y)"
},
{
"math_id": 97,
"text": "B_0(x)=1."
},
{
"math_id": 98,
"text": "z\\in\\mathbb{C}"
},
{
"math_id": 99,
"text": "|z|<1"
},
{
"math_id": 100,
"text": "\\psi_{z,k}(x)=\\sum_{n=1}^\\infty z^n \\exp i\\pi(2k+1)2^nx"
},
{
"math_id": 101,
"text": "\\mathcal{L}_T \\psi_{z,k}= z\\psi_{z,k}"
},
{
"math_id": 102,
"text": "k\\in\\mathbb{Z}."
},
{
"math_id": 103,
"text": "(2k+1)2^n."
},
{
"math_id": 104,
"text": "k=0"
},
{
"math_id": 105,
"text": "z=\\frac{1}{2}, \\frac{1}{4}, \\dots"
},
{
"math_id": 106,
"text": "\\mbox{blanc}_{w,k}(x) = \\sum_{n=0}^\\infty w^n s((2k+1)2^{n}x)"
},
{
"math_id": 107,
"text": "s(x)"
},
{
"math_id": 108,
"text": "\\mathcal{L}_T \\mbox{blanc}_{w,k} = w\\;\\mbox{blanc}_{w,k}."
},
{
"math_id": 109,
"text": "2N"
},
{
"math_id": 110,
"text": "H(\\sigma) = g \\sum_{i\\in \\mathbb{Z}_{2N}}\\sigma_i\\sigma_{i+1}. "
},
{
"math_id": 111,
"text": "C"
},
{
"math_id": 112,
"text": "\\beta"
},
{
"math_id": 113,
"text": "Z = \\sum_{\\{\\sigma_i=\\pm 1,\\, i\\in \\mathbb{Z}_{2N}\\}}\\prod_{i\\in \\mathbb{Z}_{2N}}Ce^{-\\beta g \\sigma_i\\sigma_{i+1}}. "
},
{
"math_id": 114,
"text": "Z"
},
{
"math_id": 115,
"text": "N"
},
{
"math_id": 116,
"text": "Z = \\sum_{\\{\\sigma_i=\\pm 1,\\, i\\in \\mathbb{Z}_{N}\\}}\\prod_{i\\in \\mathbb{Z}_{N}}\\mathcal{R}[C]e^{-\\mathcal{R}[\\beta g] \\sigma_i\\sigma_{i+1}}, "
},
{
"math_id": 117,
"text": "\\beta g"
},
{
"math_id": 118,
"text": "\\mathcal{R}[C]"
},
{
"math_id": 119,
"text": "\\mathcal{R}[\\beta g]"
},
{
"math_id": 120,
"text": "\\mathcal{R}[C]^2= 4\\cosh(2\\beta g)C^4,"
},
{
"math_id": 121,
"text": "e^{-2\\mathcal{R}[\\beta g]}= \\cosh(2\\beta g)."
},
{
"math_id": 122,
"text": "\\operatorname{Im}[2\\beta g]=\\frac{\\pi}{2}+\\pi n"
},
{
"math_id": 123,
"text": "n\\in \\mathbb{Z}"
},
{
"math_id": 124,
"text": "t\\in[0, 1)"
},
{
"math_id": 125,
"text": "e^{-2\\beta g}= i\\tan\\big(\\pi(t-\\frac{1}{2})\\big),"
},
{
"math_id": 126,
"text": "t"
},
{
"math_id": 127,
"text": "\\mathcal{R}[t]=2t \\bmod 1 ."
}
] |
https://en.wikipedia.org/wiki?curid=1501024
|
15012850
|
Leonardo number
|
Set of numbers used in the smoothsort algorithm
The Leonardo numbers are a sequence of numbers given by the recurrence:
formula_0
Edsger W. Dijkstra used them as an integral part of his smoothsort algorithm, and also analyzed them in some detail.
A Leonardo prime is a Leonardo number that's also prime.
Values.
The first few Leonardo numbers are
1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, ... (sequence in the OEIS)
The first few Leonardo primes are
3, 5, 41, 67, 109, 1973, 5167, 2692537, 11405773, 126491971, 331160281, 535828591, 279167724889, 145446920496281, 28944668049352441, 5760134388741632239, 63880869269980199809, 167242286979696845953, 597222253637954133837103, ... (sequence in the OEIS)
Modulo cycles.
The Leonardo numbers form a cycle in any modulo n≥2. An easy way to see it is:
The cycles for n≤8 are:
The cycle always end on the pair (1,n-1), as it's the only pair which can precede the pair (1,1).
formula_1
Expressions.
<templatestyles src="Math_proof/styles.css" />Proof
formula_2
Relation to Fibonacci numbers.
The Leonardo numbers are related to the Fibonacci numbers by the relation formula_3.
From this relation it is straightforward to derive a closed-form expression for the Leonardo numbers, analogous to Binet's formula for the Fibonacci numbers:
formula_4
where the golden ratio formula_5 and formula_6 are the roots of the quadratic polynomial formula_7.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \n L(n) = \n \\begin{cases}\n 1 & \\mbox{if } n = 0 \\\\\n 1 & \\mbox{if } n = 1 \\\\\n L(n - 1) + L(n - 2) + 1 & \\mbox{if } n > 1 \\\\\n \\end{cases}\n "
},
{
"math_id": 1,
"text": "L(n)=2L(n-1)-L(n-3)"
},
{
"math_id": 2,
"text": "L(n)=L(n-1)+L(n-2)+1=L(n-1)+L(n-2)+1+L(n-3)-L(n-3)=2L(n-1)-L(n-3)"
},
{
"math_id": 3,
"text": "L(n) = 2 F(n+1) - 1, n \\ge 0"
},
{
"math_id": 4,
"text": "L(n) = 2 \\frac{\\varphi^{n+1} - \\psi^{n+1}}{\\varphi - \\psi}- 1 = \\frac{2}{\\sqrt 5} \\left(\\varphi^{n+1} - \\psi^{n+1}\\right) - 1 = 2F(n+1) - 1"
},
{
"math_id": 5,
"text": "\\varphi = \\left(1 + \\sqrt 5\\right)/2"
},
{
"math_id": 6,
"text": "\\psi = \\left(1 - \\sqrt 5\\right)/2"
},
{
"math_id": 7,
"text": "x^2 - x - 1 = 0"
}
] |
https://en.wikipedia.org/wiki?curid=15012850
|
150159
|
Noether's theorem
|
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries of physical space.
Noether's theorem is used in theoretical physics and the calculus of variations. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.
Basic illustrations and background.
As an illustration, if a physical system behaves the same regardless of how it is oriented in space (that is, it's invariant), its Lagrangian is symmetric under continuous rotation: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.126 The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. It is the laws of its motion that are symmetric.
As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.23261
Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system.127 As an illustration, suppose that a physical theory is proposed which conserves a quantity "X". A researcher can calculate the types of Lagrangians that conserve "X" through a continuous symmetry. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory.
There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the Ward–Takahashi identities. Generalizations of Noether's theorem to superspaces also exist.
Informal statement of the theorem.
All fine technical points aside, Noether's theorem can be stated informally:
<templatestyles src="Template:Blockquote/styles.css" />If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.
A more sophisticated version of the theorem involving fields states that:
<templatestyles src="Template:Blockquote/styles.css" />To every continuous symmetry generated by local actions there corresponds a conserved current and vice versa.
The word "symmetry" in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria. The conservation law of a physical quantity is usually expressed as a continuity equation.
The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the conserved quantity is called the "Noether charge", while the flow carrying that charge is called the "Noether current". The Noether current is defined up to a solenoidal (divergenceless) vector field.
In the context of gravitation, Felix Klein's statement of Noether's theorem for action "I" stipulates for the invariants:
<templatestyles src="Template:Blockquote/styles.css" />If an integral I is invariant under a continuous group "G""ρ" with "ρ" parameters, then "ρ" linearly independent combinations of the Lagrangian expressions are divergences.
Brief illustration and overview of the concept.
The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate formula_0 and a continuous symmetry formula_1 (gray arrows on the diagram).
Consider any trajectory formula_2 (bold on the diagram) that satisfies the system's laws of motion. That is, the action formula_3 governing this system is stationary on this trajectory, i.e. does not change under any local variation of the trajectory. In particular it would not change under a variation that applies the symmetry flow formula_4 on a time segment ["t"0, "t"1] and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time formula_5 to transition between the segments gradually.
The total change in the action formula_3 now comprises changes brought by every interval in play. Parts, where variation itself vanishes, i.e outside formula_6 bring no formula_7. The middle part does not change the action either, because its transformation formula_4 is a symmetry and thus preserves the Lagrangian formula_8 and the action formula_9. The only remaining parts are the "buffering" pieces. In these regions both the coordinate formula_0 and velocity formula_10 change, but formula_10 changes by formula_11, and the change formula_12 in the coordinate is negligible by comparison since the time span formula_5 of the buffering is small (taken to the limit of 0), so formula_13. So the regions contribute mostly through their "slanting" formula_14.
That changes the Lagrangian by formula_15, which integrates to
formula_16
These last terms, evaluated around the endpoints formula_17 and formula_18, should cancel each other in order to make the total change in the action formula_7 be zero, as would be expected if the trajectory is a solution. That is
formula_19
meaning the quantity formula_20 is conserved, which is the conclusion of Noether's theorem. For instance if pure translations of formula_0 by a constant are the symmetry, then the conserved quantity becomes just formula_21, the canonical momentum.
More general cases follow the same idea:
Historical context.
A conservation law states that some quantity "X" in the mathematical description of a system's evolution remains constant throughout its motion – it is an invariant. Mathematically, the rate of change of "X" (its derivative with respect to time) is zero,
formula_22
Such quantities are said to be conserved; they are often called constants of motion (although motion "per se" need not be involved, just evolution in time). For example, if the energy of a system is conserved, its energy is invariant at all times, which imposes a constraint on the system's motion and may help in solving for it. Aside from insights that such constants of motion give into the nature of a system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the suitable conservation laws.
The earliest constants of motion discovered were momentum and kinetic energy, which were proposed in the 17th century by René Descartes and Gottfried Leibniz on the basis of collision experiments, and refined by subsequent researchers. Isaac Newton was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of Newton's laws of motion. According to general relativity, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of the stress–energy tensor (non-gravitational stress–energy) and the Landau–Lifshitz stress–energy–momentum pseudotensor (gravitational stress–energy). The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant divergence of the stress–energy tensor. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, is the Laplace–Runge–Lenz vector.
In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering invariants. A major advance came in 1788 with the development of Lagrangian mechanics, which is related to the principle of least action. In this approach, the state of the system can be described by any type of generalized coordinates q; the laws of motion need not be expressed in a Cartesian coordinate system, as was customary in Newtonian mechanics. The action is defined as the time integral "I" of a function known as the Lagrangian "L"
formula_23
where the dot over q signifies the rate of change of the coordinates q,
formula_24
Hamilton's principle states that the physical path q("t")—the one actually taken by the system—is a path for which infinitesimal variations in that path cause no change in "I", at least up to first order. This principle results in the Euler–Lagrange equations,
formula_25
Thus, if one of the coordinates, say "qk", does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that
formula_26
where the momentum
formula_27
is conserved throughout the motion (on the physical path).
Thus, the absence of the ignorable coordinate "qk" from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of "qk"; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations. This is the seed idea generalized in Noether's theorem.
Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. For example, he developed a theory of canonical transformations which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the Hamilton–Jacobi equation.
Mathematical expression.
Simple form using perturbations.
The essence of Noether's theorem is generalizing the notion of ignorable coordinates.
One can assume that the Lagrangian "L" defined above is invariant under small perturbations (warpings) of the time variable "t" and the generalized coordinates q. One may write
formula_28
where the perturbations "δt" and "δ"q are both small, but variable. For generality, assume there are (say) "N" such symmetry transformations of the action, i.e. transformations leaving the action unchanged; labelled by an index "r" = 1, 2, 3, ..., "N".
Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,
formula_29
where "ε""r" are infinitesimal parameter coefficients corresponding to each:
For translations, Q"r" is a constant with units of length; for rotations, it is an expression linear in the components of q, and the parameters make up an angle.
Using these definitions, Noether showed that the "N" quantities
formula_30
are conserved (constants of motion).
Examples.
I. Time invariance
For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes "t" → "t" + δ"t", without any change in the coordinates q. In this case, "N" = 1, "T" = 1 and Q = 0; the corresponding conserved quantity is the total energy "H"
formula_31
II. Translational invariance
Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate "q""k"; so it is invariant (symmetric) under changes "q""k" → "q""k" + "δq""k". In that case, "N" = 1, "T" = 0, and "Q""k" = 1; the conserved quantity is the corresponding linear momentum "p""k"
formula_32
In special and general relativity, these two conservation laws can be expressed either "globally" (as it is done above), or "locally" as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity defined "locally" at the space-time point: the stress–energy tensor(this will be derived in the next section).
III. Rotational invariance
The conservation of the angular momentum L = r × p is analogous to its linear momentum counterpart. It is assumed that the symmetry of the Lagrangian is rotational, i.e., that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations of an angle "δθ" about an axis n; such a rotation transforms the Cartesian coordinates by the equation
formula_33
Since time is not being transformed, "T" = 0, and "N" = 1. Taking "δθ" as the "ε" parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by
formula_34
Then Noether's theorem states that the following quantity is conserved,
formula_35
In other words, the component of the angular momentum L along the n axis is conserved. And if n is arbitrary, i.e., if the system is insensitive to any rotation, then every component of L is conserved; in short, angular momentum is conserved.
Field theory version.
Although useful in its own right, the version of Noether's theorem just given is a special case of the general version derived in 1915. To give the flavor of the general theorem, a version of Noether's theorem for continuous fields in four-dimensional space–time is now given. Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used (or most often implemented) version of Noether's theorem.
Let there be a set of differentiable fields formula_4 defined over all space and time; for example, the temperature formula_36 would be representative of such a field, being a number defined at every place and time. The principle of least action can be applied to such fields, but the action is now an integral over space and time
formula_37
(the theorem can be further generalized to the case where the Lagrangian depends on up to the "n"th derivative, and can also be formulated using jet bundles).
A continuous transformation of the fields formula_4 can be written infinitesimally as
formula_38
where formula_39 is in general a function that may depend on both formula_40 and formula_4. The condition for formula_39 to generate a physical symmetry is that the action formula_41 is left invariant. This will certainly be true if the Lagrangian density formula_42 is left invariant, but it will also be true if the Lagrangian changes by a divergence,
formula_43
since the integral of a divergence becomes a boundary term according to the divergence theorem. A system described by a given action might have multiple independent symmetries of this type, indexed by formula_44 so the most general symmetry transformation would be written as
formula_45
with the consequence
formula_46
For such systems, Noether's theorem states that there are formula_47 conserved current densities
formula_48
(where the dot product is understood to contract the "field" indices, not the formula_49 index or formula_50 index).
In such cases, the conservation law is expressed in a four-dimensional way
formula_51
which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. For example, electric charge is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.
For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, formula_52 is constant in its third argument. In that case, "N" = 4, one for each dimension of space and time. An infinitesimal translation in space, formula_53 (with formula_54 denoting the Kronecker delta), affects the fields as formula_55: that is, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point formula_40 with the value at the point formula_56 "behind" it which would be mapped onto formula_40 by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as
formula_57
The Lagrangian density transforms in the same way, formula_58, so
formula_59
and thus Noether's theorem corresponds to the conservation law for the stress–energy tensor "T""μ""ν", where we have used formula_60 in place of formula_50. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each formula_60) into a tensor formula_61, Noether's theorem gives
formula_62
with
formula_63
The conservation of electric charge, by contrast, can be derived by considering "Ψ" linear in the fields "φ" rather than in the derivatives. In quantum mechanics, the probability amplitude "ψ"(x) of finding a particle at a point x is a complex field "φ", because it ascribes a complex number to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability "p" = |"ψ"|2 can be inferred from a set of measurements. Therefore, the system is invariant under transformations of the "ψ" field and its complex conjugate field "ψ"* that leave |"ψ"|2 unchanged, such as
formula_65
a complex rotation. In the limit when the phase "θ" becomes infinitesimally small, "δθ", it may be taken as the parameter "ε", while the "Ψ" are equal to "iψ" and −"iψ"*, respectively. A specific example is the Klein–Gordon equation, the relativistically correct version of the Schrödinger equation for spinless particles, which has the Lagrangian density
formula_66
In this case, Noether's theorem states that the conserved (∂ ⋅ "j" = 0) current equals
formula_67
which, when multiplied by the charge on that species of particle, equals the electric current density due to that type of particle. This "gauge invariance" was first noted by Hermann Weyl, and is one of the prototype gauge symmetries of physics.
Derivations.
One independent variable.
Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables q are such that the action integral
formula_68
is invariant under brief infinitesimal variations in the dependent variables. In other words, they satisfy the Euler–Lagrange equations
formula_69
And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a flow, φ, which acts on the variables as follows
formula_70
where "ε" is a real variable indicating the amount of flow, and "T" is a real constant (which could be zero) indicating how much the flow shifts time.
formula_71
The action integral flows to
formula_72
which may be regarded as a function of "ε". Calculating the derivative at "ε" = 0 and using Leibniz's rule, we get
formula_73
Notice that the Euler–Lagrange equations imply
formula_74
Substituting this into the previous equation, one gets
formula_75
Again using the Euler–Lagrange equations we get
formula_76
Substituting this into the previous equation, one gets
formula_77
From which one can see that
formula_78
is a constant of the motion, i.e., it is a conserved quantity. Since φ[q, 0] = q, we get formula_79 and so the conserved quantity simplifies to
formula_80
To avoid excessive complication of the formulas, this derivation assumed that the flow does not change as time passes. The same result can be obtained in the more general case.
Field-theoretic derivation.
Noether's theorem may also be derived for tensor fields formula_81 where the index "A" ranges over the various components of the various tensor fields. These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates "x"μ where the index "μ" ranges over time ("μ" = 0) and three spatial dimensions ("μ" = 1, 2, 3). These four coordinates are the independent variables; and the values of the fields at each event are the dependent variables. Under an infinitesimal transformation, the variation in the coordinates is written
formula_82
whereas the transformation of the field variables is expressed as
formula_83
By this definition, the field variations formula_84 result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field "α""A" depends on the transformed coordinates ξμ. To isolate the intrinsic changes, the field variation at a single point "x"μ may be defined
formula_85
If the coordinates are changed, the boundary of the region of space–time over which the Lagrangian is being integrated also changes; the original boundary and its transformed version are denoted as Ω and Ω’, respectively.
Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the action, which is defined as the integral of the Lagrangian density over the given region of spacetime. Expressed mathematically, this assumption may be written as
formula_86
where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g.
formula_87
Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form
formula_88
The difference in Lagrangians can be written to first-order in the infinitesimal variations as
formula_89
However, because the variations are defined at the same point as described above, the variation and the derivative can be done in reverse order; they commute
formula_90
Using the Euler–Lagrange field equations
formula_91
the difference in Lagrangians can be written neatly as
formula_92
Thus, the change in the action can be written as
formula_93
Since this holds for any region Ω, the integrand must be zero
formula_94
For any combination of the various symmetry transformations, the perturbation can be written
formula_95
where formula_96 is the Lie derivative of
formula_81 in the "X""μ" direction. When formula_81 is a scalar or formula_97,
formula_98
These equations imply that the field variation taken at one point equals
formula_99
Differentiating the above divergence with respect to "ε" at "ε" = 0 and changing the sign yields the conservation law
formula_100
where the conserved current equals
formula_101
Manifold/fiber bundle derivation.
Suppose we have an "n"-dimensional oriented Riemannian manifold, "M" and a target manifold "T". Let formula_102 be the configuration space of smooth functions from "M" to "T". (More generally, we can have smooth sections of a fiber bundle over "M".)
Examples of this "M" in physics include:
Now suppose there is a functional
formula_107
called the action. (It takes values into formula_103, rather than formula_108; this is for physical reasons, and is unimportant for this proof.)
To get to the usual version of Noether's theorem, we need additional restrictions on the action. We assume formula_109 is the integral over "M" of a function
formula_110
called the Lagrangian density, depending on formula_4, its derivative and the position. In other words, for formula_4 in formula_102
formula_111
Suppose we are given boundary conditions, i.e., a specification of the value of formula_4 at the boundary if "M" is compact, or some limit on formula_4 as "x" approaches ∞. Then the subspace of formula_102 consisting of functions formula_4 such that all functional derivatives of formula_41 at formula_4 are zero, that is:
formula_112
and that formula_4 satisfies the given boundary conditions, is the subspace of on shell solutions. (See principle of stationary action)
Now, suppose we have an infinitesimal transformation on formula_102, generated by a functional derivation, "Q" such that
formula_113
for all compact submanifolds "N" or in other words,
formula_114
for all "x", where we set
formula_115
If this holds on shell and off shell, we say "Q" generates an off-shell symmetry. If this only holds on shell, we say "Q" generates an on-shell symmetry. Then, we say "Q" is a generator of a one parameter symmetry Lie group.
Now, for any "N", because of the Euler–Lagrange theorem, on shell (and only on-shell), we have
formula_116
Since this is true for any "N", we have
formula_117
But this is the continuity equation for the current formula_118 defined by:
formula_119
which is called the Noether current associated with the symmetry. The continuity equation tells us that if we integrate this current over a space-like slice, we get a conserved quantity called the Noether charge (provided, of course, if "M" is noncompact, the currents fall off sufficiently fast at infinity).
Comments.
Noether's theorem is an on shell theorem: it relies on use of the equations of motion—the classical path. It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that
formula_120
The quantum analogs of Noether's theorem involving expectation values (e.g., formula_121) probing off shell quantities as well are the Ward–Takahashi identities.
Generalization to Lie algebras.
Suppose we have two symmetry derivations "Q"1 and "Q"2. Then, ["Q"1, "Q"2] is also a symmetry derivation. Let us see this explicitly. Let us say
formula_122
and
formula_123
Then,
formula_124
where "f"12 = "Q"1["f"2"μ"] − "Q"2["f"1"μ"]. So,
formula_125
This shows we can extend Noether's theorem to larger Lie algebras in a natural way.
Generalization of the proof.
This applies to "any" local symmetry derivation "Q" satisfying "QS" ≈ 0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let "ε" be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. "ε" is a test function. Then, because of the variational principle (which does "not" apply to the boundary, by the way), the derivation distribution q generated by "q"["ε"][Φ("x")] = "ε"("x")"Q"[Φ("x")] satisfies "q"["ε"]["S"] ≈ 0 for every "ε", or more compactly, "q"("x")["S"] ≈ 0 for all "x" not on the boundary (but remember that "q"("x") is a shorthand for a derivation "distribution", not a derivation parametrized by "x" in general). This is the generalization of Noether's theorem.
To see how the generalization is related to the version given above, assume that the action is the spacetime integral of a Lagrangian that only depends on formula_4 and its first derivatives. Also, assume
formula_126
Then,
formula_127
for all formula_128.
More generally, if the Lagrangian depends on higher derivatives, then
formula_129
Examples.
Example 1: Conservation of energy.
Looking at the specific case of a Newtonian particle of mass "m", coordinate "x", moving under the influence of a potential "V", coordinatized by time "t". The action, "S", is:
formula_130
The first term in the brackets is the kinetic energy of the particle, while the second is its potential energy. Consider the generator of time translations "Q" = "d"/"dt". In other words, formula_131. The coordinate "x" has an explicit dependence on time, whilst "V" does not; consequently:
formula_132
so we can set
formula_133
Then,
formula_134
The right hand side is the energy, and Noether's theorem states that formula_135 (i.e. the principle of conservation of energy is a consequence of invariance under time translations).
More generally, if the Lagrangian does not depend explicitly on time, the quantity
formula_136
(called the Hamiltonian) is conserved.
Example 2: Conservation of center of momentum.
Still considering 1-dimensional time, let
formula_137
for formula_47 Newtonian particles where the potential only depends pairwise upon the relative displacement.
For formula_138, consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words,
formula_139
And
formula_140
This has the form of formula_141 so we can set
formula_142
Then,
formula_143
where formula_144 is the total momentum, "M" is the total mass and formula_145 is the center of mass. Noether's theorem states:
formula_146
Example 3: Conformal transformation.
Both examples 1 and 2 are over a 1-dimensional manifold (time). An example involving spacetime is a conformal transformation of a massless real scalar field with a quartic potential in (3 + 1)-Minkowski spacetime.
formula_147
For "Q", consider the generator of a spacetime rescaling. In other words,
formula_148
The second term on the right hand side is due to the "conformal weight" of formula_4. And
formula_149
This has the form of
formula_150
(where we have performed a change of dummy indices) so set
formula_151
Then
formula_152
Noether's theorem states that formula_153 (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side).
If one tries to find the Ward–Takahashi analog of this equation, one runs into a problem because of anomalies.
Applications.
Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example:
In quantum field theory, the analog to Noether's theorem, the Ward–Takahashi identity, yields further conservation laws, such as the conservation of electric charge from the invariance with respect to a change in the phase factor of the complex field of the charged particle and the associated gauge of the electric potential and vector potential.
The Noether charge is also used in calculating the entropy of stationary black holes.
See also.
<templatestyles src="Div col/styles.css"/>
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "q"
},
{
"math_id": 1,
"text": " \\varphi: q \\mapsto q + \\delta q "
},
{
"math_id": 2,
"text": "q(t)"
},
{
"math_id": 3,
"text": "S"
},
{
"math_id": 4,
"text": "\\varphi"
},
{
"math_id": 5,
"text": "\\tau"
},
{
"math_id": 6,
"text": "[t_0,t_1]"
},
{
"math_id": 7,
"text": "\\Delta S"
},
{
"math_id": 8,
"text": "L"
},
{
"math_id": 9,
"text": " S = \\int L "
},
{
"math_id": 10,
"text": "\\dot{q}"
},
{
"math_id": 11,
"text": "\\delta q / \\tau"
},
{
"math_id": 12,
"text": "\\delta q"
},
{
"math_id": 13,
"text": "\\delta q / \\tau\\gg \\delta q"
},
{
"math_id": 14,
"text": "\\dot{q}\\rightarrow \\dot{q}\\pm \\delta q / \\tau"
},
{
"math_id": 15,
"text": "\\Delta L \\approx \\bigl(\\partial L/\\partial \\dot{q}\\bigr)\\Delta \\dot{q} "
},
{
"math_id": 16,
"text": "\\Delta S =\n \\int \\Delta L \\approx \\int \\frac{\\partial L}{\\partial \\dot{q}}\\Delta \\dot{q} \\approx\n \\int \\frac{\\partial L}{\\partial \\dot{q}}\\left(\\pm \\frac{\\delta q}{\\tau}\\right) \\approx\n \\ \\pm\\frac{\\partial L}{\\partial \\dot{q}} \\delta q =\n \\pm\\frac{\\partial L}{\\partial \\dot{q}} \\varphi.\n"
},
{
"math_id": 17,
"text": "t_0"
},
{
"math_id": 18,
"text": "t_1"
},
{
"math_id": 19,
"text": "\n \\left(\\frac{\\partial L}{\\partial \\dot{q}} \\varphi\\right)(t_0) =\n \\left(\\frac{\\partial L}{\\partial \\dot{q}} \\varphi\\right)(t_1),\n"
},
{
"math_id": 20,
"text": "\\left(\\partial L /\\partial \\dot{q}\\right)\\varphi"
},
{
"math_id": 21,
"text": "\\left(\\partial L/\\partial \\dot{q}\\right) = p"
},
{
"math_id": 22,
"text": "\\frac{dX}{dt} = \\dot{X} = 0 ~."
},
{
"math_id": 23,
"text": "I = \\int L(\\mathbf{q}, \\dot{\\mathbf{q}}, t) \\, dt ~,"
},
{
"math_id": 24,
"text": "\\dot{\\mathbf{q}} = \\frac{d\\mathbf{q}}{dt} ~."
},
{
"math_id": 25,
"text": "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\right) = \\frac{\\partial L}{\\partial \\mathbf{q}} ~."
},
{
"math_id": 26,
"text": "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}_k} \\right) = \\frac{dp_k}{dt} = 0~,"
},
{
"math_id": 27,
"text": " p_k = \\frac{\\partial L}{\\partial \\dot{q}_k} "
},
{
"math_id": 28,
"text": "\\begin{align}\n t &\\rightarrow t^{\\prime} = t + \\delta t \\\\\n \\mathbf{q} &\\rightarrow \\mathbf{q}^{\\prime} = \\mathbf{q} + \\delta \\mathbf{q} ~,\n\\end{align}"
},
{
"math_id": 29,
"text": "\\begin{align}\n \\delta t &= \\sum_r \\varepsilon_r T_r \\\\\n \\delta \\mathbf{q} &= \\sum_r \\varepsilon_r \\mathbf{Q}_r ~,\n\\end{align}"
},
{
"math_id": 30,
"text": "\\left(\\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\cdot \\dot{\\mathbf{q}} - L \\right) T_r - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\cdot \\mathbf{Q}_r"
},
{
"math_id": 31,
"text": "H = \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\cdot \\dot{\\mathbf{q}} - L. "
},
{
"math_id": 32,
"text": "p_k = \\frac{\\partial L}{\\partial \\dot{q_k}}."
},
{
"math_id": 33,
"text": "\\mathbf{r} \\rightarrow \\mathbf{r} + \\delta\\theta \\, \\mathbf{n} \\times \\mathbf{r}."
},
{
"math_id": 34,
"text": "\\mathbf{Q} = \\mathbf{n} \\times \\mathbf{r}."
},
{
"math_id": 35,
"text": "\n\\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\cdot \\mathbf{Q} = \n\\mathbf{p} \\cdot \\left( \\mathbf{n} \\times \\mathbf{r} \\right) = \n\\mathbf{n} \\cdot \\left( \\mathbf{r} \\times \\mathbf{p} \\right) = \n\\mathbf{n} \\cdot \\mathbf{L}.\n"
},
{
"math_id": 36,
"text": "T(\\mathbf{x}, t)"
},
{
"math_id": 37,
"text": "\\mathcal{S} = \\int \\mathcal{L} \\left(\\varphi, \\partial_\\mu \\varphi, x^\\mu \\right) \\, d^4 x"
},
{
"math_id": 38,
"text": "\\varphi \\mapsto \\varphi + \\varepsilon \\Psi,"
},
{
"math_id": 39,
"text": "\\Psi"
},
{
"math_id": 40,
"text": "x^\\mu"
},
{
"math_id": 41,
"text": "\\mathcal{S}"
},
{
"math_id": 42,
"text": "\\mathcal{L}"
},
{
"math_id": 43,
"text": "\\mathcal{L} \\mapsto \\mathcal{L} + \\varepsilon \\partial_\\mu \\Lambda^\\mu,"
},
{
"math_id": 44,
"text": "r = 1, 2, \\ldots, N,"
},
{
"math_id": 45,
"text": "\\varphi \\mapsto \\varphi + \\varepsilon_r \\Psi_r,"
},
{
"math_id": 46,
"text": "\\mathcal{L} \\mapsto \\mathcal{L} + \\varepsilon_r \\partial_\\mu \\Lambda^\\mu_r."
},
{
"math_id": 47,
"text": "N"
},
{
"math_id": 48,
"text": "j^\\nu_r = \\Lambda^\\nu_r - \\frac{\\partial \\mathcal{L}}{\\partial \\varphi_{,\\nu}} \\cdot \\Psi_r"
},
{
"math_id": 49,
"text": "\\nu"
},
{
"math_id": 50,
"text": "r"
},
{
"math_id": 51,
"text": "\\partial_\\nu j^\\nu = 0,"
},
{
"math_id": 52,
"text": "L \\left(\\boldsymbol\\varphi, \\partial_\\mu{\\boldsymbol\\varphi}, x^\\mu \\right)"
},
{
"math_id": 53,
"text": "x^\\mu \\mapsto x^\\mu + \\varepsilon_r \\delta^\\mu_r"
},
{
"math_id": 54,
"text": "\\delta"
},
{
"math_id": 55,
"text": "\\varphi(x^\\mu) \\mapsto \\varphi\\left(x^\\mu - \\varepsilon_r \\delta^\\mu_r\\right)"
},
{
"math_id": 56,
"text": "x^\\mu - \\varepsilon X^\\mu"
},
{
"math_id": 57,
"text": "\\Psi_r = -\\delta^\\mu_r \\partial_\\mu \\varphi."
},
{
"math_id": 58,
"text": "\\mathcal{L}\\left(x^\\mu\\right) \\mapsto \\mathcal{L}\\left(x^\\mu - \\varepsilon_r \\delta^\\mu_r\\right)"
},
{
"math_id": 59,
"text": "\\Lambda^\\mu_r = -\\delta^\\mu_r \\mathcal{L}"
},
{
"math_id": 60,
"text": "\\mu"
},
{
"math_id": 61,
"text": "T"
},
{
"math_id": 62,
"text": "\n T_\\mu{}^\\nu =\n -\\delta^\\nu_\\mu \\mathcal{L} + \\delta^\\sigma_\\mu \\partial_\\sigma \\varphi \\frac{\\partial \\mathcal{L}}{\\partial \\varphi_{,\\nu}} =\n \\left(\\frac{\\partial \\mathcal{L}}{\\partial \\varphi_{,\\nu}}\\right) \\cdot \\varphi_{,\\mu} - \\delta^\\nu_\\mu \\mathcal{L}\n"
},
{
"math_id": 63,
"text": "T_\\mu{}^\\nu{}_{,\\nu} = 0"
},
{
"math_id": 64,
"text": "\\sigma"
},
{
"math_id": 65,
"text": "\\psi \\rightarrow e^{i\\theta} \\psi\\ ,\\ \\psi^{*} \\rightarrow e^{-i\\theta} \\psi^{*}~,"
},
{
"math_id": 66,
"text": "L = \\partial_{\\nu}\\psi \\partial_{\\mu}\\psi^{*} \\eta^{\\nu \\mu} + m^2 \\psi \\psi^{*}."
},
{
"math_id": 67,
"text": "j^\\nu = i \\left( \\frac{\\partial \\psi}{\\partial x^\\mu} \\psi^{*} - \\frac{\\partial \\psi^{*}}{\\partial x^\\mu} \\psi \\right) \\eta^{\\nu \\mu}~,"
},
{
"math_id": 68,
"text": "I = \\int_{t_1}^{t_2} L [\\mathbf{q} [t], \\dot{\\mathbf{q}} [t], t] \\, dt "
},
{
"math_id": 69,
"text": "\\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} [t] = \\frac{\\partial L}{\\partial \\mathbf{q}} [t]."
},
{
"math_id": 70,
"text": "\\begin{align}\n t &\\rightarrow t' = t + \\varepsilon T \\\\\n \\mathbf{q} [t] &\\rightarrow \\mathbf{q}' [t'] = \\varphi [\\mathbf{q} [t], \\varepsilon] = \\varphi [\\mathbf{q} [t' - \\varepsilon T], \\varepsilon]\n\\end{align}"
},
{
"math_id": 71,
"text": "\n\\dot{\\mathbf{q}} [t] \\rightarrow \\dot{\\mathbf{q}}' [t'] = \\frac{d}{dt} \\varphi [\\mathbf{q} [t], \\varepsilon] = \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} [\\mathbf{q} [t' - \\varepsilon T], \\varepsilon] \\dot{\\mathbf{q}} [t' - \\varepsilon T]\n."
},
{
"math_id": 72,
"text": "\n\\begin{align}\nI' [\\varepsilon] & = \\int_{t_1 + \\varepsilon T}^{t_2 + \\varepsilon T} L [\\mathbf{q}'[t'], \\dot{\\mathbf{q}}' [t'], t'] \\, dt' \\\\[6pt]\n& = \\int_{t_1 + \\varepsilon T}^{t_2 + \\varepsilon T} L [\\varphi [\\mathbf{q} [t' - \\varepsilon T], \\varepsilon], \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} [\\mathbf{q} [t' - \\varepsilon T], \\varepsilon] \\dot{\\mathbf{q}} [t' - \\varepsilon T], t'] \\, dt'\n\\end{align}\n"
},
{
"math_id": 73,
"text": "\n\\begin{align}\n0 = \\frac{d I'}{d \\varepsilon} [0] = {} & L [\\mathbf{q} [t_2], \\dot{\\mathbf{q}} [t_2], t_2] T - L [\\mathbf{q} [t_1], \\dot{\\mathbf{q}} [t_1], t_1] T \\\\[6pt]\n& {} + \\int_{t_1}^{t_2} \\frac{\\partial L}{\\partial \\mathbf{q}} \\left( - \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T + \\frac{\\partial \\varphi}{\\partial \\varepsilon} \\right) + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\left( - \\frac{\\partial^2 \\varphi}{(\\partial \\mathbf{q})^2} {\\dot{\\mathbf{q}}}^2 T + \\frac{\\partial^2 \\varphi}{\\partial \\varepsilon \\partial \\mathbf{q}} \\dot{\\mathbf{q}} -\n\\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\ddot{\\mathbf{q}} T \\right) \\, dt.\n\\end{align}\n"
},
{
"math_id": 74,
"text": "\n\\begin{align}\n\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T \\right) \n& = \\left( \\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\right) \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\left( \\frac{d}{dt} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\right) \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\ddot{\\mathbf{q}} \\, T \\\\[6pt]\n& = \\frac{\\partial L}{\\partial \\mathbf{q}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\left( \\frac{\\partial^2 \\varphi}{(\\partial \\mathbf{q})^2} \\dot{\\mathbf{q}} \\right) \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\ddot{\\mathbf{q}} \\, T.\n\\end{align}\n"
},
{
"math_id": 75,
"text": "\n\\begin{align}\n0 = \\frac{d I'}{d \\varepsilon} [0] = {} & L [\\mathbf{q} [t_2], \\dot{\\mathbf{q}} [t_2], t_2] T - L [\\mathbf{q} [t_1], \\dot{\\mathbf{q}} [t_1], t_1] T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_2] T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_1] T \\\\[6pt]\n& {} + \\int_{t_1}^{t_2} \\frac{\\partial L}{\\partial \\mathbf{q}} \\frac{\\partial \\varphi}{\\partial \\varepsilon} + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial^2 \\varphi}{\\partial \\varepsilon \\partial \\mathbf{q}} \\dot{\\mathbf{q}} \\, dt.\n\\end{align}\n"
},
{
"math_id": 76,
"text": "\n\\frac{d}{d t} \\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\varepsilon} \\right) \n= \\left( \\frac{d}{d t} \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\right) \\frac{\\partial \\varphi}{\\partial \\varepsilon} + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial^2 \\varphi}{\\partial \\varepsilon \\partial \\mathbf{q}} \\dot{\\mathbf{q}}\n= \\frac{\\partial L}{\\partial \\mathbf{q}} \\frac{\\partial \\varphi}{\\partial \\varepsilon} + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial^2 \\varphi}{\\partial \\varepsilon \\partial \\mathbf{q}} \\dot{\\mathbf{q}}.\n"
},
{
"math_id": 77,
"text": "\n\\begin{align}\n0 = {} & L [\\mathbf{q} [t_2], \\dot{\\mathbf{q}} [t_2], t_2] T - L [\\mathbf{q} [t_1], \\dot{\\mathbf{q}} [t_1], t_1] T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_2] T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_1] T \\\\[6pt]\n& {} + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\varepsilon} [t_2] - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\varepsilon} [t_1].\n\\end{align}\n"
},
{
"math_id": 78,
"text": "\\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} - L \\right) T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\varepsilon}"
},
{
"math_id": 79,
"text": "\\frac{\\partial \\varphi}{\\partial \\mathbf{q}} = 1"
},
{
"math_id": 80,
"text": "\\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\dot{\\mathbf{q}} - L \\right) T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\varphi}{\\partial \\varepsilon}."
},
{
"math_id": 81,
"text": "\\varphi^A"
},
{
"math_id": 82,
"text": "x^\\mu \\rightarrow \\xi^\\mu = x^\\mu + \\delta x^\\mu"
},
{
"math_id": 83,
"text": "\\varphi^A \\rightarrow \\alpha^A \\left(\\xi^\\mu\\right) = \\varphi^A \\left(x^\\mu\\right) + \\delta \\varphi^A \\left(x^\\mu\\right)\\,."
},
{
"math_id": 84,
"text": "\\delta\\varphi^A"
},
{
"math_id": 85,
"text": "\\alpha^A \\left(x^\\mu\\right) = \\varphi^A \\left(x^\\mu\\right) + \\bar{\\delta} \\varphi^A \\left(x^\\mu\\right)\\,."
},
{
"math_id": 86,
"text": "\\int_{\\Omega^\\prime} L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, \\xi^\\mu \\right) d^4\\xi - \\int_{\\Omega} L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right) d^{4}x = 0"
},
{
"math_id": 87,
"text": "{\\varphi^A}_{,\\sigma} = \\frac{\\partial \\varphi^A}{\\partial x^\\sigma}\\,."
},
{
"math_id": 88,
"text": "\n \\int_\\Omega \\left\\{ \n \\left[ L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, x^\\mu \\right) - \n L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right) \\right] +\n \\frac{\\partial}{\\partial x^\\sigma} \\left[ L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right) \\delta x^\\sigma \\right]\n \\right\\} d^4 x = 0\n\\,."
},
{
"math_id": 89,
"text": "\n \\left[\n L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, x^\\mu \\right) - \n L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right)\n \\right] = \n \\frac{\\partial L}{\\partial \\varphi^A} \\bar{\\delta} \\varphi^A + \n \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\bar{\\delta} {\\varphi^A}_{,\\sigma}\n\\,."
},
{
"math_id": 90,
"text": "\n \\bar{\\delta} {\\varphi^A}_{,\\sigma} = \n \\bar{\\delta} \\frac{\\partial \\varphi^A}{\\partial x^\\sigma} = \n \\frac{\\partial}{\\partial x^\\sigma} \\left(\\bar{\\delta} \\varphi^A\\right)\n\\,."
},
{
"math_id": 91,
"text": "\n \\frac{\\partial}{\\partial x^\\sigma} \\left( \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\right) =\n \\frac{\\partial L}{\\partial\\varphi^A}\n"
},
{
"math_id": 92,
"text": "\\begin{align}\n &\\left[ L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, x^\\mu \\right) - L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right) \\right] \\\\[4pt]\n ={} &\\frac{\\partial}{\\partial x^\\sigma} \\left( \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\right) \\bar{\\delta} \\varphi^A + \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\bar{\\delta} {\\varphi^A}_{,\\sigma}\n = \\frac{\\partial}{\\partial x^\\sigma} \\left( \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\bar{\\delta} \\varphi^A \\right).\n\\end{align}"
},
{
"math_id": 93,
"text": "\n \\int_\\Omega \\frac{\\partial}{\\partial x^\\sigma} \\left\\{\n \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\bar{\\delta} \\varphi^A + \n L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right) \\delta x^\\sigma\n \\right\\} d^{4}x = 0\n\\,."
},
{
"math_id": 94,
"text": "\n \\frac{\\partial}{\\partial x^\\sigma} \\left\\{\n \\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\bar{\\delta} \\varphi^A + \n L \\left( \\varphi^A, {\\varphi^A}_{,\\nu}, x^\\mu \\right) \\delta x^\\sigma\n \\right\\} = 0\n\\,."
},
{
"math_id": 95,
"text": "\\begin{align}\n \\delta x^{\\mu} &= \\varepsilon X^\\mu \\\\\n \\delta \\varphi^A &= \\varepsilon \\Psi^A = \\bar{\\delta} \\varphi^A + \\varepsilon \\mathcal{L}_X \\varphi^A\n\\end{align}"
},
{
"math_id": 96,
"text": "\\mathcal{L}_X \\varphi^A"
},
{
"math_id": 97,
"text": "{X^\\mu}_{,\\nu} = 0 "
},
{
"math_id": 98,
"text": "\\mathcal{L}_X \\varphi^A = \\frac{\\partial \\varphi^A}{\\partial x^\\mu} X^\\mu\\,."
},
{
"math_id": 99,
"text": "\\bar{\\delta} \\varphi^A = \\varepsilon \\Psi^A - \\varepsilon \\mathcal{L}_X \\varphi^A\\,."
},
{
"math_id": 100,
"text": "\\frac{\\partial}{\\partial x^\\sigma} j^\\sigma = 0"
},
{
"math_id": 101,
"text": "\n j^\\sigma = \n \\left[\\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\mathcal{L}_X \\varphi^A - L \\, X^\\sigma\\right]\n - \\left(\\frac{\\partial L}{\\partial {\\varphi^A}_{,\\sigma}} \\right) \\Psi^A\\,.\n"
},
{
"math_id": 102,
"text": "\\mathcal{C}"
},
{
"math_id": 103,
"text": "\\mathbb{R}"
},
{
"math_id": 104,
"text": "\\varphi_1,\\ldots,\\varphi_m"
},
{
"math_id": 105,
"text": "\\mathbb{R}^{m}"
},
{
"math_id": 106,
"text": "\\mathbb{R}^{3}"
},
{
"math_id": 107,
"text": "\\mathcal{S}:\\mathcal{C}\\rightarrow \\mathbb{R},"
},
{
"math_id": 108,
"text": "\\mathbb{C}"
},
{
"math_id": 109,
"text": "\\mathcal{S}[\\varphi]"
},
{
"math_id": 110,
"text": "\\mathcal{L}(\\varphi,\\partial_\\mu\\varphi,x)"
},
{
"math_id": 111,
"text": " \\mathcal{S}[\\varphi]\\,=\\,\\int_M \\mathcal{L}[\\varphi(x),\\partial_\\mu\\varphi(x),x] \\, d^{n}x."
},
{
"math_id": 112,
"text": "\\frac{\\delta \\mathcal{S}[\\varphi]}{\\delta \\varphi(x)}\\approx 0"
},
{
"math_id": 113,
"text": "Q \\left[ \\int_N \\mathcal{L} \\, \\mathrm{d}^n x \\right] \\approx \\int_{\\partial N} f^\\mu [\\varphi(x),\\partial\\varphi,\\partial\\partial\\varphi,\\ldots] \\, ds_\\mu "
},
{
"math_id": 114,
"text": "Q[\\mathcal{L}(x)]\\approx\\partial_\\mu f^\\mu(x)"
},
{
"math_id": 115,
"text": "\\mathcal{L}(x)=\\mathcal{L}[\\varphi(x), \\partial_\\mu \\varphi(x),x]."
},
{
"math_id": 116,
"text": "\n\\begin{align}\nQ\\left[\\int_N \\mathcal{L} \\, \\mathrm{d}^nx \\right]\n& =\\int_N \\left[\\frac{\\partial\\mathcal{L}}{\\partial\\varphi} - \\partial_\\mu \\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\varphi)} \\right]Q[\\varphi] \\, \\mathrm{d}^nx + \\int_{\\partial N} \\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\varphi)}Q[\\varphi] \\, \\mathrm{d}s_\\mu \\\\\n& \\approx\\int_{\\partial N} f^\\mu \\, \\mathrm{d}s_\\mu.\n\\end{align}\n"
},
{
"math_id": 117,
"text": "\\partial_\\mu\\left[\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\varphi)}Q[\\varphi]-f^\\mu\\right]\\approx 0."
},
{
"math_id": 118,
"text": "J^\\mu"
},
{
"math_id": 119,
"text": "J^\\mu\\,=\\,\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\varphi)}Q[\\varphi]-f^\\mu,"
},
{
"math_id": 120,
"text": "\\int_{\\partial N} J^\\mu ds_{\\mu} \\approx 0."
},
{
"math_id": 121,
"text": "\\left\\langle\\int d^{4}x~\\partial \\cdot \\textbf{J} \\right\\rangle = 0"
},
{
"math_id": 122,
"text": "Q_1[\\mathcal{L}]\\approx \\partial_\\mu f_1^\\mu"
},
{
"math_id": 123,
"text": "Q_2[\\mathcal{L}]\\approx \\partial_\\mu f_2^\\mu"
},
{
"math_id": 124,
"text": "[Q_1,Q_2][\\mathcal{L}] = Q_1[Q_2[\\mathcal{L}]]-Q_2[Q_1[\\mathcal{L}]]\\approx\\partial_\\mu f_{12}^\\mu"
},
{
"math_id": 125,
"text": "j_{12}^\\mu = \\left(\\frac{\\partial}{\\partial (\\partial_\\mu\\varphi)} \\mathcal{L}\\right)(Q_1[Q_2[\\varphi]] - Q_2[Q_1[\\varphi]])-f_{12}^\\mu."
},
{
"math_id": 126,
"text": "Q[\\mathcal{L}]\\approx\\partial_\\mu f^\\mu"
},
{
"math_id": 127,
"text": "\n\\begin{align}\nq[\\varepsilon][\\mathcal{S}] & = \\int q[\\varepsilon][\\mathcal{L}] d^{n} x \\\\[6pt]\n& = \\int \\left\\{ \\left(\\frac{\\partial}{\\partial \\varphi}\\mathcal{L}\\right) \\varepsilon Q[\\varphi]+ \\left[\\frac{\\partial}{\\partial (\\partial_\\mu \\varphi)}\\mathcal{L}\\right]\\partial_\\mu(\\varepsilon Q[\\varphi]) \\right\\} d^{n} x \\\\[6pt]\n& = \\int \\left\\{ \\varepsilon Q[\\mathcal{L}] + \\partial_{\\mu}\\varepsilon \\left[\\frac{\\partial}{\\partial \\left( \\partial_\\mu \\varphi\\right)} \\mathcal{L} \\right] Q[\\varphi] \\right\\} \\, d^{n} x \\\\[6pt]\n& \\approx \\int \\varepsilon \\partial_\\mu \\left\\{f^\\mu-\\left[\\frac{\\partial}{\\partial (\\partial_\\mu\\varphi)}\\mathcal{L}\\right]Q[\\varphi]\\right\\} \\, d^{n} x\n\\end{align}\n"
},
{
"math_id": 128,
"text": "\\varepsilon"
},
{
"math_id": 129,
"text": "\n \\partial_\\mu\\left[\n f^\\mu\n - \\left[\\frac{\\partial}{\\partial (\\partial_\\mu \\varphi)} \\mathcal{L} \\right] Q[\\varphi]\n - 2\\left[\\frac{\\partial}{\\partial (\\partial_\\mu \\partial_\\nu \\varphi)} \\mathcal{L}\\right]\\partial_\\nu Q[\\varphi]\n + \\partial_\\nu\\left[\\left[\\frac{\\partial}{\\partial (\\partial_\\mu \\partial_\\nu \\varphi)}\\mathcal{L}\\right] Q[\\varphi]\\right]\n - \\,\\dotsm\n \\right] \\approx 0.\n"
},
{
"math_id": 130,
"text": "\\begin{align}\n \\mathcal{S}[x] & = \\int L\\left[x(t),\\dot{x}(t)\\right] \\, dt \\\\\n & = \\int \\left(\\frac m 2 \\sum_{i=1}^3\\dot{x}_i^2 - V(x(t))\\right) \\, dt.\n\\end{align}"
},
{
"math_id": 131,
"text": "Q[x(t)] = \\dot{x}(t)"
},
{
"math_id": 132,
"text": "Q[L] =\n \\frac{d}{dt}\\left[\\frac{m}{2}\\sum_i\\dot{x}_i^2 - V(x)\\right] =\n m \\sum_i\\dot{x}_i\\ddot{x}_i - \\sum_i\\frac{\\partial V(x)}{\\partial x_i}\\dot{x}_i\n"
},
{
"math_id": 133,
"text": "L = \\frac{m}{2} \\sum_i\\dot{x}_i^2 - V(x)."
},
{
"math_id": 134,
"text": "\\begin{align}\n j & = \\sum_{i=1}^3\\frac{\\partial L}{\\partial \\dot{x}_i}Q[x_i] - L \\\\\n & = m \\sum_i\\dot{x}_i^2 - \\left[\\frac{m}{2}\\sum_i\\dot{x}_i^2 - V(x)\\right] \\\\[3pt]\n & = \\frac{m}{2}\\sum_i\\dot{x}_i^2 + V(x).\n\\end{align}"
},
{
"math_id": 135,
"text": "dj/dt = 0"
},
{
"math_id": 136,
"text": "\\sum_{i=1}^3 \\frac{\\partial L}{\\partial \\dot{x}_i}\\dot{x_i} - L"
},
{
"math_id": 137,
"text": "\\begin{align}\n \\mathcal{S}\\left[\\vec{x}\\right]\n & = \\int \\mathcal{L}\\left[\\vec{x}(t), \\dot{\\vec{x}}(t)\\right] dt \\\\[3pt]\n & = \\int \\left[\\sum^N_{\\alpha=1} \\frac{m_\\alpha}{2}\\left(\\dot{\\vec{x}}_\\alpha\\right)^2 - \\sum_{\\alpha<\\beta} V_{\\alpha\\beta}\\left(\\vec{x}_\\beta - \\vec{x}_\\alpha\\right)\\right] dt,\n\\end{align}"
},
{
"math_id": 138,
"text": "\\vec{Q}"
},
{
"math_id": 139,
"text": "Q_i\\left[x^j_\\alpha(t)\\right] = t \\delta^j_i."
},
{
"math_id": 140,
"text": "\\begin{align}\n Q_i[\\mathcal{L}]\n & = \\sum_\\alpha m_\\alpha \\dot{x}_\\alpha^i - \\sum_{\\alpha<\\beta}t \\partial_i V_{\\alpha\\beta}\\left(\\vec{x}_\\beta - \\vec{x}_\\alpha\\right) \\\\\n & = \\sum_\\alpha m_\\alpha \\dot{x}_\\alpha^i.\n\\end{align}"
},
{
"math_id": 141,
"text": "\\frac{d}{dt}\\sum_\\alpha m_\\alpha x^i_\\alpha"
},
{
"math_id": 142,
"text": "\\vec{f} = \\sum_\\alpha m_\\alpha \\vec{x}_\\alpha."
},
{
"math_id": 143,
"text": "\\begin{align}\n \\vec{j} & = \\sum_\\alpha \\left(\\frac{\\partial}{\\partial \\dot{\\vec{x}}_\\alpha} \\mathcal{L}\\right)\\cdot\\vec{Q}\\left[\\vec{x}_\\alpha\\right] - \\vec{f} \\\\[6pt]\n & = \\sum_\\alpha \\left(m_\\alpha \\dot{\\vec{x}}_\\alpha t - m_\\alpha \\vec{x}_\\alpha\\right) \\\\[3pt]\n & = \\vec{P}t - M\\vec{x}_{CM}\n\\end{align}"
},
{
"math_id": 144,
"text": "\\vec{P}"
},
{
"math_id": 145,
"text": "\\vec{x}_{CM}"
},
{
"math_id": 146,
"text": "\\frac{d\\vec{j}}{dt} = 0 \\Rightarrow \\vec{P} - M \\dot{\\vec{x}}_{CM} = 0."
},
{
"math_id": 147,
"text": "\\begin{align}\n \\mathcal{S}[\\varphi]\n & = \\int \\mathcal{L}\\left[\\varphi (x), \\partial_\\mu \\varphi (x)\\right] d^4 x \\\\[3pt]\n & = \\int \\left(\\frac{1}{2}\\partial^\\mu \\varphi \\partial_\\mu \\varphi - \\lambda \\varphi^4\\right) d^4 x\n\\end{align}"
},
{
"math_id": 148,
"text": "Q[\\varphi(x)] = x^\\mu\\partial_\\mu \\varphi(x) + \\varphi(x). "
},
{
"math_id": 149,
"text": "Q[\\mathcal{L}] = \\partial^\\mu\\varphi\\left(\\partial_\\mu\\varphi + x^\\nu\\partial_\\mu\\partial_\\nu\\varphi + \\partial_\\mu\\varphi\\right) - 4\\lambda\\varphi^3\\left(x^\\mu\\partial_\\mu\\varphi + \\varphi\\right)."
},
{
"math_id": 150,
"text": "\\partial_\\mu\\left[\\frac{1}{2}x^\\mu\\partial^\\nu\\varphi\\partial_\\nu\\varphi - \\lambda x^\\mu \\varphi^4 \\right] = \\partial_\\mu\\left(x^\\mu\\mathcal{L}\\right)"
},
{
"math_id": 151,
"text": "f^\\mu = x^\\mu\\mathcal{L}."
},
{
"math_id": 152,
"text": "\\begin{align}\n j^\\mu & = \\left[\\frac{\\partial}{\\partial(\\partial_\\mu\\varphi)}\\mathcal{L}\\right]Q[\\varphi]-f^\\mu \\\\\n & = \\partial^\\mu\\varphi\\left(x^\\nu\\partial_\\nu\\varphi + \\varphi\\right) - x^\\mu\\left(\\frac 1 2 \\partial^\\nu\\varphi\\partial_\\nu\\varphi - \\lambda\\varphi^4\\right).\n\\end{align}"
},
{
"math_id": 153,
"text": "\\partial_\\mu j^\\mu = 0"
}
] |
https://en.wikipedia.org/wiki?curid=150159
|
150170
|
Trigonometric tables
|
Lists of values of mathematical functions
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices.
Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics, where only modest accuracy may be required and speed is often paramount.
Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called "twiddle factors") must be evaluated many times in a given transform, especially in the common case where many transforms of the same size are computed. In this case, calling generic library routines every time is unacceptably slow. One option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a regular sequence of values is required, is to use a recurrence formula to compute the trigonometric values on the fly. Significant research has been devoted to finding accurate, stable recurrence schemes in order to preserve the accuracy of the FFT (which is very sensitive to trigonometric errors).
A trigonometry table is essentially a reference chart that presents the values of sine, cosine, tangent, and other trigonometric functions for various angles. These angles are usually arranged across the top row of the table, while the different trigonometric functions are labeled in the first column on the left. To locate the value of a specific trigonometric function at a certain angle, you would find the row for the function and follow it across to the column under the desired angle.
On-demand computation.
Modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles (Kantabutra, 1996). One common method, especially on higher-end processors with floating-point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Maintaining precision while performing such interpolation is nontrivial, but methods like Gal's accurate tables, Cody and Waite range reduction, and Payne and Hanek radian reduction algorithms can be used for this purpose. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.
The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a minimax approximation algorithm.
For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral (Brent, 1976).
Trigonometric functions of angles that are rational multiples of 2π are algebraic numbers. The values for "a/b·2π" can be found by applying de Moivre's identity for "n = a" to a "bth" root of unity, which is also a root of the polynomial "xb - 1" in the complex plane. For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the degree-37 polynomial "x"37 − 1. For this case, a root-finding algorithm such as Newton's method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however.
Half-angle and angle-addition formulas.
Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer Ptolemy, who derived them in the "Almagest", a treatise on astronomy. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which "x" lies):
formula_0
formula_1
formula_2
formula_3
These were used to construct Ptolemy's table of chords, which was applied to astronomical problems.
Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine.
A quick, but inaccurate, approximation.
A quick, but inaccurate, algorithm for calculating a table of "N" approximations "s""n" for sin(2π"n"/"N") and "c""n" for cos(2π"n"/"N") is:
"s"0 = 0
"c"0 = 1
"s""n"+1 = "s""n" + "d" × "c""n"
"c""n"+1 = "c""n" − "d" × "s""n"
for "n" = 0...,"N" − 1, where "d" = 2π/"N".
This is simply the Euler method for integrating the differential equation:
formula_4
formula_5
with initial conditions "s"(0) = 0 and "c"(0) = 1, whose analytical solution is "s" = sin("t") and "c" = cos("t").
Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/"N".
For example, for "N" = 256 the maximum error in the sine values is ~0.061 ("s"202 = −1.0368 instead of −0.9757). For "N" = 1024, the maximum error in the sine values is ~0.015 ("s"803 = −0.99321 instead of −0.97832), about 4 times smaller. If the sine and cosine values obtained were to be plotted, this algorithm would draw a logarithmic spiral rather than a circle.
A better, but still imperfect, recurrence formula.
A simple recurrence formula to generate trigonometric tables is based on Euler's formula and the relation:
formula_6
This leads to the following recurrence to compute trigonometric values "s""n" and "c""n" as above:
"c"0 = 1
"s"0 = 0
"c""n"+1 = "w""r" "c""n" − "w""i" "s""n"
"s""n"+1 = "w""i" "c""n" + "w""r" "s""n"
for "n" = 0, ..., "N" − 1, where "w""r" = cos(2π/"N") and "w""i" = sin(2π/"N"). These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing Newton's method in the complex plane to solve for the primitive root of "z""N" − 1).
This method would produce an "exact" table in exact arithmetic, but has errors in finite-precision floating-point arithmetic. In fact, the errors grow as O(ε "N") (in both the worst and average cases), where ε is the floating-point precision.
A significant improvement is to use the following modification to the above, a trick (due to Singleton) often used to generate trigonometric values for FFT implementations:
"c"0 = 1
"s"0 = 0
"c""n"+1 = "c""n" − (α "c""n" + β "s""n")
"s""n"+1 = "s""n" + (β "c""n" − α "s""n")
where α = 2 sin2(π/"N") and β = sin(2π/"N"). The errors of this method are much smaller, O(ε √"N") on average and O(ε "N") in the worst case, but this is still large enough to substantially degrade the accuracy of FFTs of large sizes.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\cos\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\tfrac{1}{2}(1 + \\cos x)}"
},
{
"math_id": 1,
"text": "\\sin\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\tfrac{1}{2}(1 - \\cos x)}"
},
{
"math_id": 2,
"text": "\\sin(x \\pm y) = \\sin(x) \\cos(y) \\pm \\cos(x) \\sin(y)\\,"
},
{
"math_id": 3,
"text": "\\cos(x \\pm y) = \\cos(x) \\cos(y) \\mp \\sin(x) \\sin(y)\\,"
},
{
"math_id": 4,
"text": "ds/dt = c"
},
{
"math_id": 5,
"text": "dc/dt = -s"
},
{
"math_id": 6,
"text": "e^{i(\\theta + \\Delta)} = e^{i\\theta} \\times e^{i\\Delta\\theta}"
}
] |
https://en.wikipedia.org/wiki?curid=150170
|
15018472
|
Szegő polynomial
|
In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product
formula_0
where dμ is a given positive measure on [−π, π]. Writing formula_1 for the polynomials, they obey a recurrence relation
formula_2
where formula_3 is a parameter, called the "reflection coefficient" or the "Szegő parameter".
|
[
{
"math_id": 0,
"text": "\\langle f|g\\rangle = \\int_{-\\pi}^{\\pi}f(e^{i\\theta})\\overline{g(e^{i\\theta})}\\,d\\mu"
},
{
"math_id": 1,
"text": "\\phi_n(z)"
},
{
"math_id": 2,
"text": "\\phi_{n+1}(z)=z\\phi_n(z) + \\rho_{n+1}\\phi_n^*(z)"
},
{
"math_id": 3,
"text": "\\rho_{n+1}"
}
] |
https://en.wikipedia.org/wiki?curid=15018472
|
1501948
|
Crystal momentum
|
Quantum-mechanical vector property in solid-state physics
In solid-state physics, crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors formula_0 of this lattice, according to
formula_1
(where formula_2 is the reduced Planck constant).
Frequently, crystal momentum is conserved like mechanical momentum, making it useful to physicists and materials scientists as an analytical tool.
Lattice symmetry origins.
A common method of modeling crystal structure and behavior is to view electrons as quantum mechanical particles traveling through a fixed infinite periodic potential formula_3 such that
formula_4
where formula_5 is an arbitrary lattice vector. Such a model is sensible because crystal ions that form the lattice structure are typically on the order of tens of thousands of times more massive than electrons,
making it safe to replace them with a fixed potential structure, and the macroscopic dimensions of a crystal are typically far greater than a single lattice spacing, making edge effects negligible. A consequence of this potential energy function is that it is possible to shift the initial position of an electron by any lattice vector formula_5 without changing any aspect of the problem, thereby defining a discrete symmetry. Technically, an infinite periodic potential implies that the lattice translation operator formula_6 commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.
These conditions imply Bloch's theorem, which states
formula_7,
or that an electron in a lattice, which can be modeled as a single particle wave function formula_8, finds its stationary state solutions in the form of a plane wave multiplied by a periodic function formula_9. The theorem arises as a direct consequence of the aforementioned fact that the lattice symmetry translation operator commutes with the system's Hamiltonian.
One of the notable aspects of Bloch's theorem is that it shows directly that steady state solutions may be identified with a wave vector formula_0, meaning that this quantum number remains a constant of motion. Crystal momentum is then conventionally defined by multiplying this wave vector by the Planck constant:
formula_10
While this is in fact identical to the definition one might give for regular momentum (for example, by treating the effects of the translation operator by the effects of a particle in free space),
there are important theoretical differences. For example, while regular momentum is completely conserved, crystal momentum is only conserved to within a lattice vector. For example, an electron can be described not only by the wave vector formula_0, but also with any other wave vector formula_11such that
formula_12
where formula_13 is an arbitrary reciprocal lattice vector. This is a consequence of the fact that the lattice symmetry is discrete as opposed to continuous, and thus its associated conservation law cannot be derived using Noether's theorem.
Physical significance.
The phase modulation of the Bloch state formula_14 is the same as that of a free particle with momentum formula_15, i.e. formula_16 gives the state's periodicity, which is not the same as that of the lattice. This modulation contributes to the kinetic energy of the particle (whereas the modulation is entirely responsible for the kinetic energy of a free particle).
In regions where the band is approximately parabolic the crystal momentum is equal to the momentum of a free particle with momentum formula_15 if we assign the particle an effective mass that's related to the curvature of the parabola.
Relation to velocity.
Crystal momentum corresponds to the physically measurable concept of velocity according to
formula_17
This is the same formula as the group velocity of a wave. More specifically, due to the Heisenberg uncertainty principle, an electron in a crystal cannot have both an exactly-defined k and an exact position in the crystal. It can, however, form a wave packet centered on momentum k (with slight uncertainty), and centered on a certain position (with slight uncertainty). The center position of this wave packet changes as the wave propagates, moving through the crystal at the velocity v given by the formula above. In a real crystal, an electron moves in this way—traveling in a certain direction at a certain speed—for only a short period of time, before colliding with an imperfection in the crystal that causes it to move in a different, random direction. These collisions, called "electron scattering", are most commonly caused by crystallographic defects, the crystal surface, and random thermal vibrations of the atoms in the crystal (phonons).
Response to electric and magnetic fields.
Crystal momentum also plays a seminal role in the semiclassical model of electron dynamics, where it follows from the acceleration theorem that it obeys the equations of motion (in cgs units):
formula_18
formula_19
Here perhaps the analogy between crystal momentum and true momentum is at its most powerful, for these are precisely the equations that a free space electron obeys in the absence of any crystal structure. Crystal momentum also earns its chance to shine in these types of calculations, for, in order to calculate an electron's trajectory of motion using the above equations, one need only consider external fields, while attempting the calculation from a set of equations of motion based on true momentum would require taking into account individual Coulomb and Lorentz forces of every single lattice ion in addition to the external field.
Applications.
Angle-resolved photo-emission spectroscopy (ARPES).
In angle-resolved photo-emission spectroscopy (ARPES), irradiating light on a crystal sample results in the ejection of an electron away from the crystal. Throughout the course of the interaction, one is allowed to conflate the two concepts of crystal and true momentum and thereby gain direct knowledge of a crystal's band structure. That is to say, an electron's crystal momentum inside the crystal becomes its true momentum after it leaves, and the true momentum may be subsequently inferred from the equation
formula_20
by measuring the angle and kinetic energy at which the electron exits the crystal, where formula_21 is a single electron's mass. Because crystal symmetry in the direction normal to the crystal surface is lost at the crystal boundary, crystal momentum in this direction is not conserved. Consequently, the only directions in which useful ARPES data can be gleaned are directions parallel to the crystal surface.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbf{k}"
},
{
"math_id": 1,
"text": "{\\mathbf{p}}_{\\text{crystal}} \\equiv \\hbar {\\mathbf{k}}"
},
{
"math_id": 2,
"text": "\\hbar"
},
{
"math_id": 3,
"text": "V(x)"
},
{
"math_id": 4,
"text": "V({\\mathbf{x}}+{\\mathbf{a}})=V({\\mathbf{x}}),"
},
{
"math_id": 5,
"text": "\\mathbf{a}"
},
{
"math_id": 6,
"text": "T(a)"
},
{
"math_id": 7,
"text": "\\psi_n({\\mathbf{x}})=e^{i{\\mathbf{k} {\\mathbf{\\cdot x}}}}u_{n{\\mathbf{k}}}({\\mathbf{x}}), \\qquad \nu_{n{\\mathbf{k}}}({\\mathbf{x}}+{\\mathbf{a}})=u_{n{\\mathbf{k}}}({\\mathbf{x}})"
},
{
"math_id": 8,
"text": "\\psi(\\mathbf{x})"
},
{
"math_id": 9,
"text": "u(\\mathbf{x})"
},
{
"math_id": 10,
"text": "{\\mathbf{p}}_{\\text{crystal}} = \\hbar {\\mathbf{k}}."
},
{
"math_id": 11,
"text": "\\mathbf{k}'"
},
{
"math_id": 12,
"text": "\\mathbf{k'} = \\mathbf{k} + \\mathbf{K},"
},
{
"math_id": 13,
"text": "\\mathbf{K}"
},
{
"math_id": 14,
"text": "\\psi_n({\\mathbf{x}})=e^{i{\\mathbf{k} {\\mathbf{\\cdot x}}}}u_{n{\\mathbf{k}}}({\\mathbf{x}})"
},
{
"math_id": 15,
"text": "\\hbar k "
},
{
"math_id": 16,
"text": " k "
},
{
"math_id": 17,
"text": "{\\mathbf{v}}_n({\\mathbf{k}}) = \\frac{1}{\\hbar} \\nabla_{\\mathbf{k}} E_n({\\mathbf{k}})."
},
{
"math_id": 18,
"text": "{\\mathbf{v}}_n({\\mathbf{k}}) = \\frac{1}{\\hbar} \\nabla_{\\mathbf{k}} E_n({\\mathbf{k}}), "
},
{
"math_id": 19,
"text": "{\\mathbf{\\dot{p}}}_{\\text{crystal}} = -e \\left( {\\mathbf{E}} -\\frac{1}{c} {\\mathbf{v}} \\times {\\mathbf{H}} \\right)"
},
{
"math_id": 20,
"text": "{\\mathbf{p_{\\parallel}}} = \\sqrt{2 m E_{\\text{kin}}}\\sin \\theta"
},
{
"math_id": 21,
"text": "m"
}
] |
https://en.wikipedia.org/wiki?curid=1501948
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.