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12991706 | Folded spectrum method | Mathematical method for solving large eigenvalue problems
In mathematics, the folded spectrum method (FSM) is an iterative method for solving large eigenvalue problems.
Here you always find a vector with an eigenvalue close to a search-value formula_0. This means you can get a vector formula_1 in the middle of the spectrum without solving the matrix.
formula_2, with formula_3 and formula_4 the Identity matrix.
In contrast to the Conjugate gradient method, here the gradient calculates by twice multiplying matrix formula_5 | [
{
"math_id": 0,
"text": "\\varepsilon"
},
{
"math_id": 1,
"text": "\\Psi"
},
{
"math_id": 2,
"text": "\\Psi_{i+1}= \\Psi_i-\\alpha( H- \\varepsilon \\mathbf{1} )^2 \\Psi_i"
},
{
"math_id": 3,
"text": "0<\\alpha^{\\,}<1"
},
{
"math_id": 4,
"text": "\\mathbf{1}"
},
{
"math_id": 5,
"text": "H:\\;G\\sim H\\rightarrow G\\sim H^2."
}
]
| https://en.wikipedia.org/wiki?curid=12991706 |
12993112 | Magic angle (EELS) | The magic angle is a particular value of the collection angle of an electron microscope at which the measured energy-loss spectrum "magically" becomes independent of the tilt angle of the sample with respect to the beam direction. The magic angle is not uniquely defined for isotropic samples, but the definition is unique in the (typical) case of small angle scattering on materials with a "c-axis", such as graphite.
The "magic" angle depends on both the incoming electron energy (which is typically fixed) and the energy loss suffered by the electron. The ratio of the magic angle formula_0 to the characteristic angle formula_1 is roughly independent of the energy loss and roughly independent of the particular type of sample considered.
Mathematical definition.
For the case of a relativistic incident electron, the "magic" angle is defined by the equality of two different functions
(denoted below by formula_2 and formula_3) of the collection angle formula_4:
formula_5
and
formula_6
where formula_7 is the speed of the incoming electron divided by the speed of light (N.B., the symbol formula_7 is also often used in the older literature to denote the collection angle instead of formula_4).
Of course, the above integrals may easily be evaluated in terms of elementary functions, but they are presented as above because in the above form it is easier to see that the former integral is due to momentum transfers which are perpendicular to the beam direction, whereas the latter is due to momentum transfers parallel to the beam direction.
Using the above definition, it is then found that
formula_8 | [
{
"math_id": 0,
"text": "\\theta_M"
},
{
"math_id": 1,
"text": "\\theta_E"
},
{
"math_id": 2,
"text": "A"
},
{
"math_id": 3,
"text": "C"
},
{
"math_id": 4,
"text": "\\alpha"
},
{
"math_id": 5,
"text": "\nA(\\alpha)=\\frac{1}{2}\\int_0^{\\alpha^2}dx\\frac{x}{(x+\\theta_E^2{(1-\\beta^2))}^2}\n"
},
{
"math_id": 6,
"text": "\nC(\\alpha)=\\theta_E^2{(1-\\beta^2)}^2\\int_0^{\\alpha^2}dx\\frac{1}{{(x+\\theta_E^2(1-\\beta^2))}^2}\n"
},
{
"math_id": 7,
"text": "\\beta"
},
{
"math_id": 8,
"text": "\n\\theta_M\\approx 2\\theta_E\n"
}
]
| https://en.wikipedia.org/wiki?curid=12993112 |
12994056 | Evolutionary invasion analysis | Mathematical modeling of phenotypic evolution
Evolutionary invasion analysis, also known as adaptive dynamics, is a set of mathematical modeling techniques that use differential equations to study the long-term evolution of traits in asexually and sexually reproducing populations. It rests on the following three assumptions about mutation and population dynamics:
Evolutionary invasion analysis makes it possible to identify conditions on model parameters for which the mutant population dies out, replaces the resident population, and/or coexists with the resident population. Long-term coexistence of the two phenotypes is known as "evolutionary branching". When branching occurs, the mutant establishes itself as a second resident in the environment.
Central to evolutionary invasion analysis is the mutant's invasion fitness. This is a mathematical expression for the long-term exponential growth rate of the mutant subpopulation when it is introduced into the resident population in small numbers. If the invasion fitness is positive (in continuous time), the mutant population can grow in the environment set by the resident phenotype. If the invasion fitness is negative, the mutant population swiftly goes extinct.
Introduction and background.
The basic principle of evolution via natural selection was outlined by Charles Darwin in his 1859 book, "On the Origin of Species". Though controversial at the time, the central ideas remain largely unchanged to this date, even though much more is now known about the biological basis of inheritance. Darwin expressed his arguments verbally, but many attempts have since then been made to formalise the theory of evolution. The best known are population genetics which models inheritance at the expense of ecological detail, quantitative genetics which incorporates quantitative traits influenced by genes at many loci, and evolutionary game theory which ignores genetic detail but incorporates a high degree of ecological realism, in particular that the success of any given strategy depends on the frequency at which strategies are played in the population, a concept known as frequency dependence.
Adaptive dynamics is a set of techniques developed during the 1990s for understanding the long-term consequences of small mutations in the traits expressing the phenotype. They link population dynamics to evolutionary dynamics and incorporate and generalise the fundamental idea of frequency-dependent selection from game theory.
Fundamental ideas.
Two fundamental ideas of adaptive dynamics are that the resident population is in a dynamical equilibrium when new mutants appear, and that the eventual fate of such mutants can be inferred from their initial growth rate when rare in the environment consisting of the resident. This rate is known as the invasion exponent when measured as the initial exponential growth rate of mutants, and as the basic reproductive number when it measures the expected total number of offspring that a mutant individual produces in a lifetime. It is sometimes called the invasion fitness of mutants.
To make use of these ideas, a mathematical model must explicitly incorporate the traits undergoing evolutionary change. The model should describe both the environment and the population dynamics given the environment, even if the variable part of the environment consists only of the demography of the current population. The invasion exponent can then be determined. This can be difficult, but once determined, the adaptive dynamics techniques can be applied independent of the model structure.
Monomorphic evolution.
A population consisting of individuals with the same trait is called monomorphic. If not explicitly stated otherwise, the trait is assumed to be a real number, and r and m are the trait value of the monomorphic resident population and that of an invading mutant, respectively.
Invasion exponent and selection gradient.
The invasion exponent formula_0 is defined as the expected growth rate of an initially rare mutant in the environment set by the resident (r), which means the frequency of each phenotype (trait value) whenever this suffices to infer all other aspects of the equilibrium environment, such as the demographic composition and the
availability of resources. For each r, the invasion exponent can be thought of as the fitness landscape experienced by an initially rare mutant. The landscape changes with each successful invasion, as is the case in evolutionary game theory, but in contrast with the classical view of evolution as an optimisation process towards ever higher fitness.
We will always assume that the resident is at its demographic attractor, and as a consequence formula_1 for all r, as otherwise the population would grow indefinitely.
The selection gradient is defined as the slope of the invasion exponent at formula_2, formula_3. If the sign of the selection gradient is positive (negative) mutants with slightly higher (lower) trait values may successfully invade. This follows from the linear approximation
formula_4
which holds whenever formula_5.
Pairwise-invasibility plots.
The invasion exponent represents the fitness landscape as experienced by a rare mutant. In a large (infinite) population only mutants with trait values formula_6 for which formula_0 is positive are able to
successfully invade. The generic outcome of an invasion is that the mutant replaces the resident, and the fitness landscape as experienced by a rare mutant changes. To determine the outcome of the resulting series of invasions pairwise-invasibility plots (PIPs) are often used. These show for each resident trait value formula_7 all mutant trait values formula_6 for which formula_0 is positive. Note that formula_0 is zero at the diagonal formula_2. In PIPs the fitness landscapes as experienced by a rare mutant correspond to the vertical lines where the resident trait value formula_7 is constant.
Evolutionarily singular strategies.
The selection gradient formula_3 determines the direction of evolutionary change. If it is positive (negative) a mutant with a slightly higher (lower) trait-value will generically invade and replace the resident. But what will happen if formula_3 vanishes? Seemingly evolution should come to a halt at such a point. While this is a possible outcome, the general situation is more complex. Traits or strategies formula_8 for which formula_9, are known as evolutionarily singular strategies. Near such points the fitness landscape as experienced by a rare mutant is locally `flat'. There are three qualitatively different ways in which this can occur. First, a degenerate case similar to the saddle point of a qubic function where finite evolutionary steps would lead past the local 'flatness'. Second, a fitness maximum which is known as an evolutionarily stable strategy (ESS) and which, once established, cannot be invaded by nearby
mutants. Third, a fitness minimum where disruptive selection will occur and the population branch into two morphs. This process is known as evolutionary branching. In a pairwise invasibility plot the singular strategies are found where the boundary of the region of positive invasion fitness intersects the diagonal.
Singular strategies can be located and classified once the selection gradient is known. To locate singular strategies, it is sufficient to find the points for which the selection gradient vanishes, i.e. to find formula_8 such that formula_10. These can be classified then using the second derivative test from basic calculus. If the second derivative evaluated at formula_8 is negative (positive) the strategy represents a local fitness maximum (minimum). Hence, for an evolutionarily stable strategy formula_8 we have
formula_11
If this does not hold the strategy is evolutionarily unstable and, provided that it is also convergence stable, evolutionary branching will eventually occur. For a singular strategy formula_8 to be convergence stable monomorphic populations with slightly lower or slightly higher trait values must be invadable by mutants with trait values closer to formula_8. That this can happen the selection gradient formula_3 in a neighbourhood of formula_8 must be positive for formula_12 and negative for formula_13. This means that the slope of formula_3 as a function of formula_7
at formula_8 is negative, or equivalently
formula_14
The criterion for convergence stability given above can also be expressed using second derivatives of the invasion exponent, and the classification can be refined to span more than the simple cases considered here.
Polymorphic evolution.
The normal outcome of a successful invasion is that the mutant replaces the resident. However, other outcomes are also possible; in particular both the resident and the mutant may persist, and the population then becomes dimorphic. Assuming that a trait persists in the population if and only if its expected growth-rate when rare is positive, the condition for coexistence among two traits formula_15 and formula_16 is
formula_17
and
formula_18
where formula_15 and formula_16 are often referred to as morphs. Such a pair is a protected dimorphism. The set of all protected dimorphisms is known as the region of coexistence. Graphically, the region consists of the overlapping parts when a pair-wise invasibility plot is mirrored over the diagonal
Invasion exponent and selection gradients in polymorphic populations.
The invasion exponent is generalised to dimorphic populations straightforwardly, as the expected growth rate formula_19 of a rare mutant in the environment set by the two morphs formula_15 and
formula_16. The slope of the local fitness landscape for a mutant close to formula_15 or formula_16 is now given by the selection gradients
formula_20
and
formula_21
In practise, it is often difficult to determine the dimorphic
selection gradient and invasion exponent analytically, and one often
has to resort to numerical computations.
Evolutionary branching.
The emergence of protected dimorphism near singular points during the course of evolution is not unusual, but its significance depends on whether selection is stabilising or disruptive. In the latter case, the traits of the two morphs will diverge in a process often referred to as evolutionary branching. Geritz 1998 presents a compelling
argument that disruptive selection only occurs near fitness minima. To understand this heuristically, consider a dimorphic population formula_15 and formula_16 near a singular point. By continuity
formula_22
and, since
formula_23
the fitness landscape for the dimorphic population must be a perturbation of that for a monomorphic resident near the singular strategy.
Trait evolution plots.
Evolution after branching is illustrated using trait evolution plots. These show the region of coexistence, the direction of evolutionary change and whether points where the selection gradient vanishes are fitness maxima or minima. Evolution may well lead the dimorphic population outside the region of coexistence, in which case one morph is extinct and the population once again becomes monomorphic.
Other uses.
Adaptive dynamics effectively combines game theory and population dynamics. As such, it can be very useful in investigating how evolution affects the dynamics of populations. One interesting finding to come out of this is that individual-level adaptation can sometimes result in the extinction of the whole population/species, a phenomenon known as evolutionary suicide.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "S_r(m)"
},
{
"math_id": 1,
"text": "S_r(r) = 0"
},
{
"math_id": 2,
"text": "m=r"
},
{
"math_id": 3,
"text": "S_r'(r)"
},
{
"math_id": 4,
"text": "S_r(m) \\approx S_r(r)+ S_r'(r) (m - r)"
},
{
"math_id": 5,
"text": "m \\approx r"
},
{
"math_id": 6,
"text": "m"
},
{
"math_id": 7,
"text": "r"
},
{
"math_id": 8,
"text": "r^*"
},
{
"math_id": 9,
"text": "S_{r^*}'(r^*)=0"
},
{
"math_id": 10,
"text": "S'_{r^*}(r^*) = 0"
},
{
"math_id": 11,
"text": "S_{r^*}''(r^*) < 0"
},
{
"math_id": 12,
"text": "r < r^*"
},
{
"math_id": 13,
"text": "r > r^*"
},
{
"math_id": 14,
"text": "\\frac{d}{dr} S_r'(r)\\Big| _{r=r^*} < 0."
},
{
"math_id": 15,
"text": "r_1"
},
{
"math_id": 16,
"text": "r_2"
},
{
"math_id": 17,
"text": "S_{r_1} (r_2) > 0 "
},
{
"math_id": 18,
"text": "S_{r_2} (r_1) > 0,"
},
{
"math_id": 19,
"text": "S_{r_1, r_2}(m)"
},
{
"math_id": 20,
"text": "S_{r_1, r_2}'(r_1)"
},
{
"math_id": 21,
"text": "S_{r_1, r_2}'(r_2)"
},
{
"math_id": 22,
"text": "S_r(m) \\approx S_{r_1,r_2}(m)"
},
{
"math_id": 23,
"text": "S_{r_1, r_2}(r_1) = S_{r_1, r_2}(r_2) = 0,"
}
]
| https://en.wikipedia.org/wiki?curid=12994056 |
1299817 | Gas mark | Temperature scale used on ovens
The gas mark is a temperature scale used on gas ovens and cookers in the United Kingdom, Ireland and some Commonwealth of Nations countries.
History.
The draft 2003 edition of the "Oxford English Dictionary" lists the earliest known usage of the concept as being in L. Chatterton's book "Modern Cookery" published in 1943: "Afternoon tea scones… Time: 20 minutes. Temperature: Gas, Regulo Mark 7". "Regulo" was a type of gas regulator used by a manufacturer of cookers; however, the scale has now become universal, and the word "Regulo" is rarely used.
The term "gas mark" was a subject of the joint BBC/OED production "Balderdash and Piffle", in May 2005. The earliest printed evidence of use of "gas mark" (with no other terms between the two words) appears to date from 1958. However, the manufacturers of the "New World" gas ranges in the mid-1930s gave away recipe books for use with their cooker, and the "Regulo" was the gas regulator. The book has no reference to degrees. All dishes to be cooked are noted to be at "Regulo Mark X".
Equivalents in Fahrenheit and Celsius.
Gas mark 1 is 275 degrees Fahrenheit (135 degrees Celsius).
Oven temperatures increase by 25 °F (13.9 °C) each time the gas mark increases by 1. Below Gas Mark 1 the scale markings halve at each step, each representing a decrease of 25 °F.
For temperatures above 135 °C (gas mark 1) to convert gas mark to degrees Celsius (formula_0) multiply the gas mark number (formula_1) by 14, then add 121:
formula_2
For the reverse conversion:
formula_3
These do not work for formula_1 less than 1. For temperatures below 135 °C (gas mark 1), to convert gas mark to degrees Celsius apply the following conversion:
formula_4
For the reverse:
formula_5
It is usual to round the results of such calculations to a round number of degrees Celsius.
Note that tables of temperature equivalents for kitchen use usually offer Celsius values rounded to the nearest 10 degrees, with steps of either 10 or 20 degrees between Gas Marks.
Other cooking temperature scales.
French ovens and recipes use a scale based on the Celsius scale:"" (abbreviated "Th"), where Thermostat 1 equals 30 °C for conventional ovens, increasing by 30 °C for each whole number along the scale.
In Germany, "" (the German word for "step") is used for gas cooking temperatures. Gas ovens are commonly marked in steps from 1 to 8, corresponding to:
Other ovens may be marked on a scale of 1–7, where Stufe <templatestyles src="Fraction/styles.css" />1⁄2 is about 125 °C in a conventional oven, Stufe 1 is about 150 °C, increasing by 25 °C for each subsequent step, up to Stufe 7 at 300 °C.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "C"
},
{
"math_id": 1,
"text": "G"
},
{
"math_id": 2,
"text": "\\left ( G \\times 14 \\right ) + 121 = C"
},
{
"math_id": 3,
"text": "G = \\frac { \\left ( C - 121 \\right ) }{14}"
},
{
"math_id": 4,
"text": " C = \\frac {243 - (25 \\times \\log_{2} (G^{-1}))} {1.8}"
},
{
"math_id": 5,
"text": " G = 2^{(1.8C - 243) / 25} "
}
]
| https://en.wikipedia.org/wiki?curid=1299817 |
1299933 | Language identification in the limit | Language identification in the limit is a formal model for inductive inference of formal languages, mainly by computers (see machine learning and induction of regular languages). It was introduced by E. Mark Gold in a technical report and a journal article with the same title.
In this model, a "teacher" provides to a "learner" some "presentation" (i.e. a sequence of strings) of some formal language. The learning is seen as an infinite process. Each time the learner reads an element of the presentation, it should provide a "representation" (e.g. a formal grammar) for the language.
Gold defines that a learner can "identify in the limit" a class of languages if, given any presentation of any language in the class, the learner will produce only a finite number of wrong representations, and then stick with the correct representation. However, the learner need not be able to announce its correctness; and the teacher might present a counterexample to any representation arbitrarily long after.
Gold defined two types of presentations:
Learnability.
This model is an early attempt to formally capture the notion of learnability.
Gold's journal article introduces for contrast the stronger models
A weaker formal model of learnability is the "Probably approximately correct learning (PAC)" model, introduced by Leslie Valiant in 1984.
Examples.
It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about.
Gold's theorem.
More formally,
Notes:
<templatestyles src="Math_theorem/styles.css" />
Gold's theorem (1967) (Theorem I.8 of (Gold, 1967)) — If a language family formula_7 contains formula_8, such that
formula_9
and formula_10, then it is not learnable.
<templatestyles src="Math_proof/styles.css" />Proof
Suppose formula_2 is a learner that can learn formula_11, then we show it cannot learn formula_12, by constructing an environment for formula_12 that "tricks" formula_2.
First, construct environments formula_13 for languages formula_11.
Next, construct environment formula_1 for formula_12 inductively as follows:
By construction, the resulting environment formula_1 contains the entirety of formula_13, thus it contains formula_21, so it is an environment for formula_12. Since the learner always switches to formula_22 for some finite formula_23, it never converges to formula_12.
Gold's theorem is easily bypassed if "negative examples" are allowed. In particular, the language family formula_24 can be learned by a learner that always guesses formula_12 until it receives the first negative example formula_25, where formula_26, at which point it always guesses formula_22.
Learnability characterization.
Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper.
If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates "tell-tales" for each language in the class (Condition 1). It is not hard to see that if an ideal learner (i.e., an arbitrary function) is allowed, then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2).
Language classes learnable in the limit.
The table shows which language classes are identifiable in the limit in which learning model. On the right-hand side, each language class is a superclass of all lower classes. Each learning model (i.e. type of presentation) can identify in the limit all classes below it. In particular, the class of finite languages is identifiable in the limit by text presentation (cf. Example 2 above), while the class of regular languages is not.
"Pattern Languages", introduced by Dana Angluin in another 1980 paper, are also identifiable by normal text presentation; they are omitted in the table, since they are above the singleton and below the primitive recursive language class, but incomparable to the classes in between.
Sufficient conditions for learnability.
Condition 1 in Angluin's paper is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class. See also "Induction of regular languages" for learnable subclasses of regular languages.
Finite thickness.
A class of languages has finite thickness if every non-empty set of strings is contained in at most finitely many languages of the class. This is exactly Condition 3 in Angluin's paper. Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit.
A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness.
Finite elasticity.
A class of languages is said to have finite elasticity if for every infinite sequence of strings formula_27 and every infinite sequence of languages in the class formula_11, there exists a finite number n such that formula_28 implies formula_22 is inconsistent with formula_29.
It is shown that a class of recursively enumerable languages is learnable in the limit if it has finite elasticity.
Mind change bound.
A bound over the number of hypothesis changes that occur before convergence.
Other concepts.
Infinite cross property.
A language L has infinite cross property within a class of languages formula_30 if there is an infinite sequence formula_31 of distinct languages in formula_30 and a sequence of finite subset formula_32 such that:
Note that L is not necessarily a member of the class of language.
It is not hard to see that if there is a language with infinite cross property within a class of languages, then that class of languages has infinite elasticity.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "L"
},
{
"math_id": 1,
"text": "E"
},
{
"math_id": 2,
"text": "f"
},
{
"math_id": 3,
"text": "a_1, a_2..., a_n"
},
{
"math_id": 4,
"text": "f(a_1, ..., a_n)"
},
{
"math_id": 5,
"text": "a_1, ..., a_n"
},
{
"math_id": 6,
"text": "E = (a_1, a_2, ...)"
},
{
"math_id": 7,
"text": "C"
},
{
"math_id": 8,
"text": "L_1, L_2, ..., L_\\infty"
},
{
"math_id": 9,
"text": "L_1 \\subsetneq L_2 \\subsetneq \\cdots"
},
{
"math_id": 10,
"text": "L_\\infty = \\cup_{n=1}^\\infty L_n"
},
{
"math_id": 11,
"text": "L_1, L_2, ..."
},
{
"math_id": 12,
"text": "L_\\infty"
},
{
"math_id": 13,
"text": "E_1, E_2, ..."
},
{
"math_id": 14,
"text": "E_1"
},
{
"math_id": 15,
"text": "L_1"
},
{
"math_id": 16,
"text": "E_2"
},
{
"math_id": 17,
"text": "L_1 \\subset L_2"
},
{
"math_id": 18,
"text": "L_2"
},
{
"math_id": 19,
"text": "E_1, E_2"
},
{
"math_id": 20,
"text": "E_3"
},
{
"math_id": 21,
"text": "\\cup_n E_n = \\cup_n L_n = L_\\infty"
},
{
"math_id": 22,
"text": "L_n"
},
{
"math_id": 23,
"text": "n"
},
{
"math_id": 24,
"text": "\\{L_1,L_2, ..., L_\\infty\\}"
},
{
"math_id": 25,
"text": "\\neg a_n"
},
{
"math_id": 26,
"text": "a_n\\in L_{n+1} \\setminus L_{n}"
},
{
"math_id": 27,
"text": "s_0, s_1, ..."
},
{
"math_id": 28,
"text": "s_n\\not\\in L_n"
},
{
"math_id": 29,
"text": "\\{s_1,...,s_{n-1}\\}"
},
{
"math_id": 30,
"text": "\\mathcal{L}"
},
{
"math_id": 31,
"text": "L_i"
},
{
"math_id": 32,
"text": "T_i"
},
{
"math_id": 33,
"text": "T_1 \\sub T_2\\sub ..."
},
{
"math_id": 34,
"text": "T_i \\in L_i"
},
{
"math_id": 35,
"text": "T_{i+1}\\not\\in L_i"
},
{
"math_id": 36,
"text": "\\lim_{n=\\infty}T_i=L"
}
]
| https://en.wikipedia.org/wiki?curid=1299933 |
13000492 | Moment distribution method | Structural analysis technique for statically indeterminate structures
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
Introduction.
In the moment distribution method, every joint of the structure to be analysed is fixed so as to develop the "fixed-end moments". Then each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved. The moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration.
The moment distribution method falls into the category of displacement method of structural analysis.
Implementation.
In order to apply the moment distribution method to analyse a structure, the following things must be considered.
Fixed end moments.
Fixed end moments are the moments produced at member ends by external loads.Spanwise calculation is carried out assuming each support to be fixed and implementing formulas as per the nature of load ,i.e. point load ( mid span or unequal) ,udl,uvl or couple.
Bending stiffness.
The bending stiffness (EI/L) of a member is represented as the flexural rigidity of the member (product of the modulus of elasticity (E) and the second moment of area (I)) divided by the length (L) of the member. What is needed in the moment distribution method is not the specific values but the ratios of bending stiffnesses between all members.
Distribution factors.
When a joint is being released and begins to rotate under the unbalanced moment, resisting forces develop at each member framed together at the joint. Although the total resistance is equal to the unbalanced moment, the magnitudes of resisting forces developed at each member differ by the members' bending stiffness. Distribution factors can be defined as the proportions of the unbalanced moments carried by each of the members. In mathematical terms, the distribution factor of member formula_0 framed at joint formula_1 is given as:
formula_2
where n is the number of members framed at the joint.
Carryover factors.
When a joint is released, balancing moment occurs to counterbalance the unbalanced moment. The balancing moment is initially the same as the fixed-end moment. This balancing moment is then carried over to the member's other end. The ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor.
Determination of carryover factors.
Let one end (end A) of a fixed beam be released and applied a moment formula_3 while the other end (end B) remains fixed. This will cause end A to rotate through an angle formula_4. Once the magnitude of formula_5 developed at end B is found, the carryover factor of this member is given as the ratio of formula_5 over formula_3:
formula_6
In case of a beam of length L with constant cross-section whose flexural rigidity is formula_7,
formula_8
formula_9
therefore the carryover factor
formula_10
Sign convention.
Once a sign convention has been chosen, it has to be maintained for the whole structure. The traditional engineer's sign convention is not used in the calculations of the moment distribution method although the results can be expressed in the conventional way. In the BMD case, the left side moment is clockwise direction and other is anticlockwise direction so the bending is positive and is called sagging.
Framed structure.
Framed structure with or without sidesway can be analysed using the moment distribution method.
Example.
The statically indeterminate beam shown in the figure is to be analysed.
The beam is considered to be three separate members, AB, BC, and CD, connected by fixed end (moment resisting) joints at B and C.
In the following calculations, clockwise moments are positive.
formula_15
formula_16
formula_17
formula_18
formula_19
formula_20
Bending stiffness and distribution factors.
The bending stiffness of members AB, BC and CD are formula_21, formula_22 and formula_23, respectively . Therefore, expressing the results in repeating decimal notation:
formula_24
formula_25
formula_26
formula_27
The distribution factors of joints A and D are formula_28 and formula_29.
Carryover factors.
The carryover factors are formula_30, except for the carryover factor from D (fixed support) to C which is zero.
Moment distribution.
Numbers in grey are balanced moments; arrows ( → / ← ) represent the carry-over of moment from one end to the other end of a member.* Step 1: As joint A is released, balancing moment of magnitude equal to the fixed end moment formula_31 develops and is carried-over from joint A to joint B.* Step 2: The unbalanced moment at joint B now is the summation of the fixed end moments formula_32, formula_33 and the carry-over moment from joint A. This unbalanced moment is distributed to members BA and BC in accordance with the distribution factors formula_34 and formula_35. Step 2 ends with carry-over of balanced moment formula_36 to joint C. Joint A is a roller support which has no rotational restraint, so moment carryover from joint B to joint A is zero.* Step 3: The unbalanced moment at joint C now is the summation of the fixed end moments formula_37, formula_38 and the carryover moment from joint B. As in the previous step, this unbalanced moment is distributed to each member and then carried over to joint D and back to joint B. Joint D is a fixed support and carried-over moments to this joint will not be distributed nor be carried over to joint C.* Step 4: Joint B still has balanced moment which was carried over from joint C in step 3. Joint B is released once again to induce moment distribution and to achieve equilibrium.* Steps 5 - 10: Joints are released and fixed again until every joint has unbalanced moments of size zero or neglectably small in required precision. Arithmetically summing all moments in each respective columns gives the final moment values.
formula_39
formula_40
formula_41
formula_42
The conventional engineer's sign convention is used here, i.e. positive moments cause elongation at the bottom part of a beam member.
Result.
For comparison purposes, the following are the results generated using a matrix method. Note that in the analysis above, the iterative process was carried to >0.01 precision. The fact that the matrix analysis results and the moment distribution analysis results match to 0.001 precision is mere coincidence.
formula_39
formula_40
formula_41
formula_42
Note that the moment distribution method only determines the moments at the joints. Developing complete bending moment diagrams require additional calculations using the determined joint moments and internal section equilibrium.
Result via displacements method.
As the Hardy Cross method provides only approximate results, with a margin of error inversely proportionate to the number of iterations, it is important to have an idea of how accurate this method might be. With this in mind, here is the result obtained by using an exact method: the displacement method
For this, the displacements method equation assumes the following form:
formula_43
For the structure described in this example, the stiffness matrix is as follows:
formula_44
The equivalent nodal force vector:
formula_45
Replacing the values presented above in the equation and solving it for formula_46 leads to the following result:
formula_47
Hence, the moments evaluated in node B are as follows:
formula_48
formula_49
The moments evaluated in node C are as follows:
formula_50
formula_51
Notes.
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{
"math_id": 0,
"text": "k"
},
{
"math_id": 1,
"text": "j"
},
{
"math_id": 2,
"text": "D_{jk} = \\frac{\\frac{E_k I_k}{L_k}}{\\sum_{i=1}^{i=n} \\frac{E_i I_i}{L_i}}"
},
{
"math_id": 3,
"text": "M_A"
},
{
"math_id": 4,
"text": "\\theta_A"
},
{
"math_id": 5,
"text": "M_B"
},
{
"math_id": 6,
"text": "C_{AB} = \\frac{M_B}{M_A}"
},
{
"math_id": 7,
"text": "EI"
},
{
"math_id": 8,
"text": "M_A = 4 \\frac{EI}{L} \\theta_A + 2 \\frac{EI}{L} \\theta_B = 4 \\frac{EI}{L} \\theta_A"
},
{
"math_id": 9,
"text": "M_B = 2 \\frac{EI}{L} \\theta_A + 4 \\frac{EI}{L} \\theta_B = 2 \\frac{EI}{L} \\theta_A"
},
{
"math_id": 10,
"text": "C_{AB} = \\frac{M_B}{M_A} = \\frac{1}{2}"
},
{
"math_id": 11,
"text": " L = 10 \\ m "
},
{
"math_id": 12,
"text": " P = 10 \\ kN "
},
{
"math_id": 13,
"text": " a = 3 \\ m "
},
{
"math_id": 14,
"text": " q = 1 \\ kN/m"
},
{
"math_id": 15,
"text": "M _{AB} ^f = - \\frac{Pb^2a }{L^2} = - \\frac{10 \\times 7^2 \\times 3}{10^2} = - 14.700 \\ kN\\cdot m"
},
{
"math_id": 16,
"text": "M _{BA} ^f = \\frac{Pa^2b}{L^2} = \\frac{10 \\times 3^2 \\times 7}{10^2} = + 6.300 \\ kN\\cdot m"
},
{
"math_id": 17,
"text": "M _{BC} ^f = - \\frac{qL^2}{12} =- \\frac{1 \\times 10^2}{12} = - 8.333 \\ kN\\cdot m"
},
{
"math_id": 18,
"text": "M _{CB} ^f = \\frac{qL^2}{12} = \\frac{1 \\times 10^2}{12} = + 8.333 \\ kN\\cdot m"
},
{
"math_id": 19,
"text": "M _{CD} ^f = - \\frac{PL}{8} = - \\frac{10 \\times 10}{8} = - 12.500 \\ kN\\cdot m"
},
{
"math_id": 20,
"text": "M _{DC} ^f =\\frac{PL}{8} =\\frac{10 \\times 10}{8} = + 12.500 \\ kN\\cdot m"
},
{
"math_id": 21,
"text": "\\frac{3EI}{L}"
},
{
"math_id": 22,
"text": "\\frac{4\\times 2EI}{L}"
},
{
"math_id": 23,
"text": "\\frac{4EI}{L}"
},
{
"math_id": 24,
"text": "D_{BA} = \\frac{\\frac{3EI}{L}}{\\frac{3EI}{L}+\\frac{4\\times 2EI}{L}} = \\frac{\\frac{3}{10}}{\\frac{3}{10}+\\frac{8}{10}} = \\frac{3}{11} = 0.(27)"
},
{
"math_id": 25,
"text": "D_{BC} = \\frac{\\frac{4\\times 2EI}{L}}{\\frac{3EI}{L}+\\frac{4\\times 2EI}{L}} = \\frac{\\frac{8}{10}}{\\frac{3}{10}+\\frac{8}{10}} = \\frac{8}{11} = 0.(72)"
},
{
"math_id": 26,
"text": "D_{CB} = \\frac{\\frac{4\\times 2EI}{L}}{\\frac{4\\times 2EI}{L}+\\frac{4EI}{L}} = \\frac{\\frac{8}{10}}{\\frac{8}{10}+\\frac{4}{10}} = \\frac{8}{12} = 0.(66)"
},
{
"math_id": 27,
"text": "D_{CD} = \\frac{\\frac{4EI}{L}}{\\frac{4\\times 2EI}{L}+\\frac{4EI}{L}} = \\frac{\\frac{4}{10}}{\\frac{8}{10}+\\frac{4}{10}} = \\frac{4}{12} = 0.(33)"
},
{
"math_id": 28,
"text": "D_{AB} = 1"
},
{
"math_id": 29,
"text": "D_{DC} = 0 "
},
{
"math_id": 30,
"text": " \\frac{1}{2} "
},
{
"math_id": 31,
"text": "M_{AB}^{f} = 14.700 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 32,
"text": "M_{BA}^{f}"
},
{
"math_id": 33,
"text": "M_{BC}^{f}"
},
{
"math_id": 34,
"text": "D_{BA} = 0.2727"
},
{
"math_id": 35,
"text": "D_{BC} = 0.7273"
},
{
"math_id": 36,
"text": "M_{BC}=3.867 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 37,
"text": "M_{CB}^{f}"
},
{
"math_id": 38,
"text": "M_{CD}^{f}"
},
{
"math_id": 39,
"text": "M_A = 0 \\ kN \\cdot m "
},
{
"math_id": 40,
"text": "M_B = -11.569 \\ kN \\cdot m "
},
{
"math_id": 41,
"text": "M_C = -10.186 \\ kN \\cdot m "
},
{
"math_id": 42,
"text": "M_D = -13.657 \\ kN \\cdot m "
},
{
"math_id": 43,
"text": "\\left[K\\right]\\left\\{d\\right\\} = \\left\\{-f\\right\\}"
},
{
"math_id": 44,
"text": "\\left[K\\right]=\\begin{bmatrix} 3\\frac{EI}{L} + 4\\frac{2EI}{L} & 2\\frac{2EI}{L} \\\\\n2\\frac{2EI}{L} & 4\\frac{2EI}{L} + 4\\frac{EI}{L} \\end{bmatrix}"
},
{
"math_id": 45,
"text": "\\left\\{f\\right\\}^T = \\left\\{-P\\frac{ab(L+a)}{2L^2}+q\\frac{L^2}{12} , -q\\frac{L^2}{12} + P\\frac{L}{8} \\right\\}\n"
},
{
"math_id": 46,
"text": "\\left\\{d\\right\\}"
},
{
"math_id": 47,
"text": "\\left\\{d\\right\\}^T=\\left\\{ 6.9368 ; -5.7845\\right\\}"
},
{
"math_id": 48,
"text": "M_{BA} = 3\\frac{EI}{L}d_1 - P\\frac{ab(L+a)}{2L^2} = -11.569"
},
{
"math_id": 49,
"text": "M_{BC} = -4\\frac{2EI}{L}d_1 -2\\frac{2EI}{L}d_2 - q\\frac{L^2}{12} = -11.569"
},
{
"math_id": 50,
"text": "M_{CB} = 2\\frac{2EI}{L}d_1 + 4\\frac{2EI}{L}d_2 - q\\frac{L^2}{12} = -10.186"
},
{
"math_id": 51,
"text": "M_{CD} = -4\\frac{EI}{L}d_2 - P\\frac{L}{8} = -10.186"
}
]
| https://en.wikipedia.org/wiki?curid=13000492 |
13001206 | Flip (mathematics) | Surgery operation in minimal model program
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions.
The minimal model program.
The minimal model program can be summarised very briefly as follows: given a variety formula_0, we construct a sequence of contractions formula_1, each of which contracts some curves on which the canonical divisor formula_2 is negative. Eventually, formula_3 should become nef (at least in the case of nonnegative Kodaira dimension), which is the desired result. The major technical problem is that, at some stage, the variety formula_4 may become 'too singular', in the sense that the canonical divisor formula_2 is no longer a Cartier divisor, so the intersection number formula_5 with a curve formula_6 is not even defined.
The (conjectural) solution to this problem is the "flip". Given a problematic formula_4 as above, the flip of formula_4 is a birational map (in fact an isomorphism in codimension 1) formula_7 to a variety whose singularities are 'better' than those of formula_4. So we can put formula_8, and continue the process.
Two major problems concerning flips are to show that they exist and to show that one cannot have an infinite sequence of flips. If both of these problems can be solved, then the minimal model program can be carried out.
The existence of flips for 3-folds was proved by . The existence of log flips, a more general kind of flip, in dimension three and four were proved by Shokurov (1993, 2003)
whose work was fundamental to the solution of the existence of log flips and other problems in higher dimension.
The existence of log flips in higher dimensions has been settled by (Caucher Birkar, Paolo Cascini & Christopher D. Hacon et al. 2010). On the other hand, the problem of termination—proving that there can be no infinite sequence of flips—is still open in dimensions greater than 3.
Definition.
If formula_10 is a morphism, and "K" is the canonical bundle of "X", then the relative canonical ring of "f" is
formula_11
and is a sheaf of graded algebras over the sheaf formula_12 of regular functions on "Y".
The blowup
formula_13
of "Y" along the relative canonical ring is a morphism to "Y". If the relative canonical ring is finitely generated (as an algebra over formula_12 ) then the morphism formula_14 is called the flip of formula_15 if formula_16 is relatively ample, and the flop of formula_15 if "K" is relatively trivial. (Sometimes the induced birational morphism from formula_0 to formula_17 is called a flip or flop.)
In applications, formula_15 is often a small contraction of an extremal ray, which implies several extra properties:
Examples.
The first example of a flop, known as the Atiyah flop, was found in .
Let "Y" be the zeros of formula_18 in formula_19, and let "V" be the blowup of "Y" at the origin.
The exceptional locus of this blowup is isomorphic to formula_20, and can be blown down to formula_21 in two different ways, giving varieties formula_9 and formula_22. The natural birational map from formula_9 to formula_22 is the Atiyah flop.
introduced Reid's pagoda, a generalization of Atiyah's flop replacing "Y" by the zeros of formula_23.
References.
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"text": "X"
},
{
"math_id": 1,
"text": "X = X_1\\rightarrow X_2 \\rightarrow \\cdots \\rightarrow X_n "
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{
"math_id": 2,
"text": "K_{X_i}"
},
{
"math_id": 3,
"text": "K_{X_n}"
},
{
"math_id": 4,
"text": "X_i"
},
{
"math_id": 5,
"text": "K_{X_i} \\cdot C"
},
{
"math_id": 6,
"text": "C"
},
{
"math_id": 7,
"text": "f\\colon X_i \\rightarrow X_i^+"
},
{
"math_id": 8,
"text": "X_{i+1} = X_i^+"
},
{
"math_id": 9,
"text": "X_1"
},
{
"math_id": 10,
"text": "f\\colon X\\to Y"
},
{
"math_id": 11,
"text": "\\bigoplus_m f_*(\\mathcal{O}_X(mK))"
},
{
"math_id": 12,
"text": "\\mathcal{O}_Y"
},
{
"math_id": 13,
"text": "f^+\\colon X^+= \\operatorname{Proj}\\big(\\bigoplus_m f_*(\\mathcal{O}_X(mK))\\big)\\to Y"
},
{
"math_id": 14,
"text": "f^+"
},
{
"math_id": 15,
"text": "f"
},
{
"math_id": 16,
"text": "-K"
},
{
"math_id": 17,
"text": "X^+"
},
{
"math_id": 18,
"text": "xy=zw"
},
{
"math_id": 19,
"text": "\\mathbb{A}^4"
},
{
"math_id": 20,
"text": "\\mathbb{P}^1\\times \\mathbb{P}^1"
},
{
"math_id": 21,
"text": "\\mathbb{P}^1"
},
{
"math_id": 22,
"text": "X_2"
},
{
"math_id": 23,
"text": "xy = (z+w^k)(z-w^k)"
}
]
| https://en.wikipedia.org/wiki?curid=13001206 |
1300358 | Fock matrix | In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors.
It is most often formed in computational chemistry when attempting to solve the Roothaan equations for an atomic or molecular system. The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Because the Fock operator is a one-electron operator, it does not include the electron correlation energy.
The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes:
formula_0
Where "i" runs over the total "N" spin orbitals. In the closed-shell case, it can be simplified by considering only the spatial orbitals. Noting that the formula_1 terms are duplicated and the exchange terms are null between different spins.
For the restricted case which assumes closed-shell orbitals and single-
determinantal wavefunctions, the Fock operator for the "i"-th electron is given by:
formula_2
where:
formula_3 is the Fock operator for the "i"-th electron in the system,
formula_4 is the one-electron Hamiltonian for the "i"-th electron,
formula_5 is the number of electrons and formula_6 is the number of occupied orbitals in the closed-shell system,
formula_7 is the Coulomb operator, defining the repulsive force between the "j"-th and "i"-th electrons in the system,
formula_8 is the exchange operator, defining the quantum effect produced by exchanging two electrons.
The Coulomb operator is multiplied by two since there are two electrons in each occupied orbital. The exchange operator is not multiplied by two since it has a non-zero result only for electrons which have the same spin as the "i"-th electron.
For systems with unpaired electrons there are many choices of Fock matrices.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\hat F(i) = \\hat h(i)+\\sum_{ j=1 }^{N} [\\hat J_j(i)-\\hat K_j(i)]"
},
{
"math_id": 1,
"text": "\\hat J"
},
{
"math_id": 2,
"text": "\\hat F(i) = \\hat h(i)+\\sum_{ j=1 }^{n/2}[2 \\hat J_j(i)-\\hat \nK_j(i)]"
},
{
"math_id": 3,
"text": "\\hat F(i)"
},
{
"math_id": 4,
"text": "{\\hat h}(i)"
},
{
"math_id": 5,
"text": "n"
},
{
"math_id": 6,
"text": " \\frac{n}{2} "
},
{
"math_id": 7,
"text": "\\hat J_j(i)"
},
{
"math_id": 8,
"text": "\\hat K_j(i)"
}
]
| https://en.wikipedia.org/wiki?curid=1300358 |
13005617 | Flat (geometry) | Affine subspace of a Euclidean space
In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space (of equal or lower dimension. Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.
The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself.
The definition of flat excludes non-straight curves and non-planar surfaces, which have different notions of distance: arc length and geodesic length, respectively.
In an n-dimensional space, there are k-flats of every dimension k from 0 to "n"; subspaces one dimension lower than the parent space, ("n" −&hairsp;1)-flats, are called "hyperplanes".
Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations.
A flat is a manifold and an algebraic variety, and is sometimes called a "linear manifold" or "linear variety" to distinguish it from other manifolds or varieties.
Descriptions.
By equations.
A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation involving x and y:
formula_0
In three-dimensional space, a single linear equation involving x, y, and z defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in n variables describes a hyperplane, and a system of linear equations describes the intersection of those hyperplanes. Assuming the equations are consistent and linearly independent, a system of k equations describes a flat of dimension "n" − "k".
Parametric.
A flat can also be described by a system of linear parametric equations. A line can be described by equations involving one parameter:
formula_1
while the description of a plane would require two parameters:
formula_2
In general, a parameterization of a flat of dimension k would require k parameters, e.g. "t"1, …, "tk".
Operations and relations on flats.
Intersecting, parallel, and skew flats.
An intersection of flats is either a flat or the empty set.
If each line from one flat is parallel to some line from another flat, then these two flats are parallel. Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
If flats do not intersect, and no line from the first flat is parallel to a line from the second flat, then these are skew flats. It is possible only if sum of their dimensions is less than dimension of the ambient space.
Join.
For two flats of dimensions "k"1 and "k"2 there exists the minimal flat which contains them, of dimension at most "k"1 + "k"2 + 1. If two flats intersect, then the dimension of the containing flat equals to "k"1 + "k"2 minus the dimension of the intersection.
Properties of operations.
These two operations (referred to as "meet" and "join") make the set of all flats in the Euclidean n-space a lattice and can build systematic coordinates for flats in any dimension, leading to Grassmann coordinates or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes.
However, the lattice of all flats is not a distributive lattice.
If two lines ℓ1 and ℓ2 intersect, then ℓ1 ∩ ℓ2 is a point. If p is a point not lying on the same plane, then (ℓ1 ∩ ℓ2) + "p" = (ℓ1 + "p") ∩ (ℓ2 + "p"), both representing a line. But when ℓ1 and ℓ2 are parallel, this distributivity fails, giving p on the left-hand side and a third parallel line on the right-hand side.
Euclidean geometry.
The aforementioned facts do not depend on the structure being that of Euclidean space (namely, involving Euclidean distance) and are correct in any affine space. In a Euclidean space:
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "3x + 5y = 8."
},
{
"math_id": 1,
"text": "x=2+3t,\\;\\;\\;\\;y=-1+t\\;\\;\\;\\;z=\\frac{3}{2}-4t"
},
{
"math_id": 2,
"text": "x=5+2t_1-3t_2,\\;\\;\\;\\; y=-4+t_1+2t_2\\;\\;\\;\\;z=5t_1-3t_2.\\,\\!"
}
]
| https://en.wikipedia.org/wiki?curid=13005617 |
13006855 | Erdős–Mordell inequality | On sums of distances in triangles
In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle "ABC" and point "P" inside "ABC", the sum of the distances from "P" to the sides is less than or equal to half of the sum of the distances from "P" to the vertices. It is named after Paul Erdős and Louis Mordell. posed the problem of proving the inequality; a proof was provided two years later by Mordell and D. F. Barrow (1937). This solution was however not very elementary. Subsequent simpler proofs were then found by , , and .
Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from "P" to the sides are replaced by the distances from "P" to the points where the angle bisectors of ∠"APB", ∠"BPC", and ∠"CPA" cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.
Statement.
Let formula_0 be an arbitrary point P inside a given triangle formula_1, and let formula_2, formula_3, and formula_4 be the perpendiculars from formula_0 to the sides of the triangles.
(If the triangle is obtuse, one of these perpendiculars may cross through a different side of the triangle and end on the line supporting one of the sides.) Then the inequality states that
formula_5
Proof.
Let the sides of ABC be "a" opposite A, "b" opposite B, and "c" opposite C; also let PA = "p", PB = "q", PC = "r", dist(P;BC) = "x", dist(P;CA) = "y", dist(P;AB) = "z". First, we prove that
formula_6
This is equivalent to
formula_7
The right side is the area of triangle ABC, but on the left side, "r" + "z" is at least the height of the triangle; consequently, the left side cannot be smaller than the right side. Now reflect P on the angle bisector at C. We find that "cr" ≥ "ay" + "bx" for P's reflection. Similarly, "bq" ≥ "az" + "cx" and "ap" ≥ "bz" + "cy". We solve these inequalities for "r", "q", and "p":
formula_8
formula_9
formula_10
Adding the three up, we get
formula_11
Since the sum of a positive number and its reciprocal is at least 2 by AM–GM inequality, we are finished. Equality holds only for the equilateral triangle, where P is its centroid.
Another strengthened version.
Let ABC be a triangle inscribed into a circle (O) and P be a point inside of ABC. Let D, E, F be the orthogonal projections of P onto BC, CA, AB. M, N, Q be the orthogonal projections of P onto tangents to (O) at A, B, C respectively, then:
formula_12
Equality hold if and only if triangle ABC is equilateral (; )
A generalization.
Let formula_13 be a convex polygon, and formula_0 be an interior point of formula_13. Let formula_14 be the distance from formula_0 to the vertex formula_15 , formula_16 the distance from formula_0 to the side formula_17, formula_18 the segment of the bisector of the angle formula_19 from formula_0 to its intersection with the side formula_17 then :
formula_20
In absolute geometry.
In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in , to the statement
that the sum of the angles of a triangle is less than or equal to two right angles. | [
{
"math_id": 0,
"text": "P"
},
{
"math_id": 1,
"text": "ABC"
},
{
"math_id": 2,
"text": "PL"
},
{
"math_id": 3,
"text": "PM"
},
{
"math_id": 4,
"text": "PN"
},
{
"math_id": 5,
"text": "PA+PB+PC\\geq 2(PL+PM+PN)"
},
{
"math_id": 6,
"text": "cr\\geq ax+by."
},
{
"math_id": 7,
"text": "\\frac{c(r+z)}2\\geq \\frac{ax+by+cz}2."
},
{
"math_id": 8,
"text": "r\\geq (a/c)y+(b/c)x,"
},
{
"math_id": 9,
"text": "q\\geq (a/b)z+(c/b)x,"
},
{
"math_id": 10,
"text": "p\\geq (b/a)z+(c/a)y."
},
{
"math_id": 11,
"text": "\n p + q + r\n\\geq\n \\left( \\frac{b}{c} + \\frac{c}{b} \\right) x +\n \\left( \\frac{a}{c} + \\frac{c}{a} \\right) y +\n \\left( \\frac{a}{b} + \\frac{b}{a} \\right) z.\n"
},
{
"math_id": 12,
"text": " PM+PN+PQ \\ge 2(PD+PE+PF)"
},
{
"math_id": 13,
"text": "A_1A_2...A_n"
},
{
"math_id": 14,
"text": "R_i"
},
{
"math_id": 15,
"text": "A_i"
},
{
"math_id": 16,
"text": "r_i"
},
{
"math_id": 17,
"text": "A_iA_{i+1}"
},
{
"math_id": 18,
"text": "w_i"
},
{
"math_id": 19,
"text": "A_iPA_{i+1}"
},
{
"math_id": 20,
"text": " \\sum_{i=1}^{n}R_i \\ge \\left(\\sec{\\frac{\\pi}{n}}\\right)\\sum_{i=1}^{n} w_i \\ge \\left(\\sec{\\frac{\\pi}{n}}\\right)\\sum_{i=1}^{n} r_i "
}
]
| https://en.wikipedia.org/wiki?curid=13006855 |
13006958 | Helix angle | Angle between a helix and an axial line
In mechanical engineering, a helix angle is the angle between any helix and an axial line on its right, circular cylinder or cone. Common applications are screws, helical gears, and worm gears.
The helix angle references the axis of the cylinder, distinguishing it from the lead angle, which references a line perpendicular to the axis. Naturally, the helix angle is the geometric complement of the lead angle. The helix angle is measured in degrees.
Concept.
In terms specific to screws, the helix angle can be found by unraveling the helix from the screw, representing the section as a right triangle, and calculating the angle that is formed. Note that while the terminology directly refers to screws, these concepts are analogous to most mechanical applications of the helix angle.
The helix angle can be expressed as:
"formula_0"
where
"l" is lead of the screw or gear
"rm" is mean radius of the screw thread or gear
Applications.
The helix angle is crucial in mechanical engineering applications that involve power transfer and motion conversion. Some examples are outlined below, though its use is much more widely spread.
Screw.
Cutting a single helical groove into a screw-stock cylinder yields what is referred to as a single-thread screw. Similarly, one may construct a double-thread screw provided that the helix angle of the two cuts is the same, and that the second cut is positioned in the uncut material between the grooves of the first. For certain applications, triple and quadruple threads are in use. The helix may be cut either right hand or left hand. In screws especially, the helix angle is essential for calculating torque in power screw applications.
The maximum efficiency for a screw is defined by the following equations:
formula_1
formula_2
Where formula_3 is the helix angle, formula_4 is the friction angle, and formula_5 is the maximum efficiency. The friction value is dependent on the materials of the screw and interacting nut, but ultimately the efficiency is controlled by the helix angle. The efficiency can be plotted versus the helix angle for a constant friction, as shown in the adjacent diagram. The maximum efficiency is a helix angle between 40 and 45 degrees, however a reasonable efficiency is achieved above 15°. Due to difficulties in forming the thread, helix angle greater than 30° are rarely used. Moreover, above 30° the friction angle becomes smaller than the helix angle and the nut is no longer self-locking and the mechanical advantage disappears.
Helical gear.
In helical and worm gears, the helix angle denotes the standard pitch circle unless otherwise specified. Application of the helix angle typically employs a magnitude ranging from 15° to 30° for helical gears, with 45° capping the safe operation limit. The angle itself may be cut with either a right-hand or left-hand orientation. In its typical parallel arrangement, meshing helical gears requires that the helix angles are of the same magnitude and cut oppositely .
Worm gear.
Worm gears resemble helical gear seats, the difference being that the shafts of a worm train are aligned perpendicularly. In this case, the helix angle of the worm meshes with the lead angle of the worm gear.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\mbox{Helix angle} = \\arctan \\left( \\frac{2 \\pi r_m}{l} \\right)"
},
{
"math_id": 1,
"text": "\\alpha = 45^o - \\frac{\\phi}{2}"
},
{
"math_id": 2,
"text": "\\eta_{max} = \\frac{1 - \\sin{\\phi}}{1 + \\sin{\\phi}} "
},
{
"math_id": 3,
"text": "\\alpha \\,"
},
{
"math_id": 4,
"text": "\\phi \\,"
},
{
"math_id": 5,
"text": "\\eta_{max}"
}
]
| https://en.wikipedia.org/wiki?curid=13006958 |
13007240 | Lead (engineering) | Lead is the axial advance of a helix or screw during one complete turn (360°) The lead for a screw thread is the axial travel for a single revolution.
"Pitch" is defined as the axial distance between adjacent threads on a helix or screw. In most screws, called ""single start" screws, which have a single helical thread along their length, the lead and pitch are equal. They only differ in "multiple start" screws, which have several intertwined threads. In these screws, the lead is equal to the pitch multiplied by the number of "starts"".
Lead angle is the angle between the helix and a plane of rotation. It is the complement of the helix angle, and is used for convenience in worms and hobs. It is understood to be at the standard pitch diameter unless otherwise specified.
The lead angle can be expressed as:
formula_0
where
"l" is lead of the helix
"dm" is mean diameter of the helix
In American literature "λ" is used to notate the Lead Angle.
In European literature, "Υ" (Greek letter gamma) may be used.
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\mbox{Lead angle} = \\arctan \\left( \\frac {l} {\\pi d_m} \\right)"
}
]
| https://en.wikipedia.org/wiki?curid=13007240 |
13007524 | Tomlinson model | Physical model in nanotribology
The Tomlinson model, also known as the Prandtl–Tomlinson Model, is one of the most popular models in nanotribology widely used as the basis for many investigations of frictional mechanisms on the atomic scale. Essentially, a nanotip is dragged by a spring over a corrugated energy landscape. A "frictional parameter" "η" can be introduced to describe the ratio between the energy corrugation and the elastic energy stored in the spring. If the tip-surface interaction is described by a sinusoidal potential with amplitude "V0" and periodicity "a" then
formula_0
where "k" is the spring constant.
If "η"<1, the tip slides continuously across the landscape (superlubricity regime). If "η">1, the tip motion consists in abrupt jumps between the minima of the energy landscape (stick-slip regime).
The name "Tomlinson model" is, however, historically incorrect: the paper by Tomlinson that is often cited in this context did not contain the model known as the "Tomlinson model" and suggests an adhesive contribution to friction. In reality it was Ludwig Prandtl who suggested in 1928 this model to describe the plastic deformations in crystals as well as the dry friction. In the meantime, many researchers still call this model the "Prandtl–Tomlinson Model".
In Russia this model was introduced by the Soviet physicists Yakov Frenkel and T. Kontorova. The Frenkel defect became firmly fixed in the physics of solids and liquids. In the 1930s, this research was supplemented with works on the theory of plastic deformation. Their theory, now known as the Frenkel–Kontorova model, is important in the study of dislocations.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\eta=\\frac{4\\pi^2 V_0}{ka^2},"
}
]
| https://en.wikipedia.org/wiki?curid=13007524 |
1300778 | DeWitt notation | Notation used in quantum field theory
Physics often deals with classical models where the dynamical variables are a collection of functions
In the DeWitt notation (named after theoretical physicist Bryce DeWitt), φ"α"("x") is written as φ"i" where "i" is now understood as an index covering both "α" and "x".
So, given a smooth functional "A", "A","i" stands for the functional derivative
formula_0
as a functional of "φ". In other words, a "1-form" field over the infinite dimensional "functional manifold".
In integrals, the Einstein summation convention is used. Alternatively,
formula_1 | [
{
"math_id": 0,
"text": "A_{,i}[\\varphi] \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\delta}{\\delta \\varphi^\\alpha(x)}A[\\varphi]"
},
{
"math_id": 1,
"text": "A^i B_i \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_M \\sum_\\alpha A^\\alpha(x) B_\\alpha(x) d^dx"
}
]
| https://en.wikipedia.org/wiki?curid=1300778 |
1301093 | Two-port network | Electric circuit with two pairs of terminals
In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two "pairs" of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port.
It is commonly used in mathematical circuit analysis.
Application.
The two-port network model is used in mathematical circuit analysis techniques to isolate portions of larger circuits. A two-port network is regarded as a "black box" with its properties specified by a matrix of numbers. This allows the response of the network to signals applied to the ports to be calculated easily, without solving for all the internal voltages and currents in the network. It also allows similar circuits or devices to be compared easily. For example, transistors are often regarded as two-ports, characterized by their h-parameters (see below) which are listed by the manufacturer. Any linear circuit with four terminals can be regarded as a two-port network provided that it does not contain an independent source and satisfies the port conditions.
Examples of circuits analyzed as two-ports are filters, matching networks, transmission lines, transformers, and small-signal models for transistors (such as the hybrid-pi model). The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz.
In two-port mathematical models, the network is described by a 2 by 2 square matrix of complex numbers. The common models that are used are referred to as z-"parameters", y-"parameters", h-"parameters", g-"parameters", and ABCD-"parameters", each described individually below. These are all limited to linear networks since an underlying assumption of their derivation is that any given circuit condition is a linear superposition of various short-circuit and open circuit conditions. They are usually expressed in matrix notation, and they establish relations between the variables
"V"1, voltage across port 1
"I"1, current into port 1
"V"2, voltage across port 2
"I"2, current into port 2
which are shown in figure 1. The difference between the various models lies in which of these variables are regarded as the independent variables. These current and voltage variables are most useful at low-to-moderate frequencies. At high frequencies (e.g., microwave frequencies), the use of power and energy variables is more appropriate, and the two-port current–voltage approach is replaced by an approach based upon scattering parameters.
General properties.
There are certain properties of two-ports that frequently occur in practical networks and can be used to greatly simplify the analysis. These include:
formula_0
Impedance parameters ("z"-parameters).
where
formula_1
All the z-parameters have dimensions of ohms.
For reciprocal networks "z"12 = "z"21. For symmetrical networks "z"11 = "z"22. For reciprocal lossless networks all the "z"mn are purely imaginary.
Example: bipolar current mirror with emitter degeneration.
Figure 3 shows a bipolar current mirror with emitter resistors to increase its output resistance. Transistor "Q"1 is "diode connected", which is to say its collector-base voltage is zero. Figure 4 shows the small-signal circuit equivalent to Figure 3. Transistor "Q"1 is represented by its emitter resistance "r"E:
formula_2
a simplification made possible because the dependent current source in the hybrid-pi model for "Q"1 draws the same current as a resistor 1 / "g"m connected across "r"π. The second transistor "Q"2 is represented by its hybrid-pi model. Table 1 below shows the z-parameter expressions that make the z-equivalent circuit of Figure 2 electrically equivalent to the small-signal circuit of Figure 4.
The negative feedback introduced by resistors "R"E can be seen in these parameters. For example, when used as an active load in a differential amplifier, "I"1 ≈ −"I"2, making the output impedance of the mirror approximately
formula_3
compared to only "r"O without feedback (that is with "R"E = 0Ω). At the same time, the impedance on the reference side of the mirror is approximately
formula_4
only a moderate value, but still larger than "r"E with no feedback. In the differential amplifier application, a large output resistance increases the difference-mode gain, a good thing, and a small mirror input resistance is desirable to avoid Miller effect.
formula_5
Admittance parameters ("y"-parameters).
where
formula_6
All the "Y"-parameters have dimensions of siemens.
For reciprocal networks "y"12 = "y"21. For symmetrical networks "y"11 = "y"22. For reciprocal lossless networks all the "y"mn are purely imaginary.
formula_7
Hybrid parameters ("h"-parameters).
where
formula_8
This circuit is often selected when a current amplifier is desired at the output. The resistors shown in the diagram can be general impedances instead.
Off-diagonal h-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another.
For reciprocal networks "h"12 = –"h"21. For symmetrical networks "h"11"h"22 – "h"12"h"21 = 1. For reciprocal lossless networks "h"12 and "h"21 are real, while "h"11 and "h"22 are purely imaginary.
Example: common-base amplifier.
Note: Tabulated formulas in Table 2 make the h-equivalent circuit of the transistor from Figure 6 agree with its small-signal low-frequency hybrid-pi model in Figure 7. Notation: "r"π is base resistance of transistor, "r"O is output resistance, and "g"m is mutual transconductance. The negative sign for "h"21 reflects the convention that "I"1, "I"2 are positive when directed "into" the two-port. A non-zero value for "h"12 means the output voltage affects the input voltage, that is, this amplifier is bilateral. If "h"12 = 0, the amplifier is unilateral.
History.
The h-parameters were initially called "series-parallel parameters". The term "hybrid" to describe these parameters was coined by D. A. Alsberg in 1953 in "Transistor metrology". In 1954 a joint committee of the IRE and the AIEE adopted the term h-"parameters" and recommended that these become the standard method of testing and characterising transistors because they were "peculiarly adaptable to the physical characteristics of transistors". In 1956, the recommendation became an issued standard; 56 IRE 28.S2. Following the merge of these two organisations as the IEEE, the standard became Std 218-1956 and was reaffirmed in 1980, but has now been withdrawn.
formula_9
Inverse hybrid parameters (g-parameters).
where
formula_10
Often this circuit is selected when a voltage amplifier is wanted at the output. Off-diagonal g-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another. The resistors shown in the diagram can be general impedances instead.
Example: common-base amplifier.
Note: Tabulated formulas in Table 3 make the g-equivalent circuit of the transistor from Figure 8 agree with its small-signal low-frequency hybrid-pi model in Figure 9. Notation: "r"π is base resistance of transistor, "r"O is output resistance, and "g"m is mutual transconductance. The negative sign for "g"12 reflects the convention that "I"1, "I"2 are positive when directed "into" the two-port. A non-zero value for "g"12 means the output current affects the input current, that is, this amplifier is bilateral. If "g"12 = 0, the amplifier is unilateral.
"ABCD"-parameters.
The ABCD-parameters are known variously as chain, cascade, or transmission parameters. There are a number of definitions given for ABCD parameters, the most common is,
formula_11
Note: Some authors chose to reverse the indicated direction of I2 and suppress the negative sign on I2.
where
formula_12
For reciprocal networks "AD" – "BC" = 1. For symmetrical networks "A" = "D". For networks which are reciprocal and lossless, A and D are purely real while B and C are purely imaginary.
This representation is preferred because when the parameters are used to represent a cascade of two-ports, the matrices are written in the same order that a network diagram would be drawn, that is, left to right. However, a variant definition is also in use,
formula_13
where
formula_14
The negative sign of –"I"2 arises to make the output current of one cascaded stage (as it appears in the matrix) equal to the input current of the next. Without the minus sign the two currents would have opposite senses because the positive direction of current, by convention, is taken as the current entering the port. Consequently, the input voltage/current matrix vector can be directly replaced with the matrix equation of the preceding cascaded stage to form a combined A'B'C'D' matrix.
The terminology of representing the ABCD parameters as a matrix of elements designated "a"11 etc. as adopted by some authors and the inverse A'B'C'D' parameters as a matrix of elements designated "b"11 etc. is used here for both brevity and to avoid confusion with circuit elements.
formula_15
Table of transmission parameters.
The table below lists ABCD and inverse ABCD parameters for some simple network elements.
Scattering parameters (S-parameters).
The previous parameters are all defined in terms of voltages and currents at ports. S-parameters are different, and are defined in terms of incident and reflected waves at ports. S-parameters are used primarily at UHF and microwave frequencies where it becomes difficult to measure voltages and currents directly. On the other hand, incident and reflected power are easy to measure using directional couplers. The definition is,
formula_16
where the ak are the incident waves and the bk are the reflected waves at port k. It is conventional to define the ak and bk in terms of the square root of power. Consequently, there is a relationship with the wave voltages (see main article for details).
For reciprocal networks "S"12 = "S"21. For symmetrical networks "S"11 = "S"22. For antimetrical networks "S"11 = –"S"22. For lossless reciprocal networks formula_17 and formula_18
Scattering transfer parameters ("T"-parameters).
Scattering transfer parameters, like scattering parameters, are defined in terms of incident and reflected waves. The difference is that T-parameters relate the waves at port 1 to the waves at port 2 whereas S-parameters relate the reflected waves to the incident waves. In this respect T-parameters fill the same role as ABCD parameters and allow the T-parameters of cascaded networks to be calculated by matrix multiplication of the component networks. T-parameters, like ABCD parameters, can also be called transmission parameters. The definition is,
formula_19
T-parameters are not as easy to measure directly as S-parameters. However, S-parameters are easily converted to T-parameters, see main article for details.
Combinations of two-port networks.
When two or more two-port networks are connected, the two-port parameters of the combined network can be found by performing matrix algebra on the matrices of parameters for the component two-ports. The matrix operation can be made particularly simple with an appropriate choice of two-port parameters to match the form of connection of the two-ports. For instance, the z-parameters are best for series connected ports.
The combination rules need to be applied with care. Some connections (when dissimilar potentials are joined) result in the port condition being invalidated and the combination rule will no longer apply. A Brune test can be used to check the permissibility of the combination. This difficulty can be overcome by placing 1:1 ideal transformers on the outputs of the problem two-ports. This does not change the parameters of the two-ports, but does ensure that they will continue to meet the port condition when interconnected. An example of this problem is shown for series-series connections in figures 11 and 12 below.
Series-series connection.
When two-ports are connected in a series-series configuration as shown in figure 10, the best choice of two-port parameter is the z-parameters. The z-parameters of the combined network are found by matrix addition of the two individual z-parameter matrices.
formula_20
As mentioned above, there are some networks which will not yield directly to this analysis. A simple example is a two-port consisting of a L-network of resistors "R"1 and "R"2. The z-parameters for this network are;
formula_21
Figure 11 shows two identical such networks connected in series-series. The total z-parameters predicted by matrix addition are;
formula_22
However, direct analysis of the combined circuit shows that,
formula_23
The discrepancy is explained by observing that "R"1 of the lower two-port has been by-passed by the short-circuit between two terminals of the output ports. This results in no current flowing through one terminal in each of the input ports of the two individual networks. Consequently, the port condition is broken for both the input ports of the original networks since current is still able to flow into the other terminal. This problem can be resolved by inserting an ideal transformer in the output port of at least one of the two-port networks. While this is a common text-book approach to presenting the theory of two-ports, the practicality of using transformers is a matter to be decided for each individual design.
Parallel-parallel connection.
When two-ports are connected in a parallel-parallel configuration as shown in figure 13, the best choice of two-port parameter is the y-parameters. The y-parameters of the combined network are found by matrix addition of the two individual y-parameter matrices.
formula_24
Series-parallel connection.
When two-ports are connected in a series-parallel configuration as shown in figure 14, the best choice of two-port parameter is the h-parameters. The h-parameters of the combined network are found by matrix addition of the two individual h-parameter matrices.
formula_25
Parallel-series connection.
When two-ports are connected in a parallel-series configuration as shown in figure 15, the best choice of two-port parameter is the g-parameters. The g-parameters of the combined network are found by matrix addition of the two individual g-parameter matrices.
formula_26
Cascade connection.
When two-ports are connected with the output port of the first connected to the input port of the second (a cascade connection) as shown in figure 16, the best choice of two-port parameter is the ABCD-parameters. The a-parameters of the combined network are found by matrix multiplication of the two individual a-parameter matrices.
formula_27
A chain of n two-ports may be combined by matrix multiplication of the n matrices. To combine a cascade of b-parameter matrices, they are again multiplied, but the multiplication must be carried out in reverse order, so that;
formula_28
Example.
Suppose we have a two-port network consisting of a series resistor R followed by a shunt capacitor C. We can model the entire network as a cascade of two simpler networks:
formula_29
The transmission matrix for the entire network [b] is simply the matrix multiplication of the transmission matrices for the two network elements:
formula_30
Thus:
formula_31
Interrelation of parameters.
Where Δ[x] is the determinant of [x].
Certain pairs of matrices have a particularly simple relationship. The admittance parameters are the matrix inverse of the impedance parameters, the inverse hybrid parameters are the matrix inverse of the hybrid parameters, and the [b] form of the ABCD-parameters is the matrix inverse of the [a] form. That is,
formula_32
Networks with more than two ports.
While two port networks are very common (e.g., amplifiers and filters), other electrical networks such as directional couplers and circulators have more than 2 ports. The following representations are also applicable to networks with an arbitrary number of ports:
For example, three-port impedance parameters result in the following relationship:
formula_33
However the following representations are necessarily limited to two-port devices:
Collapsing a two-port to a one port.
A two-port network has four variables with two of them being independent. If one of the ports is terminated by a load with no independent sources, then the load enforces a relationship between the voltage and current of that port. A degree of freedom is lost. The circuit now has only one independent parameter. The two-port becomes a one-port impedance to the remaining independent variable.
For example, consider impedance parameters
formula_0
Connecting a load, "Z"L onto port 2 effectively adds the constraint
formula_34
The negative sign is because the positive direction for "I"2 is directed into the two-port instead of into the load. The augmented equations become
formula_35
The second equation can be easily solved for "I"2 as a function of "I"1 and that expression can replace "I"2 in the first equation leaving "V"1 ( and "V"2 and "I"2 ) as functions of "I"1
formula_36
So, in effect, "I"1 sees an input impedance "Z"in and the two-port's effect on the input circuit has been effectively collapsed down to a one-port; i.e., a simple two terminal impedance.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " \\begin{bmatrix} V_1 \\\\ V_2 \\end{bmatrix} = \\begin{bmatrix} z_{11} & z_{12} \\\\ z_{21} & z_{22} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix} "
},
{
"math_id": 1,
"text": "\\begin{align}\n z_{11} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_1}{I_1} \\right|_{I_2 = 0} &\n z_{12} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_1}{I_2} \\right|_{I_1 = 0} \\\\\n z_{21} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_2}{I_1} \\right|_{I_2 = 0} &\n z_{22} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_2}{I_2} \\right|_{I_1 = 0}\n\\end{align}"
},
{
"math_id": 2,
"text": "r_\\mathrm{E} \\approx \\frac{ \\text{thermal voltage, } V_\\mathrm{T} }{ \\text{emitter current, } I_E},"
},
{
"math_id": 3,
"text": "R_{22} - R_{21} \\approx \\frac{ 2\\beta r_\\mathrm{O}R_\\mathrm{E} }{ r_\\pi + 2R_\\mathrm{E}}"
},
{
"math_id": 4,
"text": "R_{11} - R_{12} \\approx \\frac{r_\\pi}{r_\\pi + 2R_\\mathrm{E}} (r_\\mathrm{E} + R_\\mathrm{E}),"
},
{
"math_id": 5,
"text": " \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix} = \\begin{bmatrix} y_{11} & y_{12} \\\\ y_{21} & y_{22} \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ V_2 \\end{bmatrix} "
},
{
"math_id": 6,
"text": "\\begin{align}\n y_{11} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_1}{V_1} \\right|_{V_2 = 0} &\n y_{12} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_1}{V_2} \\right|_{V_1 = 0} \\\\\n y_{21} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_2}{V_1} \\right|_{V_2 = 0} &\n y_{22} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_2}{V_2} \\right|_{V_1 = 0}\n\\end{align}"
},
{
"math_id": 7,
"text": " \\begin{bmatrix} V_1 \\\\ I_2 \\end{bmatrix} = \\begin{bmatrix} h_{11} & h_{12} \\\\ h_{21} & h_{22} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ V_2 \\end{bmatrix} "
},
{
"math_id": 8,
"text": "\\begin{align}\n h_{11} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_1}{I_1} \\right|_{V_2 = 0} &\n h_{12} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_1}{V_2} \\right|_{I_1 = 0} \\\\\n h_{21} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_2}{I_1} \\right|_{V_2 = 0} &\n h_{22} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_2}{V_2} \\right|_{I_1 = 0}\n\\end{align}"
},
{
"math_id": 9,
"text": " \\begin{bmatrix} I_1 \\\\ V_2 \\end{bmatrix} = \\begin{bmatrix} g_{11} & g_{12} \\\\ g_{21} & g_{22} \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ I_2 \\end{bmatrix} "
},
{
"math_id": 10,
"text": "\\begin{align}\n g_{11} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_1}{V_1} \\right|_{I_2 = 0} &\n g_{12} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_1}{I_2} \\right|_{V_1 = 0} \\\\\n g_{21} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_2}{V_1} \\right|_{I_2 = 0} &\n g_{22} &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_2}{I_2} \\right|_{V_1 = 0}\n\\end{align}"
},
{
"math_id": 11,
"text": " \\begin{bmatrix} V_1 \\\\ I_1 \\end{bmatrix} = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\begin{bmatrix} V_2 \\\\ -I_2 \\end{bmatrix} "
},
{
"math_id": 12,
"text": "\\begin{align}\n A &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_1}{V_2} \\right|_{I_2 = 0} &\n B &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. -\\frac{V_1}{I_2} \\right|_{V_2 = 0} \\\\\n C &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{I_1}{V_2} \\right|_{I_2 = 0} &\n D &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. -\\frac{I_1}{I_2} \\right|_{V_2 = 0}\n\\end{align}"
},
{
"math_id": 13,
"text": " \\begin{bmatrix} V_2 \\\\ -I_2 \\end{bmatrix} = \\begin{bmatrix} A' & B' \\\\ C' & D' \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ I_1 \\end{bmatrix} "
},
{
"math_id": 14,
"text": "\\begin{align}\n A' &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_2}{V_1} \\right|_{I_1 = 0} &\n B' &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. \\frac{V_2}{I_1} \\right|_{V_1 = 0} \\\\\n C' &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. -\\frac{I_2}{V_1} \\right|_{I_1 = 0} &\n D' &\\mathrel{\\stackrel{\\text{def}}{=}} \\left. -\\frac{I_2}{I_1} \\right|_{V_1 = 0}\n\\end{align}"
},
{
"math_id": 15,
"text": "\\begin{align}\n \\left[\\mathbf{a}\\right] &= \\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\\\\n \\left[\\mathbf{b}\\right] &= \\begin{bmatrix} b_{11} & b_{12} \\\\ b_{21} & b_{22} \\end{bmatrix} = \\begin{bmatrix} A' & B' \\\\ C' & D' \\end{bmatrix}\n\\end{align}"
},
{
"math_id": 16,
"text": " \\begin{bmatrix} b_1 \\\\ b_2 \\end{bmatrix} = \\begin{bmatrix} S_{11} & S_{12} \\\\ S_{21} & S_{22} \\end{bmatrix} \\begin{bmatrix} a_1 \\\\ a_2 \\end{bmatrix} "
},
{
"math_id": 17,
"text": "|S_{11}| = |S_{22}|"
},
{
"math_id": 18,
"text": " |S_{11}|^2 + |S_{12}|^2 = 1."
},
{
"math_id": 19,
"text": " \\begin{bmatrix} a_1 \\\\ b_1 \\end{bmatrix} = \\begin{bmatrix} T_{11} & T_{12} \\\\ T_{21} & T_{22} \\end{bmatrix} \\begin{bmatrix} b_2 \\\\ a_2 \\end{bmatrix} "
},
{
"math_id": 20,
"text": "[\\mathbf z] = [\\mathbf z]_1 + [\\mathbf z]_2"
},
{
"math_id": 21,
"text": "[\\mathbf z]_1 = \\begin{bmatrix} R_1 + R_2 & R_2 \\\\ R_2 & R_2 \\end{bmatrix}"
},
{
"math_id": 22,
"text": "[\\mathbf z] = [\\mathbf z]_1 + [\\mathbf z]_2 = 2[\\mathbf z]_1 = \\begin{bmatrix} 2R_1 + 2R_2 & 2R_2 \\\\ 2R_2 & 2R_2 \\end{bmatrix}"
},
{
"math_id": 23,
"text": "[\\mathbf z] = \\begin{bmatrix} R_1 + 2R_2 & 2R_2 \\\\ 2R_2 & 2R_2 \\end{bmatrix}"
},
{
"math_id": 24,
"text": "[\\mathbf y] = [\\mathbf y]_1 + [\\mathbf y]_2"
},
{
"math_id": 25,
"text": "[\\mathbf h] = [\\mathbf h]_1 + [\\mathbf h]_2"
},
{
"math_id": 26,
"text": "[\\mathbf g] = [\\mathbf g]_1 + [\\mathbf g]_2"
},
{
"math_id": 27,
"text": "[\\mathbf a] = [\\mathbf a]_1 \\cdot [\\mathbf a]_2"
},
{
"math_id": 28,
"text": "[\\mathbf b] = [\\mathbf b]_2 \\cdot [\\mathbf b]_1"
},
{
"math_id": 29,
"text": "\\begin{align}[]\n [\\mathbf{b}]_1 &= \\begin{bmatrix} 1 & -R \\\\ 0 & 1 \\end{bmatrix}\\\\\n \\lbrack\\mathbf{b}\\rbrack_2 &= \\begin{bmatrix} 1 & 0 \\\\ -sC & 1 \\end{bmatrix}\n\\end{align}"
},
{
"math_id": 30,
"text": "\\begin{align}[]\n \\lbrack\\mathbf{b}\\rbrack &= \\lbrack\\mathbf{b}\\rbrack_2 \\cdot \\lbrack\\mathbf{b}\\rbrack_1 \\\\\n &= \\begin{bmatrix} 1 & 0 \\\\ -sC & 1 \\end{bmatrix} \\begin{bmatrix} 1 & -R \\\\ 0 & 1 \\end{bmatrix} \\\\\n &= \\begin{bmatrix} 1 & -R \\\\ -sC & 1 + sCR \\end{bmatrix}\n\\end{align}"
},
{
"math_id": 31,
"text": " \\begin{bmatrix} V_2 \\\\ -I_2 \\end{bmatrix} = \\begin{bmatrix} 1 & -R \\\\ -sC & 1 + sCR \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ I_1 \\end{bmatrix}"
},
{
"math_id": 32,
"text": "\\begin{align}\n \\left[\\mathbf{y}\\right] &= [\\mathbf{z}]^{-1} \\\\\n \\left[\\mathbf{g}\\right] &= [\\mathbf{h}]^{-1} \\\\\n \\left[\\mathbf{b}\\right] &= [\\mathbf{a}]^{-1} \n\\end{align}"
},
{
"math_id": 33,
"text": " \\begin{bmatrix} V_1 \\\\ V_2 \\\\V_3 \\end{bmatrix} = \\begin{bmatrix} Z_{11} & Z_{12} & Z_{13} \\\\ Z_{21} & Z_{22} &Z_{23} \\\\ Z_{31} & Z_{32} & Z_{33} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ I_2 \\\\I_3 \\end{bmatrix} "
},
{
"math_id": 34,
"text": " V_2 = -Z_\\mathrm{L} I_2 \\, "
},
{
"math_id": 35,
"text": "\\begin{align}\n V_1 &= Z_{11} I_1 + Z_{12} I_2 \\\\\n -Z_\\mathrm{L} I_2 &= Z_{21} I_1 + Z_{22} I_2\n\\end{align}"
},
{
"math_id": 36,
"text": "\\begin{align}\n I_2 &= -\\frac{Z_{21}}{Z_\\mathrm{L} + Z_{22}} I_1 \\\\[3pt]\n V_1 &= Z_{11} I_1 - \\frac{Z_{12} Z_{21}}{Z_\\mathrm{L} + Z_{22}} I_1 \\\\[2pt]\n &= \\left(Z_{11} - \\frac{Z_{12} Z_{21}}{Z_\\mathrm{L} + Z_{22}}\\right) I_1 = Z_\\text{in} I_1\n\\end{align}"
}
]
| https://en.wikipedia.org/wiki?curid=1301093 |
13012903 | Forcing (computability) | Forcing in computability theory is a modification of Paul Cohen's original set-theoretic technique of forcing to deal with computability concerns.
Conceptually the two techniques are quite similar: in both one attempts to build generic objects (intuitively objects that are somehow 'typical') by meeting dense sets. Both techniques are described as a relation (customarily denoted formula_0) between 'conditions' and sentences. However, where set-theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, computability-theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable. Therefore, some of the more difficult machinery used in set-theoretic forcing can be eliminated or substantially simplified when defining forcing in computability. But while the machinery may be somewhat different, computability-theoretic and set-theoretic forcing are properly regarded as an application of the same technique to different classes of formulas.
Terminology.
In this article we use the following terminology.
means formula_6 and formula_7 are incompatible.
Note that for Cohen forcing formula_18 is the reverse of the containment relation. This leads to an unfortunate notational confusion where some computability theorists reverse the direction of the forcing partial order (exchanging formula_19 with formula_20, which is more natural for Cohen forcing, but is at odds with the notation used in set theory).
Generic objects.
The intuition behind forcing is that our conditions are finite approximations to some object we wish to build and that formula_21 is stronger than formula_22 when formula_21 agrees with everything formula_22 says about the object we are building and adds some information of its own. For instance in Cohen forcing the conditions can be viewed as finite approximations to a real and if formula_23 then formula_21 tells us the value of the real at more places.
In a moment we will define a relation formula_24 (read formula_21 forces formula_25) that holds between conditions (elements of formula_3) and sentences, but first we need to explain the language that formula_25 is a sentence for. However, forcing is a technique, not a definition, and the language for formula_25 will depend on the application one has in mind and the choice of formula_3.
The idea is that our language should express facts about the object we wish to build with our forcing construction.
Forcing relation.
The forcing relation formula_0 was developed by Paul Cohen, who introduced forcing as a technique for proving the independence of certain statements from the standard axioms of set theory, particularly the continuum hypothesis (CH).
The notation formula_26 is used to express that a particular condition or generic set forces a certain proposition or formula formula_27 to be true in the resulting forcing extension. Here's formula_28 represents the original universe of sets (the ground model), formula_0 denotes the forcing relation, and formula_27 is a statement in set theory.
When formula_26, it means that in a suitable forcing extension, the statement formula_27 will be true. | [
{
"math_id": 0,
"text": "\\Vdash"
},
{
"math_id": 1,
"text": "2^\\omega"
},
{
"math_id": 2,
"text": "2^{<\\omega}"
},
{
"math_id": 3,
"text": "P"
},
{
"math_id": 4,
"text": "\\succ_{P}"
},
{
"math_id": 5,
"text": "0_{P}"
},
{
"math_id": 6,
"text": "p"
},
{
"math_id": 7,
"text": "q"
},
{
"math_id": 8,
"text": "q \\succ_P p"
},
{
"math_id": 9,
"text": "p,q"
},
{
"math_id": 10,
"text": "r"
},
{
"math_id": 11,
"text": "p\\mid q"
},
{
"math_id": 12,
"text": "F"
},
{
"math_id": 13,
"text": "p,q \\in F \\implies p \\nmid q"
},
{
"math_id": 14,
"text": "p \\in F \\land q \\succ_P p \\implies q \\in F"
},
{
"math_id": 15,
"text": "F'"
},
{
"math_id": 16,
"text": "C"
},
{
"math_id": 17,
"text": "(\\tau \\succ_C \\sigma \\iff \\sigma \\supset \\tau"
},
{
"math_id": 18,
"text": "\\succ_{C}"
},
{
"math_id": 19,
"text": "\\succ_P"
},
{
"math_id": 20,
"text": "\\prec_P"
},
{
"math_id": 21,
"text": "\\sigma"
},
{
"math_id": 22,
"text": "\\tau"
},
{
"math_id": 23,
"text": "\\tau \\succ_C \\sigma"
},
{
"math_id": 24,
"text": "\\sigma \\Vdash_P \\psi"
},
{
"math_id": 25,
"text": "\\psi"
},
{
"math_id": 26,
"text": "V \\Vdash \\phi"
},
{
"math_id": 27,
"text": "\\phi"
},
{
"math_id": 28,
"text": "V"
}
]
| https://en.wikipedia.org/wiki?curid=13012903 |
13012982 | Vertically integrated liquid | Vertically integrated liquid (VIL) is an estimate of the total mass of precipitation in the clouds. The measurement is obtained by observing the reflectivity of the air which is obtained with weather radar.
Definition.
Reflectivity ("Z") in dBZ represents the intensity of radar echoes returning from a clouds. According to the wavelengths used in weather radars, only precipitation can be noted (drizzle, rain, snow, hail), not the cloud droplets nor water vapor, so Z is proportional to the rain rate. Using the sum in the vertical of Z, one can find the total mass of water equivalent in and below the precipitating cloud and that is what is VIL.
From the studies of Marshall and Palmer on the drop size distribution of rain drops, it is possible to find VIL:
formula_0
Where :
To note, the unit kg/m2 multiplied by water density (1 kg/liter) gives the surface accumulation in millimeters of rain: 1 kg/m2 = 1 mm.
Usage.
The VIL measurement is usually used in determining the size of prospective hail, the potential amount of rain under a thunderstorm, and the potential downdraft strength when combined with the height of the echo tops. VIL can be used to triage storms based on their severe potential. It is sometimes still used to assess storms for their potential severity.
Multicells.
Multicells usually have alternating VIL values. Multicells can have high VIL values on one radar picture, yet much smaller values in the next radar picture.
Wet microbursts.
When VIL values quickly fall, it might mean that a downburst is imminent. This is the result of the updraft within the cell weakening, thereby losing its ability to hold the copious amounts of moisture (including hail) within the storm's structure. Downbursts of this type are referred to as 'wet microbursts' by the National Weather Service for two reasons: (1) they contain heavy rainfall and (usually) hail; (2) they have damaging winds of greater than . Microbursts are classified as being 'a swath of damaging winds not exceeding in diameter'.
Thus, wet microbursts have been sometimes mistaken for a tornado by the general public, as the damage can be quick, hard hitting, and as important or more than an EF-1 tornado. An algorithm has been developed by S. Stewart, a meteorologist for the US National Weather Service, to estimate the potential maximum gust with a descending downdraft using VIL and the Echotop on radar:
formula_1
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "VIL = \\sum_{i=0}^{i=i_{max}} 3.44 * 10^{-6} \\left[ \\left( Z_i + Z_{i+1} \\right) /2 \\right]^{4/7} dh \\qquad \\left( in\\ kg / m^2 \\right)"
},
{
"math_id": 1,
"text": "Maximum\\ Gust= \\left[ \\left( 20.628571\\ ms^{-2} \\right) * VIL - \\left( 3.125 * 10^{-6}\\ s^{-2} \\right) * Echotop^2 \\right]^{0,5} \\qquad \\left( in\\ m/s \\right)"
}
]
| https://en.wikipedia.org/wiki?curid=13012982 |
1301548 | Isocost | In economics, an isocost line shows all combinations of inputs which cost the same total amount. Although similar to the budget constraint in consumer theory, the use of the isocost line pertains to cost-minimization in production, as opposed to utility-maximization. For the two production inputs labour and capital, with fixed unit costs of the inputs, the equation of the isocost line is
formula_0
where w represents the wage rate of labour, r represents the rental rate of capital, K is the amount of capital used, L is the amount of labour used, and C is the total cost of acquiring those quantities of the two inputs.
The absolute value of the slope of the isocost line, with capital plotted vertically and labour plotted horizontally, equals the ratio of unit costs of labour and capital. The slope is:
formula_1
The isocost line is combined with the isoquant map to determine the optimal production point at any given level of output. Specifically, the point of tangency between any isoquant and an isocost line gives the lowest-cost combination of inputs that can produce the level of output associated with that isoquant. Equivalently, it gives the maximum level of output that can be produced for a given total cost of inputs. A line joining tangency points of isoquants and isocosts (with input prices held constant) is called the expansion path.
The cost-minimization problem.
The cost-minimization problem of the firm is to choose an input bundle ("K","L") feasible for the output level "y" that costs as little as possible. A cost-minimizing input bundle is a point on the isoquant for the given "y" that is on the lowest possible isocost line. Put differently, a cost-minimizing input bundle must satisfy two conditions:
The case of smooth isoquants convex to the origin.
If the "y"-isoquant is smooth and convex to the origin and the cost-minimizing bundle involves a positive amount of each input, then at a cost-minimizing input bundle an isocost line is tangent to the "y"-isoquant.
Now since the absolute value of the slope of the isocost line is the input cost ratio formula_2, and the absolute value of the slope of an isoquant is the marginal rate of technical substitution (MRTS), we reach the following conclusion:
If the isoquants are smooth and convex to the origin and the cost-minimizing input bundle involves a positive amount of each input, then this bundle satisfies the following two conditions:
The condition that the MRTS be equal to "w"/"r" can be given the following intuitive interpretation. We know that the MRTS is equal to the ratio of the marginal products of the two inputs. So the condition that the MRTS be equal to the input cost ratio is equivalent to the condition that the marginal product per dollar is equal for the two inputs. This condition makes sense: at a particular input combination, if an extra dollar spent on input 1 yields more output than an extra dollar spent on input 2, then more of input 1 should be used and less of input 2, and so that input combination cannot be optimal. Only if a dollar spent on each input is equally productive is the input bundle optimal.
An isocost line is a curve which shows various combinations of inputs that cost the same total amount . For the two production inputs labour and capital, with fixed unit costs of the inputs, the isocost curve is a straight line . The isocost line is always used to determine the optimal production combined with the isoquant line .
if w represents the wage rate of labour, r represents the rental rate of capital, K is the amount of capital used, L is the amount of labour used, and C is the total cost of the two inputs, than the isocost line can be
C=rK+wL
In the figure, the point C / w on the horizontal axis represents that all the given costs are used in labor, and the point C / r on the vertical axis represents that all the given costs are used in capital . The line connecting these two points is the isocost line.
The slope is -w/r which represents the relative price. Any point within the isocost line indicates that there are surplus after purchasing the combination of labor and capital at that point. Any point outside the isocost line indicates that the combination of labor and capital is not enough to be purchased at the given cost. Only the point in the isocost line shows the combination that can be purchased exactly at the given cost .
If the prices of the t factors change, the isocost line will also change . Suppose w rises, so that the maximum amount of labor that can be employed at the same cost will decrease, that is, the intercept of the isocost line on the L axis will decrease; and because r remains unchanged, the intercept of the isocost line on the K axis will remain unchanged. | [
{
"math_id": 0,
"text": "rK+wL = C\\,"
},
{
"math_id": 1,
"text": "-w/r. \\,"
},
{
"math_id": 2,
"text": "w/r"
}
]
| https://en.wikipedia.org/wiki?curid=1301548 |
1301559 | Leibniz formula for π | Signed odd unit fractions sum to π/4
In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that
formula_0
an alternating series.
It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), and was later independently rediscovered by James Gregory in 1671 and Leibniz in 1673. The Taylor series for the inverse tangent function, often called "Gregory's series", is
formula_1
The Leibniz formula is the special case formula_2
It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at formula_3 and therefore the value "β"(1) of the Dirichlet beta function.
Proofs.
Proof 1.
formula_4
Considering only the integral in the last term, we have:
formula_5
Therefore, by the squeeze theorem, as "n" → ∞, we are left with the Leibniz series:
formula_6
Proof 2.
Let formula_7, when formula_8, the series formula_9 converges uniformly, then
formula_10
Therefore, if formula_11 approaches formula_12 so that it is continuous and converges uniformly, the proof is complete, where, the series formula_13 to be converges by the Leibniz's test, and also, formula_11 approaches formula_12 from within the Stolz angle, so from Abel's theorem this is correct.
Convergence.
Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. Calculating π to 10 correct decimal places using direct summation of the series requires precisely five billion terms because < 10−10 for "k" > 2 × 1010 − (one needs to apply Calabrese error bound). To get 4 correct decimal places (error of 0.00005) one needs 5000 terms. Even better than Calabrese or Johnsonbaugh error bounds are available.
However, the Leibniz formula can be used to calculate π to high precision (hundreds of digits or more) using various convergence acceleration techniques. For example, the Shanks transformation, Euler transform or Van Wijngaarden transformation, which are general methods for alternating series, can be applied effectively to the partial sums of the Leibniz series. Further, combining terms pairwise gives the non-alternating series
formula_14
which can be evaluated to high precision from a small number of terms using Richardson extrapolation or the Euler–Maclaurin formula. This series can also be transformed into an integral by means of the Abel–Plana formula and evaluated using techniques for numerical integration.
Unusual behaviour.
If the series is truncated at the right time, the decimal expansion of the approximation will agree with that of π for many more digits, except for isolated digits or digit groups. For example, taking five million terms yields
formula_15
where the underlined digits are wrong. The errors can in fact be predicted; they are generated by the Euler numbers "En" according to the asymptotic formula
formula_16
where "N" is an integer divisible by 4. If "N" is chosen to be a power of ten, each term in the right sum becomes a finite decimal fraction. The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz formula.
Euler product.
The Leibniz formula can be interpreted as a Dirichlet series using the unique non-principal Dirichlet character modulo 4. As with other Dirichlet series, this allows the infinite sum to be converted to an infinite product with one term for each prime number. Such a product is called an Euler product. It is:
formula_17
In this product, each term is a superparticular ratio, each numerator is an odd prime number, and each denominator is the nearest multiple of 4 to the numerator.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\n\\frac{\\pi}{4} = 1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\frac{1}{9}-\\cdots = \\sum_{k=0}^{\\infty} \\frac{(-1)^{k}}{2k + 1},"
},
{
"math_id": 1,
"text": "\n\\arctan x = x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\frac{x^7}{7} + \\cdots\n= \\sum_{k=0}^\\infty \\frac{(-1)^kx^{2k+1}}{2k+1}.\n"
},
{
"math_id": 2,
"text": "\\arctan 1 = \\tfrac14\\pi."
},
{
"math_id": 3,
"text": "s=1,"
},
{
"math_id": 4,
"text": "\\begin{align}\n\\frac{\\pi}{4} &= \\arctan(1) \\\\ &= \\int_0^1 \\frac 1{1+x^2} \\, dx \\\\[8pt]\n& = \\int_0^1\\left(\\sum_{k=0}^n (-1)^k x^{2k}+\\frac{(-1)^{n+1}\\,x^{2n+2} }{1+x^2}\\right) \\, dx \\\\[8pt]\n& = \\left(\\sum_{k=0}^n \\frac{(-1)^k}{2k+1}\\right)\n+(-1)^{n+1} \\left(\\int_0^1\\frac{x^{2n+2}}{1+x^2} \\, dx\\right)\n\\end{align}"
},
{
"math_id": 5,
"text": "0 \\le \\int_0^1 \\frac{x^{2n+2}}{1+x^2}\\,dx \\le \\int_0^1 x^{2n+2}\\,dx = \\frac{1}{2n+3} \\;\\rightarrow 0 \\text{ as } n \\rightarrow \\infty."
},
{
"math_id": 6,
"text": "\\frac{\\pi}4 = \\sum_{k=0}^\\infty\\frac{(-1)^k}{2k+1}"
},
{
"math_id": 7,
"text": "f(z) = \\sum_{n=0}^{\\infty}\\frac{(-1)^n}{2n+1}z^{2n+1}"
},
{
"math_id": 8,
"text": "|z|<1"
},
{
"math_id": 9,
"text": " \\sum_{k=0}^\\infty (-1)^k z^{2k}"
},
{
"math_id": 10,
"text": "\\arctan(z) = \\int_{0}^{z} \\frac {1}{1+t^2} dt =\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{2n+1}z^{2n+1} = f(z) \\ (|z|<1). "
},
{
"math_id": 11,
"text": "f(z)"
},
{
"math_id": 12,
"text": "f(1)"
},
{
"math_id": 13,
"text": "\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{2n+1}"
},
{
"math_id": 14,
"text": "\\frac{\\pi}{4} = \\sum_{n=0}^{\\infty} \\left(\\frac{1}{4n+1}-\\frac{1}{4n+3}\\right) = \\sum_{n=0}^{\\infty} \\frac{2}{(4n+1)(4n+3)}"
},
{
"math_id": 15,
"text": "3.141592\\underline{4}5358979323846\\underline{4}643383279502\\underline{7}841971693993\\underline{873}058..."
},
{
"math_id": 16,
"text": "\\frac{\\pi}{2} - 2 \\sum_{k=1}^{N/2} \\frac{(-1)^{k-1}}{2k-1} \\sim \\sum_{m=0}^\\infty \\frac{E_{2m}}{N^{2m+1}}"
},
{
"math_id": 17,
"text": "\\begin{align}\n\\frac\\pi4 &= \\biggl(\\prod_{p \\,\\equiv\\, 1\\ (\\text{mod}\\ 4)}\\frac{p}{p-1}\\biggr)\n \\biggl( \\prod_{p\\,\\equiv\\, 3\\ (\\text{mod}\\ 4)}\\frac{p}{p+1}\\biggr) \\\\[7mu]\n &= \\frac{3}{4} \\cdot \\frac{5}{4} \\cdot \\frac{7}{8} \\cdot\n \\frac{11}{12} \\cdot \\frac{13}{12} \\cdot \\frac{17}{16} \\cdot\n \\frac{19}{20} \\cdot \\frac{23}{24} \\cdot \\frac{29}{28} \\cdots\n\\end{align}"
}
]
| https://en.wikipedia.org/wiki?curid=1301559 |
1301687 | Wallis product | Infinite product for pi
In mathematics, the Wallis product for π, published in 1656 by John Wallis, states that
formula_0
Proof using integration.
Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous. A modern derivation can be found by examining formula_1 for even and odd values of formula_2, and noting that for large formula_2, increasing formula_2 by 1 results in a change that becomes ever smaller as formula_2 increases. Let
formula_3
(This is a form of Wallis' integrals.) Integrate by parts:
formula_4
formula_5
Now, we make two variable substitutions for convenience to obtain:
formula_6
formula_7
We obtain values for formula_8 and formula_9 for later use.
formula_10
Now, we calculate for even values formula_11 by repeatedly applying the recurrence relation result from the integration by parts. Eventually, we end get down to formula_8, which we have calculated.
formula_12
formula_13
Repeating the process for odd values formula_14,
formula_15
formula_16
We make the following observation, based on the fact that formula_17
formula_18
formula_19
Dividing by formula_14:
formula_20, where the equality comes from our recurrence relation.
By the squeeze theorem,
formula_21
formula_22
formula_23
Proof using Laplace's method.
See the main page on Gaussian integral.
Proof using Euler's infinite product for the sine function.
While the proof above is typically featured in modern calculus textbooks, the Wallis product is, in retrospect, an easy corollary of the later Euler infinite product for the sine function.
formula_24
Let formula_25:
formula_26
Relation to Stirling's approximation.
Stirling's approximation for the factorial function formula_27 asserts that
formula_28
Consider now the finite approximations to the Wallis product, obtained by taking the first formula_29 terms in the product
formula_30
where formula_31 can be written as
formula_32
Substituting Stirling's approximation in this expression (both for formula_33 and formula_34) one can deduce (after a short calculation) that formula_31 converges to formula_35 as formula_36.
Derivative of the Riemann zeta function at zero.
The Riemann zeta function and the Dirichlet eta function can be defined:
formula_37
Applying an Euler transform to the latter series, the following is obtained:
formula_38
formula_39
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\begin{align}\n\\frac{\\pi}{2} & = \\prod_{n=1}^{\\infty} \\frac{ 4n^2 }{ 4n^2 - 1 } = \\prod_{n=1}^{\\infty} \\left(\\frac{2n}{2n-1} \\cdot \\frac{2n}{2n+1}\\right) \\\\[6pt]\n& = \\Big(\\frac{2}{1} \\cdot \\frac{2}{3}\\Big) \\cdot \\Big(\\frac{4}{3} \\cdot \\frac{4}{5}\\Big) \\cdot \\Big(\\frac{6}{5} \\cdot \\frac{6}{7}\\Big) \\cdot \\Big(\\frac{8}{7} \\cdot \\frac{8}{9}\\Big) \\cdot \\; \\cdots \\\\\n\\end{align}"
},
{
"math_id": 1,
"text": "\\int_0^\\pi \\sin^n x\\,dx"
},
{
"math_id": 2,
"text": "n"
},
{
"math_id": 3,
"text": "I(n) = \\int_0^\\pi \\sin^n x\\,dx."
},
{
"math_id": 4,
"text": "\\begin{align}\n u &= \\sin^{n-1}x \\\\\n \\Rightarrow du &= (n-1) \\sin^{n-2}x \\cos x\\,dx \\\\\n dv &= \\sin x\\,dx \\\\\n \\Rightarrow v &= -\\cos x\n\\end{align}"
},
{
"math_id": 5,
"text": "\\begin{align}\n \\Rightarrow I(n) &= \\int_0^\\pi \\sin^n x\\,dx \\\\[6pt]\n {} &= -\\sin^{n-1}x\\cos x \\Biggl|_0^\\pi - \\int_0^\\pi (-\\cos x)(n-1) \\sin^{n-2}x \\cos x\\,dx \\\\[6pt]\n {} &= 0 + (n-1) \\int_0^\\pi \\cos^2x \\sin^{n-2}x\\,dx, \\qquad n > 1 \\\\[6pt]\n {} &= (n - 1) \\int_0^\\pi (1-\\sin^2 x) \\sin^{n-2}x\\,dx \\\\[6pt]\n {} &= (n - 1) \\int_0^\\pi \\sin^{n-2}x\\,dx - (n - 1) \\int_0^\\pi \\sin^{n}x\\,dx \\\\[6pt]\n {} &= (n - 1) I(n-2)-(n-1) I(n) \\\\[6pt]\n {} &= \\frac{n-1}{n} I(n-2) \\\\[6pt]\n \\Rightarrow \\frac{I(n)}{I(n-2)}\n &= \\frac{n-1}{n} \\\\[6pt]\n\\end{align}"
},
{
"math_id": 6,
"text": "I(2n) = \\frac{2n-1}{2n}I(2n-2)"
},
{
"math_id": 7,
"text": "I(2n+1) = \\frac{2n}{2n+1}I(2n-1)"
},
{
"math_id": 8,
"text": "I(0)"
},
{
"math_id": 9,
"text": "I(1)"
},
{
"math_id": 10,
"text": "\\begin{align}\n I(0) &= \\int_0^\\pi dx = x\\Biggl|_0^\\pi = \\pi \\\\[6pt]\n I(1) &= \\int_0^\\pi \\sin x\\,dx = -\\cos x \\Biggl|_0^\\pi = (-\\cos \\pi)-(-\\cos 0) = -(-1)-(-1) = 2 \\\\[6pt]\n\\end{align}"
},
{
"math_id": 11,
"text": "I(2n)"
},
{
"math_id": 12,
"text": "I(2n)=\\int_0^\\pi \\sin^{2n}x\\,dx = \\frac{2n-1}{2n}I(2n-2) = \\frac{2n-1}{2n} \\cdot \\frac{2n-3}{2n-2}I(2n-4)"
},
{
"math_id": 13,
"text": "=\\frac{2n-1}{2n} \\cdot \\frac{2n-3}{2n-2} \\cdot \\frac{2n-5}{2n-4} \\cdot \\cdots \\cdot \\frac{5}{6} \\cdot \\frac{3}{4} \\cdot \\frac{1}{2} I(0)=\\pi \\prod_{k=1}^n \\frac{2k-1}{2k}"
},
{
"math_id": 14,
"text": "I(2n+1)"
},
{
"math_id": 15,
"text": "I(2n+1)=\\int_0^\\pi \\sin^{2n+1}x\\,dx=\\frac{2n}{2n+1}I(2n-1)=\\frac{2n}{2n+1} \\cdot \\frac{2n-2}{2n-1}I(2n-3)"
},
{
"math_id": 16,
"text": "=\\frac{2n}{2n+1} \\cdot \\frac{2n-2}{2n-1} \\cdot \\frac{2n-4}{2n-3} \\cdot \\cdots \\cdot \\frac{6}{7} \\cdot \\frac{4}{5} \\cdot \\frac{2}{3} I(1)=2 \\prod_{k=1}^n \\frac{2k}{2k+1}"
},
{
"math_id": 17,
"text": "\\sin{x} \\leq 1"
},
{
"math_id": 18,
"text": "\\sin^{2n+1}x \\le \\sin^{2n}x \\le \\sin^{2n-1}x, 0 \\le x \\le \\pi"
},
{
"math_id": 19,
"text": "\\Rightarrow I(2n+1) \\le I(2n) \\le I(2n-1)"
},
{
"math_id": 20,
"text": "\\Rightarrow 1 \\le \\frac{I(2n)}{I(2n+1)} \\le \\frac{I(2n-1)}{I(2n+1)}=\\frac{2n+1}{2n}"
},
{
"math_id": 21,
"text": "\\Rightarrow \\lim_{n\\rightarrow\\infty} \\frac{I(2n)}{I(2n+1)}=1"
},
{
"math_id": 22,
"text": "\\lim_{n\\rightarrow\\infty} \\frac{I(2n)}{I(2n+1)}=\\frac{\\pi}{2} \\lim_{n\\rightarrow\\infty} \\prod_{k=1}^n \\left(\\frac{2k-1}{2k} \\cdot \\frac{2k+1}{2k}\\right)=1"
},
{
"math_id": 23,
"text": "\\Rightarrow \\frac{\\pi}{2}=\\prod_{k=1}^\\infty \\left(\\frac{2k}{2k-1} \\cdot \\frac{2k}{2k+1}\\right)=\\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdot \\cdots"
},
{
"math_id": 24,
"text": "\\frac{\\sin x}{x} = \\prod_{n=1}^\\infty\\left(1 - \\frac{x^2}{n^2\\pi^2}\\right)"
},
{
"math_id": 25,
"text": "x = \\frac{\\pi}{2}"
},
{
"math_id": 26,
"text": "\\begin{align}\n \\Rightarrow\\frac{2}{\\pi} &= \\prod_{n=1}^\\infty \\left(1 - \\frac{1}{4n^2}\\right) \\\\[6pt]\n \\Rightarrow\\frac{\\pi}{2} &= \\prod_{n=1}^\\infty \\left(\\frac{4n^2}{4n^2 - 1}\\right) \\\\[6pt]\n &= \\prod_{n=1}^\\infty \\left(\\frac{2n}{2n-1}\\cdot\\frac{2n}{2n+1}\\right) = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdots\n\\end{align}\n"
},
{
"math_id": 27,
"text": "n!"
},
{
"math_id": 28,
"text": "n! = \\sqrt {2\\pi n} {\\left(\\frac{n}{e}\\right)}^n \\left[1 + O\\left(\\frac{1}{n}\\right) \\right]."
},
{
"math_id": 29,
"text": "k"
},
{
"math_id": 30,
"text": "p_k = \\prod_{n=1}^{k} \\frac{2n}{2n - 1}\\frac{2n}{2n + 1},"
},
{
"math_id": 31,
"text": "p_k"
},
{
"math_id": 32,
"text": "\\begin{align}\n p_k &= {1 \\over {2k + 1}} \\prod_{n=1}^{k} \\frac{(2n)^4}{[(2n)(2n - 1)]^2} \\\\[6pt]\n &= {1 \\over {2k + 1}} \\cdot {{2^{4k}\\,(k!)^4} \\over {[(2k)!]^2}}.\n\\end{align}"
},
{
"math_id": 33,
"text": "k!"
},
{
"math_id": 34,
"text": "(2k)!"
},
{
"math_id": 35,
"text": "\\frac{\\pi}{2}"
},
{
"math_id": 36,
"text": "k \\rightarrow \\infty"
},
{
"math_id": 37,
"text": "\\begin{align}\n \\zeta(s) &= \\sum_{n=1}^\\infty \\frac{1}{n^s}, \\Re(s)>1 \\\\[6pt]\n \\eta(s) &= (1-2^{1-s})\\zeta(s) \\\\[6pt]\n &= \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s}, \\Re(s)>0\n\\end{align}"
},
{
"math_id": 38,
"text": "\\begin{align}\n\n \\eta(s) &= \\frac{1}{2}+\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\left[\\frac{1}{n^s}-\\frac{1}{(n+1)^s}\\right], \\Re(s)>-1 \\\\[6pt]\n \\Rightarrow \\eta'(s) &= (1-2^{1-s})\\zeta'(s)+2^{1-s} (\\ln 2) \\zeta(s) \\\\[6pt]\n &= -\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\left[\\frac{\\ln n}{n^s}-\\frac{\\ln (n+1)}{(n+1)^s}\\right], \\Re(s)>-1\n\\end{align}"
},
{
"math_id": 39,
"text": "\\begin{align}\n \\Rightarrow \\eta'(0) &= -\\zeta'(0) - \\ln 2 = -\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\left[\\ln n-\\ln (n+1)\\right] \\\\[6pt]\n &= -\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\ln \\frac{n}{n+1} \\\\[6pt]\n &= -\\frac{1}{2} \\left(\\ln \\frac{1}{2} - \\ln \\frac{2}{3} + \\ln \\frac{3}{4} - \\ln \\frac{4}{5} + \\ln \\frac{5}{6} - \\cdots\\right) \\\\[6pt]\n &= \\frac{1}{2} \\left(\\ln \\frac{2}{1} + \\ln \\frac{2}{3} + \\ln \\frac{4}{3} + \\ln \\frac{4}{5} + \\ln \\frac{6}{5} + \\cdots\\right) \\\\[6pt]\n &= \\frac{1}{2} \\ln\\left(\\frac{2}{1}\\cdot\\frac{2}{3}\\cdot\\frac{4}{3}\\cdot\\frac{4}{5}\\cdot\\cdots\\right) = \\frac{1}{2} \\ln\\frac{\\pi}{2} \\\\\n \\Rightarrow \\zeta'(0) &= -\\frac{1}{2} \\ln\\left(2 \\pi\\right)\n\\end{align}"
},
{
"math_id": 40,
"text": "\\pi"
}
]
| https://en.wikipedia.org/wiki?curid=1301687 |
1302110 | Adaptive equalizer | An adaptive equalizer is an equalizer that automatically adapts to time-varying properties of the communication channel. It is frequently used with coherent modulations such as phase-shift keying, mitigating the effects of multipath propagation and Doppler spreading.
Adaptive equalizers are a subclass of adaptive filters. The central idea is altering the filter's coefficients to optimize a filter characteristic. For example, in case of linear discrete-time filters, the following equation can be used:
formula_0
where formula_1 is the vector of the filter's coefficients, formula_2 is the received signal covariance matrix and formula_3 is the cross-correlation vector between the tap-input vector and the desired response. In practice, the last quantities are not known and, if necessary, must be estimated during the equalization procedure either explicitly or implicitly.
Many adaptation strategies exist. They include, e.g.:
A well-known example is the decision feedback equalizer, a filter that uses feedback of detected symbols in addition to conventional equalization of future symbols. Some systems use predefined training sequences to provide reference points for the adaptation process.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " \\mathbf{w}_{opt} = \\mathbf{R}^{-1}\\mathbf{p}"
},
{
"math_id": 1,
"text": "\\mathbf{w}_{opt}"
},
{
"math_id": 2,
"text": "\\mathbf{R}"
},
{
"math_id": 3,
"text": "\\mathbf{p}"
},
{
"math_id": 4,
"text": "x"
},
{
"math_id": 5,
"text": "d(n)"
}
]
| https://en.wikipedia.org/wiki?curid=1302110 |
13021753 | Ostrowski–Hadamard gap theorem | In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence. The result is named after the mathematicians Alexander Ostrowski and Jacques Hadamard.
Statement of the theorem.
Let 0 < "p"1 < "p"2 < ... be a sequence of integers such that, for some "λ" > 1 and all "j" ∈ N,
formula_0
Let ("α""j")"j"∈N be a sequence of complex numbers such that the power series
formula_1
has radius of convergence 1. Then no point "z" with |"z"| = 1 is a regular point for "f"; i.e. "f" cannot be analytically extended from the open unit disc "D" to any larger open set—not even to a single point on the boundary of "D".
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\frac{p_{j + 1}}{p_{j}} > \\lambda."
},
{
"math_id": 1,
"text": "f(z) = \\sum_{j \\in \\mathbf{N}} \\alpha_{j} z^{p_{j}}"
}
]
| https://en.wikipedia.org/wiki?curid=13021753 |
1302341 | Luneburg lens | Spherically symmetric gradient-index lens
A Luneburg lens (original German "Lüneburg lens") is a spherically symmetric gradient-index lens. A typical Luneburg lens's refractive index "n" decreases radially from the center to the outer surface. They can be made for use with electromagnetic radiation from visible light to radio waves.
For certain index profiles, the lens will form perfect geometrical images of two given concentric spheres onto each other. There are an infinite number of refractive-index profiles that can produce this effect. The simplest such solution was proposed by Rudolf Luneburg in 1944. Luneburg's solution for the refractive index creates two conjugate foci outside the lens. The solution takes a simple and explicit form if one focal point lies at infinity, and the other on the opposite surface of the lens. J. Brown and A. S. Gutman subsequently proposed solutions which generate one internal focal point and one external focal point. These solutions are not unique; the set of solutions are defined by a set of definite integrals which must be evaluated numerically.
Designs.
Luneburg's solution.
Each point on the surface of an ideal Luneburg lens is the focal point for parallel radiation incident on the opposite side. Ideally, the dielectric constant formula_0 of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive index formula_1 falls from formula_2 to 1), according to
formula_3
where formula_4 is the radius of the lens. Because the refractive index at the surface is the same as that of the surrounding medium, no reflection occurs at the surface. Within the lens, the paths of the rays are arcs of ellipses.
Maxwell's fish-eye lens.
Maxwell's fish-eye lens is also an example of the generalized Luneburg lens. The fish-eye, which was first fully described by Maxwell in 1854 (and therefore pre-dates Luneburg's solution), has a refractive index varying according to
formula_5
where formula_6 is the index of refraction at the center of the lens and formula_4 is the radius of the lens's spherical surface. The index of refraction at the lens's surface is formula_7. The lens images each point on the spherical surface to the opposite point on the surface. Within the lens, the paths of the rays are arcs of circles.
Publication and attribution.
The properties of this lens are described in one of a number of set problems or puzzles in the 1853 "Cambridge and Dublin Mathematical Journal". The challenge is to find the refractive index as a function of radius, given that a ray describes a circular path, and further to prove the focusing properties of the lens. The solution is given in the 1854 edition of the same journal. The problems and solutions were originally published anonymously, but the solution of this problem (and one other) were included in Niven's "The Scientific Papers of James Clerk Maxwell", which was published 11 years after Maxwell's death.
Applications.
In practice, Luneburg lenses are normally layered structures of discrete concentric shells, each of a different refractive index. These shells form a stepped refractive index profile that differs slightly from Luneburg's solution. This kind of lens is usually employed for microwave frequencies, especially to construct efficient microwave antennas and radar calibration standards. Cylindrical analogues of the Luneburg lens are also used for collimating light from laser diodes.
Radar reflector.
Luneburg reflectors (the marked protrusion) on an F-35
A radar reflector can be made from a Luneburg lens by metallizing parts of its surface. Radiation from a distant radar transmitter is focussed onto the underside of the metallization on the opposite side of the lens; here it is reflected, and focussed back onto the radar station. A difficulty with this scheme is that metallized regions block the entry or exit of radiation on that part of the lens, but the non-metallized regions result in a blind-spot on the opposite side.
Removable Luneburg lens type radar reflectors are sometimes attached to military aircraft in order to make stealth aircraft visible during training operations, or to conceal their true radar signature. Unlike other types of radar reflectors, their shape doesn't affect the handling of the aircraft.
Microwave antenna.
A Luneburg lens can be used as the basis of a high-gain radio antenna. This antenna is comparable to a dish antenna, but uses the lens rather than a parabolic reflector as the main focusing element. As with the dish antenna, a "feed" to the receiver or from the transmitter is placed at the focus, the feed typically consisting of a horn antenna. The phase centre of the feed horn must coincide with the point of focus, but since the phase centre is invariably somewhat inside the mouth of the horn, it cannot be brought right up against the surface of the lens. Consequently it is necessary to use a variety of Luneburg lens that focusses somewhat beyond its surface, rather than the classic lens with the focus lying on the surface.
A Luneburg lens antenna offers a number of advantages over a parabolic dish. Because the lens is spherically symmetric, the antenna can be steered by moving the feed around the lens, without having to bodily rotate the whole antenna. Again, because the lens is spherically symmetric, a single lens can be used with several feeds looking in widely different directions. In contrast, if multiple feeds are used with a parabolic reflector, all must be within a small angle of the optical axis to avoid suffering coma (a form of de-focussing). Apart from offset systems, dish antennas suffer from the feed and its supporting structure partially obscuring the main element ("aperture blockage"); in common with other refracting systems, the Luneburg lens antenna avoids this problem.
A variation on the Luneburg lens antenna is the "hemispherical Luneburg lens antenna" or "Luneburg reflector antenna". This uses just one hemisphere of a Luneburg lens, with the cut surface of the sphere resting on a reflecting metal ground plane. The arrangement halves the weight of the lens, and the ground plane provides a convenient means of support. However, the feed does partially obscure the lens when the angle of incidence on the reflector is less than about 45°.
Path of a ray within the lens.
For any spherically symmetric lens, each ray lies entirely in a plane passing through the centre of the lens. The initial direction of the ray defines a line which together with the centre-point of the lens identifies a plane bisecting the lens. Being a plane of symmetry of the lens, the gradient of the refractive index has no component perpendicular to this plane to cause the ray to deviate either to one side of it or the other. In the plane, the circular symmetry of the system makes it convenient to use polar coordinates formula_8 to describe the ray's trajectory.
Given any two points on a ray (such as the point of entry and exit from the lens), Fermat's principle asserts that the path that the ray takes between them is that which it can traverse in the least possible time. Given that the speed of light at any point in the lens is inversely proportional to the refractive index, and by Pythagoras, the time of transit between two points formula_9 and formula_10 is
formula_11
where formula_12 is the speed of light in vacuum. Minimizing this formula_13 yields a second-order differential equation determining the dependence of formula_14 on formula_15 along the path of the ray. This type of minimization problem has been extensively studied in Lagrangian mechanics, and a ready-made solution exists in the form of the Beltrami identity, which immediately supplies the first integral of this second-order equation. Substituting formula_16 (where formula_17 represents formula_18), into this identity gives
formula_19
where formula_20 is a constant of integration. This first-order differential equation is separable, that is it can be re-arranged so that formula_14 only appears on one side, and formula_15 only on the other:
formula_21
The parameter formula_20 is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for formula_15 as a function of formula_14. In general it provides the relative rates of change of formula_15 and formula_14, which may be integrated numerically to follow the path of the ray through the lens.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\epsilon_r"
},
{
"math_id": 1,
"text": "n"
},
{
"math_id": 2,
"text": "\\sqrt{2}"
},
{
"math_id": 3,
"text": "n = \\sqrt{\\epsilon_r} = \\sqrt{2 - \\left( \\frac{r}{R} \\right)^2},"
},
{
"math_id": 4,
"text": "R"
},
{
"math_id": 5,
"text": "n(r) = \\sqrt{\\epsilon_r} = \\frac{n_0}{1 + \\left( \\frac{r}{R} \\right)^2},"
},
{
"math_id": 6,
"text": "n_0"
},
{
"math_id": 7,
"text": "n_0/2"
},
{
"math_id": 8,
"text": "(r, \\theta)"
},
{
"math_id": 9,
"text": "(r_1, \\theta_1)"
},
{
"math_id": 10,
"text": "(r_2, \\theta_2)"
},
{
"math_id": 11,
"text": "T = \\int _{(r_1, \\theta_1)}^{(r_2, \\theta_2)} \\frac{n(r)}{c} \\sqrt{(r \\,d\\theta)^2 + d r^2} = \\frac{1}{c} \\int _{\\theta_1}^{\\theta_2} n(r) \\sqrt{r^2 + \\left( \\frac{d r}{d \\theta}\\right)^2} \\,d\\theta,"
},
{
"math_id": 12,
"text": "c"
},
{
"math_id": 13,
"text": "T"
},
{
"math_id": 14,
"text": "r"
},
{
"math_id": 15,
"text": "\\theta"
},
{
"math_id": 16,
"text": "L(r, r') = n(r) \\sqrt{r'^2 + r^2}"
},
{
"math_id": 17,
"text": "r'"
},
{
"math_id": 18,
"text": "\\tfrac{d r}{d \\theta}"
},
{
"math_id": 19,
"text": "n(r) \\sqrt{r'^2 + r^2} - n(r) \\frac{r'^2}{\\sqrt{r'^2 + r^2}} = h,"
},
{
"math_id": 20,
"text": "h"
},
{
"math_id": 21,
"text": "d \\theta = \\frac{h}{r \\sqrt{\\big(n(r)\\big)^2 r^2 - h^2}} \\,dr."
}
]
| https://en.wikipedia.org/wiki?curid=1302341 |
13024295 | Iitaka dimension | In algebraic geometry, the Iitaka dimension of a line bundle "L" on an algebraic variety "X" is the dimension of the image of the rational map to projective space determined by "L". This is 1 less than the dimension of the section ring of "L"
formula_0
The Iitaka dimension of "L" is always less than or equal to the dimension of "X". If "L" is not effective, then its Iitaka dimension is usually defined to be formula_1 or simply said to be negative (some early references define it to be −1). The Iitaka dimension of "L" is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by Shigeru Iitaka (1970, 1971).
Big line bundles.
A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If "f" : "Y" → "X" is a birational morphism of varieties, and if "L" is a big line bundle on "X", then "f"*"L" is a big line bundle on "Y".
All ample line bundles are big.
Big line bundles need not determine birational isomorphisms of "X" with its image. For example, if "C" is a hyperelliptic curve (such as a curve of genus two), then its canonical bundle is big, but the rational map it determines is not a birational isomorphism. Instead, it is a two-to-one cover of the canonical curve of "C", which is a rational normal curve.
Kodaira dimension.
The Iitaka dimension of the canonical bundle of a smooth variety is called its Kodaira dimension.
Iitaka conjecture.
Consider on complex algebraic varieties in the following.
Let K be the canonical bundle on M. The dimension of H0(M,Km), holomorphic sections of Km, is denoted by Pm(M), called m-genus. Let
formula_2
then N(M) becomes to be all of the positive integer with non-zero m-genus. When N(M) is not empty, for formula_3 m-pluricanonical map formula_4 is defined as the map
formula_5
where formula_6 are the bases of H0(M,Km). Then the image of formula_4, formula_7 is defined as the submanifold of formula_8.
For certain formula_9 let formula_10 be the m-pluricanonical map where W is the complex manifold embedded into projective space PN.
In the case of surfaces with κ(M)=1 the above W is replaced by a curve C, which is an elliptic curve (κ(C)=0). We want to extend this fact to the general dimension and obtain the analytic fiber structure depicted in the upper right figure.
Given a birational map formula_11, m-pluricanonical map brings the commutative diagram depicted in the left figure, which means that formula_12, i.e. m-pluricanonical genus is birationally invariant.
It is shown by Iitaka that given n-dimensional compact complex manifold "M" with its Kodaira dimension κ(M) satisfying 1 ≤ κ(M) ≤ n-1, there are enough large "m"1,"m"2 such that formula_13 and formula_14 are birationally equivalent, which means there are the birational map formula_15. Namely, the diagram depicted in the right figure is commutative.
Furthermore, one can select formula_16 that is birational with formula_17 and formula_18 that is birational with both formula_19 and formula_19 such that
formula_20
is birational map, the fibers of formula_21 are simply connected and the general fibers of formula_21
formula_22
have Kodaira dimension 0.
The above fiber structure is called the Iitaka fiber space. In the case of the surface S ("n" = 2 = dim(S)), W* is the algebraic curve, the fiber structure is of dimension 1, and then the general fibers have the Kodaira dimension 0 i.e. elliptic curve. Therefore, S is the elliptic surface. These fact can be generalized to the general "n". Therefore The study of the higher-dimensional birational geometry decompose to the part of κ=-∞,0,n and the fiber space whose fibers is of κ=0.
The following additional formula by Iitaka, called Iitaka conjecture, is important for the classification of algebraic varieties or compact complex manifolds.
This conjecture has been only partly solved, for example in the case of
Moishezon manifolds. The classification theory might been said to be the effort to solve the Iitaka conjecture and lead another theorems that the three-dimensional variety V is abelian if and only if κ(V)=0 and q(V)=3 and its generalization so on. The minimal model program might be led from this conjecture. | [
{
"math_id": 0,
"text": "R(X, L) = \\bigoplus_{d=0}^\\infty H^0(X, L^{\\otimes d})."
},
{
"math_id": 1,
"text": "-\\infty"
},
{
"math_id": 2,
"text": "N(M)=\\{m\\ge1|P_m(M)\\ge1\\},"
},
{
"math_id": 3,
"text": "m\\in N(M)"
},
{
"math_id": 4,
"text": "\\Phi_{mK}"
},
{
"math_id": 5,
"text": "\\begin{align}\n\\Phi_{mK}: & M\\longrightarrow\\ \\ \\ \\ \\ \\ \\mathbb{P}^N \\\\\n& z\\ \\ \\ \\mapsto\\ \\ (\\varphi_0(z):\\varphi_1(z):\\cdots:\\varphi_N(z)) \n\\end{align}"
},
{
"math_id": 6,
"text": "\\varphi_i"
},
{
"math_id": 7,
"text": "\\Phi_{mK}(M)"
},
{
"math_id": 8,
"text": "\\mathbb{P}^N"
},
{
"math_id": 9,
"text": "m"
},
{
"math_id": 10,
"text": "\\Phi_{mk}:M\\rightarrow W=\\Phi_{mK}(M)\\subset \\mathbb{P}^N"
},
{
"math_id": 11,
"text": "\\varphi:M \\longrightarrow W"
},
{
"math_id": 12,
"text": "\\Phi_{mK}(M)=\\Phi_{mK}(W)"
},
{
"math_id": 13,
"text": "\\Phi_{m_1K}:M\\longrightarrow W_{m_1}(M)"
},
{
"math_id": 14,
"text": "\\Phi_{m_2K}:M\\longrightarrow W_{m_2}(M)"
},
{
"math_id": 15,
"text": "\\varphi:W_{m_1}\\longrightarrow W_{m_2}(M)"
},
{
"math_id": 16,
"text": "M^*"
},
{
"math_id": 17,
"text": "M"
},
{
"math_id": 18,
"text": "W^*"
},
{
"math_id": 19,
"text": "W_{m_1}"
},
{
"math_id": 20,
"text": "\\Phi : M^* \\longrightarrow W^* "
},
{
"math_id": 21,
"text": "\\Phi"
},
{
"math_id": 22,
"text": "M^*_w:= \\Phi^{-1}(w),\\ \\ w\\in W^*"
}
]
| https://en.wikipedia.org/wiki?curid=13024295 |
1302589 | Leakage inductance | Magnetic perturbation of imperfectly coupled transformers
Leakage inductance derives from the electrical property of an imperfectly coupled transformer whereby each winding behaves as a self-inductance in series with the winding's respective ohmic resistance constant. These four winding constants also interact with the transformer's mutual inductance. The winding leakage inductance is due to leakage flux not linking with all turns of each imperfectly coupled winding.
Leakage reactance is usually the most important element of a power system transformer due to power factor, voltage drop, reactive power consumption and fault current considerations.
Leakage inductance depends on the geometry of the core and the windings. Voltage drop across the leakage reactance results in often undesirable supply regulation with varying transformer load. But it can also be useful for harmonic isolation (attenuating higher frequencies) of some loads.
Leakage inductance applies to any imperfectly coupled magnetic circuit device including motors.
Leakage inductance and inductive coupling factor.
The magnetic circuit's flux that does not interlink both windings is the leakage flux corresponding to primary leakage inductance LPσ and secondary leakage inductance LSσ. Referring to Fig. 1, these leakage inductances are defined in terms of transformer winding open-circuit inductances and associated coupling coefficient or coupling factor formula_0.
The primary open-circuit self-inductance is given by
formula_1 ------ (Eq. 1.1a)
where
formula_2 ------ (Eq. 1.1b)
formula_3 ------ (Eq. 1.1c)
and
*formula_4 is primary self-inductance
*formula_5 is primary leakage inductance
*formula_6 is magnetizing inductance
*formula_0 is inductive coupling coefficient
Measuring basic transformer inductances & coupling factor
Transformer self-inductances formula_7 & formula_8 and mutual inductance formula_9 are, in additive and subtractive series connection of the two windings, given by,
in additive connection,
formula_10, and,
in subtractive connection,
formula_11
such that these transformer inductances can be determined from the following three equations:
formula_12
formula_13
formula_14.
The coupling factor is derived from the inductance value measured across one winding with the other winding short-circuited according to the following:
Per Eq. 2.7,
formula_15 and formula_16
Such that
formula_17
The Campbell bridge circuit can also be used to determine transformer self-inductances and mutual inductance using a variable standard mutual inductor pair for one of the bridge sides.
It therefore follows that the open-circuit self-inductance and inductive coupling factor formula_0 are given by
formula_18 ------ (Eq. 1.2), and,
formula_19, with 0 < formula_0 < 1 ------ (Eq. 1.3)
where
formula_20
formula_21
and
*formula_9 is mutual inductance
*formula_22 is secondary self-inductance
*formula_23 is secondary leakage inductance
*formula_24 is magnetizing inductance referred to the secondary
*formula_0 is inductive coupling coefficient
*formula_25 is the approximate turns ratio
The electric validity of the transformer diagram in Fig. 1 depends strictly on open-circuit conditions for the respective winding inductances considered. More generalized circuit conditions are as developed in the next two sections.
Inductive leakage factor and inductance.
A nonideal linear two-winding transformer can be represented by two mutual inductance-coupled circuit loops linking the transformer's five impedance constants as shown in Fig. 2.
where
*M is mutual inductance
*formula_26 & formula_27 are primary and secondary winding resistances
*Constants formula_9, formula_7, formula_8, formula_26 & formula_27 are measurable at the transformer's terminals
*Coupling factor formula_0 is defined as
formula_28, where 0 < formula_0 < 1 ------ (Eq. 2.1)
The winding turns ratio formula_29 is in practice given as
formula_30 ------ (Eq. 2.2).
where
*NP & NS are primary and secondary winding turns
*vP & vS and iP & iS are primary & secondary winding voltages & currents.
The nonideal transformer's mesh equations can be expressed by the following voltage and flux linkage equations,
formula_31 ------ (Eq. 2.3)
formula_32 ------ (Eq. 2.4)
formula_33 ------ (Eq. 2.5)
formula_34 ------ (Eq. 2.6),
where
*formula_35 is flux linkage
*formula_36 is derivative of flux linkage with respect to time.
These equations can be developed to show that, neglecting associated winding resistances, the ratio of a winding circuit's inductances and currents with the other winding short-circuited and at open-circuit test is as follows,
formula_37 ------ (Eq. 2.7),
where,
*ioc & isc are open-circuit and short-circuit currents
*Loc & Lsc are open-circuit and short-circuit inductances.
*formula_38 is the inductive leakage factor or Heyland factor
*formula_39 & formula_40 are primary and secondary short-circuited leakage inductances.
The transformer inductance can be characterized in terms of the three inductance constants as follows,
formula_41 ------ (Eq. 2.8)
formula_42 ------ (Eq. 2.9)
formula_43 ------ (Eq. 2.10) ,
where,
*LM is magnetizing inductance, corresponding to magnetizing reactance XM
*LPσ & LSσ are primary & secondary leakage inductances, corresponding to primary & secondary leakage reactances XPσ & XSσ.
The transformer can be expressed more conveniently as the equivalent circuit in Fig. 3 with secondary constants referred (i.e., with prime superscript notation) to the primary,
formula_44
formula_45
formula_46
formula_47.
Since
formula_48 ------ (Eq. 2.11)
and
formula_49 ------ (Eq. 2.12),
we have
formula_50 ------ (Eq. 2.13),
which allows expression of the equivalent circuit in Fig. 4 in terms of winding leakage and magnetizing inductance constants as follows,
formula_51 ------ (Eq. 2.14 formula_52 Eq. 1.1b)
formula_53 ------ (Eq. 2.15 formula_52 Eq. 1.1c).
The nonideal transformer in Fig. 4 can be shown as the simplified equivalent circuit in Fig. 5, with secondary constants referred to the primary and without ideal transformer isolation, where,
formula_54 ------ (Eq. 2.16)
*formula_55 is magnetizing current excited by flux ΦM that links both primary and secondary windings
*formula_56 is the primary current
*formula_57 is the secondary current referred to the primary side of the transformer.
Refined inductive leakage factor.
Refined inductive leakage factor derivation
a. Per Eq. 2.1 & IEC IEV 131-12-41 inductive coupling factor formula_0 is given by
formula_58 --------------------- (Eq. 2.1):
b. Per Eq. 2.7 & IEC IEV 131-12-42 Inductive leakage factor formula_38 is given by
formula_59 ------ (Eq. 2.7) & (Eq. 3.7a)
c. formula_60 multiplied by formula_61 gives
formula_62 ----------------- (Eq. 3.7b)
d. Per Eq. 2-8 & knowing that formula_63
formula_64 ---------------------- (Eq. 3.7c)
e. formula_65 multiplied by formula_66 gives
formula_67 ------------------ (Eq. 3.7d)
f. Per Eq. 3.5 formula_68 Eq. 1.1b & Eq. 2.14 and Eq. 3.6 formula_68 Eq. 1.1b & Eq. 2.14:
formula_69 --- (Eq.3.7e)
All equations in this article assume steady-state constant-frequency waveform conditions the formula_0 & formula_38 values of which are dimensionless, fixed, finite & positive but less than 1.
Referring to the flux diagram in Fig. 6, the following equations hold:
σP = ΦPσ/ΦM = LPσ/LM ------ (Eq. 3.1 formula_70 Eq. 2.7)
In the same way,
σS = ΦSσ'/ΦM = LSσ'/LM ------ (Eq. 3.2 formula_70 Eq. 2.7)
And therefore,
ΦP = ΦM + ΦPσ = ΦM + σPΦM = (1 + σP)ΦM ------ (Eq. 3.3)
ΦS' = ΦM + ΦSσ' = ΦM + σSΦM = (1 + σS)ΦM ------ (Eq. 3.4)
LP = LM + LPσ = LM + σPLM = (1 + σP)LM ------ (Eq. 3.5 formula_68 Eq. 1.1b & Eq. 2.14)
LS' = LM + LSσ' = LM + σSLM = (1 + σS)LM ------ (Eq. 3.6 formula_70 Eq. 1.1b & Eq. 2.14),
where
*σP & σS are, respectively, primary leakage factor & secondary leakage factor
*ΦM & LM are, respectively, mutual flux & magnetizing inductance
*ΦPσ & LPσ are, respectively, primary leakage flux & primary leakage inductance
*ΦSσ' & LSσ' are, respectively, secondary leakage flux & secondary leakage inductance both referred to the primary.
The leakage ratio σ can thus be refined in terms of the interrelationship of above winding-specific inductance and Inductive leakage factor equations as follows:
formula_71 ------ (Eq. 3.7a to 3.7e).
Applications.
Leakage inductance can be an undesirable property, as it causes the voltage to change with loading.
In many cases it is useful. Leakage inductance has the useful effect of limiting the current flows in a transformer (and load) without itself dissipating power (excepting the usual non-ideal transformer losses). Transformers are generally designed to have a specific value of leakage inductance such that the leakage reactance created by this inductance is a specific value at the desired frequency of operation. In this case, actually working useful parameter is not the leakage inductance value but the short-circuit inductance value.
Commercial and distribution transformers rated up to say 2,500 kVA are usually designed with short-circuit impedances of between about 3% and 6% and with a corresponding formula_72 ratio (winding reactance/winding resistance ratio) of between about 3 and 6, which defines the percent secondary voltage variation between no-load and full load. Thus for purely resistive loads, such transformers' full-to-no-load voltage regulation will be between about 1% and 2%.
High leakage reactance transformers are used for some negative resistance applications, such as neon signs, where a voltage amplification (transformer action) is required as well as current limiting. In this case the leakage reactance is usually 100% of full load impedance, so even if the transformer is shorted out it will not be damaged. Without the leakage inductance, the negative resistance characteristic of these gas discharge lamps would cause them to conduct excessive current and be destroyed.
Transformers with variable leakage inductance are used to control the current in arc welding sets. In these cases, the leakage inductance limits the current flow to the desired magnitude. Transformer leakage reactance has a large role in limiting circuit fault current within the maximum allowable value in the power system.
In addition, the leakage inductance of a HF-transformer can replace a series inductor in a resonant converter. In contrast, connecting a conventional transformer and an inductor in series results in the same electric behavior as of a leakage transformer, but this can be advantageous to reduce the eddy current losses in the transformer windings caused by the stray field.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
Bibliography.
<templatestyles src="Refbegin/styles.css" />
External links.
IEC Electropedia links:
<templatestyles src="Col-begin/styles.css"/> | [
{
"math_id": 0,
"text": "k"
},
{
"math_id": 1,
"text": "L_{oc}^{pri}=L_P=L_M+L_P^\\sigma"
},
{
"math_id": 2,
"text": "L_P^\\sigma=L_P\\cdot{(1-k)}"
},
{
"math_id": 3,
"text": "L_M=L_P\\cdot{k}"
},
{
"math_id": 4,
"text": "L_{oc}^{pri}=L_P"
},
{
"math_id": 5,
"text": "L_P^\\sigma"
},
{
"math_id": 6,
"text": "L_M"
},
{
"math_id": 7,
"text": "L_P"
},
{
"math_id": 8,
"text": "L_S"
},
{
"math_id": 9,
"text": "M"
},
{
"math_id": 10,
"text": "L_{ser}^{+}=L_P+L_S+2M"
},
{
"math_id": 11,
"text": "L_{ser}^{-}=L_P+L_S-2M"
},
{
"math_id": 12,
"text": "L_{ser}^{+}-L_{ser}^{-}=4M"
},
{
"math_id": 13,
"text": "L_{ser}^{+}+L_{ser}^{-}=2 \\cdot (L_{P}+L_{S})"
},
{
"math_id": 14,
"text": "L_P=a^2.L_S"
},
{
"math_id": 15,
"text": "L_{sc}^{pri}=L_S\\cdot{(1-k^2)}"
},
{
"math_id": 16,
"text": "L_{sc}^{sec}=L_P\\cdot{(1-k^2)}"
},
{
"math_id": 17,
"text": "k=\\sqrt{1-\\frac{L_{sc}^{pri}}{L_S}}=\\sqrt{1-\\frac{L_{sc}^{sec}}{L_P}}"
},
{
"math_id": 18,
"text": "L_{oc}^{sec}=L_S=L_{M2}+L_S^\\sigma"
},
{
"math_id": 19,
"text": "k=\\frac {\\left | M\\right|}{\\sqrt{L_PL_S}}"
},
{
"math_id": 20,
"text": "L_S^\\sigma=L_S\\cdot{(1-k)}"
},
{
"math_id": 21,
"text": "L_{M2}=L_S\\cdot {k}"
},
{
"math_id": 22,
"text": "L_{oc}^{sec}=L_S"
},
{
"math_id": 23,
"text": "L_S^\\sigma"
},
{
"math_id": 24,
"text": "L_{M2}= L_M/a^2"
},
{
"math_id": 25,
"text": "a \\equiv \\sqrt {\\frac {L_p} {L_s}} \\approx N_P/N_S"
},
{
"math_id": 26,
"text": "R_P"
},
{
"math_id": 27,
"text": "R_S"
},
{
"math_id": 28,
"text": "k=\\left | M\\right |/\\sqrt{L_PL_S}"
},
{
"math_id": 29,
"text": "a"
},
{
"math_id": 30,
"text": "a=\\sqrt{L_P/L_S}=N_P/N_S\\approx v_P/v_S \\approx i_S/i_P="
},
{
"math_id": 31,
"text": "v_P=R_P \\cdot i_P+\\frac{d\\Psi{_P}}{dt}"
},
{
"math_id": 32,
"text": "v_S=-R_S \\cdot i_S-\\frac{d\\Psi{_S}}{dt}"
},
{
"math_id": 33,
"text": "\\Psi_P=L_P \\cdot i_P-M \\cdot i_S"
},
{
"math_id": 34,
"text": "\\Psi_S=L_S \\cdot i_S-M \\cdot i_P"
},
{
"math_id": 35,
"text": "\\Psi"
},
{
"math_id": 36,
"text": "\\frac {d \\Psi}{d t}"
},
{
"math_id": 37,
"text": "\\sigma=1-\\frac{M^2}{L_PL_S}=1-k^2\\approx\\frac{L_{sc}}{L_{oc}}\\approx \\frac{L_{sc}^{sec}}{L_P}\\approx\\frac{L_{sc}^{pri}}{L_S}\\approx\\frac{i_{oc}}{i_{sc}}"
},
{
"math_id": 38,
"text": "\\sigma"
},
{
"math_id": 39,
"text": "L_{sc}^{pri}"
},
{
"math_id": 40,
"text": "L_{sc}^{sec}"
},
{
"math_id": 41,
"text": "L_M=a{M}"
},
{
"math_id": 42,
"text": "L_P^\\sigma=L_P-a{M}"
},
{
"math_id": 43,
"text": "L_S^\\sigma=L_S-{M}/a"
},
{
"math_id": 44,
"text": "L_S^{\\sigma\\prime}=a^2L_S-aM"
},
{
"math_id": 45,
"text": "R_S^\\prime=a^2R_S"
},
{
"math_id": 46,
"text": "V_S^\\prime=aV_S"
},
{
"math_id": 47,
"text": "I_S^\\prime=I_S/a"
},
{
"math_id": 48,
"text": "k=M/\\sqrt{L_PL_S}"
},
{
"math_id": 49,
"text": "a=\\sqrt{L_P/L_S}"
},
{
"math_id": 50,
"text": "aM=\\sqrt{L_P/L_S} \\cdot k \\cdot \\sqrt{L_PL_S}=kL_P"
},
{
"math_id": 51,
"text": "L_P^\\sigma=L_S^{\\sigma\\prime}=L_P \\cdot (1-k)"
},
{
"math_id": 52,
"text": "\\equiv"
},
{
"math_id": 53,
"text": "L_M=kL_P"
},
{
"math_id": 54,
"text": "i_M = i_P - i_S^'"
},
{
"math_id": 55,
"text": "i_M"
},
{
"math_id": 56,
"text": "i_P"
},
{
"math_id": 57,
"text": "i_S'"
},
{
"math_id": 58,
"text": "k=\\left | M\\right | /\\sqrt{L_PL_S}"
},
{
"math_id": 59,
"text": "\\sigma=1-k^2=1-\\frac{M^2}{L_PL_S}"
},
{
"math_id": 60,
"text": "\\frac{M^2}{L_PL_S}"
},
{
"math_id": 61,
"text": "\\frac{a^2}{a^2}"
},
{
"math_id": 62,
"text": "\\sigma=1-\\frac{a^2M^2}{L_Pa^2L_S}"
},
{
"math_id": 63,
"text": "a^2L_S=L_S^\\prime"
},
{
"math_id": 64,
"text": "\\sigma=1-\\frac{L_M^2}{L_PL_S^\\prime}"
},
{
"math_id": 65,
"text": "\\frac{L_M^2}{L_PL_S^\\prime}"
},
{
"math_id": 66,
"text": "\\frac{L_M.L_M}{L_M^2}"
},
{
"math_id": 67,
"text": "\\sigma=1-\\frac{1}{\\frac{L_P}{L_M}.\\frac{L_S^\\prime}{L_M}}"
},
{
"math_id": 68,
"text": " \\approx"
},
{
"math_id": 69,
"text": "\\sigma=1-\\frac{1}{(1+\\sigma_P)(1+\\sigma_S)}"
},
{
"math_id": 70,
"text": "\\approx"
},
{
"math_id": 71,
"text": "\\sigma=1-\\frac{M^2}{L_PL_S}=1-\\frac{a^2M^2}{L_Pa^2L_S}=1-\\frac{L_M^2}{L_PL_S{^'}}=1-\\frac{1}{\\frac{L_P}{L_M}.\\frac{L_S^'}{L_M}} =1-\\frac{1}{(1+\\sigma_P)(1+\\sigma_S)}"
},
{
"math_id": 72,
"text": "X/R"
}
]
| https://en.wikipedia.org/wiki?curid=1302589 |
13029194 | Single-linkage clustering | Agglomerative hierarchical clustering method
In statistics, single-linkage clustering is one of several methods of hierarchical clustering. It is based on grouping clusters in bottom-up fashion (agglomerative clustering), at each step combining two clusters that contain the closest pair of elements not yet belonging to the same cluster as each other.
This method tends to produce long thin clusters in which nearby elements of the same cluster have small distances, but elements at opposite ends of a cluster may be much farther from each other than two elements of other clusters. For some classes of data, this may lead to difficulties in defining classes that could usefully subdivide the data. However, it is popular in astronomy for analyzing galaxy clusters, which may often involve long strings of matter; in this application, it is also known as the friends-of-friends algorithm.
Overview of agglomerative clustering methods.
In the beginning of the agglomerative clustering process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters, until all elements end up being in the same cluster. At each step, the two clusters separated by the shortest distance are combined. The function used to determine the distance between two clusters, known as the "linkage function", is what differentiates the agglomerative clustering methods.
In single-linkage clustering, the distance between two clusters is determined by a single pair of elements: those two elements (one in each cluster) that are closest to each other. The shortest of these pairwise distances that remain at any step causes the two clusters whose elements are involved to be merged. The method is also known as "nearest neighbour clustering". The result of the clustering can be visualized as a dendrogram, which shows the sequence in which clusters were merged and the distance at which each merge took place.
Mathematically, the linkage function – the distance "D"("X","Y") between clusters "X" and "Y" – is described by the expression
formula_0
where "X" and "Y" are any two sets of elements considered as clusters, and "d"("x","y") denotes the distance between the two elements "x" and "y".
Naive algorithm.
The following algorithm is an agglomerative scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The formula_1 proximity matrix formula_2 contains all distances formula_3. The clusterings are assigned sequence numbers formula_4 and formula_5 is the level of the formula_6-th clustering. A cluster with sequence number "m" is denoted ("m") and the proximity between clusters formula_7 and formula_8 is denoted formula_9.
The single linkage algorithm is composed of the following steps:
Working example.
This working example is based on a JC69 genetic distance matrix computed from the 5S ribosomal RNA sequence alignment of five bacteria: "Bacillus subtilis" (formula_20), "Bacillus stearothermophilus" (formula_21), "Lactobacillus viridescens" (formula_22), "Acholeplasma modicum" (formula_23), and "Micrococcus luteus" (formula_24).
First step.
Let us assume that we have five elements formula_25 and the following matrix formula_26 of pairwise distances between them:
In this example, formula_27 is the lowest value of formula_26, so we cluster elements a and b.
Let u denote the node to which a and b are now connected. Setting formula_28 ensures that elements a and b are equidistant from u. This corresponds to the expectation of the ultrametricity hypothesis.
The branches joining a and b to u then have lengths formula_29 ("see the final dendrogram")
We then proceed to update the initial proximity matrix formula_26 into a new proximity matrix formula_30 (see below), reduced in size by one row and one column because of the clustering of a with b.
Bold values in formula_30 correspond to the new distances, calculated by retaining the minimum distance between each element of the first cluster formula_31 and each of the remaining elements:
formula_32
Italicized values in formula_30 are not affected by the matrix update as they correspond to distances between elements not involved in the first cluster.
Second step.
We now reiterate the three previous actions, starting from the new distance matrix formula_30 :
Here, formula_33 and formula_34 are the lowest values of formula_30, so we join cluster formula_31 with element c and with element e.
Let v denote the node to which formula_31, c and e are now connected. Because of the ultrametricity constraint, the branches joining a or b to v, and c to v, and also e to v are equal and have the following total length:
formula_35
We deduce the missing branch length:
formula_36 ("see the final dendrogram")
We then proceed to update the formula_30 matrix into a new distance matrix formula_37 (see below), reduced in size by two rows and two columns because of the clustering of formula_31 with c and with e :
formula_38
Final step.
The final formula_37 matrix is:
So we join clusters formula_39 and formula_23.
Let formula_40 denote the (root) node to which formula_39 and formula_23 are now connected.
The branches joining formula_39 and formula_23 to formula_40 then have lengths:
formula_41
We deduce the remaining branch length:
formula_42
The single-linkage dendrogram.
The dendrogram is now complete. It is ultrametric because all tips (formula_20, formula_21, formula_22, formula_24, and formula_23) are equidistant from formula_40 :
formula_43
The dendrogram is therefore rooted by formula_40, its deepest node.
Other linkages.
The naive algorithm for single linkage clustering is essentially the same as Kruskal's algorithm for minimum spanning trees. However, in single linkage clustering, the order in which clusters are formed is important, while for minimum spanning trees what matters is the set of pairs of points that form distances chosen by the algorithm.
Alternative linkage schemes include complete linkage clustering, average linkage clustering (UPGMA and WPGMA), and Ward's method. In the naive algorithm for agglomerative clustering, implementing a different linkage scheme may be accomplished simply by using a different formula to calculate inter-cluster distances in the algorithm. The formula that should be adjusted has been highlighted using bold text in the above algorithm description. However, more efficient algorithms such as the one described below do not generalize to all linkage schemes in the same way.
Faster algorithms.
The naive algorithm for single-linkage clustering is easy to understand but slow, with time complexity formula_44. In 1973, R. Sibson proposed an algorithm with time complexity formula_45 and space complexity formula_46 (both optimal) known as SLINK. The slink algorithm represents a clustering on a set of formula_47 numbered items by two functions. These functions are both determined by finding the smallest cluster formula_48 that contains both item formula_49 and at least one larger-numbered item.
The first function, formula_50, maps item formula_49 to the largest-numbered item in cluster formula_48.
The second function, formula_51, maps item formula_49 to the distance associated with the creation of cluster formula_48.
Storing these functions in two arrays that map each item number to its function value takes space formula_46, and this information is sufficient to determine the clustering itself. As Sibson shows, when a new item is added to the set of items, the updated functions representing the new single-linkage clustering for the augmented set, represented in the same way, can be constructed from the old clustering in time formula_46. The SLINK algorithm then loops over the items, one by one, adding them to the representation of the clustering.
An alternative algorithm, running in the same optimal time and space bounds, is based on the equivalence between the naive algorithm and Kruskal's algorithm for minimum spanning trees. Instead of using Kruskal's algorithm, one can use Prim's algorithm, in a variation without binary heaps that takes time formula_45 and space formula_46 to construct the minimum spanning tree (but not the clustering) of the given items and distances. Then, applying Kruskal's algorithm to the sparse graph formed by the edges of the minimum spanning tree produces the clustering itself in an additional time formula_52 and space formula_46.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "D(X,Y)=\\min_{x\\in X, y\\in Y} d(x,y),"
},
{
"math_id": 1,
"text": "N \\times N"
},
{
"math_id": 2,
"text": "D"
},
{
"math_id": 3,
"text": "d(i,j)"
},
{
"math_id": 4,
"text": "0,1, \\ldots, n-1"
},
{
"math_id": 5,
"text": "L(k)"
},
{
"math_id": 6,
"text": "k"
},
{
"math_id": 7,
"text": "(r)"
},
{
"math_id": 8,
"text": "(s)"
},
{
"math_id": 9,
"text": "d[(r),(s)]"
},
{
"math_id": 10,
"text": "L(0) = 0"
},
{
"math_id": 11,
"text": "m=0"
},
{
"math_id": 12,
"text": "(r), (s)"
},
{
"math_id": 13,
"text": "d[(r),(s)] = \\min d[(i),(j)]"
},
{
"math_id": 14,
"text": "m = m + 1"
},
{
"math_id": 15,
"text": "m"
},
{
"math_id": 16,
"text": "L(m) = d[(r),(s)]"
},
{
"math_id": 17,
"text": "(r,s)"
},
{
"math_id": 18,
"text": "(k)"
},
{
"math_id": 19,
"text": "d[(r,s),(k)] = \\min \\{d[(k),(r)], d[(k),(s)] \\}"
},
{
"math_id": 20,
"text": "a"
},
{
"math_id": 21,
"text": "b"
},
{
"math_id": 22,
"text": "c"
},
{
"math_id": 23,
"text": "d"
},
{
"math_id": 24,
"text": "e"
},
{
"math_id": 25,
"text": "(a,b,c,d,e)"
},
{
"math_id": 26,
"text": "D_1"
},
{
"math_id": 27,
"text": "D_1 (a,b)=17"
},
{
"math_id": 28,
"text": "\\delta(a,u)=\\delta(b,u)=D_1(a,b)/2"
},
{
"math_id": 29,
"text": "\\delta(a,u)=\\delta(b,u)=17/2=8.5"
},
{
"math_id": 30,
"text": "D_2"
},
{
"math_id": 31,
"text": "(a,b)"
},
{
"math_id": 32,
"text": "\\begin{array}{lllllll}\nD_2((a,b),c)&=&\\min(D_1(a,c),D_1(b,c))&=&\\min(21,30)&=&21\n\\\\\nD_2((a,b),d)&=&\\min(D_1(a,d),D_1(b,d))&=&\\min(31,34)&=&31\n\\\\\nD_2((a,b),e)&=&\\min(D_1(a,e),D_1(b,e))&=&\\min(23,21)&=&21\n\\end{array}"
},
{
"math_id": 33,
"text": "D_2 ((a,b),c)=21"
},
{
"math_id": 34,
"text": "D_2 ((a,b),e)=21"
},
{
"math_id": 35,
"text": "\\delta(a,v)=\\delta(b,v)=\\delta(c,v)=\\delta(e,v)=21/2=10.5"
},
{
"math_id": 36,
"text": "\\delta(u,v)=\\delta(c,v)-\\delta(a,u)=\\delta(c,v)-\\delta(b,u)=10.5-8.5=2"
},
{
"math_id": 37,
"text": "D_3"
},
{
"math_id": 38,
"text": "D_3(((a,b),c,e),d)=\\min(D_2((a,b),d),D_2(c,d),D_2(e,d))=\\min(31,28,43)=28"
},
{
"math_id": 39,
"text": "((a,b),c,e)"
},
{
"math_id": 40,
"text": "r"
},
{
"math_id": 41,
"text": "\\delta(((a,b),c,e),r)=\\delta(d,r)=28/2=14"
},
{
"math_id": 42,
"text": "\\delta(v,r)=\\delta(a,r)-\\delta(a,v)=\\delta(b,r)-\\delta(b,v)=\\delta(c,r)-\\delta(c,v)=\\delta(e,r)-\\delta(e,v)=14-10.5=3.5"
},
{
"math_id": 43,
"text": "\\delta(a,r)=\\delta(b,r)=\\delta(c,r)=\\delta(e,r)=\\delta(d,r)=14"
},
{
"math_id": 44,
"text": "O(n^3)"
},
{
"math_id": 45,
"text": "O(n^2)"
},
{
"math_id": 46,
"text": "O(n)"
},
{
"math_id": 47,
"text": "n"
},
{
"math_id": 48,
"text": "C"
},
{
"math_id": 49,
"text": "i"
},
{
"math_id": 50,
"text": "\\pi"
},
{
"math_id": 51,
"text": "\\lambda"
},
{
"math_id": 52,
"text": "O(n\\log n)"
}
]
| https://en.wikipedia.org/wiki?curid=13029194 |
13029322 | Vertical exaggeration | Vertical exaggeration (VE) is a scale that is used in raised-relief maps, plans and technical drawings (cross section perspectives), in order to emphasize vertical features, which might be too small to identify relative to the horizontal scale.
Scaling Factor.
The vertical exaggeration is given by:
formula_0
where "VS" is the vertical scale and "HS" is the horizontal scale, both given as representative fractions.
For example, if vertically represents and horizontally represents , the vertical exaggeration, 20×, is given by:
formula_1.
Vertical exaggeration is given as a number; for example 5× means vertical measurements appear 5 times greater than horizontal measurements. A value of 1× indicates that horizontal and vertical scales are identical, and is regarded as having "no vertical exaggeration." Vertical exaggerations less than 1 are not common, but would indicate a reduction in vertical scale (or, equivalently, a horizontal exaggeration).
Criticism.
Some scientists object to vertical exaggeration as a tool that makes an oblique visualization dramatic at the cost of misleading the viewer about the true appearance of the landscape.
In some cases, if the vertical exaggeration is too high, the map reader may get confused.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\mathit{VE} = \\mathit\\frac{VS}{HS}"
},
{
"math_id": 1,
"text": "\\mathit{VE} = \\frac{\\frac{1}{200}}{\\frac{1}{4000}} = \\frac{4000}{200} = 20"
}
]
| https://en.wikipedia.org/wiki?curid=13029322 |
1303134 | Ethyl acetate | Organic compound (CH₃CO₂CH₂CH₃)
<templatestyles src="Chembox/styles.css"/>
Chemical compound
Ethyl acetate (systematically ethyl ethanoate, commonly abbreviated EtOAc, ETAC or EA) is the organic compound with the formula , simplified to . This colorless liquid has a characteristic sweet smell (similar to pear drops) and is used in glues, nail polish removers, and the decaffeination process of tea and coffee. Ethyl acetate is the ester of ethanol and acetic acid; it is manufactured on a large scale for use as a solvent.
Production and synthesis.
Ethyl acetate was first synthesized by the Count de Lauraguais in 1759 by distilling a mixture of ethanol and acetic acid.
In 2004, an estimated 1.3 million tonnes were produced worldwide. The combined annual production in 1985 of Japan, North America, and Europe was about 400,000 tonnes. The global ethyl acetate market was valued at $3.3 billion in 2018.
Ethyl acetate is synthesized in industry mainly via the classic Fischer esterification reaction of ethanol and acetic acid. This mixture converts to the ester in about 65% yield at room temperature:
The reaction can be accelerated by acid catalysis and the equilibrium can be shifted to the right by removal of water.
It is also prepared in industry using the Tishchenko reaction, by combining two equivalents of acetaldehyde in the presence of an alkoxide catalyst:
Silicotungstic acid is used to manufacture ethyl acetate by the alkylation of acetic acid by ethylene:
Uses.
Ethyl acetate is used primarily as a solvent and diluent, being favored because of its low cost, low toxicity, and agreeable odor. For example, it is commonly used to clean circuit boards and in some nail varnish removers (acetone is also used). Coffee beans and tea leaves are decaffeinated with this solvent. It is also used in paints as an activator or hardener. Ethyl acetate is present in confectionery, perfumes, and fruits. In perfumes it evaporates quickly, leaving the scent of the perfume on the skin.
Ethyl acetate is an asphyxiant for use in insect collecting and study. In a killing jar charged with ethyl acetate, the vapors will kill the collected insect quickly without destroying it. Because it is not hygroscopic, ethyl acetate also keeps the insect soft enough to allow proper mounting suitable for a collection. However, ethyl acetate is regarded as potentially doing damage to insect DNA, making specimens processed this way less than ideal for subsequent DNA sequencing.
Laboratory uses.
In the laboratory, mixtures containing ethyl acetate are commonly used in column chromatography and extractions. Ethyl acetate is rarely selected as a reaction solvent because it is prone to hydrolysis, transesterification, and condensations.
Occurrence in wines.
Ethyl acetate is the most common ester in wine, being the product of the most common volatile organic acid – acetic acid, and the ethyl alcohol generated during the fermentation. The aroma of ethyl acetate is most vivid in younger wines and contributes towards the general perception of "fruitiness" in the wine. Sensitivity varies, with most people having a perception threshold around 120 mg/L. Excessive amounts of ethyl acetate are considered a wine fault.
Reactions.
Ethyl acetate is only weakly Lewis basic, like a typical carboxylic acid ester.
Ethyl acetate hydrolyses to give acetic acid and ethanol. Bases accelerate the hydrolysis, which is subject to the Fischer equilibrium mentioned above. In the laboratory, and usually for illustrative purposes only, ethyl esters are typically hydrolyzed in a two-step process starting with a stoichiometric amount of a strong base, such as sodium hydroxide. This reaction gives ethanol and sodium acetate, which is unreactive toward ethanol:
In the Claisen condensation, anhydrous ethyl acetate and strong bases react to give ethyl acetoacetate:
Properties.
Physical Properties.
Under normal conditions, ethyl acetate exists as a colorless, low-viscosity, and flammable liquid. Its melting point is -83 °C, with a melting enthalpy of 10.48 kJ·mol−1. At atmospheric pressure, the compound boils at 77 °C. The vaporization enthalpy at the boiling point is 31.94 kJ·mol−1. The vapor pressure function follows the Antoine equation:
formula_0
where:
This function is valid within the temperature range of 289 K (16 °C) to 349 K (76 °C).
The enthalpy of vaporization in kJ/mol is calculated according to the empirical equation by Majer and Svoboda
formula_6
where:
The table above summarizes the most important thermodynamic properties of ethyl acetate under various conditions.
Safety.
The LD50 for rats is 5620 mg/kg, indicating low acute toxicity. Given that the chemical is naturally present in many organisms, there is little risk of toxicity.
Overexposure to ethyl acetate may cause irritation of the eyes, nose, and throat. Severe overexposure may cause weakness, drowsiness, and unconsciousness. Humans exposed to a concentration of 400 ppm in 1.4 mg/L ethyl acetate for a short time were affected by nose and throat irritation. Ethyl acetate is an irritant of the conjunctiva and mucous membrane of the respiratory tract. Animal experiments have shown that, at very high concentrations, the ester has central nervous system depressant and lethal effects; at concentrations of 20,000 to 43,000 ppm (2.0–4.3%), there may be pulmonary edema with hemorrhages, symptoms of central nervous system depression, secondary anemia and liver damage. In humans, concentrations of 400 ppm cause irritation of the nose and pharynx; cases have also been known of irritation of the conjunctiva with temporary opacity of the cornea. In rare cases exposure may cause sensitization of the mucous membrane and eruptions of the skin. The irritant effect of ethyl acetate is weaker than that of propyl acetate or butyl acetate.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\log_{10}(p) = A - \\frac{B}{T + C}"
},
{
"math_id": 1,
"text": "p"
},
{
"math_id": 2,
"text": "T"
},
{
"math_id": 3,
"text": "A = 4.22809"
},
{
"math_id": 4,
"text": "B = 1245.702"
},
{
"math_id": 5,
"text": "C = -55.189"
},
{
"math_id": 6,
"text": "\\Delta H_\\text{vap} = A\\exp(-\\beta\\,T_r)\\,(1 - T_r)^\\beta "
},
{
"math_id": 7,
"text": "T_r = T/T_c"
},
{
"math_id": 8,
"text": "T_c"
},
{
"math_id": 9,
"text": "A"
},
{
"math_id": 10,
"text": "\\beta"
}
]
| https://en.wikipedia.org/wiki?curid=1303134 |
1303480 | Electrostriction | Ability of non-conductive materials to change shape under an electric field
In electromagnetism, electrostriction is a property of all electrical non-conductor or dielectrics. Electrostriction causes these materials to change their shape under the application of an electric field. It is the dual property to magnetostriction.
Explanation.
Electrostriction is a property of all dielectric materials, and is caused by displacement of ions in the crystal lattice upon being exposed to an external electric field. The cause of electrostrictive is linked to anharmonic effects. Positive ions will be displaced in the direction of the field, while negative ions will be displaced in the opposite direction. This displacement will accumulate throughout the bulk material and result in an overall strain (elongation) in the direction of the field. The thickness will be reduced in the orthogonal directions characterized by Poisson's ratio. All insulating materials consisting of more than one type of atom will be ionic to some extent due to the difference of electronegativity of the atoms, and therefore exhibit electrostriction.
The resulting strain (ratio of deformation to the original dimension) is proportional to the square of the polarization. Reversal of the electric field does not reverse the direction of the deformation.
More formally, the electrostriction coefficient is a rank four tensor (formula_0), relating the rank two strain tensor (formula_1) and the electric polarization density vector (i.e. rank one tensor; formula_2)
formula_3
The electrostrictive tensor satisfies
formula_4
The related piezoelectric effect occurs only in a particular class of dielectrics. Electrostriction applies to all crystal symmetries, while the piezoelectric effect only applies to the 20 piezoelectric point groups. Piezoelectricity is a result of electrostrictive in ferroelectric materials. Electrostriction is a quadratic effect, unlike piezoelectricity, which is a linear effect.
Materials.
Although all dielectrics exhibit some electrostriction, certain engineered ceramics, known as relaxor ferroelectrics, have extraordinarily high electrostrictive constants. The most commonly used are
Magnitude of effect.
Electrostriction can produce a strain on the order of 0.1% for some materials. This occurs at a field strength of 2 million volts per meter (2 MV/m) for the material PMN-15. Electrostriction exists in all materials, but is generally negligible.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "Q_{ijkl}"
},
{
"math_id": 1,
"text": "\\varepsilon_{ij}"
},
{
"math_id": 2,
"text": "P_k"
},
{
"math_id": 3,
"text": "\\varepsilon_{ij} = Q_{ijkl}P_k P_l."
},
{
"math_id": 4,
"text": "Q_{ijkl} = \\frac{1}{2}\\frac{\\partial^2\\varepsilon_{ij}}{\\partial P_k \\partial P_l}."
}
]
| https://en.wikipedia.org/wiki?curid=1303480 |
1303494 | Suffix array | Data structure for a string
In computer science, a suffix array is a sorted array of all suffixes of a string. It is a data structure used in, among others, full-text indices, data-compression algorithms, and the field of bibliometrics.
Suffix arrays were introduced by as a simple, space efficient alternative to suffix trees. They had independently been discovered by Gaston Gonnet in 1987 under the name "PAT array" .
gave the first in-place formula_0 time suffix array construction algorithm that is optimal both in time and space, where "in-place" means that the algorithm only needs formula_1 additional space beyond the input string and the output suffix array.
Enhanced suffix arrays (ESAs) are suffix arrays with additional tables that reproduce the full functionality of suffix trees preserving the same time and memory complexity.
The suffix array for a subset of all suffixes of a string is called sparse suffix array. Multiple probabilistic algorithms have been developed to minimize the additional memory usage including an optimal time and memory algorithm.
Definition.
Let formula_2 be an formula_3-string and let formula_4 denote the substring of formula_5 ranging from formula_6 to formula_7 inclusive.
The suffix array formula_8 of formula_5 is now defined to be an array of integers providing the starting positions of suffixes of formula_5 in lexicographical order. This means, an entry formula_9 contains the starting position of the formula_6-th smallest suffix in formula_5 and thus for all formula_10: formula_11.
Each suffix of formula_5 shows up in formula_8 exactly once. Suffixes are simple strings. These strings are sorted (as in a paper dictionary), before their starting positions (integer indices) are saved in formula_8.
Example.
Consider the text formula_5=codice_0 to be indexed:
The text ends with the special sentinel letter codice_1 that is unique and lexicographically smaller than any other character. The text has the following suffixes:
These suffixes can be sorted in ascending order:
The suffix array formula_8 contains the starting positions of these sorted suffixes:
The suffix array with the suffixes written out vertically underneath for clarity:
So for example, formula_12 contains the value 4, and therefore refers to the suffix starting at position 4 within formula_5, which is the suffix codice_2.
Correspondence to suffix trees.
Suffix arrays are closely related to suffix trees:
It has been shown that every suffix tree algorithm can be systematically replaced with an algorithm that uses a suffix array enhanced with additional information (such as the LCP array) and solves the same problem in the same time complexity.
Advantages of suffix arrays over suffix trees include improved space requirements, simpler linear time construction algorithms (e.g., compared to Ukkonen's algorithm) and improved cache locality.
Space efficiency.
Suffix arrays were introduced by in order to improve over the space requirements of suffix trees: Suffix arrays store formula_13 integers. Assuming an integer requires formula_14 bytes, a suffix array requires formula_15 bytes in total. This is significantly less than the formula_16 bytes which are required by a careful suffix tree implementation.
However, in certain applications, the space requirements of suffix arrays may still be prohibitive. Analyzed in bits, a suffix array requires formula_17 space, whereas the original text over an alphabet of size formula_18 only requires formula_19 bits.
For a human genome with formula_20 and formula_21 the suffix array would therefore occupy about 16 times more memory than the genome itself.
Such discrepancies motivated a trend towards compressed suffix arrays and BWT-based compressed full-text indices such as the FM-index. These data structures require only space within the size of the text or even less.
Construction algorithms.
A suffix tree can be built in formula_0 and can be converted into a suffix array by traversing the tree depth-first also in formula_0, so there exist algorithms that can build a suffix array in formula_0.
A naive approach to construct a suffix array is to use a comparison-based sorting algorithm. These algorithms require formula_17 suffix comparisons, but a suffix comparison runs in formula_0 time, so the overall runtime of this approach is formula_22.
More advanced algorithms take advantage of the fact that the suffixes to be sorted are not arbitrary strings but related to each other. These algorithms strive to achieve the following goals:
One of the first algorithms to achieve all goals is the SA-IS algorithm of . The algorithm is also rather simple (< 100 LOC) and can be enhanced to simultaneously construct the LCP array. The SA-IS algorithm is one of the fastest known suffix array construction algorithms. A careful implementation by Yuta Mori outperforms most other linear or super-linear construction approaches.
Beside time and space requirements, suffix array construction algorithms are also differentiated by their supported alphabet: "constant alphabets" where the alphabet size is bound by a constant, "integer alphabets" where characters are integers in a range depending on formula_13 and "general alphabets" where only character comparisons are allowed.
Most suffix array construction algorithms are based on one of the following approaches:
A well-known recursive algorithm for integer alphabets is the "DC3 / skew" algorithm of . It runs in linear time and has successfully been used as the basis for parallel and external memory suffix array construction algorithms.
Recent work by proposes an algorithm for updating the suffix array of a text that has been edited instead of rebuilding a new suffix array from scratch. Even if the theoretical worst-case time complexity is formula_17, it appears to perform well in practice: experimental results from the authors showed that their implementation of dynamic suffix arrays is generally more efficient than rebuilding when considering the insertion of a reasonable number of letters in the original text.
In practical open source work, a commonly used routine for suffix array construction was qsufsort, based on the 1999 Larsson-Sadakane algorithm. This routine has been superseded by Yuta Mori's DivSufSort, "the fastest known suffix sorting algorithm in main memory" as of 2017. It too can be modified to compute an LCP array. It uses a induced copying combined with Itoh-Tanaka. In 2021 a faster implementation of the algorithm was presented by Ilya Grebnov which in average showed 65% performance improvement over DivSufSort implementation on Silesia Corpus.
Generalized suffix array.
The concept of a suffix array can be extended to more than one string. This is called a generalized suffix array (or GSA), a suffix array that contains all suffixes for a set of strings (for example, formula_24 and is lexicographically sorted with all suffixes of each string.
Applications.
The suffix array of a string can be used as an index to quickly locate every occurrence of a substring pattern formula_25 within the string formula_5. Finding every occurrence of the pattern is equivalent to finding every suffix that begins with the substring. Thanks to the lexicographical ordering, these suffixes will be grouped together in the suffix array and can be found efficiently with two binary searches. The first search locates the starting position of the interval, and the second one determines the end position:
n = len(S)
def search(P: str) -> Tuple[int, int]:
Return indices (s, r) such that the interval A[s:r] (including the end
index) represents all suffixes of S that start with the pattern P.
# Find starting position of interval
l = 0 # in Python, arrays are indexed starting at 0
r = n
while l < r:
mid = (l + r) // 2 # division rounding down to nearest integer
# suffixAt(A[i]) is the ith smallest suffix
if P > suffixAt(A[mid]):
l = mid + 1
else:
r = mid
s = l
# Find ending position of interval
r = n
while l < r:
mid = (l + r) // 2
if suffixAt(A[mid]).startswith(P):
l = mid + 1
else:
r = mid
return (s, r)
Finding the substring pattern formula_25 of length formula_26 in the string formula_5 of length formula_13 takes formula_27 time, given that a single suffix comparison needs to compare formula_26 characters. describe how this bound can be improved to formula_28 time using LCP information. The idea is that a pattern comparison does not need to re-compare certain characters, when it is already known that these are part of the longest common prefix of the pattern and the current search interval. improve the bound even further and achieve a search time of formula_29 for constant alphabet size, as known from suffix trees.
Suffix sorting algorithms can be used to compute the Burrows–Wheeler transform (BWT). The BWT requires sorting of all cyclic permutations of a string. If this string ends in a special end-of-string character that is lexicographically smaller than all other character (i.e., $), then the order of the sorted rotated BWT matrix corresponds to the order of suffixes in a suffix array. The BWT can therefore be computed in linear time by first constructing a suffix array of the text and then deducing the BWT string: formula_30.
Suffix arrays can also be used to look up substrings in example-based machine translation, demanding much less storage than a full phrase table as used in Statistical machine translation.
Many additional applications of the suffix array require the LCP array. Some of these are detailed in the application section of the latter.
Enhanced suffix arrays.
Suffix trees are powerful data structures that have wide application in areas of pattern and string matching, indexing and textual statistics. However, it occupies a significant amount of space and thus has a drawback in many real-time applications that require processing a considerably large amount of data like genome analysis. To overcome this drawback, Enhanced Suffix Arrays were developed that are data structures consisting of suffix arrays and an additional table called the child table that contains the information about the parent-child relationship between the nodes in the suffix tree. The node branching data structure for this tree is a linked list. Enhanced suffix arrays are superior in terms of both space efficiency and time complexity and are easy to implement. Moreover, they can be applied to any algorithm that uses a suffix tree by using an abstract concept lcp-interval trees. The time complexity for searching a pattern in an enhanced suffix array is O(m|Σ|).
The suffix array of the string is an array of n integers in the range of 0 to n that represents the n+1 suffixes of the string including the special character #.
The suffix array is composed of two arrays:
Constructing the lcp-interval.
For a suffix array of S, the lcp-interval associated with the corresponding node of suffix tree of S can be defined as:
"Interval [i..j], 0 ≤ i ≤ j ≤ n is an lcp-interval of lcp-value, if"
"1. lcptab[i] < l,"
"2. lcptab[k] ≥ l for all i + 1 ≤ k ≤ j,"
"3. lcptab[k] = l for some i + 1 ≤ k ≤ j if i ≠ j and l = n − i + 1 if i = j,"
"4. lcptab[j + 1] < l."
The length of the longest common prefix of pos[i − 1] and pos[i] is stored in lcp[i],where 2 ≤ i ≤ n. The lcp-interval portrays the same parent-child relationship as that among the associated nodes in the suffix tree of S.This shows that if the corresponding node of [i..j] is a child of the corresponding node of [k..l], a lcp-interval [i..j] is a child interval of another lcp-interval [k..l]. If [k..l] is a child interval of [i..j], a lcp-interval [i..j] is the parent interval of a lcp-interval [k..l].
Constructing a child table.
The child table "cldtab" is composed of three n arrays, "up", "down" and "nextlIndex". The information about the edges of the corresponding suffix tree is stored and maintained by the "up" and "down" arrays. The "nextlIndex" array stores the links in the linked list used for node branching the suffix tree.
The "up", "down" and "nextlIndex" array are defined as follows:
By performing a bottom-up traversal of the lcp-interval of the tree, the child table can be constructed in linear time. The "up/down" values and the "nextlIndex" values can be computed separately by using two distinct algorithms.
Constructing a suffix link table.
The suffix links for an enhanced suffix array can be computed by generating the suffix link interval ["1..,r"] for each [i..j] interval during the preprocessing. The left and right elements l and r of the interval are maintained in the first index of [i..,j]. The table for this interval ranges from 0 to n. The suffix link table is constructed by the left-to-right breadth-first traversal of the lcp-interval tree. Every time an "l"-interval is computed, it is added to the list of l-intervals, which is referred to as the l-list. When the lcp-value > 0, for every "l"-interval[i..,j] in the list, link[i] is calculated. The interval ["l"..,"r"] is computed by a binary search in("l"-1)-list, where "l" is the largest left boundary amongst all the "l"-1 intervals. The suffix link interval of [i..j] is represented by this interval["l..,r"]. The values "l" and "r" are ultimately stored in the first index of [i..,j].
Notes.
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"math_id": 0,
"text": "\\mathcal{O}(n)"
},
{
"math_id": 1,
"text": "\\mathcal{O}(1)"
},
{
"math_id": 2,
"text": "S=S[1]S[2]...S[n]"
},
{
"math_id": 3,
"text": "n"
},
{
"math_id": 4,
"text": "S[i,j]"
},
{
"math_id": 5,
"text": "S"
},
{
"math_id": 6,
"text": "i"
},
{
"math_id": 7,
"text": "j"
},
{
"math_id": 8,
"text": "A"
},
{
"math_id": 9,
"text": "A[i]"
},
{
"math_id": 10,
"text": "1 \\leq i \\leq n"
},
{
"math_id": 11,
"text": "S[A[i-1],n] < S[A[i],n]"
},
{
"math_id": 12,
"text": "A[3]"
},
{
"math_id": 13,
"text": "n"
},
{
"math_id": 14,
"text": "4"
},
{
"math_id": 15,
"text": "4n"
},
{
"math_id": 16,
"text": "20n"
},
{
"math_id": 17,
"text": "\\mathcal{O}(n \\log n)"
},
{
"math_id": 18,
"text": "\\sigma"
},
{
"math_id": 19,
"text": "\\mathcal{O}(n \\log \\sigma)"
},
{
"math_id": 20,
"text": "\\sigma = 4"
},
{
"math_id": 21,
"text": "n = 3.4 \\times 10^9"
},
{
"math_id": 22,
"text": "\\mathcal{O}(n^2 \\log n)"
},
{
"math_id": 23,
"text": "\\Theta(n)"
},
{
"math_id": 24,
"text": "S=S_1,S_2,S_3,...,S_k"
},
{
"math_id": 25,
"text": "P"
},
{
"math_id": 26,
"text": "m"
},
{
"math_id": 27,
"text": "\\mathcal{O}(m \\log n)"
},
{
"math_id": 28,
"text": "\\mathcal{O}(m + \\log n)"
},
{
"math_id": 29,
"text": "\\mathcal{O}(m)"
},
{
"math_id": 30,
"text": "BWT[i] = S[A[i]-1]"
}
]
| https://en.wikipedia.org/wiki?curid=1303494 |
13034946 | Modified Dietz method | The modified Dietz method is a measure of the "ex post" (i.e. historical) performance of an investment portfolio in the presence of external flows. (External flows are movements of value such as transfers of cash, securities or other instruments in or out of the portfolio, with no equal simultaneous movement of value in the opposite direction, and which are not income from the investments in the portfolio, such as interest, coupons or dividends.)
To calculate the modified Dietz return, divide the gain or loss in value, net of external flows, by the average capital over the period of measurement. The average capital weights individual cash flows by the length of time between those cash flows until the end of the period. Flows which occur towards the beginning of the period have a higher weight than flows occurring towards the end. The result of the calculation is expressed as a percentage return over the holding period.
GIPS.
This method for return calculation is used in modern portfolio management. It is one of the methodologies of calculating returns recommended by the Investment Performance Council (IPC) as part of their Global Investment Performance Standards (GIPS). The GIPS are intended to provide consistency to the way portfolio returns are calculated internationally.
Origin.
The method is named after Peter O. Dietz. The original idea behind the work of Peter Dietz was to find a quicker, less computer-intensive way of calculating an IRR as the iterative approach using the then-quite-slow computers that were available was taking a significant amount of time; the research was produced for BAI, Bank Administration institute. The modified Dietz method is a linear IRR.
Formula.
The formula for the modified Dietz method is as follows:
formula_0
where
formula_1 is the starting market value
formula_2 is the ending market value
formula_3 is the net external inflow for the period (so contributions to a portfolio are treated as positive flows while withdrawals are negative flows)
and
formula_4 the sum of each flow formula_5 multiplied by its weight formula_6
The weight formula_6 is the proportion of the time period between the point in time when the flow formula_5 occurs and the end of the period. Assuming that the flow happens at the end of the day, formula_6 can be calculated as
formula_7
where
formula_8 is the number of calendar days during the return period being calculated, which equals end date minus start date (plus 1, unless you adopt the convention that the start date is the same as the end date of the previous period)
formula_9 is the number of days from the start of the return period until the day on which the flow formula_5 occurred.
This assumes that the flow happens at the end of the day. If the flow happens at the beginning of the day, the flow is in the portfolio for an additional day, so use the following formula for calculating the weight:
formula_10
Comparison with time-weighted return and internal rate of return.
The modified Dietz method has the practical advantage over the true time-weighted rate of return method, in that the calculation of a modified Dietz return does not require portfolio valuations at each point in time whenever an external flow occurs. The internal rate of return method shares this practical advantage with the modified Dietz method.
Conversely, if there exists a portfolio valuation at any point in time, the implied modified Dietz valuation of cashflows at that point in time is quite unlikely to agree with the actual valuation.
With the advance of technology, most systems can calculate a time-weighted return by calculating a daily return and geometrically linking in order to get a monthly, quarterly, annual or any other period return. However, the modified Dietz method remains useful for performance attribution, because it still has the advantage of allowing modified Dietz returns on assets to be combined with weights in a portfolio, calculated according to average invested capital, and the weighted average gives the modified Dietz return on the portfolio. Time weighted returns do not allow this.
The modified Dietz method also has the practical advantage over internal rate of return (IRR) method that it does not require repeated trial and error to get a result.
The modified Dietz method is based upon a simple rate of interest principle. It approximates the internal rate of return method, which applies a compounding principle, but if the flows and rates of return are large enough, the results of the Modified Dietz method will significantly diverge from the internal rate of return.
The modified Dietz return is the solution formula_11 to the equation:
formula_12
where
formula_1 is the start value
formula_2 is the end value
formula_13 is the total length of time period
and
formula_14 is the time between the start of the period and flow formula_15
Compare this with the (unannualized) internal rate of return (IRR). The IRR (or more strictly speaking, an un-annualized holding period return version of the IRR) is a solution formula_11 to the equation:
formula_16
Example.
Suppose the value of a portfolio is $100 at the beginning of the first year, and $300 at the end of the second year, and there is an inflow of $50 at the end of the first year/beginning of the second year. (Suppose further that neither year is a leap year, so the two years are of equal length.)
To calculate the gain or loss over the two-year period,
formula_17
To calculate the average capital over the two-year period,
formula_18
so the modified Dietz return is:
formula_19
The (unannualized) internal rate of return in this example is 125%:
formula_20
so in this case, the modified Dietz return is noticeably less than the unannualized IRR. This divergence between the modified Dietz return and the unannualized internal rate of return is due to a significant flow within the period, together with the fact that the returns are large. If there are no flows, there is no difference between the modified Dietz return, the unannualized IRR, or any other method of calculating the holding period return. If the flows are small, or if the returns themselves are small, then the difference between the modified Dietz return and the unannualized internal rate of return is small.
The IRR is 50% since:
formula_21
but the unannualized holding period return, using the IRR method, is 125%. Compounding an annual rate of 50% over two periods gives a holding period return of 125%:
formula_22
The simple Dietz method.
The modified Dietz method is different from the simple Dietz method, in which the cash flows are weighted equally regardless of when they occurred during the measurement period. The simple Dietz method is a special case of the Modified Dietz method, in which external flows are assumed to occur at the midpoint of the period, or equivalently, spread evenly throughout the period, whereas no such assumption is made when using the Modified Dietz method, and the timing of any external flows is taken into account.
Note that in the example above, the flow occurs midway through the overall period, which matches the assumption underlying the simple Dietz method. This means the simple Dietz return and modified Dietz return are the same in this particular example.
Adjustments.
If either the start or the end value is zero, or both, the start and/or end dates need to be adjusted to cover the period over which the portfolio has content.
Example.
Suppose we are calculating the 2016 calendar year return, and that the portfolio is empty until a transfer in of EUR 1m cash in a non-interest bearing account on Friday 30 December. By the end of the day on Saturday 31 December 2016, the exchange rate between euros and Hong Kong dollars has changed from 8.1 HKD per EUR to 8.181, which is a 1 percent increase in value, measured in Hong Kong dollar terms, so the right answer to the question of what is the return in Hong Kong dollars is intuitively 1 percent.
However, blindly applying the modified Dietz formula, using an end-of-day transaction timing assumption, the day-weighting on the inflow of 8.1m HKD on 30 December, one day before the end of the year, is 1/366, and the average capital is calculated as:
+ inflow × = 0 + 8.1m × 1/366 = 22,131.15
and the gain is:
- - = 8,181,000 - 0 - 8,100,000 = 81,000
so the modified Dietz return is calculated as:
= = 366 %
So which is the correct return, 1 percent or 366 percent?
Adjusted time interval.
The only sensible answer to the example above is that the holding period return is unambiguously 1 percent. This means the start date should be adjusted to the date of the initial external flow. Likewise, if the portfolio is empty at the end of the period, the end date should be adjusted to the final external flow. The end value is effectively the final external flow, not zero.
The return annualized using a simple method of multiplying-up 1 percent per day by the number of days in the year will give the answer 366 percent, but the holding period return is still 1 percent.
Example corrected.
The example above is corrected if the start date is adjusted to the end of the day on 30 December, and the start value is now 8.1m HKD. There are no external flows thereafter.
The corrected gain or loss is the same as before:
- = 8,181,000 - 8,100,000 = 81,000
but the corrected average capital is now:
+ = 8.1m
so the corrected modified Dietz return is now:
= = 1 %
Second example.
Suppose that a bond is bought for HKD 1,128,728 including accrued interest and commission on trade date 14 November, and sold again three days later on trade date 17 November for HKD 1,125,990 (again, net of accrued interest and commission). Assuming transactions take place at the start of the day, what is the modified Dietz holding-period return in HKD for this bond holding over the year to-date until the end-of-day on 17 November?
Answer.
The answer is that firstly, the reference to the holding period year to-date until the end-of-day on 17 November includes both the purchase and the sale. This means the effective adjusted holding period is actually from the purchase at the start of the day on 14 November until it is sold three days later on 17 November. The adjusted start value is the net amount of the purchase, the end value is the net amount of the sale, and there are no other external flows.
= 1,128,728
= 1,125,990
There are no flows, so the gain or loss is:
- = 1,125,990 - 1,128,728 = -2,738
and the average capital equals the start value, so the modified Dietz return is:
= = -0.24 % 2 d.p.
Contributions - when not to adjust the holding period.
This method of restricting the calculation to the actual holding period by applying an adjusted start or end date applies when the return is calculated on an investment in isolation. When the investment belongs inside a portfolio, and the weight of the investment in the portfolio, and the contribution of that return to that of the portfolio as a whole is required, it is necessary to compare like with like, in terms of a common holding period.
Example.
Suppose that at the beginning of the year, a portfolio contains cash, of value $10,000, in an account which bears interest without any charges. At the beginning of the fourth quarter, $8,000 of that cash is invested in some US dollar shares (in company X). The investor applies a buy-and-hold strategy, and there are no further transactions for the remainder of the year. At the end of the year, the shares have increased in value by 10% to $8,800, and $100 interest is capitalized into the cash account.
What is the return on the portfolio over the year? What are the contributions from the cash account and the shares? Furthermore, what is the return on the cash account?
Answer.
The end value of the portfolio is $2,100 in cash, plus shares worth $8,800, which is in total $10,900. There has been a 9 percent increase in value since the beginning of the year. There are no external flows in or out of the portfolio over the year.
= 0
so
= = $10,000
so the return is:
= = 9 %
This 9% portfolio return breaks down between 8 percent contribution from the $800 earned on the shares and 1 percent contribution from the $100 interest earned on the cash account, but how more generally can we calculate contributions?
The first step is to calculate the average capital in each of the cash account and the shares over the full year period. These should sum to the $10,000 average capital of the portfolio as a whole. From the average capital of each of the two components of the portfolio, we can calculate weights. The weight of the cash account is the average capital of the cash account, divided by the average capital ($10,000) of the portfolio, and the weight of the shares is the average capital of the shares over the whole year, divided by the average capital of the portfolio.
For convenience, we will assume the time weight of the outflow of $8,000 cash to pay for the shares is exactly 1/4. This means that the four quarters of the year are treated as having equal length.
The average capital of the cash account is:
= - ×
= 10,000 - × $8,000
= 10,000 - $2,000
= $8,000
The average capital of the shares over the last quarter requires no calculation, because there are no flows after the beginning of the last quarter. It is the $8,000 invested in the shares. However, the average capital in the shares over the whole year is something else. The start value of the shares at the beginning of the year was zero, and there was an inflow of $8,000 at the beginning of the last quarter, so:
= - ×
= 0 + × $8,000
= $2,000
We can see immediately that the weight of the cash account in the portfolio over the year was:
=
= 80 %
and the weight of the shares was:
=
= 20 %
which sum to 100 percent.
We can calculate the return on the cash account, which was:
= = 1.25 %
The contribution to the portfolio return is:
× return = 80 % × 1.25 % = 1 %
How about the contribution to the portfolio return from the shares?
The adjusted holding period return on the shares is 10 percent. If we multiply this by the 20 percent weight of the shares in the portfolio, the result is only 2 percent, but the correct contribution is 8 percent.
The answer is to use the return on the shares over the unadjusted full-year period to calculate the contribution:
=
= 40 %
Then the contribution from the shares to the portfolio return is:
= 20% × 40 % = 8 %
This does not mean that the correct holding period return on the shares is 40 percent, but for calculation of the contribution, use the unadjusted period return, which is the 40 percent figure, not the actual 10 percent holding period return.
Fees.
To measure returns net of fees, allow the value of the portfolio to be reduced by the amount of the fees. To calculate returns gross of fees, compensate for them by treating them as an external flow, and exclude accrued fees from valuations.
Annual rate of return.
Note that the Modified Dietz return is a holding-period return, not an annual rate of return, unless the period happens to be one year. Annualisation, which is conversion of the holding-period return to an annual rate of return, is a separate process.
Money-weighted return.
The modified Dietz method is an example of a money (or dollar) weighted methodology (as opposed to time-weighted). In particular, if the modified Dietz return on two portfolios are formula_23 and formula_24, measured over a common matching time interval, then the modified Dietz return on the two portfolios put together over the same time interval is the weighted average of the two returns:
formula_25
where the weights of the portfolios depend on the average capital over the time interval:
formula_26
Linked return versus true time-weighted return.
An alternative to the modified Dietz method is to link geometrically the modified Dietz returns for shorter periods. The linked modified Dietz method is classed as a time-weighted method, but it does not produce the same results as the true time weighted method, which requires valuations at the time of each cash flow.
Issues.
Problems with timing assumptions.
There are sometimes difficulties when calculating or decomposing portfolio returns, if all transactions are treated as occurring at a single time of day, such as the end of the day or beginning of the day. Whatever method is applied to calculate returns, an assumption that all transactions take place simultaneously at a single point in time each day can lead to errors.
For example, consider a scenario where a portfolio is empty at the start of a day, so that the start value A is zero. There is then an external inflow during that day of F = $100. By the close of the day, market prices have moved, and the end value is $99.
If all transactions are treated as occurring at the end of the day, then there is zero start value A, and zero value for average capital, because the day-weight on the inflow is zero, so no modified Dietz return can be calculated.
Some such problems are resolved if the modified Dietz method is further adjusted so as to put purchases at the open and sales at the close, but more sophisticated exception-handling produces better results.
There are sometimes other difficulties when decomposing portfolio returns, if all transactions are treated as occurring at a single point during the day.
For example, consider a fund opening with just $100 of a single stock that is sold for $110 during the day. During the same day, another stock is purchased for $110, closing with a value of $120. The returns on each stock are 10% and 120/110 - 1 = 9.0909% (4 d.p.) and the portfolio return is 20%. The asset weights "wi" (as opposed to the time weights "Wi") required to get the returns for these two assets to roll up to the portfolio return are 1200% for the first stock and a negative 1100% for the second:
w*10/100 + (1-w)*10/110 = 20/100 → w = 12.
Such weights are absurd, because the second stock is not held short.
The problem only arises because the day is treated as a single, discrete time interval.
Negative or zero average capital.
In normal circumstances, average capital is positive. When an intra-period outflow is large and early enough, average capital can be negative or zero. Negative average capital causes the Modified Dietz return to be negative when there is a profit, and positive when there is a loss. This resembles the behaviour of a liability or short position, even if the investment is not actually a liability or a short position. In cases where the average capital is zero, no Modified Dietz return can be calculated. If the average capital is close to zero, the Modified Dietz return will be large (large and positive, or large and negative).
One partial workaround solution involves as a first step, to capture the exception, detecting for example when the start value (or first inflow) is positive, and the average capital is negative. Then in this case, use the simple return method, adjusting the end value for outflows. This is equivalent to the sum of constituent contributions, where the contributions are based on simple returns and weights depending on start values.
Example.
For example, in a scenario where only part of the holdings are sold, for significantly more than the total starting value, relatively early in the period:
At the start of Day 1, the number of shares is 100
At the start of Day 1, the share price is 10 dollars
Start value = 1,000 dollars
At the end of Day 5, 80 shares are sold at 15 dollars per share
At the end of Day 40, the remaining 20 shares are worth 12.50 dollars per share
The gain or loss is end value - start value + outflow:
formula_27
formula_28
formula_29
There is a gain, and the position is long, so we would intuitively expect a positive return.
The average capital in this case is:
formula_30
formula_31
formula_32
formula_33
formula_34
The modified Dietz return in this case goes awry, because the average capital is negative, even though this is a long position. The Modified Dietz return in this case is:
formula_35
Instead, we notice that the start value is positive, but the average capital is negative. Furthermore, there is no short sale. In other words, at all times, the number of shares held is positive.
We then measure the simple return from the shares sold:
formula_36
and from the shares still held at the end:
formula_37
and combine these returns with the weights of these two portions of the shares within the starting position, which are:
formula_38 and formula_39 respectively.
This gives the contributions to the overall return, which are:
formula_40 and formula_41 respectively.
The sum of these contributions is the return:
formula_42
This is equivalent to the simple return, adjusting the end value for outflows:
formula_43
formula_44
formula_45
formula_46
formula_47
formula_48
formula_49
formula_50
Limitations.
This workaround has limitations. It is possible only if the holdings can be split up like this.
It is not ideal, for two further reasons, which are that it does not cover all cases, and it is inconsistent with the Modified Dietz method. Combined with Modified Dietz contributions for other assets, the sum of constituent contributions will fail to add up to the overall return.
Another situation in which average capital can be negative is short selling. Instead of investing by buying shares, shares are borrowed and then sold. A decline in the share price results in a profit instead of a loss. The position is a liability instead of an asset. If the profit is positive, and the average capital is negative, the Modified Dietz return is negative, indicating that although the number of shares is unchanged, the absolute value of the liability has shrunk.
In the case of a purchase, followed by a sale of more shares than had been bought, resulting in a short position (a negative number of shares), the average capital can also be negative. What was an asset at the time of the purchase became a liability after the sale. The interpretation of the Modified Dietz return varies from one situation to another.
Visual Basic.
Function georet_MD(myDates, myReturns, FlowMap, scaler)
' This function calculates the modified Dietz return of a time series
' Inputs.
' myDates. Tx1 vector of dates
' myReturns. Tx1 vector of financial returns
' FlowMap. Nx2 matrix of Dates (left column) and flows (right column)
' scaler. Scales the returns to the appropriate frequency
' Outputs.
' Modified Dietz Returns.
' Note that all the dates of the flows need to exist in the date vector that is provided.
' when a flow is entered, it only starts accumulating after 1 period.
Dim i, j, T, N As Long
Dim matchFlows(), Tflows(), cumFlows() As Double
Dim np As Long
Dim AvFlows, TotFlows As Double
' Get dimensions
If StrComp(TypeName(myDates), "Range") = 0 Then
T = myDates.Rows.Count
Else
T = UBound(myDates, 1)
End If
If StrComp(TypeName(FlowMap), "Range") = 0 Then
N = FlowMap.Rows.Count
Else
N = UBound(FlowMap, 1)
End If
' Redim arrays
ReDim cumFlows(1 To T, 1 To 1)
ReDim matchFlows(1 To T, 1 To 1)
ReDim Tflows(1 To T, 1 To 1)
' Create a vector of Flows
For i = 1 To N
j = Application.WorksheetFunction.Match(FlowMap(i, 1), myDates, True)
matchFlows(j, 1) = FlowMap(i, 2)
Tflows(j, 1) = 1 - (FlowMap(i, 1) - FlowMap(1, 1)) / (myDates(T, 1) - FlowMap(1, 1))
If i = 1 Then np = T - j
Next i
' Cumulated Flows
For i = 1 To T
If i = 1 Then
cumFlows(i, 1) = matchFlows(i, 1)
Else
cumFlows(i, 1) = cumFlows(i - 1, 1) * (1 + myReturns(i, 1)) + matchFlows(i, 1)
End If
Next i
AvFlows = Application.WorksheetFunction.SumProduct(matchFlows, Tflows)
TotFlows = Application.WorksheetFunction.Sum(matchFlows)
georet_MD = (1 + (cumFlows(T, 1) - TotFlows) / AvFlows) ^ (scaler / np) - 1
End Function
Java method for modified Dietz return.
private static double modifiedDietz (double emv, double bmv, double cashFlow[], int numCD, int numD[]) {
/* emv: Ending Market Value
* bmv: Beginning Market Value
* cashFlow[]: Cash Flow
* numCD: actual number of days in the period
* numD[]: number of days between beginning of the period and date of cashFlow[]
double md = -99999; // initialize modified dietz with a debugging number
try {
double[] weight = new double[cashFlow.length];
if (numCD <= 0) {
throw new ArithmeticException ("numCD <= 0");
for (int i=0; i<cashFlow.length; i++) {
if (numD[i] < 0) {
throw new ArithmeticException ("numD[i]<0 , " + "i=" + i);
weight[i] = (double) (numCD - numD[i]) / numCD;
double ttwcf = 0; // total time weighted cash flows
for (int i=0; i<cashFlow.length; i++) {
ttwcf += weight[i] * cashFlow[i];
double tncf = 0; // total net cash flows
for (int i=0; i<cashFlow.length; i++) {
tncf += cashFlow[i];
md = (emv - bmv - tncf) / (bmv + ttwcf);
catch (ArrayIndexOutOfBoundsException e) {
e.printStackTrace();
catch (ArithmeticException e) {
e.printStackTrace();
catch (Exception e) {
e.printStackTrace();
return md;
Excel VBA function for modified Dietz return.
Public Function MDIETZ(dStartValue As Double, dEndValue As Double, iPeriod As Integer, rCash As Range, rDays As Range) As Double
'Jelle-Jeroen Lamkamp 10 Jan 2008
Dim i As Integer: Dim Cash() As Double: Dim Days() As Integer
Dim Cell As Range: Dim SumCash As Double: Dim TempSum As Double
'Some error trapping
If rCash.Cells.Count <> rDays.Cells.Count Then MDIETZ = CVErr(xlErrValue): Exit Function
If Application.WorksheetFunction.Max(rDays) > iPeriod Then MDIETZ = CVErr(xlErrValue): Exit Function
ReDim Cash(rCash.Cells.Count - 1)
ReDim Days(rDays.Cells.Count - 1)
i = 0
For Each Cell In rCash
Cash(i) = Cell.Value: i = i + 1
Next Cell
i = 0
For Each Cell In rDays
Days(i) = Cell.Value: i = i + 1
Next Cell
SumCash = Application.WorksheetFunction.Sum(rCash)
TempSum = 0
For i = 0 To (rCash.Cells.Count - 1)
TempSum = TempSum + (((iPeriod - Days(i)) / iPeriod) * Cash(i))
Next i
MDIETZ = (dEndValue - dStartValue - SumCash) / (dStartValue + TempSum)
End Function
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\cfrac{\\text {gain or loss}}{\\text {average capital}}= \\cfrac{ B - A - F}{ A + \\sum_{i=1}^n W_i \\times F_i}"
},
{
"math_id": 1,
"text": "A"
},
{
"math_id": 2,
"text": "B"
},
{
"math_id": 3,
"text": "F = \\sum_{i=1}^n F_i"
},
{
"math_id": 4,
"text": "\\sum_{i=1}^n W_i \\times {F_i} = "
},
{
"math_id": 5,
"text": "F_i"
},
{
"math_id": 6,
"text": "W_i"
},
{
"math_id": 7,
"text": " W_i = \\frac{C-D_i}{C}"
},
{
"math_id": 8,
"text": "C"
},
{
"math_id": 9,
"text": "D_i"
},
{
"math_id": 10,
"text": " W_i = \\frac{C -D_i + 1}{C}"
},
{
"math_id": 11,
"text": "R"
},
{
"math_id": 12,
"text": "B = A \\times (1+R)+ \\sum_{i=1}^n F_i \\times \\left( 1+R \\times \\frac{T - t_i}{T} \\right)"
},
{
"math_id": 13,
"text": "T"
},
{
"math_id": 14,
"text": "t_i"
},
{
"math_id": 15,
"text": "i"
},
{
"math_id": 16,
"text": "B = A \\times (1+R)+ \\sum_{i=1}^n F_i \\times (1+R)^ \\frac{T - t_i}{T}"
},
{
"math_id": 17,
"text": "\\text {gain or loss} = B - A - F = 300 - 100 - 50 = $150\\text{.}"
},
{
"math_id": 18,
"text": "\\text {average capital} = A + \\sum \\text {weight} \\times \\text {flow} = 100 + 0.5 \\times 50 = $125\\text{,}"
},
{
"math_id": 19,
"text": "\\frac {\\text {gain or loss}}{\\text {average capital}} = \\frac {150}{125} = 120\\%"
},
{
"math_id": 20,
"text": "100 \\times (1 + 125\\%)+ 50 \\times (1+125\\%)^ \\frac{2 - 1}{2} = 225 + 50 \\times 150\\% = 225 + 75 = 300"
},
{
"math_id": 21,
"text": "100 \\times (1 + 50\\%)^2 + 50 \\times (1+50\\%)^ 1 = 225 + 50 \\times 150\\% = 225 + 75 = 300"
},
{
"math_id": 22,
"text": "(1 + 50\\%)^2 - 1 = 2.25 - 1 = 1.25 = 125\\%"
},
{
"math_id": 23,
"text": "R_1"
},
{
"math_id": 24,
"text": "R_2"
},
{
"math_id": 25,
"text": "W_1 \\times R_1+W_2 \\times R_2"
},
{
"math_id": 26,
"text": "W_i = \\frac{\\text{average capital}_i}{\\text{average capital}_1+\\text{average capital}_2}"
},
{
"math_id": 27,
"text": "20 \\times 12.50 - 100 \\times 10 + 80 \\times 15"
},
{
"math_id": 28,
"text": "= 250 - 1,000 + 1,200"
},
{
"math_id": 29,
"text": "= 450"
},
{
"math_id": 30,
"text": "\\text {start value} - \\text {time weight} \\times \\text {outflow on Day 5}"
},
{
"math_id": 31,
"text": "= 100 \\times 10 - \\frac {40 - 5}{40} \\times 80 \\times 15"
},
{
"math_id": 32,
"text": "= 1,000 - \\frac {7}{8} \\times 1,200"
},
{
"math_id": 33,
"text": "= 1,000 - 1,050"
},
{
"math_id": 34,
"text": "= -50 \\text { dollars}"
},
{
"math_id": 35,
"text": "\\frac {\\text {gain or loss}}{\\text {average capital}} = \\frac {450}{-50} = -900 \\%"
},
{
"math_id": 36,
"text": "\\frac {15 - 10}{10} = 50 \\%"
},
{
"math_id": 37,
"text": "\\frac {12.50 - 10}{10} = 25 \\%"
},
{
"math_id": 38,
"text": "\\frac {80}{100} = 80 \\%"
},
{
"math_id": 39,
"text": "\\frac {20}{100} = 20 \\%"
},
{
"math_id": 40,
"text": "50 \\% \\times 80 \\% = 40 \\%"
},
{
"math_id": 41,
"text": "25 \\% \\times 20 \\% = 5 \\%"
},
{
"math_id": 42,
"text": "40 \\% + 5 \\% = 45 \\%"
},
{
"math_id": 43,
"text": "\\text {Start value} = 1,000 \\text { dollars}"
},
{
"math_id": 44,
"text": "\\text {Adjusted end value} = \\text {end value} + \\text {outflow}"
},
{
"math_id": 45,
"text": "= 20 \\times 12.50 + 80 \\times 15"
},
{
"math_id": 46,
"text": "= 250 + 1,200"
},
{
"math_id": 47,
"text": "= 1,450"
},
{
"math_id": 48,
"text": "\\text {Simple return} = \\frac {\\text {adjusted end value} - \\text {start value}}{\\text {start value}}"
},
{
"math_id": 49,
"text": "= \\frac {1,450 - 1,000}{1,000}"
},
{
"math_id": 50,
"text": "= 45 \\%"
}
]
| https://en.wikipedia.org/wiki?curid=13034946 |
13035 | Gaussian elimination | Algorithm for solving systems of linear equations
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations:
Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the matrix is said to be in reduced row echelon form. This final form is unique; in other words, it is independent of the sequence of row operations used. For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form.
formula_0
Using row operations to convert a matrix into reduced row echelon form is sometimes called <templatestyles src="Template:Visible anchor/styles.css" />Gauss–Jordan elimination. In this case, the term "Gaussian elimination" refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes preferable to stop row operations before the matrix is completely reduced.
Definitions and example of algorithm.
The process of row reduction makes use of elementary row operations, and can be divided into two parts. The first part (sometimes called forward elimination) reduces a given system to row echelon form, from which one can tell whether there are no solutions, a unique solution, or infinitely many solutions. The second part (sometimes called back substitution) continues to use row operations until the solution is found; in other words, it puts the matrix into reduced row echelon form.
Another point of view, which turns out to be very useful to analyze the algorithm, is that row reduction produces a matrix decomposition of the original matrix. The elementary row operations may be viewed as the multiplication on the left of the original matrix by elementary matrices. Alternatively, a sequence of elementary operations that reduces a single row may be viewed as multiplication by a Frobenius matrix. Then the first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row echelon matrix.
Row operations.
There are three types of elementary row operations which may be performed on the rows of a matrix:
If the matrix is associated to a system of linear equations, then these operations do not change the solution set. Therefore, if one's goal is to solve a system of linear equations, then using these row operations could make the problem easier.
Echelon form.
For each row in a matrix, if the row does not consist of only zeros, then the leftmost nonzero entry is called the "leading coefficient" (or "pivot") of that row. So if two leading coefficients are in the same column, then a row operation of type 3 could be used to make one of those coefficients zero. Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading coefficient is to the right of the leading coefficient of the row above. If this is the case, then matrix is said to be in row echelon form. So the lower left part of the matrix contains only zeros, and all of the zero rows are below the non-zero rows. The word "echelon" is used here because one can roughly think of the rows being ranked by their size, with the largest being at the top and the smallest being at the bottom.
For example, the following matrix is in row echelon form, and its leading coefficients are shown in red:
formula_1
It is in echelon form because the zero row is at the bottom, and the leading coefficient of the second row (in the third column), is to the right of the leading coefficient of the first row (in the second column).
A matrix is said to be in reduced row echelon form if furthermore all of the leading coefficients are equal to 1 (which can be achieved by using the elementary row operation of type 2), and in every column containing a leading coefficient, all of the other entries in that column are zero (which can be achieved by using elementary row operations of type 3).
Example of the algorithm.
Suppose the goal is to find and describe the set of solutions to the following system of linear equations:
formula_2
The table below is the row reduction process applied simultaneously to the system of equations and its associated augmented matrix. In practice, one does not usually deal with the systems in terms of equations, but instead makes use of the augmented matrix, which is more suitable for computer manipulations. The row reduction procedure may be summarized as follows: eliminate x from all equations below "L"1, and then eliminate y from all equations below "L"2. This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for.
The second column describes which row operations have just been performed. So for the first step, the x is eliminated from "L"2 by adding "L"1 to "L"2. Next, x is eliminated from "L"3 by adding "L"1 to "L"3. These row operations are labelled in the table as
formula_3
Once y is also eliminated from the third row, the result is a system of linear equations in triangular form, and so the first part of the algorithm is complete. From a computational point of view, it is faster to solve the variables in reverse order, a process known as back-substitution. One sees the solution is "z" = −1, "y" = 3, and "x" = 2. So there is a unique solution to the original system of equations.
Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in "reduced" row echelon form, as it is done in the table. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss–Jordan elimination, to distinguish it from stopping after reaching echelon form.
History.
The method of Gaussian elimination appears – albeit without proof – in the Chinese mathematical text Chapter Eight: "Rectangular Arrays" of "The Nine Chapters on the Mathematical Art". Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 AD, but parts of it were written as early as approximately 150 BC. It was commented on by Liu Hui in the 3rd century.
The method in Europe stems from the notes of Isaac Newton. In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as "Arithmetica Universalis" in 1707 long after Newton had left academic life. The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject.
Some authors use the term "Gaussian elimination" to refer only to the procedure until the matrix is in echelon form, and use the term Gauss–Jordan elimination to refer to the procedure which ends in reduced echelon form. The name is used because it is a variation of Gaussian elimination as described by Wilhelm Jordan in 1888. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss–Jordan elimination independently.
Applications.
Historically, the first application of the row reduction method is for solving systems of linear equations. Below are some other important applications of the algorithm.
Computing determinants.
To explain how Gaussian elimination allows the computation of the determinant of a square matrix, we have to recall how the elementary row operations change the determinant:
If Gaussian elimination applied to a square matrix A produces a row echelon matrix B, let d be the product of the scalars by which the determinant has been multiplied, using the above rules. Then the determinant of A is the quotient by d of the product of the elements of the diagonal of B:
formula_4
Computationally, for an "n" × "n" matrix, this method needs only O("n"3) arithmetic operations, while using Leibniz formula for determinants requires formula_5 operations , and
recursive Laplace expansion requires O("n" 2"n") operations if the sub-determinants are memorized for being computed only once . Even on the fastest computers, these two methods are impractical or almost impracticable for "n" above 20.
Finding the inverse of a matrix.
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If "A" is an "n" × "n" square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the "n" × "n" identity matrix is augmented to the right of "A", forming an "n" × 2"n" block matrix ["A" | "I"]. Now through application of elementary row operations, find the reduced echelon form of this "n" × 2"n" matrix. The matrix "A" is invertible if and only if the left block can be reduced to the identity matrix "I"; in this case the right block of the final matrix is "A"−1. If the algorithm is unable to reduce the left block to "I", then "A" is not invertible.
For example, consider the following matrix:
formula_6
To find the inverse of this matrix, one takes the following matrix augmented by the identity and row-reduces it as a 3 × 6 matrix:
formula_7
By performing row operations, one can check that the reduced row echelon form of this augmented matrix is
formula_8
One can think of each row operation as the left product by an elementary matrix. Denoting by "B" the product of these elementary matrices, we showed, on the left, that "BA" = "I", and therefore, "B" = "A"−1. On the right, we kept a record of "BI" = "B", which we know is the inverse desired. This procedure for finding the inverse works for square matrices of any size.
Computing ranks and bases.
The Gaussian elimination algorithm can be applied to any "m" × "n" matrix A. In this way, for example, some 6 × 9 matrices can be transformed to a matrix that has a row echelon form like
formula_9
where the stars are arbitrary entries, and "a", "b", "c", "d", "e" are nonzero entries. This echelon matrix T contains a wealth of information about A: the rank of A is 5, since there are 5 nonzero rows in T; the vector space spanned by the columns of A has a basis consisting of its columns 1, 3, 4, 7 and 9 (the columns with "a", "b", "c", "d", "e" in T), and the stars show how the other columns of A can be written as linear combinations of the basis columns.
All of this applies also to the reduced row echelon form, which is a particular row echelon format.
Computational efficiency.
The number of arithmetic operations required to perform row reduction is one way of measuring the algorithm's computational efficiency. For example, to solve a system of "n" equations for "n" unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires "n"("n" + 1)/2 divisions, (2"n"3 + 3"n"2 − 5"n")/6 multiplications, and (2"n"3 + 3"n"2 − 5"n")/6 subtractions, for a total of approximately 2"n"3/3 operations. Thus it has a "arithmetic complexity" (time complexity, where each arithmetic operation take a unit of time, independently of the size of the inputs) of O("n"3).
This complexity is a good measure of the time needed for the whole computation when the time for each arithmetic operation is approximately constant. This is the case when the coefficients are represented by floating-point numbers or when they belong to a finite field. If the coefficients are integers or rational numbers exactly represented, the intermediate entries can grow exponentially large, so the bit complexity is exponential.
However, Bareiss' algorithm is a variant of Gaussian elimination that avoids this exponential growth of the intermediate entries; with the same arithmetic complexity of O("n"3), it has a bit complexity of O("n"5), and has therefore a strongly-polynomial time complexity.
Gaussian elimination and its variants can be used on computers for systems with thousands of equations and unknowns. However, the cost becomes prohibitive for systems with millions of equations. These large systems are generally solved using iterative methods. Specific methods exist for systems whose coefficients follow a regular pattern (see system of linear equations).
Bareiss algorithm.
The first strongly-polynomial time algorithm for Gaussian elimination was published by Jack Edmonds in 1967.37 Independently, and almost simultaneously, Erwin Bareiss discovered another algorithm, based on the following remark, which applies to a division-free variant of Gaussian elimination.
In standard Gaussian elimination, one subtracts from each row formula_10 below the pivot row formula_11 a multiple of formula_11 by formula_12 where formula_13 and formula_14 are the entries in the pivot column of formula_10 and formula_15 respectively.
Bareiss variant consists, instead, of replacing formula_10 with formula_16 This produces a row echelon form that has the same zero entries as with the standard Gaussian elimination.
Bareiss' main remark is that each matrix entry generated by this variant is the determinant of a submatrix of the original matrix.
In particular, if one starts with integer entries, the divisions occurring in the algorithm are exact divisions resulting in integers. So, all intermediate entries and final entries are integers. Moreover, Hadamard inequality provides an upper bound on the absolute values of the intermediate and final entries, and thus a bit complexity of formula_17 using soft O notation.
Moreover, as an upper bound on the size of final entries is known, a complexity formula_18 can be obtained with modular computation followed either by Chinese remaindering or Hensel lifting.
As a corollary, the following problems can be solved in strongly polynomial time with the same bit complexity:40
Numeric instability.
One possible problem is numerical instability, caused by the possibility of dividing by very small numbers. If, for example, the leading coefficient of one of the rows is very close to zero, then to row-reduce the matrix, one would need to divide by that number. This means that any error which existed for the number that was close to zero would be amplified. Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable, when using partial pivoting, even though there are examples of stable matrices for which it is unstable.
Generalizations.
Gaussian elimination can be performed over any field, not just the real numbers.
Buchberger's algorithm is a generalization of Gaussian elimination to systems of polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination, manifesting as the choice to work from left to right when selecting pivot positions.
Computing the rank of a tensor of order greater than 2 is NP-hard. Therefore, if , there cannot be a polynomial time analog of Gaussian elimination for higher-order tensors (matrices are array representations of order-2 tensors).
Pseudocode.
As explained above, Gaussian elimination transforms a given "m" × "n" matrix A into a matrix in row-echelon form.
In the following pseudocode, codice_0 denotes the entry of the matrix A in row i and column j with the indices starting from 1. The transformation is performed "in place", meaning that the original matrix is lost for being eventually replaced by its row-echelon form.
h := 1 /* "Initialization of the pivot row" */
k := 1 /* "Initialization of the pivot column" */
while h ≤ m and k ≤ n
/* "Find the k-th pivot:" */
i_max := argmax (i = h ... m, abs(A[i, k]))
if A[i_max, k] = 0
/* "No pivot in this column, pass to next column" */
k := k + 1
else
swap rows(h, i_max)
/* "Do for all rows below pivot:" */
for i = h + 1 ... m:
f := A[i, k] / A[h, k]
/* "Fill with zeros the lower part of pivot column:" */
A[i, k] := 0
/* "Do for all remaining elements in current row:" */
for j = k + 1 ... n:
A[i, j] := A[i, j] - A[h, j] * f
/* "Increase pivot row and column" */
h := h + 1
k := k + 1
This algorithm differs slightly from the one discussed earlier, by choosing a pivot with largest absolute value. Such a "partial pivoting" may be required if, at the pivot place, the entry of the matrix is zero. In any case, choosing the largest possible absolute value of the pivot improves the numerical stability of the algorithm, when floating point is used for representing numbers.
Upon completion of this procedure the matrix will be in row echelon form and the corresponding system may be solved by back substitution.
References.
<templatestyles src="Reflist/styles.css" />
Works cited.
<templatestyles src="Refbegin/styles.css" /> | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\n1 & 3 & 1 & 9 \\\\\n1 & 1 & -1 & 1 \\\\\n3 & 11 & 5 & 35\n\\end{bmatrix}\\to\n\\begin{bmatrix}\n1 & 3 & 1 & 9 \\\\\n0 & -2 & -2 & -8 \\\\\n0 & 2 & 2 & 8\n\\end{bmatrix}\\to\n\\begin{bmatrix}\n1 & 3 & 1 & 9 \\\\\n0 & -2 & -2 & -8 \\\\\n0 & 0 & 0 & 0\n\\end{bmatrix}\\to\n\\begin{bmatrix}\n1 & 0 & -2 & -3 \\\\\n0 & 1 & 1 & 4 \\\\\n0 & 0 & 0 & 0\n\\end{bmatrix} "
},
{
"math_id": 1,
"text": "\\begin{bmatrix}\n 0 & \\color{red}{\\mathbf{2}} & 1 & -1 \\\\\n 0 & 0 & \\color{red}{\\mathbf{3}} & 1 \\\\\n 0 & 0 & 0 & 0\n\\end{bmatrix}."
},
{
"math_id": 2,
"text": "\n\\begin{alignat}{4}\n 2x &{}+{}& y &{}-{}& z &{}={}& 8 & \\qquad (L_1) \\\\\n -3x &{}-{}& y &{}+{}& 2z &{}={}& -11 & \\qquad (L_2) \\\\\n -2x &{}+{}& y &{}+{}& 2z &{}={}& -3 & \\qquad (L_3)\n\\end{alignat}\n"
},
{
"math_id": 3,
"text": "\\begin{align}\n L_2 + \\tfrac32 L_1 &\\to L_2, \\\\\n L_3 + L_1 &\\to L_3.\n\\end{align}"
},
{
"math_id": 4,
"text": "\\det(A) = \\frac{\\prod\\operatorname{diag}(B)}{d}."
},
{
"math_id": 5,
"text": "(n\\, n!)"
},
{
"math_id": 6,
"text": "A =\n \\begin{bmatrix}\n 2 & -1 & 0 \\\\\n -1 & 2 & -1 \\\\\n 0 & -1 & 2\n \\end{bmatrix}.\n"
},
{
"math_id": 7,
"text": "[ A | I ] = \n \\left[\\begin{array}{ccc|ccc}\n 2 & -1 & 0 & 1 & 0 & 0 \\\\\n -1 & 2 & -1 & 0 & 1 & 0 \\\\\n 0 & -1 & 2 & 0 & 0 & 1\n \\end{array}\\right].\n"
},
{
"math_id": 8,
"text": "[ I | B ] = \n \\left[\\begin{array}{rrr|rrr}\n 1 & 0 & 0 & \\frac34 & \\frac12 & \\frac14 \\\\\n 0 & 1 & 0 & \\frac12 & 1 & \\frac12 \\\\\n 0 & 0 & 1 & \\frac14 & \\frac12 & \\frac34\n \\end{array}\\right].\n"
},
{
"math_id": 9,
"text": " T=\n\\begin{bmatrix}\na & * & * & *& * & * & * & * & * \\\\\n0 & 0 & b & * & * & * & * & * & * \\\\\n0 & 0 & 0 & c & * & * & * & * & * \\\\\n0 & 0 & 0 & 0 & 0 & 0 & d & * & * \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix},\n"
},
{
"math_id": 10,
"text": "R_i"
},
{
"math_id": 11,
"text": "R_k"
},
{
"math_id": 12,
"text": "r_{i,k}/r_{k,k},"
},
{
"math_id": 13,
"text": "r_{i,k}"
},
{
"math_id": 14,
"text": "r_{k,k}"
},
{
"math_id": 15,
"text": "R_k,"
},
{
"math_id": 16,
"text": "\\frac{r_{k,k}R_i-r_{i,k}R_k}{r_{k-1,k-1}}."
},
{
"math_id": 17,
"text": "\\tilde O(n^5),"
},
{
"math_id": 18,
"text": "\\tilde O(n^4)"
}
]
| https://en.wikipedia.org/wiki?curid=13035 |
13035709 | Hellinger distance | Metric used in probability and statistics
In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of "f"-divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger in 1909.
It is sometimes called the Jeffreys distance.
Definition.
Measure theory.
To define the Hellinger distance in terms of measure theory, let formula_0 and formula_1 denote two probability measures on a measure space formula_2 that are absolutely continuous with respect to an auxiliary measure formula_3. Such a measure always exists, e.g formula_4. The square of the Hellinger distance between formula_0 and formula_1 is defined as the quantity
formula_5
Here, formula_6 and formula_7, i.e. formula_8 and formula_9 are the Radon–Nikodym derivatives of "P" and "Q" respectively with respect to formula_3. This definition does not depend on formula_3, i.e. the Hellinger distance between "P" and "Q" does not change if formula_3 is replaced with a different probability measure with respect to which both "P" and "Q" are absolutely continuous. For compactness, the above formula is often written as
formula_10
Probability theory using Lebesgue measure.
To define the Hellinger distance in terms of elementary probability theory, we take λ to be the Lebesgue measure, so that "dP" / "dλ" and "dQ" / "d"λ are simply probability density functions. If we denote the densities as "f" and "g", respectively, the squared Hellinger distance can be expressed as a standard calculus integral
formula_11
where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain equals 1.
The Hellinger distance "H"("P", "Q") satisfies the property (derivable from the Cauchy–Schwarz inequality)
formula_12
Discrete distributions.
For two discrete probability distributions formula_13 and formula_14,
their Hellinger distance is defined as
formula_15
which is directly related to the Euclidean norm of the difference of the square root vectors, i.e.
formula_16
Also, formula_17
Properties.
The Hellinger distance forms a bounded metric on the space of probability distributions over a given probability space.
The maximum distance 1 is achieved when "P" assigns probability zero to every set to which "Q" assigns a positive probability, and vice versa.
Sometimes the factor formula_18 in front of the integral is omitted, in which case the Hellinger distance ranges from zero to the square root of two.
The Hellinger distance is related to the Bhattacharyya coefficient formula_19 as it can be defined as
formula_20
Hellinger distances are used in the theory of sequential and asymptotic statistics.
The squared Hellinger distance between two normal distributions formula_21 and formula_22 is:
formula_23
The squared Hellinger distance between two multivariate normal distributions formula_24 and formula_25 is
formula_26
The squared Hellinger distance between two exponential distributions formula_27 and formula_28 is:
formula_29
The squared Hellinger distance between two Weibull distributions formula_30 and formula_31 (where formula_32 is a common shape parameter and formula_33 are the scale parameters respectively):
formula_34
The squared Hellinger distance between two Poisson distributions with rate parameters formula_35 and formula_36, so that formula_37 and formula_38, is:
formula_39
The squared Hellinger distance between two beta distributions formula_40 and formula_41 is:
formula_42
where formula_43 is the beta function.
The squared Hellinger distance between two gamma distributions formula_44 and formula_45 is:
formula_46
where formula_47 is the gamma function.
Connection with total variation distance.
The Hellinger distance formula_48 and the total variation distance (or statistical distance) formula_49 are related as follows:
formula_50
The constants in this inequality may change depending on which renormalization you choose (formula_18 or formula_51).
These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm.
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "P"
},
{
"math_id": 1,
"text": "Q"
},
{
"math_id": 2,
"text": "\\mathcal{X}"
},
{
"math_id": 3,
"text": "\\lambda"
},
{
"math_id": 4,
"text": "\\lambda = (P + Q)"
},
{
"math_id": 5,
"text": "H^2(P,Q) = \\frac{1}{2}\\displaystyle \\int_{\\mathcal{X}} \\left(\\sqrt{p(x)} - \\sqrt{q(x)}\\right)^2 \\lambda(dx). "
},
{
"math_id": 6,
"text": "P(dx) = p(x)\\lambda(dx)"
},
{
"math_id": 7,
"text": "Q(dx) = q(x) \\lambda(dx)"
},
{
"math_id": 8,
"text": "p"
},
{
"math_id": 9,
"text": "q"
},
{
"math_id": 10,
"text": "H^2(P,Q) = \\frac{1}{2}\\int_{\\mathcal{X}} \\left(\\sqrt{P(dx)} - \\sqrt{Q(dx)}\\right)^2. "
},
{
"math_id": 11,
"text": "H^2(f,g) =\\frac{1}{2}\\int \\left(\\sqrt{f(x)} - \\sqrt{g(x)}\\right)^2 \\, dx = 1 - \\int \\sqrt{f(x) g(x)} \\, dx,"
},
{
"math_id": 12,
"text": "0\\le H(P,Q) \\le 1."
},
{
"math_id": 13,
"text": "P=(p_1, \\ldots, p_k)"
},
{
"math_id": 14,
"text": "Q=(q_1, \\ldots, q_k)"
},
{
"math_id": 15,
"text": "\n H(P, Q) = \\frac{1}{\\sqrt{2}} \\; \\sqrt{\\sum_{i=1}^k (\\sqrt{p_i} - \\sqrt{q_i})^2},\n"
},
{
"math_id": 16,
"text": "\nH(P, Q) = \\frac{1}{\\sqrt{2}} \\; \\bigl\\|\\sqrt{P} - \\sqrt{Q} \\bigr\\|_2 .\n"
},
{
"math_id": 17,
"text": "\n 1 - H^2(P,Q) = \\sum_{i=1}^k \\sqrt{p_i q_i}.\n"
},
{
"math_id": 18,
"text": "1/2"
},
{
"math_id": 19,
"text": "BC(P,Q)"
},
{
"math_id": 20,
"text": "H(P,Q) = \\sqrt{1 - BC(P,Q)}."
},
{
"math_id": 21,
"text": " P \\sim \\mathcal{N}(\\mu_1,\\sigma_1^2)"
},
{
"math_id": 22,
"text": " Q \\sim \\mathcal{N}(\\mu_2,\\sigma_2^2)"
},
{
"math_id": 23,
"text": "\n H^2(P, Q) = 1 - \\sqrt{\\frac{2\\sigma_1\\sigma_2}{\\sigma_1^2+\\sigma_2^2}} \\, e^{-\\frac{1}{4}\\frac{(\\mu_1-\\mu_2)^2}{\\sigma_1^2+\\sigma_2^2}}.\n "
},
{
"math_id": 24,
"text": " P \\sim \\mathcal{N}(\\mu_1,\\Sigma_1)"
},
{
"math_id": 25,
"text": " Q \\sim \\mathcal{N}(\\mu_2,\\Sigma_2)"
},
{
"math_id": 26,
"text": "\n H^2(P, Q) = 1 - \\frac{ \\det (\\Sigma_1)^{1/4} \\det (\\Sigma_2) ^{1/4}} { \\det \\left( \\frac{\\Sigma_1 + \\Sigma_2}{2}\\right)^{1/2} }\n \\exp\\left\\{-\\frac{1}{8}(\\mu_1 - \\mu_2)^T \n \\left(\\frac{\\Sigma_1 + \\Sigma_2}{2}\\right)^{-1}\n (\\mu_1 - \\mu_2) \n \\right\\} "
},
{
"math_id": 27,
"text": " P \\sim \\mathrm{Exp}(\\alpha)"
},
{
"math_id": 28,
"text": " Q \\sim \\mathrm{Exp}(\\beta)"
},
{
"math_id": 29,
"text": "H^2(P, Q) = 1 - \\frac{2 \\sqrt{\\alpha \\beta}}{\\alpha + \\beta}."
},
{
"math_id": 30,
"text": " P \\sim \\mathrm{W}(k,\\alpha)"
},
{
"math_id": 31,
"text": " Q \\sim \\mathrm{W}(k,\\beta)"
},
{
"math_id": 32,
"text": " k "
},
{
"math_id": 33,
"text": " \\alpha\\, , \\beta "
},
{
"math_id": 34,
"text": " H^2(P, Q) = 1 - \\frac{2 (\\alpha \\beta)^{k/2}}{\\alpha^k + \\beta^k}. "
},
{
"math_id": 35,
"text": "\\alpha"
},
{
"math_id": 36,
"text": "\\beta"
},
{
"math_id": 37,
"text": " P \\sim \\mathrm{Poisson}(\\alpha)"
},
{
"math_id": 38,
"text": " Q \\sim \\mathrm{Poisson}(\\beta)"
},
{
"math_id": 39,
"text": " H^2(P,Q) = 1-e^{-\\frac{1}{2} (\\sqrt{\\alpha} - \\sqrt{\\beta})^2}. "
},
{
"math_id": 40,
"text": " P \\sim \\text{Beta}(a_1,b_1)"
},
{
"math_id": 41,
"text": " Q \\sim \\text{Beta}(a_2, b_2)"
},
{
"math_id": 42,
"text": "H^2(P,Q) = 1 - \\frac{B\\left(\\frac{a_1 + a_2}{2}, \\frac{b_1 + b_2}{2}\\right)}{\\sqrt{B(a_1, b_1) B(a_2, b_2)}}"
},
{
"math_id": 43,
"text": "B"
},
{
"math_id": 44,
"text": " P \\sim \\text{Gamma}(a_1,b_1)"
},
{
"math_id": 45,
"text": " Q \\sim \\text{Gamma}(a_2, b_2)"
},
{
"math_id": 46,
"text": "H^2(P,Q) = 1 - \\Gamma\\left({\\scriptstyle\\frac{a_1 + a_2}{2}}\\right)\\left(\\frac{b_1+b_2}{2}\\right)^{-(a_1+a_2)/2}\\sqrt{\\frac{b_1^{a_1}b_2^{a_2}}{\\Gamma(a_1)\\Gamma(a_2)}}"
},
{
"math_id": 47,
"text": "\\Gamma"
},
{
"math_id": 48,
"text": "H(P,Q)"
},
{
"math_id": 49,
"text": "\\delta(P,Q)"
},
{
"math_id": 50,
"text": "\nH^2(P,Q) \\leq \\delta(P,Q) \\leq \\sqrt{2}H(P,Q)\\,.\n"
},
{
"math_id": 51,
"text": "1/\\sqrt{2}"
}
]
| https://en.wikipedia.org/wiki?curid=13035709 |
13036200 | Automobile drag coefficient | Resistance of a car to moving through air
The drag coefficient is a common measure in automotive design as it pertains to aerodynamics. Drag is a force that acts parallel to and in the same direction as the airflow. The drag coefficient of an automobile measures the way the automobile passes through the surrounding air. When automobile companies design a new vehicle they take into consideration the automobile drag coefficient in addition to the other performance characteristics. Aerodynamic drag increases with the square of speed; therefore it becomes critically important at higher speeds. Reducing the drag coefficient in an automobile improves the performance of the vehicle as it pertains to speed and fuel efficiency. There are many different ways to reduce the drag of a vehicle. A common way to measure the drag of the vehicle is through the drag area.
The importance of drag reduction.
The reduction of drag in road vehicles has led to increases in the top speed of the vehicle and the vehicle's fuel efficiency, as well as many other performance characteristics, such as handling and acceleration. The two main factors that impact drag are the frontal area of the vehicle and the drag coefficient. The drag coefficient is a unit-less value that denotes how much an object resists movement through a fluid such as water or air. A potential complication of altering a vehicle's aerodynamics is that it may cause the vehicle to get too much lift. Lift is an aerodynamic force that acts perpendicular to the airflow around the body of the vehicle. Too much lift can cause the vehicle to lose road traction which can be very unsafe. Lowering the drag coefficient comes from streamlining the exterior body of the vehicle. Streamlining the body requires assumptions about the surrounding airspeed and characteristic use of the vehicle.
Cars that try to reduce drag employ devices such as spoilers, wings, diffusers, and fins to reduce drag and increase speed in one direction.
Drag area.
While designers pay attention to the overall shape of the automobile, they also bear in mind that reducing the frontal area of the shape helps reduce the drag. The product of drag coefficient and area – drag area – is represented as "C"A (or CxA), a multiplication of "C" value by area.
The term "drag area" derives from aerodynamics, where it is the product of some reference area (such as cross-sectional area, total surface area, or similar) and the drag coefficient. In 2003, "Car and Driver" magazine adopted this metric as a more intuitive way to compare the aerodynamic efficiency of various automobiles.
The force F required to overcome drag is calculated with the drag equation:
formula_0
Therefore:formula_1
Where the drag coefficient and reference area have been collapsed into the drag area term. This allows direct estimation of the drag force at a given speed for any vehicle for which only the drag area is known and therefore easier comparison.
As drag area "C"A is the fundamental value that determines power required for a given cruise speed it is a critical parameter for fuel consumption at a steady speed. This relation also allows an estimation of the new top speed of a car with a tuned engine:
formula_2
Or the power required for a target top speed:
formula_3
Average full-size passenger cars have a drag area of roughly . Reported drag areas range from the 1999 Honda Insight at to the 2003 Hummer H2 at . The drag area of a bicycle (and rider) is also in the range of .
Example drag coefficients.
The average modern automobile achieves a drag coefficient of between 0.25 and 0.3. Sport utility vehicles (SUVs), with their typically boxy shapes, typically achieve a "C"=0.35–0.45. The drag coefficient of a vehicle is affected by the shape of body of the vehicle. Various other characteristics affect the coefficient of drag as well, and are taken into account in these examples. Many sports cars have a surprisingly high drag coefficient, as downforce implies drag, while others are designed to be highly aerodynamic in pursuit of a speed and efficiency, and as a result have much lower drag coefficients.
Note that the "C" of a given vehicle will vary depending on which wind tunnel it is measured in. Variations of up to 5% have been documented and variations in test technique and analysis can also make a difference. So if the same vehicle with a drag coefficient of "C"=0.30 was measured in a different tunnel it could be anywhere from "C"=0.285 to "C"=0.315.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "F = \\tfrac{1}{2} \\times \\text{air density} \\times \\text{drag coefficient} \\times \\text{reference area} \\times \\text{speed}^2 "
},
{
"math_id": 1,
"text": "F = \\tfrac{1}{2} \\times \\text{air density} \\times \\mathbf{\\text{drag area} } \\times \\text{speed}^2 "
},
{
"math_id": 2,
"text": "\\text{estimated top speed} = \\text{original top speed} \\times \\sqrt[3]{\\frac{\\text{new power}}\\text{original power}} "
},
{
"math_id": 3,
"text": "\\text{power required} = \\text{original power} \\times \\left( \\frac{\\text{target speed}}{\\text{original speed}} \\right)^3 "
}
]
| https://en.wikipedia.org/wiki?curid=13036200 |
13037119 | Scattering-matrix method | In computational electromagnetics, the scattering-matrix method (SMM) is a numerical method used to solve Maxwell's equations, related to the transfer-matrix method.
Principles.
SMM can, for example, use cylinders to model dielectric/metal objects in the domain.
The total-field/scattered-field (TF/SF) formalism where the total field is written as sum of incident and scattered at each point in the domain:
formula_0
By assuming series solutions for the total field, the SMM method transforms the domain into a cylindrical problem. In this domain total field is written in terms of Bessel and Hankel function solutions to the cylindrical Helmholtz equation. SMM method formulation, finally helps compute these coefficients of the cylindrical harmonic functions within the cylinder and outside it, at the same time satisfying EM boundary conditions.
Finally, SMM accuracy can be increased by adding (removing) cylindrical harmonic terms used to model the scattered fields.
SMM, eventually leads to a matrix formalism, and the coefficients are calculated through matrix inversion. For "N"-cylinders, each scattered field modeled using 2"M"+1 harmonic terms, SMM requires to solve a "N"(2"M" + 1) system of equations.
Advantages.
SMM, is a rigorous and accurate method deriving from first principles. Hence, it is guaranteed to be accurate within limits of model, and not show spurious effects of numerical dispersion arising in other techniques like Finite-difference time-domain (FDTD) method.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "E_{tot} = E_{inc} + E_{scatt} \\ "
}
]
| https://en.wikipedia.org/wiki?curid=13037119 |
13037166 | Dimethyldichlorosilane | <templatestyles src="Chembox/styles.css"/>
Chemical compound
Dimethyldichlorosilane is a tetrahedral organosilicon compound with the formula . At room temperature it is a colorless liquid that readily reacts with water to form both linear and cyclic Si-O chains. Dimethyldichlorosilane is made on an industrial scale as the principal precursor to dimethylsilicone and polysilane compounds.
History.
The first organosilicon compounds were reported in 1863 by Charles Friedel and James Crafts who synthesized tetraethylsilane from diethylzinc and silicon tetrachloride. However, major progress in organosilicon chemistry did not occur until Frederick Kipping and his students began experimenting with diorganodichlorosilanes () that were prepared by reacting silicon tetrachloride with Grignard reagents. Unfortunately, this method suffered from many experimental problems.
In the 1930s, the demand for silicones increased due to the need for better insulators for electric motors and sealing materials for aircraft engines, and with it the need for a more efficient synthesis of dimethyldichlorosilane. To solve the problem, General Electric, Corning Glass Works, and Dow Chemical Company began a partnership that ultimately became the Dow Corning Company. During 1941–1942, Eugene G. Rochow, a chemist from General Electric, and Richard Müller, working independently in Germany, found an alternate synthesis of dimethyldichlorosilane that allowed it to be produced on an industrial scale. This Direct Synthesis, or Direct process, which is used in today’s industry, involves the reaction of elemental silicon with methyl chloride in the presence of a copper catalyst.
Preparation.
Rochow's synthesis involved passing methyl chloride through a heated tube packed with ground silicon and copper(I) chloride. The current industrial method places finely ground silicon in a fluidized bed reactor at about 300 °C. The catalyst is applied as . Methyl chloride is then passed through the reactor to produce mainly dimethyldichlorosilane.
formula_0
The mechanism of the direct synthesis is not known. However, the copper catalyst is essential for the reaction to proceed.
In addition to dimethyldichlorosilane, products of this reaction include , , and , which are separated from each other by fractional distillation. The yields and boiling points of these products are shown in the following chart.
Main reactions.
Dimethyldichlorosilane hydrolyzes to form linear and cyclic silicones, compounds containing Si-O backbones. The length of the resulting polymer is dependent on the concentration of chain ending groups that are added to the reaction mixture. The rate of the reaction is determined by the transfer of reagents across the aqueous-organic phase boundary; therefore, the reaction is most efficient under turbulent conditions. The reaction medium can be varied further to maximize the yield of a specific product.
formula_1
Dimethyldichlorosilane reacts with methanol to produce dimethoxydimethylsilanes.
formula_2
Although the hydrolysis of dimethoxydimethylsilanes is slower, it is advantageous when the hydrochloric acid byproduct is unwanted:
formula_3
Because dimethyldichlorosilane is easily hydrolyzed, it cannot be handled in air. One method used to overcome this problem is to convert it to a less reactive bis(dimethylamino)silane.
formula_4
Another benefit to changing dimethyldichlorosilane to its bis(dimethylamino)silane counterpart is that it forms an exactly alternating polymer when combined with a disilanol comonomer.
formula_5
Sodium–potassium alloy can be used to polymerize dimethyldichlorosilane, producing polysilane chains with a Si-Si backbone. For example, dodecamethylcyclohexasilane can be prepared in this way:
formula_6
The reaction also produces polydimethylsilane and decamethylpentasilane. Diverse types of dichlorosilane precursors, such as , can be added to adjust the properties of the polymer.
In organic synthesis it (together with its close relative diphenyldichlorosilane) is used as a protecting group for "gem"-diols.
Applications.
The main purpose of dimethyldichlorosilane is for use in the synthesis of silicones, an industry that was valued at more than $10 billion per year in 2005. It is also employed in the production of polysilanes, which in turn are precursors to silicon carbide. In practical uses, dichlorodimethylsilane can be used as a coating on glass to avoid the adsorption of micro-particles.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\ce{2 CH3Cl + Si -> (CH3)2SiCl2}"
},
{
"math_id": 1,
"text": "\\begin{alignat}{4}\n n\\ \\ce{(CH3)2SiCl2}\\ +&& n\\ \\ce{H2O} \\ \\longrightarrow&& \\ce{[Si(CH3)2O]}_n \\quad\\, +&& 2n\\ \\ce{HCl} \\\\\n m\\ \\ce{(CH3)2SiCl2}\\ +&& \\ (m\\!+\\!1)\\ \\ce{H2O} \\ \\longrightarrow&& \\ \\ \\ce{HO[Si(CH3)2O]}_m\\ce{H} \\ +&&\\ 2m\\ \\ce{HCl}\n\\end{alignat}"
},
{
"math_id": 2,
"text": "\\ce{(CH3)2SiCl2 + 2CH3OH -> (CH3)2Si(OCH3)2 + 2 HCl}"
},
{
"math_id": 3,
"text": "n\\ \\ce{(CH3)2Si(OCH3)2} + n\\ \\ce{H2O -> [(CH3)2SiO]}_n + 2n\\ \\ce{CH3OH}"
},
{
"math_id": 4,
"text": "\\ce{(CH3)2SiCl2 + 4 HN(CH3)2 -> (CH3)2Si[N(CH3)2]2 + 2H2N(CH3)2Cl}"
},
{
"math_id": 5,
"text": "n\\ \\ce{(CH3)2Si[N(CH3)2]2} + n\\ \\ce{HO(CH2)2SiRSi(CH2)2OH -> [(CH3)2SiO(CH2)2SiRSi(CH2)2O]}_n + 2n\\ \\ce{HN(CH3)2}"
},
{
"math_id": 6,
"text": "\\ce{6 (CH3)2SiCl2 + 12 M -> [(CH3)2Si]6 + 12 MCl} \\ \\ce{(M = Na, K)}"
}
]
| https://en.wikipedia.org/wiki?curid=13037166 |
13038078 | Phase congruency | Phase congruency is a measure of feature significance in computer images, a method of edge detection that is particularly robust against changes in illumination and contrast.
Foundations.
Phase congruency reflects the behaviour of the image in the frequency domain. It has been noted that edgelike features have many of their frequency components in the same phase. The concept is similar to coherence, except that it applies to functions of different wavelength.
For example, the Fourier decomposition of a square wave consists of sine functions, whose frequencies are odd multiples of the fundamental frequency. At the rising edges of the square wave, each sinusoidal component has a rising phase; the phases have maximal congruency at the edges. This corresponds to the human-perceived edges in an image where there are sharp changes between light and dark.
Definition.
Phase congruency compares the weighted alignment of the Fourier components of a signal formula_0 with the sum of the Fourier components.
formula_1
where formula_2 is the local or instantaneous phase as can be calculated using the Hilbert transform and formula_0 are the local amplitude, or energy, of the signal. When all the phases are aligned, this is equal to 1.
Several ways of implementing phase congruency have been developed, of which two versions are available in open source, one written for Matlab and the other written in Java as a plugin for the ImageJ software.
Given the different notations used for its formulation, a unified version has been recently presented, where a methodology for the parameter tuning is also presented.
Advantages.
The square-wave example is naive in that most edge detection methods deal with it equally well. For example, the first derivative has a maximal magnitude at the edges. However, there are cases where the perceived edge does not have a sharp step or a large derivative. The method of phase congruency applies to many cases where other methods fail.
A notable example is an image feature consisting of a single line, such as the letter "l". Many edge-detection algorithms will pick up two adjacent edges: the transitions from white to black, and black to white. On the other hand, the phase congruency map has a single line. A simple Fourier analogy of this case is a triangle wave. In each of its crests there is a congruency of crests from different sinusoidal functions.
Disadvantages.
Calculating the phase congruency map of an image is very computationally intensive, and sensitive to image noise. Techniques of noise reduction are usually applied prior to the calculation.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "A_{\\rm n}"
},
{
"math_id": 1,
"text": "PC(t) = \\max_{\\bar{\\phi}} \\frac{\\sum_{\\rm n} A_{\\rm n} \\cos(\\phi_{\\rm n}(t)-\\bar\\phi)}{\\sum_{\\rm n}A_n}"
},
{
"math_id": 2,
"text": "\\phi_{\\rm n}"
}
]
| https://en.wikipedia.org/wiki?curid=13038078 |
1304312 | Brownian noise | Type of noise produced by Brownian motion
In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from the color, but after Robert Brown, who documented the erratic motion for multiple types of inanimate particles in water. The term "red noise" comes from the "white noise"/"white light" analogy; red noise is strong in longer wavelengths, similar to the red end of the visible spectrum.
Explanation.
The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density is inversely proportional to "f" 2, meaning it has higher intensity at lower frequencies, even more so than pink noise. It decreases in intensity by 6 dB per octave (20 dB per decade) and, when heard, has a "damped" or "soft" quality compared to white and pink noise. The sound is a low roar resembling a waterfall or heavy rainfall. See also violet noise, which is a 6 dB "increase" per octave.
Strictly, Brownian motion has a Gaussian probability distribution, but "red noise" could apply to any signal with the 1/"f" 2 frequency spectrum.
Power spectrum.
A Brownian motion, also known as a Wiener process, is obtained as the integral of a white noise signal:
formula_0
meaning that Brownian motion is the integral of the white noise formula_1, whose power spectral density is flat:
formula_2
Note that here formula_3 denotes the Fourier transform, and formula_4 is a constant. An important property of this transform is that the derivative of any distribution transforms as
formula_5
from which we can conclude that the power spectrum of Brownian noise is
formula_6
An individual Brownian motion trajectory presents a spectrum formula_7, where the amplitude formula_4 is a random variable, even in the limit of an infinitely long trajectory.
Production.
Brown noise can be produced by integrating white noise. That is, whereas (digital) white noise can be produced by randomly choosing each sample independently, Brown noise can be produced by adding a random offset to each sample to obtain the next one. As Brownian noise contains infinite spectral power at low frequencies, the signal tends to drift away infinitely from the origin. A leaky integrator might be used in audio or electromagnetic applications to ensure the signal does not “wander off”, that is, exceed the limits of the system's dynamic range. This turns the Brownian noise into Ornstein–Uhlenbeck noise, which has a flat spectrum at lower frequencies, and only becomes “red” above the chosen cutoff frequency.
Brownian noise can also be computer-generated by first generating a white noise signal, Fourier-transforming it, then dividing the amplitudes of the different frequency components by the frequency (in one dimension), or by the frequency squared (in two dimensions) etc. Matlab programs are available to generate Brownian and other power-law coloured noise in one or any number of dimensions.
Experimental Evidence.
Evidence of Brownian noise, or more accurately, of Brownian processes has been found in different fields including chemistry , electromagnetism
, fluid-dynamics , economics , and human neuromotor control .
Human Neuromotor Control.
In human neuromotor control, Brownian processes were recognized as a biomarker of human natural drift in both postural tasks - such as quietly standing or holding an object in your hand - as well as dynamic tasks. The work by Tessari et al. highlighted how these Brownian processes in humans might provide the first behavioral support to the neuroscientific hypothesis that humans encode motion in terms of descending neural velocity commands .
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " W(t) = \\int_0^t \\frac{dW}{d\\tau}(\\tau) d\\tau "
},
{
"math_id": 1,
"text": "t\\mapsto dW(t)"
},
{
"math_id": 2,
"text": "\n S_0 = \\left|\\mathcal{F}\\left[t\\mapsto\\frac{dW}{dt}(t)\\right](\\omega)\\right|^2 = \\text{const}.\n"
},
{
"math_id": 3,
"text": "\\mathcal{F}"
},
{
"math_id": 4,
"text": "S_0"
},
{
"math_id": 5,
"text": "\n \\mathcal{F}\\left[t\\mapsto\\frac{dW}{dt}(t)\\right](\\omega) = i \\omega \\mathcal{F}[t\\mapsto W(t)](\\omega),\n"
},
{
"math_id": 6,
"text": "\n S(\\omega) = \\big|\\mathcal{F}[t\\mapsto W(t)](\\omega)\\big|^2 = \\frac{S_0}{\\omega^2}.\n"
},
{
"math_id": 7,
"text": "S(\\omega) = S_0 / \\omega^2"
}
]
| https://en.wikipedia.org/wiki?curid=1304312 |
13046 | Geometric mean | N-th root of the product of n numbers
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the "n"th root of the product of n numbers, i.e., for a set of numbers "a"1, "a"2, ..., "an", the geometric mean is defined as
formula_0
or, equivalently, as the arithmetic mean in logscale:
formula_1
The geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, formula_2. The geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is, formula_3.
The geometric mean is used on a ratio scale, such as growth rates of the human population or interest rates of a financial investment over time. It also applies to benchmarking, where it is particularly useful for computing means of speedup ratios: since the mean of 0.5x (half as fast) and 2x (twice as fast) will be 1 (i.e., no speedup overall).
Suppose for example a person invests 1000 dollars in shares and achieves annual returns of +10%, -12%, +90%, -30% and +25% over 5 consecutive years to give a final investment value of 1,609 dollars. The arithmetic mean of the annual percent changes is 16.6%. However, this value is unrepresentative. If the initial investment grew by 16.6% per annum, it would be worth 2155 dollars after 5 years. In fact, to find the average percentage growth it is necessary compute the geometric mean of the successive annual growth ratios (1.1, 0.88, 1.9, 0.7, 1.25). This gives a value of 1.0998 which corresponds to an annual average growth of 9.98%. It can be readily verified that an investment of 1000 dollars which grows by 9.98% over five years would achieve a final investment value of 1,609 dollars. In this case, the geometric mean is appropriate because investment growth is multiplicative rather than additive.
The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, formula_4 and formula_5, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths formula_4 and formula_5. Similarly, the geometric mean of three numbers, formula_4, formula_5, and formula_6, is the length of one edge of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers.
The geometric mean is one of the three classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)
Formulation.
The geometric mean of a data set formula_7 is given by:
formula_8
The above figure uses capital pi notation to show a series of multiplications. Each side of the equal sign shows that a set of values is multiplied in succession (the number of values is represented by "n") to give a total product of the set, and then the "n"th root of the total product is taken to give the geometric mean of the original set. For example, in a set of four numbers formula_9, the product of formula_10 is formula_11, and the geometric mean is the fourth root of 24, or ~ 2.213. The exponent formula_12 on the left side is equivalent to the taking "n"th root. For example, formula_13.
Formulation using logarithms.
The geometric mean can also be expressed as the exponential of the arithmetic mean of logarithms. By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication:
When formula_14
formula_15
As:
formula_16
alternatively, use any positive real number base, for both the logarithms and the number you are raising to the power of the arithmetic mean of the individual logarithms at that same base.
This is sometimes called the log-average (not to be confused with the logarithmic average). It is simply computing the arithmetic mean of the logarithm-transformed values of formula_17 (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with formula_18. For example, the geometric mean of 2 and 8 can be calculated as the following, where formula_5 is any base of a logarithm (commonly 2, formula_19 or 10):
formula_20
Related to the above, it can be seen that for a given sample of points formula_21, the geometric mean is the minimizer of
formula_22,
whereas the arithmetic mean is the minimizer of
formula_23.
Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).
The log form of the geometric mean is generally the preferred alternative for implementation in computer languages because calculating the product of many numbers can lead to an arithmetic overflow or arithmetic underflow. This is less likely to occur with the sum of the logarithms for each number.
Related concepts.
Iterative means.
The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the arithmetic-geometric mean, an intersection of the two which always lies in between.
The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (formula_24) and (formula_25) are defined:
formula_26
and
formula_27
where formula_28 is the harmonic mean of the previous values of the two sequences, then formula_24 and formula_25 will converge to the geometric mean of formula_29 and formula_30. The sequences converge to a common limit, and the geometric mean is preserved:
formula_31
Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.
Comparison to arithmetic mean.
The geometric mean of a non-empty data set of positive numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.
Geometric mean of a continuous function.
If formula_32 is a positive continuous real-valued function, its geometric mean over this interval is
formula_33
For instance, taking the identity function formula_34 over the unit interval shows that the geometric mean of the positive numbers between 0 and 1 is equal to formula_35.
Applications.
Average growth rate.
In many cases the geometric mean is the best measure to determine the average growth rate of some quantity. For instance, if sales increases by 80% in one year and the next year by 25%, the end result is the same as that of a constant growth rate of 50%, since the geometric mean of 1.80 and 1.25 is 1.50. In order to determine the average growth rate, it is not necessary to take the product of the measured growth rates at every step. Let the quantity be given as the sequence formula_36, where formula_37 is the number of steps from the initial to final state. The growth rate between successive measurements formula_38 and formula_39 is formula_40. The geometric mean of these growth rates is then just:
formula_41
Normalized values.
The fundamental property of the geometric mean, which does not hold for any other mean, is that for two sequences formula_42 and formula_43 of equal length,
formula_44.
This makes the geometric mean the only correct mean when averaging "normalized" results; that is, results that are presented as ratios to reference values. This is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources (for example, life expectancy, education years, and infant mortality). In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of execution time of computer programs:
Table 1
The arithmetic and geometric means "agree" that computer C is the fastest. However, by presenting appropriately normalized values "and" using the arithmetic mean, we can show either of the other two computers to be the fastest. Normalizing by A's result gives A as the fastest computer according to the arithmetic mean:
Table 2
while normalizing by B's result gives B as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:
Table 3
and normalizing by C's result gives C as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:
Table 4
In all cases, the ranking given by the geometric mean stays the same as the one obtained with unnormalized values.
However, this reasoning has been questioned.
Giving consistent results is not always equal to giving the correct results. In general, it is more rigorous to assign weights to each of the programs, calculate the average weighted execution time (using the arithmetic mean), and then normalize that result to one of the computers. The three tables above just give a different weight to each of the programs, explaining the inconsistent results of the arithmetic and harmonic means (Table 4 gives equal weight to both programs, the Table 2 gives a weight of 1/1000 to the second program, and the Table 3 gives a weight of 1/100 to the second program and 1/10 to the first one). The use of the geometric mean for aggregating performance numbers should be avoided if possible, because multiplying execution times has no physical meaning, in contrast to adding times as in the arithmetic mean. Metrics that are inversely proportional to time (speedup, IPC) should be averaged using the harmonic mean.
The geometric mean can be derived from the generalized mean as its limit as formula_45 goes to zero. Similarly, this is possible for the weighted geometric mean.
Proportional growth.
The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.
Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.6666% and 42.8571% for each year respectively. Using the arithmetic mean calculates a (linear) average growth of 46.5079% (80% + 16.6666% + 42.8571%, that sum then divided by 3). However, if we start with 100 oranges and let it grow 46.5079% each year, the result is 314 oranges, not 300, so the linear average "over"-states the year-on-year growth.
Instead, we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.166666 and 1.428571, i.e. formula_46; thus the "average" growth per year is 44.2249%. If we start with 100 oranges and let the number grow with 44.2249% each year, the result is 300 oranges.
Financial.
The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean. It is also used in the CPI calculation and recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union.
This has the effect of understating movements in the index compared to using the arithmetic mean.
Applications in the social sciences.
Although the geometric mean has been relatively rare in computing social statistics, starting from 2010 the United Nations Human Development Index did switch to this mode of calculation, on the grounds that it better reflected the non-substitutable nature of the statistics being compiled and compared:
The geometric mean decreases the level of substitutability between dimensions [being compared] and at the same time ensures that a 1 percent decline in say life expectancy at birth has the same impact on the HDI as a 1 percent decline in education or income. Thus, as a basis for comparisons of achievements, this method is also more respectful of the intrinsic differences across the dimensions than a simple average.
Not all values used to compute the HDI (Human Development Index) are normalized; some of them instead have the form formula_47. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.
The equally distributed welfare equivalent income associated with an Atkinson Index with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes. For values other than one, the equivalent value is an Lp norm divided by the number of elements, with p equal to one minus the inequality aversion parameter.
Geometry.
In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. This property is known as the geometric mean theorem.
In an ellipse, the semi-minor axis is the geometric mean of the maximum and minimum distances of the ellipse from a focus; it is also the geometric mean of the semi-major axis and the semi-latus rectum. The semi-major axis of an ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.
Another way to think about it is as follows:
Consider a circle with radius formula_48. Now take two diametrically opposite points on the circle and apply pressure from both ends to deform it into an ellipse with semi-major and semi-minor axes of lengths formula_4 and formula_5.
Since the area of the circle and the ellipse stays the same, we have:
formula_49
The radius of the circle is the geometric mean of the semi-major and the semi-minor axes of the ellipse formed by deforming the circle.
Distance to the horizon of a sphere (ignoring the effect of atmospheric refraction when atmosphere is present) is equal to the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.
The geometric mean is used in both in the approximation of squaring the circle by S.A. Ramanujan and in the construction of the heptadecagon with "mean proportionals".
Aspect ratios.
The geometric mean has been used in choosing a compromise aspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has the aspect ratio of their geometric mean.
In aspect ratio by the SMPTE, balancing 2.35 and 4:3, the geometric mean is formula_50, and thus formula_51... was chosen. This was discovered empirically by Kerns Powers, who cut out rectangles with equal areas and shaped them to match each of the popular aspect ratios. When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1. The value found by Powers is exactly the geometric mean of the extreme aspect ratios, (1.33:1) and CinemaScope(2.35:1), which is coincidentally close to formula_52 (formula_53). The intermediate ratios have no effect on the result, only the two extreme ratios.
Applying the same geometric mean technique to 16:9 and 4:3 approximately yields the (formula_54...) aspect ratio, which is likewise used as a compromise between these ratios. In this case 14:9 is exactly the "arithmetic mean" of formula_52 and formula_55, since 14 is the average of 16 and 12, while the precise "geometric mean" is formula_56 but the two different "means", arithmetic and geometric, are approximately equal because both numbers are sufficiently close to each other (a difference of less than 2%).
Paper formats.
The geometric mean is also used to calculate B and C series paper formats. The formula_57 format has an area which is the geometric mean of the areas of formula_58 and formula_59. For example, the area of a B1 paper is formula_60, because it is the geometric mean of the areas of an A0 (formula_61) and an A1 (formula_62) paper (formula_63).
The same principle applies with the C series, whose area is the geometric mean of the A and B series. For example, the C4 format has an area which is the geometric mean of the areas of A4 and B4.
An advantage that comes from this relationship is that an A4 paper fits inside a C4 envelope, and both fit inside a B4 envelope.
See also.
<templatestyles src="Div col/styles.css"/>
Notes.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\left(\\prod_{i=1}^n a_i\\right)^\\frac{1}{n} = \\sqrt[n]{a_1 a_2 \\cdots a_n}"
},
{
"math_id": 1,
"text": "\\exp{\\left( {\\frac{1}{n}\\sum\\limits_{i=1}^{n}\\ln a_i} \\right)}"
},
{
"math_id": 2,
"text": "\\sqrt{2 \\cdot 8} = 4"
},
{
"math_id": 3,
"text": "\\sqrt[3]{4 \\cdot 1 \\cdot 1/32} = 1/2"
},
{
"math_id": 4,
"text": "a"
},
{
"math_id": 5,
"text": "b"
},
{
"math_id": 6,
"text": "c"
},
{
"math_id": 7,
"text": "\\left\\{a_1, a_2,\\, \\ldots,\\, a_n\\right\\}"
},
{
"math_id": 8,
"text": "\\left(\\prod_{i=1}^n a_i \\right)^\\frac{1}{n} = \\sqrt[n]{a_1 a_2 \\cdots a_n}."
},
{
"math_id": 9,
"text": "\\{1, 2, 3, 4\\}"
},
{
"math_id": 10,
"text": "1 \\times 2 \\times 3 \\times 4"
},
{
"math_id": 11,
"text": "24"
},
{
"math_id": 12,
"text": "\\frac{1}{n}"
},
{
"math_id": 13,
"text": "24^\\frac{1}{4} = \\sqrt[4]{24}"
},
{
"math_id": 14,
"text": "a_1, a_2, \\dots, a_n > 0"
},
{
"math_id": 15,
"text": "\\left( \\prod_{i=1}^n a_i \\right)^\\frac{1}{n} = \\exp\\left[\\frac{1}{n} \\sum_{i=1}^n \\ln a_i\\right];"
},
{
"math_id": 16,
"text": "\\begin{align}\n\\left ( \\prod_{i=1}^{n}a_i \\right )^{\\frac{1}{n}}&= \\sqrt[n]{a_1a_2\\cdots a_n}\\\\\n&=e ^{\\ln( a_1a_2\\cdots a_n)^{1/n}}\\\\\n&=e ^{\\frac{1}{n}\\left (\\ln a_1+\\ln a_2+\\cdots +\\ln a_n \\right )}\\\\\n&=e ^{{\\frac{1}{n}\\sum_{i=1}^{n}\\ln a_i}}\\\\\n \\text{geometric mean(}a\\text{)}&=e^{\\text{arithmetic mean(ln(}a\\text{))}}\n \\end{align}"
},
{
"math_id": 17,
"text": "a_i"
},
{
"math_id": 18,
"text": "f(x) = \\log x"
},
{
"math_id": 19,
"text": "e"
},
{
"math_id": 20,
"text": "b^{\\frac{1}{2}\\left[\\log_b (2) + \\log_b (8)\\right]} = 4"
},
{
"math_id": 21,
"text": "a_1, \\ldots, a_n"
},
{
"math_id": 22,
"text": "f(a) = \\sum_{i=1}^n (\\log(a_i) - \\log(a))^2 = \\sum_{i=1}^n (\\log(a_i/a) )^2"
},
{
"math_id": 23,
"text": "f(a) = \\sum_{i=1}^n (a_i - a)^2"
},
{
"math_id": 24,
"text": "a_n"
},
{
"math_id": 25,
"text": "h_n"
},
{
"math_id": 26,
"text": "a_{n+1} = \\frac{a_n + h_n}{2}, \\quad a_0 = x"
},
{
"math_id": 27,
"text": "h_{n+1} = \\frac{2}{\\frac{1}{a_n} + \\frac{1}{h_n}}, \\quad h_0 = y"
},
{
"math_id": 28,
"text": "h_{n+1}"
},
{
"math_id": 29,
"text": "x"
},
{
"math_id": 30,
"text": "y"
},
{
"math_id": 31,
"text": "\\sqrt{a_i h_i} =\n \\sqrt{\\frac{a_i + h_i}{\\frac{a_i + h_i}{h_i a_i}}} =\n \\sqrt{\\frac{a_i + h_i}{\\frac{1}{a_i} + \\frac{1}{h_i}}} =\n \\sqrt{a_{i+1} h_{i+1}}\n"
},
{
"math_id": 32,
"text": "f:[a,b]\\to(0, \\infty)"
},
{
"math_id": 33,
"text": "\\text{GM}[f] = \\exp\\left(\\frac{1}{b-a}\\int_a^b\\ln f(x)dx\\right)"
},
{
"math_id": 34,
"text": "f(x) = x"
},
{
"math_id": 35,
"text": "\\frac{1}{e}"
},
{
"math_id": 36,
"text": "a_0, a_1,..., a_n"
},
{
"math_id": 37,
"text": "n"
},
{
"math_id": 38,
"text": "a_k"
},
{
"math_id": 39,
"text": "a_{k+1}"
},
{
"math_id": 40,
"text": "a_{k+1}/a_k"
},
{
"math_id": 41,
"text": "\\left( \\frac{a_1}{a_0} \\frac{a_2}{a_1} \\cdots \\frac{a_n}{a_{n-1}} \\right)^\\frac{1}{n} = \\left(\\frac{a_n}{a_0}\\right)^\\frac{1}{n}."
},
{
"math_id": 42,
"text": "X"
},
{
"math_id": 43,
"text": "Y"
},
{
"math_id": 44,
"text": "\\operatorname{GM}\\left(\\frac{X_i}{Y_i}\\right) = \\frac{\\operatorname{GM}(X_i)}{\\operatorname{GM}(Y_i)}"
},
{
"math_id": 45,
"text": "p"
},
{
"math_id": 46,
"text": "\\sqrt[3]{1.80 \\times 1.166666 \\times 1.428571} \\approx 1.442249"
},
{
"math_id": 47,
"text": "\\left(X - X_\\text{min}\\right) / \\left(X_\\text{norm} - X_\\text{min}\\right)"
},
{
"math_id": 48,
"text": "r"
},
{
"math_id": 49,
"text": "\n\\begin{align}\n\\pi r^2 &= \\pi a b \\\\\n\n r^2 &= a b \\\\\n \nr &= \\sqrt{a b}\n\\end{align}\n"
},
{
"math_id": 50,
"text": "\\sqrt{2.35 \\times \\frac{4}{3}} \\approx 1.7701"
},
{
"math_id": 51,
"text": "16:9 = 1.77\\overline{7}"
},
{
"math_id": 52,
"text": "16:9"
},
{
"math_id": 53,
"text": "1.77\\overline{7}:1"
},
{
"math_id": 54,
"text": "1.55\\overline{5}"
},
{
"math_id": 55,
"text": "4:3 = 12:9"
},
{
"math_id": 56,
"text": "\\sqrt{\\frac{16}{9}\\times\\frac{4}{3}} \\approx 1.5396 \\approx 13.8:9,"
},
{
"math_id": 57,
"text": "B_n"
},
{
"math_id": 58,
"text": "A_n"
},
{
"math_id": 59,
"text": "A_{n-1}"
},
{
"math_id": 60,
"text": "\\frac{\\sqrt{2}}{2}\\mathrm m^2"
},
{
"math_id": 61,
"text": "1\\mathrm m^2"
},
{
"math_id": 62,
"text": "\\frac{1}{2}\\mathrm m^2"
},
{
"math_id": 63,
"text": "\\sqrt{1\\mathrm m^2 \\cdot \\frac{1}{2}\\mathrm m^2}=\\sqrt{\\frac{1}{2}\\mathrm m^4}=\\frac{1}{\\sqrt 2}\\mathrm m^2= \\frac{\\sqrt 2}{2}\\mathrm m^2"
},
{
"math_id": 64,
"text": "n_1 = \\sqrt{n_0 n_2}"
}
]
| https://en.wikipedia.org/wiki?curid=13046 |
13047079 | Slope deflection method | The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney. The slope deflection method was widely used for more than a decade until the moment distribution method was developed. In the book, "The Theory and Practice of Modern Framed Structures", written by J.B Johnson, C.W. Bryan and F.E. Turneaure, it is stated that this method was first developed "by Professor Otto Mohr in Germany, and later developed independently by Professor G.A. Maney". According to this book, professor Otto Mohr introduced this method for the first time in his book, "Evaluation of Trusses with Rigid Node Connections" or "Die Berechnung der Fachwerke mit Starren Knotenverbindungen".
Introduction.
By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined. Deformation of member is due to the bending moment.
Slope deflection equations.
The slope deflection equations can also be written using the stiffness factor formula_0 and the chord rotation formula_1:
Derivation of slope deflection equations.
When a simple beam of length formula_2 and flexural rigidity formula_3 is loaded at each end with clockwise moments formula_4 and formula_5, member end rotations occur in the same direction. These rotation angles can be calculated using the unit force method or Darcy's Law.
formula_6
formula_7
Rearranging these equations, the slope deflection equations are derived.
Equilibrium conditions.
Joint equilibrium.
Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,
formula_8
Here, formula_9 are the member end moments, formula_10 are the fixed end moments, and formula_11 are the external moments directly applied at the joint.
Shear equilibrium.
When there are chord rotations in a frame, additional equilibrium conditions, namely the shear equilibrium conditions need to be taken into account.
Example.
The statically indeterminate beam shown in the figure is to be analysed.
In the following calculations, clockwise moments and rotations are positive.
Degrees of freedom.
Rotation angles formula_16, formula_17, formula_18, of joints A, B, C, respectively are taken as the unknowns. There are no chord rotations due to other causes including support settlement.
Fixed end moments.
Fixed end moments are:
formula_19
formula_20
formula_21
formula_22
formula_23
formula_24
Slope deflection equations.
The slope deflection equations are constructed as follows:
formula_25
formula_26
formula_27
formula_28
formula_29
formula_30
Joint equilibrium equations.
Joints A, B, C should suffice the equilibrium condition. Therefore
formula_31
formula_32
formula_33
Rotation angles.
The rotation angles are calculated from simultaneous equations above.
formula_34
formula_35
formula_36
Member end moments.
Substitution of these values back into the slope deflection equations yields the member end moments (in kNm):
formula_37
formula_38
formula_39
formula_40
formula_41
formula_42
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "K=\\frac{I_{ab}}{L_{ab}}"
},
{
"math_id": 1,
"text": "\\psi =\\frac{ \\Delta}{L_{ab}}"
},
{
"math_id": 2,
"text": "L_{ab}"
},
{
"math_id": 3,
"text": "E_{ab} I_{ab}"
},
{
"math_id": 4,
"text": "M_{ab}"
},
{
"math_id": 5,
"text": "M_{ba}"
},
{
"math_id": 6,
"text": "\\theta_a - \\frac{\\Delta}{L_{ab}}= \\frac{L_{ab}}{3E_{ab} I_{ab}} M_{ab} - \\frac{L_{ab}}{6E_{ab} I_{ab}} M_{ba}"
},
{
"math_id": 7,
"text": "\\theta_b - \\frac{\\Delta}{L_{ab}}= - \\frac{L_{ab}}{6E_{ab} I_{ab}} M_{ab} + \\frac{L_{ab}}{3E_{ab} I_{ab}} M_{ba}"
},
{
"math_id": 8,
"text": "\\Sigma \\left( M^{f} + M_{member} \\right) = \\Sigma M_{joint}"
},
{
"math_id": 9,
"text": "M_{member}"
},
{
"math_id": 10,
"text": "M^{f}"
},
{
"math_id": 11,
"text": "M_{joint}"
},
{
"math_id": 12,
"text": " L = 10 \\ m "
},
{
"math_id": 13,
"text": " P = 10 \\ kN "
},
{
"math_id": 14,
"text": " a = 3 \\ m "
},
{
"math_id": 15,
"text": " q = 1 \\ kN/m"
},
{
"math_id": 16,
"text": "\\theta_A"
},
{
"math_id": 17,
"text": "\\theta_B"
},
{
"math_id": 18,
"text": "\\theta_C"
},
{
"math_id": 19,
"text": "M _{AB} ^f = - \\frac{P a b^2 }{L ^2} = - \\frac{10 \\times 3 \\times 7^2}{10^2} = -14.7 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 20,
"text": "M _{BA} ^f = \\frac{P a^2 b}{L^2} = \\frac{10 \\times 3^2 \\times 7}{10^2} = 6.3 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 21,
"text": "M _{BC} ^f = - \\frac{qL^2}{12} = - \\frac{1 \\times 10^2}{12} = - 8.333 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 22,
"text": "M _{CB} ^f = \\frac{qL^2}{12} = \\frac{1 \\times 10^2}{12} = 8.333 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 23,
"text": "M _{CD} ^f = - \\frac{PL}{8} = - \\frac{10 \\times 10}{8} = -12.5 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 24,
"text": "M _{DC} ^f = \\frac{PL}{8} = \\frac{10 \\times 10}{8} = 12.5 \\mathrm{\\,kN \\,m}"
},
{
"math_id": 25,
"text": "M_{AB} = \\frac{EI}{L} \\left( 4 \\theta_A + 2 \\theta_B \\right) = \\frac{4EI \\theta_A + 2EI \\theta_B}{L}"
},
{
"math_id": 26,
"text": "M_{BA} = \\frac{EI}{L} \\left( 2 \\theta_A + 4 \\theta_B \\right) = \\frac{2EI \\theta_A + 4EI \\theta_B}{L}"
},
{
"math_id": 27,
"text": "M_{BC} = \\frac{2EI}{L} \\left( 4 \\theta_B + 2 \\theta_C \\right) = \\frac{8EI \\theta_B + 4EI \\theta_C}{L}"
},
{
"math_id": 28,
"text": "M_{CB} = \\frac{2EI}{L} \\left( 2 \\theta_B + 4 \\theta_C \\right) = \\frac{4EI \\theta_B + 8EI \\theta_C}{L}"
},
{
"math_id": 29,
"text": "M_{CD} = \\frac{EI}{L} \\left( 4 \\theta_C \\right) = \\frac{4EI\\theta_C}{L}"
},
{
"math_id": 30,
"text": "M_{DC} = \\frac{EI}{L} \\left( 2 \\theta_C \\right) = \\frac{2EI\\theta_C}{L}"
},
{
"math_id": 31,
"text": "\\Sigma M_A = M_{AB} + M_{AB}^f = 0.4EI \\theta_A + 0.2EI \\theta_B - 14.7 = 0"
},
{
"math_id": 32,
"text": "\\Sigma M_B = M_{BA} + M_{BA}^f + M_{BC} + M_{BC}^f = 0.2EI \\theta_A + 1.2EI \\theta_B + 0.4EI \\theta_C - 2.033 = 0"
},
{
"math_id": 33,
"text": "\\Sigma M_C = M_{CB} + M_{CB}^f + M_{CD} + M_{CD}^f = 0.4EI \\theta_B + 1.2EI \\theta_C - 4.167 = 0"
},
{
"math_id": 34,
"text": "\\theta_A = \\frac{40.219}{EI} "
},
{
"math_id": 35,
"text": "\\theta_B = \\frac{-6.937}{EI} "
},
{
"math_id": 36,
"text": "\\theta_C = \\frac{5.785}{EI} "
},
{
"math_id": 37,
"text": "M_{AB} = 0.4 \\times 40.219 + 0.2 \\times \\left( -6.937 \\right) - 14.7 = 0 "
},
{
"math_id": 38,
"text": "M_{BA} = 0.2 \\times 40.219 + 0.4 \\times \\left( -6.937 \\right) + 6.3 = 11.57 "
},
{
"math_id": 39,
"text": "M_{BC} = 0.8 \\times \\left( -6.937 \\right) + 0.4 \\times 5.785 - 8.333 = -11.57 "
},
{
"math_id": 40,
"text": "M_{CB} = 0.4 \\times \\left( -6.937 \\right) + 0.8 \\times 5.785 + 8.333 = 10.19 "
},
{
"math_id": 41,
"text": "M_{CD} = 0.4 \\times -5.785 - 12.5 = -10.19 "
},
{
"math_id": 42,
"text": "M_{DC} = 0.2 \\times -5.785 + 12.5 = 13.66 "
}
]
| https://en.wikipedia.org/wiki?curid=13047079 |
13048500 | Vantieghems theorem | In number theory, Vantieghems theorem is a primality criterion. It states that a natural number "n"≥3 is prime if and only if
formula_0
Similarly, "n" is prime, if and only if the following congruence for polynomials in "X" holds:
formula_1
or:
formula_2
Example.
Let n=7 forming the product 1*3*7*15*31*63 = 615195. 615195 = 7 mod 127 and so 7 is prime<br>
Let n=9 forming the product 1*3*7*15*31*63*127*255 = 19923090075. 19923090075 = 301 mod 511 and so 9 is composite | [
{
"math_id": 0,
"text": " \\prod_{1 \\leq k \\leq n-1} \\left( 2^k - 1 \\right) \\equiv n \\mod \\left( 2^n - 1 \\right). "
},
{
"math_id": 1,
"text": " \\prod_{1 \\leq k \\leq n-1} \\left( X^k - 1 \\right) \\equiv n- \\left( X^n - 1 \\right)/\\left( X - 1 \\right) \\mod \\left( X^n - 1 \\right) "
},
{
"math_id": 2,
"text": " \\prod_{1 \\leq k \\leq n-1} \\left( X^k - 1 \\right) \\equiv n \\mod \\left( X^n - 1 \\right)/\\left( X - 1 \\right). "
}
]
| https://en.wikipedia.org/wiki?curid=13048500 |
1305071 | Bridge (graph theory) | Edge in node-link graph whose removal would disconnect the graph
In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge can uniquely determine a cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges.
This type of bridge should be distinguished from an unrelated meaning of "bridge" in graph theory, a subgraph separated from the rest of the graph by a specified subset of vertices; see bridge in the Glossary of graph theory.
Trees and forests.
A graph with formula_0 nodes can contain at most formula_1 bridges, since adding additional edges must create a cycle. The graphs with exactly formula_1 bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests.
In every undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint paths connecting them. (Every vertex is related to itself via two length-zero paths, which are identical but nevertheless edge-disjoint.) The equivalence classes of this relation are called 2-edge-connected components, and the bridges of the graph are exactly the edges whose endpoints belong to different components. The bridge-block tree of the graph has a vertex for every nontrivial component and an edge for every bridge.
Relation to vertex connectivity.
Bridges are closely related to the concept of articulation vertices, vertices that belong to every path between some pair of other vertices. The two endpoints of a bridge are articulation vertices unless they have a degree of 1, although it may also be possible for a non-bridge edge to have two articulation vertices as endpoints. Analogously to bridgeless graphs being 2-edge-connected, graphs without articulation vertices are 2-vertex-connected.
In a cubic graph, every cut vertex is an endpoint of at least one bridge.
Bridgeless graphs.
A bridgeless graph is a graph that does not have any bridges. Equivalent conditions are that each connected component of the graph has an open ear decomposition, that each connected component is 2-edge-connected, or (by Robbins' theorem) that every connected component has a strong orientation.
An important open problem involving bridges is the cycle double cover conjecture, due to Seymour and Szekeres (1978 and 1979, independently), which states that every bridgeless graph admits a multi-set of simple cycles which contains each edge exactly twice.
Tarjan's bridge-finding algorithm.
The first linear time algorithm (linear in the number of edges) for finding the bridges in a graph was described by Robert Tarjan in 1974. It performs the following steps:
Bridge-finding with chain decompositions.
A very simple bridge-finding algorithm uses chain decompositions.
Chain decompositions do not only allow to compute all bridges of a graph, they also allow to "read off" every cut vertex of "G" (and the block-cut tree of "G"), giving a general framework for testing 2-edge- and 2-vertex-connectivity (which extends to linear-time 3-edge- and 3-vertex-connectivity tests).
Chain decompositions are special ear decompositions depending on a DFS-tree "T" of "G" and can be computed very simply: Let every vertex be marked as unvisited. For each vertex "v" in ascending DFS-numbers 1..."n", traverse every backedge (i.e. every edge not in the DFS tree) that is incident to "v" and follow the path of tree-edges back to the root of "T", stopping at the first vertex that is marked as visited. During such a traversal, every traversed vertex is marked as visited. Thus, a traversal stops at the latest at "v" and forms either a directed path or cycle, beginning with v; we call this path
or cycle a "chain". The "i"th chain found by this procedure is referred to as "Ci". "C=C1,C2..." is then a "chain decomposition" of "G".
The following characterizations then allow to "read off" several properties of "G" from "C" efficiently, including all bridges of "G". Let "C" be a chain decomposition of a simple connected graph "G=(V,E)".
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "n-1"
},
{
"math_id": 2,
"text": "G"
},
{
"math_id": 3,
"text": "F"
},
{
"math_id": 4,
"text": "v"
},
{
"math_id": 5,
"text": "ND(v)"
},
{
"math_id": 6,
"text": "L(v)"
},
{
"math_id": 7,
"text": "L(w)"
},
{
"math_id": 8,
"text": "H(v)"
},
{
"math_id": 9,
"text": "H(w)"
},
{
"math_id": 10,
"text": "w"
},
{
"math_id": 11,
"text": "L(w) = w"
},
{
"math_id": 12,
"text": "H(w) < w + ND(w)"
}
]
| https://en.wikipedia.org/wiki?curid=1305071 |
13051408 | Giordano Riccati | Italian physicist (1709–1790)
Giordano Riccati or Jordan Riccati (25 February 1709 – 20 July 1790) was an Italian mathematician and physicist.
Biography.
Giordano Riccati was born in 1709 in Castelfranco Veneto, a small town about 30 km north of Padua. He was the brother of Vincenzo Riccati, and the fifth son of the theoretical mechanician Jacopo Riccati. He began his studies at the College of St. Francis Xavier in Bologna, under the guidance of Francesco Saverio Quadrio and Luigi Marchenti, a pupil of the French mathematician Pierre Varignon. In 1727, he returned to Castelfranco, where his father taught him geometry, trigonometry, calculus, statics and dynamics. He then moved to the University of Padua and attended Giovanni Poleni's lessons on hydraulics as well as the lectures of the famous physician and naturalist Antonio Vallisneri. He studied literature, philosophy, theology, architecture, acoustics and music theory. He made significant contributions in the field of physics and mathematics applied to music, publishing the "Saggio sulle leggi del contrappunto" [Essay on the laws of counterpoint], which tried to prove that music is not just an art, but it is a science as well, a "Trattato delle corde, ovvero delle Fibre Elastiche" [Treaty on chords, that is, on elastic fibers], and some studies on the works of Tartini and Rameau. Giordano helped with the improvements to the Cathedral of Treviso. He died Treviso on July 20, 1790.
Riccati was a member of the Accademia Galileiana of Padua, of the Academy of Sciences of the Institute of Bologna and of the Italian National Academy of Sciences.
Contributions.
Riccati was the first experimental mechanician to study material elastic moduli as we understand them today. His 1782 paper on determining the relative Young's moduli of steel and brass using flexural vibrations preceded Thomas Young's 1807 paper on the subject of moduli. The ratio that Riccati found was:
formula_0
Even though the experiments were performed more than 200 years ago, this value is remarkably close to accepted values found in engineering handbooks in 2007.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\n \\frac{E_{\\mbox{steel}}}{E_{\\mbox{brass}}} = 2.06\n "
}
]
| https://en.wikipedia.org/wiki?curid=13051408 |
13051535 | John Hilton Grace | British mathematician
John Hilton Grace FRS (21 May 1873 – 4 March 1958) was a British mathematician. The Grace–Walsh–Szegő theorem is named in part after him.
Early life.
He was born in Halewood, near Liverpool, the eldest of the six children of farmer William Grace and Elizabeth Hilton. He was educated at the village school and the Liverpool Institute. From there in 1892 he went up to Peterhouse, Cambridge to study mathematics.
His nephew, his younger sister's son, was the animal geneticist, Alan Robertson FRS.
Career.
He was made a Fellow of Peterhouse in 1897 and became a Lecturer of Mathematics at Peterhouse and Pembroke colleges. An example of his work was his 1902 paper on "The Zeros of a Polynomial". In 1903 he collaborated with Alfred Young on their book "Algebra of Invariants".
He was elected a Fellow of the Royal Society in 1908.
He spent 1916–1917 as visiting professor in Lahore and deputised for Professor MacDonald at Aberdeen University during the latter part of the war.
In 1922 a breakdown in health forced his retirement from academic life and he spent the next part of his life in Norfolk.
He died in Huntingdon in 1958 and was buried in the family grave at St. Nicholas Church, Halewood.
Theorem on zeros of a polynomial.
If
formula_0,
formula_1
are two polynomials that satisfy the apolarity condition, i.e. formula_2, then every neighbourhood that includes all zeros of one polynomial also includes at least one zero of the other.
Corollary.
Let formula_3 and formula_4 be defined as in the above theorem. If the zeros of both polynomials lie in the unit disk, then the zeros of the "composition" of the two, formula_5, also lie in the unit disk.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "a(z)=a_0+\\tbinom{n}{1}a_1 z+\\tbinom{n}{2}a_2 z^2+\\dots+a_n z^n"
},
{
"math_id": 1,
"text": "b(z)=b_0+\\tbinom{n}{1}b_1 z+\\tbinom{n}{2}b_2 z^2+\\dots+b_n z^n"
},
{
"math_id": 2,
"text": "a_0 b_n - \\tbinom{n}{1}a_1 b_{n-1} + \\tbinom{n}{2}a_2 b_{n-2} - \\cdots +(-1)^n a_n b_0 = 0"
},
{
"math_id": 3,
"text": "a(z)"
},
{
"math_id": 4,
"text": "b(z)"
},
{
"math_id": 5,
"text": "c(z)=a_0 b_0 + \\tbinom{n}{1}a_1 b_1 z + \\tbinom{n}{2}a_2 b_2 z^2 + \\cdots + a_n b_n z^n"
}
]
| https://en.wikipedia.org/wiki?curid=13051535 |
130526 | Riemann curvature tensor | Tensor field in Riemannian geometry
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is "flat", i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
It is a central mathematical tool in the theory of general relativity, the modern theory of gravity. The curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.
Definition.
Let ("M", g) be a Riemannian or pseudo-Riemannian manifold, and formula_0 be the space of all vector fields on M. We define the Riemann curvature tensor as a map formula_1 by the following formula where formula_2 is the Levi-Civita connection:
formula_3
or equivalently
formula_4
where ["X", "Y"] is the Lie bracket of vector fields and formula_5 is a commutator of differential operators. It turns out that the right-hand side actually only depends on the value of the vector fields formula_6 at a given point, which is notable since the covariant derivative of a vector field also depends on the field values in a neighborhood of the point. Hence, formula_7 is a formula_8-tensor field. For fixed formula_9, the linear transformation formula_10 is also called the "curvature transformation" or "endomorphism". Occasionally, the curvature tensor is defined with the opposite sign.
The curvature tensor measures "noncommutativity of the covariant derivative", and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, "flat" space).
Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant derivative
formula_11
which depends only on the values of formula_12 at a point.
The curvature can then be written as
formula_13
Thus, the curvature tensor measures the noncommutativity of the second covariant derivative. In abstract index notation, formula_14The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector formula_15 with itself:
formula_16
This formula is often called the "Ricci identity". This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor. This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows
formula_17
This formula also applies to tensor densities without alteration, because for the Levi-Civita ("not generic") connection one gets:
formula_18
where
formula_19
It is sometimes convenient to also define the purely covariant version of the curvature tensor by
formula_20
Geometric meaning.
Informally.
One can see the effects of curved space by comparing a tennis court and the Earth. Start at the lower right corner of the tennis court, with a racket held out towards north. Then while walking around the outline of the court, at each step make sure the tennis racket is maintained in the same orientation, parallel to its previous positions. Once the loop is complete the tennis racket will be parallel to its initial starting position. This is because tennis courts are built so the surface is flat. On the other hand, the surface of the Earth is curved: we can complete a loop on the surface of the Earth. Starting at the equator, point a tennis racket north along the surface of the Earth. Once again the tennis racket should always remain parallel to its previous position, using the local plane of the horizon as a reference. For this path, first walk to the north pole, then walk sideways (i.e. without turning), then down to the equator, and finally walk backwards to your starting position. Now the tennis racket will be pointing towards the west, even though when you began your journey it pointed north and you never turned your body. This process is akin to parallel transporting a vector along the path and the difference identifies how lines which appear "straight" are only "straight" locally. Each time a loop is completed the tennis racket will be deflected further from its initial position by an amount depending on the distance and the curvature of the surface. It is possible to identify paths along a curved surface where parallel transport works as it does on flat space. These are the geodesics of the space, for example any segment of a great circle of a sphere.
The concept of a curved space in mathematics differs from conversational usage. For example, if the above process was completed on a cylinder one would find that it is not curved overall as the curvature around the cylinder cancels with the flatness along the cylinder, which is a consequence of Gaussian curvature and Gauss's Theorema Egregium. A familiar example of this is a floppy pizza slice, which will remain rigid along its length if it is curved along its width.
The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be thought of as akin to capturing the curvature imposed upon someone walking in straight lines on a curved surface).
Formally.
When a vector in a Euclidean space is parallel transported around a loop, it will again point in the initial direction after returning to its original position. However, this property does not hold in the general case. The Riemann curvature tensor directly measures the failure of this in a general Riemannian manifold. This failure is known as the non-holonomy of the manifold.
Let formula_21 be a curve in a Riemannian manifold formula_22. Denote by formula_23 the parallel transport map along formula_21. The parallel transport maps are related to the covariant derivative by
formula_24
for each vector field formula_25 defined along the curve.
Suppose that formula_26 and formula_25 are a pair of commuting vector fields. Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of formula_27. Denote by formula_28 and formula_29, respectively, the parallel transports along the flows of formula_26 and formula_25 for time formula_30. Parallel transport of a vector formula_31 around the quadrilateral with sides formula_32, formula_33, formula_34, formula_35 is given by
formula_36
This measures the failure of parallel transport to return formula_37 to its original position in the tangent space formula_38. Shrinking the loop by sending formula_39 gives the infinitesimal description of this deviation:
formula_40
where formula_7 is the Riemann curvature tensor.
Coordinate expression.
Converting to the tensor index notation, the Riemann curvature tensor is given by
formula_41
where formula_42 are the coordinate vector fields. The above expression can be written using Christoffel symbols:
formula_43
(see also the list of formulas in Riemannian geometry).
Symmetries and identities.
The Riemann curvature tensor has the following symmetries and identities:
where the bracket formula_44 refers to the inner product on the tangent space induced by the metric tensor and
the brackets and parentheses on the indices denote the antisymmetrization and symmetrization operators, respectively. If there is nonzero torsion, the Bianchi identities involve the torsion tensor.
The first (algebraic) Bianchi identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the differential Bianchi identity.
The first three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has formula_45 independent components. Interchange symmetry follows from these. The algebraic symmetries are also equivalent to saying that "R" belongs to the image of the Young symmetrizer corresponding to the partition 2+2.
On a Riemannian manifold one has the covariant derivative formula_46 and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form of the last identity in the table.
Ricci curvature.
The Ricci curvature tensor is the contraction of the first and third indices of the Riemann tensor.
formula_47
Special cases.
Surfaces.
For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor has only one independent component, which means that the Ricci scalar completely determines the Riemann tensor. There is only one valid expression for the Riemann tensor which fits the required symmetries:
formula_48
and by contracting with the metric twice we find the explicit form:
formula_49
where formula_50 is the metric tensor and formula_51 is a function called the Gaussian curvature and "a", "b", "c" and "d" take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by
formula_52
Space forms.
A Riemannian manifold is a space form if its sectional curvature is equal to a constant "K". The Riemann tensor of a space form is given by
formula_53
Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function "K", then the Bianchi identities imply that "K" is constant and thus that the manifold is (locally) a space form.
Citations.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" /> | [
{
"math_id": 0,
"text": "\\mathfrak{X}(M)"
},
{
"math_id": 1,
"text": "\\mathfrak{X}(M)\\times\\mathfrak{X}(M)\\times\\mathfrak{X}(M)\\rightarrow\\mathfrak{X}(M)"
},
{
"math_id": 2,
"text": "\\nabla"
},
{
"math_id": 3,
"text": "R(X, Y)Z = \\nabla_X\\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z"
},
{
"math_id": 4,
"text": "R(X, Y) = [\\nabla_X,\\nabla_Y] - \\nabla_{[X, Y]} "
},
{
"math_id": 5,
"text": "[\\nabla_X,\\nabla_Y] "
},
{
"math_id": 6,
"text": "X, Y, Z"
},
{
"math_id": 7,
"text": "R"
},
{
"math_id": 8,
"text": "(1,3)"
},
{
"math_id": 9,
"text": "X,Y"
},
{
"math_id": 10,
"text": "Z \\mapsto R(X, Y)Z"
},
{
"math_id": 11,
"text": "\\nabla^2_{X,Y} Z = \\nabla_X\\nabla_Y Z - \\nabla_{\\nabla_X Y}Z "
},
{
"math_id": 12,
"text": "X, Y"
},
{
"math_id": 13,
"text": "R(X, Y) = \\nabla^2_{X,Y} - \\nabla^2_{Y,X} "
},
{
"math_id": 14,
"text": "R^d{}_{cab} Z^c = \\nabla_a \\nabla_b Z^d - \\nabla_b \\nabla_a Z^d . "
},
{
"math_id": 15,
"text": "A_{\\nu}"
},
{
"math_id": 16,
"text": "A_{\\nu;\\rho\\sigma} - A_{\\nu;\\sigma\\rho} = A_{\\beta} R^{\\beta}{}_{\\nu\\rho\\sigma}."
},
{
"math_id": 17,
"text": "\\begin{align}\n &\\nabla_\\delta \\nabla_\\gamma T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} - \\nabla_\\gamma \\nabla_\\delta T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} \\\\[3pt]\n ={} &R^{\\alpha_1}{}_{\\rho\\delta\\gamma} T^{\\rho\\alpha_2 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} + \\ldots +\n R^{\\alpha_r}{}_{\\rho\\delta\\gamma} T^{\\alpha_1 \\cdots \\alpha_{r-1}\\rho}{}_{\\beta_1 \\cdots \\beta_s} -\n R^{\\sigma}{}_{\\beta_1\\delta\\gamma} T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\sigma\\beta_2 \\cdots \\beta_s} - \\ldots -\n R^{\\sigma}{}_{\\beta_s\\delta\\gamma} T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_{s-1}\\sigma}\n\\end{align}"
},
{
"math_id": 18,
"text": "\\nabla_{\\mu}\\left(\\sqrt{g}\\right) \\equiv \\left(\\sqrt{g}\\right)_{;\\mu} = 0,"
},
{
"math_id": 19,
"text": "g = \\left|\\det\\left(g_{\\mu\\nu}\\right)\\right|."
},
{
"math_id": 20,
"text": "R_{\\sigma\\mu\\nu\\rho} = g_{\\rho\\zeta} R^{\\zeta}{}_{\\sigma\\mu\\nu}."
},
{
"math_id": 21,
"text": "x_t"
},
{
"math_id": 22,
"text": "M"
},
{
"math_id": 23,
"text": "\\tau_{x_t}:T_{x_0}M \\to T_{x_t}M"
},
{
"math_id": 24,
"text": "\n \\nabla_{\\dot{x}_0} Y =\n \\lim_{h\\to 0} \\frac{1}{h}\\left(Y_{x_0} - \\tau^{-1}_{x_h}\\left(Y_{x_h}\\right)\\right) =\n \\left.\\frac{d}{dt}\\left(\\tau_{x_t}^{-1}(Y_{x_t})\\right)\\right|_{t=0}\n"
},
{
"math_id": 25,
"text": "Y"
},
{
"math_id": 26,
"text": "X"
},
{
"math_id": 27,
"text": "x_0"
},
{
"math_id": 28,
"text": "\\tau_{tX}"
},
{
"math_id": 29,
"text": "\\tau_{tY}"
},
{
"math_id": 30,
"text": "t"
},
{
"math_id": 31,
"text": "Z \\in T_{x_0}M"
},
{
"math_id": 32,
"text": "tY"
},
{
"math_id": 33,
"text": "sX"
},
{
"math_id": 34,
"text": "-tY"
},
{
"math_id": 35,
"text": "-sX"
},
{
"math_id": 36,
"text": "\\tau_{sX}^{-1}\\tau_{tY}^{-1}\\tau_{sX}\\tau_{tY}Z."
},
{
"math_id": 37,
"text": "Z"
},
{
"math_id": 38,
"text": "T_{x_0}M"
},
{
"math_id": 39,
"text": "s, t \\to 0"
},
{
"math_id": 40,
"text": "\\left.\\frac{d}{ds}\\frac{d}{dt}\\tau_{sX}^{-1}\\tau_{tY}^{-1}\\tau_{sX}\\tau_{tY}Z\\right|_{s=t=0} = \\left(\\nabla_X\\nabla_Y - \\nabla_Y\\nabla_X - \\nabla_{[X,Y]}\\right)Z = R(X, Y)Z"
},
{
"math_id": 41,
"text": "R^{\\rho}{}_{\\sigma\\mu\\nu} = dx^{\\rho}\\left(R\\left(\\partial_{\\mu}, \\partial_{\\nu}\\right)\\partial_{\\sigma}\\right)"
},
{
"math_id": 42,
"text": "\\partial_{\\mu} = \\partial/\\partial x^{\\mu}"
},
{
"math_id": 43,
"text": "\n R^{\\rho}{}_{\\sigma\\mu\\nu} =\n \\partial_{\\mu}\\Gamma^{\\rho}{}_{\\nu\\sigma} -\n \\partial_{\\nu}\\Gamma^{\\rho}{}_{\\mu\\sigma} +\n \\Gamma^{\\rho}{}_{\\mu\\lambda}\\Gamma^{\\lambda}{}_{\\nu\\sigma} -\n \\Gamma^{\\rho}{}_{\\nu\\lambda}\\Gamma^{\\lambda}{}_{\\mu\\sigma}\n"
},
{
"math_id": 44,
"text": "\\langle,\\rangle"
},
{
"math_id": 45,
"text": "n^2\\left(n^2 - 1\\right)/12"
},
{
"math_id": 46,
"text": " \\nabla_u R "
},
{
"math_id": 47,
"text": "\n \\underbrace{R_{ab}}_{\\text{Ricci}}\n \\equiv \\underbrace{R^c{}_{acb}}_{\\text{Riemann}}\n = g^{cd} \\underbrace{R_{cadb}}_{\\text{Riemann}}\n"
},
{
"math_id": 48,
"text": "R_{abcd} = f(R) \\left(g_{ac}g_{db} - g_{ad}g_{cb}\\right)"
},
{
"math_id": 49,
"text": "R_{abcd} = K\\left(g_{ac}g_{db} - g_{ad}g_{cb}\\right) ,"
},
{
"math_id": 50,
"text": "g_{ab}"
},
{
"math_id": 51,
"text": "K = R/2"
},
{
"math_id": 52,
"text": "R_{ab} = Kg_{ab}."
},
{
"math_id": 53,
"text": "R_{abcd} = K\\left(g_{ac}g_{db} - g_{ad}g_{cb}\\right)."
}
]
| https://en.wikipedia.org/wiki?curid=130526 |
1305761 | Afshar experiment | The Afshar experiment is a variation of the double-slit experiment in quantum mechanics, devised and carried out by Shahriar Afshar in 2004. In the experiment, light generated by a laser passes through two closely spaced pinholes, and is refocused by a lens so that the image of each pinhole falls on a separate single-photon detector. In addition, a grid of thin wires is placed just before the lens on the dark fringes of an interference pattern.
Afshar claimed that the experiment gives information about which path a photon takes through the apparatus, while simultaneously allowing interference between the paths to be observed. According to Afshar, this violates the complementarity principle of quantum mechanics.
The experiment has been analyzed and repeated by a number of investigators. There are several theories that explain the effect without violating complementarity. John G. Cramer claims the experiment provides evidence for the transactional interpretation of quantum mechanics over other interpretations.
History.
Shahriar Afshar's experimental work was done initially at the Institute for Radiation-Induced Mass Studies (IRIMS) in Boston and later reproduced at Harvard University, while he was there as a visiting researcher. The results were first presented at a seminar at Harvard in March 2004. The experiment was featured as the cover story in the July 24, 2004 edition of the popular science magazine "New Scientist" endorsed by professor John G. Cramer of the University of Washington. The "New Scientist" feature article generated many responses, including various letters to the editor that appeared in the August 7 and August 14, 2004 issues, arguing against the conclusions being drawn by Afshar. The results were published in a SPIE conference proceedings in 2005. A follow-up paper was published in a scientific journal Foundations of Physics in January 2007 and featured in "New Scientist" in February 2007.
Experimental setup.
The experiment uses a setup similar to that for the double-slit experiment. In Afshar's variant, light generated by a laser passes through two closely spaced "circular" pinholes (not slits). After the dual pinholes, a lens refocuses the light so that the image of each pinhole falls on separate photon-detectors (Fig. 1). With pinhole 2 closed, a photon that goes through pinhole 1 impinges only on photon detector 1. Similarly, with pinhole 1 closed, a photon that goes through pinhole 2 impinges only on photon detector 2. With both pinholes open, Afshar claims, citing Wheeler in support, that pinhole 1 remains correlated to photon Detector 1 (and vice versa for pinhole 2 to photon Detector 2), and therefore that which-way information is preserved when both pinholes are open.
When the light acts as a wave, because of quantum interference one can observe that there are regions that the photons avoid, called "dark fringes". A grid of thin wires is placed just before the lens (Fig. 2) so that the wires lie in the dark fringes of an interference pattern which is produced by the dual pinhole setup. If one of the pinholes is blocked, the interference pattern will no longer be formed, and the grid of wires causes appreciable diffraction in the light and blocks some of it from detection by the corresponding photon detector. However, when both pinholes are open, the effect of the wires is negligible, comparable to the case in which there are no wires placed in front of the lens (Fig. 3), because the wires lie in the dark fringes of an interference pattern. The effect is not dependent on the light intensity (photon flux).
Afshar's interpretation.
Afshar's conclusion is that, when both pinholes are open, the light exhibits wave-like behavior when going past the wires, since the light goes through the spaces between the wires but avoids the wires themselves, but also exhibits particle-like behavior after going through the lens, with photons going to a correlated photo-detector. Afshar argues that this behavior contradicts the principle of complementarity to the extent that it shows both wave and particle characteristics in the same experiment for the same photons.
Afshar asserts that there is simultaneously high visibility "V" of interference as well as high distinguishability "D" (corresponding to which-path information), so that "V"2 + "D"2 > 1, and the wave-particle duality relation is violated.
Reception.
Specific criticism.
A number of scientists have published criticisms of Afshar's interpretation of his results, some of which reject the claims of a violation of complementarity, while differing in the way they explain how complementarity copes with the experiment. For example, one paper contests Afshar's core claim, that the Englert–Greenberger duality relation is violated. The researchers re-ran the experiment, using a different method for measuring the visibility of the interference pattern than that used by Afshar, and found no violation of complementarity, concluding "This result demonstrates that the experiment can be perfectly explained by the Copenhagen interpretation of quantum mechanics."
Below is a synopsis of papers by several critics highlighting their main arguments and the disagreements they have amongst themselves:
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\mathcal O"
}
]
| https://en.wikipedia.org/wiki?curid=1305761 |
1305787 | Fielding independent pitching | Type of baseball statistic
In baseball, fielding independent pitching (FIP) (also referred to as defense independent pitching) is intended to measure a pitcher's effectiveness based only on statistics that do not involve fielders (except the catcher). These include home runs allowed, strikeouts, hit batters, walks, and, more recently, fly ball percentage, ground ball percentage, and (to a much lesser extent) line drive percentage. By focusing on these statistics and ignoring what happens once a ball is put in play, which – on most plays – the pitcher has little control over, DIPS claims to offer a clearer picture of the pitcher's true ability.
The most controversial part of DIPS is the idea that pitchers have little influence over what happens to balls that are put into play. Some people believe this has been well-established (see below), primarily by showing the large variability of most pitchers' BABIP from year to year. However, there is a wide variation in career BABIP among pitchers, and this seems to correlate with career success. For instance, no pitcher in the Hall of Fame has a below-average career BABIP.
Formulae.
Each of the following formulae uses innings pitched (IP), a measure of the number of outs a team made while a pitcher was in the game. Since most outs rely on fielding, the results from calculations using IP are not truly independent of team defense. While the creators of DICE, FIP and similar statistics all suggest they are "defense independent", others have pointed out that their formulas involve (IP). IP is a statistical measure of how many outs were made while a pitcher was pitching. This includes those made by fielders who are typically involved in more than two thirds of the outs. These critics claim this makes pitchers' DICE or FIP highly dependent on the defensive play of their fielders.
DICE.
A simple formula, known as Defense-Independent Component ERA (DICE), was created by Clay Dreslough in 1998:
formula_0
In that equation, "HR" is home runs, "BB" is walks, "HBP" is hit batters, "K" is strikeouts, and "IP" is innings pitched. That equation gives a number that is better at predicting a pitcher's ERA in the following year than the pitcher's actual ERA in the current year.
FIP.
Tom Tango independently derived a similar formula, known as Fielding Independent Pitching, which is very close to the results of dERA and DICE.
formula_1
In that equation, "HR" is home runs, "BB" is walks, "K" is strikeouts, and "IP" is innings pitched. That equation usually gives a number that is nothing close to a normal ERA (this is the FIP core), so the equation used is more often (but not always) this one:
formula_2
where C is a constant that renders league FIP for the time period in question equal to league ERA for the same period. It is calculated as:
formula_3
where lgERA is the league average ERA, lgHR is the number of home runs in the league, lgBB is the number of walks in the league, lgK is the number of strikeouts in the league, and lgIP is the number of innings pitched in the league.
The Hardball Times, a popular baseball statistics website, uses a slightly different FIP equation, instead using 3*(BB+HBP-IBB) rather than simply 3*(BB) where "HBP" stands for batters hit by pitch and "IBB" stands for intentional base on balls.
xFIP.
Dave Studeman of The Hardball Times derived Expected Fielding Independent Pitching (xFIP), a regressed version of FIP. Calculated like FIP, it differs in that it normalizes the number of home runs the pitcher allows, replacing a pitcher's actual home run total with an expected home run total (xHR).
formula_4
where xHR is calculated using the league average home run per fly ball rate (lgHR/FB) multiplied by the number of fly balls the pitcher has allowed.
formula_5
Typically, the lgHR/FB is around 10.5%, meaning 10.5% of fly balls go for home runs. In 2015, it was 11.4%.
SIERA.
Baseball Prospectus invented this statistic, which takes into account balls in play and adjusts for balls in play. For example, if a pitcher has a high xFIP, but also induces a lot of ground balls and popups, his SIERA will be lower than his xFIP. The calculations for it are as follows:
formula_6
where SO is strikeouts, PA is plate appearances, BB is bases on balls, GB is ground ball, FB is fly ball, and PU is pop-up
Origins.
In 1999, Voros McCracken became the first to detail and publicize these effects to the baseball research community when he wrote on rec.sport.baseball, "I've been working on a pitching evaluation tool and thought I'd post it here to get some feedback. I call it 'Defensive Independent Pitching' and what it does is evaluate a pitcher base[d] strictly on the statistics his defense has no ability to affect..." Until the publication of a more widely read article in 2001, however, on Baseball Prospectus, most of the baseball research community believed that individual pitchers had an inherent ability to prevent hits on balls in play. McCracken reasoned that if this ability existed, it would be noticeable in a pitcher's 'Batting Average on Balls In Play' (BABIP). His research found the opposite to be true: that while a pitcher's ability to cause strikeouts or prevent home runs remained somewhat constant from season to season, his ability to prevent hits on balls in play did not.
To better evaluate pitchers in light of his theory, McCracken developed "Defense-Independent ERA" (dERA), the most well-known defense-independent pitching statistic. McCracken's formula for dERA is very complicated, with a number of steps. DIPS ERA is not as useful for knuckleballers and other "trick" pitchers, a factor that McCracken mentioned a few days after his original announcement of his research findings in 1999, in a posting on the rec.sport.baseball.analysis Usenet site on November 23, 1999, when he wrote: "Also to [note] is that, anecdotally, I believe pitchers with trick deliveries (e.g. Knuckleballers) might post consistently lower $H numbers than other pitchers. I looked at Tim Wakefield's career and that seems to bear out slightly".
In later postings on the rec.sport.baseball site during 1999 and 2000 (prior to the publication of his widely read article on BaseballProspectus.com in 2001), McCracken also discussed other pitcher characteristics that might influence BABIP. In 2002 McCracken created and published version 2.0 of dERA, which incorporates the ability of knuckleballers and other types of pitchers to affect the number of hits allowed on balls hit in the field of play (BHFP).
Controversy.
Controversy over DIPS was heightened when Tom Tippett at Diamond Mind published his own findings in 2003. Tippett concluded that the differences between pitchers in preventing hits on balls in play were at least partially the result of the pitcher's skill. Tippett analyzed certain groups of pitchers that appear to be able to reduce the number of hits allowed on balls hit into the field of play (BHFP). Like McCracken, Tippett found that pitchers' BABIP was more volatile on an annual basis than the rates at which they gave up home runs or walks. It was this greater volatility that had led McCracken to conclude pitchers had "little or no control" over hits on balls in play. But Tippett also found large and significant differences between pitchers' career BABIP. In many cases, it was these differences that accounted for the pitchers' relative success.
However, improvements to DIPS that look at more nuanced defense-independent stats than strikeouts, home runs, and walks (such as groundball rate), have been able to account for many of the BABIP differences that Tippet identified without reintroducing the noise from defense variability.
Despite other criticisms, the work by McCracken on DIPS is regarded by many in the sabermetric community as the most important piece of baseball research in many years. As Jonah Keri wrote in 2012, "When Voros McCracken wrote his seminal piece on pitching and defense 11 years ago, he helped change the way people—fans, writers, even general managers—think about run prevention in baseball. Where once we used to throw most of the blame for a hit on the pitcher who gave it up, McCracken helped us realize that a slew of other factors go into whether a ball hit into play falls for a hit. For many people in the game and others who simply watch it, our ability to recognize the influence of defense, park effects, and dumb luck can be traced back to that one little article".
DIPS ERA was added to ESPN.com's Sortable Stats in 2004.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "DICE=3.00 + \\frac{13HR + 3(BB + HBP) - 2K}{IP}"
},
{
"math_id": 1,
"text": "FIP=\\frac{13HR + 3BB - 2K}{IP}"
},
{
"math_id": 2,
"text": "FIP=\\frac{13HR + 3BB - 2K}{IP}+C"
},
{
"math_id": 3,
"text": "C = lgERA - {{13(lgHR) + 3(lgBB) - 2(lgK) \\over lgIP }}"
},
{
"math_id": 4,
"text": "xFIP=\\frac{13(xHR) + 3BB - 2K}{IP}+ C"
},
{
"math_id": 5,
"text": "xHR = Fly Balls * lgHR/FB"
},
{
"math_id": 6,
"text": "SIERA=6.145 - 16.986(SO/PA) + 11.434(BB/PA) - 1.858((GB-FB-PU)/PA) + 7.653((SO/PA)^2) +/- 6.664(((GB-FB-PU)/PA)^2) + 10.130(SO/PA)((GB-FB-PU)/PA) - 5.195(BB/PA)*((GB-FB-PU)/PA)"
}
]
| https://en.wikipedia.org/wiki?curid=1305787 |
1306265 | Chomp | Abstract strategy game
Chomp is a two-player strategy game played on a rectangular grid made up of smaller square cells, which can be thought of as the blocks of a chocolate bar. The players take it in turns to choose one block and "eat it" (remove from the board), together with those that are below it and to its right. The top left block is "poisoned" and the player who eats this loses.
The chocolate-bar formulation of Chomp is due to David Gale, but an equivalent game expressed in terms of choosing divisors of a fixed integer was published earlier by Frederik Schuh.
Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the minimal element (poisonous block) removed.
Example game.
Below shows the sequence of moves in a typical game starting with a 5 × 4 bar:
Player A eats two blocks from the bottom right corner; Player B eats three from the bottom row; Player A picks the block to the right of the poisoned block and eats eleven blocks; Player B eats three blocks from the remaining column, leaving only the poisoned block. Player A must eat the last block and so loses.
Note that since it is provable that player A can win when starting from a 5 × 4 bar, at least one of A's moves is a mistake.
Positions of the game.
The intermediate positions in an "m" × "n" Chomp are integer-partitions (non-increasing sequences of positive integers) λ1 ≥ λ2 ≥···≥ λr, with λ1 ≤ "n"
and "r" ≤ "m". Their number is the binomial coefficient formula_0, which grows exponentially with "m" and "n".
Winning the game.
Chomp belongs to the category of impartial two-player perfect information games, making it also analyzable by Nim because of the Sprague–Grundy theorem.
For any rectangular starting position, other than 1×1, the first player can win. This can be shown using a strategy-stealing argument: assume that the second player has a winning strategy against any initial first-player move. Suppose then, that the first player takes only the bottom right hand square. By our assumption, the second player has a response to this which will force victory. But if such a winning response exists, the first player could have played it as their first move and thus forced victory. The second player therefore cannot have a winning strategy.
Computers can easily calculate winning moves for this game on two-dimensional boards of reasonable size. However, as the number of positions grows exponentially, this is infeasible for larger boards.
For a "square" starting position (i.e., "n" × "n" for any "n" ≥ 2), the winning strategy can easily be given explicitly. The first player should present the second with an "L" shape of one row and one column only, of the same length, connected at the poisonous square. Then, whatever the second player does on one arm of the "L", the first player replies with the same move on the second arm, always presenting the second player again with a symmetric "L" shape. Finally, this "L" will degenerate into the single poisonous square, and the second player would lose.
Generalisations of Chomp.
Three-dimensional Chomp has an initial chocolate bar of a cuboid of blocks indexed as (i,j,k). A move is to take a block together with any block all of whose indices are greater or equal to the corresponding index of the chosen block. In the same way Chomp can be generalised to any number of dimensions.
Chomp is sometimes described numerically. An initial natural number is given, and players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a previously chosen divisor. This game models "n-"dimensional Chomp, where the initial natural number has "n" prime factors and the dimensions of the Chomp board are given by the exponents of the primes in its prime factorization.
Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers: for example a 2 × (ω + 4) bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block. The case of ω × ω × ω Chomp is a notable open problem; a $100 reward has been offered for finding a winning first move.
More generally, Chomp can be played on any partially ordered set with a least element. A move is to remove any element along with all larger elements. A player loses by taking the least element.
All varieties of Chomp can also be played without resorting to poison by using the misère play convention: The player who eats the final chocolate block is not poisoned, but simply loses by virtue of being the last player. This is identical to the ordinary rule when playing Chomp on its own, but differs when playing the disjunctive sum of Chomp games, where only the last final chocolate block loses. | [
{
"math_id": 0,
"text": "\\binom{m+n}{n}"
}
]
| https://en.wikipedia.org/wiki?curid=1306265 |
1306670 | Secure Remote Password protocol | Augmented password-authenticated key exchange protocol
The Secure Remote Password protocol (SRP) is an augmented password-authenticated key exchange (PAKE) protocol, specifically designed to work around existing patents.
Like all PAKE protocols, an eavesdropper or man in the middle cannot obtain enough information to be able to brute-force guess a password or apply a dictionary attack without further interactions with the parties for each guess. Furthermore, being an augmented PAKE protocol, the server does not store password-equivalent data. This means that an attacker who steals the server data cannot masquerade as the client unless they first perform a brute force search for the password.
In layman's terms, during SRP (or any other PAKE protocol) authentication, one party (the "client" or "user") demonstrates to another party (the "server") that they know the password, without sending the password itself nor any other information from which the password can be derived. The password never leaves the client and is unknown to the server.
Furthermore, the server also needs to know about the password (but not the password itself) in order to instigate the secure connection. This means that the server also authenticates itself to the client which prevents phishing without reliance on the user parsing complex URLs.
The only mathematically proven security property of SRP is that it is equivalent to Diffie-Hellman against a "passive" attacker. Newer PAKEs such as AuCPace and OPAQUE offer stronger guarantees.
Overview.
The SRP protocol has a number of desirable properties: it allows a user to authenticate themselves to a server, it is resistant to dictionary attacks mounted by an eavesdropper, and it does not require a trusted third party. It effectively conveys a zero-knowledge password proof from the user to the server. In revision 6 of the protocol only one password can be guessed per connection attempt. One of the interesting properties of the protocol is that even if one or two of the cryptographic primitives it uses are attacked, it is still secure. The SRP protocol has been revised several times, and is currently at revision 6a.
The SRP protocol creates a large private key shared between the two parties in a manner similar to Diffie–Hellman key exchange based on the client side having the user password and the server side having a cryptographic verifier derived from the password. The shared public key is derived from two random numbers, one generated by the client, and the other generated by the server, which are unique to the login attempt. In cases where encrypted communications as well as authentication are required, the SRP protocol is more secure than the alternative SSH protocol and faster than using Diffie–Hellman key exchange with signed messages. It is also independent of third parties, unlike Kerberos.
The SRP protocol, version 3 is described in RFC 2945. SRP version 6a is also used for strong password authentication in SSL/TLS (in TLS-SRP) and other standards such as EAP and SAML, and is part of IEEE 1363.2 and ISO/IEC 11770-4.
Protocol.
The following notation is used in this description of the protocol, version 6:
All other variables are defined in terms of these.
First, to establish a password "p" with server Steve, client Carol picks a random salt "s", and computes "x" = "H"("s", "p"), "v" = "g". Steve stores "v" and "s", indexed by I, as Carol's password verifier and salt. Carol must not share "x" with anybody, and must safely erase it at this step, because it is equivalent to the plaintext password "p". This step is completed before the system is used as part of the user registration with Steve. Note that the salt "s" is shared and exchanged to negotiate a session key later so the value could be chosen by either side but is done by Carol so that she can register I, "s" and "v" in a single registration request. The transmission and authentication of the registration request is not covered in SRP.
Then to perform a proof of password at a later date the following exchange protocol occurs:
Now the two parties have a shared, strong session key "K". To complete authentication, they need to prove to each other that their keys match. One possible way is as follows:
This method requires guessing more of the shared state to be successful in impersonation than just the key. While most of the additional state is public, private information could safely be added to the inputs to the hash function, like the server private key.
Alternatively, in a password-only proof the calculation of "K" can be skipped and the shared "S" proven with:
When using SRP to negotiate a shared key "K" which will be immediately used after the negotiation, it is tempting to skip the verification steps of "M"1 and "M"2. The server will reject the very first request from the client which it cannot decrypt. This can however be dangerous as demonstrated in the Implementation Pitfalls section below.
The two parties also employ the following safeguards:
Example code in Python.
An example SRP authentication
WARNING: Do not use for real cryptographic purposes beyond testing.
WARNING: This below code misses important safeguards. It does not check A, B, and U are not zero.
based on http://srp.stanford.edu/design.html
import hashlib
import random
def H(*args) -> int:
"""A one-way hash function."""
a = ":".join(str(a) for a in args)
return int(hashlib.sha256(a.encode("utf-8")).hexdigest(), 16)
def cryptrand(n: int = 1024):
return random.SystemRandom().getrandbits(n) % N
N = """00:c0:37:c3:75:88:b4:32:98:87:e6:1c:2d:a3:32:
4b:1b:a4:b8:1a:63:f9:74:8f:ed:2d:8a:41:0c:2f:
c2:1b:12:32:f0:d3:bf:a0:24:27:6c:fd:88:44:81:
97:aa:e4:86:a6:3b:fc:a7:b8:bf:77:54:df:b3:27:
c7:20:1f:6f:d1:7f:d7:fd:74:15:8b:d3:1c:e7:72:
c9:f5:f8:ab:58:45:48:a9:9a:75:9b:5a:2c:05:32:
16:2b:7b:62:18:e8:f1:42:bc:e2:c3:0d:77:84:68:
9a:48:3e:09:5e:70:16:18:43:79:13:a8:c3:9c:3d:
d0:d4:ca:3c:50:0b:88:5f:e3"""
N = int("".join(N.split()).replace(":", ""), 16)
g = 2 # A generator modulo N
k = H(N, g) # Multiplier parameter (k=3 in legacy SRP-6)
F = '#0x' # Format specifier
print("#. H, N, g, and k are known beforehand to both client and server:")
print(f'{H = }\n{N = :{F}}\n{g = :{F}}\n{k = :{F}}')
print("\n0. server stores (I, s, v) in its password database")
I = "person" # Username
p = "password1234" # Password
s = cryptrand(64) # Salt for the user
x = H(s, I, p) # Private key
v = pow(g, x, N) # Password verifier
print(f'{I = }\n{p = }\n{s = :{F}}\n{x = :{F}}\n{v = :{F}}')
print("\n1. client sends username I and public ephemeral value A to the server")
a = cryptrand()
A = pow(g, a, N)
print(f"{I = }\n{A = :{F}}") # client->server (I, A)
print("\n2. server sends user's salt s and public ephemeral value B to client")
b = cryptrand()
B = (k * v + pow(g, b, N)) % N
print(f"{s = :{F}}\n{B = :{F}}") # server->client (s, B)
print("\n3. client and server calculate the random scrambling parameter")
u = H(A, B) # Random scrambling parameter
print(f"{u = :{F}}")
print("\n4. client computes session key")
x = H(s, I, p)
S_c = pow(B - k * pow(g, x, N), a + u * x, N)
K_c = H(S_c)
print(f"{S_c = :{F}}\n{K_c = :{F}}")
print("\n5. server computes session key")
S_s = pow(A * pow(v, u, N), b, N)
K_s = H(S_s)
print(f"{S_s = :{F}}\n{K_s = :{F}}")
print("\n6. client sends proof of session key to server")
M_c = H(H(N) ^ H(g), H(I), s, A, B, K_c)
print(f"{M_c = :{F}}")
print("\n7. server sends proof of session key to client")
M_s = H(A, M_c, K_s)
print(f"{M_s = :{F}}")
Implementation pitfalls.
Offline bruteforce attack with server-first messaging in the absence of key verification.
If the server sends an encrypted message without waiting for verification from the client then an attacker is able to mount an offline bruteforce attack similar to hash cracking. This can happen if the server sends an encrypted message in the second packet alongside the salt and "B" or if key verification is skipped and the server (rather than the client) sends the first encrypted message. This is tempting as after the very first packet, the server has every information to compute the shared key "K".
The attack goes as follow:
Carol doesn't know "x" or "v". But given any password "p" she can compute:
"K""p" is the key that Steve would use if "p" was the expected password. All values required to compute "K""p" are either controlled by Carol or known from the first packet from Steve. Carol can now try to guess the password, generate the corresponding key, and attempt to decrypt Steve's encrypted message "c" to verify the key. As protocol messages tend to be structured, it is assumed that identifying that "c" was properly decrypted is easy. This allows offline recovery of the password.
This attack would not be possible had Steve waited for Carol to prove she was able to compute the correct key before sending an encrypted message. Proper implementations of SRP are not affected by this attack as the attacker would be unable to pass the key verification step.
Offline bruteforce based on timing attack.
In 2021 Daniel De Almeida Braga, Pierre-Alain Fouque and Mohamed Sabt published PARASITE, a paper in which they demonstrate practical exploitation of a timing attack over the network. This exploits non-constant implementations of modular exponentiation of big numbers and impacted OpenSSL in particular.
History.
The SRP project was started in 1997. Two different approaches to fixing a security hole in SRP-1 resulted in SRP-2 and SRP-3. SRP-3 was first published in 1998 in a conference. RFC 2945, which describes SRP-3 with SHA1, was published in 2000. SRP-6, which fixes "two-for-one" guessing and messaging ordering attacks, was published in 2002. SRP-6a appeared in the official "libsrp" in version 2.1.0, dated 2005. SRP-6a is found in standards as:
IEEE 1363.2 also includes a description of "SRP5", a variant replacing the discrete logarithm with an elliptic curve contributed by Yongge Wang in 2001. It also describes SRP-3 as found in RFC 2945. | [
{
"math_id": 0,
"text": "\\scriptstyle \\mathbb{Z}_N"
},
{
"math_id": 1,
"text": "\\scriptstyle \\mathbb{Z}_N^*"
}
]
| https://en.wikipedia.org/wiki?curid=1306670 |
13067081 | Terzaghi's principle | Theory of soil consolidation and effective stress
Terzaghi's Principle states that when stress is applied to a porous material, it is opposed by the fluid pressure filling the pores in the material.
Karl von Terzaghi introduced the idea in a series of papers in the 1920s based on his examination of building consolidation on soil. The principle states that all quantifiable changes in stress to a porous medium are a direct result of a change in effective stress. The "effective stress," formula_0, is related to "total stress," formula_1, and the "pore pressure," formula_2, by
formula_3,
where formula_4 is the identity matrix. The negative sign is there because the pore pressure serves to lessen the volume-changing stress; physically this is because there is fluid in the pores which bears a part of the total stress, so partially unloading the solid matrix from normal stresses.
Terzaghi's principle applies well to porous materials whose solid constituents are incompressible - soil, for example, is composed of grains of incompressible silica so that the volume change in soil during consolidation is due solely to the rearrangement of these constituents with respect to one another. Generalizing Terzaghi's principle to include compressible solid constituents was accomplished by Maurice Anthony Biot in the 1940s, giving birth to the theory of poroelasticity and poromechanics.
Validity.
Though the first 5 assumptions are either likely to hold, or deviation will have no discernible effect, experimental results contradict the final 3. Darcy's Law does not seem to hold at high hydraulic gradients, and both the coefficients of permeability and volume compressibility decrease during consolidation. This is due to the non-linearity of the relationship between void ratio and effective stress, although for small stress increments assumption 7 is reasonable. Finally, the relationship between void ratio and effective stress is not independent of time, again proven by experimental results.
Over the past century several formulations have been proposed for the effective stress according to several work hypotheses (e.g. compressibility of grains, their brittle or plastic behavior, high confining stress etc.). By way of example, at high pressures (e.g. in the Earth crust, at depth of some km, where the lithostatic load can reach values of several hundreds of MPa), Terzaghi’s formulation shows relevant deviation from experimental data and the formulation provided by Alec Skempton should be utilized, in order to achieve more accurate results. Substantially, the effective stress definition is conventional and related to the problem being treated. Among various effective stress formulations, Terzaghi's one seems particularly appropriate, for its simplicity and as it describes with excellent approximation a wide variety of real cases.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\boldsymbol{\\sigma}_{eff}"
},
{
"math_id": 1,
"text": "\\boldsymbol{\\sigma} "
},
{
"math_id": 2,
"text": "P"
},
{
"math_id": 3,
"text": "\\boldsymbol{\\sigma}_{eff} = \\boldsymbol{\\sigma} - P\\mathbb{I} "
},
{
"math_id": 4,
"text": "\\mathbb I "
}
]
| https://en.wikipedia.org/wiki?curid=13067081 |
13070117 | Spectral density estimation | Signal processing technique
In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.
Some SDE techniques assume that a signal is composed of a limited (usually small) number of generating frequencies plus noise and seek to find the location and intensity of the generated frequencies. Others make no assumption on the number of components and seek to estimate the whole generating spectrum.
Overview.
Spectrum analysis, also referred to as frequency domain analysis or spectral density estimation, is the technical process of decomposing a complex signal into simpler parts. As described above, many physical processes are best described as a sum of many individual frequency components. Any process that quantifies the various amounts (e.g. amplitudes, powers, intensities) versus frequency (or phase) can be called spectrum analysis.
Spectrum analysis can be performed on the entire signal. Alternatively, a signal can be broken into short segments (sometimes called "frames"), and spectrum analysis may be applied to these individual segments. Periodic functions (such as formula_0) are particularly well-suited for this sub-division. General mathematical techniques for analyzing non-periodic functions fall into the category of Fourier analysis.
The Fourier transform of a function produces a frequency spectrum which contains all of the information about the original signal, but in a different form. This means that the original function can be completely reconstructed ("synthesized") by an inverse Fourier transform. For perfect reconstruction, the spectrum analyzer must preserve both the amplitude and phase of each frequency component. These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude (amplitude) and phase in polar coordinates (i.e., as a phasor). A common technique in signal processing is to consider the squared amplitude, or power; in this case the resulting plot is referred to as a power spectrum.
Because of reversibility, the Fourier transform is called a "representation" of the function, in terms of frequency instead of time; thus, it is a frequency domain representation. Linear operations that could be performed in the time domain have counterparts that can often be performed more easily in the frequency domain. Frequency analysis also simplifies the understanding and interpretation of the effects of various time-domain operations, both linear and non-linear. For instance, only non-linear or time-variant operations can create new frequencies in the frequency spectrum.
In practice, nearly all software and electronic devices that generate frequency spectra utilize a discrete Fourier transform (DFT), which operates on samples of the signal, and which provides a mathematical approximation to the full integral solution. The DFT is almost invariably implemented by an efficient algorithm called "fast Fourier transform" (FFT). The array of squared-magnitude components of a DFT is a type of power spectrum called periodogram, which is widely used for examining the frequency characteristics of noise-free functions such as filter impulse responses and window functions. But the periodogram does not provide processing-gain when applied to noiselike signals or even sinusoids at low signal-to-noise ratios. In other words, the variance of its spectral estimate at a given frequency does not decrease as the number of samples used in the computation increases. This can be mitigated by averaging over time (Welch's method) or over frequency (smoothing). Welch's method is widely used for spectral density estimation (SDE). However, periodogram-based techniques introduce small biases that are unacceptable in some applications. So other alternatives are presented in the next section.
Techniques.
Many other techniques for spectral estimation have been developed to mitigate the disadvantages of the basic periodogram. These techniques can generally be divided into "non-parametric," "parametric," and more recently semi-parametric (also called sparse) methods. The non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure. Some of the most common estimators in use for basic applications (e.g. Welch's method) are non-parametric estimators closely related to the periodogram. By contrast, the parametric approaches assume that the underlying stationary stochastic process has a certain structure that can be described using a small number of parameters (for example, using an auto-regressive or moving-average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. When using the semi-parametric methods, the underlying process is modeled using a non-parametric framework, with the additional assumption that the number of non-zero components of the model is small (i.e., the model is sparse). Similar approaches may also be used for missing data recovery as well as signal reconstruction.
Following is a partial list of spectral density estimation techniques:
Parametric estimation.
In parametric spectral estimation, one assumes that the signal is modeled by a stationary process which has a spectral density function (SDF) formula_2 that is a function of the frequency formula_3 and formula_4 parameters formula_5. The estimation problem then becomes one of estimating these parameters.
The most common form of parametric SDF estimate uses as a model an autoregressive model formula_6 of order formula_4.392 A signal sequence formula_7 obeying a zero mean formula_6 process satisfies the equation
formula_8
where the formula_9 are fixed coefficients and formula_10 is a white noise process with zero mean and "innovation variance" formula_11. The SDF for this process is
formula_12
with formula_13 the sampling time interval and formula_14 the Nyquist frequency.
There are a number of approaches to estimating the parameters formula_15 of the formula_6 process and thus the spectral density:452-453
Alternative parametric methods include fitting to a moving-average model (MA) and to a full autoregressive moving-average model (ARMA).
Frequency estimation.
Frequency estimation is the process of estimating the frequency, amplitude, and phase-shift of a signal in the presence of noise given assumptions about the number of the components. This contrasts with the general methods above, which do not make prior assumptions about the components.
Single tone.
If one only wants to estimate the frequency of the single loudest pure-tone signal, one can use a pitch detection algorithm.
If the dominant frequency changes over time, then the problem becomes the estimation of the instantaneous frequency as defined in the time–frequency representation. Methods for instantaneous frequency estimation include those based on the Wigner–Ville distribution and higher order ambiguity functions.
If one wants to know "all" the (possibly complex) frequency components of a received signal (including transmitted signal and noise), one uses a multiple-tone approach.
Multiple tones.
A typical model for a signal formula_16 consists of a sum of formula_4 complex exponentials in the presence of white noise, formula_17
formula_18.
The power spectral density of formula_16 is composed of formula_4 impulse functions in addition to the spectral density function due to noise.
The most common methods for frequency estimation involve identifying the noise subspace to extract these components. These methods are based on eigen decomposition of the autocorrelation matrix into a signal subspace and a noise subspace. After these subspaces are identified, a frequency estimation function is used to find the component frequencies from the noise subspace. The most popular methods of noise subspace based frequency estimation are Pisarenko's method, the multiple signal classification (MUSIC) method, the eigenvector method, and the minimum norm method.
Example calculation.
Suppose formula_23, from formula_24 to formula_25 is a time series (discrete time) with zero mean. Suppose that it is a sum of a finite number of periodic components (all frequencies are positive):
formula_26
The variance of formula_23 is, for a zero-mean function as above, given by
formula_27
If these data were samples taken from an electrical signal, this would be its average power (power is energy per unit time, so it is analogous to variance if energy is analogous to the amplitude squared).
Now, for simplicity, suppose the signal extends infinitely in time, so we pass to the limit as formula_28 If the average power is bounded, which is almost always the case in reality, then the following limit exists and is the variance of the data.
formula_29
Again, for simplicity, we will pass to continuous time, and assume that the signal extends infinitely in time in both directions. Then these two formulas become
formula_30
and
formula_31
The root mean square of formula_32 is formula_33, so the variance of formula_34 is formula_35 Hence, the contribution to the average power of formula_36 coming from the component with frequency formula_37 is formula_38 All these contributions add up to the average power of formula_39
Then the power as a function of frequency is formula_40 and its statistical cumulative distribution function formula_41 will be
formula_42
formula_43 is a step function, monotonically non-decreasing. Its jumps occur at the frequencies of the periodic components of formula_44, and the value of each jump is the power or variance of that component.
The variance is the covariance of the data with itself. If we now consider the same data but with a lag of formula_45, we can take the covariance of formula_36 with formula_46, and define this to be the autocorrelation function formula_47 of the signal (or data) formula_44:
formula_48
If it exists, it is an even function of formula_49 If the average power is bounded, then formula_47 exists everywhere, is finite, and is bounded by formula_50 which is the average power or variance of the data.
It can be shown that formula_47 can be decomposed into periodic components with the same periods as formula_44:
formula_51
This is in fact the spectral decomposition of formula_47 over the different frequencies, and is related to the distribution of power of formula_44 over the frequencies: the amplitude of a frequency component of formula_47 is its contribution to the average power of the signal.
The power spectrum of this example is not continuous, and therefore does not have a derivative, and therefore this signal does not have a power spectral density function. In general, the power spectrum will usually be the sum of two parts: a line spectrum such as in this example, which is not continuous and does not have a density function, and a residue, which is absolutely continuous and does have a density function.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\sin (t)"
},
{
"math_id": 1,
"text": "(r,q)"
},
{
"math_id": 2,
"text": "S(f; a_1, \\ldots, a_p)"
},
{
"math_id": 3,
"text": "f"
},
{
"math_id": 4,
"text": "p"
},
{
"math_id": 5,
"text": "a_1, \\ldots, a_p"
},
{
"math_id": 6,
"text": "\\text{AR}(p)"
},
{
"math_id": 7,
"text": "\\{Y_t\\}"
},
{
"math_id": 8,
"text": "Y_t = \\phi_1Y_{t-1} + \\phi_2Y_{t-2} + \\cdots + \\phi_pY_{t-p} + \\epsilon_t,"
},
{
"math_id": 9,
"text": "\\phi_1,\\ldots,\\phi_p"
},
{
"math_id": 10,
"text": "\\epsilon_t"
},
{
"math_id": 11,
"text": "\\sigma^2_p"
},
{
"math_id": 12,
"text": "\n S(f; \\phi_1, \\ldots, \\phi_p, \\sigma^2_p) =\n \\frac{\\sigma^2_p\\Delta t}{\\left| 1 - \\sum_{k=1}^p \\phi_k e^{-2i\\pi f k \\Delta t}\\right|^2} \\qquad |f| < f_N,\n"
},
{
"math_id": 13,
"text": "\\Delta t"
},
{
"math_id": 14,
"text": "f_N"
},
{
"math_id": 15,
"text": "\\phi_1, \\ldots, \\phi_p,\\sigma^2_p"
},
{
"math_id": 16,
"text": "x(n)"
},
{
"math_id": 17,
"text": "w(n)"
},
{
"math_id": 18,
"text": "x(n) = \\sum_{i=1}^p A_i e^{j n \\omega_i} + w(n)"
},
{
"math_id": 19,
"text": "\\hat{P}_\\text{PHD}\\left(e^{j \\omega}\\right) = \\frac{1}{\\left|\\mathbf{e}^H \\mathbf{v}_\\text{min}\\right|^2}"
},
{
"math_id": 20,
"text": "\\hat{P}_\\text{MU}\\left(e^{j \\omega}\\right) = \\frac{1}{\\sum_{i=p+1}^M \\left|\\mathbf{e}^H \\mathbf{v}_i\\right|^2}"
},
{
"math_id": 21,
"text": "\\hat{P}_\\text{EV}\\left(e^{j \\omega}\\right) = \\frac{1}{\\sum_{i=p+1}^M \\frac{1}{\\lambda_i} \\left|\\mathbf{e}^H \\mathbf{v}_i\\right|^2}"
},
{
"math_id": 22,
"text": "\\hat{P}_\\text{MN}\\left(e^{j \\omega}\\right) = \\frac{1}{\\left|\\mathbf{e}^H \\mathbf{a}\\right|^2} ; \\ \\mathbf{a} = \\lambda \\mathbf{P}_n \\mathbf{u}_1"
},
{
"math_id": 23,
"text": "x_n"
},
{
"math_id": 24,
"text": "n=0"
},
{
"math_id": 25,
"text": "N-1"
},
{
"math_id": 26,
"text": "\\begin{align}\nx_n &= \\sum_k A_k \\sin(2\\pi\\nu_k n + \\phi_k)\\\\\n &= \\sum_k A_k \\left ( \\sin (\\phi_k) \\cos(2\\pi\\nu_k n) + \\cos(\\phi_k) \\sin(2\\pi\\nu_k n) \\right ) \\\\\n &= \\sum_k \\left(\\overbrace{a_k}^{A_k \\sin(\\phi_k)} \\cos(2\\pi\\nu_k n) + \\overbrace{b_k}^{A_k \\cos(\\phi_k)} \\sin(2\\pi\\nu_k n)\\right)\n\\end{align}"
},
{
"math_id": 27,
"text": "\\frac{1}{N} \\sum_{n=0}^{N-1} x_n^2."
},
{
"math_id": 28,
"text": "N\\to \\infty."
},
{
"math_id": 29,
"text": "\\lim_{N \\to \\infty} \\frac{1}{N} \\sum_{n=0}^{N-1} x_n^2."
},
{
"math_id": 30,
"text": "x(t) = \\sum_k A_k \\sin(2\\pi\\nu_k t + \\phi_k)"
},
{
"math_id": 31,
"text": "\\lim_{T\\to\\infty} \\frac{1}{2T} \\int_{-T}^T x(t)^2 dt."
},
{
"math_id": 32,
"text": "\\sin"
},
{
"math_id": 33,
"text": "1/\\sqrt{2}"
},
{
"math_id": 34,
"text": "A_k \\sin(2\\pi\\nu_k t + \\phi_k)"
},
{
"math_id": 35,
"text": "\\tfrac{1}{2} A_k^2."
},
{
"math_id": 36,
"text": "x(t)"
},
{
"math_id": 37,
"text": "\\nu_k"
},
{
"math_id": 38,
"text": "\\tfrac{1}{2}A_k^2."
},
{
"math_id": 39,
"text": "x(t)."
},
{
"math_id": 40,
"text": "\\tfrac{1}{2}A_k^2,"
},
{
"math_id": 41,
"text": "S(\\nu)"
},
{
"math_id": 42,
"text": "S(\\nu) = \\sum _ {k : \\nu_k < \\nu} \\frac{1}{2} A_k^2."
},
{
"math_id": 43,
"text": "S"
},
{
"math_id": 44,
"text": "x"
},
{
"math_id": 45,
"text": "\\tau"
},
{
"math_id": 46,
"text": "x(t + \\tau)"
},
{
"math_id": 47,
"text": "c"
},
{
"math_id": 48,
"text": "c(\\tau) = \\lim_{T\\to\\infty} \\frac{1}{2T} \\int_{-T}^T x(t) x(t + \\tau) dt."
},
{
"math_id": 49,
"text": "\\tau."
},
{
"math_id": 50,
"text": "c(0),"
},
{
"math_id": 51,
"text": "c(\\tau) = \\sum_k \\frac{1}{2} A_k^2 \\cos(2\\pi\\nu_k\\tau)."
}
]
| https://en.wikipedia.org/wiki?curid=13070117 |
13070361 | Fixed end moment | The fixed end moments are reaction moments developed in a beam member under certain load conditions with both ends fixed. A beam with both ends fixed is statically indeterminate to the 3rd degree, and any structural analysis method applicable on statically indeterminate beams can be used to calculate the fixed end moments.
Examples.
In the following examples, clockwise moments are positive.
The two cases with distributed loads can be derived from the case with concentrated load by integration. For example, when a uniformly distributed load of intensity formula_0 is acting on a beam, then an infinitely small part formula_1 distance formula_2 apart from the left end of this beam can be seen as being under a concentrated load of magnitude formula_3. Then,
formula_4
formula_5
Where the expressions within the integrals on the right hand sides are the fixed end moments caused by the concentrated load formula_3.
For the case with linearly distributed load of maximum intensity formula_6,
formula_7
formula_8 | [
{
"math_id": 0,
"text": "q"
},
{
"math_id": 1,
"text": "dx"
},
{
"math_id": 2,
"text": "x"
},
{
"math_id": 3,
"text": "qdx"
},
{
"math_id": 4,
"text": "M_{\\mathrm{right}}^{\\mathrm{fixed}} = \\int_{0}^{L} \\frac{q dx \\, x^2 (L-x)}{L^2} = \\frac{q L^2}{12} "
},
{
"math_id": 5,
"text": "M_{\\mathrm{left}}^{\\mathrm{fixed}} = \\int_{0}^{L} \\left \\{ - \\frac{q dx \\, x^2 (L-x) }{L^2} \\right \\}= - \\frac{q L^2}{12} "
},
{
"math_id": 6,
"text": "q_0"
},
{
"math_id": 7,
"text": "M_{\\mathrm{right}}^{\\mathrm{fixed}} = \\int_{0}^{L} q_0 \\frac{x}{L} dx \\frac{ x^2 (L-x)}{L^2} = \\frac{q_0 L^2}{20}"
},
{
"math_id": 8,
"text": "M_{\\mathrm{left}}^{\\mathrm{fixed}} = \\int_{0}^{L} \\left \\{ - q_0 \\frac{x}{L} dx \\frac{x (L-x)^2}{L^2} \\right \\} = - \\frac{q_0 L^2}{30}"
}
]
| https://en.wikipedia.org/wiki?curid=13070361 |
13075367 | Householder's method | Class of mathematical root-finding algorithm
In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order "d" + 1. Each of these methods is characterized by the number d, which is known as the "order" of the method. The algorithm is iterative and has a rate of convergence of "d" + 1.
These methods are named after the American mathematician Alston Scott Householder.
Method.
Householder's method is a numerical algorithm for solving the equation "f"("x") = 0. In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations
formula_0
beginning with an initial guess "x"0.
If f is a "d" + 1 times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood of a, the iterates "x""n" satisfy:
formula_1, for some formula_2
This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence has order "d" + 1 or better. Furthermore, when close enough to a, it commonly is the case that formula_3 for some formula_4. In particular,
Despite their order of convergence, these methods are not widely used because the gain in precision is not commensurate with the rise in effort for large d. The "Ostrowski index" expresses the error reduction in the number of function evaluations instead of the iteration count.
Motivation.
First approach.
Suppose f is analytic in a neighborhood of a and "f"("a") = 0. Then f has a Taylor series at a and its constant term is zero. Because this constant term is zero, the function "f"("x") / ("x" − "a") will have a Taylor series at a and, when "f ′" ("a") ≠ 0, its constant term will not be zero. Because that constant term is not zero, it follows that the reciprocal ("x" − "a") / "f"("x") has a Taylor series at a, which we will write as formula_9 and its constant term "c"0 will not be zero. Using that Taylor series we can write
formula_10
When we compute its d-th derivative, we note that the terms for "k" = 1, ..., "d" conveniently vanish:
formula_11
formula_12
formula_13
using big O notation. We thus get that the ratio
formula_14
formula_15
If a is the zero of f that is closest to x then the second factor goes to 1 as d goes to infinity and
formula_16 goes to a.
Second approach.
Suppose "x" = "a" is a simple root. Then near "x" = "a", (1/"f")("x") is a meromorphic function. Suppose we have the Taylor expansion:
formula_17
around a point b that is closer to a than it is to any other zero of f. By König's theorem, we have:
formula_18
These suggest that Householder's iteration might be a good convergent iteration. The actual proof of the convergence is also based on these ideas.
The methods of lower order.
Householder's method of order 1 is just Newton's method, since:
formula_19
For Householder's method of order 2 one gets Halley's method, since the identities
formula_20
and
formula_21
result in
formula_22
In the last line, formula_23 is the update of the Newton iteration at the point formula_24. This line was added to demonstrate where the difference to the simple Newton's method lies.
The third order method is obtained from the identity of the third order derivative of 1/"f"
formula_25
and has the formula
formula_26
and so on.
Example.
The first problem solved by Newton with the Newton-Raphson-Simpson method was the polynomial equation formula_27. He observed that there should be a solution close to 2. Replacing "y" = "x" + 2 transforms the equation into
formula_28.
The Taylor series of the reciprocal function starts with
formula_29
The result of applying Householder's methods of various orders at "x" = 0 is also obtained by dividing neighboring coefficients of the latter power series. For the first orders one gets the following values after just one iteration step: For an example, in the case of the 3rd order,
formula_30.
As one can see, there are a little bit more than d correct decimal places for each order d. The first one hundred digits of the correct solution are .
Let's calculate the formula_31 values for some lowest order,
formula_32
formula_33
formula_34
formula_35
And using following relations,
1st order; formula_36
2nd order; formula_37
3rd order; formula_38
Derivation.
An exact derivation of Householder's methods starts from the Padé approximation of order "d" + 1 of the function, where the approximant with linear numerator is chosen. Once this has been achieved, the update for the next approximation results from computing the unique zero of the numerator.
The Padé approximation has the form
formula_39
The rational function has a zero at formula_40.
Just as the Taylor polynomial of degree d has "d" + 1 coefficients that depend on the function f, the Padé approximation also has "d" + 1 coefficients dependent on f and its derivatives. More precisely, in any Padé approximant, the degrees of the numerator and denominator polynomials have to add to the order of the approximant. Therefore, formula_41 has to hold.
One could determine the Padé approximant starting from the Taylor polynomial of f using Euclid's algorithm. However, starting from the Taylor polynomial of 1/"f" is shorter and leads directly to the given formula. Since
formula_42
has to be equal to the inverse of the desired rational function, we get after multiplying with formula_43 in the power formula_44 the equation
formula_45.
Now, solving the last equation for the zero formula_40 of the numerator results in
formula_46.
This implies the iteration formula
formula_47.
Relation to Newton's method.
Householder's method applied to the real-valued function "f"("x") is the same as Newton's method applied to the function "g"("x"):
formula_48
with
formula_49
In particular, "d" = 1 gives Newton's method unmodified and "d" = 2 gives Halley's method.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "x_{n+1} = x_n + d\\; \\frac {\\left(1/f\\right)^{(d-1)}(x_n)} {\\left(1/f\\right)^{(d)}(x_n)} "
},
{
"math_id": 1,
"text": "| x_{n+1} - a | \\le K \\cdot {| x_n - a |}^{d+1} "
},
{
"math_id": 2,
"text": "K > 0.\\!"
},
{
"math_id": 3,
"text": "x_{n+1} - a \\approx C (x_n - a)^{d+1}"
},
{
"math_id": 4,
"text": "C \\ne 0"
},
{
"math_id": 5,
"text": "\\sqrt[d+1]{d+1}"
},
{
"math_id": 6,
"text": "\\sqrt[\\frac{(d+1)(d+2)}{2}]{d+1}"
},
{
"math_id": 7,
"text": "\\sqrt[3]{2}\\approx 1.2599"
},
{
"math_id": 8,
"text": "\\sqrt[6]{3}\\approx 1.2009"
},
{
"math_id": 9,
"text": "\\sum_{k=0}^{+\\infty} \\frac{c_k (x-a)^k}{k!}"
},
{
"math_id": 10,
"text": "\\frac{1}{f} = \\frac{c_{0}}{x-a} + \\sum_{k=1}^{+\\infty} \\frac{c_{k} (x-a)^{k-1}}{k~(k-1)!}\\,."
},
{
"math_id": 11,
"text": "\\left(\\frac{1}{f}\\right)^{(d)}\n= \\frac{(-1)^d d!~c_{0}}{(x-a)^{d+1}} + \\sum_{k=d+1}^{+\\infty} \\frac{c_{k} (x-a)^{k-d-1}}{k~(k-d-1)!}"
},
{
"math_id": 12,
"text": "= \\frac{(-1)^d d!~c_{0}}{(x-a)^{d+1}} \\left(1 + \\frac{1}{(-1)^d d!~c_{0}}\\sum_{k=d+1}^{+\\infty} \\frac{c_{k} (x-a)^{k}}{k~(k-d-1)!}\\right)"
},
{
"math_id": 13,
"text": "= \\frac{(-1)^d d!~c_{0}}{(x-a)^{d+1}} \\left(1 + \\mathcal{O}\\left((x-a)^{d+1}\\right)\\right)\n\\,,"
},
{
"math_id": 14,
"text": "d~\\frac{(1/f)^{(d-1)}}{(1/f)^{(d)}}\n= d~\\frac{(-1)^{d-1} (d-1)!~c_{0}}{(-1)^d d!~c_{0}} (x-a) \\left( \\frac{1 + \\mathcal{O}\\left((x-a)^{d}\\right)}{1 + \\mathcal{O}\\left((x-a)^{d+1}\\right)} \\right)"
},
{
"math_id": 15,
"text": "= -(x-a) \\left( 1 + \\mathcal{O}\\left((x-a)^{d}\\right) \\right)\\,."
},
{
"math_id": 16,
"text": " x + d~\\frac{(1/f)^{(d-1)}}{(1/f)^{(d)}}"
},
{
"math_id": 17,
"text": "\n(1/f)(x) = \\sum_{d=0}^{\\infty} \\frac{(1/f)^{(d)}(b)}{d!} (x-b)^d\n"
},
{
"math_id": 18,
"text": "\na-b = \\lim_{d\\rightarrow \\infty} \\frac{\\frac{(1/f)^{(d-1)}(b)}{(d-1)!}}{\\frac{(1/f)^{(d)}(b)}{d!}}=d\\frac{(1/f)^{(d-1)}(b)}{(1/f)^{(d)}(b)}.\n"
},
{
"math_id": 19,
"text": "\\begin{array}{rl}\nx_{n+1} =& x_n + 1\\,\\frac {\\left(1/f\\right)(x_n)} {\\left(1/f\\right)^{(1)}(x_n)}\\\\[.7em]\n=& x_n + \\frac{1}{f(x_n)}\\cdot\\left(\\frac{-f'(x_n)}{f(x_n)^2}\\right)^{-1}\\\\[.7em]\n=& x_n - \\frac{f(x_n)}{f'(x_n)}.\n\\end{array}\n"
},
{
"math_id": 20,
"text": "\\textstyle \n (1/f)'(x)=-\\frac{f'(x)}{f(x)^2}\\ \n"
},
{
"math_id": 21,
"text": "\\textstyle\\ \n (1/f)''(x)=-\\frac{f''(x)}{f(x)^2}+2\\frac{f'(x)^2}{f(x)^3}"
},
{
"math_id": 22,
"text": "\\begin{array}{rl}\nx_{n+1} =& x_n + 2\\,\\frac {\\left(1/f\\right)'(x_n)} {\\left(1/f\\right)''(x_n)}\\\\[1em]\n=& x_n + \\frac{-2f(x_n)\\,f'(x_n)}{-f(x_n)f''(x_n)+2f'(x_n)^2}\\\\[1em]\n=& x_n - \\frac{f(x_n)f'(x_n)}{f'(x_n)^2-\\tfrac12f(x_n)f''(x_n)}\\\\[1em]\n=& x_n + h_n\\;\\frac{1}{1+\\frac12(f''/f')(x_n)\\,h_n}.\n\\end{array}\n"
},
{
"math_id": 23,
"text": "h_n=-\\tfrac{f(x_n)}{f'(x_n)}"
},
{
"math_id": 24,
"text": "x_n"
},
{
"math_id": 25,
"text": "\\textstyle\n (1/f)'''(x)=-\\frac{f'''(x)}{f(x)^2}+6\\frac{f'(x)\\,f''(x)}{f(x)^3}-6\\frac{f'(x)^3}{f(x)^4}\n"
},
{
"math_id": 26,
"text": "\\begin{array}{rl}\nx_{n+1} =& x_n + 3\\,\\frac {\\left(1/f\\right)''(x_n)} {\\left(1/f\\right)'''(x_n)}\\\\[1em]\n=& x_n - \\frac{6f(x_n)\\,f'(x_n)^2-3f(x_n)^2f''(x_n)}{6f'(x_n)^3-6f(x_n)f'(x_n)\\,f''(x_n)+f(x_n)^2\\,f'''(x_n)}\\\\[1em]\n=& x_n + h_n\\frac{1+\\frac12(f''/f')(x_n)\\,h_n}{1+(f''/f')(x_n)\\,h_n+\\frac16(f'''/f')(x_n)\\,h_n^2}\n\\end{array}\n"
},
{
"math_id": 27,
"text": "y^3-2y-5=0"
},
{
"math_id": 28,
"text": "0=f(x)=-1 + 10 x + 6 x^2 + x^3"
},
{
"math_id": 29,
"text": "\\begin{array}{rl}\n1/f(x)=& - 1 - 10\\,x - 106 \\,x^2 - 1121 \\,x^3 - 11856 \\,x^4 - 125392 \\,x^5\\\\\n & - 1326177 \\,x^6 - 14025978 \\,x^7 - 148342234 \\,x^8 - 1568904385 \\,x^9\\\\\n & - 16593123232 \\,x^{10} +O(x^{11})\n\\end{array}"
},
{
"math_id": 30,
"text": " x_1 = 0.0 + 106/1121 = 0.09455842997324"
},
{
"math_id": 31,
"text": "x_2, x_3, x_4"
},
{
"math_id": 32,
"text": " f = -1 + 10x + 6x^2 + x^3 "
},
{
"math_id": 33,
"text": " f^\\prime = 10 + 12x + 3x^2 "
},
{
"math_id": 34,
"text": " f^{\\prime\\prime} = 12 + 6x "
},
{
"math_id": 35,
"text": " f^{\\prime\\prime\\prime} = 6 "
},
{
"math_id": 36,
"text": " x_{i+1} = x_{i} -f(x_i)/f^{\\prime}(x_i) "
},
{
"math_id": 37,
"text": " x_{i+1} = x_{i} - 2ff^{\\prime} / (2{f^{\\prime}}^2 - ff^{\\prime\\prime}) "
},
{
"math_id": 38,
"text": " x_{i+1} = x_{i} - (6f {f^{\\prime}}^2 - 3f^2 f^{\\prime\\prime}) /\n (6{f^{\\prime}}^3 -6 f f^{\\prime}f^{\\prime\\prime} + f^2f^{\\prime\\prime\\prime}) "
},
{
"math_id": 39,
"text": "f(x+h)=\\frac{a_0+h}{b_0+b_1h+\\cdots+b_{d-1}h^{d-1}}+O(h^{d+1})."
},
{
"math_id": 40,
"text": "h=-a_0"
},
{
"math_id": 41,
"text": "b_d=0"
},
{
"math_id": 42,
"text": "\n (1/f)(x+h) = \n (1/f)(x)+(1/f)'(x)h+\\cdots+(1/f)^{(d-1)}(x)\\frac{h^{d-1}}{(d-1)!}+(1/f)^{(d)}(x)\\frac{h^d}{d!}+O(h^{d+1})\n"
},
{
"math_id": 43,
"text": "a_0+h"
},
{
"math_id": 44,
"text": "h^d"
},
{
"math_id": 45,
"text": "0=b_d=a_0(1/f)^{(d)}(x)\\frac1{d!}+(1/f)^{(d-1)}(x)\\frac1{(d-1)!}"
},
{
"math_id": 46,
"text": "\\begin{align}\n h&=-a_0=\n \\frac{\\frac1{(d-1)!}(1/f)^{(d-1)}(x)}{\\frac1{d!}(1/f)^{(d)}(x)}\\\\\n &=d\\,\\frac{(1/f)^{(d-1)}(x)}{(1/f)^{(d)}(x)}\n\\end{align}"
},
{
"math_id": 47,
"text": "x_{n+1} = x_n + d\\; \\frac { \\left(1/f\\right)^{(d-1)} (x_n) } { \\left(1/f\\right)^{(d)} (x_n) } "
},
{
"math_id": 48,
"text": "x_{n+1} = x_n - \\frac{g(x_n)}{g'(x_n)}"
},
{
"math_id": 49,
"text": "g(x) = \\left|(1/f)^{(d-1)}\\right|^{-1/d}\\,."
}
]
| https://en.wikipedia.org/wiki?curid=13075367 |
1307696 | Working capital | Financial metric
Working capital (WC) is a financial metric which represents operating liquidity available to a business, organisation, or other entity, including governmental entities. Along with fixed assets such as plant and equipment, working capital is considered a part of operating capital. Gross working capital is equal to current assets. Working capital is calculated as current assets minus current liabilities. If current assets are less than current liabilities, an entity has a working capital deficiency, also called a working capital deficit and negative working capital.
A company can be endowed with assets and profitability but may fall short of liquidity if its assets cannot be readily converted into cash. Positive working capital is required to ensure that a firm is able to continue its operations and that it has sufficient funds to satisfy both maturing short-term debt and upcoming operational expenses. The management of working capital involves managing inventories, accounts receivable and payable, and cash.
Calculation.
Working capital is the difference between current assets and current liabilities. It is not to be confused with trade working capital (the latter excludes cash).
The basic calculation of working capital is based on the entity's gross current assets.
formula_0
Inputs.
Current assets and current liabilities include four accounts which are of special importance. These accounts represent the areas of the business where managers have the most direct impact:
The current portion of debt (payable within 12 months) is critical because it represents a short-term claim to current assets and is often secured by long-term assets. Common types of short-term debt are bank loans and lines of credit.
An increase in net working capital indicates that the business has either increased current assets (that it has increased its receivables or other current assets) or has decreased current liabilities—for example has paid off some short-term creditors, or a combination of both.
Working capital cycle.
Definition.
The working capital cycle (WCC), also known as the cash conversion cycle, is the amount of time it takes to turn the net current assets and current liabilities into cash. The longer this cycle, the longer a business is tying up capital in its working capital without earning a return on it. Companies strive to reduce their working capital cycle by collecting receivables quicker or sometimes stretching accounts payable. Under certain conditions, minimizing working capital might adversely affect the company's ability to realize profitability, e.g. when unforeseen hikes in demand exceed inventories, or when a shortfall in cash restricts the company's ability to acquire trade or production inputs.
Meaning.
A positive working capital cycle balances incoming and outgoing payments to minimize net working capital and maximize free cash flow. For example, a company that pays its financing is a carrying cost tinexpensive way to grow. Sophisticated buyers review closely a target's working capital cycle because it provides them with an idea of the management's effectiveness at managing their balance sheet and generating free cash flows.
As an absolute rule of funders, each of them wants to see a positive working capital because positive working capital implies there are sufficient current assets to meet current obligations. In contrast, companies risk being unable to meet current obligations with current assets when working capital is negative. While it is theoretically possible for a company to indefinitely show negative working capital on regularly reported balance sheets (since working capital may actually be positive between reporting periods), working capital will generally need to be non-negative for the business to be sustainable
Reasons why a business may show negative or low working capital over the long term while not indicating financial distress include:
Working capital management.
Decisions relating to working capital and short-term financing are referred to as "working capital management". These involve managing the relationship between a firm's short-term assets and its short-term liabilities. The goal of working capital management is to ensure that the firm is able to continue its operations and that it has sufficient cash flow to satisfy both maturing short-term debt and upcoming operational expenses.
A managerial accounting strategy focusing on maintaining efficient levels of both components of working capital, current assets, and current liabilities, in respect to each other. Working capital management ensures a company has sufficient cash flow in order to meet its short-term debt obligations and operating expenses.
Decision criteria.
By definition, working capital management entails short-term decisions—generally, relating to the next one-year period—which are "reversible". These decisions are therefore not taken on the same basis as capital-investment decisions (NPV or related, as above); rather, they will be based on cash flows, or profitability, or both.
Management of working capital.
Guided by the above criteria, management will use a combination of policies and techniques for the management of working capital. The policies aim at managing the "current assets" (generally cash and cash equivalents, inventories and debtors) and the short-term financing, such that cash flows and returns are acceptable.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " \\text{Working capital} = \\text{Current assets} - \\text{Current liabilities} "
}
]
| https://en.wikipedia.org/wiki?curid=1307696 |
1307911 | Bootstrap aggregating | Ensemble method within machine learning
<templatestyles src="Machine learning/styles.css"/>
Bootstrap aggregating, also called bagging (from bootstrap aggregating), is a machine learning ensemble meta-algorithm designed to improve the stability and accuracy of machine learning algorithms used in statistical classification and regression. It also reduces variance and helps to avoid overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the model averaging approach.
Description of the technique.
Given a standard training set formula_0 of size "n", bagging generates "m" new training sets formula_1, each of size "n′", by sampling from "D" uniformly and with replacement. By sampling with replacement, some observations may be repeated in each formula_1. If "n′"="n", then for large "n" the set formula_1 is expected to have the fraction (1 - 1/"e") (≈63.2%) of the unique examples of "D", the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, "m" models are fitted using the above "m" bootstrap samples and combined by averaging the output (for regression) or voting (for classification).
Bagging leads to "improvements for unstable procedures", which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression. Bagging was shown to improve preimage learning. On the other hand, it can mildly degrade the performance of stable methods such as K-nearest neighbors.
Process of the algorithm.
Key Terms.
There are three types of datasets in bootstrap aggregating. These are the original, bootstrap, and out-of-bag datasets. Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given.
Creating the bootstrap dataset.
The bootstrap dataset is made by randomly picking objects from the original dataset. Also, it must be the same size as the original dataset. However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below:
Suppose the original dataset is a group of 12 people. Their names are Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal.
By randomly picking a group of names, let us say our bootstrap dataset had James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore. In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore.
Creating the out-of-bag dataset.
The out-of-bag dataset represents the remaining people who were not in the bootstrap dataset. It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are Emily, Jessie, George, Rachel, and Jamal. Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done:
Importance.
Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of a random forest algorithm. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail.
Creation of Decision Trees.
The next step of the algorithm involves the generation of decision trees from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a confusion matrix, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various classification metrics based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and information gain. These features are then used to partition the samples into two sets: those who possess the top feature, and those who do not.
The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes".
This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data.
Random Forests.
The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive.
Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to overfit their data, and run quickly and efficiently even for large datasets. They are primarily useful for classification as opposed to regression, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and e-commerce where it is important to be able to predict future results based on past data. One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example.
There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability. However, they still have numerous advantages over similar data classification algorithms such as neural networks, as they are much easier to interpret and generally require less data for training. As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below.
Improving Random Forests and Bagging.
While the techniques described above utilize random forests and bagging (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest:
Algorithm (classification).
For classification, use a training set formula_0, Inducer formula_2 and the number of bootstrap samples formula_3 as input. Generate a classifier formula_4 as output
Example: ozone data.
To illustrate the basic principles of bagging, below is an analysis on the relationship between ozone and temperature (data from Rousseeuw and Leroy (1986), analysis done in R).
The relationship between temperature and ozone appears to be nonlinear in this data set, based on the scatter plot. To mathematically describe this relationship, LOESS smoothers (with bandwidth 0.5) are used. Rather than building a single smoother for the complete data set, 100 bootstrap samples were drawn. Each sample is composed of a random subset of the original data and maintains a semblance of the master set's distribution and variability. For each bootstrap sample, a LOESS smoother was fit. Predictions from these 100 smoothers were then made across the range of the data. The black lines represent these initial predictions. The lines lack agreement in their predictions and tend to overfit their data points: evident by the wobbly flow of the lines.
By taking the average of 100 smoothers, each corresponding to a subset of the original data set, we arrive at one bagged predictor (red line). The red line's flow is stable and does not overly conform to any data point(s).
Advantages and disadvantages.
Advantages:
Disadvantages:
History.
The concept of bootstrap aggregating is derived from the concept of bootstrapping which was developed by Bradley Efron.
Bootstrap aggregating was proposed by Leo Breiman who also coined the abbreviated term "bagging" (bootstrap aggregating). Breiman developed the concept of bagging in 1994 to improve classification by combining classifications of randomly generated training sets. He argued, "If perturbing the learning set can cause significant changes in the predictor constructed, then bagging can improve accuracy".
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "D"
},
{
"math_id": 1,
"text": "D_i"
},
{
"math_id": 2,
"text": "I"
},
{
"math_id": 3,
"text": "m"
},
{
"math_id": 4,
"text": "C^*"
},
{
"math_id": 5,
"text": "C_i"
}
]
| https://en.wikipedia.org/wiki?curid=1307911 |
13079354 | Drazin inverse | In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix.
Let "A" be a square matrix. The index of "A" is the least nonnegative integer "k" such that rank("A""k"+1) = rank("A""k"). The Drazin inverse of "A" is the unique matrix "A"D that satisfies
formula_0
It's not a generalized inverse in the classical sense, since formula_1 in general.
formula_4
where formula_5 is invertible with inverse formula_6 and formula_7 is a nilpotent matrix, then
formula_8
The hyper-power sequence is
formula_14 for convergence notice that formula_15
For formula_16 or any regular formula_17 with formula_18 chosen such that formula_19 the sequence tends to its Drazin inverse,
formula_20
Jordan normal form and Jordan-Chevalley decomposition.
As the definition of the Drazin inverse is invariant under matrix conjugations, writing formula_21, where J is in Jordan normal form, implies that formula_22. The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.
More generally, we may define the Drazin inverse over any perfect field, by using the Jordan-Chevalley decomposition formula_23 where formula_24 is semisimple and formula_25 is nilpotent and both operators commute. The two terms can be block diagonalized with blocks corresponding to the kernel and cokernel of formula_24. The Drazin inverse in the same basis is then defined to be zero on the kernel of formula_24, and equal to the inverse of formula_10 on the cokernel of formula_24. | [
{
"math_id": 0,
"text": "A^{k+1}A^\\text{D} = A^k,\\quad A^\\text{D}AA^\\text{D} = A^\\text{D},\\quad AA^\\text{D} = A^\\text{D}A."
},
{
"math_id": 1,
"text": "A A^\\text{D} A \\neq A"
},
{
"math_id": 2,
"text": "A^{-1}"
},
{
"math_id": 3,
"text": "A^\\text{D} = A^{-1}"
},
{
"math_id": 4,
"text": "A = \\begin{bmatrix} \n B & 0 \\\\\n 0 & N\n\\end{bmatrix}"
},
{
"math_id": 5,
"text": "B"
},
{
"math_id": 6,
"text": "B^{-1}"
},
{
"math_id": 7,
"text": "N"
},
{
"math_id": 8,
"text": "A^D = \\begin{bmatrix} \n B^{-1} & 0 \\\\\n 0 & 0\n\\end{bmatrix}"
},
{
"math_id": 9,
"text": "A^\\text{D}"
},
{
"math_id": 10,
"text": "A"
},
{
"math_id": 11,
"text": "P A^\\text{D} P^{-1}"
},
{
"math_id": 12,
"text": "PAP^{-1}"
},
{
"math_id": 13,
"text": "A^\\text{D} = 0."
},
{
"math_id": 14,
"text": "A_{i+1} := A_i + A_i\\left(I - A A_i\\right);"
},
{
"math_id": 15,
"text": "A_{i+j} = A_i \\sum_{k=0}^{2^j-1} \\left(I - A A_i\\right)^k."
},
{
"math_id": 16,
"text": "A_0 := \\alpha A"
},
{
"math_id": 17,
"text": "A_0"
},
{
"math_id": 18,
"text": "A_0 A = A A_0"
},
{
"math_id": 19,
"text": "\\left\\|A_0 - A_0 A A_0\\right\\| < \\left\\|A_0\\right\\| "
},
{
"math_id": 20,
"text": "A_i \\rightarrow A^\\text{D}."
},
{
"math_id": 21,
"text": "A = P J P^{-1}"
},
{
"math_id": 22,
"text": "A^\\text{D} = P J^\\text{D} P^{-1} "
},
{
"math_id": 23,
"text": "A = A_s + A_n "
},
{
"math_id": 24,
"text": "A_s"
},
{
"math_id": 25,
"text": "A_n "
}
]
| https://en.wikipedia.org/wiki?curid=13079354 |
1308185 | Mixing (mathematics) | Mathematical description of mixing substances
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: e.g. mixing paint, mixing drinks, industrial mixing.
The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including "strong mixing", "weak mixing" and "topological mixing", with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" condition than ergodicity).
Informal explanation.
The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, and so on. To provide the mathematical rigor, such descriptions begin with the definition of a measure-preserving dynamical system, written as &NoBreak;&NoBreak;.
The set formula_0 is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, "etc." The measure formula_1 is understood to define the natural volume of the space formula_0 and of its subspaces. The collection of subspaces is denoted by &NoBreak;}&NoBreak;, and the size of any given subset formula_2 is &NoBreak;&NoBreak;; the size is its volume. Naively, one could imagine formula_3 to be the power set of &NoBreak;&NoBreak;; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). Thus, conventionally, formula_3 consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections, unions and set complements; these can always be taken to be measurable.
The time evolution of the system is described by a map formula_4. Given some subset formula_2, its map formula_5 will in general be a deformed version of formula_6 – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the baker's map and the horseshoe map, both inspired by bread-making. The set formula_5 must have the same volume as formula_6; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).
A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be formula_7 with &NoBreak;&NoBreak;. Worse, a single point formula_8 has no size. These difficulties can be avoided by working with the inverse map &NoBreak;}&NoBreak;; it will map any given subset formula_2 to the parts that were assembled to make it: these parts are &NoBreak;}&NoBreak;. It has the important property of not "losing track" of where things came from. More strongly, it has the important property that "any" (measure-preserving) map formula_9 is the inverse of some map &NoBreak;&NoBreak;. The proper definition of a volume-preserving map is one for which formula_10 because formula_11 describes all the pieces-parts that formula_6 came from.
One is now interested in studying the time evolution of the system. If a set formula_12 eventually visits all of formula_0 over a long period of time (that is, if formula_13 approaches all of formula_0 for large formula_14), the system is said to be ergodic. If every set formula_6 behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets formula_6 wander away, never to be returned to. An example would be water running downhill—once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The ergodic decomposition theorem states that every ergodic system can be split into two parts: the conservative part, and the dissipative part.
Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets &NoBreak;&NoBreak;, and not just between some set formula_6 and &NoBreak;&NoBreak;. That is, given any two sets &NoBreak;}&NoBreak;, a system is said to be (topologically) mixing if there is an integer formula_15 such that, for all formula_16 and &NoBreak;&NoBreak;, one has that formula_17. Here, formula_18 denotes set intersection and formula_19 is the empty set.
The above definition of topological mixing should be enough to provide an informal idea of mixing (it is equivalent to the formal definition, given below). However, it made no mention of the volume of formula_6 and &NoBreak;&NoBreak;, and, indeed, there is another definition that explicitly works with the volume. Several, actually; one has both strong mixing and weak mixing; they are inequivalent, although a strong mixing system is always weakly mixing. The measure-based definitions are not compatible with the definition of topological mixing: there are systems which are one, but not the other. The general situation remains cloudy: for example, given three sets &NoBreak;}&NoBreak;, one can define 3-mixing. As of 2020, it is not known if 2-mixing implies 3-mixing. (If one thinks of ergodicity as "1-mixing", then it is clear that 1-mixing does not imply 2-mixing; there are systems that are ergodic but not mixing.)
The concept of "strong mixing" is made in reference to the volume of a pair of sets. Consider, for example, a set formula_6 of colored dye that is being mixed into a cup of some sort of sticky liquid, say, corn syrup, or shampoo, or the like. Practical experience shows that mixing sticky fluids can be quite hard: there is usually some corner of the container where it is hard to get the dye mixed into. Pick as set formula_20 that hard-to-reach corner. The question of mixing is then, can formula_6, after a long enough period of time, not only penetrate into formula_20 but also fill formula_20 with the same proportion as it does elsewhere?
One phrases the definition of strong mixing as the requirement that
formula_21
The time parameter formula_14 serves to separate formula_6 and formula_20 in time, so that one is mixing formula_6 while holding the test volume formula_20 fixed. The product formula_22 is a bit more subtle. Imagine that the volume formula_20 is 10% of the total volume, and that the volume of dye formula_6 will also be 10% of the grand total. If formula_6 is uniformly distributed, then it is occupying 10% of formula_20, which itself is 10% of the total, and so, in the end, after mixing, the part of formula_6 that is in formula_20 is 1% of the total volume. That is, formula_23 This product-of-volumes has more than passing resemblance to Bayes theorem in probabilities; this is not an accident, but rather a consequence that measure theory and probability theory are the same theory: they share the same axioms (the Kolmogorov axioms), even as they use different notation.
The reason for using formula_24 instead of formula_25 in the definition is a bit subtle, but it follows from the same reasons why
formula_26 was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner formula_20, one wants to look at where that dye "came from" (presumably, it was poured in at the top, at some time in the past). One must be sure that every place it might have "come from" eventually gets mixed into formula_20.
Mixing in dynamical systems.
Let formula_27 be a measure-preserving dynamical system, with "T" being the time-evolution or shift operator. The system is said to be strong mixing if, for any formula_28, one has
formula_29
For shifts parametrized by a continuous variable instead of a discrete integer "n", the same definition applies, with formula_30 replaced by formula_31 with "g" being the continuous-time parameter.
A dynamical system is said to be weak mixing if one has
formula_32
In other words, formula_33 is strong mixing if formula_34 in the usual sense, weak mixing if
formula_35
in the Cesàro sense, and ergodic if formula_36 in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing. The Chacon system was historically the first example given of a system that is weak-mixing but not strong-mixing.
Theorem. Weak mixing implies ergodicity.
Proof. If the action of the map decomposes into two components &NoBreak;&NoBreak;, then we have &NoBreak;&NoBreak;, so weak mixing implies &NoBreak;&NoBreak;, so one of formula_16 has zero measure, and the other one has full measure.
Covering families.
Given a topological space, such as the unit interval (whether it has its end points or not), we can construct a measure on it by taking the open sets, then take their unions, complements, unions, complements, and so on to infinity, to obtain all the Borel sets. Next, we define a measure formula_37 on the Borel sets, then add in all the subsets of measure-zero ("negligible sets"). This is how we obtain the Lebesgue measure and the Lebesgue measurable sets.
In most applications of ergodic theory, the underlying space is almost-everywhere isomorphic to an open subset of some formula_38, and so it is a Lebesgue measure space. Verifying strong-mixing can be simplified if we only need to check a smaller set of measurable sets.
A covering family formula_39 is a set of measurable sets, such that any open set is a "disjoint" union of sets in it. Compare this with base in topology, which is less restrictive as it allows non-disjoint unions.
Theorem. For Lebesgue measure spaces, if formula_33 is measure-preserving, and formula_40 for all formula_16 in a covering family, then formula_33 is strong mixing.
Proof. Extend the mixing equation from all formula_16 in the covering family, to all open sets by disjoint union, to all closed sets by taking the complement, to all measurable sets by using the regularity of Lebesgue measure to approximate any set with open and closed sets. Thus, formula_40 for all measurable &NoBreak;&NoBreak;.
"L"2 formulation.
The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system formula_27 is equivalent to the property that, for any function formula_41, the sequence formula_42 converges strongly and in the sense of Cesàro to &NoBreak;&NoBreak;, i.e.,
formula_43
A dynamical system formula_27 is weakly mixing if, for any functions formula_44 and formula_45
formula_46
A dynamical system formula_27 is strongly mixing if, for any function &NoBreak;&NoBreak;, the sequence formula_42 converges weakly to &NoBreak;&NoBreak;, i.e., for any function formula_45
formula_47
Since the system is assumed to be measure preserving, this last line is equivalent to saying that the covariance &NoBreak;&NoBreak;, so that the random variables formula_48 and formula_49 become orthogonal as formula_14 grows. Actually, since this works for any function &NoBreak;&NoBreak;, one can informally see mixing as the property that the random variables formula_48 and formula_49 become independent as formula_14 grows.
Products of dynamical systems.
Given two measured dynamical systems formula_50 and formula_51 one can construct a dynamical system formula_52 on the Cartesian product by defining formula_53 We then have the following characterizations of weak mixing:
Proposition. A dynamical system formula_50 is weakly mixing if and only if, for any ergodic dynamical system &NoBreak;&NoBreak;, the system formula_52 is also ergodic.
Proposition. A dynamical system formula_50 is weakly mixing if and only if formula_54 is also ergodic. If this is the case, then formula_54 is also weakly mixing.
Generalizations.
The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which
formula_55
holds for all measurable sets "A", "B", "C". We can define strong k-mixing similarly. A system which is strong "k"-mixing for all "k" = 2,3,4... is called mixing of all orders.
It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong "m"-mixing implies ergodicity.
Examples.
Irrational rotations of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.
Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, and the Anosov flow (the geodesic flow on the unit tangent bundle of compact manifolds of negative curvature.)
The dyadic map is "shift to left in binary". In general, for any formula_56, the "shift to left in base &NoBreak;&NoBreak;" map formula_57 is strongly mixing on the covering family &NoBreak;}&NoBreak;, therefore it is strongly mixing on &NoBreak;&NoBreak;, and therefore it is strongly mixing on &NoBreak;&NoBreak;.
Similarly, for any finite or countable alphabet &NoBreak;&NoBreak;, we can impose a discrete probability distribution on it, then consider the probability distribution on the "coin flip" space, where each "coin flip" can take results from &NoBreak;&NoBreak;. We can either construct the singly-infinite space formula_58 or the doubly-infinite space &NoBreak;&NoBreak;. In both cases, the shift map (one letter to the left) is strongly mixing, since it is strongly mixing on the covering family of cylinder sets. The Baker's map is isomorphic to a shift map, so it is strongly mixing.
Topological mixing.
A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map formula_59 is said to be topologically transitive if, for every pair of non-empty open sets formula_60, there exists an integer "n" such that
formula_61
where formula_62 is the "n"th iterate of "f". In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.
Lemma: If "X" is a complete metric space with no isolated point, then "f" is topologically transitive if and only if there exists a hypercyclic point formula_8, that is, a point "x" such that its orbit formula_63 is dense in "X".
A system is said to be topologically mixing if, given open sets formula_6 and &NoBreak;&NoBreak;, there exists an integer "N", such that, for all &NoBreak;&NoBreak;, one has
formula_64
For a continuous-time system, formula_62 is replaced by the flow &NoBreak;&NoBreak;, with "g" being the continuous parameter, with the requirement that a non-empty intersection hold for all &NoBreak;&NoBreak;.
A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.
Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.
Mixing in stochastic processes.
Let formula_65 be a stochastic process on a probability space &NoBreak;&NoBreak;. The sequence space into which the process maps can be endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology.
Define a function formula_66, called the strong mixing coefficient, as
formula_67
for all &NoBreak;&NoBreak;. The symbol formula_68, with formula_69 denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times "a" and "b", i.e. the σ-algebra generated by &NoBreak;}&NoBreak;.
The process formula_65 is said to be strongly mixing if formula_70 as &NoBreak;&NoBreak;. That is to say, a strongly mixing process is such that, in a way that is uniform over all times formula_71 and all events, the events before time formula_71 and the events after time formula_72 tend towards being independent as formula_73; more colloquially, the process, in a strong sense, forgets its history.
Mixing in Markov processes.
Suppose formula_74 were a stationary Markov process with stationary distribution formula_75 and let formula_76 denote the space of Borel-measurable functions that are square-integrable with respect to the measure formula_75. Also let
formula_77
denote the conditional expectation operator on formula_78 Finally, let
formula_79
denote the space of square-integrable functions with mean zero.
The "ρ"-mixing coefficients of the process {"xt"} are
formula_80
The process is called "ρ"-mixing if these coefficients converge to zero as "t" → ∞, and “"ρ"-mixing with exponential decay rate” if "ρt" < "e"−"δt" for some "δ" > 0. For a stationary Markov process, the coefficients "ρt" may either decay at an exponential rate, or be always equal to one.
The "α"-mixing coefficients of the process {"xt"} are
formula_81
The process is called α"-mixing if these coefficients converge to zero as "t" → ∞, it is ""α"-mixing with exponential decay rate" if "αt" < "γe"−"δt" for some "δ" > 0, and it is α"-mixing with a sub-exponential decay rate if "αt" < "ξ"("t") for some non-increasing function formula_82 satisfying
formula_83
as &NoBreak;&NoBreak;.
The "α"-mixing coefficients are always smaller than the "ρ"-mixing ones: "αt" ≤ "ρt", therefore if the process is "ρ"-mixing, it will necessarily be "α"-mixing too. However, when "ρt"
1, the process may still be "α"-mixing, with sub-exponential decay rate.
The "β"-mixing coefficients are given by
formula_84
The process is called β"-mixing if these coefficients converge to zero as "t" → ∞, it is β"-mixing with an exponential decay rate if "βt" < "γe"−"δt" for some "δ" > 0, and it is "β"-mixing with a sub-exponential decay rate if "βtξ"("t") → 0 as "t" → ∞ for some non-increasing function formula_82 satisfying
formula_83
as formula_85.
A strictly stationary Markov process is "β"-mixing if and only if it is an aperiodic recurrent Harris chain. The "β"-mixing coefficients are always bigger than the "α"-mixing ones, so if a process is "β"-mixing it will also be "α"-mixing. There is no direct relationship between "β"-mixing and "ρ"-mixing: neither of them implies the other.
References.
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{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "\\mu"
},
{
"math_id": 2,
"text": "A\\subset X"
},
{
"math_id": 3,
"text": "\\mathcal{A}"
},
{
"math_id": 4,
"text": "T:X\\to X"
},
{
"math_id": 5,
"text": "T(A)"
},
{
"math_id": 6,
"text": "A"
},
{
"math_id": 7,
"text": "x\\ne y"
},
{
"math_id": 8,
"text": "x\\in X"
},
{
"math_id": 9,
"text": "\\mathcal{A}\\to\\mathcal{A}"
},
{
"math_id": 10,
"text": "\\mu(A)=\\mu(T^{-1}(A))"
},
{
"math_id": 11,
"text": "T^{-1}(A)"
},
{
"math_id": 12,
"text": "A\\in\\mathcal{A}"
},
{
"math_id": 13,
"text": "\\cup_{k=1}^n T^k(A)"
},
{
"math_id": 14,
"text": "n"
},
{
"math_id": 15,
"text": "N"
},
{
"math_id": 16,
"text": "A, B"
},
{
"math_id": 17,
"text": "T^n(A)\\cap B\\ne\\varnothing"
},
{
"math_id": 18,
"text": "\\cap"
},
{
"math_id": 19,
"text": "\\varnothing"
},
{
"math_id": 20,
"text": "B"
},
{
"math_id": 21,
"text": "\\lim_{n\\to\\infty} \\mu\\left( T^{-n} A \\cap B\\right) = \\mu(A)\\mu(B)."
},
{
"math_id": 22,
"text": "\\mu(A)\\mu(B)"
},
{
"math_id": 23,
"text": "\\mu\\left(\\mbox{after-mixing}(A) \\cap B\\right) = \\mu(A)\\mu(B)."
},
{
"math_id": 24,
"text": "T^{-n} A"
},
{
"math_id": 25,
"text": "T^n A"
},
{
"math_id": 26,
"text": "T^{-1} A"
},
{
"math_id": 27,
"text": "(X, \\mathcal{A}, \\mu, T)"
},
{
"math_id": 28,
"text": "A,B \\in \\mathcal{A}"
},
{
"math_id": 29,
"text": "\\lim_{n\\to\\infty} \\mu \\left (A \\cap T^{-n}B \\right ) = \\mu(A)\\mu(B)."
},
{
"math_id": 30,
"text": "T^{-n}"
},
{
"math_id": 31,
"text": "T_g"
},
{
"math_id": 32,
"text": "\\lim_{n\\to\\infty} \\frac 1 n \\sum_{k=0}^{n-1} \\left |\\mu(A \\cap T^{-k}B) - \\mu(A)\\mu(B) \\right | = 0."
},
{
"math_id": 33,
"text": "T"
},
{
"math_id": 34,
"text": "\\mu (A \\cap T^{-n}B) - \\mu(A)\\mu(B) \\to 0"
},
{
"math_id": 35,
"text": " \\left |\\mu(A \\cap T^{-n} B) - \\mu(A)\\mu(B) \\right | \\to 0,"
},
{
"math_id": 36,
"text": "\\mu \\left (A \\cap T^{-n}B \\right ) \\to \\mu(A)\\mu(B)"
},
{
"math_id": 37,
"text": "\\mu "
},
{
"math_id": 38,
"text": "\\R^n"
},
{
"math_id": 39,
"text": "\\mathcal C"
},
{
"math_id": 40,
"text": "\\lim_n \\mu(T^{-n}(A)\\cap B) = \\mu(A)\\mu (B)"
},
{
"math_id": 41,
"text": "f \\in L^2 (X, \\mu)"
},
{
"math_id": 42,
"text": "(f \\circ T^n)_{n \\ge 0}"
},
{
"math_id": 43,
"text": " \\lim_{N \\to \\infty} \\left \\| {1 \\over N} \\sum_{n=0}^{N-1} f \\circ T^n - \\int_X f \\, d \\mu \\right \\|_{L^2 (X, \\mu)}= 0."
},
{
"math_id": 44,
"text": "f"
},
{
"math_id": 45,
"text": "g \\in L^2 (X, \\mu),"
},
{
"math_id": 46,
"text": " \\lim_{N \\to \\infty} {1 \\over N} \\sum_{n=0}^{N-1} \\left | \\int_X f \\circ T^n \\cdot g \\, d \\mu- \\int_X f \\, d \\mu \\cdot \\int_X g \\, d \\mu \\right |= 0."
},
{
"math_id": 47,
"text": " \\lim_{n \\to \\infty} \\int_X f \\circ T^n \\cdot g \\, d \\mu = \\int_X f \\, d \\mu \\cdot \\int_X g \\, d \\mu."
},
{
"math_id": 48,
"text": "f \\circ T^n"
},
{
"math_id": 49,
"text": "g"
},
{
"math_id": 50,
"text": "(X, \\mu, T)"
},
{
"math_id": 51,
"text": "(Y, \\nu, S),"
},
{
"math_id": 52,
"text": "(X \\times Y, \\mu \\otimes \\nu, T \\times S)"
},
{
"math_id": 53,
"text": "(T \\times S) (x,y) = (T(x), S(y))."
},
{
"math_id": 54,
"text": "(X^2, \\mu \\otimes \\mu, T \\times T)"
},
{
"math_id": 55,
"text": "\\lim_{m,n\\to\\infty} \\mu (A \\cap T^{-m}B \\cap T^{-m-n}C) = \\mu(A)\\mu(B)\\mu(C)"
},
{
"math_id": 56,
"text": "n \\in \\{2, 3, \\dots\\}"
},
{
"math_id": 57,
"text": "T(x) = nx \\bmod 1"
},
{
"math_id": 58,
"text": "\\Sigma^\\N"
},
{
"math_id": 59,
"text": "f:X\\to X"
},
{
"math_id": 60,
"text": "A,B\\subset X"
},
{
"math_id": 61,
"text": "f^n(A) \\cap B \\ne \\varnothing"
},
{
"math_id": 62,
"text": "f^n"
},
{
"math_id": 63,
"text": "\\{f^n(x): n\\in \\mathbb{N}\\}"
},
{
"math_id": 64,
"text": "f^n(A) \\cap B \\neq \\varnothing."
},
{
"math_id": 65,
"text": "(X_t)_{-\\infty < t < \\infty}"
},
{
"math_id": 66,
"text": "\\alpha"
},
{
"math_id": 67,
"text": "\\alpha(s) = \\sup \\left\\{|\\mathbb{P}(A \\cap B) - \\mathbb{P}(A)\\mathbb{P}(B)| : -\\infty < t < \\infty, A\\in X_{-\\infty}^t, B\\in X_{t+s}^\\infty \\right\\}"
},
{
"math_id": 68,
"text": "X_a^b"
},
{
"math_id": 69,
"text": "-\\infty \\le a \\le b \\le \\infty "
},
{
"math_id": 70,
"text": "\\alpha(s)\\to 0"
},
{
"math_id": 71,
"text": "t"
},
{
"math_id": 72,
"text": "t+s"
},
{
"math_id": 73,
"text": "s \\to \\infty"
},
{
"math_id": 74,
"text": "(X_t)"
},
{
"math_id": 75,
"text": "\\mathbb{Q}"
},
{
"math_id": 76,
"text": "L^2(\\mathbb{Q})"
},
{
"math_id": 77,
"text": "\\mathcal{E}_t \\varphi (x) = \\mathbb{E}[\\varphi (X_t) \\mid X_0 = x] "
},
{
"math_id": 78,
"text": "L^2(\\mathbb{Q})."
},
{
"math_id": 79,
"text": " Z = \\left \\{ \\varphi \\in L^2(\\mathbb{Q}) : \\int \\varphi \\, d\\mathbb{Q} = 0 \\right \\}"
},
{
"math_id": 80,
"text": "\\rho_t = \\sup_{\\varphi\\in Z :\\,\\|\\varphi\\|_2=1} \\| \\mathcal{E}_t\\varphi \\|_2."
},
{
"math_id": 81,
"text": "\\alpha_t = \\sup_{\\varphi \\in Z : \\|\\varphi\\|_\\infty=1} \\| \\mathcal{E}_t\\varphi \\|_1. "
},
{
"math_id": 82,
"text": "\\xi"
},
{
"math_id": 83,
"text": "\\frac{\\ln \\xi(t)}{t} \\to 0"
},
{
"math_id": 84,
"text": "\\beta_t = \\int \\sup_{0 \\le \\varphi \\le 1} \\left | \\mathcal{E}_t\\varphi(x) - \\int \\varphi \\,d\\mathbb{Q} \\right| \\,d\\mathbb{Q}."
},
{
"math_id": 85,
"text": "t \\to \\infty"
}
]
| https://en.wikipedia.org/wiki?curid=1308185 |
13084071 | Magnetic anisotropy | Directional dependence of substances' magnetic susceptibilities
In condensed matter physics, magnetic anisotropy describes how an object's magnetic properties can be different depending on direction. In the simplest case, there is no preferential direction for an object's magnetic moment. It will respond to an applied magnetic field in the same way, regardless of which direction the field is applied. This is known as magnetic isotropy. In contrast, magnetically anisotropic materials will be easier or harder to magnetize depending on which way the object is rotated.
For most magnetically anisotropic materials, there are two easiest directions to magnetize the material, which are a 180° rotation apart. The line parallel to these directions is called the easy axis. In other words, the easy axis is an energetically favorable direction of spontaneous magnetization. Because the two opposite directions along an easy axis are usually equivalently easy to magnetize along, the actual direction of magnetization can just as easily settle into either direction, which is an example of spontaneous symmetry breaking.
Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is superparamagnetic.
Sources.
The observed magnetic anisotropy in an object can happen for several different reasons. Rather than having a single cause, the overall magnetic anisotropy of a given object is often explained by a combination of these different factors:
At the molecular level.
The magnetic anisotropy of a benzene ring (A), alkene (B), carbonyl (C), alkyne (D), and a more complex molecule (E) are shown in the figure. Each of these unsaturated functional groups (A-D) create a tiny magnetic field and hence some local anisotropic regions (shown as cones) in which the shielding effects and the chemical shifts are unusual. The bisazo compound (E) shows that the designated proton {H} can appear at different chemical shifts depending on the photoisomerization state of the azo groups. The "trans" isomer holds proton {H} far from the cone of the benzene ring thus the magnetic anisotropy is not present. While the "cis" form holds proton {H} in the vicinity of the cone, shields it and decreases its chemical shift. This phenomenon enables a new set of nuclear Overhauser effect (NOE) interactions (shown in red) that come to existence in addition to the previously existing ones (shown in blue).
Single-domain magnet.
Suppose that a ferromagnet is single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the magnetic moment is formula_0 and the volume of the particle is formula_1, the magnetization is formula_2, where formula_3 is the saturation magnetization and formula_4 are direction cosines (components of a unit vector) so formula_5. The energy associated with magnetic anisotropy can depend on the direction cosines in various ways, the most common of which are discussed below.
Uniaxial.
A magnetic particle with uniaxial anisotropy has one easy axis. If the easy axis is in the formula_6 direction, the anisotropy energy can be expressed as one of the forms:
formula_7
where formula_1 is the volume, formula_8 the anisotropy constant, and formula_9 the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol formula_10 is often used to indicate the anisotropy constant, instead of formula_8. In the widely used Stoner–Wohlfarth model, the anisotropy is uniaxial.
Triaxial.
A magnetic particle with triaxial anisotropy still has a single easy axis, but it also has a hard axis (direction of maximum energy) and an intermediate axis (direction associated with a saddle point in the energy). The coordinates can be chosen so the energy has the form
formula_11
If formula_12 the easy axis is the formula_6 direction, the intermediate axis is the formula_13 direction and the hard axis is the formula_14 direction.
Cubic.
A magnetic particle with cubic anisotropy has three or four easy axes, depending on the anisotropy parameters. The energy has the form
formula_15
If formula_16 the easy axes are the formula_17 and formula_6 axes. If formula_18 there are four easy axes characterized by formula_19.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" /> | [
{
"math_id": 0,
"text": "\\boldsymbol{\\mu}"
},
{
"math_id": 1,
"text": "V"
},
{
"math_id": 2,
"text": "\\mathbf{M} = \\boldsymbol{\\mu}/V = M_s \\left(\\alpha,\\beta,\\gamma\\right)"
},
{
"math_id": 3,
"text": "M_s"
},
{
"math_id": 4,
"text": "\\alpha, \\beta, \\gamma"
},
{
"math_id": 5,
"text": "\\alpha^2 + \\beta^2 + \\gamma^2 = 1"
},
{
"math_id": 6,
"text": "z"
},
{
"math_id": 7,
"text": "E = KV \\left(1 - \\gamma^2 \\right) = KV \\sin^2\\theta,"
},
{
"math_id": 8,
"text": "K"
},
{
"math_id": 9,
"text": "\\theta"
},
{
"math_id": 10,
"text": "\\mathcal{N}"
},
{
"math_id": 11,
"text": "E = K_aV\\alpha^2 + K_bV\\beta^2."
},
{
"math_id": 12,
"text": "K_a > K_b > 0,"
},
{
"math_id": 13,
"text": "y"
},
{
"math_id": 14,
"text": "x"
},
{
"math_id": 15,
"text": "E = KV \\left(\\alpha^2\\beta^2 + \\beta^2\\gamma^2 + \\gamma^2\\alpha^2\\right)."
},
{
"math_id": 16,
"text": "K > 0,"
},
{
"math_id": 17,
"text": "x, y,"
},
{
"math_id": 18,
"text": "K < 0,"
},
{
"math_id": 19,
"text": "x = \\pm y = \\pm z"
}
]
| https://en.wikipedia.org/wiki?curid=13084071 |
13085795 | SN1CB mechanism | Chemical reactions by which metal amine complexes exchange ligands
In coordination chemistry, the SN1cB (conjugate base) mechanism describes the pathway by which many metal amine complexes undergo substitution, that is, ligand exchange. Typically, the reaction entails reaction of a polyamino metal halide with aqueous base to give the corresponding polyamine metal hydroxide:
<chem>[Co(NH3)5Cl]^2+ + OH- -> [Co(NH3)5OH]^2+ + Cl-</chem>
The rate law for the reaction is:
formula_0
The rate law is deceptive: hydroxide serves not as a nucleophile but as a base to deprotonate the coordinated ammonia. Simultaneously with deprotonation, the halide dissociates. Water binds to the coordinatively unsaturated complex followed by proton transfer to give the hydroxy complex. The conjugate base resulting from deprotonation of the amine is rarely observed. | [
{
"math_id": 0,
"text": "r=k[\\mathrm{Co(NH_3)_5Cl^{2+}}][\\mathrm{OH}^-]"
}
]
| https://en.wikipedia.org/wiki?curid=13085795 |
13087180 | Mass (mass spectrometry) | Physical quantities being measured
The mass recorded by a mass spectrometer can refer to different physical quantities depending on the characteristics of the instrument and the manner in which the mass spectrum is displayed.
Units.
The dalton (symbol: Da) is the standard unit that is used for indicating mass on an atomic or molecular scale (atomic mass). The unified atomic mass unit (symbol: u) is equivalent to the dalton. One dalton is approximately the mass of one a single proton or neutron. The unified atomic mass unit has a value of . The "amu" without the "unified" prefix is an obsolete unit based on oxygen, which was replaced in 1961.
Molecular mass.
The molecular mass (abbreviated Mr) of a substance, formerly also called molecular weight and abbreviated as MW, is the mass of one molecule of that substance, relative to the unified atomic mass unit u (equal to 1/12 the mass of one atom of 12C). Due to this relativity, the molecular mass of a substance is commonly referred to as the relative molecular mass, and abbreviated to Mr.
Average mass.
The average mass of a molecule is obtained by summing the average atomic masses of the constituent elements. For example, the average mass of natural water with formula H2O is 1.00794 + 1.00794 + 15.9994 = 18.01528 Da.
Mass number.
The mass number, also called the nucleon number, is the number of protons and neutrons in an atomic nucleus. The mass number is unique for each isotope of an element and is written either after the element name or as a superscript to the left of an element's symbol. For example, carbon-12 (12C) has 6 protons and 6 neutrons.
Nominal mass.
The nominal mass for an element is the mass number of its most abundant naturally occurring stable isotope, and for an ion or molecule, the nominal mass is the sum of the nominal masses of the constituent atoms. Isotope abundances are tabulated by IUPAC: for example carbon has two stable isotopes 12C at 98.9% natural abundance and 13C at 1.1% natural abundance, thus the nominal mass of carbon is 12. The nominal mass is not always the lowest mass number, for example iron has isotopes 54Fe, 56Fe, 57Fe, and 58Fe with abundances 6%, 92%, 2%, and 0.3%, respectively, and a nominal mass of 56 Da. For a molecule, the nominal mass is obtained by summing the nominal masses of the constituent elements, for example water has two hydrogen atoms with nominal mass 1 Da and one oxygen atom with nominal mass 16 Da, therefore the nominal mass of H2O is 18 Da.
In mass spectrometry, the difference between the nominal mass and the monoisotopic mass is the mass defect. This differs from the definition of mass defect used in physics which is the difference between the mass of a composite particle and the sum of the masses of its constituent parts.
Accurate mass.
The accurate mass (more appropriately, the measured accurate mass) is an experimentally determined mass that allows the elemental composition to be determined. For molecules with mass below 200 Da, 5 ppm accuracy is often sufficient to uniquely determine the elemental composition.
Exact mass.
The exact mass of an isotopic species (more appropriately, the calculated exact mass) is obtained by summing the masses of the individual isotopes of the molecule. For example, the exact mass of water containing two hydrogen-1 (1H) and one oxygen-16 (16O) is 1.0078 + 1.0078 + 15.9949 = 18.0105 Da. The exact mass of heavy water, containing two hydrogen-2 (deuterium or 2H) and one oxygen-16 (16O) is 2.0141 + 2.0141 + 15.9949 = 20.0229 Da.
When an exact mass value is given without specifying an isotopic species, it normally refers to the most abundant isotopic species.
Monoisotopic mass.
The monoisotopic mass is the sum of the masses of the atoms in a molecule using the unbound, ground-state, rest mass of the principal (most abundant) isotope for each element. The monoisotopic mass of a molecule or ion is the exact mass obtained using the principal isotopes. Monoisotopic mass is typically expressed in daltons.
For typical organic compounds, where the monoisotopic mass is most commonly used, this also results in the lightest isotope being selected. For some heavier atoms such as iron and argon the principal isotope is not the lightest isotope. The mass spectrum peak corresponding to the monoisotopic mass is often not observed for large molecules, but can be determined from the isotopic distribution.
Most abundant mass.
This refers to the mass of the molecule with the most highly represented isotope distribution, based on the natural abundance of the isotopes.
Isotopomer and isotopologue.
Isotopomers (isotopic isomers) are isomers having the same number of each isotopic atom, but differing in the positions of the isotopic atoms. For example, CH3CHDCH3 and CH3CH2CH2D are a pair of structural isotopomers.
Isotopomers should not be confused with isotopologues, which are chemical species that differ in the isotopic composition of their molecules or ions. For example, three isotopologues of the water molecule with different isotopic composition of hydrogen are: HOH, HOD and DOD, where D stands for deuterium (2H).
Kendrick mass.
The Kendrick mass is a mass obtained by multiplying the measured mass by a numeric factor. The Kendrick mass is used to aid in the identification of molecules of similar chemical structure from peaks in mass spectra. The method of stating mass was suggested in 1963 by the chemist Edward Kendrick.
According to the procedure outlined by Kendrick, the mass of CH2 is defined as 14.000 Da, instead of 14.01565 Da.
The Kendrick mass for a family of compounds formula_0 is given by
formula_1
For hydrocarbon analysis, formula_0 = CH2.
Mass defect (mass spectrometry).
The mass defect used in nuclear physics is different from its use in mass spectrometry. In nuclear physics, the mass defect is the difference in the mass of a composite particle and the sum of the masses of its component parts. In mass spectrometry the mass defect is defined as the difference between the exact mass and the nearest integer mass.
The Kendrick mass defect is the exact Kendrick mass subtracted from the nearest integer Kendrick mass.
Mass defect filtering can be used to selectively detect compounds with a mass spectrometer based on their chemical composition.
Packing fraction (mass spectrometry).
The term packing fraction was defined by Aston as the difference of the measured mass "M" and the nearest integer mass "I" (based on the oxygen-16 mass scale) divided by the quantity comprising the mass number multiplied by ten thousand:
formula_2.
Aston's early model of nuclear structure (prior to the discovery of the neutron) postulated that the electromagnetic fields of closely packed protons and electrons in the nucleus would interfere and a fraction of the mass would be destroyed. A low packing fraction is indicative of a stable nucleus.
Nitrogen rule.
The nitrogen rule states that organic compounds containing exclusively hydrogen, carbon, nitrogen, oxygen, silicon, phosphorus, sulfur, and the halogens either have an odd nominal mass that indicates an odd number of nitrogen atoms are present or an even nominal mass that indicates an even number of nitrogen atoms are present in the molecular ion.
Prout's hypothesis and the whole number rule.
The whole number rule states that the masses of the isotopes are integer multiples of the mass of the hydrogen atom. The rule is a modified version of Prout's hypothesis proposed in 1815, to the effect that atomic weights are multiples of the weight of the hydrogen atom.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "F"
},
{
"math_id": 1,
"text": "\\mbox{Kendrick mass}~(F) = (\\mbox{observed mass}) \\times \\frac{\\mbox{nominal mass}~(F)}{\\mbox{exact mass}~(F)}."
},
{
"math_id": 2,
"text": "f=\\frac{M-I}{10^4\\ I} "
}
]
| https://en.wikipedia.org/wiki?curid=13087180 |
1309 | Almost all | In mathematics, with negligible exceptions
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if formula_0 is a set, "almost all elements of formula_0" means "all elements of formula_0 but those in a negligible subset of formula_0". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.
In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of formula_0" means "a negligible quantity of elements of formula_0".
Meanings in different areas of mathematics.
Prevalent meaning.
Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) except for countably many".
Examples:
Meaning in measure theory.
When speaking about the reals, sometimes "almost all" can mean "all reals except for a null set". Similarly, if S is some set of reals, "almost all numbers in S" can mean "all numbers in S except for those in a null set". The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an n-dimensional space (where n is a positive integer), these definitions can be generalised to "all points except for those in a null set" or "all points in S except for those in a null set" (this time, S is a set of points in the space). Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability theory.
Examples:
Meaning in number theory.
In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in "A" below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive integers are in A.
More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or the set of primes, if A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A.
Examples:
Meaning in graph theory.
In graph theory, if A is a set of (finite labelled) graphs, it can be said to contain almost all graphs, if the proportion of graphs with n vertices that are in A tends to 1 as n tends to infinity. However, it is sometimes easier to work with probabilities, so the definition is reformulated as follows. The proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices (chosen with the uniform distribution) is in A, and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them. Therefore, equivalently to the preceding definition, the set "A" contains almost all graphs if the probability that a coin-flip–generated graph with n vertices is in A tends to 1 as n tends to infinity. Sometimes, the latter definition is modified so that the graph is chosen randomly in some other way, where not all graphs with n vertices have the same probability, and those modified definitions are not always equivalent to the main one.
The use of the term "almost all" in graph theory is not standard; the term "asymptotically almost surely" is more commonly used for this concept.
Example:
Meaning in topology.
In topology and especially dynamical systems theory (including applications in economics), "almost all" of a topological space's points can mean "all of the space's points except for those in a meagre set". Some use a more limited definition, where a subset contains almost all of the space's points only if it contains some open dense set.
Example:
Meaning in algebra.
In abstract algebra and mathematical logic, if U is an on a set X, "almost all elements of X" sometimes means "the elements of some "element" of U". For any partition of X into two disjoint sets, one of them will necessarily contain almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X, even if it isn't an ultrafilter.
Proofs.
<templatestyles src="Reflist/styles.css" />
References.
Primary sources.
<templatestyles src="Reflist/styles.css" />
Secondary sources.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "X"
}
]
| https://en.wikipedia.org/wiki?curid=1309 |
1309220 | Imputation (statistics) | Process of replacing missing data with substituted values
In statistics, imputation is the process of replacing missing data with substituted values. When substituting for a data point, it is known as "unit imputation"; when substituting for a component of a data point, it is known as "item imputation". There are three main problems that missing data causes: missing data can introduce a substantial amount of bias, make the handling and analysis of the data more arduous, and create reductions in efficiency. Because missing data can create problems for analyzing data, imputation is seen as a way to avoid pitfalls involved with listwise deletion of cases that have missing values. That is to say, when one or more values are missing for a case, most statistical packages default to discarding any case that has a missing value, which may introduce bias or affect the representativeness of the results. Imputation preserves all cases by replacing missing data with an estimated value based on other available information. Once all missing values have been imputed, the data set can then be analysed using standard techniques for complete data. There have been many theories embraced by scientists to account for missing data but the majority of them introduce bias. A few of the well known attempts to deal with missing data include: hot deck and cold deck imputation; listwise and pairwise deletion; mean imputation; non-negative matrix factorization; regression imputation; last observation carried forward; stochastic imputation; and multiple imputation.
Listwise (complete case) deletion.
By far, the most common means of dealing with missing data is listwise deletion (also known as complete case), which is when all cases with a missing value are deleted. If the data are missing completely at random, then listwise deletion does not add any bias, but it does decrease the power of the analysis by decreasing the effective sample size. For example, if 1000 cases are collected but 80 have missing values, the effective sample size after listwise deletion is 920. If the cases are not missing completely at random, then listwise deletion will introduce bias because the sub-sample of cases represented by the missing data are not representative of the original sample (and if the original sample was itself a representative sample of a population, the complete cases are not representative of that population either). While listwise deletion is unbiased when the missing data is missing completely at random, this is rarely the case in actuality.
Pairwise deletion (or "available case analysis") involves deleting a case when it is missing a variable required for a particular analysis, but including that case in analyses for which all required variables are present. When pairwise deletion is used, the total N for analysis will not be consistent across parameter estimations. Because of the incomplete N values at some points in time, while still maintaining complete case comparison for other parameters, pairwise deletion can introduce impossible mathematical situations such as correlations that are over 100%.
The one advantage complete case deletion has over other methods is that it is straightforward and easy to implement. This is a large reason why complete case is the most popular method of handling missing data in spite of the many disadvantages it has.
Single imputation.
Hot-deck.
A once-common method of imputation was hot-deck imputation where a missing value was imputed from a randomly selected similar record. The term "hot deck" dates back to the storage of data on punched cards, and indicates that the information donors come from the same dataset as the recipients. The stack of cards was "hot" because it was currently being processed.
One form of hot-deck imputation is called "last observation carried forward" (or LOCF for short), which involves sorting a dataset according to any of a number of variables, thus creating an ordered dataset. The technique then finds the first missing value and uses the cell value immediately prior to the data that are missing to impute the missing value. The process is repeated for the next cell with a missing value until all missing values have been imputed. In the common scenario in which the cases are repeated measurements of a variable for a person or other entity, this represents the belief that if a measurement is missing, the best guess is that it hasn't changed from the last time it was measured. This method is known to increase risk of increasing bias and potentially false conclusions. For this reason LOCF is not recommended for use.
Cold-deck.
Cold-deck imputation, by contrast, selects donors from another dataset. Due to advances in computer power, more sophisticated methods of imputation have generally superseded the original random and sorted hot deck imputation techniques. It is a method of replacing with response values of similar items in past surveys. It is available in surveys that measure time intervals.
Mean substitution.
Another imputation technique involves replacing any missing value with the mean of that variable for all other cases, which has the benefit of not changing the sample mean for that variable. However, mean imputation attenuates any correlations involving the variable(s) that are imputed. This is because, in cases with imputation, there is guaranteed to be no relationship between the imputed variable and any other measured variables. Thus, mean imputation has some attractive properties for univariate analysis but becomes problematic for multivariate analysis.
Mean imputation can be carried out within classes (i.e. categories such as gender), and can be expressed as formula_0 where formula_1 is the imputed value for record formula_2 and formula_3 is the sample mean of respondent data within some class formula_4. This is a special case of generalized regression imputation:
formula_5
Here the values formula_6 are estimated from regressing formula_7 on formula_8 in non-imputed data, formula_9 is a dummy variable for class membership, and data are split into respondent (formula_10) and missing (formula_11).
Non-negative matrix factorization.
Non-negative matrix factorization (NMF) can take missing data while minimizing its cost function, rather than treating these missing data as zeros that could introduce biases. This makes it a mathematically proven method for data imputation. NMF can ignore missing data in the cost function, and the impact from missing data can be as small as a second order effect.
Regression.
Regression imputation has the opposite problem of mean imputation. A regression model is estimated to predict observed values of a variable based on other variables, and that model is then used to impute values in cases where the value of that variable is missing. In other words, available information for complete and incomplete cases is used to predict the value of a specific variable. Fitted values from the regression model are then used to impute the missing values. The problem is that the imputed data do not have an error term included in their estimation, thus the estimates fit perfectly along the regression line without any residual variance. This causes relationships to be over identified and suggest greater precision in the imputed values than is warranted. The regression model predicts the most likely value of missing data but does not supply uncertainty about that value.
Stochastic regression was a fairly successful attempt to correct the lack of an error term in regression imputation by adding the average regression variance to the regression imputations to introduce error. Stochastic regression shows much less bias than the above-mentioned techniques, but it still missed one thing – if data are imputed then intuitively one would think that more noise should be introduced to the problem than simple residual variance.
Multiple imputation.
In order to deal with the problem of increased noise due to imputation, Rubin (1987) developed a method for averaging the outcomes across multiple imputed data sets to account for this. All multiple imputation methods follow three steps.
Multiple imputation can be used in cases where the data are missing completely at random, missing at random, and missing not at random, though it can be biased in the latter case. One approach is multiple imputation by chained equations (MICE), also known as "fully conditional specification" and "sequential regression multiple imputation." MICE is designed for missing at random data, though there is simulation evidence to suggest that with a sufficient number of auxiliary variables it can also work on data that are missing not at random. However, MICE can suffer from performance problems when the number of observation is large and the data have complex features, such as nonlinearities and high dimensionality.
More recent approaches to multiple imputation use machine learning techniques to improve its performance. MIDAS (Multiple Imputation with Denoising Autoencoders), for instance, uses denoising autoencoders, a type of unsupervised neural network, to learn fine-grained latent representations of the observed data. MIDAS has been shown to provide accuracy and efficiency advantages over traditional multiple imputation strategies.
As alluded in the previous section, single imputation does not take into account the uncertainty in the imputations. After imputation, the data is treated as if they were the actual real values in single imputation. The negligence of uncertainty in the imputation can lead to overly precise results and errors in any conclusions drawn. By imputing multiple times, multiple imputation accounts for the uncertainty and range of values that the true value could have taken. As expected, the combination of both uncertainty estimation and deep learning for imputation is among the best strategies and has been used to model heterogeneous drug discovery data.
Additionally, while single imputation and complete case are easier to implement, multiple imputation is not very difficult to implement. There are a wide range of statistical packages in different statistical software that readily performs multiple imputation. For example, the MICE package allows users in R to perform multiple imputation using the MICE method. MIDAS can be implemented in R with the rMIDAS package and in Python with the MIDASpy package.
See also.
<templatestyles src="Div col/styles.css"/>
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\hat{y}_i = \\bar{y}_h"
},
{
"math_id": 1,
"text": "\\hat{y}_i "
},
{
"math_id": 2,
"text": "i"
},
{
"math_id": 3,
"text": "\\bar{y}_h"
},
{
"math_id": 4,
"text": "h"
},
{
"math_id": 5,
"text": "\n\\hat{y}_{mi} = b_{r0} + \\sum_j b_{rj} z_{mij} + \\hat{e}_{mi}\n"
},
{
"math_id": 6,
"text": "b_{r0}, b_{rj}"
},
{
"math_id": 7,
"text": "y"
},
{
"math_id": 8,
"text": "x"
},
{
"math_id": 9,
"text": "z"
},
{
"math_id": 10,
"text": "r"
},
{
"math_id": 11,
"text": "m"
}
]
| https://en.wikipedia.org/wiki?curid=1309220 |
1309936 | Iron(II,III) oxide | <templatestyles src="Chembox/styles.css"/>
Chemical compound
Iron(II,III) oxide, or black iron oxide, is the chemical compound with formula Fe3O4. It occurs in nature as the mineral magnetite. It is one of a number of iron oxides, the others being iron(II) oxide (FeO), which is rare, and iron(III) oxide (Fe2O3) which also occurs naturally as the mineral hematite. It contains both Fe2+ and Fe3+ ions and is sometimes formulated as FeO ∙ Fe2O3. This iron oxide is encountered in the laboratory as a black powder. It exhibits permanent magnetism and is ferrimagnetic, but is sometimes incorrectly described as ferromagnetic. Its most extensive use is as a black pigment (see: Mars Black). For this purpose, it is synthesized rather than being extracted from the naturally occurring mineral as the particle size and shape can be varied by the method of production.
Preparation.
Heated iron metal interacts with steam to form iron oxide and hydrogen gas.
<chem>3Fe + 4H2O->Fe3O4 + 4H2 </chem>
Under anaerobic conditions, ferrous hydroxide (Fe(OH)2) can be oxidized by water to form magnetite and molecular hydrogen. This process is described by the Schikorr reaction:
<chem>\underset{ferrous\ hydroxide}{3Fe(OH)2} -> \underset{magnetite}{Fe3O4} + \underset{hydrogen}{H2} + \underset{water}{2H2O}</chem>
This works because crystalline magnetite (Fe3O4) is thermodynamically more stable than amorphous ferrous hydroxide (Fe(OH)2 ).
The Massart method of preparation of magnetite as a ferrofluid, is convenient in the laboratory: mix iron(II) chloride and iron(III) chloride in the presence of sodium hydroxide.
A more efficient method of preparing magnetite without troublesome residues of sodium, is to use ammonia to promote chemical co-precipitation from the iron chlorides: first mix solutions of 0.1 M FeCl3·6H2O and FeCl2·4H2O with vigorous stirring at about 2000 rpm. The molar ratio of the FeCl3:FeCl2 should be about 2:1. Heat the mix to 70 °C, then raise the speed of stirring to about 7500 rpm and quickly add a solution of NH4OH (10 volume %). A dark precipitate of nanoparticles of magnetite forms immediately.
In both methods, the precipitation reaction relies on rapid transformation of acidic iron ions into the spinel iron oxide structure at pH 10 or higher.
Controlling the formation of magnetite nanoparticles presents challenges: the reactions and phase transformations necessary for the creation of the magnetite spinel structure are complex. The subject is of practical importance because magnetite particles are of interest in bioscience applications such as magnetic resonance imaging (MRI), in which iron oxide magnetite nanoparticles potentially present a non-toxic alternative to the gadolinium-based contrast agents currently in use. However, difficulties in controlling the formation of the particles, still frustrate the preparation of superparamagnetic magnetite particles, that is to say: magnetite nanoparticles with a coercivity of 0 A/m, meaning that they completely lose their permanent magnetisation in the absence of an external magnetic field. The smallest values currently reported for nanosized magnetite particles is "Hc" = 8.5 A m−1, whereas the largest reported magnetization value is 87 Am2 kg−1 for synthetic magnetite.
Pigment quality Fe3O4, so called synthetic magnetite, can be prepared using processes that use industrial wastes, scrap iron or solutions containing iron salts (e.g. those produced as by-products in industrial processes such as the acid vat treatment (pickling) of steel):
C6H5NO2 + 3 Fe + 2 H2O → C6H5NH2 + Fe3O4
Reduction of Fe2O3 with hydrogen:
3Fe2O3 + H2 → 2Fe3O4 +H2O
Reduction of Fe2O3 with CO:
3Fe2O3 + CO → 2Fe3O4 + CO2
Production of nano-particles can be performed chemically by taking for example mixtures of FeII and FeIII salts and mixing them with alkali to precipitate colloidal Fe3O4. The reaction conditions are critical to the process and determine the particle size.
Iron(II) carbonate can also be thermally decomposed into Iron(II,III):
Reactions.
Reduction of magnetite ore by CO in a blast furnace is used to produce iron as part of steel production process:
<chem>{Fe3O4} + 4CO -> {3Fe} + 4CO2</chem>
Controlled oxidation of Fe3O4 is used to produce brown pigment quality γ-Fe2O3 (maghemite):
formula_0
More vigorous calcining (roasting in air) gives red pigment quality α-Fe2O3 (hematite):
formula_1
Structure.
Fe3O4 has a cubic inverse spinel group structure which consists of a cubic close packed array of oxide ions where all of the Fe2+ ions occupy half of the octahedral sites and the Fe3+ are split evenly across the remaining octahedral sites and the tetrahedral sites.
Both FeO and γ-Fe2O3 have a similar cubic close packed array of oxide ions and this accounts for the ready interchangeability between the three compounds on oxidation and reduction as these reactions entail a relatively small change to the overall structure. Fe3O4 samples can be non-stoichiometric.
The ferrimagnetism of Fe3O4 arises because the electron spins of the FeII and FeIII ions in the octahedral sites are coupled and the spins of the FeIII ions in the tetrahedral sites are coupled but anti-parallel to the former. The net effect is that the magnetic contributions of both sets are not balanced and there is a permanent magnetism.
In the molten state, experimentally constrained models show that the iron ions are coordinated to 5 oxygen ions on average. There is a distribution of coordination sites in the liquid state, with the majority of both FeII and FeIII being 5-coordinated to oxygen and minority populations of both 4- and 6-fold coordinated iron.
Properties.
Fe3O4 is ferrimagnetic with a Curie temperature of . There is a phase transition at , called Verwey transition where there is a discontinuity in the structure, conductivity and magnetic properties. This effect has been extensively investigated and whilst various explanations have been proposed, it does not appear to be fully understood.
While it has much higher electrical resistivity than iron metal (96.1 nΩ m), Fe3O4's electrical resistivity (0.3 mΩ m ) is significantly lower than that of Fe2O3 (approx kΩ m). This is ascribed to electron exchange between the FeII and FeIII centres in Fe3O4.
Uses.
Fe3O4 is used as a black pigment and is known as "C.I pigment black 11" (C.I. No.77499) or Mars Black.
Fe3O4 is used as a catalyst in the Haber process and in the water-gas shift reaction. The latter uses an HTS (high temperature shift catalyst) of iron oxide stabilised by chromium oxide. This iron–chrome catalyst is reduced at reactor start up to generate Fe3O4 from α-Fe2O3 and Cr2O3 to CrO3.
Bluing is a passivation process that produces a layer of Fe3O4 on the surface of steel to protect it from rust. Along with sulfur and aluminium, it is an ingredient in steel-cutting thermite.
Medical uses.
Nano particles of Fe3O4 are used as contrast agents in MRI scanning.
Ferumoxytol, sold under the brand names Feraheme and Rienso, is an intravenous Fe3O4 preparation for treatment of anemia resulting from chronic kidney disease. Ferumoxytol is manufactured and globally distributed by AMAG Pharmaceuticals.
Biological occurrence.
Magnetite has been found as nano-crystals in magnetotactic bacteria (42–45 nm) and in the beak tissue of homing pigeons.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\ce{\\underbrace{2Fe3O4}_{magnetite} + {1/2O2} ->}\\ {\\color{Brown}\\ce{\\underbrace{3(\\gamma-Fe2O3)}_{maghemite}}}"
},
{
"math_id": 1,
"text": "\\ce{\\underbrace{2Fe3O4}_{magnetite} + {1/2O2} ->}\\ {\\color{BrickRed}\\ce{\\underbrace{3(\\alpha-Fe2O3)}_{hematite}}}"
}
]
| https://en.wikipedia.org/wiki?curid=1309936 |
1310490 | L-space | L-space may refer to:
Topics referred to by the same term
<templatestyles src="Dmbox/styles.css" />
This page lists associated with the title . | [
{
"math_id": 0,
"text": "\\ell^p"
}
]
| https://en.wikipedia.org/wiki?curid=1310490 |
13105770 | Lambda-mu calculus | Extension of lambda calculus
In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by Michel Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction.
One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law.
Semantically these operators correspond to continuations, found in some functional programming languages.
Formal definition.
We can augment the definition of a lambda expression to gain one in the context of lambda-mu calculus. The three main expressions found in lambda calculus are as follows:
For details, see the corresponding article.
In addition to the traditional λ-variables, the lambda-mu calculus includes a distinct set of μ-variables. These μ-variables can be used to "name" or "freeze" arbitrary subterms, allowing us to later abstract on those names. The set of terms contains "unnamed" (all traditional lambda expressions are of this kind) and "named" terms. The terms that are added by the lambda-mu calculus are of the form:
Reduction.
The basic reduction rules used in the lambda-mu calculus are the following:
formula_0
formula_1, where the substitutions are to be made for all subterms of formula_2 of the form formula_3.
formula_4
formula_5, for α not freely occurring in u
These rules cause the calculus to be confluent. Further reduction rules could be added to provide us with a stronger notion of normal form, though this would be at the expense of confluence.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "(\\lambda x.u)v \\; \\triangleright_c \\; u[v/x]"
},
{
"math_id": 1,
"text": "(\\mu \\beta.u)v \\; \\triangleright_c \\; \\mu \\beta.u \\left [ [\\beta](w v)/[\\beta] w \\right ]"
},
{
"math_id": 2,
"text": "u"
},
{
"math_id": 3,
"text": "[\\beta]w"
},
{
"math_id": 4,
"text": "[\\alpha] \\mu \\beta.u \\; \\triangleright_c \\; u [\\alpha / \\beta]"
},
{
"math_id": 5,
"text": "\\mu \\alpha.[\\alpha]u \\; \\triangleright_c \\; u"
}
]
| https://en.wikipedia.org/wiki?curid=13105770 |
13110176 | Mean curvature flow | Parabolic partial differential equation
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.
Under the constraint that volume enclosed is constant, this is called surface tension flow.
It is a parabolic partial differential equation, and can be interpreted as "smoothing".
Existence and uniqueness.
The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.
Let formula_0 be a compact smooth manifold, let formula_1 be a complete smooth Riemannian manifold, and let formula_2 be a smooth immersion. Then there is a positive number formula_3, which could be infinite, and a map formula_4 with the following properties:
Necessarily, the restriction of formula_20 to formula_21 is formula_10.
One refers to formula_20 as the (maximally extended) mean curvature flow with initial data formula_22.
Convex solutions.
Following Hamilton's epochal 1982 work on the Ricci flow, in 1984 Gerhard Huisken employed the same methods for the mean curvature flow to produce the following analogous result:
Note that if formula_24 and formula_32 is a smooth hypersurface immersion whose second fundamental form is positive, then the Gauss map formula_33 is a diffeomorphism, and so one knows from the start that formula_0 is diffeomorphic to formula_34 and, from elementary differential topology, that all immersions considered above are embeddings.
Gage and Hamilton extended Huisken's result to the case formula_35. Matthew Grayson (1987) showed that if formula_36 is any smooth embedding, then the mean curvature flow with initial data formula_22 eventually consists exclusively of embeddings with strictly positive curvature, at which point Gage and Hamilton's result applies. In summary:
Properties.
The mean curvature flow extremalizes surface area, and minimal surfaces are the critical points for the mean curvature flow; minima solve the isoperimetric problem.
For manifolds embedded in a Kähler–Einstein manifold, if the surface is a Lagrangian submanifold, the mean curvature flow is of Lagrangian type, so the surface evolves within the class of Lagrangian submanifolds.
Huisken's monotonicity formula gives a monotonicity property of the convolution of a time-reversed heat kernel with a surface undergoing the mean curvature flow.
Related flows are:
Mean curvature flow of a three-dimensional surface.
The differential equation for mean-curvature flow of a surface given by formula_43 is given by
formula_44
with formula_45 being a constant relating the curvature and the speed of the surface normal, and
the mean curvature being
formula_46
In the limits formula_47 and formula_48, so that the surface is nearly planar with its normal nearly
parallel to the z axis, this reduces to a diffusion equation
formula_49
While the conventional diffusion equation is a linear parabolic partial differential equation and does not develop
singularities (when run forward in time), mean curvature flow may develop singularities because it is a nonlinear parabolic equation. In general additional constraints need to be put on a surface to prevent singularities under
mean curvature flows.
Every smooth convex surface collapses to a point under the mean-curvature flow, without other singularities, and converges to the shape of a sphere as it does so. For surfaces of dimension two or more this is a theorem of Gerhard Huisken; for the one-dimensional curve-shortening flow it is the Gage–Hamilton–Grayson theorem. However, there exist embedded surfaces of two or more dimensions other than the sphere that stay self-similar as they contract to a point under the mean-curvature flow, including the Angenent torus.
Example: mean curvature flow of "m"-dimensional spheres.
A simple example of mean curvature flow is given by a family of concentric round hyperspheres in formula_50. The mean curvature of an formula_51-dimensional sphere of radius formula_52 is formula_53.
Due to the rotational symmetry of the sphere (or in general, due to the invariance of mean curvature under isometries) the mean curvature flow equation formula_54 reduces to the ordinary differential equation, for an initial sphere of radius formula_55,
formula_56
The solution of this ODE (obtained, e.g., by separation of variables) is
formula_57,
which exists for formula_58.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "M"
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"math_id": 1,
"text": "(M',g)"
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"math_id": 2,
"text": "f:M\\to M'"
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{
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"text": "T"
},
{
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"text": "F:[0,T)\\times M\\to M'"
},
{
"math_id": 5,
"text": "F(0,\\cdot)=f"
},
{
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"text": "F(t,\\cdot):M\\to M'"
},
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"text": "t\\in[0,T)"
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"text": "t\\searrow 0,"
},
{
"math_id": 9,
"text": "F(t,\\cdot)\\to f"
},
{
"math_id": 10,
"text": "C^\\infty"
},
{
"math_id": 11,
"text": "(t_0,p)\\in(0,T)\\times M"
},
{
"math_id": 12,
"text": "t\\mapsto F(t,p)"
},
{
"math_id": 13,
"text": "t_0"
},
{
"math_id": 14,
"text": "F(t_0,\\cdot)"
},
{
"math_id": 15,
"text": "p"
},
{
"math_id": 16,
"text": "\\widetilde{F}:[0,\\widetilde{T})\\times M\\to M'"
},
{
"math_id": 17,
"text": "\\widetilde{T}\\leq T"
},
{
"math_id": 18,
"text": "\\widetilde{F}(t,p)=F(t,p)"
},
{
"math_id": 19,
"text": "(t,p)\\in [0,\\widetilde{T})\\times M."
},
{
"math_id": 20,
"text": "F"
},
{
"math_id": 21,
"text": "(0,T)\\times M"
},
{
"math_id": 22,
"text": "f"
},
{
"math_id": 23,
"text": "\\mathbb{R}^{n+1}"
},
{
"math_id": 24,
"text": "n\\geq 2"
},
{
"math_id": 25,
"text": "F(t,\\cdot)"
},
{
"math_id": 26,
"text": "t\\in(0,T)"
},
{
"math_id": 27,
"text": "c:(0,T)\\to(0,\\infty)"
},
{
"math_id": 28,
"text": "(M,(c(t)F(t,\\cdot))^\\ast g_{\\text{Euc}})"
},
{
"math_id": 29,
"text": "t"
},
{
"math_id": 30,
"text": "t\\nearrow T"
},
{
"math_id": 31,
"text": "c(t)F(t,\\cdot):M\\to\\mathbb{R}^{n+1}"
},
{
"math_id": 32,
"text": "f:M\\to\\mathbb{R}^{n+1}"
},
{
"math_id": 33,
"text": "\\nu:M\\to S^n"
},
{
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"text": "S^n"
},
{
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"text": "n=1"
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{
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"text": "f:S^1\\to\\mathbb{R}^2"
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{
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"text": "F:[0,T)\\times S^1\\to\\mathbb{R}^2"
},
{
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"text": "F(t,\\cdot):S^1\\to\\mathbb{R}^2"
},
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"text": "t_0\\in(0,T)"
},
{
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"text": "t\\in(t_0,T)"
},
{
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"text": "c"
},
{
"math_id": 42,
"text": "c(t)F(t,\\cdot):S^1\\to\\mathbb{R}^2"
},
{
"math_id": 43,
"text": "z=S(x,y)"
},
{
"math_id": 44,
"text": "\\frac{\\partial S}{\\partial t} = 2D\\ H(x,y) \\sqrt{1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2}\n"
},
{
"math_id": 45,
"text": "D"
},
{
"math_id": 46,
"text": "\n\\begin{align}\nH(x,y) & = \n\\frac{1}{2}\\frac{\n\\left(1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2\\right) \\frac{\\partial^2 S}{\\partial y^2} - \n2 \\frac{\\partial S}{\\partial x} \\frac{\\partial S}{\\partial y} \\frac{\\partial^2 S}{\\partial x \\partial y} + \n\\left(1 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2\\right) \\frac{\\partial^2 S}{\\partial x^2}\n}{\\left(1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2\\right)^{3/2}}.\n\\end{align}\n"
},
{
"math_id": 47,
"text": " \\left|\\frac{\\partial S}{\\partial x}\\right| \\ll 1 "
},
{
"math_id": 48,
"text": " \\left|\\frac{\\partial S}{\\partial y}\\right| \\ll 1 "
},
{
"math_id": 49,
"text": "\\frac{\\partial S}{\\partial t} = D\\ \\nabla^2 S\n"
},
{
"math_id": 50,
"text": "\\mathbb{R}^{m+1}"
},
{
"math_id": 51,
"text": "m"
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{
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"text": "R"
},
{
"math_id": 53,
"text": "H = m/R"
},
{
"math_id": 54,
"text": "\\partial_t F = - H \\nu"
},
{
"math_id": 55,
"text": "R_0"
},
{
"math_id": 56,
"text": "\\begin{align}\n\\frac{\\text{d}}{\\text{d}t}R(t) & = - \\frac{m}{R(t)} , \\\\\nR(0) & = R_0 .\n\\end{align}"
},
{
"math_id": 57,
"text": "R(t) = \\sqrt{R_0^2 - 2 m t}"
},
{
"math_id": 58,
"text": "t \\in (-\\infty,R_0^2/2m)"
}
]
| https://en.wikipedia.org/wiki?curid=13110176 |
13111146 | VAM (bicycling) | Metric used to measure the speed of elevation gain in cycling
VAM is the abbreviation for the Italian term velocità ascensionale media, translated in English to mean "average ascent speed" or "mean ascent velocity", but usually referred to as VAM. It is also referred to by the English backronym "Vertical Ascent in Meters". The term, which was coined by Italian physician and cycling coach Michele Ferrari, is the speed of elevation gain, usually stated in units of metres per hour.
Background.
VAM is a parameter used in cycling as a measure of fitness and speed; it is useful for relatively objective comparisons of performances and estimating a rider's power output per kilogram of body mass, which is one of the most important qualities of a cyclist who competes in stage races and other mountainous events. Dr. Michele Ferrari also stated that VAM values exponentially rise up with every gradient increase. For example, a 1180 VAM of a 64 kg rider on a 5% gradient is equivalent to a VAM of 1400 m/h on a 10 % or a VAM of 1675 m/h on a 13% gradient. Ambient conditions (e.g. friction, air resistance) have less effect on steeper slopes (absorb less power) since speeds are lower on steeper slopes
The acronym VAM is not truly expanded in English, where many think the V stands in some way for vertical, and the M represents metres, for instance "Vertical Ascent Metres/Hour." Ferrari says,
"I called this parameter Average Ascent Speed (‘VAM’ in its Italian abbreviation from Velocità Ascensionale Media)."
A direct translation of "velocità ascensionale media" is "mean (average) ascent velocity" leading to an expansion of the acronym in English as Velocity, Ascent, Mean.
Definition.
VAM is calculated the following way:
VAM = (metres ascended × 60) / minutes it took to ascend
A standard unit term with the same meaning is Vm/h, vertical metres per hour; the two are used interchangeably.
Relationship to relative power output.
Relative power means power "P" per body mass "m". Without friction and extra mass (the bicycle), the relative power would be VAM times acceleration of gravity "g":
formula_0
With "g" = 9.81 m/s2, this is equivalent to
Relative power (watts/kg) = VAM (metres/hour) formula_1 VAM (metres/hour) / 367
Including the power necessary for the extra mass and dissipated by friction leads to a lower number in the denominator. An empirical relationship is
Relative power (watts/kg) = VAM (metres/hour) / (200 + 10 × % grade)
Examples.
Examples:
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\frac P m = \\mathrm{VAM}\\cdot g"
},
{
"math_id": 1,
"text": "\\cdot\\frac{9.81}{3600}\\approx"
}
]
| https://en.wikipedia.org/wiki?curid=13111146 |
13111316 | Oyugis | Oyugis is the second-largest town (after Homa Bay town) in Homa Bay County in Nyanza Kenya. The town lies along the Kisumu-Kisii highway. It is the commercial and financial centre of Rachuonyo Sub-County in Homa Bay County of the former Nyanza Province.
Though small in comparison to neighbouring Kisii, Oyugis has recently registered fast growth especially since the creation of the district and increased use of the town by NGOs operating in the area and the Sondu Miriu hydroelectric power station.
Location.
Oyugis is located 20 kilometres north of Kisii town along the major A1 Highway. There is also an inter-county road (C26) connecting Oyugis to Kendu Bay while C18 road links Oyugis with Homa Bay Town via Rangwe. There are also other link roads like Oyugis-Gamba which heads to Nyamira and Kisii counties.
Population.
At the last census taken, Oyugis had a town council with a population of 52,433, of whom 9,084 are classified urban (1999 census ). Much of the adult population has taken to the Agrarian lifestyle. The population has a relatively low level of HIV/AIDS in comparison to the other towns located in the Nyanza province.
Electoral constituency.
Oyugis area forms Kasipul Constituency. It has five County Assembly wards: West Kamagak Ward, East Kamagak Ward, West Kasipul Ward, Central Kasipul Ward and South Kasipul Ward. Administratively, Oyugis is in Kasipul division which is one of the 19 divisions in Homabay county. The Kasipul Constituency is partitioned from the Kasipul Kabondo Constituency.
Education.
The town hosts several important academic institutions like the Agoro Sare High School and Oyugis Craft Training Center. Other notable institutions include Wire Secondary School, Nyabola Girls school, Buoye Secondary School Mititi mixed secondary school, Nyagiela secondary, DOL-Kodera mixed secondary, Nyagowa secondary school, Kwoyo-kotieno secondary school among others. It is also the home to one of the oldest primary schools in Kenya, Wire Primary School, DOL-kodera Primary established in 1912,1927 respectively.
Hearts For Kenya.
Hearts for Kenya, based in Louisville, Kentucky, exists for the purpose of combating poverty, hunger and disease in small, agrarian communities within the Nyanza province of Kenya. Its headquarters, the "Amani Daycare Center and Agrarian Institute" are located in the town of Oyugis. Ancillary projects, which are vital to the success of the agricultural project, are centered around building, nutrition, education, assistance to orphans and widows, health services, and a tree nursery. The intent of Hearts for Kenya is to enable the local citizens to carry on the projects autonomously.
The core of the venture spearheaded by Hearts for Kenya, now in its eleventh year, is a project aimed primarily at increasing agricultural productivity and establishing a niche crop. By 2017, they hope to have a medical clinic built, established and operating.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\uparrow"
}
]
| https://en.wikipedia.org/wiki?curid=13111316 |
13115105 | Caloric polynomial | In differential equations, the "m"th-degree caloric polynomial (or heat polynomial) is a "parabolically "m"-homogeneous" polynomial "P""m"("x", "t") that satisfies the heat equation
formula_0
"Parabolically "m"-homogeneous" means
formula_1
The polynomial is given by
formula_2
It is unique up to a factor.
With "t" = −1/2, this polynomial reduces to the "m"th-degree Hermite polynomial in "x". | [
{
"math_id": 0,
"text": " \\frac{\\partial P}{\\partial t} = \\frac{\\partial^2 P}{\\partial x^2}. "
},
{
"math_id": 1,
"text": " P(\\lambda x, \\lambda^2 t) = \\lambda^m P(x,t)\\text{ for }\\lambda > 0.\\, "
},
{
"math_id": 2,
"text": " P_m(x,t) = \\sum_{\\ell=0}^{\\lfloor m/2 \\rfloor} \\frac{m!}{\\ell!(m - 2\\ell)!} x^{m - 2\\ell} t^\\ell. "
}
]
| https://en.wikipedia.org/wiki?curid=13115105 |
13117099 | Hanes–Woolf plot | Graph of enzyme kinetics
In biochemistry, a Hanes–Woolf plot, Hanes plot, or plot of formula_0 against formula_1 is a graphical representation of enzyme kinetics in which the ratio of the initial substrate concentration formula_1 to the reaction velocity formula_2 is plotted against formula_1. It is based on the rearrangement of the Michaelis–Menten equation shown below:
formula_3
where formula_4 is the Michaelis constant and formula_5 is the limiting rate.
J. B. S. Haldane stated, reiterating what he and K. G. Stern had written in their book, that this rearrangement was due to Barnet Woolf. However, it was just one of three transformations introduced by Woolf. It was first published by C. S. Hanes, though he did not use it as a plot. Hanes noted that the use of linear regression to determine kinetic parameters from this type of linear transformation generates the best fit between observed and calculated values of formula_6, rather than formula_2.1415
Starting from the Michaelis–Menten equation:
formula_7
we can take reciprocals of both sides of the equation to obtain the equation underlying the Lineweaver–Burk plot:
formula_8
which can be multiplied on both sides by formula_9 to give
formula_10
Thus in the absence of experimental error data a plot of formula_11 against formula_9 yields a straight line of slope formula_12, an intercept on the ordinate of formula_13and an intercept on the abscissa of formula_14.
Like other techniques that linearize the Michaelis–Menten equation, the Hanes–Woolf plot was used historically for rapid determination of the kinetic parameters formula_4, formula_5 and formula_15, but it has been largely superseded by nonlinear regression methods that are significantly more accurate and no longer computationally inaccessible. It remains useful, however, as a means to present data graphically.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "a/v"
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{
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"text": "a"
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{
"math_id": 2,
"text": "v"
},
{
"math_id": 3,
"text": "{a \\over v } = { a \\over V } + { K_\\mathrm{m} \\over V} "
},
{
"math_id": 4,
"text": "K_\\mathrm{m}"
},
{
"math_id": 5,
"text": "V"
},
{
"math_id": 6,
"text": "1/v"
},
{
"math_id": 7,
"text": "v = {{Va} \\over {K_\\mathrm{m} + a}}"
},
{
"math_id": 8,
"text": "{1 \\over v} = {1 \\over V} + {K_\\mathrm{m} \\over V} \\cdot {1 \\over a}"
},
{
"math_id": 9,
"text": " {a}"
},
{
"math_id": 10,
"text": " {a \\over v} = {1 \\over V } \\cdot a + {K_\\mathrm{m}\\over V }"
},
{
"math_id": 11,
"text": " {a/v}"
},
{
"math_id": 12,
"text": "1/V"
},
{
"math_id": 13,
"text": "{K_\\mathrm{m}/ V }"
},
{
"math_id": 14,
"text": "-K_\\mathrm{m}"
},
{
"math_id": 15,
"text": "K_\\mathrm{m}/V"
}
]
| https://en.wikipedia.org/wiki?curid=13117099 |
13118420 | McCumber relation | The McCumber relation (or McCumber theory) is a relationship between the effective cross-sections of absorption and emission of light in the physics of solid-state lasers. It is named after Dean McCumber, who proposed the relationship in 1964.
Definition.
Let formula_0 be the effective absorption cross-section formula_1 be effective emission cross-sections at frequency formula_2, and let formula_3 be the effective temperature of the medium. The McCumber relation is
(1) formula_4
where formula_5
is thermal steady-state ratio of populations; frequency formula_6 is called "zero-line" frequency;
formula_7 is the Planck constant and
formula_8 is the Boltzmann constant. Note that the right-hand side of Equation (1) does not depend on formula_9.
Gain.
It is typical that the lasing properties of a medium are determined by the temperature and the population at the excited laser level, and are not sensitive to the method of excitation used to achieve it. In this case, the absorption cross-section
formula_0
and the emission cross-section
formula_1
at frequency formula_9 can be related to the lasers gain in such a way, that the gain at this frequency can be determined as follows:
(2) formula_10
D.E.McCumber had postulated these properties and found that the emission and absorption cross-sections are not independent; they are related with Equation (1).
Idealized atoms.
In the case of an idealized two-level atom the detailed balance for the emission and absorption which preserves the Planck formula for the black-body radiation leads to equality of cross-section of absorption and emission. In the solid-state lasers the splitting of each of laser levels leads to the broadening which greatly exceeds the natural spectral linewidth. In the case of an ideal two-level atom, the product of the linewidth and the lifetime is of order of unity, which obeys the Heisenberg uncertainty principle. In solid-state laser materials, the linewidth is several orders of magnitude larger so the spectra of emission and absorption are determined by distribution of excitation among sublevels rather than by the shape of the spectral line of each individual transition between sublevels. This distribution is determined by the effective temperature within each of laser levels. The McCumber hypothesis is that the distribution of excitation among sublevels is thermal. The effective temperature determines the spectra of emission and absorption ( The "effective temperature" is called a "temperature" by scientists even if the excited medium as whole is pretty far from the thermal state )
Deduction of the McCumber relation.
Consider the set of active centers (fig.1.). Assume fast transition between sublevels within each level, and slow transition between levels.
According to the McCumber hypothesis, the cross-sections formula_11 and formula_12 do not depend on the populations formula_13 and formula_14.
Therefore, we can deduce the relation, assuming the thermal state.
Let formula_15 be group velocity of light in the medium, the product formula_16 is spectral rate of
stimulated emission, and formula_17 is that of absorption; formula_18 is spectral rate of spontaneous emission. (Note that in this approximation, there is no such thing as a spontaneous absorption)
The balance of photons gives:
(3) formula_19
Which can be rewritten as
(4) formula_20
The thermal distribution of density of photons follows from blackbody radiation
(5) formula_21
Both (4) and (5) hold for all frequencies formula_9. For the case of idealized two-level active centers, formula_22, and formula_23, which leads to the relation between the spectral rate of spontaneous emission formula_24 and the emission cross-section formula_25. (We keep the term probability of emission for the quantity formula_26, which is probability of emission of a photon within small spectral interval formula_27 during a short time interval formula_28, assuming that at time formula_29 the atom is excited.) The relation (D2) is a fundamental property of spontaneous and stimulated emission, and perhaps the only way to prohibit a spontaneous break of the thermal equilibrium in the thermal state of excitations and photons.
For each site number formula_30, for each sublevel number formula_31, the partial spectral emission probability formula_32 can be expressed from consideration of idealized two-level atoms:
(6) formula_33
Neglecting the cooperative coherent effects, the emission is additive: for any concentration formula_34 of sites and for any partial population formula_35 of sublevels, the same proportionality between formula_36 and formula_37 holds for the effective cross-sections:
(7) formula_38
Then, the comparison of (D1) and (D2) gives the relation
(8) formula_39
This relation is equivalent of the McCumber relation (mc), if we define the zero-line frequency formula_40 as solution of equation
(9) formula_41
the subscript formula_3 indicates that the ratio of populations in evaluated in the thermal state. The zero-line frequency can be expressed as
(10) formula_42
Then (n1n2) becomes equivalent of the McCumber relation (mc).
No specific property of sublevels of active medium is required to keep the McCumber relation. It follows from the assumption about quick transfer of energy among excited laser levels and among lower laser levels. The McCumber relation (mc) has the same range of validity as the concept of the emission cross-section
itself.
Confirmation of the McCumber relation.
The McCumber relation is confirmed for various media.
In particular, relation (1) makes it possible to approximate two functions of frequency, emission and absorption cross sections, with single fit
Violation of the McCumber relation and perpetual motion.
In 2006 the strong violation of McCumber relation was observed for Yb:Gd2SiO5 and reported in 3 independent journals. Typical behavior of the cross-sections reported is shown in Fig.2 with thick curves. The emission cross-section is practically zero at wavelength
975 nm; this property makes Yb:Gd2SiO5 an excellent material for efficient solid-state lasers.
However, the property reported (thick curves) is not compatible with the second law of thermodynamics. With such a material, the perpetual motion device would be possible. It would be sufficient to fill a box with reflecting walls with Yb:Gd2SiO5 and allow it to exchange radiation with a black body through a spectrally-selective window which is transparent in vicinity of 975 nm and reflective at other wavelengths. Due to the lack of emissivity at 975 nm the medium should warm, breaking the thermal equilibrium.
On the base of the second Law of thermodynamics, the experimental results
were refuted in 2007. With the McCumber theory, the correction was suggested for the effective emission cross section (black thin curve).
Then this correction was confirmed experimentally.
References.
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{
"math_id": 0,
"text": "\\sigma_{\\rm a}(\\omega)"
},
{
"math_id": 1,
"text": "\\sigma_{\\rm e}(\\omega)"
},
{
"math_id": 2,
"text": "\\omega"
},
{
"math_id": 3,
"text": "~T~"
},
{
"math_id": 4,
"text": "\\frac{\\sigma_{\\rm e}(\\omega)}{\\sigma_{\\rm a}(\\omega)}\\exp\\!\\left( \\frac{\\hbar \\omega}{k_{\\rm B} T}\\right)\n=\\left(\\frac{N_1}{N_2}\\right)_T\n=\\exp\\!\\left( \\frac{\\hbar \\omega_{\\rm z}}{k_{\\rm B} T}\\right)"
},
{
"math_id": 5,
"text": " \\left(\\frac{N_1}{N_2}\\right)_T"
},
{
"math_id": 6,
"text": "\\omega_{\\rm z}"
},
{
"math_id": 7,
"text": "\\hbar "
},
{
"math_id": 8,
"text": "k_{\\rm B} "
},
{
"math_id": 9,
"text": "~\\omega~"
},
{
"math_id": 10,
"text": "~~~~~~~~~~~~~~~G(\\omega)=N_2 \\sigma_{\\rm e}(\\omega)-N_1 \\sigma_{\\rm a}(\\omega)"
},
{
"math_id": 11,
"text": "\\sigma_{\\rm a}"
},
{
"math_id": 12,
"text": "\\sigma_{\\rm e}"
},
{
"math_id": 13,
"text": "N_1"
},
{
"math_id": 14,
"text": "N_2"
},
{
"math_id": 15,
"text": "~v(\\omega)~"
},
{
"math_id": 16,
"text": "~n_2\\sigma_{\\rm e}(\\omega) v(\\omega)D(\\omega)~"
},
{
"math_id": 17,
"text": "~n_1\\sigma_{\\rm a}(\\omega) v(\\omega)D(\\omega)~"
},
{
"math_id": 18,
"text": "a(\\omega)n_2"
},
{
"math_id": 19,
"text": "~~~\nn_2\\sigma_{\\rm e}(\\omega) v(\\omega)D(\\omega)+n_2 a(\\omega)=\nn_1\\sigma_{\\rm a}(\\omega) v(\\omega)D(\\omega)\n~~~~~~~~~~~~~~~{\\rm (balance)}\n "
},
{
"math_id": 20,
"text": "~~~\nD(\\omega)=\n\\frac{\\frac{a(\\omega)}{\\sigma_{\\rm e}(\\omega) v(\\omega)}}\n{\\frac{n_1}{n_2} \\frac{\\sigma_{\\rm a}(\\omega)}{\\sigma_{\\rm e}(\\omega)}-1}\n~~~~~~~~~~~~~~{\\rm (D1)} "
},
{
"math_id": 21,
"text": "~~~\nD(\\omega)~=~\n\\frac{\\frac{1}{\\pi^2} \\frac{\\omega^2}{c^3}}\n{\\exp\\!\\left(\\frac{\\hbar\\omega}{k_{\\rm B}T}\\right)-1}\n~~~~~\n{\\rm (D2)}\n"
},
{
"math_id": 22,
"text": "~\\sigma_{\\rm a}(\\omega)=\\sigma_{\\rm e}(\\omega)~"
},
{
"math_id": 23,
"text": "~n_1/n_2=\\exp\\!\\left( \\frac{\\hbar\\omega}{k_{\\rm B}T} \\right)"
},
{
"math_id": 24,
"text": "a(\\omega)"
},
{
"math_id": 25,
"text": "~\\sigma_{\\rm e}(\\omega)~"
},
{
"math_id": 26,
"text": "~a(\\omega){\\rm d}\\omega{\\rm d}t~"
},
{
"math_id": 27,
"text": "~(\\omega,\\omega+{\\rm d}\\omega)~"
},
{
"math_id": 28,
"text": "~(t,t+{\\rm d}t)~"
},
{
"math_id": 29,
"text": "~t~"
},
{
"math_id": 30,
"text": "~s~"
},
{
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"text": "j"
},
{
"math_id": 32,
"text": "~a_{s,j}(\\omega)~"
},
{
"math_id": 33,
"text": "~~~\na_{s,j}(\\omega)=\\sigma_{s,j}(\\omega)\n\\frac{\\omega^2 v(\\omega)}{\\pi^2c^3}~~.\n~~~~~~~~~~~~~~~~{\\rm comparison1}\n~~{\\rm partial}\n"
},
{
"math_id": 34,
"text": "~q_{s}~"
},
{
"math_id": 35,
"text": "~n_{s,j}~"
},
{
"math_id": 36,
"text": "~a~"
},
{
"math_id": 37,
"text": "~\\sigma_{\\rm e}~"
},
{
"math_id": 38,
"text": "\n\\frac{a(\\omega)}{\\sigma_{\\rm e}(\\omega)}=\n\\frac{\\omega^2 v(\\omega)}{\\pi^2c^3}\n~~~~~~~~~~~~~~~~~~(\\rm comparison)(av)\n "
},
{
"math_id": 39,
"text": "\n\\frac{n_1}{n_2}\n\\frac{\\sigma_{\\rm a}(\\omega)}\n{\\sigma_{\\rm e}(\\omega)}=\n\\exp\\!\\left( \\frac{\\hbar\\omega}{k_{\\rm B}T}\\right)~~.\n~~~~~~~~{\\rm (n1n2) (mc1)}\n "
},
{
"math_id": 40,
"text": "\\omega_{Z}"
},
{
"math_id": 41,
"text": "~\\left(\\frac{n_1}{n_2}\\right)_{\\!T}=\n\\exp\\!\\left(\\frac{\\hbar \\omega_{\\rm Z}}{k_{\\rm B}T}\\right)~~~~,~~~\n"
},
{
"math_id": 42,
"text": " \\omega_{\\rm Z}=\\frac{k_{\\rm B}T}{\\hbar}\n\\ln \\left(\\frac{n_1}{n_2}\\right)_{T} ~~~~~~~~~~~~~~~~.~~{(\\rm oz)} "
}
]
| https://en.wikipedia.org/wiki?curid=13118420 |
1311933 | Thiele/Small parameters | Set of electromechanical parameters
Thiele/Small parameters (commonly abbreviated T/S parameters, or TSP) are a set of electromechanical parameters that define the specified low frequency performance of a . These parameters are published in specification sheets by driver manufacturers so that designers have a guide in selecting off-the-shelf drivers for loudspeaker designs. Using these parameters, a loudspeaker designer may simulate the position, velocity and acceleration of the diaphragm, the input impedance and the sound output of a system comprising a loudspeaker and enclosure. Many of the parameters are strictly defined only at the resonant frequency, but the approach is generally applicable in the frequency range where the diaphragm motion is largely pistonic, i.e., when the entire cone moves in and out as a unit without cone breakup.
Rather than purchase off-the-shelf components, loudspeaker design engineers often define desired performance and work backwards to a set of parameters and manufacture a driver with said characteristics or order it from a driver manufacturer. This process of generating parameters from a target response is known as synthesis. Thiele/Small parameters are named after A. Neville Thiele of the Australian Broadcasting Commission, and Richard H. Small of the University of Sydney, who pioneered this line of analysis for loudspeakers. A common use of Thiele/Small parameters is in designing PA system and hi-fi speaker enclosures; the TSP calculations indicate to the speaker design professionals how large a speaker cabinet will need to be and how large and long the bass reflex port (if it is used) should be.
History.
The 1925 paper of Chester W. Rice and Edward W. Kellogg, fueled by advances in radio and electronics, increased interest in direct radiator loudspeakers. In 1930, A. J. Thuras of Bell Labs patented (US Patent No. 1869178) his "Sound Translating Device" (essentially a vented box) which was evidence of the interest in many types of enclosure design at the time.
Progress on loudspeaker enclosure design and analysis using acoustic analogous circuits by academic acousticians like Harry F. Olson continued until 1954 when Leo L. Beranek of the Massachusetts Institute of Technology published "Acoustics", a book summarizing and extending the electroacoustics of the era. J. F. Novak used novel simplifying assumptions in an analysis in a 1959 paper which led to a practical solution for the response of a given loudspeaker in sealed and vented boxes, and also established their applicability by empirical measurement. In 1961, leaning heavily on Novak's work, A. N. Thiele described a series of sealed and vented box "alignments" (i.e., enclosure designs based on electrical filter theory with well-characterized behavior, including frequency response, power handling, cone excursion, etc.) in a publication in an Australian journal. This paper remained relatively unknown outside Australia until it was re-published in the "Journal of the Audio Engineering Society" in 1971. It is important to note that Thiele's work neglected enclosure losses and, although the application of filter theory is still important, his alignment tables now have little real-world utility due to neglecting enclosure losses.
Many others continued to develop various aspects of loudspeaker enclosure design in the 1960s and early 1970s. From 1968 to 1972, J. E. Benson published three articles in an Australian journal that thoroughly analyzed sealed, vented and passive radiator designs, all using the same basic model, which included the effects of enclosure, leakage and port losses. Beginning in June 1972, Richard H. Small published a series of very influential articles on direct radiator loudspeaker system analysis, including closed-box, vented-box, and passive-radiator loudspeaker systems, in the "Journal of the Audio Engineering Society", restating and extending Thiele's work. These articles were also originally published in Australia, where he had attended graduate school, and where his thesis supervisor was J. E. Benson. The work of Benson and Small overlapped considerably, but differed in that Benson performed his work using computer programs and Small used analog simulators. Small also analyzed the systems including enclosure losses. Richard H. Small and Garry Margolis, the latter of JBL, published an article in the "Journal of the Audio Engineering Society" (June 1981), which recast much of the work that had been published up till then into forms suited to the programmable calculators of the time.
Fundamental parameters.
These are the physical parameters of a loudspeaker driver, as measured at small signal levels, used in the equivalent electrical circuit models. Some of these values are neither easy nor convenient to measure in a finished loudspeaker driver, so when designing speakers using existing drive units (which is almost always the case), the more easily measured parameters listed under "Small Signal Parameters" are more practical:
Small signal parameters.
These values can be determined by measuring the input impedance of the driver, near the resonance frequency, at small input levels for which the mechanical behavior of the driver is effectively linear (i.e., proportional to its input). These values are more easily measured than the fundamental ones above. The small signal parameters are:
formula_9
formula_12
formula_14
formula_16
formula_18
where formula_19 is the density of air (1.184 kg/m3 at 25 °C), and formula_20 is the speed of sound (346.1 m/s at 25 °C). Using SI units, the result will be in cubic metres. To convert formula_17 to litres, multiply by 1000.
Large signal parameters.
These parameters are useful for predicting the approximate output of a driver at high input levels, though they are harder, sometimes extremely hard or impossible, to accurately measure. In addition, power compression, thermal, and mechanical effects due to high signal levels (e.g., high electric current and voltage, extended mechanical motion, and so on) all change driver behavior, often increasing distortion of several kinds:
formula_26
formula_31
formula_34
The expression formula_35 can be replaced by the value 5.445×10−4 m2·s/kg for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 5.365×10−4 m2·s/kg.
A version that is more easily calculated with typical published parameters is:
formula_36
The expression formula_37 can be replaced by the value 9.523×10−7 s3/m3 for dry air at 25 °C. For 25 °C air with 50% relative humidity the expression evaluates to 9.438×10−7 s3/m3.
A speaker with an efficiency of 100% (1.0) would output a watt for every watt of input. Considering the driver as a point source in an infinite baffle, at one metre this would be distributed over a hemisphere with area formula_38 m2 for an intensity of formula_39 = 0.159155 W/m2. The auditory threshold is taken to be 10–12 W/m2 (which corresponds to a pressure level of 20×10−6 Pa). Therefore a speaker with 100% efficiency would produce an SPL equal to 10log(0.159155/10–12), which is 112.02 dB.
The SPL at 1 metre for an input of 1 watt is then: dB(1 watt) = 112.02 + 10·log(formula_33)
The SPL at 1 metre for an input of 2.83 volts is then: dB(2.83 V) = dB(1 watt) + 10·log(8/formula_40) = 112.02 + 10·log(formula_33) + 10·log(8/formula_40)
Resonance frequency of driver, measured in hertz (Hz). The frequency at which the combination of the energy stored in the moving mass and suspension compliance is maximum, and results in maximum cone velocity. A more compliant suspension or a larger moving mass will cause a lower resonance frequency, and vice versa. Usually it is less efficient to produce output at frequencies below formula_8, and input signals significantly below formula_8 can cause large excursions, mechanically endangering the driver. Woofers typically have an formula_8 in the range of 13–60 Hz. Midranges usually have an formula_8 in the range of 60–500 Hz and tweeters between 500 Hz and 4 kHz. A typical factory tolerance for the value of formula_8 is ±15%.
A unitless measurement, characterizing the combined electric and mechanical damping of the driver. In electronics, formula_11 is the inverse of the damping ratio. The value of formula_15 is proportional to the energy stored, divided by the energy dissipated, and is defined at resonance (formula_8). Most drivers have formula_15 values between 0.2 and 0.5, but there are valid (if unusual) reasons to have a value outside this range.
A unitless measurement, characterizing the mechanical damping of the driver, that is, the losses in the suspension (surround and spider). It varies roughly between 0.5 and 10, with a typical value around 3. High formula_13 indicates lower mechanical losses, and low formula_13 indicates higher losses. The main effect of formula_13 is on the impedance of the driver, with high formula_13 drivers displaying a higher impedance peak. One predictor for low formula_13 is a metallic voice-coil former. These act as eddy-current brakes and increase damping, reducing formula_13. They must be designed with an electrical break in the cylinder (so no conducting loop). Some speaker manufacturers have placed shorted turns at the top and bottom of the voice coil to prevent it leaving the gap, but the sharp noise created by this device when the driver is overdriven is alarming and was perceived as a problem by owners. High formula_13 drivers are often built with nonconductive formers made from paper or various plastics.
A unitless measurement, describing the electrical damping of the loudspeaker. As the coil of wire moves through the magnetic field, it generates a current which opposes the motion of the coil. This so-called "Back-EMF" (proportional to formula_7 × velocity) decreases the total current through the coil near the resonance frequency, reducing cone movement and increasing impedance. In most drivers, formula_10 is the dominant factor in the voice coil damping. formula_10 depends on amplifier output impedance. The formula above assumes zero output impedance. When an amplifier with nonzero output impedance is used, its output impedance should be added to formula_6 for calculations involving formula_10.
Measured in tesla-metres (T·m). Technically this is formula_41 or formula_42 (a vector cross product), but the standard geometry of a circular coil in an annular voice-coil gap gives formula_43. formula_41 is also known as the 'force factor' because the force on the coil imposed by the magnet is formula_41 multiplied by the current through the coil. The higher the formula_41 product, the larger the force that is generated by a given current flowing through the voice coil. formula_41 has a very strong effect on formula_10.
Measured in litres (L) or cubic metres, it is an inverse measure of the 'stiffness' of the suspension with the driver mounted in free air. It represents the volume of air that has the same stiffness as the driver's suspension when acted on by a piston of the same area (formula_0) as the cone. Larger values mean lower stiffness, and generally require larger enclosures. formula_17 varies with the square of the diameter. A typical factory measurement tolerance for formula_17 is ±20–30%.
Measured in grams (g) or kilograms (kg), this is the mass of the cone, coil, voice-coil former and other moving parts of a driver, including the acoustic load imposed by the air in contact with the driver cone. formula_2 is the cone/coil mass without the acoustic load, and the two should not be confused. Some simulation software calculates formula_1 when formula_2 is entered. formula_2 can be very closely controlled by the manufacturer.
Units are not usually given for this parameter, but it is in mechanical 'ohms'. formula_4 is a measurement of the losses, or damping, in a driver's suspension and moving system. It is the main factor in determining formula_13. formula_4 is influenced by suspension topology, materials, and by the voice-coil former (bobbin) material.
Measured in metre per newton (m/N). Describes the compliance (i.e., the inverse of stiffness) of the suspension. The more compliant a suspension system is, the lower its stiffness, so the higher the formula_17 will be. formula_3 is proportional to formula_17 and thus has the same tolerance ranges.
Measured in ohms (Ω), this is the DC resistance (DCR) of the voice coil, best measured with the cone blocked, or prevented from moving or vibrating because otherwise the pickup of ambient sounds can cause the measurement to be unreliable. formula_6 should not be confused with the rated driver impedance, formula_6 can be tightly controlled by the manufacturer, while rated impedance values are often approximate at best. American EIA standard RS-299A specifies that formula_6 (or DCR) should be at least 80% of the rated driver impedance, so an 8-ohm rated driver should have a DC resistance of at least 6.4 ohms, and a 4-ohm unit should measure 3.2 ohms minimum. This standard is voluntary, and many 8-ohm drivers have resistances of ≈5.5 ohms, and proportionally lower for lower rated impedances.
Measured in henries (H), this is the inductance of the voice coil. The voice coil is a lossy inductor, in part due to losses in the pole piece, so the apparent inductance changes with frequency. Large formula_5 values limit the high-frequency output of the driver and cause response changes near cutoff. Simple modeling software often neglects formula_5, and so does not include its consequences. Inductance varies with excursion because the voice coil moves relative to the pole piece, acting like a sliding inductor core, increasing inductance on the inward stroke and decreasing it on the outward stroke in typical overhung voice coil arrangements. This inductance modulation is an important source of nonlinearity (distortion) in loudspeakers. Including a copper cap on the pole piece, or a copper shorting ring on it, can reduce the increase in impedance seen at higher frequencies in typical drivers, and also reduce the nonlinearity due to inductance modulation.
Measured in square metres (m2). The effective projected area of the cone or diaphragm. It is difficult to measure and depends largely on the shape and properties of the surround. Generally accepted as the cone body diameter plus one third to one half the width of the annulus (surround). Drivers with wide roll surrounds can have significantly less formula_0 than conventional types with the same frame diameter.
Specified in millimetres (mm). In the simplest form, subtract the height of the voice-coil winding from the height of the magnetic gap, take the absolute value and divide by 2. This technique was suggested by JBL's Mark Gander in a 1981 AES paper, as an indicator of a loudspeaker motor's linear range. Although easily determined, it neglects magnetic and mechanical non-linearities and asymmetry, which are substantial for some drivers. Subsequently, a combined mechanical/acoustical measure was suggested, in which a driver is progressively driven to high levels at low frequencies, with formula_21 determined by measuring excursion at a level where 10% THD is measured in the output. This method better represents actual driver performance, but is more difficult and time-consuming to determine.
Specified in watts. Frequently two power ratings are given, an "RMS" rating and a "music" (or "peak", or "system") rating, usually peak is given as ≈2 times the RMS rating. Loudspeakers have complex behavior, and a single number is really unsatisfactory. There are two aspects of power handling: thermal and mechanical. The thermal capacity is related to coil temperature and the point where adhesives and coil insulation melt or change shape. The mechanical limit comes into play at low frequencies, where excursions are largest, and involves mechanical failure of some component. A speaker that can handle 200 watts thermally at 200 Hz, may sometimes be damaged by only a few watts at some very low frequency, like 10 Hz. Power handling specifications are usually generated destructively, by long-term industry standard noise signals (IEC 268, for example) that filter out low frequencies and test only the thermal capability of the driver. Actual mechanical power handling depends greatly on the enclosure in which the driver is installed.
Specified in litres (L). The volume displaced by the cone, equal to the cone area (formula_0) multiplied by formula_21. A particular value may be achieved in any of several ways. For instance, by having a small cone with a large formula_21, or a large cone with a small formula_44. Comparing formula_24 values will give an indication of the maximum output of a driver at low frequencies. High formula_21, small cone diameter drivers are likely to be inefficient, since much of the voice-coil winding will be outside the magnetic gap at any one time and will therefore contribute little or nothing to cone motion. Likewise, large cone diameter, small formula_21 drivers are likely to be more efficient as they will not need, and so may not have, long voice coils.
Reference efficiency, specified in percent (%). Comparing drivers by their calculated reference efficiency is often more useful than using 'sensitivity' since manufacturer sensitivity figures are too often optimistic.
The sound pressure, in dB, produced by a speaker in response to a specified stimulus. Usually this is specified at an input of 1 watt or 2.83 volts (2.83 volts = 1 watt into an 8-ohm load) at a distance of one metre.
Measurement notes—large signal behavior.
Some caution is required when using and interpreting T/S parameters. Individual units may not match manufacturer specifications. Parameters values are almost never individually taken, but are at best averages across a production run, due to inevitable manufacturing variations. Driver characteristics will generally lie within a (sometimes specified) tolerance range. formula_3 is the least controllable parameter, but typical variations in formula_3 do not have large effects on the final response.
It is also important to understand that most T/S parameters are linearized small signal values. An analysis based on them is an idealized view of driver behavior, since the actual values of these parameters vary in all drivers according to drive level, voice coil temperature, over the life of the driver, etc. formula_3 decreases the farther the coil moves from rest. formula_7 is generally maximum at rest, and drops as the voice coil approaches formula_21. formula_6 increases as the coil heats and the value will typically double by 270 °C (exactly 266 °C for Cu and 254 °C for Al), at which point many voice coils are approaching (or have already reached) thermal failure.
As an example, formula_8 and formula_17 may vary considerably with input level, due to nonlinear changes in formula_3. A typical 110-mm diameter full-range driver with an formula_8 of 95 Hz at 0.5 V signal level, might drop to 64 Hz when fed a 5 V input. A driver with a measured formula_17 of 7 L at 0.5 V, may show a formula_17 increase to 13 L when tested at 4 V. formula_13 is typically stable within a few percent, regardless of drive level. formula_10 and formula_15 decrease <13% as the drive level rises from 0.5 V to 4 V, due to the changes in formula_7. Because formula_17 can rise significantly and formula_8 can drop considerably, with a trivial change in measured formula_1, the calculated sensitivity value (formula_33) can appear to drop by >30% as the level changes from 0.5 V to 4 V. Of course, the driver's actual sensitivity has not changed at all, but the calculated sensitivity is correct only under some specific conditions. From this example, it is seen that the measurements to be preferred while designing an enclosure or system are those likely to represent typical operating conditions. Unfortunately, this level must be arbitrary, since the operating conditions are continually changing when reproducing music. Level-dependent nonlinearities typically cause lower than predicted output, or small variations in frequency response.
Level shifts caused by resistive heating of the voice coil are termed power compression. Design techniques which reduce nonlinearities may also reduce power compression, and possibly distortions not caused by power compression. There have been several commercial designs that have included cooling arrangements for driver magnetic structures, which are intended to mitigate voice coil temperature rise, and the attendant rise in resistance that is the cause of the power compression. Elegant magnet and coil designs have been used to linearize formula_7 and reduce the value and modulation of formula_5. Large, linear spiders can increase the linear range of formula_3, but the large signal values of formula_7 and formula_3 must be balanced to avoid dynamic offset.
Lifetime changes in driver behavior.
The mechanical components in typical speaker drivers may change over time. Paper, a popular material in cone fabrication, absorbs moisture easily and unless treated may lose some structural rigidity over time. This may be reduced by coating with water-impregnable material such as various plastic resins. Cracks compromise structural rigidity and if large enough are generally non-repairable. Temperature has a strong, generally reversible effect; typical suspension materials become stiffer at lower temperatures. The suspension experiences fatigue, and also undergoes changes from chemical and environmental effects associated with aging such as exposure to ultraviolet light, and oxidation which affect foam and natural rubber components badly, though butyl, nitrile, SBR rubber, and rubber-plastic alloys (such as Santoprene) are more stable. The polyester type of polyurethane foam is highly prone to disintegration after 10 to 15 years. The changes in behavior from aging may often be positive, though since the environment that they are used in is a major factor the effects are not easily predicted. Gilbert Briggs, founder of Wharfedale Loudspeakers in the UK, undertook several studies of aging effects in speaker drivers in the 1950s and 1960s, publishing some of the data in his books, notably "Loudspeakers: The Why and How of Good Reproduction".
There are also mechanical changes which occur in the moving components during use. In this case, however, most of the changes seem to occur early in the life of the driver, and are almost certainly due to relaxation in flexing mechanical parts of the driver (e.g., surround, spider, etc.). Several studies have been published documenting substantial changes in the T/S parameters over the first few hours of use, some parameters changing by as much as 15% or more over these initial periods. The proprietor of the firm GR Research has publicly reported several such investigations of several manufacturers' drivers. Other studies suggest little change, or reversible changes after only the first few minutes. This variability is largely related to the particular characteristics of specific materials, and reputable manufacturers attempt to take them into account. While there are a great many anecdotal reports of the audible effects of such changes in published speaker reviews, the relationship of such early changes to subjective sound quality reports is not completely clear. Some changes early in driver life are complementary (such as a reduction in formula_8 accompanied by a rise in formula_17) and result in minimal net changes (small fractions of a dB) in frequency response. If the performance of speaker system is critical, as with high order (complex) or heavily equalized systems, it is sensible to measure T/S parameters after a period of run-in (some hours, typically, using program material), and to model the effects of normal parameter changes on driver performance.
Measurement techniques.
There are numerous methods to measure Thiele-Small parameters, but the simplest use the input impedance of the driver, measured near resonance. The impedance may be measured in free air (with the driver unhoused and either clamped to a fixture or hanging from a wire, or sometimes resting on the magnet on a surface) and/or in test baffles, sealed or vented boxes or with varying amounts of mass added to the diaphragm. Noise in the measurement environment can have an effect on the measurement, so one should measure parameters in a quiet acoustic environment.
The most common (DIY-friendly) method before the advent of computer-controlled measurement techniques is the classic free air constant current method, described by Thiele in 1961. This method uses a large resistance (e.g., formula_45 = 500 to 1000 ohms) in series with the driver and a signal generator is used to vary the excitation frequency. The voltage across the loudspeaker terminals is measured and considered proportional to the impedance. It is assumed that variations in loudspeaker impedance will have little effect on the current through the loudspeaker. This is an approximation, and the method results in formula_11 measurement errors for drivers with a high formula_25 – the measured value of formula_25 will always be somewhat low. This measurement can be corrected by measuring the total voltage across the calibration resistor and the driver (call this formula_46) at resonance and calculating the actual test current formula_47. You may then obtain a corrected formula_25 = formula_48.
A second method is the constant voltage measurement, where the driver is excited by a constant voltage, and the current passing through the coil is measured. The excitation voltage divided by the measured current equals the impedance.
A common source of error using these first two methods is the use of inexpensive AC metres. Most inexpensive metres are designed to measure residential power frequencies (50–60 Hz) and are increasingly inaccurate at other frequencies (e.g., below 40 Hz or above a few hundred hertz). In addition, distorted or non–sine wave signals can cause measurement inaccuracies. Inexpensive voltmeters are also not very accurate or precise at measuring current and can introduce appreciable series resistance, which causes measurement errors.
A third method is a response to the deficiencies of the first two methods. It uses a smaller (e.g., 10 ohm) series resistor and measurements are made of the voltage across the driver, the signal generator, and/or series resistor for frequencies around resonance. Although tedious, and not often used in manual measurements, simple calculations exist which allow the true impedance magnitude and phase to be determined. This is the method used by many computer loudspeaker measurement systems. When this method is used manually, the result of taking the three measurements is that their ratios are more important than their actual value, removing the effect of poor meter frequency response.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "S_{\\rm d}"
},
{
"math_id": 1,
"text": "M_{\\rm ms}"
},
{
"math_id": 2,
"text": "M_{\\rm md}"
},
{
"math_id": 3,
"text": "C_{\\rm ms}"
},
{
"math_id": 4,
"text": "R_{\\rm ms}"
},
{
"math_id": 5,
"text": "L_{\\rm e}"
},
{
"math_id": 6,
"text": "R_{\\rm e}"
},
{
"math_id": 7,
"text": "Bl"
},
{
"math_id": 8,
"text": "f_{\\rm s}"
},
{
"math_id": 9,
"text": "f_{\\rm s} = \\frac{1}{2 \\pi\\cdot\\sqrt{C_{\\rm ms}\\cdot M_{\\rm ms}}}"
},
{
"math_id": 10,
"text": "Q_{\\rm es}"
},
{
"math_id": 11,
"text": "Q"
},
{
"math_id": 12,
"text": "Q_{\\rm es} = \\frac{2 \\pi\\cdot f_{\\rm s}\\cdot M_{\\rm ms} \\cdot R_{\\rm e}}{(Bl)^2} = \\frac{R_{\\rm e}}{(Bl)^2} \\sqrt{\\frac{M_{\\rm ms}}{C_{\\rm ms}}}"
},
{
"math_id": 13,
"text": "Q_{\\rm ms}"
},
{
"math_id": 14,
"text": "Q_{\\rm ms} = \\frac{2 \\pi\\cdot f_{\\rm s}\\cdot M_{\\rm ms}}{R_{\\rm ms}} = \\frac{1}{R_{\\rm ms}} \\sqrt{\\frac{M_{\\rm ms}}{C_{\\rm ms}}}"
},
{
"math_id": 15,
"text": "Q_{\\rm ts}"
},
{
"math_id": 16,
"text": "Q_{\\rm ts} = \\frac{Q_{\\rm ms} \\cdot Q_{\\rm es}}{Q_{\\rm ms} + Q_{\\rm es}}"
},
{
"math_id": 17,
"text": "V_{\\rm as}"
},
{
"math_id": 18,
"text": "V_{\\rm as} = \\rho \\cdot c^2 \\cdot S_{\\rm d}^2 \\cdot C_{\\rm ms}"
},
{
"math_id": 19,
"text": "\\rho"
},
{
"math_id": 20,
"text": "c"
},
{
"math_id": 21,
"text": "X_{\\rm max}"
},
{
"math_id": 22,
"text": "X_{\\rm mech}"
},
{
"math_id": 23,
"text": "P_{\\rm e}"
},
{
"math_id": 24,
"text": "V_{\\rm d}"
},
{
"math_id": 25,
"text": "Z_{\\rm max}"
},
{
"math_id": 26,
"text": "Z_{\\rm max} = R_e\\left(1+\\frac{Q_{\\rm ms}}{Q_{\\rm es}}\\right)"
},
{
"math_id": 27,
"text": "EBP"
},
{
"math_id": 28,
"text": "EBP>100"
},
{
"math_id": 29,
"text": "EBP<50"
},
{
"math_id": 30,
"text": "50<EBP<100"
},
{
"math_id": 31,
"text": "EBP = \\frac{f_{\\rm s}}{Q_{\\rm es}}"
},
{
"math_id": 32,
"text": "Z_{\\rm nom}"
},
{
"math_id": 33,
"text": "\\eta_0"
},
{
"math_id": 34,
"text": "\\eta_0 = \\left(\\frac{\\rho \\cdot B^2 \\cdot l^2 \\cdot S_{\\rm d}^2}{2 \\cdot \\pi \\cdot c \\cdot M_{\\rm ms}^2 \\cdot R_{\\rm e}}\\right)\\times100\\%"
},
{
"math_id": 35,
"text": "\\rho/2\\pi c"
},
{
"math_id": 36,
"text": "\\eta_0 = \\left(\\frac{4 \\cdot \\pi^2 \\cdot f_{\\rm s}^3 \\cdot V_{\\rm as}}{c^3 \\cdot Q_{\\rm es}}\\right)\\times100\\%"
},
{
"math_id": 37,
"text": "4\\pi^2/c^3"
},
{
"math_id": 38,
"text": "2\\pi"
},
{
"math_id": 39,
"text": "1/(2\\pi)"
},
{
"math_id": 40,
"text": "R_e"
},
{
"math_id": 41,
"text": "B \\times l"
},
{
"math_id": 42,
"text": "B \\times l sin(\\theta)"
},
{
"math_id": 43,
"text": "sin(\\theta)=1"
},
{
"math_id": 44,
"text": "X_{\\rm max} "
},
{
"math_id": 45,
"text": "R_{\\rm test}"
},
{
"math_id": 46,
"text": "V"
},
{
"math_id": 47,
"text": "I = V/(R_{\\rm test} + Z_{\\rm max})"
},
{
"math_id": 48,
"text": "Z_{\\rm max(uncorrected)}\\times R_{\\rm test}/I"
}
]
| https://en.wikipedia.org/wiki?curid=1311933 |
13119919 | Jury stability criterion | Electrical engineering
In signal processing and control theory, the Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic polynomial. It is the discrete time analogue of the Routh–Hurwitz stability criterion. The Jury stability criterion requires that the system poles are located inside the unit circle centered at the origin, while the Routh-Hurwitz stability criterion requires that the poles are in the left half of the complex plane. The Jury criterion is named after Eliahu Ibraham Jury.
Method.
If the characteristic polynomial of the system is given by
formula_0
then the table is constructed as follows:
That is, the first row is constructed of the polynomial coefficients in order, and the second row is the first row in reverse order and conjugated.
The third row of the table is calculated by subtracting formula_1 times the second row from the first row, and the fourth row is the third row with the first "n" elements reversed (as the final element is zero).
formula_2
The expansion of the table is continued in this manner until a row containing only one non-zero element is reached.
Note the formula_3 is formula_4 for the 1st two rows. Then for 3rd and 4th row the coefficient changes (i.e. formula_5) . This can be viewed as the new polynomial which has one less degree and then continuing.
Stability test.
If formula_6 then for every value of formula_7... that is negative, the polynomial has one root outside of the unit disc. This implies that the method can be stopped after the first negative value is found when checking for stability.
Sample implementation.
This method is very easy to implement using dynamic arrays on a computer. It also tells whether all the modulus of the roots (complex and real) lie inside the unit disc. The vector v contains the real coefficients of the original polynomial in the order from highest degree to lowest degree.
/* vvd is the jury array */
vvd.push_back(v); // Store the first row
reverse(v.begin(),v.end());
vvd.push_back(v); // Store the second row
for (i=2;;i+=2)
v.clear();
double mult = vvd[i-2][vvd[i-2].size()-1]/vvd[i-2][0]; // This is an/a0 as mentioned in the article.
for (j=0; j<vvd[i-2].size()-1; j++) // Take the last 2 rows and compute the next row
v.push_back(vvd[i-2][j] - vvd[i-1][j] * mult);
vvd.push_back(v);
reverse(v.begin(), v.end()); // reverse the next row
vvd.push_back(v);
if (v.size() == 1) break;
// Check is done using
for (i=0; i<vvd.size(); i+=2)
if (vvd[i][0]<=0) break;
if (i == vvd.size())
"All roots lie inside unit disc "
else
"no"
References.
<templatestyles src="Reflist/styles.css" />
For more details please check these references:
For advanced resources:
For implementations: | [
{
"math_id": 0,
"text": "f(z) = a_n+a_{n-1}z^1+a_{n-2}z^2+\\dots+a_1z^{n-1} + a_0z^n"
},
{
"math_id": 1,
"text": "\\frac{a_n}{a_0} "
},
{
"math_id": 2,
"text": "\n\\begin{align}\na_0 \\;\\; & a_1 \\;\\; & \\dots \\;\\; & a_{n-1} \\;\\;& a_n\\\\\na_n \\;\\; & a_{n-1} \\;\\; & \\dots \\;\\; & a_1 \\;\\;& a_0\\\\\n\\left(a_0-a_n \\frac{a_n}{a_0}\\right)\\;\\;& \\left(a_{1} - a_{n-1} \\frac{a_n}{a_0}\\right) \\;\\; &\\dots\\;\\; & \\left(a_{n-1} - a_{1} \\frac{a_n}{a_0}\\right) \\;\\;& 0 \\\\\n\\left(a_{n-1} - a_{1} \\frac{a_n}{a_0}\\right)\\;\\; & \\dots \\;\\;& \\left(a_{1} - a_{n-1} \\frac{a_n}{a_0}\\right) \\;\\;& \\left(a_0-a_n \\frac{a_n}{a_0}\\right)\\;\\;&0\\\\\n\\end{align}\n"
},
{
"math_id": 3,
"text": "\\frac{a_n}{a_0}"
},
{
"math_id": 4,
"text": "a_n"
},
{
"math_id": 5,
"text": "\\frac{b_{n-1}}{b_{0}}"
},
{
"math_id": 6,
"text": "{a_0}>0"
},
{
"math_id": 7,
"text": "a_0, b_0, c_0"
}
]
| https://en.wikipedia.org/wiki?curid=13119919 |
1312042 | Salty liquorice | Variety of liquorice
Salty liquorice, salmiak liquorice or salmiac liquorice, is a variety of liquorice flavoured with salmiak salt (sal ammoniac; ammonium chloride), and is a common confection found in the Nordic countries, Benelux, and northern Germany. Salmiak salt gives salty liquorice an astringent, salty taste, akin to that of tannins—a characteristic of red wines, which adds bitterness and astringency to the flavour. Consuming salmiak liquorice can stimulate either a savoury or non-savoury palate and response. Anise oil can also be an additional main ingredient in salty liquorice. Extra-salty liquorice is additionally coated with salmiak salt or salmiak powder, or sometimes table salt.
Salty liquorice candy and pastilles are almost always black or very dark brown and can range from soft candy to hard pastille variety, and sometimes hard brittle. The other colours used are white and variants of grey. Salty liquorice or salmiak is also used as a flavouring in other products, such as ice creams, syrups, chewing gum, snus and alcoholic beverages.
History.
Sal ammoniac (ammonium chloride) has a history of being used as a cough medicine, as it works as an expectorant. Finnish author Jukka Annala speculates that salty liquorice has its origins in pharmacy stores that manufactured their own cough medicine. Where and when ammonium chloride and liquorice were first combined to produce salty liquorice is unclear, but by the 1930s it was produced in Finland, Norway, Denmark, Sweden and the Netherlands as a pastille.
Types.
Different languages often refer to salty liquorice as either "salmiak liquorice" (Swedish: "Salmiaklakrits"; Danish: "Salmiaklakrids"), or simply "salt liquorice" (Swedish: "Saltlakrits"; Danish: "Saltlakrids"). The Dutch refer to it as "" or "dubbelzoute drop" (double salted liquorice). In Germany, they are commonly known as salt liquorice ("Salzlakritz") candy and salmiak pastilles ("Salmiakpastillen") or simply Salmiak, in contrast to sweet liquorice ("Süßlakritz") candy. A traditional shape for salty liquorice pastilles is a black diamond-shaped lozenge. In Finnish, it is known as "salmiakki".
The strength of the confectionery depends on the amount of food grade ammonium chloride (salmiak salt) used, which varies by country and what's considered a safe amount. In Sweden, for example, the most popular types of salty liquorice contain an average of 7% of ammonium chloride. In 2012, there was a European Union proposal to limit the amount to 0.3%, which was met with wide opposition. Although the European Union now regulates the use of ammonium chloride to 0.3% in most foodstuffs, there is no specific restriction for it in liquorice or ice cream. At a level of up to 7.99% ammonium chloride, salmiak pastilles are considered a "traditionally-applied medicine to assist expectoration in the airways".
An antibacterial effect can be attributed to the neutralization of the slightly acidic ammonium chloride (pH about 5.5) by the relatively alkaline saliva (pH about 7), whereby ammonia is released, which has a disinfecting effect:
formula_0
Health and safety.
Germany and European Union.
Before implementation of the current European Union community-wide list of permitted flavouring substances used in food, national food legislation in Germany required that a content from above 2% ammonium chloride (salmiak salt) in salty liquorice, was required to carry the label "Adult Liquorice - Not Children's Liquorice" ("Erwachsenenlakritz - Kein Kinderlakritz") on packaging in Germany. When the ingredient content of ammonium chloride (salmiak salt) was between 4.49% and 7.99%, the declaration "extra strong" ("extra stark") was also required on packaging. More than 7.99% of ammonium chloride (salmiak salt) was not permitted in Germany at that point in time. Since then, the upper limit on ammonium chloride has been lifted.
Other uses.
In addition to being used in candy, salmiak is also used to flavour vodka, chocolate, distilled rye brandy, ice cream, cola drinks, snus, and meat.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" /> | [
{
"math_id": 0,
"text": "\\mathrm{NH_4^+ + OH^- \\longrightarrow NH_3 + H_2O}"
}
]
| https://en.wikipedia.org/wiki?curid=1312042 |
13120740 | Pulse tube refrigerator | Device using sound waves to reduce heat
The pulse tube refrigerator (PTR) or pulse tube cryocooler is a developing technology that emerged largely in the early 1980s with a series of other innovations in the broader field of thermoacoustics. In contrast with other cryocoolers (e.g. Stirling cryocooler and GM-refrigerators), this cryocooler can be made without moving parts in the low temperature part of the device, making the cooler suitable for a wide variety of applications.
Uses.
Pulse tube cryocoolers are used in niche industrial applications such as semiconductor fabrication and superconducting radio-frequency circuits. They are also used in military applications such as for the cooling of infrared sensors.
In research, PTRs are often used as precoolers of dilution refrigerators. They are also being developed for cooling of astronomical detectors where liquid cryogens are typically used, such as the Atacama Cosmology Telescope or the Qubic experiment (an interferometer for cosmology studies). Pulse tubes are particularly useful in space-based telescopes such as the James Webb Space Telescope where it is not possible to replenish the cryogens as they are depleted. It has also been suggested that pulse tubes could be used to liquefy oxygen on Mars.
Principle of operation.
Figure 1 represents the Stirling-type single-orifice pulse-tube refrigerator (PTR), which is filled with a gas, typically helium at a pressure varying from 10 to 30 bar. From left to right the components are:
The part in between X1 and X3 is thermally insulated from the surroundings, usually by vacuum. The pressure varies gradually and the velocities of the gas are low. So the name "pulse" tube cooler is misleading, since there are no pulses in the system.
The piston moves periodically from left to right and back. As a result, the gas also moves from left to right and back while the pressure within the system increases and decreases. If the gas from the compressor space moves to the right, it enters the regenerator with temperature "T"H and leaves the regenerator at the cold end with temperature "T"L, hence heat is transferred into the regenerator material. On its return, the heat stored within the regenerator is transferred back into the gas.
In the tube, the gas is thermally isolated (adiabatic), so the temperature of the gas in the tube varies with the pressure.
At the cold end of the tube, the gas enters the tube via X2 when the pressure is high with temperature "T"L and returns when the pressure is low with a temperature below "T"L, hence taking up heat from X2: this gives the desired cooling effect at X2.
To understand why the low-pressure gas returns at a lower temperature, look at figure 1 and consider gas molecules close to X3 (at the hot end), which move in and out of the tube through the orifice. Molecules flow into the tube (to the left) when the pressure in the tube is low (it is sucked into the tube via X3, coming from the orifice and the buffer). Upon entering the tube, it has the temperature "T"H. Later in the cycle, the same mass of gas is pushed out from the tube again when the pressure inside the tube is high. As a consequence, its temperature will be higher than "T"H. In the heat exchanger X3, it releases heat and cools down to the ambient temperature "T"H.
Figure 3 shows a coaxial pulse tube, which is a more useful configuration in which the regenerator surrounds the central pulse tube. This is compact and places the cold head at an end, so it is easy to integrate with whatever is to be cooled. The displacer can be passively driven, and this recovers work that would otherwise be dissipated in the orifice.
Performance.
The performance of the cooler is determined mainly by the quality of the regenerator. It has to satisfy conflicting requirements: it must have a low flow resistance (so it must be short with wide channels), but the heat exchange should also be good (so it must be long with narrow channels). The material must have a large heat capacity. At temperatures above 50K practically all materials are suitable. Bronze or stainless steel is often used. For temperatures between 10 and 50K lead is most suitable. Below 10K one uses magnetic materials which are specially developed for this application.
The so-called coefficient of performance (COP; denoted formula_1) of coolers is defined as the ratio between the cooling power formula_0 and the compressor power "P". In formula: formula_2. For a perfectly reversible cooler, formula_1 is given by Carnot's theorem:
However, a pulse-tube refrigerator is not perfectly reversible due to the presence of the orifice, which has flow resistance. Instead, the COP of an ideal PTR is given by
which is lower than that of ideal coolers.
Comparison with other coolers.
In most coolers gas is compressed and expanded periodically. Well-known coolers such as the Stirling engine coolers and the popular Gifford-McMahon coolers have a displacer that ensures that the cooling (due to expansion) takes place in a different region of the machine than the heating (due to compression). Due to its clever design, the PTR does not have such a displacer, making the construction of a PTR simpler, cheaper, and more reliable. Furthermore, there are no mechanical vibrations and no electro-magnetic interferences. The basic operation of cryocoolers and related thermal machines is described by De Waele
History.
W. E. Gifford and R. C. Longsworth, in the 1960s, invented the so-called Basic Pulse Tube Refrigerator. The modern PTR was invented in 1984 by Mikulin who introduced an orifice to the basic pulse tube. He reached a temperature of 105K. Soon after that, PTRs became better due to the invention of new variations. This is shown in figure 4, where the lowest temperature for PTRs is plotted as a function of time.
At the moment, the lowest temperature is below the boiling point of helium (4.2K). Originally this was considered to be impossible. For some time it looked as if it would be impossible to cool below the lambda point of 4He (2.17K), but the "low-temperature group" of the Eindhoven University of Technology managed to cool to a temperature of 1.73K by replacing the usual 4He as refrigerant by its rare isotope 3He. Later this record was broken by the Giessen Group that managed to get even below 1.3K. In a collaboration between the groups from Giessen and Eindhoven a temperature of 1.2K was reached by combining a PTR with a superfluid vortex cooler.
Types of pulse-tube refrigerators.
For cooling, the source of the pressure variations is unimportant. PTRs for temperatures below 20K usually operate at frequencies of 1 to 2 Hz and with pressure variations from 10 to 25 bar. The swept volume of the compressor would be very high (up to one liter and more). Therefore, the compressor is uncoupled from the cooler. A system of valves (usually a rotating valve) alternately connects the high-pressure and the low-pressure side of the compressor to the hot end of the regenerator. As the high-temperature part of this type of PTR is the same as of GM-coolers, this type of PTR is called a GM-type PTR. The gas flows through the valves are accompanied by losses which are absent in the Stirling-type PTR.
PTRs can be classified according to their shape. If the regenerator and the tube are in line (as in fig. 1) we talk about a linear PTR. The disadvantage of the linear PTR is that the cold spot is in the middle of the cooler. For many applications it is preferable that the cooling is produced at the end of the cooler. By bending the PTR we get a U-shaped cooler. Both hot ends can be mounted on the flange of the vacuum chamber at room temperature. This is the most common shape of PTRs. For some applications it is preferable to have a cylindrical geometry. In that case the PTR can be constructed in a coaxial way so that the regenerator becomes a ring-shaped space surrounding the tube.
The lowest temperature reached with single-stage PTRs is just above 10K. However, one PTR can be used to precool the other. The hot end of the second tube is connected to room temperature and not to the cold end of the first stage. In this clever way it is avoided that the heat, released at the hot end of the second tube, is a load on the first stage. In applications the first stage also operates as a temperature-anchoring platform for e.g. shield cooling of superconducting-magnet cryostats. Matsubara and Gao were the first to cool below 4K with a three-stage PTR. With two-stage PTRs temperatures of 2.1K, so just above the λ-point of helium, have been obtained. With a three-stage PTR 1.73K has been reached using 3He as the working fluid.
Prospects.
The coefficient of performance of PTRs at room temperature is low, so it is not likely that they will play a role in domestic cooling. However, below about 80K the coefficient of performance is comparable with other coolers (compare equations (1) and (2)) and in the low-temperature region the advantages get the upper hand. PTRs are commercially available for temperatures in the region of 70K and 4K. They are applied in infrared detection systems, for reduction of thermal noise in devices based on (high-"T"c) superconductivity such as SQUIDs, and filters for telecommunication. PTRs are also suitable for cooling MRI-systems and energy-related systems using superconducting magnets. In so-called dry magnets, coolers are used so that no cryoliquid is needed at all or for the recondensation of the evaporated helium. Also the combination of cryocoolers with 3He-4He dilution refrigerators for the temperature region down to 2mK is attractive since in this way the whole temperature range from room temperature to 2mK is easier to access.
For many low temperature experiments, mechanical vibrations caused by PTRs can cause microphonics on measurement lines, which is a big disadvantage of PTRs. Particularly for scanning probe microscopy uses, PTR-based scanning tunneling microscopes (STMs) have historically difficult due to the extreme vibration sensitivity of STM. Use of an exchange gas above the vibration sensitive scanning head enabled the first PTR based low temperature STMs. Now, there are commercially available PTR-based, cryogen free scanning probe systems.
References.
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| https://en.wikipedia.org/wiki?curid=13120740 |
13120745 | Specimens of Tyrannosaurus | "Tyrannosaurus" is one of the most iconic dinosaurs and is known from numerous specimens, some of which have individually acquired notability due to their scientific significance and media coverage.
"Manospondylus": AMNH 3982.
The first-named fossil specimen which can be attributed to "Tyrannosaurus rex" consists of two partial vertebrae (one of which has been lost) found by Edward Drinker Cope in 1892. Cope believed that they belonged to an "agathaumid" (ceratopsid) dinosaur, and named them "Manospondylus gigas", meaning "giant porous vertebra" in reference to the numerous openings for blood vessels he found in the bone. The "M. gigas" remains were later identified as those of a theropod rather than a ceratopsid, and H.F. Osborn recognized the similarity between "M. gigas" and "Tyrannosaurus rex" as early as 1917. However, due to the fragmentary nature of the "Manospondylus" vertebrae, Osborn did not synonymize the two genera.
"Dynamosaurus": BMNH R7994.
The holotype of "Tyrannosaurus rex", a partial skull and skeleton originally called AMNH 973 (AMNH stands for American Museum of Natural History), was discovered in the U.S. state of Montana in 1902 and excavated over the next three years. Another specimen (AMNH 5866), found in Wyoming in 1900, was described in the same paper under the name "Dynamosaurus imperiosus". At the time of their initial description and naming, these specimens had not been fully prepared and the type specimen of "T. rex" had not even been fully recovered. In 1906, after further preparation and examination, Henry Fairfield Osborn recognized both skeletons as belonging to the same species. Because the name "Tyrannosaurus rex" had appeared just one page earlier than "Dynamosaurus" in Osborn's 1905 work, it was considered the older name and has been used since. Had it not been for page order, "Dynamosaurus" would have become the official name.
Holotype: CM 9380.
CM 9380 is the type specimen used to describe "Tyrannosaurus rex". Fragments of (then) AMNH 973 were first found in 1902 by Barnum Brown, assistant curator of the American Museum of Natural History and a famous paleontologist in his own right. He forwarded news of it to Osborn; it would be three years before they found the rest of it. In 1905 when the type was described by Osborn, previous knowledge of dinosaur predators at the time were based on Jurassic carnosaurs, so the short fore-arms of the "Tyrannosaurus" were treated with extreme caution, with suspicion that bones of a smaller theropod had become jumbled with the remains of the bigger fossil. Following the 1941 entry of the United States into World War II, the holotype was sold to the Carnegie Museum of Natural History in Pittsburgh for protection against possible bombing raids. The specimen, now labeled CM 9380, is still mounted in Pittsburgh, at first with the tail acting as a tripod in the old-fashioned kangaroo pose. It has since received a modernization of its posture (mounted by Phil Fraley and crew) and can now be found balancing with tail outstretched. Along with a more lifelike posture, the specimen also now includes a composite reconstruction of the skull by Michael Holland. It has been reconstructed in recent years, it measured an estimated 11.9 meters in length and an estimated weight of 7.4–14.6 metric tonnes, 9.1 metric tonnes being the average estimate in that study, although most earlier studies have suggested lower weight figures.
AMNH 5027.
With a length of 12.1-12.2 meters, AMNH 5027 was discovered and excavated in 1908 by Barnum Brown in Montana, and described by Osborn in 1912 and 1916. At the time of discovery, a complete cervical (neck vertebrae) series for "Tyrannosaurus" was not previously known, so it was this specimen that brought the short, stocky tyrannosaur neck to light. Compared to later specimens (BMNH R7994 and FMNH PR2081, for instance) the cervical series of AMNH 5027 is much more gracile, so with later discoveries the distinction between tyrannosaurid necks and the necks of carnosaurs became more obvious. This specimen also provided the first complete skull of "Tyrannosaurus rex". In total, Brown found five partial "Tyrannosaurus" skeletons. The skeleton of this specimen was used as the iconic symbol for the Jurassic Park film series.
Osborn planned to mount the similarly sized AMNH 5027 and AMNH 973 together in dynamic poses. Designed by E.S. Christman, the scene was to depict a rearing "Tyrannosaurus" (AMNH 5027) snapping at another cowering one (AMNH 973), as they fought over the remains of a hadrosaur, described at the time as "Trachodon". However, technical difficulties prevented the mount from being executed. One obvious problem was that the Cretaceous Dinosaur Hall was too small to accommodate this dramatic display, and AMNH 5027 was already mounted by itself as the central attraction of the hall. The forearms of "Tyrannosaurus" were not well documented and the hands were unknown, so for the sake of the display, the forearms of AMNH 5027 were given three fingers, based on the forelimbs of "Allosaurus" (the more allosaur-like arms were replaced several years later when better fossils of tyrannosaurid arms were found).
The mount retained a rearing pose similar to the initial proposal. By the 1980s it was generally accepted that such a pose would have been anatomically impossible in life, and the skeleton was re-mounted in a more accurate, horizontal pose during a renovation of the museum's dinosaur halls in the early 1990s. The mount can still be seen on display on the fourth floor of the American Museum. The American Museum of Natural History features AMNH 5027 in its famed Hall of Saurischian Dinosaurs to this day.
"Nanotyrannus": CMNH 7541.
A small but nearly complete skull of "Nanotyrannus lancensis", frequently considered to be a juvenile "T. rex", was recovered from Montana in 1942. This skull, Cleveland Museum of Natural History (CMNH) 7541, measures in length and was originally classified as a species of "Gorgosaurus" ("G. lancensis") by Charles W. Gilmore in 1946. In 1988, the specimen was re-described by Robert T. Bakker, Phil Currie, and Michael Williams, then the curator of paleontology at the Cleveland Museum of Natural History, where the original specimen was housed and is now on display. Their initial research indicated that the skull bones were fused, and that it therefore represented an adult specimen. In light of this, Bakker and colleagues assigned the skull to a new genus, named "Nanotyrannus" for its apparently small adult size. The specimen is estimated to have been around long when it died. However, a detailed analysis of the specimen by Thomas Carr in 1999 showed that the specimen was, in fact, a juvenile, leading Carr and many other paleontologists to consider it a juvenile specimen of "T. rex". The current classification of CMNH 7541 is not universal, with some research suggesting the specimen belongs to a taxon distinct from "Tyrannosaurus".
LACM 23844.
In 1966, a crew working for the Natural History Museum of Los Angeles County under the direction of Harley Garbani discovered another "T. rex" (LACM 23844) which included most of the skull of a very large, mature animal. When it was put on display in Los Angeles, LACM 23844 was the largest "T. rex" skull on exhibit anywhere.
"Black Beauty": RTMP 81.6.1.
"Black Beauty" (specimen number RTMP 81.6.1) is a well-preserved fossil of "Tyrannosaurus rex". The nickname stems from the apparent shiny dark color of the fossil bones, which occurred during fossilization by the presence of minerals in the surrounding rock; it was the first "Tyrannosaurus rex" specimen to receive a nickname, beginning a trend that continues with most major "T. rex" finds. Black Beauty was found in 1980 by a high school student, Jeff Baker, while on a fishing trip with a friend in the region of the Crowsnest Pass, Alberta. A large bone was found in the riverbank and shown to their teacher. Soon afterward, the Royal Tyrrell Museum was contacted, and excavation of the sandstone matrix surrounding the fossils began in 1982. The dig site where the fossil was found is located at approximately formula_0 near the confluence of the Crowsnest and Willow Rivers, and consisted of rock belonging to the Willow Creek Formation. The specimen is housed in the Royal Tyrrell Museum in Drumheller, Alberta, Canada.
In 2009, a paper by Jack Horner and colleagues illustrated the concept of parasitic infections in dinosaurs by analysing the lesions found on the cranial bones of Black Beauty. The specimen has been used to study comparative morphology between tyrannosaurids and "Tyrannosaurus" individuals.
Replicas of Black Beauty have been shown in some exhibitions and museums, including both simple skull montages and complete skeletons. Casts are on display in museums around the world.
"Stan": BHI 3033.
Stan is the nickname given to a fossil about 11.78 m (38 ft) long found in Hell Creek Formation, South Dakota, close to Buffalo in 1987 by Stan Sacrison, who also discovered the "Tyrannosaurus" specimen nicknamed "Duffy". The original fossils are now housed at Black Hills Institute of Geological Research, Inc. center. It is a well known specimen, and one of the most complete, with 199 bones recovered. About 30 casts of the original fossil have been sold worldwide, each for a price of about $100,000. Stan's skeleton was auctioned for $31.8 million in a 2020 Christie's New York sale, making it a record-breaking dinosaur sale, with the buyer eventually being revealed as the under construction Natural History Museum Abu Dhabi in the Saadiyat Cultural District.
Like many other fossils of "Tyrannosaurus rex", the skeleton of Stan shows many broken and healed bones. These include broken ribs and damages in the skull. One of the most prominent injuries are in the neck and the skull. A piece of bone is missing at the rear, and the skull also bears a hole 1 inch wide, probably made by another "Tyrannosaurus". Also, two of the cervical vertebrae are fused, and another has additional bone growth. This could have been caused by another "Tyrannosaurus" bite. The bite marks are healed, indicating that Stan survived the wounds. Stan could also have been infected by "Trichomonas"-like parasites.
"Wankel Rex": MOR 555.
In 1988, local rancher Kathy Wankel discovered another "Tyrannosaurus rex" in Hell Creek sediments on an island in the Charles M. Russell National Wildlife Refuge of Montana. This specimen was excavated by a team from the Museum of the Rockies led by paleontologist Jack Horner, with assistance from the U.S. Army Corps of Engineers. The specimen, given the number MOR 555 but informally called the "Wankel rex," includes approximately 80-85 percent of the skeleton, including the skull, as well as what at the time was the first complete "T. rex" forelimb. It has an estimated length of around and a weight between and in newer figures. It is estimated that the "Wankel rex" was 18 years old when it died, an adult but not completely grown. The "Wankel rex" was also one of the first fossil dinosaur skeletons studied to see if biological molecules still existed within the fossilized bones. Doctoral candidate Mary Schweitzer found heme, a biological form of iron that makes up hemoglobin (the red pigment in blood).
It was long on exhibit at the Museum of the Rockies in Bozeman, Montana. In June 2013, the Corps loaned the specimen to the National Museum of Natural History, Smithsonian Institution museum in Washington, D.C., for 50 years. (The Museum of the Rockies continues to display a cast reconstruction of the skull by Michael Holland). The specimen went on temporary display on National Fossil Day, 16 October 2013, and was exhibited until the museum's dinosaur hall exhibit closed for renovation in the spring of 2014. The skeleton, named "The Nation's "T. rex" became the centerpiece of the dinosaur hall when it reopened in 2019. Casts of MOR 555 are on display at the National Museum of Scotland, the Australian Fossil and Mineral Museum, and the University of California Museum of Paleontology. A bronze cast of the specimen, known as "Big Mike", stands outside the Museum of the Rockies.
In 2022, Gregory S. Paul and colleagues argued that the Wankel rex was not actually a "T. rex", but rather the holotype for a new species: "Tyrannosaurus regina". This was heavily criticized by several other leading paleontologists, including Stephen Brusatte, Thomas Carr, Thomas Holtz, David Hone, Jingmai O'Connor, and Lindsay Zanno when they were approached by various media outlets for comment. Their criticism was subsequently published in a technical paper.
"Sue": FMNH PR2081.
Susan Hendrickson of the Black Hills Institute discovered the best-preserved "Tyrannosaurus" currently known, in the Hell Creek Formation near Faith, South Dakota, on 12 August 1990. About 90% of the skeleton was recovered, allowing the first complete description of a "Tyrannosaurus rex" skeleton. This specimen, named "Sue" in honor of its discoverer, soon became embroiled in a legal battle over its ownership. The owner of the land the fossil was found, Maurice Williams, as well the Sioux Tribe he belonged to, claimed ownership, the Institute had considered itself to have. In 1997, the suit was settled in favor of Williams and the fossil was returned to Williams' ownership. Williams quickly offered up "Sue" for auction by Sotheby's in New York, where it was sold to the Field Museum of Natural History in Chicago for US$8.4 million—the highest price ever paid for a fossil before being surpassed by Stan.
Sue has a length of , stands tall at the hips, and according to the most recent studies estimated to have weighed between 8.4 and 14 metric tons when alive. It has been hypothesised that Sue's impressive size may have been achieved due to a prolonged ontogenic development, since it is the third oldest "Tyrannosaurus" known. Sue's age at the time of death was estimated by Peter Mackovicky and the University of Florida to be 28 years old, over 6–10 years older than most big "Tyrannosaurus" specimens, like MOR 555, AMNH 5027 or BHI 3033. The only known specimen of "T. rex" that is older than Sue is Trix.
Preparation of "Sue" (FMNH PR2081) was completed at the Field Museum and the skeleton was placed on exhibit on 17 May 2000.
"Montana's "T. rex"": MOR980.
Montana's "T. rex" (also known as "Peck's rex", "Peckrex", "Rigby's rex" and "Tyrannosaurus" "imperator") is the nickname given to a fossil specimen found in Montana in 1997. The discovery was made by Louis E. Tremblay on 4 July 1997 working under the supervision of J. Keith Rigby Jr. who led the excavation and bone preparation.
The skeleton of Montana's "T. rex" includes a relatively complete skull with jaws, multiple vertebrae of the back and tail, a well preserved gastralium, and hipbone with complete ischium and pubis. The left hindleg is relatively complete with a femur, missing only some toe bones. The forelimbs include the scapula and furcula, both humeri and right hand phalanges, as well as metacarpal III. Montana's "T. rex" has been the subject of research regarding parasitic infections in dinosaurs. The forelimbs of "Montana's T. rex" have also been studied as they show evidence of use. This evidence includes the construction of metacarpal III, as well as repeated fractures in the furcula—possibly caused by heavy loads or pressure (Carpenter and Lipkin, 2005).
The fossils of Montana's "T. rex" are exhibited at Museum of the Rockies as part of a full skeletal mount completed with cast elements replacing the missing bones. This mount was installed after the Wankel Rex (now nicknamed The Nation's "T. rex") was loaned to the Smithsonian to occupy a central part in the museum's dinosaur hall, featuring a dynamic mount showing the apex predator devouring a "Triceratops" carcass.
It has been stated by Pete Makovicky, the Chicago museum's lead curator of dinosaurs, that this specimen is in the same size range as "Sue" and "Scotty".
"Bucky": TCM 2001.90.1.
Bucky is a fossil of a juvenile specimen on display at The Children's Museum of Indianapolis in Indianapolis, Indiana. It is the first juvenile "Tyrannosaurus" ever placed on permanent exhibit in a museum. The dinosaur remains were found in 1998 in the Hell Creek Formation near the town of Faith, South Dakota. The skeleton, transported by water, ended up in a low shallow valley along with bones from an "Edmontosaurus" and "Triceratops". It was discovered by rancher and cowboy Bucky Derflinger. Bucky was well preserved and easily prepared by the Black Hills Institute in South Dakota. Part of The Children's Museum of Indianapolis Dinosphere exhibit, Bucky is displayed along with Stan, an adult "Tyrannosaurus", in a hunting scene. Both dinosaurs are attacking a "Triceratops" specimen known as "Kelsey".
Bucky is one of the few dinosaur fossils found with a furcula; Bucky's furcula was the first one found for the genus "Tyrannosaurus." Bucky also has a nearly complete set of gastralia, or belly ribs, and an ulna, or lower arm bone. As of now, 101 bones, or about 34% of Bucky's skeleton, has been discovered and verified. Bucky is the sixth-most complete "Tyrannosaurus rex" out of more than 40 that have been discovered.
"E. D. Cope": BHI 6248.
E. D. Cope (named after the paleontologist of the same name) is a "Tyrannosaurus" specimen discovered in South Dakota by Bucky Derflinger in 1999 at the same site as AMNH 3982. Excavations of this 10% complete skeleton began in 2000. The known material includes a partial skull, several vertebrae, and ribs. A very wide femur with a length of 1300 mm and a circumference of 630 mm is also known.
"Jane": BMRP 2002.4.1.
Jane is a fossil specimen of small tyrannosaurid dinosaur, officially known as BMRP 2002.4.1, discovered in the Hell Creek Formation in southern Montana in 2001. Despite having a typically female name, Jane's sex is unknown—the specimen was named after Burpee Museum benefactor Jane Solem. The specimen was found in the summer of 2001 by Carol Tuck and Bill Harrison on an expedition led by Burpee Museum curator Michael Henderson. After four years of preparation, Jane was put on display at Rockford, Illinois' Burpee Museum of Natural History as the centerpiece of an exhibit called "Jane: Diary of a Dinosaur."
The Jane specimen has been central to the debate regarding the validity of the proposed tyrannosaurid genus "Nanotyrannus". However, the Jane material has yet to be properly studied and described by scientists. Although Larson (2013) saw Jane as more identical to CMNH 7541 and LACM 28471 than to adult "T. rex" in having a higher tooth count, large pneumatic foramen on the center of the quadratojugal, T-shaped postorbital, and fused shoulder blade and pelvis, Yun (2015) concurred with the opinion of most workers that "Nanotyrannus" is a juvenile "T. rex", noting that a juvenile specimen of "Tarbosaurus" described by Tsuihiji "et al." (2011) also has a T-shaped postorbital. Paleontologists who support the theory that Jane represents a juvenile believe the tyrannosaur was approximately 11 years old at its time of death, and its fully restored skeleton measured long, a bit more than half as long as the largest-known complete "T. rex" specimen, nicknamed "Sue," which measures long. According to Hutchinson "et al." (2011), the weight of the Jane specimen in life was probably between and , being the average estimate.
"B-rex": MOR 1125.
This specimen was found in the lower portion of the Hell Creek Formation near Fort Peck Lake in the Charles M. Russell National Wildlife Refuge in Garfield County, Montana. Its discoverer was Bob Harmon, a fossil preparator for the Museum of the Rockies, and was nicknamed the "B-rex" (or "Bob-rex") in honor of Harmon. The specimen was discovered in 2000, and excavated by MOR from 2001 to 2003. Although only 37 percent of the skeleton was present, this included almost all of the skull (although the skull was nearly completely disarticulated). The specimen also includes several cervical, dorsal, sacral, and caudal vertebrae; several chevrons; some cervical and dorsal ribs; left scapula and coracoid; the furcula; the left ulna; both femora, tibiae, and ulnae; the right calcaneum; right astragalus; and a number of pes phalanges.
In the March 2005 "Science magazine", Mary Higby Schweitzer of North Carolina State University and colleagues announced the recovery of soft tissue from the marrow cavity of a fossilized femur belong to B-Rex. Flexible, bifurcating blood vessels and fibrous but elastic bone matrix tissue were recognized. In addition, microstructures resembling blood cells were found inside the matrix and vessels. The structures bear resemblance to ostrich blood cells and vessels. However, since an unknown process distinct from normal fossilization seems to have preserved the material, the researchers are being careful not to claim that it is original material from the dinosaur. Paleontologist Thomas Kaye of the University of Washington in Seattle hypothesized that the soft-tissue is permineralized biofilm created by bacteria while digesting and breaking down the original specimen. He has discovered this to be true in many specimens from the same area. In 2016, it was finally confirmed by Mary Higby Schweitzer and Lindsay Zanno "et al" that the soft tissue was medullary bone tissue, like that in modern birds when they are readying to lay eggs. This confirmed the identity of the "Tyrannosaurus" MOR 1125 as a female.
"Samson".
A "T. rex" specimen was discovered on private land in Harding County, South Dakota, once in 1981 by Michael and Dee Zimmerschied, and again on 4 October 1992 (Alan and Robert Detrich re-discovered Samson after it was originally found and deemed by paleontologists that several bones had washed in and there was nothing left).
It was shortly after that when Fred Nuss and Candace Nuss of Nuss Fossils with the Detrich brothers found the most complete and undistorted "Tyrannosaurus rex" skull ever discovered. Following the sale of "Sue," another "Tyrannosaurus rex" skeleton was, the specimen was put up for auction on eBay in 2000 under the name of "Z-rex", with an asking price of over US$8 million. It failed to sell online but was purchased for an undisclosed price in 2001 by British millionaire Graham Ferguson Lacey, who renamed the skeleton "Samson" after the Biblical figure of the same name. It was prepared by the Carnegie Museum starting in May 2004. After preparation was complete in March 2006, the specimen was returned to Lacey. It, along with some other dinosaur skeletons, was sold again at auction on 3 October 2009.
Samson measured , only slightly shorter than Sue.
"Baby Bob".
On 7 July 2013, fossil hunter Robert Detrich of Wichita, Kansas, unearthed the remains of what is believed to be a 4-year-old "Tyrannosaurus rex". Detrich unearthed the fossil dubbed "Baby Bob" in a fossil-rich area near the Eastern Montana town of Jordan. Its femur measures about 25 inches, and if all the preliminary data pans out, that would make it among the smallest "T. rex" specimens ever found. Baby Bob has been fully excavated, although it will take another year to clean. Detrich said the skull, which is about 75 percent complete, and most of the major skeletal elements were found strewn across a flood plain, although very few vertebra and ribs were found.
"Scotty": RSM P2523.8.
"Scotty", cataloged as RSM P2523.8, was discovered in Saskatchewan, Canada in 1991. Since its discovery and extensive subsequent study, "Scotty" has been referred to as the largest "T. rex" ever discovered in the world, the largest of any dinosaur discovered in Canada, and as one of the oldest and most complete fossils of its kind at more than 70% bulk. "Scotty" resides at the Royal Saskatchewan Museum's T. rex Discovery Centre in Eastend, SK, Canada. In May 2019, a second mount was erected at the Royal Saskatchewan Museum in Regina, where the exhibit reflects the recent discoveries about the fossil.
"Scotty" was discovered by Robert Gebhardt, a high school principal from Eastend, SK who accompanied palaeontologists from the Royal Saskatchewan Museum on a prospective expedition into the Frenchman Formation in southwestern Saskatchewan on 16 August 1991. It wasn't until June 1994 that the Royal Saskatchewan Museum was able to begin the excavation, which was led and overseen by the Museum's Ron Borden, as well as resident paleontologists Tim Tokaryk and John Storer who were with Gebhardt when he uncovered the first fossils. The bones were deeply packed in dense, iron-laden sandstone, which took more than twenty years for the team to fully remove, excavate, and assemble the majority of the skeleton, with additional trips being made to the site to retrieve smaller bones and teeth. The entire process of excavating the skeleton was also slowed down by its considerable size.
"Scotty" is reported to be long and weighed an estimated . Despite it not being a complete fossil, paleontologists were able to create the estimation for the weight and length through measurements of important weight bearing bones such as the femur, hip, and shoulder bones that have all been measured to be larger and thicker with "Scotty" than the corresponding bones with "Sue". Going from the latest study "Scotty" exceeds "Sue" in 84.6% of the published measurements. While the reported measurements and weight for "Scotty" are larger than those of "Sue", some scientists posit that the two fossils are too close in size to officially declare "Scotty" the largest.
Like other "T. rex" fossils, "Scotty" shows signs of trichomoniasis, a parasitic infection in the jaw that left visible holes in the bone and was unique to this specific species of dinosaur. Additionally, a broken and healed rib on its right side, broken tail vertebra, as well as a hole near the eye socket are possibly the result of another "T. rex" attack. Other abnormalities, such as impacted teeth, suggest that "Scotty" was not only bitten, but also bit other animals.
"Tristan".
Commercial paleontologist Craig Pfister discovered the specimen in the lower Hell Creek Formation in Carter County, Montana, in 2010. Its excavation and preservation lasted four years. It was later sold to Danish-born investment banker Niels Nielsen, who loaned the specimen to the Museum für Naturkunde in Berlin, Germany, for research and exhibition. It has been on display at Museum für Naturkunde between 2015 and 2020, moving to the Natural History Museum of Denmark for one year, and expected back in 2021. Nielsen and his friend Jens Jensen named the specimen Tristan-Otto (short: Tristan) for their sons. The Museum für Naturkunde Berlin lists it under specimen number MB.R.91216. Several European museums have "Tyrannosaurus" casts (replicas) or parts, but Tristan is one of only two original skeletons on display in the continent (the other is "Trix" in the Netherlands). The matte-black fossilised skeleton is about long and tall at the hips. Tristan is among the most complete known "Tyrannosaurus" skeletons: It was re-assembled from about 300 separate parts, 170 of which are original (including 98% of the skull and all the teeth), the rest reproductions. It is estimated to have died when about 20 years old and it was in poor health, having several bone fractures, bite marks to the skull and signs of disease in the jaw. The disease present in Tristan's jaw was suggested to be a case of tumefactive osteomyelitis.
"Thomas".
From 2003 to 2005, Thomas was excavated by NHMLA paleontologists in southeastern Montana. At 17 years old, long and nearly , it is estimated to be a 70% complete specimen. Thomas is mounted in a "growth series" with the youngest-known "Tyrannosaurus rex" fossil, a two-year-old, specimen, and a 13-year-old, juvenile specimen.
This fossil is one of the geologically youngest "T. rex" specimens known, discovered very near the Cretaceous–Tertiary boundary.
"Victoria".
Victoria is a specimen found near Faith, South Dakota in 2013. Victoria is estimated to be around 12 ft tall and 40 ft long, and she is thought to have died in her subadult stage, between 15 and 25 years of age. Victoria has also been the subject of a traveling exhibition being displayed in places such as the Arizona Science Center. Her cause of death is unknown; however, she was believed to have been bitten in the lower jaw by another "Tyrannosaurus". The bite may have become infected, spreading and leading to sepsis.
"Ivan".
Ivan is a 65% complete "T. rex" displayed at the Museum of World Treasures in Wichita, Kansas. The specimen has the most complete tail of any "T. rex", only missing around 3 vertebrae. Ivan is around 40 ft long and 12 feet high.
"Trix": RGM 792.000.
In 2013, a team of paleontologists from the Naturalis Biodiversity Center (Leiden, Netherlands) traveled to Montana where they discovered and unearthed a large and remarkably complete "Tyrannosaurus rex" specimen that lived 67 million years ago. Black Hills institute collaborated with the team in the excavation. The bones were cleaned and assembled in a mount at Black Hills Institute's installations, with the help of both Chicago's Field Museum of Natural History and the Naturalis Museum in Leiden. Chicago's Field museum sent digital models of their famous specimen, FMNH PR 2081 (Sue) to complete the cast and Naturalis museum replicated the bones using 3D-printing technology.
The specimen was named "Trix" after former Netherlands' Queen Beatrix. At arrival in the Netherlands, it started touring on public display in an itinerant exhibition titled "T. rex in Town". The first exhibit spanned from 10 September 2016 to 5 June 2017 and was set at the only room of the Naturalis museum open to public at the time (the 17th-century building known as "Pesthuis"), due to the fact that the museum was undergoing restoration. When the Netherlands exhibition ended, it continued travelling through other European countries in 2017, 2018 and 2019. As of August 2019, Trix was returned to display at the Naturalis museum where it is installed in a special room that was under construction during Trix's European tour.
According to Peter Larson (director of Black Hills Institute), Trix is among the most complete "Tyrannosaurus" found. Between 75% and 80% of its skeletal volume was recovered. They are thought to have been at least 30 years old at death.
"Titus".
"Titus" is the name given to an obsidian-black skeleton of a "Tyrannosaurus rex" discovered in Montana's Hell Creek Formation in 2014 and excavated in 2018. It is 20% complete, and was named after the protagonist in Shakespeare's "Titus Andronicus". Exhibited in the Nottingham Natural History Museum for 13 months beginning July 2021, it was during that time only the second specimen of "Tyrannosaurus" to be on exhibit in United Kingdom, the other being the type of the junior synonym "Dynamosaurus imperosus" which has the jaw on display in the Natural History Museum, London. External bone inspection has revealed injuries to Titus' right tibia (possibly a claw or bite wound); a deformed toe on the right foot; and a bitten and healed tail. The bite wound near the end of the tail indicates a possible attack by another "Tyrannosaurus rex".
The remains of "Titus" were discovered in September 2014 by commercial paleontologist Craig Pfister near Ekalaka, Carter County, Montana. Excavation of the specimen began in 2018, and took 18 months. The bones of "Titus" were shipped to conservationist Nigel Larkin in the UK, who constructed the mount using a cast of the "Tyrannosaurus" specimen Stan to supplement the known bones of "Titus", after scanning the bones using photogrammetry to create digital models that were 3D printed for use in the exhibition, alongside the display of the real fossil skeleton, and which remained at the museum after the end of the exhibition. For the exhibit at the Nottingham Natural History Museum at Wollaton Hall, Titus was reconstructed in a walking pose.
"Tufts-Love": UWBM 99000.
In 2016 Greg Wilson, David DeMar, and a paleontology team from the Burke Museum of Natural History and Culture, the University of Washington, and the Dig Field school excavated the partial remains of a "Tyrannosaurus rex" from Montana. The partial skeleton was found by two Burke Museum volunteers, Jason Love and Luke Tufts, and was named the "Tufts-Love" rex. Paleontologists at the Burke Museum believe that the Tufts-Love rex was around 15 years old when it died. The skull is of average size for an adult "T. rex". The specimen was found in Late Cretaceous deposits and it is estimated to be 66.3 million years old. The Tufts-Love rex is undergoing preparation by Michael Holland and his team at the Burke Museum. The skeleton is estimated to be 30% complete, but it includes a complete (all of the bones of the skull and jaws are preserved) and mostly articulated skull. Holland describes the skull as minimally distorted and in an "exquisite" state of preservation.
"Peter": AWMM-IL 2022.9.
Peter is the nickname given to a specimen on loan to the Auckland War Memorial Museum by an anonymous owner, currently on display alongside “Barbara” until the end of 2023. The specimen is estimated to be 66.8 million years old and almost of adult size. He was recovered from Niobrara County, Wyoming.
“Peter” is one of only four incredibly rare and visually stunning obisidian black colored tyrannosaurus rex.
He was likely killed by exocannibalism as entire sections of bone were damaged, and some were split open by huge crushing bite-forces. The nature of the crushing on the femur and tibia, along with size of the bite marks, indicates that these bones were bitten through by another "Tyrannosaurus rex". There is also a set of smaller, parallel tooth marks nearby on the shaft that are not attributable to an adult "T. rex". Explanations for this behavior range from response to over-crowded populations, limited food supply, sexual dominance, or even play.
"Barbara": AWMM-IL 2022.21.
Barbara is the nickname given to a specimen on loan to the Auckland War Memorial Museum by the same anonymous owner as "Peter", currently on display alongside him until the end of 2023. The pair will be the first adult male and female "T. rex" to be displayed together. She is one of a few specimens believed to be pregnant.
"Barbara's" circumstances are particularly rare, more so when taken into account that she suffered and survived long after a debilitating foot injury. While she was no longer able to capture her prey, it is suggested that she got by with the help of a mate or cohort feeding her, as the injury would have rendered her immobile for upwards of 6 months. This is vaguely supported by trackway evidence that has been used to imply tyrannosaur group hunting. It is doubtful that "Barbara" ever successfully hunted again as a predator unless its prey was nearby.
"Bloody Mary": BHI 6437.
"Bloody Mary" (as named by Pete Larson) is a nickname for a nearly complete tyrannosaurid (generally considered an adolescent "T. rex" or a "Nanotyrannus") at the North Carolina Museum of Natural Sciences. This specimen, considered the most complete of any "Tyrannosaurus" specimen at more than 98% preservation, was discovered in Montana in 2006. Following years of unsuccessful attempts to sell it to museums or auction it off, the NCMNS started negotiations in 2016, which were prolonged due to a legal battle over the rights to the fossil, which was resolved in 2020. The fossils were officially displayed in 2024.
The Dueling Dinosaurs "T. rex" is an adolescent that was preserved entangled with a "Triceratops"; their combined fossil is referred to as the "Dueling Dinosaurs". Given the injuries present on both fossils, it has been theorized that both died while fighting one another. The specimen is presently undergoing study. Important biological data is likely preserved within the specimen, including body outlines, skin impressions, soft tissues, injuries, stomach contents, and even original proteins.
"Tyrannosaurus mcraeensis": NMMNH P-3698.
The remains of a tyrannosaur were discovered in 1983 in the Campanian-early Maastrichtian Hall Lake Formation in New Mexico by Donald Staton and Joe LaPoint. Reposited at the New Mexico Museum of Natural History and Science, the fossil material (NMMNH P-3698) consists of skull and lower jaw bones, in addition to isolated teeth and chevrons. Some of the bones were briefly mentioned in 1984 as belonging to "T. rex", and were described in 1986. In 2024, Sebastian G. Dalman and colleagues described this specimen as the holotype of a new "Tyrannosaurus" species, "T. mcraeensis". This species differs from "T. rex" in having smaller postorbital crests, a proportionately longer and shallower lower jaw with a less prominent chin suggestive of a weaker bite, and more laterally compressed teeth.
See also.
<templatestyles src="Stack/styles.css"/>
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "114^\\circ ~ \\textrm{W}"
}
]
| https://en.wikipedia.org/wiki?curid=13120745 |
1312654 | Advanced z-transform | In mathematics and signal processing, the advanced z-transform is an extension of the z-transform, to incorporate ideal delays that are not multiples of the sampling time. The advanced z-transform is widely applied, for example, to accurately model processing delays in digital control. It is also known as the modified z-transform.
It takes the form
formula_0
where
Properties.
If the delay parameter, "m", is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
formula_2
formula_3
formula_4
formula_5
formula_6
Example.
Consider the following example where formula_7:
formula_8
If formula_9 then formula_10 reduces to the transform
formula_11
which is clearly just the "z"-transform of formula_12.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "F(z, m) = \\sum_{k=0}^{\\infty} f(k T + m)z^{-k}"
},
{
"math_id": 1,
"text": "[0, T]."
},
{
"math_id": 2,
"text": "\\mathcal{Z} \\left\\{ \\sum_{k=1}^{n} c_k f_k(t) \\right\\} = \\sum_{k=1}^{n} c_k F_k(z, m)."
},
{
"math_id": 3,
"text": "\\mathcal{Z} \\left\\{ u(t - n T)f(t - n T) \\right\\} = z^{-n} F(z, m)."
},
{
"math_id": 4,
"text": "\\mathcal{Z} \\left\\{ f(t) e^{-a\\, t} \\right\\} = e^{-a\\, m} F(e^{a\\, T} z, m)."
},
{
"math_id": 5,
"text": "\\mathcal{Z} \\left\\{ t^y f(t) \\right\\} = \\left(-T z \\frac{d}{dz} + m \\right)^y F(z, m)."
},
{
"math_id": 6,
"text": "\\lim_{k \\to \\infty} f(k T + m) = \\lim_{z \\to 1} (1-z^{-1})F(z, m)."
},
{
"math_id": 7,
"text": "f(t) = \\cos(\\omega t)"
},
{
"math_id": 8,
"text": "\\begin{align}\nF(z, m) & = \\mathcal{Z} \\left\\{ \\cos \\left(\\omega \\left(k T + m \\right) \\right) \\right\\} \\\\\n & = \\mathcal{Z} \\left\\{ \\cos (\\omega k T) \\cos (\\omega m) - \\sin (\\omega k T) \\sin (\\omega m) \\right\\} \\\\\n & = \\cos(\\omega m) \\mathcal{Z} \\left\\{ \\cos (\\omega k T) \\right\\} - \\sin (\\omega m) \\mathcal{Z} \\left\\{ \\sin (\\omega k T) \\right\\} \\\\\n & = \\cos(\\omega m) \\frac{z \\left(z - \\cos (\\omega T) \\right)}{z^2 - 2z \\cos(\\omega T) + 1} - \\sin(\\omega m) \\frac{z \\sin(\\omega T)}{z^2 - 2z \\cos(\\omega T) + 1} \\\\\n & = \\frac{z^2 \\cos(\\omega m) - z \\cos(\\omega(T - m))}{z^2 - 2z \\cos(\\omega T) + 1}.\n\\end{align}"
},
{
"math_id": 9,
"text": "m=0"
},
{
"math_id": 10,
"text": "F(z, m)"
},
{
"math_id": 11,
"text": "F(z, 0) = \\frac{z^2 - z \\cos(\\omega T)}{z^2 - 2z \\cos(\\omega T) + 1},"
},
{
"math_id": 12,
"text": "f(t)"
}
]
| https://en.wikipedia.org/wiki?curid=1312654 |
13127410 | Oberth effect | Type of spacecraft maneuver
In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity (and so, its kinetic energy) is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, a Transylvanian Saxon physicist and a founder of modern rocketry.
Because the vehicle remains near periapsis only for a short time, for the Oberth maneuver to be most effective the vehicle must be able to generate as much impulse as possible in the shortest possible time. As a result the Oberth maneuver is much more useful for high-thrust rocket engines like liquid-propellant rockets, and less useful for low-thrust reaction engines such as ion drives, which take a long time to gain speed. Low thrust rockets can use the Oberth effect by splitting a long departure burn into several short burns near the periapsis. The Oberth effect also can be used to understand the behavior of multi-stage rockets: the upper stage can generate much more usable kinetic energy than the total chemical energy of the propellants it carries.
In terms of the energies involved, the Oberth effect is more effective at higher speeds because at high speed the propellant has significant kinetic energy in addition to its chemical potential energy. At higher speed the vehicle is able to employ the greater change (reduction) in kinetic energy of the propellant (as it is exhausted backward and hence at reduced speed and hence reduced kinetic energy) to generate a greater increase in kinetic energy of the vehicle.
Explanation in terms of work and kinetic energy.
Because kinetic energy equals "mv"2/2, this change in velocity imparts a greater increase in kinetic energy at a high velocity than it would at a low velocity. For example, considering a 2 kg rocket:
This greater change in kinetic energy can then carry the rocket higher in the gravity well than if the propellant were burned at a lower speed.
Description in terms of work.
The thrust produced by a rocket engine is independent of the rocket’s velocity relative to the surrounding atmosphere. A rocket acting on a fixed object, as in a static firing, does no useful work on the rocket; the rocket's chemical energy is progressively converted to kinetic energy of the exhaust, plus heat. But when the rocket moves, its thrust acts through the distance it moves. Force multiplied by displacement is the definition of mechanical work. The greater the velocity of the rocket and payload during the burn the greater is the displacement and the work done, and the greater the increase in kinetic energy of the rocket and its payload. As the velocity of the rocket increases, progressively more of the available kinetic energy goes to the rocket and its payload, and less to the exhaust.
This is shown as follows. The mechanical work done on the rocket (formula_0) is defined as the dot product of the force of the engine's thrust (formula_1) and the displacement it travels during the burn (formula_2):
formula_3
If the burn is made in the prograde direction, formula_4. The work results in a change in kinetic energy
formula_5
Differentiating with respect to time, we obtain
formula_6
or
formula_7
where formula_8 is the velocity. Dividing by the instantaneous mass formula_9 to express this in terms of specific energy (formula_10), we get
formula_11
where formula_12 is the acceleration vector.
Thus it can be readily seen that the rate of gain of specific energy of every part of the rocket is proportional to speed and, given this, the equation can be integrated (numerically or otherwise) to calculate the overall increase in specific energy of the rocket.
Impulsive burn.
Integrating the above energy equation is often unnecessary if the burn duration is short. Short burns of chemical rocket engines close to periapsis or elsewhere are usually mathematically modeled as impulsive burns, where the force of the engine dominates any other forces that might change the vehicle's energy over the burn.
For example, as a vehicle falls toward periapsis in any orbit (closed or escape orbits) the velocity relative to the central body increases. Briefly burning the engine (an "impulsive burn") prograde at periapsis increases the velocity by the same increment as at any other time (formula_13). However, since the vehicle's kinetic energy is related to the "square" of its velocity, this increase in velocity has a non-linear effect on the vehicle's kinetic energy, leaving it with higher energy than if the burn were achieved at any other time.
Oberth calculation for a parabolic orbit.
If an impulsive burn of Δ"v" is performed at periapsis in a parabolic orbit, then the velocity at periapsis before the burn is equal to the escape velocity ("V"esc), and the specific kinetic energy after the burn is
formula_14
where formula_15.
When the vehicle leaves the gravity field, the loss of specific kinetic energy is
formula_16
so it retains the energy
formula_17
which is larger than the energy from a burn outside the gravitational field (formula_18) by
formula_19
When the vehicle has left the gravity well, it is traveling at a speed
formula_20
For the case where the added impulse Δ"v" is small compared to escape velocity, the 1 can be ignored, and the effective Δ"v" of the impulsive burn can be seen to be multiplied by a factor of simply
formula_21
and one gets
formula_22 ≈ formula_23
Similar effects happen in closed and hyperbolic orbits.
Parabolic example.
If the vehicle travels at velocity "v" at the start of a burn that changes the velocity by Δ"v", then the change in specific orbital energy (SOE) due to the new orbit is
formula_24
Once the spacecraft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy approaches zero. Therefore, the larger the "v" at the time of the burn, the greater the final kinetic energy, and the higher the final velocity.
The effect becomes more pronounced the closer to the central body, or more generally, the deeper in the gravitational field potential in which the burn occurs, since the velocity is higher there.
So if a spacecraft is on a parabolic flyby of Jupiter with a periapsis velocity of 50 km/s and performs a 5 km/s burn, it turns out that the final velocity change at great distance is 22.9 km/s, giving a multiplication of the burn by 4.58 times.
Paradox.
It may seem that the rocket is getting energy for free, which would violate conservation of energy. However, any gain to the rocket's kinetic energy is balanced by a relative decrease in the kinetic energy the exhaust is left with (the kinetic energy of the exhaust may still increase, but it does not increase as much). Contrast this to the situation of static firing, where the speed of the engine is fixed at zero. This means that its kinetic energy does not increase at all, and all the chemical energy released by the fuel is converted to the exhaust's kinetic energy (and heat).
At very high speeds the mechanical power imparted to the rocket can exceed the total power liberated in the combustion of the propellant; this may also seem to violate conservation of energy. But the propellants in a fast-moving rocket carry energy not only chemically, but also in their own kinetic energy, which at speeds above a few kilometres per second exceed the chemical component. When these propellants are burned, some of this kinetic energy is transferred to the rocket along with the chemical energy released by burning.
The Oberth effect can therefore partly make up for what is extremely low efficiency early in the rocket's flight when it is moving only slowly. Most of the work done by a rocket early in flight is "invested" in the kinetic energy of the propellant not yet burned, part of which they will release later when they are burned.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "W"
},
{
"math_id": 1,
"text": "\\vec{F}"
},
{
"math_id": 2,
"text": "\\vec{s}"
},
{
"math_id": 3,
"text": "W = \\vec{F} \\cdot \\vec{s}."
},
{
"math_id": 4,
"text": "\\vec{F} \\cdot \\vec{s} = \\|F\\| \\cdot \\|s\\| = F \\cdot s"
},
{
"math_id": 5,
"text": "\\Delta E_k = F \\cdot s."
},
{
"math_id": 6,
"text": "\\frac{\\mathrm{d}E_k}{\\mathrm{d}t} = F \\cdot \\frac{\\mathrm{d}s}{\\mathrm{d}t},"
},
{
"math_id": 7,
"text": "\\frac{\\mathrm{d}E_k}{\\mathrm{d}t} = F \\cdot v,"
},
{
"math_id": 8,
"text": "v"
},
{
"math_id": 9,
"text": "m"
},
{
"math_id": 10,
"text": "e_k"
},
{
"math_id": 11,
"text": "\\frac{\\mathrm{d}e_k}{\\mathrm{d}t} = \\frac F m \\cdot v = a \\cdot v,"
},
{
"math_id": 12,
"text": "a"
},
{
"math_id": 13,
"text": "\\Delta v"
},
{
"math_id": 14,
"text": "\\begin{align}\n e_k &= \\tfrac{1}{2} V^2 \\\\\n &= \\tfrac{1}{2} (V_\\text{esc} + \\Delta v )^2 \\\\\n &= \\tfrac{1}{2} V_\\text{esc} ^ 2 + \\Delta v V_\\text{esc} + \\tfrac{1}{2} \\Delta v^2,\n\\end{align}"
},
{
"math_id": 15,
"text": "V = V_\\text{esc} + \\Delta v"
},
{
"math_id": 16,
"text": "\\tfrac{1}{2} V_\\text{esc}^2,"
},
{
"math_id": 17,
"text": "\\Delta v V_\\text{esc} + \\tfrac{1}{2} \\Delta v^2,"
},
{
"math_id": 18,
"text": "\\tfrac{1}{2} \\Delta v^2"
},
{
"math_id": 19,
"text": " \\Delta v V_\\text{esc}."
},
{
"math_id": 20,
"text": "V = \\Delta v \\sqrt{1 + \\frac{2 V_\\text{esc}}{\\Delta v}}."
},
{
"math_id": 21,
"text": "\\sqrt{\\frac{2 V_\\text{esc}}{\\Delta v}}"
},
{
"math_id": 22,
"text": "V"
},
{
"math_id": 23,
"text": "\\sqrt{{2 V_\\text{esc}}{\\Delta v}} ."
},
{
"math_id": 24,
"text": "v \\,\\Delta v + \\tfrac{1}{2}(\\Delta v)^2."
}
]
| https://en.wikipedia.org/wiki?curid=13127410 |
13129704 | Yuri Nesterenko (mathematician) | Soviet and Russian mathematician
Yuri Valentinovich Nesterenko (; born 5 December 1946 in Kharkov, USSR, now Ukraine) is a Soviet and Russian mathematician who has written papers in algebraic independence theory and transcendental number theory.
In 1997, he was awarded the Ostrowski Prize for his proof that the numbers π and "e"π are algebraically independent. In fact, he proved the stronger result:
He is a professor at Moscow State University, where he completed the mechanical-mathematical program in 1969, then the doctorate program (Soviet habilitation) in 1973, and became a professor of the Number Theory Department in 1992.
He studied under Andrei Borisovich Shidlovskii. Nesterenko's students have included Wadim Zudilin.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "e^{\\pi\\sqrt{3}}"
},
{
"math_id": 1,
"text": "e^{\\pi\\sqrt{n}}"
}
]
| https://en.wikipedia.org/wiki?curid=13129704 |
1313004 | Lotka's law | An application of Zipf's law describing the frequency of publication by authors in any given field
Lotka's law, named after Alfred J. Lotka, is one of a variety of special applications of Zipf's law. It describes the frequency of publication by authors in any given field. Let formula_0 be the number of publications, formula_1 be the number of authors with formula_0 publications, and formula_2 be a constants depending on the specific field. Lotka's law states that formula_3.
In Lotka's original publication, he claimed formula_4. Subsequent research showed that formula_2 varies depending on the discipline.
Equivalently, Lotka's law can be stated as formula_5, where formula_6 is the number of authors with "at least" formula_0 publications. Their equivalence can be proved by taking the derivative.
Example.
Assume that n=2 in a discipline, then as the number of articles published increases, authors producing that many publications become less frequent. There are 1/4 as many authors publishing two articles within a specified time period as there are single-publication authors, 1/9 as many publishing three articles, 1/16 as many publishing four articles, etc.
And if 100 authors wrote "exactly" one article each over a specific period in the discipline, then:
That would be a total of 294 articles and 155 writers, with an average of 1.9 articles for each writer.
Relationship to Riemann Zeta.
Lotka's law may be described using the Zeta distribution:
formula_7
for formula_8 and where
formula_9
is the Riemann zeta function. It is the limiting case of Zipf's law where an individual's maximum number of publications is infinite. | [
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "Y"
},
{
"math_id": 2,
"text": "k"
},
{
"math_id": 3,
"text": "Y \\propto X^{-k}"
},
{
"math_id": 4,
"text": "k=2"
},
{
"math_id": 5,
"text": "Y' \\propto X^{-(k-1)}"
},
{
"math_id": 6,
"text": "Y'"
},
{
"math_id": 7,
"text": "f(x) = \\frac{1}{\\zeta (s)} \\cdot \\frac{1}{x^s} "
},
{
"math_id": 8,
"text": " x = 1, 2, 3, 4, \\dots "
},
{
"math_id": 9,
"text": " \\zeta (s) = \\sum_{x=1}^\\infty \\frac{1}{x^s} "
}
]
| https://en.wikipedia.org/wiki?curid=1313004 |
13134826 | Close-up lens | Secondary lens used to enable macro photography
In photography, a close-up lens (sometimes referred to as "close-up filter" or a "macro filter") is a simple secondary lens used to enable macro photography without requiring a specialised primary lens. They work like reading glasses, allowing a primary lens to focus more closely. Bringing the focus closer allows the photographer more possibilities.
Close-up lenses typically mount on the filter thread of the primary lens, and are often manufactured and sold by suppliers of photographic filters. Nonetheless, they are lenses and not filters. Some manufacturers refer to their close-up lenses as "diopters", after the unit of measurement of their optical power.
Close-up lenses do not affect exposure, unlike extension tubes, which also can be used for macro photography with a non-macro lens.
Optical power.
Close-up lenses are often specified by their optical power in diopters, the reciprocal of the focal length in meters. For a close-up lens, the diopter value is positive: the bigger the number, the greater the effective magnification.
Higher quality achromatic lenses commonly lack a strength specification in diopters. It can be inferred as the reciprocal of the maximum specified working distance in meters (i.e., a lens with a maximum working distance of has a strength of +4 diopters).
Several close-up lenses may be used in combination; the optical power of the combination is the sum of the optical powers of the component lenses. For example, a set of lenses of +1, +2, and +4 diopters can be combined to provide a range from +1 to +7 in steps of 1.
Working distances and magnifications.
Close-up lenses change both the maximum and minimum focus distances of a lens. The range can be rather small.
Working at maximum distance.
Adding a close-up lens to a lens focused to infinity changes the focus point to the focal length of the close-up lens, that is, the inverse of its optical power. This is the combination's maximal working distance:
formula_0
That distance is sometimes given on the filter in millimeters. A +3 close-up lens has a maximal working distance of .
The magnification is the focal distance of the objective lens (f) divided by the focal distance of the close-up lens; i.e., the focal distance of the objective lens (in meters) multiplied by the diopter value (D) of the close-up lens:
formula_1
In the example above, if the lens has a focal distance, the magnification is 0.3 × 3 = 0.9.
Given the small size of most sensors (about for APS-C sensors) a insect will almost fill the frame at this magnification. Using a zoom lens makes it easy to frame the subject as desired.
Working at minimal distance.
When you add a close-up lens to a camera which is focusing at the shortest distance at which the objective lens can focus, the focus will move to a distance which is given by following formula:
formula_2
X being the shortest distance at which the objective lens can focus (in meters), and D being the diopter value of the close-up lens. This is the minimal working distance at which you will be able to take a picture with the close-up lens.
For example, a lens that can focus at combined with a +3 diopter close-up lens will give a closest working distance of 1.5 / (3 × 1.5 + 1) = 0.273 m.
The magnification reached in those conditions is given by following formula:
formula_3
MX being the magnification at distance X without the close-up lens.
In the example above, the gain of magnification at "X"min will be (3 × 1.5 + 1)
5.5.
While it would seem obvious that at this "X"min distance you will get the highest magnification, focus breathing can cause more of a difference in actual magnification than the small overall in-focus working distance range particularly for higher strength diopters.
Macro photography with a close-up lens.
Close-up lenses can make a telephoto lens function as a macro lens with a large working distance. This is useful, for example, to prevent scaring small animals or isolating the subject from messy surroundings. To use the filters for animals the size of the animal will determine the working distance (small snakes to , lizards , small butterflies, beetles ), so it is essential to know what will be the favorite subject before screwing on a close-up lens. The close-up lenses are most effective with long focal length objectives and using a zoom lens is very practical to have some flexibility in the magnification. A good technique for sharp focussing is to take a picture at a long focal length first to have optimal sharpness at the essential details and then zooming out to have the desired size in the frame.
Optical issues.
Some single-element close-up lenses produce images with severe aberrations but there are also high-quality close-up lenses composed as achromatic doublets which are capable of producing excellent images, with fairly low loss of sharpness.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "X_\\text{max} = \\frac{1}{D}"
},
{
"math_id": 1,
"text": "M_{X\\text{max}} = f D"
},
{
"math_id": 2,
"text": "X_\\text{min} = \\frac{X}{D X + 1}"
},
{
"math_id": 3,
"text": "M_{X\\text{min}} = M_X (D X + 1)"
}
]
| https://en.wikipedia.org/wiki?curid=13134826 |
1313620 | Inverse quadratic interpolation | Method of solving equations
In numerical analysis, inverse quadratic interpolation is a root-finding algorithm, meaning that it is an algorithm for solving equations of the form "f"("x") = 0. The idea is to use quadratic interpolation to approximate the inverse of "f". This algorithm is rarely used on its own, but it is important because it forms part of the popular Brent's method.
The method.
The inverse quadratic interpolation algorithm is defined by the recurrence relation
formula_0
formula_1
where "f""k" = "f"("x""k"). As can be seen from the recurrence relation, this method requires three initial values, "x"0, "x"1 and "x"2.
Explanation of the method.
We use the three preceding iterates, "x""n"−2, "x""n"−1 and "x""n", with their function values, "f""n"−2, "f""n"−1 and "f""n". Applying the Lagrange interpolation formula to do quadratic interpolation on the inverse of "f" yields
formula_2
formula_3
We are looking for a root of "f", so we substitute "y" = "f"("x") = 0 in the above equation, and this results in the above recursion formula.
Behaviour.
The asymptotic behaviour is very good: generally, the iterates "x""n" converge fast to the root once they get close. However, performance is often quite poor if the initial values are not close to the actual root. For instance, if by any chance two of the function values "f""n"−2, "f""n"−1 and "f""n" coincide, the algorithm fails completely. Thus, inverse quadratic interpolation is seldom used as a stand-alone algorithm.
The order of this convergence is approximately 1.84 as can be proved by secant method analysis.
Comparison with other root-finding methods.
As noted in the introduction, inverse quadratic interpolation is used in Brent's method.
Inverse quadratic interpolation is also closely related to some other root-finding methods.
Using linear interpolation instead of quadratic interpolation gives the secant method. Interpolating "f" instead of the inverse of "f" gives Muller's method. | [
{
"math_id": 0,
"text": " x_{n+1} = \\frac{f_{n-1}f_n}{(f_{n-2}-f_{n-1})(f_{n-2}-f_n)} x_{n-2} + \\frac{f_{n-2}f_n}{(f_{n-1}-f_{n-2})(f_{n-1}-f_n)} x_{n-1} "
},
{
"math_id": 1,
"text": " {} + \\frac{f_{n-2}f_{n-1}}{(f_n-f_{n-2})(f_n-f_{n-1})} x_n, "
},
{
"math_id": 2,
"text": " f^{-1}(y) = \\frac{(y-f_{n-1})(y-f_n)}{(f_{n-2}-f_{n-1})(f_{n-2}-f_n)} x_{n-2} + \\frac{(y-f_{n-2})(y-f_n)}{(f_{n-1}-f_{n-2})(f_{n-1}-f_n)} x_{n-1} "
},
{
"math_id": 3,
"text": "\\qquad + \\frac{(y-f_{n-2})(y-f_{n-1})}{(f_n-f_{n-2})(f_n-f_{n-1})} x_n. "
}
]
| https://en.wikipedia.org/wiki?curid=1313620 |
1313639 | Skyrmion | Type of topological solutions in non-linear sigma models
In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological soliton in the pion field, it has the remarkable property of being able to model, with reasonable accuracy, multiple low-energy properties of the nucleon, simply by fixing the nucleon radius. It has since found application in solid-state physics, as well as having ties to certain areas of string theory.
Skyrmions as topological objects are important in solid-state physics, especially in the emerging technology of spintronics. A two-dimensional magnetic skyrmion, as a topological object, is formed, e.g., from a 3D effective-spin "hedgehog" (in the field of micromagnetics: out of a so-called "Bloch point" singularity of homotopy degree +1) by a stereographic projection, whereby the positive north-pole spin is mapped onto a far-off edge circle of a 2D-disk, while the negative south-pole spin is mapped onto the center of the disk. In a spinor field such as for example photonic or polariton fluids the skyrmion topology corresponds to a full Poincaré beam (a spin vortex comprising all the states of polarization mapped by a stereographic projection of the Poincaré sphere to the real plane). A dynamical pseudospin skyrmion results from the stereographic projection of a rotating polariton Bloch sphere in the case of dynamical full Bloch beams.
Skyrmions have been reported, but not conclusively proven, to appear in Bose–Einstein condensates, thin magnetic films, and chiral nematic liquid crystals, as well as in free-space optics.
As a model of the nucleon, the topological stability of the skyrmion can be interpreted as a statement that the baryon number is conserved; i.e. that the proton does not decay. The Skyrme Lagrangian is essentially a one-parameter model of the nucleon. Fixing the parameter fixes the proton radius, and also fixes all other low-energy properties, which appear to be correct to about 30%, a significant level of predictive power.
Hollowed-out skyrmions form the basis for the chiral bag model (Cheshire Cat model) of the nucleon. The exact results for the duality between the fermion spectrum and the topological winding number of the non-linear sigma model have been obtained by Dan Freed. This can be interpreted as a foundation for the duality between a quantum chromodynamics (QCD) description of the nucleon (but consisting only of quarks, and without gluons) and the Skyrme model for the nucleon.
The skyrmion can be quantized to form a quantum superposition of baryons and resonance states. It could be predicted from some nuclear matter properties.
Topological soliton.
In field theory, skyrmions are homotopically non-trivial classical solutions of a nonlinear sigma model with a non-trivial target manifold topology – hence, they are topological solitons. An example occurs in chiral models of mesons, where the target manifold is a homogeneous space of the structure group
formula_0
where SU("N")"L" and SU("N")"R" are the left and right chiral symmetries, and SU("N")diag is the diagonal subgroup. In nuclear physics, for "N" = 2, the chiral symmetries are understood to be the isospin symmetry of the nucleon. For "N" = 3, the isoflavor symmetry between the up, down and strange quarks is more broken, and the skyrmion models are less successful or accurate.
If spacetime has the topology S3×R, then classical configurations can be classified by an integral winding number because the third homotopy group
formula_1
is equivalent to the ring of integers, with the congruence sign referring to homeomorphism.
A topological term can be added to the chiral Lagrangian, whose integral depends only upon the homotopy class; this results in superselection sectors in the quantised model. In (1 + 1)-dimensional spacetime, a skyrmion can be approximated by a soliton of the Sine–Gordon equation; after quantisation by the Bethe ansatz or otherwise, it turns into a fermion interacting according to the massive Thirring model.
Lagrangian.
The Lagrangian for the skyrmion, as written for the original chiral SU(2) effective Lagrangian of the nucleon-nucleon interaction (in (3 + 1)-dimensional spacetime), can be written as
formula_2
where formula_3, formula_4, formula_5 are the isospin Pauli matrices, formula_6 is the Lie bracket commutator, and tr is the matrix trace. The meson field (pion field, up to a dimensional factor) at spacetime coordinate formula_7 is given by formula_8. A broad review of the geometric interpretation of formula_9 is presented in the article on sigma models.
When written this way, the formula_10 is clearly an element of the Lie group SU(2), and formula_11 an element of the Lie algebra su(2). The pion field can be understood abstractly to be a section of the tangent bundle of the principal fiber bundle of SU(2) over spacetime. This abstract interpretation is characteristic of all non-linear sigma models.
The first term, formula_12 is just an unusual way of writing the quadratic term of the non-linear sigma model; it reduces to formula_13. When used as a model of the nucleon, one writes
formula_14
with the dimensional factor of formula_15 being the pion decay constant. (In 1 + 1 dimensions, this constant is not dimensional and can thus be absorbed into the field definition.)
The second term establishes the characteristic size of the lowest-energy soliton solution; it determines the effective radius of the soliton. As a model of the nucleon, it is normally adjusted so as to give the correct radius for the proton; once this is done, other low-energy properties of the nucleon are automatically fixed, to within about 30% accuracy. It is this result, of tying together what would otherwise be independent parameters, and doing so fairly accurately, that makes the Skyrme model of the nucleon so appealing and interesting. Thus, for example, constant formula_16 in the quartic term is interpreted as the vector-pion coupling ρ–π–π between the rho meson (the nuclear vector meson) and the pion; the skyrmion relates the value of this constant to the baryon radius.
Noether current.
The local winding number density is given by
formula_17
where formula_18 is the totally antisymmetric Levi-Civita symbol (equivalently, the Hodge star, in this context).
As a physical quantity, this can be interpreted as the baryon current; it is conserved: formula_19, and the conservation follows as a Noether current for the chiral symmetry.
The corresponding charge is the baryon number:
formula_20
As a conserved charge, it is time-independent: formula_21, the physical interpretation of which is that protons do not decay.
In the chiral bag model, one cuts a hole out of the center and fills it with quarks. Despite this obvious "hackery", the total baryon number is conserved: the missing charge from the hole is exactly compensated by the spectral asymmetry of the vacuum fermions inside the bag.
Magnetic materials/data storage.
One particular form of skyrmions is magnetic skyrmions, found in magnetic materials that exhibit spiral magnetism due to the Dzyaloshinskii–Moriya interaction, double-exchange mechanism or competing Heisenberg exchange interactions. They form "domains" as small as 1 nm (e.g. in Fe on Ir(111)). The small size and low energy consumption of magnetic skyrmions make them a good candidate for future data-storage solutions and other spintronics devices.
Researchers could read and write skyrmions using scanning tunneling microscopy. The topological charge, representing the existence and non-existence of skyrmions, can represent the bit states "1" and "0". Room-temperature skyrmions were reported.
Skyrmions operate at current densities that are several orders of magnitude weaker than conventional magnetic devices. In 2015 a practical way to create and access magnetic skyrmions under ambient room-temperature conditions was announced. The device used arrays of magnetized cobalt disks as artificial Bloch skyrmion lattices atop a thin film of cobalt and palladium. Asymmetric magnetic nanodots were patterned with controlled circularity on an underlayer with perpendicular magnetic anisotropy (PMA). Polarity is controlled by a tailored magnetic-field sequence and demonstrated in magnetometry measurements. The vortex structure is imprinted into the underlayer's interfacial region by suppressing the PMA by a critical ion-irradiation step. The lattices are identified with polarized neutron reflectometry and have been confirmed by magnetoresistance measurements.
A recent (2019) study demonstrated a way to move skyrmions, purely using electric field (in the absence of electric current). The authors used Co/Ni multilayers with a thickness slope and Dzyaloshinskii–Moriya interaction and demonstrated skyrmions. They showed that the displacement and velocity depended directly on the applied voltage.
In 2020, a team of researchers from the Swiss Federal Laboratories for Materials Science and Technology (Empa) has succeeded for the first time in producing a tunable multilayer system in which two different types of skyrmions – the future bits for "0" and "1" – can exist at room temperature.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\left(\\frac{\\operatorname{SU}(N)_L \\times \\operatorname{SU}(N)_R}{\\operatorname{SU}(N)_\\text{diag}}\\right),"
},
{
"math_id": 1,
"text": "\\pi_3\\left(\\frac{\\operatorname{SU}(N)_L \\times \\operatorname{SU}(N)_R}{\\operatorname{SU}(N)_\\text{diag}} \\cong \\operatorname{SU}(N)\\right)"
},
{
"math_id": 2,
"text": "\\mathcal{L} =\n \\frac{-f^2_\\pi}{4}\\operatorname{tr}(L_\\mu L^\\mu) + \\frac{1}{32g^2} \\operatorname{tr}[L_\\mu, L_\\nu] [L^\\mu, L^\\nu],\n"
},
{
"math_id": 3,
"text": "L_\\mu = U^\\dagger \\partial_\\mu U"
},
{
"math_id": 4,
"text": "U = \\exp i\\vec\\tau \\cdot \\vec\\theta"
},
{
"math_id": 5,
"text": "\\vec\\tau"
},
{
"math_id": 6,
"text": "[\\cdot, \\cdot]"
},
{
"math_id": 7,
"text": "x"
},
{
"math_id": 8,
"text": "\\vec\\theta = \\vec\\theta(x)"
},
{
"math_id": 9,
"text": "L_\\mu"
},
{
"math_id": 10,
"text": "U"
},
{
"math_id": 11,
"text": "\\vec\\theta"
},
{
"math_id": 12,
"text": "\\operatorname{tr}(L_\\mu L^\\mu)"
},
{
"math_id": 13,
"text": "-\\operatorname{tr}(\\partial_\\mu U^\\dagger \\partial^\\mu U)"
},
{
"math_id": 14,
"text": "U = \\frac{1}{f_\\pi}(\\sigma + i\\vec\\tau \\cdot \\vec\\pi),"
},
{
"math_id": 15,
"text": "f_\\pi"
},
{
"math_id": 16,
"text": "g"
},
{
"math_id": 17,
"text": "B^\\mu = \\epsilon^{\\mu\\nu\\alpha\\beta} \\operatorname{tr}L_\\nu L_\\alpha L_\\beta,"
},
{
"math_id": 18,
"text": "\\epsilon^{\\mu\\nu\\alpha\\beta}"
},
{
"math_id": 19,
"text": "\\partial_\\mu B^\\mu = 0"
},
{
"math_id": 20,
"text": "B = \\int d^3x\\, B^0(x)."
},
{
"math_id": 21,
"text": "dB/dt = 0"
}
]
| https://en.wikipedia.org/wiki?curid=1313639 |
1313664 | Selection rule | Formal constraint on the possible transitions of a system from one quantum state to another
In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in atomic nuclei, and so on. The selection rules may differ according to the technique used to observe the transition. The selection rule also plays a role in chemical reactions, where some are formally spin-forbidden reactions, that is, reactions where the spin state changes at least once from reactants to products.
In the following, mainly atomic and molecular transitions are considered.
Overview.
In quantum mechanics the basis for a spectroscopic selection rule is the value of the "transition moment integral"
formula_0
where formula_1 and formula_2 are the wave functions of the two states, "state 1" and "state 2", involved in the transition, and μ is the transition moment operator. This integral represents the propagator (and thus the probability) of the transition between states 1 and 2; if the value of this integral is "zero" then the transition is "forbidden".
In practice, to determine a selection rule the integral itself does not need to be calculated: It is sufficient to determine the symmetry of the "transition moment function" formula_3
If the transition moment function is symmetric over all of the totally symmetric representation of the point group to which the atom or molecule belongs, then the integral's value is (in general) "not" zero and the transition "is" allowed. Otherwise, the transition is "forbidden".
The transition moment integral is zero if the "transition moment function", formula_4 is anti-symmetric or odd, i.e. formula_5 holds. The symmetry of the transition moment function is the direct product of the parities of its three components. The symmetry characteristics of each component can be obtained from standard character tables. Rules for obtaining the symmetries of a direct product can be found in texts on character tables.
Examples.
Electronic spectra.
The Laporte rule is a selection rule formally stated as follows: In a centrosymmetric environment, transitions between like atomic orbitals such as "s"-"s", "p"-"p", "d"-"d", or "f"-"f," transitions are forbidden. The Laporte rule (law) applies to electric dipole transitions, so the operator has "u" symmetry (meaning "ungerade", odd). "p" orbitals also have "u" symmetry, so the symmetry of the transition moment function is given by the triple product "u"×"u"×"u", which has "u" symmetry. The transitions are therefore forbidden. Likewise, "d" orbitals have "g" symmetry (meaning "gerade", even), so the triple product "g"×"u"×"g" also has "u" symmetry and the transition is forbidden.
The wave function of a single electron is the product of a space-dependent wave function and a spin wave function. Spin is directional and can be said to have odd parity. It follows that transitions in which the spin "direction" changes are forbidden. In formal terms, only states with the same total spin quantum number are "spin-allowed". In crystal field theory, "d"-"d" transitions that are spin-forbidden are much weaker than spin-allowed transitions. Both can be observed, in spite of the Laporte rule, because the actual transitions are coupled to vibrations that are anti-symmetric and have the same symmetry as the dipole moment operator.
Vibrational spectra.
In vibrational spectroscopy, transitions are observed between different vibrational states. In a fundamental vibration, the molecule is excited from its ground state ("v" = 0) to the first excited state ("v" = 1). The symmetry of the ground-state wave function is the same as that of the molecule. It is, therefore, a basis for the totally symmetric representation in the point group of the molecule. It follows that, for a vibrational transition to be allowed, the symmetry of the excited state wave function must be the same as the symmetry of the transition moment operator.
In infrared spectroscopy, the transition moment operator transforms as either "x" and/or "y" and/or "z". The excited state wave function must also transform as at least one of these vectors. In Raman spectroscopy, the operator transforms as one of the second-order terms in the right-most column of the character table, below.
The molecule methane, CH4, may be used as an example to illustrate the application of these principles. The molecule is tetrahedral and has "Td" symmetry. The vibrations of methane span the representations A1 + E + 2T2. Examination of the character table shows that all four vibrations are Raman-active, but only the T2 vibrations can be seen in the infrared spectrum.
In the harmonic approximation, it can be shown that overtones are forbidden in both infrared and Raman spectra. However, when anharmonicity is taken into account, the transitions are weakly allowed.
In Raman and infrared spectroscopy, the selection rules predict certain vibrational modes to have zero intensities in the Raman and/or the IR. Displacements from the ideal structure can result in relaxation of the selection rules and appearance of these unexpected phonon modes in the spectra. Therefore, the appearance of new modes in the spectra can be a useful indicator of symmetry breakdown.
Rotational spectra.
The selection rule for rotational transitions, derived from the symmetries of the rotational wave functions in a rigid rotor, is Δ"J" = ±1, where "J" is a rotational quantum number.
Coupled transitions.
There are many types of coupled transition such as are observed in vibration–rotation spectra. The excited-state wave function is the product of two wave functions such as vibrational and rotational. The general principle is that the symmetry of the excited state is obtained as the direct product of the symmetries of the component wave functions. In rovibronic transitions, the excited states involve three wave functions.
The infrared spectrum of hydrogen chloride gas shows rotational fine structure superimposed on the vibrational spectrum. This is typical of the infrared spectra of heteronuclear diatomic molecules. It shows the so-called "P" and "R" branches. The "Q" branch, located at the vibration frequency, is absent. Symmetric top molecules display the "Q" branch. This follows from the application of selection rules.
Resonance Raman spectroscopy involves a kind of vibronic coupling. It results in much-increased intensity of fundamental and overtone transitions as the vibrations "steal" intensity from an allowed electronic transition. In spite of appearances, the selection rules are the same as in Raman spectroscopy.
Angular momentum.
In general, electric (charge) radiation or magnetic (current, magnetic moment) radiation can be classified into multipoles Eλ (electric) or Mλ (magnetic) of order 2λ, e.g., E1 for electric dipole, E2 for quadrupole, or E3 for octupole. In transitions where the change in angular momentum between the initial and final states makes several multipole radiations possible, usually the lowest-order multipoles are overwhelmingly more likely, and dominate the transition.
The emitted particle carries away angular momentum, with quantum number λ, which for the photon must be at least 1, since it is a vector particle (i.e., it has JP = 1− ). Thus, there is no radiation from E0 (electric monopoles) or M0 (magnetic monopoles, which do not seem to exist).
Since the total angular momentum has to be conserved during the transition, we have that
formula_6
where formula_7 and its z-projection is given by formula_8 and where formula_9 and formula_10 are, respectively, the initial and final angular momenta of the atom.
The corresponding quantum numbers λ and μ (z-axis angular momentum) must satisfy
formula_11
and
formula_12
Parity is also preserved. For electric multipole transitions
formula_13
while for magnetic multipoles
formula_14
Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles.
These considerations generate different sets of transitions rules depending on the multipole order and type. The expression "forbidden transitions" is often used, but this does not mean that these transitions "cannot" occur, only that they are "electric-dipole-forbidden". These transitions are perfectly possible; they merely occur at a lower rate. If the rate for an E1 transition is non-zero, the transition is said to be permitted; if it is zero, then M1, E2, etc. transitions can still produce radiation, albeit with much lower transitions rates. The transition rate decreases by a factor of about 1000 from one multipole to the next one, so the lowest multipole transitions are most likely to occur.
Semi-forbidden transitions (resulting in so-called intercombination lines) are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated. This is a result of the failure of LS coupling.
Summary table.
formula_15 is the total angular momentum,
formula_16 is the azimuthal quantum number,
formula_17 is the spin quantum number, and
formula_18 is the secondary total angular momentum quantum number.
Which transitions are allowed is based on the hydrogen-like atom. The symbol formula_19 is used to indicate a forbidden transition.
In hyperfine structure, the total angular momentum of the atom is formula_20 where formula_21 is the nuclear spin angular momentum and formula_22 is the total angular momentum of the electron(s). Since formula_23 has a similar mathematical form as formula_24 it obeys a selection rule table similar to the table above.
Surface.
In surface vibrational spectroscopy, the "surface selection rule" is applied to identify the peaks observed in vibrational spectra. When a molecule is adsorbed on a substrate, the molecule induces opposite image charges in the substrate. The dipole moment of the molecule and the image charges perpendicular to the surface reinforce each other. In contrast, the dipole moments of the molecule and the image charges parallel to the surface cancel out. Therefore, only molecular vibrational peaks giving rise to a dynamic dipole moment perpendicular to the surface will be observed in the vibrational spectrum.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<br> | [
{
"math_id": 0,
"text": "m_{1,2}=\\int \\psi_1^* \\, \\mu \\, \\psi_2 \\, \\mathrm{d}\\tau\\,,"
},
{
"math_id": 1,
"text": "\\psi_1"
},
{
"math_id": 2,
"text": "\\psi_2"
},
{
"math_id": 3,
"text": "\\,\\psi_1^* \\; \\mu \\; \\psi_2~."
},
{
"math_id": 4,
"text": "\\psi_1^* \\; \\mu \\; \\psi_2 \\,,"
},
{
"math_id": 5,
"text": "~y(x) = -y(-x)~"
},
{
"math_id": 6,
"text": "\\mathbf J_{\\mathrm{i}} = \\mathbf{J}_{\\mathrm{f}} + \\boldsymbol{\\lambda}"
},
{
"math_id": 7,
"text": "\\Vert \\boldsymbol{\\lambda} \\Vert = \\sqrt{\\lambda(\\lambda + 1)\\,} \\; \\hbar~,"
},
{
"math_id": 8,
"text": "\\lambda_z = \\mu \\, \\hbar~;"
},
{
"math_id": 9,
"text": "~\\mathbf J_{\\mathrm{i}}~"
},
{
"math_id": 10,
"text": "~\\mathbf J_{\\mathrm{f}}~"
},
{
"math_id": 11,
"text": "| J_{\\mathrm{i}} - J_{\\mathrm{f}} | \\le \\lambda \\le J_{\\mathrm{i}} + J_{\\mathrm{f}}"
},
{
"math_id": 12,
"text": "\\mu = M_{\\mbox{i}} - M_{\\mbox{f}}\\,."
},
{
"math_id": 13,
"text": "\\pi(\\mathrm{E}\\lambda) = \\pi_{\\mathrm{i}} \\pi_{\\mathrm{f}} = (-1)^{\\lambda}\\,"
},
{
"math_id": 14,
"text": "\\pi(\\mathrm{M}\\lambda) = \\pi_{\\mathrm{i}} \\pi_{\\mathrm{f}} = (-1)^{\\lambda+1}\\,."
},
{
"math_id": 15,
"text": "~J=L+S~"
},
{
"math_id": 16,
"text": "~L~"
},
{
"math_id": 17,
"text": "~S~"
},
{
"math_id": 18,
"text": "~M_J~"
},
{
"math_id": 19,
"text": "~ \\not \\leftrightarrow ~"
},
{
"math_id": 20,
"text": "~F=I+J~,"
},
{
"math_id": 21,
"text": "~I~"
},
{
"math_id": 22,
"text": "~J~"
},
{
"math_id": 23,
"text": "~F=I+J~"
},
{
"math_id": 24,
"text": "~J=L+S~,"
}
]
| https://en.wikipedia.org/wiki?curid=1313664 |
1314272 | Dirac comb | Periodic distribution ("function") of "point-mass" Dirac delta sampling
In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula
formula_0
for some given period formula_1. Here "t" is a real variable and the sum extends over all integers "k." The Dirac delta function formula_2 and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the formula_2s as the comb's "teeth"), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function.
The symbol formula_3, where the period is omitted, represents a Dirac comb of unit period. This implies
formula_4
Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel:
formula_5
The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and aliasing, in a single framework of continuous Fourier analysis on tempered distributions, without any reference to Fourier series. The Fourier transform of a Dirac comb is another Dirac comb. Owing to the Convolution Theorem on tempered distributions which turns out to be the Poisson summation formula, in signal processing, the Dirac comb allows modelling sampling by "multiplication" with it, but it also allows modelling periodization by "convolution" with it.
Dirac-comb identity.
The Dirac comb can be constructed in two ways, either by using the "comb" operator (performing sampling) applied to the function that is constantly formula_6, or, alternatively, by using the "rep" operator (performing periodization) applied to the Dirac delta formula_2. Formally, this yields the following:
formula_7
where
formula_8 and formula_9
In signal processing, this property on one hand allows sampling a function formula_10 by "multiplication" with formula_11, and on the other hand it also allows the periodization of formula_10 by "convolution" with formula_12.
The Dirac comb identity is a particular case of the Convolution Theorem for tempered distributions.
Scaling.
The scaling property of the Dirac comb follows from the properties of the Dirac delta function.
Since formula_13 for positive real numbers formula_14, it follows that:
formula_15
formula_16
Note that requiring positive scaling numbers formula_14 instead of negative ones is not a restriction because the negative sign would only reverse the order of the summation within formula_11, which does not affect the result.
Fourier series.
It is clear that formula_17 is periodic with period formula_1. That is,
formula_18
for all "t". The complex Fourier series for such a periodic function is
formula_19
where the Fourier coefficients are (symbolically)
formula_20
All Fourier coefficients are 1/"T" resulting in
formula_21
When the period is one unit, this simplifies to
formula_22
This is a divergent series, when understood as a series of ordinary complex numbers, but becomes convergent in the sense of distributions.
A "square root" of the Dirac comb is employed in some applications to physics, specifically:formula_23
However this is not a distribution in the ordinary sense.
Fourier transform.
The Fourier transform of a Dirac comb is also a Dirac comb. For the Fourier transform formula_24 expressed in frequency domain (Hz) the Dirac comb formula_25 of period formula_1 transforms into a rescaled Dirac comb of period formula_26 i.e. for
formula_27
formula_28
is proportional to another Dirac comb, but with period formula_29 in frequency domain (radian/s). The Dirac comb formula_30 of unit period formula_31 is thus an eigenfunction of formula_24 to the eigenvalue formula_32
This result can be established by considering the respective Fourier transforms formula_33 of the family of functions formula_34 defined by
formula_35
Since formula_34 is a convergent series of Gaussian functions, and Gaussians transform into Gaussians, each of their respective Fourier transforms formula_36 also results in a series of Gaussians, and explicit calculation establishes that
formula_37
The functions formula_34 and formula_36 are thus each resembling a periodic function consisting of a series of equidistant Gaussian spikes formula_38 and formula_39 whose respective "heights" (pre-factors) are determined by slowly decreasing Gaussian envelope functions which drop to zero at infinity. Note that in the limit formula_40 each Gaussian spike becomes an infinitely sharp Dirac impulse centered respectively at formula_41 and formula_42 for each respective formula_43 and formula_44, and hence also all pre-factors formula_45 in formula_46 eventually become indistinguishable from formula_47. Therefore the functions formula_34 and their respective Fourier transforms formula_46 converge to the same function and this limit function is a series of infinite equidistant Gaussian spikes, each spike being multiplied by the same pre-factor of one, i.e., the Dirac comb for unit period:
formula_48 and formula_49
Since formula_50, we obtain in this limit the result to be demonstrated:
formula_51
The corresponding result for period formula_1 can be found by exploiting the scaling property of the Fourier transform,
formula_52
Another manner to establish that the Dirac comb transforms into another Dirac comb starts by examining continuous Fourier transforms of periodic functions in general, and then specialises to the case of the Dirac comb. In order to also show that the specific rule depends on the convention for the Fourier transform, this will be shown using angular frequency with formula_53 for any periodic function formula_54 its Fourier transform
formula_55 obeys:
formula_56
because Fourier transforming formula_10 and formula_57 leads to formula_58 and formula_59 This equation implies that formula_60 nearly everywhere with the only possible exceptions lying at formula_61 with formula_62 and formula_63 When evaluating the Fourier transform at formula_64 the corresponding Fourier series expression times a corresponding delta function results. For the special case of the Fourier transform of the Dirac comb, the Fourier series integral over a single period covers only the Dirac function at the origin and thus gives formula_29 for each formula_65 This can be summarised by interpreting the Dirac comb as a limit of the Dirichlet kernel such that, at the positions formula_61 all exponentials in the sum formula_66 point into the same direction and add constructively. In other words, the continuous Fourier transform of periodic functions leads to
formula_67 with formula_68
and
formula_69
The Fourier series coefficients formula_70 for all formula_71 when formula_72, i.e.
formula_73
is another Dirac comb, but with period formula_74 in angular frequency domain (radian/s).
As mentioned, the specific rule depends on the convention for the used Fourier transform. Indeed, when using the scaling property of the Dirac delta function, the above may be re-expressed in ordinary frequency domain (Hz) and one obtains again:
formula_75
such that the unit period Dirac comb transforms to itself:
formula_76
Finally, the Dirac comb is also an eigenfunction of the unitary continuous Fourier transform in angular frequency space to the eigenvalue 1 when formula_77 because for the unitary Fourier transform
formula_78
the above may be re-expressed as
formula_79
Sampling and aliasing.
Multiplying any function by a Dirac comb transforms it into a train of impulses with integrals equal to the value of the function at the nodes of the comb. This operation is frequently used to represent sampling.
formula_80
Due to the self-transforming property of the Dirac comb and the convolution theorem, this corresponds to convolution with the Dirac comb in the frequency domain.
formula_81
Since convolution with a delta function formula_82 is equivalent to shifting the function by formula_83, convolution with the Dirac comb corresponds to replication or periodic summation:
formula_84
This leads to a natural formulation of the Nyquist–Shannon sampling theorem. If the spectrum of the function formula_85 contains no frequencies higher than B (i.e., its spectrum is nonzero only in the interval formula_86) then samples of the original function at intervals formula_87 are sufficient to reconstruct the original signal. It suffices to multiply the spectrum of the sampled function by a suitable rectangle function, which is equivalent to applying a brick-wall lowpass filter.
formula_88
formula_89
In time domain, this "multiplication with the rect function" is equivalent to "convolution with the sinc function." Hence, it restores the original function from its samples. This is known as the Whittaker–Shannon interpolation formula.
Remark: Most rigorously, multiplication of the rect function with a generalized function, such as the Dirac comb, fails. This is due to undetermined outcomes of the multiplication product at the interval boundaries. As a workaround, one uses a Lighthill unitary function instead of the rect function. It is smooth at the interval boundaries, hence it yields determined multiplication products everywhere, see , Theorem 22 for details.
Use in directional statistics.
In directional statistics, the Dirac comb of period formula_90 is equivalent to a wrapped Dirac delta function and is the analog of the Dirac delta function in linear statistics.
In linear statistics, the random variable formula_91 is usually distributed over the real-number line, or some subset thereof, and the probability density of formula_85 is a function whose domain is the set of real numbers, and whose integral from formula_92 to formula_93 is unity. In directional statistics, the random variable formula_94 is distributed over the unit circle, and the probability density of formula_95 is a function whose domain is some interval of the real numbers of length formula_90 and whose integral over that interval is unity. Just as the integral of the product of a Dirac delta function with an arbitrary function over the real-number line yields the value of that function at zero, so the integral of the product of a Dirac comb of period formula_90 with an arbitrary function of period formula_90 over the unit circle yields the value of that function at zero.
Notes.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t) \\ := \\sum_{k=-\\infty}^{\\infty} \\delta(t - k T) "
},
{
"math_id": 1,
"text": "T"
},
{
"math_id": 2,
"text": "\\delta"
},
{
"math_id": 3,
"text": "\\operatorname{\\text{Ш}}\\,\\,(t)"
},
{
"math_id": 4,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t) \\ = \\frac{1}{T}\\operatorname{\\text{Ш}}\\ \\!\\!\\!\\left(\\frac{t}{T}\\right)."
},
{
"math_id": 5,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t) = \\frac{1}{T}\\sum_{n=-\\infty}^{\\infty} e^{i 2 \\pi n \\frac{t}{T}}."
},
{
"math_id": 6,
"text": "1"
},
{
"math_id": 7,
"text": "\\operatorname{comb}_T \\{1\\} = \\operatorname{\\text{Ш}}_T = \\operatorname{rep}_T \\{\\delta \\}, "
},
{
"math_id": 8,
"text": "\n \\operatorname{comb}_T \\{f(t)\\} \\triangleq \\sum_{k=-\\infty}^\\infty \\, f(kT) \\, \\delta(t - kT)\n"
},
{
"math_id": 9,
"text": "\n \\operatorname{rep}_T \\{g(t)\\} \\triangleq \\sum_{k=-\\infty}^\\infty \\, g(t - kT).\n"
},
{
"math_id": 10,
"text": "f(t)"
},
{
"math_id": 11,
"text": "\\operatorname{\\text{Ш}}_{\\ T}"
},
{
"math_id": 12,
"text": "\\operatorname{\\text{Ш}}_T"
},
{
"math_id": 13,
"text": "\\delta(t) = \\frac{1}{a}\\ \\delta\\!\\left(\\frac{t}{a}\\right)"
},
{
"math_id": 14,
"text": "a"
},
{
"math_id": 15,
"text": " \\operatorname{\\text{Ш}}_{\\ T}\\left(t\\right) = \\frac{1}{T} \\operatorname{\\text{Ш}}\\,\\!\\left( \\frac{t}{T} \\right), "
},
{
"math_id": 16,
"text": "\\operatorname{\\text{Ш}}_{\\ aT}\\left(t\\right) = \\frac{1}{aT} \\operatorname{\\text{Ш}}\\,\\!\\left(\\frac{t}{aT}\\right) = \\frac{1}{a} \\operatorname{\\text{Ш}}_{\\ T}\\!\\!\\left(\\frac{t}{a}\\right)."
},
{
"math_id": 17,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t)"
},
{
"math_id": 18,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t + T) = \\operatorname{\\text{Ш}}_{\\ T}(t)"
},
{
"math_id": 19,
"text": " \\operatorname{\\text{Ш}}_{\\ T}(t) = \\sum_{n=-\\infty}^{+\\infty} c_n e^{i 2 \\pi n \\frac{t}{T}}, "
},
{
"math_id": 20,
"text": "\\begin{align}\nc_n &= \\frac{1}{T} \\int_{t_0}^{t_0 + T} \\operatorname{\\text{Ш}}_{\\ T}(t) e^{-i 2 \\pi n \\frac{t}{T}}\\, dt \\quad ( -\\infty < t_0 < +\\infty ) \\\\\n &= \\frac{1}{T} \\int_{-\\frac{T}{2}}^{\\frac{T}{2}} \\operatorname{\\text{Ш}}_{\\ T}(t) e^{-i 2 \\pi n \\frac{t}{T}}\\, dt \\\\\n &= \\frac{1}{T} \\int_{-\\frac{T}{2}}^{\\frac{T}{2}} \\delta(t) e^{-i 2 \\pi n \\frac{t}{T}}\\, dt \\\\\n &= \\frac{1}{T} e^{-i 2 \\pi n \\frac{0}{T}} \\\\\n &= \\frac{1}{T}.\n\\end{align}"
},
{
"math_id": 21,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t) = \\frac{1}{T}\\sum_{n=-\\infty}^{\\infty} \\!\\!e^{i 2 \\pi n \\frac{t}{T}}."
},
{
"math_id": 22,
"text": "\\operatorname{\\text{Ш}}\\ \\!(x) = \\sum_{n=-\\infty}^{\\infty} \\!\\!e^{i 2 \\pi n x}."
},
{
"math_id": 23,
"text": "\\delta_N^{(1 / 2)}(\\xi) = \\frac{1}{\\sqrt{NT}} \\sum_{\\nu=0}^{N-1} e^{-i \\frac{2\\pi}{T}\\xi \\nu}, \\quad \n\\lim_{N \\rightarrow \\infty}\\left|\\delta_N^{(1 / 2)}(\\xi)\\right|^2= \\sum_{k=-\\infty}^{\\infty} \\delta(\\xi - kT)."
},
{
"math_id": 24,
"text": "\\mathcal{F}"
},
{
"math_id": 25,
"text": "\\operatorname{\\text{Ш}}_{T}"
},
{
"math_id": 26,
"text": "1/T,"
},
{
"math_id": 27,
"text": "\\mathcal{F}\\left[ f \\right](\\xi)= \\int_{-\\infty}^{\\infty} dt f(t) e^{- 2 \\pi i\\xi t}, "
},
{
"math_id": 28,
"text": "\\mathcal{F}\\left[ \\operatorname{\\text{Ш}}_{T} \\right](\\xi) = \\frac{1}{T} \\sum_{k=-\\infty}^{\\infty} \\delta(\\xi-k \\frac{1}{T}) = \\frac{1}{T} \\operatorname{\\text{Ш}}_{\\ \\frac{1}{T}}(\\xi) ~"
},
{
"math_id": 29,
"text": "1/T"
},
{
"math_id": 30,
"text": "\\operatorname{\\text{Ш}}"
},
{
"math_id": 31,
"text": "T=1"
},
{
"math_id": 32,
"text": "1."
},
{
"math_id": 33,
"text": "S_{\\tau}(\\xi)=\\mathcal{F}[s_{\\tau}](\\xi)"
},
{
"math_id": 34,
"text": "s_{\\tau}(x)"
},
{
"math_id": 35,
"text": "s_{\\tau}(x) = \\tau^{-1} e^{-\\pi \\tau^2 x^2} \\sum_{n=-\\infty}^{\\infty} e^{-\\pi \\tau^{-2} ( x-n)^{2} }."
},
{
"math_id": 36,
"text": "S_\\tau(\\xi)"
},
{
"math_id": 37,
"text": "S_{\\tau}(\\xi) = \\tau^{-1} \\sum_{m=-\\infty}^{\\infty} e^{-\\pi \\tau^2 m^2} e^{-\\pi \\tau^{-2} ( \\xi-m)^{2} }."
},
{
"math_id": 38,
"text": "\\tau^{-1} e^{-\\pi \\tau^{-2} ( x-n)^{2} }"
},
{
"math_id": 39,
"text": "\\tau^{-1} e^{-\\pi \\tau^{-2} ( \\xi-m)^{2} }"
},
{
"math_id": 40,
"text": "\\tau \\rightarrow 0"
},
{
"math_id": 41,
"text": "x=n"
},
{
"math_id": 42,
"text": "\\xi=m"
},
{
"math_id": 43,
"text": "n"
},
{
"math_id": 44,
"text": "m"
},
{
"math_id": 45,
"text": " e^{-\\pi \\tau^2 m^2}"
},
{
"math_id": 46,
"text": "S_{\\tau}(\\xi)"
},
{
"math_id": 47,
"text": " e^{-\\pi \\tau^2 \\xi^2}"
},
{
"math_id": 48,
"text": "\\lim_{\\tau \\rightarrow 0} s_{\\tau}(x) = \\operatorname{\\text{Ш}}({x}),"
},
{
"math_id": 49,
"text": "\\lim_{\\tau \\rightarrow 0} S_{\\tau}(\\xi) = \\operatorname{\\text{Ш}}({\\xi})."
},
{
"math_id": 50,
"text": "S_{\\tau}=\\mathcal{F}[s_{\\tau}]"
},
{
"math_id": 51,
"text": "\\mathcal{F}[\\operatorname{\\text{Ш}}]= \\operatorname{\\text{Ш}}."
},
{
"math_id": 52,
"text": "\\mathcal{F}[\\operatorname{\\text{Ш}}_T]= \\frac{1}{T} \\operatorname{\\text{Ш}}_{\\frac{1}{T}}."
},
{
"math_id": 53,
"text": "\\omega=2\\pi \\xi :"
},
{
"math_id": 54,
"text": "f(t)=f(t+T)"
},
{
"math_id": 55,
"text": "\\mathcal{F}\\left[ f \\right](\\omega)=F(\\omega) = \\int_{-\\infty}^{\\infty} dt f(t) e^{-i\\omega t} "
},
{
"math_id": 56,
"text": "F(\\omega) (1 - e^{i \\omega T}) = 0"
},
{
"math_id": 57,
"text": "f(t+T)"
},
{
"math_id": 58,
"text": "F(\\omega)"
},
{
"math_id": 59,
"text": "F(\\omega) e^{i \\omega T}."
},
{
"math_id": 60,
"text": "F(\\omega)=0"
},
{
"math_id": 61,
"text": "\\omega= k \\omega_0,"
},
{
"math_id": 62,
"text": "\\omega_0=2\\pi / T"
},
{
"math_id": 63,
"text": "k \\in \\mathbb{Z}."
},
{
"math_id": 64,
"text": "F(k \\omega_0)"
},
{
"math_id": 65,
"text": "k."
},
{
"math_id": 66,
"text": " \\sum\\nolimits_{m=-\\infty}^{\\infty} e^{\\pm i \\omega m T} "
},
{
"math_id": 67,
"text": "F(\\omega)= 2 \\pi \\sum_{k=-\\infty}^{\\infty} c_k \\delta(\\omega-k\\omega_0) "
},
{
"math_id": 68,
"text": "\\omega_0=2 \\pi/T,"
},
{
"math_id": 69,
"text": "c_k = \\frac{1}{T} \\int_{-T/2 }^{+T/2} dt f(t) e^{-i 2 \\pi k t/T}."
},
{
"math_id": 70,
"text": "c_k=1/T"
},
{
"math_id": 71,
"text": "k"
},
{
"math_id": 72,
"text": "f \\rightarrow \\operatorname{\\text{Ш}}_{T}"
},
{
"math_id": 73,
"text": "\\mathcal{F}\\left[ \\operatorname{\\text{Ш}}_{T} \\right](\\omega) = \\frac{2 \\pi}{T} \\sum_{k=-\\infty}^{\\infty} \\delta(\\omega-k \\frac{2 \\pi}{T})"
},
{
"math_id": 74,
"text": "2 \\pi/T"
},
{
"math_id": 75,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t) \\stackrel{\\mathcal{F}}{\\longleftrightarrow} \\frac{1}{T} \\operatorname{\\text{Ш}}_{\\ \\frac{1}{T}}(\\xi) = \\sum_{n=-\\infty}^{\\infty}\\!\\! e^{-i 2\\pi \\xi n T},"
},
{
"math_id": 76,
"text": "\\operatorname{\\text{Ш}}\\ \\!(t) \\stackrel{\\mathcal{F}}{\\longleftrightarrow} \\operatorname{\\text{Ш}}\\ \\!(\\xi)."
},
{
"math_id": 77,
"text": "T=\\sqrt{2 \\pi}"
},
{
"math_id": 78,
"text": "\\mathcal{F}\\left[ f \\right](\\omega)=F(\\omega) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} dt f(t) e^{-i\\omega t}, "
},
{
"math_id": 79,
"text": "\\operatorname{\\text{Ш}}_{\\ T}(t) \\stackrel{\\mathcal{F}}{\\longleftrightarrow} \\frac{\\sqrt{2\\pi}}{T} \\operatorname{\\text{Ш}}_{\\ \\frac{2\\pi}{T}}(\\omega) = \\frac{1}{\\sqrt{2\\pi}}\\sum_{n=-\\infty}^{\\infty} \\!\\!e^{-i\\omega nT}."
},
{
"math_id": 80,
"text": " (\\operatorname{\\text{Ш}}_{\\ T} x)(t) = \\sum_{k=-\\infty}^{\\infty} \\!\\! x(t)\\delta(t - kT) = \\sum_{k=-\\infty}^{\\infty}\\!\\! x(kT)\\delta(t - kT)."
},
{
"math_id": 81,
"text": " \\operatorname{\\text{Ш}}_{\\ T} x \\ \\stackrel{\\mathcal{F}}{\\longleftrightarrow}\\ \\frac{1}{T}\\operatorname{\\text{Ш}}_\\frac{1}{T} * X"
},
{
"math_id": 82,
"text": "\\delta(t-kT)"
},
{
"math_id": 83,
"text": "kT"
},
{
"math_id": 84,
"text": " (\\operatorname{\\text{Ш}}_{\\ \\frac{1}{T}}\\! * X)(f) =\\! \\sum_{k=-\\infty}^{\\infty} \\!\\!X\\!\\left(f - \\frac{k}{T}\\right) "
},
{
"math_id": 85,
"text": "x"
},
{
"math_id": 86,
"text": "(-B, B)"
},
{
"math_id": 87,
"text": "\\tfrac{1}{2B}"
},
{
"math_id": 88,
"text": " \\operatorname{\\text{Ш}}_{\\ \\!\\frac{1}{2B}} x\\ \\ \\stackrel{\\mathcal{F}}{\\longleftrightarrow}\\ \\ 2B\\, \\operatorname{\\text{Ш}}_{\\ 2B} * X"
},
{
"math_id": 89,
"text": " \\frac{1}{2B}\\Pi\\left(\\frac{f}{2B}\\right) (2B \\,\\operatorname{\\text{Ш}}_{\\ 2B} * X) = X"
},
{
"math_id": 90,
"text": "2\\pi"
},
{
"math_id": 91,
"text": "(x)"
},
{
"math_id": 92,
"text": "-\\infty"
},
{
"math_id": 93,
"text": "+\\infty"
},
{
"math_id": 94,
"text": "(\\theta)"
},
{
"math_id": 95,
"text": "\\theta"
}
]
| https://en.wikipedia.org/wiki?curid=1314272 |
13143 | Generalized mean | N-th root of the arithmetic mean of the given numbers raised to the power n
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).
Definition.
If p is a non-zero real number, and formula_0 are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is
formula_1
(See p-norm). For "p" = 0 we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):
formula_2
Furthermore, for a sequence of positive weights wi we define the weighted power mean as
formula_3
and when "p" = 0, it is equal to the weighted geometric mean:
formula_4
The unweighted means correspond to setting all "wi" = 1/n.
Special cases.
A few particular values of p yield special cases with their own names:
<templatestyles src="Math_proof/styles.css" />Proof of formula_12 (geometric mean)
For the purpose of the proof, we will assume without loss of generality that
formula_13
and
formula_14
We can rewrite the definition of formula_15 using the exponential function as
formula_16
In the limit "p" → 0, we can apply L'Hôpital's rule to the argument of the exponential function. We assume that formula_17 but "p" ≠ 0, and that the sum of wi is equal to 1 (without loss in generality); Differentiating the numerator and denominator with respect to p, we have
formula_18
By the continuity of the exponential function, we can substitute back into the above relation to obtain
formula_19
as desired.
<templatestyles src="Math_proof/styles.css" />Proof of formula_20 and formula_21
Assume (possibly after relabeling and combining terms together) that formula_22. Then
formula_23
The formula for formula_24 follows from
formula_25
Properties.
Let formula_0 be a sequence of positive real numbers, then the following properties hold:
Generalized mean inequality.
In general, if "p" < "q", then
formula_32
and the two means are equal if and only if "x"1 = "x"2 = ... = "xn".
The inequality is true for real values of p and q, as well as positive and negative infinity values.
It follows from the fact that, for all real p,
formula_33
which can be proved using Jensen's inequality.
In particular, for p in {−1, 0, 1}, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.
Proof of the weighted inequality.
We will prove the weighted power mean inequality. For the purpose of the proof we will assume the following without loss of generality:
formula_34
The proof for unweighted power means can be easily obtained by substituting "wi" = 1/"n".
Equivalence of inequalities between means of opposite signs.
Suppose an average between power means with exponents p and q holds:
formula_35
applying this, then:
formula_36
We raise both sides to the power of −1 (strictly decreasing function in positive reals):
formula_37
We get the inequality for means with exponents −"p" and −"q", and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.
Geometric mean.
For any "q" > 0 and non-negative weights summing to 1, the following inequality holds:
formula_38
The proof follows from Jensen's inequality, making use of the fact the logarithm is concave:
formula_39
By applying the exponential function to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get
formula_40
Taking q-th powers of the xi yields
formula_41
Thus, we are done for the inequality with positive q; the case for negatives is identical but for the swapped signs in the last step:
formula_42
Of course, taking each side to the power of a negative number -1/"q" swaps the direction of the inequality.
formula_43
Inequality between any two power means.
We are to prove that for any "p" < "q" the following inequality holds:
formula_44
if p is negative, and q is positive, the inequality is equivalent to the one proved above:
formula_45
The proof for positive p and q is as follows: Define the following function: "f" : R+ → R+ formula_46. f is a power function, so it does have a second derivative:
formula_47
which is strictly positive within the domain of f, since "q" > "p", so we know f is convex.
Using this, and the Jensen's inequality we get:
formula_48
after raising both side to the power of 1/"q" (an increasing function, since 1/"q" is positive) we get the inequality which was to be proven:
formula_49
Using the previously shown equivalence we can prove the inequality for negative p and q by replacing them with −q and −p, respectively.
Generalized "f"-mean.
The power mean could be generalized further to the generalized f-mean:
formula_50
This covers the geometric mean without using a limit with "f"("x")
log("x"). The power mean is obtained for "f"("x")
"xp". Properties of these means are studied in de Carvalho (2016).
Applications.
Signal processing.
A power mean serves a non-linear moving average which is shifted towards small signal values for small p and emphasizes big signal values for big p. Given an efficient implementation of a moving arithmetic mean called codice_0 one can implement a moving power mean according to the following Haskell code.
powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
powerSmooth smooth p = map (** recip p) . smooth . map (**p)
See also.
<templatestyles src="Div col/styles.css"/>
Notes.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "x_1, \\dots, x_n"
},
{
"math_id": 1,
"text": "M_p(x_1,\\dots,x_n) = \\left( \\frac{1}{n} \\sum_{i=1}^n x_i^p \\right)^{{1}/{p}} ."
},
{
"math_id": 2,
"text": "M_0(x_1, \\dots, x_n) = \\left(\\prod_{i=1}^n x_i\\right)^{1/n} ."
},
{
"math_id": 3,
"text": "M_p(x_1,\\dots,x_n) = \\left(\\frac{\\sum_{i=1}^n w_i x_i^p}{\\sum_{i=1}^n w_i} \\right)^{{1}/{p}}"
},
{
"math_id": 4,
"text": "M_0(x_1,\\dots,x_n) = \\left(\\prod_{i=1}^n x_i^{w_i}\\right)^{1 / \\sum_{i=1}^n w_i} ."
},
{
"math_id": 5,
"text": "M_{-\\infty}(x_1,\\dots,x_n) = \\lim_{p\\to-\\infty} M_p(x_1,\\dots,x_n) = \\min \\{x_1,\\dots,x_n\\}"
},
{
"math_id": 6,
"text": "M_{-1}(x_1,\\dots,x_n) = \\frac{n}{\\frac{1}{x_1}+\\dots+\\frac{1}{x_n}}"
},
{
"math_id": 7,
"text": "M_0(x_1,\\dots,x_n) = \\lim_{p\\to0} M_p(x_1,\\dots,x_n) = \\sqrt[n]{x_1\\cdot\\dots\\cdot x_n}"
},
{
"math_id": 8,
"text": "M_1(x_1,\\dots,x_n) = \\frac{x_1 + \\dots + x_n}{n}"
},
{
"math_id": 9,
"text": "M_2(x_1,\\dots,x_n) = \\sqrt{\\frac{x_1^2 + \\dots + x_n^2}{n}}"
},
{
"math_id": 10,
"text": "M_3(x_1,\\dots,x_n) = \\sqrt[3]{\\frac{x_1^3 + \\dots + x_n^3}{n}}"
},
{
"math_id": 11,
"text": "M_{+\\infty}(x_1,\\dots,x_n) = \\lim_{p\\to\\infty} M_p(x_1,\\dots,x_n) = \\max \\{x_1,\\dots,x_n\\}"
},
{
"math_id": 12,
"text": " \\lim_{p \\to 0} M_p = M_0 "
},
{
"math_id": 13,
"text": " w_i \\in [0,1] "
},
{
"math_id": 14,
"text": " \\sum_{i=1}^n w_i = 1. "
},
{
"math_id": 15,
"text": "M_p"
},
{
"math_id": 16,
"text": "M_p(x_1,\\dots,x_n) = \\exp{\\left( \\ln{\\left[\\left(\\sum_{i=1}^n w_ix_{i}^p \\right)^{1/p}\\right]} \\right) } = \\exp{\\left( \\frac{\\ln{\\left(\\sum_{i=1}^n w_ix_{i}^p \\right)}}{p} \\right) }"
},
{
"math_id": 17,
"text": "p \\isin \\mathbb{R}"
},
{
"math_id": 18,
"text": "\\begin{align}\n \\lim_{p \\to 0} \\frac{\\ln{\\left(\\sum_{i=1}^n w_ix_{i}^p \\right)}}{p} &= \\lim_{p \\to 0} \\frac{\\frac{\\sum_{i=1}^n w_i x_i^p \\ln{x_i}}{\\sum_{j=1}^n w_j x_j^p}}{1} \\\\\n &= \\lim_{p \\to 0} \\frac{\\sum_{i=1}^n w_i x_i^p \\ln{x_i}}{\\sum_{j=1}^n w_j x_j^p} \\\\\n &= \\frac{\\sum_{i=1}^n w_i \\ln{x_i}}{\\sum_{j=1}^n w_j} \\\\\n &= \\sum_{i=1}^n w_i \\ln{x_i} \\\\\n &= \\ln{\\left(\\prod_{i=1}^n x_i^{w_i} \\right)}\n\\end{align}"
},
{
"math_id": 19,
"text": "\\lim_{p \\to 0} M_p(x_1,\\dots,x_n) = \\exp{\\left( \\ln{\\left(\\prod_{i=1}^n x_i^{w_i} \\right)} \\right)} = \\prod_{i=1}^n x_i^{w_i} = M_0(x_1,\\dots,x_n)"
},
{
"math_id": 20,
"text": "\\lim_{p \\to \\infty} M_p = M_\\infty"
},
{
"math_id": 21,
"text": "\\lim_{p \\to -\\infty} M_p = M_{-\\infty}"
},
{
"math_id": 22,
"text": "x_1 \\geq \\dots \\geq x_n"
},
{
"math_id": 23,
"text": "\\begin{align}\n \\lim_{p \\to \\infty} M_p(x_1,\\dots,x_n) &= \\lim_{p \\to \\infty} \\left( \\sum_{i=1}^n w_i x_i^p \\right)^{1/p} \\\\\n &= x_1 \\lim_{p \\to \\infty} \\left( \\sum_{i=1}^n w_i \\left( \\frac{x_i}{x_1} \\right)^p \\right)^{1/p} \\\\\n &= x_1 = M_\\infty (x_1,\\dots,x_n).\n\\end{align}"
},
{
"math_id": 24,
"text": "M_{-\\infty}"
},
{
"math_id": 25,
"text": "M_{-\\infty} (x_1,\\dots,x_n) = \\frac{1}{M_\\infty (1/x_1,\\dots,1/x_n)} = x_n."
},
{
"math_id": 26,
"text": "\\min(x_1, \\dots, x_n) \\le M_p(x_1, \\dots, x_n) \\le \\max(x_1, \\dots, x_n)"
},
{
"math_id": 27,
"text": "M_p(x_1, \\dots, x_n) = M_p(P(x_1, \\dots, x_n))"
},
{
"math_id": 28,
"text": "P"
},
{
"math_id": 29,
"text": "M_p(b x_1, \\dots, b x_n) = b \\cdot M_p(x_1, \\dots, x_n)"
},
{
"math_id": 30,
"text": "b\\cdot x_1,\\dots, b\\cdot x_n"
},
{
"math_id": 31,
"text": "M_p(x_1, \\dots, x_{n \\cdot k}) = M_p\\left[M_p(x_1, \\dots, x_{k}), M_p(x_{k + 1}, \\dots, x_{2 \\cdot k}), \\dots, M_p(x_{(n - 1) \\cdot k + 1}, \\dots, x_{n \\cdot k})\\right]"
},
{
"math_id": 32,
"text": "M_p(x_1, \\dots, x_n) \\le M_q(x_1, \\dots, x_n)"
},
{
"math_id": 33,
"text": "\\frac{\\partial}{\\partial p}M_p(x_1, \\dots, x_n) \\geq 0"
},
{
"math_id": 34,
"text": "\\begin{align}\n w_i \\in [0, 1] \\\\\n \\sum_{i=1}^nw_i = 1\n\\end{align}"
},
{
"math_id": 35,
"text": "\\left(\\sum_{i=1}^n w_i x_i^p\\right)^{1/p} \\geq \\left(\\sum_{i=1}^n w_i x_i^q\\right)^{1/q}"
},
{
"math_id": 36,
"text": "\\left(\\sum_{i=1}^n\\frac{w_i}{x_i^p}\\right)^{1/p} \\geq \\left(\\sum_{i=1}^n\\frac{w_i}{x_i^q}\\right)^{1/q}"
},
{
"math_id": 37,
"text": "\\left(\\sum_{i=1}^nw_ix_i^{-p}\\right)^{-1/p}\n= \\left(\\frac{1}{\\sum_{i=1}^nw_i\\frac{1}{x_i^p}}\\right)^{1/p}\n\\leq \\left(\\frac{1}{\\sum_{i=1}^nw_i\\frac{1}{x_i^q}}\\right)^{1/q}\n= \\left(\\sum_{i=1}^nw_ix_i^{-q}\\right)^{-1/q}"
},
{
"math_id": 38,
"text": "\\left(\\sum_{i=1}^n w_i x_i^{-q}\\right)^{-1/q} \\leq \\prod_{i=1}^n x_i^{w_i} \\leq \\left(\\sum_{i=1}^n w_i x_i^q\\right)^{1/q}."
},
{
"math_id": 39,
"text": "\\log \\prod_{i=1}^n x_i^{w_i} = \\sum_{i=1}^n w_i\\log x_i \\leq \\log \\sum_{i=1}^n w_i x_i."
},
{
"math_id": 40,
"text": "\\prod_{i=1}^n x_i^{w_i} \\leq \\sum_{i=1}^n w_i x_i."
},
{
"math_id": 41,
"text": "\\begin{align}\n&\\prod_{i=1}^n x_i^{q{\\cdot}w_i} \\leq \\sum_{i=1}^n w_i x_i^q \\\\\n&\\prod_{i=1}^n x_i^{w_i} \\leq \\left(\\sum_{i=1}^n w_i x_i^q\\right)^{1/q}.\\end{align}"
},
{
"math_id": 42,
"text": "\\prod_{i=1}^n x_i^{-q{\\cdot}w_i} \\leq \\sum_{i=1}^n w_i x_i^{-q}."
},
{
"math_id": 43,
"text": "\\prod_{i=1}^n x_i^{w_i} \\geq \\left(\\sum_{i=1}^n w_i x_i^{-q}\\right)^{-1/q}."
},
{
"math_id": 44,
"text": "\\left(\\sum_{i=1}^n w_i x_i^p\\right)^{1/p} \\leq \\left(\\sum_{i=1}^nw_ix_i^q\\right)^{1/q}"
},
{
"math_id": 45,
"text": "\\left(\\sum_{i=1}^nw_i x_i^p\\right)^{1/p} \\leq \\prod_{i=1}^n x_i^{w_i} \\leq \\left(\\sum_{i=1}^n w_i x_i^q\\right)^{1/q}"
},
{
"math_id": 46,
"text": "f(x)=x^{\\frac{q}{p}}"
},
{
"math_id": 47,
"text": "f''(x) = \\left(\\frac{q}{p} \\right) \\left( \\frac{q}{p}-1 \\right)x^{\\frac{q}{p}-2}"
},
{
"math_id": 48,
"text": "\\begin{align}\n f \\left( \\sum_{i=1}^nw_ix_i^p \\right) &\\leq \\sum_{i=1}^nw_if(x_i^p) \\\\[3pt]\n \\left(\\sum_{i=1}^n w_i x_i^p\\right)^{q/p} &\\leq \\sum_{i=1}^nw_ix_i^q\n\\end{align}"
},
{
"math_id": 49,
"text": "\\left(\\sum_{i=1}^n w_i x_i^p\\right)^{1/p} \\leq \\left(\\sum_{i=1}^n w_i x_i^q\\right)^{1/q}"
},
{
"math_id": 50,
"text": " M_f(x_1,\\dots,x_n) = f^{-1} \\left({\\frac{1}{n}\\cdot\\sum_{i=1}^n{f(x_i)}}\\right) "
}
]
| https://en.wikipedia.org/wiki?curid=13143 |
13145 | Gerolamo Cardano | Italian Renaissance mathematician, physician, astrologer (1501–1576)
Gerolamo Cardano (; also Girolamo or Geronimo; ; ; 24 September 1501– 21 September 1576) was an Italian polymath whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler. He became one of the most influential mathematicians of the Renaissance and one of the key figures in the foundation of probability; he introduced the binomial coefficients and the binomial theorem in the Western world. He wrote more than 200 works on science.
Cardano partially invented and described several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He made significant contributions to hypocycloids - published in "De proportionibus", in 1570. The generating circles of these hypocycloids, later named "Cardano circles" or "cardanic circles", were used for the construction of the first high-speed printing presses.
Today, Cardano is well known for his achievements in algebra. In his 1545 book "Ars Magna" he made the first systematic use of negative numbers in Europe, published (with attribution) the solutions of other mathematicians for cubic and quartic equations, and acknowledged the existence of imaginary numbers.
Early life and education.
Cardano was born on 24 September 1501 in Pavia, Lombardy, the illegitimate child of Fazio Cardano, a mathematically gifted jurist, lawyer, and close friend of Leonardo da Vinci. In his autobiography, Cardano wrote that his mother, Chiara Micheri, had taken "various abortive medicines" to terminate the pregnancy; he said: "I was taken by violent means from my mother; I was almost dead." She was in labour for three days. Shortly before his birth, his mother had to move from Milan to Pavia to escape the Plague; her three other children died from the disease.
After a depressing childhood, with frequent illnesses, and the rough upbringing by his overbearing father, in 1520, Cardano entered the University of Pavia against the wish of his father, who wanted his son to undertake studies of law, but Girolamo felt more attracted to philosophy and science. During the Italian War of 1521–1526, however, the authorities in Pavia were forced to close the university in 1524. Cardano resumed his studies at the University of Padua, where he graduated with a doctorate in medicine in 1525. His eccentric and confrontational style did not earn him many friends and he had a difficult time finding work after his studies had ended. In 1525, Cardano repeatedly applied to the College of Physicians in Milan, but was not admitted owing to his combative reputation and illegitimate birth. However, he was consulted by many members of the College of Physicians, because of his irrefutable intelligence.
Early career as a physician.
Cardano wanted to practice medicine in a large, rich city like Milan, but he was denied a license to practice, so he settled for the town of Piove di Sacco, where he practised without a license. There, he married Lucia Banderini in 1531. Before her death in 1546, they had three children, Giovanni Battista (1534), Chiara (1537) and Aldo Urbano (1543). Cardano later wrote that those were the happiest days of his life.
With the help of a few noblemen, Cardano obtained a teaching position in mathematics in Milan. Having finally received his medical license, he practised mathematics and medicine simultaneously, treating a few influential patients in the process. Because of this, he became one of the most sought-after doctors in Milan. In fact, by 1536, he was able to quit his teaching position, although he was still interested in mathematics. His notability in the medical field was such that the aristocracy tried to lure him out of Milan. Cardano later wrote that he turned down offers from the kings of Denmark and France, and the Queen of Scotland.
Mathematics.
Gerolamo Cardano was the first European mathematician to make systematic use of negative numbers. He published with attribution the solution of Scipione del Ferro to the cubic equation and the solution of Cardano's student Lodovico Ferrari to the quartic equation in his 1545 book "Ars Magna", an influential work on algebra. The solution to one particular case of the cubic equation formula_0 (in modern notation) had been communicated to him in 1539 by Niccolò Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long dispute) in the form of a poem, but del Ferro's solution predated Tartaglia's. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties, described for the first time by his Italian contemporary Rafael Bombelli. In "Opus novum de proportionibus" he introduced the binomial coefficients and the binomial theorem.
Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player. His book about games of chance, "Liber de ludo aleae" ("Book on Games of Chance"), written around 1564, but not published until 1663, contains the first systematic treatment of probability, as well as a section on effective cheating methods. He used the game of throwing dice to understand the basic concepts of probability. He demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes). He was also aware of the multiplication rule for independent events but was not certain about what values should be multiplied.
Other contributions.
Cardano's work with hypocycloids led him to Cardan's Movement or Cardan Gear mechanism, in which a pair of gears with the smaller being one-half the size of the larger gear is used to convert rotational motion to linear motion with greater efficiency and precision than a Scotch yoke, for example. He is also credited with the invention of the Cardan suspension or gimbal.
Cardano made several contributions to hydrodynamics and held that perpetual motion is impossible, except in celestial bodies. He published two encyclopedias of natural science which contain a wide variety of inventions, facts, and occult superstitions. He also introduced the Cardan grille, a cryptographic writing tool, in 1550.
Significantly, in the history of education of the deaf, he said that deaf people were capable of using their minds, argued for the importance of teaching them, and was one of the first to state that deaf people could learn to read and write without learning how to speak first. He was familiar with a report by Rudolph Agricola about a deaf-mute who had learned to write.
Cardano's medical writings included: a commentary on Mundinus' anatomy and of Galen's medicine, along with the treaties "Delle cause, dei segni e dei luoghi delle malattie", "Picciola terapeutica", "Degli abusi dei medici" and "Delle orine, libro quattro".
Cardano has been credited with the invention of the so-called "Cardano's Rings", also called Chinese Rings, but it is very probable that they predate Cardano. The universal joint, sometimes called "Cardan joint", was not described by Cardano.
"De Subtilitate" (1550).
As quoted from Charles Lyell's "Principles of Geology":
The title of a work of Cardano's, published in 1552, "De Subtilitate" (corresponding to what would now be called transcendental philosophy), would lead us to expect, in the chapter on minerals, many far fetched theories characteristic of that age; but when treating of petrified shells, he decided that they clearly indicated the former sojourn of the sea upon the mountains.
Scotland and Archbishop Hamilton.
In 1552 Cardano travelled to Scotland with the Spanish physician William Casanatus, via London, to treat the Archbishop of St Andrews who suffered of a disease that had left him speechless and was thought incurable. The treatment was a success and the diplomat Thomas Randolph recorded that "merry tales" about Cardano's methods were still current in Edinburgh in 1562. Cardano and Casanatus argued over the Archbishop's cure. Cardano wrote that the Archbishop had been short of breath for ten years, and after the cure was effected by his assistant, he was paid 1,400 gold crowns.
Later years and death.
Two of Cardano's children — Giovanni Battista and Aldo Urbano — came to ignoble ends. Giovanni Battista, Cardano's eldest and favourite son was arrested in 1560 for having poisoned his wife, after he had discovered that their three children were not his. Giovanni was put to trial and, when Cardano could not pay the restitution demanded by the victim's family, was sentenced to death and beheaded. Gerolamo's other son Aldo Urbano was a gambler, who stole money from his father, and so Cardano disinherited him in 1569.
Cardano moved from Pavia to Bologna, in part because he believed that the decision to execute his son was influenced by Gerolamo's battles with the academic establishment in Pavia, and his colleagues' jealousy at his scientific achievements, and also because he was beset with allegations of sexual impropriety with his students. He obtained a position as professor of medicine at the University of Bologna.
Cardano was arrested by the Inquisition in 1570 after an accusation of heresy by the Inquisitor of Como, who targeted Cardano's "De rerum varietate" (1557). The inquisitors complained about Cardano's writings on astrology, especially his claim that self-harming religiously motivated actions of martyrs and heretics were caused by the stars. In his 1543 book "De Supplemento Almanach", a commentary on the astrological work "Tetrabiblos" by Ptolemy, Cardano had also published a horoscope of Jesus. Cardano was imprisoned for several months and lost his professorship in Bologna. He abjured and was freed, probably with help from powerful churchmen in Rome. All his non-medical works were prohibited and placed on the Index.
He moved to Rome, where he received a lifetime annuity from Pope Gregory XIII (after first having been rejected by Pope Pius V, who died in 1572) and finished his autobiography. He was accepted into the Royal College of Physicians, and as well as practising medicine he continued his philosophical studies until his death in 1576.
References in literature and culture.
The seventeenth-century English physician and philosopher Sir Thomas Browne possessed the ten volumes of the Lyon 1663 edition of the complete works of Cardan in his library.
Browne critically viewed Cardan as:
that famous Physician of Milan, a great Enquirer of Truth, but too greedy a Receiver of it. He hath left many excellent Discourses, Medical, Natural, and Astrological; the most suspicious are those two he wrote by admonition in a dream, that is "De Subtilitate & Varietate Rerum". Assuredly this learned man hath taken many things upon trust, and although examined some, hath let slip many others. He is of singular use unto a prudent Reader; but unto him that only desireth Hoties, or to replenish his head with varieties; like many others before related, either in the Original or confirmation, he may become no small occasion of Error.
Richard Hinckley Allen tells of an amusing reference made by Samuel Butler in his book "Hudibras":
<templatestyles src="Template:Blockquote/styles.css" />
Alessandro Manzoni's novel "I Promessi Sposi" portrays a pedantic scholar of the obsolete, Don Ferrante, as a great admirer of Cardano. Significantly, he values him only for his superstitious and astrological writings; his scientific writings are dismissed because they contradict Aristotle, but excused on the ground that the author of the astrological works deserves to be listened to even when he is wrong.
English novelist E. M. Forster's "Abinger Harvest", a 1936 volume of essays, authorial reviews and a play, provides a sympathetic treatment of Cardano in the section titled 'The Past'. Forster believes Cardano was so absorbed in "self-analysis that he often forgot to repent of his bad temper, his stupidity, his licentiousness, and love of revenge" (212).
Works.
"Collected Works".
A chronological key to this edition is supplied by M. Fierz.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
Sources.
<templatestyles src="Refbegin/styles.css" /> | [
{
"math_id": 0,
"text": "ax^3+bx+c=0"
}
]
| https://en.wikipedia.org/wiki?curid=13145 |
13146531 | Differentiation of trigonometric functions | Mathematical process of finding the derivative of a trigonometric function
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′("a") = cos("a"), meaning that the rate of change of sin("x") at a particular angle "x = a" is given by the cosine of that angle.
All derivatives of circular trigonometric functions can be found from those of sin("x") and cos("x") by means of the quotient rule applied to functions such as tan("x") = sin("x")/cos("x"). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.
Proofs of derivatives of trigonometric functions.
Limit of sin(θ)/θ as θ tends to 0.
The diagram at right shows a circle with centre "O" and radius "r =" 1. Let two radii "OA" and "OB" make an arc of θ radians. Since we are considering the limit as "θ" tends to zero, we may assume "θ" is a small positive number, say 0 < θ < π in the first quadrant.
In the diagram, let "R"1 be the triangle "OAB", "R"2 the circular sector "OAB", and "R"3 the triangle "OAC".
The area of triangle "OAB" is:
formula_0
The area of the circular sector "OAB" is:
formula_1
The area of the triangle "OAC" is given by:
formula_2
Since each region is contained in the next, one has:
formula_3
Moreover, since sin "θ" > 0 in the first quadrant, we may divide through by sin "θ", giving:
formula_4
In the last step we took the reciprocals of the three positive terms, reversing the inequities.
We conclude that for 0 < θ < π, the quantity sin("θ")/"θ" is "always" less than 1 and "always" greater than cos(θ). Thus, as "θ" gets closer to 0, sin("θ")/"θ" is "squeezed" between a ceiling at height 1 and a floor at height cos "θ", which rises towards 1; hence sin("θ")/"θ" must tend to 1 as "θ" tends to 0 from the positive side:formula_5For the case where "θ" is a small negative number – π < θ < 0, we use the fact that sine is an odd function:
formula_6
Limit of (cos(θ)-1)/θ as θ tends to 0.
The last section enables us to calculate this new limit relatively easily. This is done by employing a simple trick. In this calculation, the sign of "θ" is unimportant.
formula_7
Using cos2"θ" – 1 = –sin2"θ",
the fact that the limit of a product is the product of limits, and the limit result from the previous section, we find that:
formula_8
Limit of tan(θ)/θ as θ tends to 0.
Using the limit for the sine function, the fact that the tangent function is odd, and the fact that the limit of a product is the product of limits, we find:
formula_9
Derivative of the sine function.
We calculate the derivative of the sine function from the limit definition:
formula_10
Using the angle addition formula sin(α+β) = sin α cos β + sin β cos α, we have:
formula_11
Using the limits for the sine and cosine functions:
formula_12
Derivative of the cosine function.
From the definition of derivative.
We again calculate the derivative of the cosine function from the limit definition:
formula_13
Using the angle addition formula cos(α+β) = cos α cos β – sin α sin β, we have:
formula_14
Using the limits for the sine and cosine functions:
formula_15
From the chain rule.
To compute the derivative of the cosine function from the chain rule, first observe the following three facts:
formula_16
formula_17
formula_18
The first and the second are trigonometric identities, and the third is proven above. Using these three facts, we can write the following,
formula_19
We can differentiate this using the chain rule. Letting formula_20, we have:
formula_21.
Therefore, we have proven that
formula_22.
Derivative of the tangent function.
From the definition of derivative.
To calculate the derivative of the tangent function tan "θ", we use first principles. By definition:
formula_23
Using the well-known angle formula tan(α+β) = (tan α + tan β) / (1 - tan α tan β), we have:
formula_24
Using the fact that the limit of a product is the product of the limits:
formula_25
Using the limit for the tangent function, and the fact that tan "δ" tends to 0 as δ tends to 0:
formula_26
We see immediately that:
formula_27
From the quotient rule.
One can also compute the derivative of the tangent function using the quotient rule.
formula_28
The numerator can be simplified to 1 by the Pythagorean identity, giving us,
formula_29
Therefore,
formula_30
Proofs of derivatives of inverse trigonometric functions.
The following derivatives are found by setting a variable "y" equal to the inverse trigonometric function that we wish to take the derivative of. Using implicit differentiation and then solving for "dy"/"dx", the derivative of the inverse function is found in terms of "y". To convert "dy"/"dx" back into being in terms of "x", we can draw a reference triangle on the unit circle, letting "θ" be y. Using the Pythagorean theorem and the definition of the regular trigonometric functions, we can finally express "dy"/"dx" in terms of "x".
Differentiating the inverse sine function.
We let
formula_31
Where
formula_32
Then
formula_33
Taking the derivative with respect to formula_34 on both sides and solving for dy/dx:
formula_35
formula_36
Substituting formula_37 in from above,
formula_38
Substituting formula_39 in from above,
formula_40
formula_41
Differentiating the inverse cosine function.
We let
formula_42
Where
formula_43
Then
formula_44
Taking the derivative with respect to formula_34 on both sides and solving for dy/dx:
formula_45
formula_46
Substituting formula_47 in from above, we get
formula_48
Substituting formula_49 in from above, we get
formula_50
formula_51
Alternatively, once the derivative of formula_52 is established, the derivative of formula_53 follows immediately by differentiating the identity formula_54 so that formula_55.
Differentiating the inverse tangent function.
We let
formula_56
Where
formula_57
Then
formula_58
Taking the derivative with respect to formula_34 on both sides and solving for dy/dx:
formula_59
Left side:
formula_60 using the Pythagorean identity
Right side:
formula_61
Therefore,
formula_62
Substituting formula_63 in from above, we get
formula_64
formula_65
Differentiating the inverse cotangent function.
We let
formula_66
where formula_67. Then
formula_68
Taking the derivative with respect to formula_34 on both sides and solving for dy/dx:
formula_69
Left side:
formula_70 using the Pythagorean identity
Right side:
formula_61
Therefore,
formula_71
Substituting formula_72,
formula_73
formula_74
Alternatively, as the derivative of formula_75 is derived as shown above, then using the identity formula_76 follows immediately thatformula_77
Differentiating the inverse secant function.
Using implicit differentiation.
Let
formula_78
Then
formula_79
formula_80
formula_82
Using the chain rule.
Alternatively, the derivative of arcsecant may be derived from the derivative of arccosine using the chain rule.
Let
formula_83
Where
formula_84 and formula_85
Then, applying the chain rule to formula_86:
formula_87
Differentiating the inverse cosecant function.
Using implicit differentiation.
Let
formula_88
Then
formula_89
formula_90
formula_91
Using the chain rule.
Alternatively, the derivative of arccosecant may be derived from the derivative of arcsine using the chain rule.
Let
formula_92
Where
formula_84 and formula_93
Then, applying the chain rule to formula_94:
formula_95
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " \\mathrm{Area}(R_1\n)\n=\\tfrac{1}{2} \\ |OA| \\ |OB| \\sin\\theta = \\tfrac{1}{2}\\sin\\theta \\, . "
},
{
"math_id": 1,
"text": "\\mathrm{Area}(R_2) \n=\\tfrac{1}{2}\\theta \\, . "
},
{
"math_id": 2,
"text": " \\mathrm{Area}(R_3\n)\n=\\tfrac{1}{2} \\ |OA| \\ |AC| = \\tfrac{1}{2} \\tan\\theta \\, . "
},
{
"math_id": 3,
"text": "\\text{Area}(R_1) < \\text{Area}(R_2) < \\text{Area}(R_3) \\implies \n\\tfrac{1}{2}\\sin\\theta < \\tfrac{1}{2}\\theta < \\tfrac{1}{2}\\tan\\theta \\, . "
},
{
"math_id": 4,
"text": "1 < \\frac{\\theta}{\\sin\\theta} < \\frac{1}{\\cos\\theta} \\implies 1 > \\frac{\\sin\\theta}{\\theta} > \\cos\\theta \\, . "
},
{
"math_id": 5,
"text": "\\lim_{\\theta \\to 0^+} \\frac{\\sin\\theta}{\\theta} = 1 \\, . "
},
{
"math_id": 6,
"text": "\\lim_{\\theta \\to 0^-}\\! \\frac{\\sin\\theta}{\\theta} \n\\ =\\ \n\\lim_{\\theta\\to 0^+}\\!\\frac{\\sin(-\\theta)}{-\\theta} \n\\ =\\ \n\\lim_{\\theta \\to 0^+}\\!\\frac{-\\sin\\theta}{-\\theta} \n\\ =\\ \n\\lim_{\\theta\\to 0^+}\\!\\frac{\\sin\\theta}{\\theta} \\ =\\ \n 1 \\, . "
},
{
"math_id": 7,
"text": " \\lim_{\\theta \\to 0}\\, \\frac{\\cos\\theta - 1}{\\theta} \n\\ =\\ \n\\lim_{\\theta \\to 0} \\left( \\frac{\\cos\\theta - 1}{\\theta} \\right)\\!\\! \\left( \\frac{\\cos\\theta + 1}{\\cos\\theta + 1} \\right) \n\\ =\\ \n\\lim_{\\theta \\to 0}\\, \\frac{\\cos^2\\!\\theta - 1}{\\theta\\,(\\cos\\theta + 1)} . "
},
{
"math_id": 8,
"text": " \\lim_{\\theta \\to 0}\\,\\frac{\\cos\\theta - 1}{\\theta}\n\\ =\\ \n\\lim_{\\theta \\to 0}\\, \\frac{-\\sin^2\\theta}{\\theta(\\cos\\theta+1)} \n\\ =\\ \n\\left( -\\lim_{\\theta \\to 0} \\frac{\\sin\\theta}{\\theta}\\right)\\! \\left( \\lim_{\\theta \\to 0}\\,\\frac{\\sin\\theta}{\\cos\\theta + 1} \\right) \n\\ =\\ \n(-1)\\left(\\frac{0}{2}\\right) = 0 \\, . "
},
{
"math_id": 9,
"text": "\n \\lim_{\\theta\\to 0} \\frac{\\tan\\theta}{\\theta}\n \\ =\\ \n \\left(\\lim_{\\theta\\to 0} \\frac{\\sin\\theta}{\\theta}\\right)\\!\n \\left( \\lim_{\\theta\\to 0} \\frac{1}{\\cos\\theta}\\right)\n \\ =\\ \n (1)(1)\n \\ =\\ \n 1 \\, . "
},
{
"math_id": 10,
"text": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\sin\\theta = \\lim_{\\delta \\to 0} \\frac{\\sin(\\theta + \\delta) - \\sin \\theta}{\\delta} . "
},
{
"math_id": 11,
"text": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\sin\\theta \n =\n\\lim_{\\delta \\to 0} \\frac{\\sin\\theta\\cos\\delta + \\sin\\delta\\cos\\theta-\\sin\\theta}{\\delta} \n =\n\\lim_{\\delta \\to 0} \\left( \\frac{\\sin\\delta}{\\delta} \\cos\\theta \n+ \\frac{\\cos\\delta -1}{\\delta}\\sin\\theta \\right) . "
},
{
"math_id": 12,
"text": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\sin\\theta \n =\n(1)\\cos\\theta + (0)\\sin\\theta \n =\n\\cos\\theta \\, . "
},
{
"math_id": 13,
"text": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\cos\\theta \n =\n\\lim_{\\delta \\to 0} \\frac{\\cos(\\theta+\\delta)-\\cos\\theta}{\\delta} . "
},
{
"math_id": 14,
"text": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\cos\\theta \n =\n\\lim_{\\delta \\to 0} \\frac{\\cos\\theta\\cos\\delta - \\sin\\theta\\sin\\delta-\\cos\\theta}{\\delta} \n =\n\\lim_{\\delta \\to 0} \\left(\\frac{\\cos\\delta -1}{\\delta}\\cos\\theta \\,-\\, \\frac{\\sin\\delta}{\\delta} \\sin\\theta \\right) . "
},
{
"math_id": 15,
"text": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\cos\\theta \n = (0) \\cos\\theta - (1) \\sin\\theta = -\\sin\\theta \\, . "
},
{
"math_id": 16,
"text": "\\cos\\theta = \\sin\\left(\\tfrac{\\pi}{2}-\\theta\\right)"
},
{
"math_id": 17,
"text": "\\sin\\theta = \\cos\\left(\\tfrac{\\pi}{2}-\\theta\\right)"
},
{
"math_id": 18,
"text": "\\tfrac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\sin\\theta = \\cos\\theta"
},
{
"math_id": 19,
"text": "\\tfrac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\cos\\theta = \\tfrac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\sin\\left(\\tfrac{\\pi}{2}-\\theta\\right)"
},
{
"math_id": 20,
"text": "f(x) = \\sin x,\\ \\ g(\\theta) =\\tfrac{\\pi}{2}-\\theta"
},
{
"math_id": 21,
"text": "\\tfrac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} f\\!\\left(g\\!\\left(\\theta\\right)\\right) = f^\\prime\\!\\left(g\\!\\left(\\theta\\right)\\right) \\cdot g^\\prime\\!\\left(\\theta\\right) = \\cos\\left(\\tfrac{\\pi}{2}-\\theta\\right) \\cdot (0-1) = -\\sin\\theta"
},
{
"math_id": 22,
"text": "\\tfrac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\cos\\theta = -\\sin\\theta"
},
{
"math_id": 23,
"text": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = \\lim_{\\delta \\to 0} \\left( \\frac{\\tan(\\theta+\\delta)-\\tan\\theta}{\\delta} \\right) .\n"
},
{
"math_id": 24,
"text": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = \\lim_{\\delta \\to 0} \\left[ \\frac{\\frac{\\tan\\theta + \\tan\\delta}{1 - \\tan\\theta\\tan\\delta} - \\tan\\theta}{\\delta} \\right]\n = \\lim_{\\delta \\to 0} \\left[ \\frac{\\tan\\theta + \\tan\\delta - \\tan\\theta + \\tan^2\\theta\\tan\\delta}{\\delta \\left( 1 - \\tan\\theta\\tan\\delta \\right)} \\right] .\n"
},
{
"math_id": 25,
"text": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = \\lim_{\\delta \\to 0} \\frac{\\tan\\delta}{\\delta} \\times \\lim_{\\delta \\to 0} \\left( \\frac{1 + \\tan^2\\theta}{1 - \\tan\\theta\\tan\\delta} \\right) .\n"
},
{
"math_id": 26,
"text": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = 1 \\times \\frac{1 + \\tan^2\\theta}{1 - 0} = 1 + \\tan^2\\theta .\n"
},
{
"math_id": 27,
"text": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = 1 + \\frac{\\sin^2\\theta}{\\cos^2\\theta}\n = \\frac{\\cos^2\\theta + \\sin^2\\theta}{\\cos^2\\theta}\n = \\frac{1}{\\cos^2\\theta}\n = \\sec^2\\theta \\, .\n"
},
{
"math_id": 28,
"text": "\\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\tan\\theta \n = \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\frac{\\sin\\theta}{\\cos\\theta}\n = \\frac{\\left(\\sin\\theta\\right)^\\prime \\cdot \\cos\\theta - \\sin\\theta \\cdot \\left(\\cos\\theta\\right)^\\prime}{ \\cos^2 \\theta }\n = \\frac{\\cos^2 \\theta + \\sin^2 \\theta}{\\cos^2 \\theta}\n"
},
{
"math_id": 29,
"text": "\\frac{1}{\\cos^2 \\theta} = \\sec^2 \\theta"
},
{
"math_id": 30,
"text": "\\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta} \\tan\\theta = \\sec^2 \\theta"
},
{
"math_id": 31,
"text": "y=\\arcsin x\\,\\!"
},
{
"math_id": 32,
"text": "-\\frac{\\pi}{2}\\le y \\le \\frac{\\pi}{2}"
},
{
"math_id": 33,
"text": "\\sin y=x\\,\\!"
},
{
"math_id": 34,
"text": "x"
},
{
"math_id": 35,
"text": "{d \\over dx}\\sin y={d \\over dx}x"
},
{
"math_id": 36,
"text": "\\cos y \\cdot {dy \\over dx} = 1\\,\\!"
},
{
"math_id": 37,
"text": " \\cos y = \\sqrt{1-\\sin^2 y}"
},
{
"math_id": 38,
"text": "\\sqrt{1-\\sin^2 y} \\cdot {dy \\over dx} =1"
},
{
"math_id": 39,
"text": "x=\\sin y"
},
{
"math_id": 40,
"text": "\\sqrt{1-x^2} \\cdot {dy \\over dx} =1"
},
{
"math_id": 41,
"text": "{dy \\over dx}=\\frac{1}{\\sqrt{1-x^2}}"
},
{
"math_id": 42,
"text": "y=\\arccos x\\,\\!"
},
{
"math_id": 43,
"text": "0 \\le y \\le \\pi"
},
{
"math_id": 44,
"text": "\\cos y=x\\,\\!"
},
{
"math_id": 45,
"text": "{d \\over dx}\\cos y={d \\over dx}x"
},
{
"math_id": 46,
"text": "-\\sin y \\cdot {dy \\over dx} =1"
},
{
"math_id": 47,
"text": "\\sin y = \\sqrt{1-\\cos^2 y}\\,\\!"
},
{
"math_id": 48,
"text": "-\\sqrt{1-\\cos^2 y} \\cdot {dy \\over dx} =1"
},
{
"math_id": 49,
"text": "x=\\cos y\\,\\!"
},
{
"math_id": 50,
"text": "-\\sqrt{1-x^2} \\cdot {dy \\over dx} =1"
},
{
"math_id": 51,
"text": "{dy \\over dx} = -\\frac{1}{\\sqrt{1-x^2}}"
},
{
"math_id": 52,
"text": "\\arcsin x"
},
{
"math_id": 53,
"text": "\\arccos x"
},
{
"math_id": 54,
"text": "\\arcsin x+\\arccos x=\\pi/2"
},
{
"math_id": 55,
"text": "(\\arccos x)'=-(\\arcsin x)'"
},
{
"math_id": 56,
"text": "y=\\arctan x\\,\\!"
},
{
"math_id": 57,
"text": "-\\frac{\\pi}{2} < y < \\frac{\\pi}{2}"
},
{
"math_id": 58,
"text": "\\tan y=x\\,\\!"
},
{
"math_id": 59,
"text": "{d \\over dx}\\tan y={d \\over dx}x"
},
{
"math_id": 60,
"text": "\n {d \\over dx}\\tan y\n = \\sec^2 y \\cdot {dy \\over dx}\n = (1 + \\tan^2 y) {dy \\over dx}\n"
},
{
"math_id": 61,
"text": "{d \\over dx}x = 1"
},
{
"math_id": 62,
"text": "(1+\\tan^2 y){dy \\over dx}=1"
},
{
"math_id": 63,
"text": "x=\\tan y\\,\\!"
},
{
"math_id": 64,
"text": "(1+x^2){dy \\over dx}=1"
},
{
"math_id": 65,
"text": "{dy \\over dx}=\\frac{1}{1+x^2}"
},
{
"math_id": 66,
"text": "y=\\arccot x"
},
{
"math_id": 67,
"text": "0<y<\\pi"
},
{
"math_id": 68,
"text": "\\cot y=x"
},
{
"math_id": 69,
"text": "\\frac{d}{dx}\\cot y=\\frac{d}{dx}x"
},
{
"math_id": 70,
"text": "\n {d \\over dx}\\cot y\n = -\\csc^2 y \\cdot {dy \\over dx}\n = -(1 + \\cot^2 y) {dy \\over dx}\n"
},
{
"math_id": 71,
"text": "-(1+\\cot^2y)\\frac{dy}{dx}=1"
},
{
"math_id": 72,
"text": "x=\\cot y"
},
{
"math_id": 73,
"text": "-(1+x^2)\\frac{dy}{dx}=1"
},
{
"math_id": 74,
"text": "\\frac{dy}{dx}=-\\frac{1}{1+x^2}"
},
{
"math_id": 75,
"text": "\\arctan x"
},
{
"math_id": 76,
"text": "\\arctan x+\\arccot x=\\dfrac{\\pi}{2}"
},
{
"math_id": 77,
"text": "\\begin{align}\n\\dfrac{d}{dx}\\arccot x \n&=\\dfrac{d}{dx}\\left(\\dfrac{\\pi}{2}-\\arctan x\\right)\\\\\n&=-\\dfrac{1}{1+x^2}\n\\end{align}"
},
{
"math_id": 78,
"text": " y = \\arcsec x\\ \\mid |x| \\geq 1"
},
{
"math_id": 79,
"text": " x = \\sec y \\mid \\ y \\in \\left [0,\\frac{\\pi}{2} \\right )\\cup \\left (\\frac{\\pi}{2},\\pi \\right]\n"
},
{
"math_id": 80,
"text": " \\frac{dx}{dy} = \\sec y \\tan y = |x|\\sqrt{x^2-1}"
},
{
"math_id": 81,
"text": "\\sqrt{x^2-1}"
},
{
"math_id": 82,
"text": " \\frac{dy}{dx} = \\frac{1}{|x|\\sqrt{x^2-1}}"
},
{
"math_id": 83,
"text": " y = \\arcsec x = \\arccos \\left(\\frac{1}{x}\\right) "
},
{
"math_id": 84,
"text": " |x| \\geq 1 "
},
{
"math_id": 85,
"text": " y \\in \\left[0, \\frac{\\pi}{2}\\right) \\cup \\left(\\frac{\\pi}{2}, \\pi\\right] "
},
{
"math_id": 86,
"text": " \\arccos \\left(\\frac{1}{x}\\right) "
},
{
"math_id": 87,
"text": " \\frac{dy}{dx} = -\\frac{1}{\\sqrt{1-(\\frac{1}{x})^2}} \\cdot \\left(-\\frac{1}{x^2}\\right)\n = \\frac{1}{x^2\\sqrt{1-\\frac{1}{x^2}}}\n = \\frac{1}{x^2\\frac{\\sqrt{x^2-1}}{\\sqrt{x^2}}}\n = \\frac{1}{\\sqrt{x^2}\\sqrt{x^2-1}}\n = \\frac{1}{|x|\\sqrt{x^2-1}} "
},
{
"math_id": 88,
"text": "y = \\arccsc x\\ \\mid |x| \\geq 1"
},
{
"math_id": 89,
"text": " x = \\csc y\\ \\mid \\ y \\in \\left [-\\frac{\\pi}{2},0 \\right )\\cup \\left (0,\\frac{\\pi}{2} \\right]"
},
{
"math_id": 90,
"text": " \\frac{dx}{dy} = -\\csc y \\cot y = -|x|\\sqrt{x^2-1}"
},
{
"math_id": 91,
"text": " \\frac{dy}{dx} = \\frac{-1}{|x|\\sqrt{x^2-1}}"
},
{
"math_id": 92,
"text": " y = \\arccsc x = \\arcsin \\left(\\frac{1}{x}\\right) "
},
{
"math_id": 93,
"text": " y \\in \\left[-\\frac{\\pi}{2}, 0\\right) \\cup \\left(0, \\frac{\\pi}{2}\\right] "
},
{
"math_id": 94,
"text": " \\arcsin \\left(\\frac{1}{x}\\right) "
},
{
"math_id": 95,
"text": " \\frac{dy}{dx} =\\frac{1}{\\sqrt{1-(\\frac{1}{x})^2}} \\cdot \\left(-\\frac{1}{x^2}\\right)\n = -\\frac{1}{x^2\\sqrt{1-\\frac{1}{x^2}}}\n = -\\frac{1}{x^2\\frac{\\sqrt{x^2-1}}{\\sqrt{x^2}}}\n = -\\frac{1}{\\sqrt{x^2}\\sqrt{x^2-1}}\n = -\\frac{1}{|x|\\sqrt{x^2-1}} "
}
]
| https://en.wikipedia.org/wiki?curid=13146531 |
1314854 | Diagonal subgroup | In the mathematical discipline of group theory, for a given group "G", the diagonal subgroup of the "n"-fold direct product "G"&hairsp;&hairsp;"n" is the subgroup
formula_0
This subgroup is isomorphic to "G".
formula_1 | [
{
"math_id": 0,
"text": "\\{(g, \\dots, g) \\in G^n : g \\in G\\}."
},
{
"math_id": 1,
"text": "(x_1, \\dots, x_n) \\cdot (g, \\dots, g) = (x_1 \\!\\cdot g, \\dots, x_n \\!\\cdot g)."
}
]
| https://en.wikipedia.org/wiki?curid=1314854 |
13150336 | Cooperative MIMO | In radio, cooperative multiple-input multiple-output (cooperative MIMO, CO-MIMO) is a technology that can effectively exploit the spatial domain of mobile fading channels to bring significant performance improvements to wireless communication systems. It is also called network MIMO, distributed MIMO, virtual MIMO, and virtual antenna arrays.
Conventional MIMO systems, known as point-to-point MIMO or collocated MIMO, require both the transmitter and receiver of a communication link to be equipped with multiple antennas. While MIMO has become an essential element of wireless communication standards, including IEEE 802.11n (Wi-Fi), IEEE 802.11ac (Wi-Fi), HSPA+ (3G), WiMAX (4G), and Long-Term Evolution (4G), many wireless devices cannot support multiple antennas due to size, cost, and/or hardware limitations. More importantly, the separation between antennas on a mobile device and even on fixed radio platforms is often insufficient to allow meaningful performance gains. Furthermore, as the number of antennas is increased, the actual MIMO performance falls farther behind the theoretical gains.
Cooperative MIMO uses distributed antennas on different radio devices to achieve close to the theoretical gains of MIMO. The basic idea of cooperative MIMO is to group multiple devices into a virtual antenna array to achieve MIMO communications. A cooperative MIMO transmission involves multiple point-to-point radio links, including links within a virtual array and possibly links between different virtual arrays.
The disadvantages of cooperative MIMO come from the increased system complexity and the large signaling overhead required for supporting device cooperation. The advantages of cooperative MIMO, on the other hand, are its capability to improve the capacity, cell edge throughput, coverage, and group mobility of a wireless network in a cost-effective manner. These advantages are achieved by using distributed antennas, which can increase the system capacity by decorrelating the MIMO subchannels and allow the system to exploit the benefits of macro-diversity in addition to micro-diversity. In many practical applications, such as cellular mobile and wireless ad hoc networks, the advantages of deploying cooperative MIMO technology outweigh the disadvantages. In recent years, cooperative MIMO technologies have been adopted into the mainstream of wireless communication standards.
Types.
Coordinated multipoint.
In coordinated multipoint (CoMP), data and channel state information (CSI) is shared among neighboring cellular base stations (BSs) to coordinate their transmissions in the downlink and jointly process the received signals in the uplink. The system architecture is illustrated in Fig. 1a. CoMP techniques can effectively turn otherwise harmful inter-cell interference into useful signals, enabling significant power gain, channel rank advantage, and/or diversity gains to be exploited. CoMP requires a high-speed backhaul network for enabling the exchange of information (e.g., data, control information, and CSI) between the BSs. This is typically achieved via an optical fiber fronthaul. CoMP has been introduced into 4G standards.
Fixed relays.
Fixed relays (illustrated in Figure 1b) are low-cost and fixed radio infrastructures without wired backhaul connections. They store data received from the BS and forward to the mobile stations (MSs), and vice versa. Fixed relay stations (RSs) typically have smaller transmission powers and coverage areas than a BS. They can be deployed strategically and cost effectively in cellular networks to extend coverage, reduce total transmission power, enhance the capacity of a specific region with high traffic demands, and/or improve signal reception. By combining the signals from the relays and possibly the source signal from the BS, the mobile station (MS) is able to exploit the inherent diversity of the relay channel. The disadvantages of fixed relays are the additional delays introduced in the relaying process, and the potentially increased levels of interference due to frequency reuse at the RSs. As one of the most mature cooperative MIMO technologies, fixed relay has attracted significant support in major cellular communication standards.
Mobile relays.
Mobile relays differ from fixed relays in the sense that the RSs are mobile and are not deployed as the infrastructure of a network. Mobile relays are therefore more flexible in accommodating varying traffic patterns and adapting to different propagation environments. For example, when a target MS temporarily suffers from poor channel conditions or requires relatively high-rate service, its neighboring MSs can help to provide multi-hop coverage or increase the data rate by relaying information to the target MS. Moreover, mobile relays enable faster and lower-cost network rollout. Similar to fixed relays, mobile relays can enlarge the coverage area, reduce the overall transmit power, and/or increase the capacity at cell edges. On the other hand, due to their opportunistic nature, mobile relays are less reliable than fixed relays since the network topology is highly dynamic and unstable.
The mobile user relays enable distributed MSs to self-organize into a wireless ad hoc network, which complements the cellular network infrastructure using multi-hop transmissions. Studies have shown that mobile user relays have a fundamental advantage in that the total network capacity, measured as the sum of the throughputs of the users, can scale linearly with the number of users given sufficient infrastructure supports. Mobile user relays are therefore a desirable enhancement to future cellular systems. However, mobile user relays face challenges in routing, radio resource management, and interference management.
Device to device (D2D) in LTE is a step toward Mobile Relays.
Cooperative subspace coding.
In Cooperative-MIMO, the decoding process involves collecting "N""R" linear combinations of "N""T" original data symbols, where "N""R" is usually the number of receiving nodes, and "N""T" is the number of transmitting nodes. The decoding process can be interpreted as solving a system of "N""R" linear equations, where the number of unknowns equals the number of data symbols ("N""T") and interference signals. Thus, in order for data streams to be successfully decoded, the number of independent linear equations (NR) must at least equal the number of data ("N""T") and interference streams.
In cooperative subspace coding, also known as linear network coding, nodes transmit random linear combinations of original packets with coefficients which can be chosen from measurements of the naturally random scattering environment. Alternatively, the scattering environment is relied upon to encode the transmissions. If the spatial subchannels are sufficiently uncorrelated from each other, the probability that the receivers will obtain linearly independent combinations (and therefore obtain innovative information) approaches 1. Although random linear network coding has excellent throughput performance, if a receiver obtains an insufficient number of packets, it is extremely unlikely that it can recover any of the original packets. This can be addressed by sending additional random linear combinations (such as by increasing the rank of the MIMO channel matrix or retransmitting at a later time that is greater than the channel coherence time) until the receiver obtains a sufficient number of coded packets to permit decoding.
Cooperative subspace coding faces high decoding computational complexity. However, in cooperative MIMO radio, MIMO decoding already employs similar, if not identical, methods as random linear network decoding. Random linear network codes have a high overhead due to the large coefficient vectors attached to encoded blocks. But in Cooperative-MIMO radio, the coefficient vectors can be measured from known training signals, which is already performed for channel estimation. Finally, linear dependency among coding vectors reduces the number of innovative encoded blocks. However, linear dependency in radio channels is a function of channel correlation, which is a problem solved by cooperative MIMO.
History.
Before the introduction of cooperative MIMO, joint processing among cellular base stations was proposed to mitigate inter-cell interference, and Cooperative diversity offered increased diversity gain using relays, but at the cost of poorer spectral efficiency. However, neither of these techniques exploits interference for spatial multiplexing gains, which can dramatically increase spectral efficiency.
In 2001, cooperative MIMO was introduced by Steve Shattil, a scientist at Idris Communications, in a provisional patent application, which disclosed Coordinated Multipoint and Fixed Relays, followed by a paper in which S. Shamai and B.M. Zaidel proposed “dirty paper” precoding in downlink co-processing for single-user cells. In 2002, Shattil introduced the Mobile Relay and Network Coding aspects of cooperative MIMO in US Pat. No. 7430257 and US Pub. No. 20080095121. Implementations of software-defined radio (SDR) and distributed computing in cooperative MIMO were introduced in US Pat. No. 7430257 (2002) and 8670390 (2004), providing the foundation for Cloud Radio Access Network (C-RAN).
Server-side implementations of cooperative MIMO were the first to be adopted into the 4G cellular specifications and are essential for 5G. CoMP and Fixed Relays pool baseband processing resources in data centers, enabling dense deployments of simple, inexpensive radio terminals (such as remote radio heads) instead of cellular base stations. This allows processing resources to easily scale to meet network demand, and the distributed antennas could enable each user device to be served by the full spectral bandwidth of the system. However, data bandwidth per user is still limited by the amount of available spectrum, which is a concern because data use per user continues to grow.
The adoption of client-side cooperative MIMO lags behind server-side cooperative MIMO. Client-side cooperative MIMO, such as mobile relays, can distribute processing loads among client devices in a cluster, which means the computational load per processor can scale more effectively as the cluster grows. While there is additional overhead for coordinating the client devices, devices in a cluster can share radio channels and spatial subchannels via short-range wireless links. This means that as the cluster grows, the available instantaneous data bandwidth per user also grows. Thus, instead of the data bandwidth per user being hard-limited by the laws of Physics (i.e., the Shannon-Hartley Theorem), data bandwidth is constrained only by computational processing power, which keeps improving according to Moore’s Law. Despite the great potential for client-side cooperative MIMO, a user-based infrastructure is more difficult for service providers to monetize, and there are additional technical challenges.
While mobile relays can reduce overall transmission energy, this savings can be offset by circuit energy required for increased computational processing. Above a certain transmission distance threshold, cooperative MIMO has been shown to achieve overall energy savings. Various techniques have been developed for handling timing and frequency offsets, which is one of the most critical and challenging issues in cooperative MIMO. Recently, research has focused on developing efficient MAC protocols.
Mathematical description.
In this section, we describe precoding using a system model of a Cooperative-MIMO downlink channel for a CoMP system. A group of BSs employs an aggregate "M" transmit antennas to communicate with "K" users simultaneously.
User "k", ("k" = 1,… , "K"), has "Nk" receive antennas. The channel model from the BSs to the "k"th user is represented by an "Nk" ×"M" channel matrix Hk.
Let sk denote the "k"th user transmit symbol vector. For user "k", a linear transmit precoding matrix, Wk, which transforms the data vector sk to the "M" ×1 transmitted vector Wk × sk, is employed by the BSs. The received signal vector at the "k"th user is given by formula_0,
where nk" = ["nk,"1, …, "nk,Nk" ]T denotes the noise vector for the "k"th user, and (.)T denotes the transpose of a matrix or vector. The components "nk,i" of the noise vector nk" are i.i.d. with zero mean and variance "σ"2 for "k" = 1,…,"K" and "i" = 1,…,"Nk". The first term, HkWks"k", represents the desired signal, and the second term, formula_1, represents interference received by user "k".
The network channel is defined as H = [H1T,…, H"K"T]T, and the corresponding set of signals received by all users is expressed by
y = HWs + n,
where H = [H1T,…, HK"T]T, y = [y1T,…, yK"T]T, W = [W1T,…, WK"T]T, s = [s1T,…, sK"T]T, and n = [n1T,…, n"K"T]T.
The precoding matrix W is designed based on channel information in order to improve performance of the Cooperative-MIMO system.
Alternatively, receiver-side processing, referred to as spatial demultiplexing, separates the transmitted symbols. Without precoding, the set of signals received by all users is expressed by
y = Hs + n
The received signal is processed with a spatial demultiplexing matrix G to recover the transmit symbols: formula_2.
Common types of precoding include zero-forcing (ZF), minimum mean squared error (MMSE) precoding, maximum ratio transmission (MRT), and block diagonalization. Common types of spatial demultiplexing include ZF, MMSE combining, and successive interference cancellation.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "\\mathrm{y}_k = \\mathrm{H}_k\\mathrm{W}_k\\mathrm{s}_k+\\mathrm{H}_k\\sum_{i\\neq k}\\mathrm{W}_i\\mathrm{s}_i+\\mathrm{n}_k"
},
{
"math_id": 1,
"text": "\\mathrm{H}_k\\sum_{i\\neq k}\\mathrm{W}_i\\mathrm{s}_i"
},
{
"math_id": 2,
"text": "\\hat{\\mathrm{s}}=\\mathrm{Gy}=\\mathrm{G}\\bigl(\\mathrm{Hs}+\\mathrm{n}\\bigr)"
}
]
| https://en.wikipedia.org/wiki?curid=13150336 |
1315049 | Downsampling (signal processing) | Resampling method
In digital signal processing, downsampling, compression, and decimation are terms associated with the process of "resampling" in a multi-rate digital signal processing system. Both "downsampling" and "decimation" can be synonymous with "compression", or they can describe an entire process of bandwidth reduction (filtering) and sample-rate reduction. When the process is performed on a sequence of samples of a "signal" or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or density, as in the case of a photograph).
"Decimation" is a term that historically means the "removal of every tenth one". But in signal processing, "decimation by a factor of 10" actually means "keeping" only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if compact disc audio at 44,100 samples/second is "decimated" by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called a "decimator". Decimation by an integer factor is also called "compression".
Downsampling by an integer factor.
Rate reduction by an integer factor "M" can be explained as a two-step process, with an equivalent implementation that is more efficient:
Step 2 alone creates undesirable aliasing (i.e. high-frequency signal components will copy into the lower frequency band and be mistaken for lower frequencies). Step 1, when necessary, suppresses aliasing to an acceptable level. In this application, the filter is called an anti-aliasing filter, and its design is discussed below. Also see undersampling for information about decimating bandpass functions and signals.
When the anti-aliasing filter is an IIR design, it relies on feedback from output to input, prior to the second step. With FIR filtering, it is an easy matter to compute only every "M"th output. The calculation performed by a decimating FIR filter for the "n"th output sample is a dot product:
formula_0
where the "h"[•] sequence is the impulse response, and "K" is its length. "x"[•] represents the input sequence being downsampled. In a general purpose processor, after computing "y"["n"], the easiest way to compute "y"["n"+1] is to advance the starting index in the "x"[•] array by "M", and recompute the dot product. In the case "M"=2, "h"[•] can be designed as a half-band filter, where almost half of the coefficients are zero and need not be included in the dot products.
Impulse response coefficients taken at intervals of "M" form a subsequence, and there are "M" such subsequences (phases) multiplexed together. The dot product is the sum of the dot products of each subsequence with the corresponding samples of the "x"[•] sequence. Furthermore, because of downsampling by "M", the stream of "x"[•] samples involved in any one of the "M" dot products is never involved in the other dot products. Thus "M" low-order FIR filters are each filtering one of "M" multiplexed "phases" of the input stream, and the "M" outputs are being summed. This viewpoint offers a different implementation that might be advantageous in a multi-processor architecture. In other words, the input stream is demultiplexed and sent through a bank of M filters whose outputs are summed. When implemented that way, it is called a polyphase filter.
For completeness, we now mention that a possible, but unlikely, implementation of each phase is to replace the coefficients of the other phases with zeros in a copy of the "h"[•] array, process the original "x"[•] sequence at the input rate (which means multiplying by zeros), and decimate the output by a factor of "M". The equivalence of this inefficient method and the implementation described above is known as the "first Noble identity". It is sometimes used in derivations of the polyphase method.
Anti-aliasing filter.
Let "X"("f") be the Fourier transform of any function, "x"("t"), whose samples at some interval, "T", equal the "x"["n"] sequence. Then the discrete-time Fourier transform (DTFT) is a Fourier series representation of a periodic summation of "X"("f"):
formula_1
When "T" has units of seconds, formula_2 has units of hertz. Replacing "T" with "MT" in the formulas above gives the DTFT of the decimated sequence, "x"["nM"]:
formula_3
The periodic summation has been reduced in amplitude and periodicity by a factor of "M". An example of both these distributions is depicted in the two traces of Fig 1.
Aliasing occurs when adjacent copies of "X"("f") overlap. The purpose of the anti-aliasing filter is to ensure that the reduced periodicity does not create overlap. The condition that ensures the copies of "X"("f") do not overlap each other is: formula_4 so that is the maximum cutoff frequency of an "ideal" anti-aliasing filter.
By a rational factor.
Let "M/L" denote the decimation factor, where: M, L ∈ formula_5; M > L.
Step 1 requires a lowpass filter after increasing ("expanding") the data rate, and step 2 requires a lowpass filter before decimation. Therefore, both operations can be accomplished by a single filter with the lower of the two cutoff frequencies. For the "M" > "L" case, the anti-aliasing filter cutoff, formula_6 "cycles per intermediate sample", is the lower frequency.
Notes.
<templatestyles src="Reflist/styles.css" />
Page citations.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "y[n] = \\sum_{k=0}^{K-1} x[nM-k]\\cdot h[k],"
},
{
"math_id": 1,
"text": "\\underbrace{\n\\sum_{n=-\\infty}^{\\infty} \\overbrace{x(nT)}^{x[n]}\\ \\mathrm e^{-\\mathrm i 2\\pi f nT}\n}_{\\text{DTFT}} = \\frac{1}{T}\\sum_{k=-\\infty}^{\\infty} X\\Bigl(f - \\frac{k}{T}\\Bigr)."
},
{
"math_id": 2,
"text": "f"
},
{
"math_id": 3,
"text": "\\sum_{n=-\\infty}^{\\infty} x(n\\cdot MT)\\ \\mathrm e^{-\\mathrm i 2\\pi f n(MT)} = \\frac{1}{MT}\\sum_{k=-\\infty}^{\\infty} X\\left(f-\\tfrac{k}{MT}\\right)."
},
{
"math_id": 4,
"text": " B < \\tfrac{0.5}{T} \\cdot \\tfrac{1}{M},"
},
{
"math_id": 5,
"text": "\\mathbb{Z}"
},
{
"math_id": 6,
"text": "\\tfrac{0.5}{M}"
}
]
| https://en.wikipedia.org/wiki?curid=1315049 |
1315510 | Upsampling | Digital signal resampling method
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. "Upsampling" can be synonymous with "expansion", or it can describe an entire process of "expansion" and filtering ("interpolation"). When upsampling is performed on a sequence of samples of a "signal" or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate (or density, as in the case of a photograph). For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.
Upsampling by an integer factor.
Rate increase by an integer factor formula_0 can be explained as a 2-step process, with an equivalent implementation that is more efficient:
In this application, the filter is called an interpolation filter, and its design is discussed below. When the interpolation filter is an FIR type, its efficiency can be improved, because the zeros contribute nothing to its dot product calculations. It is an easy matter to omit them from both the data stream and the calculations. The calculation performed by a multirate interpolating FIR filter for each output sample is a dot product:
where the formula_5 sequence is the impulse response of the interpolation filter, and formula_6 is the largest value of formula_7 for which formula_8 is non-zero.
In the case formula_10 function formula_5 can be designed as a half-band filter, where almost half of the coefficients are zero and need not be included in the dot products. Impulse response coefficients taken at intervals of formula_0 form a subsequence, and there are formula_0 such subsequences (called phases) multiplexed together. Each of formula_0 phases of the impulse response is filtering the same sequential values of the formula_11 data stream and producing one of formula_0 sequential output values. In some multi-processor architectures, these dot products are performed simultaneously, in which case it is called a polyphase filter.
For completeness, we now mention that a possible, but unlikely, implementation of each phase is to replace the coefficients of the other phases with zeros in a copy of the formula_5 array, and process the formula_12 sequence at formula_0 times faster than the original input rate. Then formula_3 of every formula_0 outputs are zero. The desired formula_13 sequence is the sum of the phases, where formula_3 terms of the each sum are identically zero. Computing formula_3 zeros between the useful outputs of a phase and adding them to a sum is effectively decimation. It's the same result as not computing them at all. That equivalence is known as the "second Noble identity". It is sometimes used in derivations of the polyphase method.
Interpolation filter design.
Let formula_14 be the Fourier transform of any function, formula_15 whose samples at some interval, formula_16 equal the formula_17 sequence. Then the discrete-time Fourier transform (DTFT) of the formula_17 sequence is the Fourier series representation of a periodic summation of formula_18
When formula_19 has units of seconds, formula_20 has units of hertz (Hz). Sampling formula_0 times faster (at interval formula_21) increases the periodicity by a factor of formula_22
which is also the desired result of interpolation. An example of both these distributions is depicted in the first and third graphs of Fig 2.
When the additional samples are inserted zeros, they decrease the sample-interval to formula_23 Omitting the zero-valued terms of the Fourier series, it can be written as:
formula_24
which is equivalent to Eq.2, regardless of the value of formula_9 That equivalence is depicted in the second graph of Fig.2. The only difference is that the available digital bandwidth is expanded to formula_25, which increases the number of periodic spectral images within the new bandwidth. Some authors describe that as new frequency components. The second graph also depicts a lowpass filter and formula_26 resulting in the desired spectral distribution (third graph). The filter's bandwidth is the Nyquist frequency of the original formula_17 sequence. In units of Hz that value is formula_27 but filter design applications usually require normalized units. (see Fig 2, table)
Upsampling by a fractional factor.
Let "L"/"M" denote the upsampling factor, where "L" > "M".
Upsampling requires a lowpass filter after increasing the data rate, and downsampling requires a lowpass filter before decimation. Therefore, both operations can be accomplished by a single filter with the lower of the two cutoff frequencies. For the "L" > "M" case, the interpolation filter cutoff, formula_28 "cycles per intermediate sample", is the lower frequency.
Notes.
<templatestyles src="Reflist/styles.css" />
Page citations.
<templatestyles src="Reflist/styles.css" />
References.
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{
"math_id": 0,
"text": "L"
},
{
"math_id": 1,
"text": "x_L[n],"
},
{
"math_id": 2,
"text": "x[n],"
},
{
"math_id": 3,
"text": "L-1"
},
{
"math_id": 4,
"text": "x_L[n] = x[n]_{\\uparrow L}."
},
{
"math_id": 5,
"text": "h"
},
{
"math_id": 6,
"text": "K"
},
{
"math_id": 7,
"text": "k"
},
{
"math_id": 8,
"text": "h[j+kL]"
},
{
"math_id": 9,
"text": "L."
},
{
"math_id": 10,
"text": "L=2,"
},
{
"math_id": 11,
"text": "x"
},
{
"math_id": 12,
"text": "x_L[n]"
},
{
"math_id": 13,
"text": "y"
},
{
"math_id": 14,
"text": "X(f)"
},
{
"math_id": 15,
"text": "x(t),"
},
{
"math_id": 16,
"text": "T,"
},
{
"math_id": 17,
"text": "x[n]"
},
{
"math_id": 18,
"text": "X(f):"
},
{
"math_id": 19,
"text": "T"
},
{
"math_id": 20,
"text": "f"
},
{
"math_id": 21,
"text": "T/L"
},
{
"math_id": 22,
"text": "L:"
},
{
"math_id": 23,
"text": "T/L."
},
{
"math_id": 24,
"text": "\\sum_{n=0, \\pm L, \\pm 2L,..., \\pm \\infty}{} x(nT/L)\\ e^{-i 2\\pi f nT/L}\n\\quad \\stackrel{m\\ \\triangleq\\ n/L}{\\longrightarrow} \n\\sum_{m=0, \\pm 1, \\pm 2,..., \\pm \\infty}{} x(mT)\\ e^{-i 2\\pi f mT},"
},
{
"math_id": 25,
"text": "L/T"
},
{
"math_id": 26,
"text": "L=3,"
},
{
"math_id": 27,
"text": "\\tfrac{0.5}{T},"
},
{
"math_id": 28,
"text": "\\tfrac{0.5}{L}"
}
]
| https://en.wikipedia.org/wiki?curid=1315510 |
1315845 | Focus (linguistics) | Grammatical category for new or contrastive information
In linguistics, focus (abbreviated FOC) is a grammatical category that conveys which part of the sentence contributes new, non-derivable, or contrastive information. In the English sentence "Mary only insulted BILL", focus is expressed prosodically by a pitch accent on "Bill" which identifies him as the only person whom Mary insulted. By contrast, in the sentence "Mary only INSULTED Bill", the verb "insult" is focused and thus expresses that Mary performed no other actions towards Bill. Focus is a cross-linguistic phenomenon and a major topic in linguistics. Research on focus spans numerous subfields including phonetics, syntax, semantics, pragmatics, and sociolinguistics.
Functional approaches.
Information structure has been described at length by a number of linguists as a grammatical phenomenon. Lexicogrammatical structures that code prominence, or focus, of some information over other information has a particularly significant history dating back to the 19th century. Recent attempts to explain focus phenomena in terms of discourse function, including those by Knud Lambrecht and Talmy Givón, often connect focus with the packaging of new, old, and contrasting information. Lambrecht in particular distinguishes three main types of focus constructions: predicate-focus structure, argument-focus structure, and sentence-focus structure. Focus has also been linked to other more general cognitive processes, including attention orientation.
In such approaches, "contrastive focus" is understood as the coding of information that is contrary to the presuppositions of the interlocutor. The topic–comment model distinguishes between the topic (theme) and what is being said about that topic (the comment, rheme, or focus).
Formalist approaches.
Standard formalist approaches to grammar argue that phonology and semantics cannot exchange information directly ("See Fig. 1"). Therefore, syntactic mechanisms including features and transformations include prosodic information regarding focus that is passed to the semantics and phonology. Focus may be highlighted either prosodically or syntactically or both, depending on the language. In syntax this can be done assigning focus markers, as shown in (1), or by preposing as shown in (2):
(1) I saw [JOHN] f.
(2) [JOHN] f, I saw.
In (1), focus is marked syntactically with the subscripted ‘f’ which is realized phonologically by a nuclear pitch accent. Clefting induces an obligatory intonation break. Therefore, in (2), focus is marked via word order and a nuclear pitch accent.
In English, focus also relates to phonology and has ramifications for how and where suprasegmental information such as rhythm, stress, and intonation is encoded in the grammar, and in particular intonational tunes that mark focus. Speakers can use pitch accents on syllables to indicate what word(s) are in focus. New words are often accented while given words are not. The accented word(s) forms the focus domain. However, not all of the words in a focus domain need be accented. (See for rules on accent placement and focus-marking). The focus domain can be either "broad", as shown in (3), or "narrow", as shown in (4) and (5):
(3) Did you see a grey dog or a cat? I saw [a grey DOG] f.
(4) Did you see a grey dog or a grey cat? I saw a grey [DOG] f.
(5) Did you see a grey dog or a black dog? I saw a [GREY] f dog.
The question/answer paradigm shown in (3)–(5) has been utilized by a variety of theorists to illustrate the range of contexts a sentence containing focus can be used felicitously. Specifically, the question/answer paradigm has been used as a diagnostic for what counts as new information. For example, the focus pattern in (3) would be infelicitous if the question was ‘Did you see a grey dog or a black dog?’.
In (3) and (4), the pitch accent is marked in bold. In (3), the pitch accent is placed on "dog" but the entire noun phrase "a grey dog" is under focus. In (4), the pitch accent is also placed on "dog" but only the noun "dog" is under focus. In (5), pitch accent is placed on "grey" and only the adjective "grey" is under focus.
Historically, generative proposals made focus a feature bound to a single word within a sentence. Chomsky and Halle formulated a Nuclear Stress Rule that proposed there to be a relation between the main stress of a sentence and a single constituent. Since this constituent is prominent sententially in a way that can contrast with lexical stress, this was originally referred to as "nuclear" stress. The purpose of this rule was to capture the intuition that within each sentence, there is one word in particular that is accented more prominently due to its importance – this is said to form the nucleus of that sentence.
Focus was later suggested to be a structural position at the beginning of the sentence (or on the left periphery) in Romance languages such as Italian, as the lexical head of a Focus Phrase (or FP, following the X-bar theory of phrase structure). Jackendoff, Selkirk, Rooth, Krifka, Schwarzschild argue that focus consists of a feature that is assigned to a node in the syntactic representation of a sentence.
Because focus is now widely seen as corresponding between heavy stress, or nuclear pitch accent, this feature is often associated with the phonologically prominent element(s) of a sentence.
Sound structure (phonological and phonetic) studies of focus are not as numerous, as relational language phenomena tend to be of greater interest to syntacticians and semanticists. But this may be changing: a recent study found that not only do focused words and phrases have a higher range of pitch compared to words in the same sentence but that words following the focus in both American English and Mandarin Chinese were lower than normal in pitch and words before a focus are unaffected. The precise usages of focus in natural language are still uncertain. A continuum of possibilities could possibly be defined between precisely enunciated and staccato styles of speech based on variations in pragmatics or timing.
Currently, there are two central themes in research on focus in generative linguistics. First, given what words or expressions are prominent, what is the meaning of some sentence? Rooth, Jacobs, Krifka, and von Stechow claim that there are lexical items and construction specific-rules that refer directly to the notion of focus. Dryer, Kadmon, Marti, Roberts, Schwarzschild, Vallduvi, and Williams argue for accounts in which general principles of discourse explain focus sensitivity. Second, given the meaning and syntax of some sentence, what words or expressions are prominent?
Prominence and meaning.
Focus directly affects the semantics, or meaning, of a sentence. Different ways of pronouncing the sentence affects the meaning, or, what the speaker intends to convey. Focus distinguishes one interpretation of a sentence from other interpretations of the same sentence that do not differ in word order, but may differ in the way in which the words are taken to relate to each other. To see the effects of focus on meaning, consider the following examples:
(6) John only introduced Bill to SUE.
In (6), accent is placed on Sue. There are two readings of (6) – broad focus shown in (7) and narrow focus shown in (8):
(7) John only [introduced Bill to SUE] f.
(8) John only introduced Bill to [SUE] f.
The meaning of (7) can be summarized as "the only thing John did was introduce Bill to Sue". The meaning of (8) can be summarized as "the only person to whom John introduced Bill is Sue".
In both (7) and (8), focus is associated with the focus sensitive expression "only". This is known as association with focus. The class of focus sensitive expressions in which focus can be associated with includes exclusives ("only", "just") non-scalar additives ("merely", "too") scalar additives ("also", "even"), particularlizers ("in particular", "for example"), intensifiers, quantificational adverbs, quantificational determiners, sentential connectives, emotives, counterfactuals, superlatives, negation and generics. It is claimed that focus operators must c-command their focus.
Alternative semantics.
In the alternative semantics approach to focus pioneered by Mats Rooth, each constituent formula_0 has both an ordinary denotation formula_1 and a focus denotation formula_2 which are composed by parallel computations. The ordinary denotation of a sentence is simply whatever denotation it would have in a non-alternative-based system while its focus denotation can be thought of as the set containing all ordinary denotations one could get by substituting the focused constituent for another expression of the same semantic type. For a sentence such as (9), the ordinary denotation will be the proposition which is true iff Mary likes Sue. Its focus denotation will be the set of each propositions such that for some contextually relevant individual 'x', that proposition is true iff Mary likes 'x'.
(9) Mary likes [SUE]f.
In formal terms, the ordinary denotation of (9) will be as shown below:
Focus denotations are computed using the "alternative sets" provided by alternative semantics. In this system, most unfocused items denote the singleton set containing their ordinary denotations.
Focused constituents denote the set of all (contextually relevant) semantic objects of the same type.
In alternative semantics, the primary composition rule is "Pointwise Functional Application". This rule can be thought of as analogous to the cross product.
Applying this rule to example (9) would give the following focus denotation if the only contextually relevant individuals are Sue, Bill, Lisa, and Mary
The focus denotation can be "caught" by focus-sensitive expressions like "only" as well as other covert items such as the squiggle operator.
Structured meanings.
Following Jacobs and Williams, Krifka argues differently. Krifka claims focus partitions the semantics into a background part and focus part, represented by the pair:
formula_13
The logical form of which represented in lambda calculus is:
formula_14
This pair is referred to as a "structured meaning". Structured meanings allow for a compositional semantic approach to sentences that involve single or multiple foci. This approach follows Frege's (1897) Principle of Compositionality: the meaning of a complex expression is determined by the meanings of its parts, and the way in which those parts are combined into structured meanings. Krifka’s structured meaning theory represents focus in a transparent and compositional fashion it encompasses sentences with more than one focus as well as sentences with a single focus. Krifka claims the advantages of structured meanings are twofold: 1) We can access the meaning of an item in focus directly, and 2) Rooth's alternative semantics can be derived from a structured meaning approach but not vice versa. To see Krifka’s approach illustrated, consider the following examples of single focus shown in (10) and multiple foci shown in (11):
(10) John introduced Bill to [SUE] f.
(11) John only introduced [BILL] f to [SUE] f.
Generally, the meaning of (10) can be summarized as "John introduced Bill to Sue and no one else", and the meaning of (11) can be summarized as "the only pair of persons such that John introduced the first to the second is Bill and Sue".
Specifically, the structured meaning of (10) is:
formula_15 where "introd" is the denotation of "introduce", j "John", b "Bill" and s "Sue".
The background part of the structured meaning is; "introd (j, b, x)"; and the focus part is "s".
Through a (modified) form of functional application (or beta reduction), the focus part of (10) and (11) is projected up through the syntax to the sentential level. Importantly, each intermediate level has distinct meaning.
Focus marking.
It has been claimed that "new" information in the discourse is accented while "given" information is not. Generally, the properties of "new" and "given" are referred to as a word's discourse status. Definitions of "new" and "given" vary. Halliday defines "given" as "anaphorically" recoverable, while "new" is defined to be "textually and situationally non-derivable information". To illustrate this point, consider the following discourse in (12) and (13):
(12) Why don’t you have some French TOAST?
(13) I’ve forgotten how to MAKE French toast.
In (13) we note that the verb "make" is not given by the sentence in (12). It is discourse new. Therefore, it is available for accentuation. However, "toast" in (13) is given in (12). Therefore, it is not available for accentuation. As previously mentioned, pitch accenting can relate to focus. Accented words are often said to be in focus or F-marked often represented by F-markers. The relationship between accent placement is mediated through the discourse status of particular syntactic nodes. The percolation of F-markings in a syntactic tree is sensitive to argument structure and head-phrase relations.
Selkirk and accent placement.
Selkirk develops an explicit account of how F-marking propagates up syntactic trees. Accenting indicates F-marking. F-marking projects up a given syntactic tree such that both lexical items, i.e. terminal nodes and phrasal levels, i.e. nonterminal nodes, can be F-marked. Specifically, a set of rules determines how and where F-marking occurs in the syntax. These rules are shown in (1) and (2):
(14) Basic Rule: An accented word is f-marked.
(15) Focus Projection:
a. F-marking the head of a phrase licenses F-marking of the phrase.
b. F-marking of the internal argument of a head licenses the F-marking of the head.
c. F-marking of the antecedent of a trace left by NP or wh-movement licenses F-marking of the trace.
To see how (14) and (15) apply, consider the following example:
Judy f [adopted f a parrot f] f] foc
Because there is no rule in (14) or (15) that licenses F-marking to the direct object from any other node, the direct object "parrot" must be accented as indicated in bold. Rule (15b) allows F-marking to project from the direct object to the head verb "adopted". Rule (15a) allows F-marking to project from the head verb to the VP "adopted a parrot". Selkirk assumes the subject "Judy" is accented if F-marked as indicated in bold.
Schwarzschild and accent placement.
Schwarzschild points out weaknesses in Selkirk’s ability to predict accent placement based on facts about the discourse. Selkirk’s theory says nothing about how accentuation arises in sentences with entirely old information. She does not fully articulate the notion of discourse status and its relation to accent marking. Schwarzschild differs from Selkirk in that he develops a more robust model of discourse status. Discourse status is determined via the entailments of the context. This is achieved through the definition in (16):
(16) Definition of given: An utterance of U counts as given if it has a salient antecedent A and
a. if U is type e, then A and U corefer;
b. otherwise: modulo formula_16-type-shifting, A entails the existential F-closure of U.
The operation in (16b) can apply to any constituent. formula_16-type-shifting "is a way of transforming syntactic constituents into full propositions so that it is possible to check whether they are entailed by the context". For example, the result of formula_16-type-shifting the VP in (17) is (18):
(17) [hums a happy tune]
(18) formula_16"x"["x" hums a happy tune]
Note that (18) is a full proposition. The existential F-closure in (16b) refers to the operation of replacing the highest F-marked node with an existentially closed variable. The operation is shown in (19) and (20):
(19) formula_16"x"["x" hums [a happy f tune f] f]
(20) formula_16"Y"formula_16"x"["x" hums "Y"]
Given the discourse context in (21a) it is possible to determine the discourse status of any syntactic node in (21b):
(21)
a. Sean [hummed a happy tune] VP
b. Angie [hummed [Chopin’s Funeral March] f] VP
If the VP in (21a) is the salient antecedent for the VP in (21b), then the VP in (21b) counts as given. formula_16-type-shifed VP in (21a) is shown in (22). The existential F-closure of the VP in (21b) is shown in (23):
(22) formula_16"x"["x" hums a happy tune]
(23) formula_16"Y"formula_16"x"["x" hums "Y"]
(22) entails (23). Therefore, the VP of (21b) counts as given. Schwarzschild assumes an optimality theoretic grammar. Accent placement is determined by a set of violable, hierarchically ranked constraints as shown in (24):
(24)
a. GIVENness: A constituent that is not F-marked is given.
b. Foc: A Foc-marked phrase contains an accent
c. AvoidF: Do not F-mark
d. HeadArg: A head is less prominent than its internal argument.
The ranking Schwarzschild proposes is seen in (25):
(25) GIVENness, Foc » AvoidF » HeadArg
As seen, GIVENness relates F-marking to discourse status. Foc relates F-marking to accent placement. Foc simply requires that a constituent(s) of an F-marked phrase contain an accent. AvoidF states that less F-marking is preferable to more F-marking. HeadArg encodes the head-argument asymmetry into the grammar directly.
Responses.
Recent empirical work by German "et al." suggests that both Selkirk’s and Schwarzschild’s theory of accentuation and F-marking makes incorrect predictions. Consider the following context:
(26) Are the children playing their game?
(27) Paul took down their tent that they play their game in.
It has been noted that prepositions are intrinsically weak and do not readily take accent. However, both Selkirk and Schwarzschild predict that in the narrow focus context, an accent will occur at most on the preposition in (27) as shown in (28):
(28) Paul took down their tent that they [play their game [in f t f] foc].
However, the production experiment reported in German "et al." showed that subjects are more likely to accent verbs or nouns as opposed to prepositions in the narrow focused context, thus ruling out accent patterns shown in (28). German "et al." argue for a stochastic constraint-based grammar similar to Anttila and Boersma that more fluidly accounts for how speakers accent words in discourse.
Notes.
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References.
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"text": "[\\![\\alpha]\\!]_f"
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"math_id": 3,
"text": " [\\![\\text{Mary likes SUE}]\\!]_o = \\textit{like}(\\textit{Mary})(\\textit{Sue})"
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"math_id": 4,
"text": " [\\![\\text{Mary}]\\!]_f = \\{Mary\\}"
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{
"math_id": 5,
"text": " [\\![\\text{likes}]\\!]_f = \\{ \\lambda x \\, . \\, \\lambda y \\, . \\, likes(y)(x) \\}"
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"math_id": 6,
"text": "\\{like(Mary)(y) |y \\in E\\}"
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"text": "\\beta"
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"text": "\\gamma"
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"text": " \\langle \\sigma, \\tau \\rangle"
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"math_id": 10,
"text": "\\sigma"
},
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"text": "[\\![\\alpha]\\!] = \\{ f(x) | f \\in [\\![\\beta]\\!], x \\in [\\![\\gamma]\\!] \\}"
},
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"math_id": 12,
"text": " [\\![\\text{Mary likes SUE}]\\!]_f = \\{like(Mary)(Sue), \\, like(Mary)(Bill), \\, like(Mary)(Lisa), \\, like(Mary)(Mary)\\}"
},
{
"math_id": 13,
"text": "\\langle B,F\\rangle"
},
{
"math_id": 14,
"text": "\\langle \\lambda x.x, A\\rangle"
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{
"math_id": 15,
"text": "\\langle introd(j, b, x), s\\rangle"
},
{
"math_id": 16,
"text": "\\exists"
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| https://en.wikipedia.org/wiki?curid=1315845 |
13160226 | Breather surface | Surface of constant negative curvature
In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature formula_0, where the curvature is well-defined. This makes them examples of generalized pseudospheres.
Mathematical background.
There is a correspondence between embedded surfaces of constant curvature -1, known as pseudospheres, and solutions to the sine-Gordon equation. This correspondence can be built starting with the simplest example of a pseudosphere, the tractroid. In a special set of coordinates, known as asymptotic coordinates, the Gauss–Codazzi equations, which are consistency equations dictating when a surface of prescribed first and second fundamental form can be embedded into three-dimensional space with the flat metric, reduce to the sine-Gordon equation.
In the correspondence, the tractroid corresponds to the static 1-soliton solution of the sine-Gordon solution. Due to the Lorentz invariance of sine-Gordon, a one-parameter family of Lorentz boosts can be applied to the static solution to obtain new solutions: on the pseudosphere side, these are known as "Lie transformations", which deform the tractroid to the one-parameter family of surfaces known as Dini's surfaces.
The method of Bäcklund transformation allows the construction of a large number of distinct solutions to the sine-Gordon equation, the multi-soliton solutions. For example, the 2-soliton corresponds to the Kuen surface. However, while this generates an infinite family of solutions, the breather solutions are not among them.
Breather solutions are instead derived from the inverse scattering method for the sine-Gordon equation. They are localized in space but oscillate in time.
Each solution to the sine-Gordon equation gives a first and second fundamental form which satisfy the Gauss-Codazzi equations. The fundamental theorem of surface theory then guarantees that there is a parameterized surface which recovers the prescribed first and second fundamental forms. Locally the parameterization is well-behaved, but extended arbitrarily the resulting surface may have self-intersections and cusps. Indeed, a theorem of Hilbert says that any pseudosphere cannot be embedded regularly (roughly, meaning without cusps) into formula_1.
Parameterization.
The parameterization formula_2 with parameter formula_3 is given by
formula_4
References.
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"text": "-1"
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"math_id": 1,
"text": "\\mathbb{R}^3"
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"math_id": 2,
"text": "\\sigma: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^3; (u,v) \\mapsto (x,y,z)"
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{
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"text": "0 < a < 1"
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"text": "\\begin{align}\nx & {} = -u+\\frac{2\\left(1-a^2\\right)\\cosh(au)\\sinh(au)}{a\\left(\\left(1-a^2\\right)\\cosh^2(au)+a^2\\,\\sin^2\\left(\\sqrt{1-a^2}v\\right)\\right)} \\\\ \\\\\ny & {} = \\frac{2\\sqrt{1-a^2}\\cosh(au)\\left(-\\sqrt{1-a^2}\\cos(v)\\cos\\left(\\sqrt{1-a^2}v\\right)-\\sin(v)\\sin\\left(\\sqrt{1-a^2}v\\right)\\right)}{a\\left(\\left(1-a^2\\right)\\cosh^2(au)+a^2\\,\\sin^2\\left(\\sqrt{1-a^2}v\\right)\\right)} \\\\ \\\\\nz & {} = \\frac{2\\sqrt{1-a^2}\\cosh(au)\\left(-\\sqrt{1-a^2}\\sin(v)\\cos\\left(\\sqrt{1-a^2}v\\right)+\\cos(v)\\sin\\left(\\sqrt{1-a^2}v\\right)\\right)}{a\\left(\\left(1-a^2\\right)\\cosh^2(au)+a^2\\,\\sin^2\\left(\\sqrt{1-a^2}v\\right)\\right)}\n\\end{align}"
}
]
| https://en.wikipedia.org/wiki?curid=13160226 |
13162 | Gelatin dessert | Dessert made with gelatin
Gelatin desserts are desserts made with a sweetened and flavoured processed collagen product (gelatin), which makes the dessert "set" from a liquid to a soft elastic solid gel. This kind of dessert was first recorded as "jelly" by Hannah Glasse in her 18th-century book "The Art of Cookery", appearing in a layer of trifle. Jelly recipes are included in the 19th-century cookbooks of English food writers Eliza Acton and Mrs Beeton.
Jelly can be made by combining plain gelatin with other ingredients or by using a premixed blend of gelatin with additives. Fully prepared gelatin desserts are sold in a variety of forms, ranging from large decorative shapes to individual serving cups.
Popular brands of premixed gelatin include: Aeroplane Jelly in Australia, Hartley's (formerly Rowntree's) in the United Kingdom, and Jell-O from Kraft Foods and Royal from Jel Sert in North America. In the United States and Canada, this dessert is known by the genericized trademark "jello".
History.
Before gelatin became widely available as a commercial product, the most typical gelatin dessert was "calf's foot jelly". As the name indicates, this was made by extracting and purifying gelatin from the foot of a calf. This gelatin was used for savory dishes in aspic, or was mixed with fruit juice and sugar for a dessert.
In the eighteenth century, gelatin from calf's feet, isinglass and hartshorn was colored blue with violet juice, yellow with saffron, red with cochineal and green with spinach and allowed to set in layers in small, narrow glasses. It was flavored with sugar, lemon juice and mixed spices. This preparation was called "jelly"; English cookery writer Hannah Glasse was the first to record the use of this jelly in trifle in her book "The Art of Cookery", first published in 1747. Preparations on making jelly (including illustrations) appear in the best selling cookbooks of English writers Eliza Acton and Isabella Beeton in the 19th century.
Due to the time-consuming nature of extracting gelatin from animal bones, gelatin desserts were a status symbol up until the mid-19th century as it indicated a large kitchen staff. Jelly molds were very common in the batteries de cuisine of stately homes.
Preparation.
To make a gelatin dessert, gelatin is dissolved in hot liquid with the desired flavors and other additives. These latter ingredients usually include sugar, fruit juice, or sugar substitutes; they may be added and varied during preparation, or pre-mixed with the gelatin in a commercial product which mainly requires the addition of hot water.
In addition to sweeteners, the prepared commercial blends generally contain flavoring agents and other additives, such as adipic acid, fumaric acid, sodium citrate, and artificial flavorings and food colors. Because the collagen is processed extensively, the final product is not categorized as a meat or animal product by the US federal government.
Prepared commercial blends may be sold as a powder or as a concentrated gelatinous block, divided into small squares. Either type is mixed with sufficient hot water to completely dissolve it, and then mixed with enough cold water to make the volume of liquid specified on the packet.
The solubility of powdered gelatin can be enhanced by sprinkling it into the liquid several minutes before heating, "blooming" the individual granules. The fully dissolved mixture is then refrigerated, slowly forming a colloidal gel as it cools.
Gelatin desserts may be enhanced in many ways, such as using decorative molds, creating multicolored layers by adding a new layer of slightly cooled liquid over the previously solidified one, or suspending non-soluble edible elements such as marshmallows or fruit. Some types of fresh fruit and their unprocessed juices are incompatible with gelatin desserts; see the Chemistry section below.
When fully chilled, the most common ratios of gelatin to liquid (as instructed on commercial packaging) usually result in a custard-like texture which can retain detailed shapes when cold but melts back to a viscous liquid when warm. A recipe calling for the addition of additional gelatin to regular jelly gives a rubbery product that can be cut into shapes with cookie cutters and eaten with fingers (called "Knox Blox" by the Knox company, makers of unflavored gelatin). Higher gelatin ratios can be used to increase the stability of the gel, culminating in gummy candies which remain rubbery solids at room temperature (see Bloom (test)).
The bloom strength of a gelatin mixture is the measure of how strong it is. It is defined by the force in grams required to press a diameter plunger into of a standard 6.67% w/v gelatin gel at . The bloom strength of a gel is useful to know when determining the possibility of substituting a gelatin of one bloom strength for a gelatin of another. One can use the following equation:
formula_0
or
formula_1
where formula_2 is the concentration, formula_3 is the bloom strength, formula_4 is a constant. For example, when making gummies, it is important to know that a 250 bloom gelatin has a much shorter (more thick) texture than a 180 bloom gelatin.
Gelatin shots.
A gelatin shot (usually called a Jell-O shot in North America and vodka jelly or jelly shot in the UK and Australia) is a shooter in which one or more liquors, usually vodka, rum, tequila, or neutral grain spirit, replaces some of the water or fruit juice that is used to congeal the gel.
The American satirist and mathematician Tom Lehrer claims to have invented the gelatin shot in the 1950s while working for the National Security Agency, where he developed vodka gelatin as a way to circumvent a restriction of alcoholic beverages on base. An early published recipe for an alcoholic gelatin drink dates from 1862, found in "How to Mix Drinks, or The Bon Vivant's Companion" by Jerry Thomas: his recipe for "Punch Jelly" calls for the addition of isinglass or other gelatin to a punch made from cognac, rum, and lemon juice.
Gelatin art desserts.
Gelatin art desserts, also known as 3D gelatin desserts, are made by injecting colorful shapes into a flavored gelatin base. Creations by Lourdes Reyes Rosas of Mexico City kicked off the growth in popularity of this 3D gelation art technique in the early 1990s, which spread to Western and Pacific countries.
These desserts are made using high quality gelatin that has a high bloom value and low odor and taste. The clear gelatin base is prepared using gelatin, water, sugar, citric acid and food flavoring.
When the clear gelatin base sets, colorful shapes are injected using a syringe.
The injected material usually consists of a sweetener (most commonly sugar), some type of edible liquid (milk, cream, water, etc.), food coloring and a thickening agent such as starch or additional gelatin.
The shapes are drawn by making incisions in the clear gelatin base using sharp objects. Colored liquid is then allowed to fill the crevice and make the cut shape visible.
Most commonly, the shapes are drawn using sterile medical needles or specialized precut gelatin art tools that allow the shape to be cut and filled with color at the same time.
Gelatin art tools are attached to a syringe and used to inject a predetermined shape into gelatin.
When combined with other ingredients, such as whipping cream or mousse, gelatin art desserts can be assembled into visually impressive formations resembling a cake.
Gelatin substitutes.
Other culinary gelling agents can be used instead of animal-derived gelatin. These plant-derived substances are more similar to pectin and other gelling plant carbohydrates than to gelatin proteins; their physical properties are slightly different, creating different constraints for the preparation and storage conditions. These other gelling agents may also be preferred for certain traditional cuisines or dietary restrictions.
Agar, a product made from red algae, is the traditional gelling agent in many Asian desserts. Agar is a popular gelatin substitute in quick jelly powder mix and prepared dessert gels that can be stored at room temperature. Compared to gelatin, agar preparations require a higher dissolving temperature, but the resulting gels congeal more quickly and remain solid at higher temperatures, , as opposed to for gelatin. Vegans and vegetarians can use agar to replace animal-derived gelatin.
Another common seaweed-based gelatin substitute is carrageenan, which has been used as a food additive since ancient times. It was first industrially-produced in the Philippines, which pioneered the cultivation of tropical red seaweed species (primarily "Eucheuma" and "Kappaphycus" spp.) from where carrageenan is extracted. The Philippines produces 80% of the world's carrageenan supply. Carrageenan gelatin substitute are traditionally known as gulaman in the Philippines. It is widely used in various traditional desserts and are sold as dried bars or in powder form. Unlike gelatin, gulaman sets at room temperature and is uniquely thermo-reversible. If melted at higher temperatures, it can revert to its original shape once cooled down. Carrageenan jelly also sets more firmly than agar and lacks agar's occasionally unpleasant smell during cooking. The use of carrageenan as a gelatin substitute has spread to other parts of the world, particularly in cuisines with dietary restrictions against gelatin, like kosher and halal cooking. It has also been used in prepackaged Jello shots to make them shelf stable at room temperatures.
Konjac is a gelling agent used in many Asian foods, including the popular konnyaku fruit jelly candies.
Chemistry.
Gelatin consists of partially hydrolyzed collagen, a protein which is highly abundant in animal tissues such as bone and skin. Collagen is a protein made up of three strands of polypeptide chains that form in a helical structure. To make a gelatin dessert, such as Jello, the collagen is mixed with water and heated, disrupting the bonds that hold the three strands of polypeptides together. As the gelatin cools, these bonds try to reform in the same structure as before, but now with small bubbles of liquid in between. This gives gelatin its semisolid, gel-like texture.
Because gelatin is a protein that contains both acid and base amino groups, it acts as an amphoteric molecule, displaying both acidic and basic properties. This allows it to react with different compounds, such as sugars and other food additives. These interactions give gelatin a versatile nature in the roles that it plays in different foods. It can stabilize foams in foods such as marshmallows, it can help maintain small ice crystals in ice cream, and it can even serve as an emulsifier for foods like toffee and margarine.
Although many gelatin desserts incorporate fruit, some fresh fruits contain proteolytic enzymes; these enzymes cut the gelatin molecule into peptides (protein fragments) too small to form a firm gel. The use of such fresh fruits in a gelatin recipe results in a dessert that never "sets".
Specifically, pineapple contains the protease (protein cutting enzyme) bromelain, kiwifruit contains actinidin, figs contain ficain, and papaya contains papain. Cooking or canning denatures and deactivates the proteases, so canned pineapple, for example, works fine in a gelatin dessert.
Legal definitions and regulations.
China.
Gelatin dessert in China is defined as edible jelly-like food prepared from a mixture of water, sugar and gelling agent. The preparation processes include concocting, gelling, sterilizing and packaging. In China, around 250 legal additives are allowed in gelatin desserts as gelling agents, colors, artificial sweeteners, emulsifiers and antioxidants.
Gelatin desserts are classified into 5 categories according to the different flavoring substances they contain. Five types of flavoring substance include artificial fruit flavored type (less than 15% of natural fruit juice), natural fruit flavored type (above 15% of natural fruit juice), natural flavored with fruit pulp type and dairy type products, which includes dairy ingredients. The last type ("others") summarizes gelatin desserts not mentioned above. It is typically sold in single-serving plastic cups or plastic food bags.
Safety.
Although eating tainted beef can lead to New Variant Creutzfeldt–Jakob disease (the human variant of mad-cow disease, bovine spongiform encephalopathy), there is no known case of BSE having been transmitted through collagen products such as gelatin.
References.
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"text": "C_1(B_1)^{\\frac12} / (B_2)^{\\frac12} = C_2"
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| https://en.wikipedia.org/wiki?curid=13162 |
13165796 | Ocean heat content | Energy stored by oceans
Ocean heat content (OHC) or ocean heat uptake (OHU) is the energy absorbed and stored by oceans. To calculate the ocean heat content, it is necessary to measure ocean temperature at many different locations and depths. Integrating the areal density of a change in enthalpic energy over an ocean basin or entire ocean gives the total ocean heat uptake. Between 1971 and 2018, the rise in ocean heat content accounted for over 90% of Earth's excess energy from global heating. The main driver of this increase was caused by humans via their rising greenhouse gas emissions. By 2020, about one third of the added energy had propagated to depths below 700 meters.
In 2023, the world's oceans were again the hottest in the historical record and exceeded the previous 2022 record maximum. The five highest ocean heat observations to a depth of 2000 meters occurred in the period 2019–2023. The North Pacific, North Atlantic, the Mediterranean, and the Southern Ocean all recorded their highest heat observations for more than sixty years of global measurements. Ocean heat content and sea level rise are important indicators of climate change.
Ocean water can absorb a lot of solar energy because water has far greater heat capacity than atmospheric gases. As a result, the top few meters of the ocean contain more energy than the entire Earth's atmosphere. Since before 1960, research vessels and stations have sampled sea surface temperatures and temperatures at greater depth all over the world. Since 2000, an expanding network of nearly 4000 Argo robotic floats has measured temperature anomalies, or the change in ocean heat content. With improving observation in recent decades, the heat content of the upper ocean has been analyzed to have increased at an accelerating rate. The net rate of change in the top 2000 meters from 2003 to 2018 was (or annual mean energy gain of 9.3 zettajoules). It is difficult to measure temperatures accurately over long periods while at the same time covering enough areas and depths. This explains the uncertainty in the figures.
Changes in ocean temperature greatly affect ecosystems in oceans and on land. For example, there are multiple impacts on coastal ecosystems and communities relying on their ecosystem services. Direct effects include variations in sea level and sea ice, changes to the intensity of the water cycle, and the migration of marine life.
Calculations.
Definition.
Ocean heat content is a term used in physical oceanography to describe a type of energy that is stored in the ocean. It is defined in coordination with a particular formulation of the thermodynamic equation of state of seawater. TEOS-10 is an international standard approved in 2010 by the Intergovernmental Oceanographic Commission.
Calculation of ocean heat content is closely aligned with that of enthalpy at an ocean surface, also called potential enthalpy. OHC changes are thus made more readily comparable to seawater heat exchanges with ice, freshwater, and humid air. OHC is always reported as a change or as an "anomaly" relative to a baseline. Positive values then also quantify ocean heat uptake (OHU) and are useful to diagnose where most of planetary energy gains from global heating are going.
To calculate the ocean heat content, measurements of ocean temperature from sample parcels of seawater gathered at many different locations and depths are required. Integrating the areal density of ocean heat over an ocean basin, or entire ocean, gives the total ocean heat content. Thus, total ocean heat content is a volume integral of the product of temperature, density, and heat capacity over the three-dimensional region of the ocean for which data is available. The bulk of measurements have been performed at depths shallower than about 2000 m (1.25 miles).
The areal density of ocean heat content between two depths is computed as a definite integral:
formula_0
where formula_1 is the specific heat capacity of sea water, h2 is the lower depth, h1 is the upper depth, formula_2 is the in-situ seawater density profile, and formula_3 is the conservative temperature profile. formula_1 is defined at a single depth h0 usually chosen as the ocean surface. In SI units, formula_4 has units of Joules per square metre (J·m−2).
In practice, the integral can be approximated by summation using a smooth and otherwise well-behaved sequence of in-situ data; including temperature (t), pressure (p), salinity (s) and their corresponding density (ρ). Conservative temperature formula_3 are translated values relative to the reference pressure (p0) at h0. A substitute known as potential temperature has been used in earlier calculations.
Measurements of temperature versus ocean depth generally show an upper mixed layer (0–200 m), a thermocline (200–1500 m), and a deep ocean layer (>1500 m). These boundary depths are only rough approximations. Sunlight penetrates to a maximum depth of about 200 m; the top 80 m of which is the habitable zone for photosynthetic marine life covering over 70% of Earth's surface. Wave action and other surface turbulence help to equalize temperatures throughout the upper layer.
Unlike surface temperatures which decrease with latitude, deep-ocean temperatures are relatively cold and uniform in most regions of the world. About 50% of all ocean volume is at depths below 3000 m (1.85 miles), with the Pacific Ocean being the largest and deepest of five oceanic divisions. The thermocline is the transition between upper and deep layers in terms of temperature, nutrient flows, abundance of life, and other properties. It is semi-permanent in the tropics, variable in temperate regions (often deepest during the summer), and shallow to nonexistent in polar regions.
Measurements.
Ocean heat content measurements come with difficulties, especially before the deployment of the Argo profiling floats. Due to poor spatial coverage and poor quality of data, it has not always been easy to distinguish between long term global warming trends and climate variability. Examples of these complicating factors are the variations caused by El Niño–Southern Oscillation or changes in ocean heat content caused by major volcanic eruptions.
Argo is an international program of robotic profiling floats deployed globally since the start of the 21st century. The program's initial 3000 units had expanded to nearly 4000 units by year 2020. At the start of each 10-day measurement cycle, a float descends to a depth of 1000 meters and drifts with the current there for nine days. It then descends to 2000 meters and measures temperature, salinity (conductivity), and depth (pressure) over a final day of ascent to the surface. At the surface the float transmits the depth profile and horizontal position data through satellite relays before repeating the cycle.
Starting 1992, the TOPEX/Poseidon and subsequent Jason satellite series altimeters have observed vertically integrated OHC, which is a major component of sea level rise. Since 2002, GRACE and GRACE-FO have remotely monitored ocean changes using gravimetry. The partnership between Argo and satellite measurements has thereby yielded ongoing improvements to estimates of OHC and other global ocean properties.
Causes for heat uptake.
Ocean heat uptake accounts for over 90% of total planetary heat uptake, mainly as a consequence of human-caused changes to the composition of Earth's atmosphere. This high percentage is because waters at and below the ocean surface - especially the turbulent upper mixed layer - exhibit a thermal inertia much larger than the planet's exposed continental crust, ice-covered polar regions, or atmospheric components themselves. A body with large thermal inertia stores a big amount of energy because of its volumetric heat capacity, and effectively transmits energy according to its heat transfer coefficient. Most extra energy that enters the planet via the atmosphere is thereby taken up and retained by the ocean.
Planetary heat uptake or heat content accounts for the entire energy added to or removed from the climate system. It can be computed as an accumulation over time of the observed differences (or imbalances) between total incoming and outgoing radiation.
Changes to the imbalance have been estimated from Earth orbit by CERES and other remote instruments, and compared against in-situ surveys of heat inventory changes in oceans, land, ice and the atmosphere. Achieving complete and accurate results from either accounting method is challenging, but in different ways that are viewed by researchers as being mostly independent of each other. Increases in planetary heat content for the well-observed 2005-2019 period are thought to exceed measurement uncertainties.
From the ocean perspective, the more abundant equatorial solar irradiance is directly absorbed by Earth's tropical surface waters and drives the overall poleward propagation of heat. The surface also exchanges energy that has been absorbed by the lower troposphere through wind and wave action. Over time, a sustained imbalance in Earth's energy budget enables a net flow of heat either into or out of greater ocean depth via thermal conduction, downwelling, and upwelling. Releases of OHC to the atmosphere occur primarily via evaporation and enable the planetary water cycle. Concentrated releases in association with high sea surface temperatures help drive tropical cyclones, atmospheric rivers, atmospheric heat waves and other extreme weather events that can penetrate far inland. Altogether these processes enable the ocean to be Earth's largest thermal reservoir which functions to regulate the planet's climate; acting as both a sink and a source of energy.
From the perspective of land and ice covered regions, their portion of heat uptake is reduced and delayed by the dominant thermal inertia of the ocean. Although the average rise in land surface temperature has exceeded the ocean surface due to the lower inertia (smaller heat-transfer coefficient) of solid land and ice, temperatures would rise more rapidly and by a greater amount without the full ocean. Measurements of how rapidly the heat mixes into the deep ocean have also been underway to better close the ocean and planetary energy budgets.
Recent observations and changes.
Numerous independent studies in recent years have found a multi-decadal rise in OHC of upper ocean regions that has begun to penetrate to deeper regions. The upper ocean (0–700 m) has warmed since 1971, while it is very likely that warming has occurred at intermediate depths (700–2000 m) and likely that deep ocean (below 2000 m) temperatures have increased. The heat uptake results from a persistent warming imbalance in Earth's energy budget that is most fundamentally caused by the anthropogenic increase in atmospheric greenhouse gases. There is very high confidence that increased ocean heat content in response to anthropogenic carbon dioxide emissions is essentially irreversible on human time scales.
Studies based on Argo measurements indicate that ocean surface winds, especially the subtropical trade winds in the Pacific Ocean, change ocean heat vertical distribution. This results in changes among ocean currents, and an increase of the subtropical overturning, which is also related to the El Niño and La Niña phenomenon. Depending on stochastic natural variability fluctuations, during La Niña years around 30% more heat from the upper ocean layer is transported into the deeper ocean. Furthermore, studies have shown that approximately one-third of the observed warming in the ocean is taking place in the 700-2000 meter ocean layer.
Model studies indicate that ocean currents transport more heat into deeper layers during La Niña years, following changes in wind circulation. Years with increased ocean heat uptake have been associated with negative phases of the interdecadal Pacific oscillation (IPO). This is of particular interest to climate scientists who use the data to estimate the "ocean heat uptake".
The upper ocean heat content in most North Atlantic regions is dominated by heat transport convergence (a location where ocean currents meet), without large changes to temperature and salinity relation. Additionally, a study from 2022 on anthropogenic warming in the ocean indicates that 62% of the warming from the years between 1850 and 2018 in the North Atlantic along 25°N is kept in the water below 700 m, where a major percentage of the ocean's surplus heat is stored.
A study in 2015 concluded that ocean heat content increases by the Pacific Ocean were compensated by an abrupt distribution of OHC into the Indian Ocean.
Although the upper 2000 m of the oceans have experienced warming on average since the 1970s, the rate of ocean warming varies regionally with the subpolar North Atlantic warming more slowly and the Southern Ocean taking up a disproportionate large amount of heat due to anthropogenic greenhouse gas emissions.
Deep-ocean warming below 2000 m has been largest in the Southern Ocean compared to other ocean basins.
Impacts.
Warming oceans are one reason for coral bleaching and contribute to the migration of marine species. Marine heat waves are regions of life-threatening and persistently elevated water temperatures. Redistribution of the planet's internal energy by atmospheric circulation and ocean currents produces internal climate variability, often in the form of irregular oscillations, and helps to sustain the global thermohaline circulation.
The increase in OHC accounts for 30–40% of global sea-level rise from 1900 to 2020 because of thermal expansion.
It is also an accelerator of sea ice, iceberg, and tidewater glacier melting. The ice loss reduces polar albedo, amplifying both the regional and global energy imbalances.
The resulting ice retreat has been rapid and widespread for Arctic sea ice, and within northern fjords such as those of Greenland and Canada.
Impacts to Antarctic sea ice and the vast Antarctic ice shelves which terminate into the Southern Ocean have varied by region and are also increasing due to warming waters. Breakup of the Thwaites Ice Shelf and its West Antarctica neighbors contributed about 10% of sea-level rise in 2020.
The ocean also functions as a sink and source of carbon, with a role comparable to that of land regions in Earth's carbon cycle. In accordance with the temperature dependence of Henry's law, warming surface waters are less able to absorb atmospheric gases including oxygen and the growing emissions of carbon dioxide and other greenhouse gases from human activity. Nevertheless the rate in which the ocean absorbs anthropogenic carbon dioxide has approximately tripled from the early 1960s to the late 2010s; a scaling proportional to the increase in atmospheric carbon dioxide.
Warming of the deep ocean has the further potential to melt and release some of the vast store of frozen methane hydrate deposits that have naturally accumulated there.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "H= c_p \\int_{h2}^{h1} \\rho(z) \\Theta(z) dz"
},
{
"math_id": 1,
"text": "c_p "
},
{
"math_id": 2,
"text": "\\rho(z)"
},
{
"math_id": 3,
"text": "\\Theta(z)"
},
{
"math_id": 4,
"text": "H"
}
]
| https://en.wikipedia.org/wiki?curid=13165796 |
1316648 | Constructive dilemma | Rule of inference of propositional logic
Constructive dilemma is a valid rule of inference of propositional logic. It is the inference that, if "P" implies "Q" and "R" implies "S" and either "P" or "R" is true, then either "Q or S" has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too. "Constructive dilemma" is the disjunctive version of modus ponens, whereas,
destructive dilemma is the disjunctive version of "modus tollens". The constructive dilemma rule can be stated:
formula_4
where the rule is that whenever instances of "formula_5", "formula_6", and "formula_7" appear on lines of a proof, "formula_8" can be placed on a subsequent line.
Formal notation.
The "constructive dilemma" rule may be written in sequent notation:
formula_9
where formula_10 is a metalogical symbol meaning that formula_8 is a syntactic consequence of formula_5, formula_6, and formula_7 in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
formula_11
where formula_0, formula_1, formula_2 and formula_3 are propositions expressed in some formal system.
If I win a million dollars, I will donate it to an orphanage.
If my friend wins a million dollars, he will donate it to a wildlife fund.
Either I win a million dollars or my friend wins a million dollars.
Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.
Natural language example.
The dilemma derives its name because of the transfer of disjunctive operator.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "P"
},
{
"math_id": 1,
"text": "Q"
},
{
"math_id": 2,
"text": "R"
},
{
"math_id": 3,
"text": "S"
},
{
"math_id": 4,
"text": "\\frac{(P \\to Q), (R \\to S), P \\lor R}{\\therefore Q \\lor S}"
},
{
"math_id": 5,
"text": "P \\to Q"
},
{
"math_id": 6,
"text": "R \\to S"
},
{
"math_id": 7,
"text": "P \\lor R"
},
{
"math_id": 8,
"text": "Q \\lor S"
},
{
"math_id": 9,
"text": "(P \\to Q), (R \\to S), (P \\lor R) \\vdash (Q \\lor S)"
},
{
"math_id": 10,
"text": "\\vdash"
},
{
"math_id": 11,
"text": "(((P \\to Q) \\land (R \\to S)) \\land (P \\lor R)) \\to (Q \\lor S)"
}
]
| https://en.wikipedia.org/wiki?curid=1316648 |
1316878 | Destructive dilemma | Rule of inference of propositional logic
Destructive dilemma is the name of a valid rule of inference of propositional logic. It is the inference that, if "P" implies "Q" and "R" implies "S" and either "Q" is false or "S" is false, then either "P" or "R" must be false. In sum, if two conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. "Destructive dilemma" is the disjunctive version of "modus tollens". The disjunctive version of "modus ponens" is the constructive dilemma. The destructive dilemma rule can be stated:
formula_4
where the rule is that wherever instances of "formula_5", "formula_6", and "formula_7" appear on lines of a proof, "formula_8" can be placed on a subsequent line.
Formal notation.
The "destructive dilemma" rule may be written in sequent notation:
formula_9
where formula_10 is a metalogical symbol meaning that formula_11 is a syntactic consequence of formula_5, formula_6, and formula_7 in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
formula_12
where formula_0, formula_1, formula_2 and formula_3 are propositions expressed in some formal system.
If it rains, we will stay inside.
If it is sunny, we will go for a walk.
Either we will not stay inside, or we will not go for a walk, or both.
Therefore, either it will not rain, or it will not be sunny, or both.
Example proof.
The validity of this argument structure can be shown by using both conditional proof (CP) and reductio ad absurdum (RAA) in the following way:
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "P"
},
{
"math_id": 1,
"text": "Q"
},
{
"math_id": 2,
"text": "R"
},
{
"math_id": 3,
"text": "S"
},
{
"math_id": 4,
"text": "\\frac{P \\to Q, R \\to S, \\neg Q \\lor \\neg S}{\\therefore \\neg P \\lor \\neg R}"
},
{
"math_id": 5,
"text": "P \\to Q"
},
{
"math_id": 6,
"text": "R \\to S"
},
{
"math_id": 7,
"text": "\\neg Q \\lor \\neg S"
},
{
"math_id": 8,
"text": "\\neg P \\lor \\neg R"
},
{
"math_id": 9,
"text": "(P \\to Q), (R \\to S), (\\neg Q \\lor \\neg S) \\vdash (\\neg P \\lor \\neg R)"
},
{
"math_id": 10,
"text": "\\vdash"
},
{
"math_id": 11,
"text": "\\neg P \\lor \\neg R"
},
{
"math_id": 12,
"text": "(((P \\to Q) \\land (R \\to S)) \\land (\\neg Q \\lor \\neg S)) \\to (\\neg P \\lor \\neg R)"
}
]
| https://en.wikipedia.org/wiki?curid=1316878 |
1317 | Antimatter | Material composed of antiparticles of the corresponding particles of ordinary matter
In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter, and can be thought of as matter with reversed charge, parity, and time, known as CPT reversal. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioactive decay, but only a tiny fraction of these have successfully been bound together in experiments to form antiatoms. Minuscule numbers of antiparticles can be generated at particle accelerators; however, total artificial production has been only a few nanograms. No macroscopic amount of antimatter has ever been assembled due to the extreme cost and difficulty of production and handling. Nonetheless, antimatter is an essential component of widely available applications related to beta decay, such as positron emission tomography, radiation therapy, and industrial imaging.
In theory, a particle and its antiparticle (for example, a proton and an antiproton) have the same mass, but opposite electric charge, and other differences in quantum numbers.
A collision between any particle and its anti-particle partner leads to their mutual annihilation, giving rise to various proportions of intense photons (gamma rays), neutrinos, and sometimes less-massive particle–antiparticle pairs. The majority of the total energy of annihilation emerges in the form of ionizing radiation. If surrounding matter is present, the energy content of this radiation will be absorbed and converted into other forms of energy, such as heat or light. The amount of energy released is usually proportional to the total mass of the collided matter and antimatter, in accordance with the notable mass–energy equivalence equation, .
Antiparticles bind with each other to form antimatter, just as ordinary particles bind to form normal matter. For example, a positron (the antiparticle of the electron) and an antiproton (the antiparticle of the proton) can form an antihydrogen atom. The nuclei of antihelium have been artificially produced, albeit with difficulty, and are the most complex anti-nuclei so far observed. Physical principles indicate that complex antimatter atomic nuclei are possible, as well as anti-atoms corresponding to the known chemical elements.
There is strong evidence that the observable universe is composed almost entirely of ordinary matter, as opposed to an equal mixture of matter and antimatter. This asymmetry of matter and antimatter in the visible universe is one of the great unsolved problems in physics. The process by which this inequality between matter and antimatter particles developed is called baryogenesis.
Definitions.
Antimatter particles carry the same charge as matter particles, but of opposite sign. That is, an antiproton is negatively charged and an antielectron (positron) is positively charged. Neutrons do not carry a net charge, but their constituent quarks do. Protons and neutrons have a baryon number of +1, while antiprotons and antineutrons have a baryon number of –1. Similarly, electrons have a lepton number of +1, while that of positrons is –1. When a particle and its corresponding antiparticle collide, they are both converted into energy.
The French term for "made of or pertaining to antimatter", , led to the initialism "C.T." and the science fiction term , as used in such novels as "Seetee Ship".
Conceptual history.
The idea of negative matter appears in past theories of matter that have now been abandoned. Using the once popular vortex theory of gravity, the possibility of matter with negative gravity was discussed by William Hicks in the 1880s. Between the 1880s and the 1890s, Karl Pearson proposed the existence of "squirts" and sinks of the flow of aether. The squirts represented normal matter and the sinks represented negative matter. Pearson's theory required a fourth dimension for the aether to flow from and into.
The term antimatter was first used by Arthur Schuster in two rather whimsical letters to "Nature" in 1898, in which he coined the term. He hypothesized antiatoms, as well as whole antimatter solar systems, and discussed the possibility of matter and antimatter annihilating each other. Schuster's ideas were not a serious theoretical proposal, merely speculation, and like the previous ideas, differed from the modern concept of antimatter in that it possessed negative gravity.
The modern theory of antimatter began in 1928, with a paper by Paul Dirac. Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of antielectrons. Although Dirac had laid the groundwork for the existence of these “antielectrons” he initially failed to pick up on the implications contained within his own equation. He freely gave the credit for that insight to J. Robert Oppenheimer, whose seminal paper “On the Theory of Electrons and Protons” (Feb 14th 1930) drew on Dirac's equation and argued for the existence of a positively charged electron (a positron), which as a counterpart to the electron should have the same mass as the electron itself. This meant that it could not be, as Dirac had in fact suggested, a proton. Dirac further postulated the existence of antimatter in a 1931 paper which referred to the positron as an "anti-electron". These were discovered by Carl D. Anderson in 1932 and named positrons from "positive electron". Although Dirac did not himself use the term antimatter, its use follows on naturally enough from antielectrons, antiprotons, etc. A complete periodic table of antimatter was envisaged by Charles Janet in 1929.
The Feynman–Stueckelberg interpretation states that antimatter and antiparticles behave exactly identical to regular particles, but traveling backward in time. This concept is nowadays used in modern particle physics, in Feynman diagrams.
Notation.
One way to denote an antiparticle is by adding a bar over the particle's symbol. For example, the proton and antiproton are denoted as and , respectively. The same rule applies if one were to address a particle by its constituent components. A proton is made up of quarks, so an antiproton must therefore be formed from antiquarks. Another convention is to distinguish particles by positive and negative electric charge. Thus, the electron and positron are denoted simply as and respectively. To prevent confusion, however, the two conventions are never mixed.
Properties.
There is no difference in the gravitational behavior of matter and antimatter. In other words, antimatter falls down when dropped, not up. This was confirmed with the thin, very cold gas of thousands of antihydrogen atoms that were confined in a vertical shaft surrounded by superconducting electromagnetic coils. These can create a magnetic bottle to keep the antimatter from coming into contact with matter and annihilating. The researchers then gradually weakened the magnetic fields and detected the antiatoms using two sensors as they escaped and annihilated. Most of the anti-atoms came out of the bottom opening, and only one-quarter out of the top.
There are compelling theoretical reasons to believe that, aside from the fact that antiparticles have different signs on all charges (such as electric and baryon charges), matter and antimatter have exactly the same properties. This means a particle and its corresponding antiparticle must have identical masses and decay lifetimes (if unstable). It also implies that, for example, a star made up of antimatter (an "antistar") will shine just like an ordinary star. This idea was tested experimentally in 2016 by the ALPHA experiment, which measured the transition between the two lowest energy states of antihydrogen. The results, which are identical to that of hydrogen, confirmed the validity of quantum mechanics for antimatter.
Origin and asymmetry.
Most matter observable from the Earth seems to be made of matter rather than antimatter. If antimatter-dominated regions of space existed, the gamma rays produced in annihilation reactions along the boundary between matter and antimatter regions would be detectable.
Antiparticles are created everywhere in the universe where high-energy particle collisions take place. High-energy cosmic rays striking Earth's atmosphere (or any other matter in the Solar System) produce minute quantities of antiparticles in the resulting particle jets, which are immediately annihilated by contact with nearby matter. They may similarly be produced in regions like the center of the Milky Way and other galaxies, where very energetic celestial events occur (principally the interaction of relativistic jets with the interstellar medium). The presence of the resulting antimatter is detectable by the two gamma rays produced every time positrons annihilate with nearby matter. The frequency and wavelength of the gamma rays indicate that each carries 511 keV of energy (that is, the rest mass of an electron multiplied by "c"2).
Observations by the European Space Agency's INTEGRAL satellite may explain the origin of a giant antimatter cloud surrounding the Galactic Center. The observations show that the cloud is asymmetrical and matches the pattern of X-ray binaries (binary star systems containing black holes or neutron stars), mostly on one side of the Galactic Center. While the mechanism is not fully understood, it is likely to involve the production of electron–positron pairs, as ordinary matter gains kinetic energy while falling into a stellar remnant.
Antimatter may exist in relatively large amounts in far-away galaxies due to cosmic inflation in the primordial time of the universe. Antimatter galaxies, if they exist, are expected to have the same chemistry and absorption and emission spectra as normal-matter galaxies, and their astronomical objects would be observationally identical, making them difficult to distinguish. NASA is trying to determine if such galaxies exist by looking for X-ray and gamma ray signatures of annihilation events in colliding superclusters.
In October 2017, scientists working on the BASE experiment at CERN reported a measurement of the antiproton magnetic moment to a precision of 1.5 parts per billion. It is consistent with the most precise measurement of the proton magnetic moment (also made by BASE in 2014), which supports the hypothesis of CPT symmetry. This measurement represents the first time that a property of antimatter is known more precisely than the equivalent property in matter.
Antimatter quantum interferometry has been first demonstrated in 2018 in the Positron Laboratory (L-NESS) of Rafael Ferragut in Como (Italy), by a group led by Marco Giammarchi.
Natural production.
Positrons are produced naturally in β+ decays of naturally occurring radioactive isotopes (for example, potassium-40) and in interactions of gamma quanta (emitted by radioactive nuclei) with matter. Antineutrinos are another kind of antiparticle created by natural radioactivity (β− decay). Many different kinds of antiparticles are also produced by (and contained in) cosmic rays. In January 2011, research by the American Astronomical Society discovered antimatter (positrons) originating above thunderstorm clouds; positrons are produced in terrestrial gamma ray flashes created by electrons accelerated by strong electric fields in the clouds. Antiprotons have also been found to exist in the Van Allen Belts around the Earth by the PAMELA module.
Antiparticles are also produced in any environment with a sufficiently high temperature (mean particle energy greater than the pair production threshold). It is hypothesized that during the period of baryogenesis, when the universe was extremely hot and dense, matter and antimatter were continually produced and annihilated. The presence of remaining matter, and absence of detectable remaining antimatter, is called baryon asymmetry. The exact mechanism that produced this asymmetry during baryogenesis remains an unsolved problem. One of the necessary conditions for this asymmetry is the violation of CP symmetry, which has been experimentally observed in the weak interaction.
Recent observations indicate black holes and neutron stars produce vast amounts of positron-electron plasma via the jets.
Observation in cosmic rays.
Satellite experiments have found evidence of positrons and a few antiprotons in primary cosmic rays, amounting to less than 1% of the particles in primary cosmic rays. This antimatter cannot all have been created in the Big Bang, but is instead attributed to have been produced by cyclic processes at high energies. For instance, electron-positron pairs may be formed in pulsars, as a magnetized neutron star rotation cycle shears electron-positron pairs from the star surface. Therein the antimatter forms a wind that crashes upon the ejecta of the progenitor supernovae. This weathering takes place as "the cold, magnetized relativistic wind launched by the star hits the non-relativistically expanding ejecta, a shock wave system forms in the impact: the outer one propagates in the ejecta, while a reverse shock propagates back towards the star." The former ejection of matter in the outer shock wave and the latter production of antimatter in the reverse shock wave are steps in a space weather cycle.
Preliminary results from the presently operating Alpha Magnetic Spectrometer ("AMS-02") on board the International Space Station show that positrons in the cosmic rays arrive with no directionality, and with energies that range from 10 GeV to 250 GeV. In September, 2014, new results with almost twice as much data were presented in a talk at CERN and published in Physical Review Letters. A new measurement of positron fraction up to 500 GeV was reported, showing that positron fraction peaks at a maximum of about 16% of total electron+positron events, around an energy of 275 ± 32 GeV. At higher energies, up to 500 GeV, the ratio of positrons to electrons begins to fall again. The absolute flux of positrons also begins to fall before 500 GeV, but peaks at energies far higher than electron energies, which peak about 10 GeV. These results on interpretation have been suggested to be due to positron production in annihilation events of massive dark matter particles.
Cosmic ray antiprotons also have a much higher energy than their normal-matter counterparts (protons). They arrive at Earth with a characteristic energy maximum of 2 GeV, indicating their production in a fundamentally different process from cosmic ray protons, which on average have only one-sixth of the energy.
There is an ongoing search for larger antimatter nuclei, such as antihelium nuclei (that is, anti-alpha particles), in cosmic rays. The detection of natural antihelium could imply the existence of large antimatter structures such as an antistar. A prototype of the "AMS-02" designated "AMS-01", was flown into space aboard the Space Shuttle "Discovery" on STS-91 in June 1998. By not detecting any antihelium at all, the "AMS-01" established an upper limit of 1.1×10−6 for the antihelium to helium flux ratio. AMS-02 revealed in December 2016 that it had discovered a few signals consistent with antihelium nuclei amidst several billion helium nuclei. The result remains to be verified, and as of 2017[ [update]], the team is trying to rule out contamination.
Artificial production.
Positrons.
Positrons were reported in November 2008 to have been generated by Lawrence Livermore National Laboratory in large numbers. A laser drove electrons through a gold target's nuclei, which caused the incoming electrons to emit energy quanta that decayed into both matter and antimatter. Positrons were detected at a higher rate and in greater density than ever previously detected in a laboratory. Previous experiments made smaller quantities of positrons using lasers and paper-thin targets; newer simulations showed that short bursts of ultra-intense lasers and millimeter-thick gold are a far more effective source.
In 2023, the production of the first electron-positron beam-plasma was reported by a collaboration led by researchers at University of Oxford working with the High-Radiation to Materials (HRMT) facility at CERN. The beam demonstrated the highest positron yield achieved so far in a laboratory setting. The experiment employed the 440 GeV proton beam, with formula_0 protons, from the Super Proton Synchrotron, and irradiated a particle converter composed of Carbon and Tantalum. This yielded a total formula_1 electron-positron pairs via a particle shower process. The produced pair beams have a volume that fills multiple Debye spheres and are thus able to sustain collective plasma oscillations.
Antiprotons, antineutrons, and antinuclei.
The existence of the antiproton was experimentally confirmed in 1955 by University of California, Berkeley physicists Emilio Segrè and Owen Chamberlain, for which they were awarded the 1959 Nobel Prize in Physics. An antiproton consists of two up antiquarks and one down antiquark (). The properties of the antiproton that have been measured all match the corresponding properties of the proton, with the exception of the antiproton having opposite electric charge and magnetic moment from the proton. Shortly afterwards, in 1956, the antineutron was discovered in proton–proton collisions at the Bevatron (Lawrence Berkeley National Laboratory) by Bruce Cork and colleagues.
In addition to antibaryons, anti-nuclei consisting of multiple bound antiprotons and antineutrons have been created. These are typically produced at energies far too high to form antimatter atoms (with bound positrons in place of electrons). In 1965, a group of researchers led by Antonino Zichichi reported production of nuclei of antideuterium at the Proton Synchrotron at CERN. At roughly the same time, observations of antideuterium nuclei were reported by a group of American physicists at the Alternating Gradient Synchrotron at Brookhaven National Laboratory.
Antihydrogen atoms.
In 1995, CERN announced that it had successfully brought into existence nine hot antihydrogen atoms by implementing the SLAC/Fermilab concept during the PS210 experiment. The experiment was performed using the Low Energy Antiproton Ring (LEAR), and was led by Walter Oelert and Mario Macri. Fermilab soon confirmed the CERN findings by producing approximately 100 antihydrogen atoms at their facilities. The antihydrogen atoms created during PS210 and subsequent experiments (at both CERN and Fermilab) were extremely energetic and were not well suited to study. To resolve this hurdle, and to gain a better understanding of antihydrogen, two collaborations were formed in the late 1990s, namely, ATHENA and ATRAP.
In 1999, CERN activated the Antiproton Decelerator, a device capable of decelerating antiprotons from to – still too "hot" to produce study-effective antihydrogen, but a huge leap forward. In late 2002 the ATHENA project announced that they had created the world's first "cold" antihydrogen. The ATRAP project released similar results very shortly thereafter. The antiprotons used in these experiments were cooled by decelerating them with the Antiproton Decelerator, passing them through a thin sheet of foil, and finally capturing them in a Penning–Malmberg trap. The overall cooling process is workable, but highly inefficient; approximately 25 million antiprotons leave the Antiproton Decelerator and roughly 25,000 make it to the Penning–Malmberg trap, which is about or 0.1% of the original amount.
The antiprotons are still hot when initially trapped. To cool them further, they are mixed into an electron plasma. The electrons in this plasma cool via cyclotron radiation, and then sympathetically cool the antiprotons via Coulomb collisions. Eventually, the electrons are removed by the application of short-duration electric fields, leaving the antiprotons with energies less than . While the antiprotons are being cooled in the first trap, a small cloud of positrons is captured from radioactive sodium in a Surko-style positron accumulator. This cloud is then recaptured in a second trap near the antiprotons. Manipulations of the trap electrodes then tip the antiprotons into the positron plasma, where some combine with antiprotons to form antihydrogen. This neutral antihydrogen is unaffected by the electric and magnetic fields used to trap the charged positrons and antiprotons, and within a few microseconds the antihydrogen hits the trap walls, where it annihilates. Some hundreds of millions of antihydrogen atoms have been made in this fashion.
In 2005, ATHENA disbanded and some of the former members (along with others) formed the ALPHA Collaboration, which is also based at CERN. The ultimate goal of this endeavour is to test CPT symmetry through comparison of the atomic spectra of hydrogen and antihydrogen (see hydrogen spectral series).
Most of the sought-after high-precision tests of the properties of antihydrogen could only be performed if the antihydrogen were trapped, that is, held in place for a relatively long time. While antihydrogen atoms are electrically neutral, the spins of their component particles produce a magnetic moment. These magnetic moments can interact with an inhomogeneous magnetic field; some of the antihydrogen atoms can be attracted to a magnetic minimum. Such a minimum can be created by a combination of mirror and multipole fields. Antihydrogen can be trapped in such a magnetic minimum (minimum-B) trap; in November 2010, the ALPHA collaboration announced that they had so trapped 38 antihydrogen atoms for about a sixth of a second. This was the first time that neutral antimatter had been trapped.
On 26 April 2011, ALPHA announced that they had trapped 309 antihydrogen atoms, some for as long as 1,000 seconds (about 17 minutes). This was longer than neutral antimatter had ever been trapped before. ALPHA has used these trapped atoms to initiate research into the spectral properties of antihydrogen.
In 2016, a new antiproton decelerator and cooler called ELENA (Extra Low ENergy Antiproton decelerator) was built. It takes the antiprotons from the antiproton decelerator and cools them to 90 keV, which is "cold" enough to study. This machine works by using high energy and accelerating the particles within the chamber. More than one hundred antiprotons can be captured per second, a huge improvement, but it would still take several thousand years to make a nanogram of antimatter.
The biggest limiting factor in the large-scale production of antimatter is the availability of antiprotons. Recent data released by CERN states that, when fully operational, their facilities are capable of producing ten million antiprotons per minute. Assuming a 100% conversion of antiprotons to antihydrogen, it would take 100 billion years to produce 1 gram or 1 mole of antihydrogen (approximately atoms of anti-hydrogen). However, CERN only produces 1% of the anti-matter Fermilab does, and neither are designed to produce anti-matter. According to Gerald Jackson, using technology already in use today we are capable of producing and capturing 20 grams of anti-matter particles per year at a yearly cost of 670 million dollars per facility.
Antihelium.
Antihelium-3 nuclei (He) were first observed in the 1970s in proton–nucleus collision experiments at the Institute for High Energy Physics by Y. Prockoshkin's group (Protvino near Moscow, USSR) and later created in nucleus–nucleus collision experiments. Nucleus–nucleus collisions produce antinuclei through the coalescence of antiprotons and antineutrons created in these reactions. In 2011, the STAR detector reported the observation of artificially created antihelium-4 nuclei (anti-alpha particles) (He) from such collisions.
The Alpha Magnetic Spectrometer on the International Space Station has, as of 2021, recorded eight events that seem to indicate the detection of antihelium-3.
Preservation.
Antimatter cannot be stored in a container made of ordinary matter because antimatter reacts with any matter it touches, annihilating itself and an equal amount of the container. Antimatter in the form of charged particles can be contained by a combination of electric and magnetic fields, in a device called a Penning trap. This device cannot, however, contain antimatter that consists of uncharged particles, for which atomic traps are used. In particular, such a trap may use the dipole moment (electric or magnetic) of the trapped particles. At high vacuum, the matter or antimatter particles can be trapped and cooled with slightly off-resonant laser radiation using a magneto-optical trap or magnetic trap. Small particles can also be suspended with optical tweezers, using a highly focused laser beam.
In 2011, CERN scientists were able to preserve antihydrogen for approximately 17 minutes. The record for storing antiparticles is currently held by the TRAP experiment at CERN: antiprotons were kept in a Penning trap for 405 days. A proposal was made in 2018 to develop containment technology advanced enough to contain a billion anti-protons in a portable device to be driven to another lab for further experimentation.
Cost.
Scientists claim that antimatter is the costliest material to make. In 2006, Gerald Smith estimated $250 million could produce 10 milligrams of positrons (equivalent to $25 billion per gram); in 1999, NASA gave a figure of $62.5 trillion per gram of antihydrogen. This is because production is difficult (only very few antiprotons are produced in reactions in particle accelerators) and because there is higher demand for other uses of particle accelerators. According to CERN, it has cost a few hundred million Swiss francs to produce about 1 billionth of a gram (the amount used so far for particle/antiparticle collisions). In comparison, to produce the first atomic weapon, the cost of the Manhattan Project was estimated at $23 billion with inflation during 2007.
Several studies funded by the NASA Institute for Advanced Concepts are exploring whether it might be possible to use magnetic scoops to collect the antimatter that occurs naturally in the Van Allen belt of the Earth, and ultimately the belts of gas giants like Jupiter, ideally at a lower cost per gram.
Uses.
Medical.
Matter–antimatter reactions have practical applications in medical imaging, such as positron emission tomography (PET). In positive beta decay, a nuclide loses surplus positive charge by emitting a positron (in the same event, a proton becomes a neutron, and a neutrino is also emitted). Nuclides with surplus positive charge are easily made in a cyclotron and are widely generated for medical use. Antiprotons have also been shown within laboratory experiments to have the potential to treat certain cancers, in a similar method currently used for ion (proton) therapy.
Fuel.
Isolated and stored antimatter could be used as a fuel for interplanetary or interstellar travel as part of an antimatter-catalyzed nuclear pulse propulsion or another antimatter rocket. Since the energy density of antimatter is higher than that of conventional fuels, an antimatter-fueled spacecraft would have a higher thrust-to-weight ratio than a conventional spacecraft.
If matter–antimatter collisions resulted only in photon emission, the entire rest mass of the particles would be converted to kinetic energy. The energy per unit mass () is about 10 orders of magnitude greater than chemical energies, and about 3 orders of magnitude greater than the nuclear potential energy that can be liberated, today, using nuclear fission (about per fission reaction or ), and about 2 orders of magnitude greater than the best possible results expected from fusion (about for the proton–proton chain). The reaction of of antimatter with of matter would produce (180 petajoules) of energy (by the mass–energy equivalence formula, "E"="mc"2), or the rough equivalent of 43 megatons of TNT – slightly less than the yield of the 27,000 kg Tsar Bomba, the largest thermonuclear weapon ever detonated.
Not all of that energy can be utilized by any realistic propulsion technology because of the nature of the annihilation products. While electron–positron reactions result in gamma ray photons, these are difficult to direct and use for thrust. In reactions between protons and antiprotons, their energy is converted largely into relativistic neutral and charged pions. The neutral pions decay almost immediately (with a lifetime of 85 attoseconds) into high-energy photons, but the charged pions decay more slowly (with a lifetime of 26 nanoseconds) and can be deflected magnetically to produce thrust.
Charged pions ultimately decay into a combination of neutrinos (carrying about 22% of the energy of the charged pions) and unstable charged muons (carrying about 78% of the charged pion energy), with the muons then decaying into a combination of electrons, positrons and neutrinos (cf. muon decay; the neutrinos from this decay carry about 2/3 of the energy of the muons, meaning that from the original charged pions, the total fraction of their energy converted to neutrinos by one route or another would be about 0.22 + (2/3)⋅0.78 = 0.74).
Weapons.
Antimatter has been considered as a trigger mechanism for nuclear weapons. A major obstacle is the difficulty of producing antimatter in large enough quantities, and there is no evidence that it will ever be feasible. Nonetheless, the U.S. Air Force funded studies of the physics of antimatter in the Cold War, and began considering its possible use in weapons, not just as a trigger, but as the explosive itself.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": "3\\times 10^{11}"
},
{
"math_id": 1,
"text": "1.5\\times 10^{13}"
}
]
| https://en.wikipedia.org/wiki?curid=1317 |
1318037 | Screw theory | Mathematical formulation of vector pairs used in physics (rigid body dynamics)
Screw theory is the algebraic calculation of pairs of vectors, such as angular and linear velocity, or forces and moments, that arise in the kinematics and dynamics of rigid bodies.
Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors.
Important theorems of screw theory include: The Transfer Principle proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws.
Chasles' theorem proves that any change between two rigid object poses can be performed by a single screw. Poinsot's theorem proves that rotations about a rigid object's major and minor -- but not intermediate -- axes are stable.
Screw theory is an important tool in robot mechanics, mechanical design, computational geometry and multibody dynamics.
This is in part because of the relationship between screws and dual quaternions which have been used to interpolate rigid-body motions. Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots).
Basic concepts.
A spatial displacement of a rigid body can be defined by a rotation about a line and a translation along the same line, called a <templatestyles src="Template:Visible anchor/styles.css" />screw motion. This is known as Chasles' theorem. The six parameters that define a screw motion are the four independent components of the Plücker vector that defines the screw axis, together with the rotation angle about and linear slide along this line, and form a pair of vectors called a screw. For comparison, the six parameters that define a spatial displacement can also be given by three Euler angles that define the rotation and the three components of the translation vector.
Screw.
A screw is a six-dimensional vector constructed from a pair of three-dimensional vectors, such as forces and torques and linear and angular velocity, that arise in the study of spatial rigid body movement. The components of the screw define the Plücker coordinates of a line in space and the magnitudes of the vector along the line and moment about this line.
Twist.
A twist is a screw used to represent the velocity of a rigid body as an angular velocity around an axis and a linear velocity along this axis. All points in the body have the same component of the velocity along the axis, however the greater the distance from the axis the greater the velocity in the plane perpendicular to this axis. Thus, the helicoidal field formed by the velocity vectors in a moving rigid body flattens out the further the points are radially from the twist axis.
The points in a body undergoing a constant twist motion trace helices in the fixed frame. If this screw motion has zero pitch then the trajectories trace circles, and the movement is a pure rotation. If the screw motion has infinite pitch then the trajectories are all straight lines in the same direction.
Wrench.
The force and torque vectors that arise in applying Newton's laws to a rigid body can be assembled into a screw called a wrench. A force has a point of application and a line of action, therefore it defines the Plücker coordinates of a line in space and has zero pitch. A torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw. The ratio of these two magnitudes defines the pitch of the screw.
Algebra of screws.
Let a "screw" be an ordered pair
formula_0
where S and V are three-dimensional real vectors. The sum and difference of these ordered pairs are computed componentwise. Screws are often called "dual vectors".
Now, introduce the ordered pair of real numbers â = ("a", "b") called a "dual scalar". Let the addition and subtraction of these numbers be componentwise, and define multiplication as
formula_1
The multiplication of a screw "S" = (S, V) by the dual scalar â = ("a", "b") is computed componentwise to be,
formula_2
Finally, introduce the dot and cross products of screws by the formulas:
formula_3
which is a dual scalar, and
formula_4
which is a screw. The dot and cross products of screws satisfy the identities of vector algebra, and allow computations that directly parallel computations in the algebra of vectors.
Let the dual scalar ẑ = ("φ", "d") define a "dual angle", then the infinite series definitions of sine and cosine yield the relations
formula_5
which are also dual scalars. In general, the function of a dual variable is defined to be "f"(ẑ) = ("f"("φ"), "df"′("φ")), where "df"′("φ") is the derivative of "f"("φ").
These definitions allow the following results:
Wrench.
A common example of a screw is the "wrench" associated with a force acting on a rigid body. Let "P" be the point of application of the force F and let P be the vector locating this point in a fixed frame. The wrench "W" = (F, P×F) is a screw. The resultant force and moment obtained from all the forces F"i", "i" = 1...,"n", acting on a rigid body is simply the sum of the individual wrenches "W""i", that is
formula_9
Notice that the case of two equal but opposite forces F and −F acting at points A and B respectively, yields the resultant
formula_10
This shows that screws of the form
formula_11
can be interpreted as pure moments.
Twist.
In order to define the "twist" of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements, D(t)=([A(t)],d(t)), where [A] is a rotation matrix and d is a translation vector. This causes a point p that is fixed in moving body coordinates to trace a curve P(t) in the fixed frame given by,
formula_12
The velocity of P is
formula_13
where v is velocity of the origin of the moving frame, that is dd/dt. Now substitute p = ["A"T](P − d) into this equation to obtain,
formula_14
where [Ω] = [d"A"/d"t"]["A"T] is the angular velocity matrix and ω is the angular velocity vector.
The screw
formula_15
is the "twist" of the moving body. The vector V = v + d × "ω" is the velocity of the point in the body that corresponds with the origin of the fixed frame.
There are two important special cases: (i) when d is constant, that is v = 0, then the twist is a pure rotation about a line, then the twist is
formula_16
and (ii) when [Ω] = 0, that is the body does not rotate but only slides in the direction v, then the twist is a pure slide given by
formula_17
Revolute joints.
For a revolute joint, let the axis of rotation pass through the point q and be directed along the vector "ω", then the twist for the joint is given by,
formula_18
Prismatic joints.
For a prismatic joint, let the vector v pointing define the direction of the slide, then the twist for the joint is given by,
formula_19
Coordinate transformation of screws.
The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plücker vector of line, which in turn are obtained from the transformations of the coordinate of points on the line.
Let the displacement of a body be defined by "D" = (["A"], d), where ["A"] is the rotation matrix and d is the translation vector. Consider the line in the body defined by the two points p and q, which has the Plücker coordinates,
formula_20
then in the fixed frame we have the transformed point coordinates P = ["A"]p + d and Q = ["A"]q + d, which yield.
formula_21
Thus, a spatial displacement defines a transformation for Plücker coordinates of lines given by
formula_22
The matrix ["D"] is the skew-symmetric matrix that performs the cross product operation, that is ["D"]y = d × y.
The 6×6 matrix obtained from the spatial displacement "D" = (["A"], d) can be assembled into the dual matrix
formula_23
which operates on a screw "s" = (s.v) to obtain,
formula_24
The dual matrix ["Â"] = (["A"], ["DA"]) has determinant 1 and is called a "dual orthogonal matrix".
Twists as elements of a Lie algebra.
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,
formula_25
This notation does not distinguish between P = ("X", "Y", "Z", 1), and P = ("X", "Y", "Z"), which is hopefully clear in context.
The velocity of this movement is defined by computing the velocity of the trajectories of the points in the body,
formula_26
The dot denotes the derivative with respect to time, and because p is constant its derivative is zero.
Substitute the inverse transform for p into the velocity equation to obtain the velocity of "P" by operating on its trajectory P("t"), that is
formula_27
where
formula_28
Recall that [Ω] is the angular velocity matrix. The matrix ["S"] is an element of the Lie algebra se(3) of the Lie group SE(3) of homogeneous transforms. The components of ["S"] are the components of the twist screw, and for this reason ["S"] is also often called a twist.
From the definition of the matrix ["S"], we can formulate the ordinary differential equation,
formula_29
and ask for the movement ["T"("t")] that has a constant twist matrix ["S"]. The solution is the matrix exponential
formula_30
This formulation can be generalized such that given an initial configuration "g"(0) in SE("n"), and a twist "ξ" in se("n"), the homogeneous transformation to a new location and orientation can be computed with the formula,
formula_31
where "θ" represents the parameters of the transformation.
Screws by reflection.
In transformation geometry, the elemental concept of transformation is the reflection (mathematics). In planar transformations a translation is obtained by reflection in parallel lines, and rotation is obtained by reflection in a pair of intersecting lines. To produce a screw transformation from similar concepts one must use planes in space: the parallel planes must be perpendicular to the screw axis, which is the line of intersection of the intersecting planes that generate the rotation of the screw. Thus four reflections in planes effect a screw transformation. The tradition of inversive geometry borrows some of the ideas of projective geometry and provides a language of transformation that does not depend on analytic geometry.
Homography.
The combination of a translation with a rotation effected by a screw displacement can be illustrated with the exponential mapping.
Since "ε"2 = 0 for dual numbers, exp("aε") = 1 + "aε", all other terms of the exponential series vanishing.
Let "F" = {1 + "εr" : "r" ∈ H}, "ε"2 = 0.
Note that "F" is stable under the rotation "q" → "p" −1 "qp" and under the translation
(1 + "εr")(1 + "εs") = 1 + "ε" ("r" + "s") for any vector quaternions "r" and "s".
"F" is a 3-flat in the eight-dimensional space of dual quaternions. This 3-flat "F" represents space, and the homography constructed, restricted to "F", is a screw displacement of space.
Let "a" be half the angle of the desired turn about axis "r", and "br" half the displacement on the screw axis. Then form "z" = exp(("a" + "bε")"r" ) and z* = exp(("a" − "bε")"r"). Now the homography is
formula_32
The inverse for "z"* is
formula_33
so, the homography sends "q" to
formula_34
Now for any quaternion vector "p", "p"* = −"p", let "q" = 1 + "pε" ∈ "F" where the required rotation and translation are effected.
Evidently the group of units of the ring of dual quaternions is a Lie group. A subgroup has Lie algebra generated by the parameters "a r" and "b s", where "a", "b" ∈ R, and "r", "s" ∈ H. These six parameters generate a subgroup of the units, the unit sphere. Of course it includes "F" and the 3-sphere of versors.
Work of forces acting on a rigid body.
Consider the set of forces F1, F2 ... F"n" 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 with rotation ["A"("t")] and the translation d("t") of a reference point in the body, given by
formula_35
where x"i" are coordinates in the moving body.
The velocity of each point Xi is
formula_36
where ω is the angular velocity vector and v is the derivative of d("t").
The work by the forces over the displacement "δ"r"i"=v"i""δt" of each point is given by
formula_37
Define the velocities of each point in terms of the twist of the moving body to obtain
formula_38
Expand this equation and collect coefficients of ω and v to obtain
formula_39
Introduce the twist of the moving body and the wrench acting on it given by
formula_40
then work takes the form
formula_41
The 6×6 matrix [Π] is used to simplify the calculation of work using screws, so that
formula_42
where
formula_43
and [I] is the 3×3 identity matrix.
Reciprocal screws.
If the virtual work of a wrench on a twist is zero, then the forces and torque of the wrench are constraint forces relative to the twist. The wrench and twist are said to be "reciprocal," that is if
formula_44
then the screws "W" and "T" are reciprocal.
Twists in robotics.
In the study of robotic systems the components of the twist are often transposed to eliminate the need for the 6×6 matrix [Π] in the calculation of work. In this case the twist is defined to be
formula_45
so the calculation of work takes the form
formula_46
In this case, if
formula_47
then the wrench "W" is reciprocal to the twist "T".
History.
The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics).
Felix Klein saw screw theory as an application of elliptic geometry and his Erlangen Program. He also worked out elliptic geometry, and a fresh view of Euclidean geometry, with the Cayley–Klein metric. The use of a symmetric matrix for a von Staudt conic and metric, applied to screws, has been described by Harvey Lipkin. Other prominent contributors include Julius Plücker, W. K. Clifford, F. M. Dimentberg, Kenneth H. Hunt, J. R. Phillips.
The homography idea in transformation geometry was advanced by Sophus Lie more than a century ago. Even earlier, William Rowan Hamilton displayed the versor form of unit quaternions as exp("a r")= cos "a" + "r" sin "a". The idea is also in Euler's formula parametrizing the unit circle in the complex plane.
William Kingdon Clifford initiated the use of dual quaternions for kinematics, followed by Aleksandr Kotelnikov, Eduard Study ("Geometrie der Dynamen"), and Wilhelm Blaschke. However, the point of view of Sophus Lie has recurred.
In 1940, Julian Coolidge described the use of dual quaternions for screw displacements on page 261 of "A History of Geometrical Methods". He notes the 1885 contribution of Arthur Buchheim. Coolidge based his description simply on the tools Hamilton had used for real quaternions.
References.
<templatestyles src="Reflist/styles.css" /> | [
{
"math_id": 0,
"text": " \\mathsf{S} = (\\mathbf{S}, \\mathbf{V}), "
},
{
"math_id": 1,
"text": " \\hat{a}\\hat{c}=(a, b)(c, d) = (ac, ad+bc). "
},
{
"math_id": 2,
"text": " \\hat{a}\\mathsf{S} = (a, b)(\\mathbf{S}, \\mathbf{V}) = (a \\mathbf{S}, a \\mathbf{V} +b \\mathbf{S})."
},
{
"math_id": 3,
"text": " \\mathsf{S}\\cdot \\mathsf{T} = (\\mathbf{S}, \\mathbf{V})\\cdot (\\mathbf{T}, \\mathbf{W}) = (\\mathbf{S}\\cdot\\mathbf{T},\\,\\, \\mathbf{S}\\cdot\\mathbf{W} +\\mathbf{V}\\cdot\\mathbf{T}), "
},
{
"math_id": 4,
"text": " \\mathsf{S}\\times \\mathsf{T} = (\\mathbf{S}, \\mathbf{V})\\times (\\mathbf{T}, \\mathbf{W}) = (\\mathbf{S}\\times \\mathbf{T},\\,\\, \\mathbf{S}\\times \\mathbf{W} +\\mathbf{V}\\times \\mathbf{T})."
},
{
"math_id": 5,
"text": " \\sin \\hat{z} = (\\sin\\varphi , d \\cos\\varphi), \\,\\,\\, \\cos\\hat{z} = (\\cos\\varphi ,- d \\sin\\varphi),"
},
{
"math_id": 6,
"text": " |\\mathsf{S}| = \\sqrt{\\mathsf{S} \\cdot \\mathsf{S}} = 1; "
},
{
"math_id": 7,
"text": " \\mathsf{S} \\cdot \\mathsf{T} = \\left|\\mathsf{S}\\right| \\left|\\mathsf{T}\\right| \\cos\\hat{z}; "
},
{
"math_id": 8,
"text": " \\mathsf{S} \\times \\mathsf{T} = \\left|\\mathsf{S}\\right| \\left|\\mathsf{T}\\right| \\sin\\hat{z} \\mathsf{N}. "
},
{
"math_id": 9,
"text": " \\mathsf{R} = \\sum_{i=1}^n \\mathsf{W}_i = \\sum_{i=1}^n (\\mathbf{F}_i, \\mathbf{P}_i\\times\\mathbf{F}_i). "
},
{
"math_id": 10,
"text": " \\mathsf{R}=(\\mathbf{F}-\\mathbf{F}, \\mathbf{A}\\times\\mathbf{F} - \\mathbf{B}\\times\\mathbf{F}) = (0, (\\mathbf{A}-\\mathbf{B})\\times\\mathbf{F})."
},
{
"math_id": 11,
"text": "\\mathsf{M}=(0, \\mathbf{M}),"
},
{
"math_id": 12,
"text": "\n\\mathbf{P}(t) = [A(t)]\\mathbf{p} + \\mathbf{d}(t).\n"
},
{
"math_id": 13,
"text": "\n\\mathbf{V}_P(t) = \\left[\\frac{dA(t)}{dt}\\right]\\mathbf{p} + \\mathbf{v}(t),\n"
},
{
"math_id": 14,
"text": "\n\\mathbf{V}_P(t) = [\\Omega]\\mathbf{P} + \\mathbf{v} - [\\Omega]\\mathbf{d}\\quad\\text{or}\\quad\\mathbf{V}_P(t) = \\mathbf{\\omega}\\times\\mathbf{P} + \\mathbf{v} + \\mathbf{d}\\times\\mathbf{\\omega},\n"
},
{
"math_id": 15,
"text": " \\mathsf{T}=(\\vec{\\omega}, \\mathbf{v} + \\mathbf{d}\\times \\vec{\\omega}),\\!"
},
{
"math_id": 16,
"text": "\\mathsf{L}=(\\omega, \\mathbf{d}\\times\\omega),"
},
{
"math_id": 17,
"text": " \\mathsf{T}=(0, \\mathbf{v})."
},
{
"math_id": 18,
"text": " \\xi = \\begin{Bmatrix} \\omega \\\\ q \\times \\omega \\end{Bmatrix}."
},
{
"math_id": 19,
"text": " \\xi = \\begin{Bmatrix} 0\\\\v \\end{Bmatrix}."
},
{
"math_id": 20,
"text": " \\mathsf{q}=(\\mathbf{q}-\\mathbf{p}, \\mathbf{p}\\times\\mathbf{q}),"
},
{
"math_id": 21,
"text": "\\mathsf{Q}=(\\mathbf{Q}-\\mathbf{P}, \\mathbf{P}\\times\\mathbf{Q}) = ([A](\\mathbf{q}-\\mathbf{p}), [A](\\mathbf{p}\\times\\mathbf{q}) + \\mathbf{d}\\times[A](\\mathbf{q}-\\mathbf{p}))"
},
{
"math_id": 22,
"text": "\n\\begin{Bmatrix} \\mathbf{Q}-\\mathbf{P} \\\\ \\mathbf{P}\\times\\mathbf{Q} \\end{Bmatrix}\n= \\begin{bmatrix} A & 0 \\\\ DA & A \\end{bmatrix}\n\\begin{Bmatrix} \\mathbf{q}-\\mathbf{p} \\\\ \\mathbf{p}\\times\\mathbf{q} \\end{Bmatrix}.\n"
},
{
"math_id": 23,
"text": "[\\hat{A}] =([A], [DA]),"
},
{
"math_id": 24,
"text": "\\mathsf{S} = [\\hat{A}]\\mathsf{s}, \\quad (\\mathbf{S}, \\mathbf{V}) = ([A], [DA])(\\mathbf{s}, \\mathbf{v}) = ([A]\\mathbf{s}, [A]\\mathbf{v}+[DA]\\mathbf{s})."
},
{
"math_id": 25,
"text": " \\textbf{P}(t)=[T(t)]\\textbf{p} = \n\\begin{Bmatrix} \\textbf{P} \\\\ 1\\end{Bmatrix}=\\begin{bmatrix} A(t) & \\textbf{d}(t) \\\\ 0 & 1\\end{bmatrix}\n\\begin{Bmatrix} \\textbf{p} \\\\ 1\\end{Bmatrix}."
},
{
"math_id": 26,
"text": " \\textbf{V}_P = [\\dot{T}(t)]\\textbf{p} =\n\\begin{Bmatrix} \\textbf{V}_P \\\\ 0\\end{Bmatrix} = \\begin{bmatrix} \\dot{A}(t) & \\dot{\\textbf{d}}(t) \\\\ 0 & 0 \\end{bmatrix}\n\\begin{Bmatrix} \\textbf{p} \\\\ 1\\end{Bmatrix}."
},
{
"math_id": 27,
"text": "\\textbf{V}_P=[\\dot{T}(t)][T(t)]^{-1}\\textbf{P}(t) = [S]\\textbf{P},"
},
{
"math_id": 28,
"text": "[S] = \\begin{bmatrix} \\Omega & -\\Omega\\textbf{d} + \\dot{\\textbf{d}} \\\\ 0 & 0 \\end{bmatrix} = \\begin{bmatrix} \\Omega & \\mathbf{d}\\times\\omega+ \\mathbf{v} \\\\ 0 & 0 \\end{bmatrix}."
},
{
"math_id": 29,
"text": "[\\dot{T}(t)] = [S][T(t)],"
},
{
"math_id": 30,
"text": "[T(t)] = e^{[S]t}."
},
{
"math_id": 31,
"text": " g(\\theta) = \\exp(\\xi\\theta) g(0),"
},
{
"math_id": 32,
"text": "[q\\ :\\ 1]\\begin{pmatrix}z & 0 \\\\ 0 & z^* \\end{pmatrix} = [q z\\ :\\ z^*] \\thicksim [(z^*)^{-1} q z\\ :\\ 1]."
},
{
"math_id": 33,
"text": " \\frac 1 {\\exp(ar - b \\varepsilon r)} = (e^{ar} e^{-br \\varepsilon} )^{-1} = e^{br \\varepsilon} e^{-ar},"
},
{
"math_id": 34,
"text": "(e^{b \\varepsilon} e^{-ar}) q (e^{ar} e^{b \\varepsilon r}) = e^{b \\varepsilon r} (e^{-ar} q e^{ar} )e^{b \\varepsilon r} = e^{2b \\varepsilon r} (e^{-ar} q e^{ar})."
},
{
"math_id": 35,
"text": " \\mathbf{X}_i(t)= [A(t)]\\mathbf{x}_i + \\mathbf{d}(t)\\quad i=1,\\ldots, n, "
},
{
"math_id": 36,
"text": "\\mathbf{V}_i = \\vec{\\omega}\\times(\\mathbf{X}_i-\\mathbf{d}) + \\mathbf{v},"
},
{
"math_id": 37,
"text": " \\delta W = \\mathbf{F}_1\\cdot\\mathbf{V}_1\\delta t+\\mathbf{F}_2\\cdot\\mathbf{V}_2\\delta t + \\cdots + \\mathbf{F}_n\\cdot\\mathbf{V}_n\\delta t."
},
{
"math_id": 38,
"text": " \\delta W = \\sum_{i=1}^n \\mathbf{F}_i\\cdot (\\vec{\\omega}\\times(\\mathbf{X}_i -\\mathbf{d}) + \\mathbf{v})\\delta t. "
},
{
"math_id": 39,
"text": "\n\\begin{align}\n\\delta W & = \\left(\\sum_{i=1}^n \\mathbf{F}_i\\right) \\cdot\\mathbf{d}\\times \\vec{\\omega}\\delta t+ \\left(\\sum_{i=1}^n \\mathbf{F}_i\\right)\\cdot\\mathbf{v}\\delta t + \\left(\\sum_{i=1}^n \\mathbf{X}_i \\times\\mathbf{F}_i\\right) \\cdot \\vec{\\omega}\\delta t \\\\[4pt]\n& = \\left(\\sum_{i=1}^n \\mathbf{F}_i\\right) \\cdot(\\mathbf{v}+\\mathbf{d}\\times \\vec{\\omega}) \\delta t + \\left(\\sum_{i=1}^n \\mathbf{X}_i \\times\\mathbf{F}_i\\right) \\cdot\\vec{\\omega}\\delta t.\n\\end{align}\n"
},
{
"math_id": 40,
"text": " \\mathsf{T} = (\\vec{\\omega},\\mathbf{d}\\times \\vec{\\omega} +\\mathbf{v})=(\\mathbf{T},\\mathbf{T}^\\circ),\\quad\\mathsf{W} = \\left(\\sum_{i=1}^n \\mathbf{F}_i, \\sum_{i=1}^n \\mathbf{X}_i \\times\\mathbf{F}_i\\right) = (\\mathbf{W},\\mathbf{W}^\\circ), "
},
{
"math_id": 41,
"text": "\\delta W = (\\mathbf{W}\\cdot\\mathbf{T}^\\circ + \\mathbf{W}^\\circ \\cdot\\mathbf{T})\\delta t."
},
{
"math_id": 42,
"text": "\\delta W = (\\mathbf{W}\\cdot\\mathbf{T}^\\circ + \\mathbf{W}^\\circ \\cdot\\mathbf{T})\\delta t = \\mathsf{W}[\\Pi]\\mathsf{T}\\delta t,"
},
{
"math_id": 43,
"text": " [\\Pi] =\\begin{bmatrix} 0 & I \\\\ I & 0 \\end{bmatrix},"
},
{
"math_id": 44,
"text": "\\delta W =\\mathsf{W}[\\Pi]\\mathsf{T}\\delta t = 0,"
},
{
"math_id": 45,
"text": "\\check{\\mathsf{T}} = (\\mathbf{d}\\times \\vec{\\omega} +\\mathbf{v},\\vec{\\omega}),"
},
{
"math_id": 46,
"text": "\\delta W =\\mathsf{W}\\cdot\\check{\\mathsf{T}}\\delta t."
},
{
"math_id": 47,
"text": "\\delta W =\\mathsf{W}\\cdot\\check{\\mathsf{T}}\\delta t= 0,"
}
]
| https://en.wikipedia.org/wiki?curid=1318037 |
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