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1516108
|
Texture (chemistry)
|
In physical chemistry and materials science, texture is the distribution of crystallographic orientations of a polycrystalline sample (it is also part of the geological fabric). A sample in which these orientations are fully random is said to have no distinct texture. If the crystallographic orientations are not random, but have some preferred orientation, then the sample has a weak, moderate or strong texture. The degree is dependent on the percentage of crystals having the preferred orientation.
Texture is seen in almost all engineered materials, and can have a great influence on materials properties. The texture forms in materials during thermo-mechanical processes, for example during production processes e.g. rolling. Consequently, the rolling process is often followed by a heat treatment to reduce the amount of unwanted texture. Controlling the production process in combination with the characterization of texture and the material's microstructure help to determine the materials properties, i.e. the "processing-microstructure-texture-property relationship". Also, geologic rocks show texture due to their thermo-mechanic history of formation processes.
One extreme case is a complete lack of texture: a solid with perfectly random crystallite orientation will have isotropic properties at length scales sufficiently larger than the size of the crystallites. The opposite extreme is a perfect single crystal, which likely has anisotropic properties by geometric necessity.
Characterization and representation.
Texture can be determined by various methods. Some methods allow a quantitative analysis of the texture, while others are only qualitative. Among the quantitative techniques, the most widely used is X-ray diffraction using texture goniometers, followed by the electron backscatter diffraction (EBSD) method in scanning electron microscopes. Qualitative analysis can be done by Laue photography, simple X-ray diffraction or with a polarized microscope. Neutron and synchrotron high-energy X-ray diffraction are suitable for determining textures of bulk materials and in situ analysis, whereas laboratory x-ray diffraction instruments are more appropriate for analyzing textures of thin films.
Texture is often represented using a pole figure, in which a specified crystallographic axis (or pole) from each of a representative number of crystallites is plotted in a stereographic projection, along with directions relevant to the material's processing history. These directions define the so-called sample reference frame and are, because the investigation of textures started from the cold working of metals, usually referred to as the rolling direction "RD", the transverse direction "TD" and the normal direction "ND". For drawn metal wires the cylindrical fiber axis turned out as the sample direction around which preferred orientation is typically observed (see below).
Common textures.
There are several textures that are commonly found in processed (cubic) materials. They are named either by the scientist that discovered them, or by the material they are most found in. These are given in Miller indices for simplification purposes.
Orientation distribution function.
The full 3D representation of crystallographic texture is given by the orientation distribution function (ODF) which can be achieved through evaluation of a set of pole figures or diffraction patterns. Subsequently, all pole figures can be derived from the ODF.
The ODF is defined as the volume fraction of grains with a certain orientation formula_0.
formula_1
The orientation formula_0 is normally identified using three Euler angles. The Euler angles then describe the transition from the sample’s reference frame into the crystallographic reference frame of each individual grain of the polycrystal. One thus ends up with a large set of different Euler angles, the distribution of which is described by the ODF.
The orientation distribution function, ODF, cannot be measured directly by any technique. Traditionally both X-ray diffraction and EBSD may collect pole figures. Different methodologies exist to obtain the ODF from the pole figures or data in general. They can be classified based on how they represent the ODF. Some represent the ODF as a function, sum of functions or expand it in a series of harmonic functions. Others, known as discrete methods, divide the ODF space in cells and focus on determining the value of the ODF in each cell.
Origins.
In wire and fiber, all crystals tend to have nearly identical orientation in the axial direction, but nearly random radial orientation. The most familiar exceptions to this rule are fiberglass, which has no crystal structure, and carbon fiber, in which the crystalline anisotropy is so great that a good-quality filament will be a distorted single crystal with approximately cylindrical symmetry (often compared to a jelly roll). Single-crystal fibers are also not uncommon.
The making of metal sheet often involves compression in one direction and, in efficient rolling operations, tension in another, which can orient crystallites in both axes by a process known as grain flow. However, cold work destroys much of the crystalline order, and the new crystallites that arise with annealing usually have a different texture. Control of texture is extremely important in the making of silicon steel sheet for transformer cores (to reduce magnetic hysteresis) and of aluminium cans (since deep drawing requires extreme and relatively uniform plasticity).
Texture in ceramics usually arises because the crystallites in a slurry have shapes that depend on crystalline orientation, often needle- or plate-shaped. These particles align themselves as water leaves the slurry, or as clay is formed.
Casting or other fluid-to-solid transitions (i.e., thin-film deposition) produce textured solids when there is enough time and activation energy for atoms to find places in existing crystals, rather than condensing as an amorphous solid or starting new crystals of random orientation. Some facets of a crystal (often the close-packed planes) grow more rapidly than others, and the crystallites for which one of these planes faces in the direction of growth will usually out-compete crystals in other orientations. In the extreme, only one crystal will survive after a certain length: this is exploited in the Czochralski process (unless a seed crystal is used) and in the casting of turbine blades and other creep-sensitive parts.
Texture and materials properties.
Material properties such as strength, chemical reactivity, stress corrosion cracking resistance, weldability, deformation behavior, resistance to radiation damage, and magnetic susceptibility can be highly dependent on the material’s texture and related changes in microstructure. In many materials, properties are texture-specific, and development of unfavorable textures when the material is fabricated or in use can create weaknesses that can initiate or exacerbate failures. Parts can fail to perform due to unfavorable textures in their component materials. Failures can correlate with the crystalline textures formed during fabrication or use of that component. Consequently, consideration of textures that are present in and that could form in engineered components while in use can be a critical when making decisions about the selection of some materials and methods employed to manufacture parts with those materials. When parts fail during use or abuse, understanding the textures that occur within those parts can be crucial to meaningful interpretation of failure analysis data.
Thin film textures.
As the result of substrate effects producing preferred crystallite orientations, pronounced textures tend to occur in thin films. Modern technological devices to a large extent rely on polycrystalline thin films with thicknesses in the nanometer and micrometer ranges. This holds, for instance, for all microelectronic and most optoelectronic systems or sensoric and superconducting layers. Most thin film textures may be categorized as one of two different types: (1) for so-called fiber textures the orientation of a certain lattice plane is preferentially parallel to the substrate plane; (2) in biaxial textures the in-plane orientation of crystallites also tend to align with respect to the sample. The latter phenomenon is accordingly observed in nearly epitaxial growth processes, where certain crystallographic axes of crystals in the layer tend to align along a particular crystallographic orientation of the (single-crystal) substrate.
Tailoring the texture on demand has become an important task in thin film technology. In the case of oxide compounds intended for transparent conducting films or surface acoustic wave (SAW) devices, for instance, the polar axis should be aligned along the substrate normal. Another example is given by cables from high-temperature superconductors that are being developed as oxide multilayer systems deposited on metallic ribbons. The adjustment of the biaxial texture in YBa2Cu3O7−δ layers turned out as the decisive prerequisite for achieving sufficiently large critical currents.
The degree of texture is often subjected to an evolution during thin film growth and the most pronounced textures are only obtained after the layer has achieved a certain thickness. Thin film growers thus require information about the texture profile or the texture gradient in order to optimize the deposition process. The determination of texture gradients by x-ray scattering, however, is not straightforward, because different depths of a specimen contribute to the signal. Techniques that allow for the adequate deconvolution of diffraction intensity were developed only recently.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\boldsymbol{g}"
},
{
"math_id": 1,
"text": "\\text{ODF}(\\boldsymbol{g})=\\frac{1}{V} \\frac{dV(\\boldsymbol{g})}{d g}."
}
] |
https://en.wikipedia.org/wiki?curid=1516108
|
151641
|
Line drawing algorithm
|
Methods of approximating line segments for pixel displays
In computer graphics, a line drawing algorithm is an algorithm for approximating a line segment on discrete graphical media, such as pixel-based displays and printers. On such media, line drawing requires an approximation (in nontrivial cases). Basic algorithms rasterize lines in one color. A better representation with multiple color gradations requires an advanced process, spatial anti-aliasing.
On continuous media, by contrast, no algorithm is necessary to draw a line. For example, cathode-ray oscilloscopes use analog phenomena to draw lines and curves.
Single color line drawing algorithms.
Single color line drawing algorithms involve drawing lines in a single foreground color onto a background. They are well-suited for usage with monochromatic displays.
The starting point and end point of the desired line are usually given in integer coordinates, so that they lie directly on the points considered by the algorithm. Because of this, most algorithms are formulated only for such starting points and end points.
Simple Methods.
The simplest method of drawing a line involves directly calculating pixel positions from a line equation. Given a starting point formula_0 and an end point formula_1, points on the line fulfill the equation formula_2, with formula_3 being the slope of the line. The line can then be drawn by evaluating this
equation via a simple loop, as shown in the following pseudocode:
dx = x2 − x1
dy = y2 − y1
m = dy/dx
for x from x1 to x2 do
y = m × (x − x1) + y1
plot(x, y)
Here, the points have already been ordered so that formula_4.
This algorithm is unnecessarily slow because the loop involves a multiplication, which is significantly slower than addition or subtraction on most devices. A faster method can be achieved by viewing the Difference between two consecutive steps:
Therefore, it is enough to simply start at the point formula_0 and then increase formula_5 by formula_6 once on every iteration of the loop. This algorithm is known as a Digital differential analyzer.
Because rounding formula_5 to the nearest whole number is equivalent to rounding formula_7 down, rounding can be avoided by using an additional control variable that is initialized with the value 0.5. formula_6 is added to this variable on every iteration. Then, if this variable exceeds 1.0, formula_5 is incremented by 1 and the control variable is decremented by 1. This allows the algorithm to avoid rounding and only use integer operations. However, for short lines, this faster loop does not make up for the expensive division formula_3, which is still necessary at the beginning.
These algorithm works just fine when formula_8 (i.e., slope is less than or equal to 1), but if formula_9 (i.e., slope greater than 1), the line becomes quite sparse with many gaps, and in the limiting case of formula_10, a division by zero exception will occur.
Issues.
In certain situations, single color line drawing algorithms run into issues:
Inconsistent brightness.
When drawing lines of the same length with differing slopes, different numbers of pixels are drawn. This leads to steeper lines being made up of fewer pixels than flatter lines of the same length, which leads to the steeper line appearing brighter than the flat line. This problem is unavoidable on monochromatic devices.
Clipping.
Clipping is an operation that limits rasterisation to a limited, usually rectangular, area. This is done by moving the start- and end points of the given line to the borders of this area if they lie outside of it. Generally, this leads to the coordinates of these points no longer being integer numbers. If these coordinates are simply rounded, the resulting line will have a different slope than intended. For this issue to be avoided, additional tests are necessary after clipping.
Antialiasing.
The biggest issue of single color line drawing algorithms is that they lead to lines with a rough, jagged appearance. On devices capable of displaying multiple levels of brightness, this issue can be avoided through antialiasing. For this, lines are usually viewed in a two-dimensional form, generally as a rectangle with a desired thickness. To draw these lines, points lying near this rectangle have to be considered.
Gupta and Sproull algorithm.
The Gupta-Sproull algorithm is based on Bresenham's line algorithm but adds antialiasing.
An optimized variant of the Gupta-Sproull algorithm can be written in pseudocode as follows:
DrawLine(x1, x2, y1, y2) {
x = x1;
y = y1;
dx = x2 − x1;
dy = y2 − y1;
d = 2 * dy − dx; // discriminator
// Euclidean distance of point (x,y) from line (signed)
D = 0;
// Euclidean distance between points (x1, y1) and (x2, y2)
length = sqrt(dx * dx + dy * dy);
sin = dx / length;
cos = dy / length;
while (x <= x2) {
IntensifyPixels(x, y − 1, D + cos);
IntensifyPixels(x, y, D);
IntensifyPixels(x, y + 1, D − cos);
x = x + 1
if (d <= 0) {
D = D + sin;
d = d + 2 * dy;
} else {
D = D + sin − cos;
d = d + 2 * (dy − dx);
y = y + 1;
The IntensifyPixels(x,y,r) function takes a radial line transformation and sets the intensity of the pixel (x,y) with the value of a cubic polynomial that depends on the pixel's distance r from the line.
Optimizations.
Line drawing algorithms can be made more efficient through approximate methods, through usage of direct hardware implementations, and through parallelization. Such optimizations become necessary when rendering a large number of lines in real time.
Approximate methods.
Boyer and Bourdin introduced an approximation algorithm that colors pixels lying directly under the ideal line. A line rendered in this way exhibits some special properties that may be taken advantage of. For example, in cases like this, sections of the line are periodical. This results in an algorithm which is significantly faster than precise variants, especially for longer lines. A worsening in quality is only visible on lines with very low steepness.
Parallelization.
A simple way to parallelize single-color line rasterization is to let multiple line-drawing algorithms draw offset pixels of a certain distance from each other. Another method involves dividing the line into multiple sections of approximately equal length, which are then assigned to different processors for rasterization. The main problem is finding the correct start points and end points of these sections.
Algorithms for massively parallel processor architectures with thousands of processors also exist. In these, every pixel out of a grid of pixels is assigned to a single processor, which then decides whether the given pixel should be colored or not.
Special memory hierarchies have been developed to accelerate memory access during rasterization. These may, for example, divide memory into multiple cells, which then each render a section of the line independently.
Rasterization involving Antialiasing can be supported by dedicated Hardware as well.
Related Problems.
Lines may not only be drawn 8-connected, but also 4-connected, meaning that only horizontal and vertical steps are allowed, while diagonal steps are prohibited. Given a raster of square pixels, this leads to every square containing a part of the line being colored. A generalization of 4-connected line drawing methods to three dimensions is used when dealing with voxel grids, for example in optimized ray tracing, where it can determine the voxels that a given ray crosses.
Line drawing algorithms distribute diagonal steps approximately evenly. Thus, line drawing algorithms may also be used to evenly distribute points with integer coordinates in a given interval.
Possible applications of this method include linear interpolation or downsampling in signal processing. There are also parallels to the Euclidean algorithm, as well as Farey sequences and a number of related mathematical constructs.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "(x_1, y_1)"
},
{
"math_id": 1,
"text": "(x_2, y_2)"
},
{
"math_id": 2,
"text": "y = m(x-x_1) + y_1"
},
{
"math_id": 3,
"text": "\\textstyle m = \\frac{\\Delta y}{\\Delta x} = \\frac{y_2-y_1}{x_2-x_1}"
},
{
"math_id": 4,
"text": "x_2 > x_1"
},
{
"math_id": 5,
"text": " y "
},
{
"math_id": 6,
"text": " m "
},
{
"math_id": 7,
"text": " y+0.5 "
},
{
"math_id": 8,
"text": "dx \\geq dy"
},
{
"math_id": 9,
"text": "dx < dy"
},
{
"math_id": 10,
"text": "dx = 0"
}
] |
https://en.wikipedia.org/wiki?curid=151641
|
15164481
|
Souček space
|
In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček. One of their main advantages is that they offer a way to deal with the fact that the Sobolev space "W"1,1 is not a reflexive space; since "W"1,1 is not reflexive, it is not always true that a bounded sequence has a weakly convergent subsequence, which is a desideratum in many applications.
Definition.
Let Ω be a bounded domain in "n"-dimensional Euclidean space with smooth boundary. The Souček space "W"1,"μ"(Ω; R"m") is defined to be the space of all ordered pairs ("u", "v"), where
formula_0
and
formula_1
weakly-∗ in the space of all R"m"×"n"-valued regular Borel measures on the closure of Ω.
formula_2
i.e. the sum of the "L"1 and total variation norms of the two components.
|
[
{
"math_id": 0,
"text": "\\lim_{k \\to \\infty} u_{k} = u \\mbox{ in } L^{1} (\\Omega; \\mathbf{R}^{m})"
},
{
"math_id": 1,
"text": "\\lim_{k \\to \\infty} \\nabla u_{k} = v"
},
{
"math_id": 2,
"text": "\\| (u, v) \\| := \\| u \\|_{L^{1}} + \\| v \\|_{M},"
}
] |
https://en.wikipedia.org/wiki?curid=15164481
|
151652
|
Normal
|
Normal(s) or The Normal(s) may refer to:
<templatestyles src="Template:TOC_right/styles.css" />
See also.
Topics referred to by the same term
<templatestyles src="Dmbox/styles.css" />
This page lists associated with the title .
|
[
{
"math_id": 0,
"text": "T_4"
}
] |
https://en.wikipedia.org/wiki?curid=151652
|
15165446
|
Nullor
|
A nullor is a theoretical two-port network consisting of a nullator at its input and a norator at its output. Nullors represent an ideal amplifier, having infinite current, voltage, transconductance and transimpedance gain. Its transmission parameters are all zero, that is, its input–output behavior is summarized with the matrix equation
formula_0
In negative-feedback circuits, the circuit surrounding the nullor determines the nullor output in such a way as to force the nullor input to zero.
Inserting a nullor in a circuit schematic imposes mathematical constraints on how that circuit must behave, forcing the circuit itself to adopt whatever arrangements are needed to meet the conditions. For example, an ideal operational amplifier can be modeled using a nullor, and the textbook analysis of a feedback circuit using an ideal op-amp uses the mathematical conditions imposed by the nullor to analyze the circuit surrounding the op-amp.
Example: voltage-controlled current sink.
Figure 1 shows a voltage-controlled current sink. The sink is intended to draw the same current "i"OUT regardless of the applied voltage "V"CC at the output. The value of current drawn is to be set by the input voltage "v"IN. Here the sink is to be analyzed by idealizing the op amp as a nullor.
Using properties of the input nullator portion of the nullor, the input voltage across the op amp input terminals is zero. Consequently, the voltage across reference resistor "R"R is the applied voltage "v"IN, making the current in "R"R simply "v"IN/"R"R. Again using the nullator properties, the input current to the nullor is zero. Consequently, Kirchhoff's current law at the emitter provides an emitter current of "v"IN/"R"R. Using properties of the norator output portion of the nullor, the nullor provides whatever current is demanded of it, regardless of the voltage at its output. In this case, it provides the transistor base current "i"B. Thus, Kirchhoff's current law applied to the transistor as a whole provides the output current drawn through resistor "R"C as
formula_1
where the base current of the bipolar transistor "i"B is normally negligible provided the transistor remains in active mode. That is, based upon the idealization of a nullor, the output current is controlled by the user-applied input voltage "v"IN and the designer's choice for the reference resistor "R"R.
The purpose of the transistor in the circuit is to reduce the portion of the current in "R"R supplied by the op-amp. Without the transistor, the current through "R"C would be "i"OUT = ("V"CC − "v"IN)/"R"C, which interferes with the design goal of independence of "i"OUT from "V"CC. Another practical advantage of the transistor is that the op amp must deliver only the small transistor base current, which is unlikely to tax the op amp's current delivery capability. Of course, only real op amps are current-limited, not nullors.
The remaining variation of the current with the voltage "V"CC is due to the Early effect, which causes the β of the transistor to change with its collector-to-base voltage "V"CB according to the relation β = β0(1 + "V"CB/"V"A), where "V"A is the so-called Early voltage. Analysis based upon a nullor leads to the output resistance of this current sink as "R"out = "r"O(β + 1) + "R"C, where "r"O is the small-signal transistor output resistance given by "r"O = ("V"A + "V"CB)/"i"out. See current mirror for the analysis.
Use of the nullor idealization allows design of the circuitry around the op-amp. The practical problem remains of designing an op-amp that behaves like a nullor.
|
[
{
"math_id": 0,
"text": " \n\\begin{pmatrix}\nv_1\\\\\ni_1\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n 0 & 0 \\\\ \n 0 & 0 \n\\end{pmatrix}\n\\begin{pmatrix}\nv_2\\\\\ni_2\n\\end{pmatrix}\n"
},
{
"math_id": 1,
"text": " i_\\text{OUT} = \\frac {v_\\text{IN}} {R_\\text{R}} - i_\\text{B} "
}
] |
https://en.wikipedia.org/wiki?curid=15165446
|
15168516
|
Bilateral filter
|
Smoothing filler for images
A bilateral filter is a non-linear, edge-preserving, and noise-reducing smoothing filter for images. It replaces the intensity of each pixel with a weighted average of intensity values from nearby pixels. This weight can be based on a Gaussian distribution. Crucially, the weights depend not only on Euclidean distance of pixels, but also on the radiometric differences (e.g., range differences, such as color intensity, depth distance, etc.). This preserves sharp edges.
Definition.
The bilateral filter is defined as
formula_0
and normalization term, formula_1, is defined as
formula_2
where
formula_3 is the filtered image;
formula_4 is the original input image to be filtered;
formula_5 are the coordinates of the current pixel to be filtered;
formula_6 is the window centered in formula_5, so formula_7 is another pixel;
formula_8 is the range kernel for smoothing differences in intensities (this function can be a Gaussian function);
formula_9 is the spatial (or domain) kernel for smoothing differences in coordinates (this function can be a Gaussian function).
The weight formula_10 is assigned using the spatial closeness (using the spatial kernel formula_9) and the intensity difference (using the range kernel formula_8). Consider a pixel located at formula_11 that needs to be denoised in image using its neighbouring pixels and one of its neighbouring pixels is located at formula_12. Then, assuming the range and spatial kernels to be Gaussian kernels, the weight assigned for pixel formula_12 to denoise the pixel formula_11 is given by
formula_13
where σd and σr are smoothing parameters, and "I"("i", "j") and "I"("k", "l") are the intensity of pixels formula_11 and formula_14 respectively.
After calculating the weights, normalize them:
formula_15
where formula_16 is the denoised intensity of pixel formula_11.
Limitations.
The bilateral filter in its direct form can introduce several types of image artifacts:
There exist several extensions to the filter that deal with these artifacts, like the scaled bilateral filter that uses downscaled image for computing the weights. Alternative filters, like the "guided filter", have also been proposed as an efficient alternative without these limitations.
Implementations.
Adobe Photoshop implements a bilateral filter in its "surface blur" tool. GIMP implements a bilateral filter in its "Filters → Blur" tools; and it is called "Selective Gaussian Blur". The free G'MIC plugin "Repair → Smooth [bilateral]" for GIMP adds more control.
A simple trick to efficiently implement a bilateral filter is to exploit Poisson-disk subsampling.
Related models.
The bilateral filter has been shown to be an application of the short time kernel of the Beltrami flow
that was introduced as an edge preserving selective smoothing mechanism before the bilateral filter.
Other edge-preserving smoothing filters include: anisotropic diffusion, weighted least squares, edge-avoiding wavelets, geodesic editing, guided filtering, and domain transforms.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\nI^\\text{filtered}(x) = \\frac{1}{W_p} \\sum_{x_i \\in \\Omega} I(x_i)f_r(\\|I(x_i) - I(x)\\|)g_s(\\|x_i - x\\|),\n"
},
{
"math_id": 1,
"text": "{W_p}"
},
{
"math_id": 2,
"text": "\nW_p = \\sum_{x_i \\in \\Omega}{f_r(\\|I(x_i) - I(x)\\|)g_s(\\|x_i - x\\|)}\n"
},
{
"math_id": 3,
"text": "I^\\text{filtered} "
},
{
"math_id": 4,
"text": "I"
},
{
"math_id": 5,
"text": "x"
},
{
"math_id": 6,
"text": "\\Omega"
},
{
"math_id": 7,
"text": "x_i \\in \\Omega"
},
{
"math_id": 8,
"text": "f_r"
},
{
"math_id": 9,
"text": "g_s"
},
{
"math_id": 10,
"text": "W_p"
},
{
"math_id": 11,
"text": "(i, j)"
},
{
"math_id": 12,
"text": "(k, l)"
},
{
"math_id": 13,
"text": "\nw(i, j, k, l) = \\exp\\left(-\\frac{(i - k)^2 + (j - l)^2}{2 \\sigma_d^2} - \\frac{\\|I(i, j) - I(k, l)\\|^2}{2 \\sigma_r^2}\\right),\n"
},
{
"math_id": 14,
"text": " (k, l)"
},
{
"math_id": 15,
"text": "\nI_D(i, j) = \\frac{\\sum_{k, l} I(k, l) w(i, j, k, l)}{\\sum_{k, l} w(i, j, k, l)},\n"
},
{
"math_id": 16,
"text": "I_D"
}
] |
https://en.wikipedia.org/wiki?curid=15168516
|
1516916
|
Magnetic core
|
Object used to guide and confine magnetic fields
A magnetic core is a piece of magnetic material with a high magnetic permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, generators, inductors, loudspeakers, magnetic recording heads, and magnetic assemblies. It is made of ferromagnetic metal such as iron, or ferrimagnetic compounds such as ferrites. The high permeability, relative to the surrounding air, causes the magnetic field lines to be concentrated in the core material. The magnetic field is often created by a current-carrying coil of wire around the core.
The use of a magnetic core can increase the strength of magnetic field in an electromagnetic coil by a factor of several hundred times what it would be without the core. However, magnetic cores have side effects which must be taken into account. In alternating current (AC) devices they cause energy losses, called core losses, due to hysteresis and eddy currents in applications such as transformers and inductors. "Soft" magnetic materials with low coercivity and hysteresis, such as silicon steel, or ferrite, are usually used in cores.
Core materials.
An electric current through a wire wound into a coil creates a magnetic field through the center of the coil, due to Ampere's circuital law. Coils are widely used in electronic components such as electromagnets, inductors, transformers, electric motors and generators. A coil without a magnetic core is called an "air core" coil. Adding a piece of ferromagnetic or ferrimagnetic material in the center of the coil can increase the magnetic field by hundreds or thousands of times; this is called a magnetic core. The field of the wire penetrates the core material, magnetizing it, so that the strong magnetic field of the core adds to the field created by the wire. The amount that the magnetic field is increased by the core depends on the magnetic permeability of the core material. Because side effects such as eddy currents and hysteresis can cause frequency-dependent energy losses, different core materials are used for coils used at different frequencies.
In some cases the losses are undesirable and with very strong fields saturation can be a problem, and an 'air core' is used. A former may still be used; a piece of material, such as plastic or a composite, that may not have any significant magnetic permeability but which simply holds the coils of wires in place.
Solid metals.
Soft iron.
"Soft" (annealed) iron is used in magnetic assemblies, direct current (DC) electromagnets and in some electric motors; and it can create a concentrated field that is as much as 50,000 times more intense than an air core.
Iron is desirable to make magnetic cores, as it can withstand high levels of magnetic field without saturating (up to 2.16 teslas at ambient temperature.) Annealed iron is used because, unlike "hard" iron, it has low coercivity and so does not remain magnetised when the field is removed, which is often important in applications where the magnetic field is required to be repeatedly switched.
Due to the electrical conductivity of the metal, when a solid one-piece metal core is used in alternating current (AC) applications such as transformers and inductors, the changing magnetic field induces large eddy currents circulating within it, closed loops of electric current in planes perpendicular to the field. The current flowing through the resistance of the metal heats it by Joule heating, causing significant power losses. Therefore, solid iron cores are not used in transformers or inductors, they are replaced by laminated or powdered iron cores, or nonconductive cores like ferrite.
Laminated silicon steel.
In order to reduce the eddy current losses mentioned above, most low frequency power transformers and inductors use laminated cores, made of stacks of thin sheets of silicon steel:
Lamination.
Laminated magnetic cores are made of stacks of thin iron sheets coated with an insulating layer, lying as much as possible parallel with the lines of flux. The layers of insulation serve as a barrier to eddy currents, so eddy currents can only flow in narrow loops within the thickness of each single lamination. Since the current in an eddy current loop is proportional to the area of the loop, this prevents most of the current from flowing, reducing eddy currents to a very small level. Since power dissipated is proportional to the square of the current, breaking a large core into narrow laminations reduces the power losses drastically. From this, it can be seen that the thinner the laminations, the lower the eddy current losses.
Silicon alloying.
A small addition of silicon to iron (around 3%) results in a dramatic increase of the resistivity of the metal, up to four times higher. The higher resistivity reduces the eddy currents, so silicon steel is used in transformer cores. Further increase in silicon concentration impairs the steel's mechanical properties, causing difficulties for rolling due to brittleness.
Among the two types of silicon steel, grain-oriented (GO) and grain non-oriented (GNO), GO is most desirable for magnetic cores. It is anisotropic, offering better magnetic properties than GNO in one direction. As the magnetic field in inductor and transformer cores is always along the same direction, it is an advantage to use grain oriented steel in the preferred orientation. Rotating machines, where the direction of the magnetic field can change, gain no benefit from grain-oriented steel.
Special alloys.
A family of specialized alloys exists for magnetic core applications. Examples are mu-metal, permalloy, and supermalloy. They can be manufactured as stampings or as long ribbons for tape wound cores. Some alloys, e.g. Sendust, are manufactured as powder and sintered to shape.
Many materials require careful heat treatment to reach their magnetic properties, and lose them when subjected to mechanical or thermal abuse. For example, the permeability of mu-metal increases about 40 times after annealing in hydrogen atmosphere in a magnetic field; subsequent sharper bends disrupt its grain alignment, leading to localized loss of permeability; this can be regained by repeating the annealing step.
Vitreous metal.
Amorphous metal is a variety of alloys (e.g. Metglas) that are non-crystalline or glassy. These are being used to create high-efficiency transformers. The materials can be highly responsive to magnetic fields for low hysteresis losses, and they can also have lower conductivity to reduce eddy current losses. Power utilities are currently making widespread use of these transformers for new installations. High mechanical strength and corrosion resistance are also common properties of metallic glasses which are positive for this application.
Powdered metals.
Powder cores consist of metal grains mixed with a suitable organic or inorganic binder, and pressed to desired density. Higher density is achieved with higher pressure and lower amount of binder. Higher density cores have higher permeability, but lower resistance and therefore higher losses due to eddy currents. Finer particles allow operation at higher frequencies, as the eddy currents are mostly restricted to within the individual grains. Coating of the particles with an insulating layer, or their separation with a thin layer of a binder, lowers the eddy current losses. Presence of larger particles can degrade high-frequency performance. Permeability is influenced by the spacing between the grains, which form distributed air gap; the less gap, the higher permeability and the less-soft saturation. Due to large difference of densities, even a small amount of binder, weight-wise, can significantly increase the volume and therefore intergrain spacing.
Lower permeability materials are better suited for higher frequencies, due to balancing of core and winding losses.
The surface of the particles is often oxidized and coated with a phosphate layer, to provide them with mutual electrical insulation.
Iron.
Powdered iron is the cheapest material. It has higher core loss than the more advanced alloys, but this can be compensated for by making the core bigger; it is advantageous where cost is more important than mass and size. Saturation flux of about 1 to 1.5 tesla. Relatively high hysteresis and eddy current loss, operation limited to lower frequencies (approx. below 100 kHz). Used in energy storage inductors, DC output chokes, differential mode chokes, triac regulator chokes, chokes for power factor correction, resonant inductors, and pulse and flyback transformers.
The binder used is usually epoxy or other organic resin, susceptible to thermal aging. At higher temperatures, typically above 125 °C, the binder degrades and the core magnetic properties may change. With more heat-resistant binders the cores can be used up to 200 °C.
Iron powder cores are most commonly available as toroids. Sometimes as E, EI, and rods or blocks, used primarily in high-power and high-current parts.
Carbonyl iron is significantly more expensive than hydrogen-reduced iron.
Carbonyl iron.
Powdered cores made of carbonyl iron, a highly pure iron, have high stability of parameters across a wide range of temperatures and magnetic flux levels, with excellent Q factors between 50 kHz and 200 MHz. Carbonyl iron powders are basically constituted of micrometer-size spheres of iron coated in a thin layer of electrical insulation. This is equivalent to a microscopic laminated magnetic circuit (see silicon steel, above), hence reducing the eddy currents, particularly at very high frequencies. Carbonyl iron has lower losses than hydrogen-reduced iron, but also lower permeability.
A popular application of carbonyl iron-based magnetic cores is in high-frequency and broadband inductors and transformers, especially higher power ones.
Carbonyl iron cores are often called "RF cores".
The as-prepared particles, "E-type"and have onion-like skin, with concentric shells separated with a gap. They contain significant amount of carbon. They behave as much smaller than what their outer size would suggest. The "C-type" particles can be prepared by heating the E-type ones in hydrogen atmosphere at 400 °C for prolonged time, resulting in carbon-free powders.
Hydrogen-reduced iron.
Powdered cores made of hydrogen reduced iron have higher permeability but lower Q than carbonyl iron. They are used mostly for electromagnetic interference filters and low-frequency chokes, mainly in switched-mode power supplies.
Hydrogen-reduced iron cores are often called "power cores".
MPP (molypermalloy).
An alloy of about 2% molybdenum, 81% nickel, and 17% iron. Very low core loss, low hysteresis and therefore low signal distortion. Very good temperature stability. High cost. Maximum saturation flux of about 0.8 tesla. Used in high-Q filters, resonant circuits, loading coils, transformers, chokes, etc.
The material was first introduced in 1940, used in loading coils to compensate capacitance in long telephone lines. It is usable up to about 200 kHz to 1 MHz, depending on vendor. It is still used in above-ground telephone lines, due to its temperature stability. Underground lines, where temperature is more stable, tend to use ferrite cores due to their lower cost.
High-flux (Ni-Fe).
An alloy of about 50–50% of nickel and iron. High energy storage, saturation flux density of about 1.5 tesla. Residual flux density near zero. Used in applications with high DC current bias (line noise filters, or inductors in switching regulators) or where low residual flux density is needed (e.g. pulse and flyback transformers, the high saturation is suitable for unipolar drive), especially where space is constrained. The material is usable up to about 200 kHz.
Sendust, KoolMU.
An alloy of 6% aluminium, 9% silicon, and 85% iron. Core losses higher than MPP. Very low magnetostriction, makes low audio noise. Loses inductance with increasing temperature, unlike the other materials; can be exploited by combining with other materials as a composite core, for temperature compensation. Saturation flux of about 1 tesla. Good temperature stability. Used in switching power supplies, pulse and flyback transformers, in-line noise filters, swing chokes, and in filters in phase-fired controllers (e.g. dimmers) where low acoustic noise is important.
Absence of nickel results in easier processing of the material and its lower cost than both high-flux and MPP.
The material was invented in Japan in 1936. It is usable up to about 500 kHz to 1 MHz, depending on vendor.
Nanocrystalline.
A nanocrystalline alloy of a standard iron-boron-silicon alloy, with addition of smaller amounts of copper and niobium. The grain size of the powder reaches down to 10–100 nanometers. The material has very good performance at lower frequencies. It is used in chokes for inverters and in high power applications. It is available under names like e.g. Nanoperm, Vitroperm, Hitperm and Finemet.
Ceramics.
Ferrite.
Ferrite ceramics are used for high-frequency applications. The ferrite materials can be engineered with a wide range of parameters. As ceramics, they are essentially insulators, which prevents eddy currents, although losses such as hysteresis losses can still occur.
Air.
A coil not containing a magnetic core is called an "air core". This includes coils wound on a plastic or ceramic form in addition to those made of stiff wire that are self-supporting and have air inside them. Air core coils generally have a much lower inductance than similarly sized ferromagnetic core coils, but are used in radio frequency circuits to prevent energy losses called core losses that occur in magnetic cores. The absence of normal core losses permits a higher Q factor, so air core coils are used in high frequency resonant circuits, such as up to a few megahertz. However, losses such as proximity effect and dielectric losses are still present. Air cores are also used when field strengths above around 2 Tesla are required as they are not subject to saturation.
Commonly used structures.
Straight cylindrical rod.
Most commonly made of ferrite or powdered iron, and used in radios especially for tuning an inductor. The coil is wound around the rod, or a coil form with the rod inside. Moving the rod in or out of the coil changes the flux through the coil, and can be used to adjust the inductance. Often the rod is threaded to allow adjustment with a screwdriver. In radio circuits, a blob of wax or resin is used once the inductor has been tuned to prevent the core from moving.
The presence of the high permeability core increases the inductance, but the magnetic field lines must still pass through the air from one end of the rod to the other. The air path ensures that the inductor remains linear. In this type of inductor radiation occurs at the end of the rod and electromagnetic interference may be a problem in some circumstances.
Single "I" core.
Like a cylindrical rod but is square, rarely used on its own.
This type of core is most likely to be found in car ignition coils.
"C" or "U" core.
"U" and "C"-shaped cores are used with "I" or another "C" or "U" core to make a square closed core, the simplest closed core shape. Windings may be put on one or both legs of the core.
"E" core.
E-shaped core are more symmetric solutions to form a closed magnetic system. Most of the time, the electric circuit is wound around the center leg, whose section area is twice that of each individual outer leg. In 3-phase transformer cores, the legs are of equal size, and all three legs are wound.
"E" and "I" core.
Sheets of suitable iron stamped out in shapes like the (sans-serif) letters "E" and "I", are stacked with the "I" against the open end of the "E" to form a 3-legged structure. Coils can be wound around any leg, but usually the center leg is used. This type of core is frequently used for power transformers, autotransformers, and inductors.
Pair of "E" cores.
Again used for iron cores. Similar to using an "E" and "I" together, a pair of "E" cores will accommodate a larger coil former and can produce a larger inductor or transformer. If an air gap is required, the centre leg of the "E" is shortened so that the air gap sits in the middle of the coil to minimize fringing and reduce electromagnetic interference.
Planar core.
A planar core consists of two flat pieces of magnetic material, one above and one below the coil. It is typically used with a flat coil that is part of a printed circuit board. This design is excellent for mass production and allows a high power, small volume transformer to be constructed for low cost. It is not as ideal as either a pot core or toroidal core but costs less to produce.
Pot core.
Usually ferrite or similar. This is used for inductors and transformers. The shape of a pot core is round with an internal hollow that almost completely encloses the coil. Usually a pot core is made in two halves which fit together around a coil former (bobbin). This design of core has a shielding effect, preventing radiation and reducing electromagnetic interference.
Toroidal core.
This design is based on a toroid (the same shape as a doughnut). The coil is wound through the hole in the torus and around the outside. An ideal coil is distributed evenly all around the circumference of the torus. The symmetry of this geometry creates a magnetic field of circular loops inside the core, and the lack of sharp bends will constrain virtually all of the field to the core material. This not only makes a highly efficient transformer, but also reduces the electromagnetic interference radiated by the coil.
It is popular for applications where the desirable features are: high specific power per mass and volume, low mains hum, and minimal electromagnetic interference. One such application is the power supply for a hi-fi audio amplifier. The main drawback that limits their use for general purpose applications is the inherent difficulty of winding wire through the center of a torus.
Unlike a split core (a core made of two elements, like a pair of "E" cores), specialized machinery is required for automated winding of a toroidal core. Toroids have less audible noise, such as mains hum, because the magnetic forces do not exert bending moment on the core. The core is only in compression or tension, and the circular shape is more stable mechanically.
Ring or bead.
The ring is essentially identical in shape and performance to the toroid, except that inductors commonly pass only through the center of the core, without wrapping around the core multiple times.
The ring core may also be composed of two separate C-shaped hemispheres secured together within a plastic shell, permitting it to be placed on finished cables with large connectors already installed, that would prevent threading the cable through the small inner diameter of a solid ring.
AL value.
The AL value of a core configuration is frequently specified by manufacturers. The relationship between inductance and AL number in the linear portion of the magnetisation curve is defined to be:
formula_0
where n is the number of turns, L is the inductance (e.g. in nH) and AL is expressed in inductance per turn squared (e.g. in nH/n2).
Core loss.
When the core is subjected to a "changing" magnetic field, as it is in devices that use AC current such as transformers, inductors, and AC motors and alternators, some of the power that would ideally be transferred through the device is lost in the core, dissipated as heat and sometimes noise. Core loss is commonly termed "iron loss" in contradistinction to copper loss, the loss in the windings. Iron losses are often described as being in three categories:
Hysteresis losses.
When the magnetic field through the core changes, the magnetization of the core material changes by expansion and contraction of the tiny magnetic domains it is composed of, due to movement of the domain walls. This process causes losses, because the domain walls get "snagged" on defects in the crystal structure and then "snap" past them, dissipating energy as heat. This is called hysteresis loss. It can be seen in the graph of the "B" field versus the "H" field for the material, which has the form of a closed loop.
The net energy that flows into the inductor expressed in relationship to the B-H characteristic of the core is shown by the equation
formula_1
This equation shows that the amount of energy lost in the material in one cycle of the applied field is proportional to the area inside the hysteresis loop. Since the energy lost in each cycle is constant, hysteresis power losses increase proportionally with frequency. The final equation for the hysteresis power loss is
formula_2
Eddy-current losses.
If the core is electrically conductive, the changing magnetic field induces circulating loops of current in it, called eddy currents, due to electromagnetic induction. The loops flow perpendicular to the magnetic field axis. The energy of the currents is dissipated as heat in the resistance of the core material. The power loss is proportional to the area of the loops and inversely proportional to the resistivity of the core material. Eddy current losses can be reduced by making the core out of thin laminations which have an insulating coating, or alternatively, making the core of a magnetic material with high electrical resistance, like ferrite. Most magnetic cores intended for power converter application use ferrite cores for this reason.
Anomalous losses.
By definition, this category includes any losses in addition to eddy-current and hysteresis losses. This can also be described as broadening of the hysteresis loop with frequency. Physical mechanisms for anomalous loss include localized eddy-current effects near moving domain walls.
Legg's equation.
An equation known as Legg's equation models the magnetic material core loss at low flux densities. The equation has three loss components: hysteresis, residual, and eddy current, and it is given by
formula_3
where
Steinmetz coefficients.
Losses in magnetic materials can be characterized by the Steinmetz coefficients, which however do not take into account temperature variability. Material manufacturers provide data on core losses in tabular and graphical form for practical conditions of use.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "L = n^2 A_L"
},
{
"math_id": 1,
"text": "W=\\int{\\left(nA_c\\frac{dB(t)}{t}\\right)\\left(\\frac{H(t)l_m}{n}\\right)dt}=(A_cl_m)\\int{HdB}"
},
{
"math_id": 2,
"text": "P_H=(f)(A_cl_m)\\int{HdB}"
},
{
"math_id": 3,
"text": "\\frac{R_{\\text{ac}}}{\\mu L} = a B_{\\text{max}} f + c f + e f^2"
},
{
"math_id": 4,
"text": "R_{ac}"
},
{
"math_id": 5,
"text": "\\mu"
},
{
"math_id": 6,
"text": "L"
},
{
"math_id": 7,
"text": "a"
},
{
"math_id": 8,
"text": "B_{\\text{max}}"
},
{
"math_id": 9,
"text": "c"
},
{
"math_id": 10,
"text": "f"
}
] |
https://en.wikipedia.org/wiki?curid=1516916
|
1517049
|
Lidstone series
|
In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can express certain types of entire functions.
Let "ƒ"("z") be an entire function of exponential type less than ("N" + 1)"π", as defined below. Then "ƒ"("z") can be expanded in terms of polynomials "A""n" as follows:
formula_0
Here "A""n"("z") is a polynomial in "z" of degree "n", "C""k" a constant, and "ƒ"("n")("a") the "n"th derivative of "ƒ" at "a".
A function is said to be of exponential type of less than "t" if the function
formula_1
is bounded above by "t". Thus, the constant "N" used in the summation above is given by
formula_2
with
formula_3
|
[
{
"math_id": 0,
"text": "f(z)=\\sum_{n=0}^\\infty \\left[ A_n(1-z) f^{(2n)}(0) + A_n(z) f^{(2n)}(1) \\right] + \\sum_{k=1}^N C_k \\sin (k\\pi z)."
},
{
"math_id": 1,
"text": "h(\\theta; f) = \\underset{r\\to\\infty}{\\limsup}\\, \\frac{1}{r} \\log |f(r e^{i\\theta})|"
},
{
"math_id": 2,
"text": "t= \\sup_{\\theta\\in [0,2\\pi)} h(\\theta; f)"
},
{
"math_id": 3,
"text": "N\\pi \\leq t < (N+1)\\pi."
}
] |
https://en.wikipedia.org/wiki?curid=1517049
|
151713
|
One-instruction set computer
|
Abstract machine that uses only one instruction
A one-instruction set computer (OISC), sometimes referred to as an ultimate reduced instruction set computer (URISC), is an abstract machine that uses only one instruction – obviating the need for a machine language opcode. With a judicious choice for the single instruction and given arbitrarily many resources, an OISC is capable of being a universal computer in the same manner as traditional computers that have multiple instructions. OISCs have been recommended as aids in teaching computer architecture and have been used as computational models in structural computing research. The first carbon nanotube computer is a 1-bit one-instruction set computer (and has only 178 transistors).
Machine architecture.
In a Turing-complete model, each memory location can store an arbitrary integer, and – depending on the model – there may be arbitrarily many locations. The instructions themselves reside in memory as a sequence of such integers.
There exists a class of universal computers with a single instruction based on bit manipulation such as bit copying or bit inversion. Since their memory model is finite, as is the memory structure used in real computers, those bit manipulation machines are equivalent to real computers rather than to Turing machines.
Currently known OISCs can be roughly separated into three broad categories:
Bit-manipulating machines.
Bit-manipulating machines are the simplest class.
FlipJump.
The FlipJump machine has 1 instruction, a;b - flips the bit a, then jumps to b. This is the most primitive OISC, but it's still useful. It can successfully do Math/Logic calculations, branching, pointers, and calling functions with the help of its standard library.
BitBitJump.
A bit copying machine, called BitBitJump, copies one bit in memory and passes the execution unconditionally to the address specified by one of the operands of the instruction. This process turns out to be capable of universal computation (i.e. being able to execute any algorithm and to interpret any other universal machine) because copying bits can conditionally modify the copying address that will be subsequently executed.
Toga computer.
Another machine, called the Toga Computer, inverts a bit and passes the execution conditionally depending on the result of inversion. The unique instruction is TOGA(a,b) which stands for TOGgle "a" And branch to "b" if the result of the toggle operation is true.
Multi-bit copying machine.
Similar to BitBitJump, a multi-bit copying machine copies several bits at the same time. The problem of computational universality is solved in this case by keeping predefined jump tables in the memory.
Transport triggered architecture.
"Transport triggered architecture" (TTA) is a design in which computation is a side effect of data transport. Usually, some memory registers (triggering ports) within common address space perform an assigned operation when the instruction references them. For example, in an OISC using a single memory-to-memory copy instruction, this is done by triggering ports that perform arithmetic and instruction pointer jumps when written to.
Arithmetic-based Turing-complete machines.
Arithmetic-based Turing-complete machines use an arithmetic operation and a conditional jump. Like the two previous universal computers, this class is also Turing-complete. The instruction operates on integers which may also be addresses in memory.
Currently there are several known OISCs of this class, based on different arithmetic operations:
Instruction types.
Common choices for the single instruction are:
Only "one" of these instructions is used in a given implementation. Hence, there is no need for an opcode to identify which instruction to execute; the choice of instruction is inherent in the design of the machine, and an OISC is typically named after the instruction it uses (e.g., an SBN OISC, the SUBLEQ language, etc.). Each of the above instructions can be used to construct a Turing-complete OISC.
This article presents only subtraction-based instructions among those that are not transport triggered. However, it is possible to construct Turing complete machines using an instruction based on other arithmetic operations, e.g., addition. For example, one variation known as DLN (Decrement and jump if not zero) has only two operands and uses decrement as the base operation. For more information see Subleq derivative languages .
Subtract and branch if not equal to zero.
The codice_0 instruction ("subtract and branch if not equal to zero") subtracts the contents at address "a" from the contents at address "b", stores the result at address "c", and then, "if the result is not 0", transfers control to address "d" (if the result is equal to zero, execution proceeds to the next instruction in sequence).
Subtract and branch if less than or equal to zero.
The subleq instruction ("subtract and branch if less than or equal to zero") subtracts the contents at address "a" from the contents at address "b", stores the result at address "b", and then, "if the result is not positive", transfers control to address "c" (if the result is positive, execution proceeds to the next instruction in sequence). Pseudocode:
Instruction subleq a, b, c
Mem[b] = Mem[b] - Mem[a]
if (Mem[b] ≤ 0)
goto c
Conditional branching can be suppressed by setting the third operand equal to the address of the next instruction in sequence. If the third operand is not written, this suppression is implied.
A variant is also possible with two operands and an internal accumulator, where the accumulator is subtracted from the memory location specified by the first operand. The result is stored in both the accumulator and the memory location, and the second operand specifies the branch address:
Instruction subleq2 a, b
Mem[a] = Mem[a] - ACCUM
ACCUM = Mem[a]
if (Mem[a] ≤ 0)
goto b
Although this uses only two (instead of three) operands per instruction, correspondingly more instructions are then needed to effect various logical operations.
Synthesized instructions.
It is possible to synthesize many types of higher-order instructions using only the subleq instruction.
Unconditional branch:
subleq Z, Z, c
Addition can be performed by repeated subtraction, with no conditional branching; e.g., the following instructions result in the content at location a being added to the content at location b:
subleq a, Z
subleq Z, b
subleq Z, Z
The first instruction subtracts the content at location a from the content at location Z (which is 0) and stores the result (which is the negative of the content at a) in location Z. The second instruction subtracts this result from b, storing in b this difference (which is now the sum of the contents originally at a and b); the third instruction restores the value 0 to Z.
A copy instruction can be implemented similarly; e.g., the following instructions result in the content at location b getting replaced by the content at location a, again assuming the content at location Z is maintained as 0:
subleq b, b
subleq a, Z
subleq Z, b
subleq Z, Z
Any desired arithmetic test can be built. For example, a branch-if-zero condition can be assembled from the following instructions:
subleq b, Z, L1
subleq Z, Z, OUT
L1:
subleq Z, Z
subleq Z, b, c
OUT:
Subleq2 can also be used to synthesize higher-order instructions, although it generally requires more operations for a given task. For example, no fewer than 10 subleq2 instructions are required to flip all the bits in a given byte:
subleq2 tmp ; tmp = 0 (tmp = temporary register)
subleq2 tmp
subleq2 one ; acc = -1
subleq2 a ; a' = a + 1
subleq2 Z ; Z = - a - 1
subleq2 tmp ; tmp = a + 1
subleq2 a ; a' = 0
subleq2 tmp ; load tmp into acc
subleq2 a ; a' = - a - 1 ( = ~a )
subleq2 Z ; set Z back to 0
Emulation.
The following program (written in pseudocode) emulates the execution of a subleq-based OISC:
int memory[], program_counter, a, b, c
program_counter = 0
while (program_counter >= 0):
a = memory[program_counter]
b = memory[program_counter+1]
c = memory[program_counter+2]
if (a < 0 or b < 0):
program_counter = -1
else:
memory[b] = memory[b] - memory[a]
if (memory[b] > 0):
program_counter += 3
else:
program_counter = c
This program assumes that memory[] is indexed by "nonnegative" integers. Consequently, for a subleq instruction (a, b, c), the program interprets a < 0, b < 0, or an executed branch to c < 0 as a halting condition. Similar interpreters written in a subleq-based language (i.e., self-interpreters, which may use self-modifying code as allowed by the nature of the subleq instruction) can be found in the external links below.
A general purpose SMP-capable 64-bit operating system called Dawn OS has been implemented in an emulated Subleq machine. The OS contains a C-like compiler. Some memory areas in the virtual machine are used for peripherals like the keyboard, mouse, hard drives, network card, etc. Basic applications written for it include a media player, painting tool, document reader and scientific calculator.
A 32-bit Subleq computer with a graphic display and a keyboard called Izhora has been constructed by Yoel Matveyev as a large cellular automation pattern.
Compilation.
There is a compiler called Higher Subleq written by Oleg Mazonka that compiles a simplified C program into subleq code.
Alternatively there is a self hosting Forth implementation written by Richard James Howe that runs on top of a Subleq VM and is capable of interactive programming of the Subleq machine
Subtract and branch if negative.
The subneg instruction ("subtract and branch if negative"), also called SBN, is defined similarly to subleq:
Instruction subneg a, b, c
Mem[b] = Mem[b] - Mem[a]
if (Mem[b] < 0)
goto c
Conditional branching can be suppressed by setting the third operand equal to the address of the next instruction in sequence. If the third operand is not written, this suppression is implied.
Synthesized instructions.
It is possible to synthesize many types of higher-order instructions using only the subneg instruction. For simplicity, only one synthesized instruction is shown here to illustrate the difference between subleq and subneg.
Unconditional branch:
subneg POS, Z, c
where Z and POS are locations previously set to contain 0 and a positive integer, respectively;
Unconditional branching is assured only if Z initially contains 0 (or a value less than the integer stored in POS). A follow-up instruction is required to clear Z after the branching, assuming that the content of Z must be maintained as 0.
subneg4.
A variant is also possible with four operands – subneg4. The reversal of minuend and subtrahend eases implementation in hardware. The non-destructive result simplifies the synthetic instructions.
Instruction subneg s, m, r, j
"(* subtrahend, minuend, result and jump addresses *)"
Mem[r] = Mem[m] - Mem[s]
if (Mem[r] < 0)
goto j
Arithmetic machine.
In an attempt to make Turing machine more intuitive, Z. A. Melzak consider the task of computing with positive numbers. The machine has an infinite abacus, an infinite number of counters (pebbles, tally sticks) initially at a special location S. The machine is able to do one operation:
Take from location X as many counters as there are in location Y and transfer them to location Z and proceed to instruction y.
If this operation is not possible because there is not enough counters in X, then leave the abacus as it is and proceed to instruction n.
In order to keep all numbers positive and mimic a human operator computing on a real world abacus, the test is performed before any subtraction. Pseudocode:
Instruction melzak X, Y, Z, n, y
if (Mem[X] < Mem[Y])
goto n
Mem[X] -= Mem[Y]
Mem[Z] += Mem[Y]
goto y
After giving a few programs: multiplication, gcd, computing the "n"-th prime number, representation in base "b" of an arbitrary number, sorting in order of magnitude, Melzak shows explicitly how to simulate an arbitrary Turing machine on his arithmetic machine.
multiply:
melzak P, ONE, S, stop ; Move 1 counter from P to S. If not possible, move to stop.
melzak S, Q, ANS, multiply, multiply ; Move q counters from S to ANS. Move to the first instruction.
stop:
where the memory location P is "p", Q is "q", ONE is 1, ANS is initially 0 and at the end "pq", and S is a large number.
He mentions that it can easily be shown using the elements of recursive functions that every number calculable on the arithmetic machine is computable. A proof of which was given by Lambek on an equivalent two instruction machine : X+ (increment X) and X− else T (decrement X if it not empty, else jump to T).
Reverse subtract and skip if borrow.
In a "reverse subtract and skip if borrow" (RSSB) instruction, the accumulator is subtracted from the memory location and the next instruction is skipped if there was a borrow (memory location was smaller than the accumulator). The result is stored in both the accumulator and the memory location. The program counter is mapped to memory location 0. The accumulator is mapped to memory location 1.
Instruction rssb x
ACCUM = Mem[x] - ACCUM
Mem[x] = ACCUM
if (ACCUM < 0)
goto PC + 2
Example.
To set x to the value of y minus z:
RSSB temp # Three instructions required to clear acc, temp [See Note 1]
RSSB temp
RSSB temp
RSSB x # Two instructions clear acc, x, since acc is already clear
RSSB x
RSSB y # Load y into acc: no borrow
RSSB temp # Store -y into acc, temp: always borrow and skip
RSSB temp # Skipped
RSSB x # Store y into x, acc
RSSB temp # Three instructions required to clear acc, temp
RSSB temp
RSSB temp
RSSB z # Load z
RSSB x # x = y - z [See Note 2]
Transport triggered architecture.
A transport triggered architecture uses only the "move" instruction, hence it was originally called a "move machine". This instruction moves the contents of one memory location to another memory location combining with the current content of the new location:
Instruction movx a, b (also written "a" -> "b")
OP = GetOperation(Mem["b"])
Mem["b"] := OP(Mem["a"], Mem["b"])
The operation performed is defined by the destination memory cell. Some cells are specialized in addition, some other in multiplication, etc. So memory cells are not simple store but coupled with an arithmetic logic unit (ALU) setup to perform only one sort of operation with the current value of the cell. Some of the cells are control flow instructions to alter the program execution with jumps, conditional execution, subroutines, if-then-else, for-loop, etc...
A commercial transport triggered architecture microcontroller has been produced called MAXQ, which hides the apparent inconvenience of an OISC by using a "transfer map" that represents all possible destinations for the "move" instructions.
Cryptoleq.
Cryptoleq is a language similar to Subleq. It consisting of one eponymous instruction and is capable of performing general-purpose computation on encrypted programs. Cryptoleq works on continuous cells of memory using direct and indirect addressing, and performs two operations "O"1 and "O"2 on three values A, B, and C:
Instruction cryptoleq a, b, c
Mem[b] = O1(Mem[a], Mem[b])
if O2(Mem[b]) ≤ 0
IP = c
else
IP = IP + 3
where a, b and c are addressed by the instruction pointer, IP, with the value of IP addressing a, IP + 1 point to b and IP + 2 to c.
In Cryptoleq operations "O"1 and "O"2 are defined as follows:
formula_0
formula_1
The main difference with Subleq is that in Subleq, "O"1("x,y") simply subtracts y from x and "O"2("x") equals to x. Cryptoleq is also homomorphic to Subleq, modular inversion and multiplication is homomorphic to subtraction and the operation of "O"2 corresponds the Subleq test if the values were unencrypted. A program written in Subleq can run on a Cryptoleq machine, meaning backwards compatibility. However, Cryptoleq implements fully homomorphic calculations and is capable of multiplications. Multiplication on an encrypted domain is assisted by a unique function G that is assumed to be difficult to reverse engineer and allows re-encryption of a value based on the "O"2 operation:
formula_2
where formula_3 is the re-encrypted value of y and formula_4 is encrypted zero. x is the encrypted value of a variable, let it be m, and formula_5 equals &NoBreak;&NoBreak;.
The multiplication algorithm is based on addition and subtraction, uses the function G and does not have conditional jumps nor branches. Cryptoleq encryption is based on Paillier cryptosystem.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\begin{array}{lcl} O_1(x,y) & = & x^{-1} y \\,\\bmod\\, N^2 \\end{array}"
},
{
"math_id": 1,
"text": "\\begin{array}{lcl} O_2(x) & = & \\left \\lfloor \\frac{x-1}{N} \\right \\rfloor \\end{array}"
},
{
"math_id": 2,
"text": "G(x,y) = \\begin{cases} \\tilde{0}, & \\text{if }O_2(\\bar{x})\\text{ }\\leq 0 \\\\ \\tilde{y}, & \\text{otherwise} \\end{cases}"
},
{
"math_id": 3,
"text": "\\tilde{y}"
},
{
"math_id": 4,
"text": "\\tilde{0}"
},
{
"math_id": 5,
"text": "\\bar{x}"
}
] |
https://en.wikipedia.org/wiki?curid=151713
|
1517134
|
Clover (telescope)
|
Canceled polarization experiment
Clover would have been an experiment to measure the polarization of the Cosmic Microwave Background. It was approved for funding in late 2004, with the aim of having the full telescope operational by 2009. The project was jointly run by Cardiff University, Oxford University, the Cavendish Astrophysics Group and the University of Manchester.
History.
The Clover Project was meant to consist of two independent telescopes, one operating at 95 GHz with the other operating at both 150 and 225 GHz. Both telescopes were to be sited near the CBI site in the Atacama Desert, Chile. The two telescope receivers would have been large format focal plane arrays of either 100 or 200 bolometric detectors.
The aim of the experiment was to measure the B-mode polarization of the Cosmic Microwave Background between multipoles of 20 and 1000 down to a sensitivity limited by the foreground contamination due to lensing. This would have allowed the detection of primordial gravitational waves in the universe so long as the ratio of scalar perturbations (caused by density fluctuations in the early universe) to the tensor perturbations caused by gravitational waves was greater than formula_0.
It was hoped that the telescope would have spent around 2 years observing a total of around 1,000 degrees of sky, made up of several patches of sky where polarized foregrounds (synchrotron and thermal dust emission) are at a minimum.
Clover was canceled in March 2009 as STFC were unable to provide the requested additional funds of 2.55 million pounds to finish the project.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "r = 0.01"
}
] |
https://en.wikipedia.org/wiki?curid=1517134
|
15175696
|
Signal-flow graph
|
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of directed graphs (also called digraphs), which includes as well that of oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications.
SFGs are most commonly used to represent signal flow in a physical system and its controller(s), forming a cyber-physical system. Among their other uses are the representation of signal flow in various electronic networks and amplifiers, digital filters, state-variable filters and some other types of analog filters. In nearly all literature, a signal-flow graph is associated with a set of linear equations.
History.
Wai-Kai Chen wrote: "The concept of a signal-flow graph was originally worked out by Shannon [1942]
in dealing with analog computers. The greatest credit for the formulation of signal-flow graphs is normally extended to Mason [1953], [1956]. He showed how to use the signal-flow graph technique to solve some difficult electronic problems in a relatively simple manner. The term signal flow graph was used because of its original application to electronic problems and the association with electronic signals and flowcharts of the systems under study."
Lorens wrote: "Previous to Mason's work, C. E. Shannon worked out a number of the properties of what are now known as flow graphs. Unfortunately, the paper originally had a restricted classification and very few people had access to the material."
"The rules for the evaluation of the graph determinant of a Mason Graph were first given and proven by Shannon [1942] using mathematical induction. His work remained essentially unknown even after Mason published his classical work in 1953. Three years later, Mason [1956] rediscovered the rules and proved them by considering the value of a determinant and how it changes as variables are added to the graph. [...]"
Domain of application.
Robichaud "et al." identify the domain of application of SFGs as follows:
"All the physical systems analogous to these networks [constructed of ideal transformers, active elements and gyrators] constitute the domain of application of the techniques developed [here]. Trent has shown that all the physical systems which satisfy the following conditions fall into this category.
Basic flow graph concepts.
The following illustration and its meaning were introduced by Mason to illustrate basic concepts:
In the simple flow graphs of the figure, a functional dependence of a node is indicated by an incoming arrow, the node originating this influence is the beginning of this arrow, and in its most general form the signal flow graph indicates by incoming arrows only those nodes that influence the processing at the receiving node, and at each node, "i", the incoming variables are processed according to a function associated with that node, say "Fi". The flowgraph in (a) represents a set of explicit relationships:
formula_0
Node "x1" is an isolated node because no arrow is incoming; the equations for "x2" and "x3" have the graphs shown in parts (b) and (c) of the figure.
These relationships define for every node a function that processes the input signals it receives. Each non-source node combines the input signals in some manner, and broadcasts a resulting signal along each outgoing branch. "A flow graph, as defined originally by Mason, implies a set of functional relations, linear or not."
However, the commonly used Mason graph is more restricted, assuming that each node simply sums its incoming arrows, and that each branch involves only the initiating node involved. Thus, in this more restrictive approach, the node "x1" is unaffected while:
formula_1
formula_2
and now the functions "fij" can be associated with the signal-flow branches "ij" joining the pair of nodes "xi, xj", rather than having general relationships associated with each node. A contribution by a node to itself like "f33" for "x3" is called a "self-loop". Frequently these functions are simply multiplicative factors (often called "transmittances" or "gains"), for example, "fij(xj)=cijxj", where "c" is a scalar, but possibly a function of some parameter like the Laplace transform variable "s". Signal-flow graphs are very often used with Laplace-transformed signals, because then they represent systems of Linear differential equations. In this case the transmittance, "c(s)", often is called a transfer function.
Choosing the variables.
<templatestyles src="Template:Blockquote/styles.css" />In general, there are several ways of choosing the variables in a complex system. Corresponding to each choice, a system of equations can be written and each system of equations can be represented in a graph. This formulation of the equations becomes direct and automatic if one has at his disposal techniques which permit the drawing of a graph directly from the schematic diagram of the system under study. The structure of the graphs thus obtained is related in a simple manner to the topology of the schematic diagram, and it becomes unnecessary to consider the equations, even implicitly, to obtain the graph. In some cases, one has simply to imagine the flow graph in the schematic diagram and the desired answers can be obtained without even drawing the flow graph.
Non-uniqueness.
Robichaud et al. wrote: "The signal flow graph contains the same information as the equations from which it is derived; but there does not exist a one-to-one correspondence between the graph and the system of equations. One system will give different graphs according to the order in which the equations are used to define the variable written on the left-hand side." If all equations relate all dependent variables, then there are "n!" possible SFGs to choose from.
Linear signal-flow graphs.
Linear signal-flow graph (SFG) methods only apply to linear time-invariant systems, as studied by their associated theory. When modeling a system of interest, the first step is often to determine the equations representing the system's operation without assigning causes and effects (this is called acausal modeling). A SFG is then derived from this system of equations.
A linear SFG consists of nodes indicated by dots and weighted directional branches indicated by arrows. The nodes are the variables of the equations and the branch weights are the coefficients. Signals may only traverse a branch in the direction indicated by its arrow. The elements of a SFG can only represent the operations of multiplication by a coefficient and addition, which are sufficient to represent the constrained equations. When a signal traverses a branch in its indicated direction, the signal is multiplied the weight of the branch. When two or more branches direct into the same node, their outputs are added.
For systems described by linear algebraic or differential equations, the signal-flow graph is mathematically equivalent to the system of equations describing the system, and the equations governing the nodes are discovered for each node by summing incoming branches to that node. These incoming branches convey the contributions of the other nodes, expressed as the connected node value multiplied by the weight of the connecting branch, usually a real number or function of some parameter (for example a Laplace transform variable "s").
For linear active networks, Choma writes: "By a 'signal flow representation' [or 'graph', as it is commonly referred to] we mean a diagram that, by displaying the algebraic relationships among relevant branch variables of network, paints an unambiguous picture of the way an applied input signal ‘flows’ from input-to-output ... ports."
A motivation for a SFG analysis is described by Chen:
"The analysis of a linear system reduces ultimately to the solution of a system of linear algebraic equations. As an alternative to conventional algebraic methods of solving the system, it is possible to obtain a solution by considering the properties of certain directed graphs associated with the system." [See subsection: Solving linear equations.] "The unknowns of the equations correspond to the nodes of the graph, while the linear relations between them appear in the form of directed edges connecting the nodes. ...The associated directed graphs in many cases can be set up directly by inspection of the physical system without the necessity of first formulating the →associated equations..."
Basic components.
A linear signal flow graph is related to a system of linear equations of the following form:
formula_3
where formula_4 = transmittance (or gain) from formula_5 to formula_6.
The figure to the right depicts various elements and constructs of a signal flow graph (SFG).
Exhibit (a) is a node. In this case, the node is labeled formula_7. A node is a vertex representing a variable or signal.
A "source" node has only outgoing branches (represents an independent variable). As a special case, an "input" node is characterized by having one or more attached arrows pointing away from the node and no arrows pointing into the node. Any open, complete SFG will have at least one input node.
An "output" or "sink" node has only incoming branches (represents a dependent variable). Although any node can be an output, explicit output nodes are often used to provide clarity. Explicit output nodes are characterized by having one or more attached arrows pointing into the node and no arrows pointing away from the node. Explicit output nodes are not required.
A "mixed" node has both incoming and outgoing branches.
Exhibit (b) is a branch with a multiplicative gain of formula_8. The meaning is that the output, at the tip of the arrow, is formula_8 times the input at the tail of the arrow. The gain can be a simple constant or a function (for example: a function of some transform variable such as formula_9, formula_10, or formula_11, for Laplace, Fourier or Z-transform relationships).
Exhibit (c) is a branch with a multiplicative gain of one. When the gain is omitted, it is assumed to be unity.
Exhibit (d) formula_12 is an input node. In this case, formula_12 is multiplied by the gain formula_8.
Exhibit (e) formula_13 is an explicit output node; the incoming edge has a gain of formula_8.
Exhibit (f) depicts addition. When two or more arrows point into a node, the signals carried by the edges are added.
Exhibit (g) depicts a simple loop. The loop gain is formula_14.
Exhibit (h) depicts the expression formula_15.
Terms used in linear SFG theory also include:
Systematic reduction to sources and sinks.
A signal-flow graph may be simplified by graph transformation rules. These simplification rules are also referred to as "signal-flow graph algebra".
The purpose of this reduction is to relate the dependent variables of interest (residual nodes, sinks) to its independent variables (sources).
The systematic reduction of a linear signal-flow graph is a graphical method equivalent to the Gauss-Jordan elimination method for solving linear equations.
The rules presented below may be applied over and over until the signal flow graph is reduced to its "minimal residual form". Further reduction can require loop elimination or the use of a "reduction formula" with the goal to directly connect sink nodes representing the dependent variables to the source nodes representing the independent variables. By these means, any signal-flow graph can be simplified by successively removing internal nodes until only the input and output and index nodes remain. Robichaud described this process of systematic flow-graph reduction:
<templatestyles src="Template:Blockquote/styles.css" />The reduction of a graph proceeds by the elimination of certain nodes to obtain a residual graph showing only the variables of interest. This elimination of nodes is called "node absorption". This method is close to the familiar process of successive eliminations of undesired variables in a system of equations. One can eliminate a variable by removing the corresponding node in the graph. If one reduces the graph sufficiently, it is possible to obtain the solution for any variable and this is the objective which will be kept in mind in this description of the different methods of reduction of the graph. In practice, however, the techniques of reduction will be used solely to transform the graph to a residual graph expressing some fundamental relationships. Complete solutions will be more easily obtained by application of Mason's rule.
The graph itself programs the reduction process. Indeed a simple inspection of the graph readily suggests the different steps of the reduction which are carried out by elementary transformations, by loop elimination, or by the use of a reduction formula.
For digitally reducing a flow graph using an algorithm, Robichaud extends the notion of a simple flow graph to a "generalized" flow graph:
<templatestyles src="Template:Blockquote/styles.css" />Before describing the process of reduction...the correspondence between the graph and a system of linear equations ... must be generalized..."The generalized graphs will represent some operational relationships between groups of variables"...To each branch of the generalized graph is associated a matrix giving the relationships between the variables represented by the nodes at the extremities of that branch...
The elementary transformations [defined by Robichaud in his Figure 7.2, p. 184] and the loop reduction permit the elimination of any node "j" of the graph by the "reduction formula":[described in Robichaud's Equation 7-1]. With the reduction formula, it is always possible to reduce a graph of any order... [After reduction] the final graph will be a cascade graph in which the variables of the sink nodes are explicitly expressed as functions of the sources. This is the only method for reducing the generalized graph since Mason's rule is obviously inapplicable.
The definition of an elementary transformation varies from author to author:
Parallel edges. Replace parallel edges with a single edge having a gain equal to the sum of original gains.
The graph on the left has parallel edges between nodes. On the right, these parallel edges have been replaced with a single edge having a gain equal to the sum of the gains on each original edge.
The equations corresponding to the reduction between N and node I1 are:
formula_16
Outflowing edges. Replace outflowing edges with edges directly flowing from the node's sources.
The graph on the left has an intermediate node N between nodes from which it has inflows, and nodes to which it flows out.
The graph on the right shows direct flows between these node sets, without transiting via N.
For the sake of simplicity, N and its inflows are not represented. The outflows from N are eliminated.
The equations corresponding to the reduction directly relating N's input signals to its output signals are:
formula_17
Zero-signal nodes.
Eliminate outflowing edges from a node determined to have a value of zero.
If the value of a node is zero, its outflowing edges can be eliminated.
Nodes without outflows.
Eliminate a node without outflows.
In this case, N is not a variable of interest, and it has no outgoing edges; therefore, N, and its inflowing edges, can be eliminated.
Self-looping edge. Replace looping edges by adjusting the gains on the incoming edges.
The graph on the left has a looping edge at node N, with a gain of g. On the right, the looping edge has been eliminated, and all inflowing edges have their gain divided by (1-g).The equations corresponding to the reduction between N and all its input signals are:
formula_18
Implementations.
The above procedure for building the SFG from an acausal system of equations and for solving the SFG's gains have been implemented as an add-on to MATHLAB 68, an on-line system providing machine aid for the mechanical symbolic processes encountered in analysis.
Solving linear equations.
Signal flow graphs can be used to solve sets of simultaneous linear equations. The set of equations must be consistent and all equations must be linearly independent.
Putting the equations in "standard form".
For M equations with N unknowns where each yj is a known value and each xj is an unknown value, there is equation for each known of the following form.
formula_19 ; the usual form for simultaneous linear equations with 1 ≤ j ≤ M
Although it is feasible, particularly for simple cases, to establish a signal flow graph using the equations in this form, some rearrangement allows a general procedure that works easily for any set of equations, as now is presented. To proceed, first the equations are rewritten as
formula_20
and further rewritten as
formula_21
and finally rewritten as
formula_22 ; form suitable to be expressed as a signal flow graph.
where δkj = Kronecker delta
The signal-flow graph is now arranged by selecting one of these equations and addressing the node on the right-hand side. This is the node for which the node connects to itself with the branch of weight including a '+1', making a "self-loop" in the flow graph. The other terms in that equation connect this node first to the source in this equation and then to all the other branches incident on this node. Every equation is treated this way, and then each incident branch is joined to its respective emanating node. For example, the case of three variables is shown in the figure, and the first equation is:
formula_23
where the right side of this equation is the sum of the weighted arrows incident on node "x1".
As there is a basic symmetry in the treatment of every node, a simple starting point is an arrangement of nodes with each node at one vertex of a regular polygon. When expressed using the general coefficients {"cin"}, the environment of each node is then just like all the rest apart from a permutation of indices. Such an implementation for a set of three simultaneous equations is seen in the figure.
Often the known values, yj are taken as the primary causes and the unknowns values, xj to be effects, but regardless of this interpretation, the last form for the set of equations can be represented as a signal-flow graph. This point is discussed further in the subsection Interpreting 'causality'.
Applying Mason's gain formula.
In the most general case, the values for all the xk variables can be calculated by computing Mason's gain formula for the path from each yj to each xk and using superposition.
formula_24
where Gkj = the sum of Mason's gain formula computed for all the paths from input yj to variable xk.
In general, there are N-1 paths from yj to variable xk so the computational effort to calculated Gkj is proportional to N-1.
Since there are M values of yj, Gkj must be computed M times for a single value of xk. The computational effort to calculate a single xk variable is proportional to (N-1)(M). The effort to compute all the xk variables is proportional to (N)(N-1)(M). If there are N equations and N unknowns, then the computation effort is on the order of N3.
Relation to block diagrams.
For some authors, a linear signal-flow graph is more constrained than a block diagram, in that the SFG rigorously describes linear algebraic equations represented by a directed graph.
For other authors, linear block diagrams and linear signal-flow graphs are equivalent ways of depicting a system, and either can be used to solve the gain.
A tabulation of the comparison between block diagrams and signal-flow graphs is provided by Bakshi & Bakshi, and another tabulation by Kumar. According to Barker "et al.":
"The signal flow graph is the most convenient method for representing a dynamic system. The topology of the graph is compact and the rules for manipulating it are easier to program than the corresponding rules that apply to block diagrams."
In the figure, a simple block diagram for a feedback system is shown with two possible interpretations as a signal-flow graph. The input "R(s)" is the Laplace-transformed input signal; it is shown as a source node in the signal-flow graph (a source node has no input edges). The output signal "C(s)" is the Laplace-transformed output variable. It is represented as a sink node in the flow diagram (a sink has no output edges). "G(s)" and "H(s)" are transfer functions, with "H(s)" serving to feed back a modified version of the output to the input, "B(s)". The two flow graph representations are equivalent.
Interpreting 'causality'.
The term "cause and effect" was applied by Mason to SFGs:
"The process of constructing a graph is one of tracing a succession of cause and effects through the physical system. One variable is expressed as an explicit effect due to certain causes; they in turn, are recognized as effects due to still other causes."
— S.J. Mason: Section IV: "Illustrative applications of flow graph technique"
and has been repeated by many later authors:
"The "signal flow graph" is another visual tool for representing causal relationships between components of the system. It is a simplified version of a block diagram introduced by S.J. Mason as a cause-and-effect representation of linear systems."
— Arthur G.O. Mutambara: "Design and Analysis of Control Systems", p.238
However, Mason's paper is concerned to show in great detail how a "set of equations" is connected to an SFG, an emphasis unrelated to intuitive notions of "cause and effect". Intuitions can be helpful for arriving at an SFG or for gaining insight from an SFG, but are inessential to the SFG. The essential connection of the SFG is to its own set of equations, as described, for example, by Ogata:
"A signal-flow graph is a diagram that represents a set of simultaneous algebraic equations. When applying the signal flow graph method to analysis of control systems, we must first transform linear differential equations into algebraic equations in [the Laplace transform variable] "s".."
— Katsuhiko Ogata: "Modern Control Engineering", p. 104
There is no reference to "cause and effect" here, and as said by Barutsky:
"Like block diagrams, signal flow graphs represent the computational, not the physical structure of a system."
— Wolfgang Borutzky, "Bond Graph Methodology", p. 10
The term "cause and effect" may be misinterpreted as it applies to the SFG, and taken incorrectly to suggest a system view of causality, rather than a "computationally" based meaning. To keep discussion clear, it may be advisable to use the term "computational causality", as is suggested for bond graphs:
"Bond-graph literature uses the term computational causality, indicating the order of calculation in a simulation, in order to avoid any interpretation in the sense of intuitive causality."
The term "computational causality" is explained using the example of current and voltage in a resistor:
"The "computational causality" of physical laws can therefore not be predetermined, but depends upon the particular use of that law. We cannot conclude whether it is the current flowing through a resistor that causes a voltage drop, or whether it is the difference in potentials at the two ends of the resistor that cause current to flow. Physically these are simply two concurrent aspects of one and the same physical phenomenon. Computationally, we may have to assume at times one position, and at other times the other."
— François Cellier & Ernesto Kofman: §1.5 "Simulation software today and tomorrow", p. 15
A computer program or algorithm can be arranged to solve a set of equations using various strategies. They differ in how they prioritize finding some of the variables in terms of the others, and these algorithmic decisions, which are simply about solution strategy, then set up the variables expressed as dependent variables earlier in the solution to be "effects", determined by the remaining variables that now are "causes", in the sense of "computational causality".
Using this terminology, it is "computational" causality, not "system" causality, that is relevant to the SFG. There exists a wide-ranging philosophical debate, not concerned specifically with the SFG, over connections between computational causality and system causality.
Signal-flow graphs for analysis and design.
Signal-flow graphs can be used for analysis, that is for understanding a model of an existing system, or for synthesis, that is for determining the properties of a design alternative.
Signal-flow graphs for dynamic systems analysis.
When building a model of a dynamic system, a list of steps is provided by Dorf & Bishop:
—RC Dorf and RH Bishop, "Modern Control Systems", Chapter 2, p. 2
In this workflow, equations of the physical system's mathematical model are used to derive the signal-flow graph equations.
Signal-flow graphs for design synthesis.
Signal-flow graphs have been used in Design Space Exploration (DSE), as an intermediate representation towards a physical implementation. The DSE process seeks a suitable solution among different alternatives. In contrast with the typical analysis workflow, where a system of interest is first modeled with the physical equations of its components, the specification for synthesizing a design could be a desired transfer function. For example, different strategies would create different signal-flow graphs, from which implementations are derived.
Another example uses an annotated SFG as an expression of the continuous-time behavior, as input to an architecture generator
Shannon and Shannon-Happ formulas.
Shannon's formula is an analytic expression for calculating the gain of an interconnected set of amplifiers in an analog computer. During World War II, while investigating the functional operation of an analog computer, Claude Shannon developed his formula. Because of wartime restrictions, Shannon's work was not published at that time, and, in 1952, Mason rediscovered the same formula.
William W. Happ generalized the Shannon formula for topologically closed systems. The Shannon-Happ formula can be used for deriving transfer functions, sensitivities, and error functions.
For a consistent set of linear unilateral relations, the Shannon-Happ formula expresses the solution using direct substitution (non-iterative).
NASA's electrical circuit software NASAP is based on the Shannon-Happ formula.
Linear signal-flow graph examples.
Simple voltage amplifier.
The amplification of a signal "V1" by an amplifier with gain "a12" is described mathematically by
formula_25
This relationship represented by the signal-flow graph of Figure 1. is that V2 is dependent on V1 but it implies no dependency of V1 on V2. See Kou page 57.
Ideal negative feedback amplifier.
A possible SFG for the asymptotic gain model for a negative feedback amplifier is shown in Figure 3, and leads to the equation for the gain of this amplifier as
formula_26 formula_27
The interpretation of the parameters is as follows: "T" = return ratio, "G∞" = direct amplifier gain, "G0" = feedforward (indicating the possible bilateral nature of the feedback, possibly deliberate as in the case of feedforward compensation). Figure 3 has the interesting aspect that it resembles Figure 2 for the two-port network with the addition of the extra "feedback relation" "x2 = T y1".
From this gain expression an interpretation of the parameters "G0" and "G∞" is evident, namely:
formula_28
There are many possible SFG's associated with any particular gain relation. Figure 4 shows another SFG for the asymptotic gain model that can be easier to interpret in terms of a circuit. In this graph, parameter β is interpreted as a feedback factor and "A" as a "control parameter", possibly related to a dependent source in the circuit. Using this graph, the gain is
formula_26 formula_29
To connect to the asymptotic gain model, parameters "A" and β cannot be arbitrary circuit parameters, but must relate to the return ratio "T" by:
formula_30
and to the asymptotic gain as:
formula_31
Substituting these results into the gain expression,
formula_32
formula_33
formula_34
which is the formula of the asymptotic gain model.
Electrical circuit containing a two-port network.
The figure to the right depicts a circuit that contains a "y"-parameter two-port network. Vin is the input of the circuit and V2 is the output. The two-port equations impose a set of linear constraints between its port voltages and currents. The terminal equations impose other constraints. All these constraints are represented in the SFG (Signal Flow Graph) below the circuit. There is only one path from input to output which is shown in a different color and has a (voltage) gain of -RLy21. There are also three loops: -Riny11, -RLy22, Riny21RLy12. Sometimes a loop indicates intentional feedback but it can also indicate a constraint on the relationship of two variables. For example, the equation that describes a resistor says that the ratio of the voltage across the resistor to the current through the resistor is a constant which is called the resistance. This can be interpreted as the voltage is the input and the current is the output, or the current is the input and the voltage is the output, or merely that the voltage and current have a linear relationship. Virtually all passive two terminal devices in a circuit will show up in the SFG as a loop.
The SFG and the schematic depict the same circuit, but the schematic also suggests the circuit's purpose. Compared to the schematic, the SFG is awkward but it does have the advantage that the input to output gain can be written down by inspection using Mason's rule.
Mechatronics : Position servo with multi-loop feedback.
This example is representative of a SFG (signal-flow graph) used to represent a servo control system and illustrates several features of SFGs. Some of the loops (loop 3, loop 4 and loop 5) are extrinsic intentionally designed feedback loops. These are shown with dotted lines. There are also intrinsic loops (loop 0, loop1, loop2) that are not intentional feedback loops, although they can be analyzed as though they were. These loops are shown with solid lines. Loop 3 and loop 4 are also known as minor loops because they are inside a larger loop.
See Mason's rule for development of Mason's Gain Formula for this example.
Terminology and classification of signal-flow graphs.
There is some confusion in literature about what a signal-flow graph is; Henry Paynter, inventor of bond graphs, writes: "But much of the decline of signal-flow graphs [...] is due in part to the mistaken notion that the branches must be linear and the nodes must be summative. Neither assumption was embraced by Mason, himself !"
This IEEE standard defines a signal-flow graph as a "network" of "directed branches" representing dependent and independent "signals" as "nodes". Incoming branches carry "branch signals" to the dependent node signals. A "dependent node" signal is the algebraic sum of the incoming branch signals at that node, i.e. nodes are summative.
State transition signal-flow graph.
A state transition SFG or state diagram is a simulation diagram for a system of equations, including the initial conditions of the states.
Closed flowgraph.
Closed flowgraphs describe closed systems and have been utilized to provide a rigorous theoretical basis for topological techniques of circuit analysis.
Nonlinear flow graphs.
Mason introduced both nonlinear and linear flow graphs. To clarify this point, Mason wrote : "A linear flow graph is one whose associated equations are linear."
Examples of nonlinear branch functions.
It we denote by xj the signal at node j, the following are examples of node functions that do not pertain to a linear time-invariant system:
formula_36
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\begin{align}\n x_\\mathrm{1} &= \\text{an independent variable} \\\\\n x_\\mathrm{2} &= F_2(x_\\mathrm{1}, x_\\mathrm{3})\\\\\n x_\\mathrm{3} &= F_3(x_\\mathrm{1}, x_\\mathrm{2}, x_\\mathrm{3})\\\\\n\\end{align}"
},
{
"math_id": 1,
"text": "x_2=f_{21}(x_1)+f_{23}(x_3) "
},
{
"math_id": 2,
"text": "x_3=f_{31}(x_1)+f_{32}(x_2)+f_{33}(x_3) \\ ,"
},
{
"math_id": 3,
"text": "\\begin{align}\n x_\\mathrm{j} &= \\sum_{\\mathrm{k}=1}^{\\mathrm{N}} t_\\mathrm{jk} x_\\mathrm{k}\n\\end{align}"
},
{
"math_id": 4,
"text": "t_{jk}"
},
{
"math_id": 5,
"text": "x_k"
},
{
"math_id": 6,
"text": "x_j"
},
{
"math_id": 7,
"text": "x"
},
{
"math_id": 8,
"text": "m"
},
{
"math_id": 9,
"text": "s"
},
{
"math_id": 10,
"text": "\\omega"
},
{
"math_id": 11,
"text": "z"
},
{
"math_id": 12,
"text": "V_{in}"
},
{
"math_id": 13,
"text": "I_{out}"
},
{
"math_id": 14,
"text": "A \\times m"
},
{
"math_id": 15,
"text": "Z = aX + bY"
},
{
"math_id": 16,
"text": "\\begin{align}\n N &= I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{1}f_\\mathrm{2} + I_\\mathrm{1} f_\\mathrm{3} + ...\\\\\n N &= I_\\mathrm{1} (f_\\mathrm{1} + f_\\mathrm{2} + f_\\mathrm{3} ) + ...\\\\\n\\end{align}"
},
{
"math_id": 17,
"text": "\\begin{align}\n N &= I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3} \\\\\n O_\\mathrm{1} &= g_\\mathrm{1} N \\\\\n O_\\mathrm{2} &= g_\\mathrm{2} N \\\\\n O_\\mathrm{3} &= g_\\mathrm{3} N \\\\\n O_\\mathrm{1} &= g_\\mathrm{1} (I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3}) \\\\\n O_\\mathrm{2} &= g_\\mathrm{2} (I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3}) \\\\\n O_\\mathrm{3} &= g_\\mathrm{3} (I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3}) \\\\\n O_\\mathrm{1} &= I_\\mathrm{1} f_\\mathrm{1}g_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2}g_\\mathrm{1} + I_\\mathrm{3} f_\\mathrm{3}g_\\mathrm{1} \\\\\n O_\\mathrm{2} &= I_\\mathrm{1} f_\\mathrm{1}g_\\mathrm{2} + I_\\mathrm{2} f_\\mathrm{2}g_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3}g_\\mathrm{2} \\\\\n O_\\mathrm{3} &= I_\\mathrm{1} f_\\mathrm{1}g_\\mathrm{3} + I_\\mathrm{2} f_\\mathrm{2}g_\\mathrm{3} + I_\\mathrm{3} f_\\mathrm{3}g_\\mathrm{3} \\\\\n\n\\end{align}"
},
{
"math_id": 18,
"text": "\\begin{align}\n N &= I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3} + N g \\\\\n N - N g &= I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3}f_\\mathrm{3} \\\\\n N (1-g) &= I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3} \\\\\n N &= (I_\\mathrm{1} f_\\mathrm{1} + I_\\mathrm{2} f_\\mathrm{2} + I_\\mathrm{3} f_\\mathrm{3}) \\div (1-g) \\\\\n N &= I_\\mathrm{1} f_\\mathrm{1}\\div (1-g) + I_\\mathrm{2} f_\\mathrm{2}\\div (1-g) + I_\\mathrm{3} f_\\mathrm{3} \\div (1-g) \\\\\n\\end{align}"
},
{
"math_id": 19,
"text": "\\begin{align}\n\\sum_{\\mathrm{k}=1}^\\mathrm{N} c_\\mathrm{jk} x_\\mathrm{k} &= y_\\mathrm{j} \n\\end{align}"
},
{
"math_id": 20,
"text": "\\begin{align}\n\\sum_{\\mathrm{k}=1}^{\\mathrm{N}} c_{\\mathrm{jk}} x_\\mathrm{k} - y_\\mathrm{j} &= 0 \\end{align} "
},
{
"math_id": 21,
"text": "\\begin{align}\n\\sum_\\mathrm{k=1}^\\mathrm{N} c_\\mathrm{jk} x_\\mathrm{k} +x_\\mathrm{j} - y_\\mathrm{j} &= x_\\mathrm{j} \\end{align} "
},
{
"math_id": 22,
"text": "\\begin{align}\n\\sum_\\mathrm{k=1}^\\mathrm{N} ( c_\\mathrm{jk} + \\delta_\\mathrm{jk}) x_\\mathrm{k} - y_\\mathrm{j} &= x_\\mathrm{j} \\end{align} "
},
{
"math_id": 23,
"text": "x_1= \\left( c_{11} +1 \\right) x_1 +c_{12} x_2 +c_{13} x_3 - y_1 \\ , "
},
{
"math_id": 24,
"text": "\\begin{align} x_\\mathrm{k} &= \\sum_{\\mathrm{j}=1}^{\\mathrm{M}} ( G_\\mathrm{kj} ) y_\\mathrm{j} \\end{align} "
},
{
"math_id": 25,
"text": "V_2 = a_{12}V_1 \\,."
},
{
"math_id": 26,
"text": "G = \\frac {y_2}{x_1}"
},
{
"math_id": 27,
"text": " = G_{\\infty} \\left( \\frac{T}{T + 1} \\right) + G_0 \\left( \\frac{1}{T + 1} \\right) \\ ."
},
{
"math_id": 28,
"text": "G_{\\infty} = \\lim_{T \\to \\infty }G\\ ; \\ G_{0} = \\lim_{T \\to 0 }G \\ . "
},
{
"math_id": 29,
"text": " = G_{0} + \\frac {A} {1 - \\beta A} \\ ."
},
{
"math_id": 30,
"text": " T = - \\beta A \\ , "
},
{
"math_id": 31,
"text": " G_{\\infty} = \\lim_{T \\to \\infty }G = G_0 - \\frac {1} {\\beta} \\ ."
},
{
"math_id": 32,
"text": "G = G_{0} + \\frac {1} {\\beta} \\frac {-T} {1 +T} "
},
{
"math_id": 33,
"text": " = G_0 + (G_0 - G_{\\infty} ) \\frac {-T} {1 +T} "
},
{
"math_id": 34,
"text": " = G_{\\infty} \\frac {T} {1 +T} + G_0 \\frac {1}{1+T} \\ ,"
},
{
"math_id": 35,
"text": " \\frac {1} {s \\mathrm{L}_\\mathrm{M}} \\, "
},
{
"math_id": 36,
"text": "\\begin{align}\n x_\\mathrm{j} &= x_\\mathrm{k} \\times x_\\mathrm{l} \\\\\n x_\\mathrm{k} &= abs(x_\\mathrm{j})\\\\\n x_\\mathrm{l} &= \\log(x_\\mathrm{k})\\\\\n x_\\mathrm{m} &= t \\times x_\\mathrm{j} \\text{ ,where } t \\text{ represents time} \\\\\n\\end{align}"
}
] |
https://en.wikipedia.org/wiki?curid=15175696
|
151783
|
Stirling's approximation
|
Approximation for factorials
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of formula_0. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.
One way of stating the approximation involves the logarithm of the factorial:
formula_1
where the big O notation means that, for all sufficiently large values of formula_0, the difference between formula_2 and formula_3 will be at most proportional to the logarithm of formula_0. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form
formula_4 The error term in either base can be expressed more precisely as formula_5, corresponding to an approximate formula for the factorial itself,
formula_6
Here the sign formula_7 means that the two quantities are asymptotic, that is, that their ratio tends to 1 as formula_0 tends to infinity. The following version of the bound holds for all formula_8, rather than only asymptotically:
formula_9
Derivation.
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum
formula_10
with an integral:
formula_11
The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating formula_12, one considers its natural logarithm, as this is a slowly varying function:
formula_13
The right-hand side of this equation minus
formula_14
is the approximation by the trapezoid rule of the integral
formula_15
and the error in this approximation is given by the Euler–Maclaurin formula:
formula_16
where formula_17 is a Bernoulli number, and "R""m","n" is the remainder term in the Euler–Maclaurin formula. Take limits to find that
formula_18
Denote this limit as formula_19. Because the remainder "R""m","n" in the Euler–Maclaurin formula satisfies
formula_20
where big-O notation is used, combining the equations above yields the approximation formula in its logarithmic form:
formula_21
Taking the exponential of both sides and choosing any positive integer formula_22, one obtains a formula involving an unknown quantity formula_23. For "m"
1, the formula is
formula_24
The quantity formula_23 can be found by taking the limit on both sides as formula_0 tends to infinity and using Wallis' product, which shows that formula_25. Therefore, one obtains Stirling's formula:
formula_26
Alternative derivations.
An alternative formula for formula_12 using the gamma function is
formula_27
(as can be seen by repeated integration by parts). Rewriting and changing variables "x"
"ny", one obtains
formula_28
Applying Laplace's method one has
formula_29
which recovers Stirling's formula:
formula_30
Higher orders.
In fact, further corrections can also be obtained using Laplace's method. From previous result, we know that formula_31, so we "peel off" this dominant term, then perform two changes of variables, to obtain:formula_32To verify this: formula_33.
Now the function formula_34 is unimodal, with maximum value zero. Locally around zero, it looks like formula_35, which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by formula_36. This equation cannot be solved in closed form, but it can be solved by serial expansion, which gives us formula_37. Now plug back to the equation to obtainformula_38notice how we don't need to actually find formula_39, since it is cancelled out by the integral. Higher orders can be achieved by computing more terms in formula_40, which can be obtained programmatically.
Thus we get Stirling's formula to two orders:formula_41
Complex-analytic version.
A complex-analysis version of this method is to consider formula_42 as a Taylor coefficient of the exponential function formula_43, computed by Cauchy's integral formula as
formula_44
This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius formula_45. The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term.
Speed of convergence and error estimates.
Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):
formula_46
An explicit formula for the coefficients in this series was given by G. Nemes. Further terms are listed in the On-Line Encyclopedia of Integer Sequences as and . The first graph in this section shows the relative error vs. formula_0, for 1 through all 5 terms listed above. (Bender and Orszag p. 218) gives the asymptotic formula for the coefficients:formula_47which shows that it grows superexponentially, and that by ratio test the radius of convergence is zero.
As "n" → ∞, the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion. It is not a convergent series; for any "particular" value of formula_0 there are only so many terms of the series that improve accuracy, after which accuracy worsens. This is shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms. More precisely, let "S"("n", "t") be the Stirling series to formula_48 terms evaluated at formula_0. The graphs show
formula_49
which, when small, is essentially the relative error.
Writing Stirling's series in the form
formula_50
it is known that the error in truncating the series is always of the opposite sign and at most the same magnitude as the first omitted term.
Other bounds, due to Robbins, valid for all positive integers formula_0 are
formula_51
This upper bound corresponds to stopping the above series for formula_2 after the formula_52 term. The lower bound is weaker than that obtained by stopping the series after the formula_53 term. A looser version of this bound is that formula_54 for all formula_8.
Stirling's formula for the gamma function.
For all positive integers,
formula_55
where Γ denotes the gamma function.
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied. If Re("z") > 0, then
formula_56
Repeated integration by parts gives
formula_57
where formula_58 is the formula_0th Bernoulli number (note that the limit of the sum as formula_59 is not convergent, so this formula is just an asymptotic expansion). The formula is valid for formula_60 large enough in absolute value, when , where ε is positive, with an error term of "O"("z"−2"N"+ 1). The corresponding approximation may now be written:
formula_61
where the expansion is identical to that of Stirling's series above for formula_12, except that formula_0 is replaced with "z" − 1.
A further application of this asymptotic expansion is for complex argument z with constant Re("z"). See for example the Stirling formula applied in Im("z")
"t" of the Riemann–Siegel theta function on the straight line + "it".
Error bounds.
For any positive integer formula_62, the following notation is introduced:
formula_63
and
formula_64
Then
formula_65
For further information and other error bounds, see the cited papers.
A convergent version of Stirling's formula.
Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series. Obtaining a convergent version of Stirling's formula entails evaluating Binet's formula:
formula_66
One way to do this is by means of a convergent series of inverted rising factorials. If
formula_67
then
formula_68
where
formula_69
where "s"("n", "k") denotes the Stirling numbers of the first kind. From this one obtains a version of Stirling's series
formula_70
which converges when Re("x") > 0.
Stirling's formula may also be given in convergent form as
formula_71
where
formula_72
Versions suitable for calculators.
The approximation
formula_73
and its equivalent form
formula_74
can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function. This approximation is good to more than 8 decimal digits for z with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory.
Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:
formula_75
or equivalently,
formula_76
An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is
formula_77
for "x" ≥ 0. The equivalent approximation for ln "n"! has an asymptotic error of and is given by
formula_78
The approximation may be made precise by giving paired upper and lower bounds; one such inequality is
formula_79
History.
The formula was first discovered by Abraham de Moivre in the form
formula_80
De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely formula_81.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "\\ln(n!) = n\\ln n - n +O(\\ln n),"
},
{
"math_id": 2,
"text": "\\ln(n!)"
},
{
"math_id": 3,
"text": "n\\ln n-n"
},
{
"math_id": 4,
"text": "\\log_2 (n!) = n\\log_2 n - n\\log_2 e +O(\\log_2 n)."
},
{
"math_id": 5,
"text": "\\tfrac12\\log(2\\pi n)+O(\\tfrac1n)"
},
{
"math_id": 6,
"text": "n! \\sim \\sqrt{2 \\pi n}\\left(\\frac{n}{e}\\right)^n."
},
{
"math_id": 7,
"text": "\\sim"
},
{
"math_id": 8,
"text": "n \\ge 1"
},
{
"math_id": 9,
"text": "\\sqrt{2 \\pi n}\\ \\left(\\frac{n}{e}\\right)^n e^{\\left(\\frac{1}{12n} - \\frac{1}{360n^3}\\right)} < n! < \\sqrt{2 \\pi n}\\ \\left(\\frac{n}{e}\\right)^n e^{\\frac{1}{12n}}. "
},
{
"math_id": 10,
"text": "\\ln(n!) = \\sum_{j=1}^n \\ln j"
},
{
"math_id": 11,
"text": "\\sum_{j=1}^n \\ln j \\approx \\int_1^n \\ln x \\,{\\rm d}x = n\\ln n - n + 1."
},
{
"math_id": 12,
"text": "n!"
},
{
"math_id": 13,
"text": "\\ln(n!) = \\ln 1 + \\ln 2 + \\cdots + \\ln n."
},
{
"math_id": 14,
"text": "\\tfrac{1}{2}(\\ln 1 + \\ln n) = \\tfrac{1}{2}\\ln n"
},
{
"math_id": 15,
"text": "\\ln(n!) - \\tfrac{1}{2}\\ln n \\approx \\int_1^n \\ln x\\,{\\rm d}x = n \\ln n - n + 1,"
},
{
"math_id": 16,
"text": "\\begin{align}\n\\ln(n!) - \\tfrac{1}{2}\\ln n & = \\tfrac{1}{2}\\ln 1 + \\ln 2 + \\ln 3 + \\cdots + \\ln(n-1) + \\tfrac{1}{2}\\ln n\\\\\n& = n \\ln n - n + 1 + \\sum_{k=2}^{m} \\frac{(-1)^k B_k}{k(k-1)} \\left( \\frac{1}{n^{k-1}} - 1 \\right) + R_{m,n},\n\\end{align}"
},
{
"math_id": 17,
"text": "B_k"
},
{
"math_id": 18,
"text": "\\lim_{n \\to \\infty} \\left( \\ln(n!) - n \\ln n + n - \\tfrac{1}{2}\\ln n \\right) = 1 - \\sum_{k=2}^{m} \\frac{(-1)^k B_k}{k(k-1)} + \\lim_{n \\to \\infty} R_{m,n}."
},
{
"math_id": 19,
"text": "y"
},
{
"math_id": 20,
"text": "R_{m,n} = \\lim_{n \\to \\infty} R_{m,n} + O \\left( \\frac{1}{n^m} \\right),"
},
{
"math_id": 21,
"text": "\\ln(n!) = n \\ln \\left( \\frac{n}{e} \\right) + \\tfrac{1}{2}\\ln n + y + \\sum_{k=2}^{m} \\frac{(-1)^k B_k}{k(k-1)n^{k-1}} + O \\left( \\frac{1}{n^m} \\right)."
},
{
"math_id": 22,
"text": "m"
},
{
"math_id": 23,
"text": "e^y"
},
{
"math_id": 24,
"text": "n! = e^y \\sqrt{n} \\left( \\frac{n}{e} \\right)^n \\left( 1 + O \\left( \\frac{1}{n} \\right) \\right)."
},
{
"math_id": 25,
"text": "e^y=\\sqrt{2\\pi}"
},
{
"math_id": 26,
"text": "n! = \\sqrt{2 \\pi n} \\left( \\frac{n}{e} \\right)^n \\left( 1 + O \\left( \\frac{1}{n} \\right) \\right)."
},
{
"math_id": 27,
"text": " n! = \\int_0^\\infty x^n e^{-x}\\,{\\rm d}x."
},
{
"math_id": 28,
"text": " n! = \\int_0^\\infty e^{n\\ln x-x}\\,{\\rm d}x = e^{n \\ln n} n \\int_0^\\infty e^{n(\\ln y -y)}\\,{\\rm d}y."
},
{
"math_id": 29,
"text": "\\int_0^\\infty e^{n(\\ln y -y)}\\,{\\rm d}y \\sim \\sqrt{\\frac{2\\pi}{n}} e^{-n},"
},
{
"math_id": 30,
"text": "n! \\sim e^{n \\ln n} n \\sqrt{\\frac{2\\pi}{n}} e^{-n}\n= \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n.\n"
},
{
"math_id": 31,
"text": "\\Gamma(x) \\sim x^x e^{-x}"
},
{
"math_id": 32,
"text": "x^{-x}e^x\\Gamma(x) = \\int_\\R e^{x(1+t-e^t)}dt"
},
{
"math_id": 33,
"text": "\\int_\\R e^{x(1+t-e^t)}dt \\overset{t \\mapsto \\ln t}{=} e^x \\int_0^\\infty t^{x-1} e^{-xt} dt \\overset{t \\mapsto t/x}{=} x^{-x} e^x \\int_0^\\infty e^{-t} t^{x-1} dt = x^{-x} e^x \\Gamma(x)"
},
{
"math_id": 34,
"text": "t \\mapsto 1+t - e^t"
},
{
"math_id": 35,
"text": "-t^2/2"
},
{
"math_id": 36,
"text": "1+t-e^t = -\\tau^2/2"
},
{
"math_id": 37,
"text": "t = \\tau - \\tau^2/6 + \\tau^3/36 + a_4 \\tau^4 + O(\\tau^5) "
},
{
"math_id": 38,
"text": "x^{-x}e^x\\Gamma(x) = \\int_\\R e^{-x\\tau^2/2}(1-\\tau/3 + \\tau^2/12 + 4a_4 \\tau^3 + O(\\tau^4)) d\\tau = \\sqrt{2\\pi}(x^{-1/2} + x^{-3/2}/12) + O(x^{-5/2})"
},
{
"math_id": 39,
"text": "a_4"
},
{
"math_id": 40,
"text": "t = \\tau + \\cdots"
},
{
"math_id": 41,
"text": " n! = \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n \\left(1 + \\frac{1}{12 n}+O\\left(\\frac{1}{n^2}\\right) \\right).\n"
},
{
"math_id": 42,
"text": "\\frac{1}{n!}"
},
{
"math_id": 43,
"text": "e^z = \\sum_{n=0}^\\infty \\frac{z^n}{n!}"
},
{
"math_id": 44,
"text": "\\frac{1}{n!} = \\frac{1}{2\\pi i} \\oint\\limits_{|z|=r} \\frac{e^z}{z^{n+1}} \\, \\mathrm dz. "
},
{
"math_id": 45,
"text": "r = r_n"
},
{
"math_id": 46,
"text": "\nn! \\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n \\left(1 +\\frac{1}{12n}+\\frac{1}{288n^2} - \\frac{139}{51840n^3} -\\frac{571}{2488320n^4}+ \\cdots \\right)."
},
{
"math_id": 47,
"text": "A_{2 j+1} \\sim(-1)^j 2(2 j) ! /(2 \\pi)^{2(j+1)}"
},
{
"math_id": 48,
"text": "t"
},
{
"math_id": 49,
"text": "\\left | \\ln \\left (\\frac{S(n, t)}{n!} \\right) \\right |, "
},
{
"math_id": 50,
"text": "\\ln(n!) \\sim n\\ln n - n + \\tfrac12\\ln(2\\pi n) +\\frac{1}{12n} - \\frac{1}{360n^3} + \\frac{1}{1260n^5} - \\frac{1}{1680n^7} + \\cdots,"
},
{
"math_id": 51,
"text": "\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n e^{\\frac{1}{12n + 1}} < n! < \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n e^{\\frac{1}{12n}}. "
},
{
"math_id": 52,
"text": "\\frac{1}{n}"
},
{
"math_id": 53,
"text": "\\frac{1}{n^3}"
},
{
"math_id": 54,
"text": "\\frac{n! e^n}{n^{n+\\frac12}} \\in (\\sqrt{2 \\pi}, e]"
},
{
"math_id": 55,
"text": "n! = \\Gamma(n + 1),"
},
{
"math_id": 56,
"text": "\\ln\\Gamma (z) = z\\ln z - z + \\tfrac12\\ln\\frac{2\\pi}{z} + \\int_0^\\infty\\frac{2\\arctan\\left(\\frac{t}{z}\\right)}{e^{2\\pi t}-1}\\,{\\rm d}t."
},
{
"math_id": 57,
"text": "\\ln\\Gamma(z) \\sim z\\ln z - z + \\tfrac12\\ln\\frac{2\\pi}{z} + \\sum_{n=1}^{N-1} \\frac{B_{2n}}{2n(2n-1)z^{2n-1}},"
},
{
"math_id": 58,
"text": "B_n"
},
{
"math_id": 59,
"text": "N \\to \\infty"
},
{
"math_id": 60,
"text": "z"
},
{
"math_id": 61,
"text": "\\Gamma(z) = \\sqrt{\\frac{2\\pi}{z}}\\,{\\left(\\frac{z}{e}\\right)}^z \\left(1 + O\\left(\\frac{1}{z}\\right)\\right)."
},
{
"math_id": 62,
"text": "N"
},
{
"math_id": 63,
"text": "\\ln\\Gamma(z) = z\\ln z - z + \\tfrac12\\ln\\frac{2\\pi}{z} + \\sum\\limits_{n=1}^{N-1}{\\frac{{B_{2n}}}{{2n\\left({2n - 1}\\right)z^{2n - 1} }}} + R_N(z)"
},
{
"math_id": 64,
"text": "\\Gamma(z) = \\sqrt{\\frac{{2\\pi}}{z}} \\left({\\frac{z}{e}}\\right)^z \\left({\\sum\\limits_{n=0}^{N-1}{\\frac{{a_n}}{{z^n}}} + \\widetilde{R}_N(z)}\\right)."
},
{
"math_id": 65,
"text": "\\begin{align}\n|R_N(z)| &\\le\n \\frac{|B_{2N}|}{2N(2N-1)|z|^{2N-1}} \\times\n \\begin{cases}\n 1 & \\text{ if } \\left|\\arg z\\right| \\leq \\frac{\\pi}{4}, \\\\\n \\left|\\csc(\\arg z)\\right| & \\text{ if } \\frac{\\pi}{4} < \\left|\\arg z\\right| < \\frac{\\pi}{2}, \\\\\n \\sec^{2N}\\left(\\tfrac{\\arg z}{2}\\right) & \\text{ if } \\left|\\arg z\\right| < \\pi,\n \\end{cases} \\\\[6pt]\n\n \\left |\\widetilde{R}_N(z) \\right | &\\le\n \\left(\\frac{\\left |a_N \\right |}{|z|^N} + \\frac{\\left |a_{N+1} \\right |}{|z|^{N+1}}\\right)\\times\n \\begin{cases}\n 1 & \\text{ if } \\left|\\arg z\\right| \\leq \\frac{\\pi}{4}, \\\\\n \\left|\\csc(2\\arg z)\\right| & \\text{ if } \\frac{\\pi}{4} < \\left|\\arg z\\right| < \\frac{\\pi}{2}.\n \\end{cases}\n\\end{align}\n"
},
{
"math_id": 66,
"text": "\\int_0^\\infty \\frac{2\\arctan\\left(\\frac{t}{x}\\right)}{e^{2\\pi t}-1}\\,{\\rm d}t = \\ln\\Gamma(x) - x\\ln x + x - \\tfrac12\\ln\\frac{2\\pi}{x}."
},
{
"math_id": 67,
"text": "z^{\\bar n} = z(z + 1) \\cdots (z + n - 1),"
},
{
"math_id": 68,
"text": "\\int_0^\\infty \\frac{2\\arctan\\left(\\frac{t}{x}\\right)}{e^{2\\pi t} - 1}\\,{\\rm d}t = \\sum_{n=1}^\\infty \\frac{c_n}{(x + 1)^{\\bar n}},"
},
{
"math_id": 69,
"text": "c_n = \\frac{1}{n} \\int_0^1 x^{\\bar n} \\left(x - \\tfrac{1}{2}\\right)\\,{\\rm d}x = \\frac{1}{2n}\\sum_{k=1}^n \\frac{k|s(n, k)|}{(k + 1)(k + 2)},"
},
{
"math_id": 70,
"text": "\\begin{align}\n\\ln\\Gamma(x) &= x\\ln x - x + \\tfrac12\\ln\\frac{2\\pi}{x} + \\frac{1}{12(x+1)} + \\frac{1}{12(x+1)(x+2)} + \\\\\n &\\quad + \\frac{59}{360(x+1)(x+2)(x+3)} + \\frac{29}{60(x+1)(x+2)(x+3)(x+4)} + \\cdots,\n\\end{align}"
},
{
"math_id": 71,
"text": "\n\\Gamma(x)=\\sqrt{2\\pi}x^{x-\\frac{1}{2}}e^{-x+\\mu(x)}\n"
},
{
"math_id": 72,
"text": "\n\\mu\\left(x\\right)=\\sum_{n=0}^{\\infty}\\left(\\left(x+n+\\frac{1}{2}\\right)\\ln\\left(1+\\frac{1}{x+n}\\right)-1\\right).\n"
},
{
"math_id": 73,
"text": "\\Gamma(z) \\approx \\sqrt{\\frac{2 \\pi}{z}} \\left(\\frac{z}{e} \\sqrt{z \\sinh\\frac{1}{z} + \\frac{1}{810z^6} } \\right)^z"
},
{
"math_id": 74,
"text": "2\\ln\\Gamma(z) \\approx \\ln(2\\pi) - \\ln z + z \\left(2\\ln z + \\ln\\left(z\\sinh\\frac{1}{z} + \\frac{1}{810z^6}\\right) - 2\\right)"
},
{
"math_id": 75,
"text": "\\Gamma(z) \\approx \\sqrt{\\frac{2\\pi}{z} } \\left(\\frac{1}{e} \\left(z + \\frac{1}{12z - \\frac{1}{10z}}\\right)\\right)^z,"
},
{
"math_id": 76,
"text": " \\ln\\Gamma(z) \\approx \\tfrac{1}{2} \\left(\\ln(2\\pi) - \\ln z\\right) + z\\left(\\ln\\left(z + \\frac{1}{12z - \\frac{1}{10z}}\\right) - 1\\right). "
},
{
"math_id": 77,
"text": "\\Gamma(1+x) \\approx \\sqrt{\\pi} \\left(\\frac{x}{e}\\right)^x \\left( 8x^3 + 4x^2 + x + \\frac{1}{30} \\right)^{\\frac{1}{6}}"
},
{
"math_id": 78,
"text": "\\ln n! \\approx n\\ln n - n + \\tfrac{1}{6}\\ln(8n^3 + 4n^2 + n + \\tfrac{1}{30}) + \\tfrac{1}{2}\\ln\\pi ."
},
{
"math_id": 79,
"text": " \\sqrt{\\pi} \\left(\\frac{x}{e}\\right)^x \\left( 8x^3 + 4x^2 + x + \\frac{1}{100} \\right)^{1/6} < \\Gamma(1+x) < \\sqrt{\\pi} \\left(\\frac{x}{e}\\right)^x \\left( 8x^3 + 4x^2 + x + \\frac{1}{30} \\right)^{1/6}."
},
{
"math_id": 80,
"text": "n! \\sim [{\\rm constant}] \\cdot n^{n+\\frac12} e^{-n}."
},
{
"math_id": 81,
"text": "\\sqrt{2\\pi} "
}
] |
https://en.wikipedia.org/wiki?curid=151783
|
1518031
|
Circular distribution
|
Type of probability distribution
In probability and statistics, a circular distribution or polar distribution is a probability distribution of a random variable whose values are angles, usually taken to be in the range [0, 2"π"). A circular distribution is often a continuous probability distribution, and hence has a probability density, but such distributions can also be discrete, in which case they are called circular lattice distributions. Circular distributions can be used even when the variables concerned are not explicitly angles: the main consideration is that there is not usually any real distinction between events occurring at the opposite ends of the range, and the division of the range could notionally be made at any point.
Graphical representation.
If a circular distribution has a density
formula_0
it can be graphically represented as a closed curve
formula_1
where the radius formula_2 is set equal to
formula_3
and where "a" and "b" are chosen on the basis of appearance.
Examples.
By computing the probability distribution of angles along a handwritten ink trace,
a lobe-shaped polar distribution emerges. The main direction of the lobe in the
first quadrant corresponds to the slant of handwriting (see: graphonomics).
An example of a circular lattice distribution would be the probability of being born in a given month of the year, with each calendar month being thought of as arranged round a circle, so that "January" is next to "December".
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "p(\\phi) \\qquad \\qquad (0\\le\\phi<2\\pi),\\,"
},
{
"math_id": 1,
"text": "[x(\\phi),y(\\phi)] = [r(\\phi)\\cos\\phi, \\, r(\\phi)\\sin\\phi], \\, "
},
{
"math_id": 2,
"text": "r(\\phi)\\,"
},
{
"math_id": 3,
"text": "r(\\phi) = a+b p(\\phi), \\, "
}
] |
https://en.wikipedia.org/wiki?curid=1518031
|
15180479
|
Projection matrix
|
Concept in statistics
In statistics, the projection matrix formula_0, sometimes also called the influence matrix or hat matrix formula_1, maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation.
Definition.
If the vector of response values is denoted by formula_2 and the vector of fitted values by formula_3,
formula_4
As formula_3 is usually pronounced "y-hat", the projection matrix formula_5 is also named "hat matrix" as it "puts a hat on formula_2".
Application for residuals.
The formula for the vector of residuals formula_6 can also be expressed compactly using the projection matrix:
formula_7
where formula_8 is the identity matrix. The matrix formula_9 is sometimes referred to as the residual maker matrix or the annihilator matrix.
The covariance matrix of the residuals formula_6, by error propagation, equals
formula_10,
where formula_11 is the covariance matrix of the error vector (and by extension, the response vector as well). For the case of linear models with independent and identically distributed errors in which formula_12, this reduces to:
formula_13.
Intuition.
From the figure, it is clear that the closest point from the vector formula_15 onto the column space of formula_14, is formula_17, and is one where we can draw a line orthogonal to the column space of formula_14. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
formula_18.
From there, one rearranges, so
formula_19.
Therefore, since formula_17 is on the column space of formula_14, the projection matrix, which maps formula_15 onto formula_16 is just formula_14, or formula_20.
Linear model.
Suppose that we wish to estimate a linear model using linear least squares. The model can be written as
formula_21
where formula_22 is a matrix of explanatory variables (the design matrix), β is a vector of unknown parameters to be estimated, and ε is the error vector.
Many types of models and techniques are subject to this formulation. A few examples are linear least squares, smoothing splines, regression splines, local regression, kernel regression, and linear filtering.
Ordinary least squares.
When the weights for each observation are identical and the errors are uncorrelated, the estimated parameters are
formula_23
so the fitted values are
formula_24
Therefore, the projection matrix (and hat matrix) is given by
formula_25
Weighted and generalized least squares.
The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. Suppose that the covariance matrix of the errors is Σ. Then since
formula_26.
the hat matrix is thus
formula_27
and again it may be seen that formula_28, though now it is no longer symmetric.
Properties.
The projection matrix has a number of useful algebraic properties. In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix formula_22. (Note that formula_29 is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized as follows:
The projection matrix corresponding to a linear model is symmetric and idempotent, that is, formula_32. However, this is not always the case; in locally weighted scatterplot smoothing (LOESS), for example, the hat matrix is in general neither symmetric nor idempotent.
For linear models, the trace of the projection matrix is equal to the rank of formula_22, which is the number of independent parameters of the linear model. For other models such as LOESS that are still linear in the observations formula_2, the projection matrix can be used to define the effective degrees of freedom of the model.
Practical applications of the projection matrix in regression analysis include leverage and Cook's distance, which are concerned with identifying influential observations, i.e. observations which have a large effect on the results of a regression.
Blockwise formula.
Suppose the design matrix formula_22 can be decomposed by columns as formula_39.
Define the hat or projection operator as formula_40. Similarly, define the residual operator as formula_41.
Then the projection matrix can be decomposed as follows:
formula_42
where, e.g., formula_43 and formula_44.
There are a number of applications of such a decomposition. In the classical application formula_14 is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the fixed effects model, where formula_14 is a large sparse matrix of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of formula_22 without explicitly forming the matrix formula_22, which might be too large to fit into computer memory.
History.
The hat matrix was introduced by John Wilder in 1972. An article by Hoaglin, D.C. and Welsch, R.E. (1978) gives the properties of the matrix and also many examples of its application.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "(\\mathbf{P})"
},
{
"math_id": 1,
"text": "(\\mathbf{H})"
},
{
"math_id": 2,
"text": "\\mathbf{y}"
},
{
"math_id": 3,
"text": "\\mathbf{\\hat{y}}"
},
{
"math_id": 4,
"text": "\\mathbf{\\hat{y}} = \\mathbf{P} \\mathbf{y}."
},
{
"math_id": 5,
"text": "\\mathbf{P}"
},
{
"math_id": 6,
"text": "\\mathbf{r}"
},
{
"math_id": 7,
"text": "\\mathbf{r} = \\mathbf{y} - \\mathbf{\\hat{y}} = \\mathbf{y} - \\mathbf{P} \\mathbf{y} = \\left( \\mathbf{I} - \\mathbf{P} \\right) \\mathbf{y}."
},
{
"math_id": 8,
"text": "\\mathbf{I}"
},
{
"math_id": 9,
"text": "\\mathbf{M} := \\mathbf{I} - \\mathbf{P}"
},
{
"math_id": 10,
"text": "\\mathbf{\\Sigma}_\\mathbf{r} = \\left( \\mathbf{I} - \\mathbf{P} \\right)^\\textsf{T} \\mathbf{\\Sigma} \\left( \\mathbf{I}-\\mathbf{P} \\right)"
},
{
"math_id": 11,
"text": "<math>\\mathbf{\\Sigma}</matH>"
},
{
"math_id": 12,
"text": "\\mathbf{\\Sigma} = \\sigma^{2} \\mathbf{I}"
},
{
"math_id": 13,
"text": "\\mathbf{\\Sigma}_\\mathbf{r} = \\left( \\mathbf{I} - \\mathbf{P} \\right) \\sigma^{2}"
},
{
"math_id": 14,
"text": "\\mathbf{A}"
},
{
"math_id": 15,
"text": "\\mathbf{b}"
},
{
"math_id": 16,
"text": "\\mathbf{x}"
},
{
"math_id": 17,
"text": "\\mathbf{Ax}"
},
{
"math_id": 18,
"text": "\\mathbf{A}^\\textsf{T}(\\mathbf{b}-\\mathbf{Ax}) = 0"
},
{
"math_id": 19,
"text": "\\begin{align}\n && \\mathbf{A}^\\textsf{T}\\mathbf{b} &- \\mathbf{A}^\\textsf{T}\\mathbf{Ax} = 0 \\\\\n \\Rightarrow && \\mathbf{A}^\\textsf{T}\\mathbf{b} &= \\mathbf{A}^\\textsf{T}\\mathbf{Ax} \\\\\n \\Rightarrow && \\mathbf{x} &= \\left(\\mathbf{A}^\\textsf{T}\\mathbf{A}\\right)^{-1}\\mathbf{A}^\\textsf{T}\\mathbf{b}\n\\end{align}"
},
{
"math_id": 20,
"text": "\\mathbf{A}\\left(\\mathbf{A}^\\textsf{T}\\mathbf{A}\\right)^{-1}\\mathbf{A}^\\textsf{T}"
},
{
"math_id": 21,
"text": "\\mathbf{y} = \\mathbf{X} \\boldsymbol\\beta + \\boldsymbol\\varepsilon,"
},
{
"math_id": 22,
"text": "\\mathbf{X}"
},
{
"math_id": 23,
"text": "\\hat{\\boldsymbol\\beta} = \\left( \\mathbf{X}^\\textsf{T} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T} \\mathbf{y},"
},
{
"math_id": 24,
"text": "\\hat{\\mathbf{y}} = \\mathbf{X} \\hat{\\boldsymbol \\beta} = \\mathbf{X} \\left( \\mathbf{X}^\\textsf{T} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T} \\mathbf{y}."
},
{
"math_id": 25,
"text": "\\mathbf{P} := \\mathbf{X} \\left(\\mathbf{X}^\\textsf{T} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T}."
},
{
"math_id": 26,
"text": "\n \\hat{\\mathbf\\beta}_{\\text{GLS}}= \\left( \\mathbf{X}^\\textsf{T} \\mathbf{\\Sigma}^{-1} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T} \\mathbf{\\Sigma}^{-1}\\mathbf{y}\n "
},
{
"math_id": 27,
"text": "\n \\mathbf{H} = \\mathbf{X}\\left( \\mathbf{X}^\\textsf{T} \\mathbf{\\Sigma}^{-1} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T} \\mathbf{\\Sigma}^{-1}\n "
},
{
"math_id": 28,
"text": "H^2 = H\\cdot H = H"
},
{
"math_id": 29,
"text": "\\left( \\mathbf{X}^\\textsf{T} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T}"
},
{
"math_id": 30,
"text": "\\mathbf{u} = (\\mathbf{I} - \\mathbf{P})\\mathbf{y},"
},
{
"math_id": 31,
"text": "\\mathbf{u} = \\mathbf{y} - \\mathbf{P} \\mathbf{y} \\perp \\mathbf{X}."
},
{
"math_id": 32,
"text": "\\mathbf{P}^2 = \\mathbf{P}"
},
{
"math_id": 33,
"text": "\\mathbf{M}"
},
{
"math_id": 34,
"text": "\\operatorname{rank}(\\mathbf{X}) = r"
},
{
"math_id": 35,
"text": "\\operatorname{rank}(\\mathbf{P}) = r"
},
{
"math_id": 36,
"text": "\\mathbf{P X} = \\mathbf{X},"
},
{
"math_id": 37,
"text": "\\left( \\mathbf{I} - \\mathbf{P} \\right) \\mathbf{X} = \\mathbf{0}"
},
{
"math_id": 38,
"text": "\\left( \\mathbf{I} - \\mathbf{P} \\right) \\mathbf{P} = \\mathbf{P} \\left( \\mathbf{I} - \\mathbf{P} \\right) = \\mathbf{0}."
},
{
"math_id": 39,
"text": "\\mathbf{X} = \\begin{bmatrix} \\mathbf{A} & \\mathbf{B} \\end{bmatrix}"
},
{
"math_id": 40,
"text": "\\mathbf{P}[\\mathbf{X}] := \\mathbf{X} \\left(\\mathbf{X}^\\textsf{T} \\mathbf{X} \\right)^{-1} \\mathbf{X}^\\textsf{T}"
},
{
"math_id": 41,
"text": "\\mathbf{M}[\\mathbf{X}] := \\mathbf{I} - \\mathbf{P}[\\mathbf{X}]"
},
{
"math_id": 42,
"text": " \\mathbf{P}[\\mathbf{X}] = \\mathbf{P}[\\mathbf{A}] + \\mathbf{P}\\big[\\mathbf{M}[\\mathbf{A}] \\mathbf{B}\\big], "
},
{
"math_id": 43,
"text": "\\mathbf{P}[\\mathbf{A}] = \\mathbf{A} \\left(\\mathbf{A}^\\textsf{T} \\mathbf{A} \\right)^{-1} \\mathbf{A}^\\textsf{T}"
},
{
"math_id": 44,
"text": "\\mathbf{M}[\\mathbf{A}] = \\mathbf{I} - \\mathbf{P}[\\mathbf{A}]"
}
] |
https://en.wikipedia.org/wiki?curid=15180479
|
1518079
|
Running angle
|
In mathematics, the running angle is the angle of consecutive vectors formula_0 with respect to the base line, i.e.
formula_1
Usually, it is more informative to compute it using a four-quadrant version of the arctan function in a mathematical software library.
|
[
{
"math_id": 0,
"text": "(Xt,Yt)"
},
{
"math_id": 1,
"text": "\\phi(t) = \\arctan\\left(\\frac {\\Delta Yt} {\\Delta Xt} \\right) ."
}
] |
https://en.wikipedia.org/wiki?curid=1518079
|
15182523
|
Stone–Geary utility function
|
The Stone–Geary utility function takes the form
formula_0
where formula_1 is utility, formula_2 is consumption of good formula_3, and formula_4 and formula_5 are parameters.
For formula_6, the Stone–Geary function reduces to the generalised Cobb–Douglas function.
The Stone–Geary utility function gives rise to the Linear Expenditure System. In case of formula_7 the demand function equals
formula_8
where formula_9 is total expenditure, and formula_10 is the price of good formula_3.
The Stone–Geary utility function was first derived by Roy C. Geary, in a comment on earlier work by Lawrence Klein and Herman Rubin. Richard Stone was the first to estimate the Linear Expenditure System.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "U = \\prod_{i} (q_i-\\gamma_i)^{\\beta_{i}}"
},
{
"math_id": 1,
"text": "U"
},
{
"math_id": 2,
"text": "q_i"
},
{
"math_id": 3,
"text": "i"
},
{
"math_id": 4,
"text": "\\beta"
},
{
"math_id": 5,
"text": "\\gamma"
},
{
"math_id": 6,
"text": "\\gamma_i = 0"
},
{
"math_id": 7,
"text": "\\sum_i \\beta_i =1 "
},
{
"math_id": 8,
"text": "q_i = \\gamma_i + \\frac{\\beta_i}{p_i} (y - \\sum_j \\gamma_j p_j) "
},
{
"math_id": 9,
"text": "y"
},
{
"math_id": 10,
"text": "p_i"
}
] |
https://en.wikipedia.org/wiki?curid=15182523
|
1518267
|
Pen tilt
|
Pen tilt refers to the angle of a writing instrument during handwriting and drawing, which can vary over time. In a coordinate system which is determined by the writing surface plane formula_0 and the vertical pen-tip movement along the formula_1 axis, all three two-dimensional planes can be discerned, and the angular signals can be delivered by a digitizer. It is part of the ISO/IEC standard 19794-7 for biometric data in signatures.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\{X,Y\\}"
},
{
"math_id": 1,
"text": "\\{Z\\}"
}
] |
https://en.wikipedia.org/wiki?curid=1518267
|
15184905
|
Log amplifier
|
Electrical circuit
A log amplifier, also known as logarithmic amplifier or logarithm amplifier or log amp, is an amplifier for which the output voltage "V"out is "K" times the natural log of the input voltage "V"in. This can be expressed as,
formula_0
where "V"ref is the normalization constant in volts and "K" is the scale factor.
The log amplifier gives an output voltage which is proportional to the logarithm of the applied input voltage.
To design a log amplifier circuit, high performance op-amps like LM1458, LM771, LM714 are commonly used and a compensated log amplifier may include more than one. In some situations, especially in RF domain, monolithic log amplifiers are also used to reduce number of components and space used, as well improve bandwidth and noise performance.
The log amplifier's operation can be inverted by an "exponentiator", such as an op-amp configured for exponential output.
Log amplifier applications.
Log amplifiers are used in many ways, such as:
Drawbacks of basic log amplifier configuration.
The reverse saturation current for the diode doubles for every ten degree Celsius rise in temperature. Similarly the emitter saturation current varies significantly from one transistor to another and also with temperature. Hence, it is very difficult to set the reference voltage for the circuit.
Basic op-amp diode circuit.
The relationship between the input voltage formula_1 and the output voltage formula_2 is given by:
formula_3
where formula_4 and formula_5 are the saturation current and the thermal voltage of the diode respectively.
The dynamic range of this basic op-amp diode circuit is limited to 40-60 dB because of non-ideal diode characteristics, but the dynamic range can be increased to over 120 dB by replacing the diode with a transistor in a "transdiode" configuration.
Transdiode configuration.
A necessary condition for successful operation of a log amplifier is that the input voltage, "V"in, is always positive. This may be ensured by using a rectifier and filter to condition the input signal before applying it to the log amplifier's input. As "V"in is positive, "V"out is obliged to be negative (since the op amp is in the inverting configuration) and is large enough to forward bias the emitter-base junction of the BJT keeping it in the active mode of operation. Now,
formula_6
where formula_7 is the saturation current of the emitter-base diode and formula_8 is the thermal voltage. Due to the virtual ground at the op amp differential input,
formula_9, and
formula_10
The output voltage is expressed as the natural log of the input voltage. Both the saturation current formula_7 and the thermal voltage formula_8 are temperature dependent, hence, temperature compensating circuits may be required.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "V_\\text{out} = K \\ln\\left(\\frac{V_\\text{in}}{V_\\text{ref}}\\right)"
},
{
"math_id": 1,
"text": "V_\\text{in}"
},
{
"math_id": 2,
"text": "V_\\text{out}"
},
{
"math_id": 3,
"text": "V_{\\text{out}} = -V_\\text{T} \\ln \\left(\\frac{V_\\text{in}}{I_\\text{S} \\, R} \\right)"
},
{
"math_id": 4,
"text": "I_\\text{S}"
},
{
"math_id": 5,
"text": "V_\\text{T}"
},
{
"math_id": 6,
"text": "\\begin{align}\n V_\\text{BE} &= -V_\\text{out} \\\\\n I_\\text{C} &= I_\\text{S}\\left(e^\\frac{V_\\text{BE}}{V_\\text{T}} - 1\\right) \\approx I_\\text{S} e^\\frac{V_\\text{BE}}{V_\\text{T}} \\\\\n \\Rightarrow V_\\text{BE} &= V_\\text{T} \\ln\\left(\\frac{I_\\text{C}}{I_\\text{S}}\\right)\n\\end{align}"
},
{
"math_id": 7,
"text": "I_\\text{S}\\,"
},
{
"math_id": 8,
"text": "V_\\text{T}\\,"
},
{
"math_id": 9,
"text": "I_\\text{C} = \\frac{V_\\text{in}}{R}"
},
{
"math_id": 10,
"text": "V_\\text{out} = -V_\\text{T} \\ln \\left(\\frac{V_\\text{in}}{I_\\text{S} R}\\right)"
}
] |
https://en.wikipedia.org/wiki?curid=15184905
|
15185443
|
Korn's inequality
|
In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative version of this statement, which intuitively says that if the gradient of a vector field is on average not far from the space of skew-symmetric matrices, then the gradient must not be far from a "particular" skew-symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity.
In (linear) elasticity theory, the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate in linear elasticity theory.
Statement of the inequality.
Let Ω be an open, connected domain in "n"-dimensional Euclidean space R"n", "n" ≥ 2. Let "H"1(Ω) be the Sobolev space of all vector fields "v" = ("v"1, ..., "v""n") on Ω that, along with their (first) weak derivatives, lie in the Lebesgue space "L"2(Ω). Denoting the partial derivative with respect to the "i"th component by ∂"i", the norm in "H"1(Ω) is given by
formula_0
Then there is a (minimal) constant "C" ≥ 0, known as the Korn constant of Ω, such that, for all "v" ∈ "H"1(Ω),
where "e" denotes the symmetrized gradient given by
formula_1
Inequality (1) is known as Korn's inequality.
|
[
{
"math_id": 0,
"text": "\\| v \\|_{H^{1} (\\Omega)} := \\left( \\int_{\\Omega} \\sum_{i = 1}^{n} | v^{i} (x) |^{2} \\, \\mathrm{d} x+\\int_{\\Omega} \\sum_{i, j = 1}^{n} | \\partial_{j} v^{i} (x) |^{2} \\, \\mathrm{d} x \\right)^{1/2}."
},
{
"math_id": 1,
"text": "e_{ij} v = \\frac1{2} ( \\partial_{i} v^{j} + \\partial_{j} v^{i} )."
}
] |
https://en.wikipedia.org/wiki?curid=15185443
|
151864
|
Divergence theorem
|
Theorem in calculus which relates the flux of closed surfaces to divergence over their volume
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the "flux" of a vector field through a closed surface to the "divergence" of the field in the volume enclosed.
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region".
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to Green's theorem.
Explanation using liquid flow.
Vector fields are often illustrated using the example of the velocity field of a fluid, such as a gas or liquid. A moving liquid has a velocity—a speed and a direction—at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Consider an imaginary closed surface "S" inside a body of liquid, enclosing a volume of liquid. The flux of liquid out of the volume at any time is equal to the volume rate of fluid crossing this surface, i.e., the surface integral of the velocity over the surface.
Since liquids are incompressible, the amount of liquid inside a closed volume is constant; if there are no sources or sinks inside the volume then the flux of liquid out of "S" is zero. If the liquid is moving, it may flow into the volume at some points on the surface "S" and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the "net" flux of liquid out of the volume is zero.
However if a "source" of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. This will cause a net outward flow through the surface "S". The flux outward through "S" equals the volume rate of flow of fluid into "S" from the pipe. Similarly if there is a "sink" or drain inside "S", such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain. The volume rate of flow of liquid inward through the surface "S" equals the rate of liquid removed by the sink.
If there are multiple sources and sinks of liquid inside "S", the flux through the surface can be calculated by adding up the volume rate of liquid added by the sources and subtracting the rate of liquid drained off by the sinks. The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the "divergence" of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by "S" equals the volume rate of flux through "S". This is the divergence theorem.
The divergence theorem is employed in any conservation law which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.
Mathematical statement.
Suppose V is a subset of formula_0 (in the case of "n"
3, "V" represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary S (also indicated with formula_1). If F is a continuously differentiable vector field defined on a neighborhood of V, then:
formula_2 formula_3 formula_4
The left side is a volume integral over the volume V, and the right side is the surface integral over the boundary of the volume V. The closed, measurable set formula_5 is oriented by outward-pointing normals, and formula_6 is the outward pointing unit normal at almost each point on the boundary formula_5. (formula_7 may be used as a shorthand for formula_8.) In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary S.
Informal derivation.
The divergence theorem follows from the fact that if a volume V is partitioned into separate parts, the flux out of the original volume is equal to the sum of the flux out of each component volume. This is true despite the fact that the new subvolumes have surfaces that were not part of the original volume's surface, because these surfaces are just partitions between two of the subvolumes and the flux through them just passes from one volume to the other and so cancels out when the flux out of the subvolumes is summed.
See the diagram. A closed, bounded volume V is divided into two volumes "V"1 and "V"2 by a surface "S"3 "(green)". The flux Φ("V"i) out of each component region "V"i is equal to the sum of the flux through its two faces, so the sum of the flux out of the two parts is
formula_9
where Φ1 and Φ2 are the flux out of surfaces "S"1 and "S"2, Φ31 is the flux through "S"3 out of volume 1, and Φ32 is the flux through "S"3 out of volume 2. The point is that surface "S"3 is part of the surface of both volumes. The "outward" direction of the normal vector formula_10 is opposite for each volume, so the flux out of one through "S"3 is equal to the negative of the flux out of the other so these two fluxes cancel in the sum.
formula_11
Therefore:
formula_12
Since the union of surfaces "S"1 and "S"2 is S
formula_13
This principle applies to a volume divided into any number of parts, as shown in the diagram. Since the integral over each internal partition "(green surfaces)" appears with opposite signs in the flux of the two adjacent volumes they cancel out, and the only contribution to the flux is the integral over the external surfaces "(grey)". Since the external surfaces of all the component volumes equal the original surface.
formula_14
The flux Φ out of each volume is the surface integral of the vector field F(x) over the surface
formula_16
The goal is to divide the original volume into infinitely many infinitesimal volumes. As the volume is divided into smaller and smaller parts, the surface integral on the right, the flux out of each subvolume, approaches zero because the surface area "S"("V"i) approaches zero. However, from the definition of divergence, the ratio of flux to volume, formula_17, the part in parentheses below, does not in general vanish but approaches the divergence div F as the volume approaches zero.
formula_18
As long as the vector field F(x) has continuous derivatives, the sum above holds even in the limit when the volume is divided into infinitely small increments
formula_19
As formula_15 approaches zero volume, it becomes the infinitesimal "dV", the part in parentheses becomes the divergence, and the sum becomes a volume integral over V
formula_20
Since this derivation is coordinate free, it shows that the divergence does not depend on the coordinates used.
Proofs.
For bounded open subsets of Euclidean space.
We are going to prove the following:
<templatestyles src="Math_theorem/styles.css" />
Theorem — Let formula_21 be open and bounded with formula_22 boundary. If formula_23 is formula_22 on an open neighborhood formula_24 of formula_25, that is, formula_26, then for each formula_27,
formula_28
where formula_29 is the outward pointing unit normal vector to formula_30.
Equivalently,
formula_31
Proof of Theorem.
For compact Riemannian manifolds with boundary.
We are going to prove the following:
<templatestyles src="Math_theorem/styles.css" />
Theorem — Let formula_25 be a formula_34 compact manifold with boundary with formula_22 metric tensor formula_33. Let formula_32 denote the manifold interior of formula_25 and let formula_30 denote the manifold boundary of formula_25. Let formula_35 denote formula_36 inner products of functions and formula_37 denote inner products of vectors. Suppose formula_38 and formula_39 is a formula_22 vector field on formula_25. Then
formula_40
where formula_41 is the outward-pointing unit normal vector to formula_30.
Proof of Theorem.
We use the Einstein summation convention. By using a partition of unity, we may assume that formula_23 and formula_39 have compact support in a coordinate patch formula_42. First consider the case where the patch is disjoint from formula_30. Then formula_24 is identified with an open subset of formula_0 and integration by parts produces no boundary terms:
formula_43
In the last equality we used the Voss-Weyl coordinate formula for the divergence, although the preceding identity could be used to define formula_44 as the formal adjoint of formula_45. Now suppose formula_24 intersects formula_30. Then formula_24 is identified with an open set in formula_46. We zero extend formula_23 and formula_39 to formula_47 and perform integration by parts to obtain
formula_48
where formula_49.
By a variant of the straightening theorem for vector fields, we may choose formula_24 so that formula_50 is the inward unit normal formula_51 at formula_30. In this case formula_52 is the volume element on formula_30 and the above formula reads
formula_53
This completes the proof.
Corollaries.
By replacing F in the divergence theorem with specific forms, other useful identities can be derived (cf. vector identities).
formula_55 formula_3 formula_56
A special case of this is formula_57, in which case the theorem is the basis for Green's identities.
formula_60 formula_3 formula_61
formula_64 formula_3 formula_65
formula_67 formula_3 formula_68
The last term on the right vanishes for constant formula_69 or any divergence free (solenoidal) vector field, e.g. Incompressible flows without sources or sinks such as phase change or chemical reactions etc. In particular, taking formula_69 to be constant:
formula_70 formula_3 formula_71
formula_73 formula_3 formula_74
By reordering the triple product on the right hand side and taking out the constant vector of the integral,
formula_75 formula_76 formula_77
Hence,
formula_78 formula_76 formula_79
Example.
Suppose we wish to evaluate
formula_3 formula_80
where S is the unit sphere defined by
formula_81
and F is the vector field
formula_82
The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:
formula_83
where W is the unit ball:
formula_84
Since the function y is positive in one hemisphere of W and negative in the other, in an equal and opposite way, its total integral over W is zero. The same is true for z:
formula_85
Therefore,
formula_3 formula_86
because the unit ball W has volume .
Applications.
Differential and integral forms of physical laws.
As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface is equal to another quantity). Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.
Continuity equations.
Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In fluid dynamics, electromagnetism, quantum mechanics, relativity theory, and a number of other fields, there are continuity equations that describe the conservation of mass, momentum, energy, probability, or other quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of "sources" or "sinks" of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).
Inverse-square laws.
Any "inverse-square law" can instead be written in a "Gauss's law"-type form (with a differential and integral form, as described above). Two examples are Gauss's law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss's law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The derivation of the Gauss's law-type equation from the inverse-square formulation or vice versa is exactly the same in both cases; see either of those articles for details.
History.
Joseph-Louis Lagrange introduced the notion of surface integrals in 1760 and again in more general terms in 1811, in the second edition of his "Mécanique Analytique." Lagrange employed surface integrals in his work on fluid mechanics. He discovered the divergence theorem in 1762.
Carl Friedrich Gauss was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem. He proved additional special cases in 1833 and 1839. But it was Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow. Special cases were proven by George Green in 1828 in "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism", Siméon Denis Poisson in 1824 in a paper on elasticity, and Frédéric Sarrus in 1828 in his work on floating bodies.
Worked examples.
Example 1.
To verify the planar variant of the divergence theorem for a region formula_87:
formula_88
and the vector field:
formula_89
The boundary of formula_87 is the unit circle, formula_90, that can be represented parametrically by:
formula_91
such that formula_92 where formula_93 units is the length arc from the point formula_94 to the point formula_95 on formula_90. Then a vector equation of formula_90 is
formula_96
At a point formula_95 on formula_90:
formula_97
Therefore,
formula_98
Because formula_99, we can evaluate formula_100, and because formula_101, formula_102. Thus
formula_103
Example 2.
Let's say we wanted to evaluate the flux of the following vector field defined by formula_104 bounded by the following inequalities:
formula_105
By the divergence theorem,
formula_106 formula_3 formula_107
We now need to determine the divergence of formula_108. If formula_109 is a three-dimensional vector field, then the divergence of formula_108 is given by formula_110.
Thus, we can set up the following flux integral formula_111 formula_112 formula_113
as follows:
formula_114
Now that we have set up the integral, we can evaluate it.
formula_115
Generalizations.
Multiple dimensions.
One can use the generalised Stokes' theorem to equate the n-dimensional volume integral of the divergence of a vector field F over a region U to the ("n" − 1)-dimensional surface integral of F over the boundary of U:
formula_116
This equation is also known as the divergence theorem.
When "n"
2, this is equivalent to Green's theorem.
When "n"
1, it reduces to the fundamental theorem of calculus, part 2.
Tensor fields.
Writing the theorem in Einstein notation:
formula_117 formula_3 formula_118
suggestively, replacing the vector field F with a rank-n tensor field T, this can be generalized to:
formula_119 formula_3 formula_120
where on each side, tensor contraction occurs for at least one index. This form of the theorem is still in 3d, each index takes values 1, 2, and 3. It can be generalized further still to higher (or lower) dimensions (for example to 4d spacetime in general relativity).
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbb{R}^n"
},
{
"math_id": 1,
"text": "\\partial V = S"
},
{
"math_id": 2,
"text": "\\iiint_V\\left(\\mathbf{\\nabla}\\cdot\\mathbf{F}\\right)\\,\\mathrm{d}V="
},
{
"math_id": 3,
"text": "\\scriptstyle S"
},
{
"math_id": 4,
"text": "(\\mathbf{F}\\cdot\\mathbf{\\hat{n}})\\,\\mathrm{d}S ."
},
{
"math_id": 5,
"text": "\\partial V"
},
{
"math_id": 6,
"text": "\\mathbf{\\hat{n}}"
},
{
"math_id": 7,
"text": "\\mathrm{d} \\mathbf{S}"
},
{
"math_id": 8,
"text": "\\mathbf{n} \\mathrm{d} S"
},
{
"math_id": 9,
"text": "\\Phi(V_\\text{1}) + \\Phi(V_\\text{2}) = \\Phi_\\text{1} + \\Phi_\\text{31} + \\Phi_\\text{2} + \\Phi_\\text{32}"
},
{
"math_id": 10,
"text": "\\mathbf{\\hat n}"
},
{
"math_id": 11,
"text": "\\Phi_\\text{31} = \\iint_{S_3} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S = -\\iint_{S_3} \\mathbf{F} \\cdot (-\\mathbf{\\hat n}) \\; \\mathrm{d}S = -\\Phi_\\text{32}"
},
{
"math_id": 12,
"text": "\\Phi(V_\\text{1}) + \\Phi(V_\\text{2}) = \\Phi_\\text{1} + \\Phi_\\text{2}"
},
{
"math_id": 13,
"text": "\\Phi(V_\\text{1}) + \\Phi(V_\\text{2}) = \\Phi(V)"
},
{
"math_id": 14,
"text": "\\Phi(V) = \\sum_{V_\\text{i}\\subset V} \\Phi(V_\\text{i})"
},
{
"math_id": 15,
"text": "|V_\\text{i}|"
},
{
"math_id": 16,
"text": "\\iint_{S(V)} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S = \\sum_{V_\\text{i}\\subset V} \\iint_{S(V_\\text{i})} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S"
},
{
"math_id": 17,
"text": "\\frac{\\Phi(V_\\text{i})}{|V_\\text{i}|} = \\frac{1}{|V_\\text{i}|} \\iint_{S(V_\\text{i})} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S"
},
{
"math_id": 18,
"text": "\\iint_{S(V)} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S = \\sum_{V_\\text{i} \\subset V} \\left(\\frac{1}{|V_\\text{i}|} \\iint_{S(V_\\text{i})} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S\\right) |V_\\text{i}|"
},
{
"math_id": 19,
"text": "\\iint_{S(V)} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S = \\lim_{|V_\\text{i}|\\to 0}\\sum_{V_\\text{i}\\subset V} \\left(\\frac{1}{|V_\\text{i}|}\\iint_{S(V_\\text{i})} \\mathbf{F} \\cdot \\mathbf{\\hat n} \\; \\mathrm{d}S\\right) |V_\\text{i}|"
},
{
"math_id": 20,
"text": "\\;\\iint_{S(V)} \\mathbf{F}\\cdot\\mathbf{\\hat n}\\; \\mathrm{d}S = \\iiint_{V} \\operatorname{div} \\mathbf{F}\\;\\mathrm{d}V\\;"
},
{
"math_id": 21,
"text": "\\Omega \\subset \\mathbb{R}^n"
},
{
"math_id": 22,
"text": "C^1"
},
{
"math_id": 23,
"text": "u"
},
{
"math_id": 24,
"text": "O"
},
{
"math_id": 25,
"text": "\\overline{\\Omega}"
},
{
"math_id": 26,
"text": "u \\in C^1(O)"
},
{
"math_id": 27,
"text": "i \\in \\{1, \\dots, n\\}"
},
{
"math_id": 28,
"text": "\\int_{\\Omega}u_{x_i}\\,dV = \\int_{\\partial \\Omega}u\\nu_i\\,dS,"
},
{
"math_id": 29,
"text": "\\nu : \\partial \\Omega \\to \\mathbb{R}^n "
},
{
"math_id": 30,
"text": "\\partial \\Omega"
},
{
"math_id": 31,
"text": "\\int_{\\Omega}\\nabla u\\,dV = \\int_{\\partial \\Omega}u\\nu\\,dS."
},
{
"math_id": 32,
"text": "\\Omega"
},
{
"math_id": 33,
"text": "g"
},
{
"math_id": 34,
"text": "C^2"
},
{
"math_id": 35,
"text": "(\\cdot, \\cdot)"
},
{
"math_id": 36,
"text": "L^2(\\overline{\\Omega})"
},
{
"math_id": 37,
"text": "\\langle \\cdot, \\cdot \\rangle"
},
{
"math_id": 38,
"text": "u \\in C^{1}(\\overline{\\Omega}, \\mathbb{R})"
},
{
"math_id": 39,
"text": "X"
},
{
"math_id": 40,
"text": "\n(\\operatorname{grad} u, X) = -(u, \\operatorname{div} X) + \\int_{\\partial \\Omega}u\\langle X, N \\rangle\\,dS,\n"
},
{
"math_id": 41,
"text": "N"
},
{
"math_id": 42,
"text": "O \\subset \\overline{\\Omega}"
},
{
"math_id": 43,
"text": "\n\\begin{align}\n (\\operatorname{grad} u, X) &= \\int_{O}\\langle \\operatorname{grad} u, X \\rangle \\sqrt{g}\\,dx \\\\\n &= \\int_{O}\\partial_j u X^j \\sqrt{g}\\,dx \\\\\n &= -\\int_{O}u \\partial_j(\\sqrt{g}X^j)\\,dx \\\\\n &= -\\int_{O} u \\frac{1}{\\sqrt{g}}\\partial_j(\\sqrt{g}X^j)\\sqrt{g}\\,dx \\\\\n &= (u, -\\frac{1}{\\sqrt{g}}\\partial_j(\\sqrt{g}X^j)) \\\\\n &= (u, -\\operatorname{div} X).\n\\end{align}\n"
},
{
"math_id": 44,
"text": "-\\operatorname{div}"
},
{
"math_id": 45,
"text": "\\operatorname{grad}"
},
{
"math_id": 46,
"text": "\\mathbb{R}_{+}^n = \\{x \\in \\mathbb{R}^n : x_n \\geq 0\\}"
},
{
"math_id": 47,
"text": "\\mathbb{R}_+^n"
},
{
"math_id": 48,
"text": "\n\\begin{align}\n (\\operatorname{grad} u, X) &= \\int_{O}\\langle \\operatorname{grad} u, X \\rangle \\sqrt{g}\\,dx \\\\\n &= \\int_{\\mathbb{R}_+^n}\\partial_j u X^j \\sqrt{g}\\,dx \\\\\n &= (u, -\\operatorname{div} X) - \\int_{\\mathbb{R}^{n - 1}}u(x', 0)X^n(x', 0)\\sqrt{g(x', 0)}\\,dx',\n\\end{align}\n"
},
{
"math_id": 49,
"text": "dx' = dx_1 \\dots dx_{n - 1}"
},
{
"math_id": 50,
"text": "\\frac{\\partial}{\\partial x_n}"
},
{
"math_id": 51,
"text": "-N"
},
{
"math_id": 52,
"text": "\\sqrt{g(x', 0)}\\,dx' = \\sqrt{g_{\\partial \\Omega}(x')}\\,dx' = dS"
},
{
"math_id": 53,
"text": "\n(\\operatorname{grad} u, X) = (u, -\\operatorname{div} X) + \\int_{\\partial \\Omega}u\\langle X, N \\rangle \\,dS.\n"
},
{
"math_id": 54,
"text": "\\mathbf{F}\\rightarrow \\mathbf{F}g"
},
{
"math_id": 55,
"text": "\\iiint_V\\left[\\mathbf{F}\\cdot \\left(\\nabla g\\right) + g \\left(\\nabla\\cdot \\mathbf{F}\\right)\\right] \\mathrm{d}V="
},
{
"math_id": 56,
"text": "g\\mathbf{F} \\cdot \\mathbf{n} \\mathrm{d}S."
},
{
"math_id": 57,
"text": "\\mathbf{F} = \\nabla f"
},
{
"math_id": 58,
"text": "\\mathbf{F}\\rightarrow \\mathbf{F}\\times \\mathbf{G}"
},
{
"math_id": 59,
"text": "\\times"
},
{
"math_id": 60,
"text": " \\iiint_V \\nabla \\cdot \\left( \\mathbf{F} \\times \\mathbf{G}\\right) \\mathrm{d}V = \\iiint_V \\left[\\mathbf{G}\\cdot\\left(\\nabla\\times\\mathbf{F}\\right) - \\mathbf{F}\\cdot \\left( \\nabla\\times\\mathbf{G}\\right)\\right]\\, \\mathrm{d}V ="
},
{
"math_id": 61,
"text": "(\\mathbf F\\times\\mathbf{G}) \\cdot \\mathbf{n} \\mathrm{d}S."
},
{
"math_id": 62,
"text": "\\mathbf{F}\\rightarrow \\mathbf{F}\\cdot \\mathbf{G}"
},
{
"math_id": 63,
"text": "\\cdot "
},
{
"math_id": 64,
"text": "\\iiint_V \\nabla \\left( \\mathbf{F} \\cdot \\mathbf{G}\\right) \\mathrm{d}V = \\iiint_V \\left[\\left(\\nabla \\mathbf{G}\\right) \\cdot \\mathbf{F} + \\left( \\nabla \\mathbf{F}\\right) \\cdot \\mathbf{G} \\right]\\, \\mathrm{d}V ="
},
{
"math_id": 65,
"text": "(\\mathbf{F} \\cdot \\mathbf{G}) \\mathbf{n} \\mathrm{d}S."
},
{
"math_id": 66,
"text": "\\mathbf{F}\\rightarrow f\\mathbf{c}"
},
{
"math_id": 67,
"text": "\\iiint_V \\mathbf{c} \\cdot \\nabla f \\, \\mathrm{d}V ="
},
{
"math_id": 68,
"text": "(\\mathbf{c} f) \\cdot \\mathbf{n} \\mathrm{d}S - \\iiint_V f (\\nabla \\cdot \\mathbf{c})\\, \\mathrm{d}V."
},
{
"math_id": 69,
"text": "\\mathbf{c}"
},
{
"math_id": 70,
"text": "\\iiint_V \\nabla f \\, \\mathrm{d}V ="
},
{
"math_id": 71,
"text": "f\\mathbf{n} \\mathrm{d}S."
},
{
"math_id": 72,
"text": "\\mathbf{F}\\rightarrow \\mathbf{c}\\times\\mathbf{F}"
},
{
"math_id": 73,
"text": "\\iiint_V\\mathbf{c} \\cdot (\\nabla\\times\\mathbf{F}) \\, \\mathrm{d}V ="
},
{
"math_id": 74,
"text": " (\\mathbf{F} \\times \\mathbf{c}) \\cdot \\mathbf{n} \\mathrm{d}S."
},
{
"math_id": 75,
"text": " \\iiint_V (\\nabla \\times\\mathbf{F}) \\, \\mathrm{d}V \\cdot \\mathbf{c} = "
},
{
"math_id": 76,
"text": " \\scriptstyle S"
},
{
"math_id": 77,
"text": " (\\mathrm{d}\\mathbf{S} \\times \\mathbf{F}) \\cdot \\mathbf{c}. "
},
{
"math_id": 78,
"text": " \\iiint_V (\\nabla \\times\\mathbf{F}) \\, \\mathrm{d}V = "
},
{
"math_id": 79,
"text": " \\mathbf{n} \\times \\mathbf{F} \\mathrm{d}S. "
},
{
"math_id": 80,
"text": "\\mathbf{F}\\cdot\\mathbf{n} \\, \\mathrm{d}S,"
},
{
"math_id": 81,
"text": "S = \\left \\{ (x,y, z) \\in \\mathbb{R}^3 \\ : \\ x^2+y^2+z^2 = 1 \\right \\},"
},
{
"math_id": 82,
"text": "\\mathbf{F} = 2x\\mathbf{i}+y^2\\mathbf{j}+z^2\\mathbf{k}."
},
{
"math_id": 83,
"text": "\\iiint_W (\\nabla \\cdot \\mathbf{F})\\,\\mathrm{d}V = 2\\iiint_W (1 + y + z)\\, \\mathrm{d}V = 2\\iiint_W \\mathrm{d}V + 2\\iiint_W y\\, \\mathrm{d}V + 2\\iiint_W z\\, \\mathrm{d}V,"
},
{
"math_id": 84,
"text": "W = \\left \\{ (x,y, z) \\in \\mathbb{R}^3 \\ : \\ x^2+y^2+z^2\\leq 1 \\right \\}."
},
{
"math_id": 85,
"text": "\\iiint_W y\\, \\mathrm{d}V = \\iiint_W z\\, \\mathrm{d}V = 0."
},
{
"math_id": 86,
"text": "\\mathbf{F}\\cdot\\mathbf{n}\\,\\mathrm{d}S = 2\\iiint_W\\, dV = \\frac{8\\pi}{3},"
},
{
"math_id": 87,
"text": "R"
},
{
"math_id": 88,
"text": "R = \\left \\{ (x, y) \\in \\mathbb{R}^2 \\ : \\ x^2 + y^2 \\leq 1 \\right \\},"
},
{
"math_id": 89,
"text": " \\mathbf{F}(x,y)= 2 y\\mathbf{i} + 5x \\mathbf{j}."
},
{
"math_id": 90,
"text": "C"
},
{
"math_id": 91,
"text": "x = \\cos(s), \\quad y = \\sin(s)"
},
{
"math_id": 92,
"text": "0 \\leq s \\leq 2\\pi"
},
{
"math_id": 93,
"text": "s"
},
{
"math_id": 94,
"text": "s = 0"
},
{
"math_id": 95,
"text": "P"
},
{
"math_id": 96,
"text": "C(s) = \\cos(s)\\mathbf{i} + \\sin(s)\\mathbf{j}."
},
{
"math_id": 97,
"text": " P = (\\cos(s),\\, \\sin(s)) \\, \\Rightarrow \\, \\mathbf{F} = 2\\sin(s)\\mathbf{i} + 5\\cos(s)\\mathbf{j}."
},
{
"math_id": 98,
"text": "\\begin{align}\n\\oint_C \\mathbf{F} \\cdot \\mathbf{n}\\, \\mathrm{d}s &= \\int_0^{2\\pi} (2 \\sin(s) \\mathbf{i} + 5 \\cos(s) \\mathbf{j}) \\cdot (\\cos(s) \\mathbf{i} + \\sin(s) \\mathbf{j})\\, \\mathrm{d}s\\\\\n&= \\int_0^{2\\pi} (2 \\sin(s) \\cos(s) + 5 \\sin(s) \\cos(s))\\, \\mathrm{d}s\\\\\n&= 7\\int_0^{2\\pi} \\sin(s) \\cos(s)\\, \\mathrm{d}s\\\\\n&= 0.\n\\end{align}"
},
{
"math_id": 99,
"text": "M = \\mathfrak{Re}(\\mathbf{F}) = 2y"
},
{
"math_id": 100,
"text": "\\frac{\\partial M}{\\partial x} = 0"
},
{
"math_id": 101,
"text": "N = \\mathfrak{Im}(\\mathbf{F}) = 5x"
},
{
"math_id": 102,
"text": "\\frac{\\partial N}{\\partial y} = 0"
},
{
"math_id": 103,
"text": "\\iint_R \\, \\mathbf{\\nabla}\\cdot\\mathbf{F} \\, \\mathrm{d}A = \\iint_R \\left (\\frac{\\partial M}{\\partial x} + \\frac{\\partial N}{\\partial y} \\right) \\, \\mathrm{d}A = 0. "
},
{
"math_id": 104,
"text": " \\mathbf{F}=2x^2 \\textbf{i} +2y^2 \\textbf{j} +2z^2\\textbf{k} "
},
{
"math_id": 105,
"text": "\\left\\{0\\le x \\le 3\\right\\}, \\left\\{-2\\le y \\le 2\\right\\}, \\left\\{0\\le z \\le 2\\pi\\right\\}"
},
{
"math_id": 106,
"text": "\\iiint_V\\left(\\mathbf{\\nabla}\\cdot\\mathbf{F}\\right) \\mathrm{d}V="
},
{
"math_id": 107,
"text": "(\\mathbf{F}\\cdot\\mathbf{n})\\, \\mathrm{d}S ."
},
{
"math_id": 108,
"text": "\\textbf{F}"
},
{
"math_id": 109,
"text": "\\mathbf{F}"
},
{
"math_id": 110,
"text": "\\nabla \\cdot \\textbf{F} = \\left( \\frac{\\partial}{\\partial x}\\textbf{i} + \\frac{\\partial}{\\partial y}\\textbf{j} + \\frac{\\partial}{\\partial z}\\textbf{k} \\right) \\cdot \\textbf{F}"
},
{
"math_id": 111,
"text": "I = "
},
{
"math_id": 112,
"text": "{\\scriptstyle S}"
},
{
"math_id": 113,
"text": "\\mathbf{F} \\cdot \\mathbf{n} \\, \\mathrm{d}S,"
},
{
"math_id": 114,
"text": "\n\\begin{align}\nI\n&=\\iiint_V \\nabla \\cdot \\mathbf{F} \\, \\mathrm{d}V\\\\[6pt]\n&=\\iiint_V \\left( \\frac{\\partial\\mathbf{F_x}}{\\partial x}+\\frac{\\partial\\mathbf{F_y}}{\\partial y}+\\frac{\\partial\\mathbf{F_z}}{\\partial z} \\right) \\mathrm{d}V\\\\[6pt]\n&=\\iiint_V (4x+4y+4z) \\, \\mathrm{d}V\\\\[6pt]\n&=\\int_0^3 \\int_{-2}^2 \\int_0^{2\\pi} (4x+4y+4z) \\, \\mathrm{d}V\n\\end{align}\n"
},
{
"math_id": 115,
"text": "\\begin{align}\n\\int_0^3 \\int_{-2}^2 \\int_0^{2\\pi} (4x+4y+4z) \\, \\mathrm{d}V &=\\int_{-2}^2 \\int_0^{2\\pi} (12y+12z+18) \\, \\mathrm{d}y \\, \\mathrm{d}z\\\\[6pt]\n&=\\int_0^{2\\pi} 24 (2z+3)\\, \\mathrm{d}z\\\\[6pt]\n&=48\\pi(2\\pi+3)\n\\end{align}\n\n"
},
{
"math_id": 116,
"text": " \\underbrace{ \\int \\cdots \\int_U }_n \\nabla \\cdot \\mathbf{F} \\, \\mathrm{d}V = \\underbrace{ \\oint_{} \\cdots \\oint_{\\partial U} }_{n-1} \\mathbf{F} \\cdot \\mathbf{n} \\, \\mathrm{d}S "
},
{
"math_id": 117,
"text": "\\iiint_V \\dfrac{\\partial \\mathbf{F}_i}{\\partial x_i} \\mathrm{d}V="
},
{
"math_id": 118,
"text": "\\mathbf{F}_i n_i\\, \\mathrm{d}S "
},
{
"math_id": 119,
"text": "\\iiint_V \\dfrac{\\partial T_{i_1i_2\\cdots i_q\\cdots i_n}}{\\partial x_{i_q}} \\mathrm{d}V="
},
{
"math_id": 120,
"text": "T_{i_1i_2\\cdots i_q\\cdots i_n}n_{i_q}\\, \\mathrm{d}S ."
}
] |
https://en.wikipedia.org/wiki?curid=151864
|
15187217
|
Vertex distance
|
Vertex distance is the distance between the back surface of a corrective lens, i.e. glasses (spectacles) or contact lenses, and the front of the cornea. Increasing or decreasing the vertex distance changes the optical properties of the system, by moving the focal point forward or backward, effectively changing the power of the lens relative to the eye. Since most refractions (the measurement that determines the power of a corrective lens) are performed at a vertex distance of 12–14 mm, the power of the correction may need to be modified from the initial prescription so that light reaches the patient's eye with the same effective power that it did through the phoropter or trial frame.
Vertex distance is important when converting between contact lens and glasses prescriptions and becomes significant if the glasses prescription is beyond ±4.00 diopters (often abbreviated D). The formula for vertex correction is formula_0, where Fc is the power corrected for vertex distance, F is the original lens power, and x is the change in vertex distance in meters.
Derivation.
The vertex distance formula calculates what power lens ("F"c) is needed to focus light on the same location if the lens has been moved by a distance "x". To focus light to the same image location:
formula_1
where "f"c is the corrected focal length for the new lens, "f" is the focal length of the original lens, and "x" is the distance that the lens was moved. The value for "x" can be positive or negative depending on the sign convention. Lens power in diopters is the mathematical inverse of focal length in meters.
formula_2
Substituting for lens power arrives at
formula_3
After simplifying the final equation is found:
formula_4
Examples.
Example 1: example prescription adjustment from glasses to contacts.
A phoropter measurement of a patient reads −8.00D sphere and −5.25D cylinder with an axis of 85° for one eye (the notation for which is typically written as −8 −5.25×85). The phoropter measurement is made at a common vertex distance of 12mm from the eye. The equivalent prescription at the patient's cornea (say, for a contact lens) can be calculated as follows (this example assumes a "negative" cylinder sign convention):
Power 1 is the spherical value, and power 2 is the "steeper" power of the astigmatic axis:
formula_5
The axis value does not change with vertex distance, so the equivalent prescription for a contact lens (vertex distance, 0mm) is −7.30D of sphere, −4.13D of cylinder with 85° of axis (−7.30 −4.13×85 or about −7.25 −4.25×85).
Example 2: example prescription adjustment from contacts to glasses.
A patient has −8D sphere contacts. What is the equivalent prescription for glasses?
formula_6
Therefore −8D contacts correspond to −8.75D or −9D glasses.
Example 3: sample plots.
The following plots show the difference in spherical power at a 0mm vertex distance (at the eye) and a 12mm vertex distance (standard eyeglasses distance). 0mm is used as the reference starting power and is one-to-one. The second plot shows the difference between the 0mm and 12mm vertex distance powers. Above around 4D of spherical power, the difference versus the corrected power becomes more than 0.25D and is clinically significant.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "F_c = \\left(F^{-1} - x\\right)^{-1}"
},
{
"math_id": 1,
"text": "f_c = f - x"
},
{
"math_id": 2,
"text": "\\begin{align}\n F &= \\frac{1}{f}; & F_\\text{c} &= \\frac{1}{f_\\text{c}}\n\\end{align}"
},
{
"math_id": 3,
"text": "\\frac{1}{F_\\text{c}} = \\frac{1}{F} - x"
},
{
"math_id": 4,
"text": "\\begin{align}\n & & \\frac{F}{F_\\text{c}} &= 1 - xF \\\\\n &\\Rightarrow & F_\\text{c} &= \\frac{F}{1 - xF} = \\frac{1}{\\frac{1}{F} - x} \\\\\n &\\Rightarrow & F &= \\frac{1}{\\frac{1}{F_\\text{c}} + x}\n\\end{align}"
},
{
"math_id": 5,
"text": "\\begin{align}\n \\text{Corrected power}_1 &= F_{\\text{c}1} = \\frac{1}{\\frac{1}{F} - x} = \\frac{1}{\\frac{1}{-8} - 0.012} = -7.30\\text{ D}, \\\\[3pt]\n \\text{Uncorrected power}_2 &= F_\\text{sphere} + F_\\text{cylinder} = -8 + -5.25 = -13.25\\text{ D}, \\\\[3pt]\n \\text{Corrected power}_2 &= F_{\\text{c}2} = \\frac{1}{\\frac{1}{F} - x} = -\\frac{13.25}{1 - 0.012 (-13.25)} = -11.43\\text{ D}, \\text{ and} \\\\[3pt]\n \\text{Corrected cylinder} &= F_{\\text{c}2} - F_{\\text{c}1} = -11.43 - (-7.30) = -4.13\\text{ D}.\n\\end{align}"
},
{
"math_id": 6,
"text": "F = \\frac{1}{\\frac{1}{F_\\text{c}} + x} = \\frac{1}{\\frac{1}{-8} + 0.012} = -8.84\\text{ D}"
}
] |
https://en.wikipedia.org/wiki?curid=15187217
|
1518742
|
Helly–Bray theorem
|
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray.
Let "F" and "F"1, "F"2, ... be cumulative distribution functions on the real line. The Helly–Bray theorem states that if "F""n" converges weakly to "F", then
formula_0
for each bounded, continuous function "g": R → R, where the integrals involved are Riemann–Stieltjes integrals.
Note that if "X" and "X"1, "X"2, ... are random variables corresponding to these distribution functions, then the Helly–Bray theorem does not imply that E("X""n") → E("X"), since "g"("x") = "x" is not a bounded function.
In fact, a stronger and more general theorem holds. Let "P" and "P"1, "P"2, ... be probability measures on some set "S". Then "P""n" converges weakly to "P" if and only if
formula_1
for all bounded, continuous and real-valued functions on "S". (The integrals in this version of the theorem are Lebesgue–Stieltjes integrals.)
The more general theorem above is sometimes taken as "defining" weak convergence of measures (see Billingsley, 1999, p. 3).
References.
"This article incorporates material from Helly–Bray theorem on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "\\int_\\mathbb{R} g(x)\\,dF_n(x) \\quad\\xrightarrow[n\\to\\infty]{}\\quad \\int_\\mathbb{R} g(x)\\,dF(x)"
},
{
"math_id": 1,
"text": "\\int_S g \\,dP_n \\quad\\xrightarrow[n\\to\\infty]{}\\quad \\int_S g \\,dP,"
}
] |
https://en.wikipedia.org/wiki?curid=1518742
|
15189
|
IEEE 754-1985
|
First edition of the IEEE 754 floating-point standard
IEEE 754-1985 is a historic industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. During its 23 years, it was the most widely used format for floating-point computation. It was implemented in software, in the form of floating-point libraries, and in hardware, in the instructions of many CPUs and FPUs. The first integrated circuit to implement the draft of what was to become IEEE 754-1985 was the Intel 8087.
IEEE 754-1985 represents numbers in binary, providing definitions for four levels of precision, of which the two most commonly used are:
The standard also defines representations for positive and negative infinity, a "negative zero", five exceptions to handle invalid results like division by zero, special values called NaNs for representing those exceptions, denormal numbers to represent numbers smaller than shown above, and four rounding modes.
Representation of numbers.
Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each.
The decimal number 0.1562510 represented in binary is 0.001012 (that is, 1/8 + 1/32). (Subscripts indicate the number base.) Analogous to scientific notation, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply multiply by the appropriate power of 2 to compensate for shifting the bits left by three positions:
formula_0
Now we can read off the fraction and the exponent: the fraction is .012 and the exponent is −3.
As illustrated in the pictures, the three fields in the IEEE 754 representation of this number are:
"sign" = 0, because the number is positive. (1 indicates negative.)
"biased exponent" = −3 + the "bias". In single precision, the bias is 127, so in this example the biased exponent is 124; in double precision, the bias is 1023, so the biased exponent in this example is 1020.
"fraction" = .01000…2.
IEEE 754 adds a bias to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed 2's-complement integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for sign and magnitude integers. If two floating-point numbers have different signs, the sign-and-magnitude comparison also works with biased exponents. However, if both biased-exponent floating-point numbers are negative, then the ordering must be reversed. If the exponent were represented as, say, a 2's-complement number, comparison to see which of two numbers is greater would not be as convenient.
The leading 1 bit is omitted since all numbers except zero start with a leading 1; the leading 1 is implicit and doesn't actually need to be stored which gives an extra bit of precision for "free."
Zero.
The number zero is represented specially:
"sign" = 0 for positive zero, 1 for negative zero.
"biased exponent" = 0.
"fraction" = 0.
Denormalized numbers.
The number representations described above are called "normalized," meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an underflow occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0. Such numbers are called denormal. They don't include as many significant digits as a normalized number, but they enable a gradual loss of precision when the result of an operation is not exactly zero but is too close to zero to be represented by a normalized number.
A denormal number is represented with a biased exponent of all 0 bits, which represents an exponent of −126 in single precision (not −127), or −1022 in double precision (not −1023). In contrast, the smallest biased exponent representing a normal number is 1 (see examples below).
Representation of non-numbers.
The biased-exponent field is filled with all 1 bits to indicate either infinity or an invalid result of a computation.
Positive and negative infinity.
Positive and negative infinity are represented thus:
"sign" = 0 for positive infinity, 1 for negative infinity.
"biased exponent" = all 1 bits.
"fraction" = all 0 bits.
NaN.
Some operations of floating-point arithmetic are invalid, such as taking the square root of a negative number. The act of reaching an invalid result is called a floating-point "exception." An exceptional result is represented by a special code called a NaN, for "Not a Number". All NaNs in IEEE 754-1985 have this format:
"sign" = either 0 or 1.
"biased exponent" = all 1 bits.
"fraction" = anything except all 0 bits (since all 0 bits represents infinity).
Range and precision.
Precision is defined as the minimum difference between two successive mantissa representations; thus it is a function only in the mantissa; while the gap is defined as the difference between two successive numbers.
Single precision.
Single-precision numbers occupy 32 bits. In single precision:
Some example range and gap values for given exponents in single precision:
As an example, 16,777,217 cannot be encoded as a 32-bit float as it will be rounded to 16,777,216. However, all integers within the representable range that are a power of 2 can be stored in a 32-bit float without rounding.
Double precision.
Double-precision numbers occupy 64 bits. In double precision:
Some example range and gap values for given exponents in double precision:
Extended formats.
The standard also recommends extended format(s) to be used to perform internal computations at a higher precision than that required for the final result, to minimise round-off errors: the standard only specifies minimum precision and exponent requirements for such formats. The x87 80-bit extended format is the most commonly implemented extended format that meets these requirements.
Examples.
Here are some examples of single-precision IEEE 754 representations:
Comparing floating-point numbers.
Every possible bit combination is either a NaN or a number with a unique value in the affinely extended real number system with its associated order, except for the two combinations of bits for negative zero and positive zero, which sometimes require special attention (see below). The binary representation has the special property that, excluding NaNs, any two numbers can be compared as sign and magnitude integers (endianness issues apply). When comparing as 2's-complement integers: If the sign bits differ, the negative number precedes the positive number, so 2's complement gives the correct result (except that negative zero and positive zero should be considered equal). If both values are positive, the 2's complement comparison again gives the correct result. Otherwise (two negative numbers), the correct FP ordering is the opposite of the 2's complement ordering.
Rounding errors inherent to floating point calculations may limit the use of comparisons for checking the exact equality of results. Choosing an acceptable range is a complex topic. A common technique is to use a comparison epsilon value to perform approximate comparisons. Depending on how lenient the comparisons are, common values include codice_0 or codice_1 for single-precision, and codice_2 for double-precision. Another common technique is ULP, which checks what the difference is in the last place digits, effectively checking how many steps away the two values are.
Although negative zero and positive zero are generally considered equal for comparison purposes, some programming language relational operators and similar constructs treat them as distinct. According to the Java Language Specification, comparison and equality operators treat them as equal, but codice_3 and codice_4 distinguish them (officially starting with Java version 1.1 but actually with 1.1.1), as do the comparison methods codice_5, codice_6 and even codice_7 of classes codice_8 and codice_9.
Rounding floating-point numbers.
The IEEE standard has four different rounding modes; the first is the default; the others are called "directed roundings".
Extending the real numbers.
The IEEE standard employs (and extends) the affinely extended real number system, with separate positive and negative infinities. During drafting, there was a proposal for the standard to incorporate the projectively extended real number system, with a single unsigned infinity, by providing programmers with a mode selection option. In the interest of reducing the complexity of the final standard, the projective mode was dropped, however. The Intel 8087 and Intel 80287 floating point co-processors both support this projective mode.
Functions and predicates.
Standard operations.
The following functions must be provided:
History.
In 1976, Intel was starting the development of a floating-point coprocessor. Intel hoped to be able to sell a chip containing good implementations of all the operations found in the widely varying maths software libraries.
John Palmer, who managed the project, believed the effort should be backed by a standard unifying floating point operations across disparate processors. He contacted William Kahan of the University of California, who had helped improve the accuracy of Hewlett-Packard's calculators. Kahan suggested that Intel use the floating point of Digital Equipment Corporation's (DEC) VAX. The first VAX, the VAX-11/780 had just come out in late 1977, and its floating point was highly regarded. However, seeking to market their chip to the broadest possible market, Intel wanted the best floating point possible, and Kahan went on to draw up specifications. Kahan initially recommended that the floating point base be decimal but the hardware design of the coprocessor was too far along to make that change.
The work within Intel worried other vendors, who set up a standardization effort to ensure a "level playing field". Kahan attended the second IEEE 754 standards working group meeting, held in November 1977. He subsequently received permission from Intel to put forward a draft proposal based on his work for their coprocessor; he was allowed to explain details of the format and its rationale, but not anything related to Intel's implementation architecture. The draft was co-written with Jerome Coonen and Harold Stone, and was initially known as the "Kahan-Coonen-Stone proposal" or "K-C-S format".
As an 8-bit exponent was not wide enough for some operations desired for double-precision numbers, e.g. to store the product of two 32-bit numbers, both Kahan's proposal and a counter-proposal by DEC therefore used 11 bits, like the time-tested 60-bit floating-point format of the CDC 6600 from 1965. Kahan's proposal also provided for infinities, which are useful when dealing with division-by-zero conditions; not-a-number values, which are useful when dealing with invalid operations; denormal numbers, which help mitigate problems caused by underflow; and a better balanced exponent bias, which can help avoid overflow and underflow when taking the reciprocal of a number.
Even before it was approved, the draft standard had been implemented by a number of manufacturers. The Intel 8087, which was announced in 1980, was the first chip to implement the draft standard.
In 1980, the Intel 8087 chip was already released, but DEC remained opposed, to denormal numbers in particular, because of performance concerns and since it would give DEC a competitive advantage to standardise on DEC's format.
The arguments over gradual underflow lasted until 1981 when an expert hired by DEC to assess it sided against the dissenters. DEC had the study done in order to demonstrate that gradual underflow was a bad idea, but the study concluded the opposite, and DEC gave in. In 1985, the standard was ratified, but it had already become the de facto standard a year earlier, implemented by many manufacturers.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "0.00101_2 = 1.01_2 \\times 2^{-3}"
}
] |
https://en.wikipedia.org/wiki?curid=15189
|
151912
|
All horses are the same color
|
Paradox arising from an incorrect proof
All horses are the same color is a falsidical paradox that arises from a flawed use of mathematical induction to prove the statement "All horses are the same color". There is no actual contradiction, as these arguments have a crucial flaw that makes them incorrect. This example was originally raised by George Pólya in a 1954 book in different terms: "Are any n numbers equal?" or "Any n girls have eyes of the same color", as an exercise in mathematical induction. It has also been restated as "All cows have the same color".
The "horses" version of the paradox was presented in 1961 in a satirical article by Joel E. Cohen. It was stated as a lemma, which in particular allowed the author to "prove" that Alexander the Great did not exist, and he had an infinite number of limbs.
The argument.
The argument is proof by induction. First, we establish a base case for one horse (formula_0). We then prove that if formula_1 horses have the same color, then formula_2 horses must also have the same color.
Base case: One horse.
The case with just one horse is trivial. If there is only one horse in the "group", then clearly all horses in that group have the same color.
Inductive step.
Assume that formula_1 horses always are the same color. Consider a group consisting of formula_2 horses.
First, exclude one horse and look only at the other formula_1 horses; all these are the same color, since formula_1 horses always are the same color. Likewise, exclude some other horse (not identical to the one first removed) and look only at the other formula_1 horses. By the same reasoning, these, too, must also be of the same color. Therefore, the first horse that was excluded is of the same color as the non-excluded horses, who in turn are of the same color as the other excluded horse. Hence, the first horse excluded, the non-excluded horses, and the last horse excluded are all of the same color, and we have proven that:
We already saw in the base case that the rule ("all horses have the same color") was valid for formula_0. The inductive step proved here implies that since the rule is valid for formula_0, it must also be valid for formula_3, which in turn implies that the rule is valid for formula_4 and so on.
Thus, in any group of horses, all horses must be the same color.
Explanation.
The argument above makes the implicit assumption that the set of formula_2 horses has the size at least 3, so that the two proper subsets of horses to which the induction assumption is applied would necessarily share a common element. This is not true at the first step of induction, i.e., when formula_5.
Let the two horses be horse A and horse B. When horse A is removed, it is true that the remaining horses in the set are the same color (only horse B remains). The same is true when horse B is removed. However, the statement "the first horse that was excluded is of the same color as the non-excluded horses, who in turn are of the same color as the other excluded horse" is meaningless, because there are no "non-excluded horses" (common elements (horses) in the two sets, since each horse is excluded once). Therefore, the above proof has a logical link broken. The proof forms a falsidical paradox; it seems to show by valid reasoning something that is manifestly false, but in fact the reasoning is flawed.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "n=1"
},
{
"math_id": 1,
"text": "n"
},
{
"math_id": 2,
"text": "n+1"
},
{
"math_id": 3,
"text": "n=2"
},
{
"math_id": 4,
"text": "n=3"
},
{
"math_id": 5,
"text": "n+1=2"
}
] |
https://en.wikipedia.org/wiki?curid=151912
|
15191637
|
CUSUM
|
Sequential analysis technique
In statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detection.
CUSUM was announced in Biometrika, in 1954, a few years after the publication of Wald's sequential probability ratio test (SPRT).
E. S. Page referred to a "quality number" formula_0, by which he meant a parameter of the probability distribution; for example, the mean. He devised CUSUM as a method to determine changes in it, and proposed a criterion for deciding when to take corrective action. When the CUSUM method is applied to changes in mean, it can be used for step detection of a time series.
A few years later, George Alfred Barnard developed a visualization method, the V-mask chart, to detect both increases and decreases in formula_0.
Method.
As its name implies, CUSUM involves the calculation of a cumulative sum (which is what makes it "sequential"). Samples from a process formula_1 are assigned weights formula_2, and summed as follows:
formula_3
formula_4
When the value of "S" exceeds a certain threshold value, a change in value has been found. The above formula only detects changes in the positive direction. When negative changes need to be found as well, the min operation should be used instead of the max operation, and this time a change has been found when the value of "S" is "below" the (negative) value of the threshold value.
Page did not explicitly say that formula_5 represents the likelihood function, but this is common usage.
This differs from SPRT by always using zero function as the lower "holding barrier" rather than a lower "holding barrier". Also, CUSUM does not require the use of the likelihood function.
As a means of assessing CUSUM's performance, Page defined the "average run length" (A.R.L.) metric; "the expected number of articles sampled before action is taken." He further wrote:
When the quality of the output is satisfactory the A.R.L. is a measure of the expense incurred by the scheme when it gives false alarms, i.e., Type I errors (Neyman & Pearson, 1936). On the other hand, for constant poor quality the A.R.L. measures the delay and thus the amount of scrap produced before the rectifying action is taken, i.e., Type II errors.
Example.
The following example shows 20 observations formula_6 of a process with a mean of 0 and a standard deviation of 0.5.
From the formula_7 column, it can be seen that formula_6 never deviates by 3 standard deviations (formula_8), so simply alerting on a high deviation will not detect a failure, whereas CUSUM shows that the formula_9 value exceeds 4 at the 17th observation.
where formula_5 is a critical level parameter (tunable, same as threshold T) that's used to adjust the sensitivity of change detection: larger formula_5 makes CUSUM less sensitive to the change and vice versa.
Variants.
Cumulative observed-minus-expected plots are a related method.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\theta"
},
{
"math_id": 1,
"text": "x_n"
},
{
"math_id": 2,
"text": "\\omega_n"
},
{
"math_id": 3,
"text": "S_0=0"
},
{
"math_id": 4,
"text": "S_{n+1}=\\max(0, S_n+x_{n+1}-\\omega_n)"
},
{
"math_id": 5,
"text": "\\omega"
},
{
"math_id": 6,
"text": "X"
},
{
"math_id": 7,
"text": "Z"
},
{
"math_id": 8,
"text": "3 \\sigma"
},
{
"math_id": 9,
"text": "S_H"
}
] |
https://en.wikipedia.org/wiki?curid=15191637
|
151925
|
Del
|
Vector differential operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a "field" (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
Definition.
In the Cartesian coordinate system formula_3 with coordinates formula_4 and standard basis formula_5, del is a vector operator whose formula_6 components are the partial derivative operators formula_7; that is,
formula_8
Where the expression in parentheses is a row vector. In three-dimensional Cartesian coordinate system formula_9 with coordinates formula_10 and standard basis or unit vectors of axes formula_11, del is written as
formula_12
As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.
More specifically, for any scalar field formula_13 and any vector field formula_14, if one "defines"
formula_15
formula_16
formula_17
formula_18
formula_19
then using the above definition of formula_20, one may write
formula_21
and
formula_22
and
formula_23
Example:
formula_24
formula_25
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.
Notational uses.
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.
Gradient.
The vector derivative of a scalar field formula_13 is called the gradient, and it can be represented as:
formula_26
It always points in the direction of greatest increase of formula_13, and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane formula_27, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
formula_28
However, the rules for dot products do not turn out to be simple, as illustrated by:
formula_29
Divergence.
The divergence of a vector field
formula_30 is a scalar field that can be represented as:
formula_31
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.
The power of the del notation is shown by the following product rule:
formula_32
The formula for the vector product is slightly less intuitive, because this product is not commutative:
formula_33
Curl.
The curl of a vector field formula_34 is a vector function that can be represented as:
formula_35
The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.
The vector product operation can be visualized as a pseudo-determinant:
formula_36
Again the power of the notation is shown by the product rule:
formula_37
The rule for the vector product does not turn out to be simple:
formula_38
Directional derivative.
The directional derivative of a scalar field formula_39 in the direction
formula_40 is defined as:
formula_41
Which is equal to the following when the gradient exists
formula_42
This gives the rate of change of a field formula_13 in the direction of formula_43, scaled by the magnitude of formula_43. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid.
Note that formula_44 is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
Laplacian.
The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
formula_45
and the definition for more general coordinate systems is given in vector Laplacian.
The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation.
Hessian matrix.
While formula_46 usually represents the Laplacian, sometimes formula_46 also represents the Hessian matrix. The former refers to the inner product of formula_20, while the latter refers to the dyadic product of formula_20:
formula_47.
So whether formula_46 refers to a Laplacian or a Hessian matrix depends on the context.
Tensor derivative.
Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field formula_48 (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as formula_49, where formula_50 represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.
For a small displacement formula_51, the change in the vector field is given by:
formula_52
Product rules.
For vector calculus:
formula_53
For matrix calculus (for which formula_54 can be written formula_55):
formula_56
Another relation of interest (see e.g. "Euler equations") is the following, where formula_57 is the outer product tensor:
formula_58
Second derivatives.
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field "f" or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:
formula_59
These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved (formula_60 in most cases), two of them are always zero:
formula_61
Two of them are always equal:
formula_62
The 3 remaining vector derivatives are related by the equation:
formula_63
And one of them can even be expressed with the tensor product, if the functions are well-behaved:
formula_64
Precautions.
Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.
Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is "not" necessarily reliable, because del does not commute in general.
A counterexample that demonstrates the divergence (formula_65) and the advection operator (formula_66) are not commutative:
formula_67
A counterexample that relies on del's differential properties:
formula_68
Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.
For that reason, identities involving del must be derived with care, using both vector identities and "differentiation" identities such as the product rule.
|
[
{
"math_id": 0,
"text": "\\operatorname{grad}f = \\nabla f"
},
{
"math_id": 1,
"text": "\\operatorname{div}\\mathbf v = \\nabla \\cdot \\mathbf v "
},
{
"math_id": 2,
"text": "\\operatorname{curl}\\mathbf v = \\nabla \\times \\mathbf v"
},
{
"math_id": 3,
"text": "\\mathbb{R}^n"
},
{
"math_id": 4,
"text": "(x_1, \\dots, x_n)"
},
{
"math_id": 5,
"text": "\\{\\mathbf e_1, \\dots, \\mathbf e_n \\}"
},
{
"math_id": 6,
"text": "x_1, \\dots, x_n"
},
{
"math_id": 7,
"text": "{\\partial \\over \\partial x_1}, \\dots, {\\partial \\over \\partial x_n}"
},
{
"math_id": 8,
"text": " \\nabla = \\sum_{i=1}^n \\mathbf e_i {\\partial \\over \\partial x_i} = \\left({\\partial \\over \\partial x_1}, \\ldots, {\\partial \\over \\partial x_n} \\right)"
},
{
"math_id": 9,
"text": "\\mathbb{R}^3"
},
{
"math_id": 10,
"text": "(x, y, z)"
},
{
"math_id": 11,
"text": "\\{\\mathbf e_x, \\mathbf e_y, \\mathbf e_z \\}"
},
{
"math_id": 12,
"text": "\\nabla = \\mathbf{e}_x {\\partial \\over \\partial x} + \\mathbf{e}_y {\\partial \\over \\partial y} + \\mathbf{e}_z {\\partial \\over \\partial z}= \\left({\\partial \\over \\partial x}, {\\partial \\over \\partial y}, {\\partial \\over \\partial z} \\right) "
},
{
"math_id": 13,
"text": "f"
},
{
"math_id": 14,
"text": "\\mathbf{F}=(F_x, F_y, F_z)"
},
{
"math_id": 15,
"text": "\\left(\\mathbf{e}_i {\\partial \\over \\partial x_i}\\right) f := {\\partial \\over \\partial x_i}(\\mathbf{e}_i f) = {\\partial f \\over \\partial x_i}\\mathbf{e}_i"
},
{
"math_id": 16,
"text": "\\left(\\mathbf{e}_i {\\partial \\over \\partial x_i}\\right) \\cdot \\mathbf{F} := {\\partial \\over \\partial x_i}(\\mathbf{e}_i\\cdot \\mathbf{F}) = {\\partial F_i \\over \\partial x_i}"
},
{
"math_id": 17,
"text": "\\left(\\mathbf{e}_x {\\partial \\over \\partial x}\\right) \\times \\mathbf{F} := {\\partial \\over \\partial x}(\\mathbf{e}_x\\times \\mathbf{F}) = {\\partial \\over \\partial x}(0, -F_z, F_y)"
},
{
"math_id": 18,
"text": "\\left(\\mathbf{e}_y {\\partial \\over \\partial y}\\right) \\times \\mathbf{F} := {\\partial \\over \\partial y}(\\mathbf{e}_y\\times \\mathbf{F}) = {\\partial \\over \\partial y}(F_z,0,-F_x)"
},
{
"math_id": 19,
"text": "\\left(\\mathbf{e}_z {\\partial \\over \\partial z}\\right) \\times \\mathbf{F} := {\\partial \\over \\partial z}(\\mathbf{e}_z\\times \\mathbf{F}) = {\\partial \\over \\partial z}(-F_y,F_x,0),"
},
{
"math_id": 20,
"text": "\\nabla"
},
{
"math_id": 21,
"text": "\n\\nabla f =\\left(\\mathbf{e}_x {\\partial \\over \\partial x}\\right)f + \\left(\\mathbf{e}_y {\\partial \\over \\partial y}\\right)f + \\left(\\mathbf{e}_z {\\partial \\over \\partial z}\\right)f = {\\partial f \\over \\partial x}\\mathbf{e}_x + {\\partial f \\over \\partial y}\\mathbf{e}_y + {\\partial f \\over \\partial z}\\mathbf{e}_z\n"
},
{
"math_id": 22,
"text": "\n\\nabla \\cdot \\mathbf{F} = \\left(\\mathbf{e}_x {\\partial \\over \\partial x}\\cdot \\mathbf{F}\\right) + \\left(\\mathbf{e}_y {\\partial \\over \\partial y}\\cdot \\mathbf{F}\\right) + \\left(\\mathbf{e}_z {\\partial \\over \\partial z}\\cdot \\mathbf{F}\\right)= {\\partial F_x \\over \\partial x} + {\\partial F_y \\over \\partial y} + {\\partial F_z \\over \\partial z}\n"
},
{
"math_id": 23,
"text": "\\begin{align}\n\\nabla \\times \\mathbf{F} &= \\left(\\mathbf{e}_x {\\partial \\over \\partial x}\\times \\mathbf{F}\\right) + \\left(\\mathbf{e}_y {\\partial \\over \\partial y}\\times \\mathbf{F}\\right) + \\left(\\mathbf{e}_z {\\partial \\over \\partial z}\\times \\mathbf{F}\\right)\\\\\n&= {\\partial \\over \\partial x}(0, -F_z, F_y) + {\\partial \\over \\partial y}(F_z,0,-F_x) + {\\partial \\over \\partial z}(-F_y,F_x,0)\\\\\n&= \\left({\\partial F_z \\over \\partial y}-{\\partial F_y \\over \\partial z}\\right)\\mathbf{e}_x + \\left({\\partial F_x \\over \\partial z}-{\\partial F_z \\over \\partial x}\\right)\\mathbf{e}_y + \\left({\\partial F_y \\over \\partial x}-{\\partial F_x \\over \\partial y}\\right)\\mathbf{e}_z\n\\end{align}"
},
{
"math_id": 24,
"text": "f(x, y, z) = x + y + z "
},
{
"math_id": 25,
"text": "\\nabla f = \\mathbf{e}_x {\\partial f \\over \\partial x} + \\mathbf{e}_y {\\partial f \\over \\partial y} + \\mathbf{e}_z {\\partial f \\over \\partial z} = \\left(1, 1, 1 \\right) "
},
{
"math_id": 26,
"text": "\\operatorname{grad}f = {\\partial f \\over \\partial x} \\hat\\mathbf x + {\\partial f \\over \\partial y} \\hat\\mathbf y + {\\partial f \\over \\partial z} \\hat\\mathbf z=\\nabla f"
},
{
"math_id": 27,
"text": "h(x,y)"
},
{
"math_id": 28,
"text": "\\nabla(f g) = f \\nabla g + g \\nabla f"
},
{
"math_id": 29,
"text": "\\nabla (\\mathbf u \\cdot \\mathbf v) = (\\mathbf u \\cdot \\nabla) \\mathbf v + (\\mathbf v \\cdot \\nabla) \\mathbf u + \\mathbf u \\times (\\nabla \\times \\mathbf v) + \\mathbf v \\times (\\nabla \\times \\mathbf u)"
},
{
"math_id": 30,
"text": " \\mathbf v(x, y, z) = v_x \\hat\\mathbf x + v_y \\hat\\mathbf y + v_z \\hat\\mathbf z "
},
{
"math_id": 31,
"text": "\\operatorname{div}\\mathbf v = {\\partial v_x \\over \\partial x} + {\\partial v_y \\over \\partial y} + {\\partial v_z \\over \\partial z} = \\nabla \\cdot \\mathbf v "
},
{
"math_id": 32,
"text": " \\nabla \\cdot (f \\mathbf v) = (\\nabla f) \\cdot \\mathbf v + f (\\nabla \\cdot \\mathbf v) "
},
{
"math_id": 33,
"text": " \\nabla \\cdot (\\mathbf u \\times \\mathbf v) = (\\nabla \\times \\mathbf u) \\cdot \\mathbf v - \\mathbf u \\cdot (\\nabla \\times \\mathbf v)"
},
{
"math_id": 34,
"text": "\\mathbf v(x, y, z) = v_x\\hat\\mathbf x + v_y\\hat\\mathbf y + v_z\\hat\\mathbf z"
},
{
"math_id": 35,
"text": "\\operatorname{curl}\\mathbf v = \\left({\\partial v_z \\over \\partial y} - {\\partial v_y \\over \\partial z} \\right) \\hat\\mathbf x + \\left({\\partial v_x \\over \\partial z} - {\\partial v_z \\over \\partial x} \\right) \\hat\\mathbf y + \\left({\\partial v_y \\over \\partial x} - {\\partial v_x \\over \\partial y} \\right) \\hat\\mathbf z = \\nabla \\times \\mathbf v"
},
{
"math_id": 36,
"text": "\\nabla \\times \\mathbf v = \\left|\\begin{matrix} \\hat\\mathbf x & \\hat\\mathbf y & \\hat\\mathbf z \\\\[2pt] {\\frac{\\partial}{\\partial x}} & {\\frac{\\partial}{\\partial y}} & {\\frac{\\partial}{\\partial z}} \\\\[2pt] v_x & v_y & v_z \\end{matrix}\\right|"
},
{
"math_id": 37,
"text": "\\nabla \\times (f \\mathbf v) = (\\nabla f) \\times \\mathbf v + f (\\nabla \\times \\mathbf v)"
},
{
"math_id": 38,
"text": "\\nabla \\times (\\mathbf u \\times \\mathbf v) = \\mathbf u \\, (\\nabla \\cdot \\mathbf v) - \\mathbf v \\, (\\nabla \\cdot \\mathbf u) + (\\mathbf v \\cdot \\nabla) \\, \\mathbf u - (\\mathbf u \\cdot \\nabla) \\, \\mathbf v"
},
{
"math_id": 39,
"text": "f(x,y,z)"
},
{
"math_id": 40,
"text": "\\mathbf a(x,y,z) = a_x \\hat\\mathbf x + a_y \\hat\\mathbf y + a_z \\hat\\mathbf z "
},
{
"math_id": 41,
"text": "(\\mathbf a\\cdot\\nabla)f=\\lim_{h \\to 0}{\\frac{f(x+a_xh,y+a_yh+z+a_zh) - f(x,y,z)}{h}}."
},
{
"math_id": 42,
"text": "\\mathbf a\\cdot\\operatorname{grad}f = a_x {\\partial f \\over \\partial x} + a_y {\\partial f \\over \\partial y} + a_z {\\partial f \\over \\partial z} = \\mathbf a \\cdot (\\nabla f) "
},
{
"math_id": 43,
"text": "\\mathbf a"
},
{
"math_id": 44,
"text": " (\\mathbf a \\cdot \\nabla) "
},
{
"math_id": 45,
"text": "\\Delta = {\\partial^2 \\over \\partial x^2} + {\\partial^2 \\over \\partial y^2} + {\\partial^2 \\over \\partial z^2} = \\nabla \\cdot \\nabla = \\nabla^2"
},
{
"math_id": 46,
"text": "\\nabla^2"
},
{
"math_id": 47,
"text": "\\nabla^2 = \\nabla \\cdot \\nabla^T"
},
{
"math_id": 48,
"text": "\\mathbf{v}"
},
{
"math_id": 49,
"text": "\\nabla \\otimes \\mathbf{v}"
},
{
"math_id": 50,
"text": "\\otimes"
},
{
"math_id": 51,
"text": "\\delta \\mathbf{r}"
},
{
"math_id": 52,
"text": " \\delta \\mathbf{v} = (\\nabla \\otimes \\mathbf{v})^T \\sdot \\delta \\mathbf{r} "
},
{
"math_id": 53,
"text": "\\begin{align}\n \\nabla (fg) &= f\\nabla g + g\\nabla f \\\\\n \\nabla(\\mathbf u \\cdot \\mathbf v) &= \\mathbf u \\times (\\nabla \\times \\mathbf v) + \\mathbf v \\times (\\nabla \\times \\mathbf u) + (\\mathbf u \\cdot \\nabla) \\mathbf v + (\\mathbf v \\cdot \\nabla)\\mathbf u \\\\\n \\nabla \\cdot (f \\mathbf v) &= f (\\nabla \\cdot \\mathbf v) + \\mathbf v \\cdot (\\nabla f) \\\\\n \\nabla \\cdot (\\mathbf u \\times \\mathbf v) &= \\mathbf v \\cdot (\\nabla \\times \\mathbf u) - \\mathbf u \\cdot (\\nabla \\times \\mathbf v) \\\\\n \\nabla \\times (f \\mathbf v) &= (\\nabla f) \\times \\mathbf v + f (\\nabla \\times \\mathbf v) \\\\\n \\nabla \\times (\\mathbf u \\times \\mathbf v) &= \\mathbf u \\, (\\nabla \\cdot \\mathbf v) - \\mathbf v \\, (\\nabla \\cdot \\mathbf u) + (\\mathbf v \\cdot \\nabla) \\, \\mathbf u - (\\mathbf u \\cdot \\nabla) \\, \\mathbf v\n\\end{align}"
},
{
"math_id": 54,
"text": "\\mathbf u \\cdot \\mathbf v"
},
{
"math_id": 55,
"text": "\\mathbf u^\\text{T} \\mathbf v"
},
{
"math_id": 56,
"text": "\\begin{align}\n \\left(\\mathbf{A}\\nabla\\right)^\\text{T} \\mathbf u &= \\nabla^\\text{T} \\left(\\mathbf{A}^\\text{T}\\mathbf u\\right) - \\left(\\nabla^\\text{T} \\mathbf{A}^\\text{T}\\right) \\mathbf u\n\\end{align}"
},
{
"math_id": 57,
"text": "\\mathbf u \\otimes \\mathbf v"
},
{
"math_id": 58,
"text": "\\begin{align}\n \\nabla \\cdot (\\mathbf u \\otimes \\mathbf v) = (\\nabla \\cdot \\mathbf u) \\mathbf v + (\\mathbf u \\cdot \\nabla) \\mathbf v\n\\end{align}"
},
{
"math_id": 59,
"text": "\\begin{align}\n \\operatorname{div}(\\operatorname{grad}f) &= \\nabla \\cdot (\\nabla f) = \\nabla^2 f \\\\\n \\operatorname{curl}(\\operatorname{grad}f) &= \\nabla \\times (\\nabla f) \\\\\n \\operatorname{grad}(\\operatorname{div}\\mathbf v) &= \\nabla (\\nabla \\cdot \\mathbf v) \\\\\n \\operatorname{div}(\\operatorname{curl}\\mathbf v) &= \\nabla \\cdot (\\nabla \\times \\mathbf v) \\\\\n \\operatorname{curl}(\\operatorname{curl}\\mathbf v) &= \\nabla \\times (\\nabla \\times \\mathbf v) \\\\\n \\Delta f &= \\nabla^2 f \\\\\n \\Delta \\mathbf v &= \\nabla^2 \\mathbf v\n\\end{align}"
},
{
"math_id": 60,
"text": " C^\\infty"
},
{
"math_id": 61,
"text": "\\begin{align}\n \\operatorname{curl}(\\operatorname{grad}f) &= \\nabla \\times (\\nabla f) = 0 \\\\\n \\operatorname{div}(\\operatorname{curl}\\mathbf v) &= \\nabla \\cdot (\\nabla \\times \\mathbf v) = 0\n\\end{align}"
},
{
"math_id": 62,
"text": " \\operatorname{div}(\\operatorname{grad}f) = \\nabla \\cdot (\\nabla f) = \\nabla^2 f = \\Delta f "
},
{
"math_id": 63,
"text": "\\nabla \\times \\left(\\nabla \\times \\mathbf v\\right) = \\nabla (\\nabla \\cdot \\mathbf v) - \\nabla^2 \\mathbf{v}"
},
{
"math_id": 64,
"text": "\\nabla (\\nabla \\cdot \\mathbf v) = \\nabla \\cdot (\\mathbf v \\otimes \\nabla )"
},
{
"math_id": 65,
"text": "\\nabla \\cdot \\mathbf v "
},
{
"math_id": 66,
"text": "\\mathbf v \\cdot \\nabla "
},
{
"math_id": 67,
"text": "\\begin{align}\n (\\mathbf u \\cdot \\mathbf v) f &\\equiv (\\mathbf v \\cdot \\mathbf u) f \\\\\n (\\nabla \\cdot \\mathbf v) f &= \\left (\\frac{\\partial v_x}{\\partial x} + \\frac{\\partial v_y}{\\partial y} + \\frac{\\partial v_z}{\\partial z} \\right)f\n = \\frac{\\partial v_x}{\\partial x}f + \\frac{\\partial v_y}{\\partial y}f + \\frac{\\partial v_z}{\\partial z}f \\\\\n (\\mathbf v \\cdot \\nabla) f &= \\left (v_x \\frac{\\partial}{\\partial x} + v_y \\frac{\\partial}{\\partial y} + v_z \\frac{\\partial}{\\partial z} \\right)f\n = v_x \\frac{\\partial f}{\\partial x} + v_y \\frac{\\partial f}{\\partial y} + v_z \\frac{\\partial f}{\\partial z} \\\\\n \\Rightarrow (\\nabla \\cdot \\mathbf v) f &\\ne (\\mathbf v \\cdot \\nabla) f \\\\\n\\end{align}"
},
{
"math_id": 68,
"text": "\\begin{align}\n (\\nabla x) \\times (\\nabla y) &= \\left (\\mathbf e_x \\frac{\\partial x}{\\partial x}+\\mathbf e_y \\frac{\\partial x}{\\partial y}+\\mathbf e_z \\frac{\\partial x}{\\partial z} \\right) \\times \\left (\\mathbf e_x \\frac{\\partial y}{\\partial x}+\\mathbf e_y \\frac{\\partial y}{\\partial y}+\\mathbf e_z \\frac{\\partial y}{\\partial z} \\right) \\\\\n &= (\\mathbf e_x \\cdot 1 +\\mathbf e_y \\cdot 0+\\mathbf e_z \\cdot 0) \\times (\\mathbf e_x \\cdot 0+\\mathbf e_y \\cdot 1+\\mathbf e_z \\cdot 0) \\\\\n &= \\mathbf e_x \\times \\mathbf e_y \\\\\n &= \\mathbf e_z \\\\\n (\\mathbf u x)\\times (\\mathbf u y) &= x y (\\mathbf u \\times \\mathbf u) \\\\\n &= x y \\mathbf 0 \\\\\n &= \\mathbf 0\n\\end{align}"
}
] |
https://en.wikipedia.org/wiki?curid=151925
|
15192512
|
Thermal time scale
|
Stellar astronomy - time scale
In astrophysics, the thermal time scale or Kelvin–Helmholtz time scale is the approximate time it takes for a star to radiate away its total kinetic energy content at its current luminosity rate. Along with the nuclear and free-fall (aka dynamical) time scales, it is used to estimate the length of time a particular star will remain in a certain phase of its life and its lifespan if hypothetical conditions are met. In reality, the lifespan of a star is greater than what is estimated by the thermal time scale because as one fuel becomes scarce, another will generally take its place – hydrogen burning gives way to helium burning, which is replaced by carbon burning.
Stellar astrophysics.
The size of a star as well as its energy output generally determine a star's thermal lifetime because the measurement is independent of the type of fuel normally found at its center. Indeed, the thermal time scale assumes that there is no fuel at all inside the star and simply predicts the length of time it would take for the resulting change in outputted energy to reach the surface of the star and become visually apparent to an outside observer.
formula_0
where "G" is the gravitational constant, M is the mass of the star, R is the radius of the star, and L is the star's luminosity. As an example, the Sun's thermal time scale is approximately 15.7 million years.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\tau_\\text{th} = \\frac{\\mbox{total kinetic energy}}{\\mbox{rate of energy loss}} \\approx \\cfrac{GM^2}{2RL}"
}
] |
https://en.wikipedia.org/wiki?curid=15192512
|
1519267
|
Copper–copper(II) sulfate electrode
|
The copper–copper(II) sulfate electrode is a reference electrode of the first kind, based on the redox reaction with participation of the metal (copper) and its salt, copper(II) sulfate.
It is used for measuring electrode potential and is the most commonly used reference electrode for testing cathodic protection corrosion control systems. The corresponding equation can be presented as follow:
Cu2+ + 2e− → Cu0(metal)
This reaction characterized by reversible and fast electrode kinetics, meaning that a sufficiently high current can be passed through the electrode with the 100% efficiency of the redox reaction (dissolution of the metal or cathodic deposition of the copper-ions).
The Nernst equation below shows the dependence of the potential of the copper-copper(II) sulfate electrode on the activity or concentration copper-ions:
formula_0
Commercial reference electrodes consist of a plastic tube holding the copper rod and saturated solution of copper sulfate. A porous plug on one end allows contact with the copper sulfate electrolyte. The copper rod protrudes out of the tube. A voltmeter negative lead is connected to the copper rod.
The potential of a copper–copper sulfate electrode is +0.314 volt with respect to the standard hydrogen electrode. Copper–copper(II) sulfate electrode is also used as one of the half cells in the galvanic Daniel-Jakobi cell.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\nE=0.337 + \\frac{RT}{2F} \\ln a_{\\rm Cu^{2+}}\n"
}
] |
https://en.wikipedia.org/wiki?curid=1519267
|
1519351
|
Nonmetricity tensor
|
Constant derivative of the metric tensor
In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be
used to study non-Riemannian spacetimes.
Definition.
By components, it is defined as follows.
formula_0
It measures the rate of change of the components of the metric tensor along the flow of a given vector field, since
formula_1
where formula_2 is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional manifold.
Relation to connection.
We say that a connection formula_3 is compatible with the metric when its associated covariant derivative of the metric tensor (call it formula_4, for example) is zero, i.e.
formula_5
If the connection is also torsion-free (i.e. totally symmetric) then it is known as the Levi-Civita connection, which is the only one without torsion and compatible with the metric tensor. If we see it from a geometrical point of view, a non-vanishing nonmetricity tensor for a metric tensor formula_6 implies that the modulus of a vector defined on the tangent bundle to a certain point formula_7 of the manifold, "changes" when it is evaluated along the direction (flow) of another arbitrary vector.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " Q_{\\mu\\alpha\\beta}=\\nabla_{\\mu}g_{\\alpha\\beta} "
},
{
"math_id": 1,
"text": "\\nabla_{\\mu}\\equiv\\nabla_{\\partial_{\\mu}} "
},
{
"math_id": 2,
"text": "\\{\\partial_{\\mu}\\}_{\\mu=0,1,2,3}"
},
{
"math_id": 3,
"text": "\\Gamma"
},
{
"math_id": 4,
"text": "\\nabla^{\\Gamma}"
},
{
"math_id": 5,
"text": " \\nabla^{\\Gamma}_{\\mu}g_{\\alpha\\beta}=0 ."
},
{
"math_id": 6,
"text": "g"
},
{
"math_id": 7,
"text": "p"
}
] |
https://en.wikipedia.org/wiki?curid=1519351
|
151959
|
W. D. Hamilton
|
British evolutionary biologist (1936–2000)
William Donald Hamilton (1 August 1936 – 7 March 2000) was a British evolutionary biologist, recognised as one of the most significant evolutionary theorists of the 20th century. Hamilton became known for his theoretical work expounding a rigorous genetic basis for the existence of altruism, an insight that was a key part of the development of the gene-centered view of evolution. He is considered one of the forerunners of sociobiology. Hamilton published important work on sex ratios and the evolution of sex. From 1984 to his death in 2000, he was a Royal Society Research Professor at Oxford University.
Richard Dawkins has written that Hamilton was "the greatest Darwinian of my lifetime".
Early life.
Hamilton was born in 1936 in Cairo, Egypt, the second of seven children. His parents were from New Zealand; his father A.M. Hamilton was an engineer, and his mother B.M. Hamilton was a physician. The Hamilton family settled in Kent. During the Second World War, Hamilton was evacuated to Edinburgh. He became interested in natural history at an early age and spent his spare time collecting butterflies and other insects. In 1946, he discovered E.B. Ford's New Naturalist book "Butterflies", which introduced him to the principles of evolution by natural selection, genetics, and population genetics.
He was educated at Tonbridge School, where he was in Smythe House. As a 12-year-old, he was seriously injured while playing with explosives his father had that were left over from making hand grenades for the Home Guard during World War II. Hamilton had to have a thoracotomy and parts of fingers on his right hand had to be amputated in King's College Hospital to save his life. He was left with scarring and needed six months to recover.
Before going up to the University of Cambridge, he travelled in France and completed two years of national service. As an undergraduate at St. John's College in Biology, he was uninspired by the "many biologists [who] hardly seemed to believe in evolution".
Hamilton's rule.
Hamilton enrolled in an MSc course in demography at the London School of Economics (LSE), under Norman Carrier, who helped secure grants for his studies. Later, when his work became more mathematical and genetical, he had his supervision transferred to John Hajnal of the LSE and Cedric Smith of University College London (UCL).
Both Fisher and J. B. S. Haldane had seen a problem in how organisms could increase the fitness of their own genes by aiding their close relatives, but not recognised its significance or properly formulated it. Hamilton worked through several examples, and eventually realised that the number that kept falling out of his calculations was Sewall Wright's coefficient of relationship. This became Hamilton's rule: in each behaviour-evoking situation, the individual assesses his neighbour's fitness against his own according to the coefficients of relationship appropriate to the situation. Algebraically, the rule posits that a costly action should be performed if:
formula_0
where "C" is the cost in fitness to the actor, "r" the genetic relatedness between the actor and the recipient, and "B" is the fitness benefit to the recipient. Fitness costs and benefits are measured in fecundity. "r" is a number between 0 and 1. His two 1964 papers entitled "The Genetical Evolution of Social Behaviour" are now widely referenced.
The proof and discussion of its consequences, however, involved detailed mathematics, and two reviewers passed over the paper. The third, John Maynard Smith, did not completely understand it either, but recognised its significance. Having his work passed over later led to friction between Hamilton and Maynard Smith, as Hamilton thought Maynard Smith had held his work back to claim credit for the idea (during the review period Maynard Smith published a paper that referred briefly to similar ideas). The Hamilton paper was printed in the "Journal of Theoretical Biology" and, when first published, was largely ignored. Recognition of its significance gradually increased to the point that it is now routinely cited in biology books.
Much of the discussion relates to the evolution of eusociality in insects of the order Hymenoptera (ants, bees and wasps) based on their unusual haplodiploid sex-determination system. This system means that females are more closely related to their sisters than to their own (potential) offspring. Thus, Hamilton reasoned, a "costly action" would be better spent in helping to raise their sisters, rather than reproducing themselves.
The supergenes notion (sometimes called the "Green-beard effect") - that organisms may evolve genes that are able to identify identical copies in others and preferentially direct social behaviours towards them - was theoretically clarified and withdrawn by Hamilton in 1987.
Spiteful behaviour.
In his 1970 paper "Selfish and Spiteful Behaviour in an Evolutionary Model" Hamilton considers the question of whether harm inflicted upon an organism must inevitably be a byproduct of adaptations for survival. What of possible cases where an organism is deliberately harming others without apparent benefit to the self? Such behaviour Hamilton calls spiteful. It can be explained as the increase in the chance of an organism's genetic alleles to be passed to the next generations by harming those that are less closely related than relationship by chance.
Spite, however, is unlikely ever to be elaborated into any complex forms of adaptation. Targets of aggression are likely to act in revenge, and the majority of pairs of individuals (assuming a panmictic species) exhibit a roughly average level of genetic relatedness, making the selection of targets of spite problematic.
Extraordinary sex ratios.
Between 1964 and 1977, Hamilton was a lecturer at Imperial College London. Whilst there he published a paper in "Science" on "extraordinary sex ratios". Fisher (1930) had proposed a model as to why "ordinary" sex ratios were nearly always 1:1 (but see Edwards 1998), and likewise extraordinary sex ratios, particularly in wasps, needed explanations. Hamilton had been introduced to the idea and formulated its solution in 1960 when he had been assigned to help Fisher's pupil A.W.F. Edwards test the Fisherian sex ratio hypothesis. Hamilton combined his extensive knowledge of natural history with deep insight into the problem, opening up a whole new area of research.
The paper introduced the concept of the "unbeatable strategy", which John Maynard Smith and George R. Price were to develop into the evolutionarily stable strategy (ESS), a concept in game theory not limited to evolutionary biology. Price had originally come to Hamilton after deriving the Price equation, and thus rederiving Hamilton's rule. Maynard Smith later peer reviewed one of Price's papers, and drew inspiration from it. The paper was not published but Maynard Smith offered to make Price a co-author of his ESS paper, which helped to improve relations between the men. Price committed suicide in 1975, and Hamilton and Maynard Smith were among the few present at the funeral.
Hamilton was a visiting professor at Harvard University and later spent nine months with the Royal Society's and the Royal Geographical Society's Xavantina-Cachimbo Expedition as a visiting professor at the University of São Paulo. From 1978 Hamilton was Professor of Evolutionary Biology at the University of Michigan. Simultaneously, he was elected a Foreign Honorary Member of American Academy of Arts and Sciences. His arrival sparked protests and sit-ins from students who did not like his association with sociobiology. There he worked with the political scientist Robert Axelrod on the prisoner's dilemma, and was a member of the BACH group with original members Arthur Burks, Robert Axelrod, Michael Cohen, and John Holland.
Hamilton was regarded as a poor lecturer. This shortcoming would not affect the recognition of his work, however, as it was popularised by Richard Dawkins in the book "The Selfish Gene" published in 1976.
Chasing the Red Queen.
Hamilton was an early proponent of the Red Queen theory of the evolution of sex (separate from the other theory of the same name previously proposed by Leigh Van Valen). This was named for a character in Lewis Carroll's "Through the Looking-Glass", who is continuously running but never actually travels any distance:
"Well, in our country," said Alice, still panting a little, "you'd generally get to somewhere else—if you ran very fast for a long time, as we've been doing."
"A slow sort of country!" said the Queen. "Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!" (Carroll, pp. 46)
This theory hypothesizes that sex evolved because new and unfamiliar combinations of genes could be presented to parasites, preventing the parasite from preying on that organism: species with sex were able to continuously "run away" from their parasites. Likewise, parasites were able to evolve mechanisms to get around the organism's new set of genes, thus perpetuating an endless race.
Return to Britain.
In 1980, he was elected a Fellow of the Royal Society, and in 1984, he was invited by Richard Southwood to be the Royal Society Research Professor in the Department of Zoology at Oxford, and a fellow of New College, where he remained until his death.
His collected papers, entitled "Narrow Roads of Gene Land", began to be published in 1996. The first volume was entitled "Evolution of Social Behaviour".
Social evolution.
The field of social evolution, of which Hamilton's Rule has central importance, is broadly defined as being the study of the evolution of social behaviours, i.e. those that impact on the fitness of individuals other than the actor. Social behaviours can be categorized according to the fitness consequences they entail for the actor and recipient. A behaviour that increases the direct fitness of the actor is mutually beneficial if the recipient also benefits, and selfish if the recipient suffers a loss. A behaviour that reduces the fitness of the actor is altruistic if the recipient benefits, and spiteful if the recipient suffers a loss. This classification was first proposed by Hamilton in 1964.
Hamilton also proposed the coevolution theory of autumn leaf color as an example of evolutionary signalling theory.
Origin of HIV.
During the 1990s, Hamilton became interested in the now-discredited hypothesis that the origin of HIV lay in Hilary Koprowski's oral polio vaccine trials in Africa during the 1950s. Hamilton's letter on the topic to "Science" journal was rejected in 1996. Despite this, he spoke to the BBC supporting the hypothesis, and wrote the foreword of Edward Hooper's 1999 book "The River". To look for evidence of the hypothesis, Hamilton went on a 2000 field trip to the Democratic Republic of the Congo to assess natural levels of simian immunodeficiency virus in primates. None of the over 60 urine and faecal samples contained detectable SIV virus.
Death.
Hamilton returned to London from Africa on 29 January 2000. He was admitted to University College Hospital, London, on 30 January 2000. He was transferred to Middlesex Hospital on 5 February 2000 and died there on 7 March 2000. An inquest was held on 10 May 2000 at Westminster Coroner's Court to inquire into rumours about the cause of his death. The coroner concluded that his death was due to "multi-organ failure due to upper gastrointestinal haemorrhage due to a duodenal diverticulum and arterial bleed through a mucosal ulcer". Following reports attributing his death to complications arising from malaria, the BBC Editorial Complaints Unit's investigation established that he had contracted malaria during his final African expedition. However, the pathologist had suggested the possibility that the ulceration and consequent haemorrhage had resulted from a pill (which might have been taken because of malarial symptoms) lodging in the diverticulum; but, even if this suggestion were correct, the link between malaria and the observed causes of death would be entirely indirect.
A secular memorial service (he was an agnostic) was held at the chapel of New College, Oxford on 1 July 2000, organised by Richard Dawkins. He was buried near Wytham Woods. He, however, had written an essay on "My intended burial and why" in which he wrote:
The second volume of his collected papers, "Evolution of Sex", was published in 2002, and the third and final volume, "Last Words", in 2005.
In 1966, he married Christine Friess; the couple had three daughters, Helen, Ruth, and Rowena. They amicably separated 26 years later. From 1994, Hamilton found companionship with Maria Luisa Bozzi, an Italian science journalist and author.
Works.
Collected papers.
Hamilton started to publish his collected papers in 1996, along the lines of Fisher's collected papers, with short essays giving each paper context. He died after the preparation of the second volume, so the essays for the third volume come from his coauthors.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "C < r \\times B "
}
] |
https://en.wikipedia.org/wiki?curid=151959
|
1519594
|
Lebesgue point
|
In mathematics, given a locally Lebesgue integrable function formula_0 on formula_1, a point formula_2 in the domain of formula_0 is a Lebesgue point if
formula_3
Here, formula_4 is a ball centered at formula_2 with radius formula_5, and formula_6 is its Lebesgue measure. The Lebesgue points of formula_0 are thus points where formula_0 does not oscillate too much, in an average sense.
The Lebesgue differentiation theorem states that, given any formula_7, almost every formula_2 is a Lebesgue point of formula_0.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f"
},
{
"math_id": 1,
"text": "\\mathbb{R}^k"
},
{
"math_id": 2,
"text": "x"
},
{
"math_id": 3,
"text": "\\lim_{r\\rightarrow 0^+}\\frac{1}{\\lambda (B(x,r))}\\int_{B(x,r)} \\!|f(y)-f(x)|\\,\\mathrm{d}y=0."
},
{
"math_id": 4,
"text": "B(x,r)"
},
{
"math_id": 5,
"text": "r > 0"
},
{
"math_id": 6,
"text": "\\lambda (B(x,r))"
},
{
"math_id": 7,
"text": "f\\in L^1(\\mathbb{R}^k)"
}
] |
https://en.wikipedia.org/wiki?curid=1519594
|
15197395
|
Return ratio
|
The return ratio of a dependent source in a linear electrical circuit is the "negative" of the ratio of "the current (voltage) returned to the site of the dependent source" to "the current (voltage) of a replacement independent source". The terms "loop gain" and "return ratio" are often used interchangeably; however, they are necessarily equivalent only in the case of a single feedback loop system with unilateral blocks.
Calculating the return ratio.
The steps for calculating the return ratio of a source are as follows:
Other Methods.
These steps may not be feasible when the dependent sources inside the devices are not directly accessible, for example when using built-in "black box" SPICE models or when measuring the return ratio experimentally.
For SPICE simulations, one potential workaround is to manually replace non-linear devices by their small-signal equivalent model, with exposed dependent sources. However this will have to be redone if the bias point changes.
A result by Rosenstark shows that return ratio can be calculated by breaking the loop at any unilateral point in the circuit. The problem is now finding how to break the loop without affecting the bias point and altering the results. Middlebrook and Rosenstark have proposed several methods for experimental evaluation of return ratio (loosely referred to by these authors as simply "loop gain"), and similar methods have been adapted for use in SPICE by Hurst. See Spectrum user note or Roberts, or Sedra, and especially Tuinenga.
Example: Collector-to-base biased bipolar amplifier.
Figure 1 (top right) shows a bipolar amplifier with feedback bias resistor "Rf" driven by a Norton signal source. Figure 2 (left panel) shows the corresponding small-signal circuit obtained by replacing the transistor with its hybrid-pi model. The objective is to find the return ratio of the dependent current source in this amplifier. To reach the objective, the steps outlined above are followed. Figure 2 (center panel) shows the application of these steps up to Step 4, with the dependent source moved to the left of the inserted source of value "it", and the leads targeted for cutting marked with an "x". Figure 2 (right panel) shows the circuit set up for calculation of the return ratio "T", which is
formula_0
The return current is
formula_1
The feedback current in "Rf" is found by current division to be:
formula_2
The base-emitter voltage "vπ" is then, from Ohm's law:
formula_3
Consequently,
formula_4
Application in asymptotic gain model.
The overall transresistance gain of this amplifier can be shown to be:
formula_5
with "R1 = RS || rπ" and "R2 = RD || rO".
This expression can be rewritten in the form used by the asymptotic gain model, which expresses the overall gain of a feedback amplifier in terms of several independent factors that are often more easily derived separately than the overall gain itself, and that often provide insight into the circuit. This form is:
formula_6
where the so-called asymptotic gain "G∞" is the gain at infinite "gm", namely:
formula_7
and the so-called feed forward or direct feedthrough "G0" is the gain for zero "gm", namely:
formula_8
For additional applications of this method, see asymptotic gain model and Blackman's theorem.
|
[
{
"math_id": 0,
"text": " T = - \\frac {i_r} {i_t} \\ . "
},
{
"math_id": 1,
"text": " i_r = g_m v_{\\pi} \\ . "
},
{
"math_id": 2,
"text": "i_f = \\frac {R_D//r_O} {R_D//r_O +R_F +r_{\\pi}// R_S} \\ i_t \\ . "
},
{
"math_id": 3,
"text": " v_{\\pi} = -i_f \\ ( r_{\\pi}// R_S ) \\ . "
},
{
"math_id": 4,
"text": " T = g_m (r_{\\pi}// R_S ) \\ \\frac {R_D//r_O} {R_D//r_O +R_F +r_{\\pi}// R_S}\\ . "
},
{
"math_id": 5,
"text": " G = \\frac {v_{out}} {i_{in}} = \\frac {(1-g_m R_F)R_1 R_2} {R_F+R_1+R_2+g_m R_1R_2} \\ , "
},
{
"math_id": 6,
"text": " G = \\ G_{ \\infty } \\frac {T} {1+T} + G_0 \\frac {1} { 1+T} \\ \\ , "
},
{
"math_id": 7,
"text": " G_{\\infty} = - R_F \\ , "
},
{
"math_id": 8,
"text": " G_{0} = \\frac { R_1 R_2 } {R_F +R_1 +R_2}\\ . "
}
] |
https://en.wikipedia.org/wiki?curid=15197395
|
1519744
|
Fade (audio engineering)
|
Gradual change in level of audio signal
In audio engineering, a fade is a gradual increase or decrease in the level of an audio signal. The term can also be used for film cinematography or theatre lighting in much the same way (see fade (filmmaking) and fade (lighting)).
In sound recording and reproduction a song may be gradually reduced to silence at its end (fade-out), or may gradually increase from silence at the beginning (fade-in). Fading-out can serve as a recording solution for pieces of music that contain no obvious ending. Quick fade-ins and -outs can also be used to change the characteristics of a sound, such as to soften the attack in vocal plosives and percussion sounds.
Professional turntablists and DJs in hip hop music use faders on a DJ mixer, notably the horizontal crossfader, in a rapid fashion while simultaneously manipulating two or more record players (or other sound sources) to create "scratching" and develop beats. Club DJs in house music and techno use DJ mixers, two or more sound sources (two record players, two iPods, etc.) along with a skill called beatmatching (aligning the beats and tempos of two records) to make seamless dance mixes for dancers at raves, nightclubs and dance parties.
History.
Origins and examples.
Possibly the earliest example of a fade-out ending can be heard in Joseph Haydn's Symphony No. 45, nicknamed the "Farewell" Symphony on account of the fade-out ending. The symphony which was written in 1772 used this device as a way of courteously asking Haydn's patron Prince Nikolaus Esterházy, to whom the symphony was dedicated, to allow the musicians to return home after a longer than expected stay. This was expressed by the players extinguishing their stand candles and leaving the stage one by one during the final adagio movement of the symphony, leaving only two muted violins playing. Esterházy appears to have understood the message, allowing the musicians to leave.
Gustav Holst's "Neptune, the mystic", part of the orchestral suite "The Planets" written between 1914 and 1916, is another early example of music having a fade-out ending during performance. Holst stipulates that the women's choruses are "to be placed in an adjoining room, the door of which is to be left open until the last bar of the piece, when it is to be slowly and silently closed", and that the final bar (scored for choruses alone) is "to be repeated until the sound is lost in the distance". Although commonplace today, the effect bewitched audiences in the era before widespread recorded sound—after the initial 1918 run-through, Holst's daughter Imogen (in addition to watching the charwomen dancing in the aisles during "Jupiter") remarked that the ending was "unforgettable, with its hidden chorus of women's voices growing fainter and fainter ... until the imagination knew no difference between sound and silence".
The technique of ending a spoken or musical recording by fading out the sound goes back to the earliest days of recording. In the era of mechanical (pre-electrical) recording, this could only be achieved by either moving the sound source away from the recording horn, or by gradually reducing the volume at which the performer(s) were singing, playing or speaking. With the advent of electrical recording, smooth and controllable fadeout effects could be easily achieved by simply reducing the input volume from the microphones using the fader on the mixing desk. The first experimental study on the effect of a fade-out showed that a version of a musical piece with fade-out in comparison to the same piece with a cold end prolonged the perceived duration by 2.4 seconds. This is called the "Pulse Continuity Phenomenon" and was measured by a tapping-along task to measure participants’ perception of pulsation.
An 1894 78 rpm record called "The Spirit of '76" features a narrated musical vignette with martial fife-and-drum that gets louder as it "nears" the listener, and quieter as it "moves away". There are early examples that appear to bear no obvious relationship to movement. One is "Barkin' Dog" (1919) by the Ted Lewis Jazz Band. Another contender is "America" (1918), a patriotic piece by the chorus of evangelist Billy Sunday. By the early 1930s, longer songs were being put on both sides of records, with the piece fading out at the end of side one and fading back in at the beginning of side two. Records at the time held only about two to five minutes of music per side. The segue allowed for longer songs (such as Count Basie's "Miss Thing"), symphonies and live concert recordings.
However, shorter songs continued to use the fade-out for unclear reasons—for example, Fred Astaire's movie theme "Flying Down to Rio" (1933). Even using fade-out as a segue device does not seem obvious, though we certainly take it for granted today. It is possible that movies were an influence here. Fade-ins and fade-outs are often used as cinematic devices that begin and end scenes; film language that developed at the same time as these early recordings. The term "fade-out" itself is of cinematic origin, appearing in print around 1918. And jazz, a favorite of early records, was a popular subject of early movies too. The same could be said for radio productions. Within a single programme of a radio production, many different types of fade can be applied. When mixing from speech to music, there are a few ways that fade can be used. Here are three examples.
Though relatively rare, songs can fade out then fade back in. Some examples of this are "Helter Skelter" and "Strawberry Fields Forever" by The Beatles, "Suspicious Minds" by Elvis Presley, "Shine On Brightly" by Procol Harum, "Sunday Bloody Sunday" by John Lennon and Yoko Ono, "That Joke Isn't Funny Anymore" by The Smiths, "Thank You" by Led Zeppelin, "In Every Dream Home A Heartache" by Roxy Music, "It's Only Money, Pt. 2" by Argent, "The Great Annihilator" by Swans, "(Reprise) Sandblasted Skin" by Pantera, "Illumination Theory" and "At Wit's End" by Dream Theater, "Future" by Paramore, "" by MF Doom, "Outro" by M83, "Cold Desert" by Kings of Leon, and "The Edge Of The World" by DragonForce.
Contemporary.
No modern recording can be reliably identified as "the first" to use the technique. In 2003, on the (now-defunct) website "Stupid Question", John Ruch listed the following recordings as possible contenders:
<templatestyles src="Template:Blockquote/styles.css" />Bill Haley's cover version of "Rocket 88" (1951) fades out to indicate the titular car driving away. There are claims that The Beatles' "Eight Days a Week" (recorded 1964) was the first song to use the reverse effect—a fade-in. In fact, The Supremes had used this effect on their single "Come See About Me", issued a little over a month before "Eight Days a Week".
More recently: "At the meta-song level, the prevalence of pre-taped sequences (for shops, pubs, parties, concert intervals, aircraft headsets) emphasizes the importance of "flow". The effect on radio pop programme form [is] a stress on continuity achieved through the use of fades, voice-over links, twin-turntable mixing and connecting jingles."
Fade.
A fade can be constructed so that the motion of the control (linear or rotary) from its start to end points affects the level of the signal in a different manner at different points in its travel. If there are no overlapping regions on the same track, regular fade (pre-fade / post-fade) should be used. A smooth fade is one that changes according to the logarithmic scale, as faders are logarithmic over much of their working range of 30-40 dB. If the engineer requires one region to gradually fade into another on the same track, a crossfade would be more suitable. If however the two regions are on different tracks, fade-ins and fade-outs will be applied. A fade-out can be accomplished without letting the sound's distance increase, however this is also something it can do. The perceived distance increase can be attributed to a diminishing level of timbral detail, not the result of a decreasing dynamic level. A listener's interest can be withdrawn from a sound that is faded at the lower end since the ear accepts a more prompt rounding off. The fade-in can be used as a device that separates the listener from the scene. An example of a mini fade out, of about a second or two, is a sustained bass note left to die down.
Shapes.
The shape of a regular fade and a crossfade can be shaped by an audio engineer. Shape implies that you can change the rate at which the level change occurs over the length of the fade. Different types of preset fades shapes include linear, logarithmic, exponential and S-curve.
Linear.
The simplest of fade curves is the linear curve and it is normally the default fade. It takes a straight line and introduces a curve. This curve represents an equal degree by which the gain increases or decreases during the length of the fade. A linear fade-in curve makes it sound as though the volume increases sharply at the beginning, and more gradually towards the end. The same principle applies on a fade-out where a gradual drop in volume can be perceived in the beginning, and the fade gets more abrupt towards the end. If in your audio there is a natural ambience or reverb present that one would like to reduce, the linear shape would be ideal because of the initial drop in perceived volume. When applied it shortens the ambience. Also if the music requires an accelerating effect, this linear curve can also be applied. This type of fade is not very natural sounding. The principle of a linear crossfade is: at the beginning of the fade the perceived volume drops more quickly, one can see at the halfway point (in the middle of the crossfade) that the perceived volume drops below 50%. This is a very noticeable drop in volume. Also if the control can move from position 0 to 100, and the percentage of the signal that is allowed to pass equals the position of the control (i.e. 25% of the signal is allowed to pass when the control is 25% of the physical distance from the 0 point to the 100 point). At the midpoint of the fade the effect of a linear crossfade is that both the sounds are below half of their maximum perceived volume; and as a result the sum of the two fades will be below the maximum level of either. This is not applicable when the two sounds are on different levels and the crossfade time is long enough. In turn if the crossfade is short (for example on a single note) the dip of the volume in the middle of the crossfade can be quite noticeable.
Logarithmic.
Another type of curve is called the logarithmic ratio (also known as "audio taper"), or an inverse-logarithmic ratio. The log/audio-taper more closely matches human hearing, with finer control at lower levels, increasing dramatically past the 50% point. Importantly, the perceived volume of a sound has a logarithmic relationship with its level in decibels. A fade that works on a logarithmic scale would thus counteract the curve of a linear fade. In fade-ins the shape of the perceived volume curve looks like the level in decibels pulled towards the middle part of the line to the bottom right corner. The fade-out in turn looks like it has been pulled toward the bottom-left corner. A logarithmic fade takes a line that has already been curved and straightens it out [15]. The logarithmic fade sounds consistent and smooth since the perceived volume is increased over the whole duration of the fade. This makes this curve very handy for fading standard pieces of music. It is best used on a long fade-out since the fade has a perceived linear nature. Also a fade-out sounds very neutral when incorporated to parts of music with natural ambience. In crossfades this type of curve sounds very natural. When this curve is applied the perceived volume of the fade's midpoint is at about 50% of the maximum – when the two sections are summed the output volume is fairly constant.
Exponential.
The exponential curve shape is in many ways the precise opposite of the logarithmic curves. The fade-in works as follows: it increases in volume slowly and then it shoots up very quickly at the end of the fade. The fade-out drops very quickly (from the maximum volume) and then declines slowly again over the duration of the fade. Simply stated a linear fade could thus be seen as an exaggerated version of an exponential fade in terms of the apparent volume. Thus the impression that would be gathered from an exponential curve's fade would sound as though the sound was rapidly accelerating towards the listener. Natural ambience can also be repressed by using an exponential fade-out. A crossfade, in the exponential shape, will have a perceivable dip in the middle, which is very undesirable in music and vocals. This depends largely on the length of the crossfade, a long crossfade on ambient sounds can sound perfectly satisfactory (the dip can add a little breath to the music). Exponential crossfades (or a curve with a similar shape) have a smaller drop in the middle of the fade.
S-curve.
The S-curve shape is interesting since it has qualities that correlate with the previously mentioned curves. The level of the sound is 50% at the midpoint, but before and after the midpoint the shape is not linear. There are also two types of S-curves. Traditional S-curve fade-in has attributes of the exponential curve can be seen at the beginning; at the midpoint to the end it is more logarithmic in nature. A traditional S-curve fade-out: is logarithmic from the beginning up to the midpoint, then its attributes are based on the exponential curve from the midpoint to the end. This is true for the situation in reverse as well (for both fade-in and fade-out). Crossfading S-curves works as follows; it diminishes the amount of time that both sounds are playing simultaneously. This ensures that the edits sound like a direct cut when the two edits meet - this adds an extra smoothness to the edited regions.
The second type of S-curve is more apt for longer crossfades, since they are smooth and have the ability to have both of the crossfades in the overall level; so that they are audible for as long as possible. There is a short period at the start of each of the crossfades where the outgoing sound drops toward 50% quickly (with the incoming sound rising just as fast to 50%). This acceleration of sound slows and both sounds will appear as if they are at the same level for most of the crossfade (in the middle) before the changeover happens.
DAW's gives one the ability to change the shape of logarithmic, exponential, and S-curve fades and crossfades. Changing the shape of a logarithmic fade will change how soon the sound will rise above 50%, and then how long it takes for the end of the fade-out to drop below 50% once again. With exponential fades the shape change will affect the shape in reverse, to the shape of the logarithmic fade. In the S-curve's traditional form the shape determines how quickly the change can occur and in the type 2 curve the change can determine the time it takes for both the sounds to get to a nearly equal level.
Fade times.
It is also possible to apply different fade times to the out and in portions; which a standard crossfade would not allow you to apply. Appropriate fade-in time for a linear fade can be around 500ms; for the fade-out 500ms would also be affective. By having this longer fade it makes sure that everything is gentle as it gives the fade time to blend in and be less abrupt. To clear up plosive sounds created through vocals a fade-in can be used, but now it has to have a very short time of around 10ms. The fade time can always be adjusted by the engineer in order to locate the best time. It is important that the fade does not change the intelligibility or character of the sound too much. When the crossfade is longer than 10ms the standard linear fades are not always the best choice for music editing.
Crossfading.
A crossfader on a DJ mixer essentially functions like two faders connected side-by-side, but in opposite directions. A crossfader is typically mounted horizontally, so that the DJ can slide the fader from the extreme left (this provides 100% of sound source A) to the extreme right (this provides 100% of sound source B), move the fader to the middle (this is a 50/50 mix of sources A and B), or adjust the fader to any point in between. It allows a DJ to fade one source out while fading another source in at the same time. This is extremely useful when beatmatching two sources of audio (or more, where channels can be mapped to one of the two sides of the crossfader individually) such as phonograph records, compact discs or digital sources.
The technique of crossfading is also used in audio engineering as a mixing technique, particularly with instrumental solos. A mix engineer will often record two or more takes of a vocal or instrumental part and create a final version which is a composite of the best passages of these takes by crossfading between each track. In the perfect case, the crossfade would keep a constant output level, an important quality for a club DJ who is creating a seamless mix of dance tracks for dancers or a radio DJ seeking to avoid "dead air" (silence) between songs, an error that can cause listeners to change channels. However, there is no standard on how this should be achieved.
There are many software applications that feature virtual crossfades, for instance, burning-software for the recording of audio-CDs. Also many DAW's (Pro Tools, Logic exc.) have this function. Crossfade is normally found on samplers and usually based on velocity. The purpose of a cross-fade it to utilize a smooth changeover between two cut pieces of audio. Velocity crossfading can be incorporated through a MIDI transformation device and where more than one note can be assigned to a given pad (note) on the MIDI keyboard; velocity crossfading may be available.
These types of crossfades (those that are based on note velocity) allow two (even more) samples to be assigned to one note or range of notes. This requires both a loud and soft sample; the reason for this is Timbre change. This type of crossfade is quite subtle depending on the proportion of the received note velocity value of the loud and soft sample.
Crossfading usually involves the sounding of a combination of one or two sounds at the same time. Crossfades can either be applied to a piece of music in real time, or can be pre-calculated. While crossfading one does not want the second part of the fade to start playing before the first part is finished; one wants the overlapping parts to be as short as possible. If edit regions are not trimmed to a zero-crossing point one will get unwelcome pops in the middle. A sound at the lowest velocity can fade into a sound of a higher velocity, in the order of: first the first sound then the second. All possible without fading out the sounds that are already present. This in turn is a form of Layering that can be used in the mix. The same effect (as was created with velocity) can be applied to a controller. This allows continued monitored control; the crossfading function can also be controlled on some instruments by the keyboard position. These sounds on the MIDI keyboard can be programmed.
A crossfade can either be used between two unrelated pieces of music or between two sounds that are similar; in both of these cases, one would like the sound to be one continuous sound without any bumps. When applying a crossfade between two very different pieces of music (relating to both tone and pitch), one could simply use a crossfade between the two pieces, make a few minor adjustments. This is because the two sounds are different from one another. In the case of a crossfade between two sounds, that are similar, phase-cancellation can become an issue. The two sounds that are crossfaded should be brought into comparison with one another. If both sounds are moving upward they will have a cumulative effect - when added together, this is what one wants. What is not desirable is when both sounds are moving in a different direction, since this can lead to cancelations. This leads to no sound on areas where the amplitudes cancel out one another; there will thus be silence in the middle of the crossfade. This occurrence is rare though since the parameters have to be the same. Commonly a crossfade will result in a gradual reduction in the amount of the sample whose pitch is lower, and an increase will be found on the pitch that is higher. The longer a crossfade, the more likely a problem will occur. One also does not want the effect of the crossfade to be very prominent in the middle of the notes, since if different notes are between the edit point there will be a time when both of the sounds can be heard simultaneously. This overlapping is not expected from a normal singing voice, no reference to Overtone singing.
While DJ pioneers such as Francis Grasso had used basic faders to transition between two records as far back as the late 1960s, they typically had separate faders for each channel. Grandmaster Flash is often credited with the invention of the first crossfader by sourcing parts from a junkyard in the Bronx. It was initially an on/off toggle switch from an old microphone that he transformed into a left/right switch which allowed him to switch from one turntable to another, thereby avoiding a break in the music. However the earliest commercial documented example was designed by Richard Wadman, one of the founders of the British company Citronic. It was called the model SMP101, made about 1977, and had a crossfader that doubled as a L/R balance control or a crossfade between two inputs.
Crossfade shapes.
When crossfading two signals that are being combined (mixed), the two fade curves can employ any of the shapes listed above (see #Shapes), such as linear, exponential, S-curve, etc. When the goal is to have the perceived loudness of the combined mix signal stay fairly constant across the full range of the mix, special shapes must be used, called "equal power" (or "constant power") shapes. Equal power shapes are based on audio power principles, particularly the fact that the power of an audio signal is proportional to the square of the amplitude. Many equal power shapes have the property that the midpoint of the mix provides an amplitude multiplier of 0.707 (square root of one half) for both signals. A variety of equal power shapes are available, and the optimal shape will generally depend on the amount of correlation between the two signals. An example pair of curves that keep power equal across the mix are formula_0 and formula_1 (where "mix" ranges from 0 to 1).
Equal power shapes typically have the sum of their curves (in the middle of the mix range) exceeding the nominal maximum amplitude (1.0), which may produce clipping in some contexts. If that is a concern, then "equal gain" (or "constant gain") shapes should be used (which may be linear or curved) that are designed so the two curves always sum to 1.
In the digital signal processing realm, the term "power curve" is often used to designate crossfade shapes, particularly for equal power shapes.
Fader.
A fader is any device used for fading, especially when it is a knob or button that "slides" along a track or slot. It is principally a variable resistance or potentiometer also called a ‘pot’. A contact can move from one end to another. As this movement takes place the resistance of the circuit can either increase or decrease. At one end the resistance of the scale is at 0 and at the other side it is infinite. A. Nisbett explains the fader law as follows in his book called The Sound studio:"The ‘law’ of the fader is near-logarithmic over much of its range, which means that a scale of decibels can be made linear (or close to it) over a working range of perhaps 60 dB. If the resistance were to increase according to the same law beyond this, it would be twice as long before reaching a point where the signal is negligible. But the range below -50 dB is of little practical use, so here the rate of fade increases rapidly to the final cut-off".
A knob which "rotates" is usually not considered a fader, although it is electrically and functionally equivalent. Some small mixers use knobs rather than faders, as do a small number of DJ mixers designed for club DJs who are creating seamless mixes of songs. A fader can be either analogue, directly controlling the resistance or impedance to the source (e.g. a potentiometer); or digital, numerically controlling a digital signal processor (DSP). Analogue faders are found on mixing consoles. A fader can also be used as a control for a voltage controlled amplifier, which has the same effect on the sound as any other fader, but the audio signal does not pass through the fader itself.
Digital.
Digital faders are also referred to as "virtual" faders, since they can be viewed on the screen of a digital audio workstation. Modern high end digital mixers often feature "flying faders", faders with piezo-electric actuators attached; such faders can be multi-use and will jump to the correct position for a selected function or saved setting. Flying faders can be automated, so that when a timecode is presented to the equipment, the fader will move according to a previously performed path. Also called an automated fader, as it recalls the movement of the channel faders in time. A full-function automation system will continuously scan the console, many times per second, in order to incorporate new settings. While this scan is in progress, the stored representation of the previous scan will be compared to that of the fader's current position. If the fader's position has changed, the new position will be identified, thus resulting in a spurt of data.
The console's computer will update the console's controls on playback. This will be done from memory at the same speed. The advantage of working with mix automation is that only one engineer can perform the job with minimal effort; it can be set up or recorded beforehand to make it even simpler. An example of this is when Ken Hamman installed linear faders that made it possible for him to alter several channels with one hand while mixing, thus he assumed an interactive role in the process of recording. This type of fader level adjustment is also called ‘riding’ the fader.
Types.
Many DJ equipment manufacturers offer different mixers for different purposes, with different fader styles, e.g., "scratching", beatmixing, and cut mixing. High-priced mixers often have crossfade curve switches allowing the DJ to select the type of crossfade necessary. Experienced DJs are also able to crossfade between tracks using the channel faders.
Pre-fader, post-fader.
On a mixer with auxiliary send mixes, the send mixes are configured pre-fader or post-fader. If a send mix is configured pre-fader, then changes to the main channel strip fader does not affect the send mix. In live sound reinforcement, this is useful for stage monitor mixes where changes in the Front of House channel levels would distract the musicians. In recording and post production, configuring a send to be pre-fader allows the amount of audio sent to the aux bus to remain unaffected by the individual track fader, thus not disturbing the stability of the feed that is being sent to the musicians. If a send mix is configured post-fader, then the level sent to the send mix follows changes to the main channel strip fader. This is useful for reverberation and other signal processor effects. An example of this is when an engineer would like to add some delay to the vocals – the fader can thus be used to adjust the amount of delay added.
Pre-fader listen (PFL), after-fader listen (AFL).
These functions will be found on a primary monitor function. This pre-fade listen is valuable since it allows one to listen through headphones in order to hear what the pre-faded part sounds like, while the studio loudspeaker is being used to monitor the rest of the program.
Pre-fade listen can also be used for talkback as well as to listen to channels before they have been faded. After-fade listen only gets its information later. The choice of listen or level will depend on the user's interest: either with the quality and/or content of the signal or with the signal's level. PFL takes place just before the fader and has a joint channel and monitoring function. PFL sends the channel's signal path to the pre-fade bus. The bus is picked up in the monitor module and made accessible as a substitute signal that is sent to the mixer output. Automatic PFL has been made available, almost universally, and no longer needs to be selected beforehand.
Pre-fade listen can also be incorporated in radio stations and serves as a vital tool. This function allows the radio presenter to listen to the source before it is faded on air; allowing the presenter to check the source's incoming level and make sure it is accurate. It is also valuable since live radio broadcasts can fall apart without it as they will not be able to monitor the sound. After-fader listen is not as useful in live programs.
References.
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"text": "\\sqrt{mix}"
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https://en.wikipedia.org/wiki?curid=1519744
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1519764
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Brake fade
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Reduction in stopping power
Brake fade (or vehicle braking system fade) is the reduction in stopping power that can occur after repeated or sustained application of the brakes of a vehicle, especially in high load or high speed conditions. Brake fade can be a factor in any vehicle that utilizes a friction braking system including automobiles, trucks, motorcycles, airplanes, and bicycles.
Brake fade is caused by a buildup of heat in the braking surfaces and the subsequent changes and reactions in the brake system components and can be experienced with both drum brakes and disc brakes. Loss of stopping power, or fade, can be caused by friction fade, mechanical fade, or fluid fade. Brake fade can be significantly reduced by appropriate equipment and materials design and selection, as well as good cooling.
Brake fade occurs most often during high performance driving or when going down a long, steep hill. It is more prevalent in drum brakes due to their configuration. Disc brakes are much more resistant to brake fade because the heat can be vented away from the rotor and pads more easily, and have come to be a standard feature in front brakes for most vehicles.
Causes of brake fade.
The reduction of friction termed brake fade is caused when the temperature reaches the "kneepoint" on the temperature-friction curve and gas builds up between disc and pad. All brake linings are cured under mechanical pressure following a heating and cooling curve backstroke, heating the friction material up to to "cure" (cross-link) the phenolic resin thermoset polymers: There is no melting of the binding resins, because phenolic resins are thermoset, not thermoplastic. In this form of fade, the brake pedal feels firm but there is reduced stopping ability. Fade can also be caused by the brake fluid boiling, with attendant release of compressible gases. In this type of fade, the brake pedal feels "spongy". This condition is worsened when there are contaminants in the fluid, such as water, which most types of brake fluids are prone to absorbing to varying degrees. For this reason brake fluid replacement is standard maintenance.
Fade in self-assisting brakes.
Various brake designs such as band brakes and many drum brakes are self-assisting: when the brake is applied, some of the braking force feeds back into the brake mechanism to further self-apply the brake. This is called "positive feedback" or "self-servo". Self-assist reduces the input force needed to apply the brake, but exaggerates fade, since a reduction in pad friction material height or thickness also reduces pad force. In contrast, for a brake without self-assist, such as a conventional disc brake, a loss of pad friction material does not change the pad force, so there is no necessary loss in the brake torque reaction for a given amount of input force.
The self-assist mechanism affects the water pump and the amount of fade. For example, the Ausco Lambert and Murphy brakes have self-assist roughly proportional to pad friction, so total braking is reduced as roughly the square of the loss in friction. Many other self-assist designs, such as band brakes and many common drum brakes, have exponential self-assist, described by formula_0, where formula_1 is the natural logarithm base, formula_2 is the coefficient of friction between shoes and drum, and formula_3 is the angle of engagement between shoes and drum. A small change in friction causes an exponential change in self assist. In many common brakes, a slight increase in friction can lead to wheel lockup with even light application. For example, on damp mornings, drum brakes can lock on first application, skidding to a stop even after the brake pedal has been released. Conversely, a slight decrease in friction can lead to severe brake fade.
Factors contributing to fade.
Brake fade failures can cascade. For example, a typical 5-axle truck/trailer combination has 10 brakes. If one brake fades, brake load is transferred to the remaining 9 brakes, causing them to work harder, get hotter, and thus fade more. Where fade is non-uniform, fade may cause a vehicle to swerve. Because of this, heavy vehicles often use disproportionately weak brakes on steered wheels, which hurts the stopping distance and causes brakes on non-steered wheels to work harder, worsening fade. An advantage of low-fade brakes such as disc brakes is steered wheels can do more braking without causing brake steer.
Brake fade typically occurs during heavy or sustained braking. Many high-speed vehicles use disc brakes, and many European heavy vehicles use disc brakes. Many U.S. and third-world heavy vehicles use drum brakes due to their lower purchase price. On heavy vehicles, air drag is often small compared to the weight, so the brakes dissipate proportionally more energy than on a typical car or motorcycle. Thus, heavy vehicles often need to use engine compression braking, and slow down so braking energy is dissipated over a longer interval. Recent studies have been performed in the United States to test the stopping distances of both drum brakes and disc brakes using a North American Standard called FMVSS-121. The results showed that when newer compounding of friction materials typically used in disc brakes is applied to drum brakes that there is virtually no difference in stopping distance or brake fade. As the United States changed its FMVSS-121 rules for Class Eight trucks built in 2012 to reduce stopping distances by about 1/3rd there was no recommendation to use either drum or disc brakes in the current law.
Newer drum technologies and turbine cooling devices inside of these drums has also eliminated any edge disc brakes may have had in heavy duty applications. By installing brake turbines inside of a specially configured drum, temperatures are many times cut in half and brake fade is nearly eliminated.
Brake failure is also caused by brake drum thermal expansion in which brake shoe clearance becomes excessive from wear. This was largely remedied in the 1950s by self-adjusting brakes. Maladjustment with wear is still a factor in trucks with drum air brakes. A Canadian survey of randomly stopped heavy trucks found over 10% of trucks using self-adjusting brakes had at least one brake out of adjustment, due either to failure of the self-adjust mechanism or wear beyond the capacity of the self adjuster. Newer brake pistons ("cans") extend stroke from about 65 mm to about 75 mm; since about 30 mm of stroke is used just putting the pads in contact with the drum, the added 10 mm of stroke is over 25% increase in useful stroke. Longer stroke reduces especially wear-related fade, but drum brakes are still fundamentally prone to fade when hot.
After cooling, faded brakes usually perform as well as before, with no visible change to the brake shoes and/or pads. However, if the brakes have been excessively hot for a prolonged period of time, glazing can occur on both of the friction linings of the shoes and pads.
When this happens, the contacting surfaces of the linings will have a smooth, shiny appearance, and will not perform as efficiently to slow the vehicle under braking.
This glazing can be easily removed by either gently using emery paper on them, or by driving the vehicle carefully whilst implementing light use of the brakes for several miles.
An incorrect explanation sometimes given for brake fade is heated brake shoes evaporate to generate gas that separate them from the drum. Such effects are easy to imagine, but physically impossible, due to the large volume of gas that would be required for such an effect. A gas bearing would need gas replenishment as fast as the disc or drum moves, since it has no gas on its surface as it approaches the pad or shoe. Also, disc brakes use much the same materials and operate well with little fade, even when the discs are glowing hot. If brake materials outgassed at drum temperatures, they would also outgas at disc temperatures and would fade substantially. Since discs have little fade, they also demonstrate outgassing is not a source of fade. Some disc brakes are drilled or slotted, but smooth discs show no more fade.
Long dual-tire skid marks on highways, made by trucks with drum brakes, are visible examples of non-linearity between brake response and pedal pressure. Large trucks still use drum brakes because they are economical and fit easily where an equivalent disc brake does not. More recently disc brakes for trucks have been promoted listing features such as no fade, possible because they have no self-assist (self-servo).
Railroads.
Railroads have been using disk brakes on passenger cars for more than 60 years, but coupled with a "Rolokron" anti-lock system to avoid the creation of "flat spots" (or “"square wheels"”) when wheels lock and skid on the rail surface (audible as steady "bang-bang-bang" noise as a train goes by—not to be confused with the bang-bang...bang-bang...bang-bang sound made by wheels rolling over a rail joint). Usually, brake disks are installed in the center of the axle, but in some applications (such as Bombardier Bi Level commuter cars), only one disk is used, mounted on the axle end outside the truck frame. High speed trains (such as the TGV) may use four disks per axle.
Freight cars (and some passenger cars like multiple-unit cars whose traction motors do not yield room on axles to allow the placement of disk brakes) are equipped with "clasp brakes" which directly grab the rolling surface of the wheels (much like the old horse buggy brakes of yesteryear). Such brakes are an external-shoe drum brake; but unlike band brakes and many internal-shoe drum brakes, there is no self-assist/self-servo effect, and so they are far less susceptible to locking than self-assist brakes. Due to high stiffness and relatively low power, these clasp brakes are even less prone to lockup than many disc brakes, and so freight cars using them are not equipped with anti-lock systems.
The first development of modern ceramic brakes was made by British engineers working in the railway industry for TGV applications in 1988. The objective was to reduce weight, the number of brakes per axle, as well as provide stable friction from very high speeds and all temperatures. The result was a carbon-fibre-reinforced ceramic process that is now used in various forms for automotive, railway, and aircraft brake applications.
Controlling fade through driving technique.
Brake fade and rotor warping can be reduced through proper braking technique; when running down a long downgrade that would require braking simply select a lower gear (this is required for many trucks on steep grades in the U.S.). Also, periodic, rather than continuous application of the brakes will allow them to cool between applications. Continuous light application of the brakes can be particularly destructive in both wear and adding heat to the brake system.
Brake modification to reduce fade.
High performance brake components provide enhanced stopping power by improving friction while reducing brake fade. Improved friction is provided by lining materials that have a higher coefficient of friction than standard brake pads, while brake fade is reduced through the use of more expensive binding resins with a higher melting point, along with slotted, drilled, or dimpled discs/rotors that reduce the gaseous boundary layer, in addition to providing enhanced heat dissipation. Heat buildup in brakes can be further addressed by body modifications that direct cold air to the brakes.
The "gaseous boundary layer" is a hot rod mechanics explanation for failing self servo effect of drum brakes because it felt like a brick under the brake pedal when it occurred. To counter this effect, brake shoes were drilled and slotted to vent gas. In spite of that, drum brakes were abandoned for their self-servo effect. Disks do not have that because application force is applied at right angles to the resulting braking force. There is no interaction.
Adherents of gas emission have carried that belief to motorcycles, bicycles and "sports" cars, while all other disk brake users from the same automotive companies have no holes through the faces of their discs, although internal radial air passages are used. Vents to release gas have not been found on railway, aircraft and passenger car brakes because there is no gas to vent. Meanwhile, heavy trucks still use drum brakes because they take up the same space. Railways have never used internal expanding drum brakes because they cause skidding, causing expensive flat spots on steel wheels.
Both disc and drum brakes can be improved by any technique that removes heat from the braking surfaces.
Drum brake fade can be reduced and overall performance enhanced somewhat by an old "hot rodder" technique of drum drilling. A carefully chosen pattern of holes is drilled through the drum working section; drum rotation centrifugally pumps a small amount air through the shoe to drum gap, removing heat; fade caused by water-wet brakes is reduced since the water is centrifugally driven out; and some brake-material dust exits the holes. Brake drum drilling requires careful detailed knowledge of brake drum physics and is an advanced technique probably best left to professionals. There are performance-brake shops that will make the necessary modifications safely.
Brake fade caused by overheating brake fluid (often called Pedal Fade) can also be reduced through the use of thermal barriers that are placed between the brake pad and the brake caliper piston, these reduce the transfer of heat from the pad to the caliper and in turn hydraulic brake fluid. Some high-performance racing calipers already include such brake heat shields made from titanium or ceramic materials. However, it is also possible to purchase aftermarket titanium brake heat shields that will fit an existing brake system to provide protection from brake heat. These inserts are precision cut to cover as much of the pad as possible. Since they are relatively cheap and easy to install, they are popular with racers and track day enthusiasts.
Another technique employed to prevent brake fade is the incorporation of fade stop brake coolers. Like titanium heat shields the brake coolers are designed to slide between the brake pad backing plate and the caliper piston. They are constructed from a high thermal conductivity, high yield strength metal composite which conducts the heat from the interface to a heat sink which is external to the caliper and in the airflow. They have been shown to decrease caliper piston temperatures by over twenty percent and to also significantly decrease the time needed to cool down. Unlike titanium heat shields, however, the brake coolers actually transfer the heat to the surrounding environment and thus keep the pads cooler.
References and sources.
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|
[
{
"math_id": 0,
"text": "e^{\\mu \\theta}"
},
{
"math_id": 1,
"text": "e"
},
{
"math_id": 2,
"text": "\\mu"
},
{
"math_id": 3,
"text": "\\theta"
}
] |
https://en.wikipedia.org/wiki?curid=1519764
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1519880
|
Collectively exhaustive events
|
In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the events 1, 2, 3, 4, 5, and 6 balls of a single outcome are collectively exhaustive, because they encompass the entire range of possible outcomes.
Another way to describe collectively exhaustive events is that their union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if
formula_0
where S is the sample space.
Compare this to the concept of a set of mutually exclusive events. In such a set no more than one event can occur at a given time. (In some forms of mutual exclusion only one event can ever occur.) The set of all possible die rolls is both mutually exclusive and collectively exhaustive (i.e., "MECE"). The events 1 and 6 are mutually exclusive but not collectively exhaustive. The events "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are also collectively exhaustive but not mutually exclusive. In some forms of mutual exclusion only one event can ever occur, whether collectively exhaustive or not. For example, tossing a particular biscuit for a group of several dogs cannot be repeated, no matter which dog snaps it up.
One example of an event that is both collectively exhaustive and mutually exclusive is tossing a coin. The outcome must be either heads or tails, or p (heads or tails) = 1, so the outcomes are collectively exhaustive. When heads occurs, tails can't occur, or p (heads and tails) = 0, so the outcomes are also mutually exclusive.
Another example of events being collectively exhaustive and mutually exclusive at same time are, event "even" (2,4 or 6) and event "odd" (1,3 or 5) in a random experiment of rolling a six-sided die. These both events are mutually exclusive because even and odd outcome can never occur at same time. The union of both "even" and "odd" events give sample space of rolling the die, hence are collectively exhaustive.
History.
The term "exhaustive" has been used in the literature since at least 1914. Here are a few examples:
The following appears as a footnote on page 23 of Couturat's text, "The Algebra of Logic" (1914):
"As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the "principle of exhaustion", inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are "exhaustive (of the universe of discourse)"." (italics added for emphasis)
In Stephen Kleene's discussion of cardinal numbers, in "Introduction to Metamathematics" (1952), he uses the term "mutually exclusive" together with "exhaustive":
"Hence, for any two cardinals M and N, the three relationships M < N, M = N and M > N are 'mutually exclusive', i.e. not more than one of them can hold. ¶ It does not appear till an advanced stage of the theory . . . whether they are " 'exhaustive' ", i.e. whether at least one of the three must hold". (italics added for emphasis, Kleene 1952:11; original has double bars over the symbols M and N).
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "A \\cup B = S"
}
] |
https://en.wikipedia.org/wiki?curid=1519880
|
15199671
|
Common-mode signal
|
Voltage common to both input terminals of an electrical device
In electrical engineering, a common-mode signal is the identical component of voltage present at both input terminals of an electrical device. In telecommunication, the common-mode signal on a transmission line is also known as longitudinal voltage.
Common-mode interference (CMI) is a type of common-mode signal. Common-mode interference is interference that appears on both signal leads, or coherent interference that affects two or more elements of a network.
In most electrical circuits, desired signals are transferred by a differential voltage between two conductors. If the voltages on these conductors are "U"1 and "U"2, the common-mode signal is the average of the voltages:
formula_0
When referenced to the local common or ground, a common-mode signal appears on both lines of a two-wire cable, in phase and with equal amplitudes. Technically, a common-mode voltage is one-half the vector sum of the voltages from each conductor of a balanced circuit to local ground or common. Such signals can arise from one or more of the following sources:
Noise induced into a cable, or transmitted from a cable, usually occurs in the common mode, as the same signal tends to be picked up by both conductors in a two-wire cable. Likewise, RF noise transmitted from a cable tends to emanate from both conductors. Elimination of common-mode signals on cables entering or leaving electronic equipment is important to ensure electromagnetic compatibility. Unless the intention is to transmit or receive radio signals, an electronic designer generally designs electronic circuits to minimise or eliminate common-mode effects.
Methods of eliminating common-mode signals.
Common-mode filtering may also be used to prevent egress of noise for electromagnetic compatibility purposes:
Common-mode rejection ratio is a measure of how well a circuit eliminates common-mode interference.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Citation/styles.css"/>
|
[
{
"math_id": 0,
"text": "U_\\text{cm} = \\frac{U_1 + U_2}{2}"
}
] |
https://en.wikipedia.org/wiki?curid=15199671
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15200706
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Mathieu group M24
|
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group "M24" is a sporadic simple group of order
210 · 33 · 5 · 7 · 11 · 23 = 244823040
≈ 2×108.
History and properties.
"M24" is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial.
The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups. It was difficult to see that M24 actually existed, that its generators did not just generate the alternating group A24. The matter was clarified when Ernst Witt constructed M24 as the automorphism (symmetry) group of an S(5,8,24) Steiner system W24 (the Witt design). M24 is the group of permutations that map every block in this design to some other block. The subgroups M23 and M22 then are easily defined to be the stabilizers of a single point and a pair of points respectively.
Construction as a permutation group.
M24 is the subgroup of S24 that is generated by the three permutations:
M24 can also be generated by two permutations:
M24 from PSL(3,4).
M24 can be built starting from PSL(3,4), the projective special linear group of 3-dimensional space over the finite field with 4 elements . This group, sometimes called M21, acts on the projective plane over the field F4, an S(2,5,21) system called W21. Its 21 blocks are called lines. Any 2 lines intersect at one point.
M21 has 168 simple subgroups of order 360 and 360 simple subgroups of order 168. In the larger projective general linear group PGL(3,4) both sets of subgroups form single conjugacy classes, but in M21 both sets split into 3 conjugacy classes. The subgroups respectively have orbits of 6, called hyperovals, and orbits of 7, called Fano subplanes. These sets allow creation of new blocks for larger Steiner systems. M21 is normal in PGL(3,4), of index 3. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F4 (the field automorphism). PGL(3,4) can therefore be extended to the group PΓL(3,4) of projective semilinear transformations, which is a split extension of M21 by the symmetric group S3. PΓL(3,4) has an embedding as a maximal subgroup of M24.
A hyperoval has no 3 points that are collinear. A Fano subplane likewise satisfies suitable uniqueness conditions.
To W21 append 3 new points and let the automorphisms in PΓL(3,4) but not in M21 permute these new points. An S(3,6,22) system W22 is formed by appending just one new point to each of the 21 lines and new blocks are 56 hyperovals conjugate under M21.
An S(5,8,24) system would have 759 blocks, or octads. Append all 3 new points to each line of W21, a different new point to the Fano subplanes in each of the sets of 120, and append appropriate pairs of new points to all the hyperovals. That accounts for all but 210 of the octads. Those remaining octads are subsets of W21 and are symmetric differences of pairs of lines. There are many possible ways to expand the group PΓL(3,4) to M24.
Automorphism group of the Golay code.
The group M24 also is the permutation automorphism group of the binary Golay code "W", i.e., the group of permutations of coordinates mapping "W" to itself. Codewords correspond in a natural way to subsets of a set of 24 objects. (In coding theory the term "binary Golay code" often refers to a shorter related length 23 code, and the length 24 code used here is called the "extended binary Golay code".) Those subsets corresponding to codewords with 8 or 12 coordinates equal to 1 are called octads or dodecads respectively. The octads are the blocks of an S(5,8,24) Steiner system and the binary Golay code is the vector space over field F2 spanned by the octads of the Steiner system.
The simple subgroups M23, M22, M12, and M11 can be defined as subgroups of M24, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate.
There is a natural connection between the Mathieu groups and the larger Conway groups, because the binary Golay code and the Leech lattice both lie in spaces of dimension 24. The Conway groups in turn are found in the Monster group. Robert Griess refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.
Polyhedral symmetries.
M24 can be constructed starting from the symmetries of the Klein quartic (the symmetries of a tessellation of the genus three surface), which is PSL(2,7), which can be augmented by an additional permutation. This permutation can be described by starting with the tiling of the Klein quartic by 56 triangles (with 24 vertices – the 24 points on which the group acts), then forming squares of out some of the 2 triangles, and octagons out of 6 triangles, with the added permutation being "interchange the two endpoints of those edges of the original triangular tiling which bisect the squares and octagons". This can be visualized by coloring the triangles – the corresponding tiling is topologically but not geometrically the , and can be (polyhedrally) immersed in Euclidean 3-space as the small cubicuboctahedron (which also has 24 vertices).
Applications.
The theory of umbral moonshine is a partly conjectural relationship between K3 surfaces and M24.
The Conway group Co1, the Fischer group Fi24, and the Janko group J4 each have maximal subgroups that are an extension of the Mathieu group M24 by a group 211. (These extensions are not all the same.)
Representations.
calculated the complex character table of M24.
The Mathieu group M24 has a 5-fold transitive permutation representation on 24 points. The corresponding linear representation over the complex numbers is the sum of the trivial representation and a 23-dimensional irreducible representation.
M24 has two rank 3 permutation representations: one on the 276 = 1+44+231 pairs of points (or duads) with stabilizer M22.2, and one on the 1288 = 1+495+792 duads, with stabilizer M12.2.
The quotient of the 24-dimensional linear representation of the permutation representation by its 1-dimensional fixed subspace gives a 23-dimensional representation, which is irreducible over any field of characteristic not 2 or 3, and gives the smallest faithful representation over such fields.
Reducing the 24-dimensional representation mod 2 gives an action on F. This has invariant subspaces of dimension 1, 12 (the Golay code), and 23. The subquotients give two irreducible representations of dimension 11 over the field with 2 elements.
Maximal subgroups.
found the 9 conjugacy classes of maximal subgroups of "M24". gave a short proof of the result, describing the 9 classes in terms of combinatorial data on the 24 points: the subgroups fix a point, duad, octad, duum, sextet, triad, trio, projective line, or octern, as described below. gave the character tables of M24 (originally calculated by ) and the 8 maximal subgroups that were known at the time.
M24 contains non-abelian simple subgroups of 13 isomorphism types: five classes of A5, four classes of PSL(3,2), two classes of A6, two classes of PSL(2,11), one class each of A7, PSL(2,23), M11, PSL(3,4), A8, M12, M22, M23, and M24. A6 is also noted below as a subquotient in the sextet subgroup.
The Mathieu group acts on the 2048 = 1+759+1288 points of the Golay code modulo the fixed space with 3 orbits, and on the 4096 = 1+24+276+2024+1771 points of the cocode with 5 orbits, and the subgroups fixing a non-trivial point of the code or cocode give 6 of the 9 classes of maximal subgroups.
The 9 classes of maximal subgroups are as follows:
Point subgroup.
The subgroup fixing a point is M23, order 10200960.
Duad subgroup.
A duad is a pair of points. The subgroup fixing a duad is
M22:2, order 887040, with orbits of 2 and 22.
Octad subgroup.
The subgroup fixing one of the 759 (= 3·11·23) octads of the Golay code or Steiner system is the octad group
24:A8, order 322560, with orbits of size 8 and 16. The linear group GL(4,2) has an exceptional isomorphism to the alternating group A8. The pointwise stabilizer O of an octad is an abelian group of order 16, exponent 2, each of whose involutions moves all 16 points outside the octad. The stabilizer of the octad is a split extension of O by A8.
Duum subgroup.
A duum is a pair of complementary dodecads (12 point sets) in the Golay code. The subgroup fixing a duad is
M12:2, order 190080, transitive and imprimitive. This subgroup was discovered by Frobenius.
The subgroup M12 acts differently on 2 sets of 12, reflecting the outer automorphism of M12.
Sextet subgroup.
26:(3.S6), order 138240: "sextet group"
Consider a tetrad, any set of 4 points in the Steiner system W24. An octad is determined by choice of a fifth point from the remaining 20. There are 5 octads possible. Hence any tetrad determines a partition into 6 tetrads, called a sextet, whose stabilizer in M24 is called a sextet group.
The total number of tetrads is 24*23*22*21/4! = 23*22*21. Dividing that by 6 gives the number of sextets, 23*11*7 = 1771. Furthermore, a sextet group is a subgroup of a wreath product of order 6!*(4!)6, whose only prime divisors are 2, 3, and 5. Now we know the prime divisors of |M24|. Further analysis would determine the order of the sextet group and hence |M24|.
It is convenient to arrange the 24 points into a 6-by-4 array:
A E I M Q U
B F J N R V
C G K O S W
D H L P T X
Moreover, it is convenient to use the elements of the field F4 to number the rows: 0, 1, u, u2.
The sextet group has a normal abelian subgroup H of order 64, isomorphic to the hexacode, a vector space of length 6 and dimension 3 over F4. A non-zero element in H does double transpositions within 4 or 6 of the columns. Its action can be thought of as addition of vector co-ordinates to row numbers.
The sextet group is a split extension of H by a group 3.S6 (a stem extension). Here is an instance within the Mathieu groups where a simple group (A6) is a subquotient, not a subgroup. 3.S6 is the normalizer in M24 of the subgroup generated by r=(BCD)(FGH)(JKL)(NOP)(RST)(VWX), which can be thought of as a multiplication of row numbers by u2. The subgroup 3.A6 is the centralizer of ⟨r⟩. Generators of 3.A6 are:
(AEI)(BFJ)(CGK)(DHL)(RTS)(VWX) (rotating first 3 columns)
(AQ)(BS)(CT)(DR)(EU)(FX)(GV)(HW)
(AUEIQ)(BXGKT)(CVHLR)(DWFJS) (product of preceding two)
(FGH)(JLK)(MQU)(NRV)(OSW)(PTX) (rotating last 3 columns).
An odd permutation of columns, say (CD)(GH)(KL)(OP)(QU)(RV)(SX)(TW), then generates 3.S6.
The group 3.A6 is isomorphic to a subgroup of SL(3,4) whose image in PSL(3,4) has been noted above as the hyperoval group.
The applet Moggie has a function that displays sextets in color.
Triad subgroup.
A triad is a set of 3 points. The subgroup fixing a triad is
PSL(3,4):S3, order 120960, with orbits of size 3 and 21.
Trio subgroup.
A trio is a set of 3 disjoint octads of the Golay code. The subgroup fixing a trio is the trio group
26:(PSL(2,7) x S3), order 64512, transitive and imprimitive.
Projective line subgroup.
The subgroup fixing a projective line structure on the 24 points is
PSL(2,23), order 6072, whose action is doubly transitive. This subgroup was observed by Mathieu.
Octern subgroup.
An octern is a certain partition of the 24 points into 8 blocks of 3. The subgroup fixing an octern is the
octern group isomorphic to PSL(2,7), of order 168, simple, transitive and imprimitive.
It was the last maximal subgroup of M24 to be found.
Conjugacy classes.
There are 26 conjugacy classes. The cycle shapes are all balanced in the sense that they remain invariant under changing length "k" cycles to length "N"/"k" cycles for some integer "N" depending on the conjugacy class.
References.
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|
[
{
"math_id": 0,
"text": "(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)"
},
{
"math_id": 1,
"text": "(3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16)"
},
{
"math_id": 2,
"text": "(1,24)(2,23)(3,12)(4,16)(5,18)(6,10)(7,20)(8,14)(9,21)(11,17)(13,22)(15,19)"
},
{
"math_id": 3,
"text": "(1,16,8,23,13,14,5)(2,7,11,19,20,24,12)(3,4,17,9,22,21,15)"
},
{
"math_id": 4,
"text": "(1,24)(2,21)(3,10)(4,22)(5,9)(6,23)(7,8)(11,18)(12,20)(13,14)(15,19)(16,17)."
}
] |
https://en.wikipedia.org/wiki?curid=15200706
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1520238
|
Embodied energy
|
Sum of all the energy required to produce any goods or services
Embodied energy is the sum of all the energy required to produce any goods or services, considered as if that energy were incorporated or 'embodied' in the product itself. The concept can be useful in determining the effectiveness of energy-producing or energy saving devices, or the "real" replacement cost of a building, and, because energy-inputs usually entail greenhouse gas emissions, in deciding whether a product contributes to or mitigates global warming. One fundamental purpose for measuring this quantity is to compare the amount of energy produced or saved by the product in question to the amount of energy consumed in producing it.
Embodied energy is an accounting method which aims to find the sum total of the energy necessary for an entire product lifecycle. Determining what constitutes this lifecycle includes assessing the relevance and extent of energy into raw material extraction, transport, manufacture, assembly, installation, disassembly, deconstruction and/or decomposition as well as human and secondary resources.
History.
The history of constructing a system of accounts which records the energy flows through an environment can be traced back to the origins of accounting itself. As a distinct method, it is often associated with the Physiocrat's "substance" theory of value, and later the agricultural energetics of Sergei Podolinsky, a Russian physician, and the ecological energetics of Vladmir Stanchinsky.
The main methods of embodied energy accounting as they are used today grew out of Wassily Leontief's input-output model and are called "Input-Output Embodied Energy analysis". Leontief's input-output model was in turn an adaptation of the neo-classical theory of general equilibrium with application to "the empirical study of the quantitative interdependence between interrelated economic activities". According to Tennenbaum Leontief's Input-Output method was adapted to embodied energy analysis by Hannon to describe ecosystem energy flows. Hannon's adaptation tabulated the total direct and indirect energy requirements (the "energy intensity") for each output made by the system. The total amount of energies, direct and indirect, for the entire amount of production was called the "embodied energy".
Methodologies.
Embodied energy analysis is interested in what energy goes to supporting a consumer, and so all energy depreciation is assigned to the final demand of consumer. Different methodologies use different scales of data to calculate energy embodied in products and services of nature and human civilization. International consensus on the appropriateness of data scales and methodologies is pending. This difficulty can give a wide range in embodied energy values for any given material. In the absence of a comprehensive global embodied energy public dynamic database, embodied energy calculations may omit important data on, for example, the rural road/highway construction and maintenance needed to move a product, marketing, advertising, catering services, non-human services and the like. Such omissions can be a source of significant methodological error in embodied energy estimations. Without an estimation and declaration of the embodied energy error, it is difficult to calibrate the , and so the value of any given material, process or service to environmental and economic processes.
Standards.
The SBTool, UK Code for Sustainable Homes was, and USA LEED still is, a method in which the embodied energy of a product or material is rated, along with other factors, to assess a building's environmental impact. Embodied energy is a concept for which scientists have not yet agreed absolute universal values because there are many variables to take into account, but most agree that products can be compared to each other to see which has more and which has less embodied energy. Comparative lists (for an example, see the University of Bath "Embodied Energy & Carbon Material Inventory") contain average absolute values, and explain the factors which have been taken into account when compiling the lists.
Typical embodied energy units used are MJ/kg (megajoules of energy needed to make a kilogram of product), tCO2 (tonnes of carbon dioxide created by the energy needed to make a kilogram of product). Converting MJ to tCO2 is not straightforward because different types of energy (oil, wind, solar, nuclear and so on) emit different amounts of carbon dioxide, so the actual amount of carbon dioxide emitted when a product is made will be dependent on the type of energy used in the manufacturing process. For example, the Australian Government gives a global average of 0.098 tCO2 = 1 GJ. This is the same as 1 MJ = 0.098 kgCO2 = 98 gCO2 or 1 kgCO2 = 10.204 MJ.
Related methodologies.
In the 2000s drought conditions in Australia have generated interest in the application of embodied energy analysis methods to water. This has led to the use of the concept of embodied water.
Data.
A range of databases exist for quantifying the embodied energy of goods and services, including materials and products. These are based on a range of different data sources, with variations in geographic and temporal relevance and system boundary completeness. One such database is the Environmental Performance in Construction (EPiC) Database developed at The University of Melbourne, which includes embodied energy data for over 250 mainly construction materials. This database also includes values for embodied water and greenhouse gas emissions.
The main reason for differences in embodied energy data between databases is due to the source of data and methodology used in their compilation. Bottom-up 'process' data is typically sourced from product manufacturers and suppliers. While this data is generally more reliable and specific to particular products, the methodology used to collect process data typically results in much of the embodied energy of a product being excluded, mainly due to the time, costs and complexity of data collection. Top-down environmentally-extended input-output (EEIO) data, based on national statistics can be used to fill these data gaps. While EEIO analysis of products can be useful on its own for initial scoping of embodied energy, it is generally much less reliable than process data and rarely relevant for a specific product or material. Hence, hybrid methods for quantifying embodied energy have been developed, using available process data and filling any data gaps with EEIO data. Databases that rely on this hybrid approach, such as The University of Melbourne's EPiC Database, provide a more comprehensive assessment of the embodied energy of products and materials.
In common materials.
Selected data from the Inventory of Carbon and Energy ('ICE') prepared by the University of Bath (UK)
In transportation.
Theoretically, embodied energy stands for the energy used to extract materials from mines, manufacture vehicles, assemble, transport, maintain, and transform them to transport energy, and ultimately recycle these vehicles. Besides, the energy needed to build and maintain transport networks, whether road or rail, should be taken into account as well. The process to be implemented is so complex that no one dares to put forward a figure.
According to the , in the field of transportation, "it is striking to note that we consume more embodied energy in our transportation expenditures than direct energy", and "we consume less energy to move around in our personal vehicles than we consume the energy we need to produce, sell and transport the cars, trains or buses we use".
Jean-Marc Jancovici advocates a carbon footprint analysis of any transportation infrastructure project, prior to its construction.
In automobiles.
Manufacturing.
According to Volkswagen, the embodied energy contents of a Golf A3 with a petrol engine amounts to 18 000 kWh (i.e. 12% of 545 GJ as shown in the report). A Golf A4 (equipped with a turbocharged direct injection) will show an embodied energy amounting to 22 000 kWh (i.e. 15% of 545 GJ as shown in the report). According to the French energy and environment agency ADEME a motor car has an embodied energy contents of 20 800 kWh whereas an electric vehicle shows an embodied energy contents amounting to 34 700 kWh.
An electric car has a higher embodied energy than a combustion engine one, owing to the battery and electronics. According to Science & Vie, the embodied energy of batteries is so high that rechargeable hybrid cars constitute the most appropriate solution, with their batteries smaller than those of an all-electric car.
Fuel.
As regards energy itself, the factor energy returned on energy invested (EROEI) of fuel can be estimated at 8, which means that to some amount of useful energy provided by fuel should be added 1/7 of that amount in embodied energy of the fuel. In other words, the fuel consumption should be augmented by 14.3% due to the fuel EROEI.
According to some authors, to produce 6 liters of petrol requires 42 kWh of embodied energy (which corresponds to approximately 4.2 liters of diesel in terms of energy content).
Road construction.
We have to work here with figures, which prove still more difficult to obtain. In the case of road construction, the embodied energy would amount to 1/18 of the fuel consumption (i.e. 6%).
Other figures available.
Treloar, "et al." have estimated the embodied energy in an average automobile in Australia as 0.27 terajoules (i.e. 75 000 kWh) as one component in an overall analysis of the energy involved in road transportation.
In buildings.
Although most of the focus for improving energy efficiency in buildings has been on their operational emissions, it is estimated that about 30% of all energy consumed throughout the lifetime of a building can be in its embodied energy (this percentage varies based on factors such as age of building, climate, and materials). In the past, this percentage was much lower, but as much focus has been placed on reducing operational emissions (such as efficiency improvements in heating and cooling systems), the embodied energy contribution has come much more into play. Examples of embodied energy include: the energy used to extract raw resources, process materials, assemble product components, transport between each step, construction, maintenance and repair, deconstruction and disposal. As such, it is important to employ a whole-life carbon accounting framework in analyzing the carbon emissions in buildings. Studies have also shown the need to go beyond the building scale and to take into account the energy associated with mobility of occupants and the embodied energy of infrastructure requirements, in order to avoid shifting energy needs across scales of the built environment.
In the energy field.
EROEI.
EROEI (Energy Returned On Energy Invested) provides a basis for evaluating the embodied energy due to energy.
Final energy has to be multiplied by formula_0 in order to get the embodied energy.
Given an EROEI amounting to eight e.g., a seventh of the final energy corresponds to the embodied energy.
Not only that, for really obtaining overall embodied energy, embodied energy due to the construction and maintenance of power plants should be taken into account, too. Here, figures are badly needed.
Electricity.
In the BP "Statistical Review of World Energy June 2018", toe are converted into kWh "on the basis of thermal equivalence assuming 38% conversion efficiency in a modern thermal power station".
In France, by convention, the ratio between primary energy and final energy in electricity amounts to 2.58, corresponding to an efficiency of 38.8%.
In Germany, on the contrary, because of the swift development of the renewable energies, the ratio between primary energy and final energy in electricity amounts to only 1.8, corresponding to an efficiency of 55.5%.
According to "EcoPassenger", overall electricity efficiency would amount to 34% in the UK, 36% in Germany and 29% in France.
Data processing.
According to association négaWatt, embodied energy related to digital services amounted to 3.5 TWh/a for networks and 10.0 TWh/a for data centres (half for the servers per se, i. e. 5 TWh/a, and the other half for the buildings in which they are housed, i. e. 5 TWh/a), figures valid in France, in 2015. The organization is optimistic about the evolution of the energy consumption in the digital field, underlining the technical progress being made. "The Shift Project", chaired by Jean-Marc Jancovici, contradicts the optimistic vision of the association négaWatt, and notes that the digital energy footprint is growing at 9% per year.
See also.
<templatestyles src="Div col/styles.css"/>
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\frac {\\hbox{1}} {\\hbox{EROEI-1}}"
}
] |
https://en.wikipedia.org/wiki?curid=1520238
|
15202655
|
Central series
|
Normal series of subgroups which indicate almost-commutativity
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal.
This article uses the language of group theory; analogous terms are used for Lie algebras.
A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable.
Definition.
A central series is a sequence of subgroups
formula_0
such that the successive quotients are central; that is, formula_1, where formula_2 denotes the commutator subgroup generated by all elements of the form formula_3, with "g" in "G" and "h" in "H". Since formula_4, the subgroup formula_5 is normal in "G" for each "i". Thus, we can rephrase the 'central' condition above as: formula_6 is normal in "G" and formula_7 is central in formula_8 for each "i". As a consequence, formula_7 is abelian for each "i".
A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action (more prosaically, a basis in which each element is represented by a strictly upper triangular matrix); compare Engel's theorem.
A group need not have a central series. In fact, a group has a central series if and only if it is a nilpotent group. If a group has a central series, then there are two central series whose terms are extremal in certain senses. Since "A"0 = {1}, the center "Z"("G") satisfies "A"1 ≤ "Z"("G"). Therefore, the maximal choice for "A"1 is "A"1 = "Z"("G"). Continuing in this way to choose the largest possible "A""i" + 1 given "Ai" produces what is called the upper central series. Dually, since "An" = "G", the commutator subgroup ["G", "G"] satisfies ["G", "G"] = ["G", "An"] ≤ "A""n" − 1. Therefore, the minimal choice for "A""n" − 1 is ["G", "G"]. Continuing to choose "Ai" minimally given "A""i" + 1 such that ["G", "A""i" + 1] ≤ "Ai" produces what is called the lower central series. These series can be constructed for any group, and if a group has a central series (is a nilpotent group), these procedures will yield central series.
Lower central series.
The lower central series (or descending central series) of a group "G" is the descending series of subgroups
"G" = "G"1 ⊵ "G"2 ⊵ ⋯ ⊵ "Gn" ⊵ ⋯,
where, for each "n",
formula_9,
the subgroup of "G" generated by all commutators formula_10 with formula_11 and formula_12. Thus, formula_13, the derived subgroup of "G", while formula_14, etc. The lower central series is often denoted formula_15. We say the series "terminates" or "stablizes" when formula_16, and the smallest such "n" is the "length" of the series.
This should not be confused with the derived series, whose terms are
formula_17,
not formula_18. The two series are related by formula_19. For instance, the symmetric group "S3" is solvable of class 2: the derived series is "S3" ⊵ {"e", (1 2 3), (1 3 2)} ⊵ {"e"}. But it is not nilpotent: its lower central series "S3" ⊵ {"e", (1 2 3), (1 3 2)} does not terminate in {"e"}. A nilpotent group is a solvable group, and its derived length is logarithmic in its nilpotency class .
For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal "λ", define
formula_20.
If formula_21 for some ordinal "λ", then "G" is said to be a hypocentral group. For every ordinal "λ", there is a group "G" such that formula_21, but formula_22 for all formula_23, .
If formula_24 is the first infinite ordinal, then formula_25 is the smallest normal subgroup of "G" such that the quotient is residually nilpotent, that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group . In the field of combinatorial group theory, it is an important and early result that free groups are residually nilpotent. In fact the quotients of the lower central series are free abelian groups with a natural basis defined by basic commutators, .
If formula_26 for some finite "n", then formula_25 is the smallest normal subgroup of "G" with nilpotent quotient, and formula_25 is called the nilpotent residual of "G". This is always the case for a finite group, and defines the formula_27 term in the lower Fitting series for "G".
If formula_28 for all finite "n", then formula_29 is not nilpotent, but it is residually nilpotent.
There is no general term for the intersection of all terms of the transfinite lower central series, analogous to the hypercenter (below).
Upper central series.
The upper central series (or ascending central series) of a group "G" is the sequence of subgroups
formula_30
where each successive group is defined by:
formula_31
and is called the "i"th center of "G" (respectively, second center, third center, etc.). In this case, formula_32 is the center of "G", and for each successive group, the factor group formula_33 is the center of formula_34, and is called an upper central series quotient. Again, we say the series terminates if it stabilizes into a chain of equalities, and its length is the number of distinct groups in it.
For infinite groups, one can continue the upper central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal "λ", define
formula_35
The limit of this process (the union of the higher centers) is called the hypercenter of the group.
If the transfinite upper central series stabilizes at the whole group, then the group is called hypercentral. Hypercentral groups enjoy many properties of nilpotent groups, such as the normalizer condition (the normalizer of a proper subgroup properly contains the subgroup), elements of coprime order commute, and periodic hypercentral groups are the direct sum of their Sylow "p"-subgroups . For every ordinal "λ" there is a group "G" with "Z""λ"("G") = "G", but "Z""α"("G") ≠ "G" for "α" < "λ", and .
Connection between lower and upper central series.
There are various connections between the lower central series (LCS) and upper central series (UCS) , particularly for nilpotent groups.
For a nilpotent group, the lengths of the LCS and the UCS agree, and this length is called the nilpotency class of the group. However, the LCS and UCS of a nilpotent group may not necessarily have the same terms. For example, while the UCS and LCS agree for the cyclic group "C2" ⊵ {"e"} and quaternion group "Q8" ⊵ {1, −1} ⊵ {1}, the UCS and LCS of their direct product "C2" × "Q8" do not agree: its LCS is "C2" × "Q8" ⊵ {"e"} × {−1, 1} ⊵ {"e"} × {1}, while its UCS is "C2" × "Q8" ⊵ "C2" × {−1, 1} ⊵ {"e"} × {1}.
A group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is the trivial subgroup), if and only if the UCS terminates at the first step (the center is the entire group).
By contrast, the LCS terminates at the zeroth step if and only if the group is perfect (the commutator is the entire group), while the UCS terminates at the zeroth step if and only if the group is centerless (trivial center), which are distinct concepts. For a perfect group, the UCS always stabilizes by the first step (Grün's lemma). However, a centerless group may have a very long LCS: a free group on two or more generators is centerless, but its LCS does not stabilize until the first infinite ordinal. This shows that the lengths of the LCS and UCS need not agree in general.
Refined central series.
In the study of "p"-groups (which are always nilpotent), it is often important to use longer central series. An important class of such central series are the exponent-"p" central series; that is, a central series whose quotients are elementary abelian groups, or what is the same, have exponent "p". There is a unique most quickly descending such series, the lower exponent-"p" central series λ defined by:
formula_36, and
formula_37.
The second term, formula_38, is equal to formula_39, the Frattini subgroup. The lower exponent-"p" central series is sometimes simply called the "p"-central series.
There is a unique most quickly ascending such series, the upper exponent-"p" central series S defined by:
S0("G") = 1
S"n"+1("G")/S"n"("G") = Ω(Z("G"/S"n"("G")))
where Ω("Z"("H")) denotes the subgroup generated by (and equal to) the set of central elements of "H" of order dividing "p". The first term, S1("G"), is the subgroup generated by the minimal normal subgroups and so is equal to the socle of "G". For this reason the upper exponent-"p" central series is sometimes known as the socle series or even the Loewy series, though the latter is usually used to indicate a descending series.
Sometimes other refinements of the central series are useful, such as the Jennings series "κ" defined by:
κ1("G") = "G", and
κ"n" + 1("G") = ["G", κ"n"("G")] (κ"i"("G"))"p", where "i" is the smallest integer larger than or equal to "n"/"p".
The Jennings series is named after Stephen Arthur Jennings who used the series to describe the Loewy series of the modular group ring of a "p"-group.
|
[
{
"math_id": 0,
"text": "\\{1\\} = A_0 \\triangleleft A_1 \\triangleleft \\dots \\triangleleft A_n = G"
},
{
"math_id": 1,
"text": "[G, A_{i+1}] \\le A_{i}"
},
{
"math_id": 2,
"text": "[G,H]"
},
{
"math_id": 3,
"text": "[g,h] = g^{-1}h^{-1}gh"
},
{
"math_id": 4,
"text": "[G,A_{i+1}] \\le A_i \\le A_{i+1}"
},
{
"math_id": 5,
"text": "A_{i+1}"
},
{
"math_id": 6,
"text": "A_{i}"
},
{
"math_id": 7,
"text": "A_{i+1}/A_{i}"
},
{
"math_id": 8,
"text": "G/A_{i}"
},
{
"math_id": 9,
"text": "G_{n+1} = [G_n,G]"
},
{
"math_id": 10,
"text": "[x,y]"
},
{
"math_id": 11,
"text": "x \\in G_n"
},
{
"math_id": 12,
"text": "y \\in G"
},
{
"math_id": 13,
"text": "G_2 = [G,G] = G^{(1)}"
},
{
"math_id": 14,
"text": "G_3 = [[G,G],G]"
},
{
"math_id": 15,
"text": "\\gamma_n(G) = G_n"
},
{
"math_id": 16,
"text": "G_n=G_{n+1}=G_{n+2}=\\cdots"
},
{
"math_id": 17,
"text": "G^{(n)} := [G^{(n-1)},G^{(n-1)}]"
},
{
"math_id": 18,
"text": "G_{n} = [G_{n-1},G]"
},
{
"math_id": 19,
"text": "G^{(n)} \\le G_n"
},
{
"math_id": 20,
"text": "G_{\\lambda} =\\bigcap \\{ G_{\\alpha} : \\alpha < \\lambda\\}"
},
{
"math_id": 21,
"text": "G_{\\lambda} = 1"
},
{
"math_id": 22,
"text": "G_{\\alpha} \\ne 1"
},
{
"math_id": 23,
"text": "\\alpha < \\lambda"
},
{
"math_id": 24,
"text": "\\omega"
},
{
"math_id": 25,
"text": "G_{\\omega}"
},
{
"math_id": 26,
"text": "G_{\\omega}=G_n"
},
{
"math_id": 27,
"text": "F_1(G)"
},
{
"math_id": 28,
"text": "G_{\\omega}\\ne G_n"
},
{
"math_id": 29,
"text": "G/G_{\\omega}"
},
{
"math_id": 30,
"text": "1 = Z_0 \\triangleleft Z_1 \\triangleleft \\cdots \\triangleleft Z_i \\triangleleft \\cdots,"
},
{
"math_id": 31,
"text": "Z_{i+1} = \\{x\\in G \\mid \\forall y\\in G:[x,y] \\in Z_i \\}"
},
{
"math_id": 32,
"text": "Z_1"
},
{
"math_id": 33,
"text": "Z_{i+ 1}/Z_i"
},
{
"math_id": 34,
"text": "G/Z_i"
},
{
"math_id": 35,
"text": "Z_\\lambda(G) = \\bigcup_{\\alpha < \\lambda} Z_\\alpha(G)."
},
{
"math_id": 36,
"text": "\\lambda_1(G) = G"
},
{
"math_id": 37,
"text": "\\lambda_{n+1}(G) = [G, \\lambda_n(G)] (\\lambda_n(G))^p"
},
{
"math_id": 38,
"text": "\\lambda_2(G)"
},
{
"math_id": 39,
"text": "[G, G]G^p = \\Phi(G)"
}
] |
https://en.wikipedia.org/wiki?curid=15202655
|
1520379
|
Nonelementary integral
|
Integrals not expressible in closed-form from elementary functions
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an "elementary function" (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations). A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.
Examples.
Examples of functions with nonelementary antiderivatives include:
Some common non-elementary antiderivative functions are given names, defining so-called special functions, and formulas involving these new functions can express a larger class of non-elementary antiderivatives. The examples above name the corresponding special functions in parentheses.
Properties.
Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can always be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence. However, even if the integrand has a convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral Taylor series.
Even if it is not possible to evaluate an indefinite integral (antiderivative) in elementary terms, one can always approximate a corresponding definite integral by numerical integration. There are also cases where there is no elementary antiderivative, but specific definite integrals (often improper integrals over unbounded intervals) can be evaluated in elementary terms: most famously the Gaussian integral formula_13
The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sqrt{1 - x^4}"
},
{
"math_id": 1,
"text": "\\frac{1}{\\ln x}"
},
{
"math_id": 2,
"text": "e^{-x^2}"
},
{
"math_id": 3,
"text": "\\sin(x^2)"
},
{
"math_id": 4,
"text": "\\cos(x^2)"
},
{
"math_id": 5,
"text": "\\frac{\\sin(x)}{x} = \\operatorname{sinc}(x) "
},
{
"math_id": 6,
"text": "\\frac{e^{-x}}{x}"
},
{
"math_id": 7,
"text": "e^{e^x} \\,"
},
{
"math_id": 8,
"text": "\\ln(\\ln x) \\,"
},
{
"math_id": 9,
"text": "{x^{c-1}}e^{-x}"
},
{
"math_id": 10,
"text": "c = 0,"
},
{
"math_id": 11,
"text": "c = \\tfrac{1}{2},"
},
{
"math_id": 12,
"text": "c = "
},
{
"math_id": 13,
"text": "\\int_{-\\infty}^{\\infty} e^{-x^2} dx = \\sqrt \\pi."
}
] |
https://en.wikipedia.org/wiki?curid=1520379
|
15204099
|
Tweedie distribution
|
Family of probability distributions
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous.
Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.
The Tweedie distributions were named by Bent Jørgensen in after Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1982 when the conference was held.
Around the same, time Bar-Lev and Enis published about the same topic.
Definitions.
The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship.
A random variable "Y" is Tweedie distributed "Twp(μ, σ2)", if formula_0 with mean formula_1, positive dispersion parameter formula_2 and
formula_3
where formula_4 is called the Tweedie power parameter.
The probability distribution "P""θ","σ"2 on the measurable sets "A", is given by
formula_5
for some σ-finite measure "νλ".
This representation uses the canonical parameter "θ" of an exponential dispersion model and cumulant function
formula_6
where we used formula_7, or equivalently formula_8.
Properties.
Additive exponential dispersion models.
The models just described are in the reproductive form. An exponential dispersion model has always a dual: the additive form. If "Y" is reproductive, then formula_9 with formula_10 is in the additive form ED*("θ","λ"), for Tweedie "Tw*p(μ, λ)". Additive models have the property that the distribution of the sum of independent random variables,
formula_11
for which "Z""i" ~ ED*("θ","λ""i") with fixed "θ" and various "λ" are members of the family of distributions with the same "θ",
formula_12
Reproductive exponential dispersion models.
A second class of exponential dispersion models exists designated by the random variable
formula_13
where "σ"2 = 1/"λ", known as reproductive exponential dispersion models. They have the property that for "n" independent random variables "Y""i" ~ ED("μ","σ"2/"w""i"), with weighting factors "wi" and
formula_14
a weighted average of the variables gives,
formula_15
For reproductive models the weighted average of independent random variables with fixed "μ" and "σ"2 and various values for "wi" is a member of the family of distributions with same "μ" and "σ"2.
The Tweedie exponential dispersion models are both additive and reproductive; we thus have the "duality transformation"
formula_16
Scale invariance.
A third property of the Tweedie models is that they are scale invariant: For a reproductive exponential dispersion model "Twp(μ, σ2)" and any positive constant "c" we have the property of closure under scale transformation,
formula_17
The Tweedie power variance function.
To define the variance function for exponential dispersion models we make use of the mean value mapping, the relationship between the canonical parameter "θ" and the mean "μ". It is defined by the function
formula_18
with cumulative function formula_19.
The variance function "V"("μ") is constructed from the mean value mapping,
formula_20
Here the minus exponent in "τ"−1("μ") denotes an inverse function rather than a reciprocal. The mean and variance of an additive random variable is then E("Z") = "λμ" and var("Z") = "λV"("μ").
Scale invariance implies that the variance function obeys the relationship "V"("μ") = "μ" "p".
The Tweedie deviance.
The unit deviance of a reproductive Tweedie distribution is given by
formula_21
The Tweedie cumulant generating functions.
The properties of exponential dispersion models give us two differential equations. The first relates the mean value mapping and the variance function to each other,
formula_22
The second shows how the mean value mapping is related to the cumulant function,
formula_23
These equations can be solved to obtain the cumulant function for different cases of the Tweedie models. A cumulant generating function (CGF) may then be obtained from the cumulant function. The additive CGF is generally specified by the equation
formula_24
and the reproductive CGF by
formula_25
where "s" is the generating function variable.
For the additive Tweedie models the CGFs take the form,
formula_26
and for the reproductive models,
formula_27
The additive and reproductive Tweedie models are conventionally denoted by the symbols "Tw"*"p"("θ","λ") and "Tw""p"("θ","σ"2), respectively.
The first and second derivatives of the CGFs, with "s" = 0, yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law,
formula_28
The Tweedie convergence theorem.
The Tweedie exponential dispersion models are fundamental in statistical theory consequent to their roles as foci of convergence for a wide range of statistical processes. Jørgensen "et al" proved a theorem that specifies the asymptotic behaviour of variance functions known as the Tweedie convergence theorem. This theorem, in technical terms, is stated thus: The unit variance function is regular of order "p" at zero (or infinity) provided that "V"("μ") ~ "c"0"μ""p" for "μ" as it approaches zero (or infinity) for all real values of "p" and "c"0 > 0. Then for a unit variance function regular of order "p" at either zero or infinity and for
formula_29
for any formula_30, and formula_31 we have
formula_32
as formula_33 or formula_34, respectively, where the convergence is through values of "c" such that "cμ" is in the domain of "θ" and "c""p"−2/"σ"2 is in the domain of "λ". The model must be infinitely divisible as "c"2−"p" approaches infinity.
In nontechnical terms this theorem implies that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behaviour, and the Tweedie distributions become foci of convergence for a wide range of data types.
Related distributions.
The Tweedie distributions include a number of familiar distributions as well as some unusual ones, each being specified by the domain of the index parameter. We have the
For 0 < "p" < 1 no Tweedie model exists. Note that all "stable" distributions mean actually "generated by stable distributions".
Occurrence and applications.
The Tweedie models and Taylor’s power law.
Taylor's law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power-law relationship. For the population count "Y" with mean "μ" and variance var("Y"), Taylor's law is written,
formula_35
where "a" and "p" are both positive constants. Since L. R. Taylor described this law in 1961 there have been many different explanations offered to explain it, ranging from animal behavior, a random walk model, a stochastic birth, death, immigration and emigration model, to a consequence of equilibrium and non-equilibrium statistical mechanics. No consensus exists as to an explanation for this model.
Since Taylor's law is mathematically identical to the variance-to-mean power law that characterizes the Tweedie models, it seemed reasonable to use these models and the Tweedie convergence theorem to explain the observed clustering of animals and plants associated with Taylor's law. The majority of the observed values for the power-law exponent "p" have fallen in the interval (1,2) and so the Tweedie compound Poisson–gamma distribution would seem applicable. Comparison of the empirical distribution function to the theoretical compound Poisson–gamma distribution has provided a means to verify consistency of this hypothesis.
Whereas conventional models for Taylor's law have tended to involve "ad hoc" animal behavioral or population dynamic assumptions, the Tweedie convergence theorem would imply that Taylor's law results from a general mathematical convergence effect much as how the central limit theorem governs the convergence behavior of certain types of random data. Indeed, any mathematical model, approximation or simulation that is designed to yield Taylor's law (on the basis of this theorem) is required to converge to the form of the Tweedie models.
Tweedie convergence and 1/"f" noise.
Pink noise, or 1/"f" noise, refers to a pattern of noise characterized by a power-law relationship between its intensities "S"("f") at different frequencies "f",
formula_36
where the dimensionless exponent "γ" ∈ [0,1]. It is found within a diverse number of natural processes. Many different explanations for 1/"f" noise exist, a widely held hypothesis is based on Self-organized criticality where dynamical systems close to a critical point are thought to manifest scale-invariant spatial and/or temporal behavior.
In this subsection a mathematical connection between 1/"f" noise and the Tweedie variance-to-mean power law will be described. To begin, we first need to introduce self-similar processes: For the sequence of numbers
formula_37
with mean
formula_38
deviations
formula_39
variance
formula_40
and autocorrelation function
formula_41
with lag "k", if the autocorrelation of this sequence has the long range behavior
formula_42
as "k"→∞ and where "L"("k") is a slowly varying function at large values of "k", this sequence is called a self-similar process.
The method of expanding bins can be used to analyze self-similar processes. Consider a set of equal-sized non-overlapping bins that divides the original sequence of "N" elements into groups of "m" equal-sized segments ("N/m" is integer) so that new reproductive sequences, based on the mean values, can be defined:
formula_43
The variance determined from this sequence will scale as the bin size changes such that
formula_44
if and only if the autocorrelation has the limiting form
formula_45
One can also construct a set of corresponding additive sequences
formula_46
based on the expanding bins,
formula_47
Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship
formula_48
Since formula_49 and formula_50 are constants this relationship constitutes a variance-to-mean power law, with "p" = 2 - "d".
The biconditional relationship above between the variance-to-mean power law and power law autocorrelation function, and the Wiener–Khinchin theorem imply that any sequence that exhibits a variance-to-mean power law by the method of expanding bins will also manifest 1/"f" noise, and vice versa. Moreover, the Tweedie convergence theorem, by virtue of its central limit-like effect of generating distributions that manifest variance-to-mean power functions, will also generate processes that manifest 1/"f" noise. The Tweedie convergence theorem thus provides an alternative explanation for the origin of 1/"f" noise, based its central limit-like effect.
Much as the central limit theorem requires certain kinds of random processes to have as a focus of their convergence the Gaussian distribution and thus express white noise, the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express 1/"f" noise.
The Tweedie models and multifractality.
From the properties of self-similar processes, the power-law exponent "p" = 2 - "d" is related to the Hurst exponent "H" and the fractal dimension "D" by
formula_51
A one-dimensional data sequence of self-similar data may demonstrate a variance-to-mean power law with local variations in the value of "p" and hence in the value of "D". When fractal structures manifest local variations in fractal dimension, they are said to be multifractals. Examples of data sequences that exhibit local variations in "p" like this include the eigenvalue deviations of the Gaussian Orthogonal and Unitary Ensembles. The Tweedie compound Poisson–gamma distribution has served to model multifractality based on local variations in the Tweedie exponent "α". Consequently, in conjunction with the variation of "α", the Tweedie convergence theorem can be viewed as having a role in the genesis of such multifractals.
The variation of "α" has been found to obey the asymmetric Laplace distribution in certain cases. This distribution has been shown to be a member of the family of geometric Tweedie models, that manifest as limiting distributions in a convergence theorem for geometric dispersion models.
Regional organ blood flow.
Regional organ blood flow has been traditionally assessed by the injection of radiolabelled polyethylene microspheres into the arterial circulation of animals, of a size that they become entrapped within the microcirculation of organs. The organ to be assessed is then divided into equal-sized cubes and the amount of radiolabel within each cube is evaluated by liquid scintillation counting and recorded. The amount of radioactivity within each cube is taken to reflect the blood flow through that sample at the time of injection. It is possible to evaluate adjacent cubes from an organ in order to additively determine the blood flow through larger regions. Through the work of J B Bassingthwaighte and others an empirical power law has been derived between the relative dispersion of blood flow of tissue samples ("RD" = standard deviation/mean) of mass "m" relative to reference-sized samples:
formula_52
This power law exponent "Ds" has been called a fractal dimension. Bassingthwaighte's power law can be shown to directly relate to the variance-to-mean power law. Regional organ blood flow can thus be modelled by the Tweedie compound Poisson–gamma distribution., In this model tissue sample could be considered to contain a random (Poisson) distributed number of entrapment sites, each with gamma distributed blood flow. Blood flow at this microcirculatory level has been observed to obey a gamma distribution, thus providing support for this hypothesis.
Cancer metastasis.
The "experimental cancer metastasis assay" has some resemblance to the above method to measure regional blood flow. Groups of syngeneic and age matched mice are given intravenous injections of equal-sized aliquots of suspensions of cloned cancer cells and then after a set period of time their lungs are removed and the number of cancer metastases enumerated within each pair of lungs. If other groups of mice are injected with different cancer cell clones then the number of metastases per group will differ in accordance with the metastatic potentials of the clones. It has been long recognized that there can be considerable intraclonal variation in the numbers of metastases per mouse despite the best attempts to keep the experimental conditions within each clonal group uniform. This variation is larger than would be expected on the basis of a Poisson distribution of numbers of metastases per mouse in each clone and when the variance of the number of metastases per mouse was plotted against the corresponding mean a power law was found.
The variance-to-mean power law for metastases was found to also hold for spontaneous murine metastases and for cases series of human metastases.
Since hematogenous metastasis occurs in direct relationship to regional blood flow and videomicroscopic studies indicate that the passage and entrapment of cancer cells within the circulation appears analogous to the microsphere experiments it seemed plausible to propose that the variation in numbers of hematogenous metastases could reflect heterogeneity in regional organ blood flow.
The blood flow model was based on the Tweedie compound Poisson–gamma distribution, a distribution governing a continuous random variable. For that reason in the metastasis model it was assumed that blood flow was governed by that distribution and that the number of regional metastases occurred as a Poisson process for which the intensity was directly proportional to blood flow. This led to the description of the Poisson negative binomial (PNB) distribution as a discrete equivalent to the Tweedie compound Poisson–gamma distribution. The probability generating function for the PNB distribution is
formula_53
The relationship between the mean and variance of the PNB distribution is then
formula_54
which, in the range of many experimental metastasis assays, would be indistinguishable from the variance-to-mean power law. For sparse data, however, this discrete variance-to-mean relationship would behave more like that of a Poisson distribution where the variance equaled the mean.
Genomic structure and evolution.
The local density of Single Nucleotide Polymorphisms (SNPs) within the human genome, as well as that of genes, appears to cluster in accord with the variance-to-mean power law and the Tweedie compound Poisson–gamma distribution. In the case of SNPs their observed density reflects the assessment techniques, the availability of genomic sequences for analysis, and the nucleotide heterozygosity. The first two factors reflect ascertainment errors inherent to the collection methods, the latter factor reflects an intrinsic property of the genome.
In the coalescent model of population genetics each genetic locus has its own unique history. Within the evolution of a population from some species some genetic loci could presumably be traced back to a relatively recent common ancestor whereas other loci might have more ancient genealogies. More ancient genomic segments would have had more time to accumulate SNPs and to experience recombination. R R Hudson has proposed a model where recombination could cause variation in the time to most common recent ancestor for different genomic segments. A high recombination rate could cause a chromosome to contain a large number of small segments with less correlated genealogies.
Assuming a constant background rate of mutation the number of SNPs per genomic segment would accumulate proportionately to the time to the most recent common ancestor. Current population genetic theory would indicate that these times would be gamma distributed, on average. The Tweedie compound Poisson–gamma distribution would suggest a model whereby the SNP map would consist of multiple small genomic segments with the mean number of SNPs per segment would be gamma distributed as per Hudson's model.
The distribution of genes within the human genome also demonstrated a variance-to-mean power law, when the method of expanding bins was used to determine the corresponding variances and means. Similarly the number of genes per enumerative bin was found to obey a Tweedie compound Poisson–gamma distribution. This probability distribution was deemed compatible with two different biological models: the microarrangement model where the number of genes per unit genomic length was determined by the sum of a random number of smaller genomic segments derived by random breakage and reconstruction of protochormosomes. These smaller segments would be assumed to carry on average a gamma distributed number of genes.
In the alternative gene cluster model, genes would be distributed randomly within the protochromosomes. Over large evolutionary timescales there would occur tandem duplication, mutations, insertions, deletions and rearrangements that could affect the genes through a stochastic birth, death and immigration process to yield the Tweedie compound Poisson–gamma distribution.
Both these mechanisms would implicate neutral evolutionary processes that would result in regional clustering of genes.
Random matrix theory.
The Gaussian unitary ensemble (GUE) consists of complex Hermitian matrices that are invariant under unitary transformations whereas the Gaussian orthogonal ensemble (GOE) consists of real symmetric matrices invariant under orthogonal transformations. The ranked eigenvalues "En" from these random matrices obey Wigner's semicircular distribution: For a "N"×"N" matrix the average density for eigenvalues of size "E" will be
formula_55
as "E"→ ∞. Integration of the semicircular rule provides the number of eigenvalues on average less than "E",
formula_56
The ranked eigenvalues can be unfolded, or renormalized, with the equation
formula_57
This removes the trend of the sequence from the fluctuating portion. If we look at the absolute value of the difference between the actual and expected cumulative number of eigenvalues
formula_58
we obtain a sequence of eigenvalue fluctuations which, using the method of expanding bins, reveals a variance-to-mean power law.
The eigenvalue fluctuations of both the GUE and the GOE manifest this power law with the power law exponents ranging between 1 and 2, and they similarly manifest 1/"f" noise spectra. These eigenvalue fluctuations also correspond to the Tweedie compound Poisson–gamma distribution and they exhibit multifractality.
The distribution of prime numbers.
The second Chebyshev function "ψ"("x") is given by,
formula_59
where the summation extends over all prime powers formula_60 not exceeding "x", "x" runs over the positive real numbers, and formula_61 is the von Mangoldt function. The function "ψ"("x") is related to the prime-counting function "π"("x"), and as such provides information with regards to the distribution of prime numbers amongst the real numbers. It is asymptotic to "x", a statement equivalent to the prime number theorem and it can also be shown to be related to the zeros of the Riemann zeta function located on the critical strip "ρ", where the real part of the zeta zero "ρ" is between 0 and 1. Then "ψ" expressed for "x" greater than one can be written:
formula_62
where
formula_63
The Riemann hypothesis states that the nontrivial zeros of the Riemann zeta function all have real part <templatestyles src="Fraction/styles.css" />1⁄2. These zeta function zeros are related to the distribution of prime numbers. Schoenfeld has shown that if the Riemann hypothesis is true then
formula_64
for all formula_65. If we analyze the Chebyshev deviations Δ("n") on the integers "n" using the method of expanding bins and plot the variance versus the mean a variance to mean power law can be demonstrated. Moreover, these deviations correspond to the Tweedie compound Poisson-gamma distribution and they exhibit 1/"f" noise.
Other applications.
Applications of Tweedie distributions include:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Y \\sim \\mathrm{ED}(\\mu, \\sigma^2)"
},
{
"math_id": 1,
"text": "\\mu = \\operatorname{E}(Y)"
},
{
"math_id": 2,
"text": "\\sigma^2"
},
{
"math_id": 3,
"text": "\\operatorname{Var}(Y) = \\sigma^2\\,\\mu^p,"
},
{
"math_id": 4,
"text": "p \\in \\mathbf{R}"
},
{
"math_id": 5,
"text": "P_{\\theta, \\sigma^2}(Y\\in A)=\\int_A \\exp\\left(\\frac{\\theta \\cdot z-\\kappa_p(\\theta)}{\\sigma^2}\\right)\\cdot \\nu_\\lambda\\, (dz),"
},
{
"math_id": 6,
"text": "\\kappa_p(\\theta)=\n\\begin{cases}\n \\frac{\\alpha-1}{\\alpha} \\left(\\frac{\\theta}{\\alpha-1}\\right)^\\alpha, & \\text{for }p\\neq 1,2\\\\\n -\\log(-\\theta), & \\text{for }p=2\\\\\n e^\\theta, & \\text{for }p=1\n\\end{cases}\n"
},
{
"math_id": 7,
"text": "\\alpha = \\frac{p-2}{p-1}"
},
{
"math_id": 8,
"text": "p = \\frac{\\alpha-2}{\\alpha-1}"
},
{
"math_id": 9,
"text": "Z=\\lambda Y"
},
{
"math_id": 10,
"text": "\\lambda = \\frac{1}{\\sigma^2}"
},
{
"math_id": 11,
"text": "Z_+ = Z_1 +\\cdots+ Z_n,"
},
{
"math_id": 12,
"text": "Z_+ \\sim \\operatorname{ED}^*(\\theta,\\lambda_1+\\cdots+\\lambda_n)."
},
{
"math_id": 13,
"text": "Y=Z/\\lambda \\sim \\operatorname{ED}(\\mu,\\sigma^2),"
},
{
"math_id": 14,
"text": "w= \\sum_{i=1}^n w_i,"
},
{
"math_id": 15,
"text": "w^{-1}\\sum_{i=1}^n w_iY_i \\sim \\operatorname{ED}(\\mu,\\sigma^2/w)."
},
{
"math_id": 16,
"text": "Y \\mapsto Z=Y/\\sigma^2."
},
{
"math_id": 17,
"text": "c \\operatorname{Tw}_p(\\mu,\\sigma^2) = \\operatorname{Tw}_p(c\\mu,c^{2-p}\\sigma^2)."
},
{
"math_id": 18,
"text": "\\tau(\\theta) = \\kappa^\\prime(\\theta) = \\mu."
},
{
"math_id": 19,
"text": "\\kappa(\\theta)"
},
{
"math_id": 20,
"text": "V(\\mu) = \\tau^\\prime[\\tau^{-1}(\\mu)]."
},
{
"math_id": 21,
"text": "d(y,\\mu) =\n\\begin{cases}\n (y-\\mu)^2, & \\text{for }p=0\\\\\n 2(y \\log(y/\\mu) + \\mu - y), & \\text{for }p=1\\\\\n 2(\\log(\\mu/y) + y/\\mu - 1), & \\text{for }p=2\\\\\n 2\\left(\\frac{\\max(y,0)^{2-p}}{(1-p)(2-p)}-\\frac{y\\mu^{1-p}}{1-p}+\\frac{\\mu^{2-p}}{2-p}\\right), & \\text{else}\n\\end{cases}\n"
},
{
"math_id": 22,
"text": "\\frac{\\partial \\tau^{-1}(\\mu)}{\\partial \\mu}= \\frac{1}{V(\\mu)}."
},
{
"math_id": 23,
"text": "\\frac{\\partial \\kappa(\\theta)}{\\partial \\theta} = \\tau(\\theta)."
},
{
"math_id": 24,
"text": "K^*(s) = \\log[\\operatorname{E}(e^{sZ})] = \\lambda[\\kappa(\\theta+s)-\\kappa(\\theta)],"
},
{
"math_id": 25,
"text": "K(s) = \\log[\\operatorname{E}(e^{sY})] = \\lambda[\\kappa(\\theta+s/\\lambda)-\\kappa(\\theta)],"
},
{
"math_id": 26,
"text": "K^*_p(s;\\theta,\\lambda) = \\begin{cases}\n\\lambda\\kappa_p(\\theta)[(1+s/\\theta)^\\alpha-1]\n & \\quad p \\ne 1,2, \\\\\n-\\lambda \\log(1+s/\\theta) & \\quad p = 2, \\\\\n\\lambda e^\\theta (e^s -1) & \\quad p = 1,\n\\end{cases}"
},
{
"math_id": 27,
"text": "K_p(s;\\theta,\\lambda) = \\begin{cases}\n\\lambda\\kappa_p(\\theta)\\left \\{ \\left[1+s/(\\theta \\lambda)\\right]^\\alpha-1 \\right \\}\n& \\quad p \\ne 1,2, \\\\[1ex]\n-\\lambda \\log[1+s/(\\theta \\lambda)] & \\quad p = 2, \\\\[1ex]\n\\lambda e^\\theta \\left(e^{s/\\lambda} -1\\right) & \\quad p = 1.\n\\end{cases}\n"
},
{
"math_id": 28,
"text": "\\mathrm{var} (Z)\\propto \\mathrm{E}(Z)^p."
},
{
"math_id": 29,
"text": "p \\notin (0,1),"
},
{
"math_id": 30,
"text": "\\mu>0"
},
{
"math_id": 31,
"text": " \\sigma^2 > 0"
},
{
"math_id": 32,
"text": "c^{-1} \\operatorname{ED}(c\\mu,\\sigma^2c^{2-p}) \\rightarrow Tw_p(\\mu,c_0 \\sigma^2)"
},
{
"math_id": 33,
"text": "c \\downarrow 0"
},
{
"math_id": 34,
"text": "c \\rightarrow \\infty"
},
{
"math_id": 35,
"text": "\\operatorname{var}(Y) = a\\mu^p,"
},
{
"math_id": 36,
"text": "S(f)\\propto \\frac 1 {f^\\gamma},"
},
{
"math_id": 37,
"text": "Y = (Y_i :i=0,1,2,\\ldots,N)"
},
{
"math_id": 38,
"text": "\\widehat{\\mu}=\\operatorname{E}(Y_i),"
},
{
"math_id": 39,
"text": "y_i = Y_i - \\widehat{\\mu}, "
},
{
"math_id": 40,
"text": "\\widehat{\\sigma}^2= \\operatorname{E}(y_i^2),"
},
{
"math_id": 41,
"text": "r(k) = \\frac{ \\operatorname{E}(y_i,y_{i+k}) }{ \\operatorname{E}(y_i^2)}"
},
{
"math_id": 42,
"text": "r(k)\\sim k^{-d} L(k) "
},
{
"math_id": 43,
"text": "Y_i^{(m)} = \\left(Y_{im-m+1}+\\cdots+Y_{im}\\right)/m."
},
{
"math_id": 44,
"text": "\\operatorname{var}[Y^{(m)}] = \\widehat{\\sigma}^2 m^{-d}"
},
{
"math_id": 45,
"text": "\\lim_{k \\to \\infty} r(k)/k^{-d} = (2-d)(1-d)/2."
},
{
"math_id": 46,
"text": "Z_i^{(m)} = m Y_i^{(m)},"
},
{
"math_id": 47,
"text": "Z_i^{(m)}=(Y_{im-m+1}+\\cdots+Y_{im})."
},
{
"math_id": 48,
"text": " \\operatorname{var}[Z_i^{(m)}] = m^2 \\operatorname{var}[Y^{(m)}] = \\left(\\frac{\\widehat{\\sigma}^2 }{ \\widehat{\\mu}^{2-d}} \\right) \\operatorname{E}[Z_i^{(m)}]^{2-d}"
},
{
"math_id": 49,
"text": "\\widehat{\\mu}"
},
{
"math_id": 50,
"text": "\\widehat{\\sigma}^2"
},
{
"math_id": 51,
"text": "D = 2-H = 2 - p/2. "
},
{
"math_id": 52,
"text": "RD(m)=RD(m_\\text{ref})\\left (\\frac{m}{m_\\text{ref}}\\right )^{1-D_s}"
},
{
"math_id": 53,
"text": "G(s) = \\exp \\left [\\lambda \\frac {\\alpha-1}{\\alpha} \\left( \\frac{\\theta} {\\alpha-1} \\right)^\\alpha \\left\\{ \\left(1- \\frac{1} {\\theta}+ \\frac {s} {\\theta}\\right)^\\alpha-1 \\right\\}\\right]"
},
{
"math_id": 54,
"text": "\\operatorname{var}(Y) = a\\operatorname{E}(Y)^b + \\operatorname{E}(Y),"
},
{
"math_id": 55,
"text": "\\bar{\\rho}(E) = \\begin{cases}\n\\sqrt{2N-E^2}/\\pi\n& \\quad \\left\\vert E \\right\\vert < \\sqrt{2N} \\\\ 0 & \\quad \\left\\vert E \\right\\vert > \\sqrt{2N}\n\\end{cases} "
},
{
"math_id": 56,
"text": "\\bar{\\eta}(E) = \\frac{1}{2\\pi}\\left [E\\sqrt{2N-E^2}+2N \\arcsin \\left( \\frac{E}{\\sqrt{2N}} \\right )+ \\pi N \\right ]. "
},
{
"math_id": 57,
"text": "e_n = \\bar{\\eta}(E)= \\int_{-\\infty}^{E_n} \\, dE' \\bar{\\rho}(E'). "
},
{
"math_id": 58,
"text": "\\left | \\bar{D}_n \\right | = \\left | n- \\bar{\\eta}(E_n) \\right | "
},
{
"math_id": 59,
"text": " \\psi(x) = \\sum_{\\widehat{p\\,}^k\\le x}\\log \\widehat{p\\,}=\\sum_{n \\leq x} \\Lambda(n) "
},
{
"math_id": 60,
"text": "\\widehat{p\\,}^k "
},
{
"math_id": 61,
"text": "\\Lambda(n)"
},
{
"math_id": 62,
"text": "\\psi_0(x) = x - \\sum_\\rho \\frac{x^\\rho}{\\rho} - \\ln 2\\pi - \\frac12 \\ln(1-x^{-2})"
},
{
"math_id": 63,
"text": "\\psi_0(x) = \\lim_{\\varepsilon \\rightarrow 0}\\frac{\\psi(x-\\varepsilon)+\\psi(x+\\varepsilon)}2."
},
{
"math_id": 64,
"text": " \\Delta(x)=\\left\\vert \\psi(x)-x \\right\\vert < \\sqrt{x} \\log^{2}(x)/(8 \\pi)"
},
{
"math_id": 65,
"text": "x>73.2"
}
] |
https://en.wikipedia.org/wiki?curid=15204099
|
15204685
|
Lithium cobalt oxide
|
<templatestyles src="Chembox/styles.css"/>
Chemical compound
Lithium cobalt oxide, sometimes called lithium cobaltate or lithium cobaltite, is a chemical compound with formula LiCoO2. The cobalt atoms are formally in the +3 oxidation state, hence the IUPAC name lithium cobalt(III) oxide.
Lithium cobalt oxide is a dark blue or bluish-gray crystalline solid, and is commonly used in the positive electrodes of lithium-ion batteries.
Structure.
The structure of LiCoO2 has been studied with numerous techniques including x-ray diffraction, electron microscopy, neutron powder diffraction, and EXAFS.
The solid consists of layers of monovalent lithium cations (Li+) that lie between extended anionic sheets of cobalt and oxygen atoms, arranged as edge-sharing octahedra, with two faces parallel to the sheet plane. The cobalt atoms are formally in the trivalent oxidation state (Co3+) and are sandwiched between two layers of oxygen atoms (O2-).
In each layer (cobalt, oxygen, or lithium), the atoms are arranged in a regular triangular lattice. The lattices are offset so that the lithium atoms are farthest from the cobalt atoms, and the structure repeats in the direction perpendicular to the planes every three cobalt (or lithium) layers. The point group symmetry is formula_0 in Hermann-Mauguin notation, signifying a unit cell with threefold improper rotational symmetry and a mirror plane. The threefold rotational axis (which is normal to the layers) is termed improper because the triangles of oxygen (being on opposite sides of each octahedron) are anti-aligned.
Preparation.
Fully reduced lithium cobalt oxide can be prepared by heating a stoichiometric mixture of lithium carbonate Li2CO3 and cobalt(II,III) oxide Co3O4 or metallic cobalt at 600–800 °C, then annealing the product at 900 °C for many hours, all under an oxygen atmosphere.
Nanometer-size particles more suitable for cathode use can also be obtained by calcination of hydrated cobalt oxalate β-CoC2O4·2H2O, in the form of rod-like crystals about 8 μm long and 0.4 μm wide, with lithium hydroxide LiOH, up to 750–900 °C.
A third method uses lithium acetate, cobalt acetate, and citric acid in equal molar amounts, in water solution. Heating at 80 °C turns the mixture into a viscous transparent gel. The dried gel is then ground and heated gradually to 550 °C.
Use in rechargeable batteries.
The usefulness of lithium cobalt oxide as an intercalation electrode was discovered in 1980 by an Oxford University research group led by John B. Goodenough and Tokyo University's Koichi Mizushima.
The compound is now used as the cathode in some rechargeable lithium-ion batteries, with particle sizes ranging from nanometers to micrometers. During charging, the cobalt is partially oxidized to the +4 state, with some lithium ions moving to the electrolyte, resulting in a range of compounds Li"x"CoO2 with 0 < "x" < 1.
Batteries produced with LiCoO2 cathodes have very stable capacities, but have lower capacities and power than those with cathodes based on (especially nickel-rich) nickel-cobalt-aluminum (NCA) or nickel-cobalt-manganese (NCM) oxides. Issues with thermal stability are better for LiCoO2 cathodes than other nickel-rich chemistries although not significantly. This makes LiCoO2 batteries susceptible to thermal runaway in cases of abuse such as high temperature operation (>130 °C) or overcharging. At elevated temperatures, LiCoO2 decomposition generates oxygen, which then reacts with the organic electrolyte of the cell, this reaction is often seen in Lithium-Ion batteries where the battery becomes highly volatile and must be recycled in a safe manner. The decomposition of LiCoO2 is a safety concern due to the magnitude of this highly exothermic reaction, which can spread to adjacent cells or ignite nearby combustible material. In general, this is seen for many lithium-ion battery cathodes.
The delithiation process is usually by chemical means, although a novel physical process has been developed based on ion sputtering and annealing cycles, leaving the material properties intact.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "R\\bar 3m"
}
] |
https://en.wikipedia.org/wiki?curid=15204685
|
1520619
|
Proper orthogonal decomposition
|
Numerical method that reduces the complexity of computationally intensive simulations<templatestyles src="Machine learning/styles.css"/>
The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics and structural analysis (like crash simulations). Typically in fluid dynamics and turbulences analysis, it is used to replace the Navier–Stokes equations by simpler models to solve.
It belongs to a class of algorithms called "model order reduction" (or in short "model reduction"). What it essentially does is to train a model based on simulation data. To this extent, it can be associated with the field of machine learning.
POD and PCA.
The main use of POD is to decompose a physical field (like pressure, temperature in fluid dynamics or stress and deformation in structural analysis), depending on the different variables that influence its physical behaviors. As its name hints, it's operating an Orthogonal Decomposition along with the Principal Components of the field. As such it is assimilated with the principal component analysis from Pearson in the field of statistics, or the singular value decomposition in linear algebra because it refers to eigenvalues and eigenvectors of a physical field. In those domains, it is associated with the research of Karhunen and Loève, and their Karhunen–Loève theorem.
Mathematical expression.
The first idea behind the Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector field u(x, t) into a set of deterministic spatial functions "Φk"("x") modulated by random time coefficients "ak"("t") so that:
formula_0
The first step is to sample the vector field over a period of time in what we call snapshots (as display in the image of the POD snapshots). This snapshot method is averaging the samples over the space dimension n, and correlating them with each other along the time samples p:
formula_1 with n spatial elements, and p time samples
The next step is to compute the covariance matrix C
formula_2
We then compute the eigenvalues and eigenvectors of C and we order them from the largest eigenvalue to the smallest.
We obtain n eigenvalues λ1...,λn and a set of n eigenvectors arranged as columns in an n × n matrix Φ:
formula_3
|
[
{
"math_id": 0,
"text": "u(x,t)=\\sum_{k=1}^\\infty a_k (t) \\phi_k(x)"
},
{
"math_id": 1,
"text": "U = \\begin{pmatrix} u(x_1,t_1) & \\cdots & u(x_n,t_1)\\\\ \\vdots & & \\vdots \\\\ u(x_1,t_p) & \\cdots & u(x_n,t_p) \\end{pmatrix}"
},
{
"math_id": 2,
"text": "C = \\frac{1}{(p-1)} U^T U"
},
{
"math_id": 3,
"text": "\\phi = \\begin{pmatrix} \\phi_{1,1} & \\cdots & \\phi_{1,n} \\\\ \\vdots & & \\vdots \\\\ \\phi_{n,1} & \\cdots & \\phi_{n,n} \\end{pmatrix}"
}
] |
https://en.wikipedia.org/wiki?curid=1520619
|
15207557
|
Gyroscopic exercise tool
|
A gyroscopic exercise tool is a specialized device used in physical therapy to improve wrist strength and promote the development of palm, wrist, forearm, and finger muscles. It can also be used as a unique demonstration of some aspects of rotational dynamics. The device consists of a tennis ball-sized plastic or metal shell surrounding a free-spinning mass, with an inner heavy core, which can be spun by a short rip string or using a self-start mechanism by means of rewinding it against a spring to give it potential energy. Once the gyroscope inside is going fast enough, the person holding the device can accelerate the spinning mass to high rotation rates by moving the wrist in a circular motion. The force enacted on the user increases as the speed of the inner gyroscope increases.
Mechanics.
Inside the outer shell, the spinning mass is fixed to a thin metal axle, each end trapped in a circular, equatorial groove in the outer shell. A lightweight ring with two notches for the axle ends rests in the groove. This ring can slip in the groove, allowing the ball to spin perpendicular to the rotational axis of the ring.
To increase the angular velocity of the ball, the sides of the groove exert forces on the ends of the axle. The normal and axial forces will have no effect, so the tangential force must be provided by the friction of the ring acting on the axle. The user can apply a torque on the ball by tilting the shell in any direction except in the plane of the groove or around an axis aligned with the axle. The tilting results in a shift of the axle ends along the groove. The direction and speed of the shift can be found from the formula for the precession of a gyroscope: the applied torque is equal to the cross product of the angular velocity of precession and the angular momentum of the spinning mass. The rate of rotation of the internal ball increases as the total amount of torque applied is increased. The direction of the torque does not matter, as long as it is perpendicular to the plane of rotation of the ball. The friction of the ring increases on the side opposite to the plane of rotation. This process obeys symmetry across the plane perpendicular to the axle. The only restriction to this process is that the relative speed of the surface of the axle and the side of the groove due to precession, formula_0, must exceed the relative speed due to the rotation of the spinning mass, formula_1. The minimum torque required to meet this condition is formula_2, where I is the moment of inertia of the spinning mass, and ω is its angular velocity.
Since angular acceleration will occur regardless of the direction of the applied torque, as long as it is large enough, the device will function without any fine-tuning of the driving motion. The tilting of the shell does not have to have a particular rhythm with the precession or even have the same frequency. Since kinetic friction is usually almost as strong as static friction for the materials typically used, it is not necessary to apply exactly the amount of torque needed for the axle to roll without slipping along the side of the groove. These factors allow beginners to learn to speed up the rotation after only a few minutes of practice.
By applying the proportionality of the kinetic force of friction to the normal force, formula_3, where formula_4 is the kinetic coefficient of friction, it can be shown that the torque spinning up the mass is a factor of formula_5 smaller than the torque applied to the shell. Since frictional force is essential for the device's operation, the groove must not be lubricated to allow for the friction of the ring to enact a force on the gyro.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathit{\\Omega}_{\\mathrm{P}} R_{\\mathrm{groove}}"
},
{
"math_id": 1,
"text": "\\omega r_{\\mathrm{axle}}"
},
{
"math_id": 2,
"text": " I \\omega^2 \\left( r_{\\mathrm{axle}} / R_{\\mathrm{groove}} \\right) "
},
{
"math_id": 3,
"text": "f_\\mathrm{k} = \\mu_\\mathrm{k} F_\\mathrm{n}"
},
{
"math_id": 4,
"text": "\\mu_\\mathrm{k}"
},
{
"math_id": 5,
"text": "\\mu_\\mathrm{k} \\left( r_{\\mathrm{axle}} / R_{\\mathrm{groove}} \\right)"
}
] |
https://en.wikipedia.org/wiki?curid=15207557
|
1520887
|
Albert Einstein Memorial
|
1979 sculpture by Robert Berks in Washington, DC, US
The Albert Einstein Memorial is a monumental bronze statue by sculptor Robert Berks, depicting Albert Einstein seated with manuscript papers in hand. It is located in central Washington, D.C., United States, in a grove of trees at the southwest corner of the grounds of the National Academy of Sciences at 2101 Constitution Avenue N.W., near the Vietnam Veterans Memorial. Two replicas exist at the Israel Academy of Sciences and Humanities and the Georgia Institute of Technology.
Life.
The memorial, situated in an elm and holly grove in the southwest corner of the grounds of the National Academy of Sciences, was unveiled at the Academy's annual meeting, April 22, 1979, in honor of the centennial of Einstein's birth. At the dedication ceremony, physicist John Archibald Wheeler described the statue as "a monument to the man who united space and time into space-time...a remembrance of the man who taught us...that the universe does not go on from everlasting to everlasting, but begins with a bang." The memorial is a popular spot for tourists visiting the national mall to pose for pictures.
The statue depicts Einstein seated in casual repose on a three-step bench of Mount Airy (North Carolina) white granite. The bronze figure weighs approximately 4 tons and is 12 feet in height. The monument is supported by three caissons, totaling 135 tons, sunk in bedrock to a depth of 23 to 25 feet., It was cast at Modern Art Foundry, Astoria Queens, NY.
The sculptor, Robert Berks, known for his portrait busts and statues (John F. Kennedy at the Kennedy Center; Mary McLeod Bethune in Lincoln Park, Washington, D.C.), based the work on a bust of Einstein he sculpted from life in 1953 at Einstein's Princeton home. Landscape architect James A. Van Sweden designed the monument landscaping.
Einstein was elected a foreign associate of the National Academy of Sciences in 1922, the year after he won the Nobel Prize in physics, and became a member of the Academy in 1942, two years after he became a naturalized American citizen.
Berks created two replicas of his 1979 monument. One of the replicas can presently be viewed in the academy garden of the Israel Academy of Sciences and Humanities; another on the campus of the Georgia Institute of Technology in Atlanta, Georgia.
Platform.
The statue and bench are at one side of a circular dais, 28 feet (8.5 m) in diameter, made from emerald-pearl granite from Larvik, Norway. Embedded in the dais are more than 2,700 metal studs representing the location of astronomical objects, including the sun, moon, planets, 4 asteroids, 5 galaxies, 10 quasars, and many stars at noon on April 22, 1979, when the memorial was dedicated. The studs are different sizes to denote the apparent magnitude of the relevant object, and different studs denote binary stars, spectroscopic binaries, pulsars, globular clusters, open clusters, and quasars. The celestial objects were accurately positioned by astronomers at the U.S. Naval Observatory. Familiar constellations are marked on the map for easy identification.
To a visitor standing at the center of the dais, Einstein appears to be making direct eye contact, and any spoken words are notably amplified.
Description.
Engraved as though written on the papers held in the statue's left hand are three equations, summarizing three of Einstein's most important scientific advances:
Along the back of the bench, behind the statue, three famous quotations from the scientist are inscribed. They were selected to reflect Einstein's sense of wonder, scientific integrity, and concern for social justice. They are :
In popular and artistic culture.
The statue was filmed and subsequently used in the opening title sequence of "Sesame Street" during the show's 20th season.
A copy of the Albert Einstein Memorial made of 100% dark and white chocolate was once on display in the Marriott Wardman Park Hotel in Washington, DC.
In July 2012, the sculpture was yarn bombed by the Polish-born artist Olek, who enclosed the entire statue in a colorful crocheted wrap of pinks, purples, and teal.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "R_{\\mu\\nu} - {1 \\over 2} g_{\\mu\\nu}R = \\kappa T_{\\mu\\nu}"
},
{
"math_id": 1,
"text": "eV=h\\nu-A\\,"
},
{
"math_id": 2,
"text": "E=mc^{2}\\,"
}
] |
https://en.wikipedia.org/wiki?curid=1520887
|
152111
|
Linear function
|
Linear map or polynomial function of degree one
In mathematics, the term linear function refers to two distinct but related notions:
As a polynomial function.
In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero).
When the function is of only one variable, it is of the form
formula_0
where "a" and "b" are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. "a" is frequently referred to as the slope of the line, and "b" as the intercept.
If "a > 0" then the gradient is positive and the graph slopes upwards.
If "a < 0" then the gradient is negative and the graph slopes downwards.
For a function formula_1 of any finite number of variables, the general formula is
formula_2
and the graph is a hyperplane of dimension "k".
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.
In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.
As a linear map.
In linear algebra, a linear function is a map "f" between two vector spaces such that
formula_3
formula_4
Here "a" denotes a constant belonging to some field "K" of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be "K" itself.
In other terms the linear function preserves vector addition and scalar multiplication.
Some authors use "linear function" only for linear maps that take values in the scalar field; these are more commonly called linear forms.
The "linear functions" of calculus qualify as "linear maps" when (and only when) "f"(0, ..., 0) = 0, or, equivalently, when the constant b equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.
|
[
{
"math_id": 0,
"text": "f(x)=ax+b,"
},
{
"math_id": 1,
"text": "f(x_1, \\ldots, x_k)"
},
{
"math_id": 2,
"text": "f(x_1, \\ldots, x_k) = b + a_1 x_1 + \\cdots + a_k x_k ,"
},
{
"math_id": 3,
"text": "f(\\mathbf{x} + \\mathbf{y}) = f(\\mathbf{x}) + f(\\mathbf{y}) "
},
{
"math_id": 4,
"text": "f(a\\mathbf{x}) = af(\\mathbf{x}). "
}
] |
https://en.wikipedia.org/wiki?curid=152111
|
1521177
|
Successor
|
Successor may refer to:
<templatestyles src="Template:TOC_right/styles.css" />
See also.
Topics referred to by the same term
<templatestyles src="Dmbox/styles.css" />
This page lists associated with the title .
|
[
{
"math_id": 0,
"text": "S(n) = n + 1"
}
] |
https://en.wikipedia.org/wiki?curid=1521177
|
1521283
|
Hardy–Ramanujan–Littlewood circle method
|
In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History.
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines. Hundreds of papers followed, and as of 2022[ [update]] the method still yields results. The method is the subject of a monograph by R. C. Vaughan.
Outline.
The goal is to prove asymptotic behavior of a series: to show that "an" ~ "F"("n") for some function. This is done by taking the generating function of the series, then computing the residues about zero (essentially the Fourier coefficients). Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle.
The circle method is specifically how to compute these residues, by partitioning the circle into minor arcs (the bulk of the circle) and major arcs (small arcs containing the most significant singularities), and then bounding the behavior on the minor arcs. The key insight is that, in many cases of interest (such as theta functions), the singularities occur at the roots of unity, and the significance of the singularities is in the order of the Farey sequence. Thus one can investigate the most significant singularities, and, if fortunate, compute the integrals.
Setup.
The circle in question was initially the unit circle in the complex plane. Assuming the problem had first been formulated in the terms that for a sequence of complex numbers "a""n" for "n"
0, 1, 2, 3, ..., we want some asymptotic information of the type "a""n" ~ "F"("n"), where we have some heuristic reason to guess the form taken by "F" (an ansatz), we write
formula_0
a power series generating function. The interesting cases are where "f" is then of radius of convergence equal to 1, and we suppose that the problem as posed has been modified to present this situation.
Residues.
From that formulation, it follows directly from the residue theorem that
formula_1
for integers "n" ≥ 0, where C is a circle of radius "r" and centred at 0, for any "r" with 0 < "r" < 1; in other words, formula_2 is a contour integral, integrated over the circle described traversed once anticlockwise. We would like to take "r"
1 directly, that is, to use the unit circle contour. In the complex analysis formulation this is problematic, since the values of "f" may not be defined there.
Singularities on unit circle.
The problem addressed by the circle method is to force the issue of taking "r"
1, by a good understanding of the nature of the singularities "f" exhibits on the unit circle. The fundamental insight is the role played by the Farey sequence of rational numbers, or equivalently by the roots of unity:
formula_3
Here the denominator "s", assuming that is in lowest terms, turns out to determine the relative importance of the singular behaviour of typical "f" near "ζ".
Method.
The Hardy–Littlewood circle method, for the complex-analytic formulation, can then be thus expressed. The contributions to the evaluation of "I""n", as "r" → 1, should be treated in two ways, traditionally called "major arcs" and "minor arcs". We divide the roots of unity "ζ" into two classes, according to whether "s" ≤ "N" or "s" > "N", where "N" is a function of "n" that is ours to choose conveniently. The integral "I""n" is divided up into integrals each on some arc of the circle that is adjacent to "ζ", of length a function of "s" (again, at our discretion). The arcs make up the whole circle; the sum of the integrals over the "major arcs" is to make up 2"πiF"("n") (realistically, this will happen up to a manageable remainder term). The sum of the integrals over the "minor arcs" is to be replaced by an upper bound, smaller in order than "F"("n").
Discussion.
Stated boldly like this, it is not at all clear that this can be made to work. The insights involved are quite deep. One clear source is the theory of theta functions.
Waring's problem.
In the context of Waring's problem, powers of theta functions are the generating functions for the sum of squares function. Their analytic behaviour is known in much more accurate detail than for the cubes, for example.
It is the case, as the false-colour diagram indicates, that for a theta function the 'most important' point on the boundary circle is at "z"
1; followed by "z"
−1, and then the two complex cube roots of unity at 7 o'clock and 11 o'clock. After that it is the fourth roots of unity "i" and −"i" that matter most. While nothing in this guarantees that the analytical method will work, it does explain the rationale of using a Farey series-type criterion on roots of unity.
In the case of Waring's problem, one takes a sufficiently high power of the generating function to force the situation in which the singularities, organised into the so-called "singular series", predominate. The less wasteful the estimates used on the rest, the finer the results. As Bryan Birch has put it, the method is inherently wasteful. That does not apply to the case of the partition function, which signalled the possibility that in a favourable situation the losses from estimates could be controlled.
Vinogradov trigonometric sums.
Later, I. M. Vinogradov extended the technique, replacing the exponential sum formulation "f"("z") with a finite Fourier series, so that the relevant integral "I""n" is a Fourier coefficient. Vinogradov applied finite sums to Waring's problem in 1926, and the general trigonometric sum method became known as "the circle method of Hardy, Littlewood and Ramanujan, in the form of Vinogradov's trigonometric sums". Essentially all this does is to discard the whole 'tail' of the generating function, allowing the business of "r" in the limiting operation to be set directly to the value 1.
Applications.
Refinements of the method have allowed results to be proved about the solutions of homogeneous Diophantine equations, as long as the number of variables "k" is large relative to the degree "d" (see Birch's theorem for example). This turns out to be a contribution to the Hasse principle, capable of yielding quantitative information. If "d" is fixed and "k" is small, other methods are required, and indeed the Hasse principle tends to fail.
Rademacher's contour.
In the special case when the circle method is applied to find the coefficients of a modular form of negative weight, Hans Rademacher found a modification of the contour that makes the series arising from the circle method converge to the exact result. To describe his contour, it is convenient to replace the unit circle by the upper half plane, by making the substitution "z"
exp(2π"iτ"), so that the contour integral becomes an integral from "τ"
"i" to "τ"
1 + "i". (The number "i" could be replaced by any number on the upper half-plane, but "i" is the most convenient choice.) Rademacher's contour is (more or less) given by the boundaries of all the Ford circles from 0 to 1, as shown in the diagram. The replacement of the line from "i" to 1 + "i" by the boundaries of these circles is a non-trivial limiting process, which can be justified for modular forms that have negative weight, and with more care can also be justified for non-constant terms for the case of weight 0 (in other words modular functions).
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f(z)= \\sum a_n z^n "
},
{
"math_id": 1,
"text": "I_n=\\oint_{C} f(z)z^{-(n+1)}\\,dz = 2\\pi ia_n"
},
{
"math_id": 2,
"text": "I_n"
},
{
"math_id": 3,
"text": " \\zeta\\ = \\exp \\left ( \\frac{2 \\pi ir}{s} \\right ). "
}
] |
https://en.wikipedia.org/wiki?curid=1521283
|
15216654
|
Vedic square
|
Multiplication table in Indian mathematics
In Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the column and row headings i.e. the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9). Numerous geometric patterns and symmetries can be observed in a Vedic square, some of which can be found in traditional Islamic art.
Algebraic properties.
The Vedic Square can be viewed as the multiplication table of the monoid formula_1 where formula_2 is the set of positive integers partitioned by the residue classes modulo nine. (the operator "formula_0" refers to the abstract "multiplication" between the elements of this monoid).
If formula_3 are elements of formula_1 then formula_4 can be defined as formula_5, where the element 9 is representative of the residue class of 0 rather than the traditional choice of 0.
This does not form a group because not every non-zero element has a corresponding inverse element; for example formula_6 but there is no formula_7 such that formula_8.
Properties of subsets.
The subset formula_9 forms a cyclic group with 2 as one choice of generator - this is the group of multiplicative units in the ring formula_2. Every column and row includes all six numbers - so this subset forms a Latin square.
From two dimensions to three dimensions.
A Vedic cube is defined as the layout of each digital root in a three-dimensional multiplication table.
Vedic squares in a higher radix.
Vedic squares with a higher radix (or number base) can be calculated to analyse the symmetric patterns that arise. Using the calculation above, formula_10. The images in this section are color-coded so that the digital root of 1 is dark and the digital root of (base-1) is light.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\circ"
},
{
"math_id": 1,
"text": "((\\mathbb{Z}/9\\mathbb{Z})^{\\times}, \\{1, \\circ\\})"
},
{
"math_id": 2,
"text": "\\mathbb{Z}/9\\mathbb{Z}"
},
{
"math_id": 3,
"text": "a,b"
},
{
"math_id": 4,
"text": "a \\circ b"
},
{
"math_id": 5,
"text": "(a \\times b) \\mod{9}"
},
{
"math_id": 6,
"text": "6\\circ 3 = 9"
},
{
"math_id": 7,
"text": "a \\in \\{ 1,\\cdots,9 \\}"
},
{
"math_id": 8,
"text": "9\\circ a = 6."
},
{
"math_id": 9,
"text": "\\{1,2,4,5,7,8\\}"
},
{
"math_id": 10,
"text": "(a \\times b)\\mod{(\\textrm{base} - 1)}"
}
] |
https://en.wikipedia.org/wiki?curid=15216654
|
1521726
|
Superquadrics
|
Family of geometric shapes
In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs.
The superquadrics include many shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation.
Some authors, such as Alan Barr, define "superquadrics" as including both the superellipsoids and the supertoroids. In modern computer vision literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from range images and point clouds are covered in several computer vision literatures.
Formulas.
Implicit equation.
The surface of the basic superquadric is given by
formula_0
where "r", "s", and "t" are positive real numbers that determine the main features of the superquadric. Namely:
Each exponent can be varied independently to obtain combined shapes. For example, if "r"="s"=2, and "t"=4, one obtains a solid of revolution which resembles an ellipsoid with round cross-section but flattened ends. This formula is a special case of the superellipsoid's formula if (and only if) "r" = "s".
If any exponent is allowed to be negative, the shape extends to infinity. Such shapes are sometimes called super-hyperboloids.
The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of scaling this basic shape by different amounts "A", "B", "C" along each axis. Its general equation is
formula_1
Parametric description.
Parametric equations in terms of surface parameters "u" and "v" (equivalent to longitude and latitude if m equals 2) are
formula_2
where the auxiliary functions are
formula_3
and the sign function sgn("x") is
formula_4
Spherical product.
Barr introduces the "spherical product" which given two plane curves produces a 3D surface. If
formula_5
are two plane curves then the spherical product is
formula_6
This is similar to the typical parametric equation of a sphere:
formula_7
which give rise to the name spherical product.
Barr uses the spherical product to define quadric surfaces, like ellipsoids, and hyperboloids as well as the torus,
superellipsoid, superquadric hyperboloids of one and two sheets, and supertoroids.
Plotting code.
The following GNU Octave code generates a mesh approximation of a superquadric:
function superquadric(epsilon,a)
n = 50;
etamax = pi/2;
etamin = -pi/2;
wmax = pi;
wmin = -pi;
deta = (etamax-etamin)/n;
dw = (wmax-wmin)/n;
[i,j] = meshgrid(1:n+1,1:n+1)
eta = etamin + (i-1) * deta;
w = wmin + (j-1) * dw;
x = a(1) .* sign(cos(eta)) .* abs(cos(eta)).^epsilon(1) .* sign(cos(w)) .* abs(cos(w)).^epsilon(1);
y = a(2) .* sign(cos(eta)) .* abs(cos(eta)).^epsilon(2) .* sign(sin(w)) .* abs(sin(w)).^epsilon(2);
z = a(3) .* sign(sin(eta)) .* abs(sin(eta)).^epsilon(3);
mesh(x,y,z);
end
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\left|x\\right|^r + \\left|y\\right|^s + \\left|z\\right|^t =1"
},
{
"math_id": 1,
"text": " \\left|\\frac{x}{A}\\right|^r + \\left|\\frac{y}{B}\\right|^s + \\left|\\frac{z}{C}\\right|^t = 1."
},
{
"math_id": 2,
"text": "\\begin{align}\n x(u,v) &{}= A g\\left(v,\\frac{2}{r}\\right) g\\left(u,\\frac{2}{r}\\right) \\\\\n y(u,v) &{}= B g\\left(v,\\frac{2}{s}\\right) f\\left(u,\\frac{2}{s}\\right) \\\\\n z(u,v) &{}= C f\\left(v,\\frac{2}{t}\\right) \\\\\n & -\\frac{\\pi}{2} \\le v \\le \\frac{\\pi}{2}, \\quad -\\pi \\le u < \\pi ,\n\\end{align}"
},
{
"math_id": 3,
"text": "\\begin{align}\n f(\\omega,m) &{}= \\sgn(\\sin \\omega) \\left|\\sin \\omega \\right|^m \\\\\n g(\\omega,m) &{}= \\sgn(\\cos \\omega) \\left|\\cos \\omega \\right|^m\n\\end{align}"
},
{
"math_id": 4,
"text": " \\sgn(x) = \\begin{cases}\n -1, & x < 0 \\\\\n 0, & x = 0 \\\\\n +1, & x > 0 .\n\\end{cases}"
},
{
"math_id": 5,
"text": "f(\\mu)=\\begin{pmatrix}f_1(\\mu) \\\\ f_2(\\mu)\\end{pmatrix},\\quad g(\\nu)=\\begin{pmatrix}g_1(\\nu)\\\\g_2(\\nu)\\end{pmatrix}"
},
{
"math_id": 6,
"text": "h(\\mu,\\nu) = f(\\mu)\\otimes g(\\nu) = \\begin{pmatrix} g_1(\\nu)\\ f_1(\\mu) \\\\ g_1(\\nu)\\ f_2(\\mu) \\\\ g_2(\\nu) \\end{pmatrix}"
},
{
"math_id": 7,
"text": "\\begin{align}\nx&=x_{0}+r\\sin \\theta \\;\\cos \\varphi \\\\\ny&=y_{0}+r\\sin \\theta \\;\\sin \\varphi \\qquad (0\\leq \\theta \\leq \\pi ,\\;0\\leq \\varphi <2\\pi )\\\\\nz&=z_{0}+r\\cos \\theta\n\\end{align}"
}
] |
https://en.wikipedia.org/wiki?curid=1521726
|
1521924
|
Jet (mathematics)
|
In mathematics, the jet is an operation that takes a differentiable function "f" and produces a polynomial, the truncated Taylor polynomial of "f", at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions.
This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables. It then gives a rigorous construction of jets and jet spaces between Euclidean spaces. It concludes with a description of jets between manifolds, and how these jets can be constructed intrinsically. In this more general context, it summarizes some of the applications of jets to differential geometry and the theory of differential equations.
Jets of functions between Euclidean spaces.
Before giving a rigorous definition of a jet, it is useful to examine some special cases.
One-dimensional case.
Suppose that formula_0 is a real-valued function having at least "k" + 1 derivatives in a neighborhood "U" of the point formula_1. Then by Taylor's theorem,
formula_2
where
formula_3
Then the "k"-jet of "f" at the point formula_1 is defined to be the polynomial
formula_4
Jets are normally regarded as abstract polynomials in the variable "z", not as actual polynomial functions in that variable. In other words, "z" is an indeterminate variable allowing one to perform various algebraic operations among the jets. It is in fact the base-point formula_1 from which jets derive their functional dependency. Thus, by varying the base-point, a jet yields a polynomial of order at most "k" at every point. This marks an important conceptual distinction between jets and truncated Taylor series: ordinarily a Taylor series is regarded as depending functionally on its variable, rather than its base-point. Jets, on the other hand, separate the algebraic properties of Taylor series from their functional properties. We shall deal with the reasons and applications of this separation later in the article.
Mappings from one Euclidean space to another.
Suppose that formula_5 is a function from one Euclidean space to another having at least ("k" + 1) derivatives. In this case, Taylor's theorem asserts that
formula_6
The "k"-jet of "f" is then defined to be the polynomial
formula_7
in formula_8, where formula_9.
Algebraic properties of jets.
There are two basic algebraic structures jets can carry. The first is a product structure, although this ultimately turns out to be the least important. The second is the structure of the composition of jets.
If formula_10 are a pair of real-valued functions, then we can define the product of their jets via
formula_11
Here we have suppressed the indeterminate "z", since it is understood that jets are formal polynomials. This product is just the product of ordinary polynomials in "z", modulo formula_12. In other words, it is multiplication in the ring formula_13, where formula_14 is the ideal generated by polynomials homogeneous of order ≥ "k" + 1.
We now move to the composition of jets. To avoid unnecessary technicalities, we consider jets of functions that map the origin to the origin. If formula_15 and formula_16 with "f"(0) = 0 and "g"(0) = 0, then formula_17. The "composition of jets" is defined by
formula_18
It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the space of jets at the origin.
In fact, the composition of "k"-jets is nothing more than the composition of polynomials modulo the ideal of polynomials homogeneous of order formula_19.
"Examples:"
formula_22
formula_23
and
formula_24
Jets at a point in Euclidean space: rigorous definitions.
Analytic definition.
The following definition uses ideas from mathematical analysis to define jets and jet spaces. It can be generalized to smooth functions between Banach spaces, analytic functions between real or complex domains, to p-adic analysis, and to other areas of analysis.
Let formula_25 be the vector space of smooth functions formula_26. Let "k" be a non-negative integer, and let "p" be a point of formula_27. We define an equivalence relation formula_28 on this space by declaring that two functions "f" and "g" are equivalent to order "k" if "f" and "g" have the same value at "p", and all of their partial derivatives agree at "p" up to (and including) their "k"-th-order derivatives. In short,formula_29 iff formula_30 to "k"-th order.
The "k"-th-order jet space of formula_25 at "p" is defined to be the set of equivalence classes of formula_31, and is denoted by formula_32.
The "k"-th-order jet at "p" of a smooth function formula_33 is defined to be the equivalence class of "f" in formula_32.
Algebro-geometric definition.
The following definition uses ideas from algebraic geometry and commutative algebra to establish the notion of a jet and a jet space. Although this definition is not particularly suited for use in algebraic geometry per se, since it is cast in the smooth category, it can easily be tailored to such uses.
Let formula_34 be the vector space of germs of smooth functions formula_26 at a point "p" in formula_27. Let formula_35 be the ideal consisting of germs of functions that vanish at "p". (This is the maximal ideal for the local ring formula_34.) Then the ideal formula_36 consists of all function germs that vanish to order "k" at "p". We may now define the jet space at "p" by
formula_37
If formula_26 is a smooth function, we may define the "k"-jet of "f" at "p" as the element of formula_32 by setting
formula_38
This is a more general construction. For an formula_39-space formula_40, let formula_41 be the stalk of the structure sheaf at formula_42 and let formula_35 be the maximal ideal of the local ring formula_41. The kth jet space at formula_42 is defined to be the ring formula_43(formula_36 is the product of ideals).
Taylor's theorem.
Regardless of the definition, Taylor's theorem establishes a canonical isomorphism of vector spaces between formula_32 and formula_44. So in the Euclidean context, jets are typically identified with their polynomial representatives under this isomorphism.
Jet spaces from a point to a point.
We have defined the space formula_32 of jets at a point formula_45. The subspace of this consisting of jets of functions "f" such that "f"("p") = "q" is denoted by
formula_46
Jets of functions between two manifolds.
If "M" and "N" are two smooth manifolds, how do we define the jet of a function formula_47? We could perhaps attempt to define such a jet by using local coordinates on "M" and "N". The disadvantage of this is that jets cannot thus be defined in an invariant fashion. Jets do not transform as tensors. Instead, jets of functions between two manifolds belong to a jet bundle.
Jets of functions from the real line to a manifold.
Suppose that "M" is a smooth manifold containing a point "p". We shall define the jets of curves through "p", by which we henceforth mean smooth functions formula_48 such that "f"(0) = "p". Define an equivalence relation formula_28 as follows. Let "f" and "g" be a pair of curves through "p". We will then say that "f" and "g" are equivalent to order "k" at "p" if there is some neighborhood "U" of "p", such that, for every smooth function formula_49, formula_50. Note that these jets are well-defined since the composite functions formula_51 and formula_52 are just mappings from the real line to itself. This equivalence relation is sometimes called that of "k"-th-order contact between curves at "p".
We now define the k"-jet of a curve "f" through "p" to be the equivalence class of "f" under formula_31, denoted formula_53 or formula_54. The k"-th-order jet space formula_55 is then the set of "k"-jets at "p".
As "p" varies over "M", formula_55 forms a fibre bundle over "M": the "k"-th-order tangent bundle, often denoted in the literature by "T""k""M" (although this notation occasionally can lead to confusion). In the case "k"=1, then the first-order tangent bundle is the usual tangent bundle: "T"1"M" = "TM".
To prove that "T""k""M" is in fact a fibre bundle, it is instructive to examine the properties of formula_55 in local coordinates. Let ("x""i")= ("x"1...,"x""n") be a local coordinate system for "M" in a neighborhood "U" of "p". Abusing notation slightly, we may regard ("x""i") as a local diffeomorphism formula_56.
"Claim." Two curves "f" and "g" through "p" are equivalent modulo formula_28 if and only if formula_57.
Indeed, the "only if" part is clear, since each of the "n" functions "x"1...,"x""n" is a smooth function from "M" to formula_58. So by the definition of the equivalence relation formula_28, two equivalent curves must have formula_59.
Conversely, suppose that formula_60; is a smooth real-valued function on "M" in a neighborhood of "p". Since every smooth function has a local coordinate expression, we may express formula_60; as a function in the coordinates. Specifically, if "q" is a point of "M" near "p", then
formula_61
for some smooth real-valued function ψ of "n" real variables. Hence, for two curves "f" and "g" through "p", we have
formula_62
formula_63
The chain rule now establishes the "if" part of the claim. For instance, if "f" and "g" are functions of the real variable "t" , then
formula_64
which is equal to the same expression when evaluated against "g" instead of "f", recalling that "f"(0)="g"(0)=p and "f" and "g" are in "k"-th-order contact in the coordinate system ("x""i").
Hence the ostensible fibre bundle "T""k""M" admits a local trivialization in each coordinate neighborhood. At this point, in order to prove that this ostensible fibre bundle is in fact a fibre bundle, it suffices to establish that it has non-singular transition functions under a change of coordinates. Let formula_65 be a different coordinate system and let formula_66 be the associated change of coordinates diffeomorphism of Euclidean space to itself. By means of an affine transformation of formula_27, we may assume without loss of generality that ρ(0)=0. With this assumption, it suffices to prove that formula_67 is an invertible transformation under jet composition. (See also jet groups.) But since ρ is a diffeomorphism, formula_68 is a smooth mapping as well. Hence,
formula_69
which proves that formula_70 is non-singular. Furthermore, it is smooth, although we do not prove that fact here.
Intuitively, this means that we can express the jet of a curve through "p" in terms of its Taylor series in local coordinates on "M".
"Examples in local coordinates:"
formula_71
Given such a tangent vector "v", let "f" be the curve given in the "x""i" coordinate system by formula_72. If "φ" is a smooth function in a neighborhood of "p" with "φ"("p") = 0, then
formula_73
is a smooth real-valued function of one variable whose 1-jet is given by
formula_74
which proves that one can naturally identify tangent vectors at a point with the 1-jets of curves through that point.
In a local coordinate system "xi" centered at a point "p", we can express the second-order Taylor polynomial of a curve "f"("t") through "p" by
formula_75
So in the "x" coordinate system, the 2-jet of a curve through "p" is identified with a list of real numbers formula_76. As with the tangent vectors (1-jets of curves) at a point, 2-jets of curves obey a transformation law upon application of the coordinate transition functions.
Let ("y""i") be another coordinate system. By the chain rule,
formula_77
Hence, the transformation law is given by evaluating these two expressions at "t" = 0.
formula_78
Note that the transformation law for 2-jets is second-order in the coordinate transition functions.
Jets of functions from a manifold to a manifold.
We are now prepared to define the jet of a function from a manifold to a manifold.
Suppose that "M" and "N" are two smooth manifolds. Let "p" be a point of "M". Consider the space formula_79 consisting of smooth maps formula_47 defined in some neighborhood of "p". We define an equivalence relation formula_31 on formula_79 as follows. Two maps "f" and "g" are said to be "equivalent" if, for every curve γ through "p" (recall that by our conventions this is a mapping formula_80 such that formula_81), we have formula_82 on some neighborhood of "0".
The jet space formula_83 is then defined to be the set of equivalence classes of formula_79 modulo the equivalence relation formula_31. Note that because the target space "N" need not possess any algebraic structure, formula_83 also need not have such a structure. This is, in fact, a sharp contrast with the case of Euclidean spaces.
If formula_47 is a smooth function defined near "p", then we define the "k"-jet of "f" at "p", formula_84, to be the equivalence class of "f" modulo formula_31.
Multijets.
John Mather introduced the notion of "multijet". Loosely speaking, a multijet is a finite list of jets over different base-points. Mather proved the multijet transversality theorem, which he used in his study of stable mappings.
Jets of sections.
Suppose that "E" is a finite-dimensional smooth vector bundle over a manifold "M", with projection formula_85. Then sections of "E" are smooth functions formula_86 such that formula_87 is the identity automorphism of "M". The jet of a section "s" over a neighborhood of a point "p" is just the jet of this smooth function from "M" to "E" at "p".
The space of jets of sections at "p" is denoted by formula_88. Although this notation can lead to confusion with the more general jet spaces of functions between two manifolds, the context typically eliminates any such ambiguity.
Unlike jets of functions from a manifold to another manifold, the space of jets of sections at "p" carries the structure of a vector space inherited from the vector space structure on the sections themselves. As "p" varies over "M", the jet spaces formula_88 form a vector bundle over "M", the "k"-th-order jet bundle of "E", denoted by "J""k"("E").
We work in local coordinates at a point and use the Einstein notation. Consider a vector field
formula_89
in a neighborhood of "p" in "M". The 1-jet of "v" is obtained by taking the first-order Taylor polynomial of the coefficients of the vector field:
formula_90
In the "x" coordinates, the 1-jet at a point can be identified with a list of real numbers formula_91. In the same way that a tangent vector at a point can be identified with the list ("vi"), subject to a certain transformation law under coordinate transitions, we have to know how the list formula_91 is affected by a transition.
So consider the transformation law in passing to another coordinate system "y""i". Let "wk" be the coefficients of the vector field "v" in the "y" coordinates. Then in the "y" coordinates, the 1-jet of "v" is a new list of real numbers formula_92. Since
formula_93
it follows that
formula_94
So
formula_95
Expanding by a Taylor series, we have
formula_96
formula_97
Note that the transformation law is second-order in the coordinate transition functions.
|
[
{
"math_id": 0,
"text": "f: {\\mathbb R}\\rightarrow{\\mathbb R}"
},
{
"math_id": 1,
"text": "x_0"
},
{
"math_id": 2,
"text": "f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(k)}(x_0)}{k!}(x-x_0)^{k}+\\frac{R_{k+1}(x)}{(k+1)!}(x-x_0)^{k+1}"
},
{
"math_id": 3,
"text": "|R_{k+1}(x)|\\le\\sup_{x\\in U} |f^{(k+1)}(x)|."
},
{
"math_id": 4,
"text": "(J^k_{x_0}f)(z)\n=\\sum_{i=0}^k \\frac{f^{(i)}(x_0)}{i!}z^i\n=f(x_0)+f'(x_0)z+\\cdots+\\frac{f^{(k)}(x_0)}{k!}z^k."
},
{
"math_id": 5,
"text": "f:{\\mathbb R}^n\\rightarrow{\\mathbb R}^m"
},
{
"math_id": 6,
"text": "\n\\begin{align}\nf(x)=f(x_0)+ (Df(x_0))\\cdot(x-x_0)+ {} & \\frac{1}{2}(D^2f(x_0))\\cdot (x-x_0)^{\\otimes 2} + \\cdots \\\\[4pt]\n& \\cdots +\\frac{D^kf(x_0)}{k!}\\cdot(x-x_0)^{\\otimes k}+\\frac{R_{k+1}(x)}{(k+1)!}\\cdot(x-x_0)^{\\otimes (k+1)}.\n\\end{align}\n"
},
{
"math_id": 7,
"text": "(J^k_{x_0}f)(z)=f(x_0)+(Df(x_0))\\cdot z+\\frac{1}{2}(D^2f(x_0))\\cdot z^{\\otimes 2} + \\cdots + \\frac{D^kf(x_0)}{k!}\\cdot z^{\\otimes k}"
},
{
"math_id": 8,
"text": "{\\mathbb R}[z]"
},
{
"math_id": 9,
"text": "z=(z_1,\\ldots,z_n)"
},
{
"math_id": 10,
"text": "f,g:{\\mathbb R}^n\\rightarrow {\\mathbb R}"
},
{
"math_id": 11,
"text": "J^k_{x_0}f\\cdot J^k_{x_0}g=J^k_{x_0}(f\\cdot g)."
},
{
"math_id": 12,
"text": "z^{k+1}"
},
{
"math_id": 13,
"text": "{\\mathbb R}[z]/(z^{k+1})"
},
{
"math_id": 14,
"text": "(z^{k+1})"
},
{
"math_id": 15,
"text": "f:{\\mathbb R}^m\\rightarrow{\\mathbb R}^\\ell"
},
{
"math_id": 16,
"text": "g:{\\mathbb R}^n\\rightarrow{\\mathbb R}^m"
},
{
"math_id": 17,
"text": "f\\circ g:{\\mathbb R}^n \\rightarrow{\\mathbb R}^\\ell"
},
{
"math_id": 18,
"text": "J^k_0 f\\circ J^k_0 g=J^k_0 (f\\circ g)."
},
{
"math_id": 19,
"text": "> k"
},
{
"math_id": 20,
"text": "f(x)=\\log(1-x)"
},
{
"math_id": 21,
"text": "g(x)=\\sin\\,x"
},
{
"math_id": 22,
"text": "(J^3_0f)(x)=-x-\\frac{x^2}{2}-\\frac{x^3}{3}"
},
{
"math_id": 23,
"text": "(J^3_0g)(x)=x-\\frac{x^3}{6}"
},
{
"math_id": 24,
"text": "\n\\begin{align}\n& (J^3_0f)\\circ (J^3_0g)=-\\left(x-\\frac{x^3}{6}\\right)-\\frac{1}{2}\\left(x-\\frac{x^3}{6}\\right)^2-\\frac{1}{3} \\left(x-\\frac{x^3}{6}\\right)^3 \\pmod{x^4} \\\\[4pt]\n= {} & -x-\\frac{x^2}{2}-\\frac{x^3}{6}\n\\end{align}\n"
},
{
"math_id": 25,
"text": "C^\\infty({\\mathbb R}^n,{\\mathbb R}^m)"
},
{
"math_id": 26,
"text": "f:{\\mathbb R}^n\\rightarrow {\\mathbb R}^m"
},
{
"math_id": 27,
"text": "{\\mathbb R}^n"
},
{
"math_id": 28,
"text": "E_p^k"
},
{
"math_id": 29,
"text": "f \\sim g \\,\\!"
},
{
"math_id": 30,
"text": " f-g = 0 "
},
{
"math_id": 31,
"text": "E^k_p"
},
{
"math_id": 32,
"text": "J^k_p({\\mathbb R}^n,{\\mathbb R}^m)"
},
{
"math_id": 33,
"text": "f\\in C^\\infty({\\mathbb R}^n,{\\mathbb R}^m)"
},
{
"math_id": 34,
"text": "C_p^\\infty({\\mathbb R}^n,{\\mathbb R}^m)"
},
{
"math_id": 35,
"text": "{\\mathfrak m}_p"
},
{
"math_id": 36,
"text": "{\\mathfrak m}_p^{k+1}"
},
{
"math_id": 37,
"text": "J^k_p({\\mathbb R}^n,{\\mathbb R}^m)=C_p^\\infty({\\mathbb R}^n,{\\mathbb R}^m)/{\\mathfrak m}_p^{k+1}"
},
{
"math_id": 38,
"text": "J^k_pf=f \\pmod {{\\mathfrak m}_p^{k+1}}"
},
{
"math_id": 39,
"text": "\\mathbb{F}"
},
{
"math_id": 40,
"text": "M"
},
{
"math_id": 41,
"text": "\\mathcal{F}_p"
},
{
"math_id": 42,
"text": "p"
},
{
"math_id": 43,
"text": "J^k_p(M)=\\mathcal{F}_p/{\\mathfrak m}_p^{k+1}"
},
{
"math_id": 44,
"text": "{\\mathbb R}^m[z_1, \\dotsc, z_n]/(z_1, \\dotsc, z_n)^{k+1}"
},
{
"math_id": 45,
"text": "p\\in {\\mathbb R}^n"
},
{
"math_id": 46,
"text": "J^k_p({\\mathbb R}^n,{\\mathbb R}^m)_q=\\left\\{J^kf\\in J^k_p({\\mathbb R}^n,{\\mathbb R}^m) \\mid f(p) = q \\right\\}"
},
{
"math_id": 47,
"text": "f:M\\rightarrow N"
},
{
"math_id": 48,
"text": "f:{\\mathbb R}\\rightarrow M"
},
{
"math_id": 49,
"text": "\\varphi : U \\rightarrow {\\mathbb R}"
},
{
"math_id": 50,
"text": "J^k_0 (\\varphi\\circ f)=J^k_0 (\\varphi\\circ g)"
},
{
"math_id": 51,
"text": "\\varphi\\circ f"
},
{
"math_id": 52,
"text": "\\varphi\\circ g"
},
{
"math_id": 53,
"text": "J^k\\! f\\,"
},
{
"math_id": 54,
"text": "J^k_0f"
},
{
"math_id": 55,
"text": "J^k_0({\\mathbb R},M)_p"
},
{
"math_id": 56,
"text": "(x^i):M\\rightarrow\\R^n"
},
{
"math_id": 57,
"text": "J^k_0\\left((x^i)\\circ f\\right)=J^k_0\\left((x^i)\\circ g\\right)"
},
{
"math_id": 58,
"text": "{\\mathbb R}"
},
{
"math_id": 59,
"text": "J^k_0(x^i\\circ f)=J^k_0(x^i\\circ g)"
},
{
"math_id": 60,
"text": "\\varphi"
},
{
"math_id": 61,
"text": "\\varphi(q)=\\psi(x^1(q),\\dots,x^n(q))"
},
{
"math_id": 62,
"text": "\\varphi\\circ f=\\psi(x^1\\circ f,\\dots,x^n\\circ f)"
},
{
"math_id": 63,
"text": "\\varphi\\circ g=\\psi(x^1\\circ g,\\dots,x^n\\circ g)"
},
{
"math_id": 64,
"text": "\\left. \\frac{d}{dt} \\left( \\varphi\\circ f \\right) (t) \\right|_{t=0}= \\sum_{i=1}^n\\left.\\frac{d}{dt}(x^i\\circ f)(t)\\right|_{t=0}\\ (D_i\\psi)\\circ f(0)"
},
{
"math_id": 65,
"text": "(y^i):M\\rightarrow{\\mathbb R}^n"
},
{
"math_id": 66,
"text": "\\rho=(x^i)\\circ (y^i)^{-1}:{\\mathbb R}^n\\rightarrow {\\mathbb R}^n"
},
{
"math_id": 67,
"text": "J^k_0\\rho:J^k_0({\\mathbb R}^n,{\\mathbb R}^n)\\rightarrow J^k_0({\\mathbb R}^n,{\\mathbb R}^n)"
},
{
"math_id": 68,
"text": "\\rho^{-1}"
},
{
"math_id": 69,
"text": "I=J^k_0I=J^k_0(\\rho\\circ\\rho^{-1})=J^k_0(\\rho)\\circ J^k_0(\\rho^{-1})"
},
{
"math_id": 70,
"text": "J^k_0\\rho"
},
{
"math_id": 71,
"text": "v=\\sum_iv^i\\frac{\\partial}{\\partial x^i}"
},
{
"math_id": 72,
"text": "x^i\\circ f(t)=tv^i"
},
{
"math_id": 73,
"text": "\\varphi\\circ f:{\\mathbb R}\\rightarrow {\\mathbb R}"
},
{
"math_id": 74,
"text": "J^1_0(\\varphi\\circ f)(t)=\\sum_itv^i \\frac{\\partial \\varphi}{\\partial x^i}(p)."
},
{
"math_id": 75,
"text": "J_0^2(x^i(f))(t)=t\\frac{dx^i(f)}{dt}(0)+\\frac{t^2}{2}\\frac{d^2x^i(f)}{dt^2}(0)."
},
{
"math_id": 76,
"text": "(\\dot{x}^i,\\ddot{x}^i)"
},
{
"math_id": 77,
"text": "\n\\begin{align}\n\\frac{d}{dt}y^i(f(t)) & = \\sum_j\\frac{\\partial y^i}{\\partial x^j}(f(t))\\frac{d}{dt}x^j(f(t)) \\\\[5pt]\n\\frac{d^2}{dt^2}y^i(f(t)) & = \\sum_{j,k}\\frac{\\partial^2 y^i}{\\partial x^j \\, \\partial x^k}(f(t))\\frac{d}{dt}x^j(f(t)) \\frac{d}{dt}x^k(f(t))+\\sum_j\\frac{\\partial y^i}{\\partial x^j}(f(t))\\frac{d^2}{dt^2}x^j(f(t))\n\\end{align}\n"
},
{
"math_id": 78,
"text": "\n\\begin{align}\n& \\dot{y}^i=\\sum_j\\frac{\\partial y^i}{\\partial x^j}(0)\\dot{x}^j \\\\[5pt]\n& \\ddot{y}^i=\\sum_{j,k}\\frac{\\partial^2 y^i}{\\partial x^j \\, \\partial x^k}(0)\\dot{x}^j\\dot{x}^k+\\sum_j\\frac{\\partial y^i}{\\partial x^j}(0)\\ddot{x}^j.\n\\end{align}\n"
},
{
"math_id": 79,
"text": "C^\\infty_p(M,N)"
},
{
"math_id": 80,
"text": "\\gamma:{\\mathbb R}\\rightarrow M"
},
{
"math_id": 81,
"text": "\\gamma(0)=p"
},
{
"math_id": 82,
"text": "J^k_0(f\\circ \\gamma)=J^k_0(g\\circ \\gamma)"
},
{
"math_id": 83,
"text": "J^k_p(M,N)"
},
{
"math_id": 84,
"text": "J^k_pf"
},
{
"math_id": 85,
"text": "\\pi:E\\rightarrow M"
},
{
"math_id": 86,
"text": "s:M\\rightarrow E"
},
{
"math_id": 87,
"text": "\\pi\\circ s"
},
{
"math_id": 88,
"text": "J^k_p(M,E)"
},
{
"math_id": 89,
"text": "v=v^i(x)\\partial/\\partial x^i"
},
{
"math_id": 90,
"text": "J_0^1v^i(x)=v^i(0)+x^j\\frac{\\partial v^i}{\\partial x^j}(0)=v^i+v^i_jx^j."
},
{
"math_id": 91,
"text": "(v^i,v^i_j)"
},
{
"math_id": 92,
"text": "(w^i,w^i_j)"
},
{
"math_id": 93,
"text": "v=w^k(y)\\partial/\\partial y^k=v^i(x)\\partial/\\partial x^i,"
},
{
"math_id": 94,
"text": "w^k(y)=v^i(x)\\frac{\\partial y^k}{\\partial x^i}(x)."
},
{
"math_id": 95,
"text": "w^k(0)+y^j\\frac{\\partial w^k}{\\partial y^j}(0)=\\left(v^i(0)+x^j\\frac{\\partial v^i}{\\partial x^j}\\right)\\frac{\\partial y^k}{\\partial x^i}(x)"
},
{
"math_id": 96,
"text": "w^k=\\frac{\\partial y^k}{\\partial x^i}(0) v^i"
},
{
"math_id": 97,
"text": "w^k_j=v^i\\frac{\\partial^2 y^k}{\\partial x^i \\, \\partial x^j}+v_j^i\\frac{\\partial y^k}{\\partial x^i}. "
}
] |
https://en.wikipedia.org/wiki?curid=1521924
|
1521971
|
Ptolemy's theorem
|
Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.
If the vertices of the cyclic quadrilateral are "A", "B", "C", and "D" in order, then the theorem states that:
formula_0
This relation may be verbally expressed as follows:
"If a quadrilateral is cyclic then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides."
Moreover, the converse of Ptolemy's theorem is also true:
"In a quadrilateral, if the sum of the products of the lengths of its two pairs of opposite sides is equal to the product of the lengths of its diagonals, then the quadrilateral can be inscribed in a circle i.e. it is a cyclic quadrilateral."
Corollaries on inscribed polygons.
Equilateral triangle.
Ptolemy's Theorem yields as a corollary a pretty theorem regarding an equilateral triangle inscribed in a circle.
Given An equilateral triangle inscribed on a circle and a point on the circle.
The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
Proof: Follows immediately from Ptolemy's theorem:
formula_1
Square.
Any square can be inscribed in a circle whose center is the center of the square. If the common length of its four sides is equal to formula_2 then the length of the diagonal is equal to formula_3 according to the Pythagorean theorem, and Ptolemy's relation obviously holds.
Rectangle.
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d2, the right hand side of Ptolemy's relation is the sum "a"2 + "b"2.
Copernicus – who used Ptolemy's theorem extensively in his trigonometrical work – refers to this result as a 'Porism' or self-evident corollary:
"Furthermore it is clear (manifestum est) that when the chord subtending an arc has been given, that chord too can be found which subtends the rest of the semicircle."
Pentagon.
A more interesting example is the relation between the length "a" of the side and the (common) length "b" of the 5 chords in a regular pentagon. By completing the square, the relation yields the golden ratio:
formula_4
Side of decagon.
If now diameter AF is drawn bisecting DC so that DF and CF are sides c of an inscribed decagon, Ptolemy's Theorem can again be applied – this time to cyclic quadrilateral ADFC with diameter "d" as one of its diagonals:
formula_5
formula_6 where formula_7 is the golden ratio.
formula_8
whence the side of the inscribed decagon is obtained in terms of the circle diameter. Pythagoras's theorem applied to right triangle AFD then yields "b" in terms of the diameter and "a" the side of the pentagon is thereafter calculated as
formula_9
As Copernicus (following Ptolemy) wrote,
"The diameter of a circle being given, the sides of the triangle, tetragon, pentagon, hexagon and decagon, which the same circle circumscribes, are also given."
Proofs.
Visual proof.
The animation here shows a visual demonstration of Ptolemy's theorem, based on Derrick & Herstein (2012).
Proof by similarity of triangles.
Let ABCD be a cyclic quadrilateral.
On the chord BC, the inscribed angles ∠BAC = ∠BDC, and on AB, ∠ADB = ∠ACB.
Construct K on AC such that ∠ABK = ∠CBD; since ∠ABK + ∠CBK = ∠ABC = ∠CBD + ∠ABD, ∠CBK = ∠ABD.
Now, by common angles △ABK is similar to △DBC, and likewise △ABD is similar to △KBC.
Thus AK/AB = CD/BD, and CK/BC = DA/BD;
equivalently, AK⋅BD = AB⋅CD, and CK⋅BD = BC⋅DA.
By adding two equalities we have AK⋅BD + CK⋅BD = AB⋅CD + BC⋅DA, and factorizing this gives (AK+CK)·BD = AB⋅CD + BC⋅DA.
But AK+CK = AC, so AC⋅BD = AB⋅CD + BC⋅DA, Q.E.D.
The proof as written is only valid for simple cyclic quadrilaterals. If the quadrilateral is self-crossing then K will be located outside the line segment AC. But in this case, AK−CK = ±AC, giving the expected result.
Proof by trigonometric identities.
Let the inscribed angles subtended by formula_10, formula_11 and formula_12 be, respectively, formula_13, formula_14 and formula_15, and the radius of the circle be formula_16, then we have formula_17, formula_18, formula_19, formula_20, formula_21 and formula_22, and the original equality to be proved is transformed to
formula_23
from which the factor formula_24 has disappeared by dividing both sides of the equation by it.
Now by using the sum formulae, formula_25 and formula_26, it is trivial to show that both sides of the above equation are equal to
formula_27
Q.E.D.
Here is another, perhaps more transparent, proof using rudimentary trigonometry.
Define a new quadrilateral formula_28 inscribed in the same circle, where formula_29 are the same
as in formula_30, and formula_31 located at a new point on the same circle, defined by formula_32,
formula_33. (Picture triangle formula_34 flipped, so that vertex formula_35 moves to vertex formula_36 and vertex formula_36 moves to vertex formula_35. Vertex formula_37 will now be located at a new point D’ on the circle.)
Then, formula_28 has the same edges lengths, and consequently the same inscribed angles subtended by
the corresponding edges, as formula_38, only in a different order. That is, formula_13, formula_14 and formula_15, for, respectively, formula_39 and formula_40.
Also, formula_30 and formula_28 have the same area. Then,
formula_41
Q.E.D.
Proof by inversion.
Choose an auxiliary circle formula_42 of radius formula_43 centered at D with respect to which the circumcircle of ABCD is inverted into a line (see figure).
Then
formula_44
Then formula_45 and formula_46 can be expressed as
formula_47, formula_48 and formula_49 respectively. Multiplying each term by formula_50 and using formula_51 yields Ptolemy's equality.
Q.E.D.
Note that if the quadrilateral is not cyclic then A', B' and C' form a triangle and hence A'B'+B'C' > A'C', giving us a very simple proof of Ptolemy's Inequality which is presented below.
Proof using complex numbers.
Embed ABCD in the complex plane formula_52 by identifying formula_53 as four distinct complex numbers formula_54. Define the cross-ratio
formula_55.
Then
formula_56
with equality if and only if the cross-ratio formula_57 is a positive real number. This proves Ptolemy's inequality generally, as it remains only to show that formula_58 lie consecutively arranged
on a circle (possibly of infinite radius, i.e. a line) in formula_52 if and only if formula_59.
From the polar form of a complex number formula_60, it follows
formula_61
with the last equality holding if and only if ABCD is cyclic, since a quadrilateral is cyclic if and only if opposite angles sum to formula_62.
Q.E.D.
Note that this proof is equivalently made by observing that the cyclicity of ABCD, i.e. the supplementarity formula_63 and formula_64, is equivalent to the condition
formula_65;
in particular there is a rotation of formula_52 in which this formula_66 is 0 (i.e. all three products are positive real numbers), and by which Ptolemy's theorem
formula_67
is then directly established from the simple algebraic identity
formula_68
Corollaries.
In the case of a circle of unit diameter the sides formula_69 of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles formula_70 and formula_71 which they subtend. Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. We may then write Ptolemy's Theorem in the following trigonometric form:
formula_72
Applying certain conditions to the subtended angles formula_70 and formula_71 it is possible to derive a number of important corollaries using the above as our starting point. In what follows it is important to bear in mind that the sum of angles formula_73.
Corollary 1. Pythagoras's theorem.
Let formula_74 and formula_75. Then formula_76
(since opposite angles of a cyclic quadrilateral are supplementary). Then:
formula_77
formula_78
formula_79
Corollary 2. The law of cosines.
Let formula_75. The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by formula_80 units where:
formula_81
It will be easier in this case to revert to the standard statement of Ptolemy's theorem:
formula_82
The cosine rule for triangle ABC.
Corollary 3. Compound angle sine (+).
Let
formula_83
Then
formula_84
Therefore,
formula_85
Formula for compound angle sine (+).
Corollary 4. Compound angle sine (−).
Let formula_86. Then formula_87. Hence,
formula_88
formula_89
formula_90
Formula for compound angle sine (−).
This derivation corresponds to the Third Theorem
as chronicled by Copernicus following Ptolemy in Almagest. In particular if the sides of a pentagon (subtending 36° at the circumference) and of a hexagon (subtending 30° at the circumference) are given, a chord subtending 6° may be calculated. This was a critical step in the ancient method of calculating tables of chords.
Corollary 5. Compound angle cosine (+).
This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest.
Let formula_91. Then formula_92. Hence
formula_93
formula_94
formula_95
Formula for compound angle cosine (+)
Despite lacking the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem (or more simply the Second Theorem) the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cognoscenti of those times to draw up accurate tables of chords (corresponding to tables of sines) and to use these in their attempts to understand and map the cosmos as they saw it. Since tables of chords were drawn up by Hipparchus three centuries before Ptolemy, we must assume he knew of the 'Second Theorem' and its derivatives. Following the trail of ancient astronomers, history records the star catalogue of Timocharis of Alexandria. If, as seems likely, the compilation of such catalogues required an understanding of the 'Second Theorem' then the true origins of the latter disappear thereafter into the mists of antiquity but it cannot be unreasonable to presume that the astronomers, architects and construction engineers of ancient Egypt may have had some knowledge of it.
Ptolemy's inequality.
The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral "ABCD", then
formula_96
where equality holds if and only if the quadrilateral is cyclic. This special case is equivalent to Ptolemy's theorem.
Related theorem about the ratio of the diagonals.
Ptolemy's theorem gives the product of the diagonals (of a cyclic quadrilateral) knowing the sides, the following theorem yields the same for the ratio of the diagonals.
formula_97
Proof: It is known that the area of a triangle formula_98 inscribed in a circle of radius formula_16 is: formula_99
Writing the area of the quadrilateral as sum of two triangles sharing the same circumscribing circle, we obtain two relations for each decomposition.
formula_100
formula_101
Equating, we obtain the announced formula.
Consequence: Knowing both the product and the ratio of the diagonals, we deduce their immediate expressions:
formula_102
Notes.
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{
"math_id": 0,
"text": "AC\\cdot BD = AB\\cdot CD+BC\\cdot AD"
},
{
"math_id": 1,
"text": " qs=ps+rs \\Rightarrow q=p+r. "
},
{
"math_id": 2,
"text": "a"
},
{
"math_id": 3,
"text": "a\\sqrt{2}"
},
{
"math_id": 4,
"text": "\\begin{array}{rl}\n b \\cdot b \\,\\;\\;\\qquad\\quad\\qquad =&\\!\\!\\!\\! a \\! \\cdot \\! a + a \\! \\cdot \\! b\n\\\\ b^2 \\;\\; - ab \\quad\\qquad =&\\!\\! a^2\n\\\\ \\frac{b^2}{a^2} \\;\\; - \\frac{ab}{a^2} \\;\\;\\;\\qquad =&\\!\\!\\! \\frac{ a^2 }{a^2}\n\\\\ \\left(\\frac{b}{a}\\right)^2 - \\frac{b}{a} + \\left(\\frac{1}{2}\\right)^2 =&\\!\\! 1 + \\left(\\frac{ 1 }{ 2}\\right)^2\n\\\\ \\left(\\frac{b}{a} - \\frac{1}{2}\\right)^2 =&\\!\\! \\quad \\frac{ 5 }{ 4}\n\\\\ \\frac{b}{a} - \\frac{1}{2} \\;\\;\\; =&\\!\\!\\!\\! \\pm \\frac{ \\sqrt{5}}{ 2}\n\\\\ \\frac{b}{a} > 0 \\, \\Rightarrow \\, \\varphi = \\frac{b}{a} =&\\!\\!\\!\\! \\frac{1 + \\sqrt{5}}{ 2}\n\\end{array}"
},
{
"math_id": 5,
"text": "ad=2bc"
},
{
"math_id": 6,
"text": "\\Rightarrow ad=2\\varphi ac"
},
{
"math_id": 7,
"text": "\\varphi"
},
{
"math_id": 8,
"text": "\\Rightarrow c=\\frac{d}{2\\varphi}."
},
{
"math_id": 9,
"text": "a = \\frac {b} {\\varphi} = b \\left( \\varphi - 1 \\right)."
},
{
"math_id": 10,
"text": "AB"
},
{
"math_id": 11,
"text": "BC"
},
{
"math_id": 12,
"text": "CD"
},
{
"math_id": 13,
"text": "\\alpha"
},
{
"math_id": 14,
"text": "\\beta"
},
{
"math_id": 15,
"text": "\\gamma"
},
{
"math_id": 16,
"text": "R"
},
{
"math_id": 17,
"text": "AB=2R\\sin\\alpha"
},
{
"math_id": 18,
"text": "BC=2R\\sin\\beta"
},
{
"math_id": 19,
"text": "CD=2R\\sin\\gamma"
},
{
"math_id": 20,
"text": "AD=2R\\sin(180^\\circ-(\\alpha+\\beta+\\gamma))"
},
{
"math_id": 21,
"text": "AC=2R\\sin(\\alpha+\\beta)"
},
{
"math_id": 22,
"text": "BD=2R\\sin(\\beta+\\gamma)"
},
{
"math_id": 23,
"text": " \\sin(\\alpha+\\beta)\\sin(\\beta+\\gamma) = \\sin\\alpha\\sin\\gamma + \\sin\\beta \\sin(\\alpha + \\beta+\\gamma)"
},
{
"math_id": 24,
"text": "4R^2"
},
{
"math_id": 25,
"text": "\\sin(x+y)=\\sin{x}\\cos y+\\cos x\\sin y"
},
{
"math_id": 26,
"text": "\\cos(x+y)=\\cos x\\cos y-\\sin x\\sin y"
},
{
"math_id": 27,
"text": "\n\\begin{align}\n& \\sin\\alpha\\sin\\beta\\cos\\beta\\cos\\gamma + \\sin\\alpha\\cos^2\\beta\\sin\\gamma \\\\ + {} & \\cos\\alpha\\sin^2\\beta\\cos\\gamma+\\cos\\alpha\\sin\\beta\\cos\\beta\\sin\\gamma.\n\\end{align}\n"
},
{
"math_id": 28,
"text": "ABCD'"
},
{
"math_id": 29,
"text": "A,B,C"
},
{
"math_id": 30,
"text": "ABCD"
},
{
"math_id": 31,
"text": "D'"
},
{
"math_id": 32,
"text": " |\\overline{AD'}| = |\\overline{CD}|"
},
{
"math_id": 33,
"text": "|\\overline{CD'}| = |\\overline{AD}|"
},
{
"math_id": 34,
"text": "ACD"
},
{
"math_id": 35,
"text": "C"
},
{
"math_id": 36,
"text": "A"
},
{
"math_id": 37,
"text": "D"
},
{
"math_id": 38,
"text": " ABCD"
},
{
"math_id": 39,
"text": "AB, BC"
},
{
"math_id": 40,
"text": "AD'"
},
{
"math_id": 41,
"text": " \n\\begin{align}\n\\mathrm{Area}(ABCD) & = \\frac{1}{2} AC\\cdot BD \\cdot \\sin(\\alpha + \\gamma); \\\\\n\\mathrm{Area}(ABCD') & = \\frac{1}{2} AB\\cdot AD'\\cdot \\sin(180^\\circ - \\alpha - \\gamma) +\n \\frac{1}{2} BC\\cdot CD' \\cdot \\sin(\\alpha + \\gamma)\\\\ & = \n\\frac{1}{2} (AB\\cdot CD + BC\\cdot AD)\\cdot \\sin(\\alpha + \\gamma).\n\\end{align}\n"
},
{
"math_id": 42,
"text": " \\Gamma "
},
{
"math_id": 43,
"text": " r "
},
{
"math_id": 44,
"text": " A'B' + B'C' = A'C'. "
},
{
"math_id": 45,
"text": " A'B', B'C' "
},
{
"math_id": 46,
"text": " A'C' "
},
{
"math_id": 47,
"text": " \\frac{AB \\cdot DB'}{DA} "
},
{
"math_id": 48,
"text": " \\frac{BC \\cdot DB'}{DC} "
},
{
"math_id": 49,
"text": " \\frac{AC \\cdot DC'}{DA} "
},
{
"math_id": 50,
"text": " \\frac{DA \\cdot DC}{DB'} "
},
{
"math_id": 51,
"text": "\\frac{DC'}{DB'} = \\frac{DB}{DC} "
},
{
"math_id": 52,
"text": "\\mathbb{C}"
},
{
"math_id": 53,
"text": "A\\mapsto z_A,\\ldots,D\\mapsto z_D"
},
{
"math_id": 54,
"text": "z_A,\\ldots,z_D\\in\\mathbb{C}"
},
{
"math_id": 55,
"text": "\\zeta:=\\frac{(z_A-z_B)(z_C-z_D)}{(z_A-z_D)(z_B-z_C)}\\in\\mathbb{C}_{\\neq0}"
},
{
"math_id": 56,
"text": "\n\\begin{align}\n\\overline{AB}\\cdot\\overline{CD}+\\overline{AD}\\cdot\\overline{BC}\n& = \\left|z_A-z_B\\right|\\left|z_C-z_D\\right| + \\left|z_A-z_D\\right|\\left|z_B-z_C\\right| \\\\\n& = \\left|(z_A-z_B)(z_C-z_D)\\right| + \\left|(z_A-z_D)(z_B-z_C)\\right| \\\\\n& = \\left(\\left|\\frac{(z_A-z_B)(z_C-z_D)}{(z_A-z_D)(z_B-z_C)}\\right| + 1\\right) \\left|(z_A-z_D)(z_B-z_C)\\right| \\\\\n& = \\left(\\left|\\zeta\\right| +1\\right) \\left|(z_A-z_D)(z_B-z_C)\\right| \\\\\n& \\geq \\left|(\\zeta +1)(z_A-z_D)(z_B-z_C)\\right| \\\\\n& = \\left|(z_A-z_B)(z_C-z_D)+(z_A-z_D)(z_B-z_C)\\right| \\\\\n& = \\left|(z_A-z_C)(z_B-z_D)\\right| \\\\\n& = \\left|z_A-z_C\\right|\\left|z_B-z_D\\right| \\\\\n& = \\overline{AC}\\cdot\\overline{BD}\n\\end{align}\n"
},
{
"math_id": 57,
"text": "\\zeta"
},
{
"math_id": 58,
"text": "z_A,\\ldots,z_D"
},
{
"math_id": 59,
"text": "\\zeta\\in\\mathbb{R}_{>0}"
},
{
"math_id": 60,
"text": "z=\\vert z\\vert e^{i\\arg(z)}"
},
{
"math_id": 61,
"text": "\n\\begin{align}\n\\arg(\\zeta) & = \\arg\\frac{(z_A-z_B)(z_C-z_D)}{(z_A-z_D)(z_B-z_C)} \\\\\n& = \\arg(z_A-z_B)+\\arg(z_C-z_D)-\\arg(z_A-z_D)-\\arg(z_B-z_C) \\pmod{2\\pi} \\\\\n& = \\arg(z_A-z_B)+\\arg(z_C-z_D)-\\arg(z_A-z_D)-\\arg(z_C-z_B) - \\arg(-1) \\pmod{2\\pi} \\\\\n& = - \\left[\\arg(z_C-z_B)-\\arg(z_A-z_B)\\right] - \\left[\\arg(z_A-z_D)-\\arg(z_C-z_D)\\right] -\\arg(-1) \\pmod{2\\pi} \\\\\n& = - \\angle ABC - \\angle CDA -\\pi \\pmod{2\\pi}\\\\\n& = 0\n\\end{align}\n"
},
{
"math_id": 62,
"text": "\\pi"
},
{
"math_id": 63,
"text": "\\angle ABC"
},
{
"math_id": 64,
"text": "\\angle CDA"
},
{
"math_id": 65,
"text": "\\arg\\left[(z_A-z_B)(z_C-z_D)\\right] = \\arg\\left[(z_A-z_D)(z_B-z_C)\\right] = \\arg\\left[(z_A-z_C)(z_B-z_D)\\right] \\pmod{2\\pi}"
},
{
"math_id": 66,
"text": "\\arg"
},
{
"math_id": 67,
"text": "\\overline{AB}\\cdot \\overline{CD}+\\overline{AD}\\cdot\\overline{BC} = \\overline{AC}\\cdot \\overline{BD}"
},
{
"math_id": 68,
"text": "(z_A-z_B)(z_C-z_D)+(z_A-z_D)(z_B-z_C)=(z_A-z_C)(z_B-z_D)."
},
{
"math_id": 69,
"text": "S_1,S_2,S_3,S_4"
},
{
"math_id": 70,
"text": "\\theta_1,\\theta_2,\\theta_3"
},
{
"math_id": 71,
"text": "\\theta_4"
},
{
"math_id": 72,
"text": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_1+\\theta_2)\\sin(\\theta_1+\\theta_4)"
},
{
"math_id": 73,
"text": "\\theta_1+\\theta_2+\\theta_3+\\theta_4=180^\\circ"
},
{
"math_id": 74,
"text": "\\theta_1=\\theta_3"
},
{
"math_id": 75,
"text": "\\theta_2=\\theta_4"
},
{
"math_id": 76,
"text": "\\theta_1+\\theta_2=\\theta_3+\\theta_4=9\n0^\\circ"
},
{
"math_id": 77,
"text": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_1+\\theta_2)\\sin(\\theta_1+\\theta_4) "
},
{
"math_id": 78,
"text": " \\sin^2\\theta_1+\\sin^2\\theta_2=\\sin^2(\\theta_1+\\theta_2) "
},
{
"math_id": 79,
"text": " \\sin^2\\theta_1+\\cos^2\\theta_1=1 "
},
{
"math_id": 80,
"text": "2x"
},
{
"math_id": 81,
"text": "x=S_2\\cos(\\theta_2+\\theta_3)"
},
{
"math_id": 82,
"text": "\\begin{array}{lcl}\nS_1 S_3 + S_2 S_4={\\overline{AC}}\\cdot{\\overline{BD}}\\\\\n\\Rightarrow S_1 S_3+{S_2}^2={\\overline{AC}}^2\\\\\n\\Rightarrow S_1[S_1-2S_2\\cos(\\theta_2+\\theta_3)]+{S_2}^2={\\overline{AC}}^2\\\\\n\\Rightarrow {S_1}^2+{S_2}^2-2S_1 S_2\\cos(\\theta_2+\\theta_3)={\\overline{AC}}^2\\\\\n\\end{array}"
},
{
"math_id": 83,
"text": "\\theta_1+\\theta_2=\\theta_3+\\theta_4=90^\\circ. "
},
{
"math_id": 84,
"text": "\n\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_3+\\theta_2)\\sin(\\theta_3+\\theta_4)\n"
},
{
"math_id": 85,
"text": " \\cos\\theta_2\\sin\\theta_3+\\sin\\theta_2\\cos\\theta_3=\\sin(\\theta_3+\\theta_2)\\times 1 "
},
{
"math_id": 86,
"text": "\\theta_1=90^\\circ"
},
{
"math_id": 87,
"text": "\\theta_2+(\\theta_3+\\theta_4)=90^\\circ"
},
{
"math_id": 88,
"text": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_3+\\theta_2)\\sin(\\theta_3+\\theta_4) "
},
{
"math_id": 89,
"text": " \\sin\\theta_3+\\sin\\theta_2\\cos(\\theta_2+\\theta_3)=\\sin(\\theta_3+\\theta_2)\\cos\\theta_2 "
},
{
"math_id": 90,
"text": "\\sin\\theta_3=\\sin(\\theta_3+\\theta_2)\\cos\\theta_2-\\cos(\\theta_2+\\theta_3)\\sin\\theta_2 "
},
{
"math_id": 91,
"text": "\\theta_3=90^\\circ"
},
{
"math_id": 92,
"text": "\\theta_1+(\\theta_2+\\theta_4)=90^\\circ "
},
{
"math_id": 93,
"text": " \\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_3+\\theta_2)\\sin(\\theta_3+\\theta_4) "
},
{
"math_id": 94,
"text": " \\cos(\\theta_2+\\theta_4)+\\sin\\theta_2\\sin\\theta_4=\\cos\\theta_2\\cos\\theta_4 "
},
{
"math_id": 95,
"text": "\\cos(\\theta_2+\\theta_4)=\\cos\\theta_2\\cos\\theta_4-\\sin\\theta_2\\sin\\theta_4 "
},
{
"math_id": 96,
"text": "\\overline{AB}\\cdot \\overline{CD}+\\overline{BC}\\cdot \\overline{DA} \\ge \\overline{AC}\\cdot \\overline{BD}"
},
{
"math_id": 97,
"text": "\n\\frac{AC}{BD}=\\frac{AB \\cdot DA + BC \\cdot CD}{AB \\cdot BC + DA \\cdot CD}"
},
{
"math_id": 98,
"text": "ABC"
},
{
"math_id": 99,
"text": "\\mathcal {A} = \\frac{AB \\cdot BC \\cdot CA}{4R}"
},
{
"math_id": 100,
"text": " \\mathcal {A}_\\text{tot} = \\frac{AB \\cdot BC \\cdot CA }{4R} + \\frac {CD \\cdot DA \\cdot AC}{4R} = \\frac {AC \\cdot (AB \\cdot BC + CD \\cdot DA)}{4R}\n"
},
{
"math_id": 101,
"text": "\\mathcal {A}_\\text{tot} = \\frac{AB \\cdot BD \\cdot DA}{4R} + \\frac {BC \\cdot CD \\cdot DB}{4R} = \\frac {BD \\cdot (AB \\cdot DA + BC \\cdot CD)}{4R}\n"
},
{
"math_id": 102,
"text": "\n\\begin{align}\nAC^2 & =AC \\cdot BD \\cdot \\frac{AC}{BD}=(AB \\cdot CD + BC \\cdot DA)\\frac{AB \\cdot DA + BC \\cdot CD}{AB \\cdot BC + DA \\cdot CD} \\\\[8pt]\nBD^2 & =\\frac {AC \\cdot BD}{\\frac{AC}{BD}}=(AB \\cdot CD + BC \\cdot DA)\\frac{AB \\cdot BC + DA \\cdot CD}{AB \\cdot DA + BC \\cdot CD}\n\\end{align}\n"
}
] |
https://en.wikipedia.org/wiki?curid=1521971
|
15220286
|
Yttria-stabilized zirconia
|
Ceramic with room temperature stable cubic crystal structure
Yttria-stabilized zirconia (YSZ) is a ceramic in which the cubic crystal structure of zirconium dioxide is made stable at room temperature by an addition of yttrium oxide. These oxides are commonly called "zirconia" (ZrO2) and "yttria" (Y2O3), hence the name.
Stabilization.
Pure zirconium dioxide undergoes a phase transformation from monoclinic (stable at room temperature) to tetragonal (at about 1173 °C) and then to cubic (at about 2370 °C), according to the scheme
monoclinic (1173 °C) ↔ tetragonal (2370 °C) ↔ cubic (2690 °C) ↔ melt.
Obtaining stable sintered zirconia ceramic products is difficult because of the large volume change, about 5%, accompanying the transition from tetragonal to monoclinic. Stabilization of the cubic polymorph of zirconia over wider range of temperatures is accomplished by substitution of some of the Zr4+ ions (ionic radius of 0.82 Å, too small for ideal lattice of fluorite characteristic for the cubic zirconia) in the crystal lattice with slightly larger ions, e.g., those of Y3+ (ionic radius of 0.96 Å). The resulting doped zirconia materials are termed "stabilized zirconias".
Materials related to YSZ include calcia-, magnesia-, ceria- or alumina-stabilized zirconias, or partially stabilized zirconias (PSZ). Hafnia-stabilized zirconia has about 25% lower thermal conductivity, making it more suitable for thermal barrier applications.
Although 8–9 mol% YSZ is known to not be completely stabilized in the pure cubic YSZ phase up to temperatures above 1000 °C.
Commonly used abbreviations in conjunction with yttria-stabilized zirconia are:
Thermal expansion coefficient.
The thermal expansion coefficients depends on the modification of zirconia as follows:
Ionic conductivity and degradation.
By the addition of yttria to pure zirconia (e.g., fully stabilized YSZ) Y3+ ions replace Zr4+ on the cationic sublattice. Thereby, oxygen vacancies are generated due to charge neutrality:
formula_0
meaning that two Y3+ ions generate one vacancy on the anionic sublattice. This facilitates moderate conductivity of yttrium-stabilized zirconia for O2− ions (and thus electrical conductivity) at elevated and high temperature. This ability to conduct O2− ions makes yttria-stabilized zirconia well suited for application as solid electrolyte in solid oxide fuel cells.
For low dopant concentrations, the ionic conductivity of the stabilized zirconias increases with increasing Y2O3 content. It has a maximum around 8–9 mol% almost independent of the temperature (800–1200 °C). Unfortunately, 8–9 mol% YSZ (8YSZ, 8YDZ) also turned out to be situated in the 2-phase field (c+t) of the YSZ phase diagram at these temperatures, which causes the material's decomposition into Y-enriched and depleted regions on the nanometre scale and, consequently, the electrical degradation during operation. The microstructural and chemical changes on the nanometre scale are accompanied by the drastic decrease of the oxygen-ion conductivity of 8YSZ (degradation of 8YSZ) of about 40% at 950 °C within 2500 hours. Traces of impurities like Ni, dissolved in the 8YSZ, e.g., due to fuel-cell fabrication, can have a severe impact on the decomposition rate (acceleration of inherent decomposition of the 8YSZ by orders of magnitude) such that the degradation of conductivity even becomes problematic at low operation temperatures in the range of 500–700 °C.
Nowadays, more complex ceramics like co-doped zirconia (e.g., with scandia) are in use as solid electrolytes.
Applications.
YSZ has a number of applications:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\text{Y}_2\\text{O}_3 \\to 2\\text{Y}_\\text{Zr}' + 3\\text{O}_\\text{O}^\\text{x} + \\text{V}_\\text{O}^{\\bullet\\bullet}\n \\text{ with } [\\text{V}_\\text{O}^{\\bullet\\bullet}] = \\frac{1}{2}[\\text{Y}_\\text{Zr}'],\n"
}
] |
https://en.wikipedia.org/wiki?curid=15220286
|
152205
|
Forcing (mathematics)
|
Technique invented by Paul Cohen for proving consistency and independence results
<templatestyles src="Template:TOC_right/styles.css" />
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe formula_0 to a larger universe formula_1 by introducing a new "generic" object formula_2.
Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing.
Intuition.
Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least formula_3 of them), identified with subsets of the set formula_4 of natural numbers, that were not there in the old universe, and thereby violate the continuum hypothesis.
In order to intuitively justify such an expansion, it is best to think of the "old universe" as a model formula_5 of the set theory, which is itself a set in the "real universe" formula_0. By the Löwenheim–Skolem theorem, formula_5 can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in formula_0) of formula_4 that are not in formula_5. Specifically, there is an ordinal formula_6 that "plays the role of the cardinal formula_3" in formula_5, but is actually countable in formula_0. Working in formula_0, it should be easy to find one distinct subset of formula_4 per each element of formula_6. (For simplicity, this family of subsets can be characterized with a single subset formula_7.)
However, in some sense, it may be desirable to "construct the expanded model formula_8 within formula_5". This would help ensure that formula_8 "resembles" formula_5 in certain aspects, such as formula_9 being the same as formula_6 (more generally, that "cardinal collapse" does not occur), and allow fine control over the properties of formula_8. More precisely, every member of formula_8 should be given a (non-unique) "name" in formula_5. The name can be thought as an expression in terms of formula_10, just like in a simple field extension formula_11 every element of formula_12 can be expressed in terms of formula_13. A major component of forcing is manipulating those names within formula_5, so sometimes it may help to directly think of formula_5 as "the universe", knowing that the theory of forcing guarantees that formula_8 will correspond to an actual model.
A subtle point of forcing is that, if formula_10 is taken to be an "arbitrary" "missing subset" of some set in formula_5, then the formula_8 constructed "within formula_5" may not even be a model. This is because formula_10 may encode "special" information about formula_5 that is invisible within formula_5 (e.g. the countability of formula_5), and thus prove the existence of sets that are "too complex for formula_5 to describe".
Forcing avoids such problems by requiring the newly introduced set formula_10 to be a generic set relative to formula_5. Some statements are "forced" to hold for any generic formula_10: For example, a generic formula_10 is "forced" to be infinite. Furthermore, any property (describable in formula_5) of a generic set is "forced" to hold under some forcing condition. The concept of "forcing" can be defined within formula_5, and it gives formula_5 enough reasoning power to prove that formula_8 is indeed a model that satisfies the desired properties.
Cohen's original technique, now called ramified forcing, is slightly different from the unramified forcing expounded here. Forcing is also equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.
The role of the model.
In order for the above approach to work smoothly, formula_5 must in fact be a standard transitive model in formula_0, so that membership and other elementary notions can be handled intuitively in both formula_5 and formula_0. A standard transitive model can be obtained from any standard model through the Mostowski collapse lemma, but the existence of any standard model of formula_14 (or any variant thereof) is in itself a stronger assumption than the consistency of formula_14.
To get around this issue, a standard technique is to let formula_5 be a standard transitive model of an arbitrary finite subset of formula_14 (any axiomatization of formula_14 has at least one axiom schema, and thus an infinite number of axioms), the existence of which is guaranteed by the reflection principle. As the goal of a forcing argument is to prove consistency results, this is enough since any inconsistency in a theory must manifest with a derivation of a finite length, and thus involve only a finite number of axioms.
Forcing conditions and forcing posets.
Each forcing condition can be regarded as a "finite" piece of information regarding the object formula_10 adjoined to the model. There are many different ways of providing information about an object, which give rise to different forcing notions. A general approach to formalizing forcing notions is to regard forcing conditions as abstract objects with a poset structure.
A forcing poset is an ordered triple, formula_15, where formula_16 is a preorder on formula_17, and formula_18 is the largest element. Members of formula_17 are the forcing conditions (or just conditions). The order relation formula_19 means "formula_20 is stronger than formula_21". (Intuitively, the "smaller" condition provides "more" information, just as the smaller interval formula_22 provides more information about the number π than the interval formula_23 does.) Furthermore, the preorder formula_16 must be atomless, meaning that it must satisfy the splitting condition:
In other words, it must be possible to strengthen any forcing condition formula_29 in at least two incompatible directions. Intuitively, this is because formula_29 is only a finite piece of information, whereas an infinite piece of information is needed to determine formula_10.
There are various conventions in use. Some authors require formula_16 to also be antisymmetric, so that the relation is a partial order. Some use the term partial order anyway, conflicting with standard terminology, while some use the term preorder. The largest element can be dispensed with. The reverse ordering is also used, most notably by Saharon Shelah and his co-authors.
Examples.
Let formula_30 be any infinite set (such as formula_4), and let the generic object in question be a new subset formula_31. In Cohen's original formulation of forcing, each forcing condition is a "finite" set of sentences, either of the form formula_32 or formula_33, that are self-consistent (i.e. formula_32 "and" formula_33 for the same value of formula_34 do not appear in the same condition). This forcing notion is usually called Cohen forcing.
The forcing poset for Cohen forcing can be formally written as formula_35, the finite partial functions from formula_36 to formula_37 under "reverse" inclusion. Cohen forcing satisfies the splitting condition because given any condition formula_29, one can always find an element formula_38 not mentioned in formula_29, and add either the sentence formula_32 or formula_33 to formula_29 to get two new forcing conditions, incompatible with each other.
Another instructive example of a forcing poset is formula_39, where formula_40 and formula_41 is the collection of Borel subsets of formula_42 having non-zero Lebesgue measure. The generic object associated with this forcing poset is a random real number formula_43. It can be shown that formula_44 falls in every Borel subset of formula_45 with measure 1, provided that the Borel subset is "described" in the original unexpanded universe (this can be formalized with the concept of "Borel codes"). Each forcing condition can be regarded as a random event with probability equal to its measure. Due to the ready intuition this example can provide, probabilistic language is sometimes used with other divergent forcing posets.
Generic filters.
Even though each individual forcing condition formula_29 cannot fully determine the generic object formula_10, the set formula_46 of all true forcing conditions does determine formula_10. In fact, without loss of generality, formula_2 is commonly considered to "be" the generic object adjoined to formula_5, so the expanded model is called formula_47. It is usually easy enough to show that the originally desired object formula_10 is indeed in the model formula_47.
Under this convention, the concept of "generic object" can be described in a general way. Specifically, the set formula_2 should be a generic filter on formula_48 relative to formula_5. The "filter" condition means that it makes sense that formula_2 is a set of all true forcing conditions:
For formula_56 to be "generic relative to formula_5" means:
Given that formula_5 is a countable model, the existence of a generic filter formula_56 follows from the Rasiowa–Sikorski lemma. In fact, slightly more is true: Given a condition formula_24, one can find a generic filter formula_56 such that formula_61. Due to the splitting condition on formula_48, if formula_56 is a filter, then formula_62 is dense. If formula_63, then formula_64 because formula_65 is a model of formula_66. For this reason, a generic filter is never in formula_65.
P-names and interpretations.
Associated with a forcing poset formula_17 is the class formula_67 of formula_17-names. A formula_17-name is a set formula_68 of the form
formula_69
Given any filter formula_56 on formula_17, the interpretation or valuation map from formula_17-names is given by
formula_70
The formula_17-names are, in fact, an expansion of the universe. Given formula_71, one defines formula_72 to be the formula_17-name
formula_73
Since formula_74, it follows that formula_75. In a sense, formula_76 is a "name for formula_77" that does not depend on the specific choice of formula_2.
This also allows defining a "name for formula_2" without explicitly referring to formula_2:
formula_78
so that formula_79.
Rigorous definitions.
The concepts of formula_48-names, interpretations, and formula_76 may be defined by transfinite recursion. With formula_80 the empty set, formula_81 the successor ordinal to ordinal formula_82, formula_83 the power-set operator, and formula_84 a limit ordinal, define the following hierarchy:
formula_85
Then the class of formula_17-names is defined as
formula_86
The interpretation map and the map formula_87 can similarly be defined with a hierarchical construction.
Forcing.
Given a generic filter formula_88, one proceeds as follows. The subclass of formula_17-names in formula_65 is denoted formula_89. Let
formula_90
To reduce the study of the set theory of formula_91 to that of formula_65, one works with the "forcing language", which is built up like ordinary first-order logic, with membership as the binary relation and all the formula_17-names as constants.
Define formula_92 (to be read as "formula_29 forces formula_93 in the model formula_65 with poset formula_17"), where formula_20 is a condition, formula_93 is a formula in the forcing language, and the formula_94's are formula_17-names, to mean that if formula_56 is a generic filter containing formula_20, then formula_95. The special case formula_96 is often written as "formula_97" or simply "formula_98". Such statements are true in formula_91, no matter what formula_56 is.
What is important is that this external definition of the forcing relation formula_99 is equivalent to an internal definition within formula_65, defined by transfinite induction (specifically formula_100-induction) over the formula_17-names on instances of formula_101 and formula_102, and then by ordinary induction over the complexity of formulae. This has the effect that all the properties of formula_91 are really properties of formula_65, and the verification of formula_66 in formula_91 becomes straightforward. This is usually summarized as the following three key properties:
Internal definition.
There are many different but equivalent ways to define the forcing relation formula_105 in formula_5. One way to simplify the definition is to first define a modified forcing relation formula_106 that is strictly stronger than formula_105. The modified relation formula_106 still satisfies the three key properties of forcing, but formula_107 and formula_108 are not necessarily equivalent even if the first-order formulae formula_109 and formula_110 are equivalent. The unmodified forcing relation can then be defined as
formula_111
In fact, Cohen's original concept of forcing is essentially formula_106 rather than formula_105.
The modified forcing relation formula_106 can be defined recursively as follows:
Other symbols of the forcing language can be defined in terms of these symbols: For example, formula_122 means formula_123, formula_124 means formula_125, etc. Cases 1 and 2 depend on each other and on case 3, but the recursion always refer to formula_48-names with lesser ranks, so transfinite induction allows the definition to go through.
By construction, formula_106 (and thus formula_105) automatically satisfies Definability. The proof that formula_106 also satisfies Truth and Coherence is by inductively inspecting each of the five cases above. Cases 4 and 5 are trivial (thanks to the choice of formula_126 and formula_127 as the elementary symbols), cases 1 and 2 relies only on the assumption that formula_2 is a filter, and only case 3 requires formula_2 to be a "generic" filter.
Formally, an internal definition of the forcing relation (such as the one presented above) is actually a transformation of an arbitrary formula formula_128 to another formula formula_129 where formula_29 and formula_48 are additional variables. The model formula_5 does not explicitly appear in the transformation (note that within formula_5, formula_130 just means "formula_131 is a formula_48-name"), and indeed one may take this transformation as a "syntactic" definition of the forcing relation in the universe formula_0 of all sets regardless of any countable transitive model. However, if one wants to force over some countable transitive model formula_5, then the latter formula should be interpreted under formula_5 (i.e. with all quantifiers ranging only over formula_5), in which case it is equivalent to the external "semantic" definition of formula_105 described at the top of this section:
For any formula formula_128 there is a theorem formula_132 of the theory formula_14 (for example conjunction of finite number of axioms) such that for any countable transitive model formula_5 such that formula_133 and any atomless partial order formula_134 and any formula_48-generic filter formula_2 over formula_5 formula_135
This the sense under which the forcing relation is indeed "definable in formula_5".
Consistency.
The discussion above can be summarized by the fundamental consistency result that, given a forcing poset formula_17, we may assume the existence of a generic filter formula_56, not belonging to the universe formula_136, such that formula_137 is again a set-theoretic universe that models formula_66. Furthermore, all truths in formula_137 may be reduced to truths in formula_136 involving the forcing relation.
Both styles, adjoining formula_56 to either a countable transitive model formula_65 or the whole universe formula_136, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of Boolean-valued analysis.
Cohen forcing.
The simplest nontrivial forcing poset is formula_138, the finite partial functions from formula_139 to formula_37 under "reverse" inclusion. That is, a condition formula_20 is essentially two disjoint finite subsets formula_140 and formula_141 of formula_139, to be thought of as the "yes" and "no" parts of formula_20, with no information provided on values outside the domain of formula_20. "formula_21 is stronger than formula_20" means that formula_142, in other words, the "yes" and "no" parts of formula_21 are supersets of the "yes" and "no" parts of formula_20, and in that sense, provide more information.
Let formula_56 be a generic filter for this poset. If formula_20 and formula_21 are both in formula_56, then formula_143 is a condition because formula_56 is a filter. This means that formula_144 is a well-defined partial function from formula_139 to formula_145 because any two conditions in formula_56 agree on their common domain.
In fact, formula_146 is a total function. Given formula_147, let formula_148. Then formula_149 is dense. (Given any formula_20, if formula_150 is not in formula_20's domain, adjoin a value for formula_150—the result is in formula_149.) A condition formula_151 has formula_150 in its domain, and since formula_152, we find that formula_153 is defined.
Let formula_154, the set of all "yes" members of the generic conditions. It is possible to give a name for formula_155 directly. Let
formula_156
Then formula_157 Now suppose that formula_158 in formula_136. We claim that formula_159. Let
formula_160
Then formula_161 is dense. (Given any formula_20, find formula_150 that is not in its domain, and adjoin a value for formula_150 contrary to the status of "formula_162".) Then any formula_163 witnesses formula_159. To summarize, formula_155 is a "new" subset of formula_139, necessarily infinite.
Replacing formula_139 with formula_164, that is, consider instead finite partial functions whose inputs are of the form formula_165, with formula_166 and formula_167, and whose outputs are formula_168 or formula_169, one gets formula_170 new subsets of formula_139. They are all distinct, by a density argument: Given formula_171, let
formula_172
then each formula_173 is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the formula_174th new set.
This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introduced which map formula_139 onto formula_175, or formula_175 onto formula_170. For example, if one considers instead formula_176, finite partial functions from formula_139 to formula_175, the first uncountable ordinal, one gets in formula_137 a bijection from formula_139 to formula_175. In other words, formula_175 has "collapsed", and in the forcing extension, is a countable ordinal.
The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does not collapse cardinals. For this, a sufficient combinatorial property is that all of the antichains of the forcing poset are countable.
The countable chain condition.
An (strong) antichain formula_68 of formula_17 is a subset such that if formula_177 and formula_178, then formula_20 and formula_21 are incompatible (written formula_179), meaning there is no formula_180 in formula_17 such that formula_181 and formula_182. In the example on Borel sets, incompatibility means that formula_183 has zero measure. In the example on finite partial functions, incompatibility means that formula_143 is not a function, in other words, formula_20 and formula_21 assign different values to some domain input.
formula_17 satisfies the countable chain condition (c.c.c.) if and only if every antichain in formula_17 is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Some mathematicians write "c.a.c." for "countable antichain condition".)
It is easy to see that formula_41 satisfies the c.c.c. because the measures add up to at most formula_169. Also, formula_184 satisfies the c.c.c., but the proof is more difficult.
Given an uncountable subfamily formula_185, shrink formula_186 to an uncountable subfamily formula_187 of sets of size formula_150, for some formula_188. If formula_189 for uncountably many formula_190, shrink this to an uncountable subfamily formula_191 and repeat, getting a finite set formula_192 and an uncountable family formula_193 of incompatible conditions of size formula_194 such that every formula_195 is in formula_196 for at most countable many formula_197. Now, pick an arbitrary formula_197, and pick from formula_193 any formula_21 that is not one of the countably many members that have a domain member in common with formula_20. Then formula_198 and formula_199 are compatible, so formula_186 is not an antichain. In other words, formula_184-antichains are countable.
The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A "maximal" antichain formula_68 is one that cannot be extended to a larger antichain. This means that every element formula_24 is compatible with some member of formula_68. The existence of a maximal antichain follows from Zorn's Lemma. Given a maximal antichain formula_68, let
formula_200
Then formula_201 is dense, and formula_60 if and only if formula_202. Conversely, given a dense set formula_201, Zorn's Lemma shows that there exists a maximal antichain formula_203, and then formula_60 if and only if formula_202.
Assume that formula_17 satisfies the c.c.c. Given formula_204, with formula_205 a function in formula_137, one can approximate formula_206 inside formula_136 as follows. Let formula_207 be a name for formula_206 (by the definition of formula_137) and let formula_20 be a condition that forces formula_207 to be a function from formula_208 to formula_209. Define a function formula_210, by
formula_211
By the definability of forcing, this definition makes sense within formula_136. By the coherence of forcing, a different formula_212 come from an incompatible formula_20. By c.c.c., formula_213 is countable.
In summary, formula_206 is unknown in formula_136 as it depends on formula_56, but it is not wildly unknown for a c.c.c.-forcing. One can identify a countable set of guesses for what the value of formula_206 is at any input, independent of formula_56.
This has the following very important consequence. If in formula_137, formula_214 is a surjection from one infinite ordinal onto another, then there is a surjection formula_215 in formula_136, and consequently, a surjection formula_216 in formula_136. In particular, cardinals cannot collapse. The conclusion is that formula_217 in formula_137.
Easton forcing.
The exact value of the continuum in the above Cohen model, and variants like formula_218 for cardinals formula_219 in general, was worked out by Robert M. Solovay, who also worked out how to violate formula_220 (the generalized continuum hypothesis), for regular cardinals only, a finite number of times. For example, in the above Cohen model, if formula_221 holds in formula_136, then formula_222 holds in formula_137.
William B. Easton worked out the proper class version of violating the formula_220 for regular cardinals, basically showing that the known restrictions, (monotonicity, Cantor's Theorem and König's Theorem), were the only formula_66-provable restrictions (see Easton's Theorem).
Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions fails to give a model of formula_66. For example, forcing with formula_223, where formula_224 is the proper class of all ordinals, makes the continuum a proper class. On the other hand, forcing with formula_225 introduces a countable enumeration of the ordinals. In both cases, the resulting formula_137 is visibly not a model of formula_66.
At one time, it was thought that more sophisticated forcing would also allow an arbitrary variation in the powers of singular cardinals. However, this has turned out to be a difficult, subtle and even surprising problem, with several more restrictions provable in formula_66 and with the forcing models depending on the consistency of various large-cardinal properties. Many open problems remain.
Random reals.
Random forcing can be defined as forcing over the set formula_226 of all compact subsets of formula_227 of positive measure ordered by relation formula_228 (smaller set in context of inclusion is smaller set in ordering and represents condition with more information). There are two types of important dense sets:
For any filter formula_2 and for any finitely many elements formula_234 there is formula_235 such that holds formula_236. In case of this ordering, this means that any filter is set of compact sets with finite intersection property. For this reason, intersection of all elements of any filter is nonempty. If formula_2 is a filter intersecting the dense set formula_237 for any positive integer formula_229, then the filter formula_2 contains conditions of arbitrarily small positive diameter. Therefore, the intersection of all conditions from formula_2 has diameter 0. But the only nonempty sets of diameter 0 are singletons. So there is exactly one real number formula_238 such that formula_239.
Let formula_240 be any Borel set of measure 1. If formula_2 intersects formula_241, then formula_242.
However, a generic filter over a countable transitive model formula_0 is not in formula_0. The real formula_238 defined by formula_2 is provably not an element of formula_0. The problem is that if formula_243, then formula_244 "formula_29 is compact", but from the viewpoint of some larger universe formula_245, formula_29 can be non-compact and the intersection of all conditions from the generic filter formula_2 is actually empty. For this reason, we consider the set formula_246 of topological closures of conditions from G (i.e., formula_247). Because of formula_248 and the finite intersection property of formula_2, the set formula_249 also has the finite intersection property. Elements of the set formula_249 are bounded closed sets as closures of bounded sets. Therefore, formula_249
is a set of compact sets with the finite intersection property and thus has nonempty intersection. Since formula_250 and the ground model formula_0 inherits a metric from the universe formula_251, the set formula_249 has elements of arbitrarily small diameter. Finally, there is exactly one real that belongs to all members of the set formula_249. The generic filter formula_2 can be reconstructed from formula_238 as formula_252.
If formula_34 is name of formula_238, and for formula_253 holds formula_244"formula_254 is Borel set of measure 1", then holds
formula_255
for some formula_256. There is name formula_34 such that for any generic filter formula_2 holds
formula_257
Then
formula_255
holds for any condition formula_29.
Every Borel set can, non-uniquely, be built up, starting from intervals with rational endpoints and applying the operations of complement and countable unions, a countable number of times. The record of such a construction is called a "Borel code". Given a Borel set formula_254 in formula_0, one recovers a Borel code, and then applies the same construction sequence in formula_1, getting a Borel set formula_258. It can be proven that one gets the same set independent of the construction of formula_259, and that basic properties are preserved. For example, if formula_260, then formula_261. If formula_254 has measure zero, then formula_258 has measure zero. This mapping formula_262 is injective.
For any set formula_240 such that formula_253 and formula_244"formula_254 is a Borel set of measure 1" holds formula_263.
This means that formula_44 is "infinite random sequence of 0s and 1s" from the viewpoint of formula_0, which means that it satisfies all statistical tests from the ground model formula_0.
So given formula_44, a random real, one can show that
formula_264
Because of the mutual inter-definability between formula_44 and formula_56, one generally writes formula_265 for formula_1.
A different interpretation of reals in formula_1 was provided by Dana Scott. Rational numbers in formula_137 have names that correspond to countably-many distinct rational values assigned to a maximal antichain of Borel sets – in other words, a certain rational-valued function on formula_266. Real numbers in formula_1 then correspond to Dedekind cuts of such functions, that is, measurable functions.
Boolean-valued models.
Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In these, any statement is assigned a truth value from some complete atomless Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model that contains this ultrafilter, which can be understood as a new model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in an appropriate way, we can get a model that has the desired property. In it, only statements that must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).
Meta-mathematical explanation.
In forcing, we usually seek to show that some sentence is consistent with formula_66 (or optionally some extension of formula_66). One way to interpret the argument is to assume that formula_66 is consistent and then prove that formula_66 combined with the new sentence is also consistent.
Each "condition" is a finite piece of information – the idea is that only finite pieces are relevant for consistency, since, by the compactness theorem, a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then we can pick an infinite set of consistent conditions to extend our model. Therefore, assuming the consistency of formula_66, we prove the consistency of formula_66 extended by this infinite set.
Logical explanation.
By Gödel's second incompleteness theorem, one cannot prove the consistency of any sufficiently strong formal theory, such as formula_66, using only the axioms of the theory itself, unless the theory is inconsistent. Consequently, mathematicians do not attempt to prove the consistency of formula_66 using only the axioms of formula_66, or to prove that formula_267 is consistent for any hypothesis formula_268 using only formula_267. For this reason, the aim of a consistency proof is to prove the consistency of formula_267 relative to the consistency of formula_66. Such problems are known as problems of relative consistency, one of which proves
The general schema of relative consistency proofs follows. As any proof is finite, it uses only a finite number of axioms:
formula_269
For any given proof, formula_66 can verify the validity of this proof. This is provable by induction on the length of the proof.
formula_270
Then resolve
formula_271
By proving the following
it can be concluded that
formula_272
which is equivalent to
formula_273
which gives (*). The core of the relative consistency proof is proving (**). A formula_66 proof of formula_274 can be constructed for any given finite subset formula_275 of the formula_66 axioms (by formula_66 instruments of course). (No universal proof of formula_274 of course.)
In formula_66, it is provable that for any condition formula_20, the set of formulas (evaluated by names) forced by formula_20 is deductively closed. Furthermore, for any formula_66 axiom, formula_66 proves that this axiom is forced by formula_18. Then it suffices to prove that there is at least one condition that forces formula_268.
In the case of Boolean-valued forcing, the procedure is similar: proving that the Boolean value of formula_268 is not formula_276.
Another approach uses the Reflection Theorem. For any given finite set of formula_66 axioms, there is a formula_66 proof that this set of axioms has a countable transitive model. For any given finite set formula_275 of formula_66 axioms, there is a finite set formula_277 of formula_66 axioms such that formula_66 proves that if a countable transitive model formula_65 satisfies formula_277, then formula_91 satisfies formula_275. By proving that there is finite set formula_278 of formula_66 axioms such that if a countable transitive model formula_65 satisfies formula_278, then formula_91 satisfies the hypothesis formula_268. Then, for any given finite set formula_275 of formula_66 axioms, formula_66 proves formula_274.
Sometimes in (**), a stronger theory formula_36 than formula_66 is used for proving formula_274. Then we have proof of the consistency of formula_267 relative to the consistency of formula_36. Note that formula_279, where formula_280 is formula_281 (the axiom of constructibility).
Notes.
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{
"math_id": 0,
"text": "V"
},
{
"math_id": 1,
"text": "V[G]"
},
{
"math_id": 2,
"text": "G"
},
{
"math_id": 3,
"text": "\\aleph_2"
},
{
"math_id": 4,
"text": "\\mathbb{N}"
},
{
"math_id": 5,
"text": "M"
},
{
"math_id": 6,
"text": "\\aleph_2^M"
},
{
"math_id": 7,
"text": "X \\subseteq \\aleph_2^M \\times \\mathbb{N}"
},
{
"math_id": 8,
"text": "M[X]"
},
{
"math_id": 9,
"text": "\\aleph_2^{M[X]}"
},
{
"math_id": 10,
"text": "X"
},
{
"math_id": 11,
"text": "L = K(\\theta)"
},
{
"math_id": 12,
"text": "L"
},
{
"math_id": 13,
"text": "\\theta"
},
{
"math_id": 14,
"text": "\\mathsf{ZFC}"
},
{
"math_id": 15,
"text": " (\\mathbb{P},\\leq,\\mathbf{1}) "
},
{
"math_id": 16,
"text": " \\leq "
},
{
"math_id": 17,
"text": " \\mathbb{P} "
},
{
"math_id": 18,
"text": " \\mathbf{1} "
},
{
"math_id": 19,
"text": " p \\leq q "
},
{
"math_id": 20,
"text": " p "
},
{
"math_id": 21,
"text": " q "
},
{
"math_id": 22,
"text": " [3.1415926,3.1415927] "
},
{
"math_id": 23,
"text": " [3.1,3.2] "
},
{
"math_id": 24,
"text": " p \\in \\mathbb{P} "
},
{
"math_id": 25,
"text": " q,r \\in \\mathbb{P} "
},
{
"math_id": 26,
"text": " q,r \\leq p "
},
{
"math_id": 27,
"text": " s \\in \\mathbb{P} "
},
{
"math_id": 28,
"text": " s \\leq q,r "
},
{
"math_id": 29,
"text": "p"
},
{
"math_id": 30,
"text": "S"
},
{
"math_id": 31,
"text": "X \\subseteq S"
},
{
"math_id": 32,
"text": "a \\in X"
},
{
"math_id": 33,
"text": "a \\notin X"
},
{
"math_id": 34,
"text": "a"
},
{
"math_id": 35,
"text": " (\\operatorname{Fin}(S,2),\\supseteq,0) "
},
{
"math_id": 36,
"text": " S "
},
{
"math_id": 37,
"text": " 2 ~ \\stackrel{\\text{df}}{=} ~ \\{ 0,1 \\} "
},
{
"math_id": 38,
"text": "a \\in S"
},
{
"math_id": 39,
"text": " (\\operatorname{Bor}(I),\\subseteq,I) "
},
{
"math_id": 40,
"text": " I = [0,1] "
},
{
"math_id": 41,
"text": " \\operatorname{Bor}(I) "
},
{
"math_id": 42,
"text": " I "
},
{
"math_id": 43,
"text": "r \\in [0, 1]"
},
{
"math_id": 44,
"text": "r"
},
{
"math_id": 45,
"text": "[0, 1]"
},
{
"math_id": 46,
"text": "G \\subseteq \\mathbb{P}"
},
{
"math_id": 47,
"text": "M[G]"
},
{
"math_id": 48,
"text": "\\mathbb{P}"
},
{
"math_id": 49,
"text": " G \\subseteq \\mathbb{P}; "
},
{
"math_id": 50,
"text": " \\mathbf{1} \\in G; "
},
{
"math_id": 51,
"text": " p \\geq q \\in G "
},
{
"math_id": 52,
"text": " p \\in G; "
},
{
"math_id": 53,
"text": " p,q \\in G "
},
{
"math_id": 54,
"text": " r \\in G "
},
{
"math_id": 55,
"text": " r \\leq p,q. "
},
{
"math_id": 56,
"text": " G "
},
{
"math_id": 57,
"text": " D \\in M "
},
{
"math_id": 58,
"text": " q \\in D "
},
{
"math_id": 59,
"text": " q \\leq p "
},
{
"math_id": 60,
"text": " G \\cap D \\neq \\varnothing "
},
{
"math_id": 61,
"text": " p \\in G "
},
{
"math_id": 62,
"text": " \\mathbb{P} \\setminus G "
},
{
"math_id": 63,
"text": " G \\in M "
},
{
"math_id": 64,
"text": " \\mathbb{P} \\setminus G \\in M "
},
{
"math_id": 65,
"text": " M "
},
{
"math_id": 66,
"text": " \\mathsf{ZFC} "
},
{
"math_id": 67,
"text": " V^{(\\mathbb{P})} "
},
{
"math_id": 68,
"text": " A "
},
{
"math_id": 69,
"text": " A \\subseteq \\{ (u,p) \\mid u ~ \\text{is a} ~ \\mathbb{P} \\text{-name and} ~ p \\in \\mathbb{P} \\}. "
},
{
"math_id": 70,
"text": " \\operatorname{val}(u,G) = \\{ \\operatorname{val}(v,G) \\mid \\exists p \\in G: ~ (v,p) \\in u \\}. "
},
{
"math_id": 71,
"text": " x \\in V "
},
{
"math_id": 72,
"text": " \\check{x} "
},
{
"math_id": 73,
"text": " \\check{x} = \\{ (\\check{y},\\mathbf{1}) \\mid y \\in x \\}. "
},
{
"math_id": 74,
"text": " \\mathbf{1} \\in G "
},
{
"math_id": 75,
"text": " \\operatorname{val}(\\check{x},G) = x "
},
{
"math_id": 76,
"text": "\\check{x}"
},
{
"math_id": 77,
"text": "x"
},
{
"math_id": 78,
"text": " \\underline{G} = \\{ (\\check{p},p) \\mid p \\in \\mathbb{P} \\} "
},
{
"math_id": 79,
"text": " \\operatorname{val}(\\underline{G},G) = \\{\\operatorname{val}(\\check p, G) \\mid p \\in G\\} = G "
},
{
"math_id": 80,
"text": "\\varnothing"
},
{
"math_id": 81,
"text": "\\alpha + 1"
},
{
"math_id": 82,
"text": "\\alpha"
},
{
"math_id": 83,
"text": "\\mathcal{P}"
},
{
"math_id": 84,
"text": "\\lambda"
},
{
"math_id": 85,
"text": " \\begin{align}\n\\operatorname{Name}(\\varnothing) & = \\varnothing, \\\\\n\\operatorname{Name}(\\alpha + 1) & = \\mathcal{P}(\\operatorname{Name}(\\alpha) \\times \\mathbb{P}), \\\\\n\\operatorname{Name}(\\lambda) & = \\bigcup \\{ \\operatorname{Name}(\\alpha) \\mid \\alpha < \\lambda \\}.\n\\end{align} "
},
{
"math_id": 86,
"text": " V^{(\\mathbb{P})} = \\bigcup \\{ \\operatorname{Name}(\\alpha) ~|~ \\alpha ~ \\text{is an ordinal} \\}. "
},
{
"math_id": 87,
"text": "x \\mapsto \\check{x}"
},
{
"math_id": 88,
"text": " G \\subseteq \\mathbb{P}"
},
{
"math_id": 89,
"text": " M^{(\\mathbb{P})} "
},
{
"math_id": 90,
"text": " M[G] = \\left\\{ \\operatorname{val}(u,G) ~ \\Big| ~ u \\in M^{(\\mathbb{P})} \\right\\}."
},
{
"math_id": 91,
"text": " M[G] "
},
{
"math_id": 92,
"text": " p \\Vdash_{M,\\mathbb{P}} \\varphi(u_1,\\ldots,u_n) "
},
{
"math_id": 93,
"text": " \\varphi "
},
{
"math_id": 94,
"text": " u_{i} "
},
{
"math_id": 95,
"text": " M[G] \\models \\varphi(\\operatorname{val}(u_1,G),\\ldots,\\operatorname{val}(u_{n},G)) "
},
{
"math_id": 96,
"text": " \\mathbf{1} \\Vdash_{M,\\mathbb{P}} \\varphi "
},
{
"math_id": 97,
"text": " \\mathbb{P} \\Vdash_{M,\\mathbb{P}} \\varphi "
},
{
"math_id": 98,
"text": " \\Vdash_{M,\\mathbb{P}} \\varphi "
},
{
"math_id": 99,
"text": " p \\Vdash_{M,\\mathbb{P}} \\varphi "
},
{
"math_id": 100,
"text": "\\in"
},
{
"math_id": 101,
"text": " u \\in v "
},
{
"math_id": 102,
"text": " u = v "
},
{
"math_id": 103,
"text": " M[G] \\models \\varphi(\\operatorname{val}(u_1,G),\\ldots,\\operatorname{val}(u_n,G)) "
},
{
"math_id": 104,
"text": " p \\Vdash_{M,\\mathbb{P}} \\varphi(u_1,\\ldots,u_n) \\land q \\leq p \\implies q \\Vdash_{M,\\mathbb{P}} \\varphi(u_1,\\ldots,u_n) "
},
{
"math_id": 105,
"text": "\\Vdash_{M,\\mathbb{P}}"
},
{
"math_id": 106,
"text": "\\Vdash_{M,\\mathbb{P}}^*"
},
{
"math_id": 107,
"text": "p \\Vdash_{M,\\mathbb{P}}^* \\varphi"
},
{
"math_id": 108,
"text": "p \\Vdash_{M,\\mathbb{P}}^* \\varphi'"
},
{
"math_id": 109,
"text": "\\varphi"
},
{
"math_id": 110,
"text": "\\varphi'"
},
{
"math_id": 111,
"text": "p\\Vdash_{M,\\mathbb P} \\varphi \\iff p\\Vdash_{M,\\mathbb P}^* \\neg \\neg \\varphi."
},
{
"math_id": 112,
"text": "p \\Vdash_{M,\\mathbb{P}}^* u \\in v"
},
{
"math_id": 113,
"text": "(\\exists (w, q) \\in v) (q \\ge p \\wedge p \\Vdash_{M,\\mathbb{P}}^* w = u)."
},
{
"math_id": 114,
"text": "p \\Vdash_{M,\\mathbb{P}}^* u \\ne v"
},
{
"math_id": 115,
"text": "(\\exists (w, q) \\in v) (q \\ge p \\wedge p \\Vdash_{M,\\mathbb{P}}^* w \\notin u) \\vee (\\exists (w, q) \\in u) (q \\ge p \\wedge p \\Vdash_{M,\\mathbb{P}}^* w \\notin v)."
},
{
"math_id": 116,
"text": "p \\Vdash_{M,\\mathbb{P}}^* \\neg \\varphi"
},
{
"math_id": 117,
"text": "\\neg (\\exists q \\le p) (q \\Vdash_{M,\\mathbb{P}}^* \\varphi)."
},
{
"math_id": 118,
"text": "p \\Vdash_{M,\\mathbb{P}}^* (\\varphi \\vee \\psi)"
},
{
"math_id": 119,
"text": "(p \\Vdash_{M,\\mathbb{P}}^* \\varphi) \\vee (p \\Vdash_{M,\\mathbb{P}}^* \\psi)."
},
{
"math_id": 120,
"text": "p \\Vdash_{M,\\mathbb{P}}^* \\exists x\\, \\varphi(x)"
},
{
"math_id": 121,
"text": "(\\exists u \\in M^{(\\mathbb{P})}) (p \\Vdash_{M,\\mathbb{P}}^* \\varphi(u))."
},
{
"math_id": 122,
"text": "u = v"
},
{
"math_id": 123,
"text": "\\neg (u \\ne v)"
},
{
"math_id": 124,
"text": "\\forall x\\, \\varphi(x)"
},
{
"math_id": 125,
"text": "\\neg \\exists x\\, \\neg \\varphi(x)"
},
{
"math_id": 126,
"text": "\\vee"
},
{
"math_id": 127,
"text": "\\exists"
},
{
"math_id": 128,
"text": "\\varphi(x_1,\\dots,x_n)"
},
{
"math_id": 129,
"text": "p\\Vdash_{\\mathbb{P}}\\varphi(u_1,\\dots,u_n)"
},
{
"math_id": 130,
"text": "u \\in M^{(\\mathbb{P})}"
},
{
"math_id": 131,
"text": "u"
},
{
"math_id": 132,
"text": "T"
},
{
"math_id": 133,
"text": "M\\models T"
},
{
"math_id": 134,
"text": "\\mathbb{P}\\in M"
},
{
"math_id": 135,
"text": "(\\forall a_1,\\ldots,a_n\\in M^{\\mathbb{P}})(\\forall p \\in\\mathbb{P})(p\\Vdash_{M,\\mathbb{P}} \\varphi(a_1,\\dots,a_n) \\,\\Leftrightarrow \\, M\\models p \\Vdash_{\\mathbb{P}}\\varphi(a_1, \\dots, a_n))."
},
{
"math_id": 136,
"text": " V "
},
{
"math_id": 137,
"text": " V[G] "
},
{
"math_id": 138,
"text": " (\\operatorname{Fin}(\\omega,2),\\supseteq,0) "
},
{
"math_id": 139,
"text": " \\omega "
},
{
"math_id": 140,
"text": " {p^{-1}}[1] "
},
{
"math_id": 141,
"text": " {p^{-1}}[0] "
},
{
"math_id": 142,
"text": " q \\supseteq p "
},
{
"math_id": 143,
"text": " p \\cup q "
},
{
"math_id": 144,
"text": " g = \\bigcup G "
},
{
"math_id": 145,
"text": " 2 "
},
{
"math_id": 146,
"text": " g "
},
{
"math_id": 147,
"text": " n \\in \\omega "
},
{
"math_id": 148,
"text": " D_{n} = \\{ p \\mid p(n) ~ \\text{is defined} \\} "
},
{
"math_id": 149,
"text": " D_{n} "
},
{
"math_id": 150,
"text": " n "
},
{
"math_id": 151,
"text": " p \\in G \\cap D_{n} "
},
{
"math_id": 152,
"text": " p \\subseteq g "
},
{
"math_id": 153,
"text": " g(n) "
},
{
"math_id": 154,
"text": " X = {g^{-1}}[1] "
},
{
"math_id": 155,
"text": " X "
},
{
"math_id": 156,
"text": " \\underline{X} = \\left \\{ \\left (\\check{n},p \\right ) \\mid p(n) = 1 \\right \\}."
},
{
"math_id": 157,
"text": "\\operatorname{val}(\\underline{X},G) = X."
},
{
"math_id": 158,
"text": " A \\subseteq \\omega "
},
{
"math_id": 159,
"text": " X \\neq A "
},
{
"math_id": 160,
"text": " D_{A} = \\{ p \\mid (\\exists n)(n \\in \\operatorname{Dom}(p) \\land (p(n) = 1 \\iff n \\notin A)) \\}."
},
{
"math_id": 161,
"text": "D_A"
},
{
"math_id": 162,
"text": " n \\in A "
},
{
"math_id": 163,
"text": " p \\in G \\cap D_A"
},
{
"math_id": 164,
"text": " \\omega \\times \\omega_{2} "
},
{
"math_id": 165,
"text": " (n,\\alpha) "
},
{
"math_id": 166,
"text": " n < \\omega "
},
{
"math_id": 167,
"text": " \\alpha < \\omega_{2} "
},
{
"math_id": 168,
"text": " 0 "
},
{
"math_id": 169,
"text": " 1 "
},
{
"math_id": 170,
"text": " \\omega_{2} "
},
{
"math_id": 171,
"text": " \\alpha < \\beta < \\omega_{2} "
},
{
"math_id": 172,
"text": " D_{\\alpha,\\beta} = \\{ p \\mid (\\exists n)(p(n,\\alpha) \\neq p(n,\\beta)) \\},"
},
{
"math_id": 173,
"text": " D_{\\alpha,\\beta} "
},
{
"math_id": 174,
"text": " \\beta "
},
{
"math_id": 175,
"text": " \\omega_{1} "
},
{
"math_id": 176,
"text": " \\operatorname{Fin}(\\omega,\\omega_{1}) "
},
{
"math_id": 177,
"text": " p,q \\in A "
},
{
"math_id": 178,
"text": " p \\ne q "
},
{
"math_id": 179,
"text": " p \\perp q "
},
{
"math_id": 180,
"text": " r "
},
{
"math_id": 181,
"text": " r \\leq p "
},
{
"math_id": 182,
"text": " r \\leq q "
},
{
"math_id": 183,
"text": " p \\cap q "
},
{
"math_id": 184,
"text": " \\operatorname{Fin}(E,2) "
},
{
"math_id": 185,
"text": " W \\subseteq \\operatorname{Fin}(E,2) "
},
{
"math_id": 186,
"text": " W "
},
{
"math_id": 187,
"text": " W_{0} "
},
{
"math_id": 188,
"text": "n < \\omega "
},
{
"math_id": 189,
"text": " p(e_{1}) = b_{1} "
},
{
"math_id": 190,
"text": " p \\in W_{0} "
},
{
"math_id": 191,
"text": " W_{1} "
},
{
"math_id": 192,
"text": " \\{ (e_{1},b_{1}),\\ldots,(e_{k},b_{k}) \\} "
},
{
"math_id": 193,
"text": " W_{k} "
},
{
"math_id": 194,
"text": " n - k "
},
{
"math_id": 195,
"text": " e "
},
{
"math_id": 196,
"text": " \\operatorname{Dom}(p) "
},
{
"math_id": 197,
"text": " p \\in W_{k} "
},
{
"math_id": 198,
"text": " p \\cup \\{ (e_{1},b_{1}),\\ldots,(e_{k},b_{k}) \\} "
},
{
"math_id": 199,
"text": "q \\cup \\{ (e_{1},b_{1}),\\ldots,(e_{k},b_{k}) \\} "
},
{
"math_id": 200,
"text": " D = \\left \\{ p \\in \\mathbb{P} \\mid (\\exists q \\in A)(p \\leq q) \\right \\}."
},
{
"math_id": 201,
"text": " D "
},
{
"math_id": 202,
"text": " G \\cap A \\neq \\varnothing "
},
{
"math_id": 203,
"text": " A \\subseteq D "
},
{
"math_id": 204,
"text": " x,y \\in V "
},
{
"math_id": 205,
"text": " f: x \\to y "
},
{
"math_id": 206,
"text": " f "
},
{
"math_id": 207,
"text": " u "
},
{
"math_id": 208,
"text": " x "
},
{
"math_id": 209,
"text": " y "
},
{
"math_id": 210,
"text": " F: x \\to \\mathcal{P}(y) "
},
{
"math_id": 211,
"text": " F(a) \\stackrel{\\text{df}}{=} \\left \\{ b \\left | (\\exists q \\in \\mathbb{P}) \\left [(q \\leq p) \\land \\left (q \\Vdash ~ u \\left (\\check{a} \\right ) = \\check{b} \\right ) \\right ] \\right \\}. \\right."
},
{
"math_id": 212,
"text": " b "
},
{
"math_id": 213,
"text": " F(a) "
},
{
"math_id": 214,
"text": " f: \\alpha \\to \\beta "
},
{
"math_id": 215,
"text": " g: \\omega \\times \\alpha \\to \\beta "
},
{
"math_id": 216,
"text": " h: \\alpha \\to \\beta "
},
{
"math_id": 217,
"text": "2^{\\aleph_{0}} \\geq \\aleph_{2} "
},
{
"math_id": 218,
"text": " \\operatorname{Fin}(\\omega \\times \\kappa,2) "
},
{
"math_id": 219,
"text": " \\kappa "
},
{
"math_id": 220,
"text": " \\mathsf{GCH} "
},
{
"math_id": 221,
"text": " \\mathsf{CH} "
},
{
"math_id": 222,
"text": " 2^{\\aleph_{0}} = \\aleph_{2} "
},
{
"math_id": 223,
"text": " \\operatorname{Fin}(\\omega \\times \\mathbf{On},2) "
},
{
"math_id": 224,
"text": " \\mathbf{On} "
},
{
"math_id": 225,
"text": " \\operatorname{Fin}(\\omega,\\mathbf{On}) "
},
{
"math_id": 226,
"text": "P"
},
{
"math_id": 227,
"text": "[0,1]"
},
{
"math_id": 228,
"text": "\\subseteq"
},
{
"math_id": 229,
"text": "n"
},
{
"math_id": 230,
"text": "D_n= \\left \\{p\\in P : \\operatorname{diam}(p)<\\frac 1n \\right \\}"
},
{
"math_id": 231,
"text": "\\operatorname{diam}(p)"
},
{
"math_id": 232,
"text": "B \\subseteq [0,1]"
},
{
"math_id": 233,
"text": "D_B=\\{p\\in P : p\\subseteq B\\}"
},
{
"math_id": 234,
"text": "p_1,\\ldots,p_n\\in G"
},
{
"math_id": 235,
"text": "q\\in G"
},
{
"math_id": 236,
"text": "q\\leq p_1,\\ldots,p_n"
},
{
"math_id": 237,
"text": "D_n"
},
{
"math_id": 238,
"text": "r_G"
},
{
"math_id": 239,
"text": "r_G\\in\\bigcap G"
},
{
"math_id": 240,
"text": "B\\subseteq[0,1]"
},
{
"math_id": 241,
"text": "D_B"
},
{
"math_id": 242,
"text": "r_G\\in B"
},
{
"math_id": 243,
"text": "p\\in P"
},
{
"math_id": 244,
"text": "V\\models"
},
{
"math_id": 245,
"text": "U\\supset V"
},
{
"math_id": 246,
"text": "C=\\{\\bar p : p \\in G \\}"
},
{
"math_id": 247,
"text": " \\bar p = p \\cup \\{\\inf(p) \\} \\cup \\{\\sup(p) \\} "
},
{
"math_id": 248,
"text": "\\bar p\\supseteq p"
},
{
"math_id": 249,
"text": "C"
},
{
"math_id": 250,
"text": "\\operatorname{diam}(\\bar p) = \\operatorname{diam}(p)"
},
{
"math_id": 251,
"text": "U"
},
{
"math_id": 252,
"text": "G=\\{p\\in P : r_G\\in\\bar p\\}"
},
{
"math_id": 253,
"text": "B\\in V"
},
{
"math_id": 254,
"text": "B"
},
{
"math_id": 255,
"text": "V[G]\\models \\left (p\\Vdash_{\\mathbb{P}}a\\in\\check{B} \\right )"
},
{
"math_id": 256,
"text": "p\\in G"
},
{
"math_id": 257,
"text": "\\operatorname{val}(a,G)\\in\\bigcup_{p\\in G}\\bar p."
},
{
"math_id": 258,
"text": "B^*"
},
{
"math_id": 259,
"text": "B "
},
{
"math_id": 260,
"text": "B \\subseteq C"
},
{
"math_id": 261,
"text": "B^* \\subseteq C^*"
},
{
"math_id": 262,
"text": "B\\mapsto B^*"
},
{
"math_id": 263,
"text": "r\\in B^*"
},
{
"math_id": 264,
"text": " G = \\left \\{ B ~ (\\text{in } V) \\mid r \\in B^* ~ (\\text{in } V[G]) \\right \\}. "
},
{
"math_id": 265,
"text": "V[r]"
},
{
"math_id": 266,
"text": "I = [0,1] "
},
{
"math_id": 267,
"text": " \\mathsf{ZFC} + H "
},
{
"math_id": 268,
"text": " H "
},
{
"math_id": 269,
"text": " \\mathsf{ZFC} + \\lnot \\operatorname{Con}(\\mathsf{ZFC} + H) \\vdash (\\exists T)(\\operatorname{Fin}(T) \\land T \\subseteq \\mathsf{ZFC} \\land (T \\vdash \\lnot H)). "
},
{
"math_id": 270,
"text": " \\mathsf{ZFC} \\vdash (\\forall T)((T \\vdash \\lnot H) \\rightarrow (\\mathsf{ZFC} \\vdash (T \\vdash \\lnot H))). "
},
{
"math_id": 271,
"text": " \\mathsf{ZFC} + \\lnot \\operatorname{Con}(\\mathsf{ZFC} + H) \\vdash (\\exists T)(\\operatorname{Fin}(T) \\land T \\subseteq \\mathsf{ZFC} \\land (\\mathsf{ZFC} \\vdash (T \\vdash \\lnot H))). "
},
{
"math_id": 272,
"text": " \\mathsf{ZFC} + \\lnot \\operatorname{Con}(\\mathsf{ZFC} + H) \\vdash (\\exists T)(\\operatorname{Fin}(T) \\land T \\subseteq \\mathsf{ZFC} \\land (\\mathsf{ZFC} \\vdash (T \\vdash \\lnot H)) \\land (\\mathsf{ZFC} \\vdash \\operatorname{Con}(T + H))), "
},
{
"math_id": 273,
"text": " \\mathsf{ZFC} + \\lnot \\operatorname{Con}(\\mathsf{ZFC} + H) \\vdash \\lnot \\operatorname{Con}(\\mathsf{ZFC}), "
},
{
"math_id": 274,
"text": " \\operatorname{Con}(T + H) "
},
{
"math_id": 275,
"text": " T "
},
{
"math_id": 276,
"text": " \\mathbf{0} "
},
{
"math_id": 277,
"text": " T' "
},
{
"math_id": 278,
"text": " T'' "
},
{
"math_id": 279,
"text": " \\mathsf{ZFC} \\vdash \\operatorname{Con}(\\mathsf{ZFC}) \\leftrightarrow \\operatorname{Con}(\\mathsf{ZFL}) "
},
{
"math_id": 280,
"text": " \\mathsf{ZFL} "
},
{
"math_id": 281,
"text": " \\mathsf{ZF} + V = L "
}
] |
https://en.wikipedia.org/wiki?curid=152205
|
152207
|
Compactness theorem
|
Theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.
The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.
The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them, except for a very limited number of examples.
History.
Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.
Applications.
The compactness theorem has many applications in model theory; a few typical results are sketched here.
Robinson's principle.
The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation.
Robinson's principle: If a first-order sentence holds in every field of characteristic zero, then there exists a constant formula_0 such that the sentence holds for every field of characteristic larger than formula_1 This can be seen as follows: suppose formula_2 is a sentence that holds in every field of characteristic zero. Then its negation formula_3 together with the field axioms and the infinite sequence of sentences
formula_4
is not satisfiable (because there is no field of characteristic 0 in which formula_5 holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset formula_6 of these sentences that is not satisfiable. formula_6 must contain formula_5 because otherwise it would be satisfiable. Because adding more sentences to formula_6 does not change unsatisfiability, we can assume that formula_6 contains the field axioms and, for some formula_7 the first formula_8 sentences of the form formula_9 Let formula_10 contain all the sentences of formula_6 except formula_11 Then any field with a characteristic greater than formula_8 is a model of formula_12 and formula_5 together with formula_10 is not satisfiable. This means that formula_2 must hold in every model of formula_12 which means precisely that formula_2 holds in every field of characteristic greater than formula_13 This completes the proof.
The Lefschetz principle, one of the first examples of a transfer principle, extends this result. A first-order sentence formula_2 in the language of rings is true in some (or equivalently, in every) algebraically closed field of characteristic 0 (such as the complex numbers for instance) if and only if there exist infinitely many primes formula_0 for which formula_2 is true in some algebraically closed field of characteristic formula_14 in which case formula_2 is true in all algebraically closed fields of sufficiently large non-0 characteristic formula_1
One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials formula_15 are surjective (indeed, it can even be shown that its inverse will also be a polynomial). In fact, the surjectivity conclusion remains true for any injective polynomial formula_16 where formula_17 is a finite field or the algebraic closure of such a field.
Upward Löwenheim–Skolem theorem.
A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'. To achieve this, let formula_18 be the initial theory and let formula_19 be any cardinal number. Add to the language of formula_18 one constant symbol for every element of formula_20 Then add to formula_18 a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of formula_21 sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of formula_22 or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least formula_19.
Non-standard analysis.
A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let formula_23 be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol formula_24 to the language and adjoining to formula_23 the axiom formula_25 and the axioms formula_26 for all positive integers formula_27 Clearly, the standard real numbers formula_28 are a model for every finite subset of these axioms, because the real numbers satisfy everything in formula_23 and, by suitable choice of formula_29 can be made to satisfy any finite subset of the axioms about formula_30 By the compactness theorem, there is a model formula_31 that satisfies formula_23 and also contains an infinitesimal element formula_30
A similar argument, this time adjoining the axioms formula_32 etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization formula_23 of the reals.
It can be shown that the hyperreal numbers formula_31 satisfy the transfer principle: a first-order sentence is true of formula_28 if and only if it is true of formula_33
Proofs.
One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem, a weak form of the axiom of choice.
Gödel originally proved the compactness theorem in just this way, but later some "purely semantic" proofs of the compactness theorem were found; that is, proofs that refer to truth but not to provability. One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:
Proof:
Fix a first-order language formula_34 and let formula_23 be a collection of formula_35-sentences such that every finite subcollection of formula_35-sentences, formula_36 of it has a model formula_37 Also let formula_38 be the direct product of the structures and formula_39 be the collection of finite subsets of formula_40 For each formula_41 let formula_42
The family of all of these sets formula_43 generates a proper filter, so there is an ultrafilter formula_44 containing all sets of the form formula_45
Now for any sentence formula_2 in formula_46
Łoś's theorem now implies that formula_2 holds in the ultraproduct formula_53 So this ultraproduct satisfies all formulas in formula_40
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "p"
},
{
"math_id": 1,
"text": "p."
},
{
"math_id": 2,
"text": "\\varphi"
},
{
"math_id": 3,
"text": "\\lnot \\varphi,"
},
{
"math_id": 4,
"text": "1 + 1 \\neq 0, \\;\\; 1 + 1 + 1 \\neq 0, \\; \\ldots"
},
{
"math_id": 5,
"text": "\\lnot \\varphi"
},
{
"math_id": 6,
"text": "A"
},
{
"math_id": 7,
"text": "k,"
},
{
"math_id": 8,
"text": "k"
},
{
"math_id": 9,
"text": "1 + 1 + \\cdots + 1 \\neq 0."
},
{
"math_id": 10,
"text": "B"
},
{
"math_id": 11,
"text": "\\lnot \\varphi."
},
{
"math_id": 12,
"text": "B,"
},
{
"math_id": 13,
"text": "k."
},
{
"math_id": 14,
"text": "p,"
},
{
"math_id": 15,
"text": "\\Complex^n \\to \\Complex^n"
},
{
"math_id": 16,
"text": "F^n \\to F^n"
},
{
"math_id": 17,
"text": "F"
},
{
"math_id": 18,
"text": "T"
},
{
"math_id": 19,
"text": "\\kappa"
},
{
"math_id": 20,
"text": "\\kappa."
},
{
"math_id": 21,
"text": "\\kappa^2"
},
{
"math_id": 22,
"text": "T,"
},
{
"math_id": 23,
"text": "\\Sigma"
},
{
"math_id": 24,
"text": "\\varepsilon"
},
{
"math_id": 25,
"text": "\\varepsilon > 0"
},
{
"math_id": 26,
"text": "\\varepsilon < \\tfrac{1}{n}"
},
{
"math_id": 27,
"text": "n."
},
{
"math_id": 28,
"text": "\\R"
},
{
"math_id": 29,
"text": "\\varepsilon,"
},
{
"math_id": 30,
"text": "\\varepsilon."
},
{
"math_id": 31,
"text": "{}^* \\R"
},
{
"math_id": 32,
"text": "\\omega > 0, \\; \\omega > 1, \\ldots,"
},
{
"math_id": 33,
"text": "{}^* \\R."
},
{
"math_id": 34,
"text": "L,"
},
{
"math_id": 35,
"text": "L"
},
{
"math_id": 36,
"text": "i \\subseteq \\Sigma"
},
{
"math_id": 37,
"text": "\\mathcal{M}_i."
},
{
"math_id": 38,
"text": "\\prod_{i \\subseteq \\Sigma}\\mathcal{M}_i"
},
{
"math_id": 39,
"text": "I"
},
{
"math_id": 40,
"text": "\\Sigma."
},
{
"math_id": 41,
"text": "i \\in I,"
},
{
"math_id": 42,
"text": "A_i = \\{j \\in I : j \\supseteq i\\}."
},
{
"math_id": 43,
"text": "A_i"
},
{
"math_id": 44,
"text": "U"
},
{
"math_id": 45,
"text": "A_i."
},
{
"math_id": 46,
"text": "\\Sigma:"
},
{
"math_id": 47,
"text": "A_{\\{\\varphi\\}}"
},
{
"math_id": 48,
"text": "j \\in A_{\\{\\varphi\\}},"
},
{
"math_id": 49,
"text": "\\varphi \\in j,"
},
{
"math_id": 50,
"text": "\\mathcal M_j"
},
{
"math_id": 51,
"text": "j"
},
{
"math_id": 52,
"text": "A_{\\{\\varphi\\}},"
},
{
"math_id": 53,
"text": "\\prod_{i \\subseteq \\Sigma} \\mathcal{M}_i/U."
}
] |
https://en.wikipedia.org/wiki?curid=152207
|
15221133
|
Euler force
|
Force arising in rotating frame of reference
In classical mechanics, the Euler force is the fictitious tangential force
that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame's axes. The Euler acceleration (named for Leonhard Euler), also known as azimuthal acceleration or transverse acceleration, is that part of the absolute acceleration that is caused by the variation in the angular velocity of the reference frame.
Intuitive example.
The Euler force will be felt by a person riding a merry-go-round. As the ride starts, the Euler force will be the apparent force pushing the person to the back of the horse; and as the ride comes to a stop, it will be the apparent force pushing the person towards the front of the horse. A person on a horse close to the perimeter of the merry-go-round will perceive a greater apparent force than a person on a horse closer to the axis of rotation.
Mathematical description.
The direction and magnitude of the Euler acceleration is given, in the rotating reference frame, by:
formula_0
where ω is the angular velocity of rotation of the reference frame and r is the vector position of the point in the reference frame. The Euler force on an object of mass "m" in the rotating reference frame is then
formula_1
|
[
{
"math_id": 0,
"text": "\n\\mathbf{a}_\\mathrm{Euler} = - \\frac{d\\boldsymbol\\omega}{dt} \\times \\mathbf{r},\n"
},
{
"math_id": 1,
"text": " \\mathbf{F}_\\mathrm{Euler} = m \\mathbf{a}_\\mathrm{Euler} = - m \\frac{d\\boldsymbol\\omega}{dt} \\times \\mathbf{r}."
}
] |
https://en.wikipedia.org/wiki?curid=15221133
|
152214
|
Zermelo–Fraenkel set theory
|
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets formula_0 and formula_1 there is a new set formula_2 containing exactly formula_0 and formula_1. Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy).
The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms (see ) and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
<templatestyles src="Template:TOC_left/styles.css" />
History.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as Russell's paradox, led to the desire for a more rigorous form of set theory that was free of these paradoxes.
In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal number formula_3 and the set formula_4 where formula_5 is any infinite set and formula_6 is the power set operation. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by John von Neumann), to Zermelo set theory yields the theory denoted by "ZF". Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
Formal language.
Formally, ZFC is a one-sorted theory in first-order logic. The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The signature has a single predicate symbol, usually denoted formula_7, which is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes a set membership relation. For example, the formula formula_8 means that formula_0 is an element of the set formula_1 (also read as formula_0 is a member of formula_1).
There are different ways to formulate the formal language. Some authors may choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as functional completeness. This section attempts to strike a balance between simplicity and intuitiveness.
The language's alphabet consists of:
With this alphabet, the recursive rules for forming well-formed formulae (wff) are as follows:
formula_17
formula_18
formula_21
formula_22
formula_23
formula_24
formula_25
A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes formula_26 and formula_27 have exactly two child nodes, while nodes formula_28, formula_29 and formula_30 have exactly one. There are countably infinitely many wff, however, each wff has a finite number of nodes.
Axioms.
There are many equivalent formulations of the ZFC axioms; for a discussion of this, see . The following particular axiom set is from . The axioms below are expressed in a mixture of first order logic and high-level abbreviations.
Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. Following , we use the equivalent well-ordering theorem in place of the axiom of choice for axiom 9.
All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below (although he notes that he does so only "for emphasis"). Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the domain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, formula_31. Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC, there are only sets, the interpretation of this logical theorem in the context of ZFC is that some "set" exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called free logic, in which it is not provable from logic alone that something exists, the axiom of infinity (below) asserts that an "infinite" set exists. This implies that "a" set exists, and so, once again, it is superfluous to include an axiom asserting as much.
1. Axiom of extensionality.
Two sets are equal (are the same set) if they have the same elements.
formula_32
The converse of this axiom follows from the substitution property of equality. ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which you are constructing set theory does not include equality "formula_14", formula_17 may be defined as an abbreviation for the following formula: formula_33
In this case, the axiom of extensionality can be reformulated as
formula_34
which says that if formula_35 and formula_36 have the same elements, then they belong to the same sets.
2. Axiom of regularity (also called the axiom of foundation).
Every non-empty set formula_35 contains a member formula_36 such that formula_35 and formula_36 are disjoint sets.
formula_37
or in modern notation:
formula_38
This (along with the axioms of pairing and union) implies, for example, that no set is an element of itself and that every set has an ordinal rank.
3. Axiom schema of specification (or of separation, or of restricted comprehension).
Subsets are commonly constructed using set builder notation. For example, the even integers can be constructed as the subset of the integers formula_39 satisfying the congruence modulo predicate formula_40:
formula_41
In general, the subset of a set formula_42 obeying a formula formula_43 with one free variable formula_35 may be written as:
formula_44
The axiom schema of specification states that this subset always exists (it is an axiom "schema" because there is one axiom for each formula_45). Formally, let formula_45 be any formula in the language of ZFC with all free variables among formula_46 (formula_36 is not free in formula_45). Then:
formula_47
Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form:
formula_48
This restriction is necessary to avoid Russell's paradox (let formula_49 then formula_50) and its variants that accompany naive set theory with unrestricted comprehension (since under this restriction formula_36 only refers to sets within" formula_42 that don't belong to themselves, and formula_51 has not" been established, even though formula_52 is the case, so formula_36 stands in a separate position from which it can't refer to or comprehend itself; therefore, in a certain sense, this axiom schema is saying that in order to build a formula_36 on the basis of a formula formula_43, we need to previously restrict the sets formula_36 will regard within a set formula_42 that leaves formula_36 outside so formula_36 can't refer to itself; or, in other words, sets shouldn't refer to themselves).
In some other axiomatizations of ZF, this axiom is redundant in that it follows from the axiom schema of replacement and the axiom of the empty set.
On the other hand, the axiom schema of specification can be used to prove the existence of the empty set, denoted formula_53, once at least one set is known to exist (see above). One way to do this is to use a property formula_45 which no set has. For example, if formula_54 is any existing set, the empty set can be constructed as
formula_55
Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on formula_54). It is common to make a definitional extension that adds the symbol "formula_53" to the language of ZFC.
4. Axiom of pairing.
If formula_35 and formula_36 are sets, then there exists a set which contains formula_35 and formula_36 as elements, for example if x = {1,2} and y = {2,3} then z will be
formula_56
The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement if we are given a set with at least two elements. The existence of a set with at least two elements is assured by either the axiom of infinity, or by the and the axiom of the power set applied twice to any set.
5. Axiom of union.
The union over the elements of a set exists. For example, the union over the elements of the set formula_57 is formula_58
The axiom of union states that for any set of sets formula_59, there is a set formula_60 containing every element that is a member of some member of formula_59:
formula_61
Although this formula doesn't directly assert the existence of formula_62, the set formula_62 can be constructed from formula_60 in the above using the axiom schema of specification:
formula_63
6. Axiom schema of replacement.
The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.
Formally, let formula_45 be any formula in the language of ZFC whose free variables are among formula_64 so that in particular formula_65 is not free in formula_45. Then:
formula_66
In other words, if the relation formula_45 represents a definable function formula_68, formula_60 represents its domain, and formula_69 is a set for every formula_70 then the range of formula_68 is a subset of some set formula_65. The form stated here, in which formula_65 may be larger than strictly necessary, is sometimes called the axiom schema of collection.
7. Axiom of infinity.
Let formula_71 abbreviate formula_72 where formula_73 is some set. (We can see that formula_74 is a valid set by applying the axiom of pairing with formula_75 so that the set z is formula_74). Then there exists a set X such that the empty set formula_53, defined axiomatically, is a member of X and, whenever a set y is a member of X then formula_76 is also a member of X.
formula_77
More colloquially, there exists a set X having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω which can also be thought of as the set of natural numbers formula_78
8. Axiom of power set.
By definition, a set formula_42 is a subset of a set formula_35 if and only if every element of formula_42 is also an element of formula_35:
formula_79
The Axiom of power set states that for any set formula_35, there is a set formula_36 that contains every subset of formula_35:
formula_80
The axiom schema of specification is then used to define the power set formula_81 as the subset of such a formula_36 containing the subsets of formula_35 exactly:
formula_82
Axioms "1–8" define ZF. Alternative forms of these axioms are often encountered, some of which are listed in . Some ZF axiomatizations include an axiom asserting that the empty set exists. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set formula_35 whose existence is being asserted are just those sets which the axiom asserts formula_35 must contain.
The following axiom is added to turn ZF into ZFC:
9. Axiom of well-ordering (choice).
The last axiom, commonly known as the axiom of choice, is presented here as a property about well-orders, as in .
For any set formula_83, there exists a binary relation formula_84 which well-orders formula_83. This means formula_84 is a linear order on formula_83 such that every nonempty subset of formula_83 has a least element under the order formula_84.
formula_85
Given axioms "1" – "8", many statements are provably equivalent to axiom "9". The most common of these goes as follows. Let formula_83 be a set whose members are all nonempty. Then there exists a function formula_68 from formula_83 to the union of the members of formula_83, called a "choice function", such that for all formula_86 one has formula_87. A third version of the axiom, also equivalent, is Zorn's lemma.
Since the existence of a choice function when formula_83 is a finite set is easily proved from axioms "1–8", AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed".
Motivation via the cumulative hierarchy.
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann. In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. The collection of all sets that are obtained in this way, over all the stages, is known as "V". The sets in "V" can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to "V".
It is provable that a set is in "V" if and only if the set is pure and well-founded. And "V" satisfies all the axioms of ZFC if the class of ordinals has appropriate reflection properties. For example, suppose that a set "x" is added at stage α, which means that every element of "x" was added at a stage earlier than α. Then, every subset of "x" is also added at (or before) stage α, because all elements of any subset of "x" were also added before stage α. This means that any subset of "x" which the axiom of separation can construct is added at (or before) stage α, and that the powerset of "x" will be added at the next stage after α. For a complete argument that "V" satisfies ZFC see .
The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations.
It is possible to change the definition of "V" so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy, which gives the constructible universe "L", which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether "V" = "L". Although the structure of "L" is more regular and well behaved than that of "V", few mathematicians argue that "V" = "L" should be added to ZFC as an additional "axiom of constructibility".
Metamathematics.
Virtual classes.
Proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC).
An alternative to proper classes while staying within ZF and ZFC is the "virtual class" notational construct introduced by , where the entire construct "y" ∈ { "x" | F"x" } is simply defined as F"y". This provides a simple notation for classes that can contain sets but need not themselves be sets, while not committing to the ontology of classes (because the notation can be syntactically converted to one that only uses sets). Quine's approach built on the earlier approach of . Virtual classes are also used in , , and in the Metamath implementation of ZFC.
Finite axiomatization.
The axiom schemata of replacement and separation each contain infinitely many instances. included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes proper classes as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem not mentioning classes and provable in one theory can be proved in the other.
Consistency.
Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpret Robinson arithmetic can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC. Hence the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using models, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
If consistent, ZFC cannot prove the existence of the inaccessible cardinals that category theory requires. Huge sets of this nature are possible if ZF is augmented with Tarski's axiom. Assuming that axiom turns the axioms of infinity, power set, and choice ("7" – "9" above) into theorems.
Independence.
Many important statements are independent of ZFC (see list of statements independent of ZFC). The independence is usually proved by forcing, whereby it is shown that every countable transitive model of ZFC (sometimes augmented with large cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
Forcing proves that the following statements are independent of ZFC:
Remarks:
A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.
Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of large cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.
Proposed additions.
The project to unify set theorists behind additional axioms to resolve the continuum hypothesis or other meta-mathematical ambiguities is sometimes known as "Gödel's program". Mathematicians currently debate which axioms are the most plausible or "self-evident", which axioms are the most useful in various domains, and about to what degree usefulness should be traded off with plausibility; some "multiverse" set theorists argue that usefulness should be the sole ultimate criterion in which axioms to customarily adopt. One school of thought leans on expanding the "iterative" concept of a set to produce a set-theoretic universe with an interesting and complex but reasonably tractable structure by adopting forcing axioms; another school advocates for a tidier, less cluttered universe, perhaps focused on a "core" inner model.
Criticisms.
"For criticism of set theory in general, see Objections to set theory"
ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set.
Many mathematical theorems can be proven in much weaker systems than ZFC, such as Peano arithmetic and second-order arithmetic (as explored by the program of reverse mathematics). Saunders Mac Lane and Solomon Feferman have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second-order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
On the other hand, among axiomatic set theories, ZFC is comparatively weak. Unlike New Foundations, ZFC does not admit the existence of a universal set. Hence the universe of sets under ZFC is not closed under the elementary operations of the algebra of sets. Unlike von Neumann–Bernays–Gödel set theory (NBG) and Morse–Kelley set theory (MK), ZFC does not admit the existence of proper classes. A further comparative weakness of ZFC is that the axiom of choice included in ZFC is weaker than the axiom of global choice included in NBG and MK.
There are numerous mathematical statements independent of ZFC. These include the continuum hypothesis, the Whitehead problem, and the normal Moore space conjecture. Some of these conjectures are provable with the addition of axioms such as Martin's axiom or large cardinal axioms to ZFC. Some others are decided in ZF+AD where AD is the axiom of determinacy, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom (see projective determinacy). The Mizar system and metamath have adopted Tarski–Grothendieck set theory, an extension of ZFC, so that proofs involving Grothendieck universes (encountered in category theory and algebraic geometry) can be formalized.
See also.
Related axiomatic set theories:
Notes.
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Works cited.
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|
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{
"math_id": 0,
"text": "a"
},
{
"math_id": 1,
"text": "b"
},
{
"math_id": 2,
"text": "\\{a,b\\}"
},
{
"math_id": 3,
"text": "\\aleph_{\\omega}"
},
{
"math_id": 4,
"text": "\\{Z_{0},\\mathcal{P}(Z_{0}),\\mathcal{P}( \\mathcal{P}(Z_{0}) ),\\mathcal{P}( \\mathcal{P}( \\mathcal{P}(Z_{0}) ) ),...\\},"
},
{
"math_id": 5,
"text": "Z_{0}"
},
{
"math_id": 6,
"text": "\\mathcal{P}"
},
{
"math_id": 7,
"text": "\\in"
},
{
"math_id": 8,
"text": "a\\in b"
},
{
"math_id": 9,
"text": "\\lnot"
},
{
"math_id": 10,
"text": "\\land"
},
{
"math_id": 11,
"text": "\\lor"
},
{
"math_id": 12,
"text": "\\forall"
},
{
"math_id": 13,
"text": "\\exists"
},
{
"math_id": 14,
"text": "="
},
{
"math_id": 15,
"text": " x "
},
{
"math_id": 16,
"text": " y "
},
{
"math_id": 17,
"text": "x=y"
},
{
"math_id": 18,
"text": "x \\in y"
},
{
"math_id": 19,
"text": " \\phi "
},
{
"math_id": 20,
"text": " \\psi "
},
{
"math_id": 21,
"text": "\\lnot \\phi "
},
{
"math_id": 22,
"text": "( \\phi \\land \\psi )"
},
{
"math_id": 23,
"text": "( \\phi \\lor \\psi )"
},
{
"math_id": 24,
"text": " \\forall x \\phi "
},
{
"math_id": 25,
"text": " \\exists x \\phi "
},
{
"math_id": 26,
"text": " \\land "
},
{
"math_id": 27,
"text": " \\lor "
},
{
"math_id": 28,
"text": "\\lnot "
},
{
"math_id": 29,
"text": " \\forall x "
},
{
"math_id": 30,
"text": " \\exists x "
},
{
"math_id": 31,
"text": "\\exists x ( x = x )"
},
{
"math_id": 32,
"text": "\\forall x \\forall y [\\forall z (z \\in x \\Leftrightarrow z \\in y) \\Rightarrow x = y]."
},
{
"math_id": 33,
"text": "\\forall z [z \\in x \\Leftrightarrow z \\in y] \\land \\forall w [x \\in w \\Leftrightarrow y \\in w]."
},
{
"math_id": 34,
"text": "\\forall x \\forall y [\\forall z (z \\in x \\Leftrightarrow z \\in y) \\Rightarrow \\forall w (x \\in w \\Leftrightarrow y \\in w)],"
},
{
"math_id": 35,
"text": "x"
},
{
"math_id": 36,
"text": "y"
},
{
"math_id": 37,
"text": "\\forall x [\\exists a ( a \\in x) \\Rightarrow \\exists y ( y \\in x \\land \\lnot \\exists z (z \\in y \\land z \\in x))]."
},
{
"math_id": 38,
"text": "\\forall x\\,(x \\neq \\varnothing \\Rightarrow \\exists y (y \\in x \\land y \\cap x = \\varnothing))."
},
{
"math_id": 39,
"text": "\\mathbb{Z}"
},
{
"math_id": 40,
"text": "x \\equiv 0 \\pmod 2"
},
{
"math_id": 41,
"text": "\\{x \\in \\mathbb{Z} : x \\equiv 0 \\pmod 2\\}."
},
{
"math_id": 42,
"text": "z"
},
{
"math_id": 43,
"text": "\\varphi(x)"
},
{
"math_id": 44,
"text": "\\{x \\in z : \\varphi(x)\\}."
},
{
"math_id": 45,
"text": "\\varphi"
},
{
"math_id": 46,
"text": "x,z,w_{1},\\ldots,w_{n}"
},
{
"math_id": 47,
"text": "\\forall z \\forall w_1 \\forall w_2\\ldots \\forall w_n \\exists y \\forall x [x \\in y \\Leftrightarrow (( x \\in z )\\land \\varphi(x,w_1,w_2,...,w_n,z) )]."
},
{
"math_id": 48,
"text": "\\{x : \\varphi(x)\\}."
},
{
"math_id": 49,
"text": "y=\\{x:x\\notin x\\}"
},
{
"math_id": 50,
"text": "y \\in y \\Leftrightarrow y \\notin y"
},
{
"math_id": 51,
"text": "y \\in z"
},
{
"math_id": 52,
"text": "y \\subseteq z"
},
{
"math_id": 53,
"text": "\\varnothing"
},
{
"math_id": 54,
"text": "w"
},
{
"math_id": 55,
"text": "\\varnothing = \\{u \\in w \\mid (u \\in u) \\land \\lnot (u \\in u) \\}."
},
{
"math_id": 56,
"text": "\\forall x \\forall y \\exists z ((x \\in z) \\land (y \\in z))."
},
{
"math_id": 57,
"text": "\\{\\{1,2\\},\\{2,3\\}\\}"
},
{
"math_id": 58,
"text": "\\{1,2,3\\}."
},
{
"math_id": 59,
"text": "\\mathcal{F}"
},
{
"math_id": 60,
"text": "A"
},
{
"math_id": 61,
"text": "\\forall \\mathcal{F} \\,\\exists A \\, \\forall Y\\, \\forall x [(x \\in Y \\land Y \\in \\mathcal{F}) \\Rightarrow x \\in A]."
},
{
"math_id": 62,
"text": "\\cup \\mathcal{F}"
},
{
"math_id": 63,
"text": "\\cup \\mathcal{F}=\\{x\\in A : \\exists Y (x \\in Y \\land Y \\in \\mathcal{F})\\}."
},
{
"math_id": 64,
"text": "x, y, A, w_1, \\dotsc, w_n,"
},
{
"math_id": 65,
"text": "B"
},
{
"math_id": 66,
"text": "\\forall A\\forall w_1 \\forall w_2\\ldots \\forall w_n \\bigl[\\forall x ( x\\in A \\Rightarrow \\exists! y\\,\\varphi ) \\Rightarrow \\exists B \\ \\forall x \\bigl(x\\in A \\Rightarrow \\exists y (y\\in B \\land \\varphi)\\bigr)\\bigr]."
},
{
"math_id": 67,
"text": "\\exists!"
},
{
"math_id": 68,
"text": "f"
},
{
"math_id": 69,
"text": "f(x)"
},
{
"math_id": 70,
"text": "x \\in A,"
},
{
"math_id": 71,
"text": "S(w)"
},
{
"math_id": 72,
"text": "w \\cup \\{w\\},"
},
{
"math_id": 73,
"text": " w "
},
{
"math_id": 74,
"text": "\\{w\\}"
},
{
"math_id": 75,
"text": "x = y = w"
},
{
"math_id": 76,
"text": "S(y)"
},
{
"math_id": 77,
"text": "\\exists X \\left [\\exists e (\\forall z \\, \\neg (z \\in e) \\land e \\in X) \\land \\forall y (y \\in X \\Rightarrow S(y) \\in X)\\right]."
},
{
"math_id": 78,
"text": "\\mathbb{N}."
},
{
"math_id": 79,
"text": "(z \\subseteq x) \\Leftrightarrow ( \\forall q (q \\in z \\Rightarrow q \\in x))."
},
{
"math_id": 80,
"text": "\\forall x \\exists y \\forall z (z \\subseteq x \\Rightarrow z \\in y)."
},
{
"math_id": 81,
"text": "\\mathcal{P}(x)"
},
{
"math_id": 82,
"text": "\\mathcal{P}(x) = \\{ z \\in y: z \\subseteq x \\}."
},
{
"math_id": 83,
"text": "X"
},
{
"math_id": 84,
"text": "R"
},
{
"math_id": 85,
"text": "\\forall X \\exists R ( R \\;\\mbox{well-orders}\\; X)."
},
{
"math_id": 86,
"text": "Y\\in X"
},
{
"math_id": 87,
"text": "f(Y)\\in Y"
}
] |
https://en.wikipedia.org/wiki?curid=152214
|
1522286
|
Schur's theorem
|
One of several theorems in different areas of mathematics
In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of . In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur.
Ramsey theory.
In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers "x", "y", "z" with
formula_0
For every positive integer "c", "S"("c") denotes the smallest number "S" such that for every partition of the integers formula_1 into "c" parts, one of the parts contains integers "x", "y", and "z" with formula_2. Schur's theorem ensures that "S"("c") is well-defined for every positive integer "c". The numbers of the form "S"("c") are called Schur's numbers.
Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers, all of whose nonempty sums belong to the same part.
Using this definition, the only known Schur numbers are "S"(n)
2, 5, 14, 45, and 161 (OEIS: ) The proof that "S"(5)
161 was announced in 2017 and required 2 petabytes of space.
Combinatorics.
In combinatorics, Schur's theorem tells the number of ways for expressing a given number as a (non-negative, integer) linear combination of a fixed set of relatively prime numbers. In particular, if formula_3 is a set of integers such that formula_4, the number of different multiples of non-negative integer numbers formula_5 such that formula_6 when formula_7 goes to infinity is:
formula_8
As a result, for every set of relatively prime numbers formula_3 there exists a value of formula_7 such that every larger number is representable as a linear combination of formula_3 in at least one way. This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins. (See Coin problem.)
Differential geometry.
In differential geometry, Schur's theorem compares the distance between the endpoints of a space curve formula_9 to the distance between the endpoints of a corresponding plane curve formula_10 of less curvature.
Suppose formula_11 is a plane curve with curvature formula_12 which makes a convex curve when closed by the chord connecting its endpoints, and formula_13 is a curve of the same length with curvature formula_14. Let formula_15 denote the distance between the endpoints of formula_10 and formula_16 denote the distance between the endpoints of formula_9. If formula_17 then formula_18.
Schur's theorem is usually stated for formula_19 curves, but John M. Sullivan has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).
Linear algebra.
In linear algebra, Schur’s theorem is referred to as either the triangularization of a square matrix with complex entries, or of a square matrix with real entries and real eigenvalues.
Functional analysis.
In functional analysis and the study of Banach spaces, Schur's theorem, due to I. Schur, often refers to Schur's property, that for certain spaces, weak convergence implies convergence in the norm.
Number theory.
In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial "p"("x") with integer coefficients, if "S" is the set of all nonzero values formula_20, then the set of primes that divide some member of "S" is infinite.
References.
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|
[
{
"math_id": 0,
"text": "x + y = z."
},
{
"math_id": 1,
"text": "\\{1,\\ldots, S(c)\\}"
},
{
"math_id": 2,
"text": "x + y = z"
},
{
"math_id": 3,
"text": "\\{a_1,\\ldots,a_n\\}"
},
{
"math_id": 4,
"text": "\\gcd(a_1,\\ldots,a_n)=1"
},
{
"math_id": 5,
"text": "(c_1,\\ldots,c_n)"
},
{
"math_id": 6,
"text": "x=c_1a_1 + \\cdots + c_na_n"
},
{
"math_id": 7,
"text": "x"
},
{
"math_id": 8,
"text": "\\frac{x^{n-1}}{(n-1)!a_1\\cdots a_n}(1+o(1))."
},
{
"math_id": 9,
"text": "C^*"
},
{
"math_id": 10,
"text": "C"
},
{
"math_id": 11,
"text": "C(s)"
},
{
"math_id": 12,
"text": "\\kappa(s)"
},
{
"math_id": 13,
"text": "C^*(s)"
},
{
"math_id": 14,
"text": "\\kappa^*(s)"
},
{
"math_id": 15,
"text": "d"
},
{
"math_id": 16,
"text": "d^*"
},
{
"math_id": 17,
"text": "\\kappa^*(s) \\leq \\kappa(s)"
},
{
"math_id": 18,
"text": "d^* \\geq d"
},
{
"math_id": 19,
"text": "C^2"
},
{
"math_id": 20,
"text": "\\begin{Bmatrix} p(n) \\neq 0 : n \\in \\mathbb{N} \\end{Bmatrix}"
}
] |
https://en.wikipedia.org/wiki?curid=1522286
|
1522373
|
Carroll's paradox
|
Paradox in physics
In physics, Carroll's paradox arises when considering the motion of a falling rigid rod that is specially constrained. Considered one way, the angular momentum stays constant; considered in a different way, it changes. It is named after Michael M. Carroll who first published it in 1984.
Explanation.
Consider two concentric circles of radius formula_0 and formula_1 as might be drawn on the face of a wall clock. Suppose a uniform rigid heavy rod of length formula_2 is somehow constrained between these two circles so that one end of the rod remains on the inner circle and the other remains on the outer circle. Motion of the rod along these circles, acting as guides, is frictionless. The rod is held in the three o'clock position so that it is horizontal, then released.
Now consider the angular momentum about the centre of the rod:
An apparent resolution of this paradox is that the physical situation cannot occur. To maintain the rod in a radial position the circles have to exert an infinite force. In real life it would not be possible to construct guides that do not exert a significant reaction force perpendicular to the rod. Victor Namias, however, disputed that infinite forces occur, and argued that a finitely thick rod experiences torque about its center of mass even in the limit as it approaches zero width.
|
[
{
"math_id": 0,
"text": "r_1"
},
{
"math_id": 1,
"text": "r_2"
},
{
"math_id": 2,
"text": "l=|r_2-r_1|"
}
] |
https://en.wikipedia.org/wiki?curid=1522373
|
15224289
|
F-divergence
|
Function that measures dissimilarity between two probability distributions
In probability theory, an formula_0-divergence is a certain type of function formula_1 that measures the difference between two probability distributions formula_2 and formula_3. Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of formula_0-divergence.
History.
These divergences were introduced by Alfréd Rényi in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease in Markov processes. "f"-divergences were studied further independently by , and and are sometimes known as Csiszár formula_0-divergences, Csiszár–Morimoto divergences, or Ali–Silvey distances.
Definition.
Non-singular case.
Let formula_2 and formula_3 be two probability distributions over a space formula_4, such that formula_5, that is, formula_2 is absolutely continuous with respect to formula_3. Then, for a convex function formula_6 such that formula_7 is finite for all formula_8, formula_9, and formula_10 (which could be infinite), the formula_0-divergence of formula_2 from formula_3 is defined as
formula_11
We call formula_0 the generator of formula_12.
In concrete applications, there is usually a reference distribution formula_13 on formula_4 (for example, when formula_14, the reference distribution is the Lebesgue measure), such that formula_15, then we can use Radon–Nikodym theorem to take their probability densities formula_16 and formula_17, giving
formula_18
When there is no such reference distribution ready at hand, we can simply define formula_19, and proceed as above. This is a useful technique in more abstract proofs.
Extension to singular measures.
The above definition can be extended to cases where formula_5 is no longer satisfied (Definition 7.1 of ).
Since formula_0 is convex, and formula_20 , the function formula_21 must nondecrease, so there exists formula_22, taking value in formula_23.
Since for any formula_24, we have formula_25 , we can extend f-divergence to the formula_26.
Properties.
Basic relations between f-divergences.
<templatestyles src="Math_proof/styles.css" />Proof
If formula_31, then formula_30 by definition.
Conversely, if formula_33, then let formula_34. For any two probability measures formula_35 on the set formula_36, since formula_37, we get
formula_38
Since each probability measure formula_35 has one degree of freedom, we can solve formula_39 for every choice of formula_40.
Linear algebra yields formula_41, which is a valid probability measure. Then we obtain formula_42.
Thus formula_43
for some constants formula_44. Plugging the formula into formula_45 yields formula_46.
Basic properties of f-divergences.
In particular, the monotonicity implies that if a Markov process has a positive equilibrium probability distribution formula_48 then formula_49 is a monotonic (non-increasing) function of time, where the probability distribution formula_50 is a solution of the Kolmogorov forward equations (or Master equation), used to describe the time evolution of the probability distribution in the Markov process. This means that all "f"-divergences formula_49 are the Lyapunov functions of the Kolmogorov forward equations. The converse statement is also true: If formula_51 is a Lyapunov function for all Markov chains with positive equilibrium formula_48 and is of the trace-form
(formula_52) then formula_53, for some convex function "f". For example, Bregman divergences in general do not have such property and can increase in Markov processes.
Analytic properties.
The f-divergences can be expressed using Taylor series and rewritten using a weighted sum of chi-type distances ().
Naive variational representation.
Let formula_54 be the convex conjugate of formula_0. Let formula_55 be the effective domain of
formula_54, that is, formula_56. Then we have two variational representations of formula_12, which we describe below.
Basic variational representation.
Under the above setup,
<templatestyles src="Math_theorem/styles.css" />
Theorem — formula_57.
This is Theorem 7.24 in.
Example applications.
Using this theorem on total variation distance, with generator formula_58 its convex conjugate is formula_59, and we obtain
formula_60
For chi-squared divergence, defined by formula_61, we obtain
formula_62
Since the variation term is not affine-invariant in formula_47, even though the domain over which formula_47 varies "is" affine-invariant, we can use up the affine-invariance to obtain a leaner expression.
Replacing formula_47 by formula_63 and taking the maximum over formula_64, we obtain
formula_65
which is just a few steps away from the Hammersley–Chapman–Robbins bound and the Cramér–Rao bound (Theorem 29.1 and its corollary in ).
For formula_66-divergence with formula_67, we have formula_68, with range formula_69. Its convex conjugate is formula_70 with range formula_71, where formula_72.
Applying this theorem yields, after substitution with formula_73,
formula_74
or, releasing the constraint on formula_75,
formula_76
Setting formula_77 yields the variational representation of formula_78-divergence obtained above.
The domain over which formula_75 varies is not affine-invariant in general, unlike the formula_78-divergence case. The formula_78-divergence is special, since in that case, we can remove the formula_79 from formula_80.
For general formula_67, the domain over which formula_75 varies is merely scale invariant. Similar to above, we can replace formula_75 by formula_81, and take minimum over formula_82 to obtain
formula_83
Setting formula_84, and performing another substitution by formula_85, yields two variational representations of the squared Hellinger distance:
formula_86
formula_87
Applying this theorem to the KL-divergence, defined by formula_88, yields
formula_89
This is strictly less efficient than the Donsker–Varadhan representation
formula_90
This defect is fixed by the next theorem.
Improved variational representation.
Assume the setup in the beginning of this section ("Variational representations").
<templatestyles src="Math_theorem/styles.css" />
Theorem — If formula_91 on
formula_92 (redefine formula_0 if necessary), then
formula_93,
where
formula_94
and formula_95, where formula_17 is the probability density function of formula_3 with respect to some underlying measure.
In the special case of formula_96, we have
formula_97.
This is Theorem 7.25 in.
Example applications.
Applying this theorem to KL-divergence yields the Donsker–Varadhan representation.
Attempting to apply this theorem to the general formula_66-divergence with formula_67 does not yield a closed-form solution.
Common examples of "f"-divergences.
The following table lists many of the common divergences between probability distributions and the possible generating functions to which they correspond. Notably, except for total variation distance, all others are special cases of formula_66-divergence, or linear sums of formula_66-divergences.
For each f-divergence formula_12, its generating function is not uniquely defined, but only up to formula_98, where formula_99 is any real constant. That is, for any formula_0 that generates an f-divergence, we have formula_100. This freedom is not only convenient, but actually necessary.
Let formula_101 be the generator of formula_66-divergence, then formula_101 and formula_102 are convex inversions of each other, so formula_103. In particular, this shows that the squared Hellinger distance and Jensen-Shannon divergence are symmetric.
In the literature, the formula_66-divergences are sometimes parametrized as
formula_104
which is equivalent to the parametrization in this page by substituting formula_105.
Relations to other statistical divergences.
Here, we compare "f"-divergences with other statistical divergences.
Rényi divergence.
The Rényi divergences is a family of divergences defined by
formula_106
when formula_107. It is extended to the cases of formula_108 by taking the limit.
Simple algebra shows that formula_109, where formula_110 is the formula_66-divergence defined above.
Bregman divergence.
The only f-divergence that is also a Bregman divergence is the KL divergence.
Integral probability metrics.
The only f-divergence that is also an integral probability metric is the total variation.
Financial interpretation.
A pair of probability distributions can be viewed as a game of chance in which one of the distributions defines the official odds and the other contains the actual probabilities. Knowledge of the actual probabilities allows a player to profit from the game. For a large class of rational players the expected profit rate has the same general form as the "ƒ"-divergence.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "f"
},
{
"math_id": 1,
"text": "D_f(P\\| Q)"
},
{
"math_id": 2,
"text": "P"
},
{
"math_id": 3,
"text": "Q"
},
{
"math_id": 4,
"text": "\\Omega"
},
{
"math_id": 5,
"text": "P\\ll Q"
},
{
"math_id": 6,
"text": "f: [0, +\\infty)\\to(-\\infty, +\\infty]"
},
{
"math_id": 7,
"text": "f(x)"
},
{
"math_id": 8,
"text": "x > 0"
},
{
"math_id": 9,
"text": "f(1)=0"
},
{
"math_id": 10,
"text": "f(0)=\\lim_{t\\to 0^+} f(t)"
},
{
"math_id": 11,
"text": " D_f(P\\parallel Q) \\equiv \\int_{\\Omega} f\\left(\\frac{dP}{dQ}\\right)\\,dQ."
},
{
"math_id": 12,
"text": "D_f"
},
{
"math_id": 13,
"text": "\\mu"
},
{
"math_id": 14,
"text": "\\Omega = \\R^n"
},
{
"math_id": 15,
"text": "P, Q \\ll \\mu"
},
{
"math_id": 16,
"text": "p"
},
{
"math_id": 17,
"text": "q"
},
{
"math_id": 18,
"text": " D_f(P\\parallel Q) = \\int_{\\Omega} f\\left(\\frac{p(x)}{q(x)}\\right)q(x)\\,d\\mu(x)."
},
{
"math_id": 19,
"text": "\\mu = P+Q"
},
{
"math_id": 20,
"text": "f(1) = 0"
},
{
"math_id": 21,
"text": "\\frac{f(x)}{x-1}"
},
{
"math_id": 22,
"text": "f'(\\infty) := \\lim_{x\\to\\infty}f(x)/x"
},
{
"math_id": 23,
"text": "(-\\infty, +\\infty]"
},
{
"math_id": 24,
"text": "p(x)>0"
},
{
"math_id": 25,
"text": "\\lim_{q(x)\\to 0} q(x)f \\left(\\frac{p(x)}{q(x)}\\right) = p(x)f'(\\infty)"
},
{
"math_id": 26,
"text": "P\\not\\ll Q"
},
{
"math_id": 27,
"text": "D_{\\sum_i a_i f_i} = \\sum_i a_i D_{f_i}"
},
{
"math_id": 28,
"text": "a_i"
},
{
"math_id": 29,
"text": "f_i"
},
{
"math_id": 30,
"text": "D_f = D_g"
},
{
"math_id": 31,
"text": "f(x) = g(x) + c(x-1)"
},
{
"math_id": 32,
"text": "c\\in \\R"
},
{
"math_id": 33,
"text": "D_f - D_g = 0"
},
{
"math_id": 34,
"text": "h = f-g"
},
{
"math_id": 35,
"text": "P, Q"
},
{
"math_id": 36,
"text": "\\{0, 1\\}"
},
{
"math_id": 37,
"text": "D_f(P\\| Q) - D_g(P\\|Q) = 0"
},
{
"math_id": 38,
"text": "h(P_1/Q_1) = -\\frac{Q_0}{Q_1}h(P_0/Q_0)"
},
{
"math_id": 39,
"text": "\\frac{P_0}{Q_0} = a, \\frac{P_1}{Q_1} = x"
},
{
"math_id": 40,
"text": "0 < a < 1 < x"
},
{
"math_id": 41,
"text": "Q_0 = \\frac{x-1}{x-a}, Q_1 = \\frac{1-a}{x-a}"
},
{
"math_id": 42,
"text": "h(x) = \\frac{h(a)}{a-1}(x-1), h(a) = \\frac{h(x)}{x-1}(a-1)"
},
{
"math_id": 43,
"text": "\nh(x)=\\begin{cases}\nc_1(x-1)\\quad\\text{if } x>1,\\\\\nc_0(x-1)\\quad\\text{if } 0<x<1,\\\\\n\\end{cases}\n"
},
{
"math_id": 44,
"text": "c_0, c_1"
},
{
"math_id": 45,
"text": "h(x) = \\frac{h(a)}{a-1}(x-1)"
},
{
"math_id": 46,
"text": "c_0 = c_1"
},
{
"math_id": 47,
"text": "g"
},
{
"math_id": 48,
"text": "P^*"
},
{
"math_id": 49,
"text": "D_f(P(t)\\parallel P^*)"
},
{
"math_id": 50,
"text": "P(t)"
},
{
"math_id": 51,
"text": "H(P)"
},
{
"math_id": 52,
"text": "H(P)=\\sum_{i}f(P_{i},P_{i}^{*})"
},
{
"math_id": 53,
"text": "H(P)= D_f(P(t)\\parallel P^*)"
},
{
"math_id": 54,
"text": "f^*"
},
{
"math_id": 55,
"text": "\\mathrm{effdom}(f^*)"
},
{
"math_id": 56,
"text": "\\mathrm{effdom}(f^*) = \\{y : f^*(y) < \\infty\\}"
},
{
"math_id": 57,
"text": "D_f(P; Q) = \\sup_{g: \\Omega\\to \\mathrm{effdom}(f^*)} E_P[g] - E_Q[f^* \\circ g]"
},
{
"math_id": 58,
"text": "f(x)= \\frac 1 2 |x-1|,"
},
{
"math_id": 59,
"text": "f^*(x^*) = \\begin{cases}\nx^* \\text{ on } [-1/2, 1/2],\\\\\n+\\infty \\text{ else.}\n\\end{cases}"
},
{
"math_id": 60,
"text": "TV(P\\| Q) = \\sup_{|g|\\leq 1/2} E_P[g(X)] - E_Q[g(X)]."
},
{
"math_id": 61,
"text": "f(x) = (x-1)^2, f^*(y) = y^2/4 + y"
},
{
"math_id": 62,
"text": "\\chi^2(P; Q) = \\sup_g E_P[g(X)] - E_Q[g(X)^2/4 + g(X)]."
},
{
"math_id": 63,
"text": "a g + b"
},
{
"math_id": 64,
"text": "a, b \\in \\R"
},
{
"math_id": 65,
"text": "\\chi^2(P; Q) = \\sup_g \\frac{(E_P[g(X)]-E_Q[g(X)])^2}{Var_Q[g(X)]},"
},
{
"math_id": 66,
"text": "\\alpha"
},
{
"math_id": 67,
"text": "\\alpha \\in (-\\infty, 0)\\cup(0, 1)"
},
{
"math_id": 68,
"text": "f_\\alpha(x) = \\frac{x^\\alpha - \\alpha x - (1-\\alpha)}{\\alpha(\\alpha-1)}"
},
{
"math_id": 69,
"text": "x\\in [0, \\infty)"
},
{
"math_id": 70,
"text": "f_\\alpha^*(y)=\\frac{1}{\\alpha}(x(y)^\\alpha - 1)"
},
{
"math_id": 71,
"text": "y\\in(-\\infty, (1-\\alpha)^{-1})"
},
{
"math_id": 72,
"text": "x(y) = ((\\alpha-1)y + 1)^{\\frac{1}{\\alpha-1}}"
},
{
"math_id": 73,
"text": "h = ((\\alpha-1)g+1)^{\\frac{1}{\\alpha-1}}"
},
{
"math_id": 74,
"text": "D_\\alpha(P\\| Q) = \\frac{1}{\\alpha(1-\\alpha)} - \\inf_{h: \\Omega\\to (0,\\infty)}\\left(\nE_Q\\left[\\frac{h^\\alpha}{\\alpha}\\right] \n+ E_P\\left[\\frac{h^{\\alpha-1}}{1-\\alpha}\\right] \n\\right),"
},
{
"math_id": 75,
"text": "h"
},
{
"math_id": 76,
"text": "D_\\alpha(P\\| Q) = \\frac{1}{\\alpha(1-\\alpha)} - \\inf_{h: \\Omega\\to \\R}\\left(\nE_Q\\left[\\frac{|h|^\\alpha}{\\alpha}\\right] \n+ E_P\\left[\\frac{|h|^{\\alpha-1}}{1-\\alpha}\\right] \n\\right)."
},
{
"math_id": 77,
"text": "\\alpha=-1"
},
{
"math_id": 78,
"text": "\\chi^2"
},
{
"math_id": 79,
"text": "|\\cdot |"
},
{
"math_id": 80,
"text": "|h|"
},
{
"math_id": 81,
"text": "a h"
},
{
"math_id": 82,
"text": "a>0"
},
{
"math_id": 83,
"text": "D_\\alpha(P\\| Q) = \\sup_{h >0} \\left[\\frac{1}{\\alpha(1-\\alpha)} \\left(\n1-\\frac{E_P[h^{\\alpha-1}]^\\alpha}{E_Q[h^\\alpha]^{\\alpha-1}}\n\\right) \\right]."
},
{
"math_id": 84,
"text": "\\alpha=\\frac 1 2"
},
{
"math_id": 85,
"text": "g=\\sqrt h"
},
{
"math_id": 86,
"text": "H^2(P\\|Q) = \\frac 1 2 D_{1/2}(P\\| Q) = 2 - \\inf_{h>0}\\left(\nE_Q\\left[h(X)\\right] \n+ E_P\\left[h(X)^{-1}\\right] \n\\right),"
},
{
"math_id": 87,
"text": "H^2(P\\|Q) = 2 \\sup_{h > 0} \\left(1-\\sqrt{E_P[h^{-1}]E_Q[h]}\\right).\n"
},
{
"math_id": 88,
"text": "f(x) = x\\ln x, f^*(y) = e^{y-1}"
},
{
"math_id": 89,
"text": "D_{KL}(P; Q) =\\sup_g E_P[g(X)] - e^{-1}E_Q[e^{g(X)}]."
},
{
"math_id": 90,
"text": "D_{KL}(P; Q) = \\sup_g E_P[g(X)]- \\ln E_Q[e^{g(X)}]."
},
{
"math_id": 91,
"text": "f(x) = +\\infty"
},
{
"math_id": 92,
"text": "x<0"
},
{
"math_id": 93,
"text": "\n D_{f}(P \\| Q)=f^{\\prime}(\\infty) P\\left[S^{c}\\right]+\\sup _{g} \\mathbb{E}_{P}\\left[g 1_{S}\\right]-\\Psi_{Q, P}^{*}(g)\n "
},
{
"math_id": 94,
"text": "\\Psi_{Q, P}^{*}(g) := \\inf _{a \\in \\mathbb{R}} \\mathbb{E}_{Q}\\left[f^{*}(g(X)-a)\\right]+a P[S]"
},
{
"math_id": 95,
"text": " S:=\\{q > 0\\} "
},
{
"math_id": 96,
"text": "f^{\\prime}(\\infty)=+\\infty"
},
{
"math_id": 97,
"text": "\n D_{f}(P \\| Q)=\\sup _{g} \\mathbb{E}_{P}[g]-\\Psi_{Q}^{*}(g), \\quad \\Psi_{Q}^{*}(g) := \\inf _{a \\in \\mathbb{R}} \\mathbb{E}_{Q}\\left[f^{*}(g(X)-a)\\right]+a\n "
},
{
"math_id": 98,
"text": "c\\cdot(t-1)"
},
{
"math_id": 99,
"text": "c"
},
{
"math_id": 100,
"text": "D_{f(t)} = D_{f(t) + c\\cdot(t-1)}"
},
{
"math_id": 101,
"text": "f_\\alpha"
},
{
"math_id": 102,
"text": "f_{1-\\alpha}"
},
{
"math_id": 103,
"text": "D_{\\alpha}(P\\| Q) = D_{1-\\alpha}(Q\\| P) "
},
{
"math_id": 104,
"text": "\\begin{cases}\n \\frac{4}{1-\\alpha^2}\\big(1 - t^{(1+\\alpha)/2}\\big), & \\text{if}\\ \\alpha\\neq\\pm1, \\\\\n t \\ln t, & \\text{if}\\ \\alpha=1, \\\\\n - \\ln t, & \\text{if}\\ \\alpha=-1\n \\end{cases}"
},
{
"math_id": 105,
"text": "\\alpha \\leftarrow \\frac{\\alpha+1}{2}"
},
{
"math_id": 106,
"text": "R_{\\alpha} (P \\| Q) = \\frac{1}{\\alpha-1}\\log\\Bigg(\nE_Q\\left[\\left(\\frac{dP}{dQ}\\right)^\\alpha\\right]\n\\Bigg) \\,"
},
{
"math_id": 107,
"text": "\\alpha \\in (0, 1)\\cup (1, +\\infty)"
},
{
"math_id": 108,
"text": "\\alpha =0, 1, +\\infty"
},
{
"math_id": 109,
"text": "R_\\alpha(P\\| Q) = \\frac{1}{\\alpha - 1}\\ln (1+\\alpha(\\alpha-1)D_\\alpha(P\\|Q))"
},
{
"math_id": 110,
"text": "D_\\alpha"
}
] |
https://en.wikipedia.org/wiki?curid=15224289
|
15224959
|
BATON Overlay
|
The BAlanced Tree Overlay Network (BATON) is a distributed tree structure designed for peer-to-peer (P2P) systems. Unlike other overlays that employ a distributed hash table, BATON organises peers in a distributed tree to facilitate range search. BATON aims to maintain a balanced tree height, similar to the AVL tree, resulting in a bounded to formula_0 for search exact and range queries as well as update operations (join/leave).
System model.
BATON is a binary tree. In each tree level, the node is named by its position in the tree.
Each node in BATON keeps four kinds of links:
The level of any node is one greater than the level of its parent. Root is on level 0. For a node at position formula_2, it will fill its left routing table by nodes at position formula_3 for any valid formula_4 and fill its right routing table by nodes at position formula_5 for any valid formula_6. The construction of the routing table has slight resemblance to the finger tables in Chord.
So according to the example structure, node "2:1" would keep links to
Height-Balanced
BATON is considered balanced if and only if the height of its two sub-trees at any node in the tree differs by at most one. If any node detects that the height-balanced constraint is violated, a restructuring process is initiated to ensure that the tree remains balanced.
Node joining and leaving.
When a new node wants to join the network in BATON, its joining request is always forwarded to the leaf node. The leaf node then checks whether its routing table is full. If the table is full, it means that the level is full of nodes, and the leaf node can accept the new node as its child to create a new level node. If the table is not full, the leaf node must forward the new node to take over one of the empty positions.
On the other hand, when a node wants to leave the network, it must update the routing tables of its parent node, child nodes, adjacent nodes, and routing nodes. If the leaving node is a leaf node, it can safely leave the network. However, if it is not a leaf node, it must find a leaf node to replace its position.
Routing.
In BATON, each node maintains a continuous key space. When a new node joins as its child, the node splits its space and assigns half of it to the child. This partitioning method allows the tree to be traversed in ascending order if we travel the tree in in-order. This is why BATON supports range queries.
To execute a range query q, BATON first locates its left bound, q.low. Then, the search process travels the tree in in-order (by adjacent link) until it reaches the upper bound, q.up. For locating a single key, BATON uses a similar routing strategy as Chord. The request is first routed to the farthest routing node that does not overshoot the key. If no such routing nodes exist, the parent link, child link, or adjacent link is used.
Restructure.
When a node x accepts a joining node y as its child and detects that the tree balance is violated, it initiates the restructuring process. Without loss of generality, let's assume that this restructuring is towards the right. Suppose that y joins as x's left child. To rebalance the system, x notifies y to replace its position and notifies its right adjacent node z that x will replace z's position. Z then checks its right adjacent node t to see if its left child is empty. If it is, and adding a child to t does not affect the tree balance, z takes the position of t's left child as its new position, and the restructuring process stops. If t's left child is full or t cannot accept x as its left child without violating the balance property, z occupies t's position, while t needs to find a new position for itself by continuing to its right adjacent node.
Load balancing.
BATON adopts two kinds of load balancing strategy. Once a node n detects that it is over loaded,
|
[
{
"math_id": 0,
"text": "O(\\log N)"
},
{
"math_id": 1,
"text": "RT := {LRT}\\cup{RRT} "
},
{
"math_id": 2,
"text": "p"
},
{
"math_id": 3,
"text": "p - 2^x"
},
{
"math_id": 4,
"text": "x \\geq 0"
},
{
"math_id": 5,
"text": "p + 2^y"
},
{
"math_id": 6,
"text": "y \\geq 0"
}
] |
https://en.wikipedia.org/wiki?curid=15224959
|
1522626
|
Wigner–Seitz cell
|
Primitive cell of crystal lattices with Voronoi decomposition applied
The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystallography.
The unique property of a crystal is that its atoms are arranged in a regular three-dimensional array called a lattice. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits discrete translational symmetry. In order to model and study such a periodic system, one needs a mathematical "handle" to describe the symmetry and hence draw conclusions about the material properties consequent to this symmetry. The Wigner–Seitz cell is a means to achieve this.
A Wigner–Seitz cell is an example of a primitive cell, which is a unit cell containing exactly one lattice point. For any given lattice, there are an infinite number of possible primitive cells. However there is only one Wigner–Seitz cell for any given lattice. It is the locus of points in space that are closer to that lattice point than to any of the other lattice points.
A Wigner–Seitz cell, like any primitive cell, is a fundamental domain for the discrete translation symmetry of the lattice. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.
Overview.
Background.
The concept of Voronoi decomposition was investigated by Peter Gustav Lejeune Dirichlet, leading to the name "Dirichlet domain". Further contributions were made from Evgraf Fedorov, ("Fedorov parallelohedron"), Georgy Voronoy ("Voronoi polyhedron"), and Paul Niggli ("Wirkungsbereich").
The application to condensed matter physics was first proposed by Eugene Wigner and Frederick Seitz in a 1933 paper, where it was used to solve the Schrödinger equation for free electrons in elemental sodium. They approximated the shape of the Wigner–Seitz cell in sodium, which is a truncated octahedron, as a sphere of equal volume, and solved the Schrödinger equation exactly using periodic boundary conditions, which require formula_0 at the surface of the sphere. A similar calculation which also accounted for the non-spherical nature of the Wigner–Seitz cell was performed later by John C. Slater.
There are only five topologically distinct polyhedra which tile three-dimensional space, ℝ3. These are referred to as the parallelohedra. They are the subject of mathematical interest, such as in higher dimensions. These five parallelohedra can be used to classify the three dimensional lattices using the concept of a projective plane, as suggested by John Horton Conway and Neil Sloane. However, while a topological classification considers any affine transformation to lead to an identical class, a more specific classification leads to 24 distinct classes of voronoi polyhedra with parallel edges which tile space. For example, the rectangular cuboid, right square prism, and cube belong to the same topological class, but are distinguished by different ratios of their sides. This classification of the 24 types of voronoi polyhedra for Bravais lattices was first laid out by Boris Delaunay.
Definition.
The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to any of the other lattice points.
It can be shown mathematically that a Wigner–Seitz cell is a primitive cell. This implies that the cell spans the entire direct space without leaving any gaps or holes, a property known as tessellation.
Constructing the cell.
The general mathematical concept embodied in a Wigner–Seitz cell is more commonly called a Voronoi cell, and the partition of the plane into these cells for a given set of point sites is known as a Voronoi diagram.
The cell may be chosen by first picking a lattice point. After a point is chosen, lines are drawn to all nearby lattice points. At the midpoint of each line, another line is drawn normal to each of the first set of lines. The smallest area enclosed in this way is called the Wigner–Seitz primitive cell.
For a 3-dimensional lattice, the steps are analogous, but in step 2 instead of drawing perpendicular lines, perpendicular planes are drawn at the midpoint of the lines between the lattice points.
As in the case of all primitive cells, all area or space within the lattice can be filled by Wigner–Seitz cells and there will be no gaps.
Nearby lattice points are continually examined until the area or volume enclosed is the correct area or volume for a primitive cell. Alternatively, if the basis vectors of the lattice are reduced using lattice reduction only a set number of lattice points need to be used. In two-dimensions only the lattice points that make up the 4 unit cells that share a vertex with the origin need to be used. In three-dimensions only the lattice points that make up the 8 unit cells that share a vertex with the origin need to be used.
Composite lattices.
For composite lattices, (crystals which have more than one vector in their basis) each single lattice point represents multiple atoms. We can break apart each Wigner–Seitz cell into subcells by further Voronoi decomposition according to the closest atom, instead of the closest lattice point. For example, the diamond crystal structure contains a two atom basis. In diamond, carbon atoms have tetrahedral sp3 bonding, but since tetrahedra do not tile space, the voronoi decomposition of the diamond crystal structure is actually the triakis truncated tetrahedral honeycomb. Another example is applying Voronoi decomposition to the atoms in the A15 phases, which forms the polyhedral approximation of the Weaire–Phelan structure.
Symmetry.
The Wigner–Seitz cell always has the same point symmetry as the underlying Bravais lattice. For example, the cube, truncated octahedron, and rhombic dodecahedron have point symmetry Oh, since the respective Bravais lattices used to generate them all belong to the cubic lattice system, which has Oh point symmetry.
Brillouin zone.
In practice, the Wigner–Seitz cell itself is actually rarely used as a description of direct space, where the conventional unit cells are usually used instead. However, the same decomposition is extremely important when applied to reciprocal space. The Wigner–Seitz cell in the reciprocal space is called the Brillouin zone, which contains the information about whether a material will be a conductor, semiconductor or an insulator.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "d \\psi/d r=0"
}
] |
https://en.wikipedia.org/wiki?curid=1522626
|
15228745
|
HGSNAT
|
Protein-coding gene in the species Homo sapiens
Heparan-α-glucosaminide "N"-acetyltransferase (also called "acetyl-CoA:heparan-α-D-glucosaminide "N"-acetyltransferase" and "acetyl-CoA:alpha-glucosaminide "N"-acetyltransferase") is an enzyme that in humans is encoded by the "HGSNAT" gene.
In enzymology, this enzyme belongs to the family of transferases, specifically those acyltransferases transferring groups other than aminoacyl groups. It is catalysed in the chemical reaction:
acetyl-CoA + heparan sulfate α-D-glucosaminide formula_0 CoA + heparan sulfate N-acetyl-α-D-glucosaminide
This enzyme participates in glycosaminoglycan degradation and glycan structures degradation. Mutations in the gene encoding this enzyme cause mucopolysaccharidosis IIIC.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=15228745
|
1522954
|
Network performance
|
Network service quality
Network performance refers to measures of service quality of a network as seen by the customer.
There are many different ways to measure the performance of a network, as each network is different in nature and design. Performance can also be modeled and simulated instead of measured; one example of this is using state transition diagrams to model queuing performance or to use a Network Simulator.
Performance measures.
The following measures are often considered important:
Bandwidth.
The available channel bandwidth and achievable signal-to-noise ratio determine the maximum possible throughput. It is not generally possible to send more data than dictated by the Shannon-Hartley Theorem.
Throughput.
"Throughput" is the number of messages successfully delivered per unit time. Throughput is controlled by available bandwidth, as well as the available signal-to-noise ratio and hardware limitations. Throughput for the purpose of this article will be understood to be measured from the arrival of the first bit of data at the receiver, to decouple the concept of throughput from the concept of latency. For discussions of this type, the terms 'throughput' and 'bandwidth' are often used interchangeably.
The "Time Window" is the period over which the throughput is measured. The choice of an appropriate time window will often dominate calculations of throughput, and whether latency is taken into account or not will determine whether the latency affects the throughput or not.
Latency.
The speed of light imposes a minimum propagation time on all electromagnetic signals. It is not possible to reduce the latency below
formula_0
where s is the distance and cm is the speed of light in the medium (roughly 200,000 km/s for most fiber or electrical media, depending on their velocity factor). This approximately means an additional millisecond round-trip delay (RTT) per 100 km (or 62 miles) of distance between hosts.
Other delays also occur in intermediate nodes. In packet switched networks delays can occur due to queueing.
Jitter.
Jitter is the undesired deviation from true periodicity of an assumed periodic signal in electronics and telecommunications, often in relation to a reference clock source. Jitter may be observed in characteristics such as the frequency of successive pulses, the signal amplitude, or phase of periodic signals. Jitter is a significant, and usually undesired, factor in the design of almost all communications links (e.g., USB, PCI-e, SATA, OC-48). In clock recovery applications it is called "timing jitter".
Error rate.
In digital transmission, the number of bit errors is the number of received bits of a data stream over a communication channel that have been altered due to noise, interference, distortion or bit synchronization errors.
The bit error rate or bit error ratio (BER) is the number of bit errors divided by the total number of transferred bits during a studied time interval. BER is a unitless performance measure, often expressed as a percentage.
The bit error probability "pe" is the expectation value of the BER. The BER can be considered as an approximate estimate of the bit error probability. This estimate is accurate for a long time interval and a high number of bit errors.
Interplay of factors.
All of the factors above, coupled with user requirements and user perceptions, play a role in determining the perceived 'fastness' or utility, of a network connection. The relationship between throughput, latency, and user experience is most aptly understood in the context of a shared network medium, and as a scheduling problem.
Algorithms and protocols.
For some systems, latency and throughput are coupled entities. In TCP/IP, latency can also directly affect throughput. In TCP connections, the large bandwidth-delay product of high latency connections, combined with relatively small TCP window sizes on many devices, effectively causes the throughput of a high latency connection to drop sharply with latency. This can be remedied with various techniques, such as increasing the TCP congestion window size, or more drastic solutions, such as packet coalescing, TCP acceleration, and forward error correction, all of which are commonly used for high latency satellite links.
TCP acceleration converts the TCP packets into a stream that is similar to UDP. Because of this, the TCP acceleration software must provide its own mechanisms to ensure the reliability of the link, taking the latency and bandwidth of the link into account, and both ends of the high latency link must support the method used.
In the Media Access Control (MAC) layer, performance issues such as throughput and end-to-end delay are also addressed.
Examples of latency or throughput dominated systems.
Many systems can be characterized as dominated either by throughput limitations or by latency limitations in terms of end-user utility or experience. In some cases, hard limits such as the speed of light present unique problems to such systems and nothing can be done to correct this. Other systems allow for significant balancing and optimization for best user experience.
Satellite.
A telecom satellite in geosynchronous orbit imposes a path length of at least 71000 km between transmitter and receiver. which means a minimum delay between message request and message receipt, or latency of 473 ms. This delay can be very noticeable and affects satellite phone service regardless of available throughput capacity.
Deep space communication.
These long path length considerations are exacerbated when communicating with space probes and other long-range targets beyond Earth's atmosphere. The Deep Space Network implemented by NASA is one such system that must cope with these problems. Largely latency driven, the GAO has criticized the current architecture. Several different methods have been proposed to handle the intermittent connectivity and long delays between packets, such as delay-tolerant networking.
Even deeper space communication.
At interstellar distances, the difficulties in designing radio systems that can achieve any throughput at all are massive. In these cases, maintaining communication is a bigger issue than how long that communication takes.
Offline data transport.
Transportation is concerned almost entirely with throughput, which is why physical deliveries of backup tape archives are still largely done by vehicle.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "t=s/c_m"
}
] |
https://en.wikipedia.org/wiki?curid=1522954
|
15230235
|
Material handling
|
Sub-discipline of mechanical engineering
Material handling involves short-distance movement within the confines of a building or between a building and a transportation vehicle. It uses a wide range of manual, semi-automated, and automated equipment and includes consideration of the protection, storage, and control of materials throughout their manufacturing, warehousing, distribution, consumption, and disposal. Material handling can be used to create "time and place utility" through the handling, storage, and control of waste, as distinct from manufacturing, which creates "form utility" by changing the shape, form, and makeup of material.
Role.
Material handling plays an important role in manufacturing and logistics. Almost every item of physical commerce has been transported on a conveyor or lift truck or another type of material handling equipment in manufacturing plants, warehouses, and retail stores. While material handling is usually required as part of every production worker's job, over 650,000 people in the U.S. work as dedicated "material moving machine operators" and have a median annual wage of $31,530 (May 2012). These operators use material handling equipment to transport various goods in a variety of industrial settings including moving construction materials around building sites or moving goods onto ships.
Design of material handling systems.
Material handling is integral to the design of most production systems since the efficient flow of material between the activities of a production system is heavily dependent on the arrangement (or "layout") of the activities. If two activities are adjacent to each other, then material might easily be handed from one activity to another. If activities are in sequence, a conveyor can move the material at low cost. If activities are separated, more expensive industrial trucks or overhead conveyors are required for transport. The high cost of using an industrial truck for material transport is due to both the labor costs of the operator and the negative impact on the performance of a production system (e.g., increased work in process) when multiple units of material are combined into a single transfer batch in order to reduce the number of trips required for transport.
The unit load concept.
A unit load is either a single unit of an item, or multiple units so arranged or restricted that they can be handled as a single unit and maintain their integrity. Although granular, liquid, and gaseous materials can be transported in bulk, they can also be contained into unit loads using bags, drums, and cylinders. Advantages of unit loads are that more items can be handled at the same time (thereby reducing the number of trips required, and potentially reducing handling costs, loading and unloading times, and product damage) and that it enables the use of standardized material handling equipment. Disadvantages of unit loads include the negative impact of batching on production system performance, and the cost of returning empty containers/pallets to their point of origin.
In-process handling.
Unit loads can be used both for in-process handling and for distribution (receiving, storing, and shipping). Unit load design involves determining the type, size, weight, and configuration of the load; the equipment and method used to handle the load; and the methods of forming (or building) and breaking down the load. For in-process handling, unit loads should not be larger than the production batch size of parts in process. Large production batches (used to increase the utilization of bottleneck activities) can be split into smaller "transfer batches" for handling purposes, where each transfer batch contains one or more unit loads, and small unit loads can be combined into a larger transfer batch to allow more efficient transport.
Distribution.
Selecting a unit load size for distribution can be difficult because containers/pallets are usually available only in standard sizes and configurations; truck trailers, rail boxcars, and airplane cargo bays are limited in width, length, and height; and the number of feasible container/pallet sizes for a load may be limited due to the existing warehouse layout and storage rack configurations and customer package/carton size and retail store shelf restrictions. Also, the practical size of a unit load may be limited by the equipment and aisle space available and the need for safe material handling.
Health and safety.
Manual material handling work contributes to a large percentage of the over half a million cases of musculoskeletal disorders reported annually in the United States. Musculoskeletal disorders often involve strains and sprains to the lower back, shoulders, and upper limbs. They can result in protracted pain, disability, medical treatment, and financial stress for those afflicted with them, and employers often find themselves paying the bill, either directly or through workers’ compensation insurance, at the same time they must cope with the loss of the full capacity of their workers.
Scientific evidence shows that effective ergonomic interventions can lower the physical demands of MMH work tasks, thereby lowering the incidence and severity of the musculoskeletal injuries they can cause. Their potential for reducing injury related costs alone make ergonomic interventions a useful tool for improving a company’s productivity, product quality, and overall business competitiveness. But very often productivity gets an additional and solid shot in the arm when managers and workers take a fresh look at how best to use energy, equipment, and exertion to get the job done in the most efficient, effective, and effortless way possible. Planning that applies these principles can result in big wins for all concerned.
Types.
Manual handling.
Manual handling refers to the use of a worker’s hands to move individual containers by lifting, lowering, filling, emptying, or carrying them. It can expose workers to physical dangers that can lead to injuries: a large percentage of the over half a million cases of musculoskeletal disorders reported in the U.S. each year arise from manual handling, and often involve strains and sprains to a person's lower back, shoulders and upper limbs.
Ergonomic improvements can be used to modify manual handling tasks to reduce injury. These improvements can include reconfiguring the task and using positioning equipment like lift/tilt/turn tables, hoists, balancers, and manipulators to reduce reaching and bending. The NIOSH (National Institute for Occupational Safety and Health) 1991 Revised Lifting Equation can be used to evaluate manual lifting tasks. Under ideal circumstances, the maximum recommended weight for manual lifting to avoid back injuries is 51 lb (23.13 kg). Using the exact conditions of the lift (height, distance lifted, weight, position of weight relative to body, asymmetrical lifts, and objects that are difficult to grasp), six multipliers are used to reduce the maximum recommended weight for less than ideal lifting tasks.
Automated handling.
Whenever technically and economically feasible, equipment can be used to reduce and sometimes replace the need to manually handle material. Most existing material handling equipment is only "semi-automated" because a human operator is needed for tasks like loading/unloading and driving that are difficult and/or too costly to fully automate. However, ongoing advances in sensing, machine intelligence, and robotics have made it possible to fully automate an increasing number of handling tasks. A rough guide to determine how much can be spent for automated equipment that would replace one material handler is to consider that, with benefits, the median moving machine operator costs a company $45,432 per year. Assuming a real interest rate of 1.7% and a service life of 5 years with no adoption/adaptation cost, no learning cost, no training cost, and no operating cost for equipment with no salvage value, a company should be willing to pay up to
formula_0
to purchase automated equipment to replace one worker. In many cases, automated equipment is not as flexible as a human operator, both with respect to not being able to do a particular task as well as a human and not being able to be as easily redeployed to do other tasks as needs change.
Notes and references.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\$45\\,432\\left (\\frac{1-1.017^{-5}}{0.017} \\right )=\\$45\\,432(4.75)=\\$219\\,019\n"
}
] |
https://en.wikipedia.org/wiki?curid=15230235
|
15231265
|
Bühlmann model
|
Random effects model in credibility theory
In credibility theory, a branch of study in actuarial science, the Bühlmann model is a random effects model (or "variance components model" or hierarchical linear model) used to determine the appropriate premium for a group of insurance contracts. The model is named after Hans Bühlmann who first published a description in 1967.
Model description.
Consider "i" risks which generate random losses for which historical data of "m" recent claims are available (indexed by "j"). A premium for the "i"th risk is to be determined based on the expected value of claims. A linear estimator which minimizes the mean square error is sought. Write
Note: formula_8 and formula_9 are functions of random parameter formula_10
The Bühlmann model is the solution for the problem:
formula_11
where formula_12 is the estimator of premium formula_13 and arg min represents the parameter values which minimize the expression.
Model solution.
The solution for the problem is:
formula_14
where:
formula_15
We can give this result the interpretation, that Z part of the premium is based on the information that we have about the specific risk, and (1-Z) part is based on the information that we have about the whole population.
Proof.
The following proof is slightly different from the one in the original paper. It is also more general, because it considers all linear estimators, while original proof considers only estimators based on average claim.
Lemma. The problem can be stated alternatively as:
formula_16
Proof:
formula_17
The last equation follows from the fact that
formula_18
We are using here the law of total expectation and the fact, that formula_19
In our previous equation, we decompose minimized function in the sum of two expressions. The second expression does not depend on parameters used in minimization. Therefore, minimizing the function is the same as minimizing the first part of the sum.
Let us find critical points of the function
formula_20
formula_21
For formula_22 we have:
formula_23
We can simplify derivative, noting that:
formula_24
Taking above equations and inserting into derivative, we have:
formula_25
formula_26
Right side doesn't depend on "k". Therefore, all formula_27 are constant
formula_28
From the solution for formula_29 we have
formula_30
Finally, the best estimator is
formula_31
References.
Citations.
<templatestyles src="Reflist/styles.css" />
Sources.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\scriptstyle =\\frac{1}{m}\\sum_{j=1}^{m}X_{ij}"
},
{
"math_id": 1,
"text": "\\Theta_i"
},
{
"math_id": 2,
"text": "m(\\vartheta)= \\operatorname E\\left [ X_{ij} |\\Theta_i = \\vartheta\\right ]"
},
{
"math_id": 3,
"text": "\\Pi=\\operatorname E(m(\\vartheta)|X_{i1},X_{i2},...X_{im})"
},
{
"math_id": 4,
"text": "\\mu = \\operatorname E(m(\\vartheta))"
},
{
"math_id": 5,
"text": "s^2(\\vartheta)=\\operatorname{Var}\\left [ X_{ij} |\\Theta_i = \\vartheta\\right ]"
},
{
"math_id": 6,
"text": "\\sigma^2=\\operatorname E\\left [ s^2(\\vartheta) \\right ]"
},
{
"math_id": 7,
"text": "v^2=\\operatorname{Var}\\left [ m(\\vartheta) \\right ]"
},
{
"math_id": 8,
"text": "m(\\vartheta)"
},
{
"math_id": 9,
"text": "s^2(\\vartheta)"
},
{
"math_id": 10,
"text": "\\vartheta"
},
{
"math_id": 11,
"text": "\\underset{a_{i0},a_{i1},...,a_{im}}{\\operatorname{arg\\,min}} \\operatorname E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi \\right)^2\\right ]"
},
{
"math_id": 12,
"text": "a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}"
},
{
"math_id": 13,
"text": "\\Pi"
},
{
"math_id": 14,
"text": "Z\\bar{X}_i+(1-Z)\\mu"
},
{
"math_id": 15,
"text": "Z=\\frac{1}{1+\\frac{\\sigma^2}{v^2m}}"
},
{
"math_id": 16,
"text": "f=\\mathbb E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-m(\\vartheta)\\right )^2\\right ]\\to \\min"
},
{
"math_id": 17,
"text": "\\begin{align}\n\\mathbb E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-m(\\vartheta)\\right )^2\\right ] &=\\mathbb E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right )^2\\right ]+\\mathbb E\\left [ \\left ( m(\\vartheta)-\\Pi\\right )^2\\right ]-2\\mathbb{E} \\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right ) \\left ( m(\\vartheta)-\\Pi\\right )\\right ] \\\\\n&=\\mathbb E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right )^2\\right ]+\\mathbb E\\left [ \\left ( m(\\vartheta)-\\Pi\\right )^2\\right ]\n\\end{align}"
},
{
"math_id": 18,
"text": "\\begin{align}\n\\mathbb E\\left [\\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right ) \\left ( m(\\vartheta)-\\Pi\\right )\\right ] &= \\mathbb E_{\\Theta}\\left[\\mathbb{E}_X\\left. \\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right ) ( m(\\vartheta)-\\Pi) \\right | X_{i1},\\ldots ,X_{im}\\right ]\\right ] \\\\\n&=\\mathbb E_{\\Theta}\\left[\\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right )\\left [ \\mathbb E_X\\left [( m(\\vartheta)-\\Pi) | X_{i1},\\ldots ,X_{im}\\right ]\\right ] \\right]\\\\\n&=0\n\\end{align}"
},
{
"math_id": 19,
"text": "\\Pi=\\mathbb E [m(\\vartheta)|X_{i1},\\ldots, X_{im}]."
},
{
"math_id": 20,
"text": "\\frac{1}{2}\\frac{\\partial f}{\\partial a_{i0}}=\\mathbb E\\left [a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-m(\\vartheta)\\right ]=a_{i0}+\\sum_{j=1}^{m}a_{ij}\\mathbb E(X_{ij})-\\mathbb E(m(\\vartheta))=a_{i0}+\\left (\\sum_{j=1}^{m}a_{ij}-1 \\right )\\mu"
},
{
"math_id": 21,
"text": "a_{i0}=\\left (1- \\sum_{j=1}^{m}a_{ij} \\right )\\mu"
},
{
"math_id": 22,
"text": "k\\neq 0"
},
{
"math_id": 23,
"text": "\\frac{1}{2}\\frac{\\partial f}{\\partial a_{ik}}=\\mathbb E\\left [ X_{ik}\\left ( a_{i0} +\\sum_{j=1}^{m}a_{ij}X_{ij}-m(\\vartheta)\\right ) \\right ]=\\mathbb E\\left [ X_{ik} \\right ]a_{i0}+\\sum_{j=1, j\\neq k}^{m}a_{ij}\\mathbb E[X_{ik}X_{ij}]+a_{ik}\\mathbb E[X^2_{ik}]-\\mathbb E[X_{ik}m(\\vartheta)]=0"
},
{
"math_id": 24,
"text": "\\begin{align}\n\\mathbb E[X_{ij}X_{ik}] & =\\mathbb E \\left [\\mathbb E [X_{ij}X_{ik}|\\vartheta] \\right ]=\\mathbb E[\\text{cov}(X_{ij}X_{ik}|\\vartheta)+\\mathbb E(X_{ij}|\\vartheta)\\mathbb E(X_{ik}|\\vartheta)]=\\mathbb E[(m(\\vartheta))^2]=v^2+\\mu^2 \\\\\n\\mathbb E[X^2_{ik}] &= \\mathbb E \\left [\\mathbb E[X^2_{ik}|\\vartheta] \\right ]=\\mathbb E[s^2(\\vartheta)+(m(\\vartheta))^2]=\\sigma^2+v^2+\\mu^2 \\\\\n\\mathbb E[X_{ik}m(\\vartheta)] & =\\mathbb E[\\mathbb E[X_{ik}m(\\vartheta)|\\Theta_i]=\\mathbb E[(m(\\vartheta))^2]=v^2+\\mu^2\n\\end{align}"
},
{
"math_id": 25,
"text": "\\frac{1}{2}\\frac{\\partial f}{\\partial a_{ik}}=\\left ( 1-\\sum_{j=1}^{m}a_{ij} \\right )\\mu^2+\\sum_{j=1,j\\neq k}^{m}a_{ij}(v^2+\\mu^2)+a_{ik}(\\sigma^2+v^2+\\mu^2)-(v^2+\\mu^2)=a_{ik}\\sigma^2-\\left ( 1-\\sum_{j=1}^{m}a_{ij} \\right )v^2=0"
},
{
"math_id": 26,
"text": "\\sigma^2a_{ik}=v^2\\left (1-\\sum_{j=1}^{m} a_{ij}\\right)"
},
{
"math_id": 27,
"text": "a_{ik}"
},
{
"math_id": 28,
"text": "a_{i1}= \\cdots =a_{im}=\\frac{v^2}{\\sigma^2+mv^2}"
},
{
"math_id": 29,
"text": "a_{i0}"
},
{
"math_id": 30,
"text": "a_{i0}=(1-ma_{ik})\\mu=\\left ( 1-\\frac{mv^2}{\\sigma^2+mv^2} \\right )\\mu"
},
{
"math_id": 31,
"text": "a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}=\\frac{mv^2}{\\sigma^2+mv^2}\\bar{X_i}+\\left ( 1-\\frac{mv^2}{\\sigma^2+mv^2} \\right )\\mu=Z\\bar{X_i}+(1-Z)\\mu"
}
] |
https://en.wikipedia.org/wiki?curid=15231265
|
15231991
|
Modified Wigner distribution function
|
"Note: the Wigner distribution function is abbreviated here as WD rather than WDF as used at Wigner distribution function"
A Modified Wigner distribution function is a variation of the Wigner distribution function (WD) with reduced or removed cross-terms.
The Wigner distribution (WD) was first proposed for corrections to classical statistical mechanics in 1932 by Eugene Wigner. The Wigner distribution function, or Wigner–Ville distribution (WVD) for analytic signals, also has applications in time frequency analysis. The Wigner distribution gives better auto term localisation compared to the smeared out spectrogram (SP). However, when applied to a signal with multi frequency components, cross terms appear due to its quadratic nature. Several methods have been proposed to reduce the cross terms. For example, in 1994 Ljubiša Stanković proposed a novel technique, now mostly referred to as S-method, resulting in the reduction or removal of cross terms. The concept of the S-method is a combination between the spectrogram and the Pseudo Wigner Distribution (PWD), the windowed version of the WD.
The original WD, the spectrogram, and the modified WDs all belong to the Cohen's class of bilinear time-frequency representations :
formula_0
where formula_1 is Cohen's kernel function, which is often a low-pass function, and normally serves to mask out the interference in the original Wigner representation.
formula_2
Mathematical definition.
Cohen's kernel function : formula_3
formula_4
where formula_5 is the short-time Fourier transform of formula_6.
formula_7
Cohen's kernel function : formula_8 which is the WD of the window function itself. This can be verified by applying the convolution property of the Wigner distribution function.
The spectrogram cannot produce interference since it is a positive-valued quadratic distribution.
formula_9
Can't solve the cross term problem, however it can solve the problem of 2 components time difference larger than window size B.
formula_10
formula_11
Where L is any integer greater than 0
Increase L can reduce the influence of cross term (however it can't eliminate completely )
For example, for L=2, the dominant third term is divided by 4 ( which is equivalent to 12dB ).
This gives a significant improvement over the Wigner Distribution.
Properties of L-Wigner Distribution:
formula_27
formula_28
formula_31
formula_32
When formula_33 and formula_34, it becomes the original Wigner distribution function.
It can avoid the cross term when the order of phase of the exponential function is no larger than formula_35
However the cross term between two components cannot be removed.
formula_36should be chosen properly such that
formula_37
formula_38
formula_39
If formula_40
when formula_41 , formula_42
formula_43
formula_44
formula_45
formula_46
Cohen's kernel function : formula_47 which is concentred on the frequency axis.
Note that the pseudo Wigner can also be written as the Fourier transform of the “spectral-correlation” of the STFT
formula_48
In the pseudo Wigner the time windowing acts as a frequency direction smoothing. Therefore, it suppresses the Wigner distribution interference components that oscillate in the frequency direction. Time direction smoothing can be implemented by a time-convolution of the PWD with a lowpass function formula_49 :
formula_50
Cohen's kernel function : formula_51 where formula_52 is the Fourier transform of the window formula_53.
Thus the kernel corresponding to the smoothed pseudo Wigner distribution has a separable form. Note that even if the SPWD and the S-Method both smoothes the WD in the time domain, they are not equivalent in general.
formula_54
Cohen's kernel function : formula_55
The S-method limits the range of the integral of the PWD with a low-pass windowing function formula_56 of Fourier transform formula_57. This results in the cross-term removal, without blurring the auto-terms that are well-concentred along the frequency axis.
The S-method strikes a balance in smoothing between the pseudo-Wigner distribution formula_58 [formula_59] and the power spectrogram formula_60 [formula_61].
Note that in the original 1994 paper, Stankovic defines the S-methode with a modulated version of the short-time Fourier transform :
formula_62
where
formula_63
Even in this case we still have
formula_64
|
[
{
"math_id": 0,
"text": "C_x(t, f)=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}W_x(\\theta,\\nu) \\Pi(t - \\theta,f - \\nu)\\, d\\theta\\, d\\nu \\quad = [W_x\\,\\ast\\,\\Pi] (t,f)"
},
{
"math_id": 1,
"text": "\\Pi \\left(t, f\\right)"
},
{
"math_id": 2,
"text": " W_x(t,f) = \\int_{-\\infty}^\\infty x(t+\\tau/2) x^*(t-\\tau/2) e^{-j2\\pi\\tau f} \\, d\\tau"
},
{
"math_id": 3,
"text": "\\Pi (t,f) = \\delta_{(0,0)} (t,f) "
},
{
"math_id": 4,
"text": "SP_x (t,f) = |ST_x (t,f)|^2 = ST_x (t,f)\\,ST_x^* (t,f)"
},
{
"math_id": 5,
"text": "ST_x"
},
{
"math_id": 6,
"text": "x"
},
{
"math_id": 7,
"text": " ST_x(t,f) = \\int_{-\\infty}^\\infty x(\\tau) w^*(t-\\tau) e^{-j2\\pi f\\tau} \\, d\\tau"
},
{
"math_id": 8,
"text": "\\Pi (t,f) = W_h(t,f) "
},
{
"math_id": 9,
"text": " W_x(t,f) = \\int_{-B}^B w(\\tau)x(t+\\tau/2) x^*(t-\\tau/2) e^{-j2\\pi\\tau f} \\, d\\tau\n"
},
{
"math_id": 10,
"text": " W_x(t,f) = \\int_{-B}^B w(\\eta)X(f+\\eta/2) X^*(f-\\eta/2) e^{j2\\pi t \\eta} \\, d\\eta\n\n"
},
{
"math_id": 11,
"text": " W_x(t,f) = \\int_{-\\infty}^\\infty w(\\tau)x^L(r+\\tau/2L) \\overline{x^{*L}(t-\\tau/2L)} e^{-j2\\pi \\tau f} \\, d\\tau\n\n"
},
{
"math_id": 12,
"text": " x(t-t0)\n\n"
},
{
"math_id": 13,
"text": " LWD: W_x(t-t0,f)\n\n"
},
{
"math_id": 14,
"text": " x(t)\\exp(j\\omega_0 t)\n\n"
},
{
"math_id": 15,
"text": " LWD: W_x(t,f-f0)\n\n"
},
{
"math_id": 16,
"text": " x(t)\n\n"
},
{
"math_id": 17,
"text": " x(t)=0 \n\n"
},
{
"math_id": 18,
"text": " for \\left\\vert t \\right\\vert >T,\n\n"
},
{
"math_id": 19,
"text": " LWD: W_x(t,f)=0\n\n"
},
{
"math_id": 20,
"text": " for\\left\\vert t \\right\\vert >T\n\n"
},
{
"math_id": 21,
"text": " f_m\n\n"
},
{
"math_id": 22,
"text": " F(f)=0\n\n"
},
{
"math_id": 23,
"text": " for \\left\\vert f \\right\\vert > f_m\n\n"
},
{
"math_id": 24,
"text": " LWD: W_x(t,f)\n\n"
},
{
"math_id": 25,
"text": " \\int_{-\\infty}^\\infty W_x(t, f)df = \\left\\vert x(t) \\right\\vert ^{2L}\n\n"
},
{
"math_id": 26,
"text": " 2L^{th}\n\n"
},
{
"math_id": 27,
"text": " \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty W_x(t,f)dtdf = \\int_{-\\infty}^\\infty \\left\\vert x(t) \\right\\vert ^{2L} dt = \\lVert x(t) \\rVert _{2L} ^{2L}\n\n"
},
{
"math_id": 28,
"text": " \\int_{-\\infty}^\\infty W_x(t,f)dt = \\left\\vert F_L(f) \\right\\vert ^2=\\left\\vert \\underbrace{ F(L_f)*F(L_f)*\\cdots*F(L_f) }_{L times} \\right\\vert ^2\n\n"
},
{
"math_id": 29,
"text": " L(L\\rightarrow \\infty)\n\n"
},
{
"math_id": 30,
"text": " (t_m, f_m)\n\n"
},
{
"math_id": 31,
"text": " \\lim_{L \\to \\infty} (W_x(t,f)/W_x(t_m,f_m)) = \\begin{cases} 0, & \\text{if }f \\neq f_m\\text{ or }t\\neq t_m\\text{ } \\\\ 1, & \\text{if }f=f_m\\text{ and } t=t_m\\end{cases}\n\n"
},
{
"math_id": 32,
"text": " W_x(t,f) = \\int_{-B}^B [\\textstyle \\prod_{l=1}^{q/2} \\displaystyle x(t+d_l \\tau) x^*(t- d_{-l} \\tau)] e^{-j2\\pi\\tau f} \\, d\\tau\n \n"
},
{
"math_id": 33,
"text": " q=2\n\n \n"
},
{
"math_id": 34,
"text": " d_l=d_{-l}=0.5\n \n"
},
{
"math_id": 35,
"text": " q/2+1\n \n"
},
{
"math_id": 36,
"text": " d_l\n \n"
},
{
"math_id": 37,
"text": " \\textstyle \\prod_{l=1}^{q/2} \\displaystyle x(t+d_l \\tau) x^*(t-d_{-l}\\tau)=\\exp\\big(j2\\pi\\textstyle \\sum_{n=1}^{q/2+1} n a_n t^{n-1}\\tau \\displaystyle\\big)\n \n"
},
{
"math_id": 38,
"text": " W_x(t,f) = \\int_{-\\infty}^\\infty \\exp\\Bigl(-j2\\pi (f-\\sum_{n=1}^{q/2+1}na_nt^{n-1})\\tau\\Bigr)d\\tau\n \n"
},
{
"math_id": 39,
"text": " \\cong \\delta\\bigl(f-\\sum_{n=1}^{q/2+1}na_nt^{n-1}\\bigr)\n \n"
},
{
"math_id": 40,
"text": " x(t)=\\exp\\bigl(j2\\pi\\sum_{n=1}^{q/2+1}a_nt^n\\bigr)\n \n"
},
{
"math_id": 41,
"text": " q=2 \n \n"
},
{
"math_id": 42,
"text": " x(t+d_l \\tau) x^*(t-d_{-l}\\tau)=\\exp\\bigl(j2\\pi\\sum_{n=1}^{q/2+1}na_nt^{n-1}\\tau\\bigr)\n \n"
},
{
"math_id": 43,
"text": " a_2(t+d_l\\tau)^2+a_1(t+d_l\\tau)-a_2(t-d_{-l}\\tau)^2-a_1(t-d_{-l}\\tau)=2a_2t\\tau+a_1\\tau\n \n"
},
{
"math_id": 44,
"text": " \\Longrightarrow d_l+d_{-l}=1, d_l-d_{-l}=0\n\n \n"
},
{
"math_id": 45,
"text": " \\Longrightarrow d_l=d_{-l}=1/2\n \n"
},
{
"math_id": 46,
"text": " PW_x(t,f) = \\int_{-\\infty}^\\infty w(\\tau/2) w^*(-\\tau/2) x(t+\\tau/2) x^*(t-\\tau/2) e^{-j2\\pi\\tau\\,f} \\, d\\tau"
},
{
"math_id": 47,
"text": "\\Pi (t,f) = \\delta_0 (t)\\,W_h(t,f) "
},
{
"math_id": 48,
"text": " PW_x(t,f) = \\int_{-\\infty}^\\infty ST_x(t, f+\\nu/2) ST_x^*(t, f-\\nu/2) e^{j2\\pi\\nu\\,t} \\, d\\nu"
},
{
"math_id": 49,
"text": "q"
},
{
"math_id": 50,
"text": " SPW_x(t,f) = [ q\\,\\ast\\, PW_x (.,f)] (t) = \\int_{-\\infty}^\\infty q(t-u) \\int_{-\\infty}^\\infty w(\\tau/2) w^*(-\\tau/2) x(u+\\tau/2) x^*(u-\\tau/2) e^{-j2\\pi\\tau\\,f} \\, d\\tau\\, du"
},
{
"math_id": 51,
"text": "\\Pi (t,f) = q(t)\\, W(f) "
},
{
"math_id": 52,
"text": "W"
},
{
"math_id": 53,
"text": "w"
},
{
"math_id": 54,
"text": " SM(t,f) = \\int_{-\\infty}^\\infty ST_x(t, f+\\nu/2) ST_x^*(t, f-\\nu/2) G(\\nu) e^{j2\\pi\\nu\\,t} \\, d\\nu"
},
{
"math_id": 55,
"text": "\\Pi (t,f) = g(t)\\, W_h(t,f) "
},
{
"math_id": 56,
"text": "g(t)"
},
{
"math_id": 57,
"text": "G(f)"
},
{
"math_id": 58,
"text": "PW_x"
},
{
"math_id": 59,
"text": "g(t) = 1"
},
{
"math_id": 60,
"text": "SP_x"
},
{
"math_id": 61,
"text": "g(t) = \\delta_0 (t)"
},
{
"math_id": 62,
"text": " SM(t,f) = \\int_{-\\infty}^\\infty \\tilde{ST}_x(t,f+\\nu) \\tilde{ST}_x^*(t,f-\\nu) P(\\nu)\\, d\\nu "
},
{
"math_id": 63,
"text": " \\tilde{ST}_x(t,f) = \\int_{-\\infty}^\\infty x(t+\\tau) w^*(\\tau) e^{-j2\\pi f\\tau} \\, d\\tau \\quad = ST_x(t,f)\\,e^{j2\\pi f t}"
},
{
"math_id": 64,
"text": "\\Pi (t,f) = p(2t)\\, W_h(t,f) "
}
] |
https://en.wikipedia.org/wiki?curid=15231991
|
15232604
|
Gabor–Wigner transform
|
The Gabor transform, named after Dennis Gabor, and the Wigner distribution function, named after Eugene Wigner, are both tools for time-frequency analysis. Since the Gabor transform does not have high clarity, and the Wigner distribution function has a "cross term problem" (i.e. is non-linear), a 2007 study by S. C. Pei and J. J. Ding proposed a new combination of the two transforms that has high clarity and no cross term problem.
Since the cross term does not appear in the Gabor transform, the time frequency distribution of the Gabor transform can be used as a filter to filter out the cross term in the output of the Wigner distribution function.
formula_0
formula_1
There are many different combinations to define the Gabor–Wigner transform. Here four different definitions are given.
Application.
The Gabor–Wigner transform performs well in image processing, filter design, signal sampling, modulation, demodulation, speech processing, and biomedical engineering.
Filter Design.
The goal of filter design is to remove unwanted portions of the signal while preserving the necessary parts. By using the Gabor–Wigner transform, we can simultaneously consider filters in both the time domain and frequency domain, representing a form of time-frequency analysis. The main concept is illustrated as follows.
Signal Modulation.
The purpose of modulation is to place a signal within a specific time or frequency range. Using the Gabor–Wigner transform, we can simultaneously consider how to introduce more or more suitable signal patterns in both the time and frequency domains. Due to the absence of cross-term issues, it performs better than the Wigner transform.
From the figure (WDF) above, it can also be observed that when using the Wigner transform (WDF), the generated cross-terms have a severe impact on modulation.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " G_x(t,f) = \\int_{-\\infty}^\\infty e^{-\\pi(\\tau-t)^2}e^{-j2\\pi f\\tau}x(\\tau) \\, d\\tau "
},
{
"math_id": 1,
"text": " W_x(t,f)=\\int_{-\\infty}^\\infty x(t+\\tau/2)x^*(t-\\tau/2)e^{-j2\\pi\\tau\\,f} \\, d\\tau"
},
{
"math_id": 2,
"text": "D_x(t,f)=G_x(t,f)\\times W_x(t,f)"
},
{
"math_id": 3,
"text": "D_x(t,f)=\\min\\left\\{|G_x(t,f)|^2,|W_x(t,f)|\\right\\}"
},
{
"math_id": 4,
"text": "D_x(t,f)=W_x(t,f)\\times \\{|G_x(t,f)|>0.25\\}"
},
{
"math_id": 5,
"text": "D_x(t,f)=G_x^{2.6}(t,f)W_x^{0.7}(t,f)"
},
{
"math_id": 6,
"text": "W_x(t,f) = \\int^{\\infty}_{-\\infty}x(t+\\frac{\\tau}{2})x^*(t-\\frac{\\tau}{2})e^{-j2\\pi \\tau f} \\cdot d\\tau"
},
{
"math_id": 7,
"text": "x"
},
{
"math_id": 8,
"text": "t"
},
{
"math_id": 9,
"text": "f\n"
},
{
"math_id": 10,
"text": "x(t) = \\alpha g(t) + \\beta s(t)"
},
{
"math_id": 11,
"text": "\\begin{align}\nW_x(t,f) &= \\int^{\\infty}_{-\\infty}x(t+\\frac{\\tau}{2})x^*(t-\\frac{\\tau}{2})e^{-j2\\pi \\tau f} \\cdot d\\tau \\\\\n &= \\int^{\\infty}_{-\\infty} \\big[ \\alpha g(t+\\frac{\\tau}{2}) + \\beta s(t+\\frac{\\tau}{2}) \\big] \\big[ \\alpha^* g^*(t-\\frac{\\tau}{2}) + \\beta^* s^*(t-\\frac{\\tau}{2}) \\big]e^{-j2\\pi \\tau f} \\cdot d\\tau \\\\\n &= \\int^{\\infty}_{-\\infty} \\big[ |\\alpha|^2 g(t+\\frac{\\tau}{2})g^*(t-\\frac{\\tau}{2}) + |\\beta|^2 s(t+\\frac{\\tau}{2})s^*(t-\\frac{\\tau}{2}) \\\\ &\\quad + \\alpha \\beta^* g(t+\\frac{\\tau}{2})s^*(t-\\frac{\\tau}{2}) + \\alpha^* \\beta g^*(t-\\frac{\\tau}{2})s(t+\\frac{\\tau}{2}) \\big] e^{-j2\\pi \\tau f} \\cdot d\\tau \\\\\n &= |\\alpha|^2 W_g(t,f)+|\\beta|^2W_s(t,f) \\\\ &\\quad + \\int^{\\infty}_{-\\infty} \\big[ \\alpha \\beta^* g(t+\\frac{\\tau}{2})s^*(t-\\frac{\\tau}{2}) + \\alpha^* \\beta g^*(t-\\frac{\\tau}{2})s(t+\\frac{\\tau}{2}) \\big] e^{-j2\\pi \\tau f} \\cdot d\\tau \\\\\n\n\\end{align}"
},
{
"math_id": 12,
"text": "W_g"
},
{
"math_id": 13,
"text": "W_s"
},
{
"math_id": 14,
"text": "G_X(t,f)\\approx 0, D_x(t,f) = G_x^\\alpha(t,f)W_x^\\beta(t,f) \\approx 0"
},
{
"math_id": 15,
"text": "x(t)"
},
{
"math_id": 16,
"text": "X(f) = X^*(-f)"
}
] |
https://en.wikipedia.org/wiki?curid=15232604
|
15233551
|
Generalized linear mixed model
|
Statistical model
In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random effects in addition to the usual fixed effects. They also inherit from generalized linear models the idea of extending linear mixed models to non-normal data.
Generalized linear mixed models provide a broad range of models for the analysis of grouped data, since the differences between groups can be modelled as a random effect. These models are useful in the analysis of many kinds of data, including longitudinal data.
Model.
Generalized linear mixed models are generally defined such that, conditioned on the random effects formula_0, the dependent variable formula_1 is distributed according to the exponential family
with its expectation
related to the linear predictor formula_2 via a link function
formula_3:
formula_4.
Here formula_5 and formula_6 are the fixed effects design matrix, and fixed effects respectively; formula_7 and formula_8 are the random effects design matrix and random effects respectively. To understand this very brief definition you will first need to understand the definition of a generalized linear model and of a mixed model.
Generalized linear mixed models are a special cases of hierarchical generalized linear models in which the random effects are normally distributed.
The complete likelihood
formula_9
has no general closed form, and integrating over the random effects is usually extremely computationally intensive. In addition to numerically approximating this integral(e.g. via Gauss–Hermite quadrature), methods motivated by Laplace approximation have been proposed. For example, the penalized quasi-likelihood method, which essentially involves repeatedly fitting (i.e. doubly iterative) a weighted normal mixed model with a working variate, is implemented by various commercial and open source statistical programs.
Fitting a model.
Fitting generalized linear mixed models via maximum likelihood (as via the Akaike information criterion (AIC)) involves integrating over the random effects. In general, those integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data sets (e.g. ungrouped binary data are particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo have increased in use, as increasing computing power and advances in methods have made them more practical.
The Akaike information criterion is a common criterion for model selection. Estimates of the Akaike information criterion for generalized linear mixed models based on certain exponential family distributions have recently been obtained.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "u"
},
{
"math_id": 1,
"text": "y"
},
{
"math_id": 2,
"text": "X\\beta+Zu"
},
{
"math_id": 3,
"text": "g"
},
{
"math_id": 4,
"text": "g(E[y\\vert u])=X\\beta+Zu"
},
{
"math_id": 5,
"text": "X"
},
{
"math_id": 6,
"text": "\\beta"
},
{
"math_id": 7,
"text": "Z"
},
{
"math_id": 8,
"text": "u"
},
{
"math_id": 9,
"text": "p(y)=\\int p(y\\vert u)\\,p(u)\\,du"
}
] |
https://en.wikipedia.org/wiki?curid=15233551
|
15234158
|
PCO2
|
Partial pressure of carbon dioxide, often used in reference to blood
"p"CO2, pCO2, or formula_0is the partial pressure of carbon dioxide (CO2), often used in reference to blood but also used in meteorology, climate science, oceanography, and limnology to describe the fractional pressure of CO2 as a function of its concentration in gas or dissolved phases. The units of "p"CO2 are mmHg, atm, torr, Pa, or any other standard unit of atmospheric pressure. The "p"CO2 of Earth's atmosphere has risen from approximately 280 ppm (parts-per-million) to a mean 2019 value of 409.8 ppm as a result of anthropogenic release of carbon dioxide from fossil fuel burning. This is the highest atmospheric concentration to have existed on Earth for at least the last 800,000 years.
Medicine.
In medicine, the partial pressure of carbon dioxide in arterial blood is called formula_1 or PaCO2. Measurement of formula_1 in the systemic circulation indicates the effectiveness of ventilation at the lungs' alveoli, given the diffusing capacity of the gas. It is a good indicator of respiratory function and the closely related factor of acid–base homeostasis, reflecting the amount of acid in the blood (without lactic acid). Normal values for humans are in the range 35–45 mmHg. Values less than this may indicate hyperventilation and (if blood pH is greater than 7.45) respiratory alkalosis. Values greater than 45 mmHg may indicate hypoventilation, and (if blood pH is less than 7.35) respiratory acidosis.
Aquatic sciences.
Oceanographers and limnologists use "p"CO2 to measure the amount of carbon dioxide dissolved in water, as well as to parameterize its flux into (influx) and out of (efflux) the atmosphere. Carbon dioxide reacts with water to form bicarbonate and carbonate ions, such that the relative solubility of carbon dioxide in water is greater than that of other unreactive gasses (e.g. Helium). As more carbon dioxide dissolves in water, its "p"CO2 rises until it equals the "p"CO2 of the overlying atmosphere. Conversely, a body of water with a "p"CO2 greater than that of the atmosphere effluxes carbon dioxide.
"p"CO2 is additionally affected by water temperature and salinity. Carbon dioxide is less soluble in warmer water than cooler water, so hot water will exhibit a larger "p"CO2 than cold water with the same concentration of carbon dioxide. "p"CO2 can be used to describe the inorganic carbon system of a body of water, together with other parameters such as pH, dissolved inorganic carbon, and alkalinity. Together, these parameters describe the concentration and speciation of inorganic carbon species (CO2 (aq), HCO3−, CO32-) in water.
Biological processes such as respiration and photosynthesis affect and can be affected by aquatic "p"CO2. Respiration degrades organic matter, releasing CO2 into the water column and increasing "p"CO2. Photosynthesis assimilates inorganic carbon, thereby decreasing aquatic "p"CO2.
References.
<templatestyles src="Reflist/styles.css" />
" This article incorporates text by Glynda Rees Doyle and Jodie Anita McCutcheon available under the license."
|
[
{
"math_id": 0,
"text": "P_\\ce{CO2}"
},
{
"math_id": 1,
"text": "P_{a_\\ce{CO2}}"
}
] |
https://en.wikipedia.org/wiki?curid=15234158
|
15234505
|
Wallman compactification
|
A compactification of T1 topological spaces
In mathematics, the Wallman compactification, generally called Wallman–Shanin compactification is a compactification of T1 topological spaces that was constructed by .
Definition.
The points of the Wallman compactification ω"X" of a space "X" are the maximal proper filters in the poset of closed subsets of "X". Explicitly, a point of ω"X" is a family formula_0 of closed nonempty subsets of "X" such that formula_0 is closed under finite intersections, and is maximal among those families that have these properties. For every closed subset "F" of "X", the class Φ"F" of points of ω"X" containing "F" is closed in ω"X". The topology of ω"X" is generated by these closed classes.
Special cases.
For normal spaces, the Wallman compactification is essentially the same as the Stone–Čech compactification.
|
[
{
"math_id": 0,
"text": "\\mathcal F"
}
] |
https://en.wikipedia.org/wiki?curid=15234505
|
15237
|
Iterative method
|
Algorithm in which each approximation of the solution is derived from prior approximations
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the "i"-th approximation (called an "iterate") is derived from the previous ones.
A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method. An iterative method is called "convergent" if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common.
In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations formula_0 by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.
Attractive fixed points.
If an equation can be put into the form "f"("x") = "x", and a solution x is an attractive fixed point of the function "f", then one may begin with a point "x"1 in the basin of attraction of x, and let "x""n"+1 = "f"("x""n") for "n" ≥ 1, and the sequence {"x""n"}"n" ≥ 1 will converge to the solution x. Here "x""n" is the "n"th approximation or iteration of "x" and "x""n"+1 is the next or "n" + 1 iteration of "x". Alternately, superscripts in parentheses are often used in numerical methods, so as not to interfere with subscripts with other meanings. (For example, "x"("n"+1) = "f"("x"("n")).) If the function "f" is continuously differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the fixed point. If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.
Linear systems.
In the case of a system of linear equations, the two main classes of iterative methods are the stationary iterative methods, and the more general Krylov subspace methods.
Stationary iterative methods.
Introduction.
Stationary iterative methods solve a linear system with an operator approximating the original one; and based on a measurement of the error in the result (the residual), form a "correction equation" for which this process is repeated. While these methods are simple to derive, implement, and analyze, convergence is only guaranteed for a limited class of matrices.
Definition.
An "iterative method" is defined by
formula_1
and for a given linear system formula_2 with exact solution formula_3 the "error" by
formula_4
An iterative method is called "linear" if there exists a matrix formula_5 such that
formula_6
and this matrix is called the "iteration matrix".
An iterative method with a given iteration matrix formula_7 is called "convergent" if the following holds
formula_8
An important theorem states that for a given iterative method and its iteration matrix formula_7 it is convergent if and only if its spectral radius formula_9 is smaller than unity, that is,
formula_10
The basic iterative methods work by splitting the matrix formula_11 into
formula_12
and here the matrix formula_13 should be easily invertible.
The iterative methods are now defined as
formula_14
From this follows that the iteration matrix is given by
formula_15
Examples.
Basic examples of stationary iterative methods use a splitting of the matrix formula_11 such as
formula_16
where formula_17 is only the diagonal part of formula_11, and formula_18 is the strict lower triangular part of formula_11.
Respectively, formula_19 is the strict upper triangular part of formula_11.
Linear stationary iterative methods are also called relaxation methods.
Krylov subspace methods.
Krylov subspace methods work by forming a basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence).
The approximations to the solution are then formed by minimizing the residual over the subspace formed.
The prototypical method in this class is the conjugate gradient method (CG) which assumes that the system matrix formula_11 is symmetric positive-definite.
For symmetric (and possibly indefinite) formula_11 one works with the minimal residual method (MINRES).
In the case of non-symmetric matrices, methods such as the generalized minimal residual method (GMRES) and the biconjugate gradient method (BiCG) have been derived.
Convergence of Krylov subspace methods.
Since these methods form a basis, it is evident that the method converges in "N" iterations, where "N" is the system size. However, in the presence of rounding errors this statement does not hold; moreover, in practice "N" can be very large, and the iterative process reaches sufficient accuracy already far earlier. The analysis of these methods is hard, depending on a complicated function of the spectrum of the operator.
Preconditioners.
The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as GMRES (alternatively, preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.
History.
Jamshīd al-Kāshī used iterative methods to calculate the sine of 1° and π in "The Treatise of Chord and Sine" to high precision.
An early iterative method for solving a linear system appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest .
The theory of stationary iterative methods was solidly established with the work of D.M. Young starting in the 1950s. The conjugate gradient method was also invented in the 1950s, with independent developments by Cornelius Lanczos, Magnus Hestenes and Eduard Stiefel, but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially the elliptic type.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "A\\mathbf{x}=\\mathbf{b}"
},
{
"math_id": 1,
"text": "\n \\mathbf{x}^{k+1} := \\Psi ( \\mathbf{x}^k ) \\,, \\quad k\\geq0\n"
},
{
"math_id": 2,
"text": " A\\mathbf x= \\mathbf b "
},
{
"math_id": 3,
"text": " \\mathbf{x}^* "
},
{
"math_id": 4,
"text": "\n \\mathbf{e}^k := \\mathbf{x}^k - \\mathbf{x}^* \\,, \\quad k\\geq0\\,.\n"
},
{
"math_id": 5,
"text": " C \\in \\R^{n\\times n} "
},
{
"math_id": 6,
"text": "\n \\mathbf{e}^{k+1} = C \\mathbf{e}^k \\quad \\forall \\, k\\geq0\n"
},
{
"math_id": 7,
"text": " C "
},
{
"math_id": 8,
"text": "\n \\lim_{k\\rightarrow \\infty} C^k=0\\,.\n"
},
{
"math_id": 9,
"text": " \\rho(C) "
},
{
"math_id": 10,
"text": "\n \\rho(C) < 1 \\,.\n"
},
{
"math_id": 11,
"text": " A "
},
{
"math_id": 12,
"text": "\n A = M - N\n"
},
{
"math_id": 13,
"text": " M "
},
{
"math_id": 14,
"text": "\n M \\mathbf{x}^{k+1} = N \\mathbf{x}^k + b \\,, \\quad k\\geq0\\,.\n"
},
{
"math_id": 15,
"text": "\n C = I - M^{-1}A = M^{-1}N\\,.\n"
},
{
"math_id": 16,
"text": "\n A = D+L+U\\,,\\quad D := \\text{diag}( (a_{ii})_i)\n"
},
{
"math_id": 17,
"text": " D "
},
{
"math_id": 18,
"text": " L "
},
{
"math_id": 19,
"text": " U "
},
{
"math_id": 20,
"text": " M:=\\frac{1}{\\omega} I \\quad (\\omega \\neq 0) "
},
{
"math_id": 21,
"text": " M:=D "
},
{
"math_id": 22,
"text": " M:=\\frac{1}{\\omega}D \\quad (\\omega \\neq 0) "
},
{
"math_id": 23,
"text": " M:=D+L "
},
{
"math_id": 24,
"text": " M:=\\frac{1}{\\omega}D+L \\quad (\\omega \\neq 0) "
},
{
"math_id": 25,
"text": " M := \\frac{1}{\\omega (2-\\omega)} (D+\\omega L) D^{-1} (D+\\omega U) \n\\quad (\\omega \\not \n\\in \\{0,2\\}) "
}
] |
https://en.wikipedia.org/wiki?curid=15237
|
15239331
|
Arithmetic hyperbolic 3-manifold
|
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space formula_0 by an arithmetic Kleinian group.
Definition and examples.
Quaternion algebras.
A quaternion algebra over a field formula_1 is a four-dimensional central simple formula_1-algebra. A quaternion algebra has a basis formula_2 where formula_3 and formula_4.
A quaternion algebra is said to be split over formula_1 if it is isomorphic as an formula_1-algebra to the algebra of matrices formula_5; a quaternion algebra over an algebraically closed field is always split.
If formula_6 is an embedding of formula_1 into a field formula_7 we shall denote by formula_8 the algebra obtained by extending scalars from formula_1 to formula_7 where we view formula_1 as a subfield of formula_7 via formula_6.
Arithmetic Kleinian groups.
A subgroup of formula_9 is said to be "derived from a quaternion algebra" if it can be obtained through the following construction. Let formula_1 be a number field which has exactly two embeddings into formula_10 whose image is not contained in formula_11 (one conjugate to the other). Let formula_12 be a quaternion algebra over formula_1 such that for any embedding formula_13 the algebra formula_14 is isomorphic to the Hamilton quaternions. Next we need an order formula_15 in formula_12. Let formula_16 be the group of elements in formula_15 of reduced norm 1 and let formula_17 be its image in formula_18 via formula_19. We then consider the Kleinian group obtained as the image in formula_9 of formula_20.
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on formula_9. Moreover, the construction above yields a cocompact subgroup if and only if the algebra formula_12 is not split over formula_1. The discreteness is a rather immediate consequence of the fact that formula_12 is only split at its complex embeddings. The finiteness of covolume is harder to prove.
An "arithmetic Kleinian group" is any subgroup of formula_9 which is commensurable to a group derived from a quaternion algebra. It follows immediately from this definition that arithmetic Kleinian groups are discrete and of finite covolume (this means that they are lattices in formula_9).
Examples.
Examples are provided by taking formula_1 to be an imaginary quadratic field, formula_21 and formula_22 where formula_23 is the ring of integers of formula_1 (for example formula_24 and formula_25). The groups thus obtained are the Bianchi groups. They are not cocompact, and any arithmetic Kleinian group which is not commensurable to a conjugate of a Bianchi group is cocompact.
If formula_12 is any quaternion algebra over an imaginary quadratic number field formula_1 which is not isomorphic to a matrix algebra then the unit groups of orders in formula_12 are cocompact.
Trace field of arithmetic manifolds.
The invariant trace field of a Kleinian group (or, through the monodromy image of the fundamental group, of an hyperbolic manifold) is the field generated by the traces of the squares of its elements. In the case of an arithmetic manifold whose fundamental groups is commensurable with that of a manifold derived from a quaternion algebra over a number field formula_1 the invariant trace field equals formula_1.
One can in fact characterise arithmetic manifolds through the traces of the elements of their fundamental group. A Kleinian group is an arithmetic group if and only if the following three conditions are realised:
Geometry and spectrum of arithmetic hyperbolic three-manifolds.
Volume formula.
For the volume an arithmetic three manifold formula_31 derived from a maximal order in a quaternion algebra formula_12 over a number field formula_32 we have the expression:
formula_33
where formula_34 are the discriminants of formula_35 respectively, formula_36 is the Dedekind zeta function of formula_1 and formula_37.
Finiteness results.
A consequence of the volume formula in the previous paragraph is that
<templatestyles src="Block indent/styles.css"/>"Given formula_38 there are at most finitely many arithmetic hyperbolic 3-manifolds with volume less than formula_39."
This is in contrast with the fact that hyperbolic Dehn surgery can be used to produce infinitely many non-isometric hyperbolic 3-manifolds with bounded volume. In particular, a corollary is that given a cusped hyperbolic manifold, at most finitely many Dehn surgeries on it can yield an arithmetic hyperbolic manifold.
Remarkable arithmetic hyperbolic three-manifolds.
The Weeks manifold is the hyperbolic three-manifold of smallest volume and the Meyerhoff manifold is the one of next smallest volume.
The complement in the three-sphere of the figure-eight knot is an arithmetic hyperbolic three-manifold and attains the smallest volume among all cusped hyperbolic three-manifolds.
Spectrum and Ramanujan conjectures.
The Ramanujan conjecture for automorphic forms on formula_40 over a number field would imply that for any congruence cover of an arithmetic three-manifold (derived from a quaternion algebra) the spectrum of the Laplace operator is contained in formula_41.
Arithmetic manifolds in three-dimensional topology.
Many of Thurston's conjectures (for example the virtually Haken conjecture), now all known to be true following the work of Ian Agol, were checked first for arithmetic manifolds by using specific methods. In some arithmetic cases the Virtual Haken conjecture is known by general means but it is not known if its solution can be arrived at by purely arithmetic means (for instance, by finding a congruence subgroup with positive first Betti number).
Arithmetic manifolds can be used to give examples of manifolds with large injectivity radius whose first Betti number vanishes.
A remark by William Thurston is that arithmetic manifolds "...often seem to have special beauty." This can be substantiated by results showing that the relation between topology and geometry for these manifolds is much more predictable than in general. For example:
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbb H^3"
},
{
"math_id": 1,
"text": "F"
},
{
"math_id": 2,
"text": "1, i, j, ij"
},
{
"math_id": 3,
"text": "i^2, j^2 \\in F^\\times"
},
{
"math_id": 4,
"text": "ij = -ji"
},
{
"math_id": 5,
"text": "M_2(F)"
},
{
"math_id": 6,
"text": "\\sigma"
},
{
"math_id": 7,
"text": "E"
},
{
"math_id": 8,
"text": "A \\otimes_\\sigma E"
},
{
"math_id": 9,
"text": "\\mathrm{PGL}_2(\\Complex)"
},
{
"math_id": 10,
"text": "\\Complex"
},
{
"math_id": 11,
"text": "\\Reals"
},
{
"math_id": 12,
"text": "A"
},
{
"math_id": 13,
"text": "\\tau: F \\to \\Reals"
},
{
"math_id": 14,
"text": "A \\otimes_\\tau \\Reals"
},
{
"math_id": 15,
"text": "\\mathcal O"
},
{
"math_id": 16,
"text": "\\mathcal O^1"
},
{
"math_id": 17,
"text": "\\Gamma"
},
{
"math_id": 18,
"text": "M_2(\\Complex)"
},
{
"math_id": 19,
"text": "\\phi"
},
{
"math_id": 20,
"text": "\\phi(\\mathcal O^1)"
},
{
"math_id": 21,
"text": "A = M_2(F)"
},
{
"math_id": 22,
"text": "\\mathcal O = M_2(O_F)"
},
{
"math_id": 23,
"text": "O_F"
},
{
"math_id": 24,
"text": "F = \\Q(i)"
},
{
"math_id": 25,
"text": "O_F = \\Z[i]"
},
{
"math_id": 26,
"text": "<matH>F</math>"
},
{
"math_id": 27,
"text": "\\gamma"
},
{
"math_id": 28,
"text": "t=\\mathrm{Trace}(\\gamma^2)"
},
{
"math_id": 29,
"text": "\\sigma: F \\to \\R"
},
{
"math_id": 30,
"text": "|\\sigma(t)| \\le 2"
},
{
"math_id": 31,
"text": "M = \\Gamma_{\\mathcal O} \\backslash \\mathbb H^3"
},
{
"math_id": 32,
"text": "f"
},
{
"math_id": 33,
"text": " \\mathrm{vol}(M) = \\frac {2 |D_F|^{\\frac 3 2} \\cdot \\zeta_F(2)} {2^{2r + 1} \\cdot \\pi^{2r}} \\cdot \\prod_{\\mathfrak p | D_A} (N(\\mathfrak p) - 1). "
},
{
"math_id": 34,
"text": "D_A,D_F"
},
{
"math_id": 35,
"text": "A,F"
},
{
"math_id": 36,
"text": "\\zeta_F"
},
{
"math_id": 37,
"text": "r = [F:\\Q]"
},
{
"math_id": 38,
"text": "v > 0"
},
{
"math_id": 39,
"text": "v"
},
{
"math_id": 40,
"text": "\\mathrm{GL}(2)"
},
{
"math_id": 41,
"text": "[1, +\\infty)"
}
] |
https://en.wikipedia.org/wiki?curid=15239331
|
152428
|
Ceva's theorem
|
Geometric relation between line segments from a triangle's vertices and their intersection
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle △"ABC", let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of △"ABC"), to meet opposite sides at D, E, F respectively. (The segments are known as cevians.) Then, using signed lengths of segments,
formula_0
In other words, the length is taken to be positive or negative according to whether X is to the left or right of Y in some fixed orientation of the line. For example, is defined as having positive value when F is between A and B and negative otherwise.
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
A slightly adapted converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
formula_1
then AD, BE, CF are concurrent, or all three parallel. The converse is often included as part of the theorem.
The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work "De lineis rectis". But it was proven much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza.
Associated with the figures are several terms derived from Ceva's name: cevian (the lines AD, BE, CF are the cevians of O), cevian triangle (the triangle △"DEF" is the cevian triangle of O); cevian nest, anticevian triangle, Ceva conjugate. ("Ceva" is pronounced Chay'va; "cevian" is pronounced chev'ian.)
The theorem is very similar to Menelaus' theorem in that their equations differ only in sign. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.
Proofs.
Several proofs of the theorem have been given.
Two proofs are given in the following.
The first one is very elementary, using only basic properties of triangle areas. However, several cases have to be considered, depending on the position of the point O.
The second proof uses barycentric coordinates and vectors, but is somehow more natural and not case dependent. Moreover, it works in any affine plane over any field.
Using triangle areas.
First, the sign of the left-hand side is positive since either all three of the ratios are positive, the case where O is inside the triangle (upper diagram), or one is positive and the other two are negative, the case O is outside the triangle (lower diagram shows one case).
To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
formula_2
Therefore,
formula_3
Similarly,
formula_4
and
formula_5
Multiplying these three equations gives
formula_6
as required.
The theorem can also be proven easily using Menelaus's theorem. From the transversal BOE of triangle △"ACF",
formula_7
and from the transversal AOD of triangle △"BCF",
formula_8
The theorem follows by dividing these two equations.
The converse follows as a corollary. Let D, E, F be given on the lines BC, AC, AB so that the equation holds. Let AD, BE meet at O and let F' be the point where CO crosses AB. Then by the theorem, the equation also holds for D, E, F'. Comparing the two,
formula_9
But at most one point can cut a segment in a given ratio so F = F’.
Using barycentric coordinates.
Given three points A, B, C that are not collinear, and a point O, that belongs to the same plane, the barycentric coordinates of O with respect of A, B, C are the unique three numbers formula_10 such that
formula_11
and
formula_12
for every point X (for the definition of this arrow notation and further details, see Affine space).
For Ceva's theorem, the point O is supposed to not belong to any line passing through two vertices of the triangle. This implies that formula_13
If one takes for X the intersection F of the lines AB and OC (see figures), the last equation may be rearranged into
formula_14
The left-hand side of this equation is a vector that has the same direction as the line CF, and the right-hand side has the same direction as the line AB. These lines have different directions since A, B, C are not collinear. It follows that the two members of the equation equal the zero vector, and
formula_15
It follows that
formula_16
where the left-hand-side fraction is the signed ratio of the lengths of the collinear line segments and .
The same reasoning shows
formula_17
Ceva's theorem results immediately by taking the product of the three last equations.
Generalizations.
The theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite ("n" – 1)-face (facet). Then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass. Moreover, the intersection point of the cevians is the center of mass of the simplex.
Another generalization to higher-dimensional simplexes extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Starting from a point in a simplex, a point is defined inductively on each k-face. This point is the foot of a cevian that goes from the vertex opposite the k-face, in a ("k" + 1)-face that contains it, through the point already defined on this ("k" + 1)-face. Each of these points divides the face on which it lies into lobes. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1.
Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.
The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century.
The theorem has also been generalized to triangles on other surfaces of constant curvature.
The theorem also has a well-known generalization to spherical and hyperbolic geometry, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\frac{\\overline{AF}}{\\overline{FB}} \\cdot \\frac{\\overline{BD}}{\\overline{DC}} \\cdot \\frac{\\overline{CE}}{\\overline{EA}} = 1."
},
{
"math_id": 1,
"text": "\\frac{\\overline{AF}}{\\overline{FB}} \\cdot \\frac{\\overline{BD}}{\\overline{DC}} \\cdot \\frac{\\overline{CE}}{\\overline{EA}} = 1,"
},
{
"math_id": 2,
"text": "\\frac{|\\triangle BOD|}{|\\triangle COD|}=\\frac{\\overline{BD}}{\\overline{DC}}=\\frac{|\\triangle BAD|}{|\\triangle CAD|}."
},
{
"math_id": 3,
"text": "\\frac{\\overline{BD}}{\\overline{DC}}=\n\\frac{|\\triangle BAD|-|\\triangle BOD|}{|\\triangle CAD|-|\\triangle COD|}\n=\\frac{|\\triangle ABO|}{|\\triangle CAO|}."
},
{
"math_id": 4,
"text": "\\frac{\\overline{CE}}{\\overline{EA}}=\\frac{|\\triangle BCO|}{|\\triangle ABO|},"
},
{
"math_id": 5,
"text": "\\frac{\\overline{AF}}{\\overline{FB}}=\\frac{|\\triangle CAO|}{|\\triangle BCO|}."
},
{
"math_id": 6,
"text": "\\left|\\frac{\\overline{AF}}{\\overline{FB}} \\cdot \\frac{\\overline{BD}}{\\overline{DC}} \\cdot \\frac{\\overline{CE}}{\\overline{EA}} \\right|= 1,"
},
{
"math_id": 7,
"text": "\\frac{\\overline{AB}}{\\overline{BF}} \\cdot \\frac{\\overline{FO}}{\\overline{OC}} \\cdot \\frac{\\overline{CE}}{\\overline{EA}} = -1"
},
{
"math_id": 8,
"text": "\\frac{\\overline{BA}}{\\overline{AF}} \\cdot \\frac{\\overline{FO}}{\\overline{OC}} \\cdot \\frac{\\overline{CD}}{\\overline{DB}} = -1."
},
{
"math_id": 9,
"text": "\\frac{\\overline{AF}}{\\overline{FB}} = \\frac{\\overline{AF'}}{\\overline{F'B}}"
},
{
"math_id": 10,
"text": "\\lambda_A, \\lambda_B, \\lambda_C"
},
{
"math_id": 11,
"text": "\\lambda_A + \\lambda_B + \\lambda_C =1,"
},
{
"math_id": 12,
"text": "\\overrightarrow{XO}=\\lambda_A\\overrightarrow{XA} + \\lambda_B\\overrightarrow{XB} + \\lambda_C\\overrightarrow{XC},"
},
{
"math_id": 13,
"text": "\\lambda_A \\lambda_B \\lambda_C\\ne 0."
},
{
"math_id": 14,
"text": "\\overrightarrow{FO}-\\lambda_C\\overrightarrow{FC}=\\lambda_A\\overrightarrow{FA} + \\lambda_B\\overrightarrow{FB}."
},
{
"math_id": 15,
"text": "\\lambda_A\\overrightarrow{FA} + \\lambda_B\\overrightarrow{FB}=0."
},
{
"math_id": 16,
"text": "\\frac{\\overline{AF}}{\\overline{FB}}=\\frac{\\lambda_B}{\\lambda_A},"
},
{
"math_id": 17,
"text": "\\frac{\\overline{BD}}{\\overline{DC}}=\\frac{\\lambda_C}{\\lambda_B}\\quad \\text{and}\\quad \\frac{\\overline{CE}}{\\overline{EA}}=\\frac{\\lambda_A}{\\lambda_C}."
}
] |
https://en.wikipedia.org/wiki?curid=152428
|
152440
|
Stellar nucleosynthesis
|
Creation of chemical elements within stars
In astrophysics, stellar nucleosynthesis is the creation of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. As a predictive theory, it yields accurate estimates of the observed abundances of the elements. It explains why the observed abundances of elements change over time and why some elements and their isotopes are much more abundant than others. The theory was initially proposed by Fred Hoyle in 1946, who later refined it in 1954. Further advances were made, especially to nucleosynthesis by neutron capture of the elements heavier than iron, by Margaret and Geoffrey Burbidge, William Alfred Fowler and Fred Hoyle in their famous 1957 B2FH paper, which became one of the most heavily cited papers in astrophysics history.
Stars evolve because of changes in their composition (the abundance of their constituent elements) over their lifespans, first by burning hydrogen (main sequence star), then helium (horizontal branch star), and progressively burning higher elements. However, this does not by itself significantly alter the abundances of elements in the universe as the elements are contained within the star. Later in its life, a low-mass star will slowly eject its atmosphere via stellar wind, forming a planetary nebula, while a higher–mass star will eject mass via a sudden catastrophic event called a supernova. The term supernova nucleosynthesis is used to describe the creation of elements during the explosion of a massive star or white dwarf.
The advanced sequence of burning fuels is driven by gravitational collapse and its associated heating, resulting in the subsequent burning of carbon, oxygen and silicon. However, most of the nucleosynthesis in the mass range "A" = 28–56 (from silicon to nickel) is actually caused by the upper layers of the star collapsing onto the core, creating a compressional shock wave rebounding outward. The shock front briefly raises temperatures by roughly 50%, thereby causing furious burning for about a second. This final burning in massive stars, called "explosive nucleosynthesis" or supernova nucleosynthesis, is the final epoch of stellar nucleosynthesis.
A stimulus to the development of the theory of nucleosynthesis was the discovery of variations in the abundances of elements found in the universe. The need for a physical description was already inspired by the relative abundances of the chemical elements in the solar system. Those abundances, when plotted on a graph as a function of the atomic number of the element, have a jagged sawtooth shape that varies by factors of tens of millions (see history of nucleosynthesis theory). This suggested a natural process that is not random. A second stimulus to understanding the processes of stellar nucleosynthesis occurred during the 20th century, when it was realized that the energy released from nuclear fusion reactions accounted for the longevity of the Sun as a source of heat and light.
History.
In 1920, Arthur Eddington, on the basis of the precise measurements of atomic masses by F.W. Aston and a preliminary suggestion by Jean Perrin, proposed that stars obtained their energy from nuclear fusion of hydrogen to form helium and raised the possibility that the heavier elements are produced in stars. This was a preliminary step toward the idea of stellar nucleosynthesis. In 1928 George Gamow derived what is now called the Gamow factor, a quantum-mechanical formula yielding the probability for two contiguous nuclei to overcome the electrostatic Coulomb barrier between them and approach each other closely enough to undergo nuclear reaction due to the strong nuclear force which is effective only at very short distances. In the following decade the Gamow factor was used by Atkinson and Houtermans and later by Edward Teller and Gamow himself to derive the rate at which nuclear reactions would occur at the high temperatures believed to exist in stellar interiors.
In 1939, in a Nobel lecture entitled "Energy Production in Stars", Hans Bethe analyzed the different possibilities for reactions by which hydrogen is fused into helium. He defined two processes that he believed to be the sources of energy in stars. The first one, the proton–proton chain reaction, is the dominant energy source in stars with masses up to about the mass of the Sun. The second process, the carbon–nitrogen–oxygen cycle, which was also considered by Carl Friedrich von Weizsäcker in 1938, is more important in more massive main-sequence stars. These works concerned the energy generation capable of keeping stars hot. A clear physical description of the proton–proton chain and of the CNO cycle appears in a 1968 textbook. Bethe's two papers did not address the creation of heavier nuclei, however. That theory was begun by Fred Hoyle in 1946 with his argument that a collection of very hot nuclei would assemble thermodynamically into iron. Hoyle followed that in 1954 with a paper describing how advanced fusion stages within massive stars would synthesize the elements from carbon to iron in mass.
Hoyle's theory was extended to other processes, beginning with the publication of the 1957 review paper "Synthesis of the Elements in Stars" by Burbidge, Burbidge, Fowler and Hoyle, more commonly referred to as the B2FH paper. This review paper collected and refined earlier research into a heavily cited picture that gave promise of accounting for the observed relative abundances of the elements; but it did not itself enlarge Hoyle's 1954 picture for the origin of primary nuclei as much as many assumed, except in the understanding of nucleosynthesis of those elements heavier than iron by neutron capture. Significant improvements were made by Alastair G. W. Cameron and by Donald D. Clayton. In 1957 Cameron presented his own independent approach to nucleosynthesis, informed by Hoyle's example, and introduced computers into time-dependent calculations of evolution of nuclear systems. Clayton calculated the first time-dependent models of the "s"-process in 1961 and of the "r"-process in 1965, as well as of the burning of silicon into the abundant alpha-particle nuclei and iron-group elements in 1968, and discovered radiogenic chronologies for determining the age of the elements.
Key reactions.
The most important reactions in stellar nucleosynthesis:
Hydrogen fusion.
Hydrogen fusion (nuclear fusion of four protons to form a helium-4 nucleus) is the dominant process that generates energy in the cores of main-sequence stars. It is also called "hydrogen burning", which should not be confused with the chemical combustion of hydrogen in an oxidizing atmosphere. There are two predominant processes by which stellar hydrogen fusion occurs: proton–proton chain and the carbon–nitrogen–oxygen (CNO) cycle. Ninety percent of all stars, with the exception of white dwarfs, are fusing hydrogen by these two processes.
In the cores of lower-mass main-sequence stars such as the Sun, the dominant energy production process is the proton–proton chain reaction. This creates a helium-4 nucleus through a sequence of reactions that begin with the fusion of two protons to form a deuterium nucleus (one proton plus one neutron) along with an ejected positron and neutrino. In each complete fusion cycle, the proton–proton chain reaction releases about 26.2 MeV. The proton–proton chain reaction cycle is relatively insensitive to temperature; a 10% rise of temperature would increase energy production by this method by 46%, hence, this hydrogen fusion process can occur in up to a third of the star's radius and occupy half the star's mass. For stars above 35% of the Sun's mass, the energy flux toward the surface is sufficiently low and energy transfer from the core region remains by radiative heat transfer, rather than by convective heat transfer. As a result, there is little mixing of fresh hydrogen into the core or fusion products outward.
In higher-mass stars, the dominant energy production process is the CNO cycle, which is a catalytic cycle that uses nuclei of carbon, nitrogen and oxygen as intermediaries and in the end produces a helium nucleus as with the proton–proton chain. During a complete CNO cycle, 25.0 MeV of energy is released. The difference in energy production of this cycle, compared to the proton–proton chain reaction, is accounted for by the energy lost through neutrino emission. The CNO cycle is very temperature sensitive, a 10% rise of temperature would produce a 350% rise in energy production. About 90% of the CNO cycle energy generation occurs within the inner 15% of the star's mass, hence it is strongly concentrated at the core. This results in such an intense outward energy flux that convective energy transfer becomes more important than does radiative transfer. As a result, the core region becomes a convection zone, which stirs the hydrogen fusion region and keeps it well mixed with the surrounding proton-rich region. This core convection occurs in stars where the CNO cycle contributes more than 20% of the total energy. As the star ages and the core temperature increases, the region occupied by the convection zone slowly shrinks from 20% of the mass down to the inner 8% of the mass. The Sun produces on the order of 1% of its energy from the CNO cycle.
The type of hydrogen fusion process that dominates in a star is determined by the temperature dependency differences between the two reactions. The proton–proton chain reaction starts at temperatures about , making it the dominant fusion mechanism in smaller stars. A self-maintaining CNO chain requires a higher temperature of approximately , but thereafter it increases more rapidly in efficiency as the temperature rises, than does the proton–proton reaction. Above approximately , the CNO cycle becomes the dominant source of energy. This temperature is achieved in the cores of main-sequence stars with at least 1.3 times the mass of the Sun. The Sun itself has a core temperature of about . As a main-sequence star ages, the core temperature will rise, resulting in a steadily increasing contribution from its CNO cycle.
Helium fusion.
Main sequence stars accumulate helium in their cores as a result of hydrogen fusion, but the core does not become hot enough to initiate helium fusion. Helium fusion first begins when a star leaves the red giant branch after accumulating sufficient helium in its core to ignite it. In stars around the mass of the Sun, this begins at the tip of the red giant branch with a helium flash from a degenerate helium core, and the star moves to the horizontal branch where it burns helium in its core. More massive stars ignite helium in their core without a flash and execute a blue loop before reaching the asymptotic giant branch. Such a star initially moves away from the AGB toward bluer colours, then loops back again to what is called the Hayashi track. An important consequence of blue loops is that they give rise to classical Cepheid variables, of central importance in determining distances in the Milky Way and to nearby galaxies. Despite the name, stars on a blue loop from the red giant branch are typically not blue in colour but are rather yellow giants, possibly Cepheid variables. They fuse helium until the core is largely carbon and oxygen. The most massive stars become supergiants when they leave the main sequence and quickly start helium fusion as they become red supergiants. After the helium is exhausted in the core of a star, helium fusion will continue in a shell around the carbon–oxygen core.
In all cases, helium is fused to carbon via the triple-alpha process, i.e., three helium nuclei are transformed into carbon via 8Be. This can then form oxygen, neon, and heavier elements via the alpha process. In this way, the alpha process preferentially produces elements with even numbers of protons by the capture of helium nuclei. Elements with odd numbers of protons are formed by other fusion pathways.
Reaction rate.
The reaction rate density between species "A" and "B", having number densities "n""A","B", is given by:
formula_0
where "k" is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process:
formula_1
here, "σ"("v") is the cross-section at relative velocity "v", and averaging is performed over all velocities.
Semi-classically, the cross section is proportional to formula_2, where formula_3 is the de Broglie wavelength. Thus semi-classically the cross section is proportional to formula_4.
However, since the reaction involves quantum tunneling, there is an exponential damping at low energies that depends on Gamow factor "E"G, giving an Arrhenius equation:
formula_5
where "S"("E") depends on the details of the nuclear interaction, and has the dimension of an energy multiplied for a cross section.
One then integrates over all energies to get the total reaction rate, using the Maxwell–Boltzmann distribution and the relation:
formula_6
where formula_7 is the reduced mass.
Since this integration has an exponential damping at high energies of the form formula_8 and at low energies from the Gamow factor, the integral almost vanished everywhere except around the peak, called Gamow peak, at "E"0, where:
formula_9
Thus:
formula_10
The exponent can then be approximated around "E"0 as:
formula_11
And the reaction rate is approximated as:
formula_12
Values of "S"("E"0) are typically 10−3 – 103 keV·b, but are damped by a huge factor when involving a beta decay, due to the relation between the intermediate bound state (e.g. diproton) half-life and the beta decay half-life, as in the proton–proton chain reaction. Note that typical core temperatures in main-sequence stars give "kT" of the order of keV.
Thus, the limiting reaction in the CNO cycle, proton capture by [<noinclude />[nitrogen-14|N]<noinclude />], has "S"("E"0) ~ "S"(0) = 3.5keV·b, while the limiting reaction in the proton–proton chain reaction, the creation of deuterium from two protons, has a much lower "S"("E"0) ~ "S"(0) = 4×10−22keV·b. Incidentally, since the former reaction has a much higher Gamow factor, and due to the relative abundance of elements in typical stars, the two reaction rates are equal at a temperature value that is within the core temperature ranges of main-sequence stars.
References.
Notes.
<templatestyles src="Reflist/styles.css" />
Citations.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "r = n_A \\, n_B \\, k "
},
{
"math_id": 1,
"text": "k = \\langle \\sigma(v)\\,v \\rangle"
},
{
"math_id": 2,
"text": "\\pi\\,\\lambda^2"
},
{
"math_id": 3,
"text": "\\lambda = h/p"
},
{
"math_id": 4,
"text": "\\frac{m}{E}"
},
{
"math_id": 5,
"text": "\\sigma(E) = \\frac{S(E)}{E} e^{-\\sqrt{\\frac{E_\\text{G}}{E}}}"
},
{
"math_id": 6,
"text": "\\frac{r}{V} = n_A n_B \\int_0^{\\infty}\\frac{S(E)}{E} \\, e^{-\\sqrt{\\frac{E_\\text{G}}{E}}} 2\\sqrt{\\frac{E}{\\pi(kT)^3}} e^{-\\frac{E}{kT}} \\,\\sqrt{\\frac{2E}{m_\\text{R}}}dE"
},
{
"math_id": 7,
"text": "m_\\text{R} = \\frac{m_1 m_2}{m_1 + m_2}"
},
{
"math_id": 8,
"text": "\\sim e^{-\\frac{E}{kT}}"
},
{
"math_id": 9,
"text": "\\frac{\\partial}{\\partial E} \\left( -\\sqrt{\\frac{E_\\text{G}}{E}} - \\frac{E}{kT}\\right) \\, = \\, 0"
},
{
"math_id": 10,
"text": "E_0 = \\left(\\frac{1}{2}kT \\sqrt{E_\\text{G}}\\right)^\\frac{2}{3}"
},
{
"math_id": 11,
"text": "e^{-\\frac{E}{kT} - \\sqrt{\\frac{E_\\text{G}}{E}}} \\approx e^{-\\frac{3E_0}{kT}} \\exp\\left(-\\frac{(E - E_0)^2}{\\frac{4}{3} E_0 kT}\\right)"
},
{
"math_id": 12,
"text": "\\frac{r}{V} \\approx n_A \\, n_B \\, \\frac{4\\sqrt{2}}{\\sqrt{3 m_\\text{R}}}\\, \\sqrt{E_0} \\frac{S(E_0)}{kT} e^{-\\frac{3E_0}{kT}} "
}
] |
https://en.wikipedia.org/wiki?curid=152440
|
1524543
|
Chinese mathematics
|
Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (binary and decimal), algebra, geometry, number theory and trigonometry.
Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like continued fractions are widely used and have been well-documented ever since. They deliberately find the principal "n"th root of positive numbers and the roots of equations. The major texts from the period, "The Nine Chapters on the Mathematical Art" and the "Book on Numbers and Computation" gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra. The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of "tian yuan shu".
As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when "The Nine Chapters on the Mathematical Art" reached its final form, while the "Book on Numbers and Computation" and "Huainanzi" are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as the Song-era polymath Shen Kuo.
Pre-imperial era.
Shang dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the "I Ching", which greatly influenced written literature during the Zhou dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers.
Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations, and negative numbers with counting rods. Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry, and the usage of decimals.
Math was one of the Six Arts students were required to master during the Zhou dynasty (1122–256 BCE). Learning them all perfectly was required to be a perfect gentleman, comparable to the concept of a "renaissance man". Six Arts have their roots in the Confucian philosophy.
The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BCE, compiled by the followers of Mozi (470–390 BCE). The "Mo Jing" described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like Euclid's first and third definitions and Plato's 'beginning of a line', the "Mo Jing" stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus, the "Mo Jing" stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved." It stated that two lines of equal length will always finish at the same place," while providing definitions for the "comparison of lengths" and for "parallels"," along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided word recognition for circumference, diameter, and radius, along with the definition of volume.
The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the "Zhoubi Suanjing" dates around 1200–1000 BC, yet many scholars believed it was written between 300 and 250 BCE. The "Zhoubi Suanjing" contains an in-depth proof of the "Gougu Theorem", a special case of the Pythagorean theorem) but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips, dated c. 305 BCE, has revealed some aspects of pre-Qin mathematics, such as the first known decimal multiplication table.
The abacus was first mentioned in the second century BC, alongside 'calculation with rods' ("suan zi") in which small bamboo sticks are placed in successive squares of a checkerboard.
Qin dynasty.
Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars, circa 213–210 BC. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering. Emperor Qin Shi Huang ordered many men to build large, life-sized statues for the palace tomb along with other temples and shrines, and the shape of the tomb was designed with geometric skills of architecture. It is certain that one of the greatest feats of human history, the Great Wall of China, required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion.
Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy, according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise.
Han dynasty.
In the Han dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called rod calculus, consisting of only nine symbols with a blank space on the counting board representing zero. Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period. The mathematical texts of the time, the "Book on Numbers and Computation" and "Jiuzhang suanshu" solved basic arithmetic problems such as addition, subtraction, multiplication and division. Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order. Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns. The value of pi is taken to be equal to three in both texts. However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used. Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment. The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume. The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life.
"Book on Numbers and Computation".
The "Book on Numbers and Computation" is approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1984 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the "Suan shu shu" is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources.
The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art. An example of the elementary mathematics in the "Suàn shù shū", the square root is approximated by using false position method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method.
"The Nine Chapters on the Mathematical Art".
"The Nine Chapters on the Mathematical Art" dates archeologically to 179 CE, though it is traditionally dated to 1000 BCE, but it was written perhaps as early as 300–200 BCE. Although the author(s) are unknown, they made a major contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure. There are no formal mathematical proofs within the text, just a step-by-step procedure. The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text.
"The Nine Chapters on the Mathematical Art" was one of the most influential of all Chinese mathematical books and it is composed of 246 problems. It was later incorporated into "The Ten Computational Canons", which became the core of mathematical education in later centuries. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. "The Nine Chapters" made significant additions to solving quadratic equations in a way similar to Horner's method. It also made advanced contributions to "fangcheng", or what is now known as linear algebra. Chapter seven solves system of linear equations with two unknowns using the false position method, similar to The Book of Computations. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns. The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution.
The version of "The Nine Chapters" that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from "Yongle Encyclopedia", he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations. His finished work would be first published in 1774, but a new revision would be published in 1776 to correct various errors as well as include a version of "The Nine Chapters" from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng. The final version of Dai Zhen's work would come in 1777, titled "Ripple Pavilion", with this final rendition being widely distributed and coming to serve as the standard for modern versions of "The Nine Chapters". However, this version has come under scrutiny from Guo Shuchen, alleging that the edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself.
Calculation of pi.
Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024. Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle. Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century.
There is no explicit method or record of how he calculated this estimate.
Division and root extraction.
Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han dynasty. "The Nine Chapters on the Mathematical Art" take these basic operations for granted and simply instruct the reader to perform them. Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of "The Nine Chapters on the Mathematical Art". Calculating the square and cube roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend ("shi") and divisor ("fa") throughout the process. This process of successive approximation was then extended to solving quadratics of the second and third order, such as formula_0, using a method similar to Horner's method. The method was not extended to solve quadratics of the nth order during the Han dynasty; however, this method was eventually used to solve these equations.
Linear algebra.
"The Book of Computations" is the first known text to solve systems of equations with two unknowns. There are a total of three sets of problems within "The Book of Computations" involving solving systems of equations with the false position method, which again are put into practical terms. Chapter Seven of "The Nine Chapters on the Mathematical Art" also deals with solving a system of two equations with two unknowns with the false position method. To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or "zi" (which are the values given for the excess and deficit) with the major terms "mu". To solve for the lesser of the two unknowns, simply add the minor terms together.
Chapter Eight of "The Nine Chapters on the Mathematical Art" deals with solving infinite equations with infinite unknowns. This process is referred to as the "fangcheng procedure" throughout the chapter. Many historians chose to leave the term "fangcheng" untranslated due to conflicting evidence of what the term means. Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Problems were done on a counting board and included the use of negative numbers as well as fractions. The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last.
Liu Hui's commentary on "The Nine Chapters on the Mathematical Art".
Liu Hui's commentary on "The Nine Chapters on the Mathematical Art" is the earliest edition of the original text available. Hui is believed by most to be a mathematician shortly after the Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint. For instance, throughout "The Nine Chapters on the Mathematical Art", the value of pi is taken to be equal to three in problems regarding circles or spheres. In his commentary, Liu Hui finds a more accurate estimation of pi using the method of exhaustion. The method involves creating successive polygons within a circle so that eventually the area of a higher-order polygon will be identical to that of the circle. From this method, Liu Hui asserted that the value of pi is about 3.14. Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles.
Three Kingdoms, Jin, and Sixteen Kingdoms.
In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium. He was the first Chinese mathematician to calculate "π"=3.1416 with his "π" algorithm. He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the infinitesimal calculus during the 3rd century CE.
In the fourth century, another influential mathematician named Zu Chongzhi, introduced the "Da Ming Li." This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in Book of Sui, we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927), which would remain the most accurate approximation of π available for the next 900 years. He also applied He Chengtian's interpolation for approximating irrational number with fraction in his astronomy and mathematical works, he obtained formula_1 as a good fraction approximate for pi; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, "the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe".
Along with his son, Zu Geng, Zu Chongzhi applied the Cavalieri's principle to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi. His work, "Zhui Shu" was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that "Zhui Shu" contains the formulas and methods for linear, matrix algebra, algorithm for calculating the value of "π", formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics.
A mathematical manual called "Sunzi mathematical classic" dated between 200 and 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. Intriguingly, "Sunzi" may have influenced the development of place-value systems and place-value systems and the associated Galley division in the West. European sources learned place-value techniques in the 13th century, from a Latin translation an early-9th-century work by Al-Khwarizmi. Khwarizmi's presentation is almost identical to the division algorithm in "Sunzi", even regarding stylistic matters (for example, using blank spaces to represent trailing zeros); the similarity suggests that the results may not have been an independent discovery. Islamic commentators on Al-Khwarizmi's work believed that it primarily summarized Hindu knowledge; Al-Khwarizmi's failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China.
In the fifth century the manual called "Zhang Qiujian suanjing" discussed linear and quadratic equations. By this point the Chinese had the concept of negative numbers.
Tang dynasty.
By the Tang dynasty study of mathematics was fairly standard in the great schools. The Ten Computational Canons was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (李淳風 602–670), as the official mathematical texts for imperial examinations in mathematics. The Sui dynasty and Tang dynasty ran the "School of Computations".
Wang Xiaotong was a great mathematician in the beginning of the Tang dynasty, and he wrote a book: Jigu Suanjing ("Continuation of Ancient Mathematics"), where numerical solutions which general cubic equations appear for the first time.
The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan, who died in 630.
The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the "Kaiyuan Zhanjing", compiled in 718 AD during the Tang dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics.
Yi Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as "chong cha", while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. Yi Xing was famed for his genius, and was known to have calculated the number of possible positions on a go board game (though without a symbol for zero he had difficulties expressing the number).
Song and Yuan dynasties.
Northern Song dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule.
Four outstanding mathematicians arose during the Song dynasty and Yuan dynasty, particularly in the twelfth and thirteenth centuries: Yang Hui, Qin Jiushao, Li Zhi (Li Ye), and Zhu Shijie. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on a form of algebraic geometry based on tiān yuán shù. His book; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians. Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with Zhu Shijie's two books "Suanxue qimeng" and the "Jade Mirror of the Four Unknowns". In one case he reportedly gave a method equivalent to Gauss's pivotal condensation.
Qin Jiushao (c. 1202 – 1261) was the first to introduce the zero symbol into Chinese mathematics." Before this innovation, blank spaces were used instead of zeros in the system of counting rods. One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?" Qin also solved a 10th order equation.
Pascal's triangle was first illustrated in China by Yang Hui in his book "Xiangjie Jiuzhang Suanfa" (詳解九章算法), although it was described earlier around 1100 by Jia Xian. Although the "Introduction to Computational Studies" (算學啓蒙) written by Zhu Shijie (fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics.
Algebra.
"Ceyuan haijing".
"Ceyuan haijing" (), or "Sea-Mirror of the Circle Measurements", is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by Li Zhi (or Li Ye) (1192–1272 AD). He used Tian yuan shu to convert intricated geometry problems into pure algebra problems. He then used "fan fa", or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations. "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His "Ts'e-yuan hai-ching" ("Sea-Mirror of the Circle Measurements") includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275).
"Jade Mirror of the Four Unknowns".
The "Jade Mirror of the Four Unknowns" was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of "fan fa", today called Horner's method, to solve these equations.
There are many summation series equations given without proof in the "Mirror". A few of the summation series are:
formula_2
formula_3
"Mathematical Treatise in Nine Sections".
The "Mathematical Treatise in Nine Sections", was written by the wealthy governor and minister Ch'in Chiu-shao (c. 1202 – c. 1261) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.
Magic squares and magic circles.
The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261–1275), who worked with magic squares of order as high as ten. "The same "Horner" device was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten." He also worked with magic circle.
Trigonometry.
The embryonic state of trigonometry in China slowly began to change and advance during the Song dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendar science and astronomical calculations. The polymath and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Joseph W. Dauben notes that in Shen's "technique of intersecting circles" formula, he creates an approximation of the arc of a circle "s" by "s" = "c" + 2"v"2/"d", where "d" is the diameter, "v" is the versine, "c" is the length of the chord "c" subtending the arc. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). Gauchet and Needham state Guo used spherical trigonometry in his calculations to improve the Chinese calendar and astronomy. Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham writes:
<templatestyles src="Template:Blockquote/styles.css" />Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).
Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of "Euclid's Elements" by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).
Ming dynasty.
After the overthrow of the Yuan dynasty, China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of botany and pharmacology. Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes:
<templatestyles src="Template:Blockquote/styles.css" />At the end of the 16th century, Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science. Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century.
Correspondingly, scholars paid less attention to mathematics; preeminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the 'increase multiply' method. Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods. To the average scholar, then, "tianyuan" seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into "The Annotations of Calculations in the Nine Chapters on the Mathematical Art", he omitted "Tian yuan shu" and the increase multiply method.
Instead, mathematical progress became focused on computational tools. In 15 century, abacus came into its "suan pan" form. Easy to use and carry, both fast and accurate, it rapidly overtook rod calculus as the preferred form of computation. "Zhusuan", the arithmetic calculation through abacus, inspired multiple new works. "Suanfa Tongzong" (General Source of Computational Methods), a 17-volume work published in 1592 by Cheng Dawei, remained in use for over 300 years. Zhu Zaiyu, Prince of Zheng used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy, a precision that enabled his development of the equal-temperament system.
In the late 16th century, Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court. With the assistance of Xu Guangqi, he was able to translate Euclid's "Elements" using the same techniques used to teach classical Buddhist texts. Other missionaries followed in his example, translating Western works on special functions (trigonometry and logarithms) that were neglected in the Chinese tradition. However, contemporary scholars found the emphasis on proofs — as opposed to solved problems — baffling, and most continued to work from classical texts alone.
Qing dynasty.
Under the Kangxi Emperor, who learned Western mathematics from the Jesuits and was open to outside knowledge and ideas, Chinese mathematics enjoyed a brief period of official support. At Kangxi's direction, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume work titled "Shuli Jingyun" ("The Essence of Mathematical Study") which was printed in 1723, and gave a systematic introduction to western mathematical knowledge. At the same time, Mei Goucheng also developed to "Meishi Congshu Jiyang" [The Compiled works of Mei]. "Meishi Congshu Jiyang" was an encyclopedic summary of nearly all schools of Chinese mathematics at that time, but it also included the cross-cultural works of Mei Wending (1633–1721), Goucheng's grandfather. The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations.
In 1773, the Qianlong Emperor decided to compile the "Complete Library of the Four Treasuries" (or "Siku Quanshu"). Dai Zhen (1724–1777) selected and proofread "The Nine Chapters on the Mathematical Art" from "Yongle Encyclopedia" and several other mathematical works from Han and Tang dynasties. The long-missing mathematical works from Song and Yuan dynasties such as "Si-yüan yü-jian" and "Ceyuan haijing" were also found and printed, which directly led to a wave of new research. The most annotated works were "Jiuzhang suanshu xicaotushuo" (The Illustrations of Calculation Process for "The Nine Chapters on the Mathematical Art" ) contributed by Li Huang and Siyuan yujian xicao (The Detailed Explanation of Si-yuan yu-jian) by Luo Shilin.
Western influences.
In 1840, the First Opium War forced China to open its door and look at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries. In 1852, the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of "Elements" and 13 volumes on "Algebra". With the assistance of Joseph Edkins, more works on astronomy and calculus soon followed. Chinese scholars were initially unsure whether to approach the new works: was study of Western knowledge a form of submission to foreign invaders? But by the end of the century, it became clear that China could only begin to recover its sovereignty by incorporating Western works. Chinese scholars, taught in Western missionary schools, from (translated) Western texts, rapidly lost touch with the indigenous tradition. Those who were self-trained or in traditionalist circles nevertheless continued to work within the traditional framework of algorithmic mathematics without resorting to Western symbolism. Yet, as Martzloff notes, "from 1911 onwards, solely Western mathematics has been practised in China."
In modern China.
Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912. Ever since then, modern Chinese mathematicians have made numerous achievements in various mathematical fields.
Some famous modern ethnic Chinese mathematicians include:
People's Republic of China.
In 1949, at the beginning of the founding of the People's Republic of China, the government paid great attention to the cause of science although the country was in a predicament of lack of funds. The Chinese Academy of Sciences was established in November 1949. The Institute of Mathematics was formally established in July 1952. Then, the Chinese Mathematical Society and its founding journals restored and added other special journals. In the 18 years after 1949, the number of published papers accounted for more than three times the total number of articles before 1949. Many of them not only filled the gaps in China's past, but also reached the world's advanced level.
During the chaos of the Cultural Revolution, the sciences declined. In the field of mathematics, in addition to Chen Jingrun, Hua Luogeng, Zhang Guanghou and other mathematicians struggling to continue their work. After the catastrophe, with the publication of Guo Moruo's literary "Spring of Science", Chinese sciences and mathematics experienced a revival. In 1977, a new mathematical development plan was formulated in Beijing, the work of the mathematics society was resumed, the journal was re-published, the academic journal was published, the mathematics education was strengthened, and basic theoretical research was strengthened.
An important mathematical achievement of the Chinese mathematician in the direction of the power system is how Xia Zhihong proved the Painleve conjecture in 1988. When there are some initial states of "N" celestial bodies, one of the celestial bodies ran to infinity or speed in a limited time. Infinity is reached, that is, there are non-collision singularities. The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895. A very important recent development for the 4-body problem is that Xue Jinxin and Dolgopyat proved a non-collision singularity in a simplified version of the 4-body system around 2013.
In addition, in 2007, Shen Weixiao and Kozlovski, Van-Strien proved the Real Fatou conjecture: Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree. This conjecture can be traced back to Fatou in the 1920s, and later Smale posed it in the 1960s. The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade.
IMO performance.
In comparison to other participating countries at the International Mathematical Olympiad, China has highest team scores and has won the all-members-gold IMO with a full team the most number of times.
In education.
The first reference to a book being used in learning mathematics in China is dated to the second century CE (Hou Hanshu: 24, 862; 35,1207). We are told that Ma Xu, who is a youth c. 110, and Zheng Xuan (127–200) both studied the "Nine Chapters on Mathematical procedures". Christopher Cullen claims that mathematics, in a manner akin to medicine, was taught orally. The stylistics of the "Suàn shù shū" from Zhangjiashan suggest that the text was assembled from various sources and then underwent codification.
References.
Citations.
<templatestyles src="Reflist/styles.css" />
Works cited.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "x^2+a=b"
},
{
"math_id": 1,
"text": "\\tfrac{355}{113}"
},
{
"math_id": 2,
"text": "1^2 + 2^2 + 3^2 + \\cdots + n^2 = {n(n + 1)(2n + 1)\\over 3!}"
},
{
"math_id": 3,
"text": "1 + 8 + 30 + 80 + \\cdots + {n^2(n + 1)(n + 2)\\over 3!} = {n(n + 1)(n + 2)(n + 3)(4n + 1)\\over 5!}"
}
] |
https://en.wikipedia.org/wiki?curid=1524543
|
15249674
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Bilinear time–frequency distribution
|
Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time; this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems.
Background.
Methods for analysing time series, in both signal analysis and time series analysis, have been developed as essentially separate methodologies applicable to, and based in, either the time or the frequency domain. A mixed approach is required in time–frequency analysis techniques which are especially effective in analyzing non-stationary signals, whose frequency distribution and magnitude vary with time. Examples of these are acoustic signals. Classes of "quadratic time-frequency distributions" (or bilinear time–frequency distributions") are used for time–frequency signal analysis. This class is similar in formulation to Cohen's class distribution function that was used in 1966 in the context of quantum mechanics. This distribution function is mathematically similar to a generalized time–frequency representation which utilizes bilinear transformations. Compared with other time–frequency analysis techniques, such as short-time Fourier transform (STFT), the bilinear-transformation (or quadratic time–frequency distributions) may not have higher clarity for most practical signals, but it provides an alternative framework to investigate new definitions and new methods. While it does suffer from an inherent cross-term contamination when analyzing multi-component signals, by using a carefully chosen window function(s), the interference can be significantly mitigated, at the expense of resolution. All these bilinear distributions are inter-convertible to each other, cf. transformation between distributions in time–frequency analysis.
Wigner–Ville distribution.
The Wigner–Ville distribution is a quadratic form that measures a local time-frequency energy given by:
formula_0
The Wigner–Ville distribution remains real as it is the fourier transform of "f"("u" + "τ"/2)·"f"*("u" − "τ"/2), which has Hermitian symmetry in "τ". It can also be written as a frequency integration by applying the Parseval formula:
formula_1
Proposition 1. for any "f" in L2(R)
formula_2
formula_3
Moyal Theorem. For "f" and "g" in L2(R),
formula_4
Proposition 2 (time-frequency support). If "f" has a compact support, then for all "ξ" the support of formula_5 along "u" is equal to the support of "f". Similarly, if formula_6 has a compact support, then for all "u" the support of formula_7 along "ξ" is equal to the support of formula_6.
Proposition 3 (instantaneous frequency). If formula_8 then
formula_9
Interference.
Let formula_10 be a composite signal. We can then write,
formula_11
where
formula_12
is the cross Wigner–Ville distribution of two signals. The interference term
formula_13
is a real function that creates non-zero values at unexpected locations (close to the origin) in the formula_14 plane. Interference terms present in a real signal can be avoided by computing the analytic part formula_15.
Positivity and smoothing kernel.
The interference terms are oscillatory since the marginal integrals vanish and can be partially removed by smoothing formula_16 with a kernel "θ"
formula_17
The time-frequency resolution of this distribution depends on the spread of kernel "θ" in the neighborhood of formula_14. Since the interferences take negative values, one can guarantee that all interferences are removed by imposing that
formula_18
The spectrogram and scalogram are examples of positive time-frequency energy distributions. Let a linear transform formula_19 be defined over a family of time-frequency atoms formula_20. For any formula_14 there exists a unique atom formula_21 centered in time-frequency at formula_14. The resulting time-frequency energy density is
formula_22
From the Moyal formula,
formula_23
which is the time frequency averaging of a Wigner–Ville distribution. The smoothing kernel thus can be written as
formula_24
The loss of time-frequency resolution depends on the spread of the distribution formula_25 in the neighborhood of formula_14.
Example 1.
A spectrogram computed with windowed fourier atoms,
formula_26
formula_27
For a spectrogram, the Wigner–Ville averaging is therefore a 2-dimensional convolution with formula_28. If g is a Gaussian window,formula_28 is a 2-dimensional Gaussian. This proves that averaging formula_16 with a sufficiently wide Gaussian defines positive energy density. The general class of time-frequency distributions obtained by convolving formula_16 with an arbitrary kernel "θ" is called a Cohen's class, discussed below.
Wigner Theorem. There is no positive quadratic energy distribution "Pf" that satisfies the following time and frequency marginal integrals:
formula_29
formula_30
Mathematical definition.
The definition of Cohen's class of bilinear (or quadratic) time–frequency distributions is as follows:
formula_31
where formula_32 is the ambiguity function (AF), which will be discussed later; and formula_33 is Cohen's kernel function, which is often a low-pass function, and normally serves to mask out the interference. In the original Wigner representation, formula_34.
An equivalent definition relies on a convolution of the Wigner distribution function (WD) instead of the AF :
formula_35
where the kernel function formula_36 is defined in the time-frequency domain instead of the ambiguity one. In the original Wigner representation, formula_37. The relationship between the two kernels is the same as the one between the WD and the AF, namely two successive Fourier transforms (cf. diagram).
formula_38
i.e.
formula_39
or equivalently
formula_40
Ambiguity function.
The class of bilinear (or quadratic) time–frequency distributions can be most easily understood in terms of the ambiguity function, an explanation of which follows.
Consider the well known power spectral density formula_41 and the signal auto-correlation function formula_42 in the case of a stationary process. The relationship between these functions is as follows:
formula_43
formula_44
For a non-stationary signal formula_45, these relations can be generalized using a time-dependent power spectral density or equivalently the famous Wigner distribution function of formula_45 as follows:
formula_46
formula_47
If the Fourier transform of the auto-correlation function is taken with respect to "t" instead of "τ", we get the ambiguity function as follows:
formula_48
The relationship between the Wigner distribution function, the auto-correlation function and the ambiguity function can then be illustrated by the following figure.
By comparing the definition of bilinear (or quadratic) time–frequency distributions with that of the Wigner distribution function, it is easily found that the latter is a special case of the former with formula_49. Alternatively, bilinear (or quadratic) time–frequency distributions can be regarded as a masked version of the Wigner distribution function if a kernel function formula_50 is chosen. A properly chosen kernel function can significantly reduce the undesirable cross-term of the Wigner distribution function.
What is the benefit of the additional kernel function? The following figure shows the distribution of the auto-term and the cross-term of a multi-component signal in both the ambiguity and the Wigner distribution function.
For multi-component signals in general, the distribution of its auto-term and cross-term within its Wigner distribution function is generally not predictable, and hence the cross-term cannot be removed easily. However, as shown in the figure, for the ambiguity function, the auto-term of the multi-component signal will inherently tend to close the origin in the "ητ"-plane, and the cross-term will tend to be away from the origin. With this property, the cross-term in can be filtered out effortlessly if a proper low-pass kernel function is applied in "ητ"-domain. The following is an example that demonstrates how the cross-term is filtered out.
Kernel properties.
The Fourier transform of formula_51 is
formula_52
The following proposition gives necessary and sufficient conditions to ensure that formula_53 satisfies marginal energy properties like those of the Wigner–Ville distribution.
Proposition: The marginal energy properties
formula_54
formula_55
are satisfied for all formula_56 if and only if
formula_57
Some time-frequency distributions.
Wigner distribution function.
Aforementioned, the Wigner distribution function is a member of the class of quadratic time-frequency distributions (QTFDs) with the kernel function formula_58. The definition of Wigner distribution is as follows:
formula_59
Modified Wigner distribution functions.
Affine invariance.
We can design time-frequency energy distributions that satisfy the scaling property
formula_60
as does the Wigner–Ville distribution. If
formula_61
then
formula_62
This is equivalent to imposing that
formula_63
and hence
formula_64
The Rihaczek and Choi–Williams distributions are examples of affine invariant Cohen's class distributions.
Choi–Williams distribution function.
The kernel of Choi–Williams distribution is defined as follows:
formula_65
where "α" is an adjustable parameter.
Rihaczek distribution function.
The kernel of Rihaczek distribution is defined as follows:
formula_66
With this particular kernel a simple calculation proves that
formula_67
Cone-shape distribution function.
The kernel of cone-shape distribution function is defined as follows:
formula_68
where "α" is an adjustable parameter. See Transformation between distributions in time-frequency analysis. More such QTFDs and a full list can be found in, e.g., Cohen's text cited.
Spectrum of non-stationary processes.
A time-varying spectrum for non-stationary processes is defined from the expected Wigner–Ville distribution. Locally stationary processes appear in many physical systems where random fluctuations are produced by a mechanism that changes slowly in time. Such processes can be approximated locally by a stationary process. Let formula_69 be a real valued zero-mean process with covariance
formula_70
The covariance operator "K" is defined for any deterministic signal formula_56 by
formula_71
For locally stationary processes, the eigenvectors of "K" are well approximated by the Wigner–Ville spectrum.
Wigner–Ville spectrum.
The properties of the covariance formula_72 are studied as a function of formula_73 and formula_74:
formula_75
The process is "wide-sense stationary" if the covariance depends only on formula_73:
formula_76
The eigenvectors are the complex exponentials formula_77 and the corresponding eigenvalues are given by the power spectrum
formula_78
For non-stationary processes, Martin and Flandrin have introduced a "time-varying spectrum"
formula_79
To avoid convergence issues we suppose that "X" has compact support so that formula_80 has compact support in formula_81. From above we can write
formula_82
which proves that the time varying spectrum is the expected value of the Wigner–Ville transform of the process "X". Here, the Wigner–Ville stochastic integral is interpreted as a mean-square integral:
formula_83
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "P_V f(u,\\xi )=\\int_{-\\infty }^\\infty f \\left (u+\\tfrac{\\tau}{2} \\right) f^* \\left(u-\\tfrac{\\tau}{2} \\right) e^{-i\\tau \\xi} \\, d\\tau "
},
{
"math_id": 1,
"text": "P_V f(u,\\xi )=\\frac{1}{2\\pi}\\int_{-\\infty }^\\infty \\hat{f} \\left(\\xi +\\tfrac{\\gamma }{2} \\right) \\hat{f}^* \\left(\\xi -\\tfrac{\\gamma}{2} \\right) e^{i\\gamma u} \\, d\\gamma "
},
{
"math_id": 2,
"text": "\\int_{-\\infty}^\\infty P_V f(u,\\xi) \\, du= | \\hat{f}(\\xi) |^2"
},
{
"math_id": 3,
"text": "\\int_{-\\infty}^\\infty P_V f(u,\\xi) \\, d\\xi =2\\pi |f(u)|^2"
},
{
"math_id": 4,
"text": "2\\pi \\left| \\int_{-\\infty }^\\infty f(t)g^*(t)\\,dt \\right|^2=\\iint{P_V f(u,\\xi )} P_Vg(u,\\xi )\\,du\\,d\\xi "
},
{
"math_id": 5,
"text": "P_V f(u,\\xi )"
},
{
"math_id": 6,
"text": "\\hat{f}"
},
{
"math_id": 7,
"text": "P_Vf(u,\\xi )"
},
{
"math_id": 8,
"text": "f_a(t)=a(t)e^{i\\phi (t)}"
},
{
"math_id": 9,
"text": "\\phi'(u)= \\frac{\\int_{-\\infty }^{\\infty }\\xi P_V f_a (u,\\xi )d\\xi}{\\int_{-\\infty}^{\\infty}P_V f_a(u,\\xi )d\\xi}"
},
{
"math_id": 10,
"text": "f= f_1 + f_2 "
},
{
"math_id": 11,
"text": "P_Vf=P_Vf_1+P_Vf_2+P_V \\left [f_1,f_2 \\right ]+P_V \\left [f_2,f_1 \\right ]"
},
{
"math_id": 12,
"text": "P_V[h,g](u,\\xi )=\\int_{-\\infty }^{\\infty } h\\left (u+\\tfrac{\\tau }{2} \\right)g^* \\left (u-\\tfrac{\\tau }{2} \\right) e^{-i\\tau \\xi }d\\tau"
},
{
"math_id": 13,
"text": "I[f_1,f_2]=P_V[f_1, f_2]+ P_V[f_2, f_1]"
},
{
"math_id": 14,
"text": "(u,\\xi )"
},
{
"math_id": 15,
"text": "f_a(t)"
},
{
"math_id": 16,
"text": "P_V f"
},
{
"math_id": 17,
"text": "P_\\theta f(u,\\xi )=\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{P_V f(u',\\xi')}} \\theta (u,u',\\xi ,\\xi') \\, du' \\, d\\xi '"
},
{
"math_id": 18,
"text": "P_\\theta f(u,\\xi )\\ge 0, \\qquad \\forall (u,\\xi )\\in {{\\mathbf{R}}^{2}}"
},
{
"math_id": 19,
"text": "Tf(\\gamma )=\\left\\langle f,\\phi_{\\gamma} \\right\\rangle "
},
{
"math_id": 20,
"text": "\\left\\{ \\phi_\\gamma \\right\\}_{\\gamma \\in \\Gamma}"
},
{
"math_id": 21,
"text": "\\phi_{\\gamma (u,\\xi )}"
},
{
"math_id": 22,
"text": "P_T f(u,\\xi ) = \\left | \\left \\langle f, \\phi_{\\gamma (u,\\xi )} \\right\\rangle \\right |^2 "
},
{
"math_id": 23,
"text": " P_T f(u,\\xi )=\\frac{1}{2\\pi} \\int_{-\\infty}^\\infty \\int_{-\\infty }^\\infty P_V f(u', \\xi') P_V \\phi_{\\gamma (u,\\xi )} (u',\\xi') \\, du' \\, d\\xi '"
},
{
"math_id": 24,
"text": "\\theta (u,u',\\xi ,\\xi')=\\frac{1}{2\\pi }P_V \\phi_{\\gamma (u,\\xi )}(u',\\xi')"
},
{
"math_id": 25,
"text": "P_V \\phi_{\\gamma (u,\\xi )} (u',\\xi')"
},
{
"math_id": 26,
"text": "\\phi_{\\gamma (u,\\xi )}(t)=g(t-u) e^{i\\xi t}"
},
{
"math_id": 27,
"text": "\\theta (u, u', \\xi, \\xi')=\\frac{1}{2\\pi} P_V \\phi_{\\gamma (u,\\xi )} (u', \\xi')=\\frac{1}{2\\pi} P_V g(u'-u, \\xi'-\\xi )"
},
{
"math_id": 28,
"text": "P_V g"
},
{
"math_id": 29,
"text": "\\int_{-\\infty }^\\infty Pf(u,\\xi ) \\, d\\xi =2\\pi |f(u)|^2"
},
{
"math_id": 30,
"text": "\\int_{-\\infty }^\\infty Pf(u,\\xi ) \\, du= |\\hat{f}(\\xi)|^2"
},
{
"math_id": 31,
"text": "C_x(t, f)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty A_x(\\eta,\\tau)\\Phi(\\eta,\\tau)\\exp (j2\\pi(\\eta t-\\tau f))\\, d\\eta \\, d\\tau,"
},
{
"math_id": 32,
"text": "A_x(\\eta,\\tau)"
},
{
"math_id": 33,
"text": "\\Phi (\\eta,\\tau)"
},
{
"math_id": 34,
"text": "\\Phi \\equiv 1"
},
{
"math_id": 35,
"text": "C_x(t, f)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty W_x(\\theta,\\nu) \\Pi(t - \\theta, f - \\nu)\\, d\\theta \\, d\\nu = [W_x \\ast \\Pi] (t,f)"
},
{
"math_id": 36,
"text": "\\Pi (t,f)"
},
{
"math_id": 37,
"text": "\\Pi = \\delta_{(0,0)}"
},
{
"math_id": 38,
"text": "\\Phi = \\mathcal{F}_t \\mathcal{F}^{-1}_f \\Pi"
},
{
"math_id": 39,
"text": "\\Phi(\\eta, \\tau) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\Pi(t,f) \\exp (-j2\\pi(t \\eta-f \\tau))\\, dt \\, df,"
},
{
"math_id": 40,
"text": "\\Pi(t,f) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\Phi(\\eta,\\tau) \\exp (j2\\pi(\\eta t-\\tau f))\\, d\\eta \\, d\\tau."
},
{
"math_id": 41,
"text": "P_x(f)"
},
{
"math_id": 42,
"text": "R_x (\\tau)"
},
{
"math_id": 43,
"text": "P_x(f)= \\int_{-\\infty}^\\infty R_x(\\tau)e^{-j2\\pi f\\tau} \\, d\\tau,"
},
{
"math_id": 44,
"text": " R_x(\\tau) = \\int_{-\\infty}^\\infty x \\left (t+ \\tfrac{\\tau}{2} \\right )x^* \\left (t- \\tfrac{\\tau}{2} \\right ) \\, dt."
},
{
"math_id": 45,
"text": "x(t)"
},
{
"math_id": 46,
"text": "W_x(t, f)= \\int_{-\\infty}^\\infty R_x(t, \\tau)e^{-j2\\pi f\\tau}\\, d\\tau,"
},
{
"math_id": 47,
"text": " R_x (t ,\\tau) = x \\left (t+ \\tfrac{\\tau}{2} \\right )x^* \\left(t- \\tfrac{\\tau}{2} \\right)."
},
{
"math_id": 48,
"text": "A_x(\\eta,\\tau)=\\int_{-\\infty}^\\infty x \\left (t+ \\tfrac{\\tau}{2} \\right )x^* \\left(t- \\tfrac{\\tau}{2} \\right)e^{j2\\pi t\\eta}\\, dt."
},
{
"math_id": 49,
"text": "\\Phi(\\eta,\\tau) = 1"
},
{
"math_id": 50,
"text": "\\Phi(\\eta,\\tau) \\neq 1"
},
{
"math_id": 51,
"text": "\\theta (u,\\xi )"
},
{
"math_id": 52,
"text": "\\hat{\\theta}(\\tau ,\\gamma )=\\int_{-\\infty }^{\\infty} \\int_{-\\infty }^{\\infty} \\theta (u,\\xi ) e^{-i(u\\gamma +\\xi \\tau )} \\, du \\, d\\xi "
},
{
"math_id": 53,
"text": "P_\\theta"
},
{
"math_id": 54,
"text": "\\int_{-\\infty }^\\infty P_\\theta f(u,\\xi ) \\, d\\xi =2\\pi | f(u) |^2,"
},
{
"math_id": 55,
"text": "\\int_{-\\infty }^\\infty P_\\theta f(u,\\xi ) \\, du= |\\hat{f}(\\xi )|^2"
},
{
"math_id": 56,
"text": "f\\in L^2(\\mathbf{R})"
},
{
"math_id": 57,
"text": "\\forall (\\tau ,\\gamma )\\in \\mathbf{R}^2: \\qquad \\hat{\\theta}(\\tau ,0)=\\hat{\\theta }(0,\\gamma )=1"
},
{
"math_id": 58,
"text": "\\Phi (\\eta,\\tau) = 1"
},
{
"math_id": 59,
"text": "W_x(t, f)= \\int_{-\\infty}^\\infty x \\left (t+ \\tfrac{\\tau}{2} \\right ) x^*\\left(t- \\tfrac{\\tau}{2} \\right)e ^{-j2\\pi f\\tau}\\, d\\tau."
},
{
"math_id": 60,
"text": "\\frac{1}{\\sqrt{s}}f\\left( \\tfrac{t}{s} \\right) \\longleftrightarrow P_V f\\left( \\tfrac{u}{s},s\\xi \\right)"
},
{
"math_id": 61,
"text": "g(t)=\\frac{1}{\\sqrt{s}}f\\left( \\tfrac{t}{s} \\right)"
},
{
"math_id": 62,
"text": " P_\\theta g(u,\\xi)= P_\\theta f\\left( \\tfrac{u}{s},s\\xi \\right)."
},
{
"math_id": 63,
"text": "\\forall s\\in \\mathbf{R}^+: \\qquad \\theta \\left( su,\\tfrac{\\xi }{s} \\right)=\\theta (u,\\xi) ,"
},
{
"math_id": 64,
"text": "\\theta (u,\\xi )=\\theta (u\\xi ,1)=\\beta (u\\xi )"
},
{
"math_id": 65,
"text": "\\Phi (\\eta,\\tau ) = \\exp (-\\alpha(\\eta \\tau)^2), "
},
{
"math_id": 66,
"text": "\\Phi (\\eta,\\tau) = \\exp \\left(-i 2\\pi \\frac{\\eta \\tau}{2} \\right), "
},
{
"math_id": 67,
"text": "C_x (t,f) = x(t) \\hat{x}^*(f) e^{i 2\\pi t f}"
},
{
"math_id": 68,
"text": "\\Phi (\\eta,\\tau) = \\frac{\\sin(\\pi \\eta \\tau)}{ \\pi \\eta \\tau }\\exp \\left(-2\\pi \\alpha \\tau^2 \\right), "
},
{
"math_id": 69,
"text": "X(t)"
},
{
"math_id": 70,
"text": "R(t,s)=E[X(t)X(s)]"
},
{
"math_id": 71,
"text": "Kf(t)=\\int_{-\\infty}^\\infty R( t,s)f(s) \\, ds "
},
{
"math_id": 72,
"text": "R(t,s)"
},
{
"math_id": 73,
"text": "\\tau =t-s"
},
{
"math_id": 74,
"text": "u=\\frac{t+s}{2}"
},
{
"math_id": 75,
"text": "R(t,s)=R\\left( u+\\tfrac{\\tau }{2},u-\\tfrac{\\tau}{2} \\right)=C( u,\\tau)"
},
{
"math_id": 76,
"text": "Kf(t)=\\int_{-\\infty}^\\infty C(t-s) f(s)\\,ds=C*f(t)"
},
{
"math_id": 77,
"text": "e^{i\\omega t}"
},
{
"math_id": 78,
"text": "P_X (\\omega)=\\int_{-\\infty }^\\infty C(\\tau) e^{-i\\omega \\tau } \\, d\\tau "
},
{
"math_id": 79,
"text": "P_X ( u,\\xi)=\\int_{-\\infty }^\\infty C(u,\\tau) e^{-i\\xi \\tau} \\, d\\tau =\\int_{-\\infty}^\\infty E\\left [ X\\left( u+\\tfrac{\\tau}{2} \\right) X\\left( u-\\tfrac{\\tau}{2} \\right) \\right ] e^{-i\\xi \\tau } \\, d\\tau "
},
{
"math_id": 80,
"text": "C(u,\\tau)"
},
{
"math_id": 81,
"text": "\\tau"
},
{
"math_id": 82,
"text": "P_X ( u,\\xi)=E[ P_V X ( u,\\xi)]"
},
{
"math_id": 83,
"text": "P_V ( u,\\xi)=\\int_{-\\infty}^\\infty \\left\\{ X\\left( u+\\tfrac{\\tau}{2}\\right)X\\left( u-\\tfrac{\\tau }{2} \\right) \\right\\} e^{-i\\xi \\tau} \\, d\\tau "
}
] |
https://en.wikipedia.org/wiki?curid=15249674
|
152518
|
Abel–Ruffini theorem
|
Equations of degree 5 or higher cannot be solved by radicals
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, "general" means that the coefficients of the equation are viewed and manipulated as indeterminates.
The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799 (which was refined and completed in 1813 and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824.
"Abel–Ruffini theorem" refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from . Galois theory implies also that
formula_0
is the simplest equation that cannot be solved in radicals, and that "almost all" polynomials of degree five or higher cannot be solved in radicals.
The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula, the cubic formula, and the quartic formula for degrees two, three, and four, respectively.
Context.
Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.
The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra, which does not provide any tool for computing exactly the solutions, although Newton's method allows approximating the solutions to any desired accuracy.
From the 16th century to beginning of the 19th century, the main problem of algebra was to search for a formula for the solutions of polynomial equations of degree five and higher, hence the name the "fundamental theorem of algebra". This meant a solution in radicals, that is, an expression involving only the coefficients of the equation, and the operations of addition, subtraction, multiplication, division, and nth root extraction.
The Abel–Ruffini theorem proves that this is impossible. However, this impossibility does not imply that a specific equation of any degree cannot be solved in radicals. On the contrary, there are equations of any degree that can be solved in radicals. This is the case of the equation formula_1 for any n, and the equations defined by cyclotomic polynomials, all of whose solutions can be expressed in radicals.
Abel's proof of the theorem does not explicitly contain the assertion that there are specific equations that cannot be solved by radicals. Such an assertion is not a consequence of Abel's statement of the theorem, as the statement does not exclude the possibility that "every particular quintic equation might be soluble, with a special formula for each equation." However, the existence of specific equations that cannot be solved in radicals seems to be a consequence of Abel's proof, as the proof uses the fact that some polynomials in the coefficients are not the zero polynomial, and, given a finite number of polynomials, there are values of the variables at which none of the polynomials takes the value zero.
Soon after Abel's publication of its proof, Évariste Galois introduced a theory, now called Galois theory that allows deciding, for any given equation, whether it is solvable in radicals. This was purely theoretical before the rise of electronic computers. With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100. Computing the solutions in radicals of solvable polynomials requires huge computations. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.
Proof.
The proof of the Abel–Ruffini theorem predates Galois theory. However, Galois theory allows a better understanding of the subject, and modern proofs are generally based on it, while the original proofs of the Abel–Ruffini theorem are still presented for historical purposes.
The proofs based on Galois theory comprise four main steps: the characterization of solvable equations in terms of field theory; the use of the Galois correspondence between subfields of a given field and the subgroups of its Galois group for expressing this characterization in terms of solvable groups; the proof that the symmetric group is not solvable if its degree is five or higher; and the existence of polynomials with a symmetric Galois group.
Algebraic solutions and field theory.
An algebraic solution of a polynomial equation is an expression involving the four basic arithmetic operations (addition, subtraction, multiplication, and division), and root extractions. Such an expression may be viewed as the description of a computation that starts from the coefficients of the equation to be solved and proceeds by computing some numbers, one after the other.
At each step of the computation, one may consider the smallest field that contains all numbers that have been computed so far. This field is changed only for the steps involving the computation of an nth root.
So, an algebraic solution produces a sequence
formula_2
of fields, and elements formula_3 such that
formula_4 for formula_5 with formula_6 for some integer formula_7 An algebraic solution of the initial polynomial equation exists if and only if there exists such a sequence of fields such that formula_8 contains a solution.
For having normal extensions, which are fundamental for the theory, one must refine the sequence of fields as follows. If formula_9 does not contain all formula_10-th roots of unity, one introduces the field formula_11 that extends formula_9 by a primitive root of unity, and one redefines formula_12 as formula_13
So, if one starts from a solution in terms of radicals, one gets an increasing sequence of fields such that the last one contains the solution, and each is a normal extension of the preceding one with a Galois group that is cyclic.
Conversely, if one has such a sequence of fields, the equation is solvable in terms of radicals. For proving this, it suffices to prove that a normal extension with a cyclic Galois group can be built from a succession of radical extensions.
Galois correspondence.
The Galois correspondence establishes a one to one correspondence between the subextensions of a normal field extension formula_14 and the subgroups of the Galois group of the extension. This correspondence maps a field K such formula_15 to the Galois group formula_16 of the automorphisms of F that leave K fixed, and, conversely, maps a subgroup H of formula_17 to the field of the elements of F that are fixed by H.
The preceding section shows that an equation is solvable in terms of radicals if and only if the Galois group of its splitting field (the smallest field that contains all the roots) is solvable, that is, it contains a sequence of subgroups such that each is normal in the preceding one, with a quotient group that is cyclic. (Solvable groups are commonly defined with abelian instead of cyclic quotient groups, but the fundamental theorem of finite abelian groups shows that the two definitions are equivalent).
So, for proving the Abel–Ruffini theorem, it remains to show that the symmetric group formula_18 is not solvable, and that there are polynomials with symmetric Galois groups.
Solvable symmetric groups.
For "n" > 4, the symmetric group formula_19 of degree n has only the alternating group formula_20 as a nontrivial normal subgroup (see ). For "n" > 4, the alternating group formula_20 is simple (that is, it does not have any nontrivial normal subgroup) and not abelian. This implies that both formula_20 and formula_19 are not solvable for "n" > 4. Thus, the Abel–Ruffini theorem results from the existence of polynomials with a symmetric Galois group; this will be shown in the next section.
On the other hand, for "n" ≤ 4, the symmetric group and all its subgroups are solvable. This explains the existence of the quadratic, cubic, and quartic formulas, since a major result of Galois theory is that a polynomial equation has a solution in radicals if and only if its Galois group is solvable (the term "solvable group" takes its origin from this theorem).
Polynomials with symmetric Galois groups.
General equation.
The "general" or "generic" polynomial equation of degree n is the equation
formula_21
where formula_22 are distinct indeterminates. This is an equation defined over the field formula_23 of the rational fractions in formula_22 with rational number coefficients. The original Abel–Ruffini theorem asserts that, for "n" > 4, this equation is not solvable in radicals. In view of the preceding sections, this results from the fact that the Galois group over F of the equation is the symmetric group formula_19 (this Galois group is the group of the field automorphisms of the splitting field of the equation that fix the elements of F, where the splitting field is the smallest field containing all the roots of the equation).
For proving that the Galois group is formula_24 it is simpler to start from the roots. Let formula_25 be new indeterminates, aimed to be the roots, and consider the polynomial
formula_26
Let formula_27 be the field of the rational fractions in formula_28 and formula_29 be its subfield generated by the coefficients of formula_30 The permutations of the formula_31 induce automorphisms of H. Vieta's formulas imply that every element of K is a symmetric function of the formula_32 and is thus fixed by all these automorphisms. It follows that the Galois group formula_33 is the symmetric group formula_34
The fundamental theorem of symmetric polynomials implies that the formula_35 are algebraic independent, and thus that the map that sends each formula_36 to the corresponding formula_35 is a field isomorphism from F to K. This means that one may consider formula_37 as a generic equation. This finishes the proof that the Galois group of a general equation is the symmetric group, and thus proves the original Abel–Ruffini theorem, which asserts that the general polynomial equation of degree n cannot be solved in radicals for "n" > 4.
Explicit example.
The equation formula_0 is not solvable in radicals, as will be explained below.
Let q be formula_38.
Let G be its Galois group, which acts faithfully on the set of complex roots of q.
Numbering the roots lets one identify G with a subgroup of the symmetric group formula_39.
Since formula_40 factors as formula_41 in formula_42, the group G contains a permutation formula_43 that is a product of disjoint cycles of lengths 2 and 3 (in general, when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials, the degrees of the factors give the lengths of the disjoint cycles in some permutation belonging to the Galois group); then G also contains formula_44, which is a transposition. Since formula_45 is irreducible in formula_46, the same principle shows that G contains a 5-cycle. Because 5 is prime, any transposition and 5-cycle in formula_39 generate the whole group; see . Thus formula_47. Since the group formula_39 is not solvable, the equation formula_0 is not solvable in radicals.
Cayley's resolvent.
Testing whether a specific quintic is solvable in radicals can be done by using Cayley's resolvent. This is a univariate polynomial of degree six whose coefficients are polynomials in the coefficients of a generic quintic. A specific irreducible quintic is solvable in radicals if and only, when its coefficients are substituted in Cayley's resolvent, the resulting sextic polynomial has a rational root.
History.
Around 1770, Joseph Louis Lagrange began the groundwork that unified the many different methods that had been used up to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrange resolvents. This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions for equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusive proof. The first person who conjectured that the problem of solving quintics by radicals might be impossible to solve was Carl Friedrich Gauss, who wrote in 1798 in section 359 of his book "Disquisitiones Arithmeticae" (which would be published only in 1801) that "there is little doubt that this problem does not so much defy modern methods of analysis as that it proposes the impossible". The next year, in his thesis, he wrote "After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory." And he added "Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place." Actually, Gauss published nothing else on this subject.
The theorem was first nearly proved by Paolo Ruffini in 1799. He sent his proof to several mathematicians to get it acknowledged, amongst them Lagrange (who did not reply) and Augustin-Louis Cauchy, who sent him a letter saying: "Your memoir on the general solution of equations is a work which I have always believed should be kept in mind by mathematicians and which, in my opinion, proves conclusively the algebraic unsolvability of general equations of higher than fourth degree." However, in general, Ruffini's proof was not considered convincing. Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini. But his memoir is so complicated that it is very difficult to determine the validity of his argument. It seems to me that his argument is not completely satisfying."
The proof also, as it was discovered later, was incomplete. Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial. To see why this is really an extra assumption, consider, for instance, the polynomial formula_48. According to Cardano's formula, one of its roots (all of them, actually) can be expressed as the sum of a cube root of formula_49 with a cube root of formula_50. On the other hand, since formula_51, formula_52, formula_53, and formula_54, the roots formula_55, formula_56, and formula_57 of formula_58 are all real and therefore the field formula_59 is a subfield of formula_60. But then the numbers formula_61 cannot belong to formula_59. While Cauchy either did not notice Ruffini's assumption or felt that it was a minor one, most historians believe that the proof was not complete until Abel proved the theorem on natural irrationalities, which asserts that the assumption holds in the case of general polynomials.
The Abel–Ruffini theorem is thus generally credited to Abel, who published a proof compressed into just six pages in 1824. (Abel adopted a very terse style to save paper and money: the proof was printed at his own expense.) A more elaborated version of the proof would be published in 1826.
Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals. Abel was working on a complete characterization when he died in 1829.
According to Nathan Jacobson, "The proofs of Ruffini and of Abel [...] were soon superseded by the crowning achievement of this line of research: Galois' discoveries in the theory of equations." In 1830, Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals, which was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois was aware of the contributions of Ruffini and Abel, since he wrote "It is a common truth, today, that the general equation of degree greater than 4 cannot be solved by radicals... this truth has become common (by hearsay) despite the fact that geometers have ignored the proofs of Abel and Ruffini..." Galois then died in 1832 and his paper "Mémoire sur les conditions de resolubilité des équations par radicaux" remained unpublished until 1846, when it was published by Joseph Liouville accompanied by some of his own explanations. Prior to this publication, Liouville announced Galois' result to the academy in a speech he gave on 4 July 1843. A simplification of Abel's proof was published by Pierre Wantzel in 1845. When Wantzel published it, he was already aware of the contributions by Galois and he mentions that, whereas Abel's proof is valid only for general polynomials, Galois' approach can be used to provide a concrete polynomial of degree 5 whose roots cannot be expressed in radicals from its coefficients.
In 1963, Vladimir Arnold discovered a topological proof of the Abel–Ruffini theorem, which served as a starting point for topological Galois theory.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x^5-x-1=0"
},
{
"math_id": 1,
"text": "x^n-1=0"
},
{
"math_id": 2,
"text": "F_0\\subseteq F_1\\subseteq \\cdots \\subseteq F_k"
},
{
"math_id": 3,
"text": "x_i\\in F_i"
},
{
"math_id": 4,
"text": "F_i=F_{i-1}(x_i)"
},
{
"math_id": 5,
"text": "i=1,\\ldots, k,"
},
{
"math_id": 6,
"text": "x_i^{n_i}\\in F_{i-1}"
},
{
"math_id": 7,
"text": "n_i>1."
},
{
"math_id": 8,
"text": "F_k"
},
{
"math_id": 9,
"text": "F_{i-1}"
},
{
"math_id": 10,
"text": "n_i"
},
{
"math_id": 11,
"text": "K_i"
},
{
"math_id": 12,
"text": "F_i"
},
{
"math_id": 13,
"text": "K_i(x_i)."
},
{
"math_id": 14,
"text": "F/E"
},
{
"math_id": 15,
"text": "E\\subseteq K \\subseteq F"
},
{
"math_id": 16,
"text": "\\operatorname{Gal}(F/K)"
},
{
"math_id": 17,
"text": "\\operatorname{Gal}(F/E)"
},
{
"math_id": 18,
"text": "S_5"
},
{
"math_id": 19,
"text": "\\mathcal S_n"
},
{
"math_id": 20,
"text": "\\mathcal A_n"
},
{
"math_id": 21,
"text": "x^n+a_1x^{n-1}+ \\cdots+ a_{n-1}x+a_n=0, "
},
{
"math_id": 22,
"text": "a_1,\\ldots, a_n"
},
{
"math_id": 23,
"text": "F=\\Q(a_1,\\ldots,a_n)"
},
{
"math_id": 24,
"text": "\\mathcal S_n,"
},
{
"math_id": 25,
"text": "x_1, \\ldots, x_n"
},
{
"math_id": 26,
"text": "P(x)=x^n+b_1x^{n-1}+ \\cdots+ b_{n-1}x+b_n= (x-x_1)\\cdots (x-x_n)."
},
{
"math_id": 27,
"text": "H=\\Q(x_1,\\ldots,x_n)"
},
{
"math_id": 28,
"text": "x_1, \\ldots, x_n,"
},
{
"math_id": 29,
"text": "K=\\Q(b_1,\\ldots, b_n)"
},
{
"math_id": 30,
"text": "P(x)."
},
{
"math_id": 31,
"text": "x_i"
},
{
"math_id": 32,
"text": "x_i,"
},
{
"math_id": 33,
"text": "\\operatorname{Gal}(H/K)"
},
{
"math_id": 34,
"text": "\\mathcal S_n."
},
{
"math_id": 35,
"text": "b_i"
},
{
"math_id": 36,
"text": "a_i"
},
{
"math_id": 37,
"text": "P(x)=0"
},
{
"math_id": 38,
"text": "x^5-x-1"
},
{
"math_id": 39,
"text": "\\mathcal S_5"
},
{
"math_id": 40,
"text": "q \\bmod 2"
},
{
"math_id": 41,
"text": "(x^2 + x + 1)(x^3 + x^2 + 1)"
},
{
"math_id": 42,
"text": "\\mathbb{F}_2[x]"
},
{
"math_id": 43,
"text": "g"
},
{
"math_id": 44,
"text": "g^3"
},
{
"math_id": 45,
"text": "q \\bmod 3"
},
{
"math_id": 46,
"text": "\\mathbb{F}_3[x]"
},
{
"math_id": 47,
"text": "G = \\mathcal S_5"
},
{
"math_id": 48,
"text": "P(x)=x^{3}-15x-20"
},
{
"math_id": 49,
"text": "10+5i"
},
{
"math_id": 50,
"text": "10-5i"
},
{
"math_id": 51,
"text": "P(-3)<0"
},
{
"math_id": 52,
"text": "P(-2)>0"
},
{
"math_id": 53,
"text": "P(-1)<0"
},
{
"math_id": 54,
"text": "P(5)>0"
},
{
"math_id": 55,
"text": "r_1"
},
{
"math_id": 56,
"text": "r_2"
},
{
"math_id": 57,
"text": "r_3"
},
{
"math_id": 58,
"text": "P(x)"
},
{
"math_id": 59,
"text": "\\mathbf{Q}(r_1,r_2,r_3)"
},
{
"math_id": 60,
"text": "\\mathbf{R}"
},
{
"math_id": 61,
"text": "10 \\pm 5i"
}
] |
https://en.wikipedia.org/wiki?curid=152518
|
15253718
|
Choi–Williams distribution function
|
Choi–Williams distribution function is one of the members of Cohen's class distribution function. It was first proposed by Hyung-Ill Choi and William J. Williams in 1989. This distribution function adopts exponential kernel to suppress the cross-term. However, the kernel gain does not decrease along the formula_0 axes in the ambiguity domain. Consequently, the kernel function of Choi–Williams distribution function can only filter out the cross-terms that result from the components that differ in both time and frequency center.
Mathematical definition.
The definition of the cone-shape distribution function is shown as follows:
formula_1
where
formula_2
and the kernel function is:
formula_3
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\eta, \\tau"
},
{
"math_id": 1,
"text": "C_x(t, f) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty A_x(\\eta,\\tau) \\Phi(\\eta,\\tau) \\exp (j2\\pi(\\eta t-\\tau f))\\, d\\eta\\, d\\tau,"
},
{
"math_id": 2,
"text": "A_x(\\eta,\\tau) = \\int_{-\\infty}^\\infty x(t+\\tau /2)x^*(t-\\tau /2) e^{-j2\\pi t\\eta}\\, dt,"
},
{
"math_id": 3,
"text": "\\Phi \\left(\\eta,\\tau \\right) = \\exp \\left[-\\alpha \\left(\\eta \\tau \\right)^2 \\right]."
}
] |
https://en.wikipedia.org/wiki?curid=15253718
|
15253921
|
Cone-shape distribution function
|
The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution, (acronymized as the ZAM distribution or ZAMD), is one of the members of Cohen's class distribution function. It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990. The distribution's name stems from the twin cone shape of the distribution's kernel function on the formula_0 plane. The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel.
Mathematical definition.
The definition of the cone-shape distribution function is:
formula_1
where
formula_2
and the kernel function is
formula_3
The kernel function in formula_0 domain is defined as:
formula_4
Following are the magnitude distribution of the kernel function in formula_0 domain.
Following are the magnitude distribution of the kernel function in formula_5 domain with different formula_6 values.
As is seen in the figure above, a properly chosen kernel of cone-shape distribution function can filter out the interference on the formula_7 axis in the formula_5 domain, or the ambiguity domain. Therefore, unlike the Choi-Williams distribution function, the cone-shape distribution function can effectively reduce the cross-term results form two component with same center frequency. However, the cross-terms on the formula_8 axis are still preserved.
The cone-shape distribution function is in the MATLAB Time-Frequency Toolbox and National Instruments' LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "t, \\tau"
},
{
"math_id": 1,
"text": "C_x(t, f)=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}A_x(\\eta,\\tau)\\Phi(\\eta,\\tau)\\exp (j2\\pi(\\eta t-\\tau f))\\, d\\eta\\, d\\tau,"
},
{
"math_id": 2,
"text": "A_x(\\eta,\\tau)=\\int_{-\\infty}^{\\infty}x(t+\\tau /2)x^*(t-\\tau /2)e^{-j2\\pi t\\eta}\\, dt,"
},
{
"math_id": 3,
"text": "\\Phi \\left(\\eta,\\tau \\right) = \\frac{\\sin \\left(\\pi \\eta \\tau \\right)}{ \\pi \\eta \\tau }\\exp \\left(-2\\pi \\alpha \\tau^2 \\right). "
},
{
"math_id": 4,
"text": "\\phi \\left(t,\\tau \\right) = \\begin{cases} \\frac{1}{\\tau} \\exp \\left(-2\\pi \\alpha \\tau^2 \\right), & |\\tau | \\ge 2|t|, \\\\ 0, & \\mbox{otherwise}. \\end{cases} "
},
{
"math_id": 5,
"text": "\\eta, \\tau"
},
{
"math_id": 6,
"text": "\\alpha"
},
{
"math_id": 7,
"text": "\\tau"
},
{
"math_id": 8,
"text": "\\eta"
}
] |
https://en.wikipedia.org/wiki?curid=15253921
|
152547
|
Bisection
|
Division of something into two equal or congruent parts
In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a "bisector". The most often considered types of bisectors are the "segment bisector", a line that passes through the midpoint of a given segment, and the "angle bisector", a line that passes through the apex of an angle (that divides it into two equal angles).
In three-dimensional space, bisection is usually done by a bisecting plane, also called the "bisector".
Perpendicular line segment bisector.
Definition.
(D)formula_2.
The proof follows from formula_3 and Pythagoras' theorem:
formula_4
Property (D) is usually used for the construction of a perpendicular bisector:
Construction by straight edge and compass.
In classical geometry, the bisection is a simple compass and straightedge construction, whose possibility depends on the ability to draw arcs of equal radii and different centers:
The segment formula_0 is bisected by drawing intersecting circles of equal radius formula_5, whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.<br>
Because the construction of the bisector is done without the knowledge of the segment's midpoint formula_6, the construction is used for determining formula_6 as the intersection of the bisector and the line segment.
This construction is in fact used when constructing a "line perpendicular to a given line" formula_7 at a "given point" formula_8: drawing a circle whose center is formula_8 such that it intersects the line formula_7 in two points formula_9, and the perpendicular to be constructed is the one bisecting segment formula_0.
Equations.
If formula_10 are the position vectors of two points formula_9, then its midpoint is formula_11 and vector formula_12 is a normal vector of the perpendicular line segment bisector. Hence its vector equation is formula_13. Inserting formula_14 and expanding the equation leads to the vector equation
(V) formula_15
With formula_16 one gets the equation in coordinate form:
(C) formula_17
Or explicitly:<br>
(E)formula_18, <br>
where formula_19, formula_20, and formula_21.
Applications.
Perpendicular line segment bisectors were used solving various geometric problems:
Perpendicular line segment bisectors in space.
Its vector equation is literally the same as in the plane case:
(V) formula_15
With formula_22 one gets the equation in coordinate form:
(C3) formula_23
Property (D) (see above) is literally true in space, too:<br>
(D) The perpendicular bisector plane of a segment formula_0 has for any point formula_1 the property: formula_24.
Angle bisector.
An angle bisector divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle.
The 'interior' or 'internal bisector' of an angle is the line, half-line, or line segment that divides an angle of less than 180° into two equal angles. The 'exterior' or 'external bisector' is the line that divides the supplementary angle (of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles.
To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.
The proof of the correctness of this construction is fairly intuitive, relying on the symmetry of the problem. The trisection of an angle (dividing it into three equal parts) cannot be achieved with the compass and ruler alone (this was first proved by Pierre Wantzel).
The internal and external bisectors of an angle are perpendicular. If the angle is formed by the two lines given algebraically as formula_25 and formula_26 then the internal and external bisectors are given by the two equations
formula_27
Triangle.
Concurrencies and collinearities.
The bisectors of two exterior angles and the bisector of the other interior angle are concurrent.
Three intersection points, each of an external angle bisector with the opposite extended side, are collinear (fall on the same line as each other).
Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
Angle bisector theorem.
The angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.
Lengths.
If the side lengths of a triangle are formula_28, the semiperimeter formula_29 and A is the angle opposite side formula_30, then the length of the internal bisector of angle A is
formula_31
or in trigonometric terms,
formula_32
If the internal bisector of angle A in triangle ABC has length formula_33 and if this bisector divides the side opposite A into segments of lengths "m" and "n", then
formula_34
where "b" and "c" are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion "b":"c".
If the internal bisectors of angles A, B, and C have lengths formula_35 and formula_36, then
formula_37
No two non-congruent triangles share the same set of three internal angle bisector lengths.
Integer triangles.
There exist integer triangles with a rational angle bisector.
Quadrilateral.
The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral (that is, the four intersection points of adjacent angle bisectors are concyclic), or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.
Rhombus.
Each diagonal of a rhombus bisects opposite angles.
Ex-tangential quadrilateral.
The excenter of an ex-tangential quadrilateral lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.
Parabola.
The tangent to a parabola at any point bisects the angle between the line joining the point to the focus and the line from the point and perpendicular to the directrix.
Bisectors of the sides of a polygon.
Triangle.
Medians.
Each of the three medians of a triangle is a line segment going through one vertex and the midpoint of the opposite side, so it bisects that side (though not in general perpendicularly). The three medians intersect each other at a point which is called the centroid of the triangle, which is its center of mass if it has uniform density; thus any line through a triangle's centroid and one of its vertices bisects the opposite side. The centroid is twice as close to the midpoint of any one side as it is to the opposite vertex.
Perpendicular bisectors.
The interior perpendicular bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices). Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side.
In an acute triangle the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an obtuse triangle the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by their respective intersecting triangle sides in equal proportions.
For any triangle the interior perpendicular bisectors are given by formula_38 formula_39 and formula_40 where the sides are formula_41 and the area is formula_42
Quadrilateral.
The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are concurrent at a point called the "vertex centroid" and are all bisected by this point.
The four "maltitudes" of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is cyclic (inscribed in a circle), these maltitudes are concurrent at (all meet at) a common point called the "anticenter".
Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.
The perpendicular bisector construction forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral.
Area bisectors and perimeter bisectors.
Triangle.
There is an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides; each of these intersects the other two sides so as to divide them into segments with the proportions formula_43. These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors.
The envelope of the infinitude of area bisectors is a deltoid (broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set). The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one.
The sides of the deltoid are arcs of hyperbolas that are asymptotic to the extended sides of the triangle. The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals formula_44 i.e. 0.019860... or less than 2%.
A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at (all pass through) the center of the Spieker circle, which is the incircle of the medial triangle. The cleavers are parallel to the angle bisectors.
A splitter of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other.
Parallelogram.
Any line through the midpoint of a parallelogram bisects the area and the perimeter.
Circle and ellipse.
All area bisectors and perimeter bisectors of a circle or other ellipse go through the center, and any chords through the center bisect the area and perimeter. In the case of a circle they are the diameters of the circle.
Bisectors of diagonals.
Parallelogram.
The diagonals of a parallelogram bisect each other.
Quadrilateral.
If a line segment connecting the diagonals of a quadrilateral bisects both diagonals, then this line segment (the Newton Line) is itself bisected by the vertex centroid.
Volume bisectors.
A plane that divides two opposite edges of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron
External links.
"This article incorporates material from Angle bisector on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "AB"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "\\quad |XA| = |XB|"
},
{
"math_id": 3,
"text": ""
},
{
"math_id": 4,
"text": "|XA|^2=|XM|^2+|MA|^2=|XM|^2+|MB|^2=|XB|^2 \\; ."
},
{
"math_id": 5,
"text": "r>\\tfrac 1 2 |AB|"
},
{
"math_id": 6,
"text": "M"
},
{
"math_id": 7,
"text": "g"
},
{
"math_id": 8,
"text": "P"
},
{
"math_id": 9,
"text": "A,B"
},
{
"math_id": 10,
"text": "\\vec a,\\vec b"
},
{
"math_id": 11,
"text": "M: \\vec m=\\tfrac{\\vec a+\\vec b}{2}"
},
{
"math_id": 12,
"text": "\\vec a -\\vec b"
},
{
"math_id": 13,
"text": "(\\vec x-\\vec m)\\cdot(\\vec a-\\vec b)=0"
},
{
"math_id": 14,
"text": "\\vec m =\\cdots"
},
{
"math_id": 15,
"text": "\\quad \\vec x\\cdot(\\vec a-\\vec b)=\\tfrac 1 2 (\\vec a^2-\\vec b^2) ."
},
{
"math_id": 16,
"text": "A=(a_1,a_2),B=(b_1,b_2)"
},
{
"math_id": 17,
"text": "\\quad (a_1-b_1)x+(a_2-b_2)y=\\tfrac 1 2 (a_1^2-b_1^2+a_2^2-b_2^2) \\; ."
},
{
"math_id": 18,
"text": "\\quad y = m(x - x_0) +y_0"
},
{
"math_id": 19,
"text": "\\; m = - \\tfrac{b_1 - a_1}{b_2 - a_2}"
},
{
"math_id": 20,
"text": "\\;x_0 = \\tfrac{1}{2}(a_1 + b_1)\\;"
},
{
"math_id": 21,
"text": "\\;y_0 = \\tfrac{1}{2}(a_2 + b_2)\\;"
},
{
"math_id": 22,
"text": "A=(a_1,a_2,a_3),B=(b_1,b_2,b_3)"
},
{
"math_id": 23,
"text": "\\quad (a_1-b_1)x+(a_2-b_2)y+(a_3-b_3)z=\\tfrac 1 2 (a_1^2-b_1^2+a_2^2-b_2^2+a_3^2-b_3^2) \\; ."
},
{
"math_id": 24,
"text": "\\;|XA| = |XB|"
},
{
"math_id": 25,
"text": "l_1x+m_1y+n_1=0"
},
{
"math_id": 26,
"text": "l_2x+m_2y+n_2=0,"
},
{
"math_id": 27,
"text": "\\frac{l_1x+m_1y+n_1}{\\sqrt{l_1^2+m_1^2}} = \\pm \\frac{l_2x+m_2y+n_2}{\\sqrt{l_2^2+m_2^2}}."
},
{
"math_id": 28,
"text": "a,b,c"
},
{
"math_id": 29,
"text": "s=(a+b+c)/2,"
},
{
"math_id": 30,
"text": "a"
},
{
"math_id": 31,
"text": " \\frac{2 \\sqrt{bcs(s-a)}}{b+c},"
},
{
"math_id": 32,
"text": "\\frac{2bc}{b+c}\\cos \\frac{A}{2}. "
},
{
"math_id": 33,
"text": "t_a"
},
{
"math_id": 34,
"text": "t_a^2+mn = bc"
},
{
"math_id": 35,
"text": "t_a, t_b,"
},
{
"math_id": 36,
"text": "t_c"
},
{
"math_id": 37,
"text": "\\frac{(b+c)^2}{bc}t_a^2+ \\frac{(c+a)^2}{ca}t_b^2+\\frac{(a+b)^2}{ab}t_c^2 = (a+b+c)^2."
},
{
"math_id": 38,
"text": "p_a=\\tfrac{2aT}{a^2+b^2-c^2},"
},
{
"math_id": 39,
"text": "p_b=\\tfrac{2bT}{a^2+b^2-c^2},"
},
{
"math_id": 40,
"text": "p_c=\\tfrac{2cT}{a^2-b^2+c^2},"
},
{
"math_id": 41,
"text": "a \\ge b \\ge c"
},
{
"math_id": 42,
"text": "T."
},
{
"math_id": 43,
"text": "\\sqrt{2}+1:1"
},
{
"math_id": 44,
"text": "\\tfrac{3}{4} \\log_e(2) - \\tfrac{1}{2},"
}
] |
https://en.wikipedia.org/wiki?curid=152547
|
1525521
|
Clapp oscillator
|
The Clapp oscillator or Gouriet oscillator is an LC electronic oscillator that uses a particular combination of an inductor and three capacitors to set the oscillator's frequency. LC oscillators use a transistor (or vacuum tube or other gain element) and a positive feedback network. The oscillator has good frequency stability.
History.
The Clapp oscillator design was published by James Kilton Clapp in 1948 while he worked at General Radio. According to Czech engineer Jiří Vackář, oscillators of this kind were independently developed by several inventors, and one developed by Gouriet had been in operation at the BBC since 1938.
Circuit.
The Clapp oscillator uses a single inductor and three capacitors to set its frequency. The Clapp oscillator is often drawn as a Colpitts oscillator that has an additional capacitor ("C"0) placed in series with the inductor.
The oscillation frequency in Hertz (cycles per second) for the circuit in the figure, which uses a field-effect transistor (FET), is
formula_0
The capacitors "C"1 and "C"2 are usually much larger than "C"0, so the 1/"C"0 term dominates the other capacitances, and the frequency is near the series resonance of "L" and "C"0. Clapp's paper gives an example where "C"1 and "C"2 are 40 times larger than "C"0; the change makes the Clapp circuit about 400 times more stable than the Colpitts oscillator for capacitance changes of "C"2.
Capacitors "C"0, "C"1 and "C"2 form a voltage divider that determines the amount of feedback voltage applied to the transistor input.
Although the Clapp circuit is used as a variable frequency oscillator (VFO) by making "C"0 a variable capacitor, Vackář states that the Clapp oscillator "can only be used for operation on fixed frequencies or at the most over narrow bands (max. about 1:1.2)." The problem is that under typical conditions, the Clapp oscillator's loop gain varies as "f" −3, so wide ranges will overdrive the amplifier. For VFOs, Vackář recommends other circuits. See Vackář oscillator.
Practical example.
The schematic shows an example with component values. Instead of field-effect transistors, other active components such as bipolar junction transistors or vacuum tubes, capable of producing gain at the desired frequency, could be used.
The common drain amplifier has a high input impedance and a low output impedance. Therefore the amplifier input, the gate, is connected to the high impedance top of the LC circuit C0, C1, C2, L1 and the amplifier output, the source, is connected to the low impedance tap of the LC circuit. The grid leak C3 and R1 sets the operating point automatically through grid leak bias. A smaller value of C3 gives less harmonic distortion, but requires a larger load resistor. The supply current for J1 flows through the radio frequency choke L2 to ground. The oscillator radio frequency current uses C2, because for the oscillator frequency this component has less reactance. The load resistor RL is part of the simulation, not part of the circuit.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\n f_0 = {1 \\over 2\\pi}\n \\sqrt{ {1 \\over L}\n \\left( {1 \\over C_0}\n + {1 \\over C_1}\n + {1 \\over C_2}\n \\right)} \\ .\n"
}
] |
https://en.wikipedia.org/wiki?curid=1525521
|
152567
|
Generalized Riemann hypothesis
|
Mathematical conjecture about zeros of L-functions
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global "L"-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these "L"-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case).
Global "L"-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet "L"-functions, it is known as the generalized Riemann hypothesis or generalised Riemann hypothesis (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label "generalized Riemann hypothesis" to cover the extension of the Riemann hypothesis to all global "L"-functions,
not just the special case of Dirichlet "L"-functions.)
Generalized Riemann hypothesis (GRH).
The generalized Riemann hypothesis (for Dirichlet "L"-functions) was probably formulated for the first time by Adolf Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.
The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function "χ" such that there exists a positive integer "k" with "χ"("n" + "k") = "χ"("n") for all "n" and "χ"("n") = 0 whenever gcd("n", "k") > 1. If such a character is given, we define the corresponding Dirichlet "L"-function by
formula_0
for every complex number "s" such that Re "s" > 1. By analytic continuation, this function can be extended to a meromorphic function (only when formula_1 is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every Dirichlet character "χ" and every complex number "s" with "L"("χ", "s") = 0, if "s" is not a negative real number, then the real part of "s" is 1/2.
The case "χ"("n") = 1 for all "n" yields the ordinary Riemann hypothesis.
Consequences of GRH.
Dirichlet's theorem states that if "a" and "d" are coprime natural numbers, then the arithmetic progression "a", "a" + "d", "a" + 2"d", "a" + 3"d", ... contains infinitely many prime numbers. Let π("x", "a", "d") denote the number of prime numbers in this progression which are less than or equal to "x". If the generalized Riemann hypothesis is true, then for every coprime "a" and "d" and for every "ε" > 0,
formula_2
where formula_3 is Euler's totient function and formula_4 is the Big O notation. This is a considerable strengthening of the prime number theorem.
If GRH is true, then every proper subgroup of the multiplicative group formula_5 omits a number less than 2(ln "n")2, as well as a number coprime to "n" less than 3(ln "n")2. In other words, formula_5 is generated by a set of numbers less than 2(ln "n")2. This is often used in proofs, and it has many consequences, for example (assuming GRH):
If GRH is true, then for every prime "p" there exists a primitive root mod "p" (a generator of the multiplicative group of integers modulo "p") that is less than formula_6
Goldbach's weak conjecture also follows from the generalized Riemann hypothesis. The yet to be verified proof of Harald Helfgott of this conjecture verifies the GRH for several thousand small characters up to a certain imaginary part to obtain sufficient bounds that prove the conjecture for all integers above 1029, integers below which have already been verified by calculation.
Assuming the truth of the GRH, the estimate of the character sum in the Pólya–Vinogradov inequality can be improved to formula_7, "q" being the modulus of the character.
Extended Riemann hypothesis (ERH).
Suppose "K" is a number field (a finite-dimensional field extension of the rationals "Q") with ring of integers O"K" (this ring is the integral closure of the integers "Z" in "K"). If "a" is an ideal of O"K", other than the zero ideal, we denote its norm by "Na". The Dedekind zeta-function of "K" is then defined by
formula_8
for every complex number "s" with real part > 1. The sum extends over all non-zero ideals "a" of O"K".
The Dedekind zeta-function satisfies a functional equation and can be extended by analytic continuation to the whole complex plane. The resulting function encodes important information about the number field "K". The extended Riemann hypothesis asserts that for every number field "K" and every complex number "s" with ζ"K"("s") = 0: if the real part of "s" is between 0 and 1, then it is in fact 1/2.
The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be "Q", with ring of integers "Z".
The ERH implies an effective version of the Chebotarev density theorem: if "L"/"K" is a finite Galois extension with Galois group "G", and "C" a union of conjugacy classes of "G", the number of unramified primes of "K" of norm below "x" with Frobenius conjugacy class in "C" is
formula_9
where the constant implied in the big-O notation is absolute, "n" is the degree of "L" over "Q", and Δ its discriminant.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\nL(\\chi,s) = \\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}\n"
},
{
"math_id": 1,
"text": " \\chi "
},
{
"math_id": 2,
"text": "\\pi(x,a,d) = \\frac{1}{\\varphi(d)} \\int_2^x \\frac{1}{\\ln t}\\,dt + O(x^{1/2+\\varepsilon})\\quad\\mbox{ as } \\ x\\to\\infty,"
},
{
"math_id": 3,
"text": "\\varphi"
},
{
"math_id": 4,
"text": "O"
},
{
"math_id": 5,
"text": "(\\mathbb Z/n\\mathbb Z)^\\times"
},
{
"math_id": 6,
"text": "O((\\ln p)^6)."
},
{
"math_id": 7,
"text": "O\\left(\\sqrt{q}\\log\\log q\\right)"
},
{
"math_id": 8,
"text": "\n\\zeta_K(s) = \\sum_a \\frac{1}{(Na)^s}\n"
},
{
"math_id": 9,
"text": "\\frac{|C|}{|G|}\\Bigl(\\operatorname{li}(x)+O\\bigl(\\sqrt x(n\\log x+\\log|\\Delta|)\\bigr)\\Bigr),"
}
] |
https://en.wikipedia.org/wiki?curid=152567
|
1525933
|
Rodrigues' rotation formula
|
Vector formula for a rotation in space, given its axis
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3).
This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to Euler, and recommended calling it "Euler's finite rotation formula." This proposal has received notable support, but some others have viewed the formula as just one of many variations of the Euler–Rodrigues formula, thereby crediting both.
Statement.
If v is a vector in ℝ3 and k is a unit vector describing an axis of rotation about which v rotates by an angle θ according to the right hand rule, the Rodrigues formula for the rotated vector vrot is
formula_0
The intuition of the above formula is that the first term scales the vector down, while the second skews it (via vector addition) toward the new rotational position. The third term re-adds the height (relative to formula_1) that was lost by the first term.
An alternative statement is to write the axis vector as a cross product a × b of any two nonzero vectors a and b which define the plane of rotation, and the sense of the angle "θ" is measured away from a and towards b. Letting "α" denote the angle between these vectors, the two angles "θ" and "α" are not necessarily equal, but they are measured in the same sense. Then the unit axis vector can be written
formula_2
This form may be more useful when two vectors defining a plane are involved. An example in physics is the Thomas precession which includes the rotation given by Rodrigues' formula, in terms of two non-collinear boost velocities, and the axis of rotation is perpendicular to their plane.
Derivation.
Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle "θ" (right hand rule, anticlockwise in the figure), producing the rotated vector formula_3.
Using the dot and cross products, the vector v can be decomposed into components parallel and perpendicular to the axis k,
formula_4
where the component parallel to k is called the vector projection of v on k,
formula_5,
and the component perpendicular to k is called the vector rejection of v from k:
formula_6,
where the last equality follows from the vector triple product formula: formula_7. Finally, the vector formula_8 is a copy of formula_9 rotated 90° around formula_10. Thus the three vectors formula_11 form a right-handed orthogonal basis of formula_12, with the last two vectors of equal length.
Under the rotation, the component formula_13 parallel to the axis will not change magnitude nor direction:
formula_14
while the perpendicular component will retain its magnitude but rotate its direction in the perpendicular plane spanned by formula_9 and formula_15, according to
formula_16
in analogy with the planar polar coordinates ("r", "θ") in the Cartesian basis e"x", e"y":
formula_17
Now the full rotated vector is:
formula_18
Substituting formula_19 or formula_20 in the last expression gives respectively:formula_21formula_22
Matrix notation.
The linear transformation on formula_23 defined by the cross product formula_24 is given in coordinates by representing v and k × v as column matrices:
formula_25
That is, the matrix of this linear transformation (with respect to standard coordinates) is the cross-product matrix:
formula_26
That is to say,
formula_27
The last formula in the previous section can therefore be written as:
formula_28
Collecting terms allows the compact expression
formula_29
where
formula_30
is the rotation matrix through an angle θ counterclockwise about the axis k, and I the 3 × 3 identity matrix. This matrix R is an element of the rotation group SO(3) of ℝ3, and K is an element of the Lie algebra formula_31 generating that Lie group (note that K is skew-symmetric, which characterizes formula_31).
In terms of the matrix exponential,
formula_32
To see that the last identity holds, one notes that
formula_33
characteristic of a one-parameter subgroup, i.e. exponential, and that the formulas match for infinitesimal θ.
For an alternative derivation based on this exponential relationship, see exponential map from formula_31 to SO(3). For the inverse mapping, see log map from SO(3) to formula_31.
The Hodge dual of the rotation formula_34 is just formula_35 which enables the extraction of both the axis of rotation and the sine of the angle of the rotation from the rotation matrix itself, with the usual ambiguity,
formula_36
where formula_37. The above simple expression results from the fact that the Hodge duals of formula_38 and formula_39 are zero, and formula_40.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbf{v}_\\mathrm{rot} = \\mathbf{v} \\cos\\theta + (\\mathbf{k} \\times \\mathbf{v})\\sin\\theta + \\mathbf{k} ~(\\mathbf{k} \\cdot \\mathbf{v}) (1 - \\cos\\theta)\\,."
},
{
"math_id": 1,
"text": "\\textbf{k}"
},
{
"math_id": 2,
"text": "\\mathbf{k} = \\frac{\\mathbf{a}\\times\\mathbf{b}}{|\\mathbf{a}\\times\\mathbf{b}|} = \\frac{\\mathbf{a}\\times\\mathbf{b}}{|\\mathbf{a}||\\mathbf{b}|\\sin\\alpha}\\,. "
},
{
"math_id": 3,
"text": "\\mathbb{v}_{\\text{rot}}"
},
{
"math_id": 4,
"text": " \\mathbf{v} = \\mathbf{v}_\\parallel + \\mathbf{v}_\\perp \\,, "
},
{
"math_id": 5,
"text": " \\mathbf{v}_\\parallel = (\\mathbf{v} \\cdot \\mathbf{k}) \\mathbf{k} "
},
{
"math_id": 6,
"text": "\\mathbf{v}_{\\perp} = \\mathbf{v} - \\mathbf{v}_{\\parallel} = \\mathbf{v} - (\\mathbf{k} \\cdot \\mathbf{v}) \\mathbf{k} = - \\mathbf{k}\\times(\\mathbf{k}\\times\\mathbf{v})"
},
{
"math_id": 7,
"text": "\\mathbf{a}\\times (\\mathbf{b} \\times \\mathbf{c})\n= (\\mathbf{a} \\cdot \\mathbf{c})\\mathbf{b} - (\\mathbf{a} \\cdot \\mathbf{b})\\mathbf{c}"
},
{
"math_id": 8,
"text": "\\mathbf{k} \\times \\mathbf{v}_{\\perp} = \\mathbf{k} \\times \\mathbf{v}"
},
{
"math_id": 9,
"text": "\\mathbf{v}_{\\perp}"
},
{
"math_id": 10,
"text": "\\mathbf{k}"
},
{
"math_id": 11,
"text": "\\mathbf{k}\\,,\\ \n\\mathbf{v}_{\\perp}\\,,\\, \n\\mathbf{k} \\times \\mathbf{v}"
},
{
"math_id": 12,
"text": "\\mathbb{R}^3"
},
{
"math_id": 13,
"text": "\\mathbf{v}_{\\parallel}"
},
{
"math_id": 14,
"text": "\\mathbf{v}_{\\parallel\\mathrm{rot}} = \\mathbf{v}_\\parallel \\,;"
},
{
"math_id": 15,
"text": "\\mathbf{k} \\times \\mathbf{v}"
},
{
"math_id": 16,
"text": " \\mathbf{v}_{\\perp\\mathrm{rot}} \n= \\cos(\\theta) \\mathbf{v}_\\perp + \\sin(\\theta) \\mathbf{k}\\times\\mathbf{v}_\\perp \n= \\cos(\\theta) \\mathbf{v}_\\perp + \\sin(\\theta) \\mathbf{k}\\times\\mathbf{v} \\,,"
},
{
"math_id": 17,
"text": "\\mathbf{r} = r\\cos(\\theta) \\mathbf{e}_x + r\\sin(\\theta) \\mathbf{e}_y \\,. "
},
{
"math_id": 18,
"text": " \\mathbf{v}_{\\mathrm{rot}}\n = \\mathbf{v}_{\\parallel\\mathrm{rot}} + \\mathbf{v}_{\\perp\\mathrm{rot}} \n = \\mathbf{v}_\\parallel + \\cos(\\theta) \\, \\mathbf{v}_\\perp + \\sin(\\theta) \\mathbf{k}\\times\\mathbf{v} .\n "
},
{
"math_id": 19,
"text": "\\mathbf{v}_{\\perp } = \\mathbf{v} - \\mathbf{v}_{\\|} "
},
{
"math_id": 20,
"text": "\\mathbf{v}_{\\| } = \\mathbf{v} - \\mathbf{v}_{\\perp}"
},
{
"math_id": 21,
"text": "\\mathbf{v}_{\\text{rot}}\n = \\cos(\\theta) \\, \\mathbf{v} + (1 - \\cos\\theta)(\\mathbf{k} \\cdot \\mathbf{v})\\mathbf{k} \n+ \\sin(\\theta) \\mathbf{k}\\times\\mathbf{v},"
},
{
"math_id": 22,
"text": "\\mathbf{v}_{\\text{rot}}\n = \\mathbf{v} \n+ (1-\\cos\\theta)\\mathbf{k}\\times(\\mathbf{k}\\times\\mathbf{v})\n+ \\sin(\\theta) \\mathbf{k}\\times\\mathbf{v}."
},
{
"math_id": 23,
"text": "\\mathbf{v}\\isin\\mathbb{R}^3 "
},
{
"math_id": 24,
"text": "\\mathbf{v} \\mapsto \\mathbf{k} \\times \\mathbf{v} "
},
{
"math_id": 25,
"text": "\\begin{bmatrix} (\\mathbf{k}\\times\\mathbf{v})_x \\\\ (\\mathbf{k}\\times\\mathbf{v})_y \n\\\\ (\\mathbf{k}\\times\\mathbf{v})_z \\end{bmatrix} \n= \\begin{bmatrix} k_y v_z - k_z v_y \\\\ k_z v_x - k_x v_z \\\\ k_x v_y - k_y v_x \\end{bmatrix} \n= \\left[\\begin{array}{rrr}\n0\\ \\, & -k_z & k_y \\\\\nk_z & 0\\ \\, & -k_x \\\\\n-k_y & k_x & 0\\ \\, \n\\end{array}\\right]\n\\begin{bmatrix} v_x \\\\ v_y \\\\ v_z \\end{bmatrix} \\,. "
},
{
"math_id": 26,
"text": "\\mathbf{K}= \n\\left[\\begin{array}{rrr}\n0\\ \\, & -k_z & k_y \\\\\nk_z & 0\\ \\, & -k_x \\\\\n-k_y & k_x & 0\\ \\, \n\\end{array}\\right]\\,.\n"
},
{
"math_id": 27,
"text": "\\mathbf{k}\\times\\mathbf{v}=\\mathbf{K}\\mathbf{v}, \n\\qquad\\qquad\n\\mathbf{k}\\times(\\mathbf{k}\\times\\mathbf{v})=\\mathbf{K}(\\mathbf{K}\\mathbf{v}) = \\mathbf{K}^2\\mathbf{v} \\,. "
},
{
"math_id": 28,
"text": "\\mathbf{v}_{\\mathrm{rot}} = \\mathbf{v} + (\\sin\\theta) \\mathbf{K}\\mathbf{v} + (1 - \\cos\\theta)\\mathbf{K}^2\\mathbf{v}\\,."
},
{
"math_id": 29,
"text": "\\mathbf{v}_\\mathrm{rot} = \\mathbf{R}\\mathbf{v}"
},
{
"math_id": 30,
"text": "\\mathbf{R} = \\mathbf{I} + (\\sin\\theta) \\mathbf{K} + (1-\\cos\\theta)\\mathbf{K}^2"
},
{
"math_id": 31,
"text": "\\mathfrak{so}(3)"
},
{
"math_id": 32,
"text": "\\mathbf{R} = \\exp (\\theta\\mathbf{K})\\,."
},
{
"math_id": 33,
"text": "\\mathbf{R}(\\theta) \\mathbf{R}(\\phi) = \\mathbf{R} (\\theta+\\phi), \\quad \\mathbf{R}(0) = \\mathbf{I}\\,, "
},
{
"math_id": 34,
"text": "\\mathbf{R}"
},
{
"math_id": 35,
"text": "\\mathbf{R}^* = -\\sin(\\theta)\\mathbf{k}"
},
{
"math_id": 36,
"text": "\\begin{align}\n \\sin(\\theta) &= \\sigma \\left|\\mathbf{R}^*\\right| \\\\[3pt]\n \\mathbf{k} &= -\\frac{\\sigma\\mathbf{R}^*}{\\left|\\mathbf{R}^*\\right|}\n\\end{align}"
},
{
"math_id": 37,
"text": "\\sigma = \\pm 1"
},
{
"math_id": 38,
"text": "\\mathbf{I}"
},
{
"math_id": 39,
"text": "\\mathbf{K}^2"
},
{
"math_id": 40,
"text": "\\mathbf{K}^* = -\\mathbf{k}"
}
] |
https://en.wikipedia.org/wiki?curid=1525933
|
15261351
|
Caltech 101
|
Dataset of images
Caltech 101 is a data set of digital images created in September 2003 and compiled by Fei-Fei Li, Marco Andreetto, Marc 'Aurelio Ranzato and Pietro Perona at the California Institute of Technology. It is intended to facilitate computer vision research and techniques and is most applicable to techniques involving image recognition classification and categorization. Caltech 101 contains a total of 9,146 images, split between 101 distinct object categories (faces, watches, ants, pianos, etc.) and a background category. Provided with the images are a set of annotations describing the outlines of each image, along with a Matlab script for viewing.
Purpose.
Most computer vision and machine learning algorithms function by training on example inputs. They require a large and varied set of training data to work effectively. For example, the real-time face detection method used by Paul Viola and Michael J. Jones was trained on 4,916 hand-labeled faces.
Cropping, re-sizing and hand-marking points of interest is tedious and time-consuming.
Historically, most data sets used in computer vision research have been tailored to the specific needs of the project being worked on. A large problem in comparing computer vision techniques is the fact that most groups use their own data sets. Each set may have different properties that make reported results from different methods harder to compare directly. For example, differences in image size, image quality, relative location of objects within the images and level of occlusion and clutter present can lead to varying results.
The Caltech 101 data set aims at alleviating many of these common problems.
However, a follow-up study demonstrated that tests based on uncontrolled natural images (like the Caltech 101 data set) can be seriously misleading, potentially guiding progress in the wrong direction.
Data set.
Images.
The Caltech 101 data set consists of a total of 9,146 images, split between 101 different object categories, as well as an additional background/clutter category.
Each object category contains between 40 and 800 images. Common and popular categories such as faces tend to have a larger number of images than others.
Each image is about 300x200 pixels. Images of oriented objects such as airplanes and motorcycles were mirrored to be left to right aligned and vertically oriented structures such as buildings were rotated to be off axis.
Annotations.
A set of annotations is provided for each image. Each set of annotations contains two pieces of information: the general bounding box in which the object is located and a detailed human-specified outline enclosing the object.
A Matlab script is provided with the annotations. It loads an image and its corresponding annotation file and displays them as a Matlab figure.
Uses.
The Caltech 101 data set was used to train and test several computer vision recognition and classification algorithms. The first paper to use Caltech 101 was an incremental Bayesian approach to one-shot learning, an attempt to classify an object using only a few examples, by building on prior knowledge of other classes.
The Caltech 101 images, along with the annotations, were used for another one-shot learning paper at Caltech.
Other Computer Vision papers that report using the Caltech 101 data set include:
Analysis and comparison.
Advantages.
Caltech 101 has several advantages over other similar data sets:
Weaknesses.
Weaknesses to the Caltech 101 data set may be conscious trade-offs, but others are limitations of the data set. Papers that rely solely on Caltech 101 are frequently rejected.
Weaknesses include:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathrm{N}_{\\mathrm{train}} \\le 30"
}
] |
https://en.wikipedia.org/wiki?curid=15261351
|
15261618
|
Constellation model
|
The constellation model is a probabilistic, generative model for category-level object recognition in computer vision. Like other part-based models, the constellation model attempts to represent an object class by a set of "N" parts under mutual geometric constraints. Because it considers the geometric relationship between different parts, the constellation model differs significantly from appearance-only, or "bag-of-words" representation models, which explicitly disregard the location of image features.
The problem of defining a generative model for object recognition is difficult. The task becomes significantly complicated by factors such as background clutter, occlusion, and variations in viewpoint, illumination, and scale. Ideally, we would like the particular representation we choose to be robust to as many of these factors as possible.
In category-level recognition, the problem is even more challenging because of the fundamental problem of intra-class variation. Even if two objects belong to the same visual category, their appearances may be significantly different. However, for structured objects such as cars, bicycles, and people, separate instances of objects from the same category are subject to similar geometric constraints. For this reason, particular parts of an object such as the headlights or tires of a car still have consistent appearances and relative positions. The Constellation Model takes advantage of this fact by explicitly modeling the relative location, relative scale, and appearance of these parts for a particular object category. Model parameters are estimated using an unsupervised learning algorithm, meaning that the visual concept of an object class can be extracted from an unlabeled set of training images, even if that set contains "junk" images or instances of objects from multiple categories. It can also account for the absence of model parts due to appearance variability, occlusion, clutter, or detector error.
History.
The idea for a "parts and structure" model was originally introduced by Fischler and Elschlager in 1973. This model has since been built upon and extended in many directions. The Constellation Model, as introduced by Dr. Perona and his colleagues, was a probabilistic adaptation of this approach.
In the late '90s, Burl et al. revisited the Fischler and Elschlager model for the purpose of face recognition. In their work, Burl et al. used manual selection of constellation parts in training images to construct a statistical model for a set of detectors and the relative locations at which they should be applied. In 2000, Weber et al. made the significant step of training the model using a more unsupervised learning process, which precluded the necessity for tedious hand-labeling of parts. Their algorithm was particularly remarkable because it performed well even on cluttered and occluded image data. Fergus et al. then improved upon this model by making the learning step fully unsupervised, having both shape and appearance learned simultaneously, and accounting explicitly for the relative scale of parts.
The method of Weber and Welling et al..
In the first step, a standard interest point detection method, such as Harris corner detection, is used to generate interest points. Image features generated from the vicinity of these points are then clustered using k-means or another appropriate algorithm. In this process of vector quantization, one can think of the centroids of these clusters as being representative of the appearance of distinctive object parts. Appropriate feature detectors are then trained using these clusters, which can be used to obtain a set of candidate parts from images.
As a result of this process, each image can now be represented as a set of parts. Each part has a type, corresponding to one of the aforementioned appearance clusters, as well as a location in the image space.
Basic generative model.
Weber & Welling here introduce the concept of "foreground" and "background". "Foreground" parts correspond to an instance of a target object class, whereas "background" parts correspond to background clutter or false detections.
Let "T" be the number of different types of parts. The positions of all parts extracted from an image can then be represented in the following "matrix,"
formula_0
where formula_1 represents the number of parts of type formula_2 observed in the image. The superscript "o" indicates that these positions are "observable", as opposed to "missing". The positions of unobserved object parts can be represented by the vector formula_3. Suppose that the object will be composed of formula_4 distinct foreground parts. For notational simplicity, we assume here that formula_5, though the model can be generalized to formula_6. A "hypothesis" formula_7 is then defined as a set of indices, with formula_8, indicating that point formula_9 is a foreground point in formula_10. The generative probabilistic model is defined through the joint probability density formula_11.
Model details.
The rest of this section summarizes the details of Weber & Welling's model for a single component model. The formulas for multiple component models are extensions of those described here.
To parametrize the joint probability density, Weber & Welling introduce the auxiliary variables formula_12 and formula_13, where formula_12 is a binary vector encoding the presence/absence of parts in detection (formula_14 if formula_15, otherwise formula_16), and formula_13 is a vector where formula_17 denotes the number of "background" candidates included in the formula_18 row of formula_10. Since formula_12 and formula_13 are completely determined by formula_7 and the size of formula_10, we have formula_19. By decomposition,
formula_20
The probability density over the number of background detections can be modeled by a Poisson distribution,
formula_21
where formula_22 is the average number of background detections of type formula_23 per image.
Depending on the number of parts formula_4, the probability formula_24 can be modeled either as an explicit table of length formula_25, or, if formula_4 is large, as formula_4 independent probabilities, each governing the presence of an individual part.
The density formula_26 is modeled by
formula_27
where formula_28 denotes the set of all hypotheses consistent with formula_12 and formula_13, and formula_29 denotes the total number of detections of parts of type formula_30. This expresses the fact that all consistent hypotheses, of which there are formula_31, are equally likely in the absence of information on part locations.
And finally,
formula_32
where formula_33 are the coordinates of all foreground detections, observed and missing, and formula_34 represents the coordinates of the background detections. Note that foreground detections are assumed to be independent of the background. formula_35 is modeled as a joint Gaussian with mean formula_36 and covariance formula_37.
Classification.
The ultimate objective of this model is to classify images into classes "object present" (class formula_38) and "object absent" (class formula_39) given the observation formula_10. To accomplish this, Weber & Welling run part detectors from the learning step exhaustively over the image, examining different combinations of detections. If occlusion is considered, then combinations with missing detections are also permitted. The goal is then to select the class with maximum a posteriori probability, by considering the ratio
formula_40
where formula_41 denotes the null hypothesis, which explains all parts as background noise. In the numerator, the sum includes all hypotheses, including the null hypothesis, whereas in the denominator, the only hypothesis consistent with the absence of an object is the null hypothesis. In practice, some threshold can be defined such that, if the ratio exceeds that threshold, we then consider an instance of an object to be detected.
Model learning.
After the preliminary step of interest point detection, feature generation and clustering, we have a large set of candidate parts over the training images. To learn the model, Weber & Welling first perform a greedy search over possible model configurations, or equivalently, over potential subsets of the candidate parts. This is done in an iterative fashion, starting with random selection. At subsequent iterations, parts in the model are randomly substituted, the model parameters are estimated, and the performance is assessed. The process is complete when further model performance improvements are no longer possible.
At each iteration, the model parameters
formula_42
are estimated using expectation maximization. formula_36 and formula_37, we recall, are the mean and covariance of the joint Gaussian formula_35, formula_24 is the probability distribution governing the binary presence/absence of parts, and formula_43 is the mean number of background detections over part types.
M-step.
EM proceeds by maximizing the likelihood of the observed data,
formula_44
with respect to the model parameters formula_45. Since this is difficult to achieve analytically, EM iteratively maximizes a sequence of cost functions,
formula_46
Taking the derivative of this with respect to the parameters and equating to zero produces the update rules:
formula_47
formula_48
formula_49
formula_50
E-step.
The update rules in the M-step are expressed in terms of sufficient statistics, formula_51, formula_52, formula_53 and formula_54, which are calculated in the E-step by considering the posterior density:
formula_55
The method of Fergus et al..
In Weber et al., shape and appearance models are constructed separately. Once the set of candidate parts had been selected, shape is learned independently of appearance. The innovation of Fergus et al. is to learn not only two, but three model parameters simultaneously: shape, appearance, and relative scale. Each of these parameters are represented by Gaussian densities.
Feature representation.
Whereas the preliminary step in the Weber et al. method is to search for the locations of interest points, Fergus et al. use the detector of Kadir and Brady to find salient regions in the image over both location (center) and scale (radius). Thus, in addition to location information formula_56 this method also extracts associated scale information formula_57. Fergus et al. then normalize the squares bounding these circular regions to 11 x 11 pixel patches, or equivalently, 121-dimensional vectors in the appearance space. These are then reduced to 10-15 dimensions by principal component analysis, giving the appearance information formula_58.
Model structure.
Given a particular object class model with parameters formula_45, we must decide whether or not a new image contains an instance of that class. This is accomplished by making a Bayesian decision,
formula_59
formula_60
formula_61
where formula_62 is the background model. This ratio is compared to a threshold formula_63 to determine object presence/absence.
The likelihoods are factored as follows:
formula_64
formula_65
Appearance.
Each part formula_66 has an appearance modeled by a Gaussian density in the appearance space, with mean and covariance parameters formula_67, independent of other parts' densities. The background model has parameters formula_68. Fergus et al. assume that, given detected features, the position and appearance of those features are independent. Thus, formula_69. The ratio of the appearance terms reduces to
formula_70
formula_71
Recall from Weber et al. that formula_7 is the hypothesis for the indices of foreground parts, and formula_12 is the binary vector giving the occlusion state of each part in the hypothesis.
Shape.
Shape is represented by a joint Gaussian density of part locations within a particular hypothesis, after those parts have been transformed into a scale-invariant space. This transformation precludes the need to perform an exhaustive search over scale. The Gaussian density has parameters formula_72. The background model formula_73 is assumed to be a uniform distribution over the image, which has area formula_74. Letting formula_30 be the number of foreground parts,
formula_75
Relative scale.
The scale of each part formula_66 relative to a reference frame is modeled by a Gaussian density with parameters formula_76. Each part is assumed to be independent of other parts. The background model formula_73 assumes a uniform distribution over scale, within a range formula_77.
formula_78
formula_79
Occlusion and statistics of feature detection.
The first factor models the number of features detected using a Poisson distribution, which has mean M. The second factor serves as a "book-keeping" factor for the hypothesis variable. The last factor is a probability table for all possible occlusion patterns.
Learning.
The task of learning the model parameters formula_80 is accomplished by expectation maximization. This is carried out in a spirit similar to that of Weber et al. Details and formulas for the E-step and M-step can be seen in the literature.
Performance.
The Constellation Model as conceived by Fergus et al. achieves successful categorization rates consistently above 90% on large datasets of motorbikes, faces, airplanes, and spotted cats. For each of these datasets, the Constellation Model is able to capture the "essence" of the object class in terms of appearance and/or shape. For example, face and motorbike datasets generate very tight shape models because objects in those categories have very well-defined structure, whereas spotted cats vary significantly in pose, but have a very distinctive spotted appearance. Thus, the model succeeds in both cases. It is important to note that the Constellation Model does not generally account for significant changes in orientation. Thus, if the model is trained on images of horizontal airplanes, it will not perform well on, for instance, images of vertically oriented planes unless the model is extended to account for this sort of rotation explicitly.
In terms of computational complexity, the Constellation Model is very expensive. If formula_81 is the number of feature detections in the image, and formula_82 the number of parts in the object model, then the hypothesis space formula_83 is formula_84. Because the computation of sufficient statistics in the E-step of expectation maximization necessitates evaluating the likelihood for every hypothesis, learning becomes a major bottleneck operation. For this reason, only values of formula_85 have been used in practical applications, and the number of feature detections formula_81 is usually kept within the range of about 20-30 per image.
Variations.
One variation that attempts to reduce complexity is the star model proposed by Fergus et al. The reduced dependencies of this model allows for learning in formula_86 time instead of formula_84. This allows for a greater number of model parts and image features to be used in training. Because the star model has fewer parameters, it is also better at avoiding the problem of over-fitting when trained on fewer images.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\n\nX^o =\n\\begin{pmatrix}\nx_{11},x_{12},{\\cdots} ,x_{1N_1} \\\\\nx_{21},x_{22},{\\cdots} ,x_{2N_2} \\\\\n\\vdots \\\\\nx_{T1},x_{T2},{\\cdots} ,x_{TN_T}\n\\end{pmatrix}\n\n"
},
{
"math_id": 1,
"text": "N_i\\,"
},
{
"math_id": 2,
"text": "i \\in \\{1,\\dots,T\\}"
},
{
"math_id": 3,
"text": "x^m\\,"
},
{
"math_id": 4,
"text": "F\\,"
},
{
"math_id": 5,
"text": "F = T\\,"
},
{
"math_id": 6,
"text": "F > T\\,"
},
{
"math_id": 7,
"text": "h\\,"
},
{
"math_id": 8,
"text": "h_i = j\\,"
},
{
"math_id": 9,
"text": "x_{ij}\\,"
},
{
"math_id": 10,
"text": "X^o\\,"
},
{
"math_id": 11,
"text": "p(X^o,x^m,h)\\,"
},
{
"math_id": 12,
"text": "b\\,"
},
{
"math_id": 13,
"text": "n\\,"
},
{
"math_id": 14,
"text": "b_i = 1\\,"
},
{
"math_id": 15,
"text": "h_i > 0\\,"
},
{
"math_id": 16,
"text": "b_i = 0\\,"
},
{
"math_id": 17,
"text": "n_i\\,"
},
{
"math_id": 18,
"text": "i^{th}"
},
{
"math_id": 19,
"text": "p(X^o,x^m,h) = p(X^o,x^m,h,n,b)\\,"
},
{
"math_id": 20,
"text": "\np(X^o,x^m,h,n,b) = p(X^o,x^m|h,n,b)p(h|n,b)p(n)p(b)\\,\n"
},
{
"math_id": 21,
"text": "\np(n) = \\prod_{i=1}^T \\frac{1}{n_i!}(M_i)^{n_i}e^{-M_i}\n"
},
{
"math_id": 22,
"text": "M_i\\,"
},
{
"math_id": 23,
"text": "i\\,"
},
{
"math_id": 24,
"text": "p(b)\\,"
},
{
"math_id": 25,
"text": "2^F\\,"
},
{
"math_id": 26,
"text": "p(h|n,b)\\,"
},
{
"math_id": 27,
"text": "\np(h|n,b) =\n\n\\begin{cases}\n\\frac{1}{ \\textstyle \\prod_{f=1}^F N_f^{b_f}}, & \\mbox{if } h \\in H(b,n) \\\\\n0, & \\mbox{for other } h\n\\end{cases}\n\n"
},
{
"math_id": 28,
"text": "H(b,n)\\,"
},
{
"math_id": 29,
"text": "N_f\\,"
},
{
"math_id": 30,
"text": "f\\,"
},
{
"math_id": 31,
"text": "\\textstyle \\prod_{f=1}^F N_f^{b_f}"
},
{
"math_id": 32,
"text": "\np(X^o,x^m|h,n) = p_{fg}(z)p_{bg}(x_{bg})\\,\n"
},
{
"math_id": 33,
"text": "z = (x^ox^m)\\,"
},
{
"math_id": 34,
"text": "x_{bg}\\,"
},
{
"math_id": 35,
"text": "p_{fg}(z)\\,"
},
{
"math_id": 36,
"text": "\\mu\\,"
},
{
"math_id": 37,
"text": "\\Sigma\\,"
},
{
"math_id": 38,
"text": "C_1\\,"
},
{
"math_id": 39,
"text": "C_0\\,"
},
{
"math_id": 40,
"text": "\n\\frac{p(C_1|X^o)}{p(C_0|X^o)} \\propto \\frac{\\sum_h p(X^o,h|C_1)}{p(X^o,h_0|C_0)}\n"
},
{
"math_id": 41,
"text": "h_0\\,"
},
{
"math_id": 42,
"text": "\n\\Theta = \\{\\mu, \\Sigma, p(b), M\\}\\,\n"
},
{
"math_id": 43,
"text": "M\\,"
},
{
"math_id": 44,
"text": "\nL(X^o|\\Theta) = \\sum_{i=1}^I \\log \\sum_{h_i} \\int p(X_i^o,x_i^m,h_i|\\Theta)dx_i^m\n"
},
{
"math_id": 45,
"text": "\\Theta\\,"
},
{
"math_id": 46,
"text": "\nQ(\\tilde{\\Theta}|\\Theta) = \\sum_{i=1}^I E[\\log p(X_i^o,x_i^m,h_i|\\tilde{\\Theta})]\n"
},
{
"math_id": 47,
"text": "\n\\tilde{\\mu} = \\frac{1}{I} \\sum_{i=1}^I E[z_i]\n"
},
{
"math_id": 48,
"text": "\n\\tilde{\\Sigma} = \\frac{1}{I} \\sum_{i=1}^I E[z_iz_i^T] - \\tilde{\\mu}\\tilde{\\mu}^T\n"
},
{
"math_id": 49,
"text": "\n\\tilde{p}(\\bar{b}) = \\frac{1}{I} \\sum_{i=1}^I E[\\delta_{b,\\bar{b}}]\n"
},
{
"math_id": 50,
"text": "\n\\tilde{M} = \\frac{1}{I} \\sum_{i=1}^I E[n_i]\n"
},
{
"math_id": 51,
"text": "E[z]\\,"
},
{
"math_id": 52,
"text": "E[zz^T]\\,"
},
{
"math_id": 53,
"text": "E[\\delta_{b,\\bar{b}}]\\,"
},
{
"math_id": 54,
"text": "E[n]\\,"
},
{
"math_id": 55,
"text": "\np(h_i,x_i^m|X_i^o,\\Theta) = \\frac{p(h_i,x_i^m,X_i^o|\\Theta)}{\\textstyle \\sum_{h_i \\in H_b} \\int p(h_i,x_i^m,X_i^o|\\Theta) dx_i^m}\n"
},
{
"math_id": 56,
"text": "X\\,"
},
{
"math_id": 57,
"text": "S\\,"
},
{
"math_id": 58,
"text": "A\\,"
},
{
"math_id": 59,
"text": "\nR = \\frac{p(\\mbox{Object}|X,S,A)}{p(\\mbox{No object}|X,S,A)}\n"
},
{
"math_id": 60,
"text": "\n= \\frac{p(X,S,A|\\mbox{Object})p(\\mbox{Object})}{p(X,S,A|\\mbox{No object})p(\\mbox{No object})}\n"
},
{
"math_id": 61,
"text": "\n\\approx \\frac{p(X,S,A|\\Theta)p(\\mbox{Object})}{p(X,S,A|\\Theta_{bg})p(\\mbox{No object})}\n"
},
{
"math_id": 62,
"text": "\\Theta_{bg}"
},
{
"math_id": 63,
"text": "T\\,"
},
{
"math_id": 64,
"text": "\np(X,S,A|\\Theta) = \\sum_{h \\in H} p(X,S,A,h|\\Theta) =\n"
},
{
"math_id": 65,
"text": "\n\\sum_{h \\in H} \\underbrace{ p(A|X,S,h,\\Theta) }_{\\mbox{Appearance}} \\underbrace{ p(X|S,h,\\Theta) }_{\\mbox{Shape}} \\underbrace{ p(S|h,\\Theta) }_{\\mbox{Rel. Scale}} \\underbrace{ p(h|\\Theta) }_{\\mbox{Other}}\n"
},
{
"math_id": 66,
"text": "p\\,"
},
{
"math_id": 67,
"text": "\\Theta_p^{app} = \\{c_p,V_p\\}"
},
{
"math_id": 68,
"text": "\\Theta_{bg}^{app} = \\{c_{bg},V_{bg}\\}"
},
{
"math_id": 69,
"text": "p(A|X,S,h,\\Theta) = p(A|h,\\Theta)\\,"
},
{
"math_id": 70,
"text": "\n\\frac{p(A|X,S,h,\\Theta)}{p(A|X,S,h,\\Theta_{bg})} = \\frac{p(A|h,\\Theta)}{p(A|h,\\Theta_{bg})}\n"
},
{
"math_id": 71,
"text": "\n= \\prod_{p=1}^P \\left ( \\frac{G(A(h_p)|c_p,V_p)}{G(A(h_p)|c_{bg},V_{bg})} \\right )^{b_p}\n"
},
{
"math_id": 72,
"text": "\\Theta^{\\mbox{shape}} = \\{\\mu,\\Sigma\\}\\,"
},
{
"math_id": 73,
"text": "\\Theta_{bg}\\,"
},
{
"math_id": 74,
"text": "\\alpha\\,"
},
{
"math_id": 75,
"text": "\n\\frac{p(X|S,h,\\Theta)}{p(X|S,h,\\Theta_{bg})} = G(X(h)|\\mu,\\Sigma)\\alpha^f\n"
},
{
"math_id": 76,
"text": "\\Theta^{\\mbox{scale}} = \\{t_p,U_p\\}\\,"
},
{
"math_id": 77,
"text": "r\\,"
},
{
"math_id": 78,
"text": "\n\\frac{p(S|h,\\Theta)}{p(S|h,\\Theta_{bg})} = \\prod_{p=1}^P G(S(h_p)|t_p,U_p)^{d_p} r^f\n"
},
{
"math_id": 79,
"text": "\n\\frac{p(h|\\Theta)}{p(h|\\Theta_{bg})} = \\frac{p_{\\mbox{Poiss}}(n|M)}{p_{\\mbox{Poiss}}(N|M)} \\frac{1}{^nC_r(N,f)} p(b|\\Theta)\n"
},
{
"math_id": 80,
"text": "\\Theta = \\{\\mu,\\Sigma,c,V,M,p(b|\\Theta),t,U\\}\\,"
},
{
"math_id": 81,
"text": "N\\,"
},
{
"math_id": 82,
"text": "P\\,"
},
{
"math_id": 83,
"text": "H\\,"
},
{
"math_id": 84,
"text": "O(N^P)\\,"
},
{
"math_id": 85,
"text": "P \\le 6"
},
{
"math_id": 86,
"text": "O(N^2P)\\,"
}
] |
https://en.wikipedia.org/wiki?curid=15261618
|
15261672
|
One-shot learning (computer vision)
|
Object categorization problem
One-shot learning is an object categorization problem, found mostly in computer vision. Whereas most machine learning-based object categorization algorithms require training on hundreds or thousands of examples, one-shot learning aims to classify objects from one, or only a few, examples. The term few-shot learning is also used for these problems, especially when more than one example is needed.
<templatestyles src="Template:TOC limit/styles.css" />
Motivation.
The ability to learn object categories from few examples, and at a rapid pace, has been demonstrated in humans. It is estimated that a child learns almost all of the 10 ~ 30 thousand object categories in the world by age six. This is due not only to the human mind's computational power, but also to its ability to synthesize and learn new object categories from existing information about different, previously learned categories. Given two examples from two object categories: one, an unknown object composed of familiar shapes, the second, an unknown, amorphous shape; it is much easier for humans to recognize the former than the latter, suggesting that humans make use of previously learned categories when learning new ones. The key motivation for solving one-shot learning is that systems, like humans, can use knowledge about object categories to classify new objects.
Background.
As with most classification schemes, one-shot learning involves three main challenges:
One-shot learning differs from single object recognition and standard category recognition algorithms in its emphasis on knowledge transfer, which makes use of previously learned categories.
Theory.
The Bayesian one-shot learning algorithm represents the foreground and background of images as parametrized by a mixture of constellation models. During the learning phase, the parameters of these models are learned using a conjugate density parameter posterior and Variational Bayesian Expectation–Maximization (VBEM). In this stage the previously learned object categories inform the choice of model parameters via transfer by contextual information. For object recognition on new images, the posterior obtained during the learning phase is used in a Bayesian decision framework to estimate the ratio of "p(object | test, train)" to "p(background clutter | test, train)" where "p" is the probability of the outcome.
Bayesian framework.
Given the task of finding a particular object in a query image, the overall objective of the Bayesian one-shot learning algorithm is to compare the probability that object is present vs the probability that only background clutter is present. If the former probability is higher, the algorithm reports the object's presence, otherwise the algorithm reports its absence. To compute these probabilities, the object class must be modeled from a set of (1 ~ 5) training images containing examples.
To formalize these ideas, let formula_0 be the query image, which contains either an example of the foreground category formula_1 or only background clutter of a generic background category formula_2. Also let formula_3 be the set of training images used as the foreground category. The decision of whether formula_4 contains an object from the foreground category, or only clutter from the background category is:
formula_5
where the class posteriors formula_6 and formula_7 have been expanded by Bayes' Theorem, yielding a ratio of likelihoods and a ratio of object category priors. We decide that the image formula_8 contains an object from the foreground class if formula_9 exceeds a certain threshold formula_10. We next introduce parametric models for the foreground and background categories with parameters formula_11 and formula_12 respectively. This foreground parametric model is learned during the learning stage from formula_3, as well as prior information of learned categories. The background model we assume to be uniform across images. Omitting the constant ratio of category priors, formula_13, and parametrizing over formula_11 and formula_12 yields
formula_14, having simplified formula_15 and formula_16 to formula_17 and formula_18
The posterior distribution of model parameters given the training images, formula_19 is estimated in the learning phase. In this estimation, one-shot learning differs sharply from more traditional Bayesian estimation models that approximate the integral as formula_20. Instead, it uses a variational approach using prior information from previously learned categories. However, the traditional maximum likelihood estimation of the model parameters is used for the background model and the categories learned in advance through training.
Object category model.
For each query image formula_8 and training images formula_21, a constellation model is used for representation. To obtain this model for a given image formula_8, first a set of N interesting regions is detected in the image using the Kadir–Brady saliency detector. Each region selected is represented by a location in the image, formula_22 and a description of its appearance, formula_23. Letting formula_24 and formula_25and formula_26 the analogous representations for training images, the expression for R becomes:
formula_27
The likelihoods formula_28 and formula_29 are represented as mixtures of constellation models. A typical constellation model has P(3 ~ 7) parts, with N(~100) interest regions. Thus a P-dimensional vector h assigns one region of interest (out of N regions) to each model part (for P parts). Thus h denotes a "hypothesis" (an assignment of interest regions to model parts) for the model and a full constellation model is represented by summing over all possible hypotheses h in the hypothesis space formula_30. Finally the likelihood is written
formula_31
The different formula_32's represent different configurations of parts, whereas the different hypotheses h represent different assignations of regions to parts, given a part model formula_32. The assumption that the shape of the model (as represented by formula_33, the collection of part locations) and appearance are independent allows one to consider the likelihood expression formula_34 as two separate likelihoods of appearance and shape.
Appearance.
The appearance of each feature is represented by a point in appearance space (discussed below in implementation). "Each part formula_35 in the constellation model has a Gaussian density within this space with mean and precision parameters formula_36." From these the appearance likelihood described above is computed as a product of Gaussians over the model parts for a give hypothesis h and mixture component formula_37.
Shape.
The shape of the model for a given mixture component formula_37 and hypothesis h is represented as a joint Gaussian density of the locations of features. These features are transformed into a scale and translation-invariant space before modelling the relative location of the parts by a 2(P - 1)-dimensional Gaussian. From this, we obtain the shape likelihood, completing our representation of formula_38 . In order to reduce the number of hypotheses in the hypothesis space formula_39, only those hypotheses that satisfy the ordering constraint that the x-coordinate of each part is monotonically increasing are considered. This eliminates formula_40 hypotheses from formula_41.
Conjugate densities.
In order to compute formula_9, the integral formula_42 must be evaluated, but is analytically intractable. The object category model above gives information about formula_43, so what remains is to examine formula_44, the posterior of formula_45, and find a sufficient approximation to render the integral tractable. Previous work approximates the posterior by a formula_46function centered at formula_47, collapsing the integral in question into formula_48. This formula_49 is normally estimated using a Maximum Likelihood (formula_50) or Maximum A Posteriori (formula_51) procedure. However, because in one-shot learning, few training examples are used, the distribution will not be well-peaked, as is assumed in a formula_46function approximation. Thus instead of this traditional approximation, the Bayesian one-shot learning algorithm seeks to "find a parametric form of formula_52 such that the learning of formula_53 is feasible". The algorithm employs a Normal-Wishart distribution as the conjugate prior of formula_53, and in the learning phase, variational Bayesian methods with the same computational complexity as maximum likelihood methods are used to learn the hyperparameters of the distribution. Then, since formula_54 is a product of Gaussians, as chosen in the object category model, the integral reduces to a multivariate Student's T distribution, which can be evaluated.
Implementation.
Feature detection and representation.
To detect features in an image so that it can be represented by a constellation model, the Kadir–Brady saliency detector is used on grey-scale images, finding salient regions of the image. These regions are then clustered, yielding a number of features (the clusters) and the shape parameter formula_55, composed of the cluster centers. The Kadir–Brady detector was chosen because it produces fewer, more salient regions, as opposed to feature detectors like multiscale Harris, which produces numerous, less significant regions.
The regions are then taken from the image and rescaled to a small patch of 11 × 11 pixels, allowing each patch to be represented in 121-dimensional space. This dimensionality is reduced using principal component analysis, and formula_56, the appearance parameter, is then formed from the first 10 principal components of each patch.
Learning.
To obtain shape and appearance priors, three categories (spotted cats, faces, and airplanes) are learned using maximum likelihood estimation. These object category model parameters are then used to estimate the hyper-parameters of the desired priors.
Given a set of training examples, the algorithm runs the feature detector on these images, and determines model parameters from the salient regions. The hypothesis index h assigning features to parts prevents a closed-form solution of the linear model, so the posterior formula_53 is estimated by variational Bayesian expectation–maximization algorithm, which is run until parameter convergence after ~ 100 iterations. Learning a category in this fashion takes under a minute on a 2.8 GHz machine with a 4-part model and < 10 training images.
Experimental results.
Motorbike example.
To learn the motorbike category:
Shared densities on transforms.
Another algorithm uses knowledge transfer by model parameters to learn a new object category that is similar in appearance to previously learned categories. An image is represented as either a texture and shape, or as a latent image that has been transformed, denoted by formula_61.
A Siamese neural network works in tandem on two different input vectors to compute comparable output vectors.
Congealing.
In this context, congealing is "the simultaneous vectorization of each of a set of images to each other". For a set of training images of a certain category, congealing iteratively transforms each image to minimize the images' joint pixelwise entropies E, where
formula_62
"where formula_63 is the binary random variable defined by the values of a particular pixel p across all of the images, formula_64 is the discrete entropy function of that variable, and formula_65 is the set of pixel indices for the image."
The congealing algorithm begins with a set of images formula_66 and a corresponding transform matrix formula_67, which at the end of the algorithm will represent the transformation of formula_66 into its latent formula_68. These latents formula_68 minimize the joint pixel-wise entropies. Thus the task of the congealing algorithm is to estimate the transformations formula_69.
Sketch of algorithm:
At the end of the algorithm, formula_75, and formula_76 transforms the latent image back into the originally observed image.
Classification.
To use this model for classification, it must be estimated with the maximum posterior probability given an observed image formula_8. Applying Bayes' rule to formula_77 and parametrization by the transformation formula_10 gives a difficult integral that must be approximated, and then the best transform formula_10 (that which maps the test image to its latent image) must be found. Once this transformation is found, the test image can be transformed into its latent, and a nearest neighbor classifier based on Hausdorff distance between images can classify the latent (and thus the test image) as belonging to a particular class formula_78.
To find formula_10, the test image I is inserted into the training ensemble for the congealing process. Since the test image is drawn from one of the categories formula_78, congealing provides a corresponding formula_79 that maps I to its latent. The latent can then be classified.
Single-example classification.
Given a set of transformations formula_80 obtained from congealing many images of a certain category, the classifier can be extended to the case where only one training formula_81 example of a new category formula_82 is allowed. Applying all the transformations formula_80 sequentially to formula_81 creates an artificial training set for formula_82. This artificial data set can be made larger by borrowing transformations from many already known categories. Once this data set is obtained, formula_8, a test instance of formula_82, can be classified as in the normal classification procedure. The key assumption is that categories are similar enough that the transforms from one can be applied to another.
Citations.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " I"
},
{
"math_id": 1,
"text": " O_{fg} "
},
{
"math_id": 2,
"text": " O_{bg} "
},
{
"math_id": 3,
"text": " I_t "
},
{
"math_id": 4,
"text": " I "
},
{
"math_id": 5,
"text": " R = \\frac{p(O_{fg}|I,I_t)}{p(O_{bg}|I, I_t)} = \\frac{p(I|I_t, O_{fg})p(O_{fg})}{p(I|I_t, O_{bg})p(O_{bg})}, "
},
{
"math_id": 6,
"text": " p(O_{fg} |I, I_t) "
},
{
"math_id": 7,
"text": "p(O_{bg}|I, I_t) "
},
{
"math_id": 8,
"text": "I"
},
{
"math_id": 9,
"text": "R"
},
{
"math_id": 10,
"text": "T"
},
{
"math_id": 11,
"text": " \\theta "
},
{
"math_id": 12,
"text": " \\theta_{bg} "
},
{
"math_id": 13,
"text": " \\frac{p(O_{fg})}{p(O_{bg})} "
},
{
"math_id": 14,
"text": " R \\propto \\frac{\\int{p(I | \\theta, O_{fg}) p(\\theta | I_t, O_{fg})} d\\theta}{\\int{p(I | \\theta_{bg}, O_{bg}) p(\\theta_{bg} | I_t, O_{bg})} d\\theta_{bg}} = \\frac{\\int{p(I | \\theta) p(\\theta | I_t, O_{fg})} d\\theta}{\\int{p(I | \\theta_{bg}) p(\\theta_{bg} | I_t, O_{bg})} d\\theta_{bg}}"
},
{
"math_id": 15,
"text": "p(I | \\theta, O_{fg}) "
},
{
"math_id": 16,
"text": "p(I | \\theta, O_{bg})"
},
{
"math_id": 17,
"text": "p(I | \\theta_{fg}) "
},
{
"math_id": 18,
"text": "p(I | \\theta_{bg})."
},
{
"math_id": 19,
"text": " p(\\theta | I_t, O_{fg}) "
},
{
"math_id": 20,
"text": " \\delta(\\theta^{ML}) "
},
{
"math_id": 21,
"text": " I_t"
},
{
"math_id": 22,
"text": "X_i"
},
{
"math_id": 23,
"text": " A_i "
},
{
"math_id": 24,
"text": " X = \\sum_{i = 1}^N X_i, A = \\sum_{i = 1}^N A_i "
},
{
"math_id": 25,
"text": "X_t "
},
{
"math_id": 26,
"text": "A_t"
},
{
"math_id": 27,
"text": " R \\propto \\frac{\\int{p(X,A | \\theta, O_{fg}) p(\\theta | X_t, A_t, O_{fg})} d\\theta}{\\int{p(X,A | \\theta_{bg}, O_{bg}) p(\\theta_{bg} | X_t, A_t, O_{bg})} d\\theta_{bg}} = \\frac{\\int{p(X,A | \\theta) p(\\theta | X_t, A_t, O_{fg})} d\\theta}{\\int{p(X,A | \\theta_{bg}) p(\\theta_{bg} | X_t, A_t, O_{bg})} \\,d\\theta_{bg}}"
},
{
"math_id": 28,
"text": " p(X, A|\\theta) "
},
{
"math_id": 29,
"text": " p(X, A|\\theta_{bg}) "
},
{
"math_id": 30,
"text": "H"
},
{
"math_id": 31,
"text": "p(X,A|\\theta) = \\sum_{\\omega=1}^{\\Omega} \\sum_{\\textbf{h} \\in H} p(X,A,\\textbf{h}, \\omega | \\theta). "
},
{
"math_id": 32,
"text": "\\omega"
},
{
"math_id": 33,
"text": "X"
},
{
"math_id": 34,
"text": "p(X,A,\\textbf{h}, \\omega | \\theta) "
},
{
"math_id": 35,
"text": " p "
},
{
"math_id": 36,
"text": " \\theta_{p,\\omega}^{A} = { \\mu_{p,\\omega}^{A}, \\Gamma_{p,\\omega}^{A} } "
},
{
"math_id": 37,
"text": " \\omega "
},
{
"math_id": 38,
"text": "p(X,A, \\textbf{h}, \\omega | \\theta)"
},
{
"math_id": 39,
"text": " H"
},
{
"math_id": 40,
"text": "P! "
},
{
"math_id": 41,
"text": " H "
},
{
"math_id": 42,
"text": " \\int{p(X,A | \\theta) p(\\theta | X_t, A_t, O_{fg})} d\\theta "
},
{
"math_id": 43,
"text": "p(X,A | \\theta)"
},
{
"math_id": 44,
"text": " p(\\theta| X_t, A_t, O)"
},
{
"math_id": 45,
"text": "\\theta"
},
{
"math_id": 46,
"text": "\\delta"
},
{
"math_id": 47,
"text": "\\theta^{*}"
},
{
"math_id": 48,
"text": "p(X, A|\\theta^{*}) "
},
{
"math_id": 49,
"text": " \\theta^{*} "
},
{
"math_id": 50,
"text": " \\theta^{*} = \\theta^{ML}"
},
{
"math_id": 51,
"text": " \\theta^{*} = \\theta^{MAP} "
},
{
"math_id": 52,
"text": "p(\\theta)"
},
{
"math_id": 53,
"text": " p(\\theta| X_t, A_t, O_{fg})"
},
{
"math_id": 54,
"text": "p(X,A|\\theta)"
},
{
"math_id": 55,
"text": " X "
},
{
"math_id": 56,
"text": " A "
},
{
"math_id": 57,
"text": " X_t "
},
{
"math_id": 58,
"text": " A_t "
},
{
"math_id": 59,
"text": " \\theta_t "
},
{
"math_id": 60,
"text": " \\Omega = 1 "
},
{
"math_id": 61,
"text": " I = T(I_L) "
},
{
"math_id": 62,
"text": " E = \\sum_{p=1}^{P}H(\\nu(p)), "
},
{
"math_id": 63,
"text": "\\nu(p)"
},
{
"math_id": 64,
"text": "H( )"
},
{
"math_id": 65,
"text": " 1\\leq p \\leq P "
},
{
"math_id": 66,
"text": " I_i"
},
{
"math_id": 67,
"text": " U_i "
},
{
"math_id": 68,
"text": " I_{L_i} "
},
{
"math_id": 69,
"text": "U_i"
},
{
"math_id": 70,
"text": "U_I"
},
{
"math_id": 71,
"text": "I_i"
},
{
"math_id": 72,
"text": "A"
},
{
"math_id": 73,
"text": "AU_i"
},
{
"math_id": 74,
"text": "U_i = AU_i"
},
{
"math_id": 75,
"text": "U_i(I) = I_{L_i}"
},
{
"math_id": 76,
"text": "T = U_i^{-1}"
},
{
"math_id": 77,
"text": "P(c_j|I)"
},
{
"math_id": 78,
"text": "c_j"
},
{
"math_id": 79,
"text": "T_{\\text{test}} = U_{\\text{test}}^{-1}"
},
{
"math_id": 80,
"text": "B_i"
},
{
"math_id": 81,
"text": "I_t"
},
{
"math_id": 82,
"text": "c"
}
] |
https://en.wikipedia.org/wiki?curid=15261672
|
15261743
|
Object categorization from image search
|
In computer vision, the problem of object categorization from image search is the problem of training a classifier to recognize categories of objects, using only the images retrieved automatically with an Internet search engine. Ideally, automatic image collection would allow classifiers to be trained with nothing but the category names as input. This problem is closely related to that of content-based image retrieval (CBIR), where the goal is to return better image search results rather than training a classifier for image recognition.
Traditionally, classifiers are trained using sets of images that are labeled by hand. Collecting such a set of images is often a very time-consuming and laborious process. The use of Internet search engines to automate the process of acquiring large sets of labeled images has been described as a potential way of greatly facilitating computer vision research.
Challenges.
Unrelated images.
One problem with using Internet image search results as a training set for a classifier is the high percentage of unrelated images within the results. It has been estimated that, when a search engine such as Google images is queried with the name of an object category (such as "airplane"), up to 85% of the returned images are unrelated to the category.
Intra-class variability.
Another challenge posed by using Internet image search results as training sets for classifiers is that there is a high amount of variability within object categories, when compared with categories found in hand-labeled datasets such as Caltech 101 and Pascal. Images of objects can vary widely in a number of important factors, such as scale, pose, lighting, number of objects, and amount of occlusion.
pLSA approach.
In a 2005 paper by Fergus et al., pLSA (probabilistic latent semantic analysis) and extensions of this model were applied to the problem of object categorization from image search. pLSA was originally developed for document classification, but has since been applied to computer vision. It makes the assumption that images are documents that fit the bag of words model.
Model.
Just as text documents are made up of words, each of which may be repeated within the document and across documents, images can be modeled as combinations of "visual words". Just as the entire set of text words are defined by a dictionary, the entire set of visual words is defined in a "codeword dictionary".
pLSA divides documents into "topics" as well. Just as knowing the topic(s) of an article allows you to make good guesses about the kinds of words that will appear in it, the distribution of words in an image is dependent on the underlying topics. The pLSA model tells us the probability of seeing each word formula_0 given the category formula_1 in terms of topics formula_2:
formula_3
An important assumption made in this model is that formula_4 and formula_1 are conditionally independent given formula_2. Given a topic, the probability of a certain word appearing as part of that topic is independent of the rest of the image.
Training this model involves finding formula_5 and formula_6 that maximizes the likelihood of the observed words in each document. To do this, the expectation maximization algorithm is used, with the following objective function:
formula_7
Application.
ABS-pLSA.
Absolute position pLSA (ABS-pLSA) attaches location information to each visual word by localizing it to one of X 揵ins?in the image. Here, formula_8 represents which of the bins the visual word falls into. The new equation is:
formula_9
formula_10 and formula_11 can be solved for in a manner similar to the original pLSA problem, using the EM algorithm
A problem with this model is that it is not translation or scale invariant. Since the positions of the visual words are absolute, changing the size of the object in the image or moving it would have a significant impact on the spatial distribution of the visual words into different bins.
TSI-pLSA.
Translation and scale invariant pLSA (TSI-pLSA). This model extends pLSA by adding another latent variable, which describes the spatial location of the target object in an image. Now, the position formula_8 of a visual word is given relative to this object location, rather than as an absolute position in the image. The new equation is:
formula_12
Again, the parameters formula_13 and formula_11 can be solved using the EM algorithm. formula_14 can be assumed to be a uniform distribution.
Implementation.
Selecting words.
Words in an image were selected using 4 different feature detectors:
Using these 4 detectors, approximately 700 features were detected per image. These features were then encoded as Scale-invariant feature transform descriptors, and vector quantized to match one of 350 words contained in a codebook. The codebook was precomputed from features extracted from a large number of images spanning numerous object categories.
Possible object locations.
One important question in the TSI-pLSA model is how to determine the values that the random variable formula_15 can take on. It is a 4-vector, whose components describe the object抯 centroid as well as x and y scales that define a bounding box around the object, so the space of possible values it can take on is enormous. To limit the number of possible object locations to a reasonable number, normal pLSA is first carried out on the set of images, and for each topic a Gaussian mixture model is fit over the visual words, weighted by formula_5. Up to formula_16 Gaussians are tried (allowing for multiple instances of an object in a single image), where formula_16 is a constant.
Performance.
The authors of the Fergus et al. paper compared performance of the three pLSA algorithms (pLSA, ABS-pLSA, and TSI-pLSA) on handpicked datasets and images returned from Google searches. Performance was measured as the error rate when classifying images in a test set as either containing the image or containing only background.
As expected, training directly on Google data gives higher error rates than training on prepared data.? In about half of the object categories tested do ABS-pLSA and TSI-pLSA perform significantly better than regular pLSA, and in only 2 categories out of 7 does TSI-pLSA perform better than the other two models.
OPTIMOL.
OPTIMOL (automatic Online Picture collection via Incremental MOdel Learning) approaches the problem of learning object categories from online image searches by addressing model learning and searching simultaneously. OPTIMOL is an iterative model that updates its model of the target object category while concurrently retrieving more relevant images.
General framework.
OPTIMOL was presented as a general iterative framework that is independent of the specific model used for category learning. The algorithm is as follows:
Note that only the most recently added images are used in each round of learning. This allows the algorithm to run on an arbitrarily large number of input images.
Model.
The two categories (target object and background) are modeled as Hierarchical Dirichlet processes (HDPs). As in the pLSA approach, it is assumed that the images can be described with the bag of words model. HDP models the distributions of an unspecified number of topics across images in a category, and across categories. The distribution of topics among images in a single category is modeled as a Dirichlet process (a type of non-parametric probability distribution). To allow the sharing of topics across classes, each of these Dirichlet processes is modeled as a sample from another 損arent?Dirichlet process. HDP was first described by Teh et al. in 2005.
Implementation.
Initialization.
The dataset must be initialized, or seeded with an original batch of images which serve as good exemplars of the object category to be learned. These can be gathered automatically, using the first page or so of images returned by the search engine (which tend to be better than the subsequent images). Alternatively, the initial images can be gathered by hand.
Model learning.
To learn the various parameters of the HDP in an incremental manner, Gibbs sampling is used over the latent variables. It is carried out after each new set of images is incorporated into the dataset. Gibbs sampling involves repeatedly sampling from a set of random variables in order to approximate their distributions. Sampling involves generating a value for the random variable in question, based on the state of the other random variables on which it is dependent. Given sufficient samples, a reasonable approximation of the value can be achieved.
Classification.
At each iteration, formula_17 and formula_18 can be obtained from model learned after the previous round of Gibbs sampling, where formula_2 is a topic, formula_19 is a category, and formula_8 is a single visual word. The likelihood of an image being in a certain class, then, is:
formula_20
This is computed for each new candidate image per iteration. The image is classified as belonging to the category with the highest likelihood.
Addition to the dataset and "cache set".
In order to qualify for incorporation into the dataset, however, an image must satisfy a stronger condition:
formula_21
Where formula_22 and formula_23 are foreground (object) and background categories, respectively, and the ratio of constants describes the risk of accepting false positives and false negatives. They are adjusted automatically at every iteration, with the cost of a false positive set higher than that of a false negative. This ensures that a better dataset is collected.
Once an image is accepted by meeting the above criterion and incorporated into the dataset, however, it needs to meet another criterion before it is incorporated into the 揷ache set敆the set of images to be used for training. This set is intended to be a diverse subset of the set of accepted images. If the model were trained on all accepted images, it might become more and more highly specialized, only accepting images very similar to previous ones.
Performance.
Performance of the OPTIMOL method is defined by three factors:
Object categorization in content-based image retrieval.
Typically, image searches only make use of text associated with images. The problem of content-based image retrieval is that of improving search results by taking into account visual information contained in the images themselves. Several CBIR methods make use of classifiers trained on image search results, to refine the search. In other words, object categorization from image search is one component of the system. OPTIMOL, for example, uses a classifier trained on images collected during previous iterations to select additional images for the returned dataset.
Examples of CBIR methods that model object categories from image search are:
|
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{
"math_id": 0,
"text": "w"
},
{
"math_id": 1,
"text": "\\displaystyle d"
},
{
"math_id": 2,
"text": "\\displaystyle z"
},
{
"math_id": 3,
"text": "\\displaystyle P(w|d) = \\sum_{z=1}^Z P(w|z)P(z|d)"
},
{
"math_id": 4,
"text": "\\displaystyle w"
},
{
"math_id": 5,
"text": "\\displaystyle P(w|z)"
},
{
"math_id": 6,
"text": "\\displaystyle P(z|d)"
},
{
"math_id": 7,
"text": "\\displaystyle L = \\prod_{d=1}^D \\prod_{w=1}^W P(w|d)^{n(w|d)}"
},
{
"math_id": 8,
"text": "\\displaystyle x"
},
{
"math_id": 9,
"text": "\\displaystyle P(w|d) = \\sum_{z=1}^Z P(w,x|z)P(z|d)"
},
{
"math_id": 10,
"text": "\\displaystyle P(w,x|z)"
},
{
"math_id": 11,
"text": "\\displaystyle P(d)"
},
{
"math_id": 12,
"text": "\\displaystyle P(w,x|d) = \\sum_{z=1}^Z \\sum_{c=1}^C P(w,x|c,z)P(c)P(z|d)"
},
{
"math_id": 13,
"text": "\\displaystyle P(w,x|c,z)"
},
{
"math_id": 14,
"text": "\\displaystyle P(c)"
},
{
"math_id": 15,
"text": "\\displaystyle C"
},
{
"math_id": 16,
"text": "\\displaystyle K"
},
{
"math_id": 17,
"text": "\\displaystyle P(z|c)"
},
{
"math_id": 18,
"text": "\\displaystyle P(x|z,c)"
},
{
"math_id": 19,
"text": "\\displaystyle c"
},
{
"math_id": 20,
"text": "\\displaystyle P(I|c) = \\prod_i \\sum_j P(x_i|z_j,c)P(z_j|c)"
},
{
"math_id": 21,
"text": "\\displaystyle \\frac{P(I|c_f)}{P(I|c_b)} > \\frac{\\lambda_{Ac_b} - \\lambda_{Rc_b}}{\\lambda_{Rc_f} - \\lambda_{Ac_f}}\\frac{P(c_b)}{P(c_f)} "
},
{
"math_id": 22,
"text": "\\displaystyle c_f"
},
{
"math_id": 23,
"text": "\\displaystyle c_b"
}
] |
https://en.wikipedia.org/wiki?curid=15261743
|
15262012
|
Finger tree
|
Purely functional data structure
In computer science, a finger tree is a purely functional data structure that can be used to efficiently implement other functional data structures. A finger tree gives amortized constant time access to the "fingers" (leaves) of the tree, which is where data is stored, and concatenation and splitting logarithmic time in the size of the smaller piece. It also stores in each internal node the result of applying some associative operation to its descendants. This "summary" data stored in the internal nodes can be used to provide the functionality of data structures other than trees.
Overview.
Ralf Hinze and Ross Paterson state a finger tree is a functional representation of persistent sequences that can access the ends in amortized constant time. Concatenation and splitting can be done in logarithmic time in the size of the smaller piece. The structure can also be made into a general purpose data structure by defining the split operation in a general form, allowing it to act as a sequence, priority queue, search tree, or priority search queue, among other varieties of abstract data types.
A "finger" is a point where one can access "part" of a data structure; in imperative languages, this is called a pointer. In a finger tree, the fingers are structures that point to the ends of a sequence, or the leaf nodes. The fingers are added on to the original tree to allow for constant time access to fingers. In the images shown below, the fingers are the lines reaching out of the spine to the nodes.
A finger tree is composed of different "layers" which can be identified by the nodes along its "spine". The spine of a tree can be thought of as the trunk in the same way trees have leaves and a root. Though finger trees are often shown with the spine and branches coming off the sides, there are actually two nodes on the spine at each level that have been paired to make this central spine. The "prefix" is on the left of the spine, while the "suffix" is on the right. Each of those nodes has a link to the next level of the spine until they reach the root.
The first level of the tree contains only values, the leaf nodes of the tree, and is of depth 0. The second level is of depth 1. The third is of depth 2 and so on. The closer to the root, the deeper the subtrees of the original tree (the tree before it was a finger tree) the nodes points to. In this way, working down the tree is going from the leaves to the root of the tree, which is the opposite of the typical tree data structure. To get this nice and unusual structure, we have to make sure the original tree has a uniform depth. To ensure that the depth is uniform, when declaring the node object, it must be parameterized by the type of the child. The nodes on the spine of depth 1 and above point to trees, and with this parameterization they can be represented by the nested nodes.
Transforming a tree into a finger tree.
We will start this process with a balanced 2–3 tree. For the finger tree to work, all the leaf nodes need to also be level.
A finger is "a structure providing efficient access to nodes of a tree near a distinguished location." To make a finger tree we need to put fingers to the right and left ends of the tree and transform it like a zipper. This gives us that constant amortized time access to the ends of a sequence.
To transform, start with the balanced 2–3 tree.
Take the leftmost and rightmost internal nodes of the tree and pull them up so the rest of the tree dangles between them as shown in the image to the right.
Combines the spines to make a standard 2–3 finger tree.
This can be described as:
data FingerTree a
= Empty
| Single a
| Deep (Digit a) (FingerTree (Node a)) (Digit a)
data Node a
= Node2 a a
| Node3 a a a
The digits in the examples shown are the nodes with letters. Each list is divided by the prefix or suffix of each node on the spine. In a transformed 2–3 tree it seems that the digit lists at the top level can have a length of two or three with the lower levels only having length of one or two. In order for some application of finger trees to run so efficiently, finger trees allow between one and four subtrees on each level.
The digits of the finger tree can be transformed into a list like so:
type Digit a = One a | Two a a | Three a a a | Four a a a a
And so on the image, the top level has elements of type "a", the next has elements of type "Node a" because the node in between the spine and leaves, and this would go on meaning in general that the "n"th level of the tree has elements of type formula_0 "a", or 2–3 trees of depth n. This means a sequence of "n" elements is represented by a tree of depth Θ(log "n"). Even better, an element "d" places from the nearest end is stored at a depth of Θ(log d) in the tree.
Reductions.
formula_1
formula_2
Deque operations.
Finger trees also make efficient deques. Whether the structure is persistent or not, all operations take Θ(1) amortized time. The analysis can be compared to Okasaki's implicit deques, the only difference being that the FingerTree type stores Nodes instead of pairs.
Application.
Finger trees can be used to build other trees. For example, a priority queue can be implemented by labeling the internal nodes by the minimum priority of its children in the tree, or an indexed list/array can be implemented with a labeling of nodes by the count of the leaves in their children. Other applications are random-access sequences, described below, ordered sequences, and interval trees.
Finger trees can provide amortized O(1) pushing, reversing, popping, O(log n) append and split; and can be adapted to be indexed or ordered sequences. And like all functional data structures, it is inherently persistent; that is, older versions of the tree are always preserved.
Random-access sequences.
Finger trees can efficiently implement random-access sequences. This should support fast positional operations including accessing the "n"th element and splitting a sequence at a certain position. To do this, we annotate the finger tree with sizes.
deriving (Eq, Ord)
instance Monoid Size where
∅ = Size 0
Size m ⊕ Size n = Size (m + n)
The "N" is for natural numbers. The new type is needed because the type is the carrier of different monoids. Another new type is still needed for the elements in the sequence, shown below.
newtype Seq a = Seq (FingerTree Size (Elem a))
instance Measured (Elem a) Size where
||Elem|| = Size 1
These lines of code show that instance works a base case for measuring the sizes and the elements are of size one. The use of newtypes doesn't cause a run-time penalty in Haskell because in a library, the "Size" and "Elem" types would be hidden from the user with wrapper functions.
With these changes, the length of a sequence can now be computed in constant time.
First publication.
Finger trees were first published in 1977 by Leonidas J. Guibas, and periodically refined since (e.g. a version using AVL trees, non-lazy finger trees, simpler 2–3 finger trees shown here, B-Trees and so on)
Implementations.
Finger trees have since been used in the Haskell core libraries (in the implementation of "Data.Sequence"), and an implementation in OCaml exists which was derived from a proven-correct Coq implementation. There is also a verified implementation in Isabelle (proof assistant) from which programs in Haskell and other (functional) languages can be generated. Finger trees can be implemented with or without lazy evaluation, but laziness allows for simpler implementations.
References.
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{
"math_id": 1,
"text": "\n\\begin{align} \n\\mathrm{instance} & \\ Reduce \\ Node \\ \\mathrm{where}& & \\\\\n&reducer\\ (\\prec)\\ (Node2 \\ a \\ b)\\ z &=&\\ a\\ \\prec(b\\ \\prec\\ z) \\\\\n&reducer\\ (\\prec)\\ (Node3 \\ a \\ b \\ c)\\ z &=&\\ a\\ \\prec(\\ b\\prec(c\\ \\prec\\ z )) \\\\\n\\\\\n&reducel\\ (\\succ)\\ z\\ (Node2 \\ b \\ a)\\ &=&\\ (z\\ \\succ\\ b)\\ \\succ\\ a \\\\\n&reducel\\ (\\succ)\\ z\\ (Node3 \\ c \\ b \\ a)\\ &=&\\ ((z\\ \\succ\\ c) \\succ\\ b) \\succ\\ a \\\\\n\\end{align}\n"
},
{
"math_id": 2,
"text": "\n\\begin{align}\n\\mathrm{instance}& \\ Reduce \\ FingerTree \\ \\mathrm{where} && \\\\\n&reducer\\ (\\prec)\\ (Empty)\\ z\\ &=&\\ z \\\\\n&reducer\\ (\\prec)\\ (Single \\ x)\\ z &=&\\ x\\ \\prec\\ z \\\\\n&reducer\\ (\\prec)\\ (Deep \\ pr \\ m \\ sf)\\ z &=&\\ pr\\ \\prec'\\ (m\\ \\prec''\\ (sf\\ \\prec'\\ z)) \\\\\n& \\ \\ \\ \\ where \\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ (\\prec')\\ = reducer\\ (\\prec) \\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ (\\prec'')\\ = reducer\\ (reducer\\ (\\prec)) \\\\\n\\\\\n& reducel\\ (\\succ)\\ z\\ (Empty)\\ &=&\\ z \\\\\n& reducel\\ (\\succ)\\ z\\ (Single\\ \\ x)\\ &=&\\ z\\ \\succ\\ x \\\\\n& reducel\\ (\\succ)\\ z\\ (Deep\\ \\ pr \\ m \\ sf)\\ &=&\\ ((z\\ \\succ'\\ pr)\\ \\succ''\\ m)\\ \\succ'\\ sf) \\\\\n& \\ \\ \\ \\ where \\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ (\\succ')\\ = reducel\\ (\\succ) \\\\\n& \\ \\ \\ \\ \\ \\ \\ \\ (\\succ'')\\ = reducel\\ (reducel\\ (\\succ))\\\\\n\n\\end{align}\n"
}
] |
https://en.wikipedia.org/wiki?curid=15262012
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MS
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MS, ms, Ms, M.S., etc. may refer to:
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See also.
Topics referred to by the same term
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This page lists associated with the title .
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[
{
"math_id": 0,
"text": "m_s"
}
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https://en.wikipedia.org/wiki?curid=152635
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Loop quantum gravity
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Theory of quantum gravity, merging quantum mechanics and general relativity
Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Albert Einstein's geometric formulation rather than the treatment of gravity as a mysterious mechanism (force). As a theory, LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network. These networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale on the order of a Planck length, approximately 10−35 meters, and smaller scales are meaningless. Consequently, not just matter, but space itself, prefers an atomic structure.
The areas of research, which involve about 30 research groups worldwide, share the basic physical assumptions and the mathematical description of quantum space. Research has evolved in two directions: the more traditional canonical loop quantum gravity, and the newer covariant loop quantum gravity, called spin foam theory. The most well-developed theory that has been advanced as a direct result of loop quantum gravity is called loop quantum cosmology (LQC). LQC advances the study of the early universe, incorporating the concept of the Big Bang into the broader theory of the Big Bounce, which envisions the Big Bang as the beginning of a period of expansion, that follows a period of contraction, which has been described as the Big Crunch.
History.
In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics, specifically Yang–Mills theory. Shortly after, Ted Jacobson and Lee Smolin realized that the formal equation of quantum gravity, called the Wheeler–DeWitt equation, admitted solutions labelled by loops when rewritten in the new Ashtekar variables. Carlo Rovelli and Smolin defined a nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions. Jorge Pullin and Jerzy Lewandowski understood that the intersections of the loops are essential for the consistency of the theory, and the theory should be formulated in terms of intersecting loops, or graphs.
In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum. That is, geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, which are graphs labelled by spins.
The canonical version of the dynamics was established by Thomas Thiemann, who defined an anomaly-free Hamiltonian operator and showed the existence of a mathematically consistent background-independent theory. The covariant, or "spin foam", version of the dynamics was developed jointly over several decades by research groups in France, Canada, UK, Poland, and Germany. It was completed in 2008, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, which is consistent with observed acceleration in the expansion of the Universe.
Background independence.
LQG is formally background independent, meaning the equations of LQG are not embedded in, or dependent on, space and time (except for its invariant topology). Instead, they are expected to give rise to space and time at distances which are 10 times the Planck length. The issue of background independence in LQG still has some unresolved subtleties. For example, some derivations require a fixed choice of the topology, while any consistent quantum theory of gravity should include topology change as a dynamical process.
Spacetime as a "container" over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the world. This is known as the relationalist interpretation of spacetime. In LQG this aspect of general relativity is taken seriously and this symmetry is preserved by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffeomorphisms. However, the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called "problem of time" in general relativity. A generally accepted calculational framework to account for this constraint has yet to be found. A plausible candidate for the quantum Hamiltonian constraint is the operator introduced by Thiemann.
Constraints and their Poisson bracket algebra.
Dirac observables.
The constraints define a constraint surface in the original phase space. The gauge motions of the constraints apply to all phase space but have the feature that they leave the constraint surface where it is, and thus the orbit of a point in the hypersurface under gauge transformations will be an orbit entirely within it. Dirac observables are defined as phase space functions, formula_0, that Poisson commute with all the constraints when the constraint equations are imposed,
formula_1
that is, they are quantities defined on the constraint surface that are invariant under the gauge transformations of the theory.
Then, solving only the constraint formula_2 and determining the Dirac observables with respect to it leads us back to the Arnowitt–Deser–Misner (ADM) phase space with constraints formula_3. The dynamics of general relativity is generated by the constraints, it can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations.
Quantization of the constraints – the equations of quantum general relativity.
Pre-history and Ashtekar new variables.
Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to those of gauge theories. The first step consists of using densitized triads formula_4 (a triad formula_5 is simply three orthogonal vector fields labeled by formula_6 and the densitized triad is defined by formula_7) to encode information about the spatial metric,
formula_8
(where formula_9 is the flat space metric, and the above equation expresses that formula_10, when written in terms of the basis formula_5, is locally flat). (Formulating general relativity with triads instead of metrics was not new.) The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices formula_11. The canonically conjugate variable is related to the extrinsic curvature by formula_12. But problems similar to using the metric formulation arise when one tries to quantize the theory. Ashtekar's new insight was to introduce a new configuration variable,
formula_13
that behaves as a complex formula_14 connection where formula_15 is related to the so-called spin connection via formula_16. Here formula_17 is called the chiral spin connection. It defines a covariant derivative formula_18. It turns out that formula_19 is the conjugate momentum of formula_17, and together these form Ashtekar's new variables.
The expressions for the constraints in Ashtekar variables; Gauss's theorem, the spatial diffeomorphism constraint and the (densitized) Hamiltonian constraint then read:
formula_20
formula_21
formula_22
respectively, where formula_23 is the field strength tensor of the connection formula_17 and where formula_24 is referred to as the vector constraint. The above-mentioned local in space rotational invariance is the original of the formula_14 gauge invariance here expressed by Gauss's theorem. Note that these constraints are polynomial in the fundamental variables, unlike the constraints in the metric formulation. This dramatic simplification seemed to open up the way to quantizing the constraints. (See the article Self-dual Palatini action for a derivation of Ashtekar's formalism).
With Ashtekar's new variables, given the configuration variable formula_25, it is natural to consider wavefunctions formula_26. This is the connection representation. It is analogous to ordinary quantum mechanics with configuration variable formula_27 and wavefunctions formula_28. The configuration variable gets promoted to a quantum operator via:
formula_29
(analogous to formula_30) and the triads are (functional) derivatives,
formula_31
(analogous to formula_32). In passing over to the quantum theory the constraints become operators on a kinematic Hilbert space (the unconstrained formula_14 Yang–Mills Hilbert space). Note that different ordering of the formula_33's and formula_34's when replacing the formula_34's with derivatives give rise to different operators – the choice made is called the factor ordering and should be chosen via physical reasoning. Formally they read
formula_35
formula_36
formula_37
There are still problems in properly defining all these equations and solving them. For example, the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with formula_38. There were serious difficulties in promoting this quantity to a quantum operator. Moreover, although Ashtekar variables had the virtue of simplifying the Hamiltonian, they are complex. When one quantizes the theory, it is difficult to ensure that one recovers real general relativity as opposed to complex general relativity.
Quantum constraints as the equations of quantum general relativity.
The classical result of the Poisson bracket of the smeared Gauss' law formula_39 with the connections is
formula_40
The quantum Gauss' law reads
formula_41
If one smears the quantum Gauss' law and study its action on the quantum state one finds that the action of the constraint on the quantum state is equivalent to shifting the argument of formula_42 by an infinitesimal (in the sense of the parameter formula_43 small) gauge transformation,
formula_44
and the last identity comes from the fact that the constraint annihilates the state. So the constraint, as a quantum operator, is imposing the same symmetry that its vanishing imposed classically: it is telling us that the functions formula_45 have to be gauge invariant functions of the connection. The same idea is true for the other constraints.
Therefore, the two step process in the classical theory of solving the constraints formula_46 (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the 'evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions formula_42 of the quantum equations formula_47. This is because it obviously solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because formula_48 is the quantum generator of gauge transformations (gauge invariant functions are constant along the gauge orbits and thus characterize them). Recall that, at the classical level, solving the admissibility conditions and evolution equations was equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in canonical quantum gravity.
Introduction of the loop representation.
It was in particular the inability to have good control over the space of solutions to Gauss's law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider the loop representation in gauge theories and quantum gravity.
LQG includes the concept of a holonomy. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after parallel transport around a closed loop; it is denoted
formula_49.
Knowledge of the holonomies is equivalent to knowledge of the connection, up to gauge equivalence. Holonomies can also be associated with an edge; under a Gauss Law these transform as
formula_50
For a closed loop formula_51 and assuming formula_52, yields
formula_53
or
formula_54
The trace of an holonomy around a closed loop is written
formula_55
and is called a Wilson loop. Thus Wilson loops are gauge invariant. The explicit form of the Holonomy is
formula_56
where formula_57 is the curve along which the holonomy is evaluated, and formula_58 is a parameter along the curve, formula_59 denotes path ordering meaning factors for smaller values of formula_58 appear to the left, and formula_60 are matrices that satisfy the formula_14 algebra
formula_61
The Pauli matrices satisfy the above relation. It turns out that there are infinitely many more examples of sets of matrices that satisfy these relations, where each set comprises formula_62 matrices with formula_63, and where none of these can be thought to 'decompose' into two or more examples of lower dimension. They are called different irreducible representations of the formula_14 algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer formula_64 according to the irreducible representation used.
The use of Wilson loops explicitly solves the Gauss gauge constraint. Loop representation is required to handle the spatial diffeomorphism constraint. With Wilson loops as a basis, any Gauss gauge invariant function expands as,
formula_65
This is called the loop transform and is analogous to the momentum representation in quantum mechanics (see Position and momentum space). The QM representation has a basis of states formula_66 labelled by a number formula_67 and expands as
formula_68
and works with the coefficients of the expansion formula_69
The inverse loop transform is defined by
formula_70
This defines the loop representation. Given an operator formula_71 in the connection representation,
formula_72
one should define the corresponding operator formula_73 on formula_74 in the loop representation via,
formula_75
where formula_76 is defined by the usual inverse loop transform,
formula_77
A transformation formula giving the action of the operator formula_73 on formula_74 in terms of the action of the operator formula_71 on formula_45 is then obtained by equating the R.H.S. of formula_78 with the R.H.S. of formula_79 with formula_80 substituted into formula_79, namely
formula_81
or
formula_82
where formula_83 means the operator formula_71 but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). The action of this operator on the Wilson loop is evaluated as a calculation in the connection representation and the result is rearranged purely as a manipulation in terms of loops (with regard to the action on the Wilson loop, the chosen transformed operator is the one with the opposite factor ordering compared to the one used for its action on wavefunctions formula_45). This gives the physical meaning of the operator formula_73. For example, if formula_84 corresponded to a spatial diffeomorphism, then this can be thought of as keeping the connection field formula_33 of formula_55 where it is while performing a spatial diffeomorphism on formula_57 instead. Therefore, the meaning of formula_73 is a spatial diffeomorphism on formula_57, the argument of formula_74.
In the loop representation, the spatial diffeomorphism constraint is solved by considering functions of loops formula_74 that are invariant under spatial diffeomorphisms of the loop formula_57. That is, knot invariants are used. This opens up an unexpected connection between knot theory and quantum gravity.
Any collection of non-intersecting Wilson loops satisfy Ashtekar's quantum Hamiltonian constraint. Using a particular ordering of terms and replacing formula_19 by a derivative, the action of the quantum Hamiltonian constraint on a Wilson loop is
formula_85
When a derivative is taken it brings down the tangent vector, formula_86, of the loop, formula_57. So,
formula_87
However, as formula_23 is anti-symmetric in the indices formula_88 and formula_89 this vanishes (this assumes that formula_57 is not discontinuous anywhere and so the tangent vector is unique).
With regard to loop representation, the wavefunctions formula_74 vanish when the loop has discontinuities and are knot invariants. Such functions solve the Gauss law, the spatial diffeomorphism constraint and (formally) the Hamiltonian constraint. This yields an infinite set of exact (if only formal) solutions to all the equations of quantum general relativity! This generated a lot of interest in the approach and eventually led to LQG.
Geometric operators, the need for intersecting Wilson loops and spin network states.
The easiest geometric quantity is the area. Let us choose coordinates so that the surface formula_90 is characterized by formula_91. The area of small parallelogram of the surface formula_90 is the product of length of each side times formula_92 where formula_93 is the angle between the sides. Say one edge is given by the vector formula_94 and the other by formula_95 then,
formula_96
In the space spanned by formula_97 and formula_98 there is an infinitesimal parallelogram described by formula_99 and formula_100. Using formula_101 (where the indices formula_33 and formula_102 run from 1 to 2), yields the area of the surface formula_90 given by
formula_103
where formula_104 and is the determinant of the metric induced on formula_90. The latter can be rewritten formula_105 where the indices formula_106 go from 1 to 2. This can be further rewritten as
formula_107
The standard formula for an inverse matrix is
formula_108
There is a similarity between this and the expression for formula_109. But in Ashtekar variables, formula_110. Therefore,
formula_111
According to the rules of canonical quantization the triads formula_112 should be promoted to quantum operators,
formula_113
The area formula_114 can be promoted to a well defined quantum operator despite the fact that it contains a product of two functional derivatives and a square-root. Putting formula_115 (formula_116-th representation),
formula_117
This quantity is important in the final formula for the area spectrum. The result is
formula_118
where the sum is over all edges formula_119 of the Wilson loop that pierce the surface formula_90.
The formula for the volume of a region formula_120 is given by
formula_121
The quantization of the volume proceeds the same way as with the area. Each time the derivative is taken, it brings down the tangent vector formula_86, and when the volume operator acts on non-intersecting Wilson loops the result vanishes. Quantum states with non-zero volume must therefore involve intersections. Given that the anti-symmetric summation is taken over in the formula for the volume, it needs intersections with at least three non-coplanar lines. At least four-valent vertices are needed for the volume operator to be non-vanishing.
Assuming the real representation where the gauge group is formula_14, Wilson loops are an over complete basis as there are identities relating different Wilson loops. These occur because Wilson loops are based on matrices (the holonomy) and these matrices satisfy identities. Given any two formula_14 matrices formula_122 and formula_123,
formula_124
This implies that given two loops formula_57 and formula_125 that intersect,
formula_126
where by formula_127 we mean the loop formula_125 traversed in the opposite direction and formula_128 means the loop obtained by going around the loop formula_57 and then along formula_125. See figure below. Given that the matrices are unitary one has that formula_129. Also given the cyclic property of the matrix traces (i.e. formula_130) one has that formula_131. These identities can be combined with each other into further identities of increasing complexity adding more loops. These identities are the so-called Mandelstam identities. Spin networks certain are linear combinations of intersecting Wilson loops designed to address the over-completeness introduced by the Mandelstam identities (for trivalent intersections they eliminate the over-completeness entirely) and actually constitute a basis for all gauge invariant functions.
As mentioned above the holonomy tells one how to propagate test spin half particles. A spin network state assigns an amplitude to a set of spin half particles tracing out a path in space, merging and splitting. These are described by spin networks formula_57: the edges are labelled by spins together with 'intertwiners' at the vertices which are prescription for how to sum over different ways the spins are rerouted. The sum over rerouting are chosen as such to make the form of the intertwiner invariant under Gauss gauge transformations.
Hamiltonian constraint of LQG.
In the long history of canonical quantum gravity formulating the Hamiltonian constraint as a quantum operator (Wheeler–DeWitt equation) in a mathematically rigorous manner has been a formidable problem. It was in the loop representation that a mathematically well defined Hamiltonian constraint was finally formulated in 1996. We leave more details of its construction to the article Hamiltonian constraint of LQG. This together with the quantum versions of the Gauss law and spatial diffeomorphism constrains written in the loop representation are the central equations of LQG (modern canonical quantum General relativity).
Finding the states that are annihilated by these constraints (the physical states), and finding the corresponding physical inner product, and observables is the main goal of the technical side of LQG.
An important aspect of the Hamiltonian operator is that it only acts at vertices (a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning). More precisely, its action is non-zero on at least vertices of valence three and greater and results in a linear combination of new spin networks where the original graph has been modified by the addition of lines at each vertex together and a change in the labels of the adjacent links of the vertex.
Chiral fermions and the fermion doubling problem.
A significant challenge in theoretical physics lies in unifying LQG, a theory of quantum spacetime, with the Standard Model of particle physics, which describes fundamental forces and particles. A major obstacle in this endeavor is the fermion doubling problem, which arises when incorporating chiral fermions into the LQG framework.
Chiral fermions, such as electrons and quarks, are fundamental particles characterized by their "handedness" or chirality. This property dictates that a particle and its mirror image behave differently under weak interactions. This asymmetry is fundamental to the Standard Model's success in explaining numerous physical phenomena.
However, attempts to integrate chiral fermions into LQG often result in the appearance of spurious, mirror-image particles. Instead of a single left-handed fermion, for instance, the theory predicts the existence of both a left-handed and a right-handed version. This "doubling" contradicts the observed chirality of the Standard Model and disrupts its predictive power.
The fermion doubling problem poses a significant hurdle in constructing a consistent theory of quantum gravity. The Standard Model's accuracy in describing the universe at the smallest scales relies heavily on the unique properties of chiral fermions. Without a solution to this problem, incorporating matter and its interactions into a unified framework of quantum gravity remains a significant challenge.
Therefore, resolving the fermion doubling problem is crucial for advancing our understanding of the universe at its most fundamental level and developing a complete theory that unites gravity with the quantum world.
Spin foams.
In loop quantum gravity (LQG), a spin network represents a "quantum state" of the gravitational field on a 3-dimensional hypersurface. The set of all possible spin networks (or, more accurately, "s-knots" – that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG Hilbert space.
In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral (functional integration) description of quantum gravity. It is closely related to loop quantum gravity.
Spin foam derived from the Hamiltonian constraint operator.
On this section see and references therein. The Hamiltonian constraint generates 'time' evolution. Solving the Hamiltonian constraint should tell us how quantum states evolve in 'time' from an initial spin network state to a final spin network state. One approach to solving the Hamiltonian constraint starts with what is called the Dirac delta function. The summation of which over different sequences of actions can be visualized as a summation over different histories of 'interaction vertices' in the 'time' evolution sending the initial spin network to the final spin network. Each time a Hamiltonian operator acts it does so by adding a new edge at the vertex.
This then naturally gives rise to the two-complex (a combinatorial set of faces that join along edges, which in turn join on vertices) underlying the spin foam description; we evolve forward an initial spin network sweeping out a surface, the action of the Hamiltonian constraint operator is to produce a new planar surface starting at the vertex. We are able to use the action of the Hamiltonian constraint on the vertex of a spin network state to associate an amplitude to each "interaction" (in analogy to Feynman diagrams). See figure below. This opens a way of trying to directly link canonical LQG to a path integral description. Just as a spin networks describe quantum space, each configuration contributing to these path integrals, or sums over history, describe 'quantum spacetime'. Because of their resemblance to soap foams and the way they are labeled John Baez gave these 'quantum spacetimes' the name 'spin foams'.
There are however severe difficulties with this particular approach, for example the Hamiltonian operator is not self-adjoint, in fact it is not even a normal operator (i.e. the operator does not commute with its adjoint) and so the spectral theorem cannot be used to define the exponential in general. The most serious problem is that the formula_132's are not mutually commuting, it can then be shown the formal quantity formula_133 cannot even define a (generalized) projector. The master constraint (see below) does not suffer from these problems and as such offers a way of connecting the canonical theory to the path integral formulation.
Spin foams from BF theory.
It turns out there are alternative routes to formulating the path integral, however their connection to the Hamiltonian formalism is less clear. One way is to start with the BF theory. This is a simpler theory than general relativity, it has no local degrees of freedom and as such depends only on topological aspects of the fields. BF theory is what is known as a topological field theory. Surprisingly, it turns out that general relativity can be obtained from BF theory by imposing a constraint, BF theory involves a field formula_134 and if one chooses the field formula_102 to be the (anti-symmetric) product of two tetrads
formula_135
(tetrads are like triads but in four spacetime dimensions), one recovers general relativity. The condition that the formula_102 field be given by the product of two tetrads is called the simplicity constraint. The spin foam dynamics of the topological field theory is well understood. Given the spin foam 'interaction' amplitudes for this simple theory, one then tries to implement the simplicity conditions to obtain a path integral for general relativity. The non-trivial task of constructing a spin foam model is then reduced to the question of how this simplicity constraint should be imposed in the quantum theory. The first attempt at this was the famous Barrett–Crane model. However this model was shown to be problematic, for example there did not seem to be enough degrees of freedom to ensure the correct classical limit. It has been argued that the simplicity constraint was imposed too strongly at the quantum level and should only be imposed in the sense of expectation values just as with the Lorenz gauge condition formula_136 in the Gupta–Bleuler formalism of quantum electrodynamics. New models have now been put forward, sometimes motivated by imposing the simplicity conditions in a weaker sense.
Another difficulty here is that spin foams are defined on a discretization of spacetime. While this presents no problems for a topological field theory as it has no local degrees of freedom, it presents problems for GR. This is known as the problem triangularization dependence.
Modern formulation of spin foams.
Just as imposing the classical simplicity constraint recovers general relativity from BF theory, it is expected that an appropriate quantum simplicity constraint will recover quantum gravity from quantum BF theory.
Progress has been made with regard to this issue by Engle, Pereira, and Rovelli, Freidel and Krasnov and Livine and Speziale in defining spin foam interaction amplitudes with better behaviour.
An attempt to make contact between EPRL-FK spin foam and the canonical formulation of LQG has been made.
Spin foam derived from the master constraint operator.
See below.
The semiclassical limit and loop quantum gravity.
The Classical limit is the ability of a physical theory to approximate classical mechanics. It is used with physical theories that predict non-classical behavior. Any candidate theory of quantum gravity must be able to reproduce Einstein's theory of general relativity as a classical limit of a quantum theory. This is not guaranteed because of a feature of quantum field theories which is that they have different sectors, these are analogous to the different phases that come about in the thermodynamical limit of statistical systems. Just as different phases are physically different, so are different sectors of a quantum field theory. It may turn out that LQG belongs to an unphysical sector – one in which one does not recover general relativity in the semiclassical limit or there might not be any physical sector.
Moreover, the physical Hilbert space formula_137 must contain enough semiclassical states to guarantee that the quantum theory obtained can return to the classical theory when formula_138 avoiding quantum anomalies; otherwise there will be restrictions on the physical Hilbert space that have no counterpart in the classical theory, implying that the quantum theory has fewer degrees of freedom than the classical theory.
Theorems establishing the uniqueness of the loop representation as defined by Ashtekar et al. (i.e. a certain concrete realization of a Hilbert space and associated operators reproducing the correct loop algebra) have been given by two groups (Lewandowski, Okolow, Sahlmann and Thiemann; and Christian Fleischhack). Before this result was established it was not known whether there could be other examples of Hilbert spaces with operators invoking the same loop algebra – other realizations not equivalent to the one that had been used. These uniqueness theorems imply no others exist, so if LQG does not have the correct semiclassical limit then the theorems would mean the end of the loop representation of quantum gravity.
Difficulties and progress checking the semiclassical limit.
There are a number of difficulties in trying to establish LQG gives Einstein's theory of general relativity in the semiclassical limit:
Difficulties in trying to examine the semiclassical limit of the theory should not be confused with it having the wrong semiclassical limit.
Concerning issue number 2 above, consider so-called weave states. Ordinary measurements of geometric quantities are macroscopic, and Planckian discreteness is smoothed out. The fabric of a T-shirt is analogous: at a distance it is a smooth curved two-dimensional surface, but on closer inspection we see that it is actually composed of thousands of one-dimensional linked threads. The image of space given in LQG is similar. Consider a large spin network formed by a large number of nodes and links, each of Planck scale. Probed at a macroscopic scale, it appears as a three-dimensional continuous metric geometry.
To make contact with low energy physics it is mandatory to develop approximation schemes both for the physical inner product and for Dirac observables; the spin foam models that have been intensively studied can be viewed as avenues toward approximation schemes for said physical inner product.
Markopoulou, et al. adopted the idea of noiseless subsystems in an attempt to solve the problem of the low energy limit in background independent quantum gravity theories. The idea has led to the possibility of matter of the standard model being identified with emergent degrees of freedom from some versions of LQG (see section below: "LQG and related research programs").
As Wightman emphasized in the 1950s, in Minkowski QFTs the formula_141 point functions
formula_142
completely determine the theory. In particular, one can calculate the scattering amplitudes from these quantities. As explained below in the section on the "Background independent scattering amplitudes", in the background-independent context, the formula_141 point functions refer to a state and in gravity that state can naturally encode information about a specific geometry which can then appear in the expressions of these quantities. To leading order, LQG calculations have been shown to agree in an appropriate sense with the formula_141point functions calculated in the effective low energy quantum general relativity.
Improved dynamics and the master constraint.
The master constraint.
Thiemann's Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Hamiltonian constraint equations in terms of a single master constraint formula_143, which involves the square of the constraints in question. An initial objection to the use of the master constraint was that on first sight it did not seem to encode information about the observables; because the Master constraint is quadratic in the constraint, when one computes its Poisson bracket with any quantity, the result is proportional to the constraint, therefore it vanishes when the constraints are imposed and as such does not select out particular phase space functions. However, it was realized that the condition
formula_144
is where formula_0 is at least a twice differentiable function on phase space is equivalent to formula_0 being a weak Dirac observable with respect to the constraints in question. So the master constraint does capture information about the observables. Because of its significance this is known as the master equation.
That the master constraint Poisson algebra is an honest Lie algebra opens the possibility of using a method, known as group averaging, in order to construct solutions of the infinite number of Hamiltonian constraints, a physical inner product thereon and Dirac observables via what is known as refined algebraic quantization, or RAQ.
The quantum master constraint.
Define the quantum master constraint (regularisation issues aside) as
formula_145
Obviously,
formula_146
for all formula_147 implies formula_148. Conversely, if formula_148 then
formula_149
implies
formula_146.
First compute the matrix elements of the would-be operator formula_150, that is, the quadratic form formula_151. formula_151 is a graph changing, diffeomorphism invariant quadratic form that cannot exist on the kinematic Hilbert space formula_152, and must be defined on formula_153. Since the master constraint operator formula_150 is densely defined on formula_154, then formula_150 is a positive and symmetric operator in formula_154. Therefore, the quadratic form formula_151 associated with formula_150 is closable. The closure of formula_151 is the quadratic form of a unique self-adjoint operator formula_155, called the Friedrichs extension of formula_150. We relabel formula_155 as formula_150 for simplicity.
Note that the presence of an inner product, viz Eq 4, means there are no superfluous solutions i.e. there are no formula_42 such that
formula_156
but for which formula_148.
It is also possible to construct a quadratic form formula_157 for what is called the extended master constraint (discussed below) on formula_152 which also involves the weighted integral of the square of the spatial diffeomorphism constraint (this is possible because formula_157 is not graph changing).
The spectrum of the master constraint may not contain zero due to normal or factor ordering effects which are finite but similar in nature to the infinite vacuum energies of background-dependent quantum field theories. In this case it turns out to be physically correct to replace formula_150 with formula_158 provided that the "normal ordering constant" vanishes in the classical limit, that is,
formula_159
so that formula_160 is a valid quantisation of formula_143.
Testing the master constraint.
The constraints in their primitive form are rather singular, this was the reason for integrating them over test functions to obtain smeared constraints. However, it would appear that the equation for the master constraint, given above, is even more singular involving the product of two primitive constraints (although integrated over space). Squaring the constraint is dangerous as it could lead to worsened ultraviolet behaviour of the corresponding operator and hence the master constraint programme must be approached with care.
In doing so the master constraint programme has been satisfactorily tested in a number of model systems with non-trivial constraint algebras, free and interacting field theories. The master constraint for LQG was established as a genuine positive self-adjoint operator and the physical Hilbert space of LQG was shown to be non-empty, a consistency test LQG must pass to be a viable theory of quantum general relativity.
Applications of the master constraint.
The master constraint has been employed in attempts to approximate the physical inner product and define more rigorous path integrals.
The Consistent Discretizations approach to LQG, is an application of the master constraint program to construct the physical Hilbert space of the canonical theory.
Spin foam from the master constraint.
The master constraint is easily generalized to incorporate the other constraints. It is then referred to as the extended master constraint, denoted formula_161. We can define the extended master constraint which imposes both the Hamiltonian constraint and spatial diffeomorphism constraint as a single operator,
formula_162.
Setting this single constraint to zero is equivalent to formula_163 and formula_164 for all formula_147 in formula_90. This constraint implements the spatial diffeomorphism and Hamiltonian constraint at the same time on the Kinematic Hilbert space. The physical inner product is then defined as
formula_165
(as formula_166). A spin foam representation of this expression is obtained by splitting the formula_167-parameter in discrete steps and writing
formula_168
The spin foam description then follows from the application of formula_169 on a spin network resulting in a linear combination of new spin networks whose graph and labels have been modified. Obviously an approximation is made by truncating the value of formula_170 to some finite integer. An advantage of the extended master constraint is that we are working at the kinematic level and so far it is only here we have access semiclassical coherent states. Moreover, one can find none graph changing versions of this master constraint operator, which are the only type of operators appropriate for these coherent states.
Algebraic quantum gravity (AQG).
The master constraint programme has evolved into a fully combinatorial treatment of gravity known as algebraic quantum gravity (AQG). The non-graph changing master constraint operator is adapted in the framework of algebraic quantum gravity. While AQG is inspired by LQG, it differs drastically from it because in AQG there is fundamentally no topology or differential structure – it is background independent in a more generalized sense and could possibly have something to say about topology change. In this new formulation of quantum gravity AQG semiclassical states always control the fluctuations of all present degrees of freedom. This makes the AQG semiclassical analysis superior over that of LQG, and progress has been made in establishing it has the correct semiclassical limit and providing contact with familiar low energy physics.
Physical applications of LQG.
Black hole entropy.
Black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. The no hair conjecture of general relativity states that a black hole is characterized only by its mass, its charge, and its angular momentum; hence, it has no entropy. It appears, then, that one can violate the second law of thermodynamics by dropping an object with nonzero entropy into a black hole. Work by Stephen Hawking and Jacob Bekenstein showed that the second law of thermodynamics can be preserved by assigning to each black hole a "black-hole entropy"
formula_171
where formula_33 is the area of the hole's event horizon, formula_172 is the Boltzmann constant, and formula_173 is the Planck length. The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.
An oversight in the application of the no-hair theorem is the assumption that the relevant degrees of freedom accounting for the entropy of the black hole must be classical in nature; what if they were purely quantum mechanical instead and had non-zero entropy? This is what is realized in the LQG derivation of black hole entropy, and can be seen as a consequence of its background-independence – the classical black hole spacetime comes about from the semiclassical limit of the quantum state of the gravitational field, but there are many quantum states that have the same semiclassical limit. Specifically, in LQG it is possible to associate a quantum geometrical interpretation to the microstates: These are the quantum geometries of the horizon which are consistent with the area, formula_33, of the black hole and the topology of the horizon (i.e. spherical). LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. These calculations have been generalized to rotating black holes.
It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes.
A recent success of the theory in this direction is the computation of the entropy of all non singular black holes directly from theory and independent of Immirzi parameter. The result is the expected formula formula_174, where formula_175 is the entropy and formula_33 the area of the black hole, derived by Bekenstein and Hawking on heuristic grounds. This is the only known derivation of this formula from a fundamental theory, for the case of generic non singular black holes. Older attempts at this calculation had difficulties. The problem was that although Loop quantum gravity predicted that the entropy of a black hole is proportional to the area of the event horizon, the result depended on a crucial free parameter in the theory, the above-mentioned Immirzi parameter. However, there is no known computation of the Immirzi parameter, so it was fixed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy.
Hawking radiation in loop quantum gravity.
A detailed study of the quantum geometry of a black hole horizon has been made using loop quantum gravity. Loop-quantization does not reproduce the result for black hole entropy originally discovered by Bekenstein and Hawking, unless one chooses the value of the Immirzi parameter to cancel out another constant that arises in the derivation. However, it led to the computation of higher-order corrections to the entropy and radiation of black holes.
Based on the fluctuations of the horizon area, a quantum black hole exhibits deviations from the Hawking spectrum that would be observable were X-rays from Hawking radiation of evaporating primordial black holes to be observed. The quantum effects are centered at a set of discrete and unblended frequencies highly pronounced on top of Hawking radiation spectrum.
Planck star.
In 2014 Carlo Rovelli and Francesca Vidotto proposed that there is a Planck star inside every black hole. Based on LQG, the theory states that as stars are collapsing into black holes, the energy density reaches the Planck energy density, causing a repulsive force that creates a star. Furthermore, the existence of such a star would resolve the black hole firewall and black hole information paradox.
Loop quantum cosmology.
The popular and technical literature makes extensive references to the LQG-related topic of loop quantum cosmology. LQC was mainly developed by Martin Bojowald. It was popularized in "Scientific American" for predicting a Big Bounce prior to the Big Bang. Loop quantum cosmology (LQC) is a symmetry-reduced model of classical general relativity quantized using methods that mimic those of loop quantum gravity (LQG) that predicts a "quantum bridge" between contracting and expanding cosmological branches.
Achievements of LQC have been the resolution of the big bang singularity, the prediction of a Big Bounce, and a natural mechanism for inflation.
LQC models share features of LQG and so is a useful toy model. However, the results obtained are subject to the usual restriction that a truncated classical theory, then quantized, might not display the true behaviour of the full theory due to artificial suppression of degrees of freedom that might have large quantum fluctuations in the full theory. It has been argued that singularity avoidance in LQC are by mechanisms only available in these restrictive models and that singularity avoidance in the full theory can still be obtained but by a more subtle feature of LQG.
Loop quantum gravity phenomenology.
Quantum gravity effects are difficult to measure because the Planck length is so small. However recently physicists, such as Jack Palmer, have started to consider the possibility of measuring quantum gravity effects mostly from astrophysical observations and gravitational wave detectors. The energy of those fluctuations at scales this small cause space-perturbations which are visible at higher scales.
Background-independent scattering amplitudes.
Loop quantum gravity is formulated in a background-independent language. No spacetime is assumed a priori, but rather it is built up by the states of theory themselves – however scattering amplitudes are derived from formula_170-point functions (Correlation function) and these, formulated in conventional quantum field theory, are functions of points of a background spacetime. The relation between the background-independent formalism and the conventional formalism of quantum field theory on a given spacetime is not obvious, and it is not obvious how to recover low-energy quantities from the full background-independent theory. One would like to derive the formula_170-point functions of the theory from the background-independent formalism, in order to compare them with the standard perturbative expansion of quantum general relativity and therefore check that loop quantum gravity yields the correct low-energy limit.
A strategy for addressing this problem has been suggested; by studying the boundary amplitude, namely a path integral over a finite spacetime region, seen as a function of the boundary value of the field. In conventional quantum field theory, this boundary amplitude is well–defined and codes the physical information of the theory; it does so in quantum gravity as well, but in a fully background–independent manner. A generally covariant definition of formula_170-point functions can then be based on the idea that the distance between physical points – arguments of the formula_170-point function is determined by the state of the gravitational field on the boundary of the spacetime region considered.
Progress has been made in calculating background-independent scattering amplitudes this way with the use of spin foams. This is a way to extract physical information from the theory. Claims to have reproduced the correct behaviour for graviton scattering amplitudes and to have recovered classical gravity have been made. "We have calculated Newton's law starting from a world with no space and no time." – Carlo Rovelli.
Gravitons, string theory, supersymmetry, extra dimensions in LQG.
Some quantum theories of gravity posit a spin-2 quantum field that is quantized, giving rise to gravitons. In string theory, one generally starts with quantized excitations on top of a classically fixed background. This theory is thus described as background dependent. Particles like photons as well as changes in the spacetime geometry (gravitons) are both described as excitations on the string worldsheet. The background dependence of string theory can have physical consequences, such as determining the number of quark generations. In contrast, loop quantum gravity, like general relativity, is manifestly background independent, eliminating the background required in string theory. Loop quantum gravity, like string theory, also aims to overcome the nonrenormalizable divergences of quantum field theories.
LQG does not introduce a background and excitations living on such a background, so LQG does not use gravitons as building blocks. Instead one expects that one may recover a kind of semiclassical limit or weak field limit where something like "gravitons" will show up again. In contrast, gravitons play a key role in string theory where they are among the first (massless) level of excitations of a superstring.
LQG differs from string theory in that it is formulated in 3 and 4 dimensions and without supersymmetry or Kaluza–Klein extra dimensions, while the latter requires both to be true. There is no experimental evidence to date that confirms string theory's predictions of supersymmetry and Kaluza–Klein extra dimensions. In a 2003 paper "A Dialog on Quantum Gravity", Carlo Rovelli regards the fact LQG is formulated in 4 dimensions and without supersymmetry as a strength of the theory as it represents the most parsimonious explanation, consistent with current experimental results, over its rival string/M-theory. Proponents of string theory will often point to the fact that, among other things, it demonstrably reproduces the established theories of general relativity and quantum field theory in the appropriate limits, which loop quantum gravity has struggled to do. In that sense string theory's connection to established physics may be considered more reliable and less speculative, at the mathematical level. Loop quantum gravity has nothing to say about the matter (fermions) in the universe.
Since LQG has been formulated in 4 dimensions (with and without supersymmetry), and M-theory requires supersymmetry and 11 dimensions, a direct comparison between the two has not been possible. It is possible to extend mainstream LQG formalism to higher-dimensional supergravity, general relativity with supersymmetry and Kaluza–Klein extra dimensions should experimental evidence establish their existence. It would therefore be desirable to have higher-dimensional Supergravity loop quantizations at one's disposal in order to compare these approaches. A series of papers have been published attempting this. Most recently, Thiemann (and alumni) have made progress toward calculating black hole entropy for supergravity in higher dimensions. It will be useful to compare these results to the corresponding super string calculations.
LQG and related research programs.
Several research groups have attempted to combine LQG with other research programs: Johannes Aastrup, Jesper M. Grimstrup et al. research combines noncommutative geometry with canonical quantum gravity and Ashtekar variables, Laurent Freidel, Simone Speziale, et al., spinors and twistor theory with loop quantum gravity, and Lee Smolin et al. with Verlinde entropic gravity and loop gravity. Stephon Alexander, Antonino Marciano and Lee Smolin have attempted to explain the origins of weak force chirality in terms of Ashketar's variables, which describe gravity as chiral, and LQG with Yang–Mills theory fields in four dimensions. Sundance Bilson-Thompson, Hackett et al., has attempted to introduce the standard model via LQGs degrees of freedom as an emergent property (by employing the idea of noiseless subsystems, a notion introduced in a more general situation for constrained systems by Fotini Markopoulou-Kalamara et al.)
Furthermore, LQG has drawn philosophical comparisons with causal dynamical triangulation and asymptotically safe gravity, and the spinfoam with group field theory and AdS/CFT correspondence. Smolin and Wen have suggested combining LQG with string-net liquid, tensors, and Smolin and Fotini Markopoulou-Kalamara quantum graphity. There is the consistent discretizations approach. Also, Pullin and Gambini provide a framework to connect the path integral and canonical approaches to quantum gravity. They may help reconcile the spin foam and canonical loop representation approaches. Recent research by Chris Duston and Matilde Marcolli introduces topology change via topspin networks.
Problems and comparisons with alternative approaches.
Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result. The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail.
Many of these problems apply to LQG, including:
The theory of LQG is one possible solution to the problem of quantum gravity, as is string theory. There are substantial differences however. For example, string theory also addresses unification, the understanding of all known forces and particles as manifestations of a single entity, by postulating extra dimensions and so-far unobserved additional particles and symmetries. Contrary to this, LQG is based only on quantum theory and general relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction. On the other hand, the consequences of LQG are radical, because they fundamentally change the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.
Presently, no semiclassical limit recovering general relativity has been shown to exist. This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit (described by general relativity with possible quantum corrections). Specifically, the dynamics of the theory are encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian. Other technical problems include finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields of quantum field theory, fate of the renormalization of the graviton in perturbation theory that lead to ultraviolet divergence beyond 2-loops (see one-loop Feynman diagram in Feynman diagram).
While there has been a proposal relating to observation of naked singularities, and doubly special relativity as a part of a program called loop quantum cosmology, there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity). Because of the above-mentioned lack of a semiclassical limit, LQG has not yet even reproduced the predictions made by general relativity.
An alternative criticism is that general relativity may be an effective field theory, and therefore quantization ignores the fundamental degrees of freedom.
ESA's INTEGRAL satellite measured polarization of photons of different wavelengths and was able to place a limit in the granularity of space that is less than 10−48m or 13 orders of magnitude below the Planck scale.
See also.
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Notes.
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Citations.
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Works cited.
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|
[
{
"math_id": 0,
"text": "O"
},
{
"math_id": 1,
"text": "\\{ G_j , O \\}_{G_j=C_a=H = 0} = \\{ C_a , O \\}_{G_j=C_a=H = 0} = \\{ H , O \\}_{G_j=C_a=H = 0} = 0,"
},
{
"math_id": 2,
"text": "G_j = 0"
},
{
"math_id": 3,
"text": "H, C_a"
},
{
"math_id": 4,
"text": "\\tilde{E}_i^a"
},
{
"math_id": 5,
"text": "E_i^a"
},
{
"math_id": 6,
"text": "i = 1,2,3"
},
{
"math_id": 7,
"text": "\\tilde{E}_i^a = \\sqrt{\\det (q)} E_i^a"
},
{
"math_id": 8,
"text": "\\det(q) q^{ab} = \\tilde{E}_i^a \\tilde{E}_j^b \\delta^{ij}."
},
{
"math_id": 9,
"text": "\\delta^{ij}"
},
{
"math_id": 10,
"text": "q^{ab}"
},
{
"math_id": 11,
"text": "i"
},
{
"math_id": 12,
"text": "K_a^i = K_{ab} \\tilde{E}^{ai} / \\sqrt{\\det (q)}"
},
{
"math_id": 13,
"text": "A_a^i = \\Gamma_a^i - i K_a^i"
},
{
"math_id": 14,
"text": "\\operatorname{SU}(2)"
},
{
"math_id": 15,
"text": "\\Gamma_a^i"
},
{
"math_id": 16,
"text": "\\Gamma_a^i = \\Gamma_{ajk} \\epsilon^{jki}"
},
{
"math_id": 17,
"text": "A_a^i"
},
{
"math_id": 18,
"text": "\\mathcal{D}_a"
},
{
"math_id": 19,
"text": "\\tilde{E}^a_i"
},
{
"math_id": 20,
"text": "G^i = \\mathcal{D}_a \\tilde{E}_i^a = 0"
},
{
"math_id": 21,
"text": "C_a = \\tilde{E}_i^b F^i_{ab} - A_a^i (\\mathcal{D}_b \\tilde{E}_i^b) = V_a - A_a^i G^i = 0,"
},
{
"math_id": 22,
"text": "\\tilde{H} = \\epsilon_{ijk} \\tilde{E}_i^a \\tilde{E}_j^b F^k_{ab} = 0"
},
{
"math_id": 23,
"text": "F^i_{ab}"
},
{
"math_id": 24,
"text": "V_a"
},
{
"math_id": 25,
"text": "A^i_a"
},
{
"math_id": 26,
"text": "\\Psi (A^i_a)"
},
{
"math_id": 27,
"text": "q"
},
{
"math_id": 28,
"text": "\\psi (q)"
},
{
"math_id": 29,
"text": "\\hat{A}_a^i \\Psi (A) = A_a^i \\Psi (A),"
},
{
"math_id": 30,
"text": "\\hat{q} \\psi (q) = q \\psi (q)"
},
{
"math_id": 31,
"text": "\\hat{\\tilde{E_i^a}} \\Psi (A) = - i {\\delta \\Psi (A) \\over \\delta A_a^i}."
},
{
"math_id": 32,
"text": "\\hat{p} \\psi (q) = -i \\hbar d \\psi (q) / dq"
},
{
"math_id": 33,
"text": "A"
},
{
"math_id": 34,
"text": "\\tilde{E}"
},
{
"math_id": 35,
"text": "\\hat{G}_j \\vert\\psi \\rangle = 0"
},
{
"math_id": 36,
"text": "\\hat{C}_a \\vert\\psi \\rangle = 0"
},
{
"math_id": 37,
"text": "\\hat{\\tilde{H}} \\vert\\psi \\rangle = 0."
},
{
"math_id": 38,
"text": "\\tilde{H} = \\sqrt{\\det (q)} H"
},
{
"math_id": 39,
"text": "G(\\lambda) = \\int d^3x \\lambda^j (D_a E^a)^j"
},
{
"math_id": 40,
"text": "\\{ G(\\lambda), A_a^i \\} = \\partial_a \\lambda^i + g \\epsilon^{ijk} A_a^j \\lambda^k = (D_a \\lambda)^i."
},
{
"math_id": 41,
"text": "\\hat{G}_j \\Psi (A) = - i D_a {\\delta \\lambda \\Psi [A] \\over \\delta A_a^j} = 0."
},
{
"math_id": 42,
"text": "\\Psi"
},
{
"math_id": 43,
"text": "\\lambda"
},
{
"math_id": 44,
"text": "\\left [ 1 + \\int d^3x \\lambda^j (x) \\hat{G}_j \\right] \\Psi (A) = \\Psi [A + D \\lambda] = \\Psi [A],"
},
{
"math_id": 45,
"text": "\\Psi [A]"
},
{
"math_id": 46,
"text": "C_I = 0"
},
{
"math_id": 47,
"text": "\\hat{C}_I \\Psi = 0"
},
{
"math_id": 48,
"text": "\\hat{C}_I"
},
{
"math_id": 49,
"text": "h_\\gamma [A]"
},
{
"math_id": 50,
"text": "(h'_e)_{\\alpha \\beta} = U_{\\alpha \\gamma}^{-1} (x) (h_e)_{\\gamma \\sigma} U_{\\sigma \\beta} (y)."
},
{
"math_id": 51,
"text": "x = y"
},
{
"math_id": 52,
"text": "\\alpha = \\beta"
},
{
"math_id": 53,
"text": "(h'_e)_{\\alpha \\alpha} = U_{\\alpha \\gamma}^{-1} (x) (h_e)_{\\gamma \\sigma} U_{\\sigma \\alpha} (x) = [U_{\\sigma \\alpha} (x) U_{\\alpha \\gamma}^{-1} (x)] (h_e)_{\\gamma \\sigma} = \\delta_{\\sigma \\gamma} (h_e)_{\\gamma \\sigma} = (h_e)_{\\gamma \\gamma}"
},
{
"math_id": 54,
"text": "\\operatorname{Tr} h'_\\gamma = \\operatorname{Tr} h_\\gamma."
},
{
"math_id": 55,
"text": "W_\\gamma [A]"
},
{
"math_id": 56,
"text": "h_\\gamma [A] = \\mathcal{P} \\exp \\left \\{-\\int_{\\gamma_0}^{\\gamma_1} ds \\dot{\\gamma}^a A_a^i (\\gamma (s)) T_i \\right \\}"
},
{
"math_id": 57,
"text": "\\gamma"
},
{
"math_id": 58,
"text": "s"
},
{
"math_id": 59,
"text": "\\mathcal{P}"
},
{
"math_id": 60,
"text": "T_i"
},
{
"math_id": 61,
"text": "[T^i ,T^j] = 2i \\epsilon^{ijk} T_k."
},
{
"math_id": 62,
"text": "(N+1) \\times (N+1)"
},
{
"math_id": 63,
"text": "N = 1,2,3,\\dots"
},
{
"math_id": 64,
"text": "N/2"
},
{
"math_id": 65,
"text": "\\Psi [A] = \\sum_\\gamma \\Psi [\\gamma] W_\\gamma [A]."
},
{
"math_id": 66,
"text": "\\exp (ikx)"
},
{
"math_id": 67,
"text": "k"
},
{
"math_id": 68,
"text": "\\psi [x] = \\int dk \\psi (k) \\exp (ikx). "
},
{
"math_id": 69,
"text": "\\psi (k)."
},
{
"math_id": 70,
"text": "\\Psi [\\gamma] = \\int [dA] \\Psi [A] W_\\gamma [A]."
},
{
"math_id": 71,
"text": "\\hat{O}"
},
{
"math_id": 72,
"text": "\\Phi [A] = \\hat{O} \\Psi [A] \\qquad Eq \\; 1,"
},
{
"math_id": 73,
"text": "\\hat{O}'"
},
{
"math_id": 74,
"text": "\\Psi [\\gamma]"
},
{
"math_id": 75,
"text": "\\Phi [\\gamma] = \\hat{O}' \\Psi [\\gamma] \\qquad Eq \\; 2,"
},
{
"math_id": 76,
"text": "\\Phi [\\gamma]"
},
{
"math_id": 77,
"text": "\\Phi [\\gamma] = \\int [dA] \\Phi [A] W_\\gamma [A] \\qquad Eq \\; 3."
},
{
"math_id": 78,
"text": "Eq \\; 2"
},
{
"math_id": 79,
"text": "Eq \\; 3"
},
{
"math_id": 80,
"text": "Eq \\; 1"
},
{
"math_id": 81,
"text": "\\hat{O}' \\Psi [\\gamma] = \\int [dA] W_\\gamma [A] \\hat{O} \\Psi [A],"
},
{
"math_id": 82,
"text": "\\hat{O}' \\Psi [\\gamma] = \\int [dA] (\\hat{O}^\\dagger W_\\gamma [A]) \\Psi [A],"
},
{
"math_id": 83,
"text": "\\hat{O}^\\dagger"
},
{
"math_id": 84,
"text": " \\hat{O}^\\dagger"
},
{
"math_id": 85,
"text": "\\hat{\\tilde{H}}^\\dagger W_\\gamma [A] = - \\epsilon_{ijk} \\hat{F}^k_{ab} \\frac{\\delta}{\\delta A_a^i} \\frac{\\delta}{\\delta A_b^j} W_\\gamma [A]."
},
{
"math_id": 86,
"text": "\\dot{\\gamma}^a"
},
{
"math_id": 87,
"text": "\\hat{F}^i_{ab} \\dot{\\gamma}^a \\dot{\\gamma}^b."
},
{
"math_id": 88,
"text": "a"
},
{
"math_id": 89,
"text": "b"
},
{
"math_id": 90,
"text": "\\Sigma"
},
{
"math_id": 91,
"text": "x^3 = 0"
},
{
"math_id": 92,
"text": "\\sin \\theta"
},
{
"math_id": 93,
"text": "\\theta"
},
{
"math_id": 94,
"text": "\\vec{u}"
},
{
"math_id": 95,
"text": "\\vec{v}"
},
{
"math_id": 96,
"text": "A = \\| \\vec{u} \\| \\| \\vec{v} \\| \\sin \\theta = \\sqrt{\\| \\vec{u} \\|^2 \\| \\vec{v} \\|^2 (1 - \\cos^2 \\theta)} = \\sqrt{\\| \\vec{u} \\|^2 \\| \\vec{v} \\|^2 - (\\vec{u} \\cdot \\vec{v})^2}"
},
{
"math_id": 97,
"text": "x^1"
},
{
"math_id": 98,
"text": "x^2"
},
{
"math_id": 99,
"text": "\\vec{u} = \\vec{e}_1 dx^1"
},
{
"math_id": 100,
"text": "\\vec{v} = \\vec{e}_2 dx^2"
},
{
"math_id": 101,
"text": "q_{AB}^{(2)} = \\vec{e}_A \\cdot \\vec{e}_B"
},
{
"math_id": 102,
"text": "B"
},
{
"math_id": 103,
"text": "A_\\Sigma = \\int_\\Sigma dx^1 dx^2 \\sqrt{\\det \\left(q^{(2)}\\right)}"
},
{
"math_id": 104,
"text": "\\det (q^{(2)}) = q_{11} q_{22} - q_{12}^2"
},
{
"math_id": 105,
"text": "\\det (q^{(2)}) = \\epsilon^{AB} \\epsilon^{CD} q_{AC} q_{BD} / 2"
},
{
"math_id": 106,
"text": "A \\dots D"
},
{
"math_id": 107,
"text": "\\det (q^{(2)}) = {\\epsilon^{3ab} \\epsilon^{3cd} q_{ac} q_{bc} \\over 2}."
},
{
"math_id": 108,
"text": "q^{ab} = {\\epsilon^{bcd} \\epsilon^{aef} q_{ce} q_{df} \\over 2!\\det (q)}."
},
{
"math_id": 109,
"text": "\\det(q^{(2)})"
},
{
"math_id": 110,
"text": "\\tilde{E}^a_i\\tilde{E}^{bi} = \\det (q) q^{ab}"
},
{
"math_id": 111,
"text": "A_\\Sigma = \\int_\\Sigma dx^1 dx^2 \\sqrt{\\tilde{E}^3_i \\tilde{E}^{3i}}."
},
{
"math_id": 112,
"text": "\\tilde{E}^3_i"
},
{
"math_id": 113,
"text": "\\hat{\\tilde{E}}^3_i \\sim {\\delta \\over \\delta A_3^i}."
},
{
"math_id": 114,
"text": "A_\\Sigma"
},
{
"math_id": 115,
"text": "N = 2J"
},
{
"math_id": 116,
"text": "J"
},
{
"math_id": 117,
"text": "\\sum_i T^i T^i = J (J+1) 1."
},
{
"math_id": 118,
"text": "\\hat{A}_\\Sigma W_\\gamma [A] = 8 \\pi \\ell_{\\text{Planck}}^2 \\beta \\sum_I \\sqrt{j_I (j_I + 1)} W_\\gamma [A]"
},
{
"math_id": 119,
"text": "I"
},
{
"math_id": 120,
"text": "R"
},
{
"math_id": 121,
"text": "V = \\int_R d^3 x \\sqrt{\\det (q)} = \\int_R dx^3 \\sqrt{\\frac{1}{3!} \\epsilon_{abc} \\epsilon^{ijk} \\tilde{E}^a_i \\tilde{E}^b_j \\tilde{E}^c_k}."
},
{
"math_id": 122,
"text": "\\mathbb{A}"
},
{
"math_id": 123,
"text": "\\mathbb{B}"
},
{
"math_id": 124,
"text": "\\operatorname{Tr}(\\mathbb{A}) \\operatorname{Tr}(\\mathbb{B}) = \\operatorname{Tr}(\\mathbb{A}\\mathbb{B}) + \\operatorname{Tr}(\\mathbb{A}\\mathbb{B}^{-1})."
},
{
"math_id": 125,
"text": "\\eta"
},
{
"math_id": 126,
"text": "W_\\gamma [A] W_\\eta [A] = W_{\\gamma \\circ \\eta} [A] + W_{\\gamma \\circ \\eta^{-1}} [A]"
},
{
"math_id": 127,
"text": "\\eta^{-1}"
},
{
"math_id": 128,
"text": "\\gamma \\circ \\eta"
},
{
"math_id": 129,
"text": "W_\\gamma [A] = W_{\\gamma^{-1}} [A]"
},
{
"math_id": 130,
"text": "\\operatorname{Tr} (\\mathbb{A} \\mathbb{B}) = \\operatorname{Tr}(\\mathbb{B} \\mathbb{A})"
},
{
"math_id": 131,
"text": "W_{\\gamma \\circ \\eta} [A] = W_{\\eta \\circ \\gamma} [A]"
},
{
"math_id": 132,
"text": "\\hat{H} (x)"
},
{
"math_id": 133,
"text": "\\int [d N] e^{i \\int d^3 x N (x) \\hat{H} (x)}"
},
{
"math_id": 134,
"text": "B_{ab}^{IJ}"
},
{
"math_id": 135,
"text": "B_{ab}^{IJ} = {1 \\over 2} \\left(E^I_a E^J_b - E^I_b E^J_a\\right)"
},
{
"math_id": 136,
"text": "\\partial_\\mu \\hat{A}^\\mu"
},
{
"math_id": 137,
"text": "H_{phys}"
},
{
"math_id": 138,
"text": "\\hbar \\to 0"
},
{
"math_id": 139,
"text": "\\mathcal{H}_{Diff}"
},
{
"math_id": 140,
"text": "\\mathcal{H}_{Phys}"
},
{
"math_id": 141,
"text": "n-"
},
{
"math_id": 142,
"text": "W (x_1, \\dots , x_n) = \\langle 0 | \\phi (x_n) \\dots \\phi (x_1) |0 \\rangle , "
},
{
"math_id": 143,
"text": "M"
},
{
"math_id": 144,
"text": "\\{ O , \\{ O , M \\} \\}_{M = 0} = 0,"
},
{
"math_id": 145,
"text": "\\hat{M} := \\int d^3x \\widehat{\\left( \\frac{H}{\\sqrt[4]{\\det (q(x))}} \\right)}^\\dagger(x) \\widehat{\\left(\\frac{H}{\\sqrt[4]{\\det (q(x))}} \\right)} (x). "
},
{
"math_id": 146,
"text": "\\widehat{\\left( \\frac{H}{\\sqrt[4]{\\det (q(x))}} \\right)} (x) \\Psi = 0"
},
{
"math_id": 147,
"text": "x"
},
{
"math_id": 148,
"text": "\\hat{M} \\Psi = 0"
},
{
"math_id": 149,
"text": "0 = \\left \\langle \\Psi , \\hat{M} \\Psi \\right \\rangle = \\int d^3x \\left\\| \\widehat{\\left( \\frac{H}{\\sqrt[4]{\\det (q(x))}} \\right)} (x) \\Psi \\right\\|^2 \\qquad Eq \\; 4"
},
{
"math_id": 150,
"text": "\\hat{M}"
},
{
"math_id": 151,
"text": "Q_M"
},
{
"math_id": 152,
"text": "H_{Kin}"
},
{
"math_id": 153,
"text": " H_{Diff}"
},
{
"math_id": 154,
"text": "H_{Diff}"
},
{
"math_id": 155,
"text": "\\hat{\\overline{M}}"
},
{
"math_id": 156,
"text": "\\widehat{\\left( \\frac{H}{\\sqrt[4]{\\det (q(x))}} \\right)} (x) \\Psi \\not= 0,"
},
{
"math_id": 157,
"text": "Q_{M_E}"
},
{
"math_id": 158,
"text": "\\hat{M}' := \\hat{M} - \\min (spec (\\hat{M})) \\hat{1}"
},
{
"math_id": 159,
"text": "\\lim_{\\hbar \\to 0} \\min (spec(\\hat{M})) = 0,"
},
{
"math_id": 160,
"text": "\\hat{M}'"
},
{
"math_id": 161,
"text": "M_E"
},
{
"math_id": 162,
"text": "M_E = \\int_\\Sigma d^3x {H (x)^2 - q^{ab} V_a (x) V_b (x) \\over \\sqrt{\\det (q)}}"
},
{
"math_id": 163,
"text": "H(x) = 0"
},
{
"math_id": 164,
"text": "V_a (x) = 0"
},
{
"math_id": 165,
"text": "\\langle\\phi, \\psi\\rangle_{\\text{Phys}} = \\lim_{T \\to \\infty} \\left\\langle\\phi, \\int_{-T}^T dt e^{i t \\hat{M}_E} \\psi\\right\\rangle"
},
{
"math_id": 166,
"text": "\\delta (\\hat{M_E}) = \\lim_{T \\to \\infty} \\int_{-T}^T dt e^{i t \\hat{M}_E}"
},
{
"math_id": 167,
"text": "t"
},
{
"math_id": 168,
"text": "e^{i t \\hat{M}_E} = \\lim_{n \\to \\infty} \\left [e^{i t \\hat{M}_E / n} \\right]^n = \\lim_{n \\to \\infty} [1 + i t \\hat{M}_E / n]^n."
},
{
"math_id": 169,
"text": "[1 + i t \\hat{M}_E / n]"
},
{
"math_id": 170,
"text": "n"
},
{
"math_id": 171,
"text": "S_{\\text{BH}} = \\frac{k_{\\text{B}}A}{4\\ell_{\\text{P}}^2},"
},
{
"math_id": 172,
"text": "k_{\\text{B}}"
},
{
"math_id": 173,
"text": "\\ell_{\\text{P}} = \\sqrt{G\\hbar/c^{3}}"
},
{
"math_id": 174,
"text": "S=A/4"
},
{
"math_id": 175,
"text": "S"
}
] |
https://en.wikipedia.org/wiki?curid=152664
|
15267398
|
Laser beam profiler
|
Measurement device
A laser beam profiler captures, displays, and records the spatial intensity profile of a laser beam at a particular plane transverse to the beam propagation path. Since there are many types of lasers—ultraviolet, visible, infrared, continuous wave, pulsed, high-power, low-power—there is an assortment of instrumentation for measuring laser beam profiles. No single laser beam profiler can handle every power level, pulse duration, repetition rate, wavelength, and beam size.
Overview.
Laser beam profiling instruments measure the following quantities:
Instruments and techniques were developed to obtain the beam characteristics listed above. These include:
As of 2002[ [update]], commercial knife-edge measurement systems cost $5,000–$12,000 USD and CCD beam profilers cost $4,000–9,000 USD. The cost of CCD beam profilers has come down in recent years, primarily driven by lower silicon CCD sensor costs, and as of 2008[ [update]] they can be found for less than $1000 USD.
Applications.
The applications of laser beam profiling include:
Measurements.
Beam width.
The beam width is the single most important characteristic of a laser beam profile. At least five definitions of beam width are in common use: D4σ, 10/90 or 20/80 knife-edge, 1/e2, FWHM, and D86. The D4σ beam width is the ISO standard definition and the measurement of the M2 beam quality parameter requires the measurement of the D4σ widths. The other definitions provide complementary information to the D4σ and are used in different circumstances. The choice of definition can have a large effect on the beam width number obtained, and it is important to use the correct method for any given application. The D4σ and knife-edge widths are sensitive to background noise on the detector, while the 1/e2 and FWHM widths are not. The fraction of total beam power encompassed by the beam width depends on which definition is used.
Beam quality.
Beam quality parameter, M2.
The M2 parameter is a measure of beam quality; a low M2 value indicates good beam quality and ability to be focused to a tight spot. The value M is equal to the ratio of the beam's angle of divergence to that of a Gaussian beam with the same D4σ waist width. Since the Gaussian beam diverges more slowly than any other beam shape, the M2 parameter is always greater than or equal to one. Other definitions of beam quality have been used in the past, but the one using second moment widths is most commonly accepted.
Beam quality is important in many applications. In fiber-optic communications beams with an M2 close to 1 are required for coupling to single-mode optical fiber. Laser machine shops care a lot about the M2 parameter of their lasers because the beams will focus to an area that is M4 times larger than that of a Gaussian beam with the same wavelength and D4σ waist width before focusing; in other words, the fluence scales as 1/M4. The rule of thumb is that M2 increases as the laser power increases. It is difficult to obtain excellent beam quality and high average power (100 W to kWs) due to thermal lensing in the laser gain medium.
The M2 parameter is determined experimentally as follows:
Complete E-field beam profiling.
Beam profilers measure the intensity, |E-field|2, of the laser beam profile but do not yield any information about the phase of the E-field. To completely characterize the E-field at a given plane, both the phase and amplitude profiles must be known. The real and imaginary parts of the electric field can be characterized using two CCD beam profilers that sample the beam at two separate propagation planes, with the application of a phase recovery algorithm to the captured data. The benefit of completely characterizing the E-field in one plane is that the E-field profile can be computed for any other plane with diffraction theory.
Power-in-the-bucket or Strehl definition of beam quality.
The M2 parameter is not the whole story in specifying beam quality. A low M2 only implies that the second moment of the beam profile expands slowly. Nevertheless, two beams with the same M2 may not have the same fraction of delivered power in a given area. Power-in-the-bucket and Strehl ratio are two attempts to define beam quality as a function of how much power is delivered to a given area. Unfortunately, there is no standard bucket size (D86 width, Gaussian beam width, Airy disk nulls, etc.) or bucket shape (circular, rectangular, etc.) and there is no standard beam to compare for the Strehl ratio. Therefore, these definitions must always be specified before a number is given and it presents much difficulty when trying to compare lasers. There is also no simple conversion between M2, power-in-the-bucket, and Strehl ratio. The Strehl ratio, for example, has been defined as the ratio of the peak focal intensities in the aberrated and ideal point spread functions. In other cases, it has been defined as the ratio between the peak intensity of an image divided by the peak intensity of a diffraction-limited image with the same total flux. Since there are many ways power-in-the-bucket and Strehl ratio have been defined in the literature, the recommendation is to stick with the ISO-standard M2 definition for the beam quality parameter and be aware that a Strehl ratio of 0.8, for example, does not mean anything unless the Strehl ratio is accompanied by a definition.
Beam divergence.
The beam divergence of a laser beam is a measure for how fast the beam expands far from the beam waist. It is usually defined as the derivative of the beam radius with respect to the axial position in the far field, i.e., in a distance from the beam waist which is much larger than the Rayleigh length. This definition yields a divergence half-angle. (Sometimes, full angles are used in the literature; these are twice as large.) For a diffraction-limited Gaussian beam, the beam divergence is λ/(πw0), where λ is the wavelength (in the medium) and w0 the beam radius (radius with 1/e2 intensity) at the beam waist. A large beam divergence for a given beam radius corresponds to poor beam quality. A low beam divergence can be important for applications such as pointing or free-space optical communications. Beams with very small divergence, i.e., with approximately constant beam radius over significant propagation distances, are called collimated beams. For the measurement of beam divergence, one usually measures the beam radius at different positions, using e.g. a beam profiler. It is also possible to derive the beam divergence from the complex amplitude profile of the beam in a single plane: spatial Fourier transforms deliver the distribution of transverse spatial frequencies, which are directly related to propagation angles. See US Laser Corps application note for a tutorial on how to measure the laser beam divergence with a lens and CCD camera.
Beam astigmatism.
Astigmatism in a laser beam occurs when the horizontal and vertical cross sections of the beam focus at different locations along the beam path. Astigmatism can be corrected with a pair of cylindrical lenses. The metric for astigmatism is the power of cylindrical lens needed to bring the focuses of the horizontal and vertical cross sections together. Astigmatism is caused by:
Astigmatism can easily be characterized by a CCD beam profiler by observing where the x and y beam waists occur as the profiler is translated along the beam path.
Beam wander or jitter.
Every laser beam wanders and jitters—albeit a small amount. The typical kinematic tip-tilt mount drifts by around 100 μrad per day in a laboratory environment (vibration isolation via optical table, constant temperature and pressure, and no sunlight that causes parts to heat). A laser beam incident upon this mirror will be translated by 100 m at a range of 1000 km. This could make the difference between hitting or not hitting a communications satellite from Earth. Hence, there is a lot of interest in characterizing the beam wander (slow time scale) or jitter (fast time scale) of a laser beam. The beam wander and jitter can be measured by tracking the centroid or peak of the beam on a CCD beam profiler. The CCD frame rate is typically 30 frames per second and therefore can capture beam jitter that is slower than 30 Hz—it cannot see fast vibrations due to one's voice, 60 Hz fan motor hum, or other sources of fast vibrations. Fortunately, this is usually not a great concern for most laboratory laser systems and the frame rates of CCDs are fast enough to capture the beam wander over the bandwidth that contains the greatest noise power. A typical beam wander measurement involves tracking the centroid of the beam over several minutes. The rms deviation of the centroid data gives a clear picture of the laser beam pointing stability. The integration time of the beam jitter measurement should always accompany the computed rms value. Even though the pixel resolution of a camera may be several micrometres, sub-pixel centroid resolution (possibly tens of nanometer resolution) is attained when the signal-to-noise ratio is good and the beam fills most of the CCD active area.
Beam wander is caused by:
Misrepresentation of beam profiler measurements for laser systems.
It is to most laser manufacturers' advantage to present specifications in a way that shows their product in the best light, even if this involves misleading the customer. Laser performance specifications can be clarified by asking questions such as:
Techniques.
Beam profilers generally fall into two classes: the first uses a simple photodetector behind an aperture which is scanned over the beam. The second class uses a camera to image the beam.
Scanning-aperture techniques.
The most common scanning aperture techniques are the knife-edge technique and the scanning-slit profiler. The former chops the beam with a knife and measures the transmitted power as the blade cuts through the beam. The measured intensity versus knife position yields a curve that is the integrated beam intensity in one direction. By measuring the intensity curve in several directions, the original beam profile can be reconstructed using algorithms developed for x-ray tomography. The measuring instrument is based on high precision multiple knife edges each deployed on a rotating drum and having a different angle with respect to beam orientation. Scanned beam is then reconstructed using tomographic algorithms and provides 2D or 3D high resolution energy distribution plots. Because of the special scanning technique the system automatically zooms in onto the current beam size enabling high resolution measurements of sub micron beams as well as relative large beams of 10 or more millimeters. To obtain measurement of various wavelength different detectors are used to allow laser beam measurements from deep UV to far IR. Unlike other camera based systems this technology also provides accurate power measurement in real time
Scanning-slit profilers use a narrow slit instead of a single knife edge. In this case, the intensity is integrated over the slit width. The resulting measurement is equivalent to the original cross section convolved with the profile of the slit.
This fusion between knife-edge technology and tomographic algorithms creates a new field of beam profiling - CKET (Computerized Knife-Edge Tomography). This creates capability of accurate measurement from a micron to over 10 millimeters with adaptable resolution over a wide spectrum range, practically if a single-surface detector exists for a certain wavelength region, then using this technology an image-like profile could be derived.
These techniques can measure very small spot sizes down to 1 μm, and can be used to directly measure high power beams. They do not offer continuous readout, although repetition rates as high as twenty hertz can be achieved. Also, the profiles give integrated intensities in the x and y directions and not the actual 2D spatial profile (integrating intensities can be hard to interpret for complicated beam profiles). They do not generally work for pulsed laser sources, because of the extra complexity of synchronizing the motion of the aperture and the laser pulses.
CCD camera technique.
The CCD camera technique is simple: attenuate and shine a laser onto a CCD and measure the beam profile directly. It is for this reason that the camera technique is the most popular method for laser beam profiling. The most popular cameras used are silicon CCDs that have sensor diameters that range up to 25 mm (1 inch) and pixel sizes down to a few micrometres. These cameras are also sensitive to a broad range of wavelengths, from deep UV, 200 nm, to near infrared, 1100 nm; this range of wavelengths encompass a broad range of laser gain media. The advantages of the CCD camera technique are:
The disadvantages of the CCD camera technique are:
Baseline subtraction for D4σ width measurements.
The D4σ width is sensitive to the beam energy or noise in the tail of the pulse because the pixels that are far from the beam centroid contribute to the D4σ width as the distance squared. To reduce the error in the D4σ width estimate, the baseline pixel values are subtracted from the measured signal. The baseline values for the pixels are measured by recording the values of the CCD pixels with no incident light. The finite value is due to dark current, readout noise, and other noise sources. For shot-noise-limited noise sources, baseline subtraction improves the D4σ width estimate as formula_5, where formula_6 is the number of pixels in the wings. Without baseline subtraction, the D4σ width is overestimated.
Averaging to get better measurements.
Averaging consecutive CCD images yields a cleaner profile and removes both CCD imager noise and laser beam intensity fluctuations. The signal-to-noise-ratio (SNR) of a pixel for a beam profile is defined as the mean value of the pixel divided by its root-mean-square (rms) value. The SNR improves as square root of the number of captured frames for shot noise processes – dark current noise, readout noise, and Poissonian detection noise. So, for example, increasing the number of averages by a factor of 100 smooths out the beam profile by a factor of 10.
Attenuation techniques.
Since CCD sensors are highly sensitive, attenuation is almost always needed for proper beam profiling. For example, 40 dB (ND 4 or 10−4) of attenuation is typical for a milliwatt HeNe laser. Proper attenuation has the following properties:
For laser beam profiling with CCD sensors, typically two types of attenuators are used: neutral density filters, and wedges or thick optical flats.
Neutral density filters.
Neutral density (ND) filters come in two types: absorptive and reflective.
Absorptive filters are usually made of tinted glass. They are useful for lower-power applications that involve up to about 100 mW average power. Above those power levels, thermal lensing may occur, causing beam size change or deformation, because of the low thermal conductivity of the substrate (usually a glass). Higher power may result in melting or cracking. Absorptive filter attenuation values are usually valid for the visible spectrum (500–800 nm) and are not valid outside of that spectral region. Some filters can be ordered and calibrated for near-infrared wavelengths, up to the long wavelength absorption edge of the substrate (around 2.2 μm for glasses). Typically, one can expect about 5-10% variation of the attenuation across a ND filter, unless specified otherwise to the manufacturer. The attenuation values of ND filters are specified logarithmically. A ND 3 filter transmits 10−3 of the incident beam power. Placing the largest attenuator last before the CCD sensor will result in the best rejection of ghost images due to multiple reflections.
Reflective filters are made with a thin metallic coating and hence operate over a larger bandwidth. An ND 3 metallic filter will be good over 200–2000 nm. The attenuation will rapidly increase outside this spectral region because of absorption in the glass substrate. These filters reflect rather than absorb the incident power, and hence can handle higher input average powers. However, they are less well suited to the high peak powers of pulsed lasers. These filters work fine to about 5 W average power (over about 1 cm2 illumination area) before heating causes them to crack. Since these filters reflect light, one must be careful when stacking multiple ND filters, since multiple reflections among the filters will cause a ghost image to interfere with the original beam profile. One way to mitigate this problem is by tilting the ND filter stack. Assuming that the absorption of the metallic ND filter is negligible, the order of the ND filter stack doesn't matter, as it does for the absorptive filters.
Diffractive beam sampler.
Diffractive beam samplers are used to monitor high power lasers where optical losses and wavefront distortions of the transmitted beam need to be kept to a minimum.
In most applications, most of the incident light must continue forward, "unaffected," in the "zero order diffracted order" while a small amount of the beam is diffracted into a higher diffractive order, providing a "sample" of the beam.
By directing the sampled light in the higher order(s) onto a detector, it is possible to monitor, in real time, not only the power levels of a laser beam, but also its profile, and other laser characteristics.
Optical wedges.
Optical wedges and reflections from uncoated optical glass surfaces are used to attenuate high power laser beams. About 4% is reflected from the air/glass interface and several wedges can be used to greatly attenuate the beam to levels that can be attenuated with ND filters. The angle of the wedge is typically selected so that the second reflection from the surface does not hit the active area of the CCD, and that no interference fringes are visible. The farther the CCD is from the wedge, the smaller the angle required. Wedges have the disadvantage of both translating and bending the beam direction — paths will no longer lie on convenient rectangular coordinates. Rather than using a wedge, an optical-quality thick glass plate tilted to the beam can also work — actually, this is the same as a wedge with a 0° angle. The thick glass will translate the beam but it will not change the angle of the output beam. The glass must be thick enough so that the beam does not overlap with itself to produce interference fringes, and if possible that the secondary reflection does not illuminate the active area of the CCD. The Fresnel reflection of a beam from a glass plate is different for the s- and p-polarizations (s is parallel to the surface of the glass, and p is perpendicular to s) and changes as a function of angle of incidence – keep this in mind if you expect that the two polarizations have different beam profiles. To prevent distortion of the beam profile, the glass should be of optical quality — surface flatness of λ/10 (λ=633 nm) and scratch-dig of 40-20 or better. A half-wave plate followed by a polarizing beam splitter form a variable attenuator and this combination is often used in optical systems. The variable attenuator made in this fashion is not recommended for attenuation for beam profiling applications because: (1) the beam profile in the two orthogonal polarizations may be different, (2) the polarization beam cube may have a low optical damage threshold value, and (3) the beam can be distorted in cube polarizers at very high attenuation. Inexpensive cube polarizers are formed by cementing two right angle prisms together. The glue does not stand up well to high powers — the intensity should be kept under 500 mW/mm2. Single-element polarizers are recommended for high powers.
Optimal beam size on the CCD detector.
There are two competing requirements that determine the optimal beam size on the CCD detector. One requirement is that the entire energy — or as much of it as possible — of the laser beam is incident on the CCD sensor. This would imply that we should focus all the energy in the center of the active region in as small a spot as possible using only a few of the central pixels to ensure that the tails of the beam are captured by the outer pixels. This is one extreme. The second requirement is that we need to adequately sample the beam profile shape. As a rule of thumb, we want at least 10 pixels across the area that encompasses most, say 80%, of the energy in the beam. Therefore, there is no hard and fast rule to select the optimal beam size. As long as the CCD sensor captures over 90% of the beam energy and has at least 10 pixels across it, the beam width measurements will have some accuracy.
Pixel size and number of pixels.
The larger the CCD sensor, the larger the size of beam that can be profiled. Sometimes this comes at the cost of larger pixel sizes. Small pixels sizes are desired for observing focused beams. A CCD with many megapixels is not always better than a smaller array since readout times on the computer can be uncomfortably long. Reading out the array in real-time is essential for any tweaking or optimization of the laser profile.
Far-field beam profiler.
A far-field beam profiler is nothing more than profiling the beam at the focus of a lens. This plane is sometimes called the Fourier plane and is the profile that one would see if the beam propagated very far away. The beam at the Fourier plane is the Fourier transform of the input field. Care must be taken in setting up a far-field measurement. The focused spot size must be large enough to span across several pixels. The spot size is approximately "f"λ/"D", where "f" is the focal length of the lens, λ is the wavelength of the light, and "D" is the diameter of the collimated beam incident upon the lens. For example, a helium-neon laser (633 nm) with 1 mm beam diameter would focus to a 317 μm spot with a 500 mm lens. A laser beam profiler with a 5.6 μm pixel size would adequately sample the spot at 56 locations.
Special applications.
The prohibitive costs of CCD laser beam profilers in the past have given way to low-cost beam profilers. Low-cost beam profilers have opened up a number of new applications: replacing irises for super-accurate alignment and simultaneous multiple port monitoring of laser systems.
Iris replacement with microradian alignment accuracy.
In the past, alignment of laser beams was done with irises. Two irises uniquely defined a beam path; the farther apart the irises and the smaller the iris holes, the better the path was defined. The smallest aperture that an iris can define is about 0.8 mm. In comparison, the centroid of a laser beam can be determined to sub-micrometre accuracy with a laser beam profiler. The laser beam profiler's effective aperture size is three orders of magnitude smaller than that of an iris. Consequently, the ability to define an optical path is 1000 times better when using beam profilers over irises. Applications that need microradian alignment accuracies include earth-to-space communications, earth-to-space ladar, master oscillator to power oscillator alignment, and multi-pass amplifiers.
Simultaneous multiple port monitoring of laser system.
Experimental laser systems benefit from the use of multiple laser beam profilers to characterize the pump beam, the output beam, and the beam shape at intermediate locations in the laser system, for example, after a Kerr-lens modelocker. Changes in the pump laser beam profile indicate the health of the pump laser, which laser modes are excited in the gain crystal, and also determine whether the laser is warmed up by locating the centroid of the beam relative to the breadboard. The output beam profile is often a strong function of pump power due to thermo-optical effects in the gain medium.
|
[
{
"math_id": 0,
"text": " \\sigma^2(z) = \\sigma_0^2 + M^4 \\left(\\frac{\\lambda}{\\pi\\sigma_0}\\right)^2(z-z_0)^2 "
},
{
"math_id": 1,
"text": " \\sigma^2(z) "
},
{
"math_id": 2,
"text": " z_0 "
},
{
"math_id": 3,
"text": " 2\\sigma_0 "
},
{
"math_id": 4,
"text": " \\sigma_0 "
},
{
"math_id": 5,
"text": " \\sqrt{N} "
},
{
"math_id": 6,
"text": "N"
}
] |
https://en.wikipedia.org/wiki?curid=15267398
|
152703
|
Hilbert's third problem
|
On dissections between polyhedra
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier writings by Carl Friedrich Gauss, David Hilbert conjectured that this is not always possible. This was confirmed within the year by his student Max Dehn, who proved that the answer in general is "no" by producing a counterexample.
The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Wallace–Bolyai–Gerwien theorem.
Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of Kraków, and was solved by Ludwik Antoni Birkenmajer with a different method than Dehn's. Birkenmajer did not publish the result, and the original manuscript containing his solution was rediscovered years later.
History and motivation.
The formula for the volume of a pyramid,
formula_0
had been known to Euclid, but all proofs of it involve some form of limiting process or calculus, notably the method of exhaustion or, in more modern form, Cavalieri's principle. Similar formulas in plane geometry can be proven with more elementary means. Gauss regretted this defect in two of his letters to Christian Ludwig Gerling, who proved that two symmetric tetrahedra are equidecomposable.
Gauss's letters were the motivation for Hilbert: is it possible to prove the equality of volume using elementary "cut-and-glue" methods? Because if not, then an elementary proof of Euclid's result is also impossible.
Dehn's answer.
Dehn's proof is an instance in which abstract algebra is used to prove an impossibility result in geometry. Other examples are doubling the cube and trisecting the angle.
Two polyhedra are called scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second. Any two scissors-congruent polyhedra have the same volume. Hilbert asks about the converse.
For every polyhedron formula_1, Dehn defines a value, now known as the Dehn invariant formula_2, with the property that,
if formula_1 is cut into polyhedral pieces formula_3, then
formula_4
In particular, if two polyhedra are scissors-congruent, then they have the same Dehn invariant. He then shows that every cube has Dehn invariant zero while every regular tetrahedron has non-zero Dehn invariant. Therefore, these two shapes cannot be scissors-congruent.
A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. If a polyhedron is cut into two, some edges are cut into two, and the corresponding contributions to the Dehn invariants should therefore be additive in the edge lengths. Similarly, if a polyhedron is cut along an edge, the corresponding angle is cut into two. Cutting a polyhedron typically also introduces new edges and angles; their contributions must cancel out. The angles introduced when a cut passes through a face add to formula_5, and the angles introduced around an edge interior to the polyhedron add to formula_6. Therefore, the Dehn invariant is defined in such a way that integer multiples of angles of formula_5 give a net contribution of zero.
All of the above requirements can be met by defining formula_2 as an element of the tensor product of the real numbers formula_7 (representing lengths of edges) and the quotient space formula_8 (representing angles, with all rational multiples of formula_5 replaced by zero). For some purposes, this definition can be made using the tensor product of modules over formula_9 (or equivalently of abelian groups), while other aspects of this topic make use of a vector space structure on the invariants, obtained by considering the two factors formula_7 and formula_8 to be vector spaces over formula_10 and taking the tensor product of vector spaces over formula_10. This choice of structure in the definition does not make a difference in whether two Dehn invariants, defined in either way, are equal or unequal.
For any edge formula_11 of a polyhedron formula_1, let formula_12 be its length and let formula_13 denote the dihedral angle of the two faces of formula_1 that meet at formula_11, measured in radians and considered modulo rational multiples of formula_5. The Dehn invariant is then defined as
formula_14
where the sum is taken over all edges formula_11 of the polyhedron formula_1. It is a valuation.
Further information.
In light of Dehn's theorem above, one might ask "which polyhedra are scissors-congruent"? Sydler (1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant. Børge Jessen later extended Sydler's results to four dimensions. In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology of certain classical groups.
Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of three-dimensional space can be tiled periodically is zero.
<templatestyles src="Unsolved/styles.css" />
Unsolved problem in mathematics:
In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?
Jessen also posed the question of whether the analogue of Jessen's results remained true for spherical geometry and hyperbolic geometry. In these geometries, Dehn's method continues to work, and shows that when two polyhedra are scissors-congruent, their Dehn invariants are equal. However, it remains an open problem whether pairs of polyhedra with the same volume and the same Dehn invariant, in these geometries, are always scissors-congruent.
Original question.
Hilbert's original question was more complicated: given any two tetrahedra "T"1 and "T"2 with equal base area and equal height (and therefore equal volume), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to "T"1 and also glued to "T"2, the resulting polyhedra are scissors-congruent?
Dehn's invariant can be used to yield a negative answer also to this stronger question.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\frac{\\text{base area} \\times \\text{height}}{3},"
},
{
"math_id": 1,
"text": "P"
},
{
"math_id": 2,
"text": "\\operatorname{D}(P)"
},
{
"math_id": 3,
"text": "P_1, P_2, \\dots P_n"
},
{
"math_id": 4,
"text": "\\operatorname{D}(P) = \\operatorname{D}(P_1)+\\operatorname{D}(P_2)+\\cdots + \\operatorname{D}(P_n)."
},
{
"math_id": 5,
"text": "\\pi"
},
{
"math_id": 6,
"text": "2\\pi"
},
{
"math_id": 7,
"text": "\\R"
},
{
"math_id": 8,
"text": "\\R/(\\Q\\pi)"
},
{
"math_id": 9,
"text": "\\Z"
},
{
"math_id": 10,
"text": "\\Q"
},
{
"math_id": 11,
"text": "e"
},
{
"math_id": 12,
"text": "\\ell(e)"
},
{
"math_id": 13,
"text": "\\theta(e)"
},
{
"math_id": 14,
"text": "\\operatorname{D}(P) = \\sum_{e} \\ell(e)\\otimes \\theta(e)"
}
] |
https://en.wikipedia.org/wiki?curid=152703
|
1527098
|
Clairaut's theorem (gravity)
|
Theorem about gravity
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatise which synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid. It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes. Today it has been largely supplanted by the Somigliana equation.
History.
Although it had been known since antiquity that the Earth was spherical, by the 17th century evidence was accumulating that it was not a perfect sphere. In 1672 Jean Richer found the first evidence that gravity was not constant over the Earth (as it would be if the Earth were a sphere); he took a pendulum clock to Cayenne, French Guiana and found that it lost <templatestyles src="Fraction/styles.css" />2+1⁄2 minutes per day compared to its rate at Paris. This indicated the acceleration of gravity was less at Cayenne than at Paris. Pendulum gravimeters began to be taken on voyages to remote parts of the world, and it was slowly discovered that gravity increases smoothly with increasing latitude, gravitational acceleration being about 0.5% greater at the poles than at the equator.
British physicist Isaac Newton explained this in his "Principia Mathematica" (1687) in which he outlined his theory and calculations on the shape of the Earth. Newton theorized correctly that the Earth was not precisely a sphere but had an oblate ellipsoidal shape, slightly flattened at the poles due to the centrifugal force of its rotation. Using geometric calculations, he gave a concrete argument as to the hypothetical ellipsoid shape of the Earth.
The goal of "Principia" was not to provide exact answers for natural phenomena, but to theorize potential solutions to these unresolved factors in science. Newton pushed for scientists to look further into the unexplained variables. Two prominent researchers that he inspired were Alexis Clairaut and Pierre Louis Maupertuis. They both sought to prove the validity of Newton's theory on the shape of the Earth. In order to do so, they went on an expedition to Lapland in an attempt to accurately measure a meridian arc. From such measurements they could calculate the eccentricity of the Earth, its degree of departure from a perfect sphere.
Clairaut confirmed that Newton's theory that the Earth was ellipsoidal was correct, but that his calculations were in error, and he wrote a letter to the Royal Society of London with his findings. The society published an article in Philosophical Transactions the following year, 1737. In it Clairaut pointed out (Section XVIII) that Newton's Proposition XX of Book 3 does not apply to the real earth. It stated that the weight of an object at some point in the earth depended only on the proportion of its distance from the centre of the earth to the distance from the centre to the surface at or above the object, so that the total weight of a column of water at the centre of the earth would be the same no matter in which direction the column went up to the surface. Newton had in fact said that this was on the assumption that the matter inside the earth was of a uniform density (in Proposition XIX). Newton realized that the density was probably not uniform, and proposed this as an explanation for why gravity measurements found a greater difference between polar regions and equatorial regions than what his theory predicted. However, he also thought this would mean the equator was further from the centre than what his theory predicted, and Clairaut points out that the opposite is true. Clairaut points out at the beginning of his article that Newton did not explain why he thought the earth was ellipsoid rather than like some other oval, but that Clairaut, and James Stirling almost simultaneously, had shown why the earth should be an ellipsoid in 1736.
Clairaut's article did not provide a valid equation to back up his argument as well. This created much controversy in the scientific community.
It was not until Clairaut wrote "Théorie de la figure de la terre" in 1743 that a proper answer was provided. In it, he promulgated what is more formally known today as Clairaut's theorem.
Formula.
Clairaut's theorem says that the acceleration due to gravity "g" (including the effect of centrifugal force) on the surface of a spheroid in hydrostatic equilibrium (being a fluid or having been a fluid in the past, or having a surface near sea level) at latitude φ is:
formula_0
where formula_1 is the value of the acceleration of gravity at the equator, "m" the ratio of the centrifugal force to gravity at the equator, and "f" the flattening of a meridian section of the earth, defined as:
formula_2
(where "a" = semimajor axis, "b" = semiminor axis). The contribution of centrifugal force is approximately formula_3 whereas the gravitational attraction itself varies approximately as formula_4 This formula holds when the surface is perpendicular to the direction of gravity (including centrifugal force), even if (as usually) the density is not constant (in which case the gravitational attraction can be calculated at any point from the shape alone, without reference to formula_5). For the earth, formula_6 and formula_7 while formula_8 so formula_9 is greater at the poles than on the equator.
Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density.
This work was subsequently pursued by Laplace, who assumed surfaces of equal density which were nearly spherical.
The English mathematician George Stokes showed in 1849 that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium. A history of more recent developments and more detailed equations for "g" can be found in Khan.
The above expression for "g" has been supplanted by the Somigliana equation (after Carlo Somigliana).
Geodesy.
The spheroidal shape of the Earth is the result of the interplay between gravity and centrifugal force caused by the Earth's rotation about its axis. In his "Principia", Newton proposed the equilibrium shape of a homogeneous rotating Earth was a rotational ellipsoid with a flattening "f" given by 1/230. As a result, gravity increases from the equator to the poles. By applying Clairaut's theorem, Laplace found from 15 gravity values that "f" = 1/330. A modern estimate is 1/298.25642. See Figure of the Earth for more detail.
For a detailed account of the construction of the reference Earth model of geodesy, see Chatfield.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " g(\\varphi) = G_e \\left[ 1 + \\left(\\frac{5}{2} m - f\\right) \\sin^2 \\varphi \\right] \\, , "
},
{
"math_id": 1,
"text": "G_e"
},
{
"math_id": 2,
"text": "f = \\frac {a-b}{a} \\, , "
},
{
"math_id": 3,
"text": "-G_em\\cos^2\\varphi,"
},
{
"math_id": 4,
"text": "G_e\\left(\\frac 3 2 m - f\\right)\\sin^2\\varphi."
},
{
"math_id": 5,
"text": "m"
},
{
"math_id": 6,
"text": "m\\approx 1/289,"
},
{
"math_id": 7,
"text": "\\frac 5 2 m\\approx 1/116,"
},
{
"math_id": 8,
"text": "f\\approx 1/300,"
},
{
"math_id": 9,
"text": "g"
}
] |
https://en.wikipedia.org/wiki?curid=1527098
|
1527151
|
Speeds and feeds
|
Two separate velocities in machine tool practice, cutting speed and feed rate
The phrase speeds and feeds or feeds and speeds refers to two separate velocities in machine tool practice, cutting speed and feed rate. They are often considered as a pair because of their combined effect on the cutting process. Each, however, can also be considered and analyzed in its own right.
"Cutting speed" (also called "surface speed" or simply "speed") is the speed difference (relative velocity) between the cutting tool and the surface of the workpiece it is operating on. It is expressed in units of distance across the workpiece surface per unit of time, typically surface feet per minute (sfm) or meters per minute (m/min). "Feed rate" (also often styled as a solid compound, "feedrate", or called simply "feed") is the relative velocity at which the cutter is advanced along the workpiece; its vector is perpendicular to the vector of cutting speed. Feed rate units depend on the motion of the tool and workpiece; when the workpiece rotates ("e.g.", in turning and boring), the units are almost always distance per spindle revolution (inches per revolution [in/rev or ipr] or millimeters per revolution [mm/rev]). When the workpiece does not rotate ("e.g.", in milling), the units are typically distance per time (inches per minute [in/min or ipm] or millimeters per minute [mm/min]), although distance per revolution or per cutter tooth are also sometimes used.
If variables such as cutter geometry and the rigidity of the machine tool and its tooling setup could be ideally maximized (and reduced to negligible constants), then only a lack of power (that is, kilowatts or horsepower) available to the spindle would prevent the use of the maximum possible speeds and feeds for any given workpiece material and cutter material. Of course, in reality those other variables are dynamic and not negligible, but there is still a correlation between power available and feeds and speeds employed. In practice, lack of rigidity is usually the limiting constraint.
The phrases "speeds and feeds" or "feeds and speeds" have sometimes been used metaphorically to refer to the execution details of a plan, which only skilled technicians (as opposed to designers or managers) would know.
Cutting speed.
Cutting speed may be defined as the rate at the workpiece surface, irrespective of the machining operation used. A cutting speed for mild steel of 100 ft/min is the same whether it is the speed of the cutter passing over the workpiece, such as in a turning operation, or the speed of the cutter moving past a workpiece, such as in a milling operation. The cutting conditions will affect the value of this surface speed for mild steel.
Schematically, speed at the workpiece surface can be thought of as the tangential speed at the tool-cutter interface, that is, how fast the material moves past the cutting edge of the tool, although "which surface to focus on" is a topic with several valid answers. In drilling and milling, the outside diameter of the tool is the widely agreed surface. In turning and boring, the surface can be defined on either side of the depth of cut, that is, either the starting surface or the ending surface, with neither definition being "wrong" as long as the people involved understand the difference. An experienced machinist summed this up succinctly as "the diameter I am turning from" versus "the diameter I am turning to." He uses the "from", not the "to", and explains why, while acknowledging that some others do not. The logic of focusing on the largest diameter involved (OD of drill or end mill, starting diameter of turned workpiece) is that this is where the highest tangential speed is, with the most heat generation, which is the main driver of tool wear.
There will be an optimum cutting speed for each material and set of machining conditions, and the spindle speed (RPM) can be calculated from this speed. Factors affecting the calculation of cutting speed are:
Cutting speeds are calculated on the assumption that optimum cutting conditions exist. These include:
The cutting "speed" is given as a set of constants that are available from the material manufacturer or supplier. The most common materials are available in reference books or charts, but will always be subject to adjustment depending on the cutting conditions. The following table gives the cutting speeds for a selection of common materials under one set of conditions. The conditions are a tool life of 1 hour, dry cutting (no coolant), and at medium feeds, so they may appear to be incorrect depending on circumstances. These cutting speeds may change if, for instance, adequate coolant is available or an improved grade of HSS is used (such as one that includes [cobalt]).
Machinability rating.
The machinability rating of a material attempts to quantify the machinability of various materials. It is expressed as a percentage or a normalized value. The American Iron and Steel Institute (AISI) determined machinability ratings for a wide variety of materials by running turning tests at 180 surface feet per minute (sfpm). It then arbitrarily assigned 160 Brinell B1112 steel a machinability rating of 100%. The machinability rating is determined by measuring the weighed averages of the normal cutting speed, surface finish, and tool life for each material. Note that a material with a machinability rating less than 100% would be more difficult to machine than B1112 and material and a value more than 100% would be easier.
Machinability ratings can be used in conjunction with the Taylor tool life equation, "VTn" = "C" in order to determine cutting speeds or tool life. It is known that B1112 has a tool life of 60 minutes at a cutting speed of 100 sfpm. If a material has a machinability rating of 70%, it can be determined, with the above knowns, that in order to maintain the same tool life (60 minutes), the cutting speed must be 70 sfpm (assuming the same tooling is used).
When calculating for copper alloys, the machine rating is arrived at by assuming the 100 rating of 600 SFM. For example, phosphorus bronze (grades A–D) has a machinability rating of 20. This means that phosphor bronze runs at 20% the speed of 600 SFM or 120 SFM. However, 165 SFM is generally accepted as the basic 100% rating for "grading steels".
Formula
Cutting Speed (V)= [πDN]/1000 m/min
Where
D=Diameter of Workpiece in meter or millimeter
N=Spindle Speed in rpm
Spindle speed.
The spindle speed is the rotational frequency of the spindle of the machine, measured in revolutions per minute (RPM). The preferred speed is determined by working backward from the desired surface speed (sfm or m/min) and incorporating the diameter (of workpiece or cutter).
The spindle may hold the:
Excessive spindle speed will cause premature tool wear, breakages, and can cause tool chatter, all of which can lead to potentially dangerous conditions. Using the correct spindle speed for the material and tools will greatly enhance tool life and the quality of the surface finish.
For a given machining operation, the cutting speed will remain constant for most situations; therefore the spindle speed will also remain constant. However, facing, forming, parting off, and recess operations on a lathe or screw machine involve the machining of a constantly changing diameter. Ideally, this means changing the spindle speed as the cut advances across the face of the workpiece, producing constant surface speed (CSS). Mechanical arrangements to effect CSS have existed for centuries, but they were never applied commonly to machine tool control. In the pre-CNC era, the ideal of CSS was ignored for most work. For unusual work that demanded it, special pains were taken to achieve it. The introduction of CNC-controlled lathes has provided a practical, everyday solution via automated CSS Machining Process Monitoring and Control. By means of the machine's software and variable speed electric motors, the lathe can increase the RPM of the spindle as the cutter gets closer to the center of the part.
Grinding wheels are designed to be run at a maximum safe speed, the spindle speed of the grinding machine may be variable but this should only be changed with due attention to the safe working speed of the wheel. As a wheel wears it will decrease in diameter, and its effective cutting speed will be reduced. Some grinders have the provision to increase the spindle speed, which corrects for this loss of cutting ability; however, increasing the speed beyond the wheels rating will destroy the wheel and create a serious hazard to life and limb.
Generally speaking, spindle speeds and feed rates are less critical in woodworking than metalworking. Most woodworking machines including power saws such as circular saws and band saws, jointers, Thickness planers rotate at a fixed RPM. In those machines, cutting speed is regulated through the feed rate. The required feed rate can be extremely variable depending on the power of the motor, the hardness of the wood or other material being machined, and the sharpness of the cutting tool.
In woodworking, the ideal feed rate is one that is slow enough not to bog down the motor, yet fast enough to avoid burning the material. Certain woods, such as black cherry and maple are more prone to burning than others. The right feed rate is usually obtained by "feel" if the material is hand fed, or by trial and error if a power feeder is used. In thicknessers (planers), the wood is usually fed automatically through rubber or corrugated steel rollers. Some of these machines allow varying the feed rate, usually by changing pulleys. A slower feed rate usually results in a finer surface as more cuts are made for any length of wood.
Spindle speed becomes important in the operation of routers, spindle moulders or shapers, and drills. Older and smaller routers often rotate at a fixed spindle speed, usually between 20,000 and 25,000 rpm. While these speeds are fine for small router bits, using larger bits, say more than or 25 millimeters in diameter, can be dangerous and can lead to chatter. Larger routers now have variable speeds and larger bits require slower speed. Drilling wood generally uses higher spindle speeds than metal, and the speed is not as critical. However, larger diameter drill bits do require slower speeds to avoid burning.
Cutting feeds and speeds, and the spindle speeds that are derived from them, are the "ideal" cutting conditions for a tool. If the conditions are less than ideal then adjustments are made to the spindle's speed, this adjustment is usually a reduction in RPM to the closest available speed, or one that is deemed (through knowledge and experience) to be correct.
Some materials, such as machinable wax, can be cut at a wide variety of spindle speeds, while others, such as stainless steel require much more careful control as the cutting speed is critical, to avoid overheating both the cutter and workpiece. Stainless steel is one material that hardens very easily under cold working, therefore insufficient feed rate or incorrect spindle speed can lead to less than ideal cutting conditions as the work piece will quickly harden and resist the tool's cutting action. The liberal application of cutting fluid can improve these cutting conditions; however, the correct selection of speeds is the critical factor.
Spindle speed calculations.
Most metalworking books have nomograms or tables of spindle speeds and feed rates for different cutters and workpiece materials; similar tables are also likely available from the manufacturer of the cutter used.
The spindle speeds may be calculated for all machining operations once the SFM or MPM is known. In most cases, we are dealing with a cylindrical object such as a milling cutter or a workpiece turning in a lathe so we need to determine the speed at the periphery of this round object. This speed at the periphery (of a point on the circumference, moving past a stationary point) will depend on the rotational speed (RPM) and diameter of the object.
One analogy would be a skateboard rider and a bicycle rider travelling side by side along the road. For a given surface speed (the speed of this pair along the road) the rotational speed (RPM) of their wheels (large for the skater and small for the bicycle rider) will be different. This rotational speed (RPM) is what we are calculating, given a fixed surface speed (speed along the road) and known values for their wheel sizes (cutter or workpiece).
The following formulae may be used to estimate this value.
Approximation.
The exact RPM is not always needed, a close approximation will work. For instance, a machinist may want to take the value of formula_0 to be 3 if performing calculations by hand.
formula_1
e.g. for a cutting speed of 100 ft/min (a plain HSS steel cutter on mild steel) and diameter of 10 inches (the cutter or the work piece)
formula_2
and, for an example using metric values, where the cutting speed is 30 m/min and a diameter of 10 mm (0.01 m),
formula_3
Accuracy.
However, for more accurate calculations, and at the expense of simplicity, this formula can be used:
formula_4
and using the same example
formula_5
and using the same example as above
formula_6
where:
Feed rate.
Feed rate is the velocity at which the cutter is fed, that is, advanced against the workpiece. It is expressed in units of distance per revolution for turning and boring (typically "inches per revolution" ["ipr"] or "millimeters per revolution"). It can be expressed thus for milling also, but it is often expressed in units of distance per time for milling (typically "inches per minute" ["ipm"] or "millimeters per minute"), with considerations of how many teeth (or flutes) the cutter has then determined what that means for each tooth.
Feed rate is dependent on the:
When deciding what feed rate to use for a certain cutting operation, the calculation is fairly straightforward for single-point cutting tools, because all of the cutting work is done at one point (done by "one tooth", as it were). With a milling machine or jointer, where multi-tipped/multi-fluted cutting tools are involved, then the desired feed rate becomes dependent on the number of teeth on the cutter, as well as the desired amount of material per tooth to cut (expressed as chip load). The greater the number of cutting edges, the higher the feed rate permissible: for a cutting edge to work efficiently it must remove sufficient material to cut rather than rub; it also must do its fair share of work.
The ratio of the spindle speed and the feed rate controls how aggressive the cut is, and the nature of the swarf formed.
Formula to determine feed rate.
This formula can be used to figure out the feed rate that the cutter travels into or around the work. This would apply to cutters on a milling machine, drill press and a number of other machine tools. This is not to be used on the lathe for turning operations, as the feed rate on a lathe is given as "feed per revolution."
formula_7
Where:
Depth of cut.
Cutting speed and feed rate come together with "depth of cut" to determine the "material removal rate", which is the volume of workpiece material (metal, wood, plastic, etc.) that can be removed per time unit.
Interrelationship of theory and practice.
Speed-and-feed selection is analogous to other examples of applied science, such as meteorology or pharmacology, in that the theoretical modeling is necessary and useful but can never fully predict the reality of specific cases because of the massively multivariate environment. Just as weather forecasts or drug dosages can be modeled with fair accuracy, but never with complete certainty, machinists can predict with charts and formulas the approximate speed and feed values that will work best on a particular job, but cannot know the exact optimal values until running the job. In CNC machining, usually the programmer programs speeds and feedrates that are as maximally tuned as calculations and general guidelines can supply. The operator then fine-tunes the values while running the machine, based on sights, sounds, smells, temperatures, tolerance holding, and tool tip lifespan. Under proper management, the revised values are captured for future use, so that when a program is run again later, this work need not be duplicated.
As with meteorology and pharmacology, however, the interrelationship of theory and practice has been developing over decades as the theory part of the balance becomes more advanced thanks to information technology. For example, an effort called the Machine Tool Genome Project is working toward providing the computer modeling (simulation) needed to predict optimal speed-and-feed combinations for particular setups in any internet-connected shop with less local experimentation and testing. Instead of the only option being the measuring and testing of the behavior of its own equipment, it will benefit from others' experience and simulation; in a sense, rather than 'reinventing a wheel', it will be able to 'make better use of existing wheels already developed by others in remote locations'.
Academic research examples.
Speeds and feeds have been studied scientifically since at least the 1890s. The work is typically done in engineering laboratories, with the funding coming from three basic roots: corporations, governments (including their militaries), and universities. All three types of institution have invested large amounts of money in the cause, often in collaborative partnerships. Examples of such work are highlighted below.
In the 1890s through 1910s, Frederick Winslow Taylor performed turning experiments that became famous (and seminal). He developed Taylor's Equation for Tool Life Expectancy.
Scientific study by Holz and De Leeuw of the Cincinnati Milling Machine Company did for milling cutters what F. W. Taylor had done for single-point cutters.
"Following World War II, many new alloys were developed. New standards were needed to increase [U.S.] American productivity. Metcut Research Associates, with technical support from the Air Force Materials Laboratory and the Army Science and Technology Laboratory, published the first Machining Data Handbook in 1966. The recommended speeds and feeds provided in this book were the result of extensive testing to determine optimum tool life under controlled conditions for every material of the day, operation and hardness."
A study on the effect of the variation of cutting parameters in the surface integrity in turning of an AISI 304 stainless steel revealed that the feed rate has the greatest impairing effect on the quality of the surface, and that besides the achievement of the desired roughness profile, it is necessary to analyze the effect of speed and feed on the creation of micropits and microdefects on the machined surface. Moreover, they found that the conventional empirical relation that relates feed rate to roughness value does not fit adequately for low cutting speeds.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " {\\pi }"
},
{
"math_id": 1,
"text": "RPM = {Cutting Speed\\times 12 \\over \\pi \\times Diameter}"
},
{
"math_id": 2,
"text": "RPM = {Cutting Speed\\times 12 \\over \\pi \\times Diameter} = {12 \\times 100 ft/min \\over 3 \\times 10 inches} = {40 revs/min}"
},
{
"math_id": 3,
"text": "RPM = {Speed \\over \\pi \\times Diameter} = { 1000 \\times 30 m/min \\over 3 \\times 10 mm} = {1000 revs/min}"
},
{
"math_id": 4,
"text": "RPM = {Speed \\over Circumference}={Speed \\over \\pi \\times Diameter}"
},
{
"math_id": 5,
"text": "RPM = {100 ft/min \\over \\pi \\times 10 \\, inches \\left ( \\frac{1 ft}{12 \\, inches} \\right )} = {100 \\over 2.62} = 38.2 revs/min"
},
{
"math_id": 6,
"text": "RPM = {30 m/min \\over \\pi \\times 10 \\, mm \\left ( \\frac{1 m}{1000 \\, mm} \\right )} = {1000*30 \\over \\pi*10} = 955 revs/min"
},
{
"math_id": 7,
"text": "FR = {RPM \\times T \\times CL} "
}
] |
https://en.wikipedia.org/wiki?curid=1527151
|
15272567
|
Yangian
|
In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse scattering method. The name "Yangian" was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang.
Initially, they were considered a convenient tool to generate the solutions of the quantum Yang–Baxter equation.
The center of the Yangian can be described by the quantum determinant.
The Yangian is a degeneration of the quantum loop algebra (i.e. the quantum affine algebra at vanishing central charge).
Description.
For any finite-dimensional semisimple Lie algebra "a", Drinfeld defined an infinite-dimensional Hopf algebra "Y"("a"), called the Yangian of "a". This Hopf algebra is a deformation of the universal enveloping algebra "U"("a"["z"]) of the Lie algebra of polynomial loops of "a" given by explicit generators and relations. The relations can be encoded by identities involving a rational "R"-matrix. Replacing it with a trigonometric "R"-matrix, one arrives at affine quantum groups, defined in the same paper of Drinfeld.
In the case of the general linear Lie algebra "gl""N", the Yangian admits a simpler description in terms of a single "ternary" (or "RTT") "relation" on the matrix generators due to Faddeev and coauthors.
The Yangian Y("gl""N") is defined to be the algebra generated by elements formula_0 with 1 ≤ "i", "j" ≤ "N" and "p" ≥ 0, subject to the relations
formula_1
Defining formula_2, setting
formula_3
and introducing the R-matrix "R"("z") = I + "z"−1 "P" on C"N"formula_4C"N",
where "P" is the operator permuting the tensor factors, the above relations can be written more simply as the ternary relation:
formula_5
The Yangian becomes a Hopf algebra with comultiplication Δ, counit ε and antipode "s" given by
formula_6
At special values of the spectral parameter formula_7, the "R"-matrix degenerates to a rank one projection. This can be used to define the quantum determinant of formula_8, which generates the center of the Yangian.
The twisted Yangian Y−("gl""2N"), introduced by G. I. Olshansky, is the co-ideal generated by the coefficients of
formula_9
where σ is the involution of "gl""2N" given by
formula_10
Applications.
Classical representation theory.
G.I. Olshansky and I.Cherednik discovered that the Yangian of "gl""N" is closely related with the branching properties of irreducible finite-dimensional representations of general linear algebras. In particular, the classical Gelfand–Tsetlin construction of a basis in the space of such a representation has a natural interpretation in the language of Yangians, studied by M.Nazarov and V.Tarasov. Olshansky, Nazarov and Molev later discovered a generalization of this theory to other classical Lie algebras, based on the twisted Yangian.
Physics.
The Yangian appears as a symmetry group in different models in physics.
Yangian appears as a symmetry group of one-dimensional exactly solvable models such as spin chains, Hubbard model and in models of one-dimensional relativistic quantum field theory.
The most famous occurrence is in planar supersymmetric Yang–Mills theory in four dimensions, where Yangian structures appear on the level of symmetries of operators, and scattering amplitude as was discovered by Drummond, Henn and Plefka.
Representation theory.
Irreducible finite-dimensional representations of Yangians were parametrized by Drinfeld in a way similar to the highest weight theory in the representation theory of semisimple Lie algebras. The role of the highest weight is played by a finite set of "Drinfeld polynomials". Drinfeld also discovered a generalization of the classical Schur–Weyl duality between representations of general linear and symmetric groups that involves the Yangian of "sl""N" and the degenerate affine Hecke algebra (graded Hecke algebra of type A, in George Lusztig's terminology).
Representations of Yangians have been extensively studied, but the theory is still under active development.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "t_{ij}^{(p)}"
},
{
"math_id": 1,
"text": " [t_{ij}^{(p+1)}, t_{kl}^{(q)}] - [t_{ij}^{(p)}, t_{kl}^{(q+1)}]= -(t_{kj}^{(p)}t_{il}^{(q)} - t_{kj}^{(q)} t_{il}^{(p)})."
},
{
"math_id": 2,
"text": "t_{ij}^{(-1)}=\\delta_{ij}"
},
{
"math_id": 3,
"text": " T(z) = \\sum_{p\\ge -1} t_{ij}^{(p)} z^{-p+1}"
},
{
"math_id": 4,
"text": "\\otimes"
},
{
"math_id": 5,
"text": "\\displaystyle{ R_{12}(z-w) T_{1}(z)T_{2}(w) = T_{2}(w) T_{1}(z) R_{12}(z-w).}"
},
{
"math_id": 6,
"text": " (\\Delta \\otimes \\mathrm{id})T(z)=T_{12}(z)T_{13}(z), \\,\\, (\\varepsilon\\otimes \\mathrm{id})T(z)= I, \\,\\, (s\\otimes \\mathrm{id})T(z)=T(z)^{-1}."
},
{
"math_id": 7,
"text": "(z-w) "
},
{
"math_id": 8,
"text": "T(z) "
},
{
"math_id": 9,
"text": "\\displaystyle{ S(z)=T(z)\\sigma T(-z),}"
},
{
"math_id": 10,
"text": "\\displaystyle{\\sigma(E_{ij}) = (-1)^{i+j}E_{2N-j+1,2N-i+1}.}"
}
] |
https://en.wikipedia.org/wiki?curid=15272567
|
15272957
|
Spectral flux density
|
Quantity that describes the rate at which energy is transferred by electromagnetic radiation
In spectroscopy, spectral flux density is the quantity that describes the rate at which energy is transferred by electromagnetic radiation through a real or virtual surface, per unit surface area and per unit wavelength (or, equivalently, per unit frequency). It is a radiometric rather than a photometric measure. In SI units it is measured in W m−3, although it can be more practical to use W m−2 nm−1 (1 W m−2 nm−1 = 1 GW m−3 = 1 W mm−3) or W m−2 μm−1 (1 W m−2 μm−1 = 1 MW m−3), and respectively by W·m−2·Hz−1, Jansky or solar flux units. The terms irradiance, radiant exitance, radiant emittance, and radiosity are closely related to spectral flux density.
The terms used to describe spectral flux density vary between fields, sometimes including adjectives such as "electromagnetic" or "radiative", and sometimes dropping the word "density". Applications include:
Flux density received from an unresolvable "point source".
For the flux density received from a remote unresolvable "point source", the measuring instrument, usually telescopic, though not able to resolve any detail of the source itself, must be able to optically resolve enough details of the sky around the point source, so as to record radiation coming from it only, uncontaminated by radiation from other sources. In this case, spectral flux density is the quantity that describes the rate at which energy transferred by electromagnetic radiation is received from that unresolved point source, per unit receiving area facing the source, per unit wavelength range.
At any given wavelength "λ", the spectral flux density, Fλ, can be determined by the following procedure:
Spectral flux density is often used as the quantity on the "y"-axis of a graph representing the spectrum of a light-source, such as a star.
Flux density of the radiative field at a measuring point.
There are two main approaches to definition of the spectral flux density at a measuring point in an electromagnetic radiative field. One may be conveniently here labelled the 'vector approach', the other the 'scalar approach'. The vector definition refers to the full spherical integral of the spectral radiance (also known as the specific radiative intensity or specific intensity) at the point, while the scalar definition refers to the many possible hemispheric integrals of the spectral radiance (or specific intensity) at the point. The vector definition seems to be preferred for theoretical investigations of the physics of the radiative field. The scalar definition seems to be preferred for practical applications.
Vector definition of flux density - 'full spherical flux density'.
The vector approach defines flux density as a vector at a point of space and time prescribed by the investigator. To distinguish this approach, one might speak of the 'full spherical flux density'. In this case, nature tells the investigator what is the magnitude, direction, and sense of the flux density at the prescribed point. For the flux density vector, one may write
formula_0
where formula_1 denotes the spectral radiance (or specific intensity) at the point formula_2 at time formula_3 and frequency formula_4, formula_5 denotes a variable unit vector with origin at the point formula_2, formula_6 denotes an element of solid angle around formula_5, and formula_7 indicates that the integration extends over the full range of solid angles of a sphere.
Mathematically, defined as an unweighted integral over the solid angle of a full sphere, the flux density is the first moment of the spectral radiance (or specific intensity) with respect to solid angle. It is not common practice to make the full spherical range of measurements of the spectral radiance (or specific intensity) at the point of interest, as is needed for the mathematical spherical integration specified in the strict definition; the concept is nevertheless used in theoretical analysis of radiative transfer.
As described below, if the direction of the flux density vector is known in advance because of a symmetry, namely that the radiative field is uniformly layered and flat, then the vector flux density can be measured as the 'net flux', by algebraic summation of two oppositely sensed scalar readings in the known direction, perpendicular to the layers.
At a given point in space, in a steady-state field, the vector flux density, a radiometric quantity, is equal to the time-averaged Poynting vector, an electromagnetic field quantity.
Within the vector approach to the definition, however, there are several specialized sub-definitions. Sometimes the investigator is interested only in a specific direction, for example the vertical direction referred to a point in a planetary or stellar atmosphere, because the atmosphere there is considered to be the same in every horizontal direction, so that only the vertical component of the flux is of interest. Then the horizontal components of flux are considered to cancel each other by symmetry, leaving only the vertical component of the flux as non-zero. In this case some astrophysicists think in terms of the astrophysical flux (density), which they define as the vertical component of the flux (of the above general definition) divided by the number π. And sometimes the astrophysicist uses the term Eddington flux to refer to the vertical component of the flux (of the above general definition) divided by the number 4π.
Scalar definition of flux density - 'hemispheric flux density'.
The scalar approach defines flux density as a scalar-valued function of a direction and sense in space prescribed by the investigator at a point prescribed by the investigator. Sometimes this approach is indicated by the use of the term 'hemispheric flux'. For example, an investigator of thermal radiation, emitted from the material substance of the atmosphere, received at the surface of the earth, is interested in the vertical direction, and the downward sense in that direction. This investigator thinks of a unit area in a horizontal plane, surrounding the prescribed point. The investigator wants to know the total power of all the radiation from the atmosphere above in every direction, propagating with a downward sense, received by that unit area. For the flux density scalar for the prescribed direction and sense, we may write
formula_8
where with the notation above, formula_9 indicates that the integration extends only over the solid angles of the relevant hemisphere, and formula_10 denotes the angle between formula_5 and the prescribed direction. The term formula_11 is needed on account of Lambert's law. Mathematically, the quantity formula_12 is not a vector because it is a positive scalar-valued function of the prescribed direction and sense, in this example, of the downward vertical. In this example, when the collected radiation is propagating in the downward sense, the detector is said to be "looking upwards". The measurement can be made directly with an instrument (such as a pyrgeometer) that collects the measured radiation all at once from all the directions of the imaginary hemisphere; in this case, Lambert-cosine-weighted integration of the spectral radiance (or specific intensity) is not performed mathematically after the measurement; the Lambert-cosine-weighted integration has been performed by the physical process of measurement itself.
Net flux.
In a flat horizontal uniformly layered radiative field, the hemispheric fluxes, upwards and downwards, at a point, can be subtracted to yield what is often called the net flux. The net flux then has a value equal to the magnitude of the full spherical flux vector at that point, as described above.
Comparison between vector and scalar definitions of flux density.
The radiometric description of the electromagnetic radiative field at a point in space and time is completely represented by the spectral radiance (or specific intensity) at that point. In a region in which the material is uniform and the radiative field is isotropic and homogeneous, let the spectral radiance (or specific intensity) be denoted by "I" (x, "t" ; r1, ν), a scalar-valued function of its arguments x, "t", r1, and ν, where r1 denotes a unit vector with the direction and sense of the geometrical vector r from the source point "P"1 to the detection point "P"2, where x denotes the coordinates of "P"1, at time "t" and wave frequency ν. Then, in the region, "I" (x, "t" ; r1, ν) takes a constant scalar value, which we here denote by "I". In this case, the value of the vector flux density at "P"1 is the zero vector, while the scalar or hemispheric flux density at "P"1 in every direction in both senses takes the constant scalar value π"I". The reason for the value π"I" is that the hemispheric integral is half the full spherical integral, and the integrated effect of the angles of incidence of the radiation on the detector requires a halving of the energy flux according to Lambert's cosine law; the solid angle of a sphere is 4π.
The vector definition is suitable for the study of general radiative fields. The scalar or hemispheric spectral flux density is convenient for discussions in terms of the two-stream model of the radiative field, which is reasonable for a field that is uniformly stratified in flat layers, when the base of the hemisphere is chosen to be parallel to the layers, and one or other sense (up or down) is specified. In an inhomogeneous non-isotropic radiative field, the spectral flux density defined as a scalar-valued function of direction and sense contains much more directional information than does the spectral flux density defined as a vector, but the full radiometric information is customarily stated as the spectral radiance (or specific intensity).
Collimated beam.
For the present purposes, the light from a star, and for some particular purposes, the light of the sun, can be treated as a practically collimated beam, but apart from this, a collimated beam is rarely if ever found in nature, though artificially produced beams can be very nearly collimated. The spectral radiance (or specific intensity) is suitable for the description of an uncollimated radiative field. The integrals of spectral radiance (or specific intensity) with respect to solid angle, used above, are singular for exactly collimated beams, or may be viewed as Dirac delta functions. Therefore, the specific radiative intensity is unsuitable for the description of a collimated beam, while spectral flux density is suitable for that purpose. At a point within a collimated beam, the spectral flux density vector has a value equal to the Poynting vector, a quantity defined in the classical Maxwell theory of electromagnetic radiation.
Relative spectral flux density.
Sometimes it is more convenient to display graphical spectra with vertical axes that show the relative spectral flux density. In this case, the spectral flux density at a given wavelength is expressed as a fraction of some arbitrarily chosen reference value. Relative spectral flux densities are expressed as pure numbers without any units.
Spectra showing the relative spectral flux density are used when we are interested in comparing the spectral flux densities of different sources; for example, if we want to show how the spectra of blackbody sources vary with absolute temperature, it is not necessary to show the absolute values. The relative spectral flux density is also useful if we wish to compare a source's flux density at one wavelength with the same source's flux density at another wavelength; for example, if we wish to demonstrate how the Sun's spectrum peaks in the visible part of the EM spectrum, a graph of the Sun's relative spectral flux density will suffice.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbf{F}(\\mathbf{x}, t;\\nu) = \\oint_\\Omega\\ I(\\mathbf{x}, t;\\mathbf{\\hat{n}},\\nu) \\,\\mathbf{\\hat{n}} \\,d\\omega(\\mathbf{\\hat{n}})"
},
{
"math_id": 1,
"text": "I(\\mathbf{x}, t;\\mathbf{\\hat{n}},\\nu)"
},
{
"math_id": 2,
"text": "\\mathbf{x}"
},
{
"math_id": 3,
"text": "t"
},
{
"math_id": 4,
"text": "\\nu\\!"
},
{
"math_id": 5,
"text": "\\mathbf{\\hat{n}}"
},
{
"math_id": 6,
"text": "d\\omega(\\mathbf{\\hat{n}})"
},
{
"math_id": 7,
"text": "\\Omega"
},
{
"math_id": 8,
"text": "F(\\mathbf{x}, t;\\nu) = \\int_{\\Omega^{^+}} I(\\mathbf{x}, t;\\mathbf{\\hat{n}},\\nu) \\,\\cos \\,(\\theta(\\mathbf{\\hat{n}})) \\,d\\omega(\\mathbf{\\hat{n}})"
},
{
"math_id": 9,
"text": "\\Omega^{^+}"
},
{
"math_id": 10,
"text": "\\theta(\\mathbf{\\hat{n}})"
},
{
"math_id": 11,
"text": "\\cos \\,(\\theta(\\mathbf{\\hat{n}}))"
},
{
"math_id": 12,
"text": "F(\\mathbf{x}, t;\\nu)"
}
] |
https://en.wikipedia.org/wiki?curid=15272957
|
152734
|
General number field sieve
|
Factorization algorithm
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of bits) is of the form
formula_0
in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots).
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve. When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order "n"1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
The size of the input to the algorithm is log2 "n" or the number of bits in the binary representation of n. Any element of the order "n""c" for a constant c is exponential in log "n". The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.
Number fields.
Suppose f is a k-degree polynomial over formula_1 (the rational numbers), and r is a complex root of f. Then, "f"("r")
0, which can be rearranged to express "r""k" as a linear combination of powers of r less than k. This equation can be used to reduce away any powers of "r" with exponent "e" ≥ "k". For example, if "f"("x")
"x"2 + 1 and r is the imaginary unit i, then "i"2 + 1
0, or "i"2
−1. This allows us to define the complex product:
formula_2
In general, this leads directly to the algebraic number field formula_3, which can be defined as the set of complex numbers given by:
formula_4
The product of any two such values can be computed by taking the product as polynomials, then reducing any powers of "r" with exponent "e" ≥ "k" as described above, yielding a value in the same form. To ensure that this field is actually k-dimensional and does not collapse to an even smaller field, it is sufficient that f is an irreducible polynomial over the rationals. Similarly, one may define the ring of integers formula_5 as the subset of formula_3 which are roots of monic polynomials with integer coefficients. In some cases, this ring of integers is equivalent to the ring formula_6. However, there are many exceptions.
Method.
Two polynomials "f"("x") and "g"("x") of small degrees "d" and "e" are chosen, which have integer coefficients, which are irreducible over the rationals, and which, when interpreted mod "n", have a common integer root "m". An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree "d" for a polynomial, consider the expansion of "n" in base "m" (allowing digits between −"m" and "m") for a number of different "m" of order "n"1/"d", and pick "f"("x") as the polynomial with the smallest coefficients and "g"("x") as "x" − "m".
Consider the number field rings Z["r"1] and Z["r"2], where "r"1 and "r"2 are roots of the polynomials "f" and "g". Since "f" is of degree "d" with integer coefficients, if "a" and "b" are integers, then so will be "b""d"·"f"("a"/"b"), which we call "r". Similarly, "s" = "b""e"·"g"("a"/"b") is an integer. The goal is to find integer values of "a" and "b" that simultaneously make "r" and "s" smooth relative to the chosen basis of primes. If "a" and "b" are small, then "r" and "s" will be small too, about the size of "m", and we have a better chance for them to be smooth at the same time. The current best-known approach for this search is lattice sieving; to get acceptable yields, it is necessary to use a large factor base.
Having enough such pairs, using Gaussian elimination, one can get products of certain "r" and of the corresponding "s" to be squares at the same time. A slightly stronger condition is needed—that they are norms of squares in our number fields, but that condition can be achieved by this method too. Each "r" is a norm of "a" − "r"1"b" and hence that the product of the corresponding factors "a" − "r"1"b" is a square in Z["r"1], with a "square root" which can be determined (as a product of known factors in Z["r"1])—it will typically be represented as an irrational algebraic number. Similarly, the product of the factors "a" − "r"2"b" is a square in Z["r"2], with a "square root" which also can be computed. It should be remarked that the use of Gaussian elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as Block Lanczos or Block Wiedemann are used.
Since "m" is a root of both "f" and "g" mod "n", there are homomorphisms from the rings Z["r"1] and Z["r"2] to the ring Z/"n"Z (the integers modulo "n"), which map "r"1 and "r"2 to "m", and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now the product of the factors "a" − "mb" mod "n" can be obtained as a square in two ways—one for each homomorphism. Thus, one can find two numbers "x" and "y", with "x"2 − "y"2 divisible by "n" and again with probability at least one half we get a factor of "n" by finding the greatest common divisor of "n" and "x" − "y".
Improving polynomial choice.
The choice of polynomial can dramatically affect the time to complete the remainder of the algorithm. The method of choosing polynomials based on the expansion of n in base m shown above is suboptimal in many practical situations, leading to the development of better methods.
One such method was suggested by Murphy and Brent; they introduce a two-part score for polynomials, based on the presence of roots modulo small primes and on the average value that the polynomial takes over the sieving area.
The best reported results were achieved by the method of Thorsten Kleinjung, which allows "g"("x")
"ax" + "b", and searches over a composed of small prime factors congruent to 1 modulo 2"d" and over leading coefficients of f which are divisible by 60.
Implementations.
Some implementations focus on a certain smaller class of numbers. These are known as special number field sieve techniques, such as used in the Cunningham project.
A project called NFSNET ran from 2002 through at least 2007. It used volunteer distributed computing on the Internet.
Paul Leyland of the United Kingdom and Richard Wackerbarth of Texas were involved.
Until 2007, the gold-standard implementation was a suite of software developed and distributed by CWI in the Netherlands, which was available only under a relatively restrictive license. In 2007, Jason Papadopoulos developed a faster implementation of final processing as part of msieve, which is in the public domain. Both implementations feature the ability to be distributed among several nodes in a cluster with a sufficiently fast interconnect.
Polynomial selection is normally performed by GPL software written by Kleinjung, or by msieve, and lattice sieving by GPL software written by Franke and Kleinjung; these are distributed in GGNFS.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\n\\begin{align}\n& \\exp\\left(\\left((64/9)^{1/3}+o(1)\\right)\\left(\\log n\\right)^{1/3} \\left(\\log\\log n\\right)^{2/3}\\right) \\\\[5pt]\n= {} & L_n\\left[1/3,(64/9)^{1/3}\\right]\n\\end{align}\n"
},
{
"math_id": 1,
"text": "\\mathbb Q"
},
{
"math_id": 2,
"text": "\n\\begin{align}\n(a+bi)(c+di) & = ac + (ad+bc)i + (bd)i^2 \\\\[4pt]\n& = (ac - bd) + (ad+bc)i.\n\\end{align}\n"
},
{
"math_id": 3,
"text": "\\mathbb Q[r]"
},
{
"math_id": 4,
"text": "a_{k-1}r^{k-1} + \\cdots + a_1 r^1 + a_0 r^0, \\text{ where } a_0,\\ldots,a_{k-1} \\in \\mathbb Q."
},
{
"math_id": 5,
"text": " \\mathbb O_{\\mathbb Q[r]} "
},
{
"math_id": 6,
"text": " \\mathbb Z[r] "
}
] |
https://en.wikipedia.org/wiki?curid=152734
|
15275
|
ISO 216
|
International standard for paper sizes, including A4
ISO 216 is an international standard for paper sizes, used around the world except in North America and parts of Latin America. The standard defines the "A", "B" and "C" series of paper sizes, which includes the A4, the most commonly available paper size worldwide. Two supplementary standards, ISO 217 and ISO 269, define related paper sizes; the ISO 269 "C" series is commonly listed alongside the A and B sizes.
All ISO 216, ISO 217 and ISO 269 paper sizes (except some envelopes) have the same aspect ratio, , within rounding to millimetres. This ratio has the unique property that when cut or folded in half widthways, the halves also have the same aspect ratio. Each ISO paper size is one half of the area of the next larger size in the same series.
History.
The oldest known mention of the advantages of basing a paper size on an aspect ratio of formula_0 is found in a letter written on 25 October 1786 by the German scientist Georg Christoph Lichtenberg to Johann Beckmann, both at the University of Göttingen. Early variants of the formats that would become ISO paper sizes A2, A3, B3, B4, and B5 then evolved in France, where they were listed in a 1798 French law on taxation of publications () that was based in part on page sizes.
Searching for a standard system of paper formats on a scientific basis at the Bridge association (), as a replacement for the vast variety of other paper formats that had been used before, in order to make paper stocking and document reproduction cheaper and more efficient, in 1911 Wilhelm Ostwald proposed, over a hundred years after the 1798 French law, a global standard – a world format () – for paper sizes based on the ratio formula_0, referring to the argument advanced by Lichtenberg's 1786 letter, but linking this to the metric system using as the width of the base format. Walter Porstmann argued in a long article published in 1918, that a firm basis for the system of paper formats, which deal with surfaces, ought not be the length but the area; that is, linking the system of paper formats to the metric system using the square metre rather than the centimetre, constrained by formula_1 and area formula_2 square metre, where formula_3 is the length of the shorter side and formula_4 is the length of the longer side, for the second equation both in metres. Porstmann also argued that formats for "containers" of paper, such as envelopes, should be 10% larger than the paper format itself.
In 1921, after a long discussion and another intervention by Porstmann, the Standardisation Committee of German Industry (, or NADI in short), which is the German Institute for Standardisation (, or DIN in short) today, published German standard "DI Norm 476" the specification of four series of paper formats with ratio formula_0, with series A as the always preferred formats and basis for the other series. All measures are rounded to the nearest millimetre. A0 has a surface area of up to a rounding error, with a width of and height of , so an actual area of ; A4 is recommended as standard paper size for business, administrative and government correspondence; and A6 for postcards. Series B is based on B0 with width of , C0 is , and D0 . Series C is the basis for envelope formats.
The DIN paper-format concept was soon introduced as a national standard in many other countries, for example, Belgium (1924), Netherlands (1925), Norway (1926), Switzerland (1929), Sweden (1930), Soviet Union (1934), Hungary (1938), Italy (1939), Finland (1942), Uruguay (1942), Argentina (1943), Brazil (1943), Spain (1947), Austria (1948), Romania (1949), Japan (1951), Denmark (1953), Czechoslovakia (1953), Israel (1954), Portugal (1954), Yugoslavia (1956), India (1957), Poland (1957), United Kingdom (1959), Venezuela (1962), New Zealand (1963), Iceland (1964), Mexico (1965), South Africa (1966), France (1967), Peru (1967), Turkey (1967), Chile (1968), Greece (1970), Zimbabwe (1970), Singapore (1970), Bangladesh (1972), Thailand (1973), Barbados (1973), Australia (1974), Ecuador (1974), Colombia (1975) and Kuwait (1975).
It finally became both an international standard (ISO 216) as well as the official United Nations document format in 1975, and it is today used in almost all countries in the world, with the exception of several countries in the Americas.
In 1977, a large German car manufacturer performed a study of the paper formats found in their incoming mail and concluded that out of 148 examined countries, 88 already used the A series formats.
Advantages.
The main advantage of this system is its scaling. Rectangular paper with an aspect ratio of formula_0 has the unique property that, when cut or folded in half midway between its shorter sides, each half has the same formula_0 aspect ratio as the whole sheet before it was divided. Equivalently, if one lays two same-sized sheets of paper with an aspect ratio of formula_0 side by side along their longer side, they form a larger rectangle with the aspect ratio of formula_0 and double the area of each individual sheet.
The ISO system of paper sizes exploits these properties of the formula_0 aspect ratio. In each series of sizes (for example, series A), the largest size is numbered 0 (so in this case A0), and each successive size (A1, A2, etc.) has half the area of the preceding sheet and can be cut by halving the length of the preceding size sheet. The new measurement is rounded down to the nearest millimetre. A folded brochure can be made by using a sheet of the next larger size (for example, an A4 sheet is folded in half to make a brochure with size A5 pages). An office photocopier or printer can be designed to reduce a page from A4 to A5 or to enlarge a page from A4 to A3. Similarly, two sheets of A4 can be scaled down to fit one A4 sheet without excess empty paper.
This system also simplifies calculating the weight of paper. Under ISO 536, paper's grammage is defined as a sheet's mass in grams (g) per area in square metres (unit symbol g/m2; the nonstandard abbreviation "gsm" is also used). One can derive the grammage of other sizes by arithmetic division. A standard A4 sheet made from 80 g/m2 paper weighs , as it is <templatestyles src="Fraction/styles.css" />1⁄16 (four halvings, ignoring rounding) of an A0 page. Thus the weight, and the associated postage rate, can be approximated easily by counting the number of sheets used.
ISO 216 and its related standards were first published between 1975 and 1995:
Properties.
A series.
Paper in the A series format has an aspect ratio of (≈ 1.414, when rounded). A0 is defined so that it has an area of before rounding to the nearest . Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the area of the preceding paper size and rounding down, so that the long side of A("n" + 1) is the same length as the short side of A"n". Hence, each next size is nearly exactly half the area of the prior size. So, an A1 page can fit two A2 pages inside the same area.
The most used of this series is the size A4, which is and thus almost exactly in area. For comparison, the letter paper size commonly used in North America () is about 6 mm (0.24 in) wider and 18 mm (0.71 in) shorter than A4. Then, the size of A5 paper is half of A4, i.e. 148 mm × 210 mm (5.8 in × 8.3 in).
The geometric rationale for using the square root of 2 is to maintain the aspect ratio of each subsequent rectangle after cutting or folding an A-series sheet in half, perpendicular to the larger side. Given a rectangle with a longer side, "x", and a shorter side, "y", ensuring that its aspect ratio, , will be the same as that of a rectangle half its size, , which means that =, which reduces to = √2; in other words, an aspect ratio of .
Any A"n" paper can be defined as A"n" = "S" × "L", where (measuring in metres)
formula_5
Therefore
formula_6, formula_7 formula_8 Etc.
B series.
The B series is defined in the standard as follows: "A subsidiary series of sizes is obtained by placing the geometrical means between adjacent sizes of the A series in sequence." The use of the geometric mean makes each step in size: B0, A0, B1, A1, B2 ... smaller than the previous one by the same factor. As with the A series, the lengths of the B series have the ratio , and folding one in half (and rounding down to the nearest millimetre) gives the next in the series. The shorter side of B0 is exactly 1 metre.
There is also an incompatible Japanese B series which the JIS defines to have 1.5 times the area of the corresponding JIS A series (which is identical to the ISO A series). Thus, the lengths of JIS B series paper are ≈ 1.22 times those of A-series paper. By comparison, the lengths of ISO B series paper are ≈ 1.19 times those of A-series paper.
Any B"n" paper (according to the ISO standard) can be defined as B"n" = "S" × "L", where (measuring in metres)
formula_9
Therefore
formula_10, formula_11 formula_12 Etc.
C series.
The C series formats are geometric means between the B series and A series formats with the same number (e.g. C2 is the geometric mean between B2 and A2). The width to height ratio of C series formats is as in the A and B series. A, B, and C series of paper fit together as part of a geometric progression, with ratio of successive side lengths of , though there is no size half-way between B"n" and A("n" − 1): A4, C4, B4, "D4", A3, ...; there is such a D-series in the Swedish extensions to the system. The lengths of ISO C series paper are therefore ≈ 1.09 times those of A-series paper.
The C series formats are used mainly for envelopes. An unfolded A4 page will fit into a C4 envelope. Due to same width to height ratio, if an A4 page is folded in half so that it is A5 in size, it will fit into a C5 envelope (which will be the same size as a C4 envelope folded in half).
Any C"n" paper can be defined as C"n" = "S" × "L", where (measuring in metres)
formula_13
Therefore
formula_14, formula_15 formula_16 Etc.
Tolerances.
The tolerances specified in the standard are:
These are related to comparison between series A, B and C.
Application.
The ISO 216 formats are organized around the ratio 1:; two sheets next to each other together have the same ratio, sideways. In scaled photocopying, for example, two A4 sheets reduced to A5 size fit exactly onto one A4 sheet, and an A4 sheet in magnified size onto an A3 sheet; in each case, there is neither waste nor want.
The principal countries not generally using the ISO paper sizes are the United States and Canada, which use North American paper sizes. Although they have also officially adopted the ISO 216 paper format, Mexico, Panama, Peru, Colombia, the Philippines, and Chile also use mostly U.S. paper sizes.
Rectangular sheets of paper with the ratio 1: are popular in paper folding, such as origami, where they are sometimes called "A4 rectangles" or "silver rectangles". In other contexts, the term "silver rectangle" can also refer to a rectangle in the proportion 1:(1 + ), known as the silver ratio.
Matching technical pen widths.
An adjunct to the ISO paper sizes, particularly the A series, are the technical drawing line widths specified in ISO 128. For example, line type A ("Continuous - thick", used for "visible outlines") has a standard thickness of 0.7 mm on an A0-sized sheet, 0.5 mm on an A1 sheet, and 0.35 mm on A2, A3, or A4.
The matching technical pen widths are 0.13, 0.18, 0.25, 0.35, 0.5, 0.7, 1.0, 1.40, and 2.0 mm, as specified in ISO 9175-1. Colour codes are assigned to each size to facilitate easy recognition by the drafter. Like the paper sizes, these pen widths increase by a factor of , so that particular pens can be used on particular sizes of paper, and then the next smaller or larger size can be used to continue the drawing after it has been reduced or enlarged, respectively.
The earlier DIN 6775 standard upon which ISO 9175-1 is based also specified a term and symbol for easy identification of pens and drawing templates compatible with the standard, called "Micronorm", which may still be found on some technical drafting equipment.
Overformats.
DIN 476 provides for formats larger than A0, denoted by a prefix factor. In particular, it lists the formats 2A0 and 4A0, which are twice and four times the size of A0 respectively:
While not formally defined, ISO 216:2007 notes them in the table of "Main series of trimmed sizes" (ISO A series) as well: "The rarely used sizes [2A0 and 4A0] which follow also belong to this series." 2A0 is also known by other unofficial names like "A00".
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sqrt{2}"
},
{
"math_id": 1,
"text": "\\tfrac{x}{y}=\\sqrt{2}"
},
{
"math_id": 2,
"text": "a = x \\times y = 1"
},
{
"math_id": 3,
"text": "x"
},
{
"math_id": 4,
"text": "y"
},
{
"math_id": 5,
"text": "\\text{A}_n = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{n + \\frac{1}{2}}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{n - \\frac{1}{2}}\n\\end{cases}"
},
{
"math_id": 6,
"text": "\\text{A0} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{0 + \\frac{1}{2}} \\approx 0.841\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{0 - \\frac{1}{2}} \\approx 1.189\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 7,
"text": "\\text{A1} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{1 + \\frac{1}{2}} \\approx 0.595\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{1 - \\frac{1}{2}} \\approx 0.841\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 8,
"text": "\\text{A2} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{2 + \\frac{1}{2}} \\approx 0.420\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{2 - \\frac{1}{2}} \\approx 0.595\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 9,
"text": "\\text{B}_n = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{n}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{n - 1}\n\\end{cases}"
},
{
"math_id": 10,
"text": "\\text{B0} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{0} = 1\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{0 - 1} \\approx 1.414\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 11,
"text": "\\text{B1} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{1} \\approx 0.707\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{1 - 1} = 1\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 12,
"text": "\\text{B2} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{2} = 0.5\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{2 - 1} \\approx 0.707\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 13,
"text": "\\text{C}_n = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{n + \\frac{1}{4}}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{n - \\frac{3}{4}}\n\\end{cases}"
},
{
"math_id": 14,
"text": "\\text{C0} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{0 + \\frac{1}{4}} \\approx 0.917\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{0 - \\frac{3}{4}} \\approx 1.297\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 15,
"text": "\\text{C1} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{1 + \\frac{1}{4}} \\approx 0.648\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{1 - \\frac{3}{4}} \\approx 0.917\\,\\text{m}\n\\end{cases}"
},
{
"math_id": 16,
"text": "\\text{C2} = \\begin{cases}\nS = \\left(\\sqrt{\\frac{1}{2}}\\right)^{2 + \\frac{1}{4}} \\approx 0.458\\,\\text{m}\\\\\nL = \\left(\\sqrt{\\frac{1}{2}}\\right)^{2 - \\frac{3}{4}} \\approx 0.648\\,\\text{m}\n\\end{cases}"
}
] |
https://en.wikipedia.org/wiki?curid=15275
|
1527574
|
Conformational isomerism
|
Different molecular structures formed only by rotation about single bonds
In chemistry, conformational isomerism is a form of stereoisomerism in which the isomers can be interconverted just by rotations about formally single bonds (refer to figure on single bond rotation). While any two arrangements of atoms in a molecule that differ by rotation about single bonds can be referred to as different conformations, conformations that correspond to local minima on the potential energy surface are specifically called conformational isomers or conformers. Conformations that correspond to local maxima on the energy surface are the transition states between the local-minimum conformational isomers. Rotations about single bonds involve overcoming a rotational energy barrier to interconvert one conformer to another. If the energy barrier is low, there is free rotation and a sample of the compound exists as a rapidly equilibrating mixture of multiple conformers; if the energy barrier is high enough then there is restricted rotation, a molecule may exist for a relatively long time period as a stable rotational isomer or rotamer (an isomer arising from hindered single-bond rotation). When the time scale for interconversion is long enough for isolation of individual rotamers (usually arbitrarily defined as a half-life of interconversion of 1000 seconds or longer), the isomers are termed atropisomers ("see:" atropisomerism). The ring-flip of substituted cyclohexanes constitutes another common form of conformational isomerism.
Conformational isomers are thus distinct from the other classes of stereoisomers (i. e. configurational isomers) where interconversion necessarily involves breaking and reforming of chemical bonds. For example, /- and "R"/"S"- configurations of organic molecules have different handedness and optical activities, and can only be interconverted by breaking one or more bonds connected to the chiral atom and reforming a similar bond in a different direction or spatial orientation. They also differ from geometric ("cis"/"trans") isomers, another class of stereoisomers, which require the π-component of double bonds to break for interconversion. (Although the distinction is not always clear-cut, since certain bonds that are formally single bonds actually have double bond character that becomes apparent only when secondary resonance contributors are considered, like the C–N bonds of amides, for instance.) Due to rapid interconversion, conformers are usually not isolable at room temperature.
The study of the energetics between different conformations is referred to as conformational analysis. It is useful for understanding the stability of different isomers, for example, by taking into account the spatial orientation and through-space interactions of substituents. In addition, conformational analysis can be used to predict and explain product selectivity, mechanisms, and rates of reactions. Conformational analysis also plays an important role in rational, structure-based drug design.
Types.
Rotating their carbon–carbon bonds, the molecules ethane and propane have three local energy minima. They are structurally and energetically equivalent, and are called the "staggered conformers". For each molecule, the three substituents emanating from each carbon–carbon bond are staggered, with each H–C–C–H dihedral angle (and H–C–C–CH3 dihedral angle in the case of propane) equal to 60° (or approximately equal to 60° in the case of propane). The three eclipsed conformations, in which the dihedral angles are zero, are transition states (energy maxima) connecting two equivalent energy minima, the staggered conformers.
The butane molecule is the simplest molecule for which single bond rotations result in two types of nonequivalent structures, known as the "anti"- and "gauche-"conformers (see figure).
For example, butane has three conformers relating to its two methyl (CH3) groups: two "gauche" conformers, which have the methyls ±60° apart and are enantiomeric, and an "anti" conformer, where the four carbon centres are coplanar and the substituents are 180° apart (refer to free energy diagram of butane). The energy difference between gauche and anti is 0.9 kcal/mol associated with the strain energy of the gauche conformer. The anti conformer is, therefore, the most stable (≈ 0 kcal/mol). The three eclipsed conformations with dihedral angles of 0°, 120°, and 240° are transition states between conformers. Note that the two eclipsed conformations have different energies: at 0° the two methyl groups are eclipsed, resulting in higher energy (≈ 5 kcal/mol) than at 120°, where the methyl groups are eclipsed with hydrogens (≈ 3.5 kcal/mol).
While simple molecules can be described by these types of conformations, more complex molecules require the use of the Klyne–Prelog system to describe the different conformers.
More specific examples of conformational isomerism are detailed elsewhere:
Free energy and equilibria of conformational isomers.
Equilibrium of conformers.
Conformational isomers exist in a dynamic equilibrium, where the relative free energies of isomers determines the population of each isomer and the energy barrier of rotation determines the rate of interconversion between isomers:
formula_0
where "K" is the equilibrium constant, Δ"G°" is the difference in standard free energy between the two conformers in kcal/mol, "R" is the universal gas constant (1.987×10−3 kcal/mol K), and "T" is the system's temperature in kelvins. In units of kcal/mol at 298 K,
formula_1
Thus, every 1.36 kcal/mol corresponds to a factor of about 10 in term of equilibrium constant at temperatures around room temperature. (The "1.36 rule" is useful in general for estimation of equilibrium constants at room temperature from free energy differences. At lower temperatures, a smaller energy difference is needed to obtain a given equilibrium constant.)
Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at different temperatures. At a free energy difference of 0 kcal/mol, this gives an equilibrium constant of 1, meaning that two conformers exist in a 1:1 ratio. The two have equal free energy; neither is more stable, so neither predominates compared to the other. A negative difference in free energy means that a conformer interconverts to a thermodynamically more stable conformation, thus the equilibrium constant will always be greater than 1. For example, the Δ"G°" for the transformation of butane from the "gauche" conformer to the "anti" conformer is −0.47 kcal/mol at 298 K. This gives an equilibrium constant is about 2.2 in favor of the "anti" conformer, or a 31:69 mixture of "gauche":"anti" conformers at equilibrium. Conversely, a positive difference in free energy means the conformer already is the more stable one, so the interconversion is an unfavorable equilibrium ("K" < 1). Even for highly unfavorable changes (large positive Δ"G°"), the equilibrium constant between two conformers can be increased by increasing the temperature, so that the amount of the less stable conformer present at equilibrium increases (although it always remains the minor conformer).
Population distribution of conformers.
The fractional population distribution of different conformers follows a Boltzmann distribution:
formula_2
The left hand side is the proportion of conformer "i" in an equilibrating mixture of "M" conformers in thermodynamic equilibrium. On the right side, "E""k" ("k" = 1, 2, ..., "M") is the energy of conformer "k", "R" is the molar ideal gas constant (approximately equal to 8.314 J/(mol·K) or 1.987 cal/(mol·K)), and "T" is the absolute temperature. The denominator of the right side is the partition function.
Factors contributing to the free energy of conformers.
The effects of electrostatic and steric interactions of the substituents as well as orbital interactions such as hyperconjugation are responsible for the relative stability of conformers and their transition states. The contributions of these factors vary depending on the nature of the substituents and may either contribute positively or negatively to the energy barrier. Computational studies of small molecules such as ethane suggest that electrostatic effects make the greatest contribution to the energy barrier; however, the barrier is traditionally attributed primarily to steric interactions.
In the case of cyclic systems, the steric effect and contribution to the free energy can be approximated by A values, which measure the energy difference when a substituent on cyclohexane in the axial as compared to the equatorial position. In large (>14 atom) rings, there are many accessible low-energy conformations which correspond to the strain-free diamond lattice.
Isolation or observation of conformational isomers.
The short timescale of interconversion precludes the separation of conformational isomers in most cases. Atropisomers are conformational isomers which can be separated due to restricted rotation. The equilibrium between conformational isomers can be observed using a variety of spectroscopic techniques.
Protein folding also generates stable conformational isomers which can be observed. The Karplus equation relates the dihedral angle of vicinal protons to their J-coupling constants as measured by NMR. The equation aids in the elucidation of protein folding as well as the conformations of other rigid aliphatic molecules. Protein side chains exhibit rotamers, whose distribution is determined by their steric interaction with different conformations of the backbone. This is evident from statistical analysis of the conformations of protein side chains in the Backbone-dependent rotamer library.
In cyclohexane derivatives, the two chair conformers interconvert with rapidly at room temperature, with cyclohexane itself undergoing the ring-flip at a rates of approximately 105 ring-flips/sec, with an overall energy barrier of 10 kcal/mol (42 kJ/mol), which precludes their separation at ambient temperatures. However, at low temperatures below the coalescence point one can directly monitor the equilibrium by NMR spectroscopy and by dynamic, temperature dependent NMR spectroscopy the barrier interconversion.
The dynamics of conformational (and other kinds of) isomerism can be monitored by NMR spectroscopy at varying temperatures. The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called "fluxional".
Besides NMR spectroscopy, IR spectroscopy is used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, νCBr differs by almost 50 cm−1.
Conformation-dependent reactions.
Reaction rates are highly dependent on the conformation of the reactants. In many cases the dominant product arises from the reaction of the "less prevalent" conformer, by virtue of the Curtin-Hammett principle. This is typical for situations where the conformational equilibration is much faster than reaction to form the product. The dependence of a reaction on the stereochemical orientation is therefore usually only visible in configurational isomers, in which a particular conformation is locked by substituents. Prediction of rates of many reactions involving the transition between sp2 and sp3 states, such as ketone reduction, alcohol oxidation or nucleophilic substitution is possible if all conformers and their relative stability ruled by their strain is taken into account.
One example with configurational isomers is provided by elimination reactions, which involve the simultaneous removal of a proton and a leaving group from vicinal or "anti"periplanar positions under the influence of a base.
The mechanism requires that the departing atoms or groups follow antiparallel trajectories. For open chain substrates this geometric prerequisite is met by at least one of the three staggered conformers. For some cyclic substrates such as cyclohexane, however, an antiparallel arrangement may not be attainable depending on the substituents which might set a conformational lock. Adjacent substituents on a cyclohexane ring can achieve antiperiplanarity only when they occupy trans diaxial positions (that is, both are in axial position, one going up and one going down).
One consequence of this analysis is that "trans"-4-"tert"-butylcyclohexyl chloride cannot easily eliminate but instead undergoes substitution (see diagram below) because the most stable conformation has the bulky "t"-Bu group in the equatorial position, therefore the chloride group is not antiperiplanar with any vicinal hydrogen (it is gauche to all four). The thermodynamically unfavored conformation has the "t"-Bu group in the axial position, which is higher in energy by more than 5 kcal/mol (see A value). As a result, the "t"-Bu group "locks" the ring in the conformation where it is in the equatorial position and substitution reaction is observed. On the other hand, "cis"-4-"tert"-butylcyclohexyl chloride undergoes elimination because antiperiplanarity of Cl and H can be achieved when the "t"-Bu group is in the favorable equatorial position.
The repulsion between an axial "t"-butyl group and hydrogen atoms in the 1,3-diaxial position is so strong that the cyclohexane ring will revert to a twisted boat conformation. The strain in cyclic structures is usually characterized by deviations from ideal bond angles (Baeyer strain), ideal torsional angles (Pitzer strain) or transannular (Prelog) interactions.
Alkane stereochemistry.
Alkane conformers arise from rotation around sp3 hybridised carbon–carbon sigma bonds. The smallest alkane with such a chemical bond, ethane, exists as an infinite number of conformations with respect to rotation around the C–C bond. Two of these are recognised as energy minimum (staggered conformation) and energy maximum (eclipsed conformation) forms. The existence of specific conformations is due to hindered rotation around sigma bonds, although a role for hyperconjugation is proposed by a competing theory.
The importance of energy minima and energy maxima is seen by extension of these concepts to more complex molecules for which stable conformations may be predicted as minimum-energy forms. The determination of stable conformations has also played a large role in the establishment of the concept of asymmetric induction and the ability to predict the stereochemistry of reactions controlled by steric effects.
In the example of staggered ethane in Newman projection, a hydrogen atom on one carbon atom has a 60° torsional angle or torsion angle with respect to the nearest hydrogen atom on the other carbon so that steric hindrance is minimised. The staggered conformation is more stable by 12.5 kJ/mol than the eclipsed conformation, which is the energy maximum for ethane. In the eclipsed conformation the torsional angle is minimised.
In butane, the two staggered conformations are no longer equivalent and represent two distinct conformers:the anti-conformation (left-most, below) and the gauche conformation (right-most, below).
Both conformations are free of torsional strain, but, in the gauche conformation, the two methyl groups are in closer proximity than the sum of their van der Waals radii. The interaction between the two methyl groups is repulsive (van der Waals strain), and an energy barrier results.
A measure of the potential energy stored in butane conformers with greater steric hindrance than the 'anti'-conformer ground state is given by these values:
The eclipsed methyl groups exert a greater steric strain because of their greater electron density compared to lone hydrogen atoms.
The textbook explanation for the existence of the energy maximum for an eclipsed conformation in ethane is steric hindrance, but, with a C-C bond length of 154 pm and a Van der Waals radius for hydrogen of 120 pm, the hydrogen atoms in ethane are never in each other's way. The question of whether steric hindrance is responsible for the eclipsed energy maximum is a topic of debate to this day. One alternative to the steric hindrance explanation is based on hyperconjugation as analyzed within the Natural Bond Orbital framework. In the staggered conformation, one C-H sigma bonding orbital donates electron density to the antibonding orbital of the other C-H bond. The energetic stabilization of this effect is maximized when the two orbitals have maximal overlap, occurring in the staggered conformation. There is no overlap in the eclipsed conformation, leading to a disfavored energy maximum. On the other hand, an analysis within quantitative molecular orbital theory shows that 2-orbital-4-electron (steric) repulsions are dominant over hyperconjugation. A valence bond theory study also emphasizes the importance of steric effects.
Nomenclature.
Naming alkanes per standards listed in the IUPAC Gold Book is done according to the Klyne–Prelog system for specifying angles (called either torsional or dihedral angles) between substituents around a single bond:
Torsional strain or "Pitzer strain" refers to resistance to twisting about a bond.
Special cases.
In "n"-pentane, the terminal methyl groups experience additional pentane interference.
Replacing hydrogen by fluorine in polytetrafluoroethylene changes the stereochemistry from the zigzag geometry to that of a helix due to electrostatic repulsion of the fluorine atoms in the 1,3 positions. Evidence for the helix structure in the crystalline state is derived from X-ray crystallography and from NMR spectroscopy and circular dichroism in solution.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " K = e^{-\\Delta G^\\circ/RT},"
},
{
"math_id": 1,
"text": " K \\approx 10^{-\\Delta G^\\circ/(1.36\\text{ kcal}/\\mathrm{mol})}. "
},
{
"math_id": 2,
"text": " \\frac{N_i}{N_\\text{total}} = \\frac {e^{-E_i/RT}} {\\sum_{k=1}^M e^{-E_k/RT}}. "
}
] |
https://en.wikipedia.org/wiki?curid=1527574
|
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