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passage: 0xEF 0xE6 1110.0110 1111.1011 1110.1110 0xFB 0xEE 0xE7 1110.0111 1111.1011 1110.1111 0xFB 0xEF 0xE8 1110.1000 1111.1110 1110.1010 0xFE 0xEA 0xE9 1110.1001 1111.1110 1110.1011 0xFE 0xEB 0xEA 1110.1010 1111.1111 1110.1010 0xFF 0xEA 0xEB 1110.1011 1111.1111 1110.1011 0xFF 0xEB 0xEC 1110.1100 1111.1110 1110.1110 0xFE 0xEE 0xED 1110.1101 1111.1110 1110.1111 0xFE 0xEF 0xEE 1110.1110 1111.1111 1110.1110 0xFF 0xEE 0xEF 1110.1111 1111.1111 1110.1111 0xFF 0xEF 0xF0 1111.0000 1111.1010 1111.1010 0xFA 0xFA 0xF1 1111.0001 1111.1010 1111.1011 0xFA 0xFB 0xF2 1111.0010 1111.1011 1111.1010 0xFB 0xFA 0xF3 1111.0011 1111.1011 1111.1011 0xFB 0xFB 0xF4 1111.0100 1111.1010 1111.1110 0xFA 0xFE 0xF5 1111.0101 1111.1010 1111.1111 0xFA 0xFF 0xF6 1111.0110 1111.1011 1111.1110 0xFB 0xFE 0xF7 1111.0111 1111.1011 1111.1111 0xFB 0xFF 0xF8 1111.1000 1111.1110 1111.1010
|
https://en.wikipedia.org/wiki/Group_coded_recording
|
passage: Therefore, this is not a particularly efficient way of calculation. To maximize the rate of convergence, choose so that
$$
\frac{|d|}{N^2} \,
$$
is as small as possible.
## Continued fraction expansion
The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.
Quadratic irrationals (numbers of the form
$$
\frac{a+\sqrt{b}}{c}
$$
, where a, b and c are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion, and hence its rational approximation. Let S be the positive number for which we are required to find the square root.
|
https://en.wikipedia.org/wiki/Square_root_algorithms
|
passage: ## Functions, homomorphisms and morphisms
Composition is a partial operation that generalizes to homomorphisms of algebraic structures and morphisms of categories into operations that are also called composition, and share many properties with function composition.
In all the case, composition is associative.
If
$$
f\colon X\to Y
$$
and
$$
g\colon Y'\to Z,
$$
the composition
$$
g\circ f
$$
is defined if and only if
$$
Y'=Y
$$
or, in the function and homomorphism cases,
$$
Y\subset Y'.
$$
In the function and homomorphism cases, this means that the codomain of
$$
f
$$
equals or is included in the domain of . In the morphism case, this means that the codomain of
$$
f
$$
equals the domain of .
There is an identity
$$
\operatorname{id}_X \colon X \to X
$$
for every object (set, algebraic structure or object), which is called also an identity function in the function case.
A function is invertible if and only if it is a bijection. An invertible homomorphism or morphism is called an isomorphism. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an inverse function. In the other cases, one talks of inverse isomorphisms.
A function has a left inverse or a right inverse if and only it is injective or surjective, respectively.
|
https://en.wikipedia.org/wiki/Inverse_element
|
passage: In Koch's postulates, he set out criteria to test if an organism is the cause of a disease, and these postulates are still used today. Both Koch and Pasteur played a role in improving antisepsis in medical treatment. In 1870–1885 the modern methods of bacteriology technique were introduced by the use of stains, and by the method of separating mixtures of organisms on plates of nutrient media.
Though it had been known since the nineteenth century that bacteria are a cause of many diseases, no effective antibacterial treatments were available until the 20th century. In 1910, Paul Ehrlich developed the first antibiotic, by changing dyes that selectively stained Treponema pallidum—the spirochaete that causes syphilis—into compounds that selectively killed the pathogen. Ehrlich was awarded a 1908 Nobel Prize for his work on immunology, and pioneered the use of stains to detect and identify bacteria, with his work being the basis of the Gram stain and the Ziehl–Neelsen stain.
In the early 20th century, there was debate about the classification of bacteria. In 1904, cyanobacteria were usually classified as a class of algae, which are eukaryotic. However, Haeckel at this time classed cyanobacteria with bacteria because they lacked nuclei. In 1938, Herbert Faulkner Copeland proposed that prokaryotes be granted their own kingdom. The development of the transmission electron microscope allowed better visualization of cell structure and helped to clarify issues.
|
https://en.wikipedia.org/wiki/Bacteriology
|
passage: Now, if we divide the time convolution above by the period
$$
T
$$
and take the limit as
$$
T \rightarrow \infty
$$
, it becomes the autocorrelation function of the non-windowed signal
$$
x(t)
$$
, which is denoted as
$$
R_{xx}(\tau)
$$
, provided that
$$
x(t)
$$
is ergodic, which is true in most, but not all, practical cases.
$$
\lim_{T\to \infty} \frac{1}{T} \left|\hat{x}_{T}(f)\right|^2 = \int_{-\infty}^\infty \left[\lim_{T\to \infty} \frac{1}{T}\int_{-\infty}^\infty x_{T}^*(t - \tau)x_{T}(t) dt \right]e^{-i 2\pi f\tau} \ d\tau = \int_{-\infty}^\infty R_{xx}(\tau)e^{-i 2\pi f\tau} d\tau
$$
Assuming the ergodicity of
$$
x(t)
$$
, the power spectral density can be found once more as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem).
Many authors use this equality to actually define the power spectral density.
|
https://en.wikipedia.org/wiki/Spectral_density
|
passage: There are two parts to proving that the Boolean satisfiability problem (SAT) is NP-complete. One is to show that SAT is an NP problem. The other is to show that every NP problem can be reduced to an instance of a SAT problem by a polynomial-time many-one reduction.
SAT is in NP because any assignment of Boolean values to Boolean variables that is claimed to satisfy the given expression can be verified in polynomial time by a deterministic Turing machine. (The statements verifiable in polynomial time by a deterministic Turing machine and solvable in polynomial time by a non-deterministic Turing machine are equivalent, and the proof can be found in many textbooks, for example Sipser's Introduction to the Theory of Computation, section 7.3., as well as in the Wikipedia article on NP).
Now suppose that a given problem in NP can be solved by the nondeterministic Turing machine
$$
M = (Q, \Sigma, s, F, \delta)
$$
, where
$$
Q
$$
is the set of states,
$$
\Sigma
$$
is the alphabet of tape symbols,
$$
s \in Q
$$
is the initial state,
$$
F \subseteq Q
$$
is the set of accepting states, and
$$
\delta \subseteq ((Q \setminus F) \times \Sigma) \times (Q \times \Sigma \times \{-1, +1\})
$$
is the transition relation.
|
https://en.wikipedia.org/wiki/Cook%E2%80%93Levin_theorem
|
passage: It mainly involves the orchestrated movements of cell sheets and of individual cells. Morphogenesis is important for creating the three germ layers of the early embryo (ectoderm, mesoderm, and endoderm) and for building up complex structures during organ development.
- Tissue growth involves both an overall increase in tissue size, and also the differential growth of parts (allometry) which contributes to morphogenesis.
### Growth
mostly occurs through cell proliferation but also through changes in cell size or the deposition of extracellular materials.
The development of plants involves similar processes to that of animals. However, plant cells are mostly immotile so morphogenesis is achieved by differential growth, without cell movements. Also, the inductive signals and the genes involved are different from those that control animal development.
### Generative biology
Generative biology is the generative science that explores the dynamics guiding the development and evolution of a biological morphological form.
## Developmental processes
Cell differentiation
Cell differentiation is the process whereby different functional cell types arise in development. For example, neurons, muscle fibers and hepatocytes (liver cells) are well known types of differentiated cells. Differentiated cells usually produce large amounts of a few proteins that are required for their specific function and this gives them the characteristic appearance that enables them to be recognized under the light microscope. The genes encoding these proteins are highly active.
|
https://en.wikipedia.org/wiki/Developmental_biology
|
passage: Seismic design is carried out by understanding the possible failure modes of a structure and providing the structure with appropriate strength, stiffness, ductility, and configuration to ensure those modes cannot occur.
### Seismic design requirements
Seismic design requirements depend on the type of the structure, locality of the project and its authorities which stipulate applicable seismic design codes and criteria. For instance, California Department of Transportation's requirements called The Seismic Design Criteria (SDC) and aimed at the design of new bridges in California incorporate an innovative seismic performance-based approach.
The most significant feature in the SDC design philosophy is a shift from a force-based assessment of seismic demand to a displacement-based assessment of demand and capacity. Thus, the newly adopted displacement approach is based on comparing the elastic displacement demand to the inelastic displacement capacity of the primary structural components while ensuring a minimum level of inelastic capacity at all potential plastic hinge locations.
In addition to the designed structure itself, seismic design requirements may include a ground stabilization underneath the structure: sometimes, heavily shaken ground breaks up which leads to collapse of the structure sitting upon it.
The following topics should be of primary concerns: liquefaction; dynamic lateral earth pressures on retaining walls; seismic slope stability; earthquake-induced settlement.
Nuclear facilities should not jeopardise their safety in case of earthquakes or other hostile external events. Therefore, their seismic design is based on criteria far more stringent than those applying to non-nuclear facilities.
|
https://en.wikipedia.org/wiki/Earthquake_engineering
|
passage: This is done by setting
$$
G(s) = G_0(s)\Vert G_1(s)
$$
, in which
$$
|G_0(s)| = |s| = k
$$
and
$$
|G_1(s)| = p(k)-k
$$
; then is a forward secure PRNG with
$$
G_0
$$
as the next state and
$$
G_1
$$
as the pseudorandom output block of the current period.
## Entropy extraction
Santha and Vazirani proved that several bit streams with weak randomness can be combined to produce a higher-quality, quasi-random bit stream.
Even earlier, John von Neumann proved that a simple algorithm can remove a considerable amount of the bias in any bit stream, which should be applied to each bit stream before using any variation of the Santha–Vazirani design.
## Designs
CSPRNG designs are divided into two classes:
1.
### Designs based on cryptographic primitives
such as ciphers and cryptographic hashes
1. Designs based on mathematical problems thought to be hard
Designs based on cryptographic primitives
- A secure block cipher can be converted into a CSPRNG by running it in counter mode using, for example, a special construct that the NIST in SP 800-90A calls CTR DRBG. CTR_DBRG typically uses Advanced Encryption Standard (AES).
- AES-CTR_DRBG is often used as a random number generator in systems that use AES encryption.
- The NIST CTR_DRBG scheme erases the key after the requested randomness is output by running additional cycles.
|
https://en.wikipedia.org/wiki/Cryptographically_secure_pseudorandom_number_generator
|
passage: The liquid thus formed may get trapped as the pores of the gas reservoir get depleted. One method to deal with this problem is to re-inject dried gas free of condensate to maintain the underground pressure and to allow re-evaporation and extraction of condensates. More frequently, the liquid condenses at the surface, and one of the tasks of the gas plant is to collect this condensate. The resulting liquid is called natural gas liquid (NGL) and has commercial value.
### Shale gas
Shale gas is natural gas produced from shale. Because shale's matrix permeability is too low to allow gas to flow in economical quantities, shale gas wells depend on fractures to allow the gas to flow. Early shale gas wells depended on natural fractures through which gas flowed; almost all shale gas wells today require fractures artificially created by hydraulic fracturing. Since 2000, shale gas has become a major source of natural gas in the United States and
### Canada
. Because of increased shale gas production the United States was in 2014 the number one natural gas producer in the world. The production of shale gas in the United States has been described as a "shale gas revolution" and as "one of the landmark events in the 21st century."
Following the increased production in the United States, shale gas exploration is beginning in countries such as Poland, China, and South Africa. Chinese geologists have identified the Sichuan Basin as a promising target for shale gas drilling, because of the similarity of shales to those that have proven productive in the United States.
|
https://en.wikipedia.org/wiki/Natural_gas
|
passage: If one wants to discuss set-theoretic operations, such as the formation of the image (range) of a function, a topos is guaranteed to be able to express this, entirely constructively.
It also produced a more accessible spin-off in pointless topology, where the locale concept isolates some insights found by treating topos as a significant development of topological space. The slogan is 'points come later': this brings discussion full circle on this page. The point of view is written up in Peter Johnstone's Stone Spaces, which has been called by a leader in the field of computer science 'a treatise on extensionality'. The extensional is treated in mathematics as ambient—it is not something about which mathematicians really expect to have a theory. Perhaps this is why topos theory has been treated as an oddity; it goes beyond what the traditionally geometric way of thinking allows. The needs of thoroughly intensional theories such as untyped lambda calculus have been met in denotational semantics. Topos theory has long looked like a possible 'master theory' in this area.
## Summary
The topos concept arose in algebraic geometry, as a consequence of combining the concept of sheaf and closure under categorical operations. It plays a certain definite role in cohomology theories. A 'killer application' is étale cohomology.
The subsequent developments associated with logic are more interdisciplinary. They include examples drawing on homotopy theory (classifying toposes).
|
https://en.wikipedia.org/wiki/History_of_topos_theory
|
passage: These properties can be restated more naturally: the category of matrices with entries in a field
$$
k
$$
with multiplication as composition is equivalent to the category of finite-dimensional vector spaces and linear maps over this field.
More generally, the set of matrices can be used to represent the -linear maps between the free modules and for an arbitrary ring with unity. When composition of these maps is possible, and this gives rise to the matrix ring of matrices representing the endomorphism ring of .
### Matrix groups
A group is a mathematical structure consisting of a set of objects together with a binary operation, that is, an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group. Since a group of every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.
Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of (that is, a smaller group contained in) their general linear group, called a special linear group. Orthogonal matrices, determined by the condition
$$
\bold M^{\rm T} \bold M = \bold I,
$$
form the orthogonal group. Every orthogonal matrix has determinant 1 or −1.
|
https://en.wikipedia.org/wiki/Matrix_%28mathematics%29
|
passage: Alternatively, the relative discriminant of K/L is the norm of the different of K/L. When L = Q, the relative discriminant ΔK/Q is the principal ideal of Z generated by the absolute discriminant ΔK . In a tower of fields K/L/F the relative discriminants are related by
$$
\Delta_{K/F} = \mathcal{N}_{L/F}\left({\Delta_{K/L}}\right) \Delta_{L/F}^{[K:L]}
$$
where
$$
\mathcal{N}
$$
denotes relative norm.
### Ramification
The relative discriminant regulates the ramification data of the field extension K/L. A prime ideal p of L ramifies in K if, and only if, it divides the relative discriminant ΔK/L. An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension.
## Root discriminant
The root discriminant of a degree n number field K is defined by the formula
$$
\operatorname{rd}_K = |\Delta_K|^{1/n}.
$$
The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.
|
https://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field
|
passage: In biology, a colony is composed of two or more conspecific individuals living in close association with, or connected to, one another. This association is usually for mutual benefit such as stronger defense or the ability to attack bigger prey.
Colonies can form in various shapes and ways depending on the organism involved. For instance, the bacterial colony is a cluster of identical cells (clones). These colonies often form and grow on the surface of (or within) a solid medium, usually derived from a single parent cell.
Colonies, in the context of development, may be composed of two or more unitary (or solitary) organisms or be modular organisms. Unitary organisms have determinate development (set life stages) from zygote to adult form and individuals or groups of individuals (colonies) are visually distinct.
### Modular organisms
have indeterminate growth forms (life stages not set) through repeated iteration of genetically identical modules (or individuals), and it can be difficult to distinguish between the colony as a whole and the modules within. In the latter case, modules may have specific functions within the colony.
In contrast, solitary organisms do not associate with colonies; they are ones in which all individuals live independently and have all of the functions needed to survive and reproduce.
Some organisms are primarily independent and form facultative colonies in reply to environmental conditions while others must live in a colony to survive (obligate).
|
https://en.wikipedia.org/wiki/Colony_%28biology%29
|
passage:
$$
a_k
$$
are all isolated singularities within the contour
$$
C
$$
.
## Calculation of residues
Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a−1 of in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
According to the residue theorem, we have:
$$
\operatorname{Res}(f,c) = {1 \over 2\pi i} \oint_\gamma f(z)\,dz
$$
where γ traces out a circle around c in a counterclockwise manner and does not pass through or contain other singularities within it. We may choose the path γ to be a circle of radius ε around c. Since ε can be as small as we desire it can be made to contain only the singularity of c due to nature of isolated singularities. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.
### Removable singularities
If the function f can be continued to a holomorphic function on the whole disk
$$
|y-c|<R
$$
, then Res(f, c) = 0. The converse is not generally true.
|
https://en.wikipedia.org/wiki/Residue_%28complex_analysis%29
|
passage: ```
It is now just a matter of implementing the Newton method using the given functions.
```c
float2 roots[3] = //Roots (solutions) of the polynomial
{
float2(1, 0),
float2(-.5, sqrt(3)/2),
float2(-.5, -sqrt(3)/2)
};
color colors[3] = //Assign a color for each root
{
red,
green,
blue
}
For each pixel (x, y) on the target, do:
{
zx = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
zy = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-2, 1))
float2 z = float2(zx, zy); //z is originally set to the pixel coordinates
for (int iteration = 0;
iteration < maxIteration;
iteration++;)
{
z -= cdiv(Function(z), Derivative(z)); //cdiv is a function for dividing complex numbers
float tolerance = 0.000001;
for (int i = 0; i < roots.Length; i++)
{
float2 difference = z - roots[i];
//If the current iteration is close enough to a root, color the pixel.
if (abs(difference.x) < tolerance && abs (difference.y) < tolerance)
{
return colors[i]; //Return the color corresponding to the root
}
}
}
return black; //If no solution is found
}
```
|
https://en.wikipedia.org/wiki/Newton_fractal
|
passage: ### Predicate completion solution
This encoding is similar to the fluent occlusion solution, but the additional predicates denote change, not permission to change. For example,
$$
\mathrm{changeopen}(t)
$$
represents the fact that the predicate
$$
\mathrm{open}
$$
will change from time
$$
t
$$
to
$$
t+1
$$
. As a result, a predicate changes if and only if the corresponding change predicate is true. An action results in a change if and only if it makes true a condition that was previously false or vice versa.
$$
\neg \mathrm{open}(0)
$$
$$
\neg \mathrm{on}(0)
$$
$$
\neg \mathrm{open}(0) \implies \mathrm{changeopen}(0)
$$
$$
\forall t. \mathrm{changeopen}(t) \iff (\neg \mathrm{open}(t) \iff \mathrm{open}(t+1))
$$
$$
\forall t. \mathrm{changeon}(t) \iff (\neg \mathrm{on}(t) \iff \mathrm{on}(t+1))
$$
The third formula is a different way of saying that opening the door causes the door to be opened. Precisely, it states that opening the door changes the state of the door if it had been previously closed.
|
https://en.wikipedia.org/wiki/Frame_problem
|
passage: In that aspect, Log Gabor filter have been shown to be a good choice to extract boundaries in natural scenes.
### Other first-order methods
Different gradient operators can be applied to estimate image gradients from the input image or a smoothed version of it. The simplest approach is to use central differences:
$$
\begin{align}
L_x(x, y) & = -\frac 1 2 L(x-1, y) + 0 \cdot L(x, y) + \frac 1 2 \cdot L(x+1, y) \\[8pt]
L_y(x, y) & = -\frac1 2 L(x, y-1) + 0 \cdot L(x, y) + \frac 1 2 \cdot L(x, y+1),
\end{align}
$$
corresponding to the application of the following filter masks to the image data:
$$
L_y = \begin{bmatrix}
+1/2 & 0 & -1/2
\end{bmatrix} L
\quad \text{and} \quad
L_x = \begin{bmatrix}
+1/2 \\
0 \\
-1/2
\end{bmatrix} L.
$$
The well-known and earlier Sobel operator is based on the following filters:
$$
L_y = \begin{bmatrix}
+1 & 0 & -1 \\
+2 & 0 & -2 \\
+1 & 0 & -1
\end{bmatrix} L
\quad \text{and} \quad
L_x = \begin{bmatrix}
+1 & +2 & +1 \\
0 & 0 & 0 \\
-1 & -2 & -1
\end{bmatrix} L.
$$
|
https://en.wikipedia.org/wiki/Edge_detection
|
passage: It looks like a solid cylinder in which any cross section is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface.
This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.
An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-dimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around any given point, it looks just like Minkowski space, the model of spacetime used in non-gravitational physics. One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory.
|
https://en.wikipedia.org/wiki/String_theory
|
passage: - †Oviraptoridae (characterized by two bony projections at the back of the mouth; exclusive to Asia)
- Paraves (avialans and their closest relatives)
- †Scansoriopterygidae (small tree-climbing theropods with membranous wings)
- †Deinonychosauria (toe-clawed dinosaurs; may not form a natural group)
- †Archaeopterygidae (small, winged theropods or primitive birds)
- †Troodontidae (omnivores; enlarged brain cavities)
- †Dromaeosauridae ("raptors")
- †Unenlagiidae (piscivores; may be dromaeosaurids)
- Avialae (modern birds and extinct relatives)
|
https://en.wikipedia.org/wiki/Dinosaur
|
passage: Superscript notation is also used for conjugation; that is, , where and are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely
$$
(g^h)^k=g^{hk}
$$
and
$$
(gh)^k=g^kh^k.
$$
### In a ring
In a ring, it may occur that some nonzero elements satisfy
$$
x^n=0
$$
for some integer . Such an element is said to be nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.
If the nilradical is reduced to the zero ideal (that is, if
$$
x\neq 0
$$
implies
$$
x^n\neq 0
$$
for every positive integer ), the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.
More generally, given an ideal in a commutative ring , the set of the elements of that have a power in is an ideal, called the radical of . The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical.
|
https://en.wikipedia.org/wiki/Exponentiation
|
passage: This normally occurs if upper-level wind shear is too strong. The storm can redevelop if the upper-level shear abates.
If a tropical wave is moving quickly, or is organized enough, it can have winds of a strength in excess of tropical storm force, but it is not considered a tropical storm unless it has a closed low-level circulation. An example of this was Hurricane Claudette in 2003, where the original wave had winds of before developing a closed low-level circulation.
## East Pacific
It has been suggested that some eastern Pacific Ocean tropical cyclones are formed out of tropical easterly waves that originate in North Africa as well. After developing into a tropical cyclone, some of those systems can then reach the Central Pacific Ocean, such as Hurricane Lane in 2018. During the summer months, tropical waves can extend northward as far as the desert of the southwestern United States, producing spells of intensified shower activity embedded within the prevailing monsoon regime.
## Screaming eagle waves
A screaming eagle is a tropical wave with a convective pattern that loosely resembles the head of an eagle. This phenomenon is caused by shearing from either westerly winds aloft or strong easterly winds at the surface. These systems are typically located within 25 degrees latitude of the equator. Rain showers and surface winds gusting to are associated with these waves. They move across the ocean at a rate of . Strong thunderstorm activity can be associated with the features when located east of a tropical upper tropospheric trough.
|
https://en.wikipedia.org/wiki/Tropical_wave
|
passage: ### Derivation
Start with the ideal gas law
$$
P = \rho R_\text{specific}T,
$$
where
$$
T
$$
the absolute temperature of the gas and specific gas constant
$$
R_\text{specific}
$$
.
Then, assuming the process is adiabatic, pressure
$$
P(\rho)
$$
can be considered a function of density
$$
\rho
$$
.
The conservation of mass and conservation of momentum can be written as a closed system of two equations
$$
\begin{align}
\rho_{t} + (\rho u)_{x} &= 0,\\
(\rho u)_{t} + (\rho u^2 + P(\rho))_{x} &=0.
\end{align}
$$
This coupled system of two nonlinear conservation laws can be written in vector form as:
$$
q_t + f(q)_x = 0,
$$
with
$$
q = \begin{bmatrix}\rho \\ \rho u\end{bmatrix} = \begin{bmatrix}q_{(1)} \\ q_{(2)}\end{bmatrix}, \quad f(q) = \begin{bmatrix}\rho u \\ \rho u^2 + P(\rho)\end{bmatrix}=\begin{bmatrix} q_{(2)} \\ q_{(2)}^2/q_{(1)} + P(q_{(1)})\end{bmatrix}.
$$
To linearize this equation, let
$$
q(x,t) = q_0 + \tilde{q}(x,t),
$$
where
$$
q_0 = ( \rho_0 , \rho_0 u_0)
$$
is the (constant) background state and
$$
\tilde{q}
$$
is a sufficiently small perturbation, i.e., any powers or products of
$$
\tilde{q}
$$
can be discarded.
|
https://en.wikipedia.org/wiki/Acoustic_wave_equation
|
passage: Signal transduction by a GPCR begins with an inactive G protein coupled to the receptor; the G protein exists as a heterotrimer consisting of Gα, Gβ, and Gγ subunits. Once the GPCR recognizes a ligand, the conformation of the receptor changes to activate the G protein, causing Gα to bind a molecule of GTP and dissociate from the other two G-protein subunits. The dissociation exposes sites on the subunits that can interact with other molecules. The activated G protein subunits detach from the receptor and initiate signaling from many downstream effector proteins such as phospholipases and ion channels, the latter permitting the release of second messenger molecules. The total strength of signal amplification by a GPCR is determined by the lifetimes of the ligand-receptor complex and receptor-effector protein complex and the deactivation time of the activated receptor and effectors through intrinsic enzymatic activity; e.g. via protein kinase phosphorylation or b-arrestin-dependent internalization.
A study was conducted where a point mutation was inserted into the gene encoding the chemokine receptor CXCR2; mutated cells underwent a malignant transformation due to the expression of CXCR2 in an active conformation despite the absence of chemokine-binding. This meant that chemokine receptors can contribute to cancer development.
|
https://en.wikipedia.org/wiki/Signal_transduction
|
passage: In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations.
|
https://en.wikipedia.org/wiki/Entropic_uncertainty
|
passage: ### Computing homothetic centers
For a given pair of circles, the internal and external homothetic centers may be found in various ways. In analytic geometry, the internal homothetic center is the weighted average of the centers of the circles, weighted by the opposite circle's radius – distance from center of circle to inner center is proportional to that radius, so weighting is proportional to the opposite radius. Denoting the centers of the circles by and their radii by and denoting the center by , this is:
$$
(x_0, y_0) = \frac{r_2}{r_1 + r_2}(x_1, y_1) + \frac{r_1}{r_1 + r_2}(x_2, y_2).
$$
The external center can be computed by the same equation, but considering one of the radii as negative; either one yields the same equation, which is:
$$
(x_e, y_e) = \frac{-r_2}{r_1 - r_2}(x_1, y_1) + \frac{r_1}{r_1 - r_2}(x_2, y_2).
$$
More generally, taking both radii with the same sign (both positive or both negative) yields the inner center, while taking the radii with opposite signs (one positive and the other negative) yields the outer center.
|
https://en.wikipedia.org/wiki/Homothetic_center
|
passage: Area and volume are thus, of course, derived from the length, but included for completeness as they occur frequently in many derived quantities, in particular densities.
Quantity SI unit Dimensions Description Symbols (Spatial) position (vector) r, R, a, d m L Angular position, angle of rotation (can be treated as vector or scalar) θ, θ rad None Area, cross-section A, S, Ω m2 L2 Vector area (Magnitude of surface area, directed normal to tangential plane of surface) m2 L2 Volume τ, V m3 L3
#### Densities, flows, gradients, and moments
Important and convenient derived quantities such as densities, fluxes, flows, currents are associated with many quantities. Sometimes different terms such as current density and flux density, rate, frequency and current, are used interchangeably in the same context; sometimes they are used uniquely.
To clarify these effective template-derived quantities, we use q to stand for any quantity within some scope of context (not necessarily base quantities) and present in the table below some of the most commonly used symbols where applicable, their definitions, usage, SI units and SI dimensions – where [q] denotes the dimension of q.
For time derivatives, specific, molar, and flux densities of quantities, there is no one symbol; nomenclature depends on the subject, though time derivatives can be generally written using overdot notation. For generality we use qm, qn, and F respectively.
|
https://en.wikipedia.org/wiki/Physical_quantity
|
passage: The mechanical role of cellulose fibers in the wood matrix responsible for its strong structural resistance, can somewhat be compared to that of the reinforcement bars in concrete, lignin playing here the role of the hardened cement paste acting as the "glue" in between the cellulose fibres. Mechanical properties of cellulose in primary plant cell wall are correlated with growth and expansion of plant cells. Live fluorescence microscopy techniques are promising in investigation of the role of cellulose in growing plant cells.
Compared to starch, cellulose is also much more crystalline. Whereas starch undergoes a crystalline to amorphous transition when heated beyond 60–70 °C in water (as in cooking), cellulose requires a temperature of 320 °C and pressure of 25 MPa to become amorphous in water.
Several types of cellulose are known. These forms are distinguished according to the location of hydrogen bonds between and within strands. Natural cellulose is cellulose I, with structures Iα and Iβ. Cellulose produced by bacteria and algae is enriched in Iα while cellulose of higher plants consists mainly of Iβ. Cellulose in regenerated cellulose fibers is cellulose II. The conversion of cellulose I to cellulose II is irreversible, suggesting that cellulose I is metastable and cellulose II is stable.
|
https://en.wikipedia.org/wiki/Cellulose
|
passage: In this case, the velocity vP is given by:
$$
\mathbf{v}_P
= \frac{\text{d}}{\text{d}t} \left(r\hat\mathbf r + z \hat\mathbf z\right)
r\omega\hat\mathbf\theta v\hat\mathbf\theta,
$$
where
$$
\omega
$$
is the angular velocity of the unit vector around the z axis of the cylinder.
The acceleration aP of the particle P is now given by:
$$
\mathbf{a}_P = \frac{\text{d}(v\hat\mathbf\theta)}{\text{d}t} = a\hat\mathbf\theta - v\theta\hat\mathbf r.
$$
The components
$$
a_r = - v\theta, \quad a_{\theta} = a,
$$
are called, respectively, the radial and tangential components of acceleration.
The notation for angular velocity and angular acceleration is often defined as
$$
\omega = \dot{\theta}, \quad \alpha = \ddot{\theta},
$$
so the radial and tangential acceleration components for circular trajectories are also written as
$$
a_r = - r\omega^2, \quad a_{\theta} = r\alpha.
$$
|
https://en.wikipedia.org/wiki/Kinematics
|
passage: The synonym self-teaching computers was also used in this time period.
Although the earliest machine learning model was introduced in the 1950s when Arthur Samuel invented a program that calculated the winning chance in checkers for each side, the history of machine learning roots back to decades of human desire and effort to study human cognitive processes. In 1949, Canadian psychologist Donald Hebb published the book The Organization of Behavior, in which he introduced a theoretical neural structure formed by certain interactions among nerve cells. Hebb's model of neurons interacting with one another set a groundwork for how AIs and machine learning algorithms work under nodes, or artificial neurons used by computers to communicate data. Other researchers who have studied human cognitive systems contributed to the modern machine learning technologies as well, including logician Walter Pitts and Warren McCulloch, who proposed the early mathematical models of neural networks to come up with algorithms that mirror human thought processes.
By the early 1960s, an experimental "learning machine" with punched tape memory, called Cybertron, had been developed by Raytheon Company to analyse sonar signals, electrocardiograms, and speech patterns using rudimentary reinforcement learning. It was repetitively "trained" by a human operator/teacher to recognise patterns and equipped with a "goof" button to cause it to reevaluate incorrect decisions. A representative book on research into machine learning during the 1960s was Nilsson's book on Learning Machines, dealing mostly with machine learning for pattern classification. Interest related to pattern recognition continued into the 1970s, as described by Duda and Hart in 1973.
|
https://en.wikipedia.org/wiki/Machine_learning
|
passage: To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be (or should be) made as close to zero as possible. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine.
The efficiency is the ratio of the power dissipated by the load resistance to the total power dissipated by the circuit (which includes the voltage source's resistance of as well as ):
$$
\eta = \frac{P_\mathrm{L}}{P_\mathrm{Total}} = \frac{I^2 \cdot R_\mathrm{L}}{I^2 \cdot (R_\mathrm{L} + R_\mathrm{S})} = \frac{R_\mathrm{L}}{R_\mathrm{L} + R_\mathrm{S}} = \frac{1}{1 + R_\mathrm{S} / R_\mathrm{L}} \, .
$$
Consider three particular cases (note that voltage sources must have some resistance):
- If
$$
R_\mathrm{L}/R_\mathrm{S} \to 0
$$
, then
$$
\eta \to 0.
$$
Efficiency approaches 0% if the load resistance approaches zero (a short circuit), since all power is consumed in the source and no power is consumed in the short.
-
|
https://en.wikipedia.org/wiki/Maximum_power_transfer_theorem
|
passage: Define
$$
A(p_i)
$$
as some function of momentum
$$
p_i
$$
only, whose total value is conserved in a collision. Assume also that the force
$$
F_i
$$
is a function of position only, and that f is zero for
$$
p_i \to \pm\infty
$$
. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as
$$
\int A \frac{\partial f}{\partial t} \,d^3\mathbf{p} = \frac{\partial }{\partial t} (n \langle A \rangle),
$$
$$
\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3\mathbf{p} = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),
$$
$$
\int A F_j \frac{\partial f}{\partial p_j} \,d^3\mathbf{p} = -n F_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle,
$$
$$
\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3\mathbf{p} = \frac{\partial }{\partial t}_\text{coll} (n \langle A \rangle) = 0,
$$
where the last term is zero, since is conserved in a collision.
|
https://en.wikipedia.org/wiki/Boltzmann_equation
|
passage: ### Safety
Uses of taxonomy in safety include:
- Safety taxonomy, a standardized set of terminologies used within the fields of safety and health care
- Human Factors Analysis and Classification System, a system to identify the human causes of an accident
- Swiss cheese model, a model used in risk analysis and risk management propounded by Dante Orlandella and James T. Reason
- A taxonomy of rail incidents in Confidential Incident Reporting & Analysis System (CIRAS)
### Other taxonomies
- Military taxonomy, a set of terms that describe various types of military operations and equipment
- Moys Classification Scheme, a subject classification for law devised by Elizabeth Moys
###
## Research
publishing
Citing inadequacies with current practices in listing authors of papers in medical research journals, Drummond Rennie and co-authors called in a 1997 article in JAMA, the Journal of the American Medical Association for
a radical conceptual and systematic change, to reflect the realities of multiple authorship and to buttress accountability. We propose dropping the outmoded notion of author in favor of the more useful and realistic one of contributor.
In 2012, several major academic and scientific publishing bodies mounted Project CRediT to develop a controlled vocabulary of contributor roles. Known as CRediT (Contributor Roles Taxonomy), this is an example of a flat, non-hierarchical taxonomy; however, it does include an optional, broad classification of the degree of contribution: lead, equal or supporting.
|
https://en.wikipedia.org/wiki/Taxonomy
|
passage: Sodium is a chemical element; it has symbol Na (from Neo-Latin ) and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable isotope is 23Na. The free metal does not occur in nature and must be prepared from compounds. Sodium is the sixth most abundant element in the Earth's crust and exists in numerous minerals such as feldspars, sodalite, and halite (NaCl). Many salts of sodium are highly water-soluble: sodium ions have been leached by the action of water from the Earth's minerals over eons, and thus sodium and chlorine are the most common dissolved elements by weight in the oceans.
Sodium was first isolated by Humphry Davy in 1807 by the electrolysis of sodium hydroxide. Among many other useful sodium compounds, sodium hydroxide (lye) is used in soap manufacture, and sodium chloride (edible salt) is a de-icing agent and a nutrient for animals including humans.
Sodium is an essential element for all animals and some plants. Sodium ions are the major cation in the extracellular fluid (ECF) and as such are the major contributor to the ECF osmotic pressure.
|
https://en.wikipedia.org/wiki/Sodium
|
passage: More precisely, since
$$
A^*A
$$
is a positive semidefinite matrix, its square root is well defined. The nuclear norm
$$
\|A\|_{*}
$$
is a convex envelope of the rank function
$$
\text{rank}(A)
$$
, so it is often used in mathematical optimization to search for low-rank matrices.
Combining von Neumann's trace inequality with Hölder's inequality for Euclidean space yields a version of Hölder's inequality for Schatten norms for
$$
1/p + 1/q = 1
$$
:
$$
\left|\operatorname{trace}(A^*B)\right| \le \|A\|_p \|B\|_q,
$$
In particular, this implies the Schatten norm inequality
$$
\|A\|_F^2 \le \|A\|_p \|A\|_q.
$$
## Monotone norms
A matrix norm
$$
\|\cdot \|
$$
is called monotone if it is monotonic with respect to the Loewner order. Thus, a matrix norm is increasing if
$$
A \preccurlyeq B \Rightarrow \|A\| \leq \|B\|.
$$
The Frobenius norm and spectral norm are examples of monotone norms.
## Cut norms
Another source of inspiration for matrix norms arises from considering a matrix as the adjacency matrix of a weighted, directed graph.
|
https://en.wikipedia.org/wiki/Matrix_norm
|
passage: ### Structured SVM
Structured support-vector machine is an extension of the traditional SVM model. While the SVM model is primarily designed for binary classification, multiclass classification, and regression tasks, structured SVM broadens its application to handle general structured output labels, for example parse trees, classification with taxonomies, sequence alignment and many more.
### Regression
A version of SVM for regression was proposed in 1996 by Vladimir N. Vapnik, Harris Drucker, Christopher J. C. Burges, Linda Kaufman and Alexander J. Smola. This method is called support vector regression (SVR). The model produced by support vector classification (as described above) depends only on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin. Analogously, the model produced by SVR depends only on a subset of the training data, because the cost function for building the model ignores any training data close to the model prediction. Another SVM version known as least-squares support vector machine (LS-SVM) has been proposed by Suykens and Vandewalle.
Training the original SVR means solving
minimize
$$
\tfrac{1}{2} \|w\|^2
$$
subject to
$$
| y_i - \langle w, x_i \rangle - b | \le \varepsilon
$$
where
$$
x_i
$$
is a training sample with target value
$$
y_i
$$
.
|
https://en.wikipedia.org/wiki/Support_vector_machine
|
passage: (r,\,\vec{v})^{-1}
&= \left(\frac{r}{r^2 + \vec{v}\cdot\vec{v}},\ \frac{-\vec{v}}{r^2 + \vec{v}\cdot\vec{v}}\right),
\end{align}
$$
where "
$$
{}\cdot{}
$$
" and "
$$
\times
$$
" denote respectively the dot product and the cross product.
|
https://en.wikipedia.org/wiki/Quaternion
|
passage: Finally random mistakes in normal DNA replication may result in cancer-causing mutations. A series of several mutations to certain classes of genes is usually required before a normal cell will transform into a cancer cell. Recent comprehensive patient-level classification and quantification of driver events in TCGA cohorts revealed that there are on average 12 driver events per tumor, of which 0.6 are point mutations in oncogenes, 1.5 are amplifications of oncogenes, 1.2 are point mutations in tumor suppressors, 2.1 are deletions of tumor suppressors, 1.5 are driver chromosome losses, 1 is a driver chromosome gain, 2 are driver chromosome arm losses, and 1.5 are driver chromosome arm gains. Mutations in genes that regulate cell division, apoptosis (cell death), and DNA repair may result in uncontrolled cell proliferation and cancer.
Cancer is fundamentally a disease of regulation of tissue growth. In order for a normal cell to transform into a cancer cell, genes that regulate cell growth and differentiation must be altered.
### Genetic and epigenetic
changes can occur at many levels, from gain or loss of entire chromosomes, to a mutation affecting a single DNA nucleotide, or to silencing or activating a microRNA that controls expression of 100 to 500 genes. There are two broad categories of genes that are affected by these changes.
### Oncogenes
may be normal genes that are expressed at inappropriately high levels, or altered genes that have novel properties. In either case, expression of these genes promotes the malignant phenotype of cancer cells.
|
https://en.wikipedia.org/wiki/Carcinogenesis
|
passage:
$$
\ P(\mathbf{i}x)x.
$$
(Inclusion)
The operator i has two more axiomatic properties:
C6.
$$
\mathbf{i}(\mathbf{i}x) = \mathbf{i}x.
$$
(Idempotence)
C7.
$$
\mathbf{i}(x \times y) = \mathbf{i}x \times \mathbf{i}y,
$$
where a×b is the mereological product of a and b, not defined when Oab is false. i distributes over product.
It can now be seen that i is isomorphic to the interior operator of topology. Hence the dual of i, the topological closure operator c, can be defined in terms of i, and Kuratowski's axioms for c are theorems. Likewise, given an axiomatization of c that is analogous to C5-7, i may be defined in terms of c, and C5-7 become theorems. Adding C5-7 to GEMT results in Casati and Varzi's preferred mereotopological theory, GEMTC.
x is self-connected if it satisfies the following predicate:
$$
SCx \leftrightarrow ((Owx \leftrightarrow (Owy \lor Owz)) \rightarrow Cyz).
$$
Note that the primitive and defined predicates of MT alone suffice for this definition.
|
https://en.wikipedia.org/wiki/Mereotopology
|
passage: -
$$
V
$$
is
$$
k
$$
- and
$$
(k+1)
$$
-acyclic at these models, which means that
$$
H_k(V(M)) = 0
$$
for all
$$
k>0
$$
and all
$$
M \in \mathcal{M}_k \cup \mathcal{M}_{k+1}
$$
.
Then the following assertions hold:
- Every natural transformation
$$
\varphi : H_0(F) \to H_0(V)
$$
induces a natural chain map
$$
f : F \to V
$$
.
- If
$$
\varphi,\psi: H_0(F)\to H_0(V)
$$
are natural transformations,
$$
f,g: F\to V
$$
are natural chain maps as before and
$$
\varphi^{M}=\psi^{M}
$$
for all models
$$
M\in\mathcal{M}_0
$$
, then there is a natural chain homotopy between
$$
f
$$
and
$$
g
$$
.
- In particular the chain map
$$
f
$$
is unique up to natural chain homotopy.
## Generalizations
### Projective and acyclic complexes
What is above is one of the earliest versions of the theorem.
|
https://en.wikipedia.org/wiki/Acyclic_model
|
passage: A finite set of
$$
n
$$
random variables
$$
\{X_1,\ldots,X_n\}
$$
is mutually independent if and only if for any sequence of numbers
$$
\{x_1, \ldots, x_n\}
$$
, the events
$$
\{X_1 \le x_1\}, \ldots, \{X_n \le x_n \}
$$
are mutually independent events (as defined above in ). This is equivalent to the following condition on the joint cumulative distribution function A finite set of
$$
n
$$
random variables
$$
\{X_1,\ldots,X_n\}
$$
is mutually independent if and only if
It is not necessary here to require that the probability distribution factorizes for all possible subsets as in the case for
$$
n
$$
events. This is not required because e.g.
$$
F_{X_1,X_2,X_3}(x_1,x_2,x_3) = F_{X_1}(x_1) \cdot F_{X_2}(x_2) \cdot F_{X_3}(x_3)
$$
implies
$$
F_{X_1,X_3}(x_1,x_3) = F_{X_1}(x_1) \cdot F_{X_3}(x_3)
$$
.
|
https://en.wikipedia.org/wiki/Independence_%28probability_theory%29
|
passage: - Ricciardi and Umezawa proposed in 1967 a general theory (quantum brain) about the possible brain mechanism of memory storage and retrieval in terms of Nambu–Goldstone bosons. This theory was subsequently extended in 1995 by Giuseppe Vitiello taking into account that the brain is an "open" system (the dissipative quantum model of the brain). Applications of spontaneous symmetry breaking and of Goldstone's theorem to biological systems, in general, have been published by E. Del Giudice, S. Doglia, M. Milani, and G. Vitiello,E. Del Giudice, S. Doglia, M. Milani, G. Vitiello (1986). Electromagnetic field and spontaneous symmetry breaking in biological matter. Nucl. Phys., B275 (FS 17), 185 - 199. and by E. Del Giudice, G. Preparata and G. Vitiello. Mari Jibu and Kunio Yasue and Giuseppe Vitiello, based on these findings, discussed the implications for consciousness.
### Theory
Consider a complex scalar field , with the constraint that
$$
\phi^* \phi= v^2
$$
, a constant. One way to impose a constraint of this sort is by including a potential interaction term in its Lagrangian density,
$$
\lambda(\phi^*\phi - v^2)^2 ~,
$$
and taking the limit as . This is called the "Abelian nonlinear σ-model".
The constraint, and the action, below, are invariant under a U(1) phase transformation, .
|
https://en.wikipedia.org/wiki/Goldstone_boson
|
passage: the following statement is true:
The four points are on a circle if and only if the angles at
$$
P_3
$$
and
$$
P_4
$$
are equal.
Usually one measures inscribed angles by a degree or radian θ, but here the following measurement is more convenient:
In order to measure the angle between two lines with equations
$$
y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2,
$$
one uses the quotient:
$$
\frac{1 + m_1 m_2}{m_2 - m_1} = \cot\theta\ .
$$
#### Inscribed angle theorem for circles
For four points
$$
P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\,
$$
no three of them on a line, we have the following (see diagram):
The four points are on a circle, if and only if the angles at
$$
P_3
$$
and
$$
P_4
$$
are equal. In terms of the angle measurement above, this means:
$$
\frac{(x_4 - x_1)(x_4 - x_2) + (y_4 - y_1)(y_4 - y_2)}
BLOCK0\frac{(x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)}
BLOCK1$$
At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.
|
https://en.wikipedia.org/wiki/Ellipse
|
passage: ### Congruences
If n is prime, then
$$
\binom {n-1}k \equiv (-1)^k \mod n
$$
for every k with
$$
0\leq k \leq n-1.
$$
More generally, this remains true if n is any number and k is such that all the numbers between 1 and k are coprime to n.
Indeed, we have
$$
\binom {n-1} k = {(n-1)(n-2)\cdots(n-k)\over 1\cdot 2\cdots k}
# \prod_{i=1}^{k}{n-i\over i}\equiv \prod_{i=1}^{k}{-i\over i}
(-1)^k \mod n.
$$
## Generating functions
|
https://en.wikipedia.org/wiki/Binomial_coefficient
|
passage:
$$
\mathrm{add}:\mathsf{nat}\to (\mathsf{nat}\to\mathsf{nat})
$$
Strictly speaking, a simple type only allows for one input and one output, so a more faithful reading of the above type is that
$$
\mathrm{add}
$$
is a function which takes in a natural number and returns a function of the form
$$
\mathsf{nat}\to\mathsf{nat}
$$
. The parentheses clarify that
$$
\mathrm{add}
$$
does not have the type
$$
(\mathsf{nat}\to \mathsf{nat})\to\mathsf{nat}
$$
, which would be a function which takes in a function of natural numbers and returns a natural number. The convention is that the arrow is right associative, so the parentheses may be dropped from
$$
\mathrm{add}
$$
's type.
#### Lambda terms
New function terms may be constructed using lambda expressions, and are called lambda terms. These terms are also defined inductively: a lambda term has the form
$$
(\lambda v .t)
$$
, where
$$
v
$$
is a formal variable and
$$
t
$$
is a term, and its type is notated
$$
\sigma\to\tau
$$
, where
$$
\sigma
$$
is the type of
$$
v
$$
, and
$$
\tau
$$
is the type of
$$
t
$$
.
|
https://en.wikipedia.org/wiki/Type_theory
|
passage: The retina transduces this image into electrical pulses using rods and cones. The optic nerve then carries these pulses through the optic canal. Upon reaching the optic chiasm the nerve fibers decussate (left becomes right). The fibers then branch and terminate in three places. Than, Ker. "How the Human
### Eye
Works." LiveScience. TechMedia Network, 10 February 2010. Web. 27 March 2016.Albertine, Kurt. Barron's Anatomy Flash Cards"Optic Chiasma." Optic Chiasm Function, Anatomy & Definition. Healthline Medical Team, 9 March 2015. Web. 27 March 2016.
### Neural
Most of the optic nerve fibers end in the lateral geniculate nucleus (LGN). Before the LGN forwards the pulses to V1 of the visual cortex (primary) it gauges the range of objects and tags every major object with a velocity tag. These tags predict object movement.
The LGN also sends some fibers to V2 and V3.
V1 performs edge-detection to understand spatial organization (initially, 40 milliseconds in, focusing on even small spatial and color changes. Then, 100 milliseconds in, upon receiving the translated LGN, V2, and V3 info, also begins focusing on global organization). V1 also creates a bottom-up saliency map to guide attention or gaze shift.
V2 both forwards (direct and via pulvinar) pulses to V1 and receives them.
|
https://en.wikipedia.org/wiki/Visual_system
|
passage: In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of
## Bravais lattices
(an infinite array of discrete points). Space groups (symmetry groups of a configuration in space) are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices.
### Crystal system
s that have space groups assigned to a common lattice system are combined into a crystal family.
The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).
## Classifications
Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
### Lattice system
A lattice system is a group of lattices with the same set of lattice point groups. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
Crystal system
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system.
|
https://en.wikipedia.org/wiki/Crystal_system
|
passage: The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions (see Kolmogorov–Smirnov test), or whether outcome frequencies follow a specified distribution (see
### Pearson's chi-square test
). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.
## Fit of distributions
In assessing whether a given distribution is suited to a data-set, the following tests and their underlying measures of fit can be used:
- Bayesian information criterion
- Kolmogorov–Smirnov test
- Cramér–von Mises criterion
- Anderson–Darling test
- Berk-Jones tests
- Shapiro–Wilk test
- Chi-squared test
- Akaike information criterion
- Hosmer–Lemeshow test
- Kuiper's test
- Kernelized Stein discrepancy
- Zhang's ZK, ZC and ZA tests
- Moran test
- Density Based Empirical Likelihood Ratio tests
|
https://en.wikipedia.org/wiki/Goodness_of_fit
|
passage: Gregory Malanowski, The Race for Wireless, AuthorHouse. 2011, p. 36
"The Hertz wave theory of wireless transmission may be kept up for a while, but I do not hesitate to say that in a short time it will be recognized as one of the most remarkable and inexplicable aberrations of the scientific mind which has ever been recorded in history."
## Feasibility
Tesla's demonstrations of wireless power transmission at Colorado Springs consisted of lighting incandescent electric lamps positioned nearby the structure housing his large experimental magnifying transmitter, with ranges out to from the transmitter. There is little direct evidence of his having transmitted power beyond these photographically documented demonstrations. He would claim afterwards that he had "carried on practical experiments in wireless transmission". He believed that he had achieved Earth electrical resonance that, according to his theory, would produce electrical effects at any terrestrial distance.
There have been varied claims over the years regarding Tesla's accomplishments with his wireless system. His own notes from Colorado Springs are unclear as to whether he was ever successful at long-range transmission. Tesla made a claim in a 1916 statement to attorney Drury W. Cooper that in 1899, he collected quantitative transmission-reception data at a distance of about .
|
https://en.wikipedia.org/wiki/World_Wireless_System
|
passage: An alternate construction is given in the article on the de Rham curve: one uses the same technique as the de Rham curves, but instead of using a binary (base-2) expansion, one uses a ternary (base-3) expansion.
### Code
Given the drawing functions `void draw_line(double distance);` and `void turn(int angle_in_degrees);`, the code to draw an (approximate) Sierpiński arrowhead curve in C++ looks like this:
```cpp
void sierpinski_arrowhead_curve(unsigned order, double length)
{
// If order is even we can just draw the curve.
if (0 == (order & 1)) {
curve(order, length, +60);
}
else /* order is odd */ {
turn(+60);
curve(order, length, -60);
}
}
```
```cpp
void curve(unsigned order, double length, int angle)
{
if (0 == order) {
draw_line(length);
} else {
curve(order - 1, length / 2, -angle);
turn(angle);
curve(order - 1, length / 2, angle);
turn(angle);
curve(order - 1, length / 2, -angle);
}
}
```
Representation as Lindenmayer system
The Sierpiński arrowhead curve can be expressed by a rewrite system (L-system).
Alphabet: X, Y
Constants: F, +, −
Axiom: XF
Production rules:
X → YF + XF + Y
Y → XF − YF − X
Here, F means "draw forward", + means "turn left 60°", and − means "turn right 60°" (see turtle graphics).
|
https://en.wikipedia.org/wiki/Sierpi%C5%84ski_curve
|
passage: The
$$
$$
denotes the convolution operator.
$$
\begin{align}
f_t &= \sigma_g(W_{f} * x_t + U_{f} * h_{t-1} + V_{f} \odot c_{t-1} + b_f) \\
i_t &= \sigma_g(W_{i} * x_t + U_{i} * h_{t-1} + V_{i} \odot c_{t-1} + b_i) \\
c_t &= f_t \odot c_{t-1} + i_t \odot \sigma_c(W_{c} * x_t + U_{c} * h_{t-1} + b_c) \\
o_t &= \sigma_g(W_{o} * x_t + U_{o} * h_{t-1} + V_{o} \odot c_{t} + b_o) \\
h_t &= o_t \odot \sigma_h(c_t)
\end{align}
$$
## Training
An RNN using LSTM units can be trained in a supervised fashion on a set of training sequences, using an optimization algorithm like gradient descent combined with backpropagation through time to compute the gradients needed during the optimization process, in order to change each weight of the LSTM network in proportion to the derivative of the error (at the output layer of the LSTM network) with respect to corresponding weight.
|
https://en.wikipedia.org/wiki/Long_short-term_memory
|
passage: The technique is largely universal on modern large radars.
### Speed measurement
Speed is the change in distance to an object with respect to time. Thus the existing system for measuring distance, combined with a memory capacity to see where the target last was, is enough to measure speed. At one time the memory consisted of a user making grease pencil marks on the radar screen and then calculating the speed using a slide rule. Modern radar systems perform the equivalent operation faster and more accurately using computers.
If the transmitter's output is coherent (phase synchronized), there is another effect that can be used to make almost instant speed measurements (no memory is required), known as the Doppler effect. Most modern radar systems use this principle into Doppler radar and pulse-Doppler radar systems (weather radar, military radar). The Doppler effect is only able to determine the relative speed of the target along the line of sight from the radar to the target. Any component of target velocity perpendicular to the line of sight cannot be determined by using the Doppler effect alone, but it can be determined by tracking the target's azimuth over time.
It is possible to make a Doppler radar without any pulsing, known as a continuous-wave radar (CW radar), by sending out a very pure signal of a known frequency. CW radar is ideal for determining the radial component of a target's velocity. CW radar is typically used by traffic enforcement to measure vehicle speed quickly and accurately where the range is not important.
|
https://en.wikipedia.org/wiki/Radar
|
passage: Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Sahni, S. "Computationally related problems," in SIAM Journal on Computing, 3, 262--279, 1974.
## Definition
### Abstract form
A convex optimization problem is defined by two ingredients:
- The objective function, which is a real-valued convex function of n variables,
$$
f :\mathcal D \subseteq \mathbb{R}^n \to \mathbb{R}
$$
;
- The feasible set, which is a convex subset
$$
C\subseteq \mathbb{R}^n
$$
.
The goal of the problem is to find some
$$
\mathbf{x^\ast} \in C
$$
attaining
$$
\inf \{ f(\mathbf{x}) : \mathbf{x} \in C \}
$$
.
|
https://en.wikipedia.org/wiki/Convex_optimization
|
passage: These are longitudinal or compression waves. The sound wave propagates in the atmosphere though a series of compressions and expansions parallel to the direction of propagation.
- internal gravity waves (require stable stratification of the atmosphere)
- inertio-gravity waves (also include a significant Coriolis effect as opposed to "normal" gravity waves)
- Rossby waves (can be seen in the troughs and ridges of 500 hPa geopotential caused by midlatitude cyclones and anticyclones)
At the equator, mixed Rossby-gravity and Kelvin waves can also be observed.
|
https://en.wikipedia.org/wiki/Atmospheric_wave
|
passage: In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
$$
D_n(x)= \sum_{k=-n}^n e^{ikx} = \left(1+2\sum_{k=1}^n\cos(kx)\right)=\frac{\sin\left(\left(n +1/2\right) x \right)}{\sin(x/2)},
$$
where is any nonnegative integer. The kernel functions are periodic with period
$$
2\pi
$$
.
The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of with any function of period 2 is the nth-degree Fourier series approximation to , i.e., we have
$$
(D_n*f)(x)=\int_{-\pi}^\pi f(y)D_n(x-y)\,dy=2\pi\sum_{k=-n}^n \hat{f}(k)e^{ikx},
$$
where
$$
\widehat{f}(k)=\frac 1 {2\pi}\int_{-\pi}^\pi f(x)e^{-ikx}\,dx
$$
is the th Fourier coefficient of . This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
|
https://en.wikipedia.org/wiki/Dirichlet_kernel
|
passage: When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by
$$
\left( \mathbb{Q}/ \{ 0 \} ,\, \cdot \right)
$$
.
## Multiplication of different kinds of numbers
Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions).
Integers
$$
N\times M
$$
is the sum of N copies of M when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by
$$
N\times (-M) = (-N)\times M = - (N\times M)
$$
and
$$
(-N)\times (-M) = N\times M
$$
The same sign rules apply to rational and real numbers.
Rational numbers
Generalization to fractions
$$
\frac{A}{B}\times \frac{C}{D}
$$
is by multiplying the numerators and denominators, respectively:
$$
\frac{A}{B}\times \frac{C}{D} = \frac{(A\times C)}{(B\times D)}
$$
.
|
https://en.wikipedia.org/wiki/Multiplication
|
passage: Minkowski space can be contracted to a point, so a TQFT applied to Minkowski space results in trivial topological invariants. Consequently, TQFTs are usually applied to curved spacetimes, such as, for example, Riemann surfaces. Most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few higher-dimensional theories exist, but they are not very well understood .
Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigations into this class of models.
(Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property. It happens to be true in most of the examples that physicists and mathematicians study, but it is not necessary. A topological sigma model targets infinite-dimensional projective space, and if such a thing could be defined it would have countably infinitely many degrees of freedom.)
## Specific models
The known topological field theories fall into two general classes:
### Schwarz-type TQFTs
and
### Witten-type TQFTs
. Witten TQFTs are also sometimes referred to as cohomological field theories. See .
Schwarz-type TQFTs
In Schwarz-type TQFTs, the correlation functions or partition functions of the system are computed by the path integral of metric-independent action functionals.
|
https://en.wikipedia.org/wiki/Topological_quantum_field_theory
|
passage: This is useful, for example, in determining the bit error rate of a digital communication system.
The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.
The error function and its approximations can be used to estimate results that hold with high probability or with low probability. Given a random variable (a normal distribution with mean and standard deviation ) and a constant , it can be shown via integration by substitution:
$$
\begin{align}
\Pr[X\leq L] &= \frac{1}{2} + \frac{1}{2} \operatorname{erf}\frac{L-\mu}{\sqrt{2}\sigma} \\
&\approx A \exp \left(-B \left(\frac{L-\mu}{\sigma}\right)^2\right)
\end{align}
$$
where and are certain numeric constants. If is sufficiently far from the mean, specifically , then:
$$
\Pr[X\leq L] \leq A \exp (-B \ln{k}) = \frac{A}{k^B}
$$
so the probability goes to 0 as .
|
https://en.wikipedia.org/wiki/Error_function
|
passage: Intuitively, we can view the hyperprior vector as pseudocounts, i.e. as representing the number of observations in each category that we have already seen. Then we simply add in the counts for all the new observations (the vector ) in order to derive the posterior distribution.
In Bayesian mixture models and other hierarchical
### Bayesian models
with mixture components, Dirichlet distributions are commonly used as the prior distributions for the categorical variables appearing in the models. See the section on applications below for more information.
### Relation to Dirichlet-multinomial distribution
In a model where a Dirichlet prior distribution is placed over a set of categorical-valued observations, the marginal joint distribution of the observations (i.e. the joint distribution of the observations, with the prior parameter marginalized out) is a Dirichlet-multinomial distribution. This distribution plays an important role in hierarchical Bayesian models, because when doing inference over such models using methods such as Gibbs sampling or variational Bayes, Dirichlet prior distributions are often marginalized out. See the article on this distribution for more details.
### Entropy
If is a
$$
\operatorname{Dir}(\boldsymbol\alpha)
$$
random variable, the differential entropy of (in nat units)
|
https://en.wikipedia.org/wiki/Dirichlet_distribution
|
passage: The hCG concentration in blood is higher than in urine. Therefore, a blood test can be positive while the urine test is still negative.
Qualitative tests (yes/no or positive/negative results) look for the presence of the beta subunit of human chorionic gonadotropin in blood or urine. For a qualitative test the thresholds for a positive test are generally determined by an hCG cut-off where at least 95% of pregnant people would get a positive result on the day of their first missed period. Qualitative urine pregnancy tests vary in sensitivity. High-sensitivity tests are more common and typically detect hCG levels between 20 and 50 milli-international units/mL (mIU/mL). Low-sensitivity tests detect hCG levels between 1500 and 2000 mIU/mL and have unique clinical applications, including confirmation of medication abortion success. Qualitative urine tests available for home use are typically designed as lateral flow tests.
Quantitative tests measure the exact amount of hCG in the sample. Blood tests can detect hCG levels as low as 1 mIU/mL, and typically clinicians will diagnose a positive pregnancy test at 5mIU/mL.
+Table 1. Human chorionic gonadotropin (hCG) detection thresholds by test type and sample typeUrine pregnancy testBlood pregnancy testDetection thresholdsHigh-sensitivity:
|
https://en.wikipedia.org/wiki/Pregnancy_test
|
passage: The double angle approach relies on the fact that the angle
$$
\theta
$$
between the normal vectors to any two physical planes passing through
$$
P
$$
(Figure 4) is half the angle between two lines joining their corresponding stress points
$$
(\sigma_\mathrm{n}, \tau_\mathrm{n})
$$
on the Mohr circle and the centre of the circle.
This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of
$$
2\theta
$$
. It can also be seen that the planes
$$
A
$$
and
$$
B
$$
in the material element around
$$
P
$$
of Figure 5 are separated by an angle
$$
\theta=90^\circ
$$
, which in the Mohr circle is represented by a
$$
180^\circ
$$
angle (double the angle).
Pole or origin of planes
The second approach involves the determination of a point on the Mohr circle called the pole or the origin of planes. Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components
$$
\sigma
$$
and
$$
\tau
$$
on any particular plane, one can draw a line parallel to that plane through the particular coordinates
$$
\sigma_\mathrm{n}
$$
|
https://en.wikipedia.org/wiki/Mohr%27s_circle
|
passage: A frequent example application of this general rule is showing that the Levi-Civita connection, which is a mapping of smooth vector fields
$$
(X,Y) \mapsto \nabla_{X} Y
$$
taking a pair of vector fields to a vector field, does not define a tensor field on M. This is because it is only
$$
\mathbb R
$$
-linear in Y (in place of full C∞(M)-linearity, it satisfies the Leibniz rule,
$$
\nabla_{X}(fY) = (Xf) Y +f \nabla_X Y
$$
)). Nevertheless, it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.
## Applications
The curvature tensor is discussed in differential geometry and the stress–energy tensor is important in physics, and these two tensors are related by Einstein's theory of general relativity.
In electromagnetism, the electric and magnetic fields are combined into an electromagnetic tensor field.
Differential forms, used in defining integration on manifolds, are a type of tensor field.
## Tensor calculus
In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus. Even to formulate such equations requires a fresh notion, the covariant derivative.
|
https://en.wikipedia.org/wiki/Tensor_field
|
passage: Building on his previous work with tangents, Fermat used the curve
$$
y = x^\frac{3}{2} \,
$$
whose tangent at x = a had a slope of
$$
{3 \over 2} a^\frac{1}{2}
$$
so the tangent line would have the equation
$$
y = {3 \over 2} a^\frac{1}{2}(x - a) + f(a).
$$
Next, he increased a by a small amount to a + ε, making segment AC a relatively good approximation for the length of the curve from A to D. To find the length of the segment AC, he used the Pythagorean theorem:
$$
\begin{align}
AC^2 &= AB^2 + BC^2 \\
BLOCK0\end{align}
$$
which, when solved, yields
$$
AC = \varepsilon \sqrt{1 + {9 \over 4} a \,}.
$$
In order to approximate the length, Fermat would sum up a sequence of short segments.
## Curves with infinite length
As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve.
|
https://en.wikipedia.org/wiki/Arc_length
|
passage: The placenta occasionally takes a form in which it comprises several distinct parts connected by blood vessels. The parts, called lobes, may number two, three, four, or more. Such placentas are described as bilobed/bilobular/bipartite, trilobed/trilobular/tripartite, and so on. If there is a clearly discernible main lobe and auxiliary lobe, the latter is called a succenturiate placenta. Sometimes the blood vessels connecting the lobes get in the way of fetal presentation during labor, which is called vasa previa.
### Gene and protein expression
About 20,000 protein coding genes are expressed in human cells and 70% of these genes are expressed in the normal mature placenta. Some 350 of these genes are more specifically expressed in the placenta and fewer than 100 genes are highly placenta specific. The corresponding specific proteins are mainly expressed in trophoblasts and have functions related to pregnancy. Examples of proteins with elevated expression in placenta compared to other organs and tissues are PEG10 and the cancer testis antigen PAGE4 and expressed in cytotrophoblasts, CSH1 and KISS1 expressed in syncytiotrophoblasts, and PAPPA2 and PRG2 expressed in extravillous trophoblasts.
## Physiology
### Development
The placenta begins to develop upon implantation of the blastocyst into the maternal endometrium, very early on in pregnancy at about week 4.
|
https://en.wikipedia.org/wiki/Placenta
|
passage: The force acts on the charged particles themselves, hence charge has a tendency to spread itself as evenly as possible over a conducting surface. The magnitude of the electromagnetic force, whether attractive or repulsive, is given by Coulomb's law, which relates the force to the product of the charges and has an inverse-square relation to the distance between them. The electromagnetic force is very strong, second only in strength to the strong interaction, but unlike that force it operates over all distances. In comparison with the much weaker gravitational force, the electromagnetic force pushing two electrons apart is 1042 times that of the gravitational attraction pulling them together.
Charge originates from certain types of subatomic particles, the most familiar carriers of which are the electron and proton. Electric charge gives rise to and interacts with the electromagnetic force, one of the four fundamental forces of nature. Experiment has shown charge to be a conserved quantity, that is, the net charge within an electrically isolated system will always remain constant regardless of any changes taking place within that system. Within the system, charge may be transferred between bodies, either by direct contact or by passing along a conducting material, such as a wire. The informal term static electricity refers to the net presence (or 'imbalance') of charge on a body, usually caused when dissimilar materials are rubbed together, transferring charge from one to the other.
Charge can be measured by a number of means, an early instrument being the gold-leaf electroscope, which although still in use for classroom demonstrations, has been superseded by the electronic electrometer.
|
https://en.wikipedia.org/wiki/Electricity
|
passage: For example, if
$$
X_n
$$
are distributed uniformly on intervals
$$
\left( 0,\frac{1}{n} \right)
$$
, then this sequence converges in distribution to the degenerate random variable
$$
X=0
$$
. Indeed,
$$
F_n(x) = 0
$$
for all
$$
n
$$
when
$$
x\leq 0
$$
, and
$$
F_n(x) = 1
$$
for all
$$
x \geq \frac{1}{n}
$$
when
$$
n > 0
$$
. However, for this limiting random variable
$$
F(0) = 1
$$
, even though
$$
F_n(0) = 0
$$
for all
$$
n
$$
. Thus the convergence of cdfs fails at the point
$$
x=0
$$
where
$$
F
$$
is discontinuous.
Convergence in distribution may be denoted as
where
$$
\scriptstyle\mathcal{L}_X
$$
is the law (probability distribution) of . For example, if is standard normal we can write
$$
X_n\,\xrightarrow{d}\,\mathcal{N}(0,\,1)
$$
.
For random vectors
$$
\left\{ X_1,X_2,\dots \right\}\subset \mathbb{R}^k
$$
the convergence in distribution is defined similarly.
|
https://en.wikipedia.org/wiki/Convergence_of_random_variables
|
passage: UBLs are structurally similar to ubiquitin and are processed, activated, conjugated, and released from conjugates by enzymatic steps that are similar to the corresponding mechanisms for ubiquitin. UBLs are also translated with C-terminal extensions that are processed to expose the invariant C-terminal LRGG. These modifiers have their own specific E1 (activating), E2 (conjugating) and E3 (ligating) enzymes that conjugate the UBLs to intracellular targets. These conjugates can be reversed by UBL-specific isopeptidases that have similar mechanisms to that of the deubiquitinating enzymes.
Within some species, the recognition and destruction of sperm mitochondria through a mechanism involving ubiquitin is responsible for sperm mitochondria's disposal after fertilization occurs.
### Prokaryotic origins
Ubiquitin is believed to have descended from bacterial proteins similar to ThiS () or MoaD (). These prokaryotic proteins, despite having little sequence identity (ThiS has 14% identity to ubiquitin), share the same protein fold. These proteins also share sulfur chemistry with ubiquitin. MoaD, which is involved in molybdopterin biosynthesis, interacts with MoeB, which acts like an E1 ubiquitin-activating enzyme for MoaD, strengthening the link between these prokaryotic proteins and the ubiquitin system. A similar system exists for ThiS, with its E1-like enzyme ThiF.
|
https://en.wikipedia.org/wiki/Ubiquitin
|
passage: Spaces that are connected but not simply connected are called non-simply connected or multiply connected.
The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of dimension, is called contractibility.
## Examples
- The Euclidean plane
$$
\R^2
$$
is simply connected, but
$$
\R^2
$$
minus the origin
$$
(0, 0)
$$
is not. If
$$
n > 2,
$$
then both
$$
\R^n
$$
and
$$
\R^n
$$
minus the origin are simply connected.
- Analogously: the n-dimensional sphere
$$
S^n
$$
is simply connected if and only if
$$
n \geq 2.
$$
- Every convex subset of
$$
\R^n
$$
is simply connected.
- A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are not simply connected.
- Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
- For
$$
n \geq 2,
$$
the special orthogonal group
$$
\operatorname{SO}(n, \R)
$$
is not simply connected and the special unitary group
$$
\operatorname{SU}(n)
$$
is simply connected.
-
|
https://en.wikipedia.org/wiki/Simply_connected_space
|
passage: Perl 3, released in October 1989, added support for binary data streams.
1990s
Originally, the only documentation for Perl was a single lengthy man page. In 1991, Programming Perl, known to many Perl programmers as the "Camel Book" because of its cover, was published and became the de facto reference for the language. At the same time, the Perl version number was bumped to 4, not to mark a major change in the language but to identify the version that was well documented by the book. Perl 4 was released in March 1991.
Perl 4 went through a series of maintenance releases, culminating in Perl 4.036 in 1993, whereupon Wall abandoned Perl 4 to begin work on Perl 5. Initial design of Perl 5 continued into 1994. The perl5-porters mailing list was established in May 1994 to coordinate work on porting Perl 5 to different platforms. It remains the primary forum for development, maintenance, and porting of Perl 5.
Perl 5.000 was released on October 17, 1994. It was a nearly complete rewrite of the interpreter, and it added many new features to the language, including objects, references, lexical (my) variables, and modules. Importantly, modules provided a mechanism for extending the language without modifying the interpreter. This allowed the core interpreter to stabilize, even as it enabled ordinary Perl programmers to add new language features. Perl 5 has been in active development since then.
Perl 5.001 was released on March 13, 1995. Perl 5.002 was released on February 29, 1996 with the new prototypes feature.
|
https://en.wikipedia.org/wiki/Perl
|
passage: ## Related problems
### Equivalent problems
Hypergraph matching is equivalent to set packing: the sets correspond to the hyperedges.
The independent set problem is also equivalent to set packing – there is a one-to-one polynomial-time reduction between them:
- Given a set packing problem on a collection
$$
\mathcal{S}
$$
, build a graph where for each set
$$
S \in \mathcal{S}
$$
there is a vertex
$$
v_S
$$
, and there is an edge between
$$
v_S
$$
and
$$
v_T
$$
iff
$$
S \cap T \neq \varnothing
$$
. Every independent set of vertices in the generated graph corresponds to a set packing in
$$
\mathcal{S}
$$
.
- Given an independent vertex set problem on a graph
$$
G(V,E)
$$
, build a collection of sets where for each vertex
$$
v
$$
there is a set
$$
S_v
$$
containing all edges adjacent to
$$
v
$$
. Every set packing in the generated collection corresponds to an independent vertex set in
$$
G(V,E)
$$
.
This is also a bidirectional PTAS reduction, and it shows that the two problems are equally difficult to approximate.
In the special case when each set contains at most k elements (the k-set packing problem), the intersection graph is (k+1)-claw-free.
|
https://en.wikipedia.org/wiki/Set_packing
|
passage: We would like to think of
$$
\xi
$$
as defining a mapping which maps
$$
\omega \in \Omega
$$
to a measure
$$
\xi_\omega \in \mathcal{M}(\mathcal{S})
$$
(namely,
$$
\Omega \mapsto \mathcal{M}(\mathcal{S})
$$
),
where
$$
\mathcal{M}(\mathcal{S})
$$
is the set of all locally finite measures on
$$
S
$$
.
Now, to make this mapping measurable, we need to define a
$$
\sigma
$$
-field over
$$
\mathcal{M}(\mathcal{S})
$$
.
This
$$
\sigma
$$
-field is constructed as the minimal algebra so that all evaluation maps of the form
$$
\pi_B: \mu \mapsto \mu(B)
$$
, where
$$
B \in \mathcal{S}
$$
is relatively compact,
are measurable. Equipped with this
$$
\sigma
$$
-field, then
$$
\xi
$$
is a random element, where for every
$$
\omega \in \Omega
$$
,
$$
\xi_\omega
$$
is a locally finite measure over
$$
S
$$
.
Now, by a point process on
$$
S
$$
|
https://en.wikipedia.org/wiki/Point_process
|
passage: - The alternating group A4 of order 12 is solvable but has no subgroups of order 6 even though 6 divides 12, showing that
## Hall's theorem
(see below) cannot be extended to all divisors of the order of a solvable group.
- If G = A5, the only simple group of order 60, then 15 and 20 are Hall divisors of the order of G, but G has no subgroups of these orders.
- The simple group of order 168 has two different conjugacy classes of Hall subgroups of order 24 (though they are connected by an outer automorphism of G).
- The simple group of order 660 has two Hall subgroups of order 12 that are not even isomorphic (and so certainly not conjugate, even under an outer automorphism). The normalizer of a Sylow of order 4 is isomorphic to the alternating group A4 of order 12, while the normalizer of a subgroup of order 2 or 3 is isomorphic to the dihedral group of order 12.
Hall's theorem
proved that if G is a finite solvable group and π
is any set of primes, then G has a Hall π-subgroup, and any two Hall are conjugate. Moreover, any subgroup whose order is
a product of primes in π is contained in some Hall . This result can be thought of as a generalization of Sylow's Theorem to Hall subgroups, but the examples above show that such a generalization is false when the group is not solvable.
|
https://en.wikipedia.org/wiki/Hall_subgroup
|
passage: The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters); the Mormon Tabernacle at Temple Square in Salt Lake City, Utah; at an exhibit on sound at the Museum of Science and Industry in Chicago; in front of the University of Illinois at Urbana–Champaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.
#### Planetary orbits
In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.
More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.
Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance.
|
https://en.wikipedia.org/wiki/Ellipse
|
passage: The ecosystems encountered by the first Americans had not been exposed to human interaction, and may have been far less resilient to human made changes than the ecosystems encountered by industrial era humans. Therefore, the actions of the Clovis people, despite seeming insignificant by today's standards could indeed have had a profound effect on the ecosystems and wild life which was entirely unused to human influence.
In the Yukon, the mammoth steppe ecosystem collapsed between 13,500 and 10,000 BP, though wild horses and woolly mammoths somehow persisted in the region for millennia after this collapse. In what is now Texas, a drop in local plant and animal biodiversity occurred during the Younger Dryas cooling, though while plant diversity recovered after the Younger Dryas, animal diversity did not. In the Channel Islands, multiple terrestrial species went extinct around the same time as human arrival, but direct evidence for an anthropogenic cause of their extinction remains lacking. In the montane forests of the Colombian Andes, spores of coprophilous fungi indicate megafaunal extinction occurred in two waves, the first occurring around 22,900 BP and the second around 10,990 BP. A 2023 study of megafaunal extinctions in the Junín Plateau of Peru found that the timing of the disappearance of megafauna was concurrent with a large uptick in fire activity attributed to human actions, implicating humans as the cause of their local extinction on the plateau.
|
https://en.wikipedia.org/wiki/Holocene_extinction
|
passage: The support vector clustering algorithm, created by Hava Siegelmann and Vladimir Vapnik, applies the statistics of support vectors, developed in the support vector machines algorithm, to categorize unlabeled data. These data sets require unsupervised learning approaches, which attempt to find natural clustering of the data into groups, and then to map new data according to these clusters.
The popularity of SVMs is likely due to their amenability to theoretical analysis, and their flexibility in being applied to a wide variety of tasks, including structured prediction problems. It is not clear that SVMs have better predictive performance than other linear models, such as logistic regression and linear regression.
## Motivation
Classifying data is a common task in machine learning.
Suppose some given data points each belong to one of two classes, and the goal is to decide which class a new data point will be in. In the case of support vector machines, a data point is viewed as a
$$
p
$$
-dimensional vector (a list of
$$
p
$$
numbers), and we want to know whether we can separate such points with a
$$
(p-1)
$$
-dimensional hyperplane. This is called a linear classifier. There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized.
|
https://en.wikipedia.org/wiki/Support_vector_machine
|
passage: From this follows
$$
x - 9 = x - 2 - \beta
$$
, so
$$
\beta \equiv 7 \pmod{11}
$$
and
$$
-\beta \equiv -7 \equiv 4 \pmod{11}
$$
.
A manual check shows that, indeed,
$$
7^2 \equiv 49 \equiv 5\pmod{11}
$$
and
$$
4^2\equiv 16 \equiv 5\pmod{11}
$$
.
## Correctness proof
The algorithm finds factorization of
$$
f_z(x)
$$
in all cases except for ones when all numbers
$$
z+\lambda_1, z+\lambda_2, \ldots, z+\lambda_n
$$
are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when
$$
\lambda_1, \ldots, \lambda_n
$$
are all residues or non-residues simultaneously (that is, when
$$
z=0
$$
would fail) may be estimated as
$$
2^{-k}
$$
where
$$
k
$$
is the number of distinct values in
$$
\lambda_1, \ldots, \lambda_n
$$
.
|
https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Rabin_algorithm
|
passage: The FOVS offers several programs: a BSc in Optometry, which takes 5 years and includes sub-specialization in orthoptics, contact lenses, ocular photography, or ocular neurology; a BSc in Ophthalmic Technology, requiring 4 years of training; and a BSc in Optical Dispensary, completed in 4 years. The FOVS also offers MSc and PhD degrees in optometry. The FOVS is the only institute of its kind in Sudan and was the first institution of higher education in Optometry in the Middle East and Africa. In 2010, Alneelain University Eye Hospital was established as part of the FOVS to expand training capacity and to serve broader Sudanese community.
Ghana
The Ghana Optometric Association (GOA) regulates the practice of Optometry in Ghana. The Kwame Nkrumah University of Science and Technology and the University of Cape Coast are the two universities that offer the degree programme in the country. After the six-year training at any of the two universities offering the course, the O.D. degree is awarded. The new optometrist must write a qualifying exam, after which the optometrist is admitted as a member of the GOA, leading to the award of the title MGOA.
#### Mozambique
The first optometry course in Mozambique was started in 2009 at Universidade Lurio, Nampula. The course is part of the Mozambique Eyecare Project. University of Ulster, Dublin Institute of Technology and Brien Holden Vision Institute are supporting partners. As of 2019, 61 Mozambican students had graduated with optometry degrees
from UniLúrio (34 male and 27 female).
|
https://en.wikipedia.org/wiki/Optometry
|
passage: ## Tangential Graeffe method
This method replaces the numbers by truncated power series of degree 1, also known as dual numbers. Symbolically, this is achieved by introducing an "algebraic infinitesimal"
$$
\varepsilon
$$
with the defining property
$$
\varepsilon^2=0
$$
. Then the polynomial
$$
p(x+\varepsilon)=p(x)+\varepsilon\,p'(x)
$$
has roots
$$
x_m-\varepsilon
$$
, with powers
$$
(x_m-\varepsilon)^{2^k}=x_m^{2^k}-\varepsilon\,{2^k}\,x_m^{2^k-1}=y_m+\varepsilon\,\dot y_m.
$$
Thus the value of
$$
x_m
$$
is easily obtained as fraction
$$
x_m=-\tfrac{2^k\,y_m}{\dot y_m}.
$$
This kind of computation with infinitesimals is easy to implement analogous to the computation with complex numbers. If one assumes complex coordinates or an initial shift by some randomly chosen complex number, then all roots of the polynomial will be distinct and consequently recoverable with the iteration.
## Renormalization
Every polynomial can be scaled in domain and range such that in the resulting polynomial the first and the last coefficient have size one.
|
https://en.wikipedia.org/wiki/Graeffe%27s_method
|
passage: In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of instantaneous, or "non-local" action at a distance. Locality evolved out of the field theories of classical physics. The idea is that for a cause at one point to have an effect at another point, something in the space between those points must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.
The special theory of relativity limits the maximum speed at which causal influence can travel to the speed of light,
$$
c
$$
. Therefore, the principle of locality implies that an event at one point cannot cause a truly simultaneous result at another point. An event at point
$$
A
$$
cannot cause a result at point
$$
B
$$
in a time less than
$$
T=D/c
$$
, where
$$
D
$$
is the distance between the points and
$$
c
$$
is the speed of light in vacuum.
The principle of locality plays a critical role in one of the central results of quantum mechanics. In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen, with their EPR paradox thought experiment, raised the possibility that quantum mechanics might not be a complete theory. They described two systems physically separated after interacting; this pair would be called entangled in modern terminology.
|
https://en.wikipedia.org/wiki/Principle_of_locality
|
passage: Since the wind speed profile is logarithmic to the water surface, the curvature has a negative sign at this point. This relation shows the wind flow transferring its kinetic energy to the water surface at their interface.
Assumptions:
1. two-dimensional parallel shear flow
1. incompressible, inviscid water and wind
1. irrotational water
1. slope of the displacement of the water surface is small
Generally, these wave formation mechanisms occur together on the water surface and eventually produce fully developed waves.
For example, if we assume a flat sea surface (Beaufort state 0), and a sudden wind flow blows steadily across the sea surface, the physical wave generation process follows the sequence:
1. Turbulent wind forms random pressure fluctuations at the sea surface. Ripples with wavelengths in the order of a few centimeters are generated by the pressure fluctuations. (The Phillips mechanism)
1. The winds keep acting on the initially rippled sea surface causing the waves to become larger. As the waves grow, the pressure differences get larger causing the growth rate to increase. Finally, the shear instability expedites the wave growth exponentially. (The Miles mechanism)
1. The interactions between the waves on the surface generate longer waves and the interaction will transfer wave energy from the shorter waves generated by the Miles mechanism to the waves which have slightly lower frequencies than the frequency at the peak wave magnitudes, then finally the waves will be faster than the crosswind speed (Pierson & Moskowitz).
|
https://en.wikipedia.org/wiki/Wind_wave
|
passage: They are special homogeneous coordinates. Barycentric coordinates are strongly related with Cartesian coordinates and, more generally, to affine coordinates ().
Barycentric coordinates are particularly useful in triangle geometry for studying properties that do not depend on the angles of the triangle, such as Ceva's theorem, Routh's theorem, and Menelaus's theorem. In computer-aided design, they are useful for defining some kinds of Bézier surfaces. Gerald Farin: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, 1990, , S. 20.
## Definition
Let
$$
A_0, \ldots, A_n
$$
be points in a Euclidean space, a flat or an affine space
$$
\mathbf A
$$
of dimension that are affinely independent; this means that there is no affine subspace of dimension that contains all the points, or, equivalently that the points define a simplex. Given any point
$$
P\in \mathbf A,
$$
there are scalars
$$
a_0, \ldots, a_n
$$
that are not all zero, such that
$$
( a_0 + \cdots + a_n ) \overset{}\overrightarrow{OP} = a_0 \overset{}\overrightarrow {OA_0} + \cdots + a_n \overset{}\overrightarrow {OA_n},
$$
for any point .
|
https://en.wikipedia.org/wiki/Barycentric_coordinate_system
|
passage: In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of check bits (or parity data), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message.
Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common channel models include memoryless models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in bursts. Consequently, error-detecting and -correcting codes can be generally distinguished between random-error-detecting/correcting and burst-error-detecting/correcting. Some codes can also be suitable for a mixture of random errors and burst errors.
If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as automatic repeat request (ARQ), and is most notably used in the
### Internet
.
|
https://en.wikipedia.org/wiki/Error_detection_and_correction
|
passage: As an example from the area of programming languages, the set of all strings denoting a floating point number can be described by an extended right-regular grammar G with N = {S,A,B,C,D,E,F}, Σ = {0,1,2,3,4,5,6,7,8,9,+,−,.,e}, where S is the start symbol, and P consists of the following rules:
{|
|-
|| S → +A || A → 0A || B → 0C || C → 0C || D → +E || E → 0F || F → 0F
|-
|| S → −A || A → 1A || B → 1C || C → 1C || D → −E || E → 1F || F → 1F
|-
|| S → A || A → 2A || B → 2C || C → 2C || D → E || E → 2F || F → 2F
|-
|| || A → 3A || B → 3C || C → 3C || || E → 3F || F → 3F
|-
|| || A → 4A || B → 4C || C → 4C || || E → 4F || F → 4F
|-
|| ||
|
https://en.wikipedia.org/wiki/Regular_grammar
|
passage: Alternatively, transforming the sample-cluster distance through a Gaussian RBF, obtains the hidden layer of a radial basis function network.
This use of k-means has been successfully combined with simple, linear classifiers for semi-supervised learning in NLP (specifically for named-entity recognition) and in computer vision. On an object recognition task, it was found to exhibit comparable performance with more sophisticated feature learning approaches such as autoencoders and restricted Boltzmann machines. However, it generally requires more data, for equivalent performance, because each data point only contributes to one "feature".
Example: In natural language processing (NLP), k-means clustering has been integrated with simple linear classifiers for semi-supervised learning tasks such as named-entity recognition (NER). By first clustering unlabeled text data using k-means, meaningful features can be extracted to improve the performance of NER models. For instance, k-means clustering can be applied to identify clusters of words or phrases that frequently co-occur in the input text, which can then be used as features for training the NER model. This approach has been shown to achieve comparable performance with more complex feature learning techniques such as autoencoders and restricted Boltzmann machines, albeit with a greater requirement for labeled data.
### Recent Developments
Recent advancements in the application of k-means clustering include improvements in initialization techniques, such as the use of k-means++ initialization to select initial cluster centroids in a more effective manner.
|
https://en.wikipedia.org/wiki/K-means_clustering
|
passage: The Arabic works attributed to Jabir ibn Hayyan introduced a systematic classification of chemical substances, and provided instructions for deriving an inorganic compound (sal ammoniac or ammonium chloride) from organic substances (such as plants, blood, and hair) by chemical means. Some Arabic Jabirian works (e.g., the "Book of Mercy", and the "Book of Seventy") were later translated into Latin under the Latinized name "Geber", and in 13th-century Europe an anonymous writer, usually referred to as pseudo-Geber, started to produce alchemical and metallurgical writings under this name. Later influential Muslim philosophers, such as Abū al-Rayhān al-Bīrūnī and Avicenna disputed the theories of alchemy, particularly the theory of the transmutation of metals.
Improvements of the refining of ores and their extractions to smelt metals was widely used source of information for early chemists in the 16th century, among them Georg Agricola (1494–1555), who published his major work De re metallica in 1556. His work, describing highly developed and complex processes of mining metal ores and metal extraction, were the pinnacle of metallurgy during that time. His approach removed all mysticism associated with the subject, creating the practical base upon which others could and would build. The work describes the many kinds of furnaces used to smelt ore, and stimulated interest in minerals and their composition. Agricola has been described as the "father of metallurgy" and the founder of geology as a scientific discipline.
|
https://en.wikipedia.org/wiki/Chemistry
|
passage: We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the class of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)
Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of asymmetry i.e. given the relation such as S (" . . . is the successor of . . . ") between two terms x and y: x S y ≠ y S x. Second, he defines the notion of transitivity for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notion of connected: "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)
He concludes: ". . . [natural] number m is said to be less than another number n when n possesses every hereditary property possessed by the successor of m.
|
https://en.wikipedia.org/wiki/Logicism
|
passage: Only available on races with 8+ runners.
- Trio Ordre (Trifecta): The bettor must correctly pick the first, second and third finishers in their finishing order. Only available on races with 4-7 runners.
- Tiercé/Tiercé Classic: First created in 1954. The bettor must correctly pick the first, second and third finishers in the race with a main dividend paid for selecting the exact order of finish, and a secondary "Désordre" dividend paid for selecting the correct three runners but in the wrong order. Essentially a combination Trio Ordre/Trio bet with the main dividend paying at least 5 times the secondary one. On popular or famous races this bet type is sometimes labelled "Tiercé Classique" but follows the same rules as the standard Tiercé.
- Quarté+:
|
https://en.wikipedia.org/wiki/Parimutuel_betting
|
passage: Fields that employ the concept of taphonomy include:
- Archaeobotany
-
### Archaeology
- Biology
-
### Forensic science
- Geoarchaeology
- Geology
- Paleoecology
-
### Paleontology
- Zooarchaeology
There are five main stages of taphonomy: disarticulation, dispersal, accumulation, fossilization, and mechanical alteration. The first stage, disarticulation, occurs as the organism decays and the bones are no longer held together by the flesh and tendons of the organism. Dispersal is the separation of pieces of an organism caused by natural events (i.e. floods, scavengers etc.). Accumulation occurs when there is a buildup of organic and/or inorganic materials in one location (scavengers or human behavior). When mineral rich groundwater permeates organic materials and fills the empty spaces, a fossil is formed. The final stage of taphonomy is mechanical alteration; these are the processes that physically alter the remains (i.e. freeze-thaw, compaction, transport, burial). These stages are not only successive, they interplay. For example, chemical changes occur at every stage of the process, because of bacteria. Changes begin as soon as the death of the organism: enzymes are released that destroy the organic contents of the tissues, and mineralised tissues such as bone, enamel and dentin are a mixture of organic and mineral components. Moreover, most often the organisms (vegetal or animal) are dead because they have been killed by a predator.
|
https://en.wikipedia.org/wiki/Taphonomy
|
passage: x[i] /= A[i][i];
}
}
/* INPUT: A,P filled in LUPDecompose; N - dimension
- OUTPUT: IA is the inverse of the initial matrix
- /
void LUPInvert(double **A, int *P, int N, double **IA) {
for (int j = 0; j < N; j++) {
for (int i = 0; i < N; i++) {
IA[i][j] = P[i] == j ? 1.0 : 0.0;
for (int k = 0; k < i; k++)
IA[i][j] -= A[i][k] * IA[k][j];
}
for (int i = N - 1; i >= 0; i--) {
for (int k = i + 1; k < N; k++)
IA[i][j] -= A[i][k] * IA[k][j];
IA[i][j] /= A[i][i];
}
}
}
/* INPUT: A,P filled in LUPDecompose; N - dimension.
- OUTPUT: Function returns the determinant of the initial matrix
- /
double LUPDeterminant(double **A, int *P, int N) {
double det = A[0][0];
for (int i = 1; i < N; i++)
det *= A[i][i];
return (P[N] - N) % 2 == 0 ? det : -det;
}
|
https://en.wikipedia.org/wiki/LU_decomposition
|
passage: Parallel transport it to get a basis
$$
\{e_i(t)\}
$$
all along
$$
\gamma
$$
.
This gives an orthonormal basis with
$$
e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|
$$
. The Jacobi field can be written in co-ordinates in terms of this basis as
$$
J(t)=y^k(t)e_k(t)
$$
and thus
$$
\frac{D}{dt}J=\sum_k\frac{dy^k}{dt}e_k(t),\quad\frac{D^2}{dt^2}J=\sum_k\frac{d^2y^k}{dt^2}e_k(t),
$$
and the Jacobi equation can be rewritten as a system
$$
\frac{d^2y^k}{dt^2}+|\dot\gamma|^2\sum_j y^j(t)\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\rangle=0
$$
for each
$$
k
$$
. This way we get a linear ordinary differential equation (ODE).
Since this ODE has smooth coefficients we have that solutions exist for all
$$
t
$$
and are unique, given
$$
y^k(0)
$$
and
$$
{y^k}'(0)
$$
, for all
$$
k
$$
.
|
https://en.wikipedia.org/wiki/Jacobi_field
|
passage: Example
Consider the field extension
$$
\mathbb{Q}(\theta)
$$
with
$$
\theta = \zeta_3\sqrt[3]{2}
$$
, where
$$
\zeta_3
$$
denotes the cube root of unity
$$
\exp(2\pi i/3).
$$
Then, we have a
$$
\mathbb{Q}
$$
-basis given by
$$
\{ 1, \zeta_3\sqrt[3]{2}, (\zeta_3\sqrt[3]{2})^2\}
$$
since any
$$
x \in \mathbb{Q}(\theta)
$$
can be expressed as some
$$
\mathbb{Q}
$$
-linear combination:
$$
x = a + b\zeta_3\sqrt[3]{2} + c(\zeta_3\sqrt[3]{2})^2 = a + b\theta + c\theta^2.
$$
We proceed to calculate the trace
$$
T(x)
$$
and norm
$$
N(x)
$$
of this number. To this end, we take an arbitrary
$$
y \in \mathbb{Q}(\theta)
$$
where
$$
y = y_0 + y_1\theta + y_2 \theta^2
$$
and compute the product
$$
x y
$$
.
|
https://en.wikipedia.org/wiki/Algebraic_number_field
|
passage: 1. It can manipulate the stack as part of performing a transition.
A pushdown automaton reads a given input string from left to right. In each step, it chooses a transition by indexing a table by input symbol, current state, and the symbol at the top of the stack. A pushdown automaton can also manipulate the stack, as part of performing a transition. The manipulation can be to push a particular symbol to the top of the stack, or to pop off the top of the stack. The automaton can alternatively ignore the stack, and leave it as it is.
Put together: Given an input symbol, current state, and stack symbol, the automaton can follow a transition to another state, and optionally manipulate (push or pop) the stack.
If, in every situation, at most one such transition action is possible, then the automaton is called a deterministic pushdown automaton (DPDA). In general, if several actions are possible, then the automaton is called a general, or nondeterministic, PDA. A given input string may drive a nondeterministic pushdown automaton to one of several configuration sequences; if one of them leads to an accepting configuration after reading the complete input string, the latter is said to belong to the language accepted by the automaton.
## Formal definition
We use standard formal language notation:
$$
\Gamma^{*}
$$
denotes the set of finite-length strings over alphabet
$$
\Gamma
$$
and
$$
\varepsilon
$$
denotes the empty string.
|
https://en.wikipedia.org/wiki/Pushdown_automaton
|
passage: Atomic, molecular, and optical physics (AMO) is the study of matter–matter and light–matter interactions, at the scale of one or a few atoms and energy scales around several electron volts. The three areas are closely interrelated. AMO theory includes classical, semi-classical and quantum treatments. Typically, the theory and applications of emission, absorption, scattering of electromagnetic radiation (light) from excited atoms and molecules, analysis of spectroscopy, generation of lasers and masers, and the optical properties of matter in general, fall into these categories.
## Atomic and molecular physics
Atomic physics is the subfield of AMO that studies atoms as an isolated system of electrons and an atomic nucleus, while molecular physics is the study of the physical properties of molecules. The term atomic physics is often associated with nuclear power and nuclear bombs, due to the synonymous use of atomic and nuclear in standard English. However, physicists distinguish between atomic physics — which deals with the atom as a system consisting of a nucleus and electrons — and nuclear physics, which considers atomic nuclei alone. The important experimental techniques are the various types of spectroscopy. Molecular physics, while closely related to atomic physics, also overlaps greatly with theoretical chemistry, physical chemistry and chemical physics.
Both subfields are primarily concerned with electronic structure and the dynamical processes by which these arrangements change.
|
https://en.wikipedia.org/wiki/Atomic%2C_molecular%2C_and_optical_physics%23Optical_physics
|
passage: If two lines and intersect, then is a point. If is a point not lying on the same plane, then , both representing a line. But when and are parallel, this distributivity fails, giving on the left-hand side and a third parallel line on the right-hand side.
## Euclidean geometry
The aforementioned facts do not depend on the structure being that of Euclidean space (namely, involving Euclidean distance) and are correct in any affine space. In a Euclidean space:
- There is the distance between a flat and a point. (See for example Distance from a point to a plane and Distance from a point to a line.)
- There is the distance between two flats, equal to 0 if they intersect. (See for example Distance between two parallel lines (in the same plane) and .)
- There is the angle between two flats, which belongs to the interval between 0 and the right angle. (See for example Dihedral angle (between two planes). See also Angles between flats.)
|
https://en.wikipedia.org/wiki/Flat_%28geometry%29
|
passage: The incenter of the triangle also lies on the Euler line, something that is not true for other triangles. If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles.
### Area
The area
$$
T
$$
of an isosceles triangle can be derived from the formula for its height, and from the general formula for the area of a triangle as half the product of base and height:
$$
T=\frac{b}{4}\sqrt{4a^2-b^2}.
$$
The same area formula can also be derived from Heron's formula for the area of a triangle from its three sides. However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between the semiperimeter and side length in those triangles.
If the apex angle
$$
(\theta)
$$
and leg lengths
$$
(a)
$$
of an isosceles triangle are known, then the area of that triangle is:
$$
T=\frac{1}{2}a^2\sin\theta.
$$
This is a special case of the general formula for the area of a triangle as half the product of two sides times the sine of the included angle.
|
https://en.wikipedia.org/wiki/Isosceles_triangle
|
passage: Review: Bulletin of the American Mathematical Society 37 (1931), 791-793,
- .
- .
- .
-
- . Review: Séquin, Carlo H. (2009), Journal of Mathematics and the Arts 3: 229–230,
## External links
-
- Ruled surface pictures from the University of Arizona
- Examples of developable surfaces on the Rhino3DE website
Category:Surfaces
Category:Differential geometry
Category:Differential geometry of surfaces
Category:Complex surfaces
Category:Algebraic surfaces
Category: Geometric shapes
Category:Analytic geometry
|
https://en.wikipedia.org/wiki/Ruled_surface
|
passage: In the same year Dmitri Ivanenko suggested that there were no electrons in the nucleus — only protons and neutrons — and that neutrons were spin particles, which explained the mass not due to protons. The neutron spin immediately solved the problem of the spin of nitrogen-14, as the one unpaired proton and one unpaired neutron in this model each contributed a spin of in the same direction, giving a final total spin of 1.
With the discovery of the neutron, scientists could at last calculate what fraction of binding energy each nucleus had, by comparing the nuclear mass with that of the protons and neutrons which composed it. Differences between nuclear masses were calculated in this way. When nuclear reactions were measured, these were found to agree with Einstein's calculation of the equivalence of mass and energy to within 1% as of 1934.
### Proca's equations of the massive vector boson field
Alexandru Proca was the first to develop and report the massive vector boson field equations and a theory of the mesonic field of nuclear forces. Proca's equations were known to Wolfgang Pauli who mentioned the equations in his Nobel address, and they were also known to Yukawa, Wentzel, Taketani, Sakata, Kemmer, Heitler, and Fröhlich who appreciated the content of Proca's equations for developing a theory of the atomic nuclei in Nuclear Physics. G. A. Proca, Alexandre Proca. Oeuvre Scientifique Publiée, S.I.A.G., Rome, 1988.
|
https://en.wikipedia.org/wiki/Nuclear_physics
|
passage: They commonly occur in monocots and pteridophytes, but also in many dicots, such as clover (Trifolium), ivy (Hedera), strawberry (Fragaria) and willow (Salix). Most aerial roots and stilt roots are adventitious. In some conifers adventitious roots can form the largest part of the root system. Adventitious root formation is enhanced in many plant species during (partial) submergence, to increase gas exchange and storage of gases like oxygen. Distinct types of adventitious roots can be classified and are dependent on morphology, growth dynamics and function.
- Aerating roots (or knee root or knee or pneumatophores): roots rising above the ground, especially above water such as in some mangrove genera (Avicennia, Sonneratia). In some plants like Avicennia the erect roots have a large number of breathing pores for exchange of gases.
- Aerial roots: roots entirely above the ground, such as in ivy (Hedera) or in epiphytic orchids. Many aerial roots are used to receive water and nutrient intake directly from the air – from fogs, dew or humidity in the air. Some rely on leaf systems to gather rain or humidity and even store it in scales or pockets. Other aerial roots, such as mangrove aerial roots, are used for aeration and not for water absorption.
|
https://en.wikipedia.org/wiki/Root
|
passage: most composites.
A PRP test is sometimes combined with a table of small pseudoprimes to quickly establish the primality of a given number smaller than some threshold.
## Variations
An Euler probable prime to base a is an integer that is indicated prime by the somewhat stronger theorem that for any prime p, a(p−1)/2 equals
$$
(\tfrac{a}{p})
$$
modulo p, where
$$
(\tfrac{a}{p})
$$
is the Jacobi symbol. An Euler probable prime which is composite is called an Euler–Jacobi pseudoprime to base a. The smallest Euler-Jacobi pseudoprime to base 2 is 561. There are 11347 Euler-Jacobi pseudoprimes base 2 that are less than 25·109.
This test may be improved by using the fact that the only square roots of 1 modulo a prime are 1 and −1. Write n = d · 2s + 1, where d is odd. The number n is a strong probable prime (SPRP) to base a if:
$$
a^d\equiv 1\pmod n,\;
$$
or
$$
a^{d\cdot 2^r}\equiv -1\pmod n\text{ for some }0\leq r\leq s-1. \,
$$
A composite strong probable prime to base a is called a strong pseudoprime to base a. Every strong probable prime to base a is also an Euler probable prime to the same base, but not vice versa.
The smallest strong pseudoprime base 2 is 2047.
|
https://en.wikipedia.org/wiki/Probable_prime
|
passage: The United States National Institute of Standards and Technology recommends storing passwords using special hashes called key derivation functions (KDFs) that have been created to slow brute force searches. Slow hashes include pbkdf2, bcrypt, scrypt, argon2, Balloon and some recent modes of Unix crypt. For KDFs that perform multiple hashes to slow execution, NIST recommends an iteration count of 10,000 or more.
|
https://en.wikipedia.org/wiki/Cryptographic_hash_function
|
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